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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 262801, 9 pageshttp://dx.doi.org/10.1155/2013/262801

Research ArticleScrew-System-Based Mobility Analysis of a Family ofFully Translational Parallel Manipulators

Ernesto Rodriguez-Leal,1 Jian S. Dai,2 and Gordon R. Pennock3

1 School of Engineering and Information Technologies, Tecnologico de Monterrey, Avenue E. Garza Sada 2501,64849, Mexico

2 School of Natural and Mathematical Sciences, King’s College London, University of London, Strand,London WC2R 2LS, UK

3 School of Mechanical Engineering, Purdue University, 610 Purdue Mall, West Lafayette, IN 47907, USA

Correspondence should be addressed to Ernesto Rodriguez-Leal; [email protected]

Received 6 July 2013; Revised 22 October 2013; Accepted 23 October 2013

Academic Editor: Hector Puebla

Copyright © 2013 Ernesto Rodriguez-Leal et al.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper investigates the mobility of a family of fully translational parallel manipulators based on screw system analysis byidentifying the common constraint and redundant constraints, providing a case study of this approach. The paper presents thebranch motion-screws for the 3-R𝑃C-Y parallel manipulator, the 3-RCC-Y (or 3-R𝑃RC-Y) parallel manipulator, and a newlyproposed 3-R𝑃C-T parallel manipulator. Then the paper determines the sets of platform constraint-screws for each of these threemanipulators. The constraints exerted on the platforms of the 3-R𝑃C architectures and the 3-RCC-Y manipulators are analyzedusing the screw system approach and have been identified as couples. A similarity has been identified with the axes of couples:they are perpendicular to the R joint axes, but in the former the axes are coplanar with the base and in the latter the axes areperpendicular to the limb. The remaining couples act about the axis that is normal to the base. The motion-screw system andconstraint-screw system analysis leads to the insightful understanding of the mobility of the platform that is then obtained bydetermining the reciprocal screws to the platform constraint screw sets, resulting in three independent instantaneous translationaldegrees-of-freedom. To validate the mobility analysis of the three parallel manipulators, the paper includes motion simulationswhich use a commercially available kinematics software.

1. Introduction

Themobility, or degrees-of-freedom, of a mechanism is com-monly defined as the number of independent coordinates thatis required to define the configuration of a kinematic chain ormechanism [1].Themobility of a particularmechanism is oneof the most fundamental issues in the kinematic analysis andsynthesis of mechanisms. One of the first contributions to thestudy of mobility was the formula presented by Chebychev[2] for calculating the independent variables in amechanism.For a comprehensive historical survey of the studies of themobility of mechanisms over the past 150 years see Gogu [3].The most commonly used approach to determine the num-ber of degrees-of-freedom of a mechanism is the Grubler-Kutzbach mobility criterion [4]. However, it is well known

that this criterion does not provide the correct result forthe mobility of a certain family of mechanisms (commonlyreferred to as overconstrained mechanisms). The primaryreason for this is that theGrubler-Kutzbachmobility criteriondoes not consider the geometry of the mechanism [5–7] thatredundant constraints are taken as effective constraints. Anadditional limitation is that the Grubler-Kutzbach criteriondoes not indicate if the identified degree-of-freedom (DOF)is a translational DOF or a rotational DOF.

The mobility of overconstrained mechanisms has beeninvestigated by several different techniques; for example,Angeles andGosselin [8] used the dimension of the nullspaceof the Jacobianmatrix. Angeles [9], Fanghella andGalletti [10,11], and Herve [12] used group theory to study the mobilityproperties of single-loop kinematic chains. Rico-Martinez

2 Mathematical Problems in Engineering

and Ravani [13] have extended the work presented in [10–12]by developing a single equation that can be used to determinethe mobility of kinematic chains using group theory. ThenRico, et al. [14] translated the mobility criterion presentedin [13] from finite kinematics, based on group theory, intoinfinitesimal kinematics, based on Lie algebra. Screw theoryhas also been used by several researchers to investigate themobility of mechanisms, in general, and parallel manipula-tors, in particular [15]. Huang and Li [16] used this geometricapproach to determine themobility and kinematic propertiesof lower mobility parallel manipulators using the constraintscrew systems of the manipulators. Zhao et al. [17] proposedthe concept of configuration degrees-of-freedom (CDOF)for a study of the mobility of spatial mechanisms. ThenZhao et al. [18] proposed a programmable algorithm toinvestigate the CDOF of a spatial parallel manipulator withreciprocal-screw theory. Dai et al. [19] made an importantcontribution to this field by investigating the screw systemsof parallel manipulators and their interrelationships, relatingthe screw systems to motion and constraints. They identifiedthe constraints as (i) platform, (ii) mechanism, and (iii)redundant constraints. This work resulted in a new approachto the mobility analysis of parallel manipulators based on thedecomposition of motion- and constraint-screw systems.

This paper adopts the screw-based method proposedby Dai et al. [19], to study the mobility of three fullytranslational parallel manipulators, namely, (i) the 3-RPC-Y parallel manipulator (where R denotes a revolute joint,P denotes a prismatic joint, C denotes a cylindric joint,the underline denotes the active joint, and Y denotes theconfiguration of the three identical legs equally distributed onthe base, referred to here as a star configuration); (ii) the 3-RCC-Y (or 3-RPRC-Y) parallel manipulator; and (iii) a newlyproposed 3-RPC-T parallelmanipulator (where T denotes theconfiguration of the legs on the base). The 3-RPC-Y parallelmanipulator was first proposed by Callegari and Tarantini[20] and the 3-RCC-Y parallelmanipulatorwas first proposedby Callegari andMarzetti [21]. However, the 3-RPC-T parallelmanipulator is a new architecture proposed by the authorsof this paper. The three parallel manipulators are illustratedin Figures 1, 2, and 3. Callegari and Tarantini studied themobility of the 3-RPC-Y parallel manipulator using thegeneral Grubler-Kutzbach mobility criterion and found thatthe platform is overconstrained. Herve and Sparacino [22]showed that the topology of this manipulator complies withthe conditions for full Cartesian motion. In other words,the platform has three translational degrees-of-freedom andcannot undergo a rotation [17]. From an inspection of theJacobian matrix of the 3-RCC-Y parallel manipulator, Herveand Sparacino showed that this manipulator also complieswith the conditions for full Cartesian motion (the platformhas three translational degrees-of-freedom). The approachpresented in this paper supports these results and is thenused to study the mobility of the newly proposed 3-RPC-Tparallel manipulator. Finally, the results are validated by theuse of motion simulations which make use of a commerciallyavailable kinematics software.

The paper is arranged as follows. Sections 2 and 3 presentthe mobility analysis of the 3-RPC-Y manipulator and the

C joint

P joint

R joint

Base

Platform

Figure 1: The CAD model of the 3-RPC-Y parallel manipulator.

R joint

C joint

C joint

Base

Platform

Figure 2: The CAD model of the 3-RCC-Y parallel manipulator.

3-RCC-Y manipulator, respectively. Then Section 4 presentsthe mobility analysis of a newly proposed 3-RPC-T parallelmanipulator. Section 5 includes the motion simulations ofall three parallel manipulators. These motion simulationsvalidate the results that were obtained from the mobilityanalysis presented in Sections 2, 3, and 4. Finally, Section 6presents some important conclusions and suggestions forfuture research, in general, and on the newly proposed 3-RPC-T parallel manipulator, in particular.

2. Mobility Analysis of the 3-RPC-YParallel Manipulator

The 3-RPC-Y parallel manipulator connects the platform tothe base by means of three legs which are RPC kinematicchains. The kinematic chain for leg i where the subscript ijdenotes joint 𝑗 (= 1, 2, 3, 4) on the leg is shown in Figure 4.Consider that the global Cartesian reference frame G (X, Y,and Z) is placed at the origin O, while a Cartesian referenceframe H (U, V, and W) is located at the centroid of theplatform E. The R joint connected to the base is denoted 𝑗

𝑖1;

the P joint is denoted as 𝑗𝑖2; and the C joint is composed of

the R and P joints 𝑗𝑖3, and 𝑗

𝑖4, respectively. Consider that the

R joint axis si1 is set to be collinearwith vector a𝑖 (representingthe position of theR joint), the P joint axis si2 is collinear withvector d

𝑖(denoting the position of the leg i), and the C joint

axes si3 and si4 are set to be collinear with vector b𝑖(which

represents the stroke of theC joint). If the platform is initiallyassembled parallel to the base, then the joint axes si1, si3, and

Mathematical Problems in Engineering 3

Platform

BaseR joint

C joint

P joint

Figure 3: The CAD model of the 3-RPC-T parallel manipulator.

si4 are invariantly parallel to each other and perpendicular tosi2. Then the 3-RPC-Y parallel manipulator is obtained whenthe three identical legs are equally distributed on the base(𝜓1

= 0∘, 𝜓2

= 120∘, and 𝜓

3= 240

∘), resulting in a starconfiguration (denoted here as Y) of the R joints on the base.Likewise, the C joints are equally distributed on the platform(where 𝜒

1= 0∘, 𝜒2= 120

∘, and 𝜒3= 240

∘).Since the P joints are selected as the active joints, then

each leg is comprised of a system of four motion-screwswhich can be written as

S𝑖1= [

s𝑖1

0 ] , S𝑖2= [

0s𝑖2

] ,

S𝑖3= [

s𝑖3

(a𝑖+ d𝑖) × s𝑖3

] , S𝑖4= [

0s𝑖4

] ,

(1)

where

d𝑖= [−𝑑𝑖c𝜃𝑖1s𝜓𝑖, 𝑑

𝑖c𝜃𝑖1c𝜓𝑖, 𝑑𝑖s𝜃𝑖1]

T,

s𝑖1= s𝑖3= s𝑖4= [c𝜓𝑖, s𝜓

𝑖, 0]

T,

s𝑖2= [−c𝜃𝑖1s𝜓𝑖, c𝜃

𝑖1c𝜓𝑖, s𝜃𝑖1]

T.

(2)

Performing the vector operations in (1) and (2) results in thegeneral branch motion-screw system for leg i; that is,

{S1} =

{{

{{

{

S𝑖1 = [c𝜓𝑖, s𝜓𝑖, 0; 0, 0, 0]T,

S𝑖2 = [0, 0, 0; −c𝜃𝑖1s𝜓𝑖, c𝜃𝑖1c𝜓𝑖, s𝜃𝑖1]T,

S𝑖3 = [c𝜓𝑖, s𝜓𝑖, 0; −𝑑𝑖s𝜃𝑖1s𝜓𝑖, 𝑑𝑖s𝜃𝑖1c𝜓𝑖, −𝑑𝑖c𝜃𝑖1]T,

S𝑖4 = [0, 0, 0; c𝜒𝑖, s𝜒𝑖, 0]T

}}

}}

}

.

(3)

Then substituting 𝜓1= 0∘, 𝜓2= 120

∘, and 𝜓3= 240

∘ into (3),the general branch motion-screw systems for legs 1, 2, and 3,respectively, are

{S1} =

{{

{{

{

S11 = [1, 0, 0; 0, 0, 0]T,

S12 = [0, 0, 0; 0, c𝜃11, s𝜃11]T,

S13 = [1, 0, 0; 0, 𝑑1s𝜃11, −𝑑1c𝜃11]T,

S14 = [0, 0, 0; 1, 0, 0]T

}}

}}

}

, (4)

{S2} =

{{

{{

{

S21=[−1, √3, 0; 0, 0, 0]T,

S22 = [0, 0, 0; −√3 c𝜃21, −c𝜃21, 2s𝜃21]T,

S23 = [−1, √3, 0; −√3𝑑2s𝜃21, −𝑑2s𝜃21, −2𝑑2c𝜃21]T,

S24 = [0, 0, 0; −1, √3, 0]T

}}

}}

}

,

(5)

{S3} =

{{

{{

{

S31 = [−1, −√3, 0; 0, 0, 0]T,

S32 = [0, 0, 0; √3 c𝜃31, −c𝜃31, 2s𝜃31]T,

S33 = [−1, −√3, 0; √3𝑑3s𝜃31, −𝑑3s𝜃31, −2𝑑3c𝜃31]T,

S34 = [0, 0, 0; −1, −√3, 0]T

}}

}}

}

.

(6)

Since the general branchmotion-screw system for leg 𝑖 is afour-screw system and their screws are independent (see (3),(4), (5), and (6)), then the branch constraint-screw system forleg i is formed by two independent reciprocal screws; that is,

{S𝑟𝑖} = {

S𝑟𝑖1= [0, 0, 0; −s𝜓

𝑖, c𝜓𝑖, 0]

T,

S𝑟𝑖2= [0, 0, 0; 0, 0, 1]

T } . (7)

Note that the reciprocal screws S𝑟𝑖1represent couples whose

axes are coplanar and perpendicular to si1, while the recipro-cal screws S𝑟

𝑖2represent couples acting about the Z-axis. The

branch constraint-screw systems for legs 1, 2, and 3 can bewritten, respectively, as

{S𝑟1} = {

S𝑟11

= [0, 0, 0; 0, 1, 0]T,

S𝑟12

= [0, 0, 0; 0, 0, 1]T } ,

{S𝑟2} = {

S𝑟21

= [0, 0, 0; √3, 1, 0]T,

S𝑟22

= [0, 0, 0; 0, 1, 0]T } ,

{S𝑟3} = {

S𝑟31

= [0, 0, 0; −√3, 1, 0]T,

S𝑟32

= [0, 0, 0; 0, 0, 1]T } .

(8)

Note that (8) provide six constraints on the platform,leading to the conclusion that this manipulator will havezero DOF. This conclusion is consistent with the observation


made by Callegari and Tarantini [20] and Callegari andMarzetti [21] concerning the overconstrained condition ofthis parallelmanipulator. Also, note that the reciprocal screwsS𝑟12, S𝑟22, and S𝑟

32in these three equations form a common

constraint that exerts the same constraint and reduces theorder of the screw system in the platform.

The platform constraint-screw system can be expressed interms of the three nonunique sets of independent reciprocalscrews in (8); that is,

{S𝑟} =

{{

{{

{

S𝑟11

= [0, 0, 0; 0, 1, 0]T,

S𝑟21

= [0, 0, 0; √3, 1, 0]T,

S𝑟12

= [0, 0, 0; 0, 0, 1]T

}}

}}

}

. (9)

Therefore, the platform motion-screw system can be writtenfrom the reciprocal screws in (9); that is,

{S𝑓} =

{{

{{

{

S𝑓1= [0, 0, 0; 1, 0, 0]

T,

S𝑓2= [0, 0, 0; 0, 1, 0]

T,

S𝑓3= [0, 0, 0; 0, 0, 1]

T

}}

}}

}

. (10)

This equation shows that the RPC-Y parallel manipulator hasthree translational DOF which are along the X-, Y-, and Z-axes. This result is consistent with the mobility predicted byCallegari and Marzetti [21].

3. Mobility Analysis of the 3-RCC-YParallel Manipulator

When an additional rotational DOF (denoted as joint 𝑗𝑖5) is

given to the P joints of the RPC-Y parallel manipulator, the

result is a RPRRP kinematic chain. Consider that the jointaxis s𝑖1of this chain is set to be collinear with vector a

𝑖, the

s𝑖3and s

𝑖4joint axes are collinear with vector b

𝑖, and the

s𝑖2and s

𝑖5joint axes are collinear with vector d

𝑖. For this

architecture, three identical legs are equally distributed onthe base (𝜓

1= 0∘, 𝜓2= 120

∘, and 𝜓3= 240

∘), while the Cjoints (i.e., joints 𝑗

𝑖3and 𝑗𝑖4) are equally distributed on the

platform (𝜒1

= 0∘, 𝜒2

= 120∘, and 𝜒

3= 240

∘). The joints𝑗𝑖2are selected as active joints. The resulting mechanism is

a 3-RPRC-Y (or 3-RCC-Y) parallel manipulator. Leg i of thismanipulator, which is comprised of a system of five screws,is shown in Figure 5. For convenience, the joint screws areexpressed in the global frame G (X, Y, and Z) and will beexpressed relative to the originO. For the sake of convenience,the platform is initially assembled parallel to the base. Notethat for this architecture the joint axes si1, si2, si3, and si4 areidentical to their corresponding joint axes of the 3-RPC-Yparallel manipulator. Therefore, the joint screws S

𝑖1, S𝑖2, S𝑖3,

and S𝑖4for the 3-RCC-Y parallel manipulator are as given by

(1). The remaining joint screw can be written as

S𝑖5= [

s𝑖5

a𝑖× s𝑖5

] . (11)

Substituting (2) into this equation and performing the vectoroperations, the joint screw can be written as

S𝑖5

= [−c𝜃𝑖1s𝜓𝑖, c𝜃𝑖1c𝜓𝑖, s𝜃𝑖1; 𝑎𝑖s𝜃𝑖1s𝜓𝑖, −𝑎𝑖s𝜃𝑖1c𝜓𝑖, 𝑎𝑖c𝜃𝑖1]

T.

(12)

Therefore, the branch motion-screw systems for the manipu-lator can be written as

{S1} =

{{{{{{{

{{{{{{{

{

S11

= [1, 0, 0; 0, 0, 0]T,

S12

= [1, 0, 0; 0, c𝜃11, s𝜃11]

T,

S13

= [1, 0, 0; 0, 𝑑1s𝜃11, −𝑑

1c𝜃11]

T,

S14

= [0, 0, 0; 1, 0, 0]T,

S15

= [0, c𝜃11, s𝜃11; 0, −𝑎s𝜃

11, 𝑎c𝜃

11]T

}}}}}}}

}}}}}}}

}

,

{S2} =

{{{{{{{{{

{{{{{{{{{

{

S21

= [−1, √3, 0; 0, 0, 0]T,

S22

= [0, 0, 0; √3c𝜃21, −c𝜃

21, 2s𝜃21]T,

S23=[−1, √3, 0; √3𝑑

2s𝜃21, −𝑑2s𝜃21, −2𝑑

2c𝜃21]T,

S24

= [0, 0, 0; −1, √3, 0]T,

S25

= [√3c𝜃21, −c𝜃

21, 2s𝜃21; √3𝑎s𝜃

21, 𝑎s𝜃21, 2𝑎c𝜃

21]T

}}}}}}}}}

}}}}}}}}}

}

,

{S3} =

{{{{{{{{{

{{{{{{{{{

{

S31

= [−1, −√3, 0; 0, 0, 0]T,

S32

= [0, 0, 0; √3 c𝜃31, −c𝜃

31, 2s𝜃31]T,

S33

= [−1, −√3, 0; √3𝑑3s𝜃31, −𝑑3s𝜃31, −2𝑑

3c𝜃31]T,

S34

= [0, 0, 0; −1, −√3, 0]T,

S35

= [√3c𝜃31, −c𝜃

31, 2s𝜃31; −√3𝑎s𝜃

31, 𝑎s𝜃31, 2𝑎c𝜃

31]T

}}}}}}}}}

}}}}}}}}}

}

.

(13)


ji3, ji4

Bi

W

U

E

V

ji2P

Z

Y

O

X

ji1

Ai𝜃i1

𝜓i

Si4Si3

si4si3

Si2

si2

di

bi

ai

si1Si1

𝜒i

Figure 4: Leg i of the 3-RPC parallel manipulator.

ji3, ji4

Bi

W

U

E

V

ji2, ji5P

Z

Y

O

X

ji1

Ai𝜃i1

𝜓i

Si4Si3

si4si3

Si5si5

Si2si2

di

bi

ai

si1Si1

𝜒i

Figure 5: Leg i of the 3-RCC-Y (3-RPRC-Y) parallel manipulator.

Since each branch motion-screw system is composed of fiveindependent screws, then the branch constraint-screw isgiven by the reciprocal screws

S𝑟𝑖1= [0, 0, 0; s𝜓

𝑖s𝜃𝑖1, −c𝜓

𝑖s𝜃𝑖1, c𝜃𝑖1]

T. (14)

The constraints exerted by these reciprocal screws representcouples whose axes are orthogonal to the vectors d

𝑖and a𝑖.

The platform constraint-screw system consists of theset of three independent reciprocal screws (namely, theconstraint-screws S𝑟

11, S𝑟21, and S𝑟

31); that is,

{S𝑟} =

{{{

{{{

{

S𝑟11

= [0, 0, 0; 0, −s𝜃11, c𝜃11]

T,

S𝑟21

= [0, 0, 0; 0, √3s𝜃21, 2c𝜃21]T,

S𝑟31

= [0, 0, 0; −√3s𝜃31, s𝜃31, 2c𝜃

31]T

}}}

}}}

}

.

(15)

The platform motion-screw system {S𝑓} can be obtained bydetermining the reciprocal screws to platform constraint-screw system given by (15). The result is a screw systemidentical to (10). This indicates that the platform has threeDOF which are translations along the X-, Y-, and Z-axes. Itis also worth noting that this manipulator has singular con-figurations in the center of the workspace, leading to directkinematics singularities when all three legs are perpendicularto the base.These conclusions and observations are consistentwith the work of Callegari and Marzetti [21].

4. Mobility Analysis of the New 3-RPC-TParallel Manipulator

The RPC kinematic chain that was described in Section 2for the 3-RPC-Y architecture (see Figure 4), is used in thissection to obtain a new fully translational parallel manipu-lator, namely, the 3-RPC-T parallel manipulator. The designconsists of three identical RPC kinematic chains distributedon the base as follows: 𝜓

1= 0∘, 𝜓2= 90∘, and 𝜓

3= 180

∘.In a similar manner, the joints 𝑗

𝑖4are distributed on the

platform by setting 𝜒1= 0∘, 𝜒2= 90∘, and 𝜒

3= 180

∘. Theconfiguration of the three identical legs equally distributedon the base results in a T configuration of the 𝑗

𝑖1joints, and

hence the name, the 3-RPC-T parallel manipulator. Note thatthe joint axes s

11and s31are collinear and coplanar with the

XY plane, while the joint axes s13, s33, s14, and s

34are collinear

and coplanar with the UV plane. Due to the geometry ofthe joint axes, legs 1 and 3 provide a synchronous motion.This section considers the selection of joints 𝑗

11, 𝑗21, and 𝑗

32

as a nonunique combination of active joints defined for themanipulator.

Since the kinematic chain of the 3-RPC-T parallel manip-ulator is identical to the kinematic chain of theRPC-Y parallelmanipulator, then the general branch motion-screw systemfor leg i is given by (3). Substituting 𝜓

1= 0∘, 𝜓2

= 90∘,

and 𝜓3= 180

∘ into this equation, the branch motion-screwsystems for legs 1, 2, and 3, respectively, are

{S1} =

{{{{

{{{{

{

S11

= [1, 0, 0; 0, 0, 0]T,

S12

= [0, 0, 0; 0, c𝜃11, s𝜃11]

T,

S13

= [1, 0, 0; 0, 𝑑1s𝜃11, −𝑑1c𝜃11]

T,

S14

= [0, 0, 0; 1, 0, 0]T

}}}}

}}}}

}

,

{S2} =

{{{{

{{{{

{

S21

= [0, 1, 0; 0, 0, 0]T,

S22

= [0, 0, 0; −c𝜃21, 0, s𝜃

21]T,

S23

= [0, 1, 0; −𝑑2s𝜃21, 0, −𝑑

2c𝜃21]

T,

S24

= [0, 0, 0; 0, 1, 0]T

}}}}

}}}}

}

,

{S3} =

{{{{

{{{{

{

S31

= [−1, 0, 0; 0, 0, 0]T,

S32

= [0, 0, 0; 0, −c𝜃31, s𝜃31]

T,

S33

= [−1, 0, 0; 0, −𝑑3s𝜃31, −𝑑3c𝜃31]

T,

S34

= [0, 0, 0; −1, 0, 1]T

}}}}

}}}}

}

.

(16)


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�x�y�z

(d)

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

(mm

/s2)

0 1 2 3 4 5Time (s)

axayaz

(e)

(mm

/s2)

0 1 2 3 4 5Time (s)

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

axayaz

(f)

Figure 6: Simulation plots for the 3-RPC-Y and the 3-RCC-Y parallel manipulators.


0 1 2 3 4 5−300

−200

−100

0

100

200

300

400

500

Time (s)

(mm

)

pxpypz

(a)

−100

−50

0

50

100

150

(mm

/s)

0 1 2 3 4 5Time (s)

�x�y�z

(b)

0 1 2 3 4 5−25

−20

−15

−10

−5

0

Time (s)

(mm

/s2)

axayaz

(c)

0 1 2 3 4 5−1

−0.5

0

0.5

1

Time (s)

(deg

)(d

eg/s

)(d

eg/s2)

𝛼

𝛽

𝛾

(d)

Figure 7: Simulation plots for the 3-RPC-T parallel manipulator.

Also, the branch constraint-screw system for leg i is given by(7). Substituting 𝜓

1= 0∘, 𝜓2= 90∘, and 𝜓

3= 180

∘ into thisequation, the branch constraint-screw system for legs 1, 2, and3, respectively, are

{S𝑟1} = {

S𝑟11

= [0, 0, 0; 0, 1, 0]T,

S𝑟12

= [0, 0, 0; 0, 0, 1]T} ,

{S𝑟2} = {

S𝑟21

= [0, 0, 0; 1, 0, 0]T,

S𝑟22

= [0, 0, 0; 0, 0, 1]T} ,

{S𝑟3} = {

S𝑟31

= [0, 0, 0; 0, 1, 0]T,

S𝑟32

= [0, 0, 0; 0, 0, 1]T} .

(17)

Equations (17) represent physical constraints exerted byeach kinematic chain on the platform. The constraints canbe interpreted geometrically as (i) couples whose axes areperpendicular to si1 and lie on the XY plane are presentedby reciprocal screws S𝑟

11and S𝑟21that provide two constraints

and by reciprocal screw S𝑟31

that is considered as providinga redundant constraint and (ii) couples acting about the Z-axis are represented by reciprocal screws S𝑟

12, S𝑟22, and S𝑟

32that


provide a common constraint that any two of these screws canbe considered redundant common constraints.

From the nonunique set of independent reciprocal screwsS𝑟11, S𝑟21, and S𝑟

12, the platform constraint-screw system can be

written as

{S𝑟} =

{{

{{

{

S𝑟11

= [0, 0, 0; 0, 1, 0]T

S𝑟21

= [0, 0, 0; 1, 0, 0]T,

S𝑟12

= [0, 0, 0; 0, 0, 1]T

}}

}}

}

. (18)

Therefore, the platform motion-screw system {S𝑓} can beobtained from the reciprocal screws of {S𝑟} in (18). The resultis a screw system which is identical to (10), indicating thatthe platform of the 3-RPC-T parallel manipulator has threetranslational DOF, that is, translations along the X-, Y-, andZ-axes.

5. Motion Simulations

To verify the results of the mobility analysis that was per-formed in Sections 2, 3, and 4, this section presents CADmodels of the 3-RPC-Y parallel manipulator, the 3-RCC-Y parallel manipulator, and the newly proposed 3-RPC-T parallel manipulator. The three models are subjected tomotion simulations. The parameters that are used in thesimulations, such as the total simulation time, the velocityof the active joints, and the dimensional parameters of thethree manipulators, are presented in Tables 1, 2, and 3.The tables include the simulation results of the initial andfinal poses of the platforms where 𝑝

𝑥, 𝑝𝑦, and 𝑝

𝑧are the

Cartesian coordinates of the platform centroid E and 𝛼, 𝛽,and 𝛾 are the projection angles (that express the orientationof the platform) about the X-, Y-, and Z-axes, respectively.Figures 6(a), 6(c), and 6(e) show the simulation results forthe platform position, velocity, and acceleration of the 3-RPC-Y parallelmanipulator. Figures 6(b), 6(d), and 6(f) showthe simulation results for the platform position, velocity, andacceleration of the 3-RCC-Y parallel manipulator. Finally,Figures 7(a), 7(b), and 7(c) show the simulation results forthe platformposition, velocity, and acceleration of the 3-RPC-T parallel manipulator. Figure 7(d) shows the results of theprojected angles of the platform. Note that for all of thesefigures the X-, Y-, and Z-axes components are shown in solid,dashed, and dotted plots, respectively.

According to (10), the platform of the 3-RPC-Y parallelmanipulator has three translational degrees-of-freedom (thatis, full Cartesian motion). Table 1 presents the simulationparameters used for obtaining the positioning profile of theplatform. From the position, velocity, and acceleration ofcentroid 𝐸 (see Figures 6(a), 6(c), and 6(e)), the platform canoccupy an arbitrary position on any axis of the global frame.Also, the results of the projected angles of the platform (seeFigure 7(d)) indicate that there is no rotation in any of theaxes of the global frame.

According to the platformmotion-screw system {S𝑓} thatis identical to (10), the platform of the 3-RCC-Y parallelmechanism also has three translational degrees-of-freedom(that is, full Cartesian motion). Table 2 presents the simu-lation parameters used for obtaining the positioning profile

Table 1: Simulation parameters and results for the 3-𝑅𝑃𝐶-Y parallelmechanism.

Simulationparameters

Initialconfiguration

Finalconfiguration

𝑎 = 433.01mm 𝑑1(mm) 446.29 519.80

Time = 5 s 𝑑2(mm) 422.69 472.14

𝑑1= 10mm/s 𝑑

3(mm) 470.34 456.30

𝑑2= 3mm/s 𝜃

11(deg) 115.01 121.51

𝑑3= −5mm/s 𝜃

12(deg) 88.73 69.82

𝜃13(deg) 63.96 76.21

𝑝𝑥(mm) 124.64 −31.26

𝑝𝑦(mm) −197.10 −271.67

𝑝𝑧(mm) 422.58 443.15

𝛼, 𝛽, 𝛾 (deg) 0 0

Table 2: Simulation parameters and results for the 3-RCC-Y parallelmechanism.



Finalconfiguration

𝑎 = 433.01mm 𝑑1(mm) 394.00 393.01

Time = 5 s 𝑑2(mm) 422.81 466.56

𝑑1= −2mm/s 𝑑

3(mm) 382.41 361.67

𝑑2= 5mm/s 𝜃

11(deg) 107.83 119.75

𝑑3= −5mm/s 𝜃

12(deg) 62.51 46.64

𝜃13(deg) 101.24 109.75

𝑝𝑥(mm) −155.69 −255.04

𝑝𝑦(mm) −120.64 −196.67

𝑝𝑧(mm) 375.08 340.27

𝛼, 𝛽, 𝛾 (deg) 0 0

Table 3: Simulation parameters and results for the 3-RPC-T parallelmechanism.



Finalconfiguration

a = 225mm 𝑑1(mm) 427.64 282.51

Time = 5 s 𝑑2(mm) 511.03 280.55

𝜃11

= −1.8 rpm 𝑑3(mm) 511.03 280.55

𝑑3= −75mm/s 𝜃

11(deg) 98.69 68.09

𝜃21

= −1 rpm 𝜃12(deg) 55.81 110.89

𝜃13(deg) 110.89 55.81

𝑝𝑥(mm) −287.15 100.05

𝑝𝑦(mm) −64.63 105.41

𝑝𝑧(mm) 422.73 262.11

𝛼, 𝛽, 𝛾 (deg) 0 0

of the platform as shown in Figure 6(b). The plots of thevelocity and acceleration of the platform are presented inFigures 6(d) and 6(f). Note that the projected angles of theplatforms of the 3-RPC-Y and 3-RCC-Y parallelmanipulators


are identical; see Figure 7(d). The simulations prove that thisparallel manipulator has fully translational motion.

According to the platform motion-screw system from(10), the newly proposed 3-RPC-T parallel manipulator (seeSection 4) has three translational degrees-of-freedom. Themobility of this manipulator is verified with a motionsimulation using the parameters listed in Table 3. Recallthat for this architecture the joints 𝑗

11, 𝑗21, and 𝑗

32are

selected as actuators. According to the simulation plots forthe position, velocity, and acceleration shown in Figures 7(a)–7(c), the platform is able to position itself along any of thethree Cartesian axes of the global frame. The simulationresults for the projection angles, angular velocity, and angularacceleration of the platform are shown in Figure 7(d). Thisfigure shows that the platform is parallel to the base for allpositions of the manipulator.

6. Conclusions

This paper presented a mobility analysis of the 3-RPC-Yparallel manipulator, the 3-RCC-Y parallel manipulator, anda newly proposed 3-RPC-T parallel manipulator.The analysiswas based on screw theory which provided geometric insightinto the investigation.Thepaper obtained the branchmotion-screws for all three architectures and then identified theplatform constraint-screw systems. The results showed thatall RPC-based architectures displayed the same constraints,namely, (i) couples whose axes are perpendicular to the Rjoint axes and coplanar with the base and (ii) couples actingabout the normal axis to the base. The constraints of the3-RCC-Y parallel manipulator are couples whose axes areorthogonal to the R joint axis and the leg of each kinematicchain. The mobility of the platform is then obtained bydetermining the reciprocal screws to the platform constraint-screw sets and the platforms are identified to have threeinstantaneous independent translational DOF. The resultsof the mobility analysis were substantiated by using CADmodels andmotion simulations.The simulations showed thatthe platforms could translate along any axis of the globalframe, while there was no rotation of the platform.

Acknowledgment

E. Rodriguez-Leal acknowledges the financial support fromTecnologico de Monterrey and CONACYT.

References

[1] T. G. Ionescu, “IFToMM terminology/English,”Mechanism andMachine Theory, vol. 38, no. 7–10, pp. 607–682, 2003.

[2] P. A. Chebychev, “Theorie des mecanismes connus sous lenom de parallelogrammes, 2eme partie,” Memoires presentesa l’Academie imperiale des sciences de Saint-Petersbourg pardivers savants, 1869 (French).

[3] G. Gogu, “Mobility of mechanisms: a critical review,” Mecha-nism and Machine Theory, vol. 40, no. 9, pp. 1068–1097, 2005.

[4] J. J. Uicker, G. R. Pennock, and J. E. Shigley,Theory of MachinesandMechanisms, Oxford University Press, New York, NY, USA,4th edition, 2011.

[5] X. Kong and C. M. Gosselin, “Type synthesis of three-degree-of-freedom spherical parallel manipulators,” The InternationalJournal of Robotics Research, vol. 23, no. 3, pp. 237–245, 2004.

[6] E. Rodriguez-Leal, J. S. Dai, and G. R. Pennock, “Kinematicanalysis of a 5-𝑅𝑆𝑃 parallel mechanism with centralizedmotion,”Meccanica, vol. 46, no. 1, pp. 221–237, 2011.

[7] E. Rodriguez-Leal, J. S. Dai, and G. R. Pennock, “A study ofthe instantaneous kinematics of the 5-𝑅𝑆𝑃 parallel mechanismusing screw theory,” in Advances in Reconfigurable Mechanismsand Robots I, pp. 355–370, Springer, London, UK, 2012.

[8] J. Angeles and C. M. Gosselin, “Determination du degre deliberte des chaınes cinematiques,” Transactions of the CanadianSociety of Mechanical Engineering, vol. 12, no. 4, pp. 219–226,1988 (French).

[9] J. Angeles, “The qualitative synthesis of parallel manipulators,”in Proceedings of the Workshop on Fundamental Issues andFuture Research Directions for Parallel Mechanisms and Manip-ulators, pp. 160–169, Quebec, Canada, October 2002.

[10] P. Fanghella, “Kinematics of spatial linkages by group algebra:a structure-based approach,” Mechanism and Machine Theory,vol. 23, no. 3, pp. 171–183, 1988.

[11] P. Fanghella and C. Galletti, “Mobility analysis of single-loopkinematic chains: an algorithmic approach based on displace-ment groups,” Mechanism and Machine Theory, vol. 29, no. 8,pp. 1187–1204, 1994.

[12] J. M. Herve, “Lie group of rigid body displacements, a funda-mental tool for mechanism design,” Mechanism and MachineTheory, vol. 34, no. 5, pp. 719–730, 1999.

[13] J. M. Rico-Martinez and B. Ravani, “On mobility analysis oflinkages using group theory,” Journal of Mechanical Design, vol.125, no. 1, pp. 70–80, 2003.

[14] J. M. Rico, J. Gallardo, and B. Ravani, “Lie algebra and themobility of kinematic chains,” Journal of Robotic Systems, vol.20, no. 8, pp. 477–499, 2003.

[15] E. Cuan-Urquizo and E. Rodriguez-Leal, “Kinematic analysisof the 3-CUP parallel mechanism,” Robotics and Computer-Integrated Manufacturing, vol. 29, no. 5, pp. 382–395, 2012.

[16] Z. Huang andQ. C. Li, “Generalmethodology for type synthesisof symmetrical lower-mobility parallel manipulators and sev-eral novel manipulators,” The International Journal of RoboticsResearch, vol. 21, no. 2, pp. 131–145, 2002.

[17] J.-S. Zhao, K. Zhou, and Z.-J. Feng, “A theory of degrees offreedom formechanisms,”Mechanism andMachineTheory, vol.39, no. 6, pp. 621–643, 2004.

[18] J.-S. Zhao, Z.-J. Feng, and J.-X. Dong, “Computation of theconfiguration degree of freedomof a spatial parallel mechanismby using reciprocal screw theory,” Mechanism and MachineTheory, vol. 41, no. 12, pp. 1486–1504, 2006.

[19] J. S. Dai, Z. Huang, and H. Lipkin, “Mobility of overconstrainedparallel mechanisms,” Journal ofMechanical Design, vol. 128, no.1, pp. 220–229, 2006.

[20] M. Callegari and M. Tarantini, “Kinematic analysis of a noveltranslational platform,” Journal of Mechanical Design, vol. 125,no. 2, pp. 308–315, 2003.

[21] M. Callegari and P. Marzetti, “Kinematics of a family of paralleltranslatingmechanisms,” in Proceedings of the 12th InternationalWorkshop on Robotics in Alpe-Adria-Danube Region (RAAD’03), Cassino, Italy, May 2003.

[22] J. M. Herve and F. Sparacino, “Structural synthesis of parallelrobots generating spatial translation,” in Proceedings of the 5thInternational Conference on Advanced Robotics, vol. 1, pp. 808–813, Pisa, Italy, 1991.

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