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STRUCTURAL SYNTHESIS OF PARALLEL ROBOTS
SOLID MECHANICS AND ITS APPLICATIONSVolume 149
Series Editor: G.M.L. GLADWELLDepartment of Civil EngineeringUniversity of WaterlooWaterloo, Ontario, Canada N2L 3GI
Aims and Scope of the SeriesThe fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchersgiving vision and insight in answering these questions on the subject of mechanics as itrelates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it includesthe foundation of mechanics; variational formulations; computational mechanics;statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids andstructures; dynamical systems and chaos; the theories of elasticity, plasticity andviscoelasticity; composite materials; rods, beams, shells and membranes; structuralcontrol and stability; soils, rocks and geomechanics; fracture; tribology; experimentalmechanics; biomechanics and machine design.
The median level of presentation is the first year graduate student. Some texts aremonographs defining the current state of the field; others are accessible to final yearundergraduates; but essentially the emphasis is on readability and clarity.
For a list of related mechanics titles, see final pages.
Structural Synthesis ofParallel Robots
Part 1: Methodology
By
GRIGORE GOGU
Mechanical Engineering Research Group,French Institute of Advanced Mechanics andBlaise Pascal University, ClermontFerrand, France
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 9781402051029 (HB)ISBN 9781402057106 (ebook)
Published by Springer,P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
www.springer.com
Printed on acidfree paper
All Rights Reserved© 2008 SpringerNo part of this work may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming, recordingor otherwise, without written permission from the Publisher, with the exceptionof any material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work.
Contents
Preface ..................................................................................................... IX Acknowledgements ...........................................................................XIV List of abbreviations and notations..................................................... XV
1 Introduction.............................................................................................1 1.1 Robot ................................................................................................1 1.2 Robotics ............................................................................................7 1.3 Parallel Robot .................................................................................10 1.4 Terminology ...................................................................................10 1.5 Structural synthesis.........................................................................23 1.6 The objectives and originality of this book ....................................25
2 Structural parameters ..........................................................................31 2.1 Critical review of mobility calculation ...........................................32
2.1.1 Chebychev’s contribution........................................................35 2.1.2 Sylvester’s contribution...........................................................36 2.1.3 Grübler’s contribution .............................................................36 2.1.4 Somov’s contribution ..............................................................37 2.1.5 Hochman’s contribution ..........................................................37 2.1.6 SomovMalytsheff’s formula ..................................................38 2.1.7 Koenigs’ formula.....................................................................39 2.1.8 Kutzbach’s mobility equation..................................................39 2.1.9 Dobrovolski’s mobility equation .............................................40 2.1.10 Contribution of Y.F. Moroskine ............................................40 2.1.11 Contribution of R. Voinea and M. Atanasiu..........................41 2.1.12 Kolchin’s mobility equation ..................................................42 2.1.13 Rössner’s contribution...........................................................42 2.1.14 Boden’s mobility equation.....................................................42 2.1.15 Manafu’s formula ..................................................................43 2.1.16 Ozol’s formula........................................................................ 43 2.1.17 Contribution of K. J. Waldron ...............................................44 2.1.18 Contribution of N. Manolescu ...............................................44 2.1.19 Contribution of C. Bagci .......................................................45
VI Contents
2.1.20 Contribution of P. Antonescu ................................................ 46 2.1.21 Contribution of F. Freudenstein and R. Alizade.................... 47 2.1.22 Hunt’s contribution................................................................ 48 2.1.23 Hervé’s contribution .............................................................. 49 2.1.24 Gronowicz’s contribution ...................................................... 50 2.1.25 Baker’s contribution .............................................................. 50 2.1.26 Davies’s contribution............................................................. 51 2.1.27 Contribution of V.P. Agrawal and J.S. Rao........................... 52 2.1.28 Contribution of J. Angeles and C. Gosselin .......................... 52 2.1.29 Contribution of F. Dudiţă and D. Diaconescu ....................... 53 2.1.30 Contribution of P. Fanghella and C. Galletti ......................... 55 2.1.31 Fayet’s contribution............................................................... 55 2.1.32 Tsai’s formula........................................................................ 56 2.1.33 McCarthy’s formula .............................................................. 56 2.1.34 Contribution of Z. Huang, L.F. Kong and Y.F. Fang ............ 57 2.1.35 Contribution of J.M. Rico, J. Gallardo and B. Ravani........... 57
2.2 ChebychevGrüblerKutzbach mobility formulae .......................... 58 2.2.1 The original ChebychevGrüblerKutzbach formula............... 58 2.2.2 The extended ChebychevGrüblerKutzbach formula............. 61 2.2.3 Limits of applicability of CGK formulae ................................ 62
2.3 Mobility and connectivity of parallel robots .................................. 78 2.3.1 General definitions and formulae for mobility and connectivity of mechanisms ............................................................. 79 2.3.2 Mobility and connectivity of simple open kinematic chains ... 82 2.3.3 Mobility and connectivity of singleloop kinematic chains..... 88 2.3.4 Connectivity between two elements of a singleloop kinematic chain................................................................................. 96 2.3.5. Mobility and connectivity of parallel robots with simple limbs ............................................................................................... 100 2.3.6. Mobility and connectivity of parallel robots with complex limbs ............................................................................................... 109 2.3.7. General formulae for robot mobility and connectivity ......... 114
2.4 Overconstraints in parallel robots................................................. 120 2.5 Redundancy in parallel robots ...................................................... 125 2.6 General formulae for structural parameters .................................. 127
3 Structural analysis .............................................................................. 131 3.1 Simple open kinematic chains ...................................................... 131 3.2 Singleloop kinematic chains........................................................ 137 3.3 Parallel mechanisms with simple limbs........................................ 148 3.4 Parallel mechanisms with complex limbs..................................... 168 3.5 Other multiloop kinematic chains ............................................... 228
VII
4 Kinematic analysis ..............................................................................235 4.1. Decoupling in axiomatic design ..................................................236 4.2. Geometric modeling ....................................................................238
4.3 Kinematic modeling .................................................................241 4.3.1 Direct and inverse kinematics matrices used in Jacobian calculation.......................................................................................242 4.3.2 Design and conventional Jacobian matrices ..........................243
4.4 Types of workspaces and singularities .........................................248 4.4.1 Types of workspaces .............................................................248 4.4.2 Types of singularities.............................................................249
4.5. Kinetostatic performance indices ................................................253 4.5.1 Crosscoupling indices ..........................................................256 4.5.2 Indices of inputoutput propensity.........................................262 4.5.3 Kinetostatic indices defined in connection with manipulability ellipsoids and polytops ...........................................266
4.6 Design Jacobian and motion decoupling ......................................274 4.6.1. Parallel robots with coupled motions ...................................276 4.6.2. Parallel robots with decoupled motions................................285 4.6.3. Parallel robots with uncoupled motions ...............................293 4.6.4. Maximally regular parallel robots ........................................296
5 Structural synthesis ............................................................................299 5.1 Structural synthesis: a systematic approach in mechanism design299 5.2. Morphological and evolutionary approaches...............................304
5.2.1 Morphological approaches ....................................................305 5.2.2 Evolutionary algorithms ........................................................307
5.3 Evolutionary morphology.............................................................310 5.3.1 Design objectives...................................................................310 5.3.2 Constituent elements..............................................................311 5.3.3 Morphological operators........................................................313 5.3.4 Set of solutions ......................................................................314 5.3.5 General structure of the evolutionary morphology................314
5.4 General approach to structural synthesis of parallel robots ..........317 5.4.1 General conditions for structural synthesis of parallel robots via theory of linear transformations.....................................317 5.4.2 General approach to structural synthesis of parallel robots via evolutionary morphology ...............................................................319
6 Limbs with two degrees of connectivity ............................................329 6.1 Limbs with two translational motions ..........................................329 6.2 Limbs with two rotational motions...............................................333 6.3 Limbs with one translational and one rotational motion ..............334
Contents
VIII Contents
6.4 Other limbs with two degrees of connectivity .............................. 335 6.5 Redundant limbs with two degrees of connectivity...................... 338
7 Limbs with three degrees of connectivity ......................................... 341 7.1 Limbs with three translational motions ........................................ 341 7.2 Planar limbs with one rotational and two translational motions... 352 7.3 Non planar limbs with one rotational and two translational motions ............................................................................................... 359 7.4 Limbs with one translational and two rotational motions............. 364 7.5 Limbs with three rotational motions............................................. 364 7.6 Other limbs with three degrees of connectivity ............................ 365 7.7 Redundant limbs with three degrees of connectivity.................... 375
8 Limbs with four degrees of connectivity........................................... 377 8.1 Limbs with Schönflies motion...................................................... 377 8.2 Limbs with two translational and two rotational motions ............ 425 8.3 Limbs with one translational and three rotational motions........... 435 8.4 Other limbs with four degrees of connectivity ............................. 436 8.5 Redundant limbs with four degrees of connectivity ..................... 442
9 Limbs with five degrees of connectivity ............................................ 445 9.1 Limbs with two rotational and three translational motions .......... 445 9.2 Limbs with two translational and three rotational motions .......... 637 9.3 Other limbs with five degrees of connectivity.............................. 644 9.4 Redundant limbs with five degrees of connectivity...................... 650
10 Limbs with six degrees of connectivity ........................................... 653 10.1 Limbs with three translational and three rotational motions ...... 653 10.2 Redundant limbs with six degrees of connectivity ..................... 661
References............................................................................................... 665
Index ....................................................................................................... 693
Preface
“…if the new theory is to lay claim to general interest, it must be capable of producing something new; it must make problems solvable which before could not be solved in a systematic way."
Reuleaux, F., Theoretische Kinematik, Braunschweig: Vieweg, 1875
Reuleaux, F., The Kinematics of Machinery, London: Macmil lan, 1876 and New York: Dover, 1963 (translated by B.W. Kennedy)
Parallel robotic manipulators can be considered a wellestablished op
tion for many different applications of manipulation, machining, guiding, testing, control, tracking, haptic force feedback, etc. A typical parallel robotic manipulator (PM) consists of a mobile platform connected to the base (fixed platform) by at least two kinematic chains called limbs. The mobile platform can achieve between one and three independent translations (T) and one to three independent rotations (R).
Parallel manipulators have been the subject of study of much robotic research during the last two decades. Early research on parallel manipulators has concentrated primarily on six degrees of freedom (DoFs) GoughStewarttype PMs introduced by Gough for a tiretesting device, and by Stewart for flight simulators. In the last decade, PMs with fewer than 6DoFs attracted researchers’ attention. Lower mobility PMs are suitable for many tasks requiring less than six DoFs.
The motion freedoms of the endeffector are usually coupled together due to the multiloop kinematic structure of the parallel manipulator. Hence, motion planning and control of the endeffector for PMs usually become very complicated. With respect to serial manipulators, such mechanisms can offer advantages in terms of stiffness, accuracy, loadtoweight ratio, dynamic performances. Their disadvantages include a smaller workspace, complex command and lower dexterity due to a high motion coupling, and multiplicity of singularities inside their workspace. Uncoupled, fullyisotropic and maximally regular PMs can overcome these disadvantages.
X Preface
Isotropy of a robotic manipulator is related to the condition number of its Jacobian matrix, which can be calculated as the ratio of the largest and the smallest singular values. A robotic manipulator is fullyisotropic if its Jacobian matrix is isotropic throughout the entire workspace, i.e., the condition number of the Jacobian matrix is equal to one. We know that the Jacobian matrix of a robotic manipulator is the matrix mapping (i) the actuated joint velocity space on the endeffector velocity space, and (ii) the static load on the endeffector and the actuated joint forces or torques. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator.
We distinguish five types of PMs (i) maximally regular PMs, if the Jacobian J is an identity matrix throughout the entire workspace, (ii) fullyisotropic PMs, if the Jacobian J is a diagonal matrix with identical diagonal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. Maximally regular and fullyisotropic PMs give a onetoone mapping between the actuated joint velocity space and the external velocity space.
The first solution for a fullyisotropic T3type translational parallel robot was developed at the same time and independently by Carricato and ParentiCastelli at University of Genoa, Kim and Tsai at University of California, Kong and Gosselin at University of Laval, and the author of this work at the French Institute of Advanced Mechanics. In 2002, the four groups published the first results of their works.
The general methods used for structural synthesis of parallel mechanisms can be divided into three approaches: the method based on displacement group theory, the methods based on screw algebra, and the method based on the theory of linear transformations. The method proposed in this book is based on the theory of linear transformations and the evolutionary morphology and allows us to obtain the structural solutions of decoupled, uncoupled, fullyisotropic and maximally regular PMs with two to six DoFs in a systematic way. The new formulae for mobility, connectivity (spatiality), redundancy and overconstraint of PMs proposed recently by the author are integrated into the synthesis approach developed in this book.
Various solutions of TaRbtype PMs are known today. In this notation, a=1,2,3 indicates the number of independent translations and b=1,2,3 the number of independent rotations of the moving platform. The parallel robots actually proposed by the robot industry have coupled and decoupled motions and just some isotropic positions in their workspace. As far as we
XI
are aware, this is the first book on robotics presenting solutions of uncoupled, fullyisotropic and maximally regular PMs.
Nonredundant/redundant, overconstrained/isostatic solutions of uncoupled and fullyisotropic/maximally regular PMs with elementary/complex limbs actuated by linear/rotary actuators with/without idle mobilities and two to six DoFs are present in a systematic approach of structural synthesis. These solutions are derived from families of PMs with decoupled motions. A serial kinematic chain is associated with each elementary limb and at least one closed loop is integrated in each complex limb.
The synthesis methodology and the solutions of PMs presented in this book represent the outcome of some recent research developed by the author in the last four years in the framework of the projects ROBEAMAX (20022003) and ROBEAMP2 (20042005) supported by the National Center for Scientific Research (CNRS). These results have been partially published by the author in the last two years. In these works the author has proposed the following for the first time in the literature:
a) new formulae for calculating the degree of mobility, the degree of connectivity(spatiality), the degree of redundancy and the number of overconstraints of parallel robotic manipulators that overcome the drawbacks of the classical ChebychevGrüblerKutzbach formulae,
b) a new approach to systematic innovation in engineering design called evolutionary morphology,
c) solutions to follow fullyisotropic parallel robots: parallel wrists with two and three degrees of mobility, planar parallel robots, parallel robots of types T2R1 and T3R2, parallel robots with Schönflies motions, hexapods with six degrees of freedom.
The various solutions proposed by the author belong to a modular family called IsoglideNTxRy with a+b=n with 2≤ n≤ 6, a=1,2,3 and b=1,2,3. The mobile platform of these robots can have any combination of n independent translations (T) and rotations (R). The IsoglidenTaRb modular family was developed by the author and his research team at the French Institute of Advanced Mechanics (IFMA) in ClermontFerrand.
This work is organized in two parts published in two distinct books. Part 1 presents the methodology proposed for structural synthesis and Part 2 the various topologies of parallel robots systematically generated by the structural synthesis approach. The originality of this work resides in combining the new formulae for mobility connectivity, redundancy and overconstraints, and the evolutionary morphology in a unified approach of structural synthesis giving interesting innovative solutions for parallel robots.
Part 1 is organized in ten chapters. The first chapter is intended to introduce the main concepts, definitions and components of the mechanical robotic system. Chapter 2 reviews the contributions in mobility calculation
Preface
XII Preface
systematized in the so called ChebychevGrüblerKutzbach mobility formulae. The drawbacks and the limitations of these formulae are discussed, and the new formulae for mobility, connectivity, redundancy and overconstraint are demonstrated via an original approach based on the theory of linear transformations. These formulae are applied in chapter 3 for the structural analysis of parallel robots with simple and complex limbs. The new formulae are also applied to calculate the mobility and other structural parameters of single and multiloop mechanisms that do not obey the classical ChebychevGrüblerKutzbach formulae, such as the mechanisms proposed by De Roberval, Sarrus, Bennett, Bricard and other so called “paradoxical mechanisms”. We have shown that these mechanisms completely obey the definitions, the theorems and the formulae proposed in the previous chapter. There is no reason to continue to consider them as “paradoxical”. Chapter 4 presents the main models and performance indices used in parallel robots. We emphasize the Jacobian matrix, which is the main issue in defining robot kinematics, singularities and performance indices. New kinetostatic performance indices are introduced in this section to define the motion decoupling and inputoutput propensity in parallel robots. Structural parameters introduced in the second chapter are integrated in the structural synthesis approach founded on the evolutionary morphology (EM) presented in chapter 5. The main paradigms of EM are presented in a closed relation with the biological background of morphological approaches and the synthetic theory of evolution. The main difference between the evolutionary algorithms and the EM are also discussed. The evolutionary algorithms are methods for solving optimizationoriented problems, and are not suited to solving conceptual designoriented problems. They always start from a given initial population of solutions and do not solve the problem of creating these solutions.
The first stage in structural synthesis of parallel robots is the generation of the kinematic chains called limbs used to give some constrained or unconstrained motion to the moving platform. The constrained motion of the mobile platform is obtained by using limbs with less than six degrees of connectivity. The various solutions of simple and complex limbs with two to six degrees of connectivity are systematically generated by the structural synthesis approach and presented in chapters 610. We focus on the solutions with a unique basis of the operational velocity space that are useful for generating various topologies of decoupled, uncoupled, fullyisotropic and maximally regular parallel robots presented in Part 2. Limbs with multiple bases of the operational velocity space and redundant limbs are also presented in these chapters. These limb solutions are systematized with respect to various combinations of independent motions of the distal link. They are defined by symbolic notations and illustrated in about 250 figures
XIII
containing more than 1500 structural diagrams. Special attention was paid to graphic quality of structural diagrams to ensure a clear correspondence between the symbolic and graphic notation of joints and the relative position of their axes. The graphic illustration of these solutions is associated with the author’s conviction that a good structural diagram really “is worth a thousand words”, especially when you are trying to disseminate the result of the structural synthesis of kinematic chains. The kinematic chains presented in chapters 610 are useful as innovative solutions of limbs in parallel, serial and hybrid robots. In fact, serial and hybrid robots may be considered as a particular case of parallel robots with only one limb which can be a simple, complex or hybrid kinematic chain. Many serial robots actually combine closed loops in their kinematic structure.
The various types of kinematic chains generated in chapters 610 are combined in Part 2 to set up innovative solutions of parallel robots with two to six degrees of mobility and various sets of independent motions of the moving platform.
Part 2 is organized in 16 chapters presenting the following: translational PMs T2type (chapter 1), T1R1type PMs with screw motion (chapter 2), other T1R1type PMs (chapter 3), R2type spherical parallel wrists (chapter 4), translational PMs T3type (chapter 5), T2R1type PMs with planar motion (chapter 6), other T2R1type PMs (chapter 7), T1R2type PMs (chapter 8), R3type spherical parallel wrists (chapter 9), T3R1type PMs with Schönflies motion (chapter 10), other T3R1type PMs (chapter 11), T2R2type PMs (chapter 12), T1R3type PMs (chapter 13), T3R2type PMs (chapter 14), T2R3type PMs (chapter 15) and PMs with six degrees of freedom (chapter 16).
Parallel robots with coupled, decoupled and uncoupled motions, along with fullyisotropic and maximally regular solutions, are systematically presented in each chapter. Innovative solutions of overconstrained or nonoverconstrained, nonredundant or redundantly actuated parallel robots with simple or complex limbs actuated by linear or rotating motors are set up in Part 2 by applying the structural synthesis methodology presented in this first part. The writing of part 2 is still in progress and will soon be finalized.
Many solutions for parallel robots obtained through this systematic approach of structural synthesis are presented here for the first time in the literature. The author had to make a difficult and challenging choice between protecting these solutions through patents, and releasing them directly into the public domain. The second option was adopted by publishing them in various recent scientific publications and mainly in this book. In this way, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products.
Preface
XIV Preface
Acknowledgements
The scientific environment of the projects ROBEAMAX (20022003) and ROBEAMP2 (20042005) supported by the CNRS was the main source of encouragement and motivation to pursue the research on the structural synthesis of parallel robots and to write this book. Deep gratitude is expressed here to Dr. François Pierrot, Deputy Director of LIRMM and coordinator of both ROBEA projects, and also to all colleagues involved in these projects from the research laboratories LIRMM, INRIA, IRCCyN LASMEA and LaMI for the valuable scientific exchanges during the joint work in these projects.
Moreover, financial support from the CNRS, FR TIMS and IFMA for developing the innovative Isoglidefamily of parallel robots are duly acknowledged.
Furthermore, Mrs. Nathalie Jacobs, Springer Dordrecht Engineering Editor is gratefully acknowledged for her availability and encouragement in pursuing this book project.
Ms. Sarah Davey and Prof. Graham M.L. Gladwell are also gratefully acknowledged for linguistic reviewing of the manuscript. Prof. Gladwell not only took great care of the book language but also made numerous suggestions for improving and clarifying some fine points of its content.
May I also acknowledge the excellent facilities and research environment provided by LaMI and IFMA which contributed actively to the completion of this project.
To conclude, I cannot forget my wife Iléana and my son Christian for their love, affection and encouragement, providing the fertile ambience for this sustained work very often prolonged late into the evening and mostly during weekends and holidays.
XV
List of abbreviations and notations
ai – axial offset of the ithlink A (1A2A…nA) and A (12…n) – simple open kinematic chain and its
links A (0≡1A2A3A4A5A) simple open kinematic chain with link 1A fixed to
base 0 A (1A≡02A…nA≡1A) – simple closed kinematic chain with link 1A fixed
to base 0 A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) simple open kinematic chain
obtained by serial concatenation of two simple chains A1 and A2 AI – artificial intelligence bi – length of the ilink B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) singleloop kinematic
chain obtained by connected the distal links (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2)
c – condition number C ← A1A2…Ak parallel robot with k simple limbs Ai (i=1, 2,…,k) C – cylindrical pair CNRS  Centre National de la Recherche Scientifique (National Center
for Scientific Research) CGK  ChebychevGrüblerKutzbach Ci  instantaneous degree of motion coupling C  global degree of motion coupling d  linear dispalcement D ← A1A2…Ak1E1E2…Ek2 parallel robot with k1 simple limbs Ai (i=1, 2,…,k1) and k2 complex limbs Ei (i=1, 2,…,k2) DoF  degreeoffreedom DPs – design parameters eA and eA1  link of A1limb (e=1,2,3,…,n) eB and eA2  link of A2limb (e=1,2,3,…,n) eC and eA3  link of A3limb (e=1,2,3,…,n) eD and eA4  link of A4limb (e=1,2,3,…,n)
1( ,..., )nE ε ε=  set of constituent elements EA – evolutionary algorithm EM – evolutionary morphology ES  Evolution strategies F  linear transformation f  degree of mobility of a joint in the general motion space with six dimensions
Preface
XVI Preface
F ← G1G2…Gk general notation for the kinematic chain associated to a serial, parallel or hybrid robot with k simple and/or complex limbs Gi (i=1, 2,…,k)
FRs – functional requirements FR TIMS  Fédération de Recherche Technologies de l’Information et
de la Mobilité, de la Sûreté GA – genetic algorithm Gi (1Gi2Gi…nGi) the kinematic chain associated to the ith limb (i=1,2,…,k). H – characteristic point of the distal link/endeffector H helical pair IFMA  Institut Français de Mécanique Avancée (French Institute of
Advanced Mechanics) INRIA  Institut National de Recherche en Informatique et en Automa
tique (The French National Institute for Research in Computer Science and Control)
IRCCyN  Institut de Recherche en Communications et Cybernétique de Nantes
n nI ×  n×n identity matrix J  Jacobian matrix JT – transpose of the Jacobian matrix J1 – inverse of the Jacobian matrix JS  direct kinematics matrix/Jacobian JP  inverse kinematics matrix/Jacobian k  total number of limbs of the parallel manipulator k1  number of simple limbs of a parallel manipulator k2  number of complex limbs of a parallel manipulator KP  kinematic pair KC  kinematic chain Q  generic notation for a kinematic chain or a mechanism LaMI  Laboratoire de Mécanique et Ingénieries (Mechanical Engineer
ing Research Group) LASMEA  Laboratoire des Sciences et Matériaux pour l'Electronique,
et d'Automatique (Laboratory of Sciences and Materials for Elec tronic, and of Automatic)
LIRMM  Laboratoire d'Informatique, de Robotique et de Microélec tronique de Montpellier (Montpellier Laboratory of Computer Sci ence, Robotics, and Microelectronics)
Lc  characteristic length m  total number of links (mechanism elements) including the fixed base mQ  number of joints in kinematic chain Q
XVII
M  mobility MQ  mobility of kinematic chain/mechanism Q iM  instantaneous mobility iMQ  instantaneous mobility of kinematic chain/mechanism Q N – number of overconstraints n – number of kinematic elements (links) excepting the fixed link
t 1 n( ,..., )Ο ο ο=  set of morphological operators applicable at each generation t O0x0y0z0  reference frame O0On  position vector of point On in fixed reference frame p  number of joints pp  number of passive joints p  operational velocity vector P  prismatic joint/pair P  actuated prismatic joint Pa  parallelogram loop Pat  telescopic parallelogram loop Pn2  planar close loop with two degrees of mobility Pn4  planar close loop with three degrees of mobility PM  parallel manipulator q  number of independent closed loops q  joint velocity vector qi  finite displacement in the ith actuated joint r  number of joint parameters that lost their independence rQ  number of joint parameters that lost their independence in kinematic chain/mechanism B rl  number of joint parameters that lost their independence in the closed
loops integrated in the complex limbs of a parallel robot Gi
lr number of joint parameters that lost their independence in the closed loops which may exist in ith limb Gi of a parallel robot
R – revolute joint/pair R – actuated revolute joint Rb – rhombus loop RF  the range of linear transformation F RQ  vector space of operational velocities of the kinematic
chain/mechanism Q Qa / bR  vector space of relative velocities between links a and b in
kinematic chain/mechanism Q ( Q
a / bR )  base of the vector space Qa / bR
R  reangularity
Preface
XVIII Preface
Qa / bS  connectivity between links a and b in a kinematic
ckain/mechanism Q SQ  connectivity of kinematic chain/mechanism Q iSQ  instantaneous connectivity of kinematic chain/mechanism Q S .  semangularity T  structural redundancy
1t n( ,..., )Τ τ τ=  set of evolution criteria from generation t to generation t+1
U  vector space of joint velocities UQ  vector space of joint velocities of kinematic chain/mechanism Q v  translational velocity vector w  general velocity vector of the moving platform w1, w2, …, w6  general coordinates W  vector space of external velocities WQ  vector space of external velocities of kinematic chain/mechanism
Q xP, yP, zP  coordinates of point P in the reference frame
, , ,α β δ ϕ  rotation angles , , ,α β δ ϕ  time derivatives of the rotation angles
∆q  vector of the infinitesimal motions of joints ∆w  vector of infinitesimal motions of the endeffector
{ }: ,i true falseι ι →  termination criterion for evolutionary morphology
1υ  conditioning index
2υ  minimum conditioning index
3υ and 4υ  manipulability indexes
iξ  square root of the eigenvalues of (JJT)1 λ  scalar / proportionality factor µ  vector of the operational forces τ  vector of the joint forces ψ  velocity transmission factor ω  angular velocity
1( ,..., )t nσ σΣ =  set of solutions at each generation t
1 n( ,..., )Φ φ φ=  set of design objectives 0  fixed base of a kinematic chain/mechanism 12…n  links (elements) of a kinematic chain/mechanism 1Q2Q…nQ  links (elements) of kinematic chain/mechanism
1 Introduction
This section introduces the terminology and presents the main objectives of the book. The family of parallel robots is placed in the general field of robots and the structural synthesis in the general context of robotics.
1.1 Robot
The origin of the term robot can be found in a science fiction play written by Karel Čapek (1920). This play titled R.U.R. was published in Czech in 1920 as an abbreviation of Rozum's Universal Robots, a collective drama in a comic prologue and three acts first performed in National Theatre in Prague on January 25, 1921. We note that in Czech “rozum” means wisdom and “robota” means servitude/forced labour/drudgery. Since the word robot was apparently coined by the author's brother, cubist painter and writer Joseph Čapek, the term is used to refer to a wide range of manmade things from devices to programs that are used to do automatically and autonomously various tasks by replacing humans.
Humanmade analogs of living systems could be found long ago in Antiquity despite the recent useage of the word robot. The pigeon conceived by the Greek mathematician Archytas of Tarentum in 425 BC and chronicled by Favorinus and by Aulus Gellius in his Attic Nights could be considered the first “flying robot” and the southseeking chariot, possibly used by the Chinesse Yellow Emperor Hunag Ti, the first “field robot”. Other legendary solutions such as the “walking machines” called wooden ox and gliding horse of Chuko Liang were invented to transport army food supplies across rough terrain during the period of China’s Three Kingdoms in 209 AC (Yan 1999). The automatic machines (AlJazary 1973) constructed by AlJazari (11361206), the designs of mechanical knight (Rosheim 2006) and flying machines (Da Vinci) of Leonardo da Vinci (14521519), the mechanical duck and the flute player – the first automatons (Balpe and Drye 1994) constructed by Jacques de Vaucanson (17091782) and the androids (the draughtsman, the musicians and the writer) constructed in 1774 by Pierre JacquetDroz (17711790) and his
2 1 Introduction
son HenriLouis (17531791) could also be considered as precursors of the current robots from the predigital era.
Hundreds of robot definitions are known today and hundreds of millions of web sites could be referenced today by a simple search on the word robot. According to the American Heritage Dictionary a robot is a mechanical device that is capable of performing a variety of often complex human tasks on command or by being programmed in advance. Other definitions of the word robot proposed by online dictionaries are systematized in Table 1.1.
Robots can be found today in the manufacturing industry, agricultural, military and domestic applications, space exploration, medicine, education, information and communication technologies, entertainment, etc.
We can see that the word robot is mainly used to refer to a wide range of mechanical devices or mechanisms, the common feature of which is that they are all capable of movement and can be used to perform physical tasks. Robots take on many different forms, ranging from humanoid, which mimic the human form and mode of movement, to industrial, whose appearance is dictated by the function they are to perform. Robots can be categorized as robotic manipulators, wheeled robots, legged robots swimming robots, flying robots, androids and self reconfigurable robots which can conform themselves to a given task. This book focuses on the parallel robotic manipulators which are the counterparts of the serial robots.
We also note that the term robot has recently morphed to also refer to “bots”, which are automated programs used in several online functions in computer science and artificial intelligence. The term has also entered into the informal language to define certain forms of human behaviour or even a traffic light in South Africa.
1.1 Robot 3
Table 1.1. Various definitions of the term robot
Dictionary Definition 1. Programmable machine for performing tasks: a mechanical device that can be programmed to carry out instructions and perform complicated tasks usually done by people. 2. Imaginary machine: a machine that resembles a human in appearance and can function like a human, especially in science fiction. 3. Person like a machine: somebody who works or behaves mechanically and emotionlessly.
Encarta® World English Dictionary, North American Edition1
4. South Africa traffic light: a set of automatic traffic lights (informal).
Compact Oxford English Dictionary2
A machine capable of carrying out a complex series of actions automatically, especially one programmable by a computer.
MerriamWebster's Online Dictionary, 10th Edition3
1a. A machine that looks like a human being and performs various complex acts (as walking or talking) of a human being; also: a similar but fictional machine whose lack of capacity for human emotions is often emphasized. 1b. An efficient insensitive person who functions automatically. 2. A device that automatically performs complicated often repetitive tasks. 3. A mechanism guided by automatic controls.
Cambridge International Dictionary of English4
1. A machine used to perform jobs automatically, which is controlled by a computer. 2. Disapproving someone who does things in a very quick and effective way but never shows their emotions. 3. A traffic light in South Africa.
Cambridge Dictionary of American English5
A mechanical device that works automatically or by computer control.
Columbia Encyclopedia, Sixth Edition6
Mechanical device designed to perform the work generally done by a human being.
1http://encarta.msn.com/encnet/features/dictionary/dictionaryhome.aspx 2http://www.AskOxford.com 3http://www.mw.com/ 4http://dictionary.cambridge.org/ 5,6http://dictionary.cambridge.org/
4 1 Introduction
Table 1.1. (cont.)
The American Heritage® Dictionary of the English Language7
1. A mechanical device that sometimes resembles a human and is capable of performing a variety of often complex human tasks on command or by being programmed in advance. 2. A machine or device that operates automatically or by remote control. 3. A person who works mechanically without original thought, especially one who responds automatically to the commands of others. 1. A mechanical device, esp. one controlled electronically, that can perform physical tasks of a human being, in industrial assembly or the like. 2. A person who moves or acts mechanically, without thought or feeling.
The Wordsmyth English DictionaryThesaurus8
Infoplease Dictionary9
1. A machine that resembles a human and does mechanical, routine tasks on command. 2. A person who acts and responds in a mechanical, routine manner, usually subject to another's will; automaton. 3. Any machine or mechanical device that operates automatically with humanlike skill.
Dictionary.com10 1. A mechanical device that sometimes resembles a human and is capable of performing a variety of often complex human tasks on command or by being programmed in advance. 2. A machine or device that operates automatically or by remote control. 3. A person who works mechanically without original thought, especially one who responds automatically to the commands of others.
Wiktionary11
1. A machine that resembles humans in shape or scope of function. 2. A machine that operates automatically. 3. A machine controlled by a fundamentally ingrained computer.
7http://www.bartleby.com/61/s0.html
8http://www.wordsmyth.net/
9http://www.infoplease.com/dictionary.html
10http://www.dictionary.com/
11http://www.wiktionary.org
1.1 Robot 5
Table 1.1. (cont.)
Rhymezone12 LookWAYup Translating Dictioary /Thesaurus13
WordNet 1.7 Vocabulary Helper14
A mechanism that can move automatically.
Wikipedia, the Free Encyclopedia15
1. A robot is a type of device. 2. Robot can also refer to Internet bot, an automated software program used on the Internet. 1. Especially in sciencefiction: a machine that vaguely resembles a human being and which can be programmed to carry out tasks. Compare android. 2. An automatic machine that can be programmed to perform specific tasks. 3. Someone who works efficiently but who lacks human warmth or sensitivity. 4. An automatic traffic signal.
AllWords.com MultiLingual Dictionary16
WebmasterWorld Webmaster and Search Engine Glossary17
Program that automatically does "some action" without user intervention. In the context of search engines, it usually refers to a program that mimics a browser to download web pages automatically. A spider is a type of robot. Some times referred to as Webbots.
Netlingo18 Traditionally, it's a device that can move and react to sensory input… The term "robot" has morphed to also refer to bots, which are automated programs used in several online functions.
Webopedia19
1. A device that responds to sensory input. 2. A program that runs automatically without human intervention.
12http://www.rhymezone.com/ 13http://lookwayup.com/free
14http://poets.notredame.ac.jp/cgibin/wn 15http://www.wikipedia.org
16http://www.allwords.com/
17http://www.webmasterworld.com/glossary/ 18http://www.netlingo.com/index.cfm 19 http://www.pcwebopaedia.com/
6 1 Introduction
Table 1.1. (cont.)
CCI Computer20 1. A mechanical device that performs a task that would otherwise be done by a human. Robots can be useful for jobs that are boring or dangerous for humans to perform. The simplest robots are capable only of repeating a programmed motion; the most sophisticated models can use sensors and artificial intelligence to distinguish between objects, understand natural language, and make decisions. Robots can be programmed or operated by remote control. 2. A computer program that performs intelligent tasks such as retrieving World Wide Web documents and indexing references. 3. A program that performs a programmed communication function such as automated email answering, responding to newsgroup message, or regulatory functions in IRC, graphical chat, and other online environments 1. A device that responds to sensory input. They are often just called "Bots". 2. A program that runs automatically without human intervention. Typically, a robot is endowed with some artificial intelligence so that it can react to different situations it may encounter. Two common types of robots are agents and spiders.
Computer Telephony & Electronics Dictionary and Glossary21
Internet Terms22 A robot is a program that is designed to automatically go
out and explore the Internet for a specific purpose. Robots that record and index all of the contents of the network to create searchable databases are sometimes called spiders or Worms. WebCrawler and Lycos are popular examples of robots.
TECHNICAL23
Robot  Besides being a mechanical device used to mimic human form, usually to accomplish some repetitive task, this term refers to a computer program that scans Web pages and links. Like a similar spider program, robots are used to scan Web pages and index them.
20http://www.currents.net/resources/dictionary/index.html
21http://www.csgnetwork.com/glossary.html
22http://www.learnthenet.com/english/glossary/glossary.htm
23http://www.ugeek.com/glossary/glossary_search.htm
1.2 Robotics 7
1.2 Robotics
The science of robotics has undergone an explosive growth since the word robotics was coined by the writer Isaac Asimov in 1950. In his science fiction book “I, Robot” he presented the famous three laws of robotics: (i) a robot may not injure a human being, or, through inaction, allow a human being to come to harm; (ii) a robot must obey the orders given it by human beings except where such orders would conflict with the first law and (iii) a robot must protect its own existence as long as such protection does not conflict with the first or second law. According to the American Heritage Dictionary, robotics is the science or study of the technology associated with the design, fabrication, theory, and application of robots. This technology combines mechanical devices, actuators, sensors, controllers, software and computers.
Today robots get closer to humans and new issues rise in matters related to human robot interaction, in advances and experiences of robots and automation at home, at work, for education, as well as in other emerging areas. The next generation of robots will be capable of interacting with humans on a sophisticated level, making decisions, helping, caring and giving companionship.
Recent research by the Japan Robotics Association (JPA), United Nations Economic Commission (UNEC) and the International Federation of Robotics (IFR) indicates that the nascent personal and service robotics market will exhibit exceptional near term growth. The service and personal robotics marketplaces together will equal the size of the industrial robotics market (the combination of manufacturing and bioindustrial) by 2005, and will be twice the size of the industrial robotics market by 2010, and almost 4X its size by 2025.
Many of the technologies required to build functional personal and service robots already exist, and markets for these products are already in place. Some of the most salient enabling technologies include advances in microprocessor technology, wireless technology, image processing, speech recognition, motion sensor technology, and embedded systems development tools.
Hundreds of universities worldwide have research programs in robotics and many are awarding degrees in robotics.
Various definitions of the word robotics are systematized in Table 1.2. We can see that they converge towards the integration of the design and the end use in the studies related to robotics. This book focuses on the conceptual design of parallel robots.
8 1 Introduction
Table 1.2. Various definitions of the term robotics
Dictionary Definition Encarta® World English Dictionary, North American Edition1
Design and use of robots: the science and technology relating to computercontrolled mechanical devices such as the automated tools commonly found on automobile assembly lines.
Compact Oxford English Dictionary2
The branch of technology concerned with the design, construction, and application of robots.
MerriamWebster's Online Dictionary, 10th Edition3
Technology dealing with the design, construction, and operation of robots in automation.
Cambridge International Dictionary of English4
The science of making and using robots.
Cambridge Dictionary of American English5
The science of designing and operating robots.
Columbia Encyclopedia, Sixth Edition6
Science and technology of general purpose, programmable machine systems.
The American Heritage® Dictionary of the English Language7,10
The science or study of the technology associated with the design, fabrication, theory, and application of robots.
The Wordsmyth English DictionaryThesaurus8
The technology of designing and using robots for various, usually industrial tasks.
Infoplease Dictionary9
The use of computercontrolled robots to perform manual tasks, especially on an assembly line.
Wiktionary11,15 The science and technology of robots, their design, manufacture, and application.
1http://encarta.msn.com/encnet/features/dictionary/dictionaryhome.aspx 2http://www.AskOxford.com 3http://www.mw.com/ 4http://dictionary.cambridge.org/ 5,6http://dictionary.cambridge.org/
7http://www.bartleby.com/61/s0.html
8http://www.wordsmyth.net/
9http://www.infoplease.com/dictionary.html
10http://www.dictionary.com/
11http://www.wiktionary.org
1.2 Robotics 9
Table 1.2. (cont.)
Rhymezone12 LookWAYup Translating Dictioary/ Thesaurus13
WordNet 1.7 Vocabulary Helper14
The area of AI concerned with the practical use of robots 1. The branch of engineering concerned with the design, construction, operation and use of industrial robots, and which incorporates many of the concepts used in artificial intelligence. 2. A form of dancing in which dancers imitate the stiff jerky movements of robots.
AllWords.com MultiLingual Dictionary16
The Online Medical Dictionary17
The application of electronic, computerised control systems to mechanical devices designed to perform human functions. Formerly restricted to industry, but nowadays applied to artificial organs controlled by bionic (bioelectronic) devices, like automated insulin pumps and other prostheses.
Netlingo18,19 The field of computer science and engineering concerned with creating robots. Robotics is a branch of artificial intelligence.
CCI Computer20 The design, manufacture, and use of robots.
Computer Telephony & Electronics Dictionary and Glossary21
The field of computer science and engineering concerned with creating robots or bots, devices that can move and react to sensory input. Robotics is one branch of artificial intelligence.
12http://www.rhymezone.com/ 13http://lookwayup.com/free
14http://poets.notredame.ac.jp/cgibin/wn 15http://www.wikipedia.org
16http://www.allwords.com/
17http://cancerweb.ncl.ac.uk/omd/ 18http://www.netlingo.com/index.cfm 19 http://www.pcwebopaedia.com/
20http://www.currents.net/resources/dictionary/index.html
21http://www.csgnetwork.com/glossary.html
10 1 Introduction
1.3 Parallel Robot
Although the appearance and capabilities of robots vary vastly, all robots share the features of a mechanical, movable structure under some form of control. The structure of a robot is usually mostly mechanical and takes the form of a mechanism having as constituent elements the links connected by kinematic joints. Serial or parallel kinematic chains are concatenated in the robot mechanism. The serial kinematic chain is formed by links connected sequentially by joints. Links are connected in series as well as in parallel making one or more closedloops in a parallel mechanism. The members of a mechanism may take various forms: bars, plates, platforms, cams, gears, sliders, sliding blocks, etc.
The mechanical architecture of parallel robots is based on parallel mechanisms in which a member called a mobile platform is connected to a reference element by at least two limbs that can be elementary (simple) or complex. The robot actuators are integrated in the limbs (also called legs) usually closed to the fixed member, also called the base or the fixed platform. The mobile platform positions the robot endeffector in space and could have between two and six degrees of freedom. Usually the number of actuators coincides with the degrees of freedom of the mobile platform, exceeding them only in the case of redundantlyactuated parallel robots.
The paradigm of parallel robots is the hexapodtype robot, which has six degrees of freedom, but recently the machine industry has discovered the potential applications of lowermobility parallel robots with just 2, 3, 4 or 5 degrees of freedom. Indeed, the study of this type of parallel manipulators is very important. They exhibit interesting features if compared to hexapods, such as simpler architecture, simpler control system, highspeed performance, low manufacturing and operating costs. Furthermore, for several parallel manipulators with decoupled or uncoupled motions the kinematic model can be easily solved to obtain algebraic expressions, which are well suited for an implementation in optimum design problems. Parallel mechanisms can be considered a wellestablished solution for many different applications of manipulation, machining, guiding, testing, control, etc.
1.4 Terminology
The terminology used in this book and defined in this section is mainly established in accordance with the terminology adopted by the International Federation for the Promotion of Mechanism and Machine Science
1.4 Terminology 11
(IFToMM) and published in (Ionescu 2003). The main terms used in this book concerning kinematic pairs (joints), kinematic chains and robot kinematics are defined in Tables 1.31.5. They are completed by some complementary remarks, notations and symbols used in this book.
In the standard terminology, a kinematic chain is an assembly of links (mechanism elements) and joints, and a mechanism is a kinematic chain in which one of its links is taken as a “frame”. In this definition the “frame” is a mechanism element deemed to be fixed. In this book, we use the notion of reference element to define the “frame” element. The reference element can be fixed or may merely be deemed to be fixed with respect to other mobile elements. The fixed base is denoted in this book by 0. A mobile element in a kinematic chain Q is denoted by nQ (n=1, 2, …). Two or more links connected together in the same link such that no relative motion can occur between them are considered as one link. The identity symbol “≡” is used between the links to indicate that they are welded together in the same link. For example, the notation 1Q≡0 is used to indicate that the first link 1Q is connected to the fixed base by a common link. A kinematic chain Q is denoted by the sequence of its links. The notation Q (1Q≡02Q…nQ) indicates a kinematic chain in which the first member is fixed and the notation Q (1Q2Q…nQ) a kinematic chain with no fixed member.
We will use the notion of mechanism to qualify the whole mechanical system, and the notion of kinematic chain to qualify the subsystems of a mechanism. So, in this book, the same assembly of links and joints Q1 is considered to be a kinematic chain when it is integrated as a subsystem in another assembly of links and joints Q2, and it is considered a mechanism when Q1 represents the whole system. The systematization, the definitions and the formulae presented in this book are also valuable for mechanisms and kinematic chains.
We use the term mechanism element or link to name a component (member) of a mechanism with any form (bar, gear, cam, etc.). In this text, unless otherwise stated, we consider all links to be rigid. We distinguish the following types of links:
a) monary link  a mechanism element connected in the kinematic chain by only one joint (a link which carries only one kinematic pairing element),
b) binary link  a mechanism element connected in the kinematic chain by two joints (a link connected to two other links),
c) polinary link  a mechanism element connected in the kinematic chain by more than two joints (ternary link  if the link is connected by three joints, quaternary link if the link is connected by four joints,…).
12 1 Introduction
Table 1.3. IFToMM terminology with respect to the kinematic pairs
Term Definition Pairing element Assembly of surfaces, lines or points of a solid body through
which it may contact with another solid body.
Mechanism element
Solid body or fluid component of a mechanism.
Frame Mechanism element deemed to be fixed.
Base Body (link) of a robot that carries the first joint(s) of the ki
nematic chain of a manipulator or pedipulator.
Rigid body Theoretical model of a solid body in which the distances between particles are considered to be constant, regardless ofany forces acting upon the body.
Link 1. Mechanism element (component) carrying kinematic pairing elements. 2. Element of a linkage.
Limit position of a link
Position of a link for which a coordinate which describes itsposition relative to an adjacent link is a maximum or a minimum.
Input (driving) link
Link whereby motion and force are imparted to a mechanism.
Output (driven) link (follower)
Link from which required forces and motions are obtained.
Bar Link that carries only revolute joints.
Crank Link able to rotate completely about a fixed axis.
Rocker Link that oscillates within a limited angle of rotation about afixed axis.
Coupler (floating link)
Link that is not connected directly to the frame.
Slider Link that forms a prismatic pair with one link and a revolute pair with another link.
1.4 Terminology 13
Table 1.3. (cont.)
Sliding block Compact element of a prismatic pair which slides along aguiding element (e.g. in a slot).
Guide Element of a prismatic pair that is fixed to the frame and constrains the motion of a sliding block.
Crosshead Component between a piston and a connecting rod which, byforming a prismatic joint with the frame, provides a reaction to the component of force in the connecting rod normal to theline of stroke of the piston.
Connecting rod Coupler between a piston or a crosshead and a crank shaft.
Cam Component with a curved profile or surface whereby it imparts a displacement either by point or line contact with acam follower.
Kinematic pair Mechanical model of the connection of two pairing elements having relative motion of a certain type and degree of freedom. Note: see synonym at joint.
Joint 1. Physical realization of a kinematic pair, including connection via intermediate mechanism elements. 2. Kinematic pair.
Degree of freedom (connectivity) of a kinematic pair
The number of independent coordinates needed to describe the relative positions of pairing elements.
Closure of a kinematic pair
Process of constraining two rigid bodies to form a kinematic pair by force (force closure), geometric shape (form closure), or flexible materials (material closure).
Forceclosed (open) pair
Kinematic pair the elements of which are held in contact bymeans of external forces.
Formclosed pair Kinematic pair the elements of which are constrained to contact each other by means of particular geometric shapes.
Lower pair Kinematic pair that is formed by surface contact between itselements.
14 1 Introduction
Table 1.3. (cont.)
Higher pair Kinematic pair that is formed by point or line contact between its elements.
Revolute pair (hinge)
Pair that allows only a rotary motion between two links.
Prismatic pair Pair that allows only a rectilinear translation between twolinks.
Helical (screw) pair
Pair that allows only a screw motion between two links.
Cylindrical pair Pair for which the degree of freedom is two and that allows a rotation about a particular axis together with an independent translation in the direction of this axis.
Spherical pair Pair for which the degree of freedom is three and that allows independent relative rotations about three mutually orthogonal axes.
Planar contact (sandwich) pair
Pair for which the degree of freedom is three and that allows relative motion in parallel planes.
Universal (Cardan, Hooke’s) joint (universal coupling)
Kinematic joint connecting two shafts with intersecting axes.
Pin joint Joint using a pin as the connecting component between tworigid bodies.
Joint variable Quantity that describes the relative motion between two consecutive links of a manipulator.
Joint space Space defined by a vector whose components are joint variables.
1.4 Terminology 15
Table 1.4. IFToMM terminology with respect to the kinematic chains
Term Definition Kinematic chain Assembly of links and joints.
Closed kinematic chain
Kinematic chain each link of which is connected with at leasttwo other links.
Open kinematic chain
Kinematic chain in which there is at least one link which carries only one kinematic pairing element.
Loop Subset of links that forms a closed circuit.
Tree (mobile) Kinematic chain that contains no loops.
Linkage Kinematic chain whose joints are equivalent to lower pairsonly.
Fourbar linkage Linkage with four binary links.
Kinematic joint Kinematic chain whose kinematical properties are equivalent in some respects to those of a kinematic pair.
Mechanism 1. System of bodies designed to convert motions of, andforces on, one or several bodies into constrained motions of,and forces on, other bodies. 2. Kinematic chain with one of its components (links) taken as a frame.
Kinematic inversion
Transformation of one mechanism into another by choosing a different member to be the frame.
Limit position of a mechanism
Configuration of a mechanism in which one of its links is in alimit position
Planar mechanism Mechanism in which all points of its links describe paths located in parallel planes.
Spherical mechanism
Mechanism in which all points of its links describe paths located on concentric spheres.
16 1 Introduction
Table 1.4. (cont.)
Spatial mechanism
Mechanism in which some points of some of its links describe nonplanar paths, or paths located in nonparallel planes.
Guidance mechanism
Mechanism that guides a link through a prescribed sequence of positions.
Fourbar mechanism
Mechanism with four binary links.
Structure (of a mechanism)
Number and kinds of elements in a mechanism (members andjoints) and the sequence of their contact.
Isomorphism Equality of structures in respect of the numbers of members and joints, and the sequence of their interconnections.
Equivalent mechanism
Mechanism whose kinematical properties are equivalent insome respects to those of another mechanism with a different structure.
Degree of freedom (mobility) of a kinematic chain or mechanism
Number of independent coordinates needed to define the configuration of a kinematic chain or mechanism.
Constraint Any condition that reduces the degree of freedom of a system.
Drive System of mutually connected devices for setting in motion one or several parts of a machine or a mechanism.
Actuator Device that creates direct physical action on the process,other mechanical devices or the surrounding environment toperform some useful function.
1.4 Terminology 17
Table 1.5. IFToMM terminology with respect to the robot kinematics
Term Definition Robot
Mechanical system under automatic control that performs operations such as handling and locomotion.
Manipulator Device for gripping and the controlled movement of objects.
Anthropomorphic robot
Robot with a manipulator containing rotary joints similar to those of a human arm.
Endeffector Device attached to the distal end of a robot arm by which objects can be grasped or acted upon.
Gripper Endeffector which grasps, holds and releases objects.
Module Selfcontained assembly of robot elements which can be connected in various ways with other, not necessarily identical, modules to form a robot.
Parallel manipulator
Manipulator that controls the motion of its endeffector by means of at least two kinematic chains joining the endeffector to the frame.
Walking machine Robot that performs locomotion functions similar to human beings or animals.
Pedipulator Articulated leg of a walking machine.
Proximal Close to the body (away from the endeffector) of a robot arm.
Distal Away from the base (towards the endeffector) of a robot.
Motion Changing position of a body relative to a frame of reference.
Absolute motion Motion with respect to a fixed frame of reference.
Relative motion Motion with respect to a moving frame of reference.
Inverse motion Motion of a frame of reference relative to a moving body.
Frame motion (transportation)
Motion of a moving frame of reference.
18 1 Introduction
Table 1.5. (cont.)
Displacement Change of position of a body with respect to a fixed frame of reference.
Path (trajectory) Line that a moving point describes in a given system of reference.
Axis of rotation Straight line in a rotating rigid body whose points have zero displacement either in a finite time interval, or in an infinitesimal time interval, with respect to a frame of reference.
Screw axis Straight line in rigid body whose points are displaced relative to a frame of reference, either in a finite or an infinitesimal time interval, coaxial with the line itself.
Translation Motion (or component of the motion) of a rigid body in which each straight line rigidly connected with the body remains parallel to its initial direction.
Rectilinear translation
Translation in which the paths of points of a rigid body are straight lines.
Rotation Motion (or a component of the motion of) a rigid body in which all its points move on circular arcs centred on the same axis.
Angular displacement
Displacement of a rigid body in rotation.
Angle of rotation Angle turned through by any line rigidly connected to a rotating body and perpendicular to the axis of rotation.
Precession Rotation of a rigid body about an axis fixed in space when combined with rotation about an axis fixed in the body, the two axes being concurrent.
Nutation Motion of a rigid body occurring simultaneously with a precession, when the angle between the axes of rotation and precession varies in time.
Orientation Movement or manipulation of a rigid body into a predetermined attitude.
Pose Combination of position and attitude.
1.4 Terminology 19
Table 1.5. (cont.)
Reference point Point chosen for position reference in defining a pose.
Planar (plane) motion
Motion of a rigid body in which its points describe curves located in parallel planes.
Spatial motion Motion of a body in which at least one of its points describes a spatial curve.
Screw motion Motion consisting of a rotation combined with a translationthat is parallel to the rotation.
Spherical motion Spatial motion of a body in which all points of the bodymove on concentric spheres.
Limit position Configuration of a mechanism in which the position of a particular link, such as the output, is in some significant respecta maximum or a minimum.
Dead point Configuration of a mechanism in which the input cannot be moved without assistance to the motion of another link.
Velocity Rate of displacement with respect to time.
Absolute velocity Velocity with respect to a fixed frame of reference.
Relative velocity Velocity with respect to a moving frame of reference.
Frame (transportation) velocity
Absolute velocity of a particular point of a moving frame ofreference.
Angular velocity Rate of angular displacement with respect to time.
Instantaneous screw axis
Locus of the points in a rigid body in general spatial motion, whose linear velocity is parallel to the angular velocity vectorof the body at the given instant.
Working range Range of any variable for normal operation of a robot.
Working space (volume)
Totality of points that can be reached by the reference point of a robot arm.
20 1 Introduction
Table 1.5. (cont.)
Singular configuration
Special position of the robot links which implies a reduction of mobility of the endeffector.
Direct task Computation of the pose, motion and forces at the endeffector of a robot arm from given actuator forces, displacements, velocities and accelerations.
Inverse task Computation of actuator forces, displacements, velocities andaccelerations from given forces, pose and motion of the endeffector of a robot.
Redundant mobility (of a robot)
Amount by which the degree of freedom of a robot exceeds the number of independent variables that are necessary to define the task to be performed.
Redundancy Existence of more than one means to accomplish a givenfunction.
Manoeuvrability Ability of a robot with redundant mobility to solve a task byusing various combinations of movements of its links.
Kinematics Branch of theoretical mechanics dealing with the geometry of motion, irrespective of the causes that produce the motion.
Kinematic analysis Analysis of the kinematic aspects of mechanisms.
The IFToMM terminology defines open/closed kinematic chains and
mechanisms, but it does not introduce the notions of simple (elementary) and complex kinematic chains and mechanisms. A closed kinematic chain is a kinematic chain in which each link is connected with at least two other links, and an open kinematic chain is a kinematic chain in which there is at least one link which carries only one kinematic pairing element (see Tables 1.3 and 1.4). In a simple open kinematic chain (openloop mechanism) only monary and binary links are connected. In a complex kinematic chain at least one ternary element exists. We designate in each mechanism two extreme elements called reference element and final element. They are also called distal links. In an open kinematic chain, these elements are situated at the extremities of the chain. In a singleloop kinematic chain, the final element can be any element of the chain except the reference element. In a parallel mechanism, the two extreme elements are the mobile and the reference platform. The two platforms are connected by at least two elemen
1.4 Terminology 21
tary or complex kinematic chains called limbs. Each limb contains at least one joint. An elementary limb is composed of a simple open kinematic chain in which the final element is the mobile platform. A complex limb is composed of a complex kinematic chain in which the final element is also the mobile platform.
As we can see in Table 1.3, IFToMM terminology (Ionescu 2003) use the term kinematic pair to define the mechanical model of the connection of two pairing elements having relative motion of a certain type and degree of freedom. The word joint is used as a synonym for the kinematic pair and also to define the physical realization of a kinematic pair, including connection via intermediate mechanism elements. Both synonymous terms are used in this text.
Usually, in parallel robots, lower pairs are used: revolute R, prismatic P, helical H, cylindrical C, spherical S and planar pair E. The definitions of these kinematic pairs are presented in Table 1.3. The graphical representations used in this book for the lower pairs are presented in Fig. 1.1(a)(f). Universal joints and homokinetic joints are also currently used in the mechanical structure of the parallel robots to transmit the rotation motion between two shafts with intersecting axes. If the instantaneous velocities of the two shafts are always the same, the kinematic joint is homokinetic (from the Greek “homos” and “kinesis” meaning “same” and “movement”). We know that the universal joint (Cardan joint or Hooke’s joint) are heterokinetic joints. Various types of homokinetic joints are known today: Tracta, Weiss, Bendix, Dunlop, Rzeppa, Birfield, Glaenzer, Thompson, Triplan, Tripode, UF (undercutfree) ball joint, AC (angular contact) ball joint, VL plunge ball joint, DO (double offset) plunge ball joint, AAR (angular adjusted roller), helical flexure Ujoints, etc. (Dudiţǎ et al. 2001a,
Fig. 1.1. Symbols used to represent the lower kinematic pairs and the kinematic joints: (a) revolute pair, (b) prismatic pair, (c) helical pair, (d) cylindrical pair, (e) spherical pair, (f) planar contact pair, (g) universal joint and (h) homokinetic joint
22 1 Introduction
Fig. 1.2. Various representations of a GoughStewart type parallel robot: physical implementation (a), CAD model (b), structural diagram (c) and its associated graph (d), Alimb (e) and its associated graph (f)
1.5 Structural synthesis 23
2001b). The graphical representations used in this book for the universal homokinetic joints are presented in Fig. 1.1(g)(h).
A parallel robot can be illustrated by a physical implementation or by an abstract representation. The physical implementation is usually illustrated by robot photography and the abstract representation by CAD model, structural diagram and structural graph. Figure 1.2 gives an example of the various representations of a GoughStewart type parallel robot largely used today in industrial applications. The physical implementation in Fig. 1.2a is a photograph of the parallel robot built by Deltalab (http://www.deltalab.fr/). In a CAD model (Fig. 1.2b) the links and the joints are represented as being as close as possible to the physical implementation (Fig. 1.2a). In a structural diagram (Fig. 1.2c) they are represented by simplified symbols, such as those introduced in Fig. 1.1, respecting the geometric relations defined by the relative positions of joint axes. A structural graph (Fig. 1.2d) is a network of vertices or nodes connected by edges or arcs with no geometric relations. The links are noted in the nodes and the joints on the edges. We can see that the GoughStewart type parallel robot has six identical limbs denoted in Fig. 1.2c by A, B, C, D , E and F. The final link is the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F and the reference member is the fixed platform 1A≡1B≡1C≡1D≡1E≡1F≡0. Each limb is connected to both platforms by spherical pairs. A prismatic pair is actuated in each limb. The spherical pairs are not actuated and are called passive pairs. The two platforms are polinary links, the other two links of each limb are binary links. The mechanism associated with the GoughStewart type parallel robot is a complex mechanism with a multiloop associated graph (Fig. 1.2d). The simple open kinematic chain associated with Alimb is denoted by A (1A≡02A3A4A≡4) – Fig. 1.2e and its associated graph is treetype (Fig. 1.2f).
1.5 Structural synthesis
Recent advances in research on parallel robots have contributed mainly to expanding their potential use to both terrestrial and space applications including areas such as high speed manipulation, material handling, motion platforms, machine tools, medical applications, planetary and underwater exploration. Therefore, the need for methodologies devoted to the systematic design of highly performing parallel robots is continually increasing. Structural synthesis is directly related to the conceptual phase of robot design, and represents one of the highly challenging subjects in recent robotics research. One of the most important activities in the invention and the
24 1 Introduction
design of parallel robots is to propose the most suitable solutions to increase the performance characteristics.
The challenging and difficult objective of structural synthesis is to find a method to set up the mechanical architecture achieving the required structural parameters. The mechanical architecture is defined by number, type and relative position of joint axes in the parallel robot. The structural parameters are mobility, connectivity, redundancy and degree of overconstraint. They define the number of actuators, the degrees of freedom and the motiontype of the moving platform.
In general, parallel manipulators’ performances are highly dependent on their mechanical architecture, so that structural synthesis becomes the central problem in the conceptual design phase, but only a few works can be found in the literature on this topic. We note that this is the first book focusing on the structural synthesis of the mechanical architecture of parallel robots. The topic of this book addresses the problem of structural, also called topological, synthesis of parallel robotic manipulators in a systematic way. This is an “urgent need” put forward by robot manufacturers and scientists.
Torgny Brogårdh (2002) from ABB Automation Technology Products / Robotics, in his conference paper entitled “PKM Research  Important Issues, presented as seen from a Product Development Perspective at ABB Robotics” delivered in the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, said:
“This paper will address some of the most important PKM research issues as seen from a robot manufacturer’s point of view. In section 2 the importance of linking research activities to the application requirements is discussed and in section 3, which is the most important part of this paper, the urgent need for a systematic topology synthesis is put forward. The rest of the paper is structured according to a proposed development process for industrial parallel kinematic robots.”
In his keynote speech entitled “Still a long way to go on the road for parallel mechanisms” presented in ASME 2002 DETC Conference, JeanPierre Merlet from INRIA Sophia Antipolis, France, the author of the first books on parallel robots (Merlet 1990, 1997, 2006) has stated (Merlet 2002): “Synthesis of parallel robot is an open field (there is a very limited number of papers addressing this issue) and, in my opinion, one of the main issue for the development of parallel robots in practice”.
The importance of design synthesis defined by structural and dimensional synthesis is also reiterated by JP. Merlet (2005): “ Synthesis of robots may be decomposed into two processes: structural synthesis (determine the general arrangement of the mechanical struc
1.6 The objectives and originality of this book 25
ture such as the type and number of joints and the way they will be connected) and dimensional synthesis (determine the length of the links, the axis and location of the joints, the necessary maximal joint forces/torques). The performances that may be obtained for a robot are drastically dependent on both synthesis.”
In this paper JP. Merlet has put the emphasis on dimensional synthesis. In this book, we look mainly at structural synthesis which we consider as an essential step in robot design and optimization.
1.6 The objectives and originality of this book
Early research on parallel manipulators (PMs) concentrated on the six degrees of freedom GoughStewarttype mechanism introduced by Gough for a tiretesting device (Gough and Whitehall, 1962) and by Stewart for flight simulators (Stewart, 1965). In the last decade, parallel manipulators with fewer than six degrees of freedom have attracted researchers’ attention. Lower mobility solutions are suitable for many tasks requiring less than six independent motions of the endeffector.
Parallel mechanisms have been the subject of study of much robotic research during the last two decades. Some important studies are now available to support the design of parallel manipulators, such as the pioneering works of Hunt (1973, 1978, 1983) that showed various kinematic architectures of parallel robots, the comprehensive enumerations presented by Merlet (1990, 1997, 2000, 2006) and Tsai (1999, 2001), and the various methods for systematic synthesis of PMs presented in the latest editions of Advances in Robot Kinematics (Lenarčič J, ParentiCastelli V 1996; Lenarčič and Husty 1998; Lenarčič J, Stanišić MM 2000; Lenarčič and Thomas 2002; Lenarčič and Galletti, 2004; Lenarčič and Roth, 2006).
The motion freedoms of the endeffector are usually coupled together due to the multiloop kinematic structure of the parallel manipulator(PM). Hence, motion planning and control of the endeffector for PMs usually become very complicated. With respect to serial manipulators, such mechanisms can offer advantages in terms of stiffness, accuracy, loadtoweight ratio, dynamic performances. Their disadvantages include a smaller workspace, complex command and lower dexterity due to a high motion coupling, and multiplicity of singularities inside their workspace. Uncoupled, fullyisotropic and maximally regular PMs can overcome these disadvantages.
Redundancy in parallel manipulators is used to eliminate some singular configurations (Wang and Gosselin 2004; Kurtz and Hayward 1992; Mer
26 1 Introduction
let 1997; Firmani and Podhorodeski 2004, Alberich et al. 2006), to minimize the joint rates, to optimize the joint torques/forces (Dasgupta and Mruthyunjaya 1998b; Bruckman et al 2006; Nokleby et al. 2005) to increase dexterity workspace (Marquet et al 2001a,b), stiffness (Chakarov 2004), eigenfrequencies, kinematic and dynamic accuracy (Valasek et al. 2004), to improve velocity performances (Krut et al. 2004) and both kinematic and dynamic control algorithms (Liu et al. 2001; Cheng at al. 2003), to develop large forces in micro electromechanical systems (Mukherjee and Murlidhar 2001), decoupling the orientations and the translations (Jin et al. 2004), to obtain reconfigurable platforms (Mohamed and Gosselin 2005) and limbs (Fischer et al. 2004) or combined advantages (Nahon and Angeles 1991; Zanganach and Angeles 1994a,b; Kim 1997; Kock and Scumacher 1998, 2000; Mohamed 2003a,b).
For the first time in the literature, the author recently presented new formulae for calculating the degree of redundancy of parallel robots, along with the use of redundancy for motion decoupling of parallel robots (Gogu 2006a). Redundancy is achieved by introducing additional actuated joints within the limbs beyond the minimum necessary for actuating the manipulator.
Isotropy of a robotic manipulator is related to the condition number of its Jacobian matrix, which can be calculated as the ratio of the largest and the smallest singular values. A robot manipulator is said to be isotropic if the condition number of its Jacobian matrix is equal to one. A robot manipulator is said to be fullyisotropic if it is isotropic throughout the entire workspace. We know that the Jacobian matrix of a robotic manipulator is the matrix mapping (i) the actuated joint velocity space on the endeffector velocity space, and (ii) the static load on the endeffector and the actuated joint forces or torques. Thus, the condition number of the Jacobian matrix is a useful performance indicator characterizing the distortion of a unit hypersphere under this linear mapping. The condition number of the Jacobian matrix was first used by Salisbury and Craig (1982) to design mechanical fingers and developed by Angeles (1997) as a kinetostatic performance index of robotic mechanical systems. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator (Zanganeh and Angeles 1997; Takeda and Funabashi 1998; Baron and Barnier 2001; Baron et al. 2002; Mohammadi Daniali and ZsomborMurray 1999; Badescu et al. 2002; Fattah and Ghasemi 2002; Krut et al. 2002; Caro et al. 2003; Fassi et al. 2003, 2005; Tsai and Hunag 2003; Carricato and ParentiCastelli 2002; Carricato 2005; Liu et al. 2005; Pashkevich et al. 2005).
The author has proposed a systematization of PMs based on the form of the Jacobian matrix J. We distinguish five types of PMs (Gogu, 2004a; Gogu, 2007a): (i) maximally regular PMs, if the Jacobian is an identity
1.6 The objectives and originality of this book 27
matrix throughout the entire workspace, (ii) fullyisotropic PMs, if the Jacobian is a diagonal matrix with identical diagonal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. Maximally regular and fullyisotropic PMs give a onetoone mapping between the actuated joint velocity space and the external velocity space.
The first solution for a fullyisotropic T3type translational parallel robot was developed at the same time and independently by Carricato and ParentiCastelli at University of Genoa, Kim and Tsai at University of California, Kong and Gosselin at University of Laval, and the author of this work at the French Institute of Advanced Mechanics. In 2002, the four groups published the first results of their works (Carricato and ParentiCastelli 2002; Kim and Tsai 2002; Kong and Gosselin 2002b; Gogu 2002b). Each of the last three groups has built a prototype of this robot in their research laboratories and called this robot CPM (Kim and Tsai 2002), Orthogonal Tripteron (Gosselin et al. 2004) or Isoglide3T3 (Gogu 2004b). The first practical implementation of this robot was the CPM developed at the University of California.
The systematic synthesis of spatial mechanisms used in parallel robots is mainly founded on four approaches: group theory, screw theory, velocityloop equations and theory of linear transformations (Table 1.6).
The method proposed in this book is founded on the theory of linear transformations and evolutionary morphology (Gogu 2005a) and allows us to obtain the structural solutions of decoupled, uncoupled, fullyisotropic and maximally regular PMs with two to six DoFs in a systematic way. The new formulae for mobility, connectivity (spatiality), redundancy and overconstraint of PMs proposed recently by the author, Gogu (2005a, 2005b, 2005c, 2005d) are integrated into the synthesis approach developed in this book.
Various solutions of TaRbtype PMs are known today. In this notation, a=1,2,3 indicates the number of independent translations and b=1,2,3 the number of independent rotations of the moving platform. The parallel robots actually proposed by the robot industry have coupled and decoupled motions and just some isotropic positions in their workspace. As far as we are aware, this is the first book of robotics presenting solutions of uncoupled, fullyisotropic and maximally regular PMs.
28 1 Introduction
Table 1.6. Approaches used in the systematic synthesis of spatial mechanisms and parallel robots
Approach References Group theory
Hervé 1978, 1995, 1999, 2004; Hervé and Sparacino 1991; Karouia and Hervé 2000, 2002, 2004; Angeles 2004; Li et al. 2004; Huynh and Hervé 2005; Lee and Hervé 2006; Rico et al. 2006.
Screw theory Tsai 1999; Fang and Tsai 2002; Frisoli et al. 2000; Kong and Gosselin 2001, 2004a, 2004b, 2004c, 2004d, 2006; Huang and Li 2002, 2003; Carricato 2005
Velocityloop equations Di Gregorio and ParentiCastelli 1998; Di Gregorio 2002, Carricato and ParentiCastelli 2002, 2003
Theory of linear transformations
Gogu 2002, 2003, 2004a, 2004b, 2004c, 2004d, 2005e, 2005f, 2005g, 2005h
Nonredundant/redundant, overconstrained/isostatic solutions of uncou
pled and fullyisotropic PMs with elementary/complex limbs actuated by linear/rotary actuators with/without idle mobilities and two to six DoFs are present in a systematic approach of structural synthesis. These solutions are derived from families of PMs with decoupled motions. A serial kinematic chain is associated with each elementary limb, and at least one closed loop is integrated in each complex limb.
The synthesis methodology and the solutions of PMs presented in this book represent the outcome of some recent research developed by the author in the last four years in the framework of the projects ROBEAMAX (20022003) and ROBEAMP2 (20042005) supported by the French National Council of Scientific Research (CNRS). These results have been partially published by the author in the last two years. In these works the author has proposed the following for the first time in the literature:
a) new formulae for calculating the degree of mobility, the degree of connectivity(spatiality), the degree of redundancy and the number of over
1.6 The objectives and originality of this book 29
constraints of parallel robotic manipulators that overcome the drawbacks of the formulae founded on the ChebychevGrüblerKutzbach (CGK) mobility formula (Gogu 2005be),
b) a new approach to systematic innovation in engineering design called evolutionary morphology (Gogu 2005a),
c) solutions to follow fullyisotropic parallel robots: parallel wrists with two (Gogu 2005f) and three (Gogu 2007c) degrees of mobility, planar parallel robots (Gogu 2004c), parallel robots of T1R2type (Gogu 2005i), parallel robots with Schönflies motions (Gogu 2004a, 2005g, 2006c, 2007a), parallel robots of T3R2 type (Gogu 2006a,b,d), hexapods with six degrees of freedom (Gogu 2006e).
The various solutions proposed by the author belong to a modular family called IsoglideNTxRy with a+b=n with 2≤ n≤ 6, a=1,2,3 and b=1,2,3. The mobile platform of these robots (Gogu 2007b) can have any combination of n independent translations (T) and rotations (R).The IsoglidenTaRb modular family was developed by the author and his research team at the French Institute of Advanced Mechanics (IFMA) in ClermontFerrand.
The two parts of this work gather the author’s contributions, and develop them in a general and organic treatment of structural synthesis of parallel robotic manipulators, including new solutions presented for the first time in the literature. The main objectives of this book are as follows:
• to present an original method of defining and calculating the structural parameters of parallel robots (mobility, connectivity, redundancy, degree of overconstraint) founded on the theory of linear transformations,
• to present an original method for structural synthesis of parallel robots founded on the theory of linear transformations and evolutionary morphology,
• to propose innovative solutions of parallel manipulators with 2 to 6 degrees of freedom and various degrees of motion coupling, with the emphasis on decoupled, uncoupled, fullyisotropic and maximally regular solutions.
The innovative solutions presented in this book are useful in the design of new parallel robots for many applications: haptic devices, machinetools called parallelkinematicsmachines (PKM), robotassisted surgery, surveillance, telescope and pointing devices, motion simulation, testing machines, etc. The book will be useful for students, engineers and researchers working in associated fields of engineering.
Many solutions of parallel robots obtained through this systematic approach of structural synthesis are presented here for the first time in the lit
30 1 Introduction
erature. The author had to make a difficult and challenging choice between protecting these solutions through patents, and releasing them directly into the public domain. The second option was adopted by publishing them in various recent scientific publications and mainly in this book. In this way, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products.
2 Structural parameters
This section introduces the main structural parameters used in the structural synthesis of parallel robots: connectivity, mobility, redundancy, degree of overconstraint. We recall that IFToMM definitions of mobility, connectivity, redundancy and constraint are presented in Tables 1.31.5. In this chapter, we present an original approach based on the theory of linear transformations to define and to demonstrate new formulae for the calculation of these parameters along with the state of the art before the first publication of this new approach by the author of this work (Gogu 2005be).
We will show that the classical ChebychevGrüblerKutzbach formula for global mobility calculation of multiloop mechanisms does not fit many parallel manipulators. We know that a mobility formula is an explicit relationship between mobility and the structural parameters of the mechanism. Usually, these structural parameters are easily determined by inspection without the need to develop the set of constraint equations used in instantaneous mobility calculation. We introduce a mathematical framework for defining and calculating the global mobility of parallel robots. With this formal structure, combined with ideas from the theory of linear transformations, we define a new mobility formula for parallel manipulators with simple or complex limbs. Limb connectivity and mobile platform connectivity are the key parameters of the new mobility formula. These parameters are used also to calculate the degree of redundancy and the degree of overconstraint of a parallel manipulator. They are also useful to determine the necessary conditions to obtain a nonoverconstrained solution.
As far as we are aware, this is the first book to present the general relations between mobility, redundancy and overconstraint of parallel manipulators and the connectivity of the mobile platform. These relations are demonstrated via the theory of linear transformations and are used mainly in the structural analysis and synthesis presented in the following chapters. Examples of practical applications of the new formulae in structural analysis of parallel robots and other single and multiloop mechanisms are presented in chapter 3.
32 2 Structural parameters
2.1 Critical review of mobility calculation
Mobility is the main structural parameter of a mechanism and also one of the most fundamental concepts in the kinematic and the dynamic modelling of mechanisms. IFToMM terminology defines the mobility or the degree of freedom as the number of independent coordinates required to define the configuration of a kinematic chain or mechanism (Ionescu 2003). Mobility M is used to verify the existence of a mechanism (M>0), to indicate the number of independent parameters in robot modelling and to determine the number of inputs needed to drive the mechanism.
Earlier works on the mobility of mechanisms go back to the second half of the nineteenth century. During the twentieth century, sustained efforts were made to find general methods for the determination of the mobility of any rigid body mechanism. Various formulae and approaches were derived and presented in the literature. Contributions have continued to emerge in the last years. Mobility calculation still remains a central subject in the theory of mechanisms.
In the calculation of a mechanism’s mobility, the following key controlling parameters are used: the number of joints, p; the number of mechanism elements n (n=m1; m denotes the total number of elements, including the fixed base); the mobility f; the degree of constraint, c ,of the joint; the motion parameter,b; the constraint parameter d=6b of the mechanism; and the number of independent closed loops of the mechanism, q.
The various methods proposed in the literature for mobility calculation of the closed loop mechanisms fall into two basic categories:
a) approaches for mobility calculation based on setting up the kinematic constraint equations and calculating their rank for a given position of the mechanism with specific joint locations,
b) formulae for a quick calculation of mobility without the need to develop the set of constraint equations.
The approaches used for mobility calculation based on setting up the kinematic constraint equations and their rank calculation are valid without exception. The major drawback of these approaches is that the mobility cannot be determined quickly without setting up the kinematic model of the mechanism. Usually this model is expressed by the closure equations that must be analyzed for dependency. There is no way to derive information about mechanism mobility without performing position, velocity or static analysis by using analytical tools (screw system theory, linear algebra, affine geometry, Lie algebra, etc). For this reason, the real and practical value of these approaches is very limited in spite of their valuable theoretical foundations. The rank of the constraint equations is calculated in a
2.1 Critical review of mobility calculation 33
given position of the mechanism with specific joint locations. The mobility calculated in relation to a given configuration of the mechanism is an instantaneous mobility which can be different from the general mobility (global mobility or gross mobility). Global mobility has a single value for a given mechanism; it is a global parameter characterizing the mechanism in all its configurations except its singular ones. Instantaneous mobility is a local parameter characterizing the mechanism in a given configuration including singular ones. In a singular configuration the instantaneous mobility could be different from the global mobility.
The formula for a quick calculation of mobility is an explicit relationship between structural parameters of the mechanism: the number of links and joints, the motion/constraint parameters of joints and of the mechanism. Usually, these structural parameters are easily determined by inspection without any need to develop the set of kinematic constraint equations.
We note that the classical formulae for a quick calculation of mobility do not fit many classical mechanisms such as the mechanisms proposed by De Roberval 1670; Sarrus 1853; Delassus 1900, 1902, 1922; Bennett 1902; Bricard 1927; Myard 1931a,b; Goldberg 1943; Altman 1952; Baker 1978; Baker et al. 1982, Waldron 1979; Wohlhart 1987, 1991, 1995; etc., or for many recent parallel robots. Special geometric conditions play a significant role in the determination of the mobility of these mechanisms. When these mechanisms were limited to special examples, considered as “curiosities”, they were called by various names: paradoxical mechanisms (Bricard, 1927), paradoxical chains (Hervé 1999, 1978a,b; Norton 1999), exceptions (Myszka 2001), special cases (Mabie 1987); linkage with a paradox between “practical degrees of freedom” and “computed degrees of freedom” (Eckhardt 1998), overconstrained yet mobile linkage (Phillips 1984, 1990; Waldron and Kinzel 1999), linkages with anomalous mobility (Waldron and Kinzel 1999).
The formulae known in the literature for the determination of mobility are not applicable to many types of recent parallel robots, see for example the robots Delta (Clavel 1988, 1990), Star (Hervé and Sparacino 1992, 1993), H4 (Pierrot and Company 1999), Orthoglide (Wenger and Chablat 2000), CPM (Kim and Tsai 2002), Tripteron (Gosselin et al. 2004) and other parallel manipulators presented in the literature (Kong and Gosselin 2002, Carricato and ParentiCastelli 2002) or recently proposed by the author belonging to Isoglide family (Gogu 2002b, 2004ad, 2005fj, 2006ae, 2007ac). We cannot continue to consider the mechanisms that do not fit the various methods or equations for determining mobility as having structural flaws. We have to review the formulae and consider the flaw to be that of the formula rather than of the mechanism. In this chapter we will
34 2 Structural parameters
establish limits of these formulae, and propose a new formula for quick calculation of the mobility of parallel robots.
Many formulae/approaches proposed in the last 150 years for the calculation of the mechanism mobility can be reduced to the same original formula. Previous works on mobility calculation (Wittenbauer 1923; Federhofer 1932; Artobolevskii 1953; Boden 1962; Bogolyubov 1964; Mruthyunjaya 2003) have mentioned various formulae but they have not analysed their genesis and similarities. In this section, we focus on a brief presentation and a critical analysis of these formulae/approaches to clearly identify the different contributions in this very important subject of the theory of mechanisms. Thirty five contributions are reviewed. To ensure unity of presentation, the original notations of some of the authors were not maintained; this also made it easier to distinguish between the different formulae. These contributions had been published before the first publication, in 2005, of the new approach proposed by the author (Gogu 2005be).
We apply each formula to determine the degree of mobility of a recent parallel manipulator with simple limbs, the parallel Cartesian robotic manipulator CPM (Kim and Tsai 2002; Kong and Gosselin 2002, Carricato and ParentiCastelli 2002, Gogu 2002b) presented in Fig. 2.1, which only contains revolute (R) and prismatic (P) joints. A serial kinematic chain is associated with each simple limb linking the mobile platform to the fixed base. The mechanism has n=10 kinematic elements (links), p0=3 prismatic joints adjacent to the fixed element, pn=9 revolute joints nonadjacent to the fixed element and q=2 independent closed loops. Each loop contains the same number and the same type of joints. The three limbs are identical
Fig. 2.1. Parallel Cartesian robotic manipulator: (a) kinematic chain; (b) associated graph
2.1 Critical review of mobility calculation 35
from a structural point of view. In each limb the direction of the prismatic joint and the axes of the revolute joints are parallel (PRRR). Kim and Tsai (2002) have mentioned that this mechanism has 3 degrees of freedom, in spite of the fact that the general degreesoffreedom equation presented by Tsai (2000) predicts that the mechanism has zero degrees of freedom.
We analyse the results obtained by using the different formulae to determine the degree of mobility of the CPM robotic manipulator, and we draw conclusions on the limits to the various formulae.
2.1.1 Chebychev’s contribution
In the second half of the nineteenth century, the mathematician P. A. Chebychev published an article (concerning a 4bar linkage) in two parts in the proceedings of the Russian Royal Academy of Science of SaintPetersburg. Chebychev was the first scientist who proposed a mathematical formalisation for the calculation of mechanism mobility. The first part of the article (Chebychev 1854) was published under the impulse of Watt’s invention, known at the end of the eighteenth century. In this article, Chebychev applied his method of polynomial function approximation to find the lengths of the elements of the 4bar mechanism so that it could describe a given couple curve with the least error. In the second part of the article, Chebychev (1869) proposed the first formula for the calculation of the number of independent variables in a mechanism. This contribution was quickly integrated into the first treatises on the theory of mechanisms published at the end of the nineteenth century (DwelshauvresDery 1876, Laboulaye 1878).
Chebychev expressed his formula in the form:
3n 2(p0+pn)=1 (2.1)
where 3n represents the number of variables required to describe the position and the orientation of the n kinematic (mobile) bars in the plane and 2(p0+pn) is the number of constraint equations imposed by the p=(p0+pn) revolute joints of the mechanism that can be adjacent (p0) or nonadjacent (pn) to the fixed base. It is known that each revolute joint introduces two constraint equations in a planar mechanism. Chebychev applied this formula to simple planar mechanisms with p0=2 and to complex planar mechanisms (p0=3) having only revolute joints and one degree of mobility. He did not extend the use of Eq. (2.1) to other types of mechanisms. This formula could be extended in the form
M= 3n 2(p0+pn) (2.2)
36 2 Structural parameters
to any planar mechanism with one degree of freedom joints (helical, prismatic and revolute joints).
The wellknown λ 4bar mechanisms were obtained by Chebychev to describe couple curves containing a straight line segment by using Eq. (2.1).
Supposing that we try to extrapolate Eq. (2.2), we obtain M=6 for the mechanism presented in Fig. 2.1 (with n=10, p0=3 and pn=9). This result is obviously erroneous, Chebychev’s formula (2.2) being limited to the planar mechanisms (mechanisms in which the axes of all the revolute joints are parallel and perpendicular to the plane containing the directions of the prismatic pairs).
2.1.2 Sylvester’s contribution
J. J. Sylvester (1874) presented a modified form of Eq. (2.1) as a structural condition for one degree of freedom pinconnected planar mechanisms:
3m2p4=0. (2.3)
In Eq. (2.3), m is the total number of links of a mechanism including the fixed element. Sylvester stated Eq. (2.3) in the following words: “In order for a combination of links to fulfil this so to say fatalistic condition of constrained motion and to entitle it to the name of a linkage in the speaker’s sense, a numerical relation must be satisfied between the number of links and the number of joints, viz., three times the number of links must be four greater than twice the number of joints. In applying this rule it must be understood that, if three links are jointed together, the junction counts for two joints; if four are jointed together, for three joints; and so on.” We can see that Eq. (2.3) can be obtained by simply replacing n by m1 in the Chebychev’s formula and, in the general case, joints connecting a links count as a1 joints.
2.1.3 Grübler’s contribution
M. Grübler (1883, 1885) presented a structural condition for one degree of freedom planar mechanisms identical to Eq. (2.3). Later, Grübler proposed a structural condition for one degree of freedom spatial complex mechanisms with helical joints (Grübler 1916, 1917):
5h6m+7=0 (2.4)
were h is the total number of helical joints.
2.1 Critical review of mobility calculation 37
2.1.4 Somov’s contribution
P. O. Somov (1887) proposed the following structural condition for one degree of freedom mechanisms:
mq(b1)=2. (2.5)
Somov introduced for the first time the motion parameter (b), called mobility number, for planar (b=3) and for spatial mechanisms (b=6).
2.1.5 Hochman’s contribution
Hochman made an important contribution to the structural analysis of mechanisms in his book on the kinematics of machinery (Hochman 1890). A detailed presentation of Hochman’s contribution can be found in (Bogolyubov 1964; Dudiţă and Diaconescu 1987).
He proposed a relation between the total number of links m, the total number of joints p and the number of independent closed loops q in a complex mechanism (m+qp=1) .This is Euler’s formula in the theory of graphs.
He also proposed various structural conditions for the existence of one degree of freedom mechanisms:
b(m1)C=1 (2.6)
F qb=1 (2.7)
b (pq)C=1 (2.8)
with b 1
bi
i 1C iC
−
=
=∑ (2.9)
b 1b
ii 1
F iF−
=
=∑ (2.10)
F+C=pb (2.11)
where C is the total number of constraints imposed by the joints and F – the total number of degrees of freedom of the joints. b
iC represents the number of joints having the degree of constraint b
ic i= , and biF is the
number of pairs having the degree of connectivity bif i= . Hochman de
38 2 Structural parameters
fined the motion parameter b as the mechanism category given by the number of simple existing motions between any two links (b<6). He marked the difference between degree of constraint of a joint in the motion space with dimension b, denoted by b bc b f= − , and the degree of constraint of a joint in the motion space with dimension b=6 , also called joint class and denoted by c=6f.
We can see that Eq. (2.6) extends Eq. (2.1) proposed by Chebychev. For the general case of a mechanism with mobility M, Eqs. (2.62.8) become:
M=b(m1)C (2.12)
M=F qb (2.13)
M=b(pq)C. (2.14)
Hochman considered that a mechanism with M=1 has determined motions. If M<1 the mechanism is a fixed structure without relative motions and if M>1 the motions of the mechanisms are not determined. We can see that Hochman limited the mechanisms with determined motions to one degree of freedom mechanisms. He considered that Eqs. (2.122.14) are applicable to any kind of mechanism having various types of joints and the same b for each independent loop (b=3 for planar and spherical mechanisms, b=6 for spatial mechanisms).
By applying Eqs. (2.12)( 2.14) to the mobility calculation of the mechanism presented in Fig. 2.1, with m=11, q=2, p=12, we obtain: M=6 (if we presume b=3 and consequently C=12*2=24), M=4 (if b=4 and C=12*3=36), M=2 (if b=5 and C=12*4=48) and M=0 (if b=6 and C=12*5=60).
We can see that the results obtained for this mechanism are erroneous for all values presumed for b (b=3,…,6). It is obvious that the mechanism presented in Fig. 2.1 has the same b for all the closed loops, but the results obtained by using the formulae proposed by Hochman are erroneous. So, we can say that Eqs. (2.122.14) do not work for all mechanisms even if the independent loops have the same motion coefficient.
2.1.6 SomovMalytsheff’s formula
SomovMalytsheff formula (Somov, 1887; Malytsheff 1923) 5
ii 1
M=6n iC=∑
(2.15)
2.1 Critical review of mobility calculation 39
represents a particular case of Eqs. (2.12) and (2.9) proposed by Hochman (with the notations n=m1 and 6
iC =C). Eq. (2.15) is applicable only to spatial mechanisms with b=6.
2.1.7 Koenigs’ formula
Gabriel Koenigs (1905) published an introduction to a new theory of mechanisms in which he presented an equation for the calculation of the degrees of freedom of a mechanism
M=6(m1)C. (2.16)
where 6(m1) is the number of parameters necessary to define the relative positions of the n=m1 kinematic links of the mechanism and C is the number of independent constraint equations given by the joints. We can see that Eq. (2.16) represents the particular case of Eq. (2.12), proposed by Hochman, in which b=6.
2.1.8 Kutzbach’s mobility equation
The mobility equation proposed by Kutzbach (1929)
M=(6d)(m1)p
ii 1
( 6 d f )=
− −∑ (2.17)
also represents the particular case of Eq. (2.12) for a spatial mechanism with b=6d, 6
i if f= and
p
ii 1
C (6 d f )=
= − −∑ . (2.18)
In the literature, Eq. (2.17) is considered to be applicable to all mechanisms having the same d for each independent loop.
By applying Eqs. (2.17) to the mobility calculation of the mechanism presented in Fig. 2.1 (for which m=11, p=12 and fei=1, i=1,2,…,12), we obtain : M=6 (if we assume d=3), M=4 (if d=2), M=2 (if d=1) and M=0 (if d=0). This shows that the results obtained for the mechanism analysed are also erroneous for all values presumed for d (d=0,…,3). So, we can say that Eq. (2.17) does not work for all mechanisms even if the independent loops have the same motion coefficient.
40 2 Structural parameters
2.1.9 Dobrovolski’s mobility equation
Dobrovolski named the constraint parameter d as the mechanism family and indicated the following mobility equation (Dobrovolski 1949, 1951):
5
d ii d 1
M =(6d)n ( i d )C= +
−∑ (2.19)
where 6 di iC C −= represents the number of joints having the degree of con
straint 6 dic i d− = − . We can see that Eqs. (2.19) and (2.12) are similar by
taking into consideration that n=m1 and 5
ii d 1
C ( i d )C= +
= −∑ (2.20)
Dobrovolski considered d=6b=0,…,4, but he did not indicate any method for setting up d.
Eq. (2.19) written in the form 6 d 1
d ii 1
M (6 d )( m 1) (6 d i )F− −
=
= − − − − −∑ (2.21)
is also known as the ArtobolevskiDobrovolski mobility equation (Bagci 1971). In Eq. (2.21), Fi= 6 d
iF − is the number of pairs having the degree of connectivity 6 d
if i− = . Eqs. (2.19) and (2.21) are also considered in the literature to be applicable to any mechanism having the same d for each independent loop.
By applying Eqs. (2.19) and (2.21) to the mobility calculation of the mechanism presented in Fig. 2.1 we obtain the same erroneous results that were obtained by using Eqs. (2.17) and (2.12): M=6 (if we presume d=3), M=4 (if d=2), M=2 (if d=1) and M=0 (if d=0). We can see that Eqs. (2.19) and (2.21) do not work for all mechanisms, even if the independent loops have the same motion coefficient, as with Eqs. (2.12) and (2.17) from which they can be derived.
2.1.10 Contribution of Y.F. Moroskine
To calculate the mobility of a planar or spatial mechanism, Y. F. Moroskine (1954, 1958) proposed the formulae:
M=Fr (2.22)
2.1 Critical review of mobility calculation 41
5
1i
iM iC r
=
= −∑ (2.23)
where F is the total number of kinematic parameters of the mechanism, r  the rank of the linear homogeneous set of equations defining the kinematic constraints, Ci  the number of joints of class i (joints with the degree of freedom 6i). The two formulae proposed by Moroskine are valid without exception, but these formulae allow a quick calculation of mobility only after developing the kinematic equations and finding their rank. These calculations will be presented in Section 2.2.
2.1.11 Contribution of R. Voinea and M. Atanasiu
R. Voinea and M. Atanasiu (1960) proposed the following formulae for mobility calculation of singleloop closed mechanisms:
1 2 eM F r F ( r +r w )= − = − (2.24)
where F is the total number of degrees of freedom in the joints given by the total number of kinematic pairs with a connectivity of one (revolute, prismatic and helical joints) equivalent to the joints of the mechanism, r is the rank of the screw system equivalent to the joints of the singleloop mechanism, r1 and r2 are the ranks of the screw systems equivalent to the two open subchains connecting a link e to the fixed base. In the singleloop mechanism the link e has the connectivity we with respect to the fixed base.
To calculate the mobility of a complex mechanism, R. Voinea and M. Atanasiu (1960) proposed the formula:
1
q
j pj
M F r p=
= − +∑ (2.25)
where rj is the rank of the screw system of joints with connectivity one equivalent to the joints of the jth independent closed loop of the mechanism, and pp is the total number of passive joints in the mechanism. To calculate the rank rj, R. Voinea and M. Atanasiu used the theory of the instantaneous screw axis proposed by Ball (1900). An important contribution was provided by R. Voinea and M. Atanasiu by identifying the geometrical configurations of screw systems with 1≤r≤6 and by proposing an analytical method for computing the motion parameter bj=rj (Voinea and Atanasiu 1959, 1960).
42 2 Structural parameters
By applying Eq. (2.25) to the mobility calculation of the mechanism presented in Fig. 2.1 (for which F=p=12, r1=r2=5 and pp=0) we obtain M=2, which is an erroneous result. We can see that all the closed loops of the mechanism are identical and have the same rank (rj=5).
Eq. (2.25) is the first mobility formula applicable to mechanisms having independent closed loops with different values for the motion coefficient bj=rj, but this formula does not work for all complex mechanisms, even if rj is the same for all independent loops.
2.1.12 Kolchin’s mobility equation
Kolchin (1960) modified the ArtobolevskiDobrovolski mobility equation (2.21) and expressed it in the form:
6 d 1
ii 1
M 6( m 1) (6 i )F dq− −
=
= − − − +∑ (2.26)
Kolchin’s contribution was also summarized in (Erdman 1993). By applying Eq. (2.26) to the mobility calculation of the mechanism
presented in Fig. 2.1, we obtain the same erroneous results that were obtained by using Eqs. (2.12), (2.17), (2.19) and (2.21).
2.1.13 Rössner’s contribution
The mobility equation proposed by W. Rössner (1961)
M=p
ii 1
f=∑ 6(pm)6
(2.27)
also represents a particular case of Eq. (2.13) for the spatial mechanism with b=6 (by taking into consideration Euler’s formula q=pm+1).
Rössner also mentioned the existence of so called “half open” kinematic chain containing at least one link which must have at least four joints. The “half open” kinematic chain is also known in the literature as a kinematic chain with fractionated freedom.
2.1.14 Boden’s mobility equation
H. Boden (1962) extended Eq. (2.27) proposed by Rössner to make it applicable to spatial mechanisms (b=6) containing planar loops (b=3):
2.1 Critical review of mobility calculation 43
M=p
ii 1
f=∑ 6(pm)6+D
(2.28)
where
[ ]z
Ek Ekk 1
D 3( p m ) 3=
= − +∑ . (2.29)
z is the number of distinct directions of the joint axes belonging to planar loops, pEk and mEk are the number of joints and the number of links in the planar loops having the axes of the revolute joints in kth direction. D takes into consideration the planar loops existing in the spatial mechanism.
2.1.15 Manafu’s formula
In order to take into consideration the fact that the degree of mobility of a complex kinematic chain depends on the choice of the independent closed loops, N. Manolescu and V. Manafu (1963) proposed the following mobility equation, presented as Manafu’s mobility equation (Manolescu 1968):
1
q
i ci
M max( M M )=
= −∑ (2.30)
where Mi is the mobility of the simple closed loop i, and Mc is the mobility of the “connecting mechanism” (the mechanism formed with the common joints between different closed loops). To calculate the degree of mobility of an simple closed loop, Manolescu and Manafu used Dobrovolski’s mobility equation (2.19). The mobility of the connecting mechanism is given by the sum of the degrees of freedom of the common joints.
Each simple closed loop of the mechanism presented in Fig. 2.1 has d=1 and Mi=3 (i=1,2). We can see that any two simple closed loops have four common joints (Mc=4). So, by using Eq. (2.30) we obtain the erroneous value of M=2 for the mobility of the mechanism presented in Fig. 2.1.
2.1.16 Ozol’s formula
Ozol’s formula (Ozol 1963) 5
cc 1
M= (6 c )C 6q=
− −∑ (2.31)
44 2 Structural parameters
represents a particular case of Eqs. (2.13) and (2.10) proposed by Hochman for b=6 and the notation 6
iF =Fi and 5cc=1
(6 c )C−∑ = 5ii 1
iF=∑ ,
where cC represents the number of joints of class c and iF the number of pairs having the degree of connectivity if i= .
2.1.17 Contribution of K. J. Waldron
To calculate the mobility of a closed loop, Waldron (1966) proposed the formula
M=F b− (2.32)
which also represents a particular case of Eqs. (2.13) and (2.10) proposed by Hochman (for q=1). Waldron called b the order of the equivalent screw system of the closed loop and he defined it as the order of the screw system of a series chain containing the same joints as the loop in the same geometrical relationship. To calculate b, Waldron used the theory of the instantaneous screw axis proposed by Ball (1900), and developed later by Voinea and Atanasiu (1959, 1960), Phillips and Hunt (1964), Hunt and Phillips (1965).
Waldron delineates the series and parallel laws for screw system theory and shows how to determine both the relative freedom between particular links and the mobility of a certain class of multiloop linkages in which some members (links) are shared by two or more loops. He shows that such linkages may be handled by considering one loop to be a complex joint in the other. In such cases, the values of b applicable to each closed loop may vary with the order in which the closed loops are considered. He emphasized the importance of relative freedom between two particular members of a linkage in mobility calculation. An example is given in (Waldron, 1966) for the mechanism known as the De Roberval Scale.
2.1.18 Contribution of N. Manolescu
N. Manolescu and V. Manafu (1963) also proposed the mobility equation q5
i ji 1 j 1
M=6n iC max d= =
+∑ ∑ (2.33)
that extends formula (2.15), proposed by SomovMalytsheff, by adding a supplementary term. In this term, dj is the family of the simple closed loop j. By applying Eq. (2.33) to the mobility calculation of the mechanism pre
2.1 Critical review of mobility calculation 45
sented in Fig. 2.1, with n=10, q=2, 5ii 1
C iC=
= ∑ =12*5=60, we obtain: M=0 (if we presume d1= d2=0), M=2 (if d1= d2=1), M=4 (if d1= d2=2), M=6 (if d1= d2=3), M=8 (if d1= d2=4) and M=10 (if d1= d2=5).
The mechanism presented in Fig. 2.1 has the same dj for all the closed loops, but the results obtained by using the formula proposed by Manolescu are erroneous even if the independent loops have the same motion coefficient.
Manolescu (1968) proposed the mobility formula 5
cc d 1
M= (6 c )C (6 d )q= +
− − −∑ (2.34)
that extends formula (2.31), proposed by Ozol. By taking into consideration the notation 6d=b, 6
iF =Fi and 5 5c ic=1 i 1
(6 c )C iF=
− =∑ ∑ , where
cC represents the number of joints of class c and iF the number of pairs having the degree of connectivity if i= . We can note that Manolescu’s formula (2.34) is identical to Eq. (2.13) proposed by Hochman.
Manolescu (1964) described three kinds of kinematic chains having total, partial and fractional mobility. A multiloop kinematic chain has a total mobility M>0 if each of its loops have a mobility Mi≥M and a partial mobility if it has at least one loop of mobility Mi<M. The multiloop kinematic chain of mobility M is said to have a fractional mobility if it contains a socalled “separation link” which when cut into two split the chain into two separate subchains of mobilities M1 and M2 such that M=M1+M2.
2.1.19 Contribution of C. Bagci
Bagci (1971) proposed the following mobility equation to calculate the degrees of freedom of motion in a mechanism of m links and q loops:
M=Mi+Mr=M0+D+McMp (2.35)
in which Mi is the number of inputs required to drive the mechanism, 1
1
mr j rj
M f−
==∑ is the total number of redundant freedoms in the mecha
nism, frj is the number of redundant freedoms of the link j. r
0 ii 1M b( m 1) ( b i )F
== − − −∑ (2.36)
where r=bdmin1 is the maximum number of degrees of freedom that a joint can have in the motion space of the mechanism and dmin is the small
46 2 Structural parameters
est number of general constraints in the loops of the mechanism. By taking into consideration the notation b=6d, Eq. (2.36) represents ArtobolevskiDobrovolski’s mobility equation (2.21) when dmin=d.
qjj 1
D d=
=∑ is the total number of general constraints in the mechanism.
This term is similar to the supplementary term introduced by Manolescu and Manafu in Eq. (2.33) to extend the formula proposed by SomovMalytsheff.
Ngc cjj 1
M ( q 1)=
= −∑ is the total number of overclosing constraints in the
mechanism, gN  the number of groups of loops introducing overclosing constraints, cjq  the number of loops in the group j introducing overclosing constraints.
pp pjj 1
M f=
=∑ is the total number of passive mobilities in the joints of the
mechanism, pjf  the number of passive mobilities in joint j. The mechanism presented in Fig. 2.1 have Mc=Mp=0. By applying Eq.
(2.35) to the mobility calculation of this mechanism, we obtain the same erroneous results that were obtained by using Eq. (2.33).
2.1.20 Contribution of P. Antonescu
P. Antonescu proposed two mobility formulae applicable to mechanisms having independent closed loops with different values for the motion coefficient bi. The first formula (Antonescu 1973)
5
da a a ii 1
M =(6d )(m1) ( i d )C=
−∑ (2.37)
represents a generalization of Eq. (2.18) proposed by Dobrovolski, in which m1=n and da is the apparent family calculated as the average of the constraint parameters di (i=1,2,..q) of the independent loops
q
a ii 1
1d dq =
= ∑ . (2.38)
By applying Eq. (2.37) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain the same erroneous results that were obtained by using Eq. (2.33).
The second formula proposed by Antonescu (1999)
2.1 Critical review of mobility calculation 47
5 6
i ji 1 j 2
M= iF jR= =
−∑ ∑ (2.39)
represents a particular case of Eq. (2.25) proposed by Voinea and Atanasiu where pp=0. We see that 5
ii=1F= iF∑ and q 6
j jj=1 j 2r jR
==∑ ∑ , where Fi
represents the number of joints having the degree of connectivity if i= and Rj is the number of closed loops having the motion coefficient bj=j. By applying Eq. (2.39) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain the same erroneous results that were obtained by using Eq. (2.25) from which Eq. (2.39) can be derived.
2.1.21 Contribution of F. Freudenstein and R. Alizade
F. Freudenstein and R. Alizade (1975) proposed a mobility equation that “applies to mechanisms without exception” as the authors stated in their paper :
qE
i ii 1 k 1
M e b= =
= −∑ ∑ (2.40)
in which ei is ith independent, scalar, displacement variable of the mechanism (associated with the relative displacements of the joint), and E is the total number of independent, scalar, displacement variables. The authors called bi the “mobility number”; it is given by the number of independent, scalar, differential loopclosure equations for the ith independent loop. In the general case (absence of metric restrictions) this number is equal to the degree of freedom of the space within which the mechanism operates. If the value of bi is not identical in each independent loop, the mechanism is said to have mixed mobility.
Three special cases of Eq. (2.40) are also presented (Freudenstein and Alizade 1975):
a) the displacement variables are in 1:1 correspondence with the degrees of freedom of the relative motion in joints
p q
i ki 1 k 1
M f b= =
= −∑ ∑ (2.41)
b) the number of independent scalar loopclosure equations is identical in each independent loop
48 2 Structural parameters
1
E
ii
M e qb=
= −∑ , (2.42)
c) general kinematic chains satisfying simultaneously the two previous conditions
1
p
ii
M f qb=
= −∑ . (2.43)
We can see that Eqs. (2.40)( 2.43) represent particular cases of Eq. (2.25) proposed by Voinea and Atanasiu where pp=0, E
ii=1F= e∑ and
rj=bk. Eq. (2.43) is identical to Eq. (2.13) proposed by Hochman, by taking into consideration that p
ii=1F= f∑ .
By applying Eqs. (2.40)( 2.43) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain obviously the same erroneous results that were obtained by using Eq. (2.25) from which these equations can be derived.
In a recent work, K. J. Waldron, G. L. Kinzel (1999) presented a formula similar to Eq. (2.41) for the mobility calculation that accommodates overconstrained closures of arbitrary type. They critically analysed the utility of this equation by mentioning: “Unfortunately, unless the value of bk associated with the different closures can be identified by inspection, such expressions have no value. The reason is that the mobility equation gives a quick check of the number of position variables and independent equations without the need to develop these equations. However, the only way to verify an overconstraint closure of a type not identifiable by inspection is to develop the closure equations and analyze them for dependency. Therefore the quickcheck advantage of the mobility equation disappears, and there is no way to derive information about the linkage without performing a complete position analysis.”
By applying Eqs. (2.41) and (2.43) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain obviously the same erroneous results that were obtained by using Eq. (2.21) from which these equations can be derived.
2.1.22 Hunt’s contribution
The formula proposed by Hunt (1978)
2.1 Critical review of mobility calculation 49
p
ii 1
M b( m p 1) f=
= − − +∑ (2.44)
is identical to Eq. (2.13) proposed by Hochman, by taking into consideration that p
ii=1F= f∑ and q=pm+1.
Hunt (1978) also presented a formula similar to Eq. (2.41). A warning is added that this equation poses further questions of how to be sure of identifying suitable independent loops and how to assign the correct values to bk.
2.1.23 Hervé’s contribution
J.M. Hervé (1978a,b) proposed a mobility formula based on the algebraic group structure of the displacement set
p
ii 1
M b( m 1) ( b f )=
= − − −∑ (2.45)
where b is given by the dimension of the displacement subgroup associated with the kinematic chain and fi by the dimension of the displacement subset associated with the kinematic pair between the links i and i+1. This formula is identical to Kutzbach’s mobility equation (2.17) by taking into consideration the fact that b=6d. As with Eq. (2.17), Eq. (2.45) also represents a particular case of Eq. (2.12).
J.M. Hervé (1978a,b) considered the kinematic chains that obey to Eq. (2.45) as “trivial chains” and added a warning concerning the application of this formula to “exceptional chains” and to “paradoxical chains”. An “exceptional chain” is a kinematic chain which involves several intersecting subgroups with different dimensions. The dimension of the displacement subgroup associated with a “paradoxical chain” depends on specific geometric constraints defined by linklength and twist angle relationships. In a more recent paper J.M. Hervé (1999) considered that “paradoxical chains are still an enigma”.
By applying Eq. (2.45) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain obviously the same erroneous results that were obtained by using Eqs. (2.12) and (2.17) from which Eq. (2.45) can be derived. We note that the mechanism presented in Fig. 2.1 does not fit Hervé’s definitions of exceptional and paradoxical chains.
50 2 Structural parameters
2.1.24 Gronowicz’s contribution
A. Gronowicz (1981) presented a method for identification of mobility properties for multiloop kinematic chains and he proposed the following formula for mobility calculation:
1
1 1 1
q q q
i iji i j i
M M F−
= = = +
= −∑ ∑ ∑ (2.46)
where Mi is the mobility of the loop i and Fij is the mobility of the connecting mechanism made of the joints common between any two loops i and j (the mobility of the joints which have been counted twice in the q loops of the mechanism).
We can see that Eq. (2.46) can be derived from Eq. (2.30) proposed by Manafu, by taking into account that 1
1 1
q qiji j i
F−
= = +∑ ∑ = Mc and ignoring the
fact that the degree of mobility of a complex kinematic chain depends on the choice of the independent closed loops, as in (Manolescu and Manafu 1963).
By applying Eq. (2.46) to the mobility calculation of the mechanism presented in Fig. 2.1, we obtain the same erroneous results that were obtained by using Eqs. (2.30), from which Eq. (2.46) can be derived.
2.1.25 Baker’s contribution
J.E. Baker (1980a) proposed an algorithm for determining the freedom between links which are separated by crossjointing. The algorithm incorporates quantitative relationships and embraces earlier results, particularly the series and parallel laws of screw system theory presented by Waldron (1966) and Davies and Primrose (1971). Baker (1980b) used the motor notations of screw system algebra to represent all six coordinates of a screw.
Baker (1981) placed the notion of mobility in perspective with the algorithm developed previously (Baker 1980b) and he pointed out that the determination of gross mobility without regard to relative freedom between specific links is not possible by the known techniques. By placing the calculation of gross mobility as a secondary stage after the study of relative freedom, Baker shows that the number of independent parameters of the linkage is obtained by considering the relative motion alone between individual members, for any pair of which we can write down the appropriate screw motor. He did not present any explicit formulae for the calculation of mechanism mobility, but said that “we may find the chain’s mobility as a consequence simply by counting the number of independent parameters”.
2.1 Critical review of mobility calculation 51
2.1.26 Davies’s contribution
T.H. Davies (1981) adapted Kirchhoff’s circulation law for potential difference to find a set of independent instantaneous screw motors associated with any two links in a kinematic chain when the configuration of the kinematic chain is given. The procedure leads to a constraint matrix formulation and the rank calculation and is applicable to any kinematic chain: it does not require special cases to be identified. Davies (1983ac) used a procedure based on the constraint matrix to determine mobility, passivity and redundancy of mobile and immobile assemblies called mechanical networks.
Davies proposed the following formulae for the calculation of the degree of mobility (Davies 1983a,b)
M=er (2.47)
q
kk 1
M e b=
= −∑ (2.48)
where e represents the number of joints with one degree of freedom associated to the mechanical network, r is the rank of the coefficient matrix of constraint equations and bk is the order of the screw system of the loop (circuit) k.
As mentioned in (Davies 1983b), Eq. (2.48) is identical to Eq. (2.22) proposed by Moroskine (by taking into consideration that e=F) and Eq. (2.48) is identical to Eq (2.41) proposed by Freudenstein and Alizade (1975) and by Hunt (1978).
A warning is added in (Davies 1983b) that Eq. (2.48) poses further questions of how to assign the correct values to bk. The danger lies in the interpretation of the words “bk is the order of the screw system of the circuit k”. Davies stated that the interpretation that most would assume, and evidently the one intended in (Hunt 1978), is that bk is the order of the screw system to which the joints of loop (circuit) k belong when no other loop is closed. With this interpretation, Eq. (2.48) underestimates the value of M for some mechanical networks, whichever set of independent loop is chosen. To give Eq. (2.48) universal applicability, Davies mentioned that bk must be defined as the order of the screw system of loop k when the constraints imposed by all previously closed loops have been taken into account. The loops can be considered in any sequence but the constraints imposed by loops closed earlier in the sequence must be taken into account. He also noticed in (Davies 1983b) that this interpretation of bk is unfortunately rather more difficult to apply but there may be circumstances in
52 2 Structural parameters
which this stepwise approach proves easier than finding the rank of the constraint matrix of the mechanical network in its entirety.
2.1.27 Contribution of V.P. Agrawal and J.S. Rao
V.P. Agrawal and J.S. Rao (1987a) developed analytical tests for fractional degree of freedom kinematic chains based on the path loop connectivity matrix. They used loop connectivity properties of multiloop kinematic chains to develop a sevenstep hierarchical classification scheme of kinematic structures (Agrawal and Rao 1987b). A formula is also proposed for calculating the mobility of a multiloop mechanism with simple or multiple joints in terms of the mobility of its independent loops and connecting mechanism (Agrawal and Rao 1987c)
1 22 2N Nq q 1 qi i i i
i ij mi nii 1 i 1 j i 1 i 1 i 1
( m m 2 ) ( n 3n 2 )M M F F F2 2
−
= = = + = =
− − − += − + +∑ ∑ ∑ ∑ ∑
(2.49)
where: mi is the number of links forming the ith internal multiple joint, ni  the number of links forming the ith external multiple joint, N1  the total number of internal multiple joints, N2  the total number of external multiple joints, Fmi  the mobility of simple joint forming the ith internal multiple joint, Fni  the mobility of simple joint forming the ith external multiple joint.
A multiple joint formed by mi links is equivalent to (mi1) simple joints. An internal multiple joint is completely inside the mechanism, i.e. it is common to independent loops, and not to the peripheral loop. An external multiple joint is on the peripheral loop of the kinematic chain and hence it is common to both independent loops and the peripheral loop.
We can see that Eq. (2.49) extends Eq. (2.46) proposed by Gronowicz to multiple jointed mechanisms. Agrawal and Rao (1987c) considered that Eq. (2.49) is applicable to any general mechanism with constant or variable general constraints with simple or multiple joints. In spite of this statement, by applying Eq. (2.49) to the mobility calculation of the mechanisms presented in Fig. 2.1 (which has only simple joints) we obtain the same erroneous results that were obtained by using Eq. (2.46).
2.1.28 Contribution of J. Angeles and C. Gosselin
J. Angeles and C. Gosselin (1988) addressed the problem of finding the mobility of an overconstraint mechanism, which is done by using the Jacobian matrix of a simple or multiloop closed kinematic chain. They
2.1 Critical review of mobility calculation 53
have shown that, once the Jacobian matrix J of a kinematic chain coupled by either revolute or prismatic pairs is suitably defined, the mobility of the chain can be uniquely computed as the dimension of the nullspace of J (the nullity of J):
M=nullity(J). (2.50)
The authors mentioned that this idea was originally suggested by Freudenstein (1962) in a very general manner. We can also see that Eq. (2.50) can be derived from Eq. (2.22) proposed by Moroskine. We take into account that in a linear transformation (with a Jacobian matrix J) of a finitedimensional vector space V, the following equation can be written:
rank(J) + nullity(J)=dim(V) (2.51)
and the fact that r=rank(J) and N= dim(V). The Jacobian matrix produces the mobility of the kinematic chain, re
gardless of its type (single or multiloop). Five full examples, including the Bennet mechanism and a planar three degrees of mobility parallel robotic manipulator are given in (Angeles and Gosselin 1988). As with Moroskine’s equation, Eq. (2.50) is valid without exception, but it does not give us a quick calculation of mobility without the need to develop the kinematic equations and to calculate the nullity of the Jacobian matrix. These calculations will be more disused in Section 2.2.
2.1.29 Contribution of F. Dudiţă and D. Diaconescu
F. Dudiţă and D. Diaconescu (1987) proposed mobility formulae applicable to simple or complex mechanisms. They demonstrated that the passive mobilities in joints do not influence the mobility of a simple (single loop) mechanism; consequently the following two formulae give the same results:
1
p
ii
M f b=
= −∑ (2.52)
1
pe e
ii
M f b=
= −∑ (2.53)
where eif denotes the effective (active) mobilities in the ith joint, and be is
the dimension of the active motion space of the mechanism (the motion parameter obtained by taking into consideration only the active mobilities of the joints). We can see that Eq. (2.52) also represents a particular case
54 2 Structural parameters
of Eqs. (2.13) and (2.10) proposed by Hochman (for q=1). Dudiţă and Diaconescu called b the kinematic rank of the mechanism (the dimension of the motion space). To calculate the kinematic rank, they used the screw coordinates and the system of generators from the screw algebra.
To calculate the degree of mobility of a complex (multiloop) mechanism, Dudiţă and Diaconescu (1987) proposed the following formulae:
1 1
p q
i ji j
M f b= =
= −∑ ∑ (2.54)
1 1
p qe e
i ji j
M f b= =
= −∑ ∑ (2.55)
1
q
i ci
M M M=
= −∑ (2.56)
with e
c j com. jj
M ( q 1) f= −∑ (2.57
were Mc is the mobility of the common joints between different closed loops, e
com. jf is the active degree of mobility of the jth common joint and qj is the number of loops to which the common joint j belongs.
We can see that Eq. (2.54) represents a particular case of Eq. (2.25) proposed by Voinea and Atanasiu when pp=0 ( p
ii=1F= f∑ and
q qj jj=1 j 1
r b=
=∑ ∑ ). By applying Eq. (2.54) to the mobility calculation of
the mechanisms presented in Fig. 2.1, we obtain the same erroneous results that were obtained by using Eq. (2.25) from which Eq. (2.54) can be derived. The same erroneous results are also obtained by using Eq. (2.55). In the mechanisms presented in Fig. 2.1 all the joint mobilities are effective ( p p e
i ii=1 i=1 f f=∑ ∑ and q q e
j jj 1 j 1b b
= ==∑ ∑ ). By applying Eq. (2.56) to the
mobility calculation of the mechanisms presented in Fig. 2.1, we obtain the same erroneous results that were obtained by using Eq. (2.30) proposed by Manafu.
2.1 Critical review of mobility calculation 55
2.1.30 Contribution of P. Fanghella and C. Galletti
P. Fanghella and C. Galletti (1994) developed a systematic approach to determining the mobility properties of singleloop kinematic chains based on displacement groups introduced by Hervé (1978a). Such properties are defined by the following: the mobility of the chain, the number of degrees of freedom of the relative motion between any two links in the chain (link connectivity), the type of relative displacements between any two links in the chain (the displacement group containing all their relative displacements), and the set of invariant properties (Fanghella 1988) of every displacement subgroup.
The algorithm uses the following equation for determining mobility of singleloop kinematic chains:
p
i iii 1
M f min( co ,i 1,...,m )=
= − =∑ (2.58)
where coii is the loop connectivity defined as the connectivity of the open chain obtained by cutting the link i of the closed loop. Any link i (i=1,…,m) of the closed chain can be cut, and, due to nonassociative characteristics of the kinematic constraints composition, different values of the connectivities coii can be obtained. The minimum value coii gives the correct loop connectivity.
The approach is applicable to singleloop kinematic chains which can be defined by the composition of the ten displacement groups defined in [46]. Any mechanism not complying with this scheme cannot be considered. The approach being independent of local singularities cannot be used to determine the local or permanent changes in mechanism mobility due to singular configurations.
When taking into account the fact that min(coii, i=1,…,m)=b, we can see that Eq. (2.58) also represents a particular case of Eq. (2.13 ) proposed by Hochman (for q=1). The main contribution of P. Fanghella and C Galletti (1994) is to give a systematic approach to determining the loop connectivity b based on the displacement groups.
2.1.31 Fayet’s contribution
M. Fayet (1995a) proposed a general iterative process to obtain the spaces of twists between any two bodies of a multiloop mechanism. A linear application represented by a “generative matrix” is set up by a triangular projective method to generalize the sum of dependent spaces in a multiloop mechanism.
56 2 Structural parameters
The iterative method proposed to obtain the space of twists between any two bodies (Fayet 1995a) is used by Fayet and Bonnet (1995) in order to obtain the rank of constraint equations of a multiloop mechanism. They bring out a general proof for the stepwise approach put forward by Davies (1983b) to determine the order of the screw system of loop k when the constraints imposed by all previously closed loops have been taken into account. Three examples are given in (Fayet and Bonnet 1995), including the mechanism analysed by Davies (1983b), to emphasize the applicability of the iterative approach. This method has universal applicability but it gives the instantaneous mobility (mobility in a given position of the mechanism). The rank of the constraint equations is obtained by the rank of the twists consistent with the iterative approach defined for the given position.
2.1.32 Tsai’s formula
To take into consideration the fact that passive (internal) degrees of freedom cannot be used to transmit motion or torque about an axis, Tsai (1999) proposed subtracting the number of passive degrees of freedom, fp, from the degree of freedom equation:
p
i pi 1
M b( m p 1) f f=
= − − + −∑ . (2.59)
By applying Eq. (2.59) to the mobility calculation of the mechanisms presented in Fig. 2.1 having fp=0, we obtain the same erroneous results that were obtained by using Eq. (2.13) from which Eq. (2.59) can be derived.
2.1.33 McCarthy’s formula
To calculate the mobility of the platform linkages, in which the platform is connected to the ground by serial chains, McCarthy (2000) proposed the mobility formula
cpci
i 1M b ( b M )
=
= − −∑ (2.60)
where pc represents the number of serial open chains connecting the platform to the ground and c
iM is the mobility of the ith chain including its part of the platform. By applying the formula (2.60) to the mobility calcu
2.1 Critical review of mobility calculation 57
lation of the mechanism presented in Fig. 2.1, having ciM 4= , we obtain
the erroneous results M=0 (if we consider the motion parameter b=6), M=2 (if we consider b=5), M=4 (if we consider b=4).
2.1.34 Contribution of Z. Huang, L.F. Kong and Y.F. Fang
Huang, Kong and Fang proposed the formula p
ii 1
M b( m p 1) f υ=
= − − + +∑ (2.61)
considered as a “correct application of the GrüblerKutzbach formula” in the case of parallel mechanisms (Huang et al. 2003). In Eq. (2.61) b=6d is called order of the mechanism and d is the number of independent common constraints in the mechanism. A common constraint is defined in (Huang et al. 2003) as a screw reciprocal to the unit twists associated with all kinematic pairs in the mechanism. The corrective term υ represents the number of redundant constraints that are linearly dependent on other constraints. Eq. (2.61) gives the instantaneous mobility (mobility in a given position of the mechanism). The order of the mechanism and the number of redundant constraints are calculated for each given position.
2.1.35 Contribution of J.M. Rico, J. Gallardo and B. Ravani
Rico and Ravani (2003) have extended previous works (Hervé 1978a,b, 1999; Fanghella 1988; Fanghella and Galletti 1994) in the application of group theory to analysis of kinematic chains. They proposed the following formula to calculate the mobility of a singleloop closed mechanism:
p
i c cc ai 1
M f dim( H ( i, j )) dim( H ( i, j )) dim( H ( i, j ))=
= − − +∑ (2.62)
where Hc(i,j) and Hcc(i,j) are the clockwise and the counterclockwise composite subgroups associated with the two open chains connecting two links i and j in the singleloop mechanism, and Ha(i,j) is the absolute composite subgroup between i and j (the connectivity between links i and j). Dim stands for the dimension of the respective subgroup of the Euclidean group. In a kinematic chain with n links, the clockwise kinematic chain is defined by i, i+1,…,j1, j and the counterclockwise kinematic chain by i, i1,…, 2, 1, n,n1,…,j+1, j.
58 2 Structural parameters
We can see that Eq. (2.62) is equivalent to Eq. (2.24) proposed by Voinea and Atanasiu by taking into account that p
ii 1N f
==∑ ,
1 cr =dim(H (i,j)) , r2=dim(Hcc(i,j)), and we= dim(Ha(i,j)). Rico, Gallardo and Ravani (2003) demonstrated that mobility formula
can be obtained either by using arguments of finite kinematics or by using arguments of infinitesimal kinematics. The authors translated the mobility formula (2.62) from the finite kinematics language of group theory into the infinitesimal kinematics language of Lie algebra. The instantaneous form of the mobility formula presented here is based on the theory of subspaces and subalgebras of the Lie algebra of the Euclidean group and their possible intersections.
2.2 ChebychevGrüblerKutzbach mobility formulae
The most widely known formulae, presented in the previous section, can be grouped together into a ChebychevGrüblerKutzbach (CGK) formula and the extensions of this formula proposed by Voinea and Atanasiu, and by Manafu. In this section we explain why these formulae for quick calculation of mobility do not work for all mechanisms, and we demonstrate the limits of their applicability. We will point out that some motion parameters used in mobility formulae cannot be associated with a loop independently of any other closed loop. Loop independence is also revisited in this section.
2.2.1 The original ChebychevGrüblerKutzbach formula
As we have presented in Section 2.1, the development of a formula for mobility calculation can be dated back to about 150 years ago. Earlier works on the mobility of mechanisms go back to the second half of the nineteenth century (Chebychev 1869; Sylvester 1874; Grübler 1883, 1885; Somov 1887; Hochman, 1890) and the beginning of the twentieth century (Koenigs 1905; Grübler 1917; Malytsheff 1923; Kutzbach 1929). At that time the main formulae defining what we call the CBK formula for multiloop mechanisms were already set up. Versions of these formulae were proposed all through the twentieth century (Dobrovolski 1949, 1951; Artobolevski 1953; Kolchin 1960; Rössner 1961; Boden 1962; Ozol 1963; Manolescu and Manafu 1963; Manolescu 1968; Bagci 1971; Hunt 1978; Hervé 1978a,b; Tsai 1999).
2.2 ChebychevGrüblerKutzbach mobility formulae 59
The original CGK formula gave the mobility M of a mechanism with q≥1 independent loops having the same motion parameter b for each loop, and could be expressed by
1
p
ii
M f qb=
= −∑ . (2.63)
The first term of Eq. (2.63) represents the number of independent motion parameters of the joints before loop closures provide further constraints, and the second term the number of joint parameters that lose their independence after closing the q loops. As presented in the previous section, the motion parameter b was introduced by Somov (1887) to characterize the motion space of the mechanism, and is called the mobility number.
Table 2.1. Terminology used for parameter b in the CGK formula
Terminology References Motion parameter Somov 1887 Mechanism category Hochman 1890 Rank of linear set of screws Voinea and Atanasiu 1959 Link mobility Voinea and Atanasiu 1960 Relative infinitesimal displacements of two bodies
Hunt 1967
Connectivity of the complex joint between two bodies
Waldron 1966
Connectivity of the instantaneous screw system of two bodies
Davies and Primrose 1971
Degree of freedom of the complex joint between two bodies
Hervé, 1978a,b
Relative freedom between links Baker 1980b Complex connectivity Baker 1980b Internal freedom Baker 1981 Connectivity between a pair of members Phillips 1984 Freedom of the complex joint between a pair of members
Phillips 1984
Rank of the kinematic space of the mechanism Dudiţǎ and Diaconescu 1987 Kinematic constraints between two bodies Fanghella 1988 Connectivity Fanghella and Galletti 1994 Dimension of the space of twists between two bodies
Fayet 1995a
Link connectivity Shoham and Roth 1997 Degree of freedom of the mechanism with an output member
Zhao et al. 2004
Spatiality Gogu 2005be
60 2 Structural parameters
Table 2.2. Tools used to define and to calculate instantaneous values of parameter b in a given position of the mechanism
Tools References AronholdKennedy theorem of three axes in planar motion of three bodies
Aronhold 1872; Kennedy 1886
PhillipsHunt theorem of three axes in spatial motion of three bodies Phillips and Hunt 1964
ChaslesBall theory of the instantaneous screw axes Chasles 1830; Ball 1900; Waldron 1966; Davies and Primrose 1971; Dudiţǎ and Diaconescu 1987;
Schoenflies’s geometry of linear complex Schoenflies 1893; Hunt 1967, 1978
Klein’s projective space of linear set of screws Voinea and Atanasiu 1959; Rico and Ravani 2004b
Reciprocal screw theory Zhao et al. 2004 Screw motor Baker 1980b Twist theory and triangular projective method Fayet 1995a Mathematical group structure of the set of displacements
Hervé 1978a,b; Fanghella 1988; Fanghella and Galletti 1994; Rico and Ravani, 2003
Lie group of rigid body displacement Hervé 1999; Rico et al. 2003, 2004a
In the literature, various terminologies are used to define this parameter (Table 2.1). Various tools can be used to define and to calculate instantaneous values of this parameter for a given position of the mechanism (Table 2.2). This book presents a development of these previous contributions. We determine general connectivity, also called spatiality in the author’s previous publications, without the calculation of instantaneous connectivity. We base our approach on the theory of linear transformations (Greenberg 1998; Gogu, 2002b).
Dobrovolski (1949, 1951) introduced “mechanism family” d=6b to characterize the dimension of the constraint space of a mechanism (Manolescu and Manafu 1963; Manolescu 1968; Antonescu 1973). The constraint parameter of the mechanism represents the number of general motion components suppressed by the mechanism or the number of general constraints of the mechanism.
Various expressions of CGK mobility formula, known in the literature (Dobrovolski 1949, 1951; Artobolevski 1953; Kolchin 1960; Rössner 1961; Boden 1962; Ozol 1963; Manolescu and Manafu 1963; Manolescu 1968; Hunt 1978; Hervé 1978a) can be obtained from Eq. (2.63) by taking into consideration Euler’s formula, particular values of b (b=3 for planar
2.2 ChebychevGrüblerKutzbach mobility formulae 61
or spherical mechanisms, b=6 for spatial mechanisms) or the relations between:
a) the degree of mobility f and the degree of constraint c of a joint (c=6f),
b) the motion parameter b and the constraint parameter d of the mechanism (d=6b),
c) the total number of kinematic links n and the total number of links m including the fixed base (n=m1).
2.2.2 The extended ChebychevGrüblerKutzbach formula
A slightly different formula, that we call the extended CGK formula, was proposed in the middle of the twentieth century by Voinea and Atanasiu (1960) and by Manafu (Manolescu and Manafu 1963).
The original CGK mobility formula was extended by Voinea and Atanasiu (1960) to mechanisms having q≥1 independent loops with different motion parameters bk (k=1,…,q):
p q
i ki 1 k 1
M f b= =
= −∑ ∑ . (2.64)
We can see that Eq. (2.63) represents a particular case of Eq. (2.64) where bk=b (k=1,2,…,q).
This extended formula was developed later by Antonescu (1973), Freudenstein and Alizade (1975), Dudiţă and Diaconescu (1987). Some additional terms were introduced in the CGK mobility formula to take into consideration passive mobilities, redundant freedoms and overclosing constraints existing in the mechanism (Voinea and Atanasiu 1960; Bagci 1971; Tsai 1999).
The extended formula in the form proposed by Manafu was developed later by Gronowicz (1981), Davies (1983b), Agrawal and Rao (1987c), Dudiţă and Diaconescu (1987), Rico and Ravani (2004a):
1
q
i ci
M M M=
= −∑ (2.65)
where Mi is the mobility of the ith simple closed loop and Mc is the mobility of the common joints between different closed loops (the mobility of the joints that appears in mobility calculation of two or more closed loops). We can see that Eq. (2.65) is directly derived from Eq. (2.64) by taking into account the fact that
62 2 Structural parameters
1
ip
i j ij
M f b=
= −∑ (2.66)
and
1 1 1
ipq p
c j ji j j
M f f= = =
= −∑∑ ∑ (2.67)
where pi is the number of joints of ith simple closed loop. We can see that Eq. (2.66) represents a particular case of Eq. (2.64) when q=1 and pi=p.
For a given multiloop mechanisms Eq. (2.65) can give different results depending of the choice of the independent closed loops. To overcome this flaw Manolescu and Manfu (1963) proposed taking into consideration only the maximum value of mobility given for the same mechanism by Eq. (2.65). Dudiţă and Diaconescu (1987) indicated that passive mobilities of common joints must not be taken into consideration in the calculation of Mc. Agrawal and Rao (1987c) introduced some additional terms in Eq. (2.65) to take into consideration the existence of multiple joints in the multiloop mechanism.
From Fig. 2.1 we can simply determine by inspection the values of the structural parameters necessary to calculate the mobility of the parallel robot CPM. This mechanism has p=12 joints (3 prismatic and 9 revolute joints) and q=2 independent loops. Each joint has one degree of mobility (fi=1). Passive mobilities do not exist in the joints of this mechanism. The two independent loops have the same motion parameter b=b1=b2=5. The CGK mobility formula, defined by Eqs. (2.632.65) or by any other derived formula proposed by different authors mentioned in this section, gives M=2. This is an erroneous result. Kim and Tsai (2002) have shown that by considering b=6 Eq. (2.63) gives M=0; that also represent an erroneous result. In fact any other value of b gives an erroneous result. We can see that the CGK mobility formula is not applicable to this mechanism nor to many other multiloop mechanisms. The main objective of the following section is to explain this flaw and to propose accurate limits of applicability of the original and the extended CGK mobility formulae.
2.2.3 Limits of applicability of CGK formulae
Let us consider a mechanism with p joints and q≥1 closed loops. Each joint i introduces fi independent motion parameters. The closed loops cancel the independence of a part of the joint’s independent parameters. Eq. (2.63) shows that the number of independent coordinates (parameters)
2.2 ChebychevGrüblerKutzbach mobility formulae 63
needed to define the configuration of a mechanism with closed loops is the difference between the number of independent motion parameters of the joints (
1
pii
f=∑ ) before loop closures provide further constraints, and the
number r of joint parameters that lose their independence after loop closures. So, Eq. (2.63) can be rewritten as:
1
p
ii
M f r=
= −∑ . (2.68)
Eq. (2.68) is valid without exception. Moroskine (1954) indicated that the number of joint parameters that lose their independence in the closed loops of the mechanism is given by the rank of the homogeneous linear set of velocity equations of the mechanism. To calculate this rank we need to set up the velocity constraint equations of the mechanism. These calculations are not simple in the case of multiloop mechanisms. The numerical calculation of the rank in the proximity of the singular positions introduces certain ambiguities in the interpretation of the results. It is therefore very difficult to tell the difference between a real zero and a very small numerical value close to zero in the calculation of the determinants. This drawback is specific to all methods for mobility calculation based on rank calculation of constraint equations in a given position of the mechanism when joint location is defined numerically. Symbolic calculation of the rank can overcome these limits if the constraint equations are defined symbolically in a generic position, as will be presented in this section.
The basic kinematic structure of the parallel Cartesian robotic manipulator presented in Fig. 2.1 is obtained by concatenating three limbs denoted by A (0≡1A2A3A4A5A), B (0≡1B2B3B4B5B) and C (0≡1C2C3C4C5C). The first link 0≡1X of each limb is the fixed platform and the final link is the moving platform 5≡5X (X=A,B,C). The first joint of each limb is actuated. We denote by d2X and 2 Xd (X=A,B,C) the finite displacements and the velocities of the actuated prismatic joints and by iXϕ and iXϕ (i=3,4,5 and X=A,B,C) the finite displacements and the angular velocities of the passive joints. We note that all revolute joints are passive joints. The closure equations of the parallel Cartesian robotic manipulator presented in Fig. 2.1 can be established by the condition that the linear ( 0
Hv ) and angular ( 0
Hω ) velocity of point H situated on the mobile platform (expressed in the reference system O0x0y0z0 – see Fig. 2.2) must be the same in the three limbs (H≡HA≡HB≡HC):
64 2 Structural parameters
00 0
00 0
HCHA HB
HCHA HB
vv v= =
ωω ω,
(2.69)
or
[ ] −
0 0HA HB
0 0HA HB
v v= 0
ω ω
(2.70)
and
[ ]=
−
00HCHA
00HCHA
vv0
ωω.
(2.71)
Eqs. (2.70) and (2.71) represents the closure equations of the closed loops 0≡1A2A…5A≡5B…2B1B≡0 and 0≡1A2A…5A≡5C…2C1C≡0 . These loops can be considered as two independent closed loops of the Cartesian robotic manipulator, presented in Fig. 2.1. By calculating the velocity of point H in the three limbs of the parallel robotic manipulator Eqs. (2.70) and (2.71) lead to the following sets of linear equations:
[ ] [ ]T
1 2 A 3 A 4 A 5 A 2B 3B 4 B 5B 8 1D d d 0ϕ ϕ ϕ ϕ ϕ ϕ
× = (2.72)
[ ] [ ]T
2 2 A 3 A 4 A 5 A 2C 3C 4C 5C 8 1D d d 0ϕ ϕ ϕ ϕ ϕ ϕ
× = (2.73)
[ ]T
2 A 3 A 4 A 5 A 2B 3B 4 B 5B 2C 3C 4C 5CD d d dϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ
[ ]12 10
×=
(2.74)
where
[ ]1
3B 3B 4B 43B 4B 43B
3A 3A 4A 43A 4A 43A
3A 3A 4A 43A 4A 43A 3B 3B 4B 43B 4B 43B
D
1 0 0 0 0 b C b C b C 00 b S b S b S 0 1 0 0 00 b C +b C b C 0 0 b S +b S b S 00 1 1 1 0 0 0 00 0 0 0 0 1 1 10 0 0 0 0 0 0 0
=
− − − −
(2.75)
2.2 ChebychevGrüblerKutzbach mobility formulae 65
[ ]2
3C 3C 4C 43C 4C 43C
3A 3A 4A 43A 4A 43A 3C 3C 4C 43C 4C 43C
3A 3A 4A 43A 4A 43A
D
1 0 0 0 0 b S +b S b S 00 b S b S b S 0 0 b C b C b C 00 b C +b C b C 0 1 0 0 00 1 1 1 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 1 1 1
=
− − −
(2.76)
[ ] [ ]3 4D D D=
3B 3B 4B 43B
3A 3A 4A 43A 4A 43A
3A 3A 4A 43A 4A 43A 3B 3B 4B 43B
3
3A 3A 4A 43A 4A 43A
3A 3A 4A 43A 4A 43A
1 0 0 0 0 b C b C0 b S b S b S 0 1 00 b C +b C b C 0 0 b S +b S0 1 1 1 0 00 0 0 0 0 10 0 0 0 0 0
D1 0 0 0 0 00 b S b S b S 0 0 00 b C +b C b C 0 0 00 1 1 1 0 00 0 0 0 0 00 0 0 0 0 0
− −=
4B 43B
4B 43B
43C 3C 4C 43C 4C 43C
3C 3C 4C 43C 4C 43C
b C 0 0 0 0 00 0 0 0 0 0
b S 0 0 0 0 00 0 0 0 0 01 1 0 0 0 0
0 0 0 0 0 0D
0 0 0 b S +b S b S 00 0 0 b C b C b C 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 1 1 1
− − = − − −
(2.77)
66 2 Structural parameters
and 3 A 3 AS sin( )ϕ= , 3 A 3 AC cos( )ϕ= , 43 A 4 A 3 AS sin( )ϕ ϕ= + ,
43 A 4 A 3 AC cos( )ϕ ϕ= + , etc. Fig. 2.2 presents the notations associated with the three limbs. By sym
bolic calculation of the rank of these matrices with MAPLE® we obtain: 1 A B 1r r rank( D ) 5−= = = , 1 A C 2r r rank( D ) 5−= = = and A B Cr r rank( D )− −= = =
9. We can see that 1 2r r r≠ + . This result indicates that the structurally independent closed loops 0≡1A2A…5A≡5B…2B1B≡0 and 0≡1A2A…5A≡5C…2C1C≡0 are not independent from a kinematic point of view. In the general case, rank r of the linear set of velocity equations of a multiloop mechanism cannot be obtained by summing up the ranks (rj, j=1,2,…,q) of the linear sets of velocity equations of the q structurally independent closed loops.
We recall that in the theory of graphs, the number of independent closed loops q is given by Euler formula q=pm+1, where p represents the number of edges (arcs) and m the number of vertices (nodes). In section 2.1.5, we have seen that Hochman (1890) applied Euler formula to calculate the number of independent closed loops q in a multiloop mechanism. In this case, p is associated with the total number of joints and m with the total number of links including the fixed base. In section 1.4, we have defined the structural graph associated with the multiloop mechanism by networking the joints and links with no geometric relations.
We can establish the following properties in relation with the structural limitation of the independence of the closed loops.
Property 1. The independence of the q closed loops of a multiloop mechanism given by Euler’s formula is restricted, in the general case, to the structural independence
Property 2. In the general case, q structural independent closed loops in a multiloop mechanism are not implicitly kinematically independent.
We note that the structural independence is defined in the sense of the theory of graphs and the definition of the structural graph by ignoring any geometric and kinematic relation between the links and joints of the multiloop mechanism. In the general case, the motions of the q closedloops are not independent.
By using Eq. (2.68) we get the correct value of mobility of the parallel Cartesian robotic manipulator presented in Fig. 1 (M=129=3).
We recall that the rank of the matrices [D1], [D2] and [D] can be calculated numerically or symbolically. The numerical calculation gives us the instantaneous rank in a given position of the mechanism defined by the numerical values of joint variables and geometric parameters. The symbolic calculation gives us the global rank in an arbitrary position of the mechanism without indicating the numerical values of the joint variables
2.2 ChebychevGrüblerKutzbach mobility formulae 67
Fig. 2.2. Simple open kinematic chains associated with the three limbs of parallel Cartesian robotic manipulator and the notations used in the kinematic constraint equations: (a) limb A (0A≡1A2A3A4A5A); (b) limb B (0B≡1B2B3B4B5B); (c) limb C (0C≡1C2C3C4C5C)
and the geometric parameters. We just use the symbolic definition of the matrices given by (2.75)(2.76). The global rank represents the maximum value of the instantaneous ranks. The mobility calculation based on the instantaneous rank gives us the instantaneous mobility of the mechanism. By using symbolic calculation of the rank we can calculate global or full cycle mobility of mechanisms; by using numerical calculation of the rank we can calculate instantaneous mobility.
The following three theorems can be demonstrated in relation to the rank of the homogeneous linear set of kinematic constraint equations of a multiloop mechanism.
68 2 Structural parameters
Let us consider the general case of a multiloop mechanism with q structurally independent closed loops and N=F independent motion parameters of the joints, before loop closures provide further constraints (
1
pii
N F f=
= =∑ ), defined by the vector x=[X] N 1× . The homogeneous linear set of kinematic constraint equations of the multiloop mechanism can be written as
[ ] [ ] [ ]6 q N N 1 6 q 1A X 0
× × ×= . (2.78)
The matrix [ ]6 q NA
× from Eq. (2.78) can be partitioned in many forms:
[ ]
1
2
6 q N
q 6 q N
AA
A
A
×
×
⋅
= ⋅ ⋅
,
(2.79)
[ ]
1 2
3
6 q N
q 6 q N
AA
A
A
−
×
×
⋅
= ⋅ ⋅
(2.80)
[ ]
1 2 ... i
i 1
6 q N
q 6 q N
AA
A
A
− − −
+
×
×
⋅
= ⋅ ⋅
(2.81)
where
j 6 NA
× is the partition of [ ]6 q N
A×
corresponding to the loop jth,
2.2 ChebychevGrüblerKutzbach mobility formulae 69
[ ]
1
2
1 2 ... i 6 i N
i 6 i N
AA
A
A
− − − ×
×
⋅
= ⋅ ⋅
is the partition of [ ]6 q NA
× corresponding to the
first i loops (i=2,3,…, q1),
[ ] 11 2 12 N
2 12 N
AA
A− ××
=
is the partition of [ ]6 q NA
× corresponding to the first
two loops. We notice that the loops can be considered in any sequence. We denote
by rj the rank of the matrix j 6 NA
× (j=1,2,…,q).
We consider :U W
( )→F
x F x
the matrix transformation from the finite dimensional vector space U (called joint vector space) into the finite dimensional vector space W (called external vector space). The linear transformation F is the transformation implied by multiplication by the matrix [ ]6 q N
A×
, that is to each
vector x in the joint vector space U the linear transformation F assigns a uniquely determined vector F(x) in the external vector space W
F(x)=[A][X]. (2.82)
The vector F(x) in W is the image of vector x in U under transformation F. If SU is a subset of U, then the set of all vectors F(x) in W, for which x is in SU, represents the image of SU under F. Conversely, if SW is a subset of W, then the set of all vectors x in U, for which F(x) is in SW, represents the inverse image of SW. Since U and W are vector spaces, each must contain a zero vector. We will denote these vectors as 0U and 0W. The image of U in W represents the range RF of F and the inverse image of 0W in U is the nullspace or kernel KF of F, that is
KF { }WU ( )= ∈ =x F x 0 (2.83)
The rank of F is the dimension of the range RF, and the nullity of F is the dimension of kernel KF
dim(RF)=rank(F) (2.84)
70 2 Structural parameters
and
dim(KF)=nullity(F). (2.85)
As F is a matrix transformation, with matrix [A], then
dim(RF)=rank(A) (2.86)
and
dim(KF)= nullity(A). (2.87)
We know that in a matrix transformation F, with matrix [A], we have:
dim(RF)+dim(KF)=dim(U) (2.88)
rank(F)+nullity(F)=dim(U) (2.89)
rank(A)+nullity(A)=dim(U). (2.90)
Eqs. (2.88)(2.90) express a sort of “conservation of dimension” in the matrix transformation F (Greenberg, 1989).
Similarly, we consider Fj: jU W→ (j=1,2,…,q) and F12…i:
1 2 ... iU W − − −→ (i=2,…,q1) the matrix transformations with the matrices
j 6 NA
× and [ ]1 2 ... i 6 i N
A − − − × . We note the range and kernel of Fj by RF(j)
and KF(j), and the range and kernel of F12…i by RF(12…i ) and KF(12…i). Theorem 1. The rank r of the homogeneous linear set of kinematic con
straint equations of a mechanism with q structurally independent closed loops is given by
( 1 2 ... i )
q 1
1 ( i 1 ) / Ki 1
r r dim( R )− − −
−
+=
= +∑ FF (2.91)
where r1 is the rank of [ ]1 6 NA
× and
( 1 2 ... i )( i 1 ) / KR− − −+ FF is the range of the re
striction of Fi+1 to the kernel of F12…i (i=1,2,…,q1): i 1 F( 1 2 ... i ) F ( 1 2 ... i ) i 1
i 1
/ : W
( )+ − − − − − − +
+
→F K K x F x
and
( 1 2 ... i )( i 1 ) / K
( 1 2 ... i ) ( 1 2 ... i ) ( i 1 )
dim( R )
dim( K ) dim( K K )− − −+
− − − − − − +
=
− ∩FF
F F F
(2.92)
Theorem 2. The rank r of the homogeneous linear set of kinematic constraint equations of a mechanism with q structurally independent closed
2.2 ChebychevGrüblerKutzbach mobility formulae 71
loops is less than, or equal to, the sum of the ranks (ri , i=1,2,…,q) of the linear sets of kinematic constraint equations of the q loops
q
ii 1
r r=
≤∑ . (2.93)
Theorem 3. The rank r of the homogeneous linear set of kinematic constraint equations of a mechanism with q structurally independent closed loops is equal to the sum of the ranks ri (i=1,2,…,q) of the linear sets of kinematic constraint equations of the q loops if and only if the rank of the linear set of kinematic constraint equations of (i+1)th loop is equal to the dimension of the range of the restriction of Fi+1 to the kernel of F12…i
F ( 1 2 ... i )i 1 F( i 1 ) / Kr dim( R ), i=1,2,..., q1− − −+ += (2.94)
The proof of Eqs. (2.91) and (2.92) is obtained by recurrence on the basis of the demonstration presented by Fayet (1995, b) for q=2.
Let us consider the matrix [ ]6 q NA
× from Eq. (2.78) partitioned as
[ ] 1 2 ... ( q 1 )6 q N
q 6 q N
AA
A− − − −
××
=
. (2.95)
Eq. (2.95) results from Eq. (2.81) with i=q1. In this case, the rank of [ ]6 q FA
×is equal to the rank of 1 2 ... ( q 1 ) 6( q 1 ) N
A − − − − − × augmented by the di
mension of the range of the restriction of Fq to the kernel of F12…(q1), that is
( 1 2 ... ( q 1 ))1 2 ... ( q 1 ) ( q ) / Kr rank( A ) dim( R )− − − −− − − −= +
FF . (2.96)
The matrix 1 2 ... ( q 1 ) 6( q 1 ) NA − − − − − × can be partitioned as
[ ] 1 2 ... ( q 2 )
6( q 1 ) Nq 1 6( q 1 ) N
AA
A− − − −
− ×− − ×
=
(2.97)
and
( 1 2 ... ( q 2 ))1 2 ... ( q 1 ) 1 2 ... ( q 2 ) ( q 1 ) / Krank( A ) rank( A ) dim( R )− − − −− − − − − − − − −= +
FF (2.98)
We can see that by inverse recursion we obtain Eq. (2.91). Eq. (2.92) is obtained by taking into account the fact that Eq. (2.78) is
equivalent to the following equations:
72 2 Structural parameters
[ ] [ ]1 2 ... ( q 2 )
N 1 6( q 1 ) 1q 1 6( q 1 ) N
AX 0
A− − − −
× − ×− − ×
=
(2.99)
[ ] [ ][ ] [ ]
1 2 1 1 6 1 16 1
1 6 16
0
0
... ( q ) N ( q )( q ) N
q NN
A X
A X
− − − − × − ×− ×
× ××
= =
(2.100)
That is equivalent to saying that
KF = KF(12…(q1))∩KFq. (2.101)
Let 'qF =
( 1 2 ... ( q 1 ))q / K − − − −FF be the restriction of Fq to the kernel of F12…(q1).
Similarly with Eqs. (2.83) and (2.88), we can write
KF’(q) { }( 1 2 ... ( q 1 )) WqK ( )− − − −= ∈ =F qx F x 0 . (2.102)
The “conservation of dimension” in the matrix transformation 'qF gives
dim(KF(12…(q1)))=dim(KF’(q))+dim(RF’(q)). (2.103)
We can also express
KF’q { }1 2 ... ( q 1 ) W ( 1 2 ... ( q 1 )) q WqU ( ) ( )− − − − − − − −= ∈ = =x F x 0 , F x 0 , (2.104)
that is
KF’(q) = KF(12…(q1))∩KF(q)= KF . (2.105)
From Eqs. (2.103) and (2.105) we can write
dim(RF’(q))= dim(KF(12…(q1)))dim(KF(12…(q1))∩KF(q)) (2.106)
Eq. (2.92) is equivalent to Eq. (2.106) with i=q1. The proof of Eq. (2.93) can be derived directly from Eq. (2.91) by tak
ing into consideration that
( 1 2 ... i )i 1 ( i 1 ) / Kr dim( R ), i 1,2,..., q  1− − −+ +≥ =
FF . (2.107)
The proof of Eq. (2.94) results directly from Eqs. (2.91), (2.93) and (2.107).
As we have seen, for the parallel Cartesian robotic manipulator presented in Fig. 2.1, the symbolic calculation of the rank of the matrix (2.77) using MAPLE® gives A B Cr r rank( A ) 9− −= = = . This result can also be obtained by using Eq. (2.91):
2.2 ChebychevGrüblerKutzbach mobility formulae 73
( 1 2 ... i ) ( 1 )
q 1
1 ( i 1 ) / K 1 ( 2 ) / Ki 1
1 ( 1 ) ( 1 ) ( 2 )
r r dim( R ) r dim( R )
r dim( K ) dim( K K )
− − −
−
+=
= + = + =
+ − ∩ =
∑ F FF F
F F F 5+73=9
where r1=5 is the rank of [ ]1 6 FA
× and
( 1 )( 2 ) / Kdim( R ) =FF
( 1 ) ( 1 ) ( 2 )dim( K ) dim( K K )− ∩F F F =73=4 is the dimension of the range of the restriction of F2 to the kernel of F1. The dimension of the kernel of F1 and the dimension of the intersections of the kernels of F1 and F2 are determined by using symbolic calculation with MAPLE®. The rank of [ ]1D gives the motion parameter b1 of the first loop (0≡1A2A…5A≡5B…2B1B≡0) before the closure of the second loop. The dimension of the range of the restriction of F2 to the kernel of F1 gives the number of independent constraints associated to the second loop by taking into account the closure of the first loop. The value of dim(
( 1 )( 2 ) / KRFF )=4 is different from the rank
of [ ]2D (r2=rank( [ ]2D =5). The parallel Cartesian robotic manipulator presented in Fig. 2.1 does not obey Eq. (2.94). For this reason, mobility calculation using the same motion parameter for both loops fails. The following interpretation could be given to this result. The rank of [ ]2D gives the motion parameter b2=5 of the second loop (0≡1A2A…5A≡5C…2C1C≡0) before the closure of the first loop. In fact, we have to consider the number of independent constraints associated to the second loop by taking into account the closure of the first loop. We have to take into account the dimension of the range of the restriction of F2 to the kernel of F1 instead of the motion parameter b2=5. The motion parameters b1 and b2 are easily obtained by inspection with no need to calculate the rank of the matrices [ ]1D and [ ]2D . The motion parameter b1 is given by the number of independent motions between the extreme links 1B and 0 in the serial open kinematic chain 0≡1A2A…5A≡5B…2B1B associated with the first loop when no other loop is closed (Fig. 2.3). We can observe that five independent motions ( , , , ,x y z x yv v v ω ω ) exist between the extreme links 1B and 0 (Fig. 2.3). These velocities form the basis of RF(1). The motion parameter b2 is given by the number of independent motions between the extreme links 1C and 0 in the serial open kinematic chain 0≡1A2A…5A≡5C…2C1C associated with the second loop when no other loop is closed (Fig. 2.4). Five independent motions ( , , , ,x y z x zv v v ω ω ) exist between the extreme links 1C and 0 (Fig. 2.4). These velocities form the basis of RF(2). The dimension of the range of the restriction of F2 to the kernel of F1 can
74 2 Structural parameters
Fig. 2.3. Simple open kinematic chain (0A≡1A2A…5A≡5B…2B1B) associated with the loop AB of parallel Cartesian robotic manipulator: (a) kinematic chain; (b) associated graph
Fig. 2.4. Simple open kinematic chain (0A≡1A2A…5A≡5C…2C1C) associated with the loop AC of parallel Cartesian robotic manipulator: (a) kinematic chain; (b) associated graph
2.2 ChebychevGrüblerKutzbach mobility formulae 75
Fig. 2.5. Complex open kinematic chain obtained by concatenating the closed loop AB and the simple open kinematic chain associated with limb C of the parallel Cartesian robotic manipulator: (a) kinematic chain; (b) associated graph
also be obtained by inspection. Dim(( 1 )( 2 ) / KR
FF ) is given by the number of
independent motions between the extreme links 1C and 0 in the complex open kinematic chain from Fig. 2.5 obtained by concatenating the closed loop AB and the simple open kinematic chain associated with limb C. Only four independent motions ( , , ,x y z zv v v ω ) exist in this case between the extreme links 1C and 0 (Fig. 2.5). These velocities form the basis of
( 1 )( 2 ) / KRFF .
Two other examples of parallel mechanisms with translational motions and with Schönflies motions proposed by the author of this work (Gogu, 2002) are presented to illustrate the applicability of Eq. (2.91).
Fig. 2.6 presents a parallel robotic manipulator (Gogu 2002) with three limbs derived from the solution presented in Fig. 1.1 by eliminating the actuated prismatic joint from limb C. In this case, limb C just has a guiding role, constraining the mobile platform to a planar motion. This mechanism has p=11 joints (2 prismatic and 9 revolute joints) and q=2 independent loops. Each joint has one degree of mobility (fi=1). The two independent loops also have the same motion parameter b1=b2=5 whichever set of independent loops is chosen. The CGK mobility formula gives M=11(5+5)=1. This is an erroneous result. In fact Eq. (2.91) gives r=5+4=9 and Eq. (2.68) indicates two degrees of mobility M=119=2. This is the correct result.
76 2 Structural parameters
Fig. 2.6. Translational robotic manipulator with two degrees of mobility: (a) kinematic chain; (b) associated graph
Fig. 2.7. Fullyparallel robotic manipulator with Schönflies motions: (a) kinematic chain; (b) associated graph
2.2 ChebychevGrüblerKutzbach mobility formulae 77
Fig. 2.7 presents a parallel robotic manipulator with decoupled Schönflies motions (Gogu 2002). The mobile platform has four degrees of freedom: three independent translations and one rotation about an axis of fixed direction. The parallel mechanism has four limbs integrating p=20 joints (4 prismatic and 16 revolute joints) and q=3 independent loops. The three independent loops have the same motion parameter b=b1=b2=6 whichever set of independent loops is chosen. The CGK mobility formula gives M=20(6+6+6)=2. This is also an erroneous result. In fact Eq. (2.91) gives r=6+5+5=16 and Eq. (2.68) indicates the right value of the degrees of mobility: M=2016=4.
We can conclude that original and extended CKG formulae for mobility calculation of multiloop mechanisms are applicable just to the mechanisms that have the rank r of the homogeneous linear set of kinematic constraint equations equal to the sum of the motion parameters bi associated with the q structurally independent closed loops (i=1,2,…,q) when no other loops are closed.
q
ii 1
r b=
=∑ . (2.108)
The CGK formulae are appropriate only for the mechanisms that obey Eqs. (2.94) or (2.108). Equation (2.108) provides a simple criterion under which the CGK formulae are applicable.
For example, the simple planar jointed systems presented in Fig. 2.8 has p=6 revolute joints and q=2 structuralal independent loops with b1=b2=3. The linkage in Fig. 2.8b is derived from Fig. 2.8a by intyroducing the following geometric constraints: AB=CD=EF, AD=BC and DE=CF. The CGK formulae give M=66=0. This is a correct result for the jointed system in Fig. 2.8a which is an immobile structure with r=6, and an eroneous
Fig. 2.8. Planar jointed systems with M=0 (a) and M=1 (b), and their associated graph (c)
78 2 Structural parameters
result for the double parallelogram linkage in Fig 2.8b. In Chapter 3 example 39 we will see that r=5 and M=1 in the case of the double parallelogram linkage. The jointed system in Fig. 2.8a obeys Eq. (2.108) and the linkage in Fig. 2.8b fails to obey this criterion. The CGK formulae can not take into account the geometric constraints mentioned above.
Critical analysis of the CGK formula was previously discussed by Waldron (1966), Davies (1983b) and Fayet and Bonnet (1995), but they did not formalize the limits of applicability of this formula.
2.3 Mobility and connectivity of parallel robots
We have seen that in the last 150 years, sustained efforts have been made to find general mobility formulae for a quick calculation of mobility of any multiloop mechanism, but the “magic formula” had not yet been found. We have shown in the previous sections that several dozen approaches proposed for the calculation of mechanism mobility can be reduced to the same original formula that we call the ChebychevGrüblerKutzbach formula in its original or extended forms. The CGK formula is still largely used for mobility calculation of multiloop mechanisms. Even so, this formula does not fit many classical mechanisms or recent parallel robots. We have explained why this wellknown formula does not work for some multiloop mechanisms, and we have set up a criterion governing mechanisms to which this formula can be applied. We founded our demonstration on two properties related to structural limitation of the independence of the closed loops and on three theorems related to the rank of the homogeneous linear set of constraint equations of a multiloop mechanism.
The lack of mobility formula for quick calculation of mobility of multiloop mechanism with wider applicability represents a stumbling block in structural analysis and type synthesis of spatial mechanisms. To overcome this, the author has recently proposed new formulae (Gogu 2005be) to enable quick calculation of the mobility of parallel mechanisms obeying Eq. (2.91). We have demonstrated new formulae to calculate r for various types of parallel mechanisms without calculating the rank of the homogeneous linear set of constraint equations associated with loop closure or without calculating the rank of the complete screw system associated to the joints of the mechanism. The parameters used in these new formulae can be easily obtained by inspection. An analytical method to compute these parameters has also been developed just for verification and for a better understanding of the meaning of these parameters. This new approach
2.3 Mobility and connectivity of parallel robots 79
based on the theory of linear transformations is presented in the following sections.
2.3.1 General definitions and formulae for mobility and connectivity of mechanisms
In this section we present some general definitions and formulae for mobility and connectivity of mechanisms by using the theory of linear transformations. For a mechanism (simple or complex, open or closed) with holonomic joints, we can define the mobility from a geometric and from a kinematic point of view. We take into consideration the fact that the differential joint constraint equations are linear and homogeneous. So, this set of equations can be integrated to obtain the finite displacements in joints.
Alternative definitions, theorems, properties and conditions are presented in this chapter in order to illustrate the dual (infinitesimal and finite) nature of mobility and connectivity, their correlations and various ways to express them. They are formalized for various types of kinematic chains used in parallel robots in the logical order of increasing complexity.
It is common in scientific approaches to choose a number of hypotheses (postulates) that are assumed to be true within a given theory, and then the theory consists of all theorems and propositions provable using those hypotheses as assumptions. The logic of the propositions presented in this section is:
If A is a given type of kinematic chain than various relations Bi exist and can be demonstrated between the structural parameters of the chain.
In this case the given theory is the theory of structural parameters of kinematic chains in which we assume, as a postulate, that any kinematic chain gives a linear transformation between joint velocity space and external velocity space. The various definitions, properties and propositions presented in this chapter with respect to mobility, connectivity, redundancy and overconstraint form the bases of this approach based on the theory of linear transformations.
Definition 2.1A (Geometric definition of mechanism mobility). Mobility represents the number of independent finite displacements in the joints needed to define the configuration of the mechanism.
Definition 2.1B (Kinematic definition of mechanism mobility). Mobility represents the number of independent infinitesimal displacements in the joints needed to define the infinitesimal displacements of any element of the mechanism.
We distinguish instantaneous mobility iM and general mobility M. The instantaneous mobility characterises the mechanism in a given position i.
80 2 Structural parameters
The general mobility represents the minimum value of the instantaneous mobility
M=min(iM) (2.109)
General mobility is also called fullcycle mobility. A given mechanism has an infinite number of positions and, for each position, we can calculate an instantaneous mobility. In these conditions, general mobility calculation is theoretically impossible by using Eq. (2.109). In this section we will present methods for calculating general mobility without the calculation of the instantaneous mobility. In the following sections, general mobility will be called simply mobility. Simple open mechanisms have M=iM in any position of the mechanism. Complex open kinematic chains and simple/complex closed mechanisms have M≤ iM.
Mobility proposition 1. The mobility of a mechanism with p joints and q≥1 structurally independent closed loops is given by the difference between the total number of independent motion parameters of the joints
1
pii
f=∑ before loop closure provides further constraints, and the r joint pa
rameters that lose their independence after closing the loops. This proposition could be expressed by Eq. (2.68) and the proof is ob
tained directly by taking into account the fact that: a) each joint i with a degree of mobility (connectivity) fi, introduces fi independent motion parameters before all closure provides further constraints, and b) the closed loops cancel the independence of r joint parameters.
Eq. (2.68) was proposed by Moroskine (1954, 1958). He indicated that r is given by the rank of the homogeneous linear set of constraint equations of the mechanism.
Mechanism existence condition 1. A mechanism with q≥1 structurally independent closed loops exists if and only if the total number of independent motion parameters of joints
1
pii
f=∑ before closing the loops is
higher than the number r of joint parameters that lose their independence in the closed loops:
1
p
ii
f r=
>∑ . (2.110)
Eq. (2.110) follows directly from Eq. (2.68) by taking into consideration that a jointed system is a mechanism if M>0. In this case, the jointed system is mobile. If the jointed system is immobile (M=0) it is called a jointed structure or simply a structure (see Fig. 2.8a).
In this section we will define M and r by using the theory of linear transformations. As we have seen in section 2.2.3, the kinematic equation of an
2.3 Mobility and connectivity of parallel robots 81
simple open chain can be described by a matrix transformation from the finite dimensional vector space U (called joint velocity space) into the finite dimensional vector space W (called external velocity space) of basis (W)=( x y z x y zv ,v ,v , , ,ω ω ω ):
:U W
( )→F
q F q .
The linear transformation F is the transformation multiplication by the matrix [ ]6 N
J×
, where 1
pii
N f=
=∑ . That is to each vector q , in the joint vector space U, the linear transformation F assigns a uniquely determined vector ( )F q in the vector space W
F( q )= [ ] [ ]6 N N 1J
× ×q . (2.111)
Matrix [ ]6 NJ
×is usually called the Jacobian matrix. Vector q represents
the joint velocity vector and it is composed of the relative velocities in the joints of the mechanism. Vector ( )F q in W is the image of vector q in U under transformation F. The image of U in W represents the range RF of F and we call it the operational space. For a mechanism Q, its operational space will be denoted by QRF or simply by RQ or RQ.
As F is a matrix transformation, with matrix[ ]6 NJ
×, then we can write
1
p
ii
dim(U ) N f=
= =∑ (2.112)
dim(RF)=rank(F)=rank( [ ]6 NJ
×). (2.113)
For closed chains with 1
pii
N f=
=∑ freedoms in joints and q independent loops, the closure equations of the q loops involve ( )F q =0 and the Jacobian becomes a 6q×N matrix.
For a closed simple (q=1) or complex (q>1) mechanism, the number of joint parameters that lose their independence after loop closures is
r= dim(RF)=rank(F)=rank( [ ]6 q NJ
×). (2.114)
Let us consider a general mechanism Q (that can be simple or complex, open or closed).
82 2 Structural parameters
Definition 2.2A (Geometric definition of connectivity between two links of a mechanism). Connectivity Q
a / bS between two links a and b in a mechanism Q represents the number of independent finite displacements allowed by the mechanism between the two links.
Definition 2.2B (Kinematic definition of connectivity between two links of a mechanism). Connectivity Q
a / bS between two links a and b in a mechanism Q represents the number of independent infinitesimal displacements allowed by the mechanism between the two links.
Connectivity proposition 1. Connectivity Qa / bS between two links (a
and b) in a mechanism Q is given by the dimension of the vector space Qa / bR of the relative velocities between links a and b in mechanism Q
Q Qa / b a / bS dim( R )= . (2.115)
The proof of Eq. (2.115) results directly from Definition 2.2 and the definition of the vector space Q
a / bR . We notice that Qa / bR is a subspace of
QR and implicitly a subspace of W. Any mechanism gives a linear transformation F and consequently Eq.
(2.68) is valid without exception. It is applicable to any type of mechanism (simple or complex, open or closed). In the following section we will present and demonstrate formulae for calculating the dimension of the range RF for simple open mechanisms without calculating the rank of the screw system associated with the joints of the mechanism. The new formulae, demonstrated by using linear transformations, are based on connectivity of a simple open kinematic chain. This important structural parameter is easily determined by inspection. An analytical method will also be presented just for verification and for a better understanding of the meaning of this parameter.
2.3.2 Mobility and connectivity of simple open kinematic chains
A simple open kinematic chain A (12…n) is a serial kinematic chain with n links and p joints (Fig. 2.9). We denote the reference link by 1 and the final link by n. The reference link can be fixed (1≡0) or mobile. All intermediate links (2, …,n1) are binary links. We denote by M=MA the mobility of a simple open kinematic chain given by Eq. (2.68) with r=0. For a given simple open kinematic chain, MA has a constant positive value.
By analogy with the definition of mobility, we can define connectivity from both a geometrical and a kinematical point of view. Connectivity SA of a simple open kinematic chain A (12…n) is given by the connectivity
2.3 Mobility and connectivity of parallel robots 83
Fig. 2.9. General case of a simple open kinematic chain
between the final and reference links n and 1. Links n and 1 are the extreme links of the simple open kinematic chain.
Definition 2.3A. (Geometric definition of connectivity of a simple open kinematic chain). Connectivity of a simple open kinematic chain A (12…n) represents the number of independent finite displacements between the extreme links n and 1 of A.
Definition 2.3B. (Kinematic definition of connectivity of a simple open kinematic chain). Connectivity of a simple open kinematic chain A (12…n) represents the number of independent infinitesimal displacements between the extreme links n and 1 of A
/1A
A nS S= . (2.116)
By analogy with mobility, we distinguish instantaneous connectivity iSA and general connectivity SA. The instantaneous connectivity characterizes the kinematic chain in a given position i. The general connectivity represents the maximum value of the instantaneous connectivity
SA=max(iSA). (2.117)
A given kinematic chain has an infinite number of positions, and with each position we can associate an instantaneous connectivity. In these conditions, general connectivity calculation is theoretically impossible by using Eq. (2.117). In this section we will present methods for calculating the general connectivity without calculating the instantaneous connectivity. General connectivity will be called simply connectivity. For the simple open kinematic chain, SA≥ iSA.
For a simple open kinematic chain A (1 2 …n), the linear transformation FA and Eq. (1.111) becomes (Gogu 2002a):
v=
ω[ ]J [ ]q
(2.118)
where: v and ω are the translation and angular velocity vectors of the distal link n, q is the joint velocity vector composed of the relative velocities in the joints of the mechanism and [ ] [ ]6 N
J J×
= is the Jacobian matrix.
84 2 Structural parameters
Vectors v and ω belong to the velocity vector space An / 1R . This space
defines the operational space of the simple open kinematic chain and
dim( An / 1R )=dim(RA) (2.119)
Mobility proposition 2. The mobility of the simple open kinematic chain A with p joints is given by the dimension of its joint space
MA=dim (UA) =1
p
ii
f=∑ .
(2.120)
Eq. (2.120) results directly from Eqs. (2.68). and (2.112). Connectivity proposition 2A. The connectivity of the simple open ki
nematic chain A (1 2 …n) is given by the dimension of its operational space
SA=dim (RA). (2.121)
Connectivity proposition 2B. The connectivity of the simple open kinematic chain A (1 2 …n) is given by the rank of the Jacobian matrix of the linear transformation FA associated with A
SA=rank (JA). (2.122)
Eqs. (2.121) and (2.122) result directly from Eqs. (2.113) and (2.116). Connectivity indicates the number of independent velocities of the final
link related to the reference link. For a given simple open kinematic chain, SA depends on the geometric configuration of the chain, defined by the finite displacements in the joints, and on the geometric parameters of the links. We recall that the rank of the Jacobian matrix can be calculated numerically or symbolically. The numerical calculation gives us the instantaneous connectivity in a given position of the kinematic chain defined by numerical values of joint variables and geometric parameters. The symbolic calculation gives us the general connectivity in an arbitrary position of the kinematic chain without indicating numerical values of joint variables and geometric parameters.
Connectivity proposition 3. The connectivity of the simple open kinematic chain A is less than or equal to its mobility:
SA≤MA . (2.123)
This is a direct result of Eqs. (2.118), (2.120) and (2.122). The joint velocity space UA is a product space
2.3 Mobility and connectivity of parallel robots 85
p
A i 1 2 pi 1
U U U U U=
= = × ×⋅ ⋅ ⋅×∏ (2.124)
where Ui is the velocity space of the ith joint (the joint between links i and i1).
Let us consider that the simple open kinematic chain A (12…i…n) is obtained by serial concatenation of two simple open kinematic chains A1 (12…i) and A2.(i…n). In this case, A is denoted by A≡A1A2. We consider the subspaces UA1 and UA2 of the joint velocity space UA and their corresponding operational subspaces RA1 and RA2 of the operational space RA≡A1A2 related to the linear transformation FA≡FA≡A1A2. The subspaces UA1 and UA2 have the property { }A1 A2 UU U 0∩ = , that is the intersection of the two subspaces UA1 and UA2 is the zero vector of the linear space UA.
The operational space RA≡A1A2 is an addition space
RA≡A1A2=RA1+RA2 . (2.125)
The vector spaces UA, Ui, RA, RA≡A1A2, RA1 and RA2 have finite dimensions. To determine the dimensions of the joint space UA and of the operational spaces RA and RA≡A1A2, the following properties of the vector spaces are useful:
Property 3 (Dimension of the joint space). The dimension of the joint space is given by the dimension of the product space UA
p
A Aii 1
dim(U ) dim(U )=
=∑ . (2.126)
From Eqs. (2.68) and (1.126) we can see, as a particular case, that the dimension of the velocity space of a joint is given by its degree of mobility (dim(Ui)= fi).
Property 4 (Dimension of the operational space). The dimension of the operational space RA≡A1A2 is given by the dimension of the addition space:
A A1 A2 A1 A2
A1 A2 A1 A2
dim( R ) dim( R R ) dim( R ) dim( R ) dim( R R )
= − = + =+ − ∩
(2.127)
where A1 A2R R∩ is the common range of the subspaces RA1 and RA2 of the operational space RA=A1A2 associated with the simple open mechanism A≡A1A2. We call A1 A2R R∩ the common operational space of A1 and A2.
Connectivity proposition 4A. The connectivity of a simple open kinematic chain A≡A1A2, obtained by serial concatenation of two open simple
86 2 Structural parameters
kinematic chains A1 and A2, is given by the dimension of its addition operational space
A A1 A2 A A1 A2 A1 A2
A1 A2 A1 A2
S dim( R ) dim( R R ) dim( R ) dim( R ) dim( R R )
≡ − ≡ −= = + =+ − ∩
. (2.128)
Connectivity proposition 4B. The connectivity of a simple open kinematic chain A≡A1A2, obtained by serial concatenation of two open simple kinematic chains A1 and A2, is given by the rank of the Jacobian matrix of the linear transformation FA≡A1A2 associated to A≡A1A2
A A1 A2 A A1 A2S rank( J )≡ − ≡ −= . (2.129)
Eqs. (2.128) and (2.129) result directly from Eqs. (2.121), (2.122) and (2.127).
Singular configurations and redundancy of a simple open kinematic chain can be defined by using its mobility and connectivity.
Definition 2.4 (Structural singularity of a simple open kinematic chain). When the general connectivity of a simple open kinematic chain A is smaller than its general mobility (SA<MA), the kinematic chain A has a structural (architectural) singularity.
Definition 2.5 (Instantaneous singularity of a simple open kinematic chain). When the instantaneous connectivity of a simple open kinematic chain in a given position i is smaller than its general connectivity (iSA<SA)., the kinematic chain A is in an instantaneous (kinematic) singularity. The position i is called a singular configuration.
Definition 2.6. (Structural redundancy of a simple open kinematic chain). The structural redundancy TA of the simple open kinematic chain A represents the difference between the dimension of the joint space and the dimension of the operational space
TA=MASA (2.130)
Structural redundancy proposition 1A. The structural redundancy of a simple open kinematic chain A (1 2 …n) is given by the dimension of the kernel of the linear transformation FA associated with A
TA= dim(KF(A)) (2.131)
Structural redundancy proposition 1B. The structural redundancy of a simple open kinematic chain A (1 2 …n) is given by the nullity of the linear transformation FA associated with A
TA = nullity(FA)=nullity (JA). (2.132)
2.3 Mobility and connectivity of parallel robots 87
Eqs. (2.131) and (2.132) follow directly from Definition 2.6 and Eqs. (2.85) and (2.87).
Similar redundancy propositions can be set up for a simple open kinematic chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) obtained by connecting in the same link the extreme links (1A1≡1A2 and nA1≡nA2) of the two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2).
Structural redundancy proposition 2A. The structural redundancy of a simple open kinematic chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) is given by the dimension of the kernel of the linear transformation FA≡A1A2 associated with A≡A1A2
TA= dim(KF(A≡A1A2)) (2.133)
Structural redundancy proposition 2B. The structural redundancy of a simple open kinematic chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) is given by the nullity of the linear transformation FA≡A1A2 associated with A≡A1A2
TA = nullity(FA≡A1A2)=nullity (JA≡A1A2). (2.134)
Eqs. (2.133) and (2.134) follows directly from Definition 2.6 and Eqs. (2.85) and (2.87).
Note 1. Formulae (2.122), (2.123), (2.131) and (2.132) apply to the simple open kinematic chain A (12…n(n+1)).
Note 2. Formulae (2.128), (2.129), (2.133) and (2.134) apply to the simple open kinematic chain A1A2 (1A12A1…nA1≡nA2…2A21A2) obtained by serial concatenation of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2) having the extreme elements joined in the same link nA1≡nA2.
Note 3. Formulae (2.122), (2.129), (2.131)(2.134) involve the calculation of the Jacobian matrices. Numerical calculation of the rank of the Jacobian matrix allows us to calculate only instantaneous values of the structural parameters. Symbolic calculation of the rank could overcome this drawback but both cases are very timeconsuming. For this reason, these formulae have limited practical value.
Note 4. Formulae (2.121), (2.128) and (2.130) are based on the connectivity of the simple open chain. This parameter can be easily determined by inspection by observing the independent motions allowed by the kinematic chain between its extreme links.
Connectivity of a simple open kinematic chain is the main parameter used to calculate mobility of closedloop kinematic chains as will be presented in the following sections.
88 2 Structural parameters
2.3.3 Mobility and connectivity of singleloop kinematic chains
A singleloop kinematic chain is a simple closed kinematic chain with only binary links. We denote it by the letter B. There are two possibilities to obtain a singleloop kinematic chain B having p pairs and n links :
a) by connecting in the same link the extreme links 1 and n+1 of a simple open kinematic chain A (12…n(n+1)), we obtain the closed kinematic chain B (12…n(n+1)≡1) – see Fig. 2.10(a),
b) by connecting in the same link the extreme elements (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2), we obtain the singleloop kinematic chain (1A12A1…nA1≡nA2…2A21A2≡1A1) which we denote by B ← A1A2 – see Fig. 2.10(b).
In the first case, the closure condition indicates the fact that the velocity of link (n+1) related to reference element 1 in the singleloop kinematic chain B is zero:
B
vω
=
[0]. (2.135)
In the second case, the closure condition indicates the fact that velocity of the connecting point situated on the common link (nA1≡nA2) related to the reference link 1A1≡1A2 is the same in the two simple open kinematic chains A1 and A2
A1 A2
=
v vω ω
. (2.136)
The connecting point is denoted by H, and is called the characteristic point.
Eq. (2.136) can be written in the form
= =
A1 A2
v v ω ω
[0]. (2.137)
or
+ =
A1 A2 B
v v v=
ω ω ω[0].
(2.138)
If the open kinematic chain A (12…n(n+1)) is identical to the open kinematic chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2), we can see that the closure equations (2.135) and (2.138) are also identical.
2.3 Mobility and connectivity of parallel robots 89
Fig. 2.10. Simple closed kinematic chains obtained by: (a) connecting in the same link the extreme links 1 and (n+1) of a simple open kinematic chain A (12…n(n+1)) and (b) connecting in the same links the extreme links (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2)
Reciprocally, to any singleloop kinematic chain B with n links, we can associate by splitting up a link:
a) a simple open kinematic chain A (12…n(n+1)) having n+1 links, if we split up the reference element, Fig. 2.10(a), and we keep one part as a reference element and the other part becomes the (n+1)th link.
b) two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2), having n+2 links , if we split up the links 1A1≡1A2 and nA≡nB, Fig 2.12(b).
The simple open kinematic chains A (12…n(n+1)), A1 (1A12A1…nA1), A2 (1A22A2…nA2) and A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) are called simple open kinematic chains associated with the singleloop mechanism B.
We denote by /1A
A nS S= , A1A1 nA1 / 1A1S S= , A2
A2 nA2 / 1A2S S= , A A1 A2A A1 A2 1A2 / 1A1S S ≡ −≡ − =
the connectivity of the simple open kinematic chains A (12…n(n+1)), A1 (1A12A1…nA1), A2 (1A22A2…nA2) and A≡A1A2 (1A12A1…nA1≡nA2…2A21A2). We also denote by UA, UA1, UA2, UA≡A1A2 and by RA, RA1, RA2,
90 2 Structural parameters
RA≡A1A2 the joint velocity vector spaces and the operational velocity vector spaces associated with the simple open kinematic chains A, A1, A2, A≡A1A2.
Mobility proposition 3. The mobility of the singleloop mechanism B with p joints and one closed loop is given by the difference between the number of independent motion parameters of the joints
1
pii
f=∑ before
loop closure provide further constraints and the number of joint parameters rB that lose their independence after loop closure:
1
p
B i Bi
M f r=
= −∑ . (2.139)
Eq. (2.139) represents a particular case for single loop mechanisms of Eq. (2.68).
The following connectivity propositions can be set up for a singleloop kinematic chain B (12…n(n+1)≡1) obtained by connecting the extreme links 1 and (n+1) of a simple open kinematic chain A (12…n(n+1)) in the same link.
Connectivity proposition 5A. The number of joint parameters that lose their independence in a singleloop kinematic chain B (12…n(n+1)≡1) is given by the connectivity of the open simple chain A (12…nn+1) associated with the singleloop chain B
B Ar S= (2.140)
Connectivity proposition 5B. The number of joint parameters that lose their independence in a singleloop kinematic chain B (12…n(n+1)≡1) is given by the dimension of the operational space of the simple open chain A (12…n(n+1)) associated with B
B Ar dim( R )= . (2.141)
Connectivity proposition 5C. The number of joint parameters that lose their independence in a singleloop kinematic chain B (12…n(n+1)≡1) is given by the rank of the Jacobian matrix of the linear transformation FA corresponding to the simple open chain A (12…n(n+1)) associated with B
B Ar rank( J )= (2.142)
Proof. The kinematic equation of the simple open kinematic chain A (12…nn+1) is expressed by Eq. (2.118). By connecting the last link n+1
2.3 Mobility and connectivity of parallel robots 91
with the reference link 1, the relative velocity between links n+1 and 1 becomes zero, and (2.118) becomes a homogeneous linear set
[ ]AJ [ ]q =[0]. (2.143)
In the homogeneous linear set (2.143), the rank of the Jacobian matrix [ ]AJ indicates the number of dependent joint velocities in the singleloop kinematic chain B (12…n(n+1)≡1). But at the same time the rank of the Jacobian matrix [ ]AJ indicates the connectivity and the dimension of the operational space of the simple open chain A (12…n(n+1)) associated with singleloop chain B – see connectivity propositions 2A and 2B and Eqs. (2.121) and (2.122).
Similar connectivity propositions can be set up for a singleloop kinematic chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) in Fig. 2.10b, obtained by connecting the extreme elements (1A1≡1A2 and nA1≡nA2) of the two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2).
Connectivity proposition 6A. The number of joint parameters that lose their independence in a singleloop kinematic chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is given by the connectivity of the open simple chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2
rB=SA≡A1A2 . (2.144)
Connectivity proposition 6B. The number of joint parameters that lose their independence in a singleloop kinematic chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is given by the dimension of the operational space of the open simple chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2
B B A1 A2 A A1 A2 A1 A2
A1 A2 A1 A2
r dim( R ) dim( R ) dim( R R ) dim( R ) dim( R ) dim( R R )
← − ≡ −= = = + =+ − ∩
(2.145)
Connectivity proposition 6C. The number of joint parameters that lose their independence in a singleloop kinematic chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is given by the rank of the Jacobian matrix of the linear transformation FA1A2 corresponding to the simple open simple chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2
B A A1 A2r rank( J )≡ −= . (2.146)
92 2 Structural parameters
Proof. By connecting the distal link 1A2 and the reference link 1A1, the relative velocity between 1A2 and 1A1 becomes zero in the singleloop kinematic chain B ← A1A2 ,and Eq. (2.118) becomes
[ ]A1A2J [ ] [ ]A1 A2q 0
−= . (2.147)
In the homogeneous linear set (2.147), the rank of the Jacobian matrix [ ]A A1A2J ≡ indicates the number of dependent joint velocities in the closed simple kinematic chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1). At the same time the rank of the Jacobian matrix [ ]A A1A2J ≡ indicates the connectivity and the dimension of the operational space of the open simple chain A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2 – see connectivity propositions 4A and 4B and Eqs. (2.128) and (2.129).
The following mobility propositions can be set up for a singleloop kinematic chain B (12…n(n+1)≡1) obtained by connecting in the same link the extreme links 1 and (n+1) of a simple open kinematic chain A (12…n(n+1)).
Mobility proposition 4A. The mobility of a singleloop kinematic chain B (12…n(n+1)≡1) is given by the difference between the mobility and the connectivity of the open simple chain A (12…n(n+1)) associated with B:
B A AM M S= − (2.148)
Eq. (2.148) results directly from Eqs. (2.120), (2.139) and (2.140). Mobility proposition 4B. The mobility of a singleloop kinematic chain
B (12…n(n+1)≡1) is given by the difference between the dimensions of joint and operational spaces of the simple open chain A (12…n(n+1)) associated with B
B A AM dim(U ) dim( R )= − (2.149)
Eq. (2.149) results directly from Eqs. (2.120), (2.139) and (2.141). Mobility proposition 4C. The mobility of a singleloop kinematic chain
B (12…n(n+1)≡1) is given by the difference between the dimension of the joint space and the rank of the Jacobian matrix of the linear transformation FA corresponding to the simple open chain A (12…n(n+1)) associated with B
B A AM dim(U ) rank( J )= − . (2.150)
Eq. (2.150) results directly from Eqs. (2.120), (2.139) and (2.142). Mobility proposition 4D. The mobility of a singleloop kinematic chain
B (12…n(n+1)≡1) is given by the nullity of the linear transformation
2.3 Mobility and connectivity of parallel robots 93
FA corresponding to the simple open chain A (12…n(n+1)) associated with B
MB= nullity(FA)=nullity(JA) (2.151)
Eq. (2.151) results from Eq. (2.149) by taking into consideration the conservation of the dimension in the matrix transformation FA
dim(RA)+nullity(FA)=dim(UA) (2.152)
and
rank(JA)+nullity(JA)=dim(UA). (2.153)
Mobility proposition 4E. The mobility of a singleloop kinematic chain B (12…n(n+1)≡1) is given by the structural redundancy of the simple open chain A (12…n(n+1)) associated with B
MB= TA . (2.154)
Eq. (2.154) results directly from Eqs. (2.132) and (2.151). Similar mobility propositions can be set up for a singleloop kinematic
chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) in Fig. 2.10b, obtained by connecting the extreme elements (1A1≡1A2 and nA1≡nA2) of the two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2).
Mobility proposition 5A. The mobility of a singleloop kinematic chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is given by the difference between the mobility and the connectivity of the simple open chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2:
B A A1 A2 A A1 A2M M S≡ − ≡ −= − (2.155)
Eq. (2.155) follows directly from Eqs. (2.120), (2.139) and (2.144). Mobility proposition 5B. The mobility of a singleloop kinematic chain
B B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is given by the difference between the dimensions of joint and operational spaces of the simple open chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2
B A A1 A2 A A1 A2M dim(U ) dim( R )≡ − ≡ −= − (2.156)
Eq. (2.156) follows directly from Eqs. (2.120), (2.139) and (2.145). Mobility proposition 5C. The mobility of a singleloop kinematic chain
B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is given by the difference between the dimension of the joint space and the rank of the Jacobian matrix of the linear transformation FA corresponding to the simple open chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2
94 2 Structural parameters
B A A1 A2 A A1 A2M dim(U ) rank( J )≡ − ≡ −= − (2.157)
Eq. (2.157) follows directly from Eqs. (2.120), (2.139) and (2.146). Mobility proposition 5D. The mobility of a singleloop kinematic chain
B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is given by the nullity of the linear transformation FA≡A1A2 corresponding to the simple open chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2
MB= nullity(FA≡A1A2)=nullity(J A≡A1A2) (2.158)
Eq. (2.158) results from Eq. (2.156) by taking into consideration the conservation of the dimension in the matrix transformation FA≡A1A2.
Mobility proposition 5E. The mobility of a singleloop kinematic chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is given by the structural redundancy of the simple open chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2
MB= T A≡A1A2 (2.159)
Eq. (2.159) follows from Eq. (2.134) and (2.158). Various combinations of Eqs. (2.139) and (2.140)(2.142) can be used to
obtain equivalent formulae for mobility calculation of a singleloop kinematic chain B (12…n(n+1)≡1) obtained by connecting the extreme links 1 and (n+1) of a simple open kinematic chain A (12…n(n+1)) – Fig. 2.10(a):
p
B i Ai 1
M f S=
= −∑ (2.160)
p
B i Ai 1
M f dim( R )=
= −∑ (2.161)
p
B i Ai 1
M f rank( J )=
= −∑ . (2.162)
Various combinations of Eqs. (2.139) and (2.144)(2.146) can be used to obtain equivalent formulae for mobility calculation of a singleloop kinematic chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) obtained by connecting the extreme elements (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2). – Fig. 2.10(b):
2.3 Mobility and connectivity of parallel robots 95
p
B i A=A1A2i 1
M f S=
= −∑ (2.163)
p
B i A A1 A2i 1
M f dim( R )≡ −=
= −∑ (2.164)
p
B i A1 A2 A1 A2i 1
M f dim( R ) dim( R ) dim( R R )=
= − − + ∩∑ (2.165)
p
B i A A1 A2i 1
M f rank( J ) ≡ −=
= −∑ . (2.166)
The following propositions can be set up concerning the existence of a singleloop mechanism by expressing the fact that MB>0.
Mechanism existence condition 2. A singleloop mechanism B exists if and only if the total number of independent motion parameters of joints
1
pii
f=∑ before loop closure is higher than the number of joint parameters
rB that lose their independence after loop closure:
1
p
i Bi
f r=
>∑ . (2.167)
Eq. (2.167) is a particular case of (2.110) when q=1. Mechanism existence condition 3A. A singleloop mechanism B (12
…n(n+1)≡1) exists if and only if the mobility of the simple open chain A (12…n(n+1)) associated with B is higher than the connectivity of A
MA>SA . (2.168)
Eq. (2.168) results directly from Eqs. (2.120), (2.140) and (2.167). Mechanism existence condition 3B. A singleloop mechanism B ← A1
A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) exists if and only if the mobility of the simple open chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2 is higher than the connectivity of A≡A1A2
MA>SA≡A1A2 . (2.169
Eq. (2.169) results directly from Eqs. (2.120), (2.144) and (2.167). Mechanism existence condition 4A. A singleloop mechanism B (12
…n(n+1)≡1) exists if and only if the simple open chain A (12…n(n+1)) associated with B is structurally redundant
96 2 Structural parameters
TA>0 . (2.170)
Eq. (2.170) results directly from (2.154). Mechanism existence condition 4B. A singleloop mechanism B ← A1
A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) exists if and only if the simple open chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with B ← A1A2 is structurally redundant
TA≡A1A2 >0 . (2.171)
Eq. (2.171) results directly from (2.159). Note 5. Formulae (2.140)(2.142), (2.148)(2.154), (2.160)(2.162),
(2.168) and (2.170) apply to the singleloop kinematic chain B (12…n(n+1)≡1) obtained by connecting in the same link the extreme links 1 and (n+1) of the simple open kinematic chain A (12…n(n+1)) associated with B.
Note 6. Formulae (2.144)(2.146), (2.155)(2.159), (2.163)(2.166), (2.169) and (2.171) apply to the singleloop kinematic chain B ← A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) obtained by connecting the extreme elements (1A1≡1A2 and nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2). In this case the three chains A1, A2 and A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) are called simple open kinematic chains associated with B ← A1A2.
Note 7. Formulae (2.142), (2.146), (2.150)(2.153), (2.157), (2.158), (2.162), (2.166) involve the calculation of the Jacobian matrices. Numerical calculation of the rank and nullity of the Jacobian matrix allows us to calculate only instantaneous values of the structural parameters. Symbolic calculation of the rank and nullity could overcome this drawback but both cases are very timeconsuming. For this reason, these formulae also have limited practical value for the quick calculation of mobility.
Note 8. Formulae (2.140), (2.141), (2.144), (2.145), (2.148), (2.149), (2.154)(2.156), (2.159)(2.161), (2.163)(2.165) and (2.168)(2.171) are based on the connectivity of the simple open chain associated with the closed mechanism. This parameter can be easily determined through inspection by observing the independent motions given by the associated open chain between its extreme links.
2.3.4 Connectivity between two elements of a singleloop kinematic chain
Let us consider a singleloop kinematic chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) obtained by connecting the extreme elements
2.3 Mobility and connectivity of parallel robots 97
(1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2) to the same link  Fig. 2.10b. We consider UA1 and UA2 the joint velocity subspaces of the open simple kinematic chains A1 and A2 and their corresponding operational spaces RA1 and RA2. The subspaces UA1 and UA2 have the property { }A1 A2 UU U 0∩ = . The common space of RA1 and RA2 is the vector space B
n / 1 A1 A2R R R= ∩ of relative velocities between links n≡nA1≡nA2 and 1≡1A1≡1A2 in the singleloop kinematic chain B ←A1A2.
Connectivity proposition 7A. The connectivity Bn / 1S between links
n≡nA1≡nA2 and 1≡1A1≡1A2 in the singleloop kinematic chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) obtained by connecting the extreme elements 1≡1A1≡1A2 and n≡nA1≡nA2 of two open simple kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2), to the same link, is given by the dimension of the common vector space of A1 and A2
Bn / 1 A1 A2S dim( R R )= ∩ . (2.172)
Proof. In the open simple kinematic chain A1, the number of relative independent infinitesimal displacements or velocities between extreme links nA1 and 1A1 is given by SA1=dim (RA1). The number of relative independent velocities between extreme links nA2 and 1A2 in the simple open kinematic chain A2 is given by SA2=dim (RA2). As we have considered 1A1≡1A2 and nA1≡nA2, the number of relative independent velocities between two elements 1≡1A1≡1A2 and n≡nA1≡nA2 in the singleloop mechanism B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is given by the dimension of the intersection of operational vector spaces RA1 and RA2.
Connectivity proposition 7B. The connectivity Bn / 1S between links
n≡nA1≡nA2 and 1≡1A1≡1A2 in the singleloop kinematic chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1), obtained by connecting the extreme elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2), to the same link is given by the difference between the sum of the connectivities of A1 and A2 and the connectivity of the simple open chain A=A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with the closed chain B ←A1A2
Bn / 1 A1 A2 A A1 A2S S S S ≡ −= + − . (2.173)
Proof. According to Eq. (2.172), Eq. (2.128) can be written as B
A A1 A2 A1 A2 n / 1S S S S≡ − = + − . (2.174)
98 2 Structural parameters
Eq. (2.174) expresses the fact that connectivity of the simple open chain A=A1A2 associated with the closed chain B ←A1A2 is given by the difference between the sum of connectivities of A1 and A2 and the connectivity between links n≡nA1≡nA2 and 1≡1A1≡1A2 in singleloop kinematic chain B ←A1A2.
Eq. (2.173) follows directly from (2.174). A similar equation to (2.173) was proposed by Voinea and Atanasiu (1960) to define the connectivity of a link in a singleloop mechanism by using screw theory.
Connectivity proposition 8A. Two links n≡nA1≡nA2 and 1≡1A1≡1A2 in the singleloop kinematic chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1), obtained by connecting the extreme elements 1≡1A1≡1A2 and n≡nA1≡nA2 of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2) to the same link, have relative movements if the dimension of the common operational space of A1 and A2 is larger than zero
Bn / 1S >0 if A1 A2dim( R R ) 0∩ > . (2.175)
If
A1 A2dim( R R ) 0∩ = (2.176)
two links n≡nA1≡nA2 and 1≡1A1≡1A2 are relatively immobile ( Bn / 1S =0).
Eqs. (2.175) and (2.176) result directly from Eq. (2.172). Connectivity proposition 8B. Two links n≡nA1≡nA2 and 1≡1A1≡1A2 in
the singleloop kinematic chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1), obtained by connecting the extreme elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2) to the same link, have relative movements if the sum of connectivities of A1 and A2 is greater than the connectivity of the simple open chain A=A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with the closed chain B ←A1A2
Bn / 1S >0 if A1 A2 A A1 A2S S S ≡ −+ > . (2.177)
If
A1 A2 A A1 A2S S S ≡ −+ = (2.178)
two links n≡nA1≡nA2 and 1≡1A1≡1A2 are relatively immobile ( Bn / 1S =0).
Eqs. (2.177) and (2.178) result directly from Eq. (2.173). Connectivity proposition 9. Connectivity B
n / 1S between two links n≡nA1≡nA2 and 1≡1A1≡1A2 in the singleloop kinematic chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1), obtained by connecting the extreme
2.3 Mobility and connectivity of parallel robots 99
elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2) to the same link, is less than or equal to the mobility MB of the singleloop mechanism B ←A1A2
Bn / 1 BS M≤ . (2.179)
The proof of Eq. (2.179) is founded on definitions 1 and 2 of mobility and connectivity. If B
n / 1 BS M> , MB independent parameters are not sufficient to describe the relative motion between two elements 1≡1A1≡1A2 and n≡nA1≡nA2 of the singleloop kinematic chain B ←A1A2 ;this contradicts the definition of mobility.
Connectivity proposition 10A. The basis of the vector space Bn / 1R of
relative velocities between the links n≡nA1≡nA2 and 1≡1A1≡1A2 in the singleloop kinematic chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) does not vary with with the position of the characteristic point on link n≡nA1≡nA2.
Proof. We recall that the characteristic point is the point where link n≡nA1≡nA2 is split up to get associated open chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2). The independent velocities of this point define the basis ( B
n / 1R ). Link n≡nA1≡nA2 has an infinite number of points and the velocity of each point can be expressed by a linear combination of the independent velocities of the same basis. The basis ( B
n / 1R ) is valid for any point of the mobile platform including the points with the most restrictive motion.
Connectivity proposition 10B. The connectivity Bn / 1S between the
links n≡nA1≡nA2 and 1≡1A1≡1A2 in the singleloop kinematic chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) obtained by connecting the extreme elements (1≡1A1≡1A2 and n≡nA1≡nA2) of two simple open kinematic chains A1 (1A12A1…nA1) and A2 (1A22A2…nA2) to the same link, does not vary with the position of the characteristic point on link n≡nA1≡nA2.
Connectivity proposition 10B is a direct consequence of proposition 10A.
Connectivity proposition 11A. The basis of operational space RA≡A1A2 of simple open kinematic chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated to singleloop chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) does not vary with the choice of split up links 1 and n.
Proof. We have seen that the simple open kinematic chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with the singleloop kinematic chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) could be obtained by splitting up the link 1A1≡1A2 into 1A1 and 1A2. Two simple open chains A1
100 2 Structural parameters
(1A12A1…nA1) and A2 (1A22A2…nA2) can also be associated with A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) by splitting up the link nA1≡nA2 into nA1 and nA2. The operational vector space RA≡A1A2 is an addition space RA≡A1
A2=R1A+RA2 independently of the choice of 1A1≡1A2 and nA1≡nA2 to be split up.
Connectivity proposition 11B. The connectivity SA≡A1A2 of the simple open kinematic chain A≡A1A2 (1A12A1…nA1≡nA2…2A21A2) associated with the singleloop chain B ←A1A2 (1A12A1…nA1≡nA2…2A21A2≡1A1) is does not vary with the choice of split up links 1 and n.
Connectivity proposition 11B is a direct consequence of proposition 11A.
The following remark is very important for the choice of the bases of operational vector spaces RA1 and RA2 when there are various ways to choose them.
Remark 1. (Choice of the bases of operational vector spaces RA1 and RA2). If there are different ways to choose the bases of the operational vector spaces RA1 and/or RA2, the choice must be made by respecting the following:
a) connectivity propositions 79 stating that Bn / 1 B0 S M< ≤ (2.180)
b) connectivity propositions 10A and B and 11A and B, c) propositions 2 and 3 of the existence of the mobile singleloop
mechanism and the corresponding conditions (2.167)(2.171), d) definition of connectivity (in the sense of general connectivity) as the
maximum value of the instantaneous connectivity  Eq. (2.117), e) permanent constraints in the relative motions of the links generated
by the closed loop with special geometric conditions. The formalisation for mobility and connectivity of simple open kine
matic chains and singleloop mechanisms will be used in the next sections to demonstrate new formulae for quick mobility calculation of parallel robots with simple and complex limbs.
2.3.5. Mobility and connectivity of parallel robots with simple limbs
In this section we consider parallel robots with simple limbs. They are complex closed mechanisms composed of two polinary links and many binary links. The two polinary links are the reference link and the final link. They represent the reference and mobile platforms.
2.3 Mobility and connectivity of parallel robots 101
Fig. 2.11. The parallel mechanism C=A1A2...AK with k elementary limbs
Definition 2.7 (Parallel robot with simple limbs). A parallel robot with simple limbs is a complex mechanism C ← A1A2…Ak with two polinary elements 1C≡1Ai and nC≡nAi connected by k simple open chains Ai (1Ai2Ai…nAi), i=1,2,…k (Fig. 2.11). The two polinary elements are the reference platform 1C and the mobile platform nC. The simple open kinematic chain Ai is called a simple limb.
The complex mechanism C ← A1A2…Ak has p joints and k1 structurally independent closed loops concatenated inparallel. We denote by UAi and RAi the joint velocity space and the operational velocity space of the simple open kinematic chain Ai, and by UA1Ai and RA1Ai the joint velocity space and the operational velocity space of the simple open kinematic chain A1Ai (1A12A1…nA1≡nAi(n1)Ai(n2)Ai…2Ai1Ai) associated with each closed loop Bi ← A1Ai (1A12A1…nA1≡nAi(n1)Ai(n2)Ai…2Ai1Ai≡1A1), i=2,3,… ,k. We denote by r=rC the number of joint parameters that lose their independence in the k1 structurally independent closed loops concatenated inparallel in the complex mechanism C ← A1A2…Ak.
We note that the simple limbs Ai are detached from the parallel mechanism C ← A1A2…Ak when defining the associated parameters UAi, RAi and SAi. Todisconnect them the mobile platform is cut up in the proximity of each limb.
Definition 2.8. (Connectivity of the mobile platform in the parallel robot with simple limbs). The connectivity C
n / 1S of the mobile platform n≡nAi in a parallel mechanism C ← A1A2…Ak with k simple limbs Ai (1Ai2Ai…nAi), i=1,2,…k is given by the number of independent velocities be
102 2 Structural parameters
tween the mobile platform n≡nAi and the reference platform 1=1Ai (i=1,2,…k) in the parallel mechanism C.
Connectivity proposition 12 (Connectivity of the mobile platform in the parallel robot with simple limbs). The connectivity C
n / 1S of the mobile platform n≡nAi in a parallel mechanism C ← A1A2…Ak with k simple limbs Ai (1Ai2Ai…nAi), i=1,2,…k is given by the dimension of the common operational spaces of the simple limbs Ai associated with the parallel mechanism C
C Cn / 1 n / 1 A1 A2 AkS dim( R ) dim( R R ... R )= = ∩ ∩ ∩ (2.181)
where C A1 A2 ... Akn / 1 n / 1S S − − −= denotes the connectivity between the mobile platform
n≡nAi and the reference platform 1≡1Ai in the parallel mechanism C ← A1A2…Ak,
C A1 A2 ... Akn / 1 n / 1R R − − −=  operational velocity vector space of parallel mecha
nism C ← A1A2…Ak defined by the relative velocities between the mobile platform n≡nAi and the reference platform 1≡1Ai in the parallel mechanism C ← A1A2…Ak.
The proof of the Eq. (2.181) is obtained by recurrence A1 A2 A1 A2n / 1 n / 1 A1 A2S dim( R ) dim( R R )− −= = ∩ (2.182)
A1 A2 A3 A1 A2 A3 A1 A2n / 1 n / 1 n / 1 A3
A1 A2 A3
S dim( R ) dim( R R ) dim( R R R )
− − − − −= = ∩ =∩ ∩
(2.183)
A1 A2 ... Ai A1 A2 ... Ai A1 A2 ... A( i 1 )n / 1 n / 1 n / 1 Ai
A1 A2 A( i 1 ) Ai
S dim( R ) dim( R R ) dim( R R ... R R )
− − − − − − − − − −
−
= = ∩ =∩ ∩ ∩ ∩
(2.184)
where A1 A2n / 1S − is the connectivity and the between the mobile platform n≡nAi
and the reference platform 1≡1Ai (i=1,2) in the parallel mechanism C ← A1A2 with only two limbs,
A1 A2 A3n / 1S − − is the connectivity between the mobile platform n≡nAi and the
reference platform 1≡1Ai (i=1,2,3) in the parallel mechanism C ← A1A2A3 with only three limbs,
2.3 Mobility and connectivity of parallel robots 103
A1 A2 ... Ain / 1S − − − is the connectivity between the mobile platform n≡nAi and
the reference platform 1≡1Ai (i=1,2,…k) in the parallel mechanism C ← A1A2…Ai with i limbs,
A1 A2n / 1R − is the operational velocity space of the parallel mechanism
C ← A1A2 with only two limbs, A1 A2 A3n / 1R − −  operational velocity space of the parallel mechanism C ← A1
A2A3 with only three limbs, A1 A2 ... Ain / 1R − − −  operational velocity space of the parallel mechanism
C ← A1A2…Ai with i limbs. Conectivity proposition 13A. The number of joint parameters that lose
their independence in a parallel mechanism C ← A1A2…Ak with k simple limbs Ai (1Ai2Ai…nAi) is given by the difference between the sum of the dimensions of the operational spaces RAi of the simple open chains Ai (i=1,2,…,k) associated with the simple limbs and the connectivity of the mobile platform C
n / 1S in the parallel mechanism C ← A1A2…Ak k
C A1 A2 ... Ak Ai A1 A2 Aki 1
r r dim( R ) dim( R R ... R )− − −=
= = − ∩ ∩ ∩∑ . (2.185)
Proof. The proof of Eq. (2.185) is also obtained by recurrence. We initially consider the complex open mechanism associated with the parallel mechanism by considering that the reference links 1Ai of the simple limbs Ai are not integrated into the reference platform 1Ai≠1Aj (i,j=1,2,…k, i≠j). At this initial step the reference platform does not exist. It is divided into small parts corresponding to the links 1Ai of each limb. Only the links nAi of the simple limbs Ai are connected in the mobile platform n=nAi (i=1,2,…k)  see Fig. 2.12. The complex open kinematic chain has no closed loops. All mobility parameters of the joints are independent.
In the first step, we connect only links 1A1 and 1A2 of limbs A1 and A2 to form the reference platform 1 (1≡1A1≡1A2). The complex kinematic chain in Fig. 2.13 has only one closed loop A1A2. By taking into consideration connectivity proposition 6B, the number of joint parameters that lose their independence in the closed loop B ← A1A2 (1A12A1…(n1)A1nA1≡nA2(n1)A2 …2A21A2≡1A1) is given by the connectivity of the simple open chain A1A2 (1A12A1…(n1)A1nA1≡nA2(n1)A2 …2A21A2) associated with the closed loop B ← A1A2
A1 A2 A1 A2 A1 A2
A1 A2 A1 A2
r dim( R ) dim( R R ) dim( R ) dim( R ) dim( R R )
− −= = + =+ − ∩
. (2.186)
104 2 Structural parameters
In the second step, we connect link 1A3 of limb A3 with the reference platform 1 (1≡1A1≡1A2≡1A3)  see Fig. 2.14. The number of joint parameters that lose their independence in the closed loops A1A2A3 is
Fig. 2.12. Complex open kinematic chain with k branches
Fig. 2.13. Complex kinematic chain with one closed loop and k2 open branches
2.3 Mobility and connectivity of parallel robots 105
A1 A2A1 A2 A3 A1 A2 ( A1 A2 ) A3 A1 A2 n / 1 A3
A1 A2 A1 A2A1 A2 n / 1 A3 n / 1 A3
A1 A2 A1 A2 A3 A1 A2 A3
A1 A2 A3 A1 A2 A3
r r S r dim( R R )
r dim( R ) dim( R ) dim( R R )r dim( R R ) dim( R ) dim( R R R )dim( R ) dim( R ) dim( R ) dim( R R R ).
−− − − − − −
− −−
−
= + = + + =
+ + − ∩ =+ ∩ + − ∩ ∩ =
+ + − ∩ ∩
(2.187)
In Eq. (2.187), we denoted by S(A1A2)A3 the connectivity of the complex open kinematic chain (A1A2)A3 obtained by serial concatenation of the simple closed kinematic chain B ← A1A2 and the simple open kinematic chain A3 in Fig. 2.13. S(A1A2)A3 is given by the connectivity between the link 1A3 and the reference link 1≡1A1≡1A2 in the complex kinematic chain (A1A2)A3 by taking into consideration the existence of the closed loop B ← A1A2
A1 A2( A1 A2 ) A3 n / 1 A3
A1 A2 A1 A2n / 1 A3 n / 1 A3
A1 A2 A3 A1 A2 A3
S dim( R R )
dim( R ) dim( R ) dim( R R )dim( R R ) dim( R ) dim( R R R ).
−− −
− −
= + =
+ − ∩ =∩ + − ∩ ∩
(2.188
In the third step, we connect link 1A4 of limb A4 with the reference platform 1≡1A1≡1A2≡1A3≡1A4  see Fig. 2.15. The number of joint parameters that lose their independence in the closed loops A1A2A3A4 is
A1 A2 A3 A4 A1 A2 A3 ( A1 A2 A3 ) A4
A1 A2 A3A1 A2 A3 n / 1 A4
A1 A2 A3 A1 A2 A3 A4A1 A2 A3 n / 1 A4 n / 1 n / 1
A1 A2 A3 A1 A2 A3 A4
A1 A2 A3 A4
A1
r r S
r dim( R R )
r dim( R ) dim( R ) dim( R R )r dim( R R R ) dim( R )dim( R R R R )dim( R ) di
− − − − − − − −
− −− −
− − − −− −
− −
= + =
+ + =
+ + − ∩ =+ ∩ ∩ + −∩ ∩ ∩ =+ A2 A3 A4
A1 A2 A3 A4
m( R ) dim( R ) dim( R )dim( R R R R ).
+ +− ∩ ∩ ∩
(2.189)
The connectivity of the complex open kinematic chain (A1A2A3)A4 obtained by serial concatenation of the complex closed kinematic chain A1A2A3 and the simple open kinematic chain A4 in Fig. 2.14 is denoted by S(A1A2A3)A4 in Eq. (2.189). S(A1A2A3)A4 is given by the connectivity between link 1A4 and the reference link 1≡1A1≡1A2≡1A3 in the complex open kinematic chain (A1A2A3)A4 by taking into consideration the closed loops of the complex closed kinematic chain A1A2A3
106 2 Structural parameters
A1 A2 A3( A1 A2 A3 ) A4 n / 1 A4
A1 A2 A3 A1 A2 A3n / 1 A4 n / 1 A4
A1 A2 A3 A4 A1 A2 A3 A4
S dim( R R )
dim( R ) dim( R ) dim( R R )dim( R R R ) dim( R ) dim( R R R R )
− −− − −
− − − −
= + =
+ − ∩ =∩ ∩ + − ∩ ∩ ∩
.
(2.190)
Fig. 2.14. Complex kinematic chain with two structurally independent closed loops and k3 open branches
Fig. 2.15. Complex open kinematic chain with three structurally independent closed loops and k4 open branches
2.3 Mobility and connectivity of parallel robots 107
At the end of the recursive procedure we obtain Eq. (2.185). Connectivity proposition 13B. The number of joint parameters that
lose their independence in the parallel mechanism C ← A1A2…Ak with k simple limbs Ai (1Ai2Ai…nAi) is given by the difference between the sum of the connectivities SAi of the open simple chains Ai (i=1,2,…,k) associated with the simple limbs Ai ,and the platform connectivity C
n / 1S in the parallel mechanism C ← A1A2…Ak
kC
C A1 A2 ... Ak Ai n / 1i 1
r r S S− − −=
= = −∑ (2.191
where SAi=dim(RAi), Cn / 1 A1 A2 AkS d im( R R ... R )= ∩ ∩ ∩ , RAi is the opera
tional vector space of the simple open kinematic chain Ai associated with the simple limb Ai.
The proof of Eq. (2.191) is obtained directly from Eq. (2.185) and from the definition of the connectivity of a simple open chain as the relative connectivity between the extreme links nAi and 1 of the elementary open chain Ai (SAi= Ai
nAi / 1S ). We can see that Eq. (2.174) represents a particular case of Eq. (2.191) when k=2. In this case the parallel mechanism with two limbs becomes a singleloop mechanism.
Mobility proposition 6. The mobility of a parallel mechanism C ← A1A2…Ak with p joints and k simple limbs Ai (1Ai2Ai…nAi), i=1,2,…k, is given by
k k
C Ai Ai A1 A2 Aki 1 i 1
M dim(U ) dim( R ) dim( R R ... R )= =
= − + ∩ ∩ ∩∑ ∑ (2.192)
or p k
CC i Ai n / 1
i 1 i 1M f S S
= =
= − +∑ ∑ . (2.193)
where: UAi and RAi are the joint velocity space and the operational velocity
space of the simple open kinematic chain Ai (1Ai2Ai…nAi) associated with Ai limb,
fi is the mobility of the ith joint, SAi is the connectivity of simple open kinematic chain Ai associated with
Ailimb, Cn / 1S is the connectivity of mobile platform in the parallel mechanism C.
Eqs. (2.192) and (2.193) are obtained directly by integrating Eqs. (2.126), (2.185) and (2.191) in Eq. (2.68).
108 2 Structural parameters
Connectivity proposition 14. Mobile and reference platforms n≡nAi and 1≡1Ai of a parallel mechanism C ← A1A2…Ak with k simple limbs Ai (1Ai2Ai…nAi), i=1,2,…k have relative movements if the dimension of the common operational space of Ai is greater than zero
Cn / 1S >0 if A1 A2 Akdim( R R ... R ) 0∩ ∩ ∩ > . (2.194)
If
A1 A2 Akdim( R R ... R ) 0∩ ∩ ∩ = (2.195)
two platforms are relatively immobile ( Cn / 1S =0).
Eqs. (2.194) and (2.195) result directly from Eq. (2.181). Connectivity proposition 15.The connectivity C
n / 1S between the mobile and reference platforms n≡nAi and 1≡1Ai of the parallel mechanism C ← A1A2…Ak with k simple limbs Ai (1Ai2Ai…nAi), i=1,2,…k is less than or equal to the mobility MC of the parallel mechanism C ← A1A2…Ak
Cn / 1 CS M≤ . (2.196)
The proof of Eq. (2.196) is founded on definitions 1 and 2 of mobility and connectivity. If c
n / 1 cS M> , Mc independent parameters are not sufficient to describe the relative motion between 1≡1Ai and n≡nAi of the parallel mechanism C ← A1A2…Ak which is contradictory to the definition of mobility.
Connectivity proposition 16. The basis of the vector space Cn / 1R of
relative velocities between the mobile and reference platforms n≡nAi and 1≡1Ai in the parallel mechanism C ← A1A2…Ak with k simple limbs Ai (1Ai2Ai…nAi), i=1,2,…k does not vary with the position of the characteristic point on the mobile platform n≡nAi.
Proof. The independent velocities of the characteristic point define the basis ( C
n / 1R ). The mobile platform n≡nAi has an infinite number of points, and the velocity of each point can be expressed by a linear combination of the independent velocities of the same basis. The basis ( C
n / 1R ) is valid for any point of the mobile platform including the points with the most restrictive motion.
The following remark is very important for the choice of the bases of operational vector spaces RAi when multiple possibilities exist to define them.
Remark 2. (Choice of the bases of operational vector spaces RAi). If there are different ways to choose the bases of the operational vector spaces RAi ,the choice must be made by respecting the following:
2.3 Mobility and connectivity of parallel robots 109
a) connectivity propositions 14 and 15 stating that Cn / 1 C0 S M< ≤ (2.197)
b) connectivity proposition 16, c) definition of mobility (in the sense of general mobility) as the mini
mum value of the instantaneous mobility  Eq. (2.109), d) permanent constraints in the relative motions of the links generated
by the closed loops with special geometric conditions. We note that condition c) implies that the bases in Eq. (2.181) are se
lected so that the minimum value of Cn / 1S is obtained. By this choice, Eqs.
(2.192) and (2.193) fit in with Eq. (2.109) setting up general mobility as the minimum value of instantaneous mobility.
2.3.6. Mobility and connectivity of parallel robots with complex limbs
In this section we consider parallel robots with complex limbs. They are complex closed mechanisms integrating more than two polynary links. Two polynary links are the mobile and the reference platforms. The remaining polinary links represents intermediary platforms integrated into the complex limbs.
Definition 2.9 (Parallel robot with complex limbs). A parallel robot with complex limbs denoted by D ← A1A2…Ak1E1E2…Ek2 is a complex mechanism with p joints in which the mobile platform n≡nAi≡nEj is connected to the reference platform 1≡1Ai≡1Ej by k≥2 limbs, of which at least one limb is complex.
A complex limb contains at least one closed loop. We denoted by Ai (i=1,2,…k1) the elementary limbs, and by Ej (j=1,2,…k2) the complex limbs (k1+k2=k).
We can see that parallel mechanisms with simple limbs represents a particular case of D when k2=0. If k1=0 all limbs are complex. We denote by UAi, UEj and RAi, REj the joint velocity spaces and the operational velocity spaces associated with limbs Ai and Ej.
We also denote by r=rD the number of joint parameters that lose their independence in the closed loops of the complex mechanism D ← A1A2…Ak1E1E2…Ek2.
The joint velocity space UEj and the operational velocity space REj associated with the complex limb Ej can be introduced by analogy with the corresponding parameters UAi and RAi defined for the simple open kinematic chains in section 2.3.2.
110 2 Structural parameters
Definition 2.10. (Joint velocity space associated with a complex limb). Joint velocity space UEj associated with complex limb Ej is the joint space of the complex open kinematic chain associated with the limb by opening all closed loops integrated in the limb.
Definition 2.11. (Operational velocity space associated with a complex limb). The operational velocity space REj associated with the complex limb Ej is defined by the relative velocities between the mobile platform n≡nEj and the reference platform 1≡1Ej in the complex cinematic chain associated with limb Ej.
We note that the constraints generated by the closed loops integrated in the complex limb Ej are taken into account in the operational velocity space REj ,and are ignored in the joint velocity space UEj.
Definition 2.12. (Connectivity of a complex limb) The connectivity SEj of a complex limb is given by the number of independent motions between the mobile platform n≡nEj and the reference platform 1≡1Ej in the complex cinematic chain associated with limb Ej.
We note that simple and complex limbs Ai and Ej are detached from the parallel mechanism D ← A1A2…Ak1E1E2…Ek2 when defining the associated parameters UAi, RAi, SAi and UEj, REj, SEj. To disconnect them, the mobile platform is cut up near each limb.
Definition 2.13. (Connectivity of the mobile platform in a parallel robot with complex limbs). The connectivity D
n / 1S of the mobile platform n≡nAi≡nEj in a parallel mechanism D ← A1A2…Ak1E1E2…Ek2 with k1 simple limbs Ai (1Ai2Ai…nAi), i=1,2,…k1 and and k2 complex limbs Ej (j=1,2,…k2) is defined as the number of independent motions between the mobile platform n≡nAi≡nEj and the reference platform 1≡1Ai≡1Ej in the parallel mechanism D.
Connectivity proposition 17 (Connectivity of the mobile platform in a parallel mechanism with complex limbs). The connectivity of the mobile platform n≡nAi≡nEj in the parallel mechanism D ← A1A2…Ak1E1E2…Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is defined as the dimension of the common operational spaces of the simple and complex limbs associated with the parallel mechanism D
D Dn / 1 n / 1 A1 Ak1 E1 Ek 2S dim( R ) dim( R ... R R ... R )= = ∩ ∩ ∩ ∩ ∩ . (2.198)
where D A1 A2 ... Ak1 E1 E2 ... Ek 2n / 1 n / 1S S − − − − − − −= denotes the connectivity between the mo
bile platform n≡nAi≡nEj and the reference platform 1≡1Ai≡1Ej in the parallel mechanism D ← A1A2…Ak1E1E2…Ek2,
2.3 Mobility and connectivity of parallel robots 111
D A1 A2 ... Ak1 E1 E 2 ... Ek 2n / 1 n / 1R R − − − − − − −= is the operational velocity vector space of
the parallel mechanism D ← A1A2…Ak1E1E2…Ek2 defined by the relative velocities between the mobile platform n≡nAi≡nEj and the reference platform 1≡1Ai≡1Ej in the parallel mechanism D ← A1A2…Ak1E1E2…Ek2,
Eq. (2.198) is obtained by recurrence, as in Eq. (2.181). Connectivity proposition 18A. The number of joint parameters that
lose their independence in the parallel mechanism D ← A1A2…Ak1E1E2…Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is given by
k1 k 2D
D A1 ... Ak1 E1 ... Ek 2 Ai Ej n / 1 li 1 j 1
r r dim( R ) dim( R ) dim( R ) r− − − − −= =
= = + − +∑ ∑ (2.199)
Eq. (2.199) follows from Eq. (2.185) by taking into account the fact that rl=
2
1
k Ejlj
r=∑ additional joint parameters lose their independence in the
closed loops integrated in the complex limbs. By Ejlr we have denoted the
joint parameters that lose their independence in the closed loops integrated in complex limb Ej. We note that in complex limb Ej, the closed loops could be concatenated serially or in parallel. Two closed loops are serially concatenated if they do not have any common joint. One or more common joints could be shared by closed loops concatenated in parallel. The number of joint parameters that lose their independence in a single closed loop or in a closed loop serially concatenated in Ejlimb is given by connectivity propositions 5A, B, C and 6A, B, C – see Eqs. (2.140)(2.142) and (2.144)(2.146). The number of joint pareameters that lose their independence in closed loops concatenated in parallel is given by connectivity propositions 13A and B – see Eqs. (2.185) and (2.191).
Connectivity proposition 18B. The number of joint parameters that lose their independence in the parallel mechanism D ← A1A2…Ak1E1E2…Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is given by
k1 k 2D
D A1 ... Ak1 E1 ... Ek 2 Ai Ej n / 1 li 1 j 1
r r S S S r− − − − −= =
= = + − +∑ ∑ (2.200)
where: SAi and SEj are the connectivities of limbs Ai and Ej. Eq. (2.200) is obtained directly from Eq. (2.199) and from the definition of the connectivity of simple and complex chains associated with the limbs, as the number of independent motions between the extreme links (nAi and 1Ai of limb Ai and nEj and 1Ej of limb Ej). We can see that Eqs. (2.181), (2.185) and
112 2 Structural parameters
(2.191) represents particular cases of Eqs. (2.198), (2.199) and (2.200) when k2=0. In this case the parallel mechanism with complex limbs D ← A1A2…Ak1E1E2…Ek2 becomes a parallel mechanism with simple limbs C ← A1A2…Ak1.
Serially and parallel concatenated closed loops can be integrated in a limb. A closed loop is serially concatenated in a limb if the joints of the loop do not belong to other closed loops of the same limb. A group of closed loops are parallel concatenated in a limb if they have at least one common joint. The additional number of joint parameters that lose their independence in a closed loop serially concatenated in a complex limb is easily obtained by inspection by using the connectivity propositions 5AC and 6AC and their associated equations (2.140)(2.142) and (2.144)(2.146). The additional number of joint parameters losing their independence in a group of closed loops that are parallel concatenated in a complex limb is also easily obtained by inspection by using the connectivity propositions 13AB and their associated equations (2.185) and (2.191). We note that the closed loops parallel concatenated form a parallel kinematic chain integrated in the complex limb.
Mobility proposition 7. The mobility of a parallel mechanism D ← A1A2…Ak1E1E2…Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is given by
k1 k 2 k1 k 2
D Ai Ej Ai Eji 1 j 1 i 1 j 1
Dn / 1 l
M dim(U ) dim(U ) dim( R ) dim( R )
dim( R ) r= = = =
= + − − +
−
∑ ∑ ∑ ∑ (2.201)
or p k1 k 2
DD i Ai Ej n / 1 l
i 1 i 1 j 1M f S S S r
= = =
= − − + −∑ ∑ ∑ (2.202)
where: UAi and RAi are the joint velocity space and the operational velocity
space of the simple kinematic chain Ai associated with Ailimb, UEj and REj are the joint velocity space and the operational velocity
space of complex kinematic chain Ej associated with Ejlimb, fi is the mobility of the ith joint and p the total number of joints of the
parallel mechanism D ← A1A2…Ak1E1E2…Ek2 SAi is the connectivity of simple kinematic chain Ai associated with Ai
limb, SEj is the connectivity of complex kinematic chain Ej associated with Ej
limb,
2.3 Mobility and connectivity of parallel robots 113
Dn / 1S is the connectivity of the mobile platform in the parallel mecha
nism D ← A1A2…Ak1E1E2…Ek2. Eqs. (2.201) and (2.202) are obtained directly by integrating Eqs.
(2.199) and (2.200) in Eq. (2.68). By analogy with connectivity propositions 14, 15 and 16 applicable to
paralle robots with simple limbs, the following propositions can be formulated for parallel robots with complex limbs.
Connectivity proposition 19. The mobile and reference platforms n≡nAi≡nEj and 1≡1Ai≡1Ej of a parallel mechanism D ← A1A2…Ak1E1E2…Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) have relative motions if the dimension of the common operational space of Ai and Ej is greater than zero
Dn / 1S >0 ⇔ A1 Ak1 E1 Ek 2dim( R ... R R ... R ) 0∩ ∩ ∩ ∩ ∩ > . (2.203)
If
A1 Ak1 E1 Ek 2dim( R ... R R ... R ) 0∩ ∩ ∩ ∩ ∩ = (2.204)
two platforms are relatively immobile ( Dn / 1S =0).
Eqs. (2.203) and (2.204) result directly from Eq. (2.198). Connectivity proposition 20. The connectivity D
n / 1S between the mobile and reference platforms n≡nAi≡nEj and 1≡1Ai≡1Ej of a parallel mechanism D ← A1A2…Ak1E1E2…Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) is less than or equal to the mobility MD of the parallel mechanism D ← A1A2…Ak1E1E2…Ek2
Dn / 1 DS M≤ . (2.205)
The proof of Eq. (2.205) is founded on the definitions 1 and 2 of mobility and connectivity. If Eq. (2.205) is not satisfied, MD independent parameters are not enough to describe the relative motion between n≡nAi≡nEj and 1≡1Ai≡1Ej of parallel mechanism D ← A1A2…Ak1E1E2…Ek2 which contradicts the definition of mobility.
Connectivity proposition 21. The basis of vector space Dn / 1R of relative
velocities between the mobile and reference platforms n≡nAi≡nEj and 1≡1Ai≡1Ej in the parallel mechanism D ← A1A2…Ak1E1E2…Ek2 with k1 elementary limbs Ai (i=1,2,…k1) and k2 complex limbs Ej (j=1,2,…k2) does not vary with the position of the characteristic point on the mobile platform n≡nAi≡nEj.
Proof. The independent velocities of the characteristic point define the basis ( D
n / 1R ). The mobile platform n≡nAi has an infinite number of points,
114 2 Structural parameters
and the velocity of each point can be expressed by a linear combination of the independent velocities of the same basis. The basis ( D
n / 1R ) is valid for any point of the mobile platform including the points with the most restrictive motion.
The following remark is also very important for the choice of the bases of operational vector spaces RAi and REj when when there are various ways to choose them.
Remark 3. (Choice of the bases of operational vector spaces RAi and REj). If there are different ways to choose the bases of the operational vector spaces RAi and REj , the choice must be made by respecting the following:
a) connectivity propositions 19 and 20 stating that Dn / 1 D0 S M< ≤ (2.206)
b) connectivity proposition 21, c) definition of mobility (in the sense of general mobility) as the mini
mum value of the instantaneous mobility  Eq. (2.109), d) permanent constraints in the relative motions of the links generated
by the closed loops with special geometric conditions. We note that condition c) implies that the bases in Eq. (2.198) are se
lected so that the minimum value of Dn / 1S is obtained. By making this
choice, Eqs. (2.201) and (2.202) fit in with Eq. (2.109) setting up the general mobility as the minimum value of the instantaneous mobility.
2.3.7. General formulae for robot mobility and connectivity
The formulae for mobility and connectivity demonstrated in the previous section for parallel robots with complex limbs can be easily applied to parallel robots with simple limbs (k2=0), to single loop mechanisms (k2=0 and k1=2) or to simple open kinematic chains (k2=0 and k1=1). In this way, the corresponding formulae for mobility and connectivity introduced in sections 2.3.22.3.5 can be obtained as particular cases of general formulae demonstrated in section 2.3.6. To facilitate the correspondences and to bring together the different notations, in this section we present the formulae demonstrated in section 2.3.6 for parallel robots with complex limbs in a unified approach applicable to any kind of serial, parallel or hybrid serialparallel robot.
We consider the general case of a robot in which the endeffector is connected to the reference link by k≥1 kinematic chains. We recall that the endeffector is a polynary link called a mobile platform in the case of par
2.3 Mobility and connectivity of parallel robots 115
allel robots, and a monary link for serial robots. The reference link may either be the fixed base or may be deemed to be fixed. The kinematic chains connecting the endeffector to the reference link can be simple or complex. They are called limbs or legs in the case of parallel robots. A serial robot can be considered to be a parallel robot with just one simple limb, and a hybrid robot a parallel robot with just one complex limb.
We denote by F ← G1G2…Gk the kinematic chain associated with a general serial, parallel or hybrid robot, and by Gi (1Gi2Gi…nGi) the kinematic chain associated with the ith limb (i=1,2,…,k). The end effector is n≡nGi and the reference link 1≡1Gi. If the reference link is the fixed base, it is denoted by 1≡1Gi≡0. The total number of robot joints is denoted by p.
Definition 2.14 (Serial robot). A serial robot F ← G1 is a robot in which endeffector n≡nG1 is connected to reference link 1≡1G1 by just one simple open kinematic chain Gi (1Gi2Gi…nGi) called a serial kinematic chain.
Definition 2.15 (Parallel robot). A parallel robot F ← G1G2…Gk is a robot in which endeffector n≡nGi is connected in parallel to reference link 1≡1Gi by k≥2 kinematic chains Gi (1Gi2Gi…nGi) called limbs or legs.
Definition 2.16 (Hybrid serialparallel robot). A hybrid serialparallel robot F ← G1 is a robot in which endeffector n≡nG1 is connected to reference link 1≡1G1 by just one complex kinematic chain Gi (1Gi2Gi…nGi) called complex limb or complex legs.
Definition 2.17 (Fullyparallel robot). A fullyparallel robot F ← G1G2…Gk is a parallel robot in which the number of limbs is equal to the robot mobility (k=M≥2), and each limb integrates just one actuator.
Definition 2.18. (Joint velocity space associated with a kinematic chain). The joint velocity space UGi associated with a kinematic chain Gi is the joint velocity space of the open kinematic chain associated with Gi by opening all closed loops which may exist in Gi.
Definition 2.19. (Operational velocity space associated with a kinematic chain). Operational velocity space RGi associated with a kinematic chain Gi (1Gi2Gi…nGi) is the vector space of relative velocities between extreme links nGi and 1Gi.
We note that the constraints generated by the closed loops which may exist in Gi are taken into account in the operational velocity space RGi and are ignored in the joint velocity space UGi.
Definition 2.20. (Operational velocity space associated with a robot). The operational velocity space RF associated with a robot is defined by the relative velocities between the endeffector and the reference link in the mechanism F ← G1G2…Gk in which the endeffector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi).
116 2 Structural parameters
Definition 2.21. (Connectivity of a kinematic chain) Connectivity SGi of a kinematic chain Gi (1Gi2Gi…nGi) is given by the number of independent motions between extreme links nGi and 1Gi.
We note that the kinematic chain Gi is detached from the mechanism F ← G1G2…Gk when defining the associated parameters UGi, RGi and SGi. To disconnect them the mobile platform is cut up near each limb. For serial robots Gi and F represent the same kinematic chain.
Definition 2.22. (Connectivity of robot endeffector). The connectivity SF of the robot endeffector in the mechanism F ← G1G2…Gk in which the endeffector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi), is defined as the number of independent motions between the endeffector and the reference link in the mechanism F. The connectivity SF of the robotend effector is also called the robot connectivity, and represents the connectivity of the kinematic chain F ← G1G2…Gk.
Connectivity proposition 22. (Connectivity of robot endeffector). The connectivity SF of the robot endeffector in the mechanism F ← G1G2…Gk in which the endeffector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi), is given by the dimension of the common operational spaces of the kinematic chains Gi
F F G1 G2 GkS dim( R ) dim( R R ... R )= = ∩ ∩ ∩ . (2.207)
where RGi is the operational velocity space of the kinematic chain Gi ,and RF is the operational velocity space of the kinematic chain F.
Eq. (2.207) indicates that the constraints introduced by all k≥1 kinematic chains Gi (1Gi2Gi…nGi) are taken into account when defining the connectivity of robot endeffector. Eq. (2.207) represents a generalization of Eqs. (2.121), (2.172), (2.181) and (2.198) demonstrated in the previous sections.
Connectivity proposition 23A. The number of joint parameters that lose their independence in themechanism F ← G1G2…Gk in which the endeffector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi) is given by
k
Gi F li 1
r dim( R ) dim( R ) r=
= − +∑ (2.208)
where RGi and RF are the operational velocity spaces of the kinematic chains Gi and F ,and rl represents the additional joint parameters that lose their independence in the closed loops which may exist in the kinematic chains Gi.
2.3 Mobility and connectivity of parallel robots 117
We note that k
Gil l
i 1r r
=
=∑ , where Gilr represents the additional joint pa
rameters that lose their independence in the closed loops which may exist in ith kinematic chain Gi.
Eq. (2.208) represents a generalization of of Eqs. (2.145), (2.185) and (2.199) demonstrated in the previous sections.
Connectivity proposition 23B. The number of joint parameters that lose their independence in the mechanism F ← G1G2…Gk in which the endeffector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi) is given by
k
Gi F li 1
r S S r=
= − +∑ (2.209)
where: SGi and SF are the connectivities of the kinematic chains Gi ,and F and rl represents the additional joint parameters that lose their independence in the closed loops which may exist in the kinematic chains Gi.
Eq. (2.209) represents a generalization of Eqs. (2.140), (2.191) and (2.200) demonstrated in the previous sections.
We recall that serially and parallel concatenated closed loops can be integrated in the kinematic chain Gi. A closed loop is serially concatenated in Gi if the joints of the loop do not belong to other closed loops of Gi. A group of closed loops are concatenated in parallel in Gi if they have at least one common joint. The additional number of joint parameters losing their independence in a closed loop serially concatenated in Gi is easily obtained by inspection by using the connectivity propositions 5AC and 6AC and their associated equations (2.140)(2.142) and (2.144)(2.146). The additional number of joint parameters losing their independence in a group of closed loops that are concatenated in parallel in a complex limb is also easily obtained by inspection by using the connectivity propositions 13AB and their associated equations (2.185) and (2.191). We note that the closed loops concatenated in parallel form a parallel kinematic chain integrated in Gi.
Mobility proposition 8A. The mobility of a mechanism F ← G1G2…Gk in which the endeffector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi) is given by
k k
Gi Gi F li 1 i 1
M dim(U ) dim( R ) dim( R ) r= =
= − + −∑ ∑ . (2.210)
118 2 Structural parameters
Mobility proposition 8A. The mobility of a mechanism F ← G1G2…Gk in which the endeffector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi) is given by
p k
i Gi F li 1 i 1
M f S S r= =
= − + −∑ ∑ (2.211)
where: UGi and RGi are the joint velocity space and operational velocity space of
the kinematic chain Gi, RF is the operational velocity space of the kinematic chain F ← G1G2
…Gk, fi is the mobility of the ith joint, and p the total number of joints of the
mechanism F, SGi is the connectivity of the kinematic chain Gi, SF is the connectivity of the kinematic chain F ← G1G2…Gk, rl represents the additional joint parameters that lose their independence
in the closed loops which may exist in the kinematic chains Gi. Eqs. (2.210) and (2.211) represent the generalization of of Eqs. (2.120),
(2.160), (2.161), (2.192), (2.193), (2.201) and (2.202) demonstrated in the previous sections.
Connectivity proposition 24. A mechanism F ← G1G2…Gk in which the endeffector is connected to the reference link by k≥1 kinematic chains Gi (1Gi2Gi…nGi) permits relative motions between endeffector n≡nGi and reference link 1≡1Gi if the dimension of the common operational space of Gi (i=1,2,…,k) is greater than zero
SF>0 ⇔ G1 G2 Gkdim( R R ... R ) 0∩ ∩ ∩ > . (2.212)
If
G1 G2 Gkdim( R R ... R ) 0∩ ∩ ∩ = (2.213)
the endeffector and reference link are relatively immobile (SF=0). Eqs. (2.212) and (2.213) represent the generalization of Eqs. (2.175),
(2.176), (2.194), (2.195), (2.203) and (2.204) demonstrated in the previous sections.
Connectivity proposition 25. The connectivity SF of a mechanism F ← G1G2…Gk in which the endeffector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi) is less than or equal to the mobility M of the mechanism F ← G1G2…Gk
FS M≤ . (2.214)
2.3 Mobility and connectivity of parallel robots 119
Eq. (2.214) represents the generalisation of Eqs. (2.179), (2.196) and (2.205) demonstrated in the previous sections.
Connectivity proposition 26. The basis of the vector space RF of relative velocities between the endeffector and the reference link in the mechanism F ← G1G2…Gk in which the endeffector n≡nGi is connected to the reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi), does not vary with the position of the characteristic point on the endeffector.
This connectivity proposition represents a generalization of connectivity propositions 10A, 16 and 21 demonstrated in sections 2.3.42.3.6.
The following remark is also very important for the choice of the bases of operational vector spaces RGi when there are various ways to choose them.
Remark 4. (Choice of the bases of operational vector spaces RGi). If there are various ways to choose the bases of the operational vector spaces RGi ,the choice must be made by respecting the following:
a) connectivity propositions 24 and 25 stating that
F0 S M< ≤ (2.215)
b) connectivity proposition 26, c) definition of mobility (in the sense of general mobility) as the mini
mum value of the instantaneous mobility  Eq. (2.109), d) permanent constraints in the relative motions of the links generated
by the closed loops with special geometric conditions. We note that condition c) implies that the bases in Eq. (2.207) are se
lected so that the minimum value of FS is obtained. By this choice, Eqs. (2.210) and (2.211) fit in with Eq. (2.109) setting up the general mobility as the minimum value of the instantaneous mobility.
Remark 4 represents a generalization of previous remarks presented in sections 2.3.42.3.6.
We note that the new formulae presented in this section are applicable to the following:
(i) Serial robots, when k=1, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 ,and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
(ii) Parallel robots with simple limbs, when k≥2, rl≠0 and all Gi (i=1,2,…,k) are simple open kinematic chains.
(iii) Parallel robots with complex limbs, when k≥2, rl≠0 and at least one of Gi (i=1,2,…,k) is a complex kinematic chain.
(iv) Serial robots with closed loops and parallel hybrid serialparallel robots, when k=1, rl≠0 and G1 is a complex open kine
120 2 Structural parameters
matic chain. In this case F and G1 also represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
The application of the new formulae proposed in this section will be illustrated in chapter 3 for the various types of robots. In the following chapters these formulae will be used for structural synthesis of parallel robots with simple and complex limbs. Along with the morphological approach, they represent the cornerstone of the structural synthesis method.
2.4 Overconstraints in parallel robots
Usually overconstrained mechanisms are defined as mechanisms that, without satisfying the ChebychevGrüblerKutzbach mobility criterion, have full cycle mobility. Overconstrained mechanisms have mobility over a finite range of motions even though they should be structures (M≤0) according to the CGK formula given in Eq. (2.64). This definition was introduced to designate single and multiloop mechanisms that do not obey the wellknown mobility formulae presented in section 2.2 (see for examples the mechanisms presented in Figs. 2.1, 2.6, 2.7 and 2.8b).
The first overconstrained multiloop mechanism which appeared in the literature was proposed by the French mathematician Gilles Personne (or Personier) De Roberval and dates from 1669. The original drawings presented by De Roberval in “Mémoires de l'Académie royale des sciences: depuis 1666, jusqu'à 1699” are presented in Fig. 2.16 (De Roberval 1669). This mechanism is known as the De Roberval scale, and is used in the standard laboratory doublepan balance. The horizontal arms together with both vertical pan supports make a double parallelogram mechanism. In this way the pans will stay horizontal in the full range of movement. More importantly, it prevents the pans from rotating as they move up and down, and this is the principle that ensures that the position of weights on the pans has no effect on the balance.
De Roberval’s scale (Fig. 2.8b) could be considered both as the first overconstrained mechanism and as the first “parallel mechanism with one degree of mobility” and three simple limbs. We will analyse this mechanism in the next chapter.
Since than, other overconstrained mechanisms have been proposed (Sarrus 1853; Delassus 1900, 1902, 1922; Bennett 1902, 1914; Bricard 1927; Myard 1931a,b; Goldberg1943; Altman 1952; Baker 1978; Waldron 1968,1969 1979; Wohlhart 1987, 1991, 1995; Dietmaier 1995; Fang and Tsai 2004a; Mavroidis and Roth 1995) et al . References can be found to
2.4 Overconstraints in parallel robots 121
Fig. 2.16. De Roberval scale published in 1669: front cover of «Mémoires de l'Académie royale des sciences: depuis 1666, jusqu'à 1699» (a), original drawings (b) and (c)
122 2 Structural parameters
almost all known overconstrained mechanisms in (Baker 1980a, 1984; Waldron 1973a,b; Phillips 1990).
In this section we revisit the definition of overconstrained mechanisms independently of the CGK formulae. The author presented the new general formulae for the calculation of the number of overconstrains in parallel robots for the first time in (Gogu 2005e).
We consider a mechanism with p joints, n mobile links (reference link not included) and q structurally independent loops.
Overconstraint proposition 1A. The number of overconstraints N of a mechanism with mobility M is given by the difference between the number of kinematic constraints introduced by the joints of the mechanism before and after closing the loops
1(6 )
p
ii
N c n M=
= − −∑ (2.216)
Proof. The number of kinematic constraints introduced by the joints of the mechanism before closing the loops is given by the sum of the degrees of joint constraints
1
pii
c=∑ . The total number of constraints existing in the
mechanism after closing the loops is given by the difference between the number of independent motions of the n mobile members considered free in the general kinematic space with six dimensions, and the number of independent motion parameters remaining after connecting the members by joints and closing the q independent loops of the mechanism.
Eq. (2.216) could also be interpreted from the static point of view. In this case,
1
pii
c=∑ represents the number of independent forces acting in the
joints of the mechanism, 6n indicates the number of static equations for the n mobile members, and M indicates the number of forces in the actuated joints.
Overconstraint proposition 1B. The number of overconstraints N of a mechanism with mobility M is given by the difference between the maximum number of joint kinematic parameters that could lose their independence in the q closed loops, and the number of joint kinematic parameters that actually lose their independence in the closed loops:
N=6qr (2.217)
Proof. Eq. (2.217) is obtained from Eq. (2.216) by taking into consideration Eq. (2.68), the definition of q given by the theory of graphs (q=pn) and the relation between the degree of mobility and the degree of constraint of a joint (f+c=6).
2.4 Overconstraints in parallel robots 123
If the number of overconstraints is zero (N=0) the mechanism is called nonoverconstrained or isostatic.
Eqs. (2.216) and (2.217) are applicable to single and multiloop mechanisms.
For a singleloop mechanism (q=1) we obtain
N=6r. (2.218)
We note that a simple open kinematic chain cannot be overconstrained. Let us consider the case of the mechanism F ← G1G2…Gk in which end
effector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi). The operational vector spaces of F and Gi are RGi and RF with dim(RF)=SF and dim(RGi)=SGi. We recall that RGi is the vector space of relative velocities between extreme links nGi and 1Gi in kinematic chain Gi, and RF the vector space of relative velocities between extreme links n≡nGi and 1≡1Gi in the mechanism F ← G1G2…Gk. The dimensions of these spaces give connectivities SF and SGi of F and Gi (see definitions 1922 in section 2.3.7 for more details).
Overconstraint proposition 2A. The number of overconstraints of a mechanism F ← G1G2…Gk in which endeffector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi) is given by
k
Gi F li 1
N 6q dim( R ) dim( R ) r=
= − + −∑ (2.219)
where RGi and RF are the operational velocity spaces of kinematic chains Gi and F and rl represent the additional joint parameters that lose their independence in the closed loops which may exist in kinematic chains Gi.
Eq. (2.219) results directly from Eqs. (2.217) and (2.185) by taking into consideration the fact that the number of structurally independent loops formed between k kinematic chains inparallel concatenated is q=k1. Additional loops may exist in kinematic chains Gi.
Overconstraint proposition 2A. The number of overconstraints of a mechanism F ← G1G2…Gk in which endeffector n≡nGi is connected to reference link 1≡1Gi by k≥1 kinematic chains Gi (1Gi2Gi…nGi) is given by
k
Gi F li 1
N 6q S S r=
= − + −∑ (2.220)
where: SGi and SF are the connectivities of kinematic chains Gi and F, and rl represents the additional joint parameters that lose their independence in the closed loops which may exist in kinematic chains Gi.
124 2 Structural parameters
Eq. (2.220) follows directly from Eq. (2.219) by taking into account the definition of connectivities SGi and SF as the dimensions of velocity vector spaces SGi and SF.
The number of overconstraints gives the degree of overconstraint of the mechanism.
To obtain a nonoverconstrained mechanism, idle mobilities are introduced in the joints of the mechanism.
Definition 2.23. (Idle mobility). Idle mobility is a joint mobility that does not affect a mechanism’s mobility.
Idle mobilities could change just SGi, SF, and consequently the number of overconstraints N.
In theoretical conditions, when no errors exist with respect to parallel and perpendicular positions of joint axes, motion amplitude in an idle mobility is zero. Real life manufacturing and assembling processes introduce errors in the relative positions of the joint axes and, in this case, the idle mobilities become effective mobilities usually with small amplitudes, depending on the precision of the parallel robot.
We note that formulae (2.219) and (2.220) are applicable to the following:
(i) Serial robots, when k=1, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
(ii) Parallel robots with simple limbs, when k≥2, rl≠0 and all Gi (i=1,2,…,k) are simple open kinematic chains.
(iii) Parallel robots with complex limbs when k≥2, rl≠0 and at least one of Gi (i=1,2,…,k) is a complex kinematic chain.
(v) Serial robots with closed loops and parallel hybrid serialparallel robots, when k=1, rl≠0 and G1 is a complex open kinematic chain. In this case F and G1 also represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
The application of these new formulae along with the selection of idle mobilities to obtain nonoverconstrained mechanisms will be illustrated in chapter 3 for the various types of robots. In the following chapters these formulae will be used for structural synthesis of overconstrained and non overconstrained parallel robots with simple and complex limbs.
2.5 Redundancy in parallel robots 125
2.5 Redundancy in parallel robots
Redundant robots are mechanismas with more degrees of freedom than required for doing the prescribed task in the task space. Redundant actuation in parallel manipulators can be achieved by: (i) actuating some of the passive joints within the existing limbs, (ii) introducing additional actuated limbs beyond the minimum necessary to actuate the manipulator and (iii) introducing some additional actuated joints within the limbs beyond the minimum necessary to actuate the manipulator. We note that actuator redundancy does not affect the connectivity of the endeffector and only increases the number of actuators. In all cases, only M actuators have independent motion. The other actuators have dependent motions. In extreme cases, dependency consists in actuator locking. A selective choice of actuators with independent motion could have many advantages.
Redundancy in parallel manipulators is used to eliminate some singular configurations (Wang and Gosselin 2004; Kurtz and Hayward 1992; Merlet 1997; Firmani and Podhorodeski 2004, Alberich et al. 2006), to minimize the joint rates, to optimize the joint torques/forces (Dasgupta and Mruthyunjaya 1998b; Bruckman et al 2006; Nokleby et al. 2005) to increase dexterity workspace (Marquet et al 2001a,b), stiffness (Chakarov 2004), eigenfrequencies, kinematic and dynamic accuracy (Valasek et al. 2004), to improve velocity performances (Krut et al. 2004) and both kinematic and dynamic control algorithms (Liu et al. 2001; Cheng at al. 2003), to develop large forces in micro electromechanical systems (Mukherjee et al. 2001), decoupling the orientations and the translations (Jin et al. 2004), to obtain reconfigurable platforms (Mohamed and Gosselin 2005) and limbs (Fischer et al. 2004) or combined advantages (Nahon and Angeles 1991; Zanganach and Angeles 1994a,b; Kim 1997; Kock and Scumacher 1998, 2000; Mohamed 2003a,b).
The author was the first to present new formulae for calculating the degree of redundancy of parallel robots (Gogu 2006a). Redundancy is achieved by introducing additional actuated joints within the limbs, beyond the minimum necessary to actuate the manipulator. In this way we reduce motion coupling in parallel manipulators.
Let us consider again the case of the mechanism F ← G1G2…Gk in which endeffector n≡nGi is connected to reference link 1≡1Gi by k simple or complex kinematic chains Gi (1Gi2Gi…nGi). The mechanism F ← G1G2…Gk is characterized by:
(i) UGi  the vector space of joint velocities in kinematic chain Gi, (ii) RGi  the vector space of relative velocities between extreme
links nGi and 1Gi in kinematic chain Gi,
126 2 Structural parameters
(iii) RF  the vector space of relative velocities between extreme links n≡nGi and 1≡1Gi in parallel mechanism F ← G1G2…Gk,
(iv) SGi=dim(RGi)  the connectivity of Gi, (v) SF=dim(RF)  the connectivity of F, (see definitions 1822 in section 2.3.7 for more details). Definition 2.24. (Structural redundancy of a kinematic chain). The
structural redundancy of the kinematic chain represents the difference between its mobility and connectivity.
Structural redundancy proposition 3A. The structural redundancy of a mechanism F ← G1G2…Gk in which endeffector n≡nGi is connected to reference link 1≡1Gi by k kinematic chains Gi (1Gi2Gi…nGi) is given by
k k
Gi Gi li 1 i 1
T dim(U ) dim( R ) r= =
= − −∑ ∑ (2.221)
Structural redundancy proposition 3B. The structural redundancy of a mechanism F ← G1G2…Gk in which endeffector n≡nGi is connected to reference link 1≡1Gi by k kinematic chains Gi (1Gi2Gi…nGi) is given by
p k
i Gi li 1 i 1
T f S r= =
= − −∑ ∑ (2.222)
where: UGi and RGi are the joint velocity space and operational velocity space of
kinematic chain Gi, fi is the mobility of the ith joint, and p the total number of joints of the
mechanism F, SGi is the connectivity of the kinematic chain Gi, rl represents the additional joint parameters losing their independence in
the closed loops which may exist in kinematic chains Gi. Eqs. (2.221) and (2.222) results directly from structural redundancy
definition and Eqs. (2.210) and (2.211). In a parallel robot, the structural redundancy gives the internal mobili
ties in the limbs. We note that formulae (2.221) and (2.211) are applicable to the follow
ing: (iv) Serial robots, when k=1, rl=0 and G1 is a simple open kinematic
chain. In this case F and G1 represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
(v) Parallel robots with simple limbs, when k≥2, rl≠0 and all Gi (i=1,2,…,k) are simple open kinematic chains.
2.6 General formulae for structural parameters 127
(vi) Parallel robots with complex limbs, when k≥2, rl≠0 and at least one of Gi (i=1,2,…,k) is a complex kinematic chain.
(vi) Serial robots with closed loops and parallel hybrid serialparallel robots, when k=1, rl≠0 and G1 is a complex open kinematic chain. In this case F and G1 also represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
The application of these new formulae will be illustrated in chapter 3 for the various types of robots. In the following chapters these formulae will be used for structural synthesis of redundant and non redundant parallel robots with simple and complex limbs.
2.6 General formulae for structural parameters
The propositions and formulae demonstrated in the previous sections for mobility, connectivity, overconstraint and redundancy of various types of robots can be extrapolated to single and multiloop mechanisms. In this case the endeffector n≡nGi is replaced by a link called a characteristic link, and the reference link is the fixed base 1≡1Gi≡0. Theoretically any link except the reference one can be considered as a characteristic link. In this way, a mechanism open or closed, single or multiloop can be defined as a kinematic chain F ← G1G2…Gk with p≥2 joints.
Definition 2.25. (Mechanism). A mechanism F ← G1G2…Gk is a kinematic chain in which a characteristic link denoted by n≡nGi (i=1,2,…k) is connected to reference link 1≡1Gi≡0 by k≥1 subkinematic chains Gi (1Gi2Gi…nGi).
To simplfy the terminology, subkinematic chains Gi that are parts of the mechanism F ← G1G2…Gk are simply called kinematic chains.
Definition 2.26. (Simple mechanism). A simple mechanism is a mechanism F ← G1G2…Gk with just monary and binary links.
Definition 2.27. (Complex mechanism). A complex mechanism is a mechanism F ← G1G2…Gk with at least one ternary link.
We recall the following:(i) a monary link is a link which carries only one kinematic pairing element, (ii) a binary link is a link connected in the kinematic chain by two joints and (iii) a polynary link is a link connected in the kinematic chain by more than two joints –see section 1.4.
Definition 2.28. (Serial mechanism). A serial mechanism F ← G1 is a kinematic chain in which characteristic link n≡nG1 is connected to reference link 1≡1G1≡0 by just one simple open kinematic chain G1 (1G12G1…nG1).
128 2 Structural parameters
Definition 2.29. (Singleloop mechanism). A singleloop mechanism F ← G1G2 is a kinematic chain in which characteristic link n≡nGi (i=1,2) is connected to reference link 1≡1Gi≡0 by k=2 serial kinematic chains Gi (1Gi2Gi…nGi).
Definition 2.30. (Multiloop mechanism). A multiloop mechanism F ← G1G2…Gk is a kinematic chain, containing at least one ternary link, in which characteristic link n≡nGi (i=1,2,…k) is connected to reference link 1≡1Gi≡0 by k≥2 kinematic chains Gi (1Gi2Gi…nGi).
The ternary link can be the characteristic link or any other link of the kinematic chain.
We can see that serial and singleloop mechanisms are simple, and multiloop mechanisms are complex mechanisms.
We note that joint velocity space UGi, ,operational velocity space RGi , and connectivity SGi of the kinematic chain Gi (1Gi2Gi…nGi) are defined in the previous section (see definitions 18, 19 and 21). The kinematic chain Gi is detached from the mechanism F ← G1G2…Gk when defining associated parameters UGi, RGi and SGi. To disconnect it the characteristic link is cut after the last joint of Gi.
Definition 2.31. (Connectivity of the characteristic link). Connectivity SF of characteristic link n≡nGi in the mechanism F ← G1G2…Gk is defined by the number of independent motions between n≡nGi and reference link 1≡1Gi≡0 allowed by the mechanism F.
A mechanism F ← G1G2…Gk permits relative motions between endeffector n≡nGi and reference link 1≡1Gi≡0 when n≡nGi and 1≡1Gi≡0 are connected by k≥1 kinematic chains Gi (1Gi2Gi…nGi) if and only if the dimension of the common operational space of Gi (i=1,2,…,k) is higher than zero – see Eq. (2.212).
The general formule demonstrated in this chapter for mobility M, degree of overconstraint N and redundancy T of parallel robots are also applicable to the mechanism F ← G1G2…Gk.
To conclude this chapter we recall these formulae and the meaning of the parameters used to define them.
p
ii 1
M f r=
= −∑
(2.223)
N=6qr (2.224)
T=MSF (2.225)
where
2.6 General formulae for structural parameters 129
k
Gi F li 1
r S S r=
= − +∑ (2.226)
F F G1 G2 GkS dim( R ) dim( R R ... R )= = ∩ ∩ ∩ (2.227)
and k
Gil l
i 1r r
=
=∑ . (2.228)
We recall that p represents the total number of joints, k the total number of independent closed loops in the sense of graph theory, fi the mobility of the ith joint, r the number of joint parameters that lose their independence in the mechanism F, Gi
lr the number of joint parameters that lose their independence in the closed loops of limb Gi, rl the total number of joint parameters that lose their independence in the closed loops that may exist in the limbs of mechanism F, SF connectivity of characteristic link n≡nGi in the mechanism F, SGi the connectivity of characteristic link n≡nGi in limb Gi disconnected from the mechanism F, RG operational velocity space of the mechanism F ← G1G2…Gk and RGi the operational velocity space of limb Gi disconnected from the mechanism F.
We note that the intersection in Eq. (2.227) is consistent if the operational velocity spaces RGi are defined by the velocities of the same point situated on the characteristic link. This point is called the characteristic point, and denoted in this work by H. It is a point with the most restrictive motions of the characteristic link.
The connectivity SF of the characteristic link n≡nGi in the mechanism F ← G1G2…Gk is less than or equal to the mobility M of the mechanism F.
The basis of the vector space RF of relative velocities between the characteristic link n≡nGi and the reference link 1≡1Gi≡0 in the mechanism F ← G1G2…Gk dos not vary with the position of the characteristic point on n≡nGi.
When there are various ways to choose the basis of the operational spaces, the bases of RGi in Eq. (2.227) are selected such as the minimum value of FS is obtained. By this choice, the result of Eq. (2.223) fits in with Eq. (2.109) setting up the general mobility as the minimum value of the instantaneous mobility.
Formulae (2.223)(2.228) are applicable to any kind of mechanism F ← G1G2…Gk:
130 2 Structural parameters
(i) Serial mechanisms, when k=1, q=0, rl=0 and G1 is a simple open kinematic chain. In this case F and G1 represent the same kinematic chain F ← G1 and RF and RG1 the same vector spaces with SF=SGi=dim(RF)=dim(RGi).
(ii) Singleloop mechanisms, when k=2, q=1, rl≠0 and Gi (i=1,2) are simple open kinematic chains.
(iii) Multiloop mechanisms, when k≥2, q≥2, rl≠0 and mechanism F ← G1G2…Gk contains at least one ternary link.
Kinematic inversion and change of the distal link may be applied to associate a parallel mechanism with any multiloop mechanism. Although relative motions between components remain the same, structural parameters (mobility, degree of overconstraint and degree of structural redundancy) of mechanisms are invariant with respect to kinematic inversion or the change of the distal link. Kinematic inversion changes the reference link by repositionning the “observer”. The following two properties may be formulated:
Property 5 (Invariance of structural parameters with the change of the reference link). Mobility, overconstraints and redundancy of the mechanism are invariants under the change of the reference link.
Property 6 (Invariance of structural parameters with the change of the distal link). Mobility, overconstraints and redundancy of the mechanism are invariants under the change of the distal link.
The application of these new formulae will be illustrated in chapter 3 for the various types of mechanism to overcome the drawbacks of the CGK mobility formulae widely used for mobility calculation of multiloop mechanisms. These new formulae do not involve setting up the instantaneous constraint systems and are successfully applied in chapter 3 to various mechanisms including so called “paradoxical” mechanisms.
3 Structural analysis
Structural analysis of parallel robotic manipulators aims at determining the main structural parameters: connectivity, mobility, redundancy, degree of overconstraint. The following examples will illustrate the definitions, the theorems and the properties presented in the previous chapter. We emphasise the difference between mobility M and connectivity S; these are often confused in the literature under the name of degree of freedom (DoF).
As we have seen, mobility and connectivity are the key structural parameters also used to calculate the degree of redundancy and the degree of overconstraint. We begin with simple examples of open and closedloop kinematic chains associated with the limbs of parallel robots and we finish with structural analysis of various types of parallel robots with simple or complex limbs. We illustrate the fact that the formulae demonstrated in the previous chapter for parallel robots are also applicable to single or multiloop mechanisms that do not obey the classical ChebychevGrüblerKutzbach (CGK) formulae.
3.1 Simple open kinematic chains
We recall that an open kinematic chain is a chain in which there is at least one link which carries only one kinematic pairing element, called a monary link. In a simple open kinematic chain (openloop mechanism) a link is connected by one, or a maximum of two, joints, which means only monary and binary links exist. Simple open kinematic chains are associated with simple limbs of parallel robotic manipulators.
Example 1. The simple open kinematic chain A (12…n+1) presented in Fig. 3.1a has n+1 links, and p=n cylindrical joints with the same axis. The mobility of each joint is fi=2 (i=1,2,…,p). This kinematic chain has MA=2p=2n, SA=2 and TA=MASA=2p2. The mobility is given by Eq.(2.120). The independent finite displacements in the joints necessary to define the configuration of the simple open kinematic chain are i 1,iϕ + and
i 1,id + (i=1,2,…,n,n+1)  see definition 1 in chapter 2. The independent velocities in the joints required to define the velocities of the distal link (n+1)
132 3 Structural analysis
are i 1,i i 1 ,iω ϕ+ += and i 1 ,i i 1,iv d+ += (i=1,2,…,n,n+1)  see definition 2. They form the basis of the joint space (UA)=( 21ϕ , 21d , …, i 1,iϕ + , i 1,id + ,…, n 1,nϕ + ,
n+1,nd ) having the cardinal 2m. The dimension of this joint space gives the
mobility of the mechanism mA A i 1
M dim(U ) f 2m=
= = =∑ . The connectivity is given by Eq.(2.121). The dimension of the operational space can easily be determined by inspection. We can see that the independent finite displacements between the distal links 1 and n+1 are x and xϕ (see definition 3). The independent velocities between the distal links are xv and xω (see definition 4). They form the basis of the operational space (RA)=( x xv ,ω ) having the cardinal 2. The dimension of this space gives the connectivity SA=dim(RA)=2. The kinematic chain achieves the addition of linear and angular velocities of the joints ( 1 21 32 1+= = + + +x n n ,nv v v v ... v and
1 21 32 1+= = + + +x n n ,n...ω ω ω ω ω ). Structural redundancy TA=2(m1) is given by Eq. (2.130).
Figure 3.1b presents the particular case where the p cylindrical joints are replaced by p revolute joints with the same axis. The mobility of each joint
Fig. 3.1. Simple open kinematic chains with n+1 links and p=n joints: (a) MA=2p=2n, SA=2, TA=2(n1); (b and c) MA=p=n, SA=1, TA=n1
3.1 Simple open kinematic chains 133
is fi=1 (i=1,2,…,p). The mechanism has MA=p=n, SA=1 and TA=MASA= n1. The only independent velocity between the distal links is xω , and (RA)=( xω ), SA=dim(RA)=1. To control this velocity we have to know the MA=p independent velocities in the revolute joints of the mechanism. The mechanism achieves the addition of joint angular velocities ( 1 21 32 1+= = + + +x n n ,n...ω ω ω ω ω ).
The kinematic chain presented in Fig. 3.1c represents another particular case in which the p cylindrical joints are replaced by p prismatic joints having the same direction. The mobility of each joint is fi=1 (i=1,2,…,p). The mechanism has MA=p=n, S=1 and TA=MASA=n1. The only relative independent velocity between the distal and proximal links is xv . We obtain (RA)=( xv ) and SA=dim(RA)=1. To control this velocity we have to know the MA=p independent velocities in the prismatic joints of the mechanism. The mechanism achieves the addition of joint linear velocities ( 1 21 32 1+= = + + +x n n ,nv v v v ... v ).
Example 2. The simple open kinematic chain in Fig. 3.2 has n+1 links and p=n (n≥3) revolute joints R with parallel axes. The mobility of each joint is fi=1 (i=1,2,…,n). The mechanism has MA=p=n, SA=3 and TA=MASA=n3. The relative independent velocities between the distal links 1 and n+1 are x y zv ,v ,ω , than (RA)=( x y zv ,v ,ω ) and SA=dim(RA)=3. To control these velocities we have to know MA=p=n independent velocities in the revolute joints of the mechanism.
Fig. 3.2. Simple open kinematic chain with n+1 links and p=n revolute joints with parallel axes: MA=p=n, SA=3, TA=n3
134 3 Structural analysis
Example 3. Each simple open kinematic chain associated with the three limbs of the parallel Cartesian robotic manipulator presented in Fig. 2.1 has MA1=MA2=MA3=4, SA1=SA2=SA3=4 and TA1=TA2=TA3=0. The independent finite displacements in the joints needed to define the configuration of the simple open kinematic chain associated with limb A1 (0A≡1A2A3A4A5A) are 2 Ad , 3 Aϕ , 4 Aϕ and 5 Aϕ  Fig. 2.2a. The independent velocities in the joints needed to define the velocities of any link of A1limb are 2 A 3A 4 Ad , ,ϕ ϕ and 5 Aϕ . They form the basis of the joint velocity space
(UA1)=( 2A 3A 4 A 5 Ad , , ,ϕ ϕ ϕ ) having the cardinal 4. The dimension of this space gives the mobility of the simple open mechanism MA1=dim (UA1)=4. The relative independent velocities between the distal links (0A≡1A and 5) of the simple open kinematic chain associated with A1limb are x y zv ,v ,v
and xω . They form the basis of the operational velocity space (RA1)=( x y z xv ,v ,v ,ω ) having the cardinal 4. The dimension of this operational space gives the connectivity of the simple open kinematic chain associated with A1limb, SA1=dim (RA1)=4. Similar results are obtained for the A2limb (0B≡1B2B3B4B5B)  Fig. 2.2b  and for A3limb (0C≡1C2C3C4C5C)  Fig. 2.2c. We have (RA2)=( x y z yv ,v ,v ,ω ) and (RA3)=( x y z zv ,v ,v ,ω ) and SA1=SA2=SA3=4. These results can easily be verified by calculating the rank of the Jacobian matrix associated with each limb.
If we express the translation velocity of point H and the angular velocity of link 5A in the reference coordinate system O0x0y0z0 (Fig. 2.2a) we obtain the following kinematic equation for the chain associated with A1limb
[ ]TT
x y z x y z 2 A 3 A 4 A 5 AA1v v v J dω ω ω ϕ ϕ ϕ = (3.1)
with
[ ]3A 3A 4A 43A 4A 43A
3A 3A 4A 43A 4A 43AA1
1 0 0 00 b S b S b S 00 b C +b C b C 0
J0 1 1 10 0 0 00 0 0 0
=
. (3.2)
3.1 Simple open kinematic chains 135
Similar equations can be written for the simple open kinematic chains associated with A2limb (Fig. 2.2a) and with A3limb (Fig. 2.2b):
[ ]TT
x y z x y z 2B 3B 4 B 5BA2v v v J dω ω ω ϕ ϕ ϕ = (3.3)
and
[ ]TT
x y z x y z 2C 3C 4C 5CA3v v v J dω ω ω ϕ ϕ ϕ = (3.4)
with the following Jacobian matrices:
[ ]
3B 3B 4B 43B 4B 43B
3B 3B 4B 43B 4B 43BA2
0 b C +b C b C 01 0 0 00 b S b S b S 0
J0 0 0 00 1 1 10 0 0 0
=
(3.5)
and
[ ]
3C 3C 4C 43C 4C 43C
3C 3C 4C 43C 4C 43C
A3
0 b S b S b S 00 b C +b C b C 01 0 0 0
J0 0 0 00 0 0 00 1 1 1
=
(3.6)
The connectivity of the simple open mechanism associated with each limb of the parallel Cartesian robotic manipulator in Fig. 2.2 can also be obtained by using the rank of the Jacobian matrices: SA1=rank ([ ]A1
J )=4, SA2=rank ( [ ]A2
J )=4, SA3=rank ( [ ]A3J )=4. By analyzing the Jacobian ma
trices (3.2), (3.5) and (3.6) we can see that each simple kinematic chain in Fig. 2.2 has an operational space with a unique basis. After eliminating the two lines containing only zero terms, we can calculate the determinants of
136 3 Structural analysis
Fig. 3.3. Singular configurations of simple open kinematic chain A1: structural singularities (a), (b) and (c) and instantaneous singularity (d)
Jacobian matrices [ ]Ai( 4 4 )J
× (i=1,2,3). For example, the determinant of the
Jacobian matrix [ ]A1( 4 4 )J
× is d=b3Ab4AS4A. This simple open kinematic chain
has structural singularities when b3A=0 and/or b4A=0 (see Fig. 3.3ac) and an instantaneous singularity when S4A=0, that is 4 A 0ϕ = (see Fig. 3.3d). These singularities do not affect the mobility that remains MA1=4 for all cases in Fig. 3.3. The singularities affect only the connectivity of this simple open kinematic chain that become SA1=3 (Fig. 3.3a, b and d) and SA1=2  Fig. 3.3c.
Example 4. The simple open kinematic chains presented in Fig. 2.3 are obtained by serial concatenation of two limbs of the parallel Cartesian robotic manipulator from Fig. 2.1. The independent velocities of distal link 1B with respect to reference link 1A≡0A in the simple open chain A≡A1A2 (0A≡1A2A…5A≡5A…2B1B) are: x y z xv ,v ,v ,ω , and yω . They form the basis of the operational space (RA≡A1A2)=( x y z x yv ,v ,v , ,ω ω ) having the cardinal 5. The dimension of this operational space gives the connectivity of the open
3.2 Singleloop kinematic chains 137
simple chain A≡A1A2: SA1A2=dim (RA1A2)=5. This result can also be obtained by using Eq. (2.127) :
SA≡A1A2=dim(RA≡A1A2)= dim(RA1)+dim(RA2)dim(RA1∩RA2)=4+43=5. This kinematic chain has the following structural parameters: MA=8,
SA=5, TA=3. Similar results may be obtained for the simple open kinematic chain in
Fig. 2.4. The independent velocities of the distal link 1C relative to the reference link 1A≡0A in the simple open chain A≡A1A3 (0A≡1A2A…5A≡5C…2C1C) are: x y z xv ,v ,v ,ω , and zω . They form the basis of the operational space (RA≡A1A3)=( x y z x zv ,v ,v , ,ω ω ) also having the cardinal 5. Implicitly, the connectivity of the open chain A≡A1A3 is SA≡A1A3= dim(RA≡A1A3)=5. This result can also be obtained as follows: SA≡A1A3=dim(RA≡A1A3)= dim(RA1)+dim(RA3)dim(RA1∩RA3)=4+43=5.
We can see that, in general, the dimensions of the velocity joint space and the operational velocity space can easily be determined by inspection for the simple open kinematic chains. The calculation of the rank of the Jacobian matrix is useful just for verification, if necessary.
3.2 Singleloop kinematic chains
We recall that, in a singleloop kinematic chain each link is connected with two other links, which means only binary links exist in a singleloop mechanism.
Example 5. The singleloop kinematic chain in Fig. 3.4 has p=8 onedegreeoffreedom joints and m=8 links, including the fixed reference link (1A≡1B≡0). The singleloop kinematic chain B1 (1A≡02A…5A≡5B…2B1B≡0) can be obtained by fixing together the distal links (1A≡0 and 1B) of the simple open kinematic chain A≡A1A2 (1A≡02A…5A≡5B…2B1B) from Fig. 2.3. For the singleloop kinematic chain B1, Eq. (2.140) gives rB1=SA≡A1A2=dim(RA≡A1A2)=5 (see example 4) and the mobility given by Eq. (2.160) is MB1=85=3. We may obtain the same result by considering that the singleloop kinematic chain B1 ←A1A2 (1A≡02A…5A≡5B…2B1B≡0) is obtained by connecting together the distal links 1A≡0 with 1B≡0 and 5A with 5B of simple open kinematic chains A1 (0A≡1A2A3A4A5A) and A2 (0B≡1B2B3B4B5B) in Fig. 2.2ab and by applying Eqs. (2.144) and (2.164).
The unique basis of the operational spaces of simple open kinematic chains A1 and A2 are (RA1)=( x y z xv ,v ,v ,ω ) and (RA2)=( x y z yv ,v ,v ,ω ). We recall that (RA1) and (RA2) represent the bases of the vector spaces of the ve
138 3 Structural analysis
locities of point H in open kinematic chains A1 and A2 in Fig. 2.2ab. The connectivity between links 5≡5A≡5B and 1≡1A≡1B≡0 in the singleloop mechanism B1 ←A1A2 given by Eq. (2.172) is B1
5 / 1 A1 A2S dim( R R ) 3= ∩ = . The relative independent velocities allowed by the singleloop mechanism B1 ←A1A2 between links 5≡5A≡5B and 1≡1A≡1B≡0 are vx, vy and vz.
Similar results are obtained for the singleloop kinematic chain B2 (0A≡1A2A…5A≡5C…2C1C≡0) in Fig. 3.5: rB2=SA≡A1A3=5 and MB2=85=3. This kinematic chain may be obtained by fixing together the distal links (1A≡0 and 1C) of the simple open kinematic chain A≡A1A3 (0A≡1A2A…5A≡5C…2C1C) from Fig. 2.4 or by connecting together the distal
Fig. 3.4 Singleloop kinematic chain B1 ←A1A2 associated with limbs A and B of the parallel Cartesian robotic manipulator: (a) structural diagram, (b) associated graph
Fig. 3.5. Singleloop kinematic chain B1 ←A1A3 associated with limbs A and C of the parallel Cartesian robotic manipulator: (a) structural diagram, (b) associated graph
3.2 Singleloop kinematic chains 139
links 1A≡0 with 1C≡0 and 5A with 5C of simple open kinematic chains A1 (0A≡1A2A3A4A5A) and A2 (0B≡1B2B3B4B5B) in Fig. 2.2a and c (see example 4 for the calculation of SA≡A1A3 and dim(RA≡A1A3)).
The basis of the operational space of open simple kinematic chain A3 is (RA3)=( x y z zv ,v ,v ,ω ) and B2
5 / 1 A1 A3S dim( R R ) 3= ∩ = . The relative independent velocities allowed by singleloop mechanism B2 ←A1A3 between the links 5 and 1≡0 are also the translational velocities vx, vy and vz.
The degree of overconstraint of these singleloop mechanisms given by Eq. (2.218) is N=5. Five joint parameters lose their independence in each closed loop (r=rB1=rB2=5).
Example 6. The singleloop mechanism B (1A≡02A…4A≡4B…2B1B≡0) in Fig. 3.6a has p=6 onedegreeoffreedom revolute joints and m=6 links including the fixed reference link (1A≡1B≡0). It represents an orthogonal version of the Sarrus mechanism (Sarrus 1853), and may be obtained by eliminating the prismatic joints from the kinematic chain in Fig. 3.4. The simple open kinematic chain A≡A1A2 (1A≡02A…4A≡4B…2B1B) associated with the closed loop mechanism B has rB=SA≡A1
A2=dim(RA≡A1A2)=5 (Fig. 3.6b). The same five relative independent motions ( , , , ,x y z x yv v v ω ω ) exist between the distal links 1B and 1A≡0 in the simple open kinematic chains presented in Figs. 2.3 and 3.6b. The mobility of the singleloop mechanism B given by Eq. (2.160) is MB=65=1. We could obtain the same result by considering that the singleloop kinematic chain B ←A1A2 (1A≡02A…4A≡4B…2B1B≡0) is obtained by connecting together the distal links 1A≡0 with 1B≡0 and 4A with 4B of the simple open kinematic chains A1 (0A≡1A2A3A4A) and A2 (0B≡1B2B3B4B) in Fig. 3.6cd. Three relative independent motions ( , ,y z xv v ω ) exist between distal links 4A and 1A≡0 in the simple open kinematic chain A1 (Fig. 3.6c) and ( , ,x z yv v ω ) between distal links 4B and 1B≡0 in the simple open kinematic chain A2 (Fig. 3.6d). Eq. (2.144) gives rB=SA≡A1A2= dim(RA≡A1
A2)=dim(RA1)+dim(RA2)dim(RA1∩RA2)=3+31=5. We can see that, in general, the mobility of a simple closedloop mecha
nism can easily be calculated if we know the dimension of the velocity operational space of the simple open kinematic chain associated with the closedloop mechanism.
The unique bases of the operational spaces of simple open kinematic chains A1 and A2 are (RA1)=( y z xv ,v ,ω ) and (RA2)=( x z yv ,v ,ω ). The connectivity between links 4≡4A≡4B and 1≡1A≡1B≡0 in singleloop mechanism B ←A1A2 given by Eq. (2.172) is B
4 / 1 A1 A2S dim( R R ) 1= ∩ = . The relative
140 3 Structural analysis
Fig. 3.6. Orthogonal version of Sarrus mechanism: structural diagram and graph of the single loopmechanism (a), associated simple open chain A≡A1A2 (b), associated simple open chain A1 (c) and associated simple open chain A1 (d)
3.2 Singleloop kinematic chains 141
independent velocity allowed by the singleloop mechanism B ←A1A2 between links 4≡4A≡4B and 1≡1A≡1B≡0 is vz. The degree of overconstraint of this singleloop mechanism given by Eq. (2.218) is N=5. Five joint parameters lose their independence in the closed loop (r=rB=5).
Note. The following examples illustrate the application of the formulae presented in this section by emphasizing the choice of the bases of operational vector spaces RA1 and RA2 when there are various ways to choose them.
Example 7. Fourbar planar mechanism B (1≡02345≡1) presented in Fig. 3.7a, has p=4 revolute joints with parallel axes and m=4 links
Fig. 3.7. Fourbar mechanism B (1≡02345≡1) (a) and its associated simple open kinematic chain A (1≡02345) (b)
Fig. 3.8. Fourbar planar mechanism B (1≡02345≡1) (a) and its associated simple open chains A1 and A2 obtained by splitting up link 4 (b), link 3 (c) and link 2 (d)
142 3 Structural analysis
Fig. 3.9. Fourbar planar parallelogram mechanism B (1≡02345≡1) (a) and its associated simple open chains A1 (1A≡02A3A) and A2 (1B≡02B3B) (b)
Table 3.1. Valid bases of operational vector spaces RA1 and RA2 for the fourbar planar mechanism B ←A1A2
No. A1 and A2 (RA1) (RA2) (RA1 ∩ RA2) 1 Fig. 3.8b ( x y zv ,v ,ω ) ( zω ) ( zω ) 2 Fig. 3.8c ( x yv ,v ) ( y zv ,ω ) ( yv ) 3 Fig. 3.8c ( x yv ,v ) ( x zv ,ω ) ( xv ) 4 Fig. 3.8d ( zω ) ( x y zv ,v ,ω ) ( zω )
including the fixed reference link (1≡5≡0). It can be obtained by connecting in the same link the distal links of the simple open kinematic chain A (1≡02345) in Fig. 3.7b. We can simply observe by inspection that rA=SA=3 and (RA)=( x y zv ,v ,ω ). The mobility given by Eq. (2.160) is MB=43=1. The same result can be obtained by considering that the singleloop fourbar mechanism B (1≡02345≡1) can be obtained by connecting in the same links the distal links of the two simple open kinematic chains A1 and A2 (Fig. 3.8). The simple chains A1 and A2 can be obtained by splitting up links 2, 3 or 4. The bases of vector spaces (RA1) and (RA2) used in Eq. (2.174) to calculate SA1, SA2 and B
n / 1S are systematized in Table 3.1. These bases respect Eq. (2.180) and the conditions mentioned in Remark 1. We note that these bases are also valid for a fourbar planar parallelogram mechanism (Fig. 3.9). Special geometric conditions concerning the lengths of opposite links do not allow link 3 to rotate with respect to reference link 1≡0. In the parallelogram planar mechanism, link 3 makes a translational circular motion around the reference link with a constant orientation.
Example 8. The fourbar spatial mechanism B (1≡02345≡1) presented in Fig. 3.10a has p=4 spherical joints and m=4 links including the fixed reference link (1≡5≡0). It can be obtained by connecting in the same
3.2 Singleloop kinematic chains 143
link the distal links of the simple open kinematic chain A (1≡02345) in Fig. 3.10b. We can simply observe by inspection that six independent motions exist between distal links 5 and 1 of the open kinematic chain A (Fig. 3.10b). We thus obtain (RA)=( x y z x y zv ,v ,v , , ,ω ω ω ) and rB=SA=6. The mobility given by Eq. (2.160) is MB=126=6.
We may obtain the same result by considering that the singleloop fourbar spatial mechanism B (1≡02341) is obtained by connecting in the same link the distal links of two simple open kinematic chains A1 (1A≡02A3A) and A2 (1B≡02B3B). In Fig. 3.10c, the simple chains A1 and A2 are obtained by splitting up link 3. The bases of the vector spaces (RA1) and (RA2) used in Eq. (2.174) to calculate SA1, SA2 and B
3 / 1S are: (RA1)=( x y z x yv ,v ,v , ,ω ω ) and (RA2)=( x z x y zv ,v , , ,ω ω ω ) or (RA2)= ( y z x y zv ,v , , ,ω ω ω ). These bases respect Eq. (2.180) and the conditions mentioned in Remark 1. We recall that (RA1) and (RA2) represent the bases of the vector spaces of the velocities of point H in the open kinematic chains A1 and A2. We thus obtain B
3 / 1S =dim(RA1∩RA2)=4. Eq. (2.144) gives rB=SA≡A1A2=dim(RA≡A1A2)= dim(RA1)+dim(RA2)dim(RA1∩RA2)=5+54=6.
Fig. 3.10. Fourbar spatial mechanism B (1≡02345≡1) (a), its associated simple open chains (b) and (c) and the particular case of fourbar spatial parallelogram (d)
144 3 Structural analysis
Link 3 may have four independent motions in a fourbar spatial mechanism: two translations and two rotations: x z x yv ,v , ,ω ω or y z x yv ,v , ,ω ω . If we consider link 3 as the characteristic link we obtain SF= B
3 / 1S =4, and Eq. (2.225) indicates two degrees of redundancy for this mechanism. They are given by the rotations of links 2 and 4 about the axes passing through the centres of the spherical joints adjacent to each link. We may consider that this mechanism has two internal and four external mobilities. We note that these structural analyses are also valid for the fourbar spatial mechanism with equal lengths of the opposite links (Fig. 10c). This mechanism is also called a fourbar spatial parallelogram.
Example 9. The slidercrank spatial mechanism B (1≡02345≡1) presented in Fig. 3.11a has p=4 joints RSSPtype and m=4 links including the fixed reference link (1≡5≡0). It can be obtained by connecting in the same link the distal links of the simple open kinematic chain A (1≡02345) in Fig. 2.11b. We can simply observe by inspection that six independent motions exist between distal links 5 and 1 of the open kinematic chain A (Fig. 3.11b) and (RA)=( x y z x y zv ,v ,v , , ,ω ω ω ) and rB=SA=6.
Fig. 3.11. The slidercrank spatial mechanism B (1≡02345≡1) of type RSSP (a) and its associated simple open chains (b) and (c)
3.2 Singleloop kinematic chains 145
The mobility given by Eq. (2.160) is MB=86=2. We can obtain the same result by considering that the singleloop slidercrank spatial mechanism B (1≡02341) is obtained by connecting in the same link the distal links of two simple open kinematic chains A1 (1A≡02A3A) and A2 (1B≡02B3B). In Fig. 3.11c, the simple chains A1 and A2 are obtained by splitting up link 3. The vector space RA1 may have one of the following basis: (RA1)=( x y z xv ,v ,v ,ω ), (RA1)=( x y z yv ,v ,v ,ω ) or (RA1)=( x y z zv ,v ,v ,ω ). The vector space RA2 has the unique basis (RA1)=( y x y zv , , ,ω ω ω ). These bases respect Eq. (2.180) and the conditions mentioned in Remark 1. We obtain
B3 / 1S =dim(RA1∩RA2)=2. Equation (2.144) gives rB=SA≡A1A2=dim(RA≡A1
A2)= dim(RA1)+dim(RA2)dim(RA1∩RA2)=4+42=6. Link 3 may have two independent motions in this spatial mechanism: one translation and one rotation. The translation is yv and the rotation may be any component ( xω ,
yω or zω ) of the angular velocity ω of link 3 about the axis passing through the centre of the two spherical joints. If we consider link 3 as the characteristic link, we obtain SF= B
3 / 1S =2 and Eq. (2.225) indicates zero degrees of redundancy. Two independent parameters have to be known to control the motion of all links. One of these parameters is the rotation of link 3 about the axis passing through the centre of the two spherical joints. The second parameter may be the translation of slider 4, or the rotation of crank 2. We can consider that this mechanism has one internal and one external mobility.
Example 10. The Bennett mechanism B (1≡02345≡1) presented in Fig. 3.12a has p=4 joints and m=4 links including the fixed reference link (1≡5≡0). It can be obtained by connecting distal links 2A and 4B of the simple open kinematic chains A1 (1≡02A) and A2 (1B≡02B3B4B) in Fig. 3.12b. We can simply observe by inspection that SA1=1 and SA2=3. In accordance with connectivity theorem 8a and Eq. (2.175), link 2 of the Bennett fourlink mechanism B can have relative motions with respect to the reference link only if the bases of the operational spaces RA1 and RA2 have at least one common velocity, that is B
2 / 1S 1= . We may consider that (RA1)=( zω ) and (RA2) is any basis of cardinal 3 obtained by combining zω with two other rotational and translational velocities. Eq. (2.174) gives SA=A1A2 =1+31=3. The mobility of the Bennett mechanism given by Eq. (2.163) is MB=43=1. One degree of freedom motion of the Bennett mechanism is due to its special geometric configuration (see Fig. 3.12a): (i) the opposite links (i.e., links that are not concurrent) have the same length denoted by a and b, (ii) the twist angles α and β are equal on opposite sides but with different signs and (iii) the link length and twist angles
146 3 Structural analysis
Fig. 3.12. Bennett mechanism B (1≡02345≡1) (a) and its associated simple open chains A1 (1A≡02A) and A2 (1B≡02B3B4B) (b)
3.2 Singleloop kinematic chains 147
Fig. 3.13. Bricard mechanism B (1≡0234567≡1) (a) and its associated simple open chains A1 (1A≡02A) and A2 (1B≡02B3B4B5B6B) (b)
148 3 Structural analysis
must satisfy the relation: asinβ =bsinα. We can see that the Bennett mechanism completely obeys the definitions, the theorems and the formulae presented in chapter 2 for quick mobility calculation of singleloop mechanisms. There is no reason to continue to consider this mechanism as “paradoxical” more than one hundred years after its invention (Bennett 1902).
Example 11. The general trihedral 6R singleloop mechanism of Bricard B (1≡0234567≡1) in Fig. 3.13a has p=6 joints and m=6 links including the fixed reference link (1≡7≡0). It can be obtained by connecting distal links 2A and 6B of the open simple kinematic chains A1 (1≡02A) and A2 (1B≡02B3B4B5B6B) in Fig. 3.13b. We can simply observe by inspection that SA1=1 and SA2=5. In accordance with connectivity theorem 8a and Eq. (2.175), link 2 of the Bricard mechanism B can have relative motions related to the reference link only if the bases of the operational spaces RA1 and RA2 have at least one common velocity, that is B
2 / 1S 1= . We may consider that (RA1)=( zω ), and (RA2) is any basis of cardinal 5 obtained by combining three rotational and two translational velocities. Eq. (2.174) gives SA=A1A2 =1+51=5. The mobility of the Bricard mechanism given by Eq. (2.163) is MB=65=1. One degree of freedom motion of the Bricard mechanism is due to its special geometric configuration defined by the existence of two pairs of three joint axes intersecting in two different points. In this way, a common line intersects all joint axes. We can see that the Bricard mechanism completely obeys the definitions, the theorems and the formulae presented in this section for quick mobility calculation of singleloop mechanisms. As with the Bennett mechanism, there is no reason to continue to consider the Bricard mechanism as “paradoxical” for so long (Bricard 1927).
3.3 Parallel mechanisms with simple limbs
We recall that a parallel mechanism with simple limbs is a complex mechanism C ← A1A2…Ak with two polinary links 1≡1Ai and n≡nAi connected by k simple chains Ai (1Ai2Ai…nAi), i=1,2,…k. The two polinary links are the reference platform 1 and the mobile platform n. The simple open kinematic chain Ai is called a simple limb. When the reference platform is fixed to the base, we use the notation 1≡1Ai≡0.
We also illustrate the calculation of the structural parameters of various complex closed mechanisms that can be considered as parallel mechanisms with simple limbs. We emphasize examples of robotic manipulators and
3.3 Parallel mechanisms with simple limbs 149
complex closed mechanisms that do not obey the classical criteria for quick calculation of mobility presented in sections 2.1 and 2.2.
Example 12. The parallel mechanism C ← A1A2A3 in Fig. 3.14a is obtained by parallel concatenation of 3 simple limbs A1 (1A≡02A3A4A5A), A2 (1B≡02B3B4B5B) and A3 (1C≡02C3C4C). The mobile platform is 5≡5A≡5B≡4C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). The mechanism has m=10 links including the fixed base, p=11 joints, two independent loops (q=1110+1=2), k1=3 and k2=0. The limbs are composed of revolute (R) and prismatic (P) joints. Limbs A1 and A2 are of type PRRR and limb A3 of type RRR. The axes of the revolute pairs are parallel to the O0x0axis in limb A1, O0y0axis in limb A2 and O0z0axis in limb A3.
Fig. 3.14. Translational parallel mechanism Isoglide2T2 of type 2PRRRRRR: structural diagram (a), associated graph (b) and the simple open kinematic chains associated with its simple limbs A1 (c), A2 (d) and A3(e)
150 3 Structural analysis
Fig. 3.15. CAD model of Isoglide2T2 of type 2PRRRRRR
This robot belongs to a family of innovative and modular parallel robotic manipulators with 26 degrees of mobility and decoupled motions, proposed by the author (Gogu 2002b), and developed under the name of IsoglidenTaRb at the French Institute of Advanced Mechanics (IFMA). In this notation, a represents the number of independent translational motions and b the number of independent rotational motions of the endeffector. The integer numbers a, b and n satisfy the relation n=a+b with 0≤a≤3, 0≤b≤3 and 2≤n≤6. When a or b equal zero, the corresponding notation T or R is missing in IsoglidenTaRb. This is the case for Isoglide2T2 with a= 2 and b=0. Figure 3.15 presents a CAD model of this parallel robotic manipulator. Other solutions of parallel robots of this family will be analyzed in the following examples.
We simply observe by inspection that the simple limbs (Fig. 3.14b, c, d) have the following unique bases of their operational velocity spaces: (RA1)=( x y z xv ,v ,v ,ω ), (RA2)=( x y z yv ,v ,v ,ω ) and (RA3)=( x y zv ,v ,ω ). The dimensions of these three operational spaces give the connectivity of the simple open chains associated with each limb Ai of the parallel mechanism C ← A1A2A3 (SA1= SA2=4 and SA3=3). The connectivity of the mobile platform in parallel mechanism C ← A1A2A3, given by Eq. (2.181), is C
5 / 1S =
A1 A2 A3dim( R R R ) 2∩ ∩ = . Only two relative independent velocities ( xv and yv ) exist between mobile platform 5 and reference base 1≡0 in parallel mechanism C ← A1A2A3. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =11. The number of joint parameters that lose their independence in the parallel mechanism C ← A1A2A3 given by Eq. (2.185) is rC=112=9. The mobility of the parallel mechanism C ← A1A2A3 given by Eq. (2.193) is MC=1111+2=2. Two independent parameters are necessary
3.3 Parallel mechanisms with simple limbs 151
to define the position of any link in the mechanism. In the parallel mechanism C ← A1A2A3 in Fig. 3.14a, these parameters are the two displacements q1 and q2 in the linear actuators mounted on the fixed base. We have seen in section 2.2 that this mechanism does not obey the classical CGK formulae but it perfectly complies with the formulae presented in this section for mobility calculation of parallel mechanisms with simple limbs.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=129=3 and the degree of structural redundancy given by Eq. (2.222) is TC=0. We note that in a parallel robot with simple limbs, rl=0.
Example 13. Planar parallel mechanism C ← A1A2A3 in Fig. 3.16a is obtained by parallel concatenation of 3 simple limbs A1 (1A≡02A3A4A), A2
Fig. 3.16. Translational parallel mechanism 3PPR: structural diagram (a), associated graph (b) and the simple open kinematic chains associated with its simple limbs A1 (c), A2 (d) and A3(e)
152 3 Structural analysis
(1B≡02B3B4B) and A3 (1C≡02C3C4C). The mobile platform is 4≡4A≡4B≡4C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). The mechanism has m=8 links including the fixed base, p=9 joints, two independent loops (q=98+1=2), k1=3 and k2=0. The mobile platform is 4≡4A≡4B≡4C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). The limbs are composed of revolute R and prismatic P joints. The three limbs are planar P⊥ P⊥ Rtype. The axes of the revolute pairs are parallel to O0z0.
We simply observe by inspection that the simple open chains associated with each limb Ai of the parallel mechanism C ← A1A2A3 (Fig. 3.16ce) have the same basis for their operational velocity spaces (RA1)=(RA2)=(RA3)= ( x y zv ,v ,ω ) and the same connectivity SA1=SA2= SA3=3. The connectivity of the mobile platform in parallel mechanism C ← A1A2A3, given by Eq. (2.181), is C
4 / 1S = A1 A2 A3dim( R R R ) 3∩ ∩ = . Three relative independent velocities x yv ,v and zω exist between mo
bile platform 4 and the reference base 1≡0 in the parallel mechanism C ← A1A2A3. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =9. The number of joint parameters that lose their independence in the parallel mechanism C ← A1A2A3, given by Eq. (2.191) is rC=93=6.
The mobility of the parallel mechanism C ← A1A2A3, given by Eq. (2.193) is MC=99+3=3. Three independent parameters are necessary to define the position of any link in the mechanism. In the parallel mechanism C ← A1A2A3 in Fig. 3.16a, these parameters may be the three displacements q1, q2 and q3 in the linear actuators mounted on the fixed base. This parallel mechanism obeys theorem 3 and Eq. (2.94). For this reason, the mobility of this parallel mechanism can also be obtained by using the classical CGK formulae presented in section 2.2.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=126=6 and the degree of structural redundancy given by Eq. (2.222) is TC=99=0. We note that rl=0.
Example 14. Results similar to those in example 13 are also obtained for the planar parallel mechanism C ← A1A2A3 in Fig. 3.17. This mechanism has only revolute pairs with parallel axes. It is used in planar parallel robots actuated by three rotating motors usually mounted on the fixed base.
Example 15. The parallel Cartesian robotic manipulator in Fig. 2.1 is a parallel mechanism C ← A1A2A3 obtained by parallel concatenation of 3 simple limbs A1 (1A≡02A3A4A5A), A2 (1B≡02B3B4B5B) and A3 (1C≡0
3.3 Parallel mechanisms with simple limbs 153
Fig. 3.17. Planar parallel mechanism with rotation actuators C ← A1A2A3 (a) and the simple open kinematic chains associated with its simple limbs A1 (b), A2 (c) and A3(d)
154 3 Structural analysis
Fig. 3.18. CAD model of Isoglide3T3
2C3C4C5C). As we have seen in example 3 – section 2.2, we simply observe by inspection (Fig. 2.2) that (RA1)=( x y z xv ,v ,v ,ω ), (RA2)= ( x y z yv ,v ,v ,ω ), (RA3)=( x y z zv ,v ,v ,ω ) and SAi=4 (i=1,2,3). The connectivity of the mobile platform 5≡5A≡5B≡5B in the parallel mechanism C ← A1A2A3, given by Eq. (2.181), is C
5 / 1S = A1 A2 A3dim( R R R ) 3∩ ∩ = . Three relative independent translational velocities ( xv , yv and zv ) exist between the mobile and fixed platforms. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =12. The number of joint parameters that lose their independence in the parallel mechanism C ← A1A2A3 given by Eq. (2.191) is rC=123=9. The mobility of parallel mechanism C ← A1A2A3 given by Eq. (2.193) is MC=1212+3=3. Three independent parameters are necessary to define the position of any link in the mechanism, for example the displacements q1, q2 and q3 in the linear actuators mounted on the fixed base. We have seen in section 2.2 that this mechanism does not obey the classical CGK formulae but it completely obeys to new formulae for mobility calculation of parallel mechanisms with simple limbs.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=129=3 and the degree of structural redundancy given by Eq. (2.222) is TC=1212=0 with rl=0.
The parallel robotic manipulator in Fig. 2.1 also belongs to the Isoglide family. It represents Isoglide3T3 of type T3 (Fig. 3.18).
Example 16. Four versions of the parallel robotic manipulators with simple limbs Isoglide4T3R1 are analysed in this example. They are obtained from the solution in Fig. 3.19 by changing the type of joints
3.3 Parallel mechanisms with simple limbs 155
(a)
(b)
Fig. 3.19. The parallel manipulator Isoglide4T3R1 developed at the French Institute of Advanced Mechanics in a modular design approach: partial view (a) and general view (b)
156 3 Structural analysis
Fig. 3.20. Parallel mechanism with simple limbs Isoglide4T3R1 of type 3PRRRR+1PRRR: structural diagram (a), associated graph (b), simple open chains associated with limbs A1 (c), A2 (d), A3 (e) and A4 (f)
3.3 Parallel mechanisms with simple limbs 157
Fig. 3.21. CAD model of 3PRRRR+1PRRRtype Isoglide4T3R1 parallel robot
connecting the four limbs to the moving platform. We can see that the solutions in Figs. 3.15 and 3.18 are particular cases derived from this solution.
In the four solutions of Isoglide4T3R1, the mobile platform has Schönflies motions that are three independent translations T and one rotation R about a fixed direction axis.
Isoglide4T3R1 (3PRRRR+1PRRRtype) is a parallel mechanism C ← A1A2A3A4 obtained by parallel concatenation of 4 simple limbs A1 (1A≡02A3A4A5A6A), A2 (1B≡02B3B4B5B), A3 (1C≡02C3C4C5C6C) and A4 (1D≡02D3D4D5D6D)  see Figs. 3.20 and 3.21. The mechanism has three limbs of type PRRR⊥R and one limb of type PRRR. In each limb, the axes of the first three revolute joints are parallel to the direction of the prismatic joint. The last revolute joints of the four limbs have parallel axes. The actuated prismatic joints of limbs A1, A2 and A3 have orthogonal directions. The actuated prismatic joints of limbs A3 and A4 have parallel directions.
For ( / 2, / 2)yϕ π π∈ − + the simple open kinematic chains associated with the four limbs have (Fig. 3.20cf): (RA1)=( x y z x yv ,v ,v , ,ω ω ), (RA3)=(RA4)= ( x y z y zv ,v ,v , ,ω ω ), (RA2)= ( x y z yv ,v ,v ,ω ), and SA1=SA3=SA4
=5, SA2=4. The connectivity of the platform 6≡6A≡5B≡6C≡6D in parallel mechanism C ← A1A2A3A4 is C
6 / 1 A1 A2 A3 A4S dim( R R R R ) 4= ∩ ∩ ∩ = . Four relative independent velocities x yv ,v , zv and yω exist between the
158 3 Structural analysis
(a) (b)
(c) Fig. 3.22. Parallel mechanism with simple limbs Isoglide4T3R1 of type 4PRRRR (a), associated graph (b), CAD model (c)
3.3 Parallel mechanisms with simple limbs 159
mobile and the fixed platforms. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =19. The number of joint parameters that lose their independence in the parallel robot given by Eq. (2.191) is rC=194=15, and the mobility given by Eq. (2.193) is MC=1915=4. Four independent joint parameters are necessary to completely describe the motion of the mechanism. In Isoglide4T3R1 presented in Figs. 3.193.24 these parameters are the displacements of four linear actuators qi (i=1,2,…,4) mounted on the fixed base.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=1815=3 and the degree of structural redundancy given by Eq. (2.222) is TC=1919=0 with rl=0.
Figures 3.22 and 3.23 present solutions of Isoglide4T3R1 with two and, respectively, one degree of overconstraint. The solution presented in Fig. 3.22 is not overconstrained (NC=0).
Isoglide4T3R1 of type 4PRRRR see Fig. 3.22 has four limbs of type PRRR⊥R and the following structural parameters: (RA1)=(RA2)= ( x y z x yv ,v ,v , ,ω ω ), (RA3)= (RA4)=( x y z y zv ,v ,v , ,ω ω ), SAi=5 (i=1,2,3,4),
C6 / 1 A1 A2 A3 A4S dim( R R R R ) 4= ∩ ∩ ∩ = , r=5+5+5+54=16, MC=206=4,
NC=1816=2. The last revolute joint of limb A2 has an idle mobility denoted by R¤. This limb is of type PRRRR¤.
Fig. 3.23. Parallel mechanism with simple limbs Isoglide4T3R1 of type 2RRS+1PRRRR+1PRRR (a) and associated graph (b)
160 3 Structural analysis
(a) (b)
(c) Fig. 3.24. Parallel mechanism with simple limbs Isoglide4T3R1 of type 3PRRS+1PRRR (a), associated graph (b), CAD model (c)
3.3 Parallel mechanisms with simple limbs 161
Isoglide4T3R1 of type 2PRRS+1PRRRR+1PRRR see Fig. 3.23 has two idle mobilities (one in each spherical joint S=RRR¤) and the following structural parameters: (RA1)=( x y z x yv ,v ,v , ,ω ω ), (RA2)=( x y z yv ,v ,v ,ω ), (RA3)=( x y z x y zv ,v ,v , , ,ω ω ω ), (RA4)=( x y z x y zv ,v ,v , , ,ω ω ω ), SA1=5,
SA2=4, SAi=6 (i=3,4), C6 / 1 A1 A2 A3 A4S dim( R R R R ) 4= ∩ ∩ ∩ = , rC=21
4=17, MC=2117=4, NC=1817=1. Isoglide4T3R1 of type 3PRRS+1PRRR  see Fig. 3.24 has three idle
mobilities (one in each spherical joint S=RRR¤) and the following structural parameters: (RA1)=( x y z x y zv ,v ,v , , ,ω ω ω ), (RA2)=( x y z yv ,v ,v ,ω ), (RA3)=( x y z x y zv ,v ,v , , ,ω ω ω ), (RA4)=( x y z x y zv ,v ,v , , ,ω ω ω ), SA2=4, SAi=6
(i=1,3,4), C5 / 1 A1 A2 A3 A4S dim( R R R R ) 4= ∩ ∩ ∩ = , rC=224=18 MC=22
18=4, NC=1818=0. Example 17. The parallel robotic manipulator Isoglide5T3R2 (Fig.
3.25) is another solution in the Isoglide family.
Fig. 3.25. Parallel mechanism with simple limbs Isoglide5T3R2 (a) and its associated graph (b)
162 3 Structural analysis
Fig. 3.26. Simple open chains associated with limbs A1 (a), A2 (b), A3 (c), A4 (d) and A5 (e) of Isoglide5T3R2
Parallel mechanism C ← A1A2A3A4A5 is obtained by parallel concatenation of 5 simple limbs A1 (1A≡02A3A4A 5A6A7A), A2 (1B≡02B3B4B5B6B), A3 (1C≡02C3C4C5C6C7C), A4 (1D≡02D3D4D5D6D) and A5 (1E≡02E3E4E5E6E7E).
3.3 Parallel mechanisms with simple limbs 163
For (0, / 2)α π∈ and ( / 2, / 2)β π π∈ − + the simple open kinematic chains associated with the five limbs have (Fig. 3.26ce): (RA1)=( x y z xv ,v ,v , , ,α βω ω ω ), (RA2)=(RA4)=( x y zv ,v ,v , ,α βω ω ), (RA3)= (RA5)=( x y z zv ,v ,v , , ,α βω ω ω ) and SA1=SA3=SA5 =6, SA2= SA4=5.
The connectivity of the platform 7≡7A≡6B≡7C≡6D≡7E is C7 / 1 A1 A2 A3 A4 A4S dim( R R R R R ) 5= ∩ ∩ ∩ ∩ = . Five relative independent
velocities x y zv ,v ,v , αω and βω exist between the mobile and the fixed
platforms. The dimension of the joint vector space U is dim(U)= pii 1
f=∑
=28. The number of joint parameters that lose their independence in the parallel robot is rC=285=23 and the mobility MC=2823=5. Five independent joint parameters are necessary to completely describe the motion of the mechanism. In the Isoglide5T3R2 these parameters are the displacements of the five linear actuators qi (i=1,2,…,5) mounted on the fixed base.
The degree of overconstraint of this parallel robot given by Eq. (2.217) is NC=2423=1 and the degree of structural redundancy given by Eq. (2.222) is TC=2828=0 with rl=0.
Example 18. The GoughStewart parallel robot 6UPStype in Fig. 3.27a is a parallel mechanism C ← A1A2A3A4A5A6 obtained by parallel concatenation of 6 simple identical limbs A1 (1A≡02A3A4A), A2 (1B≡02B3B4B), A3 (1C≡02C3C4C), A4 (1D≡02D3D4D), A5 (1E≡02E3E4E) and A6 (1F≡02F3F4F). Each telescopic limb is connected to the fixed platform by a universal joint U and to the mobile platform by the spherical S joint (Fig. 3.27c). We have (RAi)=( x y z x y zv ,v ,v , , ,ω ω ω ) and SAi=6 (i=1,2,…,6). The connectivity of the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F in the parallel mechanism C ← A1A2A3A4A5A6 is C
4 / 1S 6= . Six relative independent velocities x y z x yv ,v ,v , ,ω ω and zω exist between the mobile and the fixed platforms. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =36. The number of joint parameters that lose their independence in the parallel robot is rC=366=30 and the mobility M=3630=6. Six independent joint parameters are necessary to completely describe the motion of the mechanism. In GoughStewart type parallel robots these parameters are the displacements of the six linear actuators qi (i=1,2,…,6) integrated in the prismatic joints. We can see that for this type of parallel robot the actuators are not mounted on the fixed base. This solution is not overconstrained (NA=3030=0) and non redundant (TA=3636=0).
164 3 Structural analysis
Fig. 3.27. GoughStewart parallel mechanism of type 6UPS (a), associated graph (b), simple open chain associated with limbs A1 (c)
The GoughStewart parallel robot 6SPStype in Fig. 1.2 is obtained by parallel concatenation of 6 simple identical limbs SPStype. Each telescopic limb is connected to the fixed and moving platforms by spherical joints. In this case, both the limb connectivity and the connectivity of the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F in the parallel mechanism C ← A1A2A3A4A5A6 are unchanged (RAi)=( x y z x y zv ,v ,v , , ,ω ω ω ), SAi=6
(i=1,2,…,6) and C4 / 1S 6= . The dimension of the joint vector space U is
dim(U)= pii 1
f=∑ =42. The number of joint parameters that lose their inde
pendence in the parallel robot is rC=30 and the mobility MC=4230=12. Twelve independent joint parameters are necessary to completely describe the motion of all links in this mechanism. We note that, among these 12 joint parameters, 6 represent internal limb mobilities. Each limb has an internal mobility that is a rotation around the axis directed to the centers of its spherical joints. These internal mobilities do not influence the motion of the mobile platform. For this reason, just six joint parameters are in fact used to actuate the robot. This non overconstarined solution (NC=0) has six degrees of redundancy (TC=4236=6).
Example 19. The parallel mechanism C ← A1A2A3 in Fig. 3.28a is obtained by parallel concatenation of 3 simple limbs A1 (1A≡02A3A4A), A2 (1B≡02B 3B4B) and A3 (1C≡02C3C4C). The mobile platform is 4≡4A≡4B≡4C and the reference platform is fixed to the base 1≡1A≡1B≡1C≡0.
3.3 Parallel mechanisms with simple limbs 165
Fig. 3.28. General case of 3RPS mechanism (a), associated graph (b), simple open chains associated with limbs A1 (c), A2 (d) and A3 (e)
166 3 Structural analysis
Each telescopic limb is connected to the base by a revolute pair and to the mobile platform by a spherical pair. This parallel mechanism called 3RPS was proposed by Hunt (1983) in his pioneering article on structural synthesis of parallel mechanisms. We can observe by inspection that RA1 has a unique basis (RA1)=( y z x y zv ,v , , ,ω ω ω ), and RA2 and RA3 have multiple bases. The basis of RA2 is formed by any combination of five independent velocities of the point H in the open chain associated with limb A2. We may consider one of the following sets of bases for RA2: ( y z x y zv ,v , , ,ω ω ω ), ( x z x y zv ,v , , ,ω ω ω ), ( x y x y zv ,v , , ,ω ω ω ), ( x y z x zv ,v ,v , ,ω ω ), ( x y z y zv ,v ,v , ,ω ω ) or ( x y z x yv ,v ,v , ,ω ω ). The same five possibilities exist for the basis of RA3. In accordance with Remark 2 and Eq. (2.197), the bases of RA2 and RA3 are selected such that the minimum value of C
4 / 1S is obtained. We have C C4 / 1 4 / 1S dim( R )= = A1 A2 A3dim( R R R )∩ ∩ =3, SA1=SA2=SA3=5, rC=3
and MC=3. The complete set of possible bases for C4 / 1R is: ( x y z, ,ω ω ω ),
( z x yv , ,ω ω ), ( z x zv , ,ω ω ), ( z y zv , ,ω ω ), ( y x yv , ,ω ω ), ( y x zv , ,ω ω ), ( y y zv , ,ω ω ), ( y z xv ,v ,ω ), ( y z yv ,v ,ω ), ( y z zv ,v ,ω ). As we can see, the proposed method gives us, in a very simple way, a complete and correct result concerning the mobility and connectivity of a 3RPS parallel manipulator without studying the instantaneous constraint system. As previously mentioned, a given parallel mechanism has an infinite number of positions and for each position we can associate an instantaneous constraint system. In these conditions, an instantaneous constraint study could offer just some partial results on the instantaneous connectivity of the mobile platform depending on the configuration of the mechanism (Huang and Fang 1996; Huang et al. 1996; Fang and Huang 1997; Huynh 2004).
This parallel manipulator is not overconstrained (N=0) and is non redundant (T=0).
Example 20. The parallel mechanism C ← A1A2A3 in Fig. 3.29a is obtained by parallel concatenation of 3 simple limbs A1 (1A≡02A3A4A5A6A), A2 (1B≡02B3B4B5B6B) and A3 (1C≡02C3C4C5C6C). The mobile platform is 6≡6A≡6B≡6C and the reference platform is fixed to the base 1≡1A≡1B≡1C≡0. Each telescopic limb is connected to the base and to the mobile platform by two intersecting revolute pairs (Cardan joints). This parallel mechanism called 3RRPRR or 3UPU was proposed by (Tsai 1996; Tsai and Joshi 2000) as a translational parallel manipulator. In order to keep the moving platform from changing its orientation, Tsai pointed out the following two geometric conditions for the four revolute joint axes
3.3 Parallel mechanisms with simple limbs 167
Fig. 3.29. 3UPU mechanism (a), associated graph (b), simple open chains associated with limbs A1 (c), A2 (d) and A3 (e)
168 3 Structural analysis
of each limb: (i) the first revolute joint axis is parallel to the last revolute joint axis and (ii) the two intermediate revolute joint axes are parallel to one another. These geometric conditions do not imply the intersection of the first two revolute joint axes specific to the universal joint.
Hence, any 3RRPRR parallel manipulator whose revolute joint axes satisfy the above conditions will be a translational manipulator (Tsai and Joshi 2000). These geometric conditions were generalized later by Di Gregorio and ParentiCastelli (1998, 2002). This parallel manipulator has multiple bases for RA1, RA2 and RA3. We can observe by inspection (Fig. 3.22ce) that RA1 could have two different bases, RA2 and RA3 three different bases each. The two possible bases of RA1 are: ( x y z x zv ,v ,v , ,ω ω ) or ( x y z x yv ,v ,v , ,ω ω ). The three bases of RA2 and RA3 are: ( x y z x zv ,v ,v , ,ω ω ), ( x y z x yv ,v ,v , ,ω ω ), ( x y z y zv ,v ,v , ,ω ω ). In accordance with remark 2 and Eq. (2.197), the bases of RA1, RA2 and RA3 are selected so that the minimum value of C
6 / 1S is obtained. We thus obtain SA1=SA2=SA3=5, C6 / 1( R )=
A1 A2 A3( R R R )∩ ∩ =( x y zv ,v ,v ), C C4 / 1 4 / 1S dim( R )= =3, rC=12, MC=3,
NC=0, TC=0. The three independent translations of the mobile platform are obtained by actuating the prismatic pairs integrated in the limbs.
3.4 Parallel mechanisms with complex limbs
We recall that a parallel robotic manipulator with complex limbs denoted by D ← A1A2…Ak1E1E2…Ek2 is a complex mechanism with p joints in which the mobile platform n≡nAi≡nEj is connected to the reference platform 1≡1Ai≡1Ej by k≥2 limbs, of which at least one limb is complex. A complex limb contains at least one closed loop. The elementary limbs are denoted by Ai (i=1,2,…k1) and the complex limbs by Ej (j=1,2,…k2) with k1+k2=k. When the reference platform is fixed to the base, we use the notation 1≡1Ai≡1Ej≡0. We have denoted by k1 the total number of simple limbs, and by k2 the total number of complex limbs.
We illustrate the calculation of the structural parameters of parallel robotic manipulators with complex limbs which contain various combinations of planar or spatial closed loops. We emphasize examples which do not comply with the classical CGK formulae.
Example 21. The parallel mechanism 2PPatype in Fig 3.30a represents a maximallyregular and fullyisotropic translational robotic manipulator used in pick and place operations (Gogu 2002). The parallel mechanism D ← E1E2 in Fig. 3.30 is obtained by parallel concatenation of two complex
3.4 Parallel mechanisms with complex limbs 169
Fig. 3.30. Fullyisotropic overconstrained translational parallel mechanism 2PPatype: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)
170 3 Structural analysis
Fig. 3.31. Fullyisotropic non overconstrained translational parallel mechanism PPassPCPasstype: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)
3.4 Parallel mechanisms with complex limbs 171
limbs E1 (1A≡02A3A4A5A) and E2 (1B≡02B3B4B5B). A planar parallelogram closed loop Pa is included in each complex limb. The mechanism has the following structural characteristics: m=8 links including the fixed base, p=10 joints, three independent loops (q=108+1=3), k1=0 and k2=2. The mobile platform is 5≡5A≡5B and the reference platform is fixed to the base (1≡1A≡1B≡0). The two limbs of type P⊥Pa are identical. Each limb is composed of a prismatic pair P and a planar parallelogram loop Pa. The axes of the revolute joints in the parallelogram loops are parallel to the O0zaxis, and the directions of the two prismatic pairs are parallel to the axes O0x and O0y.
We simply observe by inspection (Fig. 3.30c,d) that the bases of the operational velocity spaces of two limbs are (RE1)=(RE2)=( x yv ,v ). The dimensions of these operational spaces give the connectivity SE1= SE2=2 of the kinematic chains associated with each limb of parallel mechanism D ← E1E2. The connectivity of the mobile platform in the parallel mechanism D ← E1E2, given by Eq. (2.198), is D
5 / 1S = E1 E 2dim( R R ) 2∩ = . Just two independent translational velocities xv and yv exist between mobile platform 5 and reference base1≡1A≡1B≡0 in the parallel mechanism D ← E1E2. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =10. Three joint parameters lose their independence in each planar parallelogram (rl=6).
The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=42+6=8. The mobility of the parallel mechanism D ← E1E2 given by Eq. (2.202) is MD=104+26=2. Two independent parameters are necessary to define the position of any link in the mechanism. These parameters are the two displacements q1 and q2 in the prismatic actuators mounted on the fixed base (underlined joints).
Equations (2.217) and (2.222) indicate that the parallel mechanism has ten degrees of overconstraint (ND=184+26=10) and zero degrees of redundancy (TD=1046=0). We note that the classical CGK formulae give an erroneous mobility (MD=1) for this parallel mechanism.
Figure 3.31 presents a non overconstrained solution of type PPassPCPass. Two spherical joints S are included in each parallelogram loop.
A cylindrical joint C is also integrated in the E2limb. We have denoted by Pass the parallelogram loop containing two spherical joints. Six joint parameters lose their independence in a Passtype parallelogram loop. This solution has the following structural parameters: m=9, p=11, q=3, k1=0, k2=2, (RE1)=( x y yv ,v ,ω ), (RE2)=( x y z x zv ,v ,v , ,ω ω ), SE1=3, SE2=5, D
5 / 1S = 2, dim(U)=20, rl=12, rD=18, MD=2, ND=0, TD=0.
172 3 Structural analysis
Example 22. The parallel mechanism 2PaPatype in Fig 3.32a also represents a maximallyregular and fullyisotropic translational robotic manipulator used in pick and place operations (Gogu 2002). The parallel mechanism D ← E1E2 in Fig. 3.32a is obtained by parallel concatenation of two complex limbs E1 (1A≡02A3A4A5A6A7A) and E2 (1B≡02B3B4B5B6A7A). Two planar parallelogram loops Pa are included in each complex limb. The mechanism has the following structural characteristics: m=12 links including the fixed base, p=16 joints, five independent loops (q=1612+1=5), k1=0 and k2=2. The mobile platform is 7≡7A≡7B and the reference platform is fixed to the base (1≡1A≡1B≡0). The two limbs of type PaPa are identical. Each limb is composed of two planar parallelogram loops Pa. The axes of the revolute joints in the parallelogram loops are parallel to parallel to O0z.
Fig. 3.32. Fullyisotropic overconstrained translational parallel mechanism 2PaPatype: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)
3.4 Parallel mechanisms with complex limbs 173
We simply observe by inspection (Fig. 3.32c,d) that the bases of the operational velocity spaces of both limbs are (RE1)=(RE2)=( x yv ,v ). The dimensions of these operational spaces give the connectivity SE1= SE2=2 of the kinematic chains associated with each limb of the parallel mechanism D ← E1E2. The connectivity of the mobile platform in the parallel mechanism D ← E1E2, given by Eq. (2.198), is D
5 / 1S = E1 E 2dim( R R ) 2∩ = . Just two independent translational velocities xv and yv exist between mobile platform 7 and reference base 1≡1A≡1B≡0 in parallel mechanism D ← E1E2. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =16.
Fig. 3.33. Fullyisotropic non overconstrained translational parallel mechanism Pa*Pa*CPa*Pa*Stype: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c) and A2 (d)
174 3 Structural analysis
Three joint parameters lose their independence in each planar parallelogram (rl=12). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=42+12=14. The mobility of the parallel mechanism D ← E1E2 given by Eq. (2.202) is MD=164+212=2. Two independent parameters are necessary to define the position of any link in the mechanism. These parameters are the two displacements q1 and q2 in the revolute actuators mounted on the fixed base (underlined joints). Equations (2.217) and (2.222) indicate that the parallel mechanism has 16 degrees of overconstraint (ND=304+212=16) and zero degrees of redundancy (TD=16412=0).
We note that the classical CGK formulae give an erroneous mobility (MD=1) for this parallel mechanism.
Figure 3.33 presents a non overconstrained solution Pa*Pa*CPa*Pa*Stype. One cylindrical and one spherical joint are introduced in each parallelogram loop denoted by Pa*. A cylindrical joint C is also integrated in the E1limb, and a spherical joint in the E2limb. Six joint parameters lose their independence in a Pa*type parallelogram loop. This solution has the following structural parameters: m=14, p=18, q=5, k1=0, k2=2, (RE1)=( x y zv ,v ,v ), (RE2)=( x y x y zv ,v , , ,ω ω ω ), SE1=3, SE2=5, D
5 / 1S = 2, dim(U)=32, rl=24, rD=30, MD=2, ND=0, TD=0.
Example 23. The parallel mechanism presented in Fig 3.34a represents an innovative maximallyregular and fullyisotropic spherical parallel wrist proposed by the author in (Gogu 2005f). The parallel mechanism D ← A1E1
in Fig. 3.34 is obtained by parallel concatenation of a simple limb A1 (1A≡02A3A) and a complex limb E1 (1B≡02B3B4B5B6B7B8B). A planar parallelogram closed loop Pa is included in the complex limb E1.
The mechanism has the following structural characteristics: m=9 links including the fixed base, p=10 joints, two independent loops (q=109+1=2), one simple (k1=1) and one complex (k2=1) limb. The mobile platform is 8≡3A≡8B and the reference platform is fixed to the base (1≡1A≡1B≡0). The simple limb A1 is composed of two revolute pairs with orthogonal and intersecting axes R⊥Rtype. The complex limb E1 is of typePa⊥RRR⊥R. The axes of the last revolute joints of the two limbs are parallel. The axes of the revolute joints in the parallelogram loop are parallel to the x0y0plane and the three revolute pairs have axes parallel to O0z. We simply observe by inspection that the limbs (Fig. 3.34c,d) have the following bases of their operational velocity spaces: (RA1)=( xy z,ω ω ) and (RE1)=( x y z xy zv ,v ,v , ,ω ω ). The rotation axis of velocity xyω lies in the x0y0plane. The dimensions of these operational spaces SA1=2 and
3.4 Parallel mechanisms with complex limbs 175
Fig. 3.34. Fullyisotropic overconstrained parallel wrist RRPaRRRRtype with two degrees of freedom: kinematic structure (a), associated graph (b), kinematic chains associated with limbs A1 (c) and A2 (d)
SE1=5 give the connectivity of the kinematic chains associated with each limb of parallel mechanism D ← A1E1. The connectivity of mobile platform in the parallel mechanism D ← A1E1, given by Eq. (2.198), is
D8 / 1S = E1 E 2dim( R R ) 2∩ = . Just two independent angular velocities xyω and zω exist between the
mobile platform 8 and the reference base 1≡0 in the parallel mechanism D ← A1E1. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =10. Three joint parameters lose their independence in the planar parallelogram loop incorporated in limb E1 (rl=3). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=72+3=8. The mobility of parallel mechanism D ← A1E1 given by Eq. (2.202) is MD=107+23=2. Two independent parameters are necessary to define the position and orientation of any link in the mechanism. These parameters are the two displacements q1 and q2 in the revolute actuators mounted on the fixed base (underlined joints). Equations (2.217) and (2.222) indicate that the parallel mechanism has four degrees of overconstraint (ND=128=4) and zero degrees of redundancy (TD=1073=0).
176 3 Structural analysis
Fig. 3.35. Fullyisotropic non overconstrained parallel wrist RRPa*RRStype with two degrees of freedom: kinematic structure (a), associated graph (b), kinematic chains associated with limbs A1 (c) and A2 (d)
We note that the classical CGK formulae give an erroneous mobility (MD=1) for this parallel mechanism.
Figure 3.35 presents a non overconstrained solution RRPa*RRStype. One cylindrical and one spherical joint are introduced in the parallelogram loop denoted by Pa*. The complex limb E2 of type Pa*RRS includes a Pa*type parallelogram loop, two revolute joints with parallel axes and a spherical joint. This solution has the following structural parameters: m=8, p=9, q=2, k1=1, k2=1, (RE1)=( xy z,ω ω ), (RE2)=( x y x y zv ,v , , ,ω ω ω ), SA1=2,
SE1=6, D5 / 1S = 2, dim(U)=14, rl=6, rD=12, MD=2, ND=0, TD=0.
Example 24. Orthoglide (Fig. 3.36) is a translational parallel robot (Wanger and Chablat 2000; Chablat and Wanger 2003) developed by IRCCyN (Institut de Recherche en Communications et Cybernétique de Nantes). The parallel mechanism D ← E1E2E3 in Figs. 3.36a and 3.37a is obtained by parallel concatenation of 3 complex limbs E1 (1A≡02A3A4A5A), E2 (1B≡02B3B4B5B) and E3 (1C≡02C3C4C5C). The mechanism has the following structural characteristics: m=17 links including the fixed base, p=21 revolute and prismatic joints, five independent loops (q=2117+1=5), three complex limbs (k1=0, k2=3). The mobile platform is
3.4 Parallel mechanisms with complex limbs 177
5≡5A≡5B≡5C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). Each complex limb P⊥R⊥Pa⊥Rtype is composed of one prismatic and two revolute pairs combined with a planar parallelogram loop. The axes of the revolute joints are perpendicular to the axes of the parallelogram loop. In each limb, the prismatic pair is actuated and its direction is perpendicular to the axis of the revolute pair. The directions of the three actuated prismatic pairs are also perpendicular to each other.
We simply observe by inspection that the complex limbs (Fig. 3.37ce) have the following unique bases of their operational velocity spaces: (RE1)=( x y z yv ,v ,v ,ω ), (RE2)=( x y z zv ,v ,v ,ω ) and (RE3)=( x y z xv ,v ,v ,ω ). The dimensions of these three operational spaces SEi=4 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of the parallel mechanism D ← E1E2E3. The connectivity of mobile platform in
Fig. 3.36. Translational parallel robot Orthoglide (courtesy of CNRSIRCCyN)
178 3 Structural analysis
Fig. 3.37. Parallel robot Orthoglide: structural diagram (a), associated graph (b), complex open chains associated with limbs A1 (c), A2 (d) and A3 (e)
3.4 Parallel mechanisms with complex limbs 179
the parallel mechanism D ← E1E2E3, given by Eq. (2.198), is D7 / 1S =
E1 E 2 E3dim( R R R ) 3∩ ∩ = . Just three translational velocities xv , yv and
zv exist between the mobile platform 7 and the reference base 1≡0 in the parallel mechanism D ← E1E2E3.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =21. Three
joint parameters lose their independence in each planar parallelogram loop (rl=9). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=123+9=18. The mobility of the parallel mechanism D ← E1E2E3 given by Eq. (2.202) is MD=2112+39=3. Three independent parameters are necessary to define the position of any link in the mechanism. In the Orthoglide robot, these parameters are the three displacements q1, q2 and q3 of the linear actuators mounted on the fixed base. Equation (2.217) indicates that the Orthoglide parallel mechanism has twelve degrees of overconstraint (ND=3012+39=12) and Eq. (2.222) gives zero degrees of redundancy (TD=21129=0).
Example 25. The fullyisotropic translational parallel mechanism 3PPaPatype (Gogu 2004b) in Fig. 3.38 is obtained by parallel concatenation of 3 complex limbs E1 (1A≡02A3A4A5A6A7A8A), E2 (1B≡02B3B4B5B6B7B8B) and E3 (1C≡02C3C4C5C6C7C8C). The mechanism has the following structural characteristics: m=20 links including the fixed base, p=27 revolute and prismatic joints, eight independent loops (q=2720+1=8) and three complex limbs (k1=0, k2=3). The mobile platform is 8≡8A≡8B≡8C and the reference platform is fixed to the base (1≡1A≡1B≡1C≡0). Each complex limb PPaRtype is composed of one prismatic joint and two planar parallelogram loops. The prismatic joints of the three limbs are actuated and their directions are mutually perpendicular.
We simply observe by inspection that the complex limbs (Fig. 3.38ce) have the following unique bases of their operational velocity spaces: (RE1)=( x y zv ,v ,v ), (RE2)=( x y zv ,v ,v ) and (RE3)=( x y zv ,v ,v ). The dimensions of these three operational spaces SEi=3 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1E2E3.
The connectivity of the mobile platform in the parallel mechanism D ← E1E2E3, given by Eq. (2.198), is D
7 / 1S = E1 E 2 E3dim( R R R ) 3∩ ∩ = . Just three translational velocities xv , yv and zv exist between the mobile platform 8 and the reference base 1≡0 in the parallel mechanism D ← E1E2E3.
180 3 Structural analysis
Fig. 3.38. Fullyisotropic overconstrained translational parallel mechanism 3PPaPatype: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c), A2 (d) and A3 (e)
3.4 Parallel mechanisms with complex limbs 181
Fig. 3.39. Fullyisotropic non overconstrained translational parallel mechanism 3PRPa*Pa*type: kinematic structure (a), associated graph (b)
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =27. Three
joint parameters lose their independence in each planar parallelogram loop (rl=18). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=93+18=24.
The mobility of the parallel mechanism D ← E1E2E3 given by Eq. (2.202) is MD=279+318=3. Three independent parameters are necessary to define the position of any link in the mechanism. These parameters are the three displacements q1, q2 and q3 of the linear actuators mounted on the fixed base. Equation (2.217) indicates that the parallel mechanism in Fig. 3.38 has 24 degrees of overconstraint (ND=489+318=24) and Eq. (2.222) gives zero degrees of redundancy (TD=27918=0).
182 3 Structural analysis
Fig. 3.40. Complex chains associated with limbs A1 (a), A2 (b) and A3 (c) of the fullyisotropic non overconstrained translational parallel mechanism 3PRPa*Pa*type
3.4 Parallel mechanisms with complex limbs 183
Figure 3.39 presents a non overconstrained solution 3PRPa*Pa*Rtype. One cylindrical and one spherical joint are introduced in the parallelogram loop denoted by Pa*. Two revolute joints with perpendicular axes are introduced in each complex limb outside the parallelogram loops. The first revolute joints of the three limbs are mutually orthogonal. The last revolute joints of the three limbs are also mutually orthogonal. We simply observe by inspection that the complex limbs isolated from the parallel mechanism (Fig. 3.40ac) have the following bases of their operational velocity spaces: (RE1)=( x y z x yv ,v ,v , ,ω ω ), (RE2)=( x y z y zv ,v ,v , ,ω ω ) and (RE3)= ( x y z x zv ,v ,v , ,ω ω ). Six joint parameters lose their independence in each parallelogram loop. This solution has the following structural parameters: m=26, p=33, q=8, k1=0, k2=3, SE1= SE2=SE3=5, D
10 / 1S = 3, dim(U)=51, rl=36, rD=48, MD=3, ND=0, TD=0.
Example 26. The Delta parallel mechanism 3RPa4stype invented by Clavel (1990) is widely used today in translational parallel robots. The solution presented in Fig. 3.41a (Clavel 1990) represent a hybrid robot in which a translational mechanism Deltatype with rotating actuators (Fig. 3.41a) is serially concatenated with a revolute joint. The translational parallel mechanism (Fig. 3.42a) is obtained by parallel concatenation of 3 complex limbs E1 (1A≡02A3A4A5A), E2 (1B≡02B3B4B5B) and E3 (1C≡02C3C4C5C). The mechanism has the following structural characteristics: m=11 links including the fixed base, p=15 joints, five independent loops (q=1511+1=5) and three complex limbs (k1=0, k2=3). The mobile platform is 5≡5A≡5B≡5C and the reference platform is fixed to the base
Fig. 3.41. Hybrid mechanism Deltatype invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990) (a) and its associated translational parallel mechanism Deltatype with rotating actuators (b)
184 3 Structural analysis
Fig. 3.42. Translational Delta mechanism 3RPa4stype: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c), A2 (d) and A3 (e)
(1≡1A≡1B≡1C≡0). Each complex limb RPa4stype is composed of one revolute and a spatial parallelogram loop containing 4 spherical joints. The revolute joints of the three limbs are actuated and the directions of their rotation axes lie in a plane. Moving platform 5 in each complex limb isolated from the Delta parallel mechanism (Fig. 3.42ce) may realize three translation velocities x y zv ,v ,v and two angular velocities iω and i+1ω (i=1 for Alimb, i=3 for Blimb and i=5 for Climb): (RE1)= ( x y z 1 2v ,v ,v , ,ω ω ) (RE2)=( x y z 3 4v ,v ,v , ,ω ω ) and (RE3)= ( x y z 5 6v ,v ,v , ,ω ω ),
3.4 Parallel mechanisms with complex limbs 185
The dimensions of these three operational spaces SEi=5 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1E2E3. The connectivity of the mobile platform in the parallel mechanism D ← E1E2E3, given by Eq. (2.198), is D
5 / 1S =
E1 E 2 E3dim( R R R ) 3∩ ∩ = . Just three translational velocities xv , yv and
zv exist between the mobile platform 5 and the reference base 1≡0 in the parallel mechanism D ← E1E2E3.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =39. Six
joint parameters lose their independence in each parallelogram loop Pa4Stype (rl=18). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=153+18=30. The mobility of parallel mechanism D ← E1E2E3 given by Eq. (2.202) is MD=3915+318=9. Nine independent parameters have to be known to define the position and the orientation of each link of the parallel mechanism. Equation (2.222) gives six degrees of redundancy (TD=391518=6) for the Delta parallel mechanism. Each parallelogram loop Pa4Stype contains two internal mobilities introduced by the rotation of links 3 and 4 of each limb along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These six degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining three degrees of mobility are external mobilities that are transmitted from the input to the output of the parallel mechanism. They represent the independent parameters
Fig. 3.43. Parallel mechanism linear Deltatype invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990) (a) and its associated translational parallel mechanism Deltatype with linear actuators (b)
186 3 Structural analysis
Fig. 3.44. Translational Delta mechanism 3PPa4stype: kinematic structure (a), associated graph (b), complex chains associated with limbs A1 (c), A2 (d) and A3 (e)
that are necessary to define the position of the moving platform. These parameters are the three angular displacements q1, q2 and q3 of the rotating actuators mounted on the fixed base. Equation (2.217) indicates that the parallel mechanism in Fig. 3.42 is not overconstraint (ND=3015+318=0).
The same structural analysis and parameters are applicable to the translational parallel Deltatype mechanism with linear actuators in Figs. 3.43b and 3.44a. The only difference is that the three degrees of external mobilities are associated with three linear displacements in the linear actuators mounted on the fixed base instead of rotating actuators (Fig. 3.44).
3.4 Parallel mechanisms with complex limbs 187
The hybrid mechanism in Fig. 3.41a may be considered a complex open kinematic chain F with the following structural characteristics: m=12 links including the fixed base, p=16 joints, five independent loops (q=1511+1=5) and just one complex limb (k1=0, k2=1, k=1). In this case, SG1=SF and Eqs. (2.226) and (2.228) give r=rl=rG1. The number of joint parameters that lose their independence in the five closed loops of this mechanism is r=rD=30, where rD was calculated above for the mechanism in Fig 3.42a. Equations (2.223)(2.225) give the following structural parameters for the mechanism in Fig. 3.44: M=10, N=0, T=6. The six degrees of redundancy represent internal mobilities in the limbs and the remaining four degrees of mobility are external mobilities given by the angular displacements of the four revolute actuated joints.
Example 27. The Delta parallel mechanism in Fig. 3.45 represents another solution invented by R. Clavel [Clavel US patent 4976582]. This robot may be considered as a parallel mechanism D ← E1E2 obtained by parallel concatenation of 2 complex limbs E1 and E2 (Fig. 3.46a). Limb E1 (Fig. 3.46b) is the hybrid Delta mechanism analysed in Example 26, and limb E2 of type RUPU (Fig. 3.46c) is composed of a homokinetic double universal joint U with a telescopic intermediary shaft (1D≡02D3D4D5D6D7D). As we have seen in Example 26, limb E1 of type 3RPa4sR is composed by serial concatenation of a translational parallel mechanism 3RPa4s and a revolute joint. Five independent closed loops exist in limb E1, and
Fig. 3.45. Parallel mechanism Deltatype with rotating actuators invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990)
188 3 Structural analysis
Fig. 3.46. Parallel mechanism Deltatype with rotating actuators: kinematic structure (a), associated graph (b), complex chains associated with limbs E1 (c) and E2 (d)
four in limb E2. We have seen in the previous example that 30 joint parameters lose their independence in the closed loops of limb E1 ( E1
lr =30). The connectivity theorem 5A and Eq. (2.140) in chapter 2, and example 1 in this chapter also indicate that one joint parameter loses its independence
3.4 Parallel mechanisms with complex limbs 189
Fig. 3.47. Parallel mechanism Deltatype with linear and rotating actuators: kinematic structure (a), associated graph (b), complex chains associated with limbs E1 (c) and E2 (d)
in each closed loop composed of two revolute joints of same axis of the universal joints ( E 2
lr =4). The parallel mechanism in Fig. 3.46a has the following structural characteristics: m=17 links including the fixed base, p=26 joints, ten independent loops (q=2617+1=10) and two complex limbs (k1=0, k2=2). The endeffector is 7≡6A≡6B≡6C≡7D, and the reference platform is fixed to the base (1≡1A≡1B≡1C≡1D≡0). The bases and the dimensions of the operational velocities spaces associated with the two com
190 3 Structural analysis
plex limbs are: (RE1)=( x y z zv ,v ,v ,ω ), (RE2)=( x y z x y zv ,v ,v , , ,ω ω ω ), SE1=4 and SE2=6. The connectivity of the endeffector in parallel mechanism D ← E1E2, given by Eq. (2.198), is D
7 / 1S = E1 E 2dim( R R ) 4∩ = . Three translational velocities xv , yv and zv and one angular velocity zω exist between the endeffector 7 and the reference base 1≡0 in the parallel mechanism D ← E1E2.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =50. The
number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=104+34=40. The mobility of parallel mechanism D ← E1E2 in Fig. 3.46a given by Eq. (2.202) is MD=5010+434=10. Ten joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism. Equation (2.222) gives six degrees of redundancy (TD=501034=6) for Delta parallel mechanism in Fig. 3.46a. Each parallelogram loop Pa4Stype contains two internal mobilities introduced by the rotation of the links 3 and 4 of each limb along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These six degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining four degrees of mobility are external mobilities that are transmitted from the input to the output of the parallel mechanism. They represent the independent parameters that are necessary to define the position of the moving platform. These parameters are the three angular displacements qi (i=1,…,4) in the rotating actuators mounted on the fixed base.
Equation (2.217) indicates that the parallel mechanism in Fig. 3.46a has ND=6010+434=20. Five overconstraints are introduced in each closed loop with two revolute joints of same axis of the universal joints. Usually these overconstraints in the universal joints are eliminated by a precise alignment of rotation axes and the technical solutions used for the revolute joints. To avoid these overconstraints in static and dynamic analysis of the mechanisms with universal joints, just one revolute joint kinematically equivalent to the two revolute joints of same axis is taken into account. In this case, the mechanism in Fig. 46a may be considered as a parallel mechanism D ← E1A1 with one complex limb E1 as in the previous analysis, and a simple limb A1 (1D≡02D3D4D5D6D7D). The following structural parameters are associated with mechanism D ← E1A1: m=17, p=22, q=6, k1=1, k2=1, (RE1)=( x y z zv ,v ,v ,ω ), (RE2)=( x y z x y zv ,v ,v , , ,ω ω ω ),
SE1=4, SE2=6, D7 / 1S = 4 dim(U)= p
ii 1f
=∑ =46, rl=30, rD=36, MD=10, TD=6
3.4 Parallel mechanisms with complex limbs 191
and ND=0. With this hypothesis, mechanism D ← E1A1 in Fig. 46a may be considered non overconstrained.
The same structural analysis and parameters are applicable to the parallel Deltatype mechanism in Figs. 3.43a and 3.47a. The only difference is that the four degrees of external mobilities are associated with three linear displacements in the linear actuators and one angular displacement in the rotating motor mounted on the fixed base (Fig. 3.47).
Example 28. The mechanism in Fig. 3.48 represents a translational Deltatype parallel robot with universal joints also invented by Clavel (1990). This translational parallel mechanism D ← E1E2E3 is obtained by parallel concatenation of 3 complex limbs E1 (1A≡02A3A4A5A6A), E2 (1B≡02B3B4B5B6B) and E3 (1C≡02C3C4C5C6C) – Fig. 3.49a. The mechanism has the following structural characteristics: m=14 links including the fixed base, p=27 revolute joints, 14 independent closed loops (q=2714+1=14) and three complex limbs (k1=0, k2=3). The mobile platform is 6≡6A≡6B≡6C and the reference platform is
Fig. 3.48. Translational Delta parallel mechanism 3RUUtype invented by R. Clavel and presented in US Patent 4976582 (Clavel 1990)
Table 3.2. Valid bases of operational vector spaces RE1, RE2 and RE3 of complex limbs of the Delta parallel mechanism
No. (RE1) (RE2) (RE3) (3
Eii 1R
=∩ )
1 ( x y z x yv ,v ,v , ,ω ω ) ( x y z y zv ,v ,v , ,ω ω ) ( x y z z xv ,v ,v , ,ω ω ) ( x y zv ,v ,v ) 2 ( x y z y zv ,v ,v , ,ω ω ) ( x y z z xv ,v ,v , ,ω ω ) ( x y z x yv ,v ,v , ,ω ω ) ( x y zv ,v ,v ) 3 ( x y z z xv ,v ,v , ,ω ω ) ( x y z x yv ,v ,v , ,ω ω ) ( x y z y zv ,v ,v , ,ω ω ) ( x y zv ,v ,v )
192 3 Structural analysis
Fig. 3.49. Delta Parallel mechanism 3RUUtype: kinematic structure (a), associated graph (b), complex chain associated with a limb (c)
fixed to the base (1≡1A≡1B≡1C≡0). Each complex limb RUUtype is composed of nine revolute joints. Two closed loops with two revolute joints with the same axis exist in each universal joint.
The moving platform 6 in each complex limb isolated from the Delta parallel mechanism (Fig. 3.49c) may realize three translation velocities x y zv ,v ,v and two angular velocities iω and i+1ω (i=1 for Alimb, i=3 for
Blimb and i=5 for Climb). In the general case, each angular velocity has
3.4 Parallel mechanisms with complex limbs 193
three components x y z, ,ω ω ω but just one can be independent. Various bases of the operational velocity space of each complex limb are systematized in Table 3.2. They are set up by taking into account Eq. (2.206) and Remark 3 in chapter 2. The dimensions of these three operational spaces SEi=5 (i=1,2,3) give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1E2E3. The connectivity of the mobile platform in the parallel mechanism D ← E1E2E3, given by Eq. (2.198), is D
6 / 1S = E1 E 2 E3dim( R R R ) 3∩ ∩ = . Just three translational velocities xv , yv and zv exist between the mobile platform 6 and the reference base 1≡0 in the parallel mechanism D ← E1E2E3.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =27.One
joint parameter loses its independence in each closed loop of the universal joints (rl=12). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=153+12=24. The mobility of the parallel mechanism D ← E1E2E3 given by Eq. (2.202) is MD=2715+312=3. Three independent parameters have to be known to define the position and the orientation of each link of the parallel mechanism. These parameters are the three angular displacements q1, q2 and q3 in the rotating actuators mounted on the fixed base. Equation (2.222) gives zero degrees of redundancy (TD=271512=0).
Equation (2.217) gives ND=8415+312=60. Each closed loop in the universal joint introduces five overconstraints. Usually these overconstraints in the universal joints are eliminated by a precise alignment of rotation axes and the technical solutions used for the revolute joints.
As we have shown in the previous example, to avoid these overconstraints in static and dynamic analysis of the mechanisms with universal joints, just one revolute joint kinematically equivalent to the two revolute joints of same axis is taken into account. With this hypothesis, the mechanism in Fig. 3.49a may be considered as a parallel mechanism C ← A1A2A3 with three simple limbs A1 (1A≡02A3A4A5A), A2 (1B≡02B3B4B5B) and A3 (1C≡02C3C4C5C). The following structural parameters are associated with mechanism C ← A1A2A3: m=14, p=15, q=2, k1=3, k2=0, SA1=SA2=SA3=5, C
6 / 1S = 3, dim(U)= pii 1
f=∑ =15, rl=0, rC=12, MC=3, TC=0
and NC=0. In this case, mechanism C ← A1A2A3 in Fig. 49a may be considered non overconstrained.
Example 29. The mechanism in Fig. 3.50 represents a translational Tricepttype parallel robot invented by Neumann (1998). This translational parallel mechanism D ← E1E2E3E4 is obtained by parallel concatenation of 4 complex limbs E1 (1A≡02A3A4A5A),
194 3 Structural analysis
Fig. 3.50. Tricept parallel mechanism invented by KE. Neumann and presented in US Patent 4732525 (Neumann 1988)
Fig. 3.51. Tricept parallel mechanism: CAD model (a), kinematic structure (b), associated graph (c) – courtesy of Vincent Robin for the CAD model (a)
3.4 Parallel mechanisms with complex limbs 195
E2 (1B≡02B3B4B5B), E3 (1C≡02C3C4C5C) and E4 (1D≡02D3D4D) – Fig. 3.51a. The mechanism has the following structural characteristics: m=13 links including the fixed base, p=20 joints, 8 independent closedloops (q=2013+1=8) and four complex limbs (k1=0, k2=4). The mobile platform is 5≡5A≡5B≡5C≡4D, and the reference platform is fixed to the base (1≡1A≡1B≡1C≡1D≡0). Five closed loops with two revolute joints with the same axis exist in the mechanism, one in each limb Ai (i=1,2,3) and two in the universal joint of limb A4.
The moving platform 5 in each limb A1, A2 and A3 isolated from the Tricept parallel mechanism may realize six independent velocities, x y z x yv ,v ,v , ,ω ω and zω . In limb A4 isolated from the Tricept parallel
mechanism, the moving platform 5 may realize just three independent motions which may be three translations, or combinations of rotations and translations. Various bases may be set up for the operational velocity space of limb A4 isolated from the Tricept parallel mechanism: ( x y zv ,v ,v ), ( x y xv ,v ,ω ), ( x y yv ,v ,ω ), ( y z zv ,v ,ω ), ( x x zv , ,ω ω ), ( y x zv , ,ω ω ), ( z x zv , ,ω ω ), ( x y zv , ,ω ω ), ( y y zv , ,ω ω ) or ( z y zv , ,ω ω ).
The dimensions of the four operational spaces give the connectivity of the kinematic chains associated with each limb Ei of parallel mechanism D ← E1E2E3E4: SEi=6 (i=1,2,3) and SE4=3. The connectivity of the mobile platform in the parallel mechanism D ← E1E2E3E4, given by Eq. (2.198), is D
5 / 1S = E1 E 2 E3 E4dim( R R R R ) 3∩ ∩ ∩ = . Just three independent velocities exist between the mobile platform 5 and the reference base 1≡0 in parallel mechanism D ← E1E2E3E4. The basis of the operational velocity space of the moving platform 5 in the Tricept parallel mechanism coincides with the basis of the operational velocity space of limb A4 isolated from the Tricept parallel mechanism above mentioned.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =25. One
joint parameter loses its independence in each closed loop, with just two revolute joints of same axis (rl=5). The number of joint parameters that lose their independence in the parallel mechanism given by Eq. (2.200) is rD=213+5=23. The mobility of the parallel mechanism D ← E1E2E3E4 given by Eq. (2.202) is MD=2621+35=3. Three independent parameters have to be known to define the position and the orientation of each link of the parallel mechanism. These parameters are the three displacements q1, q2 and q3 of the linear actuators mounted in the telescopic limbs A1, A2 and A3. These actuators are not adjacent to the fixed base. Equation (2.222) gives zero degrees of redundancy (TD=26215=0). Equation (2.217) gives ND=4821+35=25. These overconstraints are introduced in the five
196 3 Structural analysis
Fig. 3.52. Kinematic structure of Tricept robot considered as a parallel mechanism with four simple limbs
closed loops with two revolute joints of same axis. Each closed loop introduces 5 overconstraints. Usually these overconstraints are eliminated by a precise alignment of rotation axes, and by the technical solutions used for the revolute joints.
As we have shown in the previous example, to avoid these overconstraints in static and dynamic analysis of such a mechanisms, just one revolute joint kinematically equivalent to the two revolute joints of same axis is taken into account. With this hypothesis, the Tricept parallel mechanism (Fig. 3.52) may be considered as a parallel mechanism C ← A1A2A3A4 with four simple limbs A1 (1A≡02A3A4A5A), A2 (1B≡02B3B4B5B), A3 (1C≡02C3C4C5C) and E4 (1D≡02D3D4D). The following structural parameters are associated with mechanism C ← A1A2A3A4: m=13, p=15, q=3, k1=0, k2=1, k=1, C
5 / 1S = 3, dim(U)= pii 1
f=∑ =21, rl=0, rC=18, MC=3,
TC=0 and NC=0. The parallel mechanism C ← A1A2A3 in Fig. 3.52 is non overconstrained.
Example 30. The mechanism in Fig. 3.53 represents the hybrid robot IRB 940 fabricated by ABB by concatenating a Tricept parallel mechanism and a serial wrist RRRtype. This hybrid mechanism may be considered to be a complex open kinematic chain F with the following structural characteristics: m=16 links including the fixed base, p=23 joints, eight independent loops (q=2316+1=6) and just one complex limb (k1=0, k2=1, k=1). In this case, SG1=SF and Eqs. (2.226) and (2.228) give r=rl=rG1. The number of joint parameters that lose their independence in the eight closed
3.4 Parallel mechanisms with complex limbs 197
Fig. 3.53. Hybrid mechanism obtained by concatenating a Tricept parallel mechanism and a serial wrist: CAD model (a), kinematic structure (b), associated graph (c) – courtesy of Vincent Robin for the CAD model (a)
loops of this mechanism is r=rD=23, where rD was calculated above for the mechanism in Fig 3.51 in the previous example. Equations (2.223)(2.225) give the following structural parameters for the mechanism in Fig. 3.53: M=6, N=25, T=0. As in the previous example, the 25 overconstraints are introduced by the five closed loops with just two revolute joints with the same axis. If we take into account just one revolute joint kinematically equivalent to the two revolute joints of same axis (Fig. 3.54) the following structural parameters are associated: m=13, p=15, q=3, k1=0, k2=1, k=1, SG1=SF=3, dim(U)= p
ii 1f
=∑ =21. The number of joint parameters that lose their independence in the eight closed loops is r=rC=18,
198 3 Structural analysis
Fig. 3.54. A hybrid mechanism obtained by concatenating a Tricept parallel mechanism with simple limbs and a serial wrist: kinematic structure (a), associated graph (b)
where rC was calculated above for the mechanism in Fig 3.52 in the previous example.
Equations (2.223)(2.225) give the following structural parameters for the mechanism in Fig. 3.54: M=3, T=0 and N=0. The mechanism is non overconstrained.
Example 31. The parallel robot H4 (Fig. 3.55) was developed by Montpellier Laboratory of Computer Science, Robotics, and Microelectronics –LIRMM (Pierrot and Company 1999; Company and Pierrot 1999; Pierrot et al. 2003, Company et al. 2003, 2006). This mechanism F ← G1G2 is obtained by parallel concatenation of 2 hybrid limbs G1 and G2 with the moving platform 6 (Fig. 3.56a). Each limb Gi ← DiRi (i=1,2) is obtained by concatenating a complex kinematic chain Di with a revolute joint Ri. Complex chains D1 ← E1E2 and D2 ← E3E4 represent parallel mechanisms with two complex limbs E1E2 (Fig. 3.56b) and E3E4 (Fig. 3.56c) connecting the fixed base to two intermediary moving platforms 5≡5A≡5B and 5’≡5C≡5D. Each complex limb Ei combines a revolute joint and a spatial parallelogram loop with four spherical joints Pa4s: E1 (1A≡02A3A4A5A), A2 (1B≡02B3B4B5B), A3 (1C≡02C3C4C5C) and E4 (1D≡02D3D4D5D).
The mechanism has the following structural characteristics: m=16 links including the fixed base, p=22 joints, 7 independent closed loops (q=2216+1=7) and two complex limbs (k1=0, k2=2, k=2).
3.4 Parallel mechanisms with complex limbs 199
Fig. 3.55. H4 parallel robot developed by Montpellier Laboratory of Computer Science, Robotics, and Microelectronics (a) and CAD models (b) and (c) (courtesy LIRMM)
The four complex limbs Ei isolated from the parallel robot (Figs. 3.57 and 3.58) have the following bases and dimensions for their operational velocities spaces: (RE1)=( x y z 1 2v ,v ,v , ,ω ω ),(RE2)=( x y z 3 4v ,v ,v , ,ω ω ), (RE1)=( x y z 5 6v ,v ,v , ,ω ω ),(RE2)=( x y z 7 8v ,v ,v , ,ω ω ) and SEi=5 (i=1,...,4).
The connectivity of the intermediary platforms 5 in the parallel mechanisms D1 ← E1E2, given by Eq. (2.198), is D1
5 / 1S = 3.
200 3 Structural analysis
Fig. 3.56. H4 parallel robot, kinematic structure (a), associated graph (b), limbs G1 (c) and G2 (d) isolated from the parallel robot
Just three independent translational velocities xv , yv and zv exist between 5 and reference base 1A≡1B ≡0 in parallel mechanism D1 ← E1E2 isolated from the parallel robot. Equation (2.209) indicates that 19 joint parameters lose their independence in the closed loops of parallel mechanism D1 ← E1E2 (rD1=103+12=19). We have taken into account that 6 joint parameters
3.4 Parallel mechanisms with complex limbs 201
Fig. 3.57. Parallel mechanism D1 ← E1E2 (a) and its complex limbs E1 (b) and E2 (c) isolated from the parallel mechanism D1
lose their independence in each parallelogram closed loop integrated in complex limbs E1 and E2. The same results are obtained for limb D2:
D25'/ 1S 3= and rD2=19. In each hybrid limb G1 and G2 isolated from the H4 parallel robot (Fig.
3.56b and c), the moving platform 6 may realize four independent velocities, x y zv ,v ,v and zω . We have: (RG1)=(RG2)=( x y z zv ,v ,v ,ω ) and SG1=SG2=4.
202 3 Structural analysis
Fig. 3.58. Parallel mechanism D2 ← E3E4 (a) and its complex limbs E3 (b) and E4 (c) isolated from the parallel mechanism D2
The connectivity of mobile platform 6 in the parallel robot H4 (Fig. 3.56a) is F
6 / 1S = G1 G2dim( R R ) 4∩ = . Four independent velocities exist between mobile platform 6 and reference base 1≡0 in parallel mechanism F ← G1G2 associated with H4 robot. The basis of the operational velocity space of the moving platform 6 in H4 parallel robot coincides with the ba
3.4 Parallel mechanisms with complex limbs 203
sis of the operational velocity space of limbs G1 and G1 isolated from the parallel mechanism F ← G1G2 .
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =54. The
number of joint parameters that lose their independence in the parallel mechanism F ← G1G2 given by Eq. (2.226) is r=84+38=42. We have taken into account in Eq. (2.228) that 19 joint parameters lose their independence in each hybrid limb Gi (i=1,2) as we have calculated above. The mobility of the parallel mechanism F ← G1G2 given by Eq. (2.223) is M=5442=12. Twelve joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism F ← G1G2. Equation (2.225) gives eight degrees of redundancy (T=124=8) for H4 parallel robot. Each parallelogram loop Pa4Stype contains two internal mobilities introduced by the rotation of the links 3 and 4 of each limb along a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These eight degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining four degrees of mobility are external mobilities that are transmitted from the input to the output of the parallel mechanism. They represent the independent parameters that are necessary to define the position and the orientation of the moving platform 6. These parameters are the angular displacements in the four rotating actuators mounted on the fixed base.
Equation (2.224) indicates that H4 parallel robot is not overconstrained N=4242=0.
Example 32. Isoglide4T3R1 in Fig. 3.59 is the translational parallel robot with decoupled motions proposed in (Gogu 2002; 2005e). This mechanism F ← G1G2 is obtained by parallel concatenation of a hybrid limb G1 and a simple limb G2 to the moving platform 6 (Fig. 3.59a,b). The hybrid limb G1 ← CR (Fig. 3.59c) is obtained by serial concatenation of a complex kinematic chain C with a revolute joint R. The complex chain C ← A1A2A3 represents a translational parallel mechanisms with three simple limbs Isoglide3T3 of type 3PRRR analysed in example 15. The intermediary moving platform 5 of Isoglide3T3 is connected to the moving platform 6 by a revolute joint. The simple limb G2 (1D≡02D3D4D5D6) is of type PRRR⊥R (Fig. 3.49d). The revolute joints connecting the limbs G1 and G2 to the moving platform have parallel axes.
The mechanism has the following structural characteristics: m=16 links including the fixed base, p=18 joints, 3 independent closed loops (q=1816+1=3), one simple limb and one complex limb (k1=1, k2=1, k=2).
204 3 Structural analysis
Fig. 3.59. Parallel robot Isoglide4T3R1 with one simple and one complex limb (a), associated graph (b), limbs G1 (c) and G2 (d) isolated from the parallel mechanism
3.4 Parallel mechanisms with complex limbs 205
Fig. 3.60. CAD model of parallel robot Isoglide4T3R1 with one simple and one complex limb
The two limbs isolated from the parallel robot (Figs. 3.59c and d) have the following bases and dimensions for their operational velocities spaces: (RG1)=( x y z yv ,v ,v ,ω ), (RG2)=( x y z y zv ,v ,v , ,ω ω ), and SG1=4, SG2=5.
As we have seen in Example 15, nine joint parameters lose their independence in the closed loops of the translational parallel mechanism C ← A1A2A3 included in the hybrid limb G1 (rl= G1
lr =rC=9). The connectivity of the mobile platform 6 in Isoglide4T3R1 (Fig. 3.59a) is
F6 / 1S = G1 G2dim( R R ) 4∩ = . Four independent velocities exist between the
mobile platform 6 and the reference base 1≡1A≡1B≡1C≡1D≡0 in the parallel mechanism F ← G1G2 associated with this parallel robot of the Isoglide family.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =18. The
number of joint parameters that lose their independence in the parallel mechanism F ← G1G2 given by Eq. (2.226) is r=94+9=14. The mobility of the parallel mechanism F ← G1G2 given by Eq. (2.223) is M=1814=4. Four joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism F ← G1G2. These parameters are the linear displacements of the four linear actuators mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=44=0) and Eq. (2.224) indicates that parallel robot Isoglide4T3R1 in Fig. 3.59a has four degrees of overconstraints (N=1814=4).
206 3 Structural analysis
Figure 3.60 presents a CAD model of the parallel robot Isoglide4T3R1 with one hybrid and one simple limb.
Example 33. Isoglide3T2R1 in Fig. 3.61 is a fullyisotropic planar parallel robot proposed in (Gogu 2004c). This mechanism F ← G1G2G3 is obtained by parallel concatenation of two simple limbs G1 and G2 and a hybrid limb G3. The moving platform is denoted by 8, and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.61a,b). Simple limbs G1 (1A≡02A3A4A5A8), and G2 (1B≡02B3B4B5B8) are of type PRRR⊥R (Figs. 3.61c and d). Hybrid limb G3 (Fig. 3.61e) is obtained by concatenating a revolute pair with two interconnected planar parallelogram loops (2C3C4C6C) and (7C6C4C5C 8). The revolute pairs connecting the three limbs to the moving platform have the same axis.
The mechanism has the following structural characteristics: m=16 links including the fixed base, p=19 joints, 4 independent closed loops (q=1916+1=4), two simple limbs and one complex limb (k1=2, k2=1, k=3).
The three limbs isolated from the parallel robot (Figs. 3.61c,d,e) have the following bases and dimensions for their operational velocities spaces: (RG1)=( x y z x zv ,v ,v , ,ω ω ), (RG2)=( x y z y zv ,v ,v , ,ω ω ), (RG3)=( x y zv ,v ,ω ), and SG1= SG2=5, SG3=3.
To calculate the number of joint parameters that lose their independence in the two closed loops integrated in hybrid limb G3 ← RC, we consider the
Fig. 3.61. Fullyisotropic planar parallel robot Isoglide3T2R1 with one hybrid and two simple limbs (a), associated graph (b)
3.4 Parallel mechanisms with complex limbs 207
Fig. 3.62. Limbs G1 (c), G2 (d) and G3 (e) isolated from the parallel mechanism of Isoglide3T2R1 in Fig. 3.61
kinematic chain formed by these two closed loops as a parallel kinematic chain C ← A1A2A3 with three simple limbs connecting the ternary links 6C and 4C (Fig. 3. 63). The three simple limbs are: A1 (6C4C) of type RR (Fig. 3.63b), A2 (6C2C3C4C) of type RRR (Fig. 3.63c) and A3 (6C7C85C4C) of type RRRR (Fig. 3.63d).
208 3 Structural analysis
Fig. 3.63. Parallel kinematic chain C ← A1A2A3 with distal links 4C and 6C and associated with the closed loops in limb G3 ← RC (a), limb A1 (b), limb A2 (c) and limb A3 (d)
The three limbs isolated from the parallel kinematic chain C ← A1A2A3 (Figs. 3.63b,c,d) have the following bases and dimensions for their operational velocities spaces: (RA1)=( zω ), (RA2)= ( x y zv ,v ,ω ), (RA3)=( x y zv ,v ,ω ) and SA1=1, SA2= SA3=3. The connectivity of link 4C with respect to link 6C in parallel kinematic chain C ← A1A2A3 (Fig. 3.62a) is C
4C / 6 CS =
A1 A2 A3dim( R R R ) 1∩ ∩ = . Just one rotation about the zaxis is possible between the two links. The number of joint parameters that lose their independence in the parallel kinematic chain C ← A1A2A3 is rC=71=6. This value is given by Eq. (2. 191) by taking by taking into account that in the
3.4 Parallel mechanisms with complex limbs 209
parallel kinematic chain C ← A1A2A3 we have considered 6C as the reference link, and 4C as the distal link. We thus obtain G3
l l Cr r r 6= = = . The connectivity of the mobile platform 8 in Isoglide3T2R1 with two
simple and one hybrid limb in Fig. 3.61a is F8 / 1S = G1 G2 G3dim( R R R )∩ ∩
=3. Three independent velocities ( x yv ,v and zω ) exist between mobile platform 8 and reference base 1≡1A≡1B≡1C≡0 in parallel mechanism F ← G1G2G3 associated with this parallel robot of Isoglide family.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =19. The
number of joint parameters that lose their independence in the parallel mechanism F ← G1G2G3 given by Eq. (2.226) is r=133+6=16. The mobility of the parallel mechanism F ← G1G2G3 given by Eq. (2.223) is M=1916=3. Three joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism F ← G1G2G3. These parameters are the displacements (qi, i=1,2,3) in the two linear and one rotating actuators mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=33=0) and Eq. (2.224) indicates that parallel robot Isoglide3T2R1 in Fig. 3.61a has eight degrees of overconstraint (N=2416=8). Six of these overconstraints are in the two planar parallelogram loops.
Example 34. Isoglide3T2R1 in Fig. 3.64 is a slightly modified version of the fullyisotropic planar parallel robot presented in Fig. 3.61 (Gogu, 2004c). This mechanism F ← G1G2 is obtained by parallel concatenation of two hybrid limbs G1 and G2. The moving platform is denoted by 8 and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.64a,b). This solution is derived from the solution in Fig. 3.61 by superposing two revolute joints in just one joint. Limb G1 ← BR is obtained by serially concatenating a closed loop B ← A1A2 with a revolute joint. Limbs G2 in Fig. 3.64a and G3 in Fig. 3.61a are identical. The revolute pairs connecting the two limbs to the moving platform have the same axis. The mechanism has the following structural characteristics: m=15 links including the fixed base, p=18 joints, 4 independent closed loops (q=1815+1=4) and two hybrid limbs (k=2).
The two limbs isolated from the parallel robot (Figs. 3.64c,d) have the following bases and dimensions for their operational velocities spaces: (RG1)=( x y z zv ,v ,v ,ω ), (RG2)=( x y zv ,v ,ω ) and SG1=4, SG2=3.
Five joint parameters lose their independence in the closed loop of limb G1 (see Example 5), G1
l Br r 5= = , and six in the closed loops of limb G2 (see Example 33), G2
lr 6= .
210 3 Structural analysis
Fig. 3.64. Fullyisotropic planar parallel robot Isoglide3T2R1 with two hybrid limbs (a), associated graph (b), limb G1 (c) and limb G2 (d)
The connectivity of mobile platform 8 in Isoglide3T2R1 with two hybrid limbs in (Fig. 3.64a) is F
8 / 1S = G1 G2 G3dim( R R R ) 3∩ ∩ = . Three independent velocities ( x yv ,v and zω ) exist between the mobile platform 8 and the reference base 1≡1A≡1B≡1C≡0 in the parallel mechanism F ← G1G2 associated with this parallel robot of the Isoglide family.
3.4 Parallel mechanisms with complex limbs 211
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =18. The
number of joint parameters that lose their independence in the parallel mechanism F ← G1G2 given by Eq. (2.226) is r=73+11=15. We have taken into account that Eq. (2.228) gives rl=11. The mobility of the parallel mechanism F ← G1G2 given by Eq. (2.223) is M=1815=3. Three joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism F ← G1G2. These parameters are the displacements (qi, i=1,2,3) in the two linear and one rotating actuators mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=33=0) and Eq. (2.224) indicates that the parallel robot Isoglide4T3R1 in Fig. 3.63a has nine degrees of overconstraint (N=2415=9). Six of these overconstraints are in the two planar parallelogram loops.
Example 35. Isoglide3T2R1 in Fig. 3.65 is a slightly modified version of the fullyisotropic planar parallel robot presented in Fig. 3.64 (Gogu, 2004c). The three limbs of this mechanism are interconnected in one complex kinematic chain F. The moving platform is denoted by 8 and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.65a,b). The mechanism has the following structural characteristics: m=15 links including the fixed base, p=18 joints, 4 independent closed loops (q=1815+1=4) and one complex limb (k=1). The connectivity of mobile platform 8 in Isoglide3T2R1 with one complex limb in (Fig. 3.65a) is F
8 / 1S = G1S = 3. Three independent velocities ( x yv ,v and zω ) exist between mobile platform 8 and reference base
Fig. 3.65. Fullyisotropic planar parallel robot Isoglide3T2R1 with three interconnected limbs (a), associated graph (b)
212 3 Structural analysis
Fig. 3.66. Parallel kinematic chain F ← G1G2 with distal link 7 associated with fullyisotropic parallel robot Isoglide3T2R1 with three interconnected limbs (a), limb G1 (b) and limb G2 (c)
3.4 Parallel mechanisms with complex limbs 213
Fig. 3.67. CAD model of Isoglide3T2R1 with interconnected limbs
1≡1A≡1B≡1C≡0 in the mechanism F associated with this parallel robot of the Isoglide family.
To calculate the number of joint parameters that lose their independence in the four closed loops, we may consider F ← G1G2 as a parallel mechanism with two hybrid limbs G1 and G2 connecting reference link 1 and distal link 7 (Fig. 3.66a). As we have seen in chapter 2, the number of joint parameters that lose their independence in the closed loops of a mechanism does not depend of the choice of the distal links. The two hybrid limbs G1 and G2 in Figs. 3.66c,d have the same structural parameters with limbs G1 and G2 in Figs. 3.64c,d analysed in Example 34: G1
lr 5= and G2lr 6= . Eq.
(2.228) gives rl=11 as in the previous example. The number of joint parameters that lose their independence in the parallel mechanism F ← G1G2 given by Eq. (2.226) is r=73+11=15.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =18 and
the mobility of the mechanism F in Fig. 3.65a, given by Eq. (2.223) is M=1815=3. Three joint parameters have to be known to define the position and the orientation of each link of the mechanism F. These parameters are the displacements (qi, i=1,2,3) in the two linear and one rotating actuators mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=33=0), and Eq. (2.224) indicates that the parallel robot Isoglide3T2R1 with interconnected limbs in Fig. 3.65a has nine degrees of overconstraint (N=2415=9). Six of these overconstraints are in the two planar parallelogram loops.
The two versions of Isoglide4T3R1 in Figs. 3.64a and 3.65a have the same structural parameters. A CAD model of Isoglide3T2R1 with interconnected limbs is presented in Fig. 3.67.
214 3 Structural analysis
Example 36. Isoglide4T3R1 in Fig. 3.68 is a fullyisotropic parallel robot with Schönflies motions (Gogu 2004a, 2007a). The three limbs of this mechanism are interconnected in one complex kinematic chain F. The moving platform is denoted by 9, and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.68a,b). The mechanism has the following structural characteristics: m=16 links including the fixed base, p=19 joints, 4 independent closed loops (q=1916+1=4) and one complex limb (k=1). The connectivity of mobile platform 8 in the fullyisotropic Isoglide4T3R1 in Fig. 3.68a is
F9 / 1S = G1S = 4. Four independent velocities ( x y zv ,v ,v and zω ) exist be
tween the mobile platform 9 and the reference base in the mechanism F associated with this parallel robot of the Isoglide family.
To calculate the number of joint parameters that lose their independence in the four closed loops, we may consider F ← G1G2 as a parallel mechanism with two hybrid limbs G1 and G2 connecting the reference link 1 with the distal link 8 (Fig. 3.69a). As we have seen in chapter 2 and in the previous example, the number of joint parameters that lose their independence in the closed loops of a mechanism does not depend of the choice of the distal link. The hybrid limb G1 (Figs. 3.69b) has the same structural parameters as limb G1 in Figs. 3.64c and 3.66b analysed in Examples 34 and 35: (RG1)=( x y z zv ,v ,v ,ω ), SG1=4, and G1
lr 5= . The hybrid limb G2 (Fig. 3.69c) is obtained by concatenating a prismatic joint with two planar parallelogram loops. We simply observe by inspection that (RG2)=( x y z zv ,v ,v ,ω ) and SG2=4.
Fig. 3.68. Fullyisotropic parallel robot Isoglide4T3R1 with three interconnected limbs (a), associated graph (b)
3.4 Parallel mechanisms with complex limbs 215
Fig. 3.69. The parallel kinematic chain F ← G1G2 with distal link 8 associated with the fullyisotropic parallel robot Isoglide4T3R1 with three interconnected limbs (a), limb G1 (b) and limb G2 (c)
216 3 Structural analysis
Fig. 3.70. The parallel kinematic chain C ← A1A2A3 with reference link 7C and distal link 5C associated with the closed loops in limb G2 ← PC (a), limb A1 (b), limb A2 (c) and limb A3 (d)
To calculate the number of joint parameters that lose their independence in the two closed loops integrated in the hybrid limb G3 ← PC, we consider the kinematic chain formed by these two closed loops as a parallel kinematic chain C ← A1A2A3 with three simple limbs connecting the ternary links 5C and 7C (Fig. 3. 70a). The three simple limbs are: A1 (7C5C) of type RR (Fig. 3.70b), A2 (7C896C5C) of type RRRR (Fig. 3.70c) and A3 (7C2C3C4C5C) of type RRRR (Fig. 3.70d). The three limbs isolated from the parallel kinematic chain C ← A1A2A3 (Figs. 3.70b,c,d) have the following bases and dimensions for their operational velocities spaces: (RA1)=( zω ), (RA2)= ( x y zv ,v ,ω ), (RA3)=( x y zv ,v ,ω ) and SA1=1, SA2= SA3=3.
3.4 Parallel mechanisms with complex limbs 217
Fig. 3.71. CAD model of Isoglide4T3R1 with interconnected limbs
The connectivity of link 5C with respect to link 7C in the parallel kinematic chain C ← A1A2A3 is C
5C / 7CS = A1 A2 A3dim( R R R ) 1∩ ∩ = (Fig. 3.62a). Just one rotation about the zaxis is possible between the two links. The number of joint parameters that lose their independence in the parallel kinematic chain C ← A1A2A3 is rC=71=6. This value is given by Eq. (2. 191) by taking into account that in the parallel kinematic chain C ← A1A2A3 we have considered 7C as the reference link, and 5C as the distal link. We thus obtain G2
l Cr r 6= = . The number of joint parameters that lose their independence in the par
allel mechanism F ← G1G2 given by Eq. (2.226) is r=84+11=15. We have taken into account that the number of joint parameters that lose their independence in the two limbs of the parallel mechanism F ← G1G2, given by Eq. (2.228) is G1 G2
l l lr r r 11= + = . The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =19, and the mobility of the mechanism F in Fig. 3.68a, given by Eq. (2.223) is M=1915=4. Four joint parameters have to be known to define the position and the orientation of each link of the mechanism F. These parameters are the displacements (qi, i=1,2,3) of the three linear actuators mounted on the fixed base, and one rotating actuator mounted on link 2C in the proximity of the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=44=0) and Eq. (2.224) indicates that the parallel robot Isoglide4T3R1 with three interconnected limbs in Fig. 3.68a has nine degrees of overconstraint (N=2415=9). Six of these overconstraints are in the two planar parallelogram loops. Figure 3.71 presents the CAD model of this solution.
218 3 Structural analysis
Example 37. Isoglide4T3R1 in Fig. 3.72 represents the first solution of a fullyisotropic parallel robot with Schönflies motions presented in the literature (Gogu 2004a). The limbs of this mechanism are interconnected in one complex kinematic chain F. The moving platform is denoted by 10, and the fixed base by 1≡1A≡1B≡1C≡0 (Fig. 3.72a,b). The mechanism has the following structural characteristics: m=17 links including the fixed base, p=21 joints, 5 independent closed loops (q=2117+1=5) and one complex limb (k=1). The connectivity of the mobile platform 10 in the fullyisotropic Isoglide4T3R1 in Fig. 3.72a is F
10 / 1S = G1S = 4. Four independent velocities ( x y zv ,v ,v and zω ) exist between mobile platform 10 and the reference base in the mechanism F associated with this parallel robot of the Isoglide family.
To calculate the number of joint parameters that lose their independence in the five closed loops, we may consider F ← G1G2 as a parallel mechanism with two complex limbs G1 and G2 connecting reference link 1 with distal link 9 (Fig. 3.73a). As we have seen in chapter 2 and in the previous examples, the number of joint parameters that lose their independence in the closed loops of a mechanism does not depend of the choice of the distal link. The hybrid limb G1 (Figs. 3.73b) has the same structural parameters as the limb G1 in Figs. 3.64c, 3.66b and 3.69b analysed in Examples 3436: (RG1)=( x y z zv ,v ,v ,ω ), SG1=4, and G1
lr 5= . Three closed loops are combined in complex limb G2 (Fig. 3.73c). We simply observe by inspection that (RG2)=( x y z zv ,v ,v ,ω ) and SG2=4.
Fig. 3.72. The fullyisotropic parallel robot Isoglide4T3R1 with three interconnected limbs and the actuators mounted on the fixed base (a), associated graph (b)
3.4 Parallel mechanisms with complex limbs 219
Fig. 3.73. The parallel kinematic chain F ← G1G2 with distal link 8 associated with the fullyisotropic parallel robot Isoglide4T3R1 with three interconnected limbs and actuators on the fixed base (a), limb G1 (b) and limb G2 (c)
220 3 Structural analysis
Fig. 3.74. Parallel kinematic chain D ← A1A2A3E1 with reference link 8C and distal link 6C associated with the closed loops in limb G2 (a), limb A1 (b), limb A2 (c) and limb E1 (d)
3.4 Parallel mechanisms with complex limbs 221
To calculate the number of joint parameters that lose their independence in the three closed loops integrated in the complex limb G3, we consider the kinematic chain formed by these three closed loops as a parallel kinematic chain D ← A1A2E1 with two simple limbs (A1 and A2) and one complex limb E1 connecting the ternary links 6C and 8C (Fig. 3. 74a). The two simple limbs are: A1 (8C6C) of type RR (Fig. 3.74b) and A2 (8C9107C6C) of type RRRR (Fig. 3.74c). Link 6C is connected to link 8C by the complex limb E1 (Fig. 3.74d) including a closed loop B (12C3C4C1). The three limbs isolated from the parallel kinematic chain D ← A1A2E1 (Figs. 3.74b,c,d) have the following bases and dimensions for their operational velocities spaces: (RA1)=( zω ), (RA2)= ( x y zv ,v ,ω ), (RE1)=( x y zv ,v ,ω ) and SA1=1, SA2= SE1=3.
We observe that the operational velocity space of relative velocities between links 6C and 8C in the complex limb E1 is not influenced by closed loop B. The three independent velocities in (RE1) are in fact given by the three revolute pairs with parallel axes outside the closed loop B. This closed loop B ←
B B1 2A A− (Fig. 3.75a) is obtained by putting together the
distal links of two simple chains B1A and B
2A of type PR and RP respectively (Fig. 3.75b,c). Chains B
1A and B2A have (RA1B)=(RA1B)=( z zv ,ω ) and
SAIB=SA2B= 2. Two joint parameters lose their independence in the closed loop B (rB=2+22=2).
The connectivity of link 6C with respect to link 8C in the parallel kinematic chain D ← A1A2E1 is D
6C / 8CS = A1 A2 E1dim( R R R ) 1∩ ∩ = (Fig. 3.73a). Just one rotation about the zaxis is possible between the two links. The number of joint parameters that lose their independence in the parallel kinematic chain D ← A1A2E1 is rD=71+2=8. This value is given by Eq. (2. 191) by taking by taking into account that in the parallel kinematic chain D ← A1A2E1 we have considered 8C as the reference link, and 6C as the distal link. We thus obtain G2
l Dr r 8= = . The number of joint parameters that lose their independence in the par
allel mechanism F ← G1G2 given by Eq. (2.226) is r=84+13=17. We have taken into account that the number of joint parameters that lose their indpendence in the two limbs of parallel mechanism F ← G1G2, given by Eq. (2.228) is G1 G2
l l lr r r 13= + = . The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =21 and the mobility of the mechanism F in Fig. 3.68a, given by Eq. (2.223) is M=2117=4. Four joint parameters have to be known to define the position and the orientation of each link of the mechanism F. These parameters
222 3 Structural analysis
Fig. 3.75. Closed loop B ←
B B1 2A A− integrated in limb G2 (a) and its associated
simple open kinematic chains B1A (b) and B
2A (c)
are the displacements (qi, i=1,2,3,4) in the three linear and one rotating actuators mounted on the fixed base.
Equation (2.225) gives zero degrees of redundancy (T=44=0) and Eq. (2.224) indicates that the parallel robot Isoglide4T3R1 with three interconnected limbs and actuators on the fixed base in Fig. 3.72a has nine degrees of overconstraint (N=3017=13). Six of these overconstraints are in the two planar parallelogram loops, and four in the closed loop B integrated in limb G2.
Example 38. I4R in Fig. 3.76 is an innovative solution for a parallel robot with Schönflies motions for highspeed pickandplace operations, developed by LIRMM  Montpellier Laboratory of Computer Science, Robotics, and Microelectronics (Krut et al. 2004b, Company et al. 2006). The limbs of this mechanism are interconnected in one complex kinematic chain F ← G1. The robot uses a twopart moving platform 5 and 5’ connected by a prismatic joint (Fig. 3.77).
3.4 Parallel mechanisms with complex limbs 223
(a) (b)
(c) Fig. 3.76. The I4R parallel robot developed by Montpellier Laboratory of Computer Science, Robotics, and Microelectronics (a) and CAD models (b) and (c) (courtesy LIRMM)
224 3 Structural analysis
Fig. 3.77. I4R parallel robot: kinematic structure (a) and associated graph (b)
3.4 Parallel mechanisms with complex limbs 225
Fig. 3.78. Pulleycable system used in I4R parallel robot as technological solution for rolling without sliding joint (courtesy LIRMM)
The relative translation between the two parts of the moving platform is transformed in rotational motion of an endeffector 6 by a rollingwithout sliding joint denoted by Rw. (Fig. 3.77). This solution gives a high tilting angle of the endeffector connected to hafplatform 5 by a revolute joint and to halfplatform 5’ by the rollingwithoutsliding joint. A pulleycable system is used as a technological solution for the rolling without sliding joint (Fig. 3.78)
Each halfplatform is connected to two complex limbs RPa4Stype (Fig. 3.77). The complex mechanism associated with the I4R robot may be considered to be a parallel mechanism D ← A1A2E1 connecting a reference link 5 and a distal link 5’ (Fig. 3.79). This mechanism is obtained by kinematic inversion with respect to link 5. This inversion conserves the structural parameters of the mechanism. It is equivalent to positioning the “observer” on link 5 instead on the fixed base 1≡1A≡1B≡1C≡1D≡0. The parallel mechanism D ← A1A2E1 has one complex and two simple limbs. The two simple limbs are: A1 (565’) of type RRw (Fig. 3.80a) and A2 (55’) reduced to a prismatic joint (Fig. 3.80b). The complex limb E1 combines 6 closed loops (Fig. 3.80c). In Example 31, we have demonstrated that 38 joint parameters lose their independence in the closed loops combined in limbs G1 and G2 of the parallel robot H4. These closed loops are structurally identical with those of the complex limb E1 of I4R. Subsequently, rl= E1
lr =38 for the I4R parallel robot. The three limbs isolated from the parallel mechanism D ← A1A2E1 (Figs. 3.80) have the following bases and dimensions for their operational velocities spaces: (RA1)=( x zv ,ω ), (RA2)= ( xv ), (RE1)=( x y zv ,v ,v ) and SA1=2, SA2= 1, SE3=3.
The connectivity of link 5’ with respect to link 5 in the parallel mechanism D ← A1A2E1 (Fig. 3.79) is D
5'/ 5S = A1 A2 E1dim( R R R ) 1∩ ∩ = . This
226 3 Structural analysis
Fig. 3.79. Parallel mechanism D ← A1A2E1 with reference link 5 and distal link 5’ associated with parallel robot I4R by kinematic inversion with respect to link 5 (a) and its associated graph (b)
3.4 Parallel mechanisms with complex limbs 227
Fig. 3.80. Limbs isolated from parallel mechanism D ← A1A2E1 associated with parallel robot I4R by kinematic inversion with respect to link 5: simple limbs A1 (a) and A2 (b) and complex limb E1(c)
228 3 Structural analysis
result holds because just one translation about the xaxis is possible between the two halfplatforms 5 and 5’. The number of joint parameters that lose their independence in the parallel kinematic chain D ← A1A2E1 given by Eq. (2. 200) is r=rD=61+38=43.
The dimension of the joint vector space U is dim(U)= pii 1
f=∑ =55 and
the mobility of the mechanism associated with I4R parallel robot given by Eq. (2.223) is M=5543=12. Twelve joint parameters have to be known to define the position and the orientation of each link of the parallel mechanism I4R. Equation (2.225) gives eight degrees of redundancy (T=124=8). As for the H4 robot analysed in Example 31, each parallelogram loop Pa4Stype contains two internal mobilities introduced by the rotation of the links 3 and 4 of each limb about a rotation axis passing through the centre of the two spherical joints adjacent to links 3 and 4. These eight degrees of redundancy represent internal mobilities in the limbs that cannot be transmitted from the input to the output of the parallel mechanism. The remaining four degrees of mobility are external mobilities that are transmitted from the input to the output of the parallel mechanism. They represent the independent parameters that are necessary to define the position and the orientation of the endeffector 6. These parameters are the angular displacements of the four rotating actuators mounted on the fixed base.
Equation (2.224) indicates that the I4R parallel robot has five degrees of overconstraint (N=4843=5).
3.5 Other multiloop kinematic chains
The structural formulae demonstrated in chapter 2 for parallel robots may also be applied to various multiloop mechanisms that do not obey to the classical CGK formulae. The three following examples illustrate the application of the new formulae presented in chapter 2 to complex mechanisms that may be considered to be parallel mechanisms with simple limbs.
Example 39. The mechanism proposed by De Roberval (1670) may be considered as a parallel mechanism F ← G1G2G3 (Fig. 3.81) obtained by the concatenation of two planar parallelogram loops. The six revolute joints have parallel axes. The CGK formulae give zero degrees of mobility for this mechanism, which is clearly wrong.
3.5 Other multiloop kinematic chains 229
Fig. 3.81. The De Roberval mechanism (a), associated graph (b), simple open chains associated with limbs G1 (c), G2 (d) and G3 (e) isolated from the mechanism
The ternary links 1 and 3 may be considered as the fixed and moving platforms. They are connected by three simple limbs RRtype: G1(1A2A3A), G2(1B2B3B) and G3(1C2C3C). The mechanism has the following structural characteristics: m=5 links including the fixed base, p=6 revolute joints, 2 structurally independent closed loops (q=65+1=2) and three simple limbs (k=3). We may observe by inspection that the three simple limbs have the following bases for their operational spaces: (RG1)=( x yv ,v ) (RG2)=( x xv ,ω ) or (RG2)=( y xv ,ω ) and (RG3)=( x yv ,v ) – see Fig. 3.81ce. The dimensions of these operational vector spaces give the connectivity of the kinematic chains associated with each limb Gi of the parallel mechanism: SGi=2, i=1,2,3. We have considered the characteristic point H belonging to the moving platform, point O0 as the reference point, and the velocities of H are expressed in the coordinate system O0x0y0z0. In accordance with remark 2 and Eq. (2.197), the point H and the bases of RGi
(i=1,2,3) are selected such that the minimum value of F3 / 1S is obtained. By
this choice, the connectivity of the mobile platform 3 in the parallel mechanism F, given by Eq. (2.227), is SF= F
3 / 1S = G1 G2 G3dim( R R R )∩ ∩ =1. Only one relative independent velocity xv or yv exists between the mobile platform 3 and the reference link 1≡0. The dimension of the joint
230 3 Structural analysis
Fig. 3.82. Parallel mechanism equivalent to De Roberval mechanism by kinematic inversion with respect to link 1 (a), associated graph (b)
vector space U is dim(U)= pii 1
f=∑ =6. The number of joint parameters that
lose their independence in De Roberval’s mechanism, given by Eq.(2.226) is r=61=5. The mobility of the mechanism, given by Eq. (2.223), is M=65=1. Equation (2.224) gives N=7 and T=0. De Roberval mechanism in Fig. 3.81 is non redundant with seven degrees of overconstraint.
It is not necessary to treat this mechanism as a special mechanism with passive freedoms or mechanisms with over closed loops. No special equations are necessary to calculate the mobility of this mechanism; it obeys the general formulae proposed in Chapter 2 for calculating the degree of mobility of parallel mechanisms with simple limbs.
The same results are obtained for the De Roberval mechanism in Fig. 3.82. This may be transformed into the mechanism in Fig. 3.81 by kinematic inversion with respect to link 1. As we have seen in chapter 2 kinematic inversion does not alter the mobility.
Example 40. The mechanism F ← G1G2 in Fig. 3.83 is obtained by interconnecting two De Roberval mechanisms. The eight revolute joints have parallel axes. The CGK formula gives a negative mobility for this mechanism, which is wrong.
The ternary links 1 and 3 may be considered as the fixed and moving platforms. They are connected by one simple limb G1(1A2A3A) of type RR, and a complex limb G2 analysed in Example 38. The mechanism has the following structural characteristics: m=6 links including the fixed base, p=8 revolute joints, 3 structurally independent closed loops (q=86+1=3) and two limbs (k=2). We may simply observe by inspection that the two limbs have the following bases and dimensions of their operational spaces: (RG1)=( x yv ,v ), (RG2)=( xv ) or (RG2)=( yv ) and SG1=2, SG2=1– see Fig. 3.83c, d. We have considered the characteristic point H belonging to the moving platform, point O0 as the reference point, and the velocities of H are
3.5 Other multiloop kinematic chains 231
Fig. 3.83. Complex mechanism with one simple and one complex limb F ← G1G2 (a), associated graph (b), kinematic chains associated with limbs G1 (c) and G2 (d) isolated from the mechanism
expressed in the coordinate system O0x0y0z0. In accordance with remark 2 and Eq. (2.197), point H and the bases of RGi (i=1,2,3) are selected so that the minimum value of F
3 / 1S is obtained. By this choice, the connectivity of the mobile platform 3 in the parallel mechanism F, given by Eq. (2.227), is SF= F
3 / 1S = G1 G2dim( R R )∩ =1. Only one relative independent velocity xv or yv exists between the mobile platform 3 and the reference link 1≡0. As we have seen in Example 38, five joint parameters lose their independence in the two closed loops in limb G2 (rl=5). The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =8. The number of joint parameters that lose their independence in the mechanism in Fig. 3.83 given by Eq.(2.226) is r=31+5=7. The mobility of the mechanism given by Eq. (2.223), is M=87=1. Equation (2.224) gives N=11 and T=0. The mechanism in Fig. 3.83 is non redundant with 11 degrees of overconstraint.
Example 41. The complex mechanism presented in Fig. 3.84a can also be considered a parallel mechanism F ← G1G2G3 with three simple limbs,
232 3 Structural analysis
in which ternary links 1 and 4 are the reference and the “distal” links. They are connected by three simple limbs (Fig. 3.84a,b): G1(1A2A3A4A) of type RRR , G2(1B2B) reduced to a single revolute joint and G3(1C2C3C4C) of type CCR. All revolute (R) and cylindrical (C) joints have parallel axes.
The mechanism has the following structural characteristics: m=6 links including the fixed base, p=7 joints, 2 structurally independent closed loops (q=76+1=2) and three simple limbs (k=3). The CGK formulae give different results (M=1 or M=2) depending on the choice of the two structurally independent loops. If the closed loops (12A3A43C2C1) and (143C2C1) are selected, these formulae give the erroneous result M=1; the other two combination of two closed loops give M=2. Theoretically, any two loops of the graph presented in Fig. 3.84d can be considered as structurally independent. In fact, the two loops can be independent from a structural point of view, in the sense of graph theory, but they are not always independent from a kinematic point of view, as we have shown in chapter 2. This ambiguity can be avoided by using the new formulae.
By inspection, we can see that the operational spaces RGi (i=1,2,3) of the simple open kinematic chains associated with the three limbs isolated from the mechanism (Fig. 3.84ce) have the following bases and dimensions: (RG1)=( x y zv ,v ,ω ), (RG2)=( zω ), (RG3)=( x y z zv ,v ,v ,ω ) and SG1=3, SG2=1,
Fig. 3.84. The complex mechanism F ← G1G2G3 with three simple limbs (a), associated graph (b), simple open chains associated with limbs G1 (c), G2 (d) and G3 (e) isolated from the mechanism
3.5 Other multiloop kinematic chains 233
SG3=4. We have considered the characteristic point H situated on link 4 on the axis of the revolute joint connecting this link to the fixed base.
The connectivity of link 4 in the parallel mechanism F ← G1G2G3 is F4 / 1 A1 A2 A3S dim( R R R ) 1= ∩ ∩ = . Only one relative independent velocity
( zω ) exists between links 4 and 1. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =9. The number of joint parameters that lose their independence in this complex mechanism is r=81=7 and the correct value of the mobility is given by Eq. (2.223) is M=97=2. The mechanism has one degree of redundancy (T=21=1) and five degrees of overconstraint (N=127=5). We can see that this mechanism also obeys Eq. (2.223), and the mobility is determined without any ambiguity.
Example 42. The complex mechanism presented in Fig. 3.85a was proposed by Baker (1980b) and analysed by Davies (1981) and Fayet (1995a). It may be considered a parallel mechanism F ← G1G2G3 with three simple limbs, in which ternary links 1 and 4 are the reference and the distal links respectively. They are connected by three simple limbs (Fig. 3.85a): G1(124) of type CS, G2(14) reduced to a planar joint Pn and G3(134) of
Fig. 3.85. Complex mechanism F ← G1G2G3 with three simple proposed by Baker (1980b) and analysed by Davies (1981) and Fayet (1995a)  (a), simple open chains associated with limbs G1 (b), G2 (c) and G3 (d) isolated from the mechanism
234 3 Structural analysis
type CS. The mechanism has the following structural characteristics: m=4 links including the reference link, p=5 joints, 2 structurally independent closed loops (q=54+1=2) and three simple limbs (k=3).
By inspection, we can see that the operational spaces RGi (i=1,2,3) of the simple open kinematic chains associated with the three limbs isolated from the mechanism (Fig. 3.85bd) have the following bases and dimensions: (RG1)=( y z x y zv ,v , , ,ω ω ω ), (RG2)=( x y zv ,v ,ω ), (RG3)=( y x y zv , , ,ω ω ω ) and SG1=5, SG2=3, SG3=4. In accordance with remark 2 and Eq. (2.197), point H and the bases of RGi (i=1,2,3) have been selected such that the minimum value of F
4 / 1S to be obtained. By this choice, the connectivity of the mobile platform 3 in the parallel mechanism F, given by Eq. (2.227), is SF= F
4 / 1S = G1 G2 G3dim( R R R )∩ ∩ =2. Two independent velocity yv and zω exist between the mobile platform 4 and the reference link 1. The dimension of the joint vector space U is dim(U)= p
ii 1f
=∑ =13. The number of joint parameters that lose their independence in the
mechanism in Fig. 3.85 given by Eq.(2.226) is r=122=10. The mobility of the mechanism given by Eq. (2.223), is M=1310=3. Equation (2.224) gives N=2 and T=1. The mechanism in Fig. 3.85 has two degrees of overconstraint and one degree of redundancy. One internal mobility exists in limb G3 – the rotation of the link 3 that cannot be transmitted to link 4. The remaining two degrees of mobility are external mobilities that have to be actuated to obtain the translation and the rotation of distal link 4. Translation about the yaxis may be actuated in one of the three joints connected to the fixed base. Rotation about the zaxis may be actuated by a circular translation in the planar joint.
4 Kinematic analysis
Kinematics of parallel robots is very important for structural synthesis. We have seen in chapter 2 that the structural parameters of parallel robots are mainly kinematically defined. The duality between velocity and force allows us to give alternative definitions of structural parameters based on statics. The author has preferred the kinematic definition, considering that the kinematic nature of the structural parameters facilitates the understanding of their meaning. Apparently, we are more skilled in imagining the motion than imaging the static or dynamic forces.
In this chapter, we will introduce the main models and performance indices used in parallel robots. In general, to study robot motions, geometric and kinematic models are used. We will present the general approach used to set up these models and we emphasize the Jacobian matrix which is the main issue in defining robot kinematics, singularities and performance indices. We saw in chapter 2 that the Jacobian is the matrix of the linear transformation between the velocity joint space and the operational space. This matrix contains very important information concerning robot behavior from a kinematic and static (kinetostatic) point of view.
The Jacobian will be used to define and to characterize motion decoupling in parallel robots. In modern design theories, decoupling the functional requirements has become a basic design principle (Suh 1990). In robotics, this principle leads to significant advantages in design, modeling, motion planning and control (Vukobratovic and Kircanski 1983; Konstantinov et al. 1987; Patarinski et al. 1991; Patarinski and Uichiyama 1993; YoucefToumi 1992; Galabov et al. 1999).
In many robotic applications the possibility of decoupling endeffector motion becomes a cornerstone criterion in the choice of the robot mechanism. The problem is that in the early stage of parallel robot design the Jacobian matrix is not yet known. For structural synthesis of parallel robots we use an alternative method to define motion decoupling and singularities, which does not need prior knowledge of the Jacobian matrix. This method is based on the main structural parameters introduced in chapter 2 via the theory of linear transformation.
236 4 Kinematic analysis
4.1. Decoupling in axiomatic design
Axiomatic design is a new design theory developed by Suh (1990) to establish a scientific foundation for the design field, and provide a fundamental basis for the creation of products, processes, systems, software and organizations. This is a significant departure from the conventional design processes largely dominated by empiricism and intuition.
Axiomatic design considers that the design objective is always stated in the functional domain, whereas the physical solution is always generated in the physical domain. These two domains are in permanent interconnection at every hierarchical level of the design process. To satisfy the design objectives defined in terms of functional requirements (FRs), a physical embodiment defined in terms of design parameters (DPs) must be created. In this way, the design is formally defined in axiomatic design as the creation of synthesized solutions in the form of products, processes or systems that satisfy the perceived needs through the mapping between the functional requirements in the functional domain and the design parameters of the physical domain, through the proper selection of DPs that satisfy FRs. This mapping process in not unique, and the actual outcome depends on the designer’s individual creative process. Therefore, there can be a large number of plausible design solutions and mapping techniques. The design axioms, theorems and corollaries formulated in axiomatic design provide the principles that the mapping techniques must satisfy to produce a good design, and offer a basis for comparing and selecting design solutions. Axiomatic design is based on two axioms that govern good design: the independence axiom and the information axiom. The independence axiom states that “in an acceptable design, the DPs and the FRs are related in such a way that a specific DP can be adjusted to satisfy its corresponding FR without affecting other functional requirements”. The information axiom states that “the best design is a functionally uncoupled design that has minimum information content” – see page 48 of “Principles of Design” by Suh (1990). We can see that decoupling is the keyword of both axioms.
The independence axiom is mathematically represented by the design equation
{FR}=[D]{DP} (4.1)
where {FR} is the set of functional requirements, {DP} is the set of design parameters and [D] is a m×n matrix called the design matrix. The elements of this matrix are denoted by aij (i=1,2,…n and j=1,2,…,m). The lefthand side of the design equation represents “what we want in terms of design goals” and the righthand side represents “how we hope to satisfy
4.1. Decoupling in axiomatic design 237
these goals in terms of a design solution”. We mention that in (Suh 1990), {FR} and {DP} are called “vectors”. In this work we call them “sets” because they may not always satisfy the mathematical definition of a vector.
A design that can be represented by Eq. (4.1) with [D] being a diagonal matrix, i.e. aij=0 unless i=j, satisfies the first axiom since the independence of FRs is assured when each DP is changed. That is, FRi can be satisfied independently by simply changing the corresponding DPi without affecting any other FR. Such a design is called an uncoupled design. If the design matrix is triangular the design can be decoupled and is called a decoupled or quasicoupled design. A truly coupled design is one in which the design matrix is neither diagonal nor triangular.
Two general quantitative measures are proposed in (Suh 1990) for functional independence: reangularity R and semangularity S . They are the following functions of the design matrix elements aij:
R
12 2n
ki kjk 1
n ni 1,n 1 2 2j 1 i ,n ki kj
k 1 k 1
a a1
a a
=
= −= +
= =
= −
∑∏
∑ ∑
(4.2)
and
S n
jj1
j 1 n 22kj
k 1
a
a=
=
=
∏∑
.
(4.3)
Reangularity measures the degree of orthogonality between the DPs; . the maximum value of the reangularity occurs when the DPs are orthogonal. As the degree of coupling increases, so the value of R decreases. The sums in the denominator of Eq. (4.2) are the squares of the magnitudes of the columns in the design matrix. These terms are used to normalize the columns of the matrix. When the columns are already normalized, the sums are equal to one. Normalization is necessary in order to make the measure insensitive to changes in the scale of any of the DPs. The sum in the numerator is the dot product of the ith and jth columns of the design matrix. This sum, when properly normalized by the denominator, can be thought of as the cosine of the angle between the vectors in the ith and jth columns of the design matrix. Since this cosine term is squared and sub
238 4 Kinematic analysis
tracted from unity, the argument of the product operator is the square of the sine of the same angle.
Semangularity characterizes the alignment of DPs with their corresponding FRs. The denominator of Eq. (4.3) is a normalizing factor that is equal to unity if the columns of the design matrix are normalized. When the design matrix is normalized, semangularity is simply the product of the absolute values of the diagonal elements of the design matrix. If the design matrix is properly normalized, then the diagonals are unity when all offdiagonal elements are zero. When the semangularity is unity, the DPs are aligned with the FRs. An uncoupled design has R =1 and S =1.
By analogy with the three types of design, in the next section we will define parallel robots with coupled, decoupled and uncoupled motions by considering the Jacobian as the design matrix.
4.2. Geometric modeling
In general in robot analysis, geometric modeling is set up in terms of finite motions; it is also called geometric analysis or pose analysis.
Definition 4.1 (Geometric problem). The geometric problem in robot analysis involves setting up the implicit relation between the finite motions of the actuated joints and the moving platform.
Usually, the geometric model is expressed in terms of linear and/or angular finite displacements of the actuated joints; moving platform pose is defined by its spatial position and orientation with respect to the reference frame. This model can be direct or inverse. In the direct geometric model the inputs are the finite displacement of the actuated joints, and the output is the pose of the moving platform. In the inverse geometric model the input is the pose of the moving platform and the outputs are the finite displacements of the actuated joints.
Let us consider the general case of a parallel robot F ← G1G2…Gk in which the moving platform n≡nGi is connected to the reference link 1≡1Gi by k limbs Gi (1Gi2Gi…nGi), i=1,2,…k. The robot has mobility M, and the connectivity of the moving platform is SF. The kinematic chains associated with limbs Gi can be simple or complex, and each limb Gi could integrate gi linear and/or rotation actuators. We have seen in chapter 2 that the total number of actuators is equal to the mobility
k
ii 1
g M=
=∑ . (4.4)
4.2. Geometric modeling 239
The geometric constraints imposed by the limbs on the motion of the moving platform give, for the ith limb, the following set of gi general relations
( )ij ij 1 2 6f q ,w ,w ,...,w 0= , i=1,2,…,k ; j=1,2,…,gi. (4.5)
where qij (i=1,2,…,k ; j=1,2,…,gi) represent the linear or angular displacements in the actuated joints and w1, w2, …, w6 represent the parameters used to describe the spatial position and orientation of the moving platform; these are called the general coordinates of the moving platform.
We note that various parameters can be used to represent the spatial pose of the moving platform: quaternions, biquaternions, homogeneous operators, rotation matrices, complex matrices, etc. More details on these parameters are presented in the book (Gogu and Coiffet 1996).
In general, relations (4.5) can be obtained by expressing the vector loop equation of each limb in the fixed coordinate frame, and eliminating the variables of the passive joints. Setting up analytical closedform relations (4.5) is possible when an analytical inverse geometric model is known for each limb. In the general case, setting up analytical closedform relations (4.5) may not always be possible for all limb architectures. However, in parallel robots, chains Gi are usually very simple, and such relations can be easily obtained.
A very common way to set up closedform relations (4.5) is to start from the equalities
( ) ( ) ( ) ( )2 2 2Ai Bi Ai Bi Ai Bi ij ij 1 2 6x x y y z z h q ,w ,w ,...,w− + − + − = (4.6)
written for each limb Gi (i=1,2,…,M). The distance constraints can be solved algebraically or geometrically
(Thomas et al. 2004). The lefthand of Eq. (4.6) represents the square of the distance AiBi be
tween the extreme points of each limb (Fig. 4.1). Points Ai are situated on the fixed base and have coordinates xAi, yAi and zAi. Points Bi are situated on the moving platform and have coordinates xBi, yBi and zBi. The coordinates of point Ai are the components of position vector O0Ai expressed in the fixed reference frame 0 0 0 0O x y z . The coordinates of point Bi are the components of vector O0Bi also expressed in the fixed reference frame by
O0Bi=O0On+R0nOnBi (4.7)
where OnBi is the position vector of point Bi in the mobile frame Onxnynzn attached to the moving platform, O0On is the position vector of point On in the fixed reference frame 0 0 0 0O x y z , and R0n is the 3× 3 rotation matrix
240 4 Kinematic analysis
giving the relative orientation of the frame n n n nO x y z with respect to
0 0 0 0O x y z . Vectors O0Ai and OnBi are constant and known for a given geometry of the fixed and moving platforms. In the general case, vector O0On depends on three generalized coordinates, these define the position of the characteristic point of a moving platform. The rotation matrix R0n depends on three generalized coordinates used to define the orientation of the moving platform. In this way, the coordinates of point Bi in the fixed reference frame depend on six general coordinates.
The righthand of Eq. (4.6) is a function depending on at least qij. Usually, in the parallel robots with telescopic limbs this function depends only on qij, that is hij=hij(qij). For example in Goughtype parallel robots gi=1 (i=1,2,…,6) and hij= 2
i1q . In other cases, this function could also depend on up to six general coordinates.
Eqs. (4.5) could be used to solve the direct and inverse geometric problem of parallel manipulators.
Definition 4.2 (Direct geometric problem). The direct geometric problem involves the calculation of the general coordinates w1, w2, …, w6 when the actuated joint parameters qij (i=1,2,…,k ; j=1,2,…,gi) are known in Eq. (4.5).
Definition 4.3 (Inverse geometric problem). The inverse geometric problem involves the calculation of the actuated joint parameters qij (i=1,2,…,k ; j=1,2,…,gi) when the general coordinates w1, w2, …, w6 of the moving platform are known in Eq. (4.5).
Fig. 4.1. General case of a parallel robot with k limbs
241
We note that contrary to the situation in serial robots, the direct geometric problem is generally more difficult than the inverse one. The inverse geometric problem can be easily solved analytically by using Eqs. (4.6). The direct geometric problem is usually solved numerically. Various methods can be used: elimination, homotopy, Gröbner basis, interval analysis or other adhoc methods (Merlet 2005). We will see in the examples presented at the end of this chapter that parallel robots with decoupled motions can have analytical solutions for both inverse and direct problems obtained in a very straightforward manner. This is an important advantage both for kinematic and dynamic modelling and path planning and control.
4.3 Kinematic modeling
In general in robot analysis, kinematic modeling is set up in terms of infinitesimal motions; it is also called differential motion analysis.
Definition 4.4 (Kinematic problem). The kinematic problem in robot analysis involves setting up the implicit relations between the infinitesimal motions of the actuated joints and the moving platform.
Usually, the kinematic models are expressed in term of robot velocity and eventually robot acceleration. The kinematic model can also be direct and inverse. In the direct kinematic model the input is the velocity of the actuated joints and the output the velocity of the moving platform. In the inverse kinematic model the input is the velocity of the moving platform and the outputs are the velocities of the actuated joints. In robot kinematics, the linear mapping between the velocity of the actuated joints and moving platform is set up by using the Jacobian matrix. Various methods can be used to calculate the Jacobian of a parallel robot: velocity vectorloop equations (Mohamed et al. 1983, Khalil and Dombre 2002), theory of reciprocal screws (Mohamed and Duffy 1985), motor algebra (Sugimoto 1987), Grassmann geometry (Merlet 1989), Plücker line coordinates (Hunt et al. 1991), etc. Various types of Jacobian matrices are known: conventional Jacobian (Tsai 1999), screwbased Jacobian (Tsai 1998), and overall Jacobian (Joshi and Tsai 2002). The screwbased Jacobian is developed by using the concept of reciprocal screws introduced by Ball (1900). The screwbased Jacobian becomes the conventional Jacobian after interchanging the first three columns with the last three columns in the parallel Jacobian. The overall Jacobian or full Jacobian (Merlet 2005) was proposed by Joshi and Tsai (2002) to provide information about both architecture and constraint singularities of parallel robots with less than six degrees of freedom. In this section, we introduce the design and conventional Jacobian and we emphasize the difference between these two Jacobian matrices.
4.3. Kinematic modeling
242 4 Kinematic analysis
The design Jacobian is useful for defining and understanding: (i) the mechanical nature of motion decoupling in parallel robots, (ii) the nature of conventional Jacobian matrix, and (iii) the formulation nature of certain singularities in parallel manipulators.
4.3.1 Direct and inverse kinematics matrices used in Jacobian calculation
By differentiating Eq. (4.5) with respect to time, we obtain
ij ij ij ijij 1 2 6
ij 1 2 6
f f f fq w w ... w 0
q w w w∂ ∂ ∂ ∂
+ + + + =∂ ∂ ∂ ∂
, (4.8)
or
ij ij ij ijij 1 2 6
ij 1 2 6
f f f fq w w ... w
q w w w∂ ∂ ∂ ∂
= − − − −∂ ∂ ∂ ∂
,
i=1,2,…,k; j=1,2,…,gi.
(4.9)
In matrix form, Eq. (4.9) can be written as follows
Js q =Jp w (4.10)
where q =1 2 k
T
11 1g 21 2 g k1 kgq ... q q ... q ... q ... q is the veloc
ity vector of actuated joints, w = [ ]T1 2 6w w ... w is the velocity vector of the moving platform,
k
k
11
11
12
12s
kg
kg
f 0 0q
f0 0qJ
f0 0
q
∂ ∂ ∂ ∂=
∂ ∂
,
(4.11)
is the direct kinematic matrix and
4.3. Kinematic modeling 243
k k k
11 11 11
1 2 6
12 12 12
1 2 6p
kg kg kg
1 2 6
f f fw w wf f fw w wJ
f f fw w w
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= − ∂ ∂ ∂ ∂ ∂ ∂
(4.12)
is the parallel kinematic matrix. In the literature, Js and Jp are also named serial and parallel Jacobian matrices (Wanger and Chablat 2000) by analogy with the corresponding matrix of Eq. (4.8) written in matrix form. The latter is called Jacobian, although this is not correct from a strictly mathematical point of view. Matrices Js and Jp were used by Gosselin and Angeles (1990) for singularity analysis and were also called direct kinematics and inverse kinematics Jacobians in (Tsai, 1999). We use the names of direct and inverse kinematics matrices coined by Chablat and Wenger (1998).
Eq. (4.10) represents the general kinematic model of parallel robots in which Js is a M×M matrix and Jp is a 6×M matrix. If the mobility M is greater than connectivity SF, the parallel robot is structurally redundant. If SF<6 just SF generalized coordinates are independent and (6SF) are dependent variables. If the number of limbs is equal to the mobility, and each limb is actuated (k=M), the robot is fullyparallel.
4.3.2 Design and conventional Jacobian matrices
Usually robot mobility satisfies 2≤M≤6 and M=SF. In this case, we can eliminate (6SF) rows of JS corresponding to dependent velocities of the moving platform and JS becomes a M×M square matrix. As M=SF, the M×M parallel Jacobian is nonsingular and Eq. (4.10) gives
w =J q (4.13)
where
J= 1pJ − sJ (4.14)
is the M×M Jacobian of the parallel robot. It represents the matrix of the linear transformation between the Mdimensional velocity vector q of the
244 4 Kinematic analysis
actuated joints and the Mdimensional operational velocity vector w of the moving platform. The element of this matrix situated in line i and column j indicate how an infinitesimal change in qj affects the infinitesimal change in wi.
Eq. (4.13) is the mathematical expression of the direct kinematic model and
q = 1J − w (4.15)
is the mathematical expression of the inverse kinematic model. We can see that
1J − = 1sJ − pJ (4.16)
Usually the inverse Jacobian matrix can be calculated easier than the Jacobian matrix by taking into account of the fact that sJ is easier to invert than pJ .
Definition 4.5 (Direct kinematic problem). The direct kinematic problem involves the calculation of the Jacobian matrix in Eq. (4.13). This matrix represents the linear transformation from the velocity joint space of actuated joints to the operational velocity space of the moving platform.
Definition 4.6 (Inverse kinematic problem). The inverse kinematic problem involves the calculation of the inverse Jacobian matrix in Eq. (4.15). This matrix represents the linear transformation from the operational velocity space of the moving platform to the velocity joint space of the actuated joints.
Design Jacobian
Introducing the Jacobian matrices by differentiating Eq. (4.5) we implicitly consider that the velocities of the moving platform result from direct differentiation of general coordinates w1, w2, …, w6 used to define the pose of the moving platform. To respect this hypothesis, the orientation of the moving platform must be defined by three consecutive rotation angles
,α β and δ used in the rotation matrix R0n. They could be rollpitchyaw angles ( nz′  ny′′  nx′′′ type), Euler angles ( nz′  nx′′  nz′′′ type or nz′  ny′′  nz′′′ type) or Bryant angles ( nx′  ny′′  nz′′′ type) or any other combination of three ordered rotations (Gogu, Coiffet, 1996). The axes mentioned in the brackets indicate the rotation axes na′  nb′′  nc′′′ of the three rotations ,α β and δ . They are the axes of the three intermediary coordinate systems used to de
245
fine the relative position and orientation of n n n nO x y z with respect to
0 0 0 0O x y z
0 0 0 0 n n n n n n n n n n n n n n n nO x y z O x y z O x y z O x y z O x y z′ ′ ′ ′ ′′ ′′ ′′ ′′ ′′′ ′′′ ′′′ ′′′→ → → → (4.17)
in the following order of transformations a b c
0n' 0 n n n n n n nT ( O O ) R ( ) R ( ) R ( )α β δ′ ′′ ′′′′ ′′ ′ ′′ ′ ′′− − − . (4.18)
The notation 0n' 0 nT ( O O ) indicates that the reference coordinate system
0 0 0 0O x y z is translated by the vector 0 nO O into the coordinate system
n n n nO x y z′ ′ ′ ′ . The notation an nR ( )α′′ ′′ indicates that n n n nO x y z′ ′ ′ ′ is superposed
on the coordinate system n n n nO x y z′′ ′′ ′′ ′′ by a rotation through an angle α , and so on. For rollpitchyaw and Euler angles the first rotation axis na′ is
nz′ ; for Bryant angles na′ is nx′ . The second rotation axis nb′′ is ny′′ except for the Euler angles nz′  nx′′  nz′′′ type when nb′′ is nx′′ . The third rotation axis
nc′′′ is nz′′′ except for rollpitchyaw angles, when nc′′′ is nx′′′ . The rotation matrix R0n giving the orientation of the moving frame
n n n nO x y z with respect to the fixed frame 0 0 0 0O x y z is obtained by multiplying the rotation matrices associated with the transformations defined in (4.18)
a b c0n n n n n n nR R ( )R ( )R ( )α β δ′ ′′ ′′′
′ ′′ ′ ′′ ′ ′′= . (4.19)
In the general case of a parallel robot with six degrees of freedoms, we denote the translational velocities of the characteristic point of the moving platform by iw (i=1,2,3), and the angular velocities of the moving platform by iw (i=4,5,6). The reference frame origin On is usually chosen as the characteristic point of the moving platform.
Usually in robot kinematics, iw (i=1,2,3) are associated with the translational velocities x yv ,v and zv of the characteristic point in the directions
parallel to the x0, y0 and z0axis, that is 0v = [ ]T1 2 3w ,w ,w = T
x y zv ,v ,v .
The other three velocities iw (i=4,5,6) in Eq. (4.13) are associated with
the angular velocity of the moving platform ω = [ ]T4 5 6w ,w ,w =T
, ,α β δ
=T
, ,α β δω ω ω . In this case, the Jacobian matrices in Eqs. (4.13)(4.16) are called design Jacobian matrices.
4.3. Kinematic modeling
246 4 Kinematic analysis
Definition 4.7 (Design Jacobian). The design Jacobian of a parallel robot is a matrix used in setting up the relation between joint and operational velocities when the angular velocities of the moving platform are defined as the time derivatives of the rotation angles of the moving platform.
The direct kinematic model of a parallel robot with SF=6 founded on the design Jacobian is
0vω
= J q (4.20)
with [ 0v ,ω ]T=[ x y zv ,v ,v , , ,α β δω ω ω ]T=[ x, y,z, , ,α β δ ]T and
q =[ 1 2 3 4 5 6q ,q ,q ,q ,q ,q ]T. The direct kinematic model of a lowerconnectivity parallel robot with
1<SF<6 founded on the design Jacobian is defined by analogy with Eq. (4.27) and taking into account the fact that the base of the operational velocity space of such a robot can be represented by any combination of two or more independent translational or angular velocities of the moving platform.
The design Jacobian J is always a square matrix SF× SF with 1<SF<≤ 6. We have derived its name from the design matrix used in axiomatic design. The design Jacobian is used mainly in the structural synthesis approach presented in this book.
Conventional Jacobian
The conventional Jacobian is used when the angular velocity of the moving platform is expressed by its components in the reference frame. We denote this velocity by 0ω =
T
x y z, ,ω ω ω . Definition 4.8 (Conventional Jacobian). The conventional Jacobian of
a parallel robot is a matrix used in setting up the relation between joint and operational velocities when the angular velocity of the moving platform is expressed in the reference frame.
In this case, the angular velocity components cannot be directly associated with the time derivatives of the rotation angles of the moving platform. For this reason, the conventional Jacobian does not represent a Jacobian matrix in the strict mathematical sense. This might be the meaning of “conventional” in the original name of this matrix.
Vector ω = [ ]T4 5 6w ,w ,w =T
, ,α β δ =T
, ,α β δω ω ω can be used to cal
culate the angular velocity vector of the moving platform with respect to the fixed reference frame, by using the following mapping
247
0ω = rJ ω , (4.21)
where rJ is the Jacobian of angular velocity mapping. Angular velocity 0ω and moving platform velocities [ 0v , 0ω ]T=
[ x y z x y zv ,v ,v , , ,ω ω ω ]T can also be calculated directly without calculating ω . Eqs. (4.10) and (4.13) become
sJ q = pJ 0
0
vω
, (4.22)
and
0
0
vω
= J q (4.23)
In this case, matrices Js and Jp and J represent conventional Jacobian matrices and are calculated without time derivatives of rotation angles ,α β and δ by formulating a velocity vectorloop equation for each limb. This equation is obtained by differentiating, with respect to time, the geometrical loopclosure equation of each limb (Fig. 4.1) written as
O0Ai+ AiBi =O0On+OnBi . (4.24)
When we differentiate the geometric loopclosure equation (4.24) we must take into account the fact that the time derivative of a vector is the sum of its “apparent” time derivative in the rotating frame and the vector cross product of the angular vector of the frame and the vector itself.
For example, for a parallel robot with telescopic limbs, the time derivative of Eq. (4.24) gives the following velocity vectorloop equation
iq ai + qi i iaω × = 0v + 0 ibω × (4.25)
where qi=AiBi and ai denote the variable length of ith limb, and a unit vector along AiBi, respectively, ib = OnBi, and iω denotes the angular velocity of the ith limb with respect to the fixed reference frame.
To eliminate iω , we dotmultiply both sides of Eq. (4.25) by ai
iq = i 0a v⋅ + ( )×i i 0b a ω . (4.26)
For Goughtype parallel robot with six telescopic limbs, Eq. (4.26) written six times (i=1,2,…6) yields six scalar equations which can be assembled in matrix form (4.22) or (4.23) with
4.3. Kinematic modeling
248 4 Kinematic analysis
pJ =
( )( )
( )
× × ×
TT1 1 1
TT2 2 2
TT6 6 6
a b a
a b a
a b a
(4.27)
sJ = 6 6I × and 0J = 1pJ − , where 6 6I × denotes the 6× 6 identity matrix.
Property 4.1. The design and conventional Jacobian matrices are identical for all parallel robots with up to three translational and one rotational degrees of freedom of the moving platform. Differences appear between the two matrices, just when the moving platform has two or three rotational degrees of freedom.
4.4 Types of workspaces and singularities
Workspace, singularity and performance indices are three connected subjects in parallel robot analysis. Various performance indices characterize robot behaviour and its workspace in close relation with its singular configurations.
4.4.1 Types of workspaces
An important parameter in robot design is the dextrous workspace, defined as all possible locations of an operating point for which all orientations are possible. In general, a major difficulty in the workspace and dexterity analysis of parallel robots is the strong coupling of positioning and orientation parameters (Dasgupta and Mruthyunjaya 2000; Merlet 2006b). To overcome this difficulty, various types of partial representations of the workspace of parallel robots are used: (i) translation workspace with constant orientation of the moving platform, (ii) orientation workspace with a fixed location of an operation point of the moving platform, (iii) reachable workspace with a range of defined orientations.
In addition, dexterity workspace analysis should preferably be done in association with singularities, accuracy and rigidity. Near singularities, the accuracy and the rigidity change drastically, especially in high dynamics conditions. The accuracy of parallel robots is also strongly dependent on the static and dynamic rigidities of the limbs. Dimensions, shape, tolerances, mechanical properties and technological solutions for links and
4.4 Types of workspaces and singularities 249
joints play a crucial role in robot accuracy and dexterity. In the design process, they represent the main parameters for robot dimensioning and optimization along with other factors, including reliability, safety, cost, ease of operation, and settling time. In the early stages of structural synthesis not all of these parameters can be integrated. Great efforts are made to integrate all these parameters into workspace analysis. More details on this topic may be found in the textbooks specialising in parallel robot analysis (Merlet 2006b; Tsai 1999).
4.4.2 Types of singularities
In a general mathematical sense, a singularity is a point at which a given mathematical model is not defined or fails to be wellbehaved in some particular way. In engineering, a singularity is a configuration of a system in which the subsequent behaviour cannot be predicted. In a singular configuration, the underlying equations of the mechanical model exhibit a mathematical singularity. The corresponding motions, forces, or other physical parameters modelled by these equations become undetermined.
Numerous investigations have been conducted on singular configurations of mechanisms, with recent emphasis on parallel robotic manipulators. The identification of singular configurations in parallel robots is of great importance. Near singular configuration, infinitesimal changes in input/output velocities or forces can produce huge variations of outputs/inputs. In singular configurations, the mechanism loses the ability to transmit resolutely motion and force, and becomes uncontrollable; the kinematic and static behaviour of the mechanism change dramatically; the mechanism gains extra degrees of freedom and loses its stiffness. Furthermore, the actuator forces may become very intense; this might lead to robot failure or permanent damage of the robot and surrounding equipment, including a breakdown of the mechanism. Therefore, much effort has been devoted to avoiding singularities by a proper design. When they do exist in the workspace, they must be circumvented by proper path planning (Dasgupta and Mruthyunjaya 1998a). Singularity avoidance and a singularityfree workspace represent important design objectives (Carricato and ParentiCastelli 2002; Carricato 2005; Gogu 2005g; Wolf and Shohan 2003; BenHorin and Shoham 2006a,b).
Various tools have been used in singularity analysis of parallel robots: screw theory (Hunt 1978; Kumar 1992; Huang et al. 1999; Joshi and Tsai 2002; Fang and Tsai 2003; Zhao 2005) Grassmann geometry or line geometry (Merlet 1989, 2006; Collins and Long 1995; Hao and McCarty, 1998, Notash 1998; Simaan and Shoham 2001; Gosselin and Angeles
250 4 Kinematic analysis
1990; Pottman et al 1999; EbertUphoff et al. 2000), Study’s parameterization of Euclidean displacements (Husty and Karger 2000, 2002; Karger and Husty 1996; Karger 1998, 2003) Clifford algebra (Colins and McCarty 1998), GrassmannCayley algebra (Ben Horin ans Shoham 2006a,b,c), Morse function theory and differential forms associated with the constraint functions (Liu at al. 2003), Rodrigues parameters (Di Gregorio 2001a), mixed product of vectors (Di Gregorio 2002), linear decomposition of the determinant (Li et al. 2006), differentialgeometric approaches (Park and Kim 1999; Ghosal and Ravani 1998; Kong and Gosselin 2001a; Takeda 2005) geometric approach based on the concept of common tangent (Basu and Ghosal 1997), etc. These tools have been applied to investigate the singular configurations based on the reciprocal screws, the linear transformation between joint and operational velocity spaces and Jacobian matrix calculation, the equilibrium equations governing the forces exerted on the output link by the environment and connecting chains, velocity vectorloop equation, simple geometric rules, characteristic tetrahedron associated with the wrench, the pressure angle in the mechanism, constraint equations, selfmotions of the moving platform, local mobilities in the connecting chains, etc.
Beginning with the pioneering work of Hunt (1978) and Merlet (1989), line geometry has been successfully applied by several researchers to the kinematics and statics of parallel manipulators with extendable limbs. Line geometry is used because the rows of the Jacobian matrix in a linearly actuated fullyparallel manipulator are the Plücker line coordinates of the axes of its extendable links (Hunt 1991). Hence, linear dependency of these lines determines the conditions for instability and singularity of a parallel manipulator.
Much effort was made to understand the nature of singularities in parallel robots and to categorize them.
Singularities of parallel robots fall into two basic classes (Kong Gosselin 2001a): the first one, called stationary, kinematic, twist, or inverse kinematic singularity, in which the moving platform loses one or more degrees of freedom; and, a second one, called uncertainty, static, wrench, or forward kinematic singularity, in which the moving platform may undergo infinitesimal or finite motion when all inputs are locked. These basic classes have been analysed in depth and more complex classifications have been proposed.
Gosselin and Angeles (1990) have suggested a classification of singularities in closed kinematic chains by considering the matrices Jp and Js introduced in the previous section. They have identified three types of singularities: (i) singularity of type I, when matrix Js is singular, (ii) singularity of type II, when matrix Jp is singular, and (iii) singularity of type III when
4.4 Types of workspaces and singularities 251
matrices Js and Jp are both singular. Tsai (1999) terms the three types of singularities: inverse, forward and combined singularities. In accordance with the name of Js and Jp, Wanger and Chablat (2000) use serial and parallel singularities to define the first two types. Serial singularities occur when the determinant of Js becomes zero at the boundary of the operational space. Parallel singularities occur when the determinant of Jp is zero. They are hard to pin down by simple geometric analysis. In a serial singularity, the velocity of the moving platform can be zero for a nonzero actuator velocity vector. In a parallel singularity, the velocity of the moving platform can be nonzero when the actuator velocity vector is zero.
Singularities of type III have been redefined many times in the literature. This singularitytype was considered in (Ma and Angeles 1991) as a subset of architecture singularities. In (Sefrioui and Gosselin 1995) it is considered that this type of singularity occurs when the nonlinear kinematic constraints degenerate. It is mentioned in (Daniali at al. 1995a,b), that this type of singularity arises merely from the way in which the kinematic relations are formulated, representing in fact a formulation singularity that is not necessarily architecturedependent, contrary to the earlier claim (Ma and Angeles 1991). Later it was observed that configurations in which both matrices JS and Jp are singular exists for general designs of most 3DOF planar parallel manipulators (Bonev et al. 2003). Generally, there is not uncontrollable finite motion in these configurations. These singularities do not occur just for particular designs and do not necessarily correspond to self motions (Bonev et al. 2003), although, type III singularity continue to be mentioned in the literature in its initial definition.
Zlatanov et al. (1994a,b, 1995a,b) have further refined the previous classification by introducing six singularity types; each singularity configuration belongs to at least two types. The six singularities are: (i) redundant input (RItype) which occurs when a nonzero input is possible with zero output, (ii) redundant output (ROtype) which occurs when a nonzero output is possible with zero input, (iii) impossible input (IItype) when a certain input is not feasible for any output, (iv) impossible output (IOtype) when certain output is not feasible for any input, (v) redundant passive motion (RPMtype) which occurs when a nonzero instantaneous motion is possible with both the input and output being equal to zero, and (vi) increased instantaneous mobility (IIMtype) which occurs when the transitory or instantaneous mobility is higher than the fullcycle mobility of the kinematic chain. This classification is based on a slightly modified definition of singular configurations by taking into consideration the forward and inverse relations between the input and the output of the mechanism, and by considering mobility and singularity as input/outputdependent parameters. This classification is applied to nonredundant mechanisms; the
252 4 Kinematic analysis
input defines the actuated joint velocities and the output defines the instantaneous velocities of the output link. We can see that type I and type II singularities are in fact RI and ROtype singularities. The definition of kinematic singularity adopted in (Merlet 2006a,b) corresponds to a IIMtype singularity and is called an uncertain configuration by Hunt (1978).
Later on, Park and his coworkers (Han et al. 2002) identified a new singularity in 3UPU parallel mechanisms that did not obeye the previous classifications. Zlatanov et al. (2001) called it a constraint singularity.
Many other classifications have been proposed to systematize the singular configurations of parallel robots: formulation singularities, configuration singularities and architecture singularities (Ma and Angeles 1991), rotational and translational singularities (Di Gregorio and ParentiCastelli, 2002), parameterization singularities with endeffector and actuator singularities as two special cases (Liu et al. 2003).
Ma and Angeles (1991) proposed three groups to make a distinction between the singularities which arise merely from the way in which the kinematic equations are formulated, the instantaneous singularities or the permanent singularities. A platform which is always in a singular position within the realm of specific displacement is called by Wohlhart (2000) an architecturally shaky platform with regard to these displacements. Complete lists of architecture singularities have been recently given by (Karger and Husty 1996), Karger (1998, 2003) and Husty and Karger (2000, 2002).
Sen and Mruthyunjaya (1998) have presented a centrobased characterization of singularities in planar closedloop manipulators. The singularities have been classified into five types: cusp, ordinary, fixed, force and intermediate. The first two are grouped as point singularities, and the last three as link singularities. Singularities of planar parallel robots have also been extensively studied in (Daniali et al. 1995a,b; Sefrioui and Gosselin 1995; Bonev et al. 2003).
Fang and Tsai (2002) divided the singularity cases into limb singularity, manipulator singularity and actuation singularity.
Ider (2005) studied drive singularities; these occur when the actuators lose control of the endeffector in certain directions. If certain conditions related to the consistency of the dynamic equations are satisfied by a proper design of the trajectory, the manipulator can pass through singular positions while the actuator forces and the system motion remain stable.
Three general solutions exist for coping with singular configurations inside the workspace: (i) developing special control strategies (Bhattacharya et al 1998; Dash et al. 2003; Dasgupata and Mruthyunjaya 1998a, 2000), (ii) using redundantly actuated robots (Wang and Gosselin 2004; Kurtz and Hayward 1992; Merlet 1997; Firmani and Podhorodeski 2004; Notash and Podhorodeski 1994; Kim et al. 2001) or (iii) designing robots with
4.5. Kinetostatic performance indices 253
freeofsingularity workspace (Chablat and Wenger 2003), (Calimbari and Tarantini 2003) Ottaviano and Cecarelli 2002; Gallant and Boudreau 2002; Carricato and ParentiCastelli 2002; Kong and Gosselin 2002a, Kim and Tsai 2002, Gogu 2004ad, 2005fi, 2006ae, 2007ac). The last two solutions will be integrated in the structural synthesis approach proposed in this book.
4.5. Kinetostatic performance indices
Various performance characteristics are used in robot analysis (Chedmail et al. 1998, Dombre 2001, Merlet 2006). These characteristics are commonly called performance indices, and are usually calculated for a given robot architecture by using the kinematic, static or dynamic models. They are defined as the robot’s capability to achieve some design criteria related to motion and force transmission, repeatability, singularity avoidance, accuracy, workspace, etc. In this section, we discuss the kinematic and static (kinetostatic) instantaneous performance indices of parallel robots based on Jacobian matrix. We emphasize their correlation with motion decoupling. These indices are related to motion and force decoupling, manipulability, transmission factor, minimal singular value, condition number, conditioning ellipsoid, minimum singular value, and isotropy. These indices can be defined on the basis of the conventional or the design Jacobian. We will integrate them in the structural synthesis approach via motion decoupling. In the early engineering design stage of structural synthesis, it is especially important that the designer understand the formal nature of these indices, their interconnections and their effect on robot behaviour.
Eqs. (4.20) and (4.23) can be generalized to express the direct kinematics problem
p= J q (4.28)
where p represents the vector of operational velocities, q the vector of joint velocities, and J could be the conventional or the design Jacobian.
The corresponding inverse kinematics problem can be written in the form
q = 1J − p. (4.29)
The Jacobian matrix can be used to describe various kinetostatic relationships related to direct and inverse problems between the following:
a) joint velocity vector and operational velocity vector, as in Eqs. (4.28) and (4.29);
254 4 Kinematic analysis
b) infinitesimal motion of joints ∆q and infinitesimal motion of the endeffector ∆w
∆w =J∆q (4.30)
or
∆q = 1J − ∆w (4.31)
c) the vector of the joint forces τ , and the vector µ of the forces developed on the endeffector (vector of operational forces)
µ = TJ − τ (4.32)
or
τ = TJ µ (4.33)
Eqs. (4.28), (4.30) and (4.32) are associated with the direct problems in which the input is defined in the joint space, and the output in the operational space. Eqs. (4.29), (4.31) and (4.33) are associated with the inverse problems in which the input is defined in the operational space, and the output in the joint space. We note Eq. (4.33) expressing the inverse static problem follows from the principle of power conservation in robot mechanical system:
⋅Tµ p = ⋅Tτ q (4.34)
and by taking into account Eq. (2.28). Eqs. (4.28), (4.29) and (4.30)(4.33) show that the Jacobian matrix con
tains important information concerning the kinetostatic characteristics of the robot.
The various kinetostatic indices used to qualify robot behaviour are defined in a Euclidean vector space in which vector norms play an essential role. We have seen that Eq. (4.28) represents a linear transformation between joint and operational velocity spaces. In defining this linear transformation there is no need to consider the two vector spaces to be Euclidean.
When joint velocity vector q or operational velocity vector p contain components with different physical units, the norms of these two vectors become physically inconsistent, however Eq. (4.28) remains physically consistent. Physical consistency refers to the physical validity of equations combining quantities of different units in a meaningful way. The simplest
4.5. Kinetostatic performance indices 255
example of physically inconsistency is the attempt to add quantities with different units.
The translation and angular velocities of the moving platform given by Eq. (4.28) will always have the same units (length/time and 1/time) as the velocities of the actuators. The linear transformation between actuated and operational vector spaces expressed by this equation does not alter velocity units. For this reason Eq. (4.28) is always physically consistent, although the norms of the velocity vectors of the moving platform and actuated joint could be physically inconsistent. The two vector norms are physically consistent just for four types of parallel robots: (iii) translational parallel robots with two or three degrees of mobility actuated by linear motors and (iiiiv) parallel wrists with two or three degrees of mobility actuated by rotation motors. All other parallel robots have inconsistent norms for the two velocity vectors, because these vectors include terms with dimension length/time and 1/time. The vector velocity of the moving platform is not Euclidian since its elements do not share the same physical unit. For a manipulator with mixed revolute and prismatic actuated joints, the jointrate vector is not Euclidian either. As a consequence, operational velocity space and joint velocity space are Euclidian vector spaces just for the four types of above mentioned parallel robots. The Jacobian matrix contains only dimensionless numbers when the norms of the two velocity vectors are physically consistent and a mix of dimensionless numbers and terms with length and/or time units when the norms of the two velocity vectors are physically inconsistent.
This warning is also applies to the forces and moments of the actuated joints, and the operational forces in Eq. (4.32) and (4.33).
A general study of incommensurate systems and physical consistency in robotics have been presented by Schwartz at al. (2002, 2003) and we will follow their lead with respect to physical consistency. An incommensurate system involves physical quantities with different units that are nonetheless described and controlled by physically consistent equations.
The kinetostatic performance indices are based on the norm of velocity vectors, the determinant and eigenvalues of the Jacobian matrix J, or other matrices such as TJ , 1J − , TJ − , TJJ , TJ J , T 1( JJ )− , T 1( J J )− . We have denoted T 1( J )− = 1 T( J )− by TJ − . The mathematical equations used to calculate these parameters often obscure the interaction between units, and lead to results that may be misinterpreted, erroneous, or simply arbitrary. We must always check the physical consistency of these mathematical entities. The general definitions and formulae presented in this section for kinetostatic indices are introduced under the hypothesis of respecting physical consistency.
256 4 Kinematic analysis
In the structural synthesis approach, we will integrate the notion of motion decoupling, and in this section we will emphasize its connection with the kinetostatic performance indices.
4.5.1 Crosscoupling indices
Crosscoupling indices represent measures of decoupling between the column vectors of the matrices used in formalizing the direct and inverse problems in robot kinematics and statics.
These indices take values between zero and one. A unit value of one of these indices indicates that the two column vectors are orthogonal. When all crosscoupling indices have maximum values, the robot kinematics/statics are completely decoupled.
We have seen that Eqs. (4.28)(4.33) represent linear transformations between vectors of the joint space and operational space. In the general case, these equations can be written in the form
y=Ax. (4.35)
Vector x could be q or p and vector y could also be q or p in any combination with the condition that x≠ y. Matrix A could be the Jacobian J, its transpose JT or an inverse matrix 1J − or TJ − .
For n× n square matrix A and a unit vector x, Eq. (4.35) becomes
y=n
ii i=∑A
(4.36)
where Ai is the vector associated with of the ith column of matrix A. The angular relationship ijθ between two columns Ai and Aj of matrix A can be obtained by taking the dot product
cos ijθ = i j
i j
⋅A AA A
. (4.37)
If we denote by aij the element of matrix A situated in the ith row and jth column, Eq. (4.37) becomes
4.5. Kinetostatic performance indices 257
cos ijθ
n
ki kjk 1
1n n 2
2 2ki kj
k 1 k 1
a a
a a
=
= =
=
∑
∑ ∑.
(4.38)
Definition 4.9. (Local crosscoupling index). The local crosscoupling index ijκ is a measure of decoupling between two column vectors Ai and Aj of the matrix A and is given by the sine of intersection angle ijθ between Ai and Aj
ijκ =sin ijθ . (4.39)
By taking into account Eqs. (4.37) and (4.38), the crosscoupling index can be expressed by
ijκ =
12 2
i j
i j
1 ⋅ −
A AA A
(4.40)
or
ijκ
12 2n
ki kjk 1
n n2 2ki kj
k 1 k 1
a a1
a a
=
= =
= −
∑
∑ ∑.
(4.41)
Since the angle of interaction between column vectors is in the range of (0,π ) we can see that 0≤ ijκ ≤ 1. For any nonsingular matrix A, all n columns of A are linearly independent. When two column vectors iA and jA are linearly dependent, i.e. iA =λ jA , matrix A is singular and ijκ =0. In this case we say that matrix A is rank deficient; the two column vectors are aligned, that is ijθ =0. When ijκ =1, the two column vectors iA and jA of matrix A are mutually orthogonal and the robot joint motions are completely decoupled. The name of this index is derived from the fact that its maximum value corresponds to a decoupled motion.
A more general crosscoupling index can be defined by extending the partial index to the entire matrix A.
258 4 Kinematic analysis
Definition 4.10. (Crosscoupling index). The crosscoupling index is a measure of decoupling between all the column vectors of matrix A, and is given by the product of the local crosscoupling indices
iji 1,n 1j 1 i ,n
κ κ= −= +
= ∏ (4.42)
or 1
2 2n
ki kjk 1
n ni 1,n 1 2 2j 1 i ,n ki kj
k 1 k 1
a a1
a aκ =
= −= +
= =
= −
∑∏
∑ ∑.
(4.43)
We can see that the crosscoupling index defined by Eq. (4.43) is equivalent to the reangularity index in axiomatic design  see Eq. (4.2).
We derive various crosscoupling indices from the two indices defined above by associating the matrix A with the transformation matrices of the kinematics and statics problems defined in Eqs. (4.28), (4.29), (4.32), (4.33). These Jacobian matrices are n× n square matrices with n=SF=M.
Kinematics crosscoupling indices
Crosscoupling indices in direct and inverse kinematics problems are derived from the local and general indices defined above by associating the matrix A with the Jacobian J and its inverse 1J − . We denote two column vectors of the Jacobian J by Ji and Jj, and two column vectors of the inverse Jacobian 1J − by 1
i−J and 1
j−J .
Definition 4.11. (Local crosscoupling index of the direct kinematics problem). The local crosscoupling index of the direct kinematic problem is a measure of decoupling between two column vectors Ji and Jj of the Jacobian matrix J given by the sine of the intersection angle between Ji and Jj in an instantaneous pose of the moving platform.
This index is given by Eqs. (4.40) or (4.41) in which A=J and Ai=Ji (i=1,2,…,n).
This instantaneous index is named the local kinematic crosscoupling index in (Lee et al. 1993) and is used to measure the degree of linear dependency between column vectors of the conventional Jacobian matrix.
For any nonsingular pose of the moving platform, all n=SF columns of the Jacobian matrix are linearly independent. These column vectors be
4.5. Kinetostatic performance indices 259
come linearly dependent only in singular poses as we have mentioned above.
Definition 4.12. (Local crosscoupling index of the inverse kinematics problem). The local crosscoupling index of the inverse kinematic problem is a measure of decoupling between two column vectors 1
i−J and
1j−J of inverse Jacobian matrix 1J − , and is given by the sine of intersec
tion angle between 1i−J and 1
j−J in an instantaneous pose of the moving
platform. This index is given by Eqs. (4.40) or (4.41) in which A= 1J − and
iA = 1i−J (i=1,2,…,n).
Definition 4.13. (Crosscoupling index of the direct kinematics problem). The crosscoupling index of the direct kinematic problem is a measure of decoupling between the column vectors of Jacobian matrix J, and is given by the product of the corresponding local crosscoupling indices.
This index is given by Eqs. (4.42) or (4.43) in which A=J and Ai=Ji (i=1,2,…,n).
Definition 4.14. (Crosscoupling index of the inverse kinematics problem). The crosscoupling index of the inverse kinematic problem is a measure of decoupling between the column vectors of inverse Jacobian matrix J1 and is given by the product of the corresponding local crosscoupling indices.
It is given by Eqs. (4.42) or (4.43) in which A= 1J − and iA = 1i−J
(i=1,2,…,n).
Statics crosscoupling indices
The crosscoupling indices in direct and inverse statics problems are derived from the local and general indices defined above by associating matrix A with Jacobian TJ and its inverse TJ − . We denote two column vectors of the transpose Jacobian TJ by T
iJ and TjJ , and two column vectors
of the inverse transpose Jacobian TJ − by Ti−J and T
j−J .
Definition 4.15A. (Local crosscoupling index of the inverse statics problem). The local crosscoupling index of the inverse statics problem is a measure of decoupling between two column vectors T
iJ and TjJ of
transpose Jacobian TJ , and is given by the sine of intersection angle between T
iJ and TjJ in an instantaneous pose of the moving platform.
260 4 Kinematic analysis
This index is given by Eqs. (4.40) or (4.41) in which A= TJ and
iA = TiJ (i=1,2,…,n).
The local crosscoupling index of the inverse statics problem can also be defined and calculated by using Jacobian matrix J and taking into account that the columns of the transpose Jacobian TJ are the rows of J.
Definition 4.15B. (Local crosscoupling index of the inverse statics problem). The local crosscoupling index of the inverse statics problem is a measure of decoupling between two row vectors i
*J and j*J of the
Jacobian matrix J and is given by the sine of intersection angle between
i*J and j
*J in an instantaneous pose of the moving platform
ijκ =
12 2* *
i j* *
i j
1 ⋅ −
J JJ J
(4.44)
or
ijκ
12 2n
ik jkk 1
n n2 2ik jk
k 1 k 1
a a1
a a
=
= =
= −
∑
∑ ∑.
(4.45)
Definition 4.16A. (Local crosscoupling index of the direct statics problem). The local crosscoupling index of the direct statics problem is a measure of decoupling between two column vectors T
i−J and T
j−J of the
inverse transpose Jacobian matrix TJ − given by the sine of the intersection angle between T
iJ and TjJ in an instantaneous pose of the moving plat
form. This index can be calculated by Eqs. (4.40) or (4.41) in which A= TJ −
and iA = Ti−J (i=1,2,…,n).
The local crosscoupling index of the direct statics problem can also be defined and calculated by using the inverse Jacobian matrix by taking into account that the columns of TJ − are the rows of 1J − .
Definition 4.16B. (Local crosscoupling index of the direct statics problem). The local crosscoupling index of the direct statics problem is a
4.5. Kinetostatic performance indices 261
measure of decoupling between two row vectors 1i−*J and 1
j−*J of the in
verse Jacobian 1J − , and is given by the sine of intersection angle between 1
i−*J and 1
j−*J in an instantaneous pose of the moving platform
ijκ =
12 21 * 1
i j1 * 1i j
1 ⋅ −
*
*
J JJ J
(4.46)
or
ijκ
12 2n
ik jkk 1
n n2 2ik jk
k 1 k 1
b b1
b b
=
= =
= −
∑
∑ ∑
(4.47)
where bij is the element situated on ith row and jth column of matrix 1J − . Definition 4.17A. (Crosscoupling index of the inverse statics prob
lem). The crosscoupling index of the inverse kinematic problem is a measure of decoupling between the column vectors of the transpose Jacobian matrix TJ given by the product of the corresponding local crosscoupling indices.
It can be calculated by Eqs. (4.42) or (4.43) in which A= TJ and
iA = TiJ (i=1,2,…,n).
Definition 4.17B. (Crosscoupling index of the inverse statics problem). The crosscoupling index of the inverse kinematic problem is a measure of decoupling between the row vectors of the transpose Jacobian matrix TJ given by the product of the corresponding local crosscoupling indices
12 2n
ik jkk 1
n ni 1,n 1 2 2j 1 i ,n ik jk
k 1 k 1
a a1
a aκ =
= −= +
= =
= −
∑∏
∑ ∑.
(4.48)
Definition 4.18A. (Crosscoupling index of the direct statics problem). The crosscoupling index of the direct statics problem is a measure
262 4 Kinematic analysis
of decoupling between the column vectors of the inverse transpose Jacobian matrix TJ − given by the product of the corresponding local crosscoupling indices.
It can be calculated by Eqs. (4.42) or (4.43) in which A= TJ − and
iA = Ti−J (i=1,2,…,n).
Definition 4.18B. (Crosscoupling index of the direct statics problem). The crosscoupling index of the direct static problem is a measure of decoupling between the row vectors of the inverse Jacobian matrix 1J − given by the product of the corresponding local crosscoupling indices
12 2n
ik jkk 1
n ni 1,n 1 2 2j 1 i ,n ik jk
k 1 k 1
b b1
b bκ =
= −= +
= =
= −
∑∏
∑ ∑.
(4.49)
where bij is the element situated in the ith row and jth column of matrix 1J − .
4.5.2 Indices of inputoutput propensity
The indices of inputoutput propensity represent a measure of the predilection of input components for their corresponding output components defined by the equations of the direct and inverse problems in robot kinematics and statics. The word propensity is used here in the sense of predilection of a “personalized” input for its corresponding output.
These indices also take values between zero and one. The unit value of these indices indicates that the correspondence between the inputand output has a maximum intensity. When all indices of inputoutput propensity have unit value robot kinematic/statics are completely decoupled.
We first define these indices for the general case of the linear transformation set up by Eq. (4.34) in which we consider vector x as input and y as output. A pair of corresponding inputoutput components xi, yi is formed by the ith components of both x and y vectors.
Definition 4.19. (Local index of inputoutput propensity). The local index of inputoutput propensity is a measure of the predilection of the ith component of input vector for the ith component of output vector, and is given by the ratio between the absolute value of the element aii and the norm of the column vector iA of matrix A
4.5. Kinetostatic performance indices 263
iη = ii1
n 22ki
k 1
a
a=
∑
.
(4.50)
We can see that iη is a measure of the relative value of aii in the column vector iA of matrix A.
A more general index of inputoutput propensity can be defined by extending the partial index to n=SF=M inputoutput pairs.
Definition 4.20. (Index of inputoutput propensity). The index of inputoutput propensity is a measure of the of the predilection of n=SF=M input components for their corresponding output components, and is given by the product of the local corresponding indices
n nii
i 1i 1 i 1 n 2
2ki
k 1
a
a
η η= =
=
= =
∏ ∏∑
.
(4.51)
We can see that the index of inputoutput propensity defined by Eq. (4.51) is equivalent to the semangularity in axiomatic design  see Eq. (4.3).
We derive various indices of inputoutput propensity from the two indices defined above by associating matrix A with the transformation matrices of the kinematics and statics problems defined in Eqs. (4.28), (4.29), (4.32), (4.33).
Kinematics indices of inputoutput propensity
Indices of inputoutput propensity in direct and inverse kinematics problems are derived from the local and general indices defined above by associating matrix A with Jacobian J and its inverse 1J − .
Definition 4.21. (Local index of inputoutput propensity in direct kinematics). The local index of inputoutput propensity in direct kinematics is a measure of the predilection of the ith component of the joint velocity vector for the ith component of operational velocity vector, and is given by the ratio between the absolute value of the element aii and the norm of the column vector iJ of the Jacobian matrix J.
This index is given by Eq. (4.50) in which A=J and iA = iJ Definition 4.22. (Local index of inputoutput propensity in inverse
kinematics). The local index of inputoutput propensity in inverse kine
264 4 Kinematic analysis
matics is a measure of the predilection of the ith component of operational velocity vector for the ith component of joint velocity vector, and is given by the ratio between the absolute value of the element aii and the norm of the column vector 1
i−J of Jacobian matrix J.
It is given by Eq. (4.50) in which A= 1J − and iA = 1i−J .
Definition 4.23. (Index of inputoutput propensity in direct kinematics). The index of inputoutput propensity in direct kinematics is a measure of the of the predilection of the components of joint velocity vector for their corresponding components of operational velocity vector, and is given by the product of the local corresponding indices.
It is given by Eq. (4.51) in which A=J and iA = iJ . Definition 4.24. (Index of inputoutput propensity in inverse kine
matics). The index of inputoutput propensity in inverse kinematics is a measure of the predilection of components of operational velocity vector for their corresponding components of joint velocity vector, and is given by the product of the local corresponding indices.
It is given by Eq. (4.51) in which A= 1J − and iA = 1i−J .
Statics indices of inputoutput propensity
Indices of inputoutput propensity in direct and inverse statics problems are derived from the local and general indices defined above by associating matrix A with transpose Jacobian TJ and its inverse TJ − .
Definition 4.25A. (Local index of inputoutput propensity in inverse statics). The local index of inputoutput propensity in inverse statics is a measure of the predilection of the ith component of operational force vector for the ith component of joint force vector and is given by the ratio between the absolute value of the element aii and the norm of the column vector T
iJ of transpose Jacobian matrix TJ . It is given by Eq. (4.50) in which A= TJ and iA = T
iJ . Definition 4.25B. (Local index of inputoutput propensity in inverse
statics). The local index of inputoutput propensity in inverse statics is a measure of the predilection of the ith component of operational force vector for the ith component of joint force vector, and is given by the ratio between the absolute value of the element aii and the norm of the row vector
i*J of Jacobian matrix J
4.5. Kinetostatic performance indices 265
iη = ii1
n 22ik
k 1
a
a=
∑
.
(4.52)
Definition 4.26A. (Local index of inputoutput propensity in direct statics). The local index of inputoutput propensity in direct statics is a measure of the predilection of the ith component of joint force vector for the ith component of operational force vector, and is given by the ratio between the absolute value of the element aii and the norm of the column vector T
i−J of inverse transpose Jacobian TJ − .
It is given by Eq. (4.50) in which A= TJ − and iA = Ti−J .
Definition 4.26B. (Local index of inputoutput propensity in direct statics). The local index of inputoutput propensity in direct statics is a measure of the predilection of the ith component of joint force vector for the ith component of operational force vector, and is given by the ratio between the absolute value of the element bii and the norm of the row vector
1i
*J of Jacobian matrix 1J −
iη = ii1
n 22ik
k 1
b
b=
∑
.
(4.53)
where bii is the element situated on ith row and jth column of matrix 1J − . Definition 4.27. (Index of inputoutput propensity in inverse statics).
The index of inputoutput propensity in inverse statics is a measure of the predilection of components of operational force vector for their corresponding components of joint force vector, and is given by the product of the local corresponding indices.
This index can be calculated by Eq. (4.51) in which A= TJ and iA = TiJ
or by n n
iii 1
i 1 i 1 n 22ik
k 1
a
a
η η= =
=
= =
∏ ∏∑
.
(4.54)
Definition 4.28. (Index of inputoutput propensity in direct statics). The index of inputoutput propensity in direct statics is a measure of the predilection of components of joint force vector for their corresponding
266 4 Kinematic analysis
components of operational force vector, and is given by the product of the local corresponding indices.
It can be calculated by Eq. (4.51) in which A= TJ − and iA = Ti−J or by
n nii
i 1i 1 i 1 n 2
2ik
k 1
b
b
η η= =
=
= =
∏ ∏∑
.
(4.55)
where bii is the element situated in the ith row and jth column of matrix 1J − .
4.5.3 Kinetostatic indices defined in connection with manipulability ellipsoids and polytops
Manipulability ellipsoids and polytops of the Jacobian matrix of robotic manipulators were introduced by Salisbury and Craig (1982) and Yoshikawa (1985a,b, 1990) to characterize robot behaviour. They can be defined from the viewpoints of kinematics, statics and dynamics. The shape and volume of this ellipsoid/polytop is often used to characterize the manipulator dexterity and motion/force amplification. In this section, we simply present the kinetostatic indices defined in connection with manipulability ellipsoids and polytops and we emphasise their connection with motion decoupling that we integrate in the structural synthesis approach.
The manipulability ellipsoid/polytop is associated with the mapping between the joint and operational velocity/force spaces. We mention that ndimensional ellipsoids are the images of spheres under invertible linear transformations and the polytop represents the generalization to any dimension of a polygon in two dimensions, and a polyhedron in three dimensions. The difference between manipulability ellipsoid and manipulability polytop resides in the norm used for the vectors of the joint and operational spaces (Yoshikawa 1985a,b, 1990; Merlet 2006a,b). Numerous types of norms can be used to represent the manipulability ellipsoid and polytop in an ndimensional space. For a nonredundant robot the dimension of this space is equal to robot mobility and endeffector connectivity (n=M=SF). The most common are L1, L2 and L∞ norms. The L1norm is the sum of absolute magnitudes of each element. The L2norm, also called Euclidean norm is the square root of the sum of the squares of the magnitudes of each element, and the infinity norm is the maximum of the magnitudes of the elements.
4.5. Kinetostatic performance indices 267
The eigenvalues and the singular values of Jacobian matrix are used to define several kinetostatic indices in connection with the ndimensional ellipsoid. In this section, we present the force and velocity manipulability ellipsoids and their associated indices.
Velocity and force ellipsoids
Definition 4.29. (Velocity ellipsoid). The velocity ellipsoid represents the image in robot operational velocity space of an ndimensional sphere of unit radius defined in the joint velocity space.
This map between joint and operational velocity spaces is expressed by the linear transformation with the Jacobian matrix defined in Eq. (4.28).
Let us consider the set of all operational velocities corresponding to all joint velocities with Euclidean norm bounded by unity:
1≤q . (4.56)
We can write 2 2 2 2
1 2 nq q ... q 1= = + + + ≤Tq q q . (4.57)
In the ndimensional joint velocity space, the upper bound of Eq. (4.57) can be associated with an ndimensional sphere of unit radius.
Eq. (4.28) maps this hypersphere, defined in the joint velocity space, to an ndimensional ellipsoid in the operational space
( ) 1T TJJ 1−
≤p p . (4.58)
The endeffector velocity is amplified in the direction of the major axes and reduced in the direction of the minor axis of this hyperellipsoid for a unit actuated joint velocity. If the ellipsoid is a sphere, the endeffector moves uniformly in all directions. Yoshikawa have called this the manipulability ellipsoid (Yshikawa 1985a). This ellipsoid was later called the velocity eigenellipsoid (Lee et al. 1993).
The eigenvectors of matrix T 1( JJ )− define the directions of the principal axes of this manipulability ellipsoid. The magnitudes of singular values of J represent the lengths of the principal axes. We recall that the singular values of J are the square roots of the eigenvalues of TJJ .
By taking into account the reciprocity forcevelocity we can also define a force ellipsoid.
268 4 Kinematic analysis
Definition 4.30. (Force ellipsoid). The force ellipsoid represents the image in robot operational force space of an ndimensional sphere of unit radius defined in the actuated joint force space.
This map between joint and operational force spaces is expressed by the linear transformation with the Jacobian matrix defined in Eq. (4.33).
Let us consider the set of all operational forces statically exerted by the endeffector on an object corresponding to all actuated joint forces with Euclidean norm bounded by unity
1≤τ . (4.59)
We can write 2 T 2 2 2
1 2 n... 1τ τ τ= = + + + ≤τ τ τ . (4.60)
In the ndimensional joint velocity space, the upper bound of Eq. (4.60) can be associated with an ndimensional sphere of unit radius.
Eq. (4.33) maps this force hypersphere, defined in the joint space of actuator forces, to an ndimensional force ellipsoid in the operational force space described by
T TJJ 1≤τ τ . (4.61)
The endeffector exerts high forces in the direction of the major axes and low forces in the direction of the minor axes of this hyperellipsoid. If the ellipsoid is a sphere, the endeffector exerts forces in all directions uniformly. Yoshikawa (1990) have called it manipulatingforce ellipsoid. This ellipsoid was also called later forceeigenellipsoid (Lee et al. 1993).
The eigenvectors of matrix JJT define the directions of the principal axes of this force ellipsoid. The magnitudes of inverse singular values of J represent the lengths of the principal axes.
Velocity and force polytops
Similar representations of manipulability can be obtained if we use the infinity norm instead of the Euclidean norm when bounding the joint velocities and forces.
Definition 4.31. (Velocity polytop). The velocity polytop represents the image in robot operational velocity space of an ndimensional cube of unit edgelength defined in the joint velocity space.
This map between joint and operational velocity spaces is also expressed by the linear transformation with the Jacobian matrix defined in Eq. (4.28).
For the manipulability polytop, each joint velocity is bounded by unity
4.5. Kinetostatic performance indices 269
iq 1≤ , i=1,2,…,n. (4.62)
In the ndimensional joint velocity space, the upper bound of Eq. (4.62) can be associated with an ndimensional cube of unit edgelength.
Eq. (4.28) maps this hypercube, defined in the joint velocity space, to an ndimensional polytop in the operational velocity space. This polytop was introduced by Yoshikawa (1990) and called the manipulability parallelepiped in ndimensional space, and later the manipulability polyhedron (Merlet 2006a,b) or simply the operational polytop that can include the largest manipulability ellipsoid (Krut et 2004). A possible advantage of using the polytop instead of the ellipsoid in manipulability analysis resides in the facility of building and intersecting polytops.
Definition 4.32. (Force polytop). The force polytop represents the image in robot operational force space of an ndimensional cube of unit edgelength in the actuated joint force space.
This map between joint and operational force spaces is also expressed by the linear transformation with the Jacobian matrix defined in Eq. (4.33).
For the manipulatingforce polytop, each joint force is bounded to unity
i 1≤τ , i=1,2,…,n. (4.63)
In the ndimensional force space of the actuated joints, Eq. (4.63) can also be associated with an ndimensional cube of unit edgelength.
Eq. (4.33) maps this hypercube, defined in the force space of the actuated joints, to an ndimensional polytop in the operational space of forces exerted by the endeffector.
Several indices are defined by using the manipulability ellipsoid or polytop. We denote by iυ (i=1,2,…5) and iψ (i=1,2, …,n) these various indices. These indices are applicable for both serial and parallel robots when the physical consistency is respected, as mentioned at the beginning of this section. They are defined by using the volume of the manipulability ellipsoid and the length of its principal axes, and are mainly used in dimensional synthesis (Bouzgarrou et al. 2000; Chablat et al. 2000).
Isotropy and conditioning indices
A robotic manipulator is called isotropic if its Jacobian matrix is isotropic in at least one configuration (Salisbury and Craig 1982; Angeles and Rojas 1987; Angeles and LópezCajún 1988; Angeles 1992). A given matrix B is called isotropic if its singular values are all equal and nonzero. Isotropy in robotic manipulators is closely related to condition number of the Jacobian
270 4 Kinematic analysis
matrix. The condition number of the matrix B is defined as the ratio of the greatest and the smallest singular values of J (Golub and Van Loan 1983)
max
min
scs
= . (4.64)
If B is a square matrix than c is the product of the norms of B and B1: 1c B B−= . (4.65)
The magnitude of the condition number depends on the choice of the matrix norm. Eqs. (4.64) and (4.65) give similar result if 2norm is used in Eq. (4.65). When 1norm, Frobenius norm or infinity norm are used the two equations give different values of the condition number. We recall that: (i) the 1norm of matrix B is the largest column sum, where the column sum is the sum of the magnitudes of the elements in a given column, (ii) the 2norm of matrix B is the square root of the largest eigenvalue of BBT, (iii) the Frobenius norm of matrix B is the square root of the sum of the squares of the magnitudes of each element and (iv) the infinity norm of matrix B is the maximum row sum, where the row sum is the sum of the magnitudes of the elements in a given row.
Isotropic matrices have a condition number of unity. Furthermore, the condition number of either square singular matrices or rankdeficient matrices is infinity, and hence isotropy and singularity are apparently at the opposite side of the spectrum with regard to matrix invertibility. We recall that for a m×n matrix B with m≤n, its singular values are the eigenvalues of the m×m symmetric matrix BBT. Hence matrix B is isotropic if all its singularvalues are identical and nonzero. This is equivalent to saying that, if B is isotropic, than the product BBT is proportional to the identity matrix; i.e., a positive scalar exists in this case for which
BBT=λ I (4.66)
where I is m×m identity matrix. The condition number of matrix B is configuration dependent and is
used by Angeles (1992) to measure the distance of the given pose to a singular configuration.
A parallel robot is fullyisotropic if the condition number is unity in the entire workspace.
Definition 4.33. (Conditioning index 1υ ). The conditioning index of a parallel manipulator is the inverse of the condition number in a given pose, and is given by the ratio of the smallest and the greatest singular values of its Jacobian matrix
4.5. Kinetostatic performance indices 271
1υ = min
max
ss
(4.67)
Definition 4.34. (Minimum conditioning index 2υ ). The minimum conditioning index of a parallel manipulator is the minimum value of the conditioning index corresponding to the entire robot workspace.
2υ =min( 1υ ) (4.68)
As we have already mentioned, the dimensional in homogeneity of the elements of Jacobian matrix leads to physical inconsistency when evaluating the manipulator conditioning index. There are various ways of coping with this problem:
a) Redefine the manipulator Jacobian as the matrix mapping joint rates into velocities of a set of points on the endeffector, thereby eliminating the endeffector angular velocity and hence producing a dimensionally homogeneous Jacobian (Gosselin 1990; Kim and Ryu 2003). Although this inverse Jacobian matrix is homogeneous in terms of units, the condition number will no longer describe the intrinsic behaviour of the robot as its value depends on the choice of points on the endeffector (Merlet 2006a,b).
b) Write the pointvelocity manipulator equations in dimensionless form by suitably dividing both sides of the velocity equation by a natural length of the robotic manipulator (Ma and Angeles 1991; Angeles 1992; Fattah and Hasan Ghasemi 2002). This length, called a characteristic length, is defined as that minimizing the condition number of the arising dimensionless Jacobian so obtained.
c) Calculate two condition numbers by splitting the Jacobian matrix into two parts corresponding to linear mapping of translational and angular velocities, or forces and moments (Kosuge et al 1993; Ahn and Hong 1996; Hong and Kim 2000).
Manipulability indices
Definition 4.35. (Manipulability index 3υ ). The manipulability index 3υ represents the absolute value of the determinant of the parallel robot Jacobian matrix in a current pose
3υ = det J . (4.69)
272 4 Kinematic analysis
The manipulability index 3υ is closely related to the singular values si (i=1,2,…,n) of Jacobian matrix J and to the volume Ve of the manipulability ellipsoid
3υ =n
ii 1
s=∏
(4.70)
and
3υ = e
m
Ve
(7.71)
where
( )( )
( )( )
n2
m n 12
2if n is even
2 4 6 n 2 ne
2 2if n is odd
1 3 5 n 2 n
−
⋅ ⋅ ⋅ ⋅ ⋅ − ⋅=
⋅ ⋅ ⋅ ⋅ ⋅ − ⋅
π
π.
Definition 4.36. (Manipulability index 4υ ). The manipulability index
4υ represents the ratio between the absolute value of the Jacobian determinant in a current pose and its maximum absolute value corresponding to the overall workspace
34
3maxmax
det Jdet J
υυυ
= = . (4.72)
This normalized index was proposed in (Lee et al. 1998) and extended in (Zhang et al. 2000) to redundant parallel robots.
Definition 4.37. (Performance index 5υ ). The performance index 5υ represents the minimum singular value of the Jacobian matrix
5 minsυ = (4.73)
This performance index was proposed in (Klein and Blaho 1987) to characterize the distance to singular configuration.
Transmission factors
Definition 4.38. (Velocity transmission factors iψ , i=1,2,…,n) The velocity transmission factors in the directions of the principal axes, iψ
4.5. Kinetostatic performance indices 273
(i=1,2,…,n) are the singular values of 1J − ; that is, the inverses of the square roots of the eigenvalues of TJJ
ii
1s
ψ = , i=1,2,…,n. (4.73)
We recall that the singular values of J are the square roots of the eigenvalues of TJJ and the singular values of 1J − are the inverses of the singular values of J.
The velocity transmission factors are used in (Chablat and Wenger, 2003) to determine the link length and the linear joint limits of the translational parallel robot Orthoglide.
We note that Eqs. (4.67), (4.68), (4.70), (4.71) and (4.73) are applicable to nonredundant and redundant robotic manipulators. For redundant robotic manipulators, Eqs. (4.69) and (4.72) become
( )T3 det JJυ = (4.74)
and
( )( )
T3
4 T3maxmax
det JJ
det JJυ
υυ
= = . (4.75)
We note that the manipulability index 3υ also called the manipulability measure was introduced by Yoshikawa (1985a,b, 1990) to calculate a kind of distance of the robot’s current configuration from singular ones, starting from the observation that in a singular position the determinant of the Jacobian matrix and one or more eigenvalues become zero, leading to
3υ =0. Yoshikawa considered that when the manipulability index 3υ reaches its maximum, the robot is furthest away from the singular configuration. The use of manipulability indices is critically reviewed by Angeles (1992) pointing out that the determinant of a square matrix, or that of the product of a rectangular matrix and its transpose does not measure how far from singularity a matrix is (Forsythe and Moler 1967). In fact, the non zero determinant of a square matrix merely indicates that a matrix is invertible, but it does not measure the invertibility of the matrix.
The determinant of the Jacobian matrix has the advantage of being invariant with respect to changes of reference frame and, by normalization, it can be made invariant with respect to changes of scale and physical units. In this case, the manipulability index could exhibit consistent values with
274 4 Kinematic analysis
respect to positioning errors, as in the example presented in (Merlet, 2006). The use of condition number along with manipulability as kinetostatic performance indices is critically reviewed by Merlet (2006) when accuracy of a parallel robot is analysed. This analysis emphasizes the sensitivity of the condition number of a normalized Jacobian with respect to various types of norms used for its calculation.
Since there are no natural norms in the spaces used to represent generalized velocities and forces (Park 1995), kinetostatic performance indices are not invariant under changes of the norms used. This noninvariance property introduces certain applicability limitations of these indices as measures of distance from singularity, as presented in (Staffetti et al. 2002).
4.6 Design Jacobian and motion decoupling
Motion decoupling in parallel robots can be defined by analogy with design decoupling in axiomatic design. We consider the design Jacobian as a design matrix and we use it to introduce various decoupling measures and types of parallel robots.
We have seen in 4.3.2 that the angular velocity 0ω =T
x y z, ,ω ω ω cannot be calculated by direct differentiation of the rotation angles of the moving platform. The corresponding rotation angles do not have a physical meaning when using 0ω . Moreover, Eqs. (4.20)(4.22) used to calculate 0ω with or without time derivatives of the rotation angles ,α β and δ , introduce some hypothetical complementary coupling between ω and 0ω and implicitly between the velocity of the actuated joints and the velocity of the moving platform. This coupling has no physical correspondence in the mechanical architecture of the kinematic chain associated with the robot. It is due merely to the theoretical approach used in modelling robot kinematics by using 0ω . This hypothetical coupling is unavoidable in calculating
0ω for any robot with two or three rotational degrees of freedom of the endeffector. Even the simplest orientation mechanism with just two orthogonal revolute pairs is affected by this hypothetical coupling. For this reason, in motion coupling analysis we use ω =
T, ,α β δ =
T, ,α β δω ω ω
and [ 0v ,ω ]T= [ x y zv ,v ,v , , ,α β δω ω ω ]T to define the velocity of the moving platform. This choice allows us to characterise the real nature of motion coupling due to robot mechanical architecture.
4.6 Design Jacobian and motion decoupling 275
Parallel robots with coupled, decoupled, or uncoupled motions, and maximally regular robots can be defined by using the design Jacobian defined in connection with the direct kinematics problem of nonredundant parallel robots
J=w q (4.76)
where [ ]T1 nw ... w=w is the vector of operational velocities of the mov
ing platform, [ ]T1 nq ... q=q are the velocity vectors of the actuated joints, and J is the design Jacobian. We recall that n is equal to robot mobility and the connectivity of the moving platform n=M=SF.
The degree of motion coupling in a non redundant parallel robot defines the number of interconnections existing between the joint and operational motions. We define an instantaneous and a global degree of motion coupling and we denote them by Ci and C .
Definition 4.39 (Degree of instantaneous motion coupling). The degree of instantaneous motion coupling of a non redundant parallel robot is given by the number of non zero offdiagonal terms of the design Jacobian in a given nonsingular pose of the moving platform.
We have seen that the design Jacobian is a square n× n matrix. The maximum value of the absolute motion coupling is n(n1) and indicates that all n=M actuators participate in n=SF independent motions of the moving platform (n=M=SF). In non redundant parallel robots the degree of motion coupling could vary from 2 to 30; the maximum value corresponds to parallel robots with n=M=SF=6, and the minimum to robots with n=M=SF=2.
Definition 4.40 (Global degree of motion coupling). The global degree of motion coupling of a non redundant parallel robot is given by the maximum value of the instantaneous degree of motion coupling
C =min(Ci). (4.77)
The degree of instantaneous motion coupling is defined numerically for the design Jacobian. The global degree can be easily determined on the Jacobian defined symbolically with no need to use Eq. (4.77). In this case, the global degree of motion coupling of a non redundant parallel robot is given by the number of non zero offdiagonal terms in the symbolical design Jacobian.
In defining various types of parallel robots we use the global degree that will simply be called degree of motion coupling.
276 4 Kinematic analysis
Definition 4.41 (Parallel robot with fullycoupled motions). A non redundant parallel robot has fullycoupled motions if the design Jacobian symbolically defined contains no elements with zero values.
4.6.1. Parallel robots with coupled motions
A parallel robot with coupled motions represents the archetype of inparallel actuated robot arms beginning with the first industrial parallel robots developed in the first half of the twelfth century and the first comprehensive study of their structural synthesis made by Hunt (1983). The majority of parallel robots used today have coupled motions.
Definition 4.42A (Parallel robot with coupled motions). A parallel robot has coupled motions if the design Jacobian cannot be set up, in a nonsingular pose, as a diagonal or triangular matrix under alternative reordering of lines and columns, as long as ordering preserves the association of each actuated joint velocity with its corresponding operational velocity.
In the general case of direct kinematics problems, the component iw (i=1,2,…,M) of the operational velocity vector of a parallel robot with
coupled motions is a function of M components of the actuated velocity space iq (i=1,2,…,M): i i 1 Mw w ( q ,...,q )= , i=1,2,…,M, where M is the degree of mobility of the parallel robot.
To preserve the association of each actuated joint velocity with its corresponding operational velocity, interchanging between two lines of the design Jacobian must be accompanied by a corresponding interchange in the sequence of components in the operational velocity vector. In the same way, interchanging two columns must be accompanied by a corresponding interchange in the sequence of components in the actuated velocity vector.
Reordering lines and columns by preserving the association between each actuated joint velocity and its corresponding operational velocity does not affect the nature of the linear mapping set up by the Jacobian matrix. For example, such a reordering is the usual way to switch from the operational velocity vector [ 0v , 0ω ]T= [ x y z x y zv ,v ,v , , ,ω ω ω ]T, calculated by the conventional Jacobian, and the corresponding twist $=[ x y z x y z, , ,v ,v ,v ,ω ω ω ], calculated by the screwbased Jacobian.
In general for a parallel robot, the inverse Jacobian is easier to calculate than the direct Jacobian. Motion coupling could be also defined with respect to inverse design Jacobian in connection with the inverse kinematics problem of nonredundant parallel robots
4.6 Design Jacobian and motion decoupling 277
1J −=q w . (4.78)
The property of being diagonal or triangular is conserved by matrix inversion in nonsingular poses. In a singular pose, some elements of the Jacobian matrix may become zero and the instantaneous degree of motion coupling could change. For this reason, the degrees of motion coupled are defined in nonsingular poses.
Definition 4.42B (Parallel robot with coupled motions). A parallel robot has coupled motions if the inverse design Jacobian cannot be set up, in a nonsingular pose, as a diagonal or triangular matrix under alternative reordering of lines and columns, as long as ordering preserves the association of each actuated joint velocity with its corresponding operational velocity.
In the general case of the inverse kinematics problem, the component iq (i=1,2,…,M) of the actuated velocity space of a parallel robot with cou
pled motions is a function of M components iw (i=1,2,…,M) of the operational velocity vector: i i 1 Mq q ( w ,...,w )= , i=1,2,…,M.
Orthoglide parallel robot
We investigate the motion coupling of the parallel robot Orthoglide (Wanger and Chablat 2000; Chablat and Wanger 2003) developed by IRCCyN and analysed in example 24 in chapter 3. For convenience, the kinematic structure of the robot is completed with the necessary notations and presented in Fig. 4.2. We develop design and conventional Jacobian matrices by using the procedures presented in section 4.3 and we show that the two Jacobian matrices are the same for this robot.
The moving platform of Orthoglide moves in the Cartesian space with fixed orientation. We denote by x, y and z the three spatial coordinates of the characteristic point H, that is w1=x, w2=y, w3=z. The operational velocity of the moving platform is w = 0v = [ ]Tx, y,z =
T
x y yv ,v ,v . The robot is actuated by three linear motors which are mounted or
thogonally. The velocity vector of the actuated joints is q = [ ]T1 2 3q ,q ,q . The distance di=CiBi (i=1,2,3) is maintained constant by the parallelogram loops used in the three limbs. For each limb we can write:
( ) ( ) ( )2 2 22i Ci Bi Ci Bi Ci Bid x x y y z z= − + − + − , i=1,2,3. (4.79)
The coordinates of points Ci and Bi in the fixed reference frame 0 0 0 0O x y z are: C1 (q1+a, 0, 0), C2 (0, q2+b, 0), C3 (0, 0, q1+a) and B1 (x+a,
278 4 Kinematic analysis
Fig. 4.2. Orthoglide and the notations used in kinematic modelling
y, z), B2 (x, y+b, z), B3 (x, y, z+c) with a, b and c as constant lengths. Introducing these coordinates in Eq. (4.79) and taking its derivatives with respect to time three times, once for each i=1,2 and 3, yields three scalar equations, which can be arranged in matrix form
sJ q = pJ v0 (4.80)
or
q = 1sJ −
pJ v0= 1J − v0 (4.81)
with the following matrices
sJ1
2
3
x q 0 00 y q 00 0 z q
− = − −
,
(4.82)
pJ1
2
3
x q y zx y q zx y z q
− = − −
(4.83)
4.6 Design Jacobian and motion decoupling 279
and
1J −
1 1
2 2
3 3
y z1x q x q
x z1y q y q
x y 1z q z q
− −
= − −
− −
(4.84)
The same result can be obtained by using a loopclosure equation written for each limb
O0Ci+CiBi=O0H+HBi , i=1,2,3. (4.85)
A velocity vectorloop equation is obtained by taking the derivative of Eq. (4.85) with respect to time
iq ni+ di i iaω × = 0v (4.86)
where ia and in denotes the unit vectors along CiBi and O0Ci, respectively, and iω denotes the angular velocity of a virtual link CiBi with respect to fixed reference frame. This virtual link is associated with the parallelogram loop in ith limb.
Eq. (4.86) written three times, once for each i=1,2 and 3, yields three scalar equations, which can be arranged in matrix form (4.80) with
sJ =0 0
0 00 0
⋅ ⋅ ⋅
1 1
2 2
3 3
n a n a
n a,
(4.87)
and
pJ =
T1T2T3
aaa
,
(4.88)
To eliminate iω , we dotmultiply both sides of Eq. (4.86) by ia :
iq ni ia⋅ = 0v ia⋅ , i=1,2,3. (4.88)
280 4 Kinematic analysis
1J − =
T1
1 1
T2
2 2
T3
3 3
⋅
⋅
⋅
an a
an a
an a
(4.90)
Taking into account the fact that the unit vectors ni point in the positive directions of the axes of the fixed reference frame 0 0 0 0O x y z , that is n1=i0, n2=j0 and n3=k0, we can see that the expressions of the matrices defined by Eqs. (4.88)(4.90) and (482)(484) are the same.
The inverse geometric solution of Orthoglide can be obtained by introducing the coordinates of points Ci and Bi in Eq. (4.79)
2 2 21 1
2 2 22 1
2 2 23 1
q x d y z
q y d x z
q z d x y
= + − −
= + − −
= + − −
.
(4.91)
By taking into account Eqs. (4.91), the Jacobian matrix (4.84) becomes
1J −
2 2 2 2 2 21 1
2 2 2 2 2 22 2
2 2 2 2 2 23 3
y z1d y z d y z
x z1d x z d x z
x y 1d x y d x y
− − − − − − = − − − − − − − − − − − −
.
(4.92)
In a general position of the workspace the Jacobian matrix J1 is a full matrix indicating that Orthoglide is a parallel robot with fullycoupled motions (C=6). There is just one point in the workspace of Orthoglide, defined by x=y=z=0, where the inverse Jacobians (4.83) or (4.89) become instantaneously the identity matrix. This point is reached when the characteristic point H of the moving platform is situated in the origin of the fixed reference frame, as in Fig. 4.3, and ia and in point in opposite directions. In this pose instantaneous degree of motion coupling is Ci =0.
4.6 Design Jacobian and motion decoupling 281
Fig. 4.3. Paricular position of Orthoglide with instantaneous degree of motion coupling Ci =0
Structural solutions derived from parallel robots with fullycoupled motions
Various solutions of parallel robots can be derived from parallel robots with fullycoupled motions aiming to decrease the degree of motion coupling. In this section, we illustrate two solutions (Figs. 4.4 and 4.5) derived from Orthoglide (Gogu 2004b,d). The moving platform of each solution also moves in the Cartesian space, 0v = [ ]Tx, y,z =
T
x y yv ,v ,v , with fixed orientation, and the actuators are mounted on the fixed base.
The first solution 2PR1PaR11PR1R1R1type is derived by replacing one limb (PR1PaR1type) with a limb of PR1R1R1type used in the CPM robot in Fig. 2.1. We recall that the revolute pairs with the same index have parallel axes. As with Orthoglide, the first solution is actuated by three linear motors which are mounted orthogonally on the fixed base with q = [ ]T1 2 3q ,q ,q  see Fig. 4.4.
Eqs. (4.79) are applicable just for the limbs of PRPaRtype for i=2 and 3. Referring to Fig. 4.4a, the following simple geometric relation can be written for the limb of PRRRtype:
282 4 Kinematic analysis
Fig. 4.4. Translational parallel robot of 2PR1PaR11PR1R1R1type with decoupled motions in a general position with C =4 (a) and a particular position when Ci =0 (b)
4.6 Design Jacobian and motion decoupling 283
x = q1a. (4.93)
Eq. (4.93) replaces Eq. (4.79) for i=1. Matrices sJ , pJ and 1J − in Eqs. (4.80) and (4.81) have the following expressions
sJ 2
3
1 0 00 y q 00 0 z q
= − −
(4.94)
pJ 2
3
1 0 0x y q zx y z q
= − −
,
(4.95)
1J −
2 2
3 3
1 0 0x z1
y q y qx y 1
z q z q
= − −
− −
.
(4.96)
In a general position (Fig. 4.4a) in the workspace the global degree of motion coupling is C =4. When characteristic point H of the moving platform is situated in the origin of the fixed reference frame (x=y=z=0), as in Fig. 4.4b, the instantaneous degree of motion coupling is zero.
The second solution 2PR1PaR11R2R2R1R1R1type is derived by replacing one limb PR1PaR1type with a limb 1R2R2R1R1R1type. This solution is actuated by a rotation motor and two linear motors (Fig. 4.5).
Eqs. (4.79) are also applicable just for the limbs of PRPaRtype for i=2 and 3. Referring to Fig. 4.5a, the following simple geometric relation can be written for the limb 1R2R2R1R1R1type
x=d1sinq1. (4.97)
Eq. (4.97) replaces Eq. (4.79) for i=1. Matrices Js, Jp and J in Eqs. (4.80) and (4.81) have the following expressions:
284 4 Kinematic analysis
Fig. 4.5. Translational parallel robot of 2PR1PaR11R2R2R1R1R1type with decoupled motions in a general position with C =4 (a) and a particular position when Ci =0 (b)
4.6 Design Jacobian and motion decoupling 285
sJ1 1
2
3
d cos q 0 00 y q 00 0 z q
= − −
(4.98)
pJ 2
3
1 0 0x y q zx y z q
= − −
(4.99)
and
1J −
1 1
2 2
3 3
1 0 0d cos q
x z1y q y q
x y 1z q z q
= − −
− −
.
(4.100)
This robot has also Ci=0 when characteristic point H of the moving platform is situated in the origin of the fixed reference frame (x=y=z=0), as in Fig. 4.5b.
4.6.2. Parallel robots with decoupled motions
The first solutions of parallel robots with decoupled motions have been proposed recently (Kim and Tsai 2002; Kong and Gosselin 2002b; Gogu 2004ad, 2005fi, 2006ae, 2007ac).
Definition 4.43 (Parallel robot with decoupled motions). A parallel robot has decoupled motions if the design Jacobian (or its inverse) can be set up as a triangular matrix under alternative reordering of lines and columns, as long as ordering preserves the association of each actuated joint velocity with its corresponding operational velocity.
In the direct kinematics problem, the component iw (i=1,2,…,M) of operational velocity vector of a parallel robot with decoupled motions is a function of just the first i components iq of the actuated velocity space:
i i 1 iw w ( q ,...,q )= , i=1,2,…,M, where M is the degree of mobility of the parallel robot.
286 4 Kinematic analysis
In the inverse kinematics problem, the component iq (i=1,2,…,M) of the actuated velocity space of a parallel robot with decoupled motions is a function of just the first i components iw of operational velocity vector:
i i 1 Mq q ( w ,...,w )= , i=1,2,…,M. Definition 4.44 (Parallel robot with halfdecoupled motions). A par
allel robot has halfdecoupled motions if: (i) the design Jacobian (or its inverse) cannot be set up as a diagonal or triangular matrix under alternative reordering of lines and columns and (ii) the translational motions are decoupled with respect to rotational motions of the moving platform.
In this case, the design Jacobian can be partitioned in four blocks. The two blocks situated on the main diagonal are translation and rotationrelated (sub)matrices. The other two blocks contains only zero elements.
Various solutions of parallel robots with decoupled motions can be derived from their counterparts with coupled motions. In this section, we illustrate two solutions derived from previous solutions with coupled motions (Gogu 2004b,d). These solutions are also translational parallel robots with 0v = [ ]Tx, y,z =
T
x y yv ,v ,v , and the actuators are mounted on the fixed base.
The first solution 1PR1PaR12PR1R1R1type is also derived from Orthoglide by replacing a second limb PR1PaR1type with a limb PR1R1R1type. Orthoglide, this solution is also actuated by three linear motors which are mounted orthogonally on the fixed base with q = [ ]T1 2 3q ,q ,q  see Fig. 4.6.
Eqs. (4.79) are applicable just for the limb PRPaRtype for i=3. Referring to Fig. 4.6a, the following simple geometric relation can be written for the two limbs PRRRtype:
x = q1a (4.101)
and
y = q2b. (4.102)
Eqs. (4.101) and (4.102) replace Eq. (4.79) for i=1 and 3. Matrices sJ ,
pJ and 1J − in Eqs. (4.80) and (4.81) have the following expressions
sJ
3
1 0 00 1 00 0 z q
= −
(4.103)
4.6 Design Jacobian and motion decoupling 287
Fig. 4.6. Translational parallel robot of 1PR1PaR12PR1R1R1type with decoupled motions in a general position with C =2 (a) and a particular position when Ci =0 (b)
288 4 Kinematic analysis
pJ
3
1 0 00 1 0x y z q
= −
, (4.104)
1J −
3 3
1 0 00 1 0x y 1
z q z q
= − −
. (4.105)
By taking into account the inverse geometric solution, Eq. (4.105) becomes
1J −
2 2 2 2 2 23 3
1 0 00 1 0x y 1
d x y d x y
= − − − − − −
.
(4.106)
The various kinetostatic indices of translational parallel robot 1PR1PaR12PR1R1R1type with decoupled motions are presented in Figs. 4.74.11.
The values of the conditioning index are 0.101≤ 1υ ≤1 and velocity transmission factors satisfy 0.318≤ iψ ≤3.146 (Fig. 4.7).
Local crosscoupling indices 12κ , 13κ , 23κ and crosscoupling index κ of the direct kinematics problem for this parallel robot are the same corresponding indices of the direct kinematics. For this reason, we name both groups as the kinematics crosscoupling indices. The diagrams presented in Fig. 4.8 indicate a strong coupling at the extremities of the working space and a complete uncoupled kinematics in the central point of the working space (x=y=0) where 12 13 23 1κ κ κ κ= = = = . The length d3 must be correlated with the dimensions of the working space such as
2 23d x y> + . When ( )2 2 2
3x y d+ → , the parallel robot is in the vicinity of workspace limit singularity and the crosscoupling indices tend to zero. We have considered d3=B3C3 =3m.
4.6 Design Jacobian and motion decoupling 289
Fig.4 .7. Conditioning index 1υ (a) and velocity transmission factors iψ (b)(d) of parallel robot 2PRRR1PRPaRtype
The corresponding indices of the direct and inverse statics problem for this parallel robot are also the same and we call both groups as the statics crosscoupling indices. The diagrams presented in Fig. 4.9 show that, for this robot, crosscoupling in the statics problems is more important than in kinematics problems.
For this parallel robot, the indices of inputoutput propensity 1η , 2η , 3η and η of the direct and inverse kinematics problems are also the same, and we called both groups as the indices of kinematics inputoutput propensity. The diagrams presented in Fig. 4.10 indicate a constant inputoutput kinematics propensity in the case of 3η and the strongest propensity in the central point of the working space (x=y=0) where 1η = 2η = 3η =η . The inputoutput propensity is more important in statics than in kinematics problems.
The second solution 1PR1PaR12R2R2R1R1R1type is also derived by Ortyhoglide by replacing a second limb PR1PaR1type with a limb R2R2R1R1R1type. This solution is actuated by two rotation motors and one linear motor (Fig. 4.12).
Eqs. (4.79) are applicable just for the limb PRPaRtype for i=3. Referring to Fig. 4.7a, the following simple geometric relation can be written for the two limbs of 1R2R2R1R1R1type
290 4 Kinematic analysis
Fig. 4.8. Kinematics crosscoupling indices of parallel robot 2PRRR1PRPaRtype
Fig. 4.9. Statics crosscoupling indices of parallel robot 2PRRR1PRPaRtype
4.6 Design Jacobian and motion decoupling 291
Fig. 4.10. Indices of kinematics inputoutput propensity of parallel robot 2PRRR1PRPaRtype
Fig. 4.11. Indices of statics inputoutput propensity of parallel robot 2PRRR1PRPaRtype
292 4 Kinematic analysis
Fig.4.12. Translational parallel robot 1PR1PaR12R2R2R1R1R1type with decoupled motions in a general position with C =2 (a) and a particular position when Ci =0 (b)
4.6 Design Jacobian and motion decoupling 293
x=d1cosq1 (4.107)
and
y=d2cosq2. (4.108)
Eqs. (4.107) and (4.108) replace Eq. (4.79) for i=1 and 3. Matrices sJ ,
pJ and 1J − in Eqs. (4.80) and (4.81) have the following expressions
sJ1 1
2 2
3
d cos q 0 00 d cos q 00 0 z q
= −
(4.109)
pJ
3
1 0 00 1 0x y z q
= −
(4.110)
and
1 1
1
2 2
3 3
1 0 0d cos q
1J 0 0d cos q
x y 1z q z q
−
=
− −
.
(4.111)
Both solutions presented in Figs. 4.6 and 4.12 have a global degree of motion coupling C =2. When the characteristic point H of the moving platform is situated in the origin of the fixed reference frame (x=y=z=0), as in Figs. 4.6b and 4.12b, the instantaneous degree of motion coupling becomes Ci=0.
4.6.3. Parallel robots with uncoupled motions
Along with parallel robot with decoupled motions, parallel robots with uncoupled motions have been proposed recently (Kong and Gosselin 2002b, Gogu 2004ad, 2005fi, 2006ae, 2007ac).
294 4 Kinematic analysis
Fig. 4.13. Translational parallel robot 3R2R2R1R1R1type with uncoupled motions in a general position (a) and the initial position when qi=0, i=1,2,3 (b)
4.6 Design Jacobian and motion decoupling 295
Definition 4.45 (Parallel robot with uncoupled motions). A parallel robot has uncoupled motions if the design Jacobian (or its inverse) can be set up as a diagonal matrix.
In the kinematics problem, the component iw (i=1,2,…,M) of operational velocity vector of a parallel robot with uncoupled motions is merely a function of ith component iq of the actuated velocity space: i i iw w ( q )= , i=1,2,…,M, where M is the degree of mobility of the parallel robot.
In a parallel robot with uncoupled motions, pJ becomes the identity matrix and the design and direct kinematics matrix have the same expression
J= sJ . (4.112)
The parallel robots with uncoupled motions have unity value for the crosscoupling indices and for the indices of inputoutput propensity. Various solutions of parallel robots with uncoupled motions can be derived from their counterparts with decoupled motions. In this section, we illustrate a solution derived from the parallel robot with decoupled motions 1PR1PaR12R2R2R1R1R1type analysed in the previous section (Gogu 2004d). This solution, 3R2R2R1R1R1type, is a translational parallel robot with 0v = [ ]Tx, y,z =
T
x y yv ,v ,v and has three rotation actuators mounted on the fixed base (Fig. 4.13). It is derived from its counterpart with decoupled motions in Fig. 4.12 by replacing the remaining limb of type 1PR1PaR1 by a limb of type 2R2R2R1R1R1. In this way, the parallel robot with uncoupled motions 3R2R2R1R1R1type has three structurally identical limbs.
Referring to Fig. 4.13a, the following simple geometric relation can be written for the three limbs of 1R2R2R1R1R1type
x=d1cosq1 (4.113)
y=d2cosq2 (4.114)
and
z=d3cosq3. (4.115)
Matrices sJ and J in Eqs. (4.80) and (4.81) have the same simple expression
296 4 Kinematic analysis
sJ1 1
2 2
3 3
d cos q 0 0J 0 d cos q 0
0 0 d cos q
= =
(4.116)
The Jacobian is a diagonal matrix and C =Ci=0 in any nonsingular position with / 2iq π≠ ± including the initial pose when characteristic point H of the moving platform is situated at the origin of the fixed reference frame (x=y=z=0) and qi=0 (i=1,2,3)  Fig. 4.13b.
4.6.4. Maximally regular parallel robots
Maximally regular parallel robots represent a particular case within the category of parallel robots with uncoupled motions; the design Jacobian is the identity matrix. Along with parallel robots with decoupled and uncoupled motions, maximally regular parallel robots have quite recently been developed as fullyisotropic parallel robots. The term maximally regular parallel robot was recently proposed by Merlet (2006) to define isotropic robots. We extend this term, and we use it to define parallel robots with unity value for all kinetostatic performance indices presented in the previous section.
Definition 4.46 (Maximally regular parallel robot). A parallel robot with connectivity SF of the moving platform is maximally regular if the design Jacobian is the SF× SF unit matrix.
In the kinematics problem, the component iw (i=1,2,…,M) of the operational velocity vector of a maximally regular parallel robot is equal to its corresponding component iq of the actuated velocity space: i iw q= , i=1,2,…,M, where M is the degree of mobility of the parallel robot.
We recall that connectivity indicates the number of independent motions of the moving platform and is also known as the degree of freedom.
Maximally regular parallel robots have unit values for all kinetostatic performance indices presented in the previous sections: crosscoupling indices, indices of inputoutput propensity, conditioning indices, manipulability indices, and transmission factors. The manipulability ellipsoid of maximally regular parallel robots is a hypersphere of unit radius.
The first solution of such a robot was the translational parallel robot presented in Fig. 2.1. This type of robot was developed at the same time and independently by Carricato and ParentiCastelli at University of Genoa, Kim and Tsai at University of California, Kong and Gosselin at University of Laval, and the author at the French Institute of Advanced Mechanics. In
4.6 Design Jacobian and motion decoupling 297
2002, the four groups published the first results of their works (Carricato and ParentiCastelli 2002; Kim and Tsai 2002; Kong and Gosselin 2002b; Gogu 2002b). Each of the last three groups has built a prototype of this robot in their research laboratories and has called this robot CPM (Kim and Tsai 2002), Orthogonal Tripteron (Gosselin et al. 2004) or Isoglide3T3 (Gogu 2004b). The first practical implementation of this robot was the CPM developed at University of California.
The author has also published the first solutions of the following fullyisotropic parallel robots: parallel wrists with two (Gogu 2005f) and three (Gogu 2007c) degrees of mobility, planar parallel robots (Gogu 2004c), parallel robots of T2R1type (Gogu 2005i), parallel robots with Schönflies motions (Gogu 2004a, 2005g, 2006c, 2007a), parallel robots of T3R2 type (Gogu 2006a,b,d), hexapodes with six degrees of freedom (Gogu 2006e). These solutions belong to a modular family called IsoglideNTxRy with a+b=n with 2≤ n≤ 6, a=1,2,3 and b=1,2,3. The moving platform of the robots belonging to the IsoglidenTaRb family (Gogu 2007b) can have any combination of n independent translations (T) and rotations (R). This family is composed of three groups marked by A, B and C:
a) parallel robots with decoupled motions: IsoglidenTaRbAx, b) parallel robots with uncoupled motions: IsoglidenTaRbBx, c) maximally regular parallel robots: IsoglidenTaRbCx,
where x=1,2,… is used to make a distinction between the different solutions in each group.
The IsoglidenTaRb modular family was developed by the author and his research team at the French Institute of Advanced Mechanics in ClermontFerrand.
The presentation of the systematic approach of the structural synthesis of the innovative IsoglidenTaRb family with simple or complex limbs, whether fullyparallel or not, whether redundant or not redundant, whether overconstrained or not, is included in the second part of this book.
5 Structural synthesis
The method proposed in this book for structural synthesis of parallel mechanisms is founded on the theory of linear transformations and evolutionary morphology. The structural parameters defined via the theory of linear transformations in chapter 2 are integrated in an original approach to innovative design proposed in (Gogu 2005a, 2007a) and called evolutionary morphology. This method allows us to obtain the structural solutions of decoupled, uncoupled, fullyisotropic and maximally regular parallel robots in a systematic way. Overconstrained/isostatic solutions with elementary/complex limbs actuated by linear/rotary actuators with/without idle mobilities could be obtained.
The first part of this chapter presents the general paradigms of structural synthesis and evolutionary morphology as structured approaches to inventive engineering design. In the second part, the general bases are set up in applying evolutionary morphology to structural synthesis of parallel robots with coupled, decoupled, uncoupled and maximally regular parallel robots.
5.1 Structural synthesis: a systematic approach in mechanism design
Mechanism design has always fascinated mankind. The fascination arises from the fact that understanding how a mechanism functions is fairly easy, but comprehending how it originated and why it was designed in a particular form in which it exists is more difficult (Kota 1992).
Leonardo da Vinci in his works The Madrid Codex and Atlantic Codex was the first to advocate the necessity of a science of mechanisms, and the wide application of mathematical methods to the construction of machines. He had the vision that any machine can be built with a set of mechanisms, and he even listed 22 elements from which machines can be constructed.
Willis (1841) called for a science “that provides us with methods by which all forms and arrangements that are applicable to the desired purposes can be obtained”. Instead of selecting mechanisms based on intuition and experience the designer could choose the most suitable solutions from
300 5 Structural synthesis
this list of possibilities. Willis developed a classification scheme of mechanisms according to type of motion transmission.
Reuleaux (1870) laid the foundation for a systematic study of mechanisms by proposing a formal classification of mechanisms and their symbolic representation. In this classification, he recognized the preeminence of joints and identified them as kinematic pairs. While Reuleaux’s systematization of kinematic pairs into lower and higher pairs, and his classification of mechanisms based on their types of joints, are still used today, his symbolic notation is regarded as too unwieldy for use in structural synthesis procedures (Erdman 1992). Davies and Crossley (1966) remarked that “although Reuleaux’s work has done much to clarify thinking about mechanisms, his notation is rarely used, probably because in making it both comprehensive and concise, the graphic quality of a drawing is entirely lost.” This drawback is still valid for modernday systematization of mechanisms based on symbolic notations without graphical counterparts, or with a poor graphical representation in which the types of joints and the relative position of their axes are represented ambiguously.
Researches in design formalization have found that the fundamental task of conceptualizing devices is a mixture of art and science. Researchers in mechanism theory could have conceded that mechanism design is an art and then left its pursuit to the gifted few. Instead, they continue to search for a scientific basis for creative design, attempting to offer a gift of creativity to the engineering community. (Kota 1992).
Creative design, in the context of mechanisms, is what is more widely known as systematics, and consists in generating the entire set of solutions defined by certain structural parameters without regard to the dimensions of the components. Sometimes systematics is associated with systematization, classification end simple enumeration of the known solutions without giving emphasis to the generating process (Artobolevskii 1976). The systematic generating process of mechanisms is generally called structural synthesis. Various other terms are also used to define this systematic synthesis approach as shown in Table 5.1.
Type synthesis and number synthesis were identified by Hain (1961) and Crosley (1964) as two main approaches in the systematics of mechanisms. Type synthesis is based on the selection of a particular type of mechanism (linkage, cam, gear, etc.). Number synthesis is the process of finding the arrangements of a given number of bodies and joints which result in kinematic chains with the desired mobility (Crosley 1964). Crossley (1980) also identified type synthesis, that is the “determination of the simplest type(s) of mechanisms that are capable of producing the desired motion” as the first stage in mechanism synthesis. He introduced the synthesis
5.1 Structural synthesis: a systematic approach in mechanism design 301
Table 5.1. Terminology used to define the systematic synthesis of mechanisms
Terminology
References
Structural synthesis
Manolescu and Tempea 1967a,b; Manolescu 1968; Mruthyunjaya 1979, 1984a,b,c 2003; Yang 1985; Dudiţă and Diaconescu 1986; Dudiţă et al. 1989; 2001a,b; Hervé and Sparacino 1991; Hwang and Hwang 1992; Wenhui et al. 1995; Rao 1997; Rao and Deshmukh 2001;Rao et al. 2001; Alizade and Bayram 2004; Alizade et al. 200 Gogu 2004b, 2007a; Sunkari and Schmidt 2006.
Structure synthesis Linda et al. 2000; Schmitd et al. 2000; Jin and Yang 2001, 2002; Fang and Tsai 2002, 2004b.
Structural and kinematic synthesis
Manolescu and Tempea 1967
Structural kinematics
Hunt 1983
Kinematic synthesis
Denavit and Hartenberg 1964; Vucina and Freudenstein 1991; Sandor and Erdman 1984; Erdman and Sandor 1996
Topological synthesis
Sardain 1997; Murphy et al. 1996; DarZen and KangLi 2000; Carricato and ParentiCastelli 2004.
Topology synthesis
Jin and Yang 2004.
Type synthesis Hain 1961;Crosley 1964; Alizade et al. 1985; Olson et al. 1985; Dijksman and Timmermans 1994; Huang and Li 2002a, 2003; Huang 2004; Kong and Gosselin 2002b, 2004ae, 2005, 2006; Lu and Leinonen 2005.
Qualitative synthesis
Angeles 2004.
Number synthesis Hain 1961; Crosley 1964; Yang 1983; Raghavan 1996.
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of permutations based on the study of the permutations of given combinations of links and joints as a second stage. Both stages are founded on the systematic generation of all possible mechanisms that can give rise to the desired kinematic performance. Freudenstein (1967) also associated type synthesis with “the determination of the structure of the mechanism from desired kinematic performance”. Type and number synthesis does not necessarily mean identifying a single mechanism type from a multitude of possibilities (Kota 1992).
Erdman (1987) pointed out that an improper choice of kinematic configuration may demand sophisticated control algorithms to undo poor type selection. Kota (1992) emphasized that a rational approach to mechanism design requires an exhaustive survey of all possibilities before the various solutions are compared and analyzed.
Eckhardt (1998) pointed out that the design of a machine or mechanism or any moving mechanical system always starts with kinematic synthesis defined by (i) the functional relationship between the parts, (ii) how those parts are interconnected, (iii) how these parts move relative to each other. Only after choices have been made regarding these factors, can matters such as materials, strengths, manufacturing, and costs be seriously addressed. Failure to devote the proper attention to kinematics “up front” can, and often does, lead to the design of a system with substandard or suboptimal performance and/or unsatisfactory reliability.
The main issues in systematic synthesis are the following: proper classification of mechanisms, systematic representation of mechanisms (symbolically or otherwise) and means for generating different possibilities in a systematic way (Kota 1992). The process of generating all possible distinct kinematic chains is dificult and time consuming (Rao and Deshmukh 2001). Nevertheless, it is desirable to enumerate the kinematic chains systematically and also to know the inherent characteristics of a chain related to its structure so that all the acceptable chains can be evaluated in depth before making the final selection for a specified task.
The common point of the various methods developed for generating systematically the possible solutions is the use of a mobility criterion as the main structural parameter for the synthesis approach of mechanisms. The main difference between these methods resides in the tool used to define the generating process.
Various approaches are reported in the literature for the synthesis of mechanisms from intuitive and empirical knowledge to formalized methods, as shown in Table 5.2. The large part of these methods was applied to synthesis of planar mechanism.
5.1 Structural synthesis: a systematic approach in mechanism design 303
Table 5.2. Approaches used for the synthesis of mechanisms
Approach
References
Visual inspection and intuition
Grübler 1883, 1885; Alt 1921; Hain 1955; Crossley 1964.
Assur groups
Assur 1913, 1914; Manolescu et al. 1964; Manolescu 1964, 1965, 1968; Manolescu and Tempea 1967a,b.
Franke’s notation
Franke 1958. Davies and Crossley 1966; Haas and Crossley 1969.
Structural graphs
Paul 1960; Freudenstein and Dobrjanskyj 1964; Manolescu 1964; Crossley 1965; Woo 1967; Dobrjanskyj and Freudenstein 1967;Freudenstein 1967; Huang and Soni; Kiper and Schian 1975; Turtle 1987. Dudiţă and Diaconescu 1986; Dudiţă et al. 1989; 2001a,b;
Adjacency matrices
Hunag and Soni 1972; Hwang and Hwnag 1992; Wenhui et al. 1995; Murphy et al. 1996, Lu and Leinonen 2005.
Baranov trusses
Manolescu 1973.
Joint substitution
Jensen 1991.
Associated linkage
Johnson 1978.
Transformation of binary chains
Mruthyunjaya 1979, 1984a,b,c.
Finite groups
Tuttle and Peterson 1987; Tuttle et al. 1987, 1988, 1989.
Prime chain
Dijksman and Timmermans 1994.
Melbourne method
Tischler et al. 1995a,b.
Structural groups
Mitrouchev, 2001; Alizade and Bayram 2004, Alizade et al. 2006.
Evolutionary morphology
Gogu 2005a, 2005a, 2007a.
Virtual chain Kong and Gosselin 2006.
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As it was shown in Table 1.6, the systematic synthesis of spatial mechanisms used in parallel robots is founded on: four main methods: (i) group theory, (ii) screw theory, (iii) velocityloop equations, and (iv) theory of linear transformations. The four methods determine, by using different mathematical tools, the set of kinematic chains defined by a given number and type of independent motions of a distal link with respect to a reference link. The various limb topologies and solutions for their connection in parallel mechanisms are generated in two sequential stages.
This chapter focuses on the presentation of a new systematic approach to structural synthesis of parallel mechanisms, integrating evolutionary morphology and the complete set of structural parameters (mobility, connectivity, overconstraint and redundancy) demonstrated via the theory of linear transformations.
5.2. Morphological and evolutionary approaches
In general, any abstract task can be thought of as solving a problem, which, in turn, can be perceived as a search through a space of potential solutions. Generation of this space is an inventive activity inherent to any innovationoriented synthesis approach. Too much innovation is guided by mystery. An inventive structured approach eliminates mystery, errors and time bombs, and reveals new solutions while remaining adaptable to future change. In an inventive structured approach, the inventive effort becomes an inherent judgment process, traceable and reproducible. An inventive structured approach is ideally suited to engineering design and for inventive problem solving (Ritchey 1998; Gogu 2000). Evolutionary morphology is such a structured approach to inventive design, combining features of morphological research and evolutionary algorithms. The morphological approach is a methodology, first proposed by Zwicky (1969) to enhance the systems engineering process. The morphological approach determines the requirements and conceptual design solutions of a system by considering the "totality" of all solutions included in a morphological box. This is an initial phase before the optimization process looks for “the best solution”. The evolutionary algorithms are stochastic techniques whose search methods for an optimal solution model some natural phenomena: genetic inheritance and Darwinian striving for survival.
We use “morphosis” as the morphological terminology for change, replacing one system by the next. This concept is based on the fundamental scientific method of alternating between analysis and synthesis. It can be trusted as a useful, qualitative method for investigating complex problems
5.2. Morphological and evolutionary approaches 305
which cannot be treated by direct mathematical formalization (equations), causal modelling or simulation (Ritchey 1998). We differentiate between static, dynamic and evolutionary morphosis. In static morphosis, a system is disassembled into very tiny components, and reassembled into a completely different system that has no obvious similarity with the previous system. In dynamic morphosis, an existing system changes by enhancing, adding or reducing existing properties (parameters). Evolutionary morphosis has the characteristics of both static and dynamic morphosis.
5.2.1 Morphological approaches
The term morphology comes from the ancient Greek (morphe) and means shape or form. The general definition of morphology is "the study of form or pattern", i.e. the shape and arrangement of parts of an object, and how these "conform" to create a “whole” or Gestalt. The "objects" in question can be physical objects (e.g. an organism, an anatomy, an ecosystem) or mental objects (e.g. linguistic forms, concepts or systems of ideas).
The term morphology, as an explicitly defined scientific method, was first mentioned by J. W. Goethe (17491832). In addition to poetry, dramas, and novels, Goethe continued his work in science, studying structures and shapes in mineralogy, geology, and osteology (the study of bones). In 1784, he had made the discovery, by methods which foreshadowed the science of comparative morphology, that the human jawbone contained traces of a structure similar to the intermaxillary bone in other mammals. In 1790, he wrote Versuch, die Metamorphose der Pflanzen zu erklären (Essay on the Metamorphosis of Plants), which further developed his ideas on comparative morphology and to some extent foreshadowed Darwin's ideas on organic evolution. His "comparative morphology" of botany describing how animals and plants changed shape and evolved into new species was a breakthrough in thinking.
Today, morphology is associated with a number of scientific disciplines. In linguistics, morphology is the study of word formation and how the parts of words relate to each other. Morphology is used in disciplines where formal structure and structural interrelations, and not necessarily quantity, are the central issues, e.g. geology, anatomy, botany, zoology etc.
Zwicky (1969) proposed a generalised form of morphological research. He proposed to generalize and systematize the concept of morphological research and include not only the study of the shapes of geometrical, geological, biological, and generally material structures, but also to study the more abstract structural interrelations among phenomena, concepts, and ideas, whatever their character might be.
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Essentially, general morphological analysis is a method for identifying and investigating the total set of possible relationships or “configurations” contained in a given complex problem. In this sense, it is closely related to typology construction (Bailey 1994), although it is more generalised in form and conceptual range.
Although Zwicky coined the term morphological analysis, the technique predates him and can be traced back to the “universal method of thinking” presented by the Catalan philosopher Ramon Lull (12331315) in his work Ars Magna (The Great Art) published in 1275. Zwicky was the first to use the technique in modernday applications. He used morphological research in new product and process ideation such as jet engines (Zwicky 1947), propulsive power generation (Zwicky 1962) and astronomical investigation (Zwicky 1957). After his main work on the subject (Zwicky 1969) originally published in 1966 by Droemersche Verlagsanstalt Th. Knaur Nachf., München/Zürich, under the title Entdecken, Erfinden, Forschen im morphologischen Weltblid, morphological research was extensively applied to engineering design (Norris 1963), manufacturing and process design (Royston 1969; Willis and Hawthone 1969), systems engineering (Hall 1969), technological forecasting and planning (Ayres 1969; Gerardin 1973).
Zwicky presented three morphological methods based on systematic field coverage, morphological box, and negation and construction (Zwicky 1969). The method of complete field coverage searches all solutions of a welldefined problem. Starting from known pegs of knowledge, infiltrating the surrounding fields, and using a sufficient number of “principles of thought” we strive to discover new facts, formulate new problems, and if necessary invent and produce needed new materials, instruments and procedures. Some of the major fundamental principles of thought that we need are based on our ability to count and recognize identities, equalities, differences and coincidences in time and space. The method is applicable to three types of cases; these involve interrelations among certain objects, interrelations among certain natural phenomena, and the interplay between certain abstract concepts.
Among the methods and procedures that have been conceived and perfected through the morphological approach, the construction of the morphological box of all possible aspects and solutions of a given problem, as well as the unbiased evaluation of the solutions found, are two of the most effective. A morphological box – also fittingly known as a ”Zwicky box” – is constructed by setting the parameters against each other in an ndimensional morphological box. Each cell of the ndimensional box contains one particular ”value” or condition from each of the parameters, and thus marks out a particular state or configuration of the problem. Zwicky
5.2. Morphological and evolutionary approaches 307
(1969) summarises the five iterative steps of the morphological box method:
1. The problem to be solved must be concisely formulated. 2. All of the parameters that might be of importance for the solution of
the given problem must be localized and analysed. 3. The morphological box or multidimensional matrix, which contains
all of the potential solutions of the given problem, is constructed. 4. All the solutions contained in the morphological box are closely scru
tinized and evaluated with respect to the purposes that are to be achieved. 5. The optimally suitable solutions are selected and applied, provided
the necessary means are available. Of course, the matrixing of parameters, in order to uncover the multi
plicity of relationships associated with a complex problem, is nothing new. The virtually universal use of “fourfold tables” and the study of typology construction as a classification technique attest to this fact (Bayley 1994). The morphological box represents a static morphosis based on a simple combinatorial approach. However, Zwicky’s highly systematic approach to this field should not be underestimated. Used properly – and on the right types of problems – the method is both deceptively complex and rich.
The method of negation and construction, the third morphological method presented by Zwicky, is inspired by pure mathematics. The insight gained as a result of any negation is immediately used for the purpose of future construction. Following up the wellreasoning negation of apparent truths and of the so called absolute facts with the constructive use of the vistas that thus open themselves, we succeed not only in making sporadic discoveries and inventions but whole groups and entire classes of them.
The three morphological methods proposed by Zwicky have little theoretical base. They combine analysis and synthesis with some general creativity paradigms, called “principles of thought” such as combination, inversion and hypothesis smashing.
5.2.2 Evolutionary algorithms
The beginnings of the evolutionary algorithms can be traced back to the early 1950s when several biologists used computers to simulate biological systems (Goldberg 1989). There are several variants of evolutionary algorithms (genetic algorithms, evolution strategies, evolutionary programming, genetic programming) and there are also many hybrid systems which incorporate various features of these paradigms. However, the structures of evolutionary methods are very much the same (Dasgupta and Michalewicz 1997).
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An evolutionary algorithm maintains a population of individuals, { }1( ) , . . . ,t t
nP t x x= for iteration t. Each individual represents a
potential solution to the problem at hand, and is implemented as a data structure S. Each solution 1
tx is evaluated to give some measure of its fitness. Then, a new population (iteration t+1) is formed by selecting the fittest individuals. Some members of the new population undergo transformations by means of genetic operators to form new solutions. There are unary transformations im (mutation type), which create new individuals by small changes in a single individual ( :im S S→ ), and higher order transformations jc (crossover type), which create new individuals by combining parts from several (two or more) individuals ( : ...jc S S S× × → ). After a number of generations, the algorithm converges to a “best individual”. It is hoped that the best individual represents a nearoptimum (reasonable) solution.
Despite powerful similarities between various evolutionary algorithms, there are also many differences between them, often hidden on a lower level of abstraction. They use different data structures for their chromosomal representation; consequently, the genetic operators are also different. They may or may not incorporate some complementary information to control the search process. There are many methods for selecting individuals for survival and reproduction. These methods include: a) proportional selection, where the probability of selection is proportional to the individual’s fitness, b) ranking methods, where all individuals in a population are sorted from the best to the worst and probabilities of their selection are unchanged for the whole evolution process, and c) tournament selection, where some number of individuals (usually two) compete for selection to the next generation. This competition (tournament) step is repeated throughout the population a number of times. The generational policy is also an important strategy. It is possible to replace the whole population by a population of offspring, or it is possible to select the best individuals from two populations, a population of parents and a population of offspring. This selection can be done in a deterministic or nondeterministic way. It is also possible to produce a few (in particular a single) offspring which replace some individuals in a steady state generation policy. Also we can use an elitist model which keeps the best individual from one generation to another.
The representations used for a particular problem together with the set of genetic operators constitute the essential components of any evolution
5.2. Morphological and evolutionary approaches 309
ary algorithms. These are the key elements which allow us to distinguish between various paradigms of evolutionary methods.
Genetic algorithms, as they are known today, are closely related to the work done in the late 1960s and early 1970s at the University of Michigan under the direction of John Holland (1975). Genetic algorithms (GAs) were conceived to model adaptation processes, mainly operated on binary strings and used a recombination operator with mutation as a background operator. Mutation flips a bit in a chromosome, and a crossover exchanges genetic material between two parents. The fitness of an individual is assigned proportionally to the value of the objective function for the individual. Individuals are selected for the next generation on the basis of their fitness. The combined effect of selection, crossover and mutation gives a socalled reproductive schema growth equation. The crossover operator enables structured, yet random information exchange. The mutation operator introduces greater variability into the population. To apply a genetic algorithm to a particular problem, it is necessary to design a mapping between a space of potential solutions for the problem and a space of binary strings of some length. This is not a trivial task and, often, it has an important impact on the final solution (Michalewicz 1996), (Bäck 1996). Evolutionary morphology contributes to solving this task. Conceptual aspects inspired by evolutionary theory are integrated into evolutionary morphology to generate qualitative diversification of the initial population.
Evolution strategies (ESs) were developed to solve parameter optimization problems (Schwefel 1994). The earliest evolution strategies were based on a population consisting of one individual only. There was also only one genetic operator used in the evolution process: a mutation. The interesting idea (not presented in GAs) was to use a chromosome that represents an individual as a pair of floatvalued vectors i.e ( , )v v x σ= . The first vector x represents a point in the search space and the second vector is a vector of the standard deviations. Mutations are realized by replacing x by 1 (0, )t tx x N σ+ = + , where (0, )N σ is a vector of independent random Gaussian numbers with a mean of zero and standard deviation σ , which remains unchanged during the evolution process. This is in accordance with the biological observation that smaller changes occur more often than larger ones. The offspring, as a mutated individual, is accepted as a new member of the population, and it replaces its parents if it has better fitness and all imposed constraints are satisfied. Otherwise, the offspring is eliminated and the population remains unchanged (Schwefel 1995).
Evolutionary algorithms are methods for solving optimizationoriented problems which are not suited for solving conceptual design oriented prob
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lems. They always start from a given initial population of solutions and do not solve the problem of creating these solutions.
5.3 Evolutionary morphology
Evolutionary morphology (EM) is a new structured approach to inventive design proposed by the author (Gogu 2005a).
In this section we emphasize the structure of EM and its specific differences with respect to morphological research and evolutionary algorithms. To help understanding of the proposed formalization, the main paradigms of EM are presented in a closed relation with the biological background of the synthetic theory of evolution.
EM is defined as a 6tuple
t t t tEM ( ,E, , , , )Φ Ο Ψ Σ ι= (5.1)
applicable to each generation t of the morphological evolutionary process, in which :
1 n( ,..., )Φ φ φ= is the set of design objectives,
1( ,..., )nE ε ε=  the set of constituent elements,
t 1 n( ,..., )Ο ο ο=  the set of morphological operators applicable at each generation t,
1t n( ,..., )Τ τ τ=  the set of evolution criteria from generation t to generation t+1
1( ,..., )t nσ σΣ =  the set of solutions at each generation t, called morphological products or morphologies at generation t,
{ }: ,i true falseι ι →  a termination criterion for EM.
5.3.1 Design objectives
The set of design objectives 1 n( ,..., )Φ φ φ= represents the required characteristics of the final solution; these could be expressed by any kind of formal language: mathematical (equalities, inequalities), semantic (textual description), graphical (parametric design), etc. The design objective is not necessarily expressed by an objective function as in genetic algorithms. EM is not an optimisation technique; it is an approach towards generating the total set of possible solutions respecting the complete set of design objectives. To help generate the total set of possible solutions, the EM oper
5.3 Evolutionary morphology 311
ates preferably with qualitative characteristics or with quantified characteristics limited to few discrete values.
5.3.2 Constituent elements
The set of constituent elements 1( ,..., )nE ε ε= represents qualitative “protoelements” (from the Greek “proteios” or primary and element), used to build the solutions during the morphological evolutionary process.
The constituent elements belong to homogeneous groups, called primary groups, which could be one of the following:
a) material parts (subsystems) of the final system, b) geometrical, physical or chemical characteristics, c) geometrical, physical, chemical or organisational principles, d) natural phenomena, e) human behaviour, f) interrelations between certain parts, characteristics, principles, natural
phenomena and human behaviour, …etc. This list of primary groups is not exhaustive; it could be completed with
problemspecific groups. The protoelements are qualitative elements. For instance distance, as a geometrical characteristic, could be a qualitative constituent element when it is important as a qualitative dimension and not as a specific value with a specific unit of measurement. Distance, as a constituent element, is equivalent to a length in parametric design. In conceptual design the most important thing is to consider or ignore the existence of this distance as a design parameter. The absolute value and the unit of measurement are not important at this design stage.
The constituent elements represent the evolving units in EM. They are equivalent to nucleic acids and proteins of a cell structure. EM is equivalent to chromosome generation, a common gene pool included in the genotype of the individuals. The genetic algorithms manipulate the genotype of the initial population to find the “best” individual. EM generates the initial population genotype.
The identification and study of genes are of great interest to biologists, and are also of medical importance when a particular gene is involved in disease. All the genes of an organism, distributed over several chromosomes, are summarized under the term genome. Biological research has revealed that the human genome contains approximately 100,000 genes, of which about 4,000 may be associated with disease. A globally coordinated effort called the Human Genome Project was started in 1990 to characterize the entire human genome. The primary goal of the Human Genome
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Project is the generation of various genome maps, including the entire nucleotide sequence of the human genome. The Human Genome Project has been greatly assisted by the ability to clone large fragments of DNA into artificial yeast chromosome vectors for further analysis, and the automation of many techniques such as DNA sequencing. The identification and study of constituent elements in EM is equivalent to the generation of the “genome of the technical system” in engineering conceptual design. This generation could help both in understanding and avoiding functional flaws as well as in diversifying and improving the technical systems designed by the engineer.
A constituent element has a monary or a binary structure. The monary structure ( iε ) indicates that the constituent element maintains the same feature. The binary structure indicates that the constituent element has two different features. In conceptual design we often use ( iε =0 or 0iε ≠ ) which correspond to the absence ( iε =0) or the presence ( 0iε ≠ ) of a feature a in a solution jσ . Any other two distinct features could also be used; for instance the parallel and perpendicular geometric position ( iε ⇒ or iε ⇒⊥ ). The solutions built up exclusively from constituent elements having monary structure are equivalent to haploid organisms in evolutionary theory. The solutions built up exclusively from constituent elements having binary structure are equivalent to diploid organisms. Haploid organisms are modelled by one set of genetic information per individual which is unicellular from a biological point of view. In the diploid organisms (e.g., in the case of humans) each body cell includes two sets of chromosomes, such that for each chromosome two homologous forms exist. The structure of both forms of chromosomes is identical, but their genetic information content may be different. In contrast, haploid organisms possess only one set of chromosomes per body cell. For diploid organisms, corresponding genes in homologous chromosomes are called alleles. The fact that alleles of diploid organisms may be identical (homozygotic) or different (heterozygotic) is worthy of mention here. The constituent element with a monary structure in EM is homozygotic for a particular gene in evolutionary theory. A constituent element with a binary structure is heterozygotic for the particular gene. Both alleles are carried in the feature of the constituent element, as in the genetic material of the individual, but if one is dominant, only that one will be expressed. The other allele is called recessive.
5.3 Evolutionary morphology 313
5.3.3 Morphological operators
The morphological operators t 1 n( ,..., )Ο ο ο= are (re)combination, mutation, migration and selection. These operators are deterministic and can be applied at each generation t of the EM.
Morphological combination
Morphological combination is applied directly to the constituent elements to obtain the complete set of solutions at the first generation. Morphological recombination is used to combine solutions of generation t with solutions of the previous generations, including the constituent elements, to obtain the complete set of solutions at generation t+1. We use the morphological box to generate the total set of possible solutions at each generation. In EM, (re)combination is a deterministic operator and not a probabilistic one as in genetic algorithms. The mechanism of morphological (re)combination is combinatorial. The evolution is made by adding parts/characteristics to the previous morphology, and not by interchanging (crossingover) characteristics as in intrachromosomous recombination (crossover) specific to genetic algorithms. The (re)combination in EM is equivalent to chromosome generation in ontogenesis.
We know that chromosomes contain the genetic blueprints for a specific organism. Several thousand genes are arranged in a single line on a chromosome, a threadlike structure of nucleic acids and protein. Different groups of organisms have different numbers of chromosomes; for example, human beings have 23 pairs (46 total) of chromosomes, divided into 8 different sets according to their size and shape. In humans 23 of the chromosomes come from the mother and 23 from the father. The variation present in individuals is a reflection of the genetic recombination of these sets of chromosomes from generation to generation in phylogeny.
Morphological mutation
Morphological mutation modifies variability in the process of solution generation, as in genetic algorithms; the difference is that morphological mutation is a deterministic operator (all mutations are predefined) and not a probabilistic one. Morphological mutations act in two ways:
a) increasing variability by interchanging a dominant allele by a recessive one,
b) diminishing variability by changing a binary constituent element into a monary one (by cutting up the dominant or the recessive allele).
314 5 Structural synthesis
Morphological migration
Morphological migration acts by replacing dominant constituent element ( iε ) by a new dominant constituent element ( jε ). The constituent elements
iε and jε must belong to the same group of homogeneous elements. Morphological migration, like mutation in genetic algorithms, introduces more variability in the result of the generation process.
Morphological selection and evolution criteria
Morphological selection acts to eliminate the nonevolving intermediary solutions and the incompatible final solutions. Nonevolving intermediary solutions are the solutions in a generation t that did not evolve with respect to the previous generation. Incompatible final solutions are solutions which do not conform to the set of final objectives. The set of evolution criteria from generation t to t+1 allow us to eliminate the nonevolving solutions at generation t, and to stop them spreading to the following generations.
5.3.4 Set of solutions
The set of solutions 1( ,..., )t nσ σΣ = at each generation t is also the called morphology at generation t. The set of solutions in the final generation represents the result of EM which we call the morphological product. It corresponds to the first generation that satisfies all the final objectives. When the morphological product is obtained, the termination criterion stops the evolution algorithm.
5.3.5 General structure of the evolutionary morphology
The general structure of evolutionary morphology is presented in Fig. 5.1. EM evolves from the constituent elements to the final generation t=n in the following steps:
• Define the set of final objectives. • Define the set of constituent elements and identify their struc
ture (monary or binary constituent element) and the dominant allele for the binary elements.
• Define the set of morphological operators and identify the constituent elements and the morphologies on which they act.
5.3 Evolutionary morphology 315
Fig. 5.1. General structure of the evolutionary morphology
• Verify the termination criterion, if { }: i falseι ϕ → continue else go to Procedure 1.
• Apply the morphological combination of constituent elements to obtain the total set of morphologies of the first generation.
• Verify the termination criterion, if { }: i falseι ϕ → continue else go to Procedure 1.
• Apply the set of evolution criteria from generation 1 to generation 2 and eliminate the nonevolving solutions.
• Apply the morphological recombination of the constituent elements and the morphologies issued in generation 1 to generate the total set of morphologies of generation 2.
• Verify the termination criterion, if { }: i falseι ϕ → continue else go to Procedure 1.
• Apply the set of evolution criteria from generation 2 to generation 3 and eliminate the nonevolving solutions from generation 2.
… • Apply the morphological recombination of the constituent ele
ments and the morphologies issued in generation t=n1 to generate the total set of morphologies of generation t=n.
• Verify the termination criterion, { }: i falseι ϕ → , that means t=n is the last generation.
316 5 Structural synthesis
Procedure 1 while (not terminationcriterion) do while (migration operator defined) do apply migration operator by interchanging a domi
nant constituent element ( iε ) by a new dominant constituent element ( jε )
apply the selection criterion to eliminate incompatible solutions
end while (mutation operator defined) do apply mutation operator by interchanging a domi
nant constituent element by the recessive one apply the selection criterion to eliminate
incompatible solutions end end The inventive structured approach is a key ingredient in conceptual en
gineering design. It allows the designer to examine all possible options at minimum cost by an inventive approach. EM is based on the fundamental scientific method of alternating between analysis and synthesis. For this reason, it can be trusted as a useful, qualitative method for investigating inventive problems which cannot be treated by formal mathematical methods, causal modelling or simulation. Furthermore, EM gives a fairly clear “audit trail”, which makes the judgmental processes relatively traceable, reproducible and inventive. The “solution space” of EM consists of the subset of configurations which satisfy some criteria (design objectives).
Of course, as is the case with everything else, the output of an EM is no better than the quality of its input. The final solutions depend a lot on the choice of constituent elements. This choice is a problemdependent task. The main advantage of this approach consists in finding the complete set of possible solutions in an evolutionary sequence based on a set of protoelements. EM may help us to invent new relationships or configurations, which may not be so evident, or which we might have overlooked by other less structured methods. Most importantly, it encourages the identification and investigation of boundary conditions, i.e. the limits and extremes of different contexts and factors. The evolutionary sequence limits the generation and the proliferation of incompatible solutions.
The main paradigms of EM are inspired from the synthetic theory of evolution developed by modern biology. EM integrates some features of morphological research and evolutionary algorithms but it represents a
5.4 General approach to structural synthesis of parallel robots 317
radically different approach. Evolutionary algorithms start from a given initial population to obtain an optimum solution with respect to a fitness function. EM creates this initial population to enhance the chance to obtain a “more global optimum” by qualitative diversification of the initial population. EM and evolutionary algorithms are complementary methods. Evolutionary algorithms are optimized oriented methods; EM is a conceptual design oriented method.
5.4 General approach to structural synthesis of parallel robots
We consider the general case of a parallel robot F ← G1G2…Gk in which the endeffector n≡nGi is connected to the fixed base 1≡1Gi≡0 by k≥2 simple or complex kinematic chains Gi (1Gi2Gi…nGi) i=1,2,…,k. The parallel robot has p joints. The limbs can be identical or different. The first link 1Gi of each limb is fixed to the base 1Gi≡0, and the distal link is the moving platform nAi≡n. Link 1≡1Gi≡0 is also called the fixed platform. We denote by iq (i=1,…,MF) the velocities of the actuated joints, and by iw (i=1,…, SF) the translational and angular velocities of the moving platform. MF represents the mobility of the parallel robot F, and F
n / 1S the connectivity between mobile platform n≡nGi and the fixed base 1≡1Gi≡0 in the parallel robot F. The parallel robot F ← G1G2…Gk is TxRytype, with x+y=SF and 2≤ SF≤ 6, for x=1,2,3 and y=1,2,3. The mobile platform of the parallel robot TxRytype can make x independent translations and y independent rotations.
5.4.1 General conditions for structural synthesis of parallel robots via theory of linear transformations
In chapter 2 we saw that a mechanism F corresponds to a linear transformation F from joint velocity vector space U into external velocity vector space W. Velocity spaces U and W are finite dimensional vector spaces. The Jacobian matrix J is the matrix of this linear mapping. To each vector q , in the joint vector space U, the linear transformation F assigns a uniquely determined vector ( )F q in the vector space W. In parallel robots, the vector q represents the actuated joint velocity vector, and it is composed of the relative velocities of the actuated joints. The vector ( )F q in W is the image of the vector q in U under the transformation F. The
318 5 Structural synthesis
image of U in W represents the range RF of F, and is called the operational velocity vector space. The connectivity /
Fa bS between two links a and b in
the mechanism F represents the number of independent relative velocities allowed by the mechanism between the two links. It is given by the dimension of the vector space /
Fa bR of relative velocities between links a and b in
mechanism F. The range /Fa bR is a subspace of RF, and implicitly a sub
space of W. The connectivity characterizes the degree of freedom of links a and b in the mechanism F. The number of overconstraints of F is denoted by NF, and its degree of structural redundancy by TF. All these parameters are defined in chapter 2.
The existence of F ← G1G2…Gk involves the following set of general conditions for any position of the mechanism when 1q ≠0 (i=1,2,…,MF):
F G1 G 2 Gk( R ) ( R R ... R )= ∩ ∩ ∩ (5.2)
SF≤MF≤ MGi (i=1,…,k), (5.3)
MGi≥ SGi≤6 (i=1,…,k), (5.4)
MF≥p
ii 1
f 6q=
−∑ , (5.5)
where: RGi is the operational velocity vector space of limb Gi isolated from the parallel robot F, MGi is the mobility of limb Gi isolated from F, SGi= Gi
nGi / 1S =dim(RGi) is the connectivity between the mobile platform n≡nGi and the fixed base 1≡1Gi≡0 in limb Gi isolated from F, SF=dim(RF) is the connectivity between the mobile platform n≡nGi and the fixed base 1≡1Gi≡0 in the parallel robot F, q is the number of independent closed loops of F, fi is the mobility of the ith joint. Conditions (5.3)(5.5) follow from Eqs. (2.211), (2.220) and (2.222).
Conditions (5.3)(5.4) become
SF=MF≤ MGi (i=1,…,k), (5.6)
MGi=SGi≤6 (i=1,…,k), (5.7)
for non redundant parallel robots (TF=0); and
SF<MF≤ MGi (i=1,…,k), (5.8)
5.4 General approach to structural synthesis of parallel robots 319
MGi>SGi≤6 (i=1,…,k), (5.9)
for redundant parallel robots with TF>0. Condition (5.5) becomes
MF=p
ii 1
f 6q=
−∑ , (5.10)
for non overconstrained parallel robots (NF=0); and
MF>p
ii 1
f 6q=
−∑ , (5.11)
for overconstrained parallel robots with NF>0. We recall that MF, NF, TF are given by Eqs. (2.211), (2.220), (2.222) and
Gip
Gi i Gii 1
M f r=
= −∑ (5.12)
where pGi represent the number of joints of the limb Gi and rGi the number of joint parameters that lose their independence in the closed loops of limb Gi. For simple (elementary) limbs, rGi=0; and for complex limbs rGi is given by Eq. (2.209) – see also section 2.3.8.
Eqs. (5.2)(5.11) are applicable to all parallel robots with any degree of motion coupling.
Parallel robots with uncoupled motions obey some additional conditions. In this case, the operational velocity vector space RF has (i) a unique basis (RF) of dimension SF for any position of the mechanism when iq ≠0 (i=1,2,…,MF) and (ii) to each actuator locked, i.e. iq =0, a specific unique basis of dimension SF1 is associated with the parallel robot. These conditions express the possibility of direct actuation of each independent motion of the mobile platform by just one specific actuator.
5.4.2 General approach to structural synthesis of parallel robots via evolutionary morphology
Evolutionary morphology is used for structural synthesis of each limb Gi (i=1,…,k) by taking into account the general conditions defined in the previous chapter via the theory of linear transformations.
320 5 Structural synthesis
Set of design objectives
In structural synthesis of TxRytype PMs with x+y=SF and 2≤ SF≤ 6, for x=1,2,3 and y=1,2,3 we adopt the general set of design objectives
1 8 i( ,..., )Φ φ φ += i=1,2,…,MF presented in Table 5.3. Design objectives 1 8,...,φ φ are applicable to the structural design of all
parallel robots, and 9 8 i,...,φ φ + (i=1,2,…,MF) represent additional objectives for PMs with uncoupled motions. Design objectives 1 2,φ φ and 3φ are represented by qualitative characteristics. In the first stage of structural synthesis, PMs with orthogonal non redundant limbs are generated. In such a limb, a link is connected by revolute (R) and prismatic pairs (P) with parallel (║) or orthogonal (⊥ ) axes/directions. The revolute and prismatic pairs with orthogonal position are adopted for ease of manufacture. We know that any kind of motion of the mobile platform can be generated by concatenating revolute and prismatic pairs. Various multidegreesof freedom joints can be obtained by combining revolute and prismatic joints; for example a cylindrical joint (C) is equivalent to a combination R║P or P║R, a spherical joint (S) is equivalent to three revolute pairs with intersecting axes (R⊥R⊥R), a combination of a revolute and a helical joint (H) with the same axis (R║H
Table 5.3. Set of design objectives
No. Notation Objective Related equations
1 1φ k≥2 elementary limbs or complex limbs integrating c≥1 closed loops in a limb

2 2φ Generic limbs with revolute (R) and prismatic (P) pairs

3 3φ Orthogonal limbs  4 4φ Connectivity of the moving platform SF Eq. (2.207) 5 5φ Unique basis of the operational velocity
space (RF) Eq. (5.2)
6 6φ Mobility MF>2 Eq. (2.211) 7 7φ Number of overconstraints QN 0≥ Eqs. (2.220),
8 8φ Degree of structural redundancy FT 0≥ Eq. (2.222) 9 8 iφ + Direct actuation of unidirectional mo
tion iw by iq (i=1,2,…,MF) in PMs with
decoupled motions
Eqs. (2.207), (5.2)
5.4 General approach to structural synthesis of parallel robots 321
or H║R) is equivalent to a cylindrical joint and implicitly to a combination R║P or P║R. The same is true for a planar pair or any other pair with more than one degree of freedom.
Set of constituent elements
The following set of constituent elements 1i 4iE ( ,..., )ε ε= is used: a) 1iε  kinematic pair (KPi) between the links i 1 and i, if i iKP P⇒ (prismatic pair) than 1i 1ε = , and if i iKP R⇒ (revolute pair) then 1iε =0, b) 2iε  axial offset (ai) of the link i, if ia 0≠ than 2i 1ε = , and if ai=0 than 2iε =0, c) 3iε  length (bi) of the link i, if 0ib ≠ than 3i 1ε = , and if bi=0 than 3iε =0, d) 4iε  relative position pi of the axes/directions of the two joints adja
cent to the same link i, if ip ⇒⊥ than 4i 1ε = , and if ip ⇒ than 4iε =0. We consider ji 1ε = as dominant and ji 0ε = as recessive alleles. In accordance with the TCS method for geometric modeling of robot
manipulators (Gogu et al. 1997), a link i could have two dimensions in an orthogonal kinematic chain: axial offset ai, measured on the axis of the pair (i1,i), and link length bi, defined by the normal to the axes of the two pairs adjacent to element i (Fig. 5.2, ae).
In general, the previous constituent elements have binary structure, except for the following elements, which have a monary structure:
a) 21ε =0, we make a1=0 by a proper choice of reference frame used in geometric modelling and Jacobian matrix calculation,
b) 2iε =0 and 3iε =0 if KPi between the links i 1 and i is a prismatic pair, that is the prismatic pair has only a direction and not a defined axis (Fig. 5.2, gj).
The constituent elements with monary structure are not affected by the morphological mutation operators.
Fig. 5.2 illustrates the geometrical meaning of the constituent elements included in the “genetic code” of link i. Kinematic pair KPi belongs to a constituent group representing material subsystems of the parallel
322 5 Structural synthesis
Fig. 5.2. Geometrical meaning of the constituent elements included in the “genetic code” of link i
manipulator (the joints). Dimensions ai and bi belong to a constituent group representing geometrical characteristics of the links, and relative position pi belongs to a constituent group representing interrelations among the axes of the revolute joints. We note that the constituent elements are qualitative. In this case, distance is important as a qualitative dimension and not as a specific value with a specific unit of measurement. Distance, as a constituent element, is equivalent to length in parametric design. In conceptual design the most important thing is to consider or ignore the existence of distance as a design parameter. The absolute value and the unit of measurement are not important at this design stage.
Other types of constituent elements could be introduced in the evolutionary process by the mutation operator, as we will see in the next sections, to obtain the diversification of the initial solutions.
5.4 General approach to structural synthesis of parallel robots 323
Set of evolution criteria
The evolutionary process from generation t to generation t+1 is analyzed by the set of evolution criteria t 1 2T ( , )τ τ= where 1τ and 2τ are connectivity criteria defined by
St<St+1 for Mt+1<MGi (5.13)
and
St=St+1= GiS for Mt+1=MGi. (5.14)
Evolution criterion (5.13) is applicable to the generations of non redundant limbs with Mt+1<MAi. The evolutionary process from generation t to generation t+1 with Mt+1=MGi is analyzed by the connectivity criterion 2τ defined by (5.14). This criterion indicates that the process is stationary. The solutions of the generation t+1 that do not obey Eqs. (5.13) and (5.14) are eliminated from the evolutionary process. This allows us to avoid the proliferation of nonevolving solutions in the next generations. These nonevolving solutions represent kinematic chains with structural singularities. For example, open kinematic chains with two parallel prismatic pairs, three prismatic pairs with directions parallel to the same plane, four parallel revolute pairs, etc., are nonevolving solutions with structural singularities that are not allowed to proliferate in the evolutionary formation of nonredundant limbs.
Set of morphological operators
The set of morphological operators t 1 2 3 4( , , , )Ο ο ο ο ο= is composed of 1ο  (re)combination, 2ο  selection, 3ο  mutation and 4ο  migration.
The mechanism of morphological (re)combination is combinatorial. Evolution is achieved by adding structural components (kinematic pairsKPi) and geometric characteristics (ai, bi and pi) to the previous morphology and not by interchanging (crossingover) characteristics as in crossover specific to genetic algorithms.
At least MF=SF generations are necessary to evolve by successive combinations from the first generation of protoelements with MGi=SGi=1 to a first solution of elementary limb with MF=SF satisfying the set of design objectives.
Morphological migration could introduce new constituent elements formed by new joints or combinations of joints into the evolutionary process. Some examples are presented in Table 5.4. The new and the replaced joints/combination of joints have the same mobility. We note that the num
324 5 Structural synthesis
number of joint parameters that lose their independence in a planar closed loop integrated in a complex limb is r=3.
The notations ⊥ and ║ used in Table 5.4 indicate the perpendicular or parallel positions of the joint axes. For example the notation R P⊥ ⊥ ║R
Table 5.4. Examples of new joints or combinations of joints that could be introduced in the evolutionary process by morphological migration
No. New joint or combination of joints
Replaced joint or combination of joints
1 Helical joint (H) Revolute joint (R) 2 Cylindrical joint (C) Two consecutive joints:
(i) a revolute and a prismatic joint with parallel axes/directions, (ii) two helical joints with parallel axes, (iii) a helical and a prismatic joint with parallel axes/directions, (iv) a helical and a revolute joint with parallel axes
3 Universal joint (U) Two orthogonal intersecting revolute joints ( R R⊥ ) adjacent to the link i when
4i 1ε = ( ip ⇒⊥ ), 2iε =0 (ai=0) and
3iε =0 (bi=0), 4 Spherical joint (S) Three orthogonal revolute joints with inter
secting axes ( R R R⊥ ⊥ ) 5 Planar joint (Pl) Open planar kinematic chain with three de
grees of freedom 6 Planar parallelogram
closed loop (Pa) with one degree of freedom
Prismatic joint (P)
7 R║Rb║Rtype or R║Rb1║…║Rbn║Rtype planar kinematic chains integrating one or more rhombus loops (Rb)
R P⊥ ⊥ ║Rtype planr kinematic chain
8 Pn2║Rtype planar kinematic chain integrating a planar loop with two degrees of freedom (Pn2)
P P R⊥⊥ ⊥ type planar kinematic chain
9 3Pn planar loop with three degrees of freedom
Open planar kinematic chain with three degrees of freedom
5.4 General approach to structural synthesis of parallel robots 325
indicates that the axis of the first pair is perpendicular to the direction of the second pair, the axis of the third pair is also perpendicular to the direction of the second pair, and at the same time parallel to the direction of the first pair. The notation R║Rb║R indicates that the axes of the first and last revolute joints are parallel to the revolute axes of the rhombus loop.
At each stage, morphological selection acts to eliminate the nonevolving intermediate solutions and the incompatible solutions. The nonevolving intermediate solutions are the solutions for a generation t that did not evolve with respect to the previous generation. The incompatible solutions are the solutions which do not satisfy the set of design objectives.
The set of evolution criteria from generation t to t+1 allows us to eliminate the nonevolving solutions at generation t, and to stop them spreading to the following generations.
Set of final solutions
The set of solutions 1( ,..., )t nσ σΣ = at each generation t is also called the morphology at generation t. The set of solutions that satisfies all design objectives represents the result of the EM; we call this the morphological product. When the morphological product is obtained, the termination criterion stops the evolution algorithm.
The various types of limbs generated by the general approach to structural synthesis can be systematized by taking into account their structural parameters: the type (simple or complex), mobility MG, connectivity SG, basis of operational velocity space (RG), degree of overconstraint (NG) and degree of structural redundancy (TG). If SG≥SF than a limb with connectivity SG could be used in a parallel robots F with connectivity SF of the moving platform. This allows the same limb to be used in different types of parallel robots. For this reason, limb synthesis is a key issue in the structural synthesis of parallel robots.
The various types of limbs generated by this systematic synthesis approach could combine simple or complex kinematic chains with prismatic or revolute pairs connected to the fixed base. The prismatic pair could accommodate a linear actuator, and the revolute pair a rotary actuator mounted on the fixed base. Usually the complex limbs integrate planar closed loops with one, two or three degrees of mobility. The simple limbs are not overconstrained (NG=0). Between three and six joint parameters could lose their independence in a planar closed loop. Usually the limbs are not redundant (TG=MGSG=0). Redundant limbs with MG>SG could be used in order to obtain simple solutions for parallel robots which have uncoupled motions, and are maximally regular.
326 5 Structural synthesis
The following chapters of this book point out on the general systematization of limbs which satisfy the set of design objectives
1 8( ,..., )Φ φ φ= defined in section 5.4.2. Limb connectivity is the main criterion for this systematization.
We focus on solutions, with a unique basis of the operational velocity space, that are useful for generating various topologies of parallel robots with decoupled, uncoupled, fullyisotropic and maximally regular parallel robots. Limbs with multiple bases of the operational velocity space and redundant limbs are also presented in these chapters. These limb solutions are systematized with respect to various combinations of independent motions of the distal link.
Other solutions with nonorthogonal intersecting axes may be also used in parallel robots. As we have already mentioned, only solutions with orthogonal and/or parallel axes of the kinematic joints are presented in this work. Such links and joints are easy to make.
Limbs with two degrees of connectivity are systematized in chapter 6. They are defined by the following independent motions of the distal link: two translational motions (section 6.1), two rotational motions (section 6.2) and one translational and one rotational motion (section 6.3). Examples of limbs with two degrees of connectivity and multiple bases of the operational velocity space are presented in section 6.4, and redundant limbs in section 6.5.
Limbs with three degrees of connectivity are systematized in chapter 7. They are defined by the following independent motions of the distal link: three translational motions (section 7.1), one rotational and two translational motions (sections 7.2 and 7.3), one translational and two rotational motions (section 7.4), and three rotational motions (section 7.5). Examples of limbs with three degrees of connectivity and multiple bases of the operational velocity space are presented in section 7.6, and redundant limbs in section 7.7.
Limbs with four degrees of connectivity are systematized in chapter 8. They are defined by the following independent motions of the distal link: one rotational and three translational motions (section 8.1), two translational and two rotational motions (section 8.2) and one translational and three rotational motions (section 8.3). Examples of limbs with four degrees of connectivity and multiple bases of the operational velocity space are presented in section 8.4, and redundant limbs in section 8.5.
Limbs with five degrees of connectivity are systematized in chapter 9. They are defined by three translational motions and two rotational motions (section 9.1) and two translational and three rotational motions (section 9.2). Examples of limbs with five degrees of connectivity and multiple
5.4 General approach to structural synthesis of parallel robots 327
bases of the operational velocity space are also presented in section 9.3 and redundant limbs in section 9.4.
The last chapter of this book presents some examples of nonredundant and redundant limbs with six degrees of connectivity. The complete set of solutions of nonredundant limbs has been generated by the systematic synthesis approach presented in this chapter. Due to space limitation the entire set of solutions cannot be presented. In fact, these limbs do not introduce restrictions in the motion of the distal link. For this reason, just the simplest solutions of limbs with six degrees of connectivity are used in parallel robots. They are presented in chapter 10.
The limbs presented in chapters 610 are defined by symbolic notations and illustrated in about 250 figures containing more than 1500 structural diagrams. Special attention was paid to graphic quality of structural diagrams to ensure a clear correspondence between the symbolic and graphic notation of joints and the relative position of axes.
The kinematic chains presented in chapters 610 are useful as innovative solutions of limbs in parallel, serial or hybrid robots. In fact, serial and hybrid robots may be considered as particular case of parallel robots, with just one limb which can be a simple, complex or hybrid kinematic chain, as we have seen in chapter 3 of this book.
The work is organized in two parts published in two distinct books. Part 1 presents the methodology proposed for structural synthesis and Part 2 the various topologies of parallel robots systematically generated by the structural synthesis approach. The limbs presented in chapters 610, will be used in Part 2 of this work for structural synthesis of TxRytype parallel robots with x+y=SF and 2≤ SF≤ 6, for x=1,2,3 and y=1,2,3 . Part 2 of this work will present the following topologies: translational PMs T2type (chapter 1), T1R1type PMs with screw motion (chapter 2), other T1R1type PMs (chapter 3), R2type spherical parallel wrists (chapter 4), translational PMs T3type (chapter 5), T2R1type PMs with planar motion (chapter 6), other T2R1type PMs (chapter 7), T1R2type PMs (chapter 8), R3type spherical parallel wrists (chapter 9), T3R1type PMs with Schönflies motion (chapter 10), other T3R1type PMs (chapter 11), T2R2type PMs (chapter 12), T1R3type PMs (chapter 13), T3R2type PMs (chapter 14), T2R3type PMs (chapter 15) and PMs with six degrees of freedom (chapter 16).
Parallel robots with coupled, decoupled and uncoupled motions, along with fullyisotropic and maximally regular solutions, are systematically presented in each chapter of Part 2. Innovative solutions of overconstrained or nonoverconstrained, nonredundant or redundantly actuated parallel robots with simple or complex limbs actuated by linear or rotating
328 5 Structural synthesis
motors are set up in Part 2 by applying the structural synthesis methodology presented in this chapter.
As we have already mentioned, many solutions for parallel robots obtained through this systematic approach of structural synthesis are presented here for the first time in the literature. The author had to make a difficult and challenging choice between protecting these solutions through patents, and releasing them directly into the public domain. By publishing them in this work, the author hopes to contribute to a rapid and widespread implementation of these solutions in future industrial products such as haptic devices, parallel kinematic machines, guiding, testing, control and tracking devices.
6 Limbs with two degrees of connectivity
Limbs with two degrees of connectivity (SG=2) are used in parallel robots F with the connectivity of the moving platform SF=2. The basis of the operational velocity space of these limbs contains just two independent velocities that can be two orthogonal translations, two orthogonal rotations or combination of one translation and one rotation. In this chapter and the following chapters as well, we denote by iv and iω (i=1,2,3) the independent translational and angular velocities of a characteristic point H situated on the distal link of a kinematic chain.
Nonredundant solutions generated by the evolutionary morphology approach are presented in the first four sections of this chapter. Some examples of redundant solutions are also presented in the last section. The first four sections present solutions of limbs defined by the following: (i) two translational motions, (ii) two rotational motions, (iii) one translational and one rotational motion, and (iv) multiple bases of the operational velocity space.
6.1 Limbs with two translational motions
These limbs give two translational velocities v1 and v2 in the basis of operational velocity space (RG)=( 1 2,v v ) for any position of the characteristic point H on the distal link. They give rise to two independent translations along with a constant orientation of the moving platform.
The simplest structural solutions for both simply and complex non redundant limbs with only revolute R and prismatic P joints are presented in Figs. 6.1 and 6.2. Prismatic or revolute joints may be connected to the fixed base. The following structural parameters MG=SG=2, (RG)=( 1 2,v v ) and TG=0 characterize these solutions.
The complex limbs integrate one or two planar parallelogram loops. As presented in chapter 3, a closed parallelogram loop Pa could combine up to three idle mobilities that may be two orthogonal rotations and one translation. The direction of the translation is perpendicular to the two idle
330 6 Limbs with two degrees of connectivity
Fig. 6.1. Simple limb PPtype (a) and complex limbs of types PPa (b) and PaPat (c) defined by MG=SG=2, (RG)=( 1 2,v v )
Fig. 6.2. Complex limbs of types PaP (a)(b) and PaPa (c) defined by MG=SG=2, (RG)=( 1 2,v v )
6.1 Limbs with two translational motions 331
Fig. 6.3. Parallelogram loops of types Pa (a), Pac (b), Pau (c), Pas (d), Pauu (e), Pacu (f), Pa* (g), Pass (h)
332 6 Limbs with two degrees of connectivity
Table 6.1. Structural parameters NG and rG of various types of limbs defined by (RG)=( 1 2,v v )
No. Limbtype NG rG 1 P ⊥ P 0 0 2 P ⊥ Pa and Pa ⊥ P 3 3 3 P ⊥ Pac, Pac ⊥ P, P ⊥ Pau and Pau ⊥ P 2 4 4 P ⊥ Pas, Pas ⊥ P, P ⊥ Pauu, Pauu ⊥ P, P ⊥ Pacu and Pacu ⊥ P 1 5 5 P ⊥ Pa* and Pa* ⊥ P 0 6 6 Pa ⊥ Pa 6 6 7 Pa ⊥ Pac, Pac ⊥ Pa, Pa ⊥ Pau and Pau ⊥ Pa 5 7 8 Pa ⊥ Pas, Pas ⊥ Pa, Pa ⊥ Pauu, Pauu ⊥ Pa, Pa ⊥ Pacu , Pacu ⊥ Pa,
Pac ⊥ Pac, Pau ⊥ Pau, Pac ⊥ Pau and Pau ⊥ Pac 4 8
9 Pa ⊥ Pa*, Pa* ⊥ Pa, Pac ⊥ Pas, Pas ⊥ Pac, Pac ⊥ Pauu, Pauu ⊥ Pac, Pac ⊥ Pacu, Pacu ⊥ Pac, Pau ⊥ Pas, Pas ⊥ Pau, Pau ⊥ Pauu, Pauu ⊥ Pau, Pau ⊥ Pacu and Pacu ⊥ Pau
3 9
10 Pa* ⊥ Pac, Pac ⊥ Pa*, Pa* ⊥ Pau, Pau ⊥ Pa*, Pas ⊥ Pas, Pas ⊥ Pauu, Pauu ⊥ Pas, Pas ⊥ Pacu, Pacu ⊥ Pas, Pauu ⊥ Pas, Pas ⊥ Pauu, Pauu ⊥ Pauu, Pauu ⊥ Pacu, Pacu ⊥ Pauu, Pacu ⊥ Pas, Pas ⊥ Pacu, Pacu ⊥ Pauu, Pauu ⊥ Pacu and Pacu ⊥ Pacu
2 10
11 Pa* ⊥ Pas, Pas ⊥ Pa*, Pa* ⊥ Pauu, Pauu ⊥ Pa*, Pa* ⊥ Pacu and Pacu ⊥ Pa*
1 11
10 Pa* ⊥ Pa* 0 12 NG degree of overconstraint, rG number of joint parameters that lose their independence in the closed loops that might be integrated in Glimb, (RG) basis of the operational velocity space
rotations and parallel to the rotation axes of the revolute pairs of the parallelogram loop. The idle mobilities are usually introduced in the parallelogram loop by replacing a revolute pair with: (i) a cylindrical pair with the same axis, (ii) a universal pair that has one revolute axis superposed on the axis of the replaced revolute joint or (iii) a spherical pair.
A closed parallelogram loop integrating idle mobilities is denoted by: (i) Pac, when an idle mobility of translation is integrated in a cylindrical joint (Fig. 6.3b), (ii) Pau, when an idle mobility of rotation is integrated in a universal joint (Fig.6.3c), (iii) Pas, when two idle mobilities of rotation are integrated in a spherical joint (Fig. 6.3d), (iv) Pauu, when two idle mobilities of rotation are integrated in two universal joints (Fig. 6.3e), (v) Pacu, when one idle mobility of translation is integrated in a cylindrical joint and one idle mobility of rotation is integrated in a universal joint (Fig. 6.3f), (vi) Pacs (also denoted by Pa*), when the three idle mobilities are integrated in a cylindrical and a spherical pair (Fig. 6.3g) and (vii) Pass (also denoted by Pa+R), when the three idle mobilities are integrated in two
6.2 Limbs with two rotational motions 333
Table 6.2. Joint arrangement and structural parametersa of the limbs defined by MG=SG=2, (RG)=( 1 2,v v ) and TG=0 presented in Figs. 6.1 and 6.2
Limbtype Joint Structural parameters No. arrangement m p q rG NG
Examples
1 PP P⊥ P 3 2    Fig. 6.1a 2 PPa P⊥ Pa 5 5 1 3 3 Fig. 6.1b 3 PaPa PaPat 7 8 2 6 6 Fig. 6.1c 4 PaP Pa⊥ P 5 5 1 3 3 Fig. 6.2a,b 5 PaPa PaPa 7 8 2 6 6 Fig. 6.2c MG degree of mobility, SG connectivity, (RG) basis of the operational velocity space, TG degree of structural redundancy, m number of links including the fixed base, p number of joints, q number of independent closed loops, NG degree of overconstraint, rG number of joint parameters that lose their independence in the closed loops integrated in the limb. a Structural parameters rG, MG, NG and TG are given by Eqs. 2.209, 2.211, 2.220 and 2.222 taking into account the fact that for an isolated limb r=rG=rl, M=MG, N=NG, T=TG, k=1 and SG=SF.
spherical joints adjacent to the same link (Fig. 6.3g). In the latter, an additional mobility of rotation is introduced by the internal mobility of the link adjacent to the two spherical joints. The solution in Fig. 6.1c combines a telescopic parallelogram loop denoted by Pat with a constant length parallelogram loop Pa. A revolute and a prismatic joint are connected to the fixed base.
Three joint parameters lose their independence in a planar parallelogram loop without idle mobilities, that is r=3 in Pa and Pattype loops.
In a parallelogram loop with idle mobilities 4≤ r≤ 6: r=4 for Pac and Pau, r=5 for Pas, Pauu and Pacu, r=6 for Pa* and Pass. The degree of overconstraint and the number of joint parameters that lose their independence in the closed loops that might be integrated in the various types of limbs with MG=SG=2, (RG)=( 1 2,v v ) and TG=0 are systematized in Table 6.1.
The joint arrangement and structural parameters of the solutions presented in Figs. 6.1 and 6.2 are systematized in Table 6.2.
6.2 Limbs with two rotational motions
of non redundant limb contains just two revolute joints. If the characteristic point H of the distal link is situated at the intersection of the axes of the
These limbs give two rotation velocities 1ω and 2ω in the basis of the operational velocity space (RG)=( 1 2,ω ω ). This simplest structural solution
334 6 Limbs with two degrees of connectivity
Fig. 6.4. Simple limb RRtype defined by MG=SG=2 and (RG)=( 1 2,ω ω )
two revolute pairs, this limb gives two rotations along with a constant position of the characteristic point (Fig. 6.4). The limb is Utype and has MG=SG=2, (RG)=( 1 2,ω ω ), NG=0, TG=0, rG=0.
Other solutions with nonorthogonal intersecting axes may also be used. As we have mentioned in chapter 5, just solutions with orthogonal and/or parallel axes of the kinematic joints are presented in this work. They are the easiest to make.
If the characteristic point H is not situated at the intersection of the axes of the revolute joints, the basis of the operational velocity space may contain (i) two angular velocities, (ii) two translational velocities or (iii) one rotational and two translational velocities.
6.3 Limbs with one translational and one rotational motion
These limbs give the translational and angular velocities 1v and 1ω in the basis of the operational velocity space (RG)=( 1 1,v ω ). The simplest structural solutions of non redundant limbs contain just one revolute and one prismatic joint. These limbs give a rotation of the moving platform and a translation of the characteristic point H situated on the rotation axes (Figs. 6.5 and 6.6). They have the following structural parameters: MG=SG=2, (RG)=( 1 1,v ω ), NG=0, TG=0, rG=0.
The direction of the translation could be either parallel (Fig. 6.5) or perpendicular (Fig. 6.6) to the rotation axis. Both translation and rotation have fixed directions. The order of the rotation and the translation is not important when the direction of the translation is parallel to the rotation axis (Fig. 6.5).
6.4 Other limbs with two degrees of connectivity 335
Fig. 6.5. Simple limbs PRtype and RPtype defined by MG=SG=2, (RG)=( 1 1,v ω ) and the direction of the translation parallel to the rotation axis
Fig. 6.6. Simple limb PRtype defined by MG=SG=2, (RG)=( 1 1,v ω ) and the direction of the translation perpendicular to the rotation axis
If the characteristic point H is not situated on the rotation axis, the basis of the operational velocity space may contain (i) two translational velocities or (ii) one translational and one rotational velocity.
6.4 Other limbs with two degrees of connectivity
The limbs presented in this section may be defined by multiple bases of the operational velocity space for any position of the characteristic point on the distal link. These bases combine translational and angular velocities. For example, the basis of the operational velocity space of the limb in Fig. 6.7 may be defined by two translational velocities ( 1 2,v v ) or by one translational and one rotational velocity ( i 1,v ω ), i=1,2.
Examples of simple and complex limbs with two degrees of connectivity and multiple bases of the operational velocity space are presented in
336 6 Limbs with two degrees of connectivity
Figs. 6.76.9. The complex limbs may contain planar or spatial closed loops. The planar loops are of type parallelogram Pa (Fig. 6.9af) rhombus Rb (Fig. 6.8a,b) or other planar loops Pn2 (Fig. 6.9g,h).
Various solutions of planar bimobile closed loops Pn2 containing five revolute or prismatic joints may be used, for example: RRRRR, P⊥RRRR, R⊥P⊥ RRR, RR⊥P⊥ RR, P ⊥ P ⊥⊥ RRR, R⊥P ⊥⊥ P ⊥⊥ RR, P⊥R ⊥⊥ P⊥ RR, R⊥P⊥ R⊥P⊥ R, P⊥RR⊥P⊥ R, P⊥RRR⊥P, P⊥P ⊥⊥ RR⊥P.
Fig. 6.7. Simple limbs RPtype (a) and RRtype (b) and (c) defined by MG=SG=2 and multiple bases of the operational velocity space
Fig. 6.8. Complex limbs RRbtype (a) and RRbRbtype (b) defined by MG=SG=2 and multiple bases of the operational velocity space
6.4 Other limbs with two degrees of connectivity 337
Fig. 6.9. Complex limbs of types RPa (a)(c), PaR (d)(f), and Pn2 (g), (h) defined by MG=SG=2 and multiple bases of the operational velocity space
338 6 Limbs with two degrees of connectivity
Table 6.3. Structural parametersa of the limbs with MG=SG=2 and TG=0 presented in Figs. 6.76.9
Structural parameters
No. Limbtype Joint arrangement
m p q rG NG
Examples
1 RP R⊥ P 3 2    Fig. 6.7a 2 RR RR 3 2    Fig. 6.7b 3 RR R⊥ R 3 2    Fig. 6.7c 4 RRb RRb 5 5 1 3 3 Fig. 6.8a 5 RRbRb RRbRb 7 8 2 6 6 Fig. 6.8b 6 RPa R⊥ Pa 5 5 1 3 3 Fig. 6.9a,b 7 RPa RPa 5 5 1 3 3 Fig. 6.9c 8 PaR Pa⊥ R 5 5 1 3 3 Fig. 6.9de 9 PaR PaR 5 5 1 3 3 Fig. 6.9f 9 Pa2 Pa2 5 5 1 3 3 Figs. 6.9g,h aSee footnote of Table 6.2 for the nomenclature of structural parameters.
The solutions of trimobile closed loops Pn3 contain six revolute and prismatic joints, for example: RRRRRR, P⊥RR RRR, R⊥P⊥ RRRR, RR⊥P⊥ RRR, P⊥P ⊥⊥ RRRR, R⊥P ⊥⊥ P ⊥⊥ RRR, P⊥R ⊥⊥ P⊥ RRR, R⊥P⊥ R⊥P⊥ RR, P⊥RR⊥P⊥ RR, P⊥RRR⊥P⊥ R, RR⊥P ⊥⊥ P ⊥⊥ RR, P⊥RRRR⊥P, R⊥P⊥ RR⊥P⊥ R, P⊥P ⊥⊥ RRR⊥P.
The revolute joints integrated in the closed loops Pn2 and Pn3 have parallel axes and the directions of the prismatic joints are contained in a plane perpendicular to the revolute axes.
A complex limb may integrate one, two or more rhombus (Rb) loops. In Fig. 8.8 just two examples with one and two rhombus loops are presented.
Three joint parameterslose their independence in each planar closed loop Rb, Pn2 or Pn3type.
The structural parameters of the limbs presented in Figs.6.76.9 are systematized in Table 6.3.
6.5 Redundant limbs with two degrees of connectivity
A large variety of redundant limbs with SG=2 and MG>2 can be obtained by continuing the evolutionary morphology approach until a certain degree of redundancy TG≥1 is reached. Due to space limitation, just some examples are presented in Fig. 6.10. These examples are obtained from the non
6.5 Redundant limbs with two degrees of connectivity 339
Fig. 6.10. Example of translational limbs of types PPP (a) PPaP (b) and PaPatP (c) with one degree of redundancy defined by MG=3, SG=2 and (RG)=( 1 2,v v )
redundant solutions in Fig. 6.1 by adding a prismatic joint in the plane of motion.
The structural parameters are defined as: (i) m=4, p=3 q=0, rG=0, SG=2, MG=3, (RG)=( 1 2,v v ), NG=0 and TG=1, for the example in Fig. 6.10a, (ii) m=6, p=6 q=1, rG=3, SG=2, MG=3, (RG)=( 1 2,v v ), NG=3 and TG=1, for the example in Fig. 6.10b, and (iii) m=8, p=9 q=2, rG=6, SG=2, MG=3, (RG)=( 1 2,v v ), NG=6 and TG=1, for the example in Fig. 6.10c. Three joint parameters lose their independence in each parallelogram loop Pa and Pattype. Each solution has one degree of redundancy which may be used to increase the work space or to increase the precision by setting up a twostage mechanism with a macro and a micro level with different precision. For example the redundant translation may be used for the micro level stage to achieve a high precision or a compliance motion.
7 Limbs with three degrees of connectivity
Limbs with three degrees of connectivity (SG=3) are used in parallel robots F with the connectivity of the moving platform SF≤ 3. The base of the operational space (RG) of these limbs contains three independent velocities. Nonredundant solutions generated by the evolutionary morphology approach are presented in the first six sections of this chapter. Some examples of redundant solutions are presented in the last section. The first six sections present solutions of limbs defined by the following: (i) three translational motions, (ii)(iii) one rotational and two translational motions, (iv) one translational and two rotational motions, (v) three rotational motions, and (vi) multiple bases of the operational velocity space.
7.1 Limbs with three translational motions
These limbs give three translational velocities v1, v2 and v3 in the basis of the operational velocity space (RG)=( 1 2 3, ,v v v ) along with a constant orientation of the distal link. The characteristic point H may have any position on the distal link. The simplest structural solutions for simple and complex nonredundant limbs with only prismatic and revolute pairs are presented in Figs. 7.17.6.
The complex limbs combine up to three planar parallelogram loops. As presented in section 6.1, each closed parallelogram loop could combine up to three idle mobilities that may be two orthogonal rotations and one translation.
The degree of overconstraint and the number of joint parameters that lose their independence in the closed loops combined in the various types of limbs defined by MG=SG=3, (RG)=( 1 2 3, ,v v v ) and TG=0 are systematized in Table 7.1.
In the solutions presented in Figs. 7.2b,d, 7.4b and 7.5a,d,f, the number of joints may be reduced by replacing the sequence of joints PPa or PaP by a parallelogram loop Pacctype with two cylindrical joints which may be adjacent to the same link or to different links of the parallelogram loop (Fig. 7.7). Parallelogram loop Pacctype has the following structural
342 7 Limbs with three degrees of connectivity
Table 7.1. Structural parameters NG and rG of the complex limbs with parallelogram loops
No. Complex limbs containing: NG rG 1 one loop Patype 3 3 2 one loop Pac or Pautype 2 4 3 one loop Pas, Pauu or Pacutype 1 5 4 one loop Pa*type 0 6 5 one loop Pa4utype 0 6 6 two loops Patype 6 6 7 two loops of type: (i) Pa and Pac, (ii) Pa and Pau 5 7 8 a) two loops Pac or Pautype
b) two loops of different types: (i) Pa and Pas, (ii) Pa and Pauu, (iii) Pa and Pacu , (iv) Pac and Pau
4 8
9 two loops of different types: (i) Pa and Pa*, (ii) Pas and Pac, (iii) Pas and Pau, (iv) Pauu and Pac, (v) Pauu and Pau, (vi) Pacu and Pac, (vii) Pacu and Pau
3 9
10 a) two loops Pas, Pacu or Pauutype b) two loops of different types: (i) Pas and Pauu, (ii) Pas and Pacu, (iii) Pauu and Pacu, (vii) Pa* and Pac, (viii) Pa* and Pau,
2 10
11 two loops of different types: (i) Pa* and Pauu, (ii) Pa* and Pacu , (iii) Pa* and Pas
1 11
12 two loops Pa*type 0 12 13 three loops Patype 9 9 14 two loops Patype and one loop Pac or Pautype 8 10 15 a) two loops Pac or Pautype and one loop Patype
b) two loops Patype and one loop Pas, Pauu or Pacutype 7 11
16 a) three loops Pac or Pautype b) two loops Patype and one loop Pa*type c) one loop Pac or Pautype, one loop Pas, Pauu or Pacutype and one loop Patype
6 12
17 a) one loop Pas, Pauu or Pacutype and the other two loops of types Pac or Pau b) one loop Patype and the other two loops of types Pas, Pauu or Pacu c) one loop Pac or Pautype, one loop Pa*type and one loop Patype
5 13
18 a) one loop Pa*type and the other two loops of types Pac or Pau b) one loop Pac or Pautype and the other two loops of types Pas, Pauu or Pacu c) one loop Pa*type, one loop Patype and one loop Pas, Pauu or Pacutype
4 14
19 a) one loop Pa*type, one loop Pac or Pautype and one loop Pas, Pauu or Pacutype b) three loops of types Pas, Pauu or Pacu c) one loop Pa*type, one loop Patype and one loop Pas, Pauu or Pacutype
3 15
7.1 Limbs with three translational motions 343
Table 7.1. (cont.)
20 a) one loop Pa*type and the other two loops of types Pas, Pauu or Pacu b) two loops Pa*type and one loop Pac or Pautype
2 16
21 two loops Pa*type and one loop Pas, Pauu or Pacutype 1 17 22 three loops of Pa*type 0 18 NG degree of overconstraint, rG number of joint parameters that lose their independence in the closed loops that may be integrated in Glimb
parameters: MG=2, NG=2 and rG=4. Limbs defined by MG=SG=3 and (RG)= ( 1 2 3, ,v v v ) may be obtained by using parallelogram loops Pacctype (Figs. 7.8 and 7.11).
The various solutions illustrated in Figs. 7.17.11 may be systematized in by taking into account various criteria: a) limb type a1) simple limbs composed by a simple open kinematic chain (Fig. 7.1) a2) complex limbs containing at least one closed loop (Figs. 7.27.6, 7.87.11) b) type of kinematic joint connected to the fixed base b1) prismatic joint (Figs. 7.1, 7.2, 7.3a, 7.8) b2) revolute joint (Figs. 7.47.6, 7.97.11) b3) prismatic and revolute joints (Figs. 7.3b) c) type of joints c1) with only prismatic and/or revolute joints (Figs. 7.17.6) c2) with prismatic, revolute and cylindrical joints (Figs. 7.87.11) d) number of closed loops integrated in the limb (q≥ 1) d1) with one closed loop (Figs. 7.2, 7.4, 7.8, 7.9) d2) with two closed loops (Figs. 7.3, 7.5, 7.10) d3) with three closed loops (Fig. 7.6) e) number of joint parameters that lose their independence in the closed loops integrated in the limb (3≤ rG≤ 12) e1) rG=3 (Figs. 7.2ad, 7.4) e2) rG=4 (Figs. 7.8, 7.9) e3) rG=6 (Figs. 7.3, 7.5) e4) rG=7 (Figs. 7.10, 7.11)
e5) rG=9 (Fig. 7.6). The joint arrangement and structural parameters of the solutions pre
sented in Figs. 7.17.11 are systematized in Table 7.2.
344 7 Limbs with three degrees of connectivity
Table 7.2. Joint arrangement and structural parametersa of the limbs defined by MG=SG=3, (RG)=( 1 2 3, ,v v v ) and TG=0 presented in Figs. 7.17.11
Structural parameters No. Limbtype Joint arrangement m p q rG NG
Examples
1 PPP P⊥ P⊥ P 4 3    Fig. 7.1 2 PPPa P⊥ P⊥ Pa 6 6 1 3 3 Fig. 7.2a 3 PPPa P⊥ PPa 6 6 1 3 3 Fig. 7.2b 4 PPaP P⊥ PaP 6 6 1 3 3 Fig. 7.2c 5 PPaP PPa⊥ P 6 6 1 3 3 Fig. 7.2d,e 6 PPaPa PPaPa 8 9 2 6 6 Fig. 7.3a 7 PPaPa PaPat P 8 9 2 6 6 Fig. 7.3b 8 PaPP Pa⊥ P⊥ P 6 6 1 3 3 Fig. 7.4a,b 9 PaPP PaP⊥ P 6 6 1 3 3 Fig. 7.4c,d 10 PaPaP Pa⊥ PaP 8 9 2 6 6 Fig. 7.5a 11 PaPaP Pa⊥ Pa⊥  P 8 9 2 6 6 Fig. 7.5b 12 PaPaP Pa⊥ Pa ⊥⊥ P 8 9 2 6 6 Fig. 7.5c 13 PaPaP PaPaP 8 9 2 6 6 Fig. 7.5d 14 PaPPa PaP⊥ Pa 8 9 2 6 6 Fig. 7.5e 15 PaPPa Pa⊥ P ⊥⊥ Pa 8 9 2 6 6 Fig. 7.5f 16 PaPPa Pa⊥ PPa 8 9 2 6 6 Fig. 7.5g 17 PaPPa PaPPa 8 9 2 6 6 Fig. 7.5h 18 PaPaPa Pa⊥ PaPa 10 12 3 9 9 Fig. 7.6a 19 PaPaPa PaPa⊥ Pa 10 12 3 9 9 Fig. 7.6b 20 PaPaPa Pa⊥ Pa⊥ Pa 10 12 3 9 9 Fig. 7.6c 21 PPacc P⊥ Pacc 5 5 1 4 3 Fig. 7.8ac 22 PaccP Pacc ⊥ P 5 5 1 4 2 Fig. 7.9ad 23 PaccPa PaccPa 7 8 2 7 5 Fig. 7.10a,b 24 PaccPa Pacc ⊥ Pa 7 8 2 7 5 Fig. 7.11a,b 25 PaPacc PaPacc 7 8 2 7 5 Fig. 7.10ce 26 PaPacc Pa⊥ Pacc 7 8 2 7 5 Fig. 7.11ce aSee footnote of Table 6.2 for the nomenclature of structural parameters
7.1 Limbs with three translational motions 345
Fig. 7.1. Simple limb PPPtype defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
Fig. 7.2. Complex limbs of types PPPa (ac), PPaP (de) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
346 7 Limbs with three degrees of connectivity
Fig. 7.3. Complex limbs of type PPaPa (a) and PaPatP (b) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
Fig. 7.4. Complex limbs PaPPtype defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
7.1 Limbs with three translational motions 347
Fig. 7.5. Complex limbs of types PaPaP (a)(d) and PaPPa (e)(h) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
348 7 Limbs with three degrees of connectivity
Fig. 7.6. Complex limbs PaPaPatype defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
Fig. 7.7. Parallelogram loop with two cylindrical joints Pacctype used to replace a kinematic chain PPa or PaPtype
7.1 Limbs with three translational motions 349
Fig. 7.8. Complex limbs PPacctype defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
Fig. 7.9. Complex limbs PaccPtype defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
350 7 Limbs with three degrees of connectivity
Fig. 7.10. Complex limbs of types PaccP (a, b) and PPacc (ce) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
7.1 Limbs with three translational motions 351
Fig. 7.11. Complex limbs of types PaccPa (a)(c) and PaPacc (d, e) defined by MG=SG=3 and (RG)=( 1 2 3, ,v v v )
352 7 Limbs with three degrees of connectivity
7.2 Planar limbs with one rotational and two translational motions
These limbs give one rotational and two translational velocities in the basis of the operational velocity space (RG)=( 1 2 1, ,v v ω ). The axis of angular velocity 1ω is perpendicular to the plane of translational velocities v1 and v2.
Structural solutions for nonredundant limbs with only revolute and prismatic pairs defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω ) are presented in Figs. 7.127.18.
These solutions have no idle mobilities and the plane of translational velocities has a constant orientation. Solutions for simple limbs are illustrated in Figs. 7.127.14. They have prismatic (Fig. 7.12 and 7.13) or revolute (Fig. 7.14) pairs adjacent to the fixed base. The complex limbs may contain one (Figs. 7.13, 7.15 and 7.16) or two (Fig. 7.17) parallelogram loops and have prismatic (Fig. 7.13) or revolute (Fig. 7.157.18) pairs adjacent to the fixed base. The solution in Fig. 7.17e integrates a parallelogram loop Patype connected to an extensible parallelogram loop denoted by Pat.
In the complex limbs, the revolute joints connected to the fixed base may be integrated in a parallelogram loop (Figs. 7.16 and 7.17a and b) or outside the closed loop (Figs. 7.15, 7.17ce and 7.18).
Each planar parallelogram loop could integrate up to three idle mobilities as we have seen in the previous sections. The degree of overconstraint and the number of joint parameters that lose their independence in the parallelogram loops integrated in the various types of limbs are given in Table 7.1. The complex limb in Fig. 7.17e are defined by NG=6 and rG=6.
Fig. 7.12. Simple limbs of types PPR (a), PRR (b) and PRP (c) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.2 Planar limbs with one rotational and two translational motions 353
Table 7.3. Joint arrangement and structural parametersa of the limbs defined by MG=SG=3, (RG)=( 1 2 1, ,v v ω ) and TG=0 presented in Figs. 7.127.18
Structural parameters No. Limbtype Joint arrangement m p q rG NG
Examples
1 PPR P⊥ P ⊥⊥ R 4 3    Fig. 7.12a 2 PRR P⊥ RR 4 3    Fig. 7.12b 3 PRP P⊥ R⊥ P 4 3    Fig. 7.12c 4 PPaR P⊥ PaR 6 6 1 3 3 Fig. 7.13a 5 PRPa P⊥ RPa 6 6 1 3 3 Fig. 7.13b 6 RRR RRR 4 3    Fig. 7.14a 7 RPR R⊥ P⊥ R 4 3    Fig. 7.14b 8 RRP RR⊥ P 4 3    Fig. 7.14c 9 RPP R⊥ P ⊥⊥ P 4 3    Fig. 7.14d 10 RPPa R⊥ P⊥  Pa 6 6 1 3 3 Fig. 7.15a 11 RPaP RPa⊥ P 6 6 1 3 3 Fig. 7.15b,c 12 RRPa RRPa 6 6 1 3 3 Fig. 7.15d 13 PaPR Pa⊥ P⊥ R 6 6 1 3 3 Fig. 7.16a,b 14 PaRP PaR⊥ P 6 6 1 3 3 Fig. 7.16c 15 PaRR PaRR 6 6 1 3 3 Fig. 7.16d 16 PaPaR PaPaR 8 9 2 6 6 Fig. 7.17a 17 PaRPa PaRPa 8 9 2 6 6 Fig. 7.17b 18 RPaPa RPaPa 8 9 2 6 6 Fig. 7.17c,d 19 RPaPat RPaPat 8 9 2 6 6 Fig. 7.17e 20 RPaPat RPaPat 8 9 2 6 6 Fig. 7.17e 21 RRbR R Rb R 6 6 1 3 3 Fig. 7.18a 22 RRbRbR R Rb Rb R 8 9 2 6 6 Fig. 7.18b 23 Pn2 R Pn2 R 6 6 1 3 3 Fig. 7.18c,d 24 Pn3 Pn3 6 6 1 3 3 Fig. 7.18e,f aSee footnote of Table 6.2 for the nomenclature of structural parameters
The joint arrangement and structural parameters of the solutions pre
sented in Figs. 7.127.18 are systematized in Table 7.3.
354 7 Limbs with three degrees of connectivity
Fig. 7.13. Complex limbs of types PPaR (a) and PRPa (b) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.14. Simple limbs of types RRR (a), RPR (b), RRP (c) and RPP (d) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.2 Planar limbs with one rotational and two translational motions 355
Fig. 7.15. Complex limbs of types PRPa (a), RPaP (b), (c) and RRPa (d) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
356 7 Limbs with three degrees of connectivity
Fig. 7.16. Complex limbs of types PaPR (a), (b), PaRP (c) and PaRR (d) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.2 Planar limbs with one rotational and two translational motions 357
Fig. 7.17. Complex limbs of types PaPaR (a), PaRPa (b), RPaPa (c, d) and RPaPat (e) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
358 7 Limbs with three degrees of connectivity
Fig. 7.18. Complex limbs of types RRbR (a), RRbRbR (b) Pn2R (c, d) and Pn3 (e, f) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.3 Non planar limbs with one rotational and two translational motions 359
7.3 Non planar limbs with one rotational and two translational motions
In this case, the axis of angular velocity 1ω lies in the plane of translational velocities v1 and v2. If the characteristic point H of the distal link is situated on the rotation axis and the plane of translations has a constant orientation, the basis of the operational velocities space is (RG)= ( 1 2 1, ,v v ω ).
Structural solutions fort nonredundant limbs with only revolute and prismatic pairs are presented in Figs. 7.197.22. These solutions have no idle mobilities and the plane of translational velocities has a constant orientation. The rotation axis also has a constant direction in the plane of translations. Solutions for simple limbs with prismatic pairs adjacent to the fixed base are illustrated in Figs. 7.19. The complex limbs may contain one (Figs. 7.20 and 7.21) or two (Fig. 7.22) parallelogram loops and have prismatic (Fig. 7.20) or revolute (Fig. 7.21 and 7.22a, b) pairs adjacent to the fixed base. The solutions in Fig. 7.22c and have prismatic and revolute joints connected to the fixed base.
Each planar parallelogram loop could integrate up to three idle mobilities as we have seen in the previous sections. The degree of overconstraint and the number of joint parameters that lose their independence in the parallelogram loops integrated in the various types of limbs presented in Figs. 7.197.22 are given in Table 7.1.
In the solutions presented in Figs. 7.197.22, the number of joints may be reduced by replacing:
a) two consecutive revolute and prismatic joints with the same axes/direction (RP or PR) by a cylindrical joint such as in Fig. 7.23,
b) a sequence of joints R⊥Pa or Pa⊥Rtype by a parallelogram loop Passtype with two spherical joints adjacent to the distal link such as in Fig. 7.24.
Solutions with a cylindrical joint and a parallelogram loop Passtype are presented in Figs. 7.25 and 7.26. The solution presented in Fig. 7.26a has NG=0, rG=6 and the solution in Fig. 7.26b are defined by NG=3 and rG=9.
The joint arrangement and structural parameters of the solutions presented in Figs. 7.197.26 are systematized in Table 7.4.
If the characteristic pint of the distal link is not on situated on the rotation axis the basis of the operational velocities space can be ( 1 2 1, ,v v ω ) or ( 1 2 3, ,v v v ).
360 7 Limbs with three degrees of connectivity
Table 7.4. Joint arrangement and structural parametersa of the limbs defined by MG=SG=3, (RG)=( 1 2 1, ,v v ω ) and TG=0 presented in Figs. 7.197.26
Structural parameters No. Limbtype Joint arrangement m p q rG NG
Examples
1 PPR P⊥ P⊥  R 4 3    Fig. 7.19a 2 PPR P⊥ PR 4 3    Fig. 7.19b 3 PPaR P⊥ Pa⊥  R 6 6 1 3 3 Fig. 7.20a 4 PPaR P⊥ Pa ⊥⊥ R 6 6 1 3 3 Fig. 7.20b 5 PaPR Pa⊥ P ⊥⊥ R 6 6 1 3 3 Fig. 7.21a,b 6 PaPR Pa⊥ PR 6 6 1 3 3 Fig. 7.21c,d 7 PaRP Pa⊥ RP 6 6 1 3 3 Fig. 7.21e,f 8 PaPaR PaPa⊥ R 8 9 2 6 6 Fig. 7.22a,b 9 PaPaR PaPat ⊥ R 8 9 2 6 6 Fig. 7.22c,d 10 PC P⊥ C 3 2    Fig. 7.25a 11 PaC Pa⊥ C 5 5 1 3 3 Fig. 7.25b 12 PPass P⊥ Pass 5 5 1 6 0 Fig. 7.26a 13 PPass PPass 7 8 2 9 3 Fig. 7.26b aSee footnote of Table 6.2 for the nomenclature of structural parameters
Fig. 7.19. Non planar simple limbs of types PPR (a, b) and PRP (c) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
7.3 Non planar limbs with one rotational and two translational motions 361
Fig. 7.20. Non planar complex limbs PPaRtype defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.21. Non planar complex limbs of types PaPR (ad) and PaRP (e, f) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
362 7 Limbs with three degrees of connectivity
Fig. 7.22. Non planar complex limbs of types PaPaR (a, b) and PaPatR (c, d) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.23. Cylindrical joint used to replace a kinematic chain RP or PRtype
7.3 Non planar limbs with one rotational and two translational motions 363
Fig. 7.24. Parallelogram loop Passtype used to replace a sequence of joints Pa⊥R or R⊥ Patype
Fig. 7.25. Non planar simple limb PCtype (a) and complex limbs PaCtype (b) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.26. Non planar complex limbs of type PPass (a) and PaPass (b) defined by MG=SG=3 and (RG)=( 1 2 1, ,v v ω )
364 7 Limbs with three degrees of connectivity
Fig. 7.27. Simple limbs of types PRR (a, b) and RPR (c) defined by MG=SG=3 and (RG)=( 1 1 2, ,v ω ω )
7.4 Limbs with one translational and two rotational motions
These limbs have one translational 1v and two rotation velocities 1ω and
2ω in the basis of the operational velocity space (RG)=( 1 1 2, ,v ω ω ). They allow two rotations along with a variable position of characteristic point H of the distal link. Point H is situated at the intersection of the rotation axes on a line with a fixed direction (Fig. 7.27). The simplest structural solutions for nonredundant limbs with only prismatic and revolute pairs have the following structural parameters: MG=SG=3, (RG)=( 1 1 2, ,v ω ω ), NG=0, TG=0, rG=0.
If the characteristic point H is not situated at the intersection of the axes of the revolute joints, the basis of the operational velocity space may contain (i) one translational and two angular velocities, (ii) one rotational and two translational velocities or (iii) three translational velocities.
7.5 Limbs with three rotational motions
These limbs has three angular velocities 1ω , 2ω and 3ω in the basis of the operational velocity space (RG)=( 1 2 3, ,ω ω ω ). The simplest structural solution for nonredundant limbs has three revolute pairs with intersecting axes and are defined by MG=SG=3, (RG)=( 1 2 3, ,ω ω ω ), NG=0, TG=0, rG=0.
7.6 Other limbs with three degrees of connectivity 365
Fig. 7.28. Simple limb RRRtype defined by MG=SG=3 and (RG)=( 1 2 3, ,ω ω ω )
The solution in Fig. 7.28 gives three rotations of the distal link along with a constant position of the characteristic point H situated at the intersecting axes. Other solutions with nonorthogonal intersecting axes may also be used. As we have mentioned in the previous chapters, just solutions with orthogonal and/or parallel axes of the kinematic joints are presented in this work. They are the easiest to make.
If the characteristic point of the distal link is not situated at the intersecting axes of the revolute joints, the basis of the operational velocity spacemay contain : (i) three angular velocities, (ii) one translational and two angular velocities, (iii) one rotational and two translational velocities or (iv) three translational velocities.
7.6 Other limbs with three degrees of connectivity
The limbs presented in this section have multiple bases for the operational velocity space for any position of the characteristic point on the distal link. These bases combine translational and angular velocities. For example, the basis of the operational velocity space of the limb in Fig. 7.29a may be defined by three translational velocities ( 1 2 3, ,v v v ) or by two translational and one angular velocities ( 1 i 1, ,v v ω ), and the basis of the operational velocity space of the limb in Fig. 7.29b by two translational and one angular velocities ( 1 i i, ,v v ω ) or by one translational and two angular velocities ( 1 1 i, ,v ω ω ), i=2,3. These bases are valid for any position of the characteristic point H on distal link 4.
Examples of simple and complex limbs with three degrees of connectivity and multiple bases of the operational velocity space are presented in Figs. 7.297.35. The complex limbs may contain planar or spatial loops.
366 7 Limbs with three degrees of connectivity
Fig. 7.29. Simple limbs of types PRP (a), PRR (bd), RPP (e, f) and RRR (gi) defined by MG=SG=3 and multiple bases of the operational velocity space
7.6 Other limbs with three degrees of connectivity 367
Fig. 7.30. Simple limbs of types RPR (ad) and RRP (eg) defined by MG=SG=3 and multiple bases of the operational velocity space
368 7 Limbs with three degrees of connectivity
Fig. 7.31. Complex limbs of types PPaR (ac) and PRPa (dg) defined by MG=SG=3 and multiple bases of the operational velocity space
7.6 Other limbs with three degrees of connectivity 369
Fig. 7.32. Complex limbs of types RRPa (ac) and RPaR (di) defined by MG=SG=3 and multiple bases of the operational velocity space
370 7 Limbs with three degrees of connectivity
Fig. 7.33. Complex limbs of types PaRR (ac) and PaRP (d, e) defined by MG=SG=3 and multiple bases of the operational velocity space
7.6 Other limbs with three degrees of connectivity 371
Fig. 7.34. Complex limbs of types RPaP (ah) and RPPa (il) defined by MG=SG=3 and multiple bases of the operational velocity space
372 7 Limbs with three degrees of connectivity
Fig. 7.35. Complex limbs of types PRRb (a), PRRbRb (b), and PPa2 (cd) defined by MG=SG=3 and multiple bases of the operational velocity space
7.6 Other limbs with three degrees of connectivity 373
Fig. 7.36. Complex limbs of types RRRb (a), RRRbRb (b), and RPa2 (cf) defined by MG=SG=3 and multiple bases of the operational velocity space
374 7 Limbs with three degrees of connectivity
Table 7.5. Joint arrangement and structural parametersa of the limbs defined by MG=SG=3 and TG=0 presented in Figs. 7.297.36
Structural parameters No. Limbtype Joint arrangement m p q rG NG
Examples
1 PRP PR⊥ P 4 3    Fig. 7.29a 2 PRR P R⊥ P 4 3    Fig. 7.29b 3 PRR PRP 4 3    Fig. 7.29c 4 PRR P⊥ R⊥ R 4 3    Fig. 7.29d 5 RPP R⊥ P⊥  P 4 3    Fig. 7.29e 6 RPP RP⊥ P 4 3    Fig. 7.29f 7 RRR R⊥ RR 4 3    Fig. 7.29g 8 RRR RR⊥ R 4 3    Fig. 7.29h 9 RRR R⊥ R⊥ R 4 3    Fig. 7.29i 10 RPR R⊥ P ⊥⊥ R 4 3    Fig. 7.30a 11 RPR R⊥ PR 4 3    Fig. 7.30b 12 RPR RPR 4 3    Fig. 7.30c 13 RPR RP⊥ R 4 3    Fig. 7.30d 14 RRP R⊥ RP 4 3    Fig. 7.30e 15 RRP R⊥ R⊥ P 4 3    Fig. 7.30f 16 RRP RRP 4 3    Fig. 7.30g 17 PPaR PPa⊥ R 6 6 1 3 3 Fig. 7.31a,b 18 PPaR PPaR 6 6 1 3 3 Fig. 7.31c 19 PRPa P⊥ R⊥ Pa 6 6 1 3 3 Fig. 7.31d,e 20 PRPa PR⊥ Pa 6 6 1 3 3 Fig. 7.31f 21 PRPa PRPa 6 6 1 3 3 Fig. 7.31g 22 RRPa R⊥ RPa 6 6 1 3 3 Fig. 7.32a 23 RRPa R⊥ R⊥ Pa 6 6 1 3 3 Fig. 7.32b,c 24 RPaR R⊥ PaR 6 6 1 3 3 Fig. 7.32d,e 25 RPaR R⊥ Pa⊥ R 6 6 1 3 3 Fig. 7.32f,g 26 RPaR RPaR 6 6 1 3 3 Fig. 7.32h 27 RPaR RPa⊥ R 6 6 1 3 3 Fig. 7.32i 28 PaRR Pa⊥ R⊥ R 6 6 1 3 3 Fig. 7.33a,b 29 PaRR PaR⊥ R 6 6 1 3 3 Fig. 7.33c 30 PaRP Pa⊥ R⊥ P 6 6 1 3 3 Fig. 7.33d,e 31 RPaP RPaP 6 6 1 3 3 Fig. 7.34a,g 32 RPaP R⊥ Pa⊥  P 6 6 1 3 3 Fig. 7.34b,h 33 RPaP R⊥ PaP 6 6 1 3 3 Fig. 7.34c 34 RPaP R⊥ Pa ⊥⊥ P 6 6 1 3 3 Fig. 7.34d,f 35 RPPa R⊥ PPa 6 6 1 3 3 Fig. 7.34i 36 RPPa RPPa 6 6 1 3 3 Fig. 7.34j 37 RPPa RP⊥ Pa 6 6 1 3 3 Fig. 7.34k 38 RPPa R⊥ P⊥ Pa 6 6 1 3 3 Fig. 7.34l 39 PRRb PRRb 6 6 1 3 3 Fig. 7.35a 40 PRRbRb PRRbRb 8 9 2 6 6 Fig. 7.35b 42 PPn2 PPn2 6 6 1 3 3 Fig. 7.35c,d
7.7 Redundant limbs with three degrees of connectivity 375
Table 7.5. (cont.)
43 RRRb P⊥ RRb 6 6 1 3 3 Fig. 7.36a 44 RRRbRb P⊥ RRbRb 8 9 2 6 6 Fig. 7.36b 45 PPn2 R⊥ Pn2 6 6 1 3 3 Fig. 7.36cf aSee footnote of Table 6.2 for the nomenclature of structural parameters
The planar loops are of type parallelogram Pa (Fig. 6.9af) rhombus Rb (Fig. 6.8a,b) or other planar loops Pn2 (Fig. 6.9h,i). The structural parameters of these limbs are systematized in Table 7.5.
7.7 Redundant limbs with three degrees of connectivity
A large variety of redundant limbs with SG=3 and MG>3 can be obtained by continuing the evolutionary morphology approach until a certain degree of redundancy TG≥1 is reached. Due to space limitation, just two examples are presented in Figs. 7.37 and 7.38.
Figure 7.37 represents a particular case of example 2 analyzed in chapter 3 with p=4 revolute joints with parallel axes. This solution is defined by the following structural parameters: m=5, q=0, rG=0, SG=3, MG=4 (RG)=( 1 2 1, ,v v ω ), NG=0 and TG=1. The kinematic chain has four degrees of mobility but distal link 5 can realize just three independent planar motions.
The complex limb in Fig. 7.38 combines a parallelogram loop Pa, a telescopic parallelogram loop Pat and a serial chain R║R type. Three joint parameters lose their independence in each parallelogram loop Pa and Pattype. This solution is defined by the following structural parameters: m=9, p=10, q=2, rG=6, SG=3, MG=4, (RG)=( 1 2 1, ,v v ω ), NG=6 and TG=1.
As in the previous example, this kinematic chain has four degrees of mobility but distal link 9 can realize just three independent planar motions.
This degree of redundancy gives more dexterity in avoiding singularities and obstacles. Structural redundancy can also contribute to increasing the working space of the robot as we will present in Part 2 of this work.
376 7 Limbs with three degrees of connectivity
Fig. 7.37. Example of simple planar limb RRRRtype defined by MG=4, SG=3 and (RG)=( 1 2 1, ,v v ω )
Fig. 7.38. Example of complex planar limb PaPatRRtype defined by MG=4, SG=3 and (RG)=( 1 2 1, ,v v ω )
8 Limbs with four degrees of connectivity
Limbs with four degrees of connectivity (SG=4) are used in parallel robots F where the connectivity of moving platform SF≤ 4. The basis of the operational space of these limbs contains four independent velocities. Nonredundant solutions generated by the evolutionary morphology approach are presented in the first four sections of this chapter. Some examples of redundant solutions are also presented in the last section. The first four sections present solutions of limbs defined by the following: (i) one rotational and three translational motions (Schönflies motion), (ii) two translational and two rotational motions, (iii) one translational and three rotational motions, and (iv) multiple bases of the operational velocity space.
8.1 Limbs with Schönflies motion
These limbs give three translational velocities, v1, v2, v3, and one angular velocity 1ω about an axis of fixed direction. The basis of the operational velocity space of these limbs is (RG)=( 1 2 3 1, , ,v v v ω ). The characteristic point H may have any position on the distal link.
This motion was studied by the German mathematician Arthur Moritz Schönflies (Schoenflies) and is usually called Schönflies motion. The referenced book of Schönflies (1893) is the translation from German into French of “Geometrie der Bewegung in synthetischer Darstellung”, Teubner, Leipzig, 1886 where the author’s name is spelt Schoenflies. This is why, probably for the first time, Bottema and Roth (1979) called this special motion type Schönflies motion. In Schönflies’ book, as well as in the Bottema and Roth book, a socalled Schoenflies motion is described as being a rigidbody motion having axodes (surfaces generated by the instantaneous screw axes in any timedependent motion) that are cylinders (or prismatic surfaces) rather than being a fourdegreesoffreedom (4dof) motion. Schönflies is better known for his important contribution to the theory of 230 discrete (noncontinuous) subgroups of displacements as a basic tool in crystallography developed independently by Schönflies and
378 8 Limbs with four degrees of connectivity
Fedorov. An initial enumeration of 4dof serial kinematic chains generating Schönflies motions composed of 1dof kinematic pairs and planehinged parallelogram is presented by Hervé and Sparacino (1991).
The simplest structural solutions of non redundant simple and complex limbs with only revolute and prismatic pairs defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) are presented in Figs. 8.18.50. These solutions have no idle mobilities.
In the solutions presented in Figs. 8.18.50 the number of joints may be reduced by replacing as follows:
a) two consecutive revolute and prismatic joints with the same axes/direction (RP or PR) by a cylindrical joint such as in Fig. 7.23,
b) a sequence of joints R⊥Pa or Pa⊥R by a parallelogram loop Passtype with two spherical joints adjacent to the distal link such as in Fig. 7.24,
c) a sequence of joints PPa or PaP by a parallelogram loop Pacctype with two cylindrical joints which may be adjacent to the same link or to different links of the parallelogram loop such as in Fig. 7.7.
The complex limbs may combine up to three planar parallelogram loops. As presented in section 6.1, each closed parallelogram loop could contain up to three idle mobilities that may be two orthogonal rotations and one translation.
The degree of overconstraint and the number of joint parameters that lose their independence in the parallelogram closed loops that may be combined in the various types of limbs with MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and TG=0 are systematized in Table 7.1.
The various solutions illustrated in Figs. 8.18.50 may be systematized in by taking into account various criteria: a) limb type a1) simple limbs composed by a simple open kinematic chain (Figs. 8.18.7)
a2) complex limbs containing at least one closed loop (Figs. 8.8 8.55) b) position of the rotation axis called rotation axis of the Schönflies motion
b1) fixed position and orientation (Figs. 8.1, 8.5, 8.8, 8.16, 8.21 8.23, 8.268.28, 8.43)
b2) lying in a plane of fixed position and orientation (Figs. 8.2, 8.10, 8.11, 8.18, 8.33, 8.41, 8.42, 8.45)
b3) lying on a cylinder of fixed position, orientation and radius (Figs. 8.4, 8.6, 8.9, 8.17, 8.24, 8.34, 8.38, 8.39, 8.47)
8.1 Limbs with Schönflies motion 379
b4) passing by the characteristic point H of the distal link (Figs. 8.3, 8.7, 8.128.14, 8.20, 8.25, 8.298.31, 8.358.37, 8.44, 8.488.50)
b5) unchanging orientation (Figs. 8.15, 8.19, 8.32, 8.40, 8.46) c) type of kinematic joint connected to the fixed base c1) prismatic joint (Figs. 8.18.4, 8.88.20, 8.488.50) c2) revolute joint (Figs. 8.58.7, 8.218.47) d) type of closed loops
d1) parallelogram loops (Figs. 8.88.47) d2) extensible parallelogram loop (Fig. 8.25b) d3) rhombus loops (Fig. 8.48) d4) other loops (Figs. 8.49, 8.50)
e) number of closed loops combined in the limb (q≥ 1) e1) with one closed loop (Figs. 8.88.15, 8.218.23, 8.24, 8.29 8.34, 8.48a, 8.49, 8.50) e2) with two closed loops (Figs. 8.168.19, 8.258.28, 8.35 8.42, 8.48b) e3) with three closed loops (Figs. 8.438.47) f) number of parameters that lose their independence in the closed loops combined in the limb (3≤ rG≤ 12) f1) rG=3 (Figs. 8.88.15, 8.218.24, 8.298.34, 8.48a, 8.49, 8.50) f3) rG=6 (Figs. 8.168.20, 8.25, 8.268.28, 8.358.42, 8.48b) f5) rG=9 (Figs. 8.438.47). g) degree of mobility of a planar loops combined in the complex limbs g1) one degree of mobility (Figs. 8.88.47) g2) two degrees of mobility (Figs. 8.48, 8.49) g3) three degrees of mobility (Fig.8.50).
We note that in all cases, the rotation axis of the Schönflies keep a constant orientation. For this reason, we consider case b5 mentioned above as the general case. We denote it by S. The particular positions corresponding to cases b1b4 are denoted by S1 S4. Rotation axis of the Schönflies motion may be: (i) in a fixed position  denoted by S1, (ii) lying in a plane of fixed position and orientation  denoted by S2, (iii) lying on a cylinder of fixed positionorientation and radius  denoted by S3, and (iv) passing by the characteristic point of the distal link – denoted by S4.
Various solutions of planar bimobile closed loops Pn2type and Pn3type may be used as presented in chapter 6. The revolute joints combined in the closed loops Pn2 and Pn3 have parallel axes and the directions of the prismatic joints are contained in a plane perpendicular to the revolute axes.
380 8 Limbs with four degrees of connectivity
Table 8.1. Joint arrangement and structural parametersa of the limbs defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and TG=0 presented in Figs. 8.18.55
Structural parameters No. Limbtype Rotation axis of the Schönflies motion b m p q rG NG
Figure
1 PPaRP S 7 7 1 3 3 Fig. 8.15a,b 2 PPRPa S 7 7 1 3 3 Fig. 8.15c 3 PPaRPa S 9 10 2 6 6 Fig. 8.19a,b 4 PaPaRPa S 11 13 3 9 9 Fig. 8.46af 5 PRPP S1 5 4    Fig. 8.1 6 RPPP S1 5 4    Fig. 8.5ac 7 PRPPa S1 7 7 1 3 3 Fig. 8.8a,b 8 PRPaP S1 7 7 1 3 3 Fig. 8.8ce 9 PRPaPa S1 9 10 2 6 6 Fig. 8.16 10 RPPaP S1 7 7 1 3 3 Fig. 8.21ai 11 RPaPP S1 7 7 1 3 3 Fig. 8.22ak 12 RPPPa S1 7 7 1 3 3 Fig. 8.23af 13 RPaPaP S1 9 10 2 6 6 Fig. 8.26al 14 RPaPPa S1 9 10 2 6 6 Fig. 8.27al 15 RPPaPa S1 9 10 2 6 6 Fig. 8.28ac 16 RPaPaPa S1 11 13 3 9 9 Fig. 8.43ai 17 PPRP S2 5 4    Fig. 8.2a,b 18 PRPPa S2 5 4    Fig. 8.2c,d 19 PPRPa S2 7 7 1 3 3 Fig. 8.10ad 20 PRPPa S2 7 7 1 3 3 Fig. 8.10eh 21 PRPaP S2 7 7 1 3 3 Fig. 8.11af 22 PPaRP S2 7 7 1 3 3 Fig. 8.11g,h 23 PRPaPa S2 9 10 2 6 6 Fig. 8.18a,b 24 PaPRP S2 7 7 1 3 3 Fig. 8.33ac 25 PaRPP S2 7 7 1 3 3 Fig. 8.33df 26 PaRPaP S2 9 10 2 6 6 Fig. 8.41af 27 PaPRPa S2 9 10 2 6 6 Fig. 8.41gi 28 PaRPPa S2 9 10 2 6 6 Fig. 8.42ae 29 PaRPaP S2 9 10 2 6 6 Fig. 8.42f 30 PaPaRP S2 9 10 2 6 6 Fig. 8.42g 31 PaPRPa S2 9 10 2 6 6 Fig. 8.42h 32 PaRPPa S2 9 10 2 6 6 Fig. 8.42i 33 PaPaRPa S2 11 13 3 9 9 Fig. 8.45a,b 34 PaRPaPa S2 11 13 3 9 9 Fig. 8.45cg
8.1 Limbs with Schönflies motion 381
Table 8.1. (cont.)
35 PRRP S3 5 4    Fig. 8.4 36 RPRP S3 5 4    Fig. 8.6a,b 37 RRPP S3 5 4    Fig. 8.6c 38 PPaRP S3 7 7 1 3 3 Fig. 8.9a 39 PRRPa S3 7 7 1 3 3 Fig. 8.9b 40 PPaRPa S3 9 10 2 6 6 Fig. 8.17 41 RPRPa S3 7 7 1 3 3 Fig. 8.24a,b 42 RRPPa S3 7 7 1 3 3 Fig. 8.24c 43 RRPaP S3 7 7 1 3 3 Fig. 8.24d 44 PaPRP S3 7 7 1 3 3 Fig. 8.34a 45 PaRPP S3 7 7 1 3 3 Fig. 8.34b,c 46 PaRPaP S3 9 10 2 6 6 Fig. 8.38ad 47 PaPRPa S3 9 10 2 6 6 Fig. 8.38e,f 48 PaRPPa S3 9 10 2 6 6 Fig. 8.39ad 49 PaRPaPa S3 11 13 3 9 9 Fig. 8.47ac 50 PPPR S4 5 4    Fig. 8.3ac 51 PPRP S4 5 4    Fig. 8.3d 52 PPRR S4 5 4    Fig. 8.3e,f 53 PRPR S4 5 4    Fig. 8.3h 54 PRRR S4 5 4    Fig. 8.3g 55 RPPR S4 5 4    Fig. 8.7a,b 56 RPRP S4 5 4    Fig. 8.7c 57 RPRR S4 5 4    Fig. 8.7d,e 58 RRPR S4 5 4    Fig. 8.7f,g 59 RRRP S4 5 4    Fig. 8.7h 60 PPPaR S4 7 7 1 3 3 Fig. 8.12af 61 PPaRP S4 7 7 1 3 3 Fig. 8.12gi 62 PPaPR S4 7 7 1 3 3 Fig. 8.13ai 63 PPaRR S4 7 7 1 3 3 Fig. 8.14a 64 PRPaR S4 7 7 1 3 3 Fig. 8.14b 65 PPaPaR S4 9 10 2 6 6 Fig. 8.20ac 66 RPPaPa S4 9 10 2 6 6 Fig. 8.25a 67 RPPaPat S4 9 10 2 6 6 Fig. 8.25b 68 PaPPR S4 7 7 1 3 3 Fig. 8.29al 69 PaPRP S4 7 7 1 3 3 Fig. 8.30ad 70 PaRRP S4 7 7 1 3 3 Fig. 8.30e 71 PaRPR S4 7 7 1 3 3 Fig. 8.31ac
382 8 Limbs with four degrees of connectivity
Table 8.1. (cont.)
72 PaRRR S4 7 7 1 3 3 Fig. 8.31d 73 PaPRR S4 7 7 1 3 3 Fig. 8.31e 74 PaPRP S4 7 7 1 3 3 Fig. 8.32a,b 75 PaPaPR S4 9 10 2 6 6 Fig. 8.35ak 76 PaPPaR S4 9 10 2 6 6 Fig. 8.36al 77 PaPaRP S4 9 10 2 6 6 Fig. 8.37ad 78 PaPaRP S4 9 10 2 6 6 Fig. 8.40ac 79 PaPRPa S4 9 10 2 6 6 Fig. 8.40dh 80 PaPaPaR S4 11 13 3 9 9 Fig. 8.44ai 81 PRRbR S4 7 7 1 3 3 Fig. 8.48a 82 PRRbRbR S4 9 10 2 6 6 Fig. 8.48b 83 PPn2R S4 7 7 1 3 3 Fig. 8.49a,b 84 PPn3 S4 7 7 1 3 3 Fig. 8.50a,b aSee footnote of Table 6.2 for the nomenclature of structural parameters bS1 fixed position and orientation, S2 lying in a plane of fixed position and orientation, S3 lying on a cylinder of fixed position, orientation and radius, S4 passing by the characteristic point of the distal link, S fixed orientation.
A complex limb may contain one, two or more rhombus (Rb) loops. In
Fig. 8.48 just two examples with one and two rhombus loops are presented.
Three joint parameterslose their independence in each planar closed loop Rb, Pn2 or Pn3type.
The structural parameters of the solutions presented in Figs. 8.18.55 are systematized in Table 8.1.
8.1 Limbs with Schönflies motion 383
Fig. 8.1. Simple limb PRPPtype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
Fig. 8.2. Simple limbs of types PPRP (a, b) and PRPP (c, d) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying in a plane
384 8 Limbs with four degrees of connectivity
Fig. 8.3. Simple limbs of types PPPR (ac), PPRP (d), PPRR (e, f), PRRR (g) and PRPR (h) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 385
Fig. 8.4. Simple limb PRRPtype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cylinder
Fig. 8.5. Simple limbs RPPPtype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
Fig. 8.6. Simple limbs of types RPRP (a, b) and RRPP (c) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cylinder
386 8 Limbs with four degrees of connectivity
Fig. 8.7. Simple limbs of types RPPR (a, b), RPRP (c), RPRR (d, e), RRPR (f, g) and RRRP (h) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 387
Fig. 8.8. Complex limbs of types PRPPa (a, b) and PRPaP (ce) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
Fig. 8.9. Complex limbs of types PPaRP (a) and PRRPa (b) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cylinder
388 8 Limbs with four degrees of connectivity
Fig. 8.10. Complex limbs of types PPRPa (ad) and PRPPa (eh) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying in a plane
8.1 Limbs with Schönflies motion 389
Fig. 8.11. Complex limbs of types PRPaP (af) and PPaRP (g, h) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying in a plane
390 8 Limbs with four degrees of connectivity
Fig. 8.12. Complex limbs of types PPPaR (af) and PPaRP (gi) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 391
Fig. 8.13. Complex limbs PPaPRtype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
392 8 Limbs with four degrees of connectivity
Fig. 8.14. Complex limbs of types PPaRR (a) and PRPaR (b) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
Fig. 8.15. Complex limbs of types PPaRP (a, b) and PPRPa (c) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and fixed orientation of the rotation axis of Schönflies motion
8.1 Limbs with Schönflies motion 393
Fig. 8.16. Complex limb PRPaPatype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
Fig. 8.17. Complex limbs PPaRPatype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cylinder
Fig. 8.18. Complex limbs PRPaPatype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying in a plane
394 8 Limbs with four degrees of connectivity
Fig. 8.19. Complex limbs PPaRPatype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and fixed orientation of the rotation axis of Schönflies motion
Fig. 8.20. Complex limbs PPaPaRtype defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 395
Fig. 8.21. Complex limbs RPPaPtype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
396 8 Limbs with four degrees of connectivity
Fig. 8.22. Complex limbs RPaPPtype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
8.1 Limbs with Schönflies motion 397
Fig. 8.22 (cont.)
Fig. 8.23. Complex limbs RPPPatype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
398 8 Limbs with four degrees of connectivity
Fig. 8.24. Complex limbs of types RPRPa (a, b), RRPPa (c), and RRPaP (d) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying on a cylinder
Fig. 8.25. Complex limbs of types RPPaPa (a) and RPPaPat (b) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 399
Fig. 8.26. Complex limbs RPaPaPtype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
400 8 Limbs with four degrees of connectivity
Fig. 8.26 (cont.)
8.1 Limbs with Schönflies motion 401
Fig. 8.27. Complex limbs RPaPPatype defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
402 8 Limbs with four degrees of connectivity
Fig. 8.27 (cont.)
8.1 Limbs with Schönflies motion 403
Fig. 8.28. Complex limbs RPPaPatype defined by MG=SG=4, (RG)= ( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion in a fixed position
404 8 Limbs with four degrees of connectivity
Fig. 8.29. Complex limbs PaPPRtype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
8.1 Limbs with Schönflies motion 405
Fig. 8.29 (cont.)
Fig. 8.30. Complex limbs of types PaPRP (ad) and PaRRP (e) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
406 8 Limbs with four degrees of connectivity
Fig. 8.31. Complex limbs of types PaRPR (ac), PaRRR (d) and PaPRR (e) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion passing by the characteristic point of the distal link
Fig. 8.32. Complex limbs PaPRPtype defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and fixed orientation of the rotation axis of Schönflies motion
8.1 Limbs with Schönflies motion 407
Fig. 8.33. Complex limbs of types PaPRP (ac) and PRPP (df) defined by MG=SG=4, (RG)=( 1 2 3 1, , ,v v v ω ) and the rotation axis of the Schönflies motion lying in a plane