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Concurrent Design of Reconfigurable Robots using a Robotic Hardware-in-the-loop Simulation
by
Robin Chhabra
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science
University of Toronto Institute for Aerospace Studies University of Toronto
© Copyright by Robin Chhabra 2008
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Concurrent Design of Reconfigurable Robots using a Robotic
Hardware-in-the-loop Simulation
Robin Chhabra
Master of Applied Science
University of Toronto Institute for Aerospace Studies University of Toronto
2008
Abstract
This thesis discusses a practical approach to the concurrent analysis and synthesis of
reconfigurable robot manipulators, which is based on the alternative design methodology of
Linguistic Mechatronics (LM) as well as the utilization of a modular Robotic Hardware-in-the-
loop Simulation (RHILS) platform. Linguistic Mechatronics is a systematic design methodology
for mechatronic systems, which formalizes subjective notions and simplifies the optimization
process, in the hope that communication between designers with various backgrounds and clients
is enhanced and numerous design variables with different natures can be considered
concurrently. The methodology redefines the ultimate goal of design based on the qualitative
notions of wish and must satisfactions, and formalizes the effect of designer’s subjective attitude
in the process, which can be adjusted based on the reality of system performance. The underlying
concepts of LM is investigated through a simulation case study. In addition, the RHILS platform
involves physical joint modules and the control unit to reduce modeling complexities while
taking into account various physical phenomena. This platform is employed to the design
architecture in order to evaluate the design attributes during a design process. Ultimately, the
new approach is applied to redesigning kinematic, dynamic and control parameters of an
industrial manipulator.
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Acknowledgments
I would like to thank my supervisor Dr. M. R. Emami for his support, patience and guidance
throughout this thesis. He was always available to answer my questions and give the direction to
my research. Besides, he gave me the opportunity to develop and impose my raw ideas in this
work. Beyond that, he also familiarized me with the academic world as a researcher and
introduced me the importance of ethics in research.
I would also like to thank my family for their help and support during this thesis.
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Table of Contents Abstract.......................................................................................................................................... ii
Acknowledgments ........................................................................................................................ iii
Table of Contents ......................................................................................................................... iv
List of Notations .......................................................................................................................... vii
List of Figures.............................................................................................................................. xii
List of Tables .............................................................................................................................. xiv
List of Appendices....................................................................................................................... xv
Chapter 1: Introduction and Background.................................................................................. 1
1.1 Motivation................................................................................................................................. 1
1.2 Objectives ................................................................................................................................. 2
1.2.1 LM Methodology............................................................................................................... 2
1.2.2 Implementation ................................................................................................................. 2
1.3 Thesis Outline ........................................................................................................................... 3
1.4 Literature Review...................................................................................................................... 3
1.4.1 Reconfigurable Robotics................................................................................................... 3
1.4.2 Concurrent Design ............................................................................................................ 6
1.4.3 Robotic Hardware-in-the-loop Simulation........................................................................ 8
Chapter 2: Theoretical Background ......................................................................................... 10
2.1 Fuzzy Sets and Fuzzy Logic ................................................................................................... 10
2.1.1 Fuzzy-Logic Modeling.................................................................................................... 10
2.1.2 Fuzzy Connectives .......................................................................................................... 11
2.1.2.1 Parameterized operators........................................................................................ 13
2.1.3 Takagi-Sugeno-Kang (TSK) Inference Mechanism ....................................................... 15
2.1.4 Fuzzy Rule-Base Generation........................................................................................... 16
v
2.1.4.1 Fuzzy C-Means Clustering ................................................................................... 17
2.1.4.2 Selecting Significant Input Variable..................................................................... 18
2.1.4.3 Assigning Input Membership Functions............................................................... 19
2.2 Bond Graphs Modeling........................................................................................................... 19
2.2.1 Basic Elements ................................................................................................................ 20
2.2.1.1 Single-Port Elements ............................................................................................ 20
2.2.1.2 Double-Port Elements (Transformers and Gyrators) ........................................... 21
2.2.1.3 Multi-Port Elements (Junctions)........................................................................... 22
2.2.2 Causality.......................................................................................................................... 22
2.2.2.1 Fixed Causality ..................................................................................................... 23
2.2.2.2 Constrained Causality ........................................................................................... 23
2.2.2.3 Preferred Causality................................................................................................ 23
2.2.2.4 Indifferent Causality ............................................................................................. 24
Chapter 3: Linguistic Mechatronics.......................................................................................... 25
3.1 Design Problem in LM Framework......................................................................................... 27
3.2 Calculation of Overall Satisfaction......................................................................................... 29
3.2.1 Aggregation of Must Design Attributes .......................................................................... 29
3.2.2 Aggregation of Wish Design Attributes .......................................................................... 30
3.2.3 Aggregation of Overall Wish and Must Satisfactions ..................................................... 31
3.3 Primary Phase of LM .............................................................................................................. 32
3.4 Secondary Phase of LM .......................................................................................................... 33
3.5 Performance Supercriterion.................................................................................................... 33
Chapter 4: Application, Simulation Results ............................................................................. 38
4.1 Design Problem....................................................................................................................... 38
4.1.1 Design Variables ............................................................................................................. 38
4.1.2 Design Attributes ............................................................................................................ 39
vi
4.1.2.1 Must Design Attributes ......................................................................................... 39
4.1.2.2 Wish Design Attributes ......................................................................................... 40
4.1.3 Assigning Satisfactions ................................................................................................... 42
4.2 Primary Phase of LM .............................................................................................................. 43
4.3 Secondary Phase of LM .......................................................................................................... 45
4.4 Performance Supercriterion .................................................................................................... 46
4.5 Results and Discussion ........................................................................................................... 48
Chapter 5: Application, Experimental Results ........................................................................ 51
5.1 Design Architecture Including RHILS .................................................................................... 51
5.1.1 Host Workstation ............................................................................................................ 51
5.1.2 Target Workstation ......................................................................................................... 52
5.1.3 Hardware Emulation ....................................................................................................... 52
5.2 Design Problem....................................................................................................................... 53
5.2.1 Design Variables ............................................................................................................. 53
5.2.2 Design Attributes ............................................................................................................ 54
5.3 Design Process ................................................................................................................... 55
5.4 Results and Discussion ........................................................................................................... 57
Chapter 6: Conclusions and Future Work ............................................................................... 59
6.1 Conclusions............................................................................................................................. 59
6.2 Future work............................................................................................................................. 61
References or Bibliography .......................................................................................................... 62
Figures.......................................................................................................................................... 67
Tables ........................................................................................................................................... 80
Appendix A (Pareto-optimality of the overall satisfaction optimization).................................... 84
Appendix B (First and second laws of thermodynamics from a different perspective) .............. 85
vii
List of Notations A – Design attributes set
Ai – A design attribute
ai – A design attribute satisfaction
Bij – Antecedent fuzzy set
bij – Linear coefficient in TSK
C – Set of suitable intervals for design variables
C – Complication operator, storage element
c – Number of clusters
Cj – Suitable interval for a design variable
cj – Membership function over the suitable interval of a design variable
CS – Set of optimally satisfactory design candidates
CoA – Center of area defuzzification method
)(•cond – Condition number
D – Design availabilities set
di – Offset of link
Dik – Consequent fuzzy set
Dj – A design availability
E – Effective work, end-effector error
er – Effort vector
iE – Voltage source of motor
fr
– Flow vector
Fi – An attribute function
11iii f ,+ – Force vector in frame i
G – Generalized mean operator
ig – Gravitational acceleration
(M)GY – (Modulated) gyrator element
H – Gyrator ratio matrix
I – Storage element
iCI – Moment of inertia matrix
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Ii – Integral gain
Jm – Objective function in FCM
imj – Moment of inertia of motor
K – Dynamic energy
imK – Torque coefficient of motor
ifbKa , – Acceleration feedback gain
iffKa , – Acceleration feedforward gain
ifbKv , – Velocity feedback gain
iffKv , – Velocity feedforward gain
L – Length sum
li – Length of link
iml – Inductor coefficient of motor
M – Must attributes set
M – Manipulability
Mi – A must attibute
m – Weighting exponent
mi – A must attribute satisfaction, mass
N – Number of data points, transformation ratio matrix, number of working points
n – Number of design variables
n0 – Number of significant design variables
NM – Number of must attributes
NW – Number of wish attributes
±WN – Number of positive/negative-differential wish attributes
ndof – Number of D.O.F
P – Power
pr – Generalized momentum
Pi – Proportional gain
p, q, α – Attitude parameters
qr – Generalized displacement
QL – Structural length index
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R – dissipative element, reachability
r – Number of input variables
r0 – Number of significant inputs
iCi r – Center of mass position
ri – Radius of link
1ii R − – Rotation matrix
ii r – Distance between two frames
imr – Total resistance of motor
Rmax – Maximum reachability
S – T-conorm operator, source(sink), supplied energy
SB – Fuzzy between-cluster scatter matrix
scs – Cluster validity index
(M)Se/f – (Modulated) effort/flow source (sink)
SW – Fuzzy within-cluster scatter matrix
sT – Trace of fuzzy total scatter matrix
T – T-norm operator, response time
tf – Final time
i1iiiT ,+ – Torque vector in frame i
(M)TF – (Modulated) transformation element
)(•tr – Trace of a matrix
U – Membership matrix
Ui – Input fuzzy variable
iau – Voltage after driver
uik – A member of membership matrix
uj – Crisp input variable
V – Set of cluster centers, workspace volume
v – Weighted mean of dataset
iCi v – Center of mass linear velocity
ii v – Linear velocity
2ij
1ij vv – Range of input corresponding to output equals zero
x
Vk – Output fuzzy variable
vk – A cluster center
W – Wish attributes set
Wi – A wish attribute
wi – A wish attribute satisfaction ±W – Positive/negative-differential wish attributes set
X – Design variables set
X – Set of data points
X0 – Set of initial design variables
xi – a data point, an availability satisfaction
Xj – Fuzzy model input space, a design variable
Xj0 – an initial design variable maxmin/
kX – Minimum/maximum of a design variable
Y – Union of X and A
y – Cost function
yi – Crisp output variable *y – Crisp output of TSK
Yk – Fuzzy model output space
ii z – Joint axis
αi – Twist of link
ijΓ – The range where input is one
jΓ – The entire range of input
jΔ – Maximum permitted translational error
jδ – Maximum permitted rotational error
jxΔ – x direction position error
jyΔ – y direction position error
jzΔ – z derection position error
jxδ – Rotational error about x axis
jyδ – Rotational error about y axis
xi
jzδ – Rotational error about z axis
iη – Transmission ratio
jiθ – Angle between two links
maxmin/iθ – Minimum/maximum of an angle
imθ – Angle of motor
μ – Satisfaction ),,( αμ qp – Overall satisfaction
ilμ – Output overall satisfaction of rule i in TSK
*lμ – Defuzzified overall satisfaction in TSK
)( pMμ – Overall must satisfaction
),( qW
αμ – Overall wish satisfaction
)(q
W ±μ – Overall positive/negative-differential wish attributes satisfaction
jπ – Measure of non-significance
iτ – Degree of fire of rule i in TSK, torque along joint axis
j
iτmax – Maximum absolute value of torque
maxiτ – Maximum allowed amount of torque
jTτ – Total required torque
iiω – Angular velocity
ff / – Superior/loosely superior
A• – Inner product norm using A matrix
xii
List of Figures Figure 1.1 (a) HILS software flowchart, (b) HILS Block, (c) Real-time optimization with HILS architecture.................................................................................................................................... 67
Figure 2.1 (a) scalar bond, (b) vector bond connecting two power ports of components A and B
....................................................................................................................................................... 68
Figure 2.2 Block diagrams for all possible causality assignments of bond graphs elements ....... 68
Figure 3.1 Linguistic Mechatronics flowchart.............................................................................. 69
Figure 4.1 Design Architecture including simulation................................................................... 70
Figure 4.2 Satisfactions on design variables and attributes .......................................................... 71
Figure 4.3 Trace of total scatter matrix for identifying m............................................................. 71
Figure 4.4 Specification of c......................................................................................................... 72
Figure 4.5 Antecedents membership functions for the most satisfactory rule.............................. 73
Figure 4.6 Bond graph representation of a serial link manipulator .............................................. 74
Figure 4.7 Bond graph representation of an electric motor .......................................................... 75
Figure 4.8 Circuit schematic of an electric motor......................................................................... 75
Figure 4.9 Block diagram model of the controller........................................................................ 75
Figure 4.10 Simulink® model of a 5 D.O.F manipulator based on bond graphs .......................... 76
Figure 4.11 Simulink® model of a link based on bond graphs ..................................................... 77
Figure 4.12 Simulink® model of an electric motor based on bond graphs ................................... 77
Figure 5.1 The design architecture including RHILS.................................................................... 78
Figure 5.2 The schematic of CRS DM Master Controller............................................................. 78
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Figure 5.3 (a) CRS CataLyst-5 robot (b) RHILS platform............................................................ 79
Figure 5.4 Satisfactions on design variables and attributes .......................................................... 79
xiv
List of Tables Table 2.1 Bond graphs elements in various energy domains........................................................ 80
Table 4.1 Design variables and attributes ranges.......................................................................... 80
Table 4.2 Measure of non-significance for design variables ........................................................ 81
Table 4.3 Significant design variables .......................................................................................... 81
Table 4.4 Numbering of significant design variables ................................................................... 81
Table 4.5 Antecedents and consequent parameters of the most satisfactory rule......................... 81
Table 4.6 Initial values of design variables .................................................................................. 82
Table 4.7 Motors coefficients used in the simulation ................................................................... 82
Table 4.8 Design results................................................................................................................ 82
Table 5.1 Design variables and attributes ranges.......................................................................... 83
Table 5.2 Design Results .............................................................................................................. 83
xv
List of Appendices Appendix A: Pareto-optimality of the overall satisfaction optimization ...................................... 84
Appendix B: First and second laws of thermodynamics from a different perspective................. 85
1
Chapter 1: Introduction and Background
Designers occasionally employ a subsystem-partitioning approach to synthesizing complex
engineering systems. Reconfigurable robots are good examples of such systems. Their design
methodology is traditionally based on the sequential decomposition of mechanical,
electromechanical, and control/instrumentation subsystems, so that at each step a subset of
design variables is considered separately [1]. Although conventional decoupled or loosely-
coupled approaches of design seem intuitively practical, they undermine the interconnection
between various subsystems that may indeed play a significant role in multidisciplinary systems.
The necessity of communication and collaboration between the subsystems implies that such
systems ought to be synthesized concurrently. In the concurrent design process, design
knowledge is accumulated from all the participating disciplines, and they are offered equal
opportunities to contribute to the current state of the design in parallel. The synergy resulting
from integrating different disciplines in concurrent design has been documented in several case
studies, to the effect that the outcome is a new and previously unattainable set of performance
characteristics [2]. However, the challenge in a concurrent design process is that the
multidisciplinary system model can become prohibitively complicated; hence computationally
demanding, plus a large number of multidisciplinary objective and constraint functions must be
taken into account simultaneously with a great number of design variables.
This thesis addresses the above challenge through a combination of two solutions: a) an
alternative design methodology, namely Linguistic Mechatronics (LM), which takes into account
subjective notions of design and transforms a multi-objective constrained optimization problem
to a single-objective unconstrained formulation to simplify the concurrent design computations;
b) an efficient system modeling technique that generates real-time models that account for
complex phenomena such as sensor noise, actuator limitation, transmission flexibility, etc., by
utilizing real hardware modules in the loop.
1.1 Motivation The main goal of this research was to formalize a practical design methodology for concurrent
synthesis of multidisciplinary systems and take advantage of a Robotic Hardware-In-the-Loop
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Simulation (RHILS) in order to evaluate the design attributes in the design process. The design
framework namely Linguistic Mechatronics (LM) was developed and an RHILS was employed in
parallel with LM to tackle a design problem.
1.2 Objectives This thesis followed the following objectives: i) formalize a design framework for mechatronic
systems, i.e., LM, ii) implement LM and RHILS in parallel in order to redesign CRS CataLyst-5
that is an industrial manipulator.
1.2.1 LM Methodology Linguistic Mechatronics, formulated based on fuzzy logic, systemizes the qualitative and
subjective aspects of conceptual design through a practical multiattribute concurrent approach,
yet takes into account objective performances using a holistic mechatronic system modeling. The
methodology considers subjective aspects of design in the form of the designer’s preferences and
attitude, and reduces the complicated multi-objective constrained optimization problem to a
single-objective unconstrained optimization utilizing fuzzy-logic operators to aggregate
satisfactions corresponding to the design attributes. Hence, not only does the suggested approach
ease the communication between designers of different backgrounds and clients, but it also
makes the multiattribute design solution more practical.
1.2.2 Implementation The combination of LM and RHILS is a flexible, sufficiently accurate and rapidly changing
design platform for concurrent synthesis of robotics systems specially reconfigurable robots that
play an important role in aerospace industry. On one hand, LM design framework simplifies the
optimization problem by redefining the main goal of design based on satisfaction instead of pure
optimization and on the other hand, RHILS takes into account phenomena that are difficult to
model and make the analysis of the system real time, efficient in terms of time and money and
more reliable. This design platform allows the designer to consider a large number of design
variables in multidisciplinary systems and include both physical objectivity and human
subjectivity aspects of design.
3
1.3 Thesis Outline The remainder of this chapter is devoted to a thorough literature study in the fields of concurrent
design, hardware-in the-loop simulation and reconfigurable robotics. Chapter 2 details the
required theoretical background including fuzzy set theory and bond graph modeling in order to
develop the notion of Linguistic Mechatronics. The LM methodology is formulated in the third
chapter and forth chapter demonstrates a simulation case study. Chapter 5 addresses another case
study utilizing LM and RHILS in order to redesign an industrial robotic manipulator called CRC
CataLyst-5. The conclusion of this thesis is presented in Chapter 6 along with the summary of
the research and the discussion on the design results.
1.4 Literature Review
1.4.1 Reconfigurable Robotics Reconfigurability has been the subject of interest for both research and industrial communities in
the past two decades. The need for reconfigurable production lines is felt in almost every sector
of industry. For example, in manufacturing, on the one hand, products are various and rapidly
changing, and manufacturing time has been reduced significantly (e.g. car industries) [3]. On the
other hand, customers except high quality, long durability and low price of the manufacturing
products [4]. The best way of treating this dilemma would be to modularize the products, and
design manufacturing systems that are capable of accommodating as many process variations as
possible [4]. Nowadays, major production lines are run by robots and manipulators. Thus, the
solution to the above challenge would be in improving the robotic systems. What is a Robot?
According to the Robot Institute of America (1979), a robot is a reprogrammable,
multifunctional manipulator designed to move material, parts, tools, or specialized devices
through various programmed motions for the performance of a variety of tasks [5].
Therefore, the necessity of adaptive manufacturing systems leads to fabricating and utilizing
reconfigurable robotic systems. This group of robots are fabricated based on modularity.
Modularity consists of any number of identical interconnected units or modules. Modular
systems have advantages in terms of manufacturing and robustness, due to their homogeneity
and redundancy [6]. The modularity idea has also come into manipulator design based on the
assembly of different specialized modules sharing common connection interface that can be
4
manually combined in different ways to accomplish various tasks. For example a 14 Degree-Of
Freedom (D.O.F) reconfigurable robot manipulator has been developed as a part of the
Dockwelder EU project to perform in ship manufacturing industry [7].
Exploration of planets, moon and bodies in space is a clear goal for NASA and other space
agencies. Robotic approach to this task has the benefit of being able to perform the task at lower
cost and without endangering human life. For this purpose, Modular reconfigurable robots have
advantages such as saving weight and space, increasing robustness, self-repair ability and
adaptability [8]. Consider the robotic construction of a radio antenna on the moon’s surface. It
should be able to excavate soil, transport material, assemble parts, inspect constructed
assemblies, etc. [9]. It is impossible to design a robot which is strong enough for all of these
tasks. Along achieving this goal, Planetary Surface Vehicles (PSVs) have also been designed
which may be used in the future for manned or un-manned exploration of other planets. It has
been shown that reconfigurability improves the performance of these vehicles by 30% [10].
Furthermore, imagine that a space station is going to be assembled in space. It is a very huge,
complex and expensive structure. Is it possible to be assembled by men? And is it worth to do
so? The most recent research works present the innovative notion of self-assembling in space. In
near future, There will be free-flying intelligent fiber/rope, match-maker robots with self-
reconfigurability and self-adjustability tethering for autonomous ducking [11].
Researches have also been conducted on heterogeneous self-reconfigurable robots which are
believed that would offer the same benefits of homogeneous ones with the added ability to match
not only structure to task but also capability to task [12]. Hence, a number of ad hoc techniques
for modular self-reconfigurable robots are presented in the literature. They are innovative
methods that can be useful only for specific applications. For example, several design
approaches have been suggested for engineering robotic systems, such as, Metamorphic Robotic
System [13], Molecule [14], Miniaturised Self-Reconfigurable System [15], Crystalline [16] and
Semi-Cylindrical Reconfigurable Robot [17]. It shows that designing modular reconfigurable
robots depends on the specific applications and lack of a systematic design methodology is felt in
the literature.
Robots can be divided into three groups: (a) serial robots, (b) parallel robots, and (c) hybrid
robots [3]. The modularity notion has also been employed to parallel and hybrid robots that are
5
known mostly because of their fault tolerance. For example a novel four D.O.F hybrid robot,
named Bicept, can be integrated with a 1 D.O.F feed mechanism or a fixed base in order to form
a set of reconfigurations with parallel-serial architecture [18]. A few other researches have been
launched on reconfigurable parallel robots. One of them that is studied experimentally is
discussed in [19].
A smaller number of types of the modular manipulators than self-reconfigurable robots has also
been developed. Reconfigurable Modular Manipulator System (RMMS) built in the Carnegie
Mellon University [20] that consists of joint and link modules and the Modular Robotic System
(MRS) developed at the University of Toronto [3] that is now commercially available are
examples of the modular manipulators. Guelph is also a reconfigurable manipulator in the
neurophysiologic industry. Previously, exercise bicycle had been used in this field of medicine to
perform a couple of tests, but the bicycle has been substituted by Guelph, because of
compactness, reconfigurability, length variability of links, tolerating large torque and low overall
mass [21].
Modular reconfigurable robots consist of joint and link modules that are like cells [22] of our
body. One of the most significant issues in modular robotics is concurrent design and
development of these modules [23]. The design and manufacturing technology of the modules is
an important issue in designing and fabricating modular robots.
The major problems in modular robots can be listed as [24,25,26]:
a) Finding an algorithm to shift from any shape to a specified one.
b) Finding a reconfiguration path.
c) Finding a suitable and optimal configuration under a given set of conditions (in which all
functional requirements are met [4]).
Since, the space of configurations grows exponentially with the number of modules, how to
select among configurations is becoming a critical problem. In addition to that there are a lot of
research works about motion planning and optimal reconfiguration of modular reconfigurable
robots [27].
6
Problem c is the most important issue in the case of modular robots. There are two ways of
addressing this problem.
i) Finding an optimal configuration of a modular reconfigurable robot with having a finite
number of modules for a specific task (Configuration design with a given architecture)
ii) Finding an optimal architecture of a modular reconfigurable robot with respect of the
tasks that have to be done (Architecture design for a given number of tasks).
The following research is in the direction of solving the second case.
1.4.2 Concurrent Design Many solutions have been proposed by researchers to solve the two aforementioned problems at
the end of the previous section. For example in [4,28,29,30] solutions have been suggested for
the first category based on the Genetic Algorithm (GA) which is a powerful approach for the
optimization problems with mixed discrete and continuous variables.
Bi et. al. [31] introduce a basic architecture to design modular robotic systems. To define this
architecture, he uses Axiomatic Design Theory (ADT) [32]. According to ADT, the functional
requirements should be introduced, first, subsequently, a set of design parameters that can
evaluate the functional requirements is defined, and finally the modular robot architecture is
identified by a set of features [3]. Thus, in [30] Bi et. al. develop an alternative methodology in
order to optimize the configuration of a modular system. The advantage of this method is the
concurrent consideration of both type and dimension synthesis. They show that the design of
modular architectures necessitates a multi-objective optimization because of high coupling of
kinematic, dynamic, and modular design variables. Previously, Paredis in [33] discussed the
importance of concurrent design in fault tolerance of the modular reconfigurable robotics. In [34]
Bi et. al. define the volume of the configuration space expressed using Denevit-Hartenberg (D-
H) notation to be a measure of system adaptability, and they demonstrate that system
reconfigurability is not sufficient for system adaptability. In continuation of this research they
focus on using FEM-based method to obtain the kinematic and dynamic models for
reconfiguration variations where their design parameters are specified; since it combines
kinematic, and dynamic specifications [35].
7
Concurrent design is certainly a more suitable way of engineering reconfigurable robotic
systems. However, the question would be how? Several practical approaches of concurrent
design have been suggested in the literature. Some of them are based on fuzzy set theory;
nevertheless, they are just useful for the first steps of design and can not complete the design
procedure. Although, the most crucial stage of system synthesis is preliminary design where cost,
functionality and many other performance factors are estimated, decision making in this stage is
mostly done in an environment where the goals and constraints and consequences of actions are
not precisely known [36]. The theory of fuzzy sets, presented by Zadeh in 1965, exhibits a
potential of quantifying the imprecise and vague notions of design [37]. Therefore, it appears that
it is proficient to apply fuzzy set theory in order to tackle this decision making problem. A
preliminary design method in [36], based on fuzzy set theory, has been employed to helicopter
design that the results show a better performance in compare with the crisp design approaches.
Method of Imprecision (MoI) is another fuzzy design framework that evaluates imprecision and
uncertainty with the preference that in this method designer’s judgment is included [38,39]. Later
on, Otto & Antonsson, in [40], also argue that one of the significant issues influencing the
decision-making process is designer’s attitude. They introduce design strategies including
conservative, aggressive and moderate using theory of fuzzy sets.
Current design evaluation methods are limited into two respects:
a) The direct measurement of attribute performance levels that does not reflect the
subsequent worth to the designer.
b) The ad hoc methods for determining the relative importance or priority of attributes that
do not accurately quantify beneficial attribute trade offs.
In [41], a formal Methodology for the Evaluation of Design Alternatives (MEDA) is presented in
order to explicitly determine the designer’s subjective evaluation function over several attributes
that can be continuously used in the iterative design process.
Previously, it was assumed that it was possible to evaluate a design only by a scalar, such as
weight, performance or cost, but it has been investigated that a design problem is way more
complex than a single-objective optimization. Therefore, in a novel approach, Rao et. al. [42]
combined the fuzzy set theory and game theory and suggested the cooperative fuzzy game theory
8
to handle a multi-objective design problem. It yields a new optimization method that can solve
problems of single and Multiple Objective Optimization (MOO). Nonetheless, this method is
occasionally utilized in preliminary decision making of design process.
All mentioned methods of design are not based on pure optimization and attempt to take into
account human subjectivity, however, employing optimization methods is inevitable. The
existing optimization methods used in any of the aforementioned approaches can be categorized
as [43]:
a) The random search methods or stochastic optimization e.g. Genetic or Evolutionary
Algorithms (GA/EA), simulated annealing, tuba search, particle swarm and ant colony
method.
b) Sampling methods: They are somehow opposite of direct optimization schemes that only
include the information when improvement is obtained (e.g. Pattern Search). These
methods also include the objective function values as quantitative information (e.g.
Nadler-Mead).
c) Surrogate optimization method: In this approach, optimization is not performed on the
objective function but on an approximation of that. This is based on sequential stochastic
approximation in combination with sequential quadratic programming.
The first two methods are often applied to simulation based optimization problems.
In addition to these methods there are direct methods of optimization which are suitable for
simple optimization problems (e.g. gradient descent based & Newton’s and Quasi-Newton
methods).
1.4.3 Robotic Hardware-in-the-loop Simulation Generally, Hardware-In-the-Loop Simulation (HILS) is an approach of system modeling that
involves physical components in the simulation instead of or in addition to their mathematical
model. The concept behind HILS is to use physical hardware for system components that are
difficult to model and link them to a computer model that simulates the other aspects of the
system in order to reduce the simulation complexity. The flowcharts in Figure 1.1 illustrate the
9
hardware and software requirements of HILS. This technique has been successfully applied to
development and testing in a wide range of engineering fields, including: aerospace [44],
automotive [45], controls [46], manufacturing [47], and naval and defense [48].
In robotics, the interest in HILS has been growing among researchers in the last decade and it has
been applied from a number of different perspectives. Robot-In-the-Loop Simulation (RILS) is an
instance of HILS that has been introduced lately as a device in order to allow real robots and
model robots to work together for system-wide measurement and testing [49]. It permits the real
robots to be experimented in a virtual environment. This approach seems promising in large-
scale cooperative robots or when the real field is not available e.g. Mars Rovers.
Space manipulators, such as, Special Purpose Dexterous Manipulator (SPDM) that has been
developed by Canadian Space Agency, are not manufactured to work in gravity. In [50] an HILS
has been proposed that make the free motion and dynamic emulation of SPDM’s end point
possible with a setup linking a computer simulation of the SPDM to a physical hydraulic robot.
In addition to manipulators, HILS has been also applied to test and simulate vehicles and mobile
robots [51], employ analytical equations to simplify and accelerate the simulation process [52]
and recently, optimize systems like a humanoid robot with having enough stability in walking
process [43]. In the last case HILS is employed to feedback system quality for the optimization
procedure.
Also HILS is utilized in other fields of robotics such as controller-in-the-loop simulations where
a real control system interacts with a computer model of the robot [53], joint-in-the-loop
simulations that use a computer model to compute the dynamic loads seen at each joint and then
emulate those loads on the real actuators [54], and joint/controller-in-the-loop simulations where
both joint and control hardware units can be run in the simulation loop [55]. Each of these
approaches applies the HILS concept slightly differently, but all have produced promising
results.
10
Chapter 2: Theoretical Background
In this chapter a brief review is conducted over the theoretical basis of the tools employed in the
rest of the thesis. In the next chapter a systematic design methodology is developed based on
fuzzy set theory and bond graph prototyping technique. In this methodology first, fuzzy-logic
modeling is used to find the starting configuration for the design process, subsequently, fuzzy
operators are employed in order to formalize the design problem and finally, to capture the
objective notions of design an energy-based modeling scheme, named bond graphs, is utilized.
In the following sections, fuzzy set theory, fuzzy modeling and fuzzy operators are briefly
discussed and bond graphs theory as an energy-based, object-oriented system modeling scheme
is detailed.
2.1 Fuzzy Sets and Fuzzy Logic Since the exact boundaries for sets defined by vague notions can not be determined, fuzzy set
theory replaces the decision of whether or not an element belongs to a set, by a measure of scale.
Every element is evaluated by a measure of expressing its place in the class. A smaller measure
expresses that the given element is closer to the edge of the class and vise versa. Therefore, the
key idea in fuzzy set theory is that an element has a degree of membership to a fuzzy set, and it
can be a member of a fuzzy set to some extend. This measure is usually assigned in the interval
[0,1].
From another perspective, based on the same notion, a proposition need not be simply true or
false, but may be true to some degree. This degree can be again assumed to be a real number in
the interval [0,1]. This generalization of the binary (yes-no) logic to an interval is called fuzzy
logic. In other words, fuzzy set theory based on fuzzy logic is the generalization of the abstract
set theory based on two-state logic. Thus, fuzzy set theory has a wider scope of applicability than
abstract set theory in solving the problems involving subjectivity.
2.1.1 Fuzzy-Logic Modeling Fuzzy-logic modeling is an approach to forming the system model by using a descriptive
language based on fuzzy-logic with fuzzy propositions. This linguistic approach of system
11
modeling can be formulated by three distinct features: (a) the use of linguistic variables instead
of, or in addition to, numerical variables; (b) the characterization of simple relations between
variables by IF-THEN fuzzy rules; (c) the formulation of complex relations by fuzzy reasoning
algorithms.
The decision-making ability of the fuzzy models depends on the existence of a set of rules and a
fuzzy reasoning mechanism. In general, the clustered knowledge of a system can be interpreted
by fuzzy models consisting of IF-THEN rules with multi-antecedent and multi-consequent
variables.(with r antecedents, s consequents, and n rules):
IF U1 is B11 AND…AND Ur is B1r THEN V1 is D11 AND…AND Vs is D1s ALSO … (2.1) ALSO IF U1 is Bn1 AND…AND Ur is Bnr THEN V1 is Dn1 AND…AND Vs is Dns
where Uj (j=1,…,r) is jth input variable and Vk (k=1,…,s) is kth output variable, Bij (i=1,..,n,
j=1,…,r) and Dik (i=1,…,n, k=1,…,s) are fuzzy sets over the input space Xj and output space Yk.
However, conceptually, a Multi-Input Multi-Output (MIMO) fuzzy system can be always broken
down into s Multi-Input Single-Output (MISO) fuzzy systems. Although the number of rules is
increased, modeling and inference would be more straightforward for MISO fuzzy systems. That
is why in the literature and also in this thesis MISO fuzzy systems are considered as generic
presentation of fuzzy systems.
2.1.2 Fuzzy Connectives One major step in fuzzy-logic modeling is to decide about the reasoning mechanism. The basic
elements of reasoning are connective operators that must be transformed to algebraic functions in
order to apply them at the computational level. In abstract set theory linguistic connectives AND,
OR and NOT are intuitively transformed into functions such as Min and Max, however, in fuzzy
set theory classes of triangular norms (t-norm), triangular conorms (t-conorm) and
complementation operators are employed instead. A t-norm (T) or t-conorm (S) is a binary
operator on the unit interval [0,1] that is commutative, associative, non-decreasing, with a neutral
element. The difference between T and S operators is that the neutral element is 1 for t-norm and
0 for t-conorm. Although these functions are binary operators, their associativity property allows
them to be extended to n-ary operations. The triangular norm and conorm operators are used
12
widely in fuzzy set theory, however, they were independently introduced by Schweitzer and
Sklar [56] in the context of statistical metric space. Furthermore, a fuzzy complementation
operator C is defined on the unit interval [0,1] such that it is strictly decreasing, involutive with
the following boundary properties: (a) C(0)=1 and (b) C(1)=0. Unlike the classical set theory in
which the connective functions are uniquely defined, in fuzzy set theory the interpretation of
logical connectives is neither unique nor so obvious. Therefore, there exist a large number of
different classes of T, S and C. Some of the well-known operators that are going to be mentioned
in this thesis are defined as follows:
Generalized Max-Min Operators (Zadeh Operators)
Generalized Algebraic Product and Sum
Generalized Drastic Product and Sum
Fuzzy Complementation
A suitable parametric complementation operator is suggested by Yager in [57]:
n1n1n1 aaaaaaT ∧∧== ...),...,min(),...,(min . (2.2)
n1n1n1 aaaaaaS ∨∨== ...),...,max(),...,(max . (2.3)
∏=
=n
1iin1prod aaaT ),...,( . (2.4)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
±−+−= ∏∑∑∑∑∑∑==
≠=
≠≠==
≠==
n
1ii
n
1i
n
ij1j
n
ikjk1k
kji
n
1i
n
ij1j
ji
n
1iin1sum aaaaaaaaaS ...),...,( . (2.5)
⎩⎨⎧ =∨∨∧∧
=otherwise
...if...),...,(
01aaaa
aaT n1n1n1W . (2.6)
⎩⎨⎧ =∧∧∨∨
=otherwise
...if...),...,(
10aaaa
aaS n1n1n1W . (2.7)
13
For q=1 this function is the standard complement of crisp set theory which can be also adopted in
fuzzy set theory.
2.1.2.1 Parameterized operators
By using properties of t-norm and t-conorm operators, it can be easily proved that for any
arbitrary T and S and for all ],[ 10ai ∈ :
In order to cover various types of intersection-union operators, parameterized families of T and S
have been suggested among which Emami, Turksen and Goldenberg have introduced a class of
parameterized operators that is adopted for further investigation in this thesis[58].
In the extreme cases this class of parameterized operators approaches Generalized Min-Max
operators as +∞→p , Generalized Algebraic Product and Sum as 1p → , and Generalized
Drastic Product and Sum as 0p → . Thus, by varying parameter p all the range of t-norm and t-
conorm that is between Min-Max and Drastic operators is covered.
It is worth mentioning that the proposed t-norm possesses the tendency of being strictly
increasing with respect to the parameter p, i.e.,
q1
qa1aC )()( −= . ],[, 10a0q ∈> (2.8)
),...,(),...,(),...,(),...,(),...,(),...,(
min
min
n1n1n1W
n1n1n1W
aaSaaSaaSaaTaaTaaT
≤≤≤≤
. (2.9)
p1pn
p1n
p1n
p2n
p2n
p1
p1n21
p bb1bb1bb1bbbbS /)( ]]...]]])()[()[...[([),...,,( −−−− −+−+−+= ; (2.10)
where ],[ 10bi ∈ and ),( +∞∈ 0p . The corresponding t-norm operator is defined based on De
Morgan laws using standard complementation operator, as:
))(),...,(),((),...,,( )()(n21
pn21
p a1a1a1S1aaaT −−−−= . (2.11)
)],...,,(),...,,([][ )()(n21
pn21
p21 aaaTaaaTpp 21 >⇔> ; (2.12)
14
while
However, in the following conditions at the boundaries the function is monotonically non-
decreasing, i.e.,
while
and in the rest of the boundary points it is again strictly increasing. The same proposition may be
true for the corresponding t-conorm, however, wherever T is strictly increasing or monotonically
non-decreasing S, instead, is strictly decreasing or monotonically non-increasing, respectively.
In many cases of aggregation of fuzzy sets the type of aggregation required is neither the pure
AND (t-norm) with its complete lack of compensation nor the pure OR (t-conorm). Therefore,
the type of aggregation operator desired lies somewhere between these two extremes. These
types of operators are called mean aggregation and possess commutativity, monotonicity and
idempotency. An appropriate parametric operator, termed generalized mean operator is defined
by Yager [57].
where ),( +∞−∞∈α . It appears that this type of aggregation varies between Min operator while
−∞→α and Max operator as +∞→α .
It is worth mentioning that, this operator possesses tendency of being strictly increasing with
respect to α, i.e.,
],[:),...,,( 1a0n1iaaa in21 <<∈∀ . (2.13)
)],...,,(),...,,([][ )()(n21
pn21
p21 aaaTaaaTpp 21 =⇔> ; (2.14)
)],,[()],[(:),...,,( 1aiin1i0an1iaaa i0in21 =≠∈∀∨=∈∃ ; (2.15)
α
αα/
)( ),...,,(1n
1iin21 a
n1aaaG ⎟
⎠⎞
⎜⎝⎛= ∑
=
. (2.16)
)],...,,(),...,,([][ )()(n21n2121 aaaGaaaG 21 αααα >⇔> ; (2.17)
15
while
However, in the following conditions at the boundaries the function is monotonically non-
decreasing, i.e.,
while
and in the rest of the boundary points it is again strictly increasing.
2.1.3 Takagi-Sugeno-Kang (TSK) Inference Mechanism A known disadvantage of the fuzzy-logic models discussed above is that they do not deal with an
explicit form of the objective knowledge about the system if such information cannot be
expressed or incorporated into the fuzzy set framework. This kind of knowledge is often
available in the form of a database, extracted from a system model, experimental prototype or
mathematical equations. Sugeno and his co-researchers proposed an alternative type of fuzzy
reasoning, termed Takagi-Sugeno-Kang (TSK) type of reasoning. The TSK reasoning method is
associated with a rule-base of a special format that is characterized with functional type
consequents instead of the fuzzy consequents:
IF u1 is B11 AND…AND ur is B1r THEN rr1111101 ububby +++= ...
ALSO … (2.21) ALSO IF u1 is Bn1 AND…AND ur is Bnr THEN rnr11n0nn ububby +++= ... s
where uj (j=1,…,r) is the crisp value of jth input variable and yi (i=1,…,n) is the crisp value of the
output variable corresponding to the ith rule, Bij (i=1,..,n, j=1,…,r) are fuzzy sets over the input
space Xj. Each linear function in rule consequent can be regarded as a linear model of the system
with the parameters bij (i=1,…,n, j=1,…,r). The crisp output *y inferred by the fuzzy model
],[:),...,,( 1a0n1iaaa in21 <<∈∀ . (2.18)
)],...,,(),...,,([][ )()(n21n2121 aaaGaaaG 21 αααα =⇔> ; (2.19)
or,],,[:),...,,( 01a0aiin1iaaa0ii0n21 ==≠∈∀ ; (2.20)
16
under the TSK method is defined by the weighted average of the crisp outputs yi of individual
linear subsystem (rule):
and iτ is the degree of fire of the ith rule:
Geometrically, the rules in TSK mechanism correspond to approximations of the mapping
YXX r1 →×× ... by piecewise linear functions. In a more general case, these linear functions
can be replaced by nonlinear costume functions.
As it is observable in (2.22), TSK method of reasoning is compact and computationally efficient.
Therefore, it is widely used in fuzzy-logic modeling specially when tuning techniques are
employed to customize the membership functions so that the fuzzy system best models the data.
Hence, TSK method is used for modeling a robotic system in the next chapter.
2.1.4 Fuzzy Rule-Base Generation In the heuristic approach to fuzzy modeling, it is assumed that expert information including the
definition of the rule antecedents and consequents is available. Seeking more objectivity in
constructing fuzzy models, some more formal techniques have been developed in order to use
available input-output data to augment human knowledge. The most critical step of fuzzy system
modeling is fuzzy rule-base generation that is performed in the following sequence: (a)
clustering output data and assigning output membership functions, (b) finding the non-significant
input variables and assigning the membership functions to the rest of them, and (c) tuning the
input and output membership functions. In order to carry out the process of clustering the output
space, one of the most applicable fuzzy clustering algorithms, i.e., Fuzzy C-Means (FCM)
clustering is used.
∑∑
∑∑ =
=
=
=
+++==n
1irir11i0in
1jj
in
1iin
1jj
i ububbyy )...(*
τ
τ
τ
τ; (2.22)
))(),...,((i rir11i uBuBT=τ . (2.23)
17
2.1.4.1 Fuzzy C-Means Clustering
Clustering methods are occasionally based on the optimization of an objective function in order
to find the optimum membership matrix, U=[uik], that contains the membership value of ith data
point, Xxi ∈ , to the kth partition. In FCM, this function, Jm, is defined as the weighted sum of
the squared errors of data points and the minimization problem is formulated as:
where c21 vvvV ,...,,= is the set of unknown cluster centers and AXXX T
A= is any inner
product norm in which A is an rr× positive definite matrix that specifies the shape of the
clusters. The N, c and r are the number of data points, clusters and input dimensions,
respectively. The FCM clustering is carried out through an iterative optimization of (2.24).
A prerequisite for FCM is to assign the number of clusters (c) and the weighting exponent (m). In
many practical cases there is no a priori information about the optimum values of them. The
main criteria for the specification of optimal clustering are based on two requirements: (a)
separation between the resulting clusters; (b) compactness of the clusters. There exist a number
of investigations in the literature in order to define cluster validity criteria. An algorithm is
suggested based on the fuzzy within-cluster, i.e., Equation (2.25), and between-cluster, i.e.,
Equation (2.26), scatter matrices in [59] that is employed to find the optimum values for m and
c in this thesis.
where the fuzzy total mean vector, v , is a weighted mean of data considering their membership
to each of the clusters in fuzzy partition defined as:
⎥⎦⎤
⎢⎣⎡ −= ∑∑
= =
N
1i
c
1k
2
Akim
ikmVUvxuXVUJ )();,(min
),(. (2.24)
∑∑= =
−−=c
1i
N
1k
Tikik
mikW vxvxuS ))(()( ; (2.25)
∑ ∑= =
−−⎟⎠⎞
⎜⎝⎛=
c
1i
Tii
N
1k
mikB vvvvuS ))(()( ; (2.26)
18
SB represents the separation between the fuzzy clusters and SW is an expressive index for the
compactness of fuzzy clusters. Hence, for obtaining the best clusters, tr(SW) should be minimized
to increase the compactness of clusters and tr(SB) should be maximized to increase the separation
between clusters. Therefore, scs is defined to be minimized as:
where tr() is the trace of a matrix. This criterion identifies the optimum number of clusters, c.
Another parameter whose value should be decided in fuzzy clustering is the weighting exponent
(m). This parameter varies in the range of (1, +∞) and the larger m is the fuzzier are the
membership assignments to each data point. In order to have a reliable index for cluster validity,
scs, m should be far enough from both extremes. The trace of fuzzy total scatter matrix (ST) that is
a monotonically decreasing function of m is employed to find a reliable value for m.
Therefore, for a dataset to be clustered, an appropriate value for m is what holds sT somewhere in
the middle of its domain. Since sT and scs are both functions of m and c, the process of choosing
the parameters should be performed by a few iterations.
2.1.4.2 Selecting Significant Input Variable
After assigning the appropriate fuzzy clusters and membership functions for the output sample
data, the next step in fuzzy modeling is to project the output membership functions on the input
spaces to form the input membership functions corresponding to each rule. However, for the
systems with a large number of input variables in order to have a more efficient fuzzy model it is
better to first reduce the dimension of the input data by introducing a quantitative index jπ as an
overall measure of the non-significance of input variable xj in the fuzzy model as follows [59]:
∑∑∑∑ = =
= =
=c
1i
N
1kk
mikc
1i
N
1k
mik
xuu
1v )()(
; (2.27)
)()( BWcs StrStrs −= . (2.28)
)()( BWTT SStrStrs +== . (2.29)
19
where ijΓ is the range in which membership function )( jij xB is one , jΓ is the entire range of
the variable xj and n and r are the number of rules and input variables, respectively. The smaller
the value of jπ the more effective jth variable is and vice versa.
2.1.4.3 Assigning Input Membership Functions
Once the significant input variables are selected, a clustering method, called line fuzzy clustering,
is employed to map the output membership functions on input spaces. This method works based
on the distance of each point xjk (j=1,…,r0, k=1,…,N), while r0 is the number of significant input
variables, located on the axis xj, to the line ij2
ij1 vv that is the range of jth input variable
corresponding to the output membership function equal or close to one. Finally, for kth data point
the membership value of jth input variable in ith rule is calculated as [59]:
Finally, based on an algorithm proposed by Emami et. al. in [59] these membership functions
and the consequent parameters are tuned.
2.2 Bond Graphs Modeling Several attempts have been launched to unify the modeling approaches corresponding to
different physical systems. In the early 60’s Henry Paynter noticed that the notion of energy and
energy exchange can be considered as a common notion to all systems with different physical
disciplines [60]. He extended the concept of port introduced in electrical circuits to an arbitrary
power port and applied it domain-independently. Hence, he presented an innovative notion based
on energy exchange between two ports along a line that he called power bond. The concept of
bond graph modeling was completed when he finally introduced the notion of junction in 1959
∏=
=n
1i j
ijj Γ
Γπ ; j=1,2,…,r (2.30)
⎪⎪
⎩
⎪⎪
⎨
⎧
≤≤
><
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
=
−
−
=
−
−
=∑∑
ij2
jkij1
ij2
jkij1
jk
1
1m1
n
1l lj2
jk
ij2
jk
1
1m1
n
1l lj1
jk
ij1
jk
jik
vxv1
vxvxvx
vx
vx
vxMax
uor,
(2.31)
20
[61]. Therefore, In bond graphs modeling approach all components of a system are recognized
by the energy they supply or absorb, store or dissipate, and reversibly or irreversibly transform
[62].
2.2.1 Basic Elements Bond graphs are labeled and directed graphs, in which the nodes represent subsystems, system
components or elements and the edges, called (power) bonds, denote an ideal energy flow
between power ports. The amount of power transmitted at each instant can be determined by
scalar production of the two power variables (or power conjugates), termed flow )( fr
and effort
)(er , that are vectors, in general.
As the power can travel back and forth between two power ports, a half arrow is added to each
bond indicating a reference direction of the energy flow [62]. Table 2.1 lists the analogies used
for effort and flow in different physical domains. For example for a translational mechanical
phenomenon effort and flow are translated to force and linear velocity and in electronics voltage
and current, respectively. Other analogies may be used based on different applications. Note that
in different physical domains the power conjugates may be vectors or scalars. Thus, in order to
distinguish between them single and double line half arrows are employed for scalar and vector
bond graphs representation, respectively (see Figure 2.1). Although the power conjugates can be
different physical entities in various domains, the product of them is always power that is a
common scalar quantity between all physical disciplines. In order to complete the model,
physical components should be mapped to conceptual elements that can simulate the dominant
dynamic behavior of them [61]. These elements are basically divided into three groups: (a)
single-port, (b) double-port and (c) multi-port elements.
2.2.1.1 Single-Port Elements
Ideal single-port components consist of three generic types: source (sink) elements (S), storage
elements (I, C), and dissipative elements (R). The S is an ideal source (sink) of either flow (Sf) or
effort (Se). Hence, this element contains a power port that the bond is either coming out of the
element, for a source, or going toward it for a sink. Moreover, sources (sinks) may also have a
fePrr.= . (2.32)
21
signal port for e.g. feedback control and they can vary during the simulation. In this case the
source (sink) is identified by MSf or MSe, standing for modulated source (sink).
Storage elements accumulate energy and release it back into the system. Depending on whether
pr (generalized momentum) or qr (generalized displacement) can be conserved the storage
elements can be divided into two types. Where
In the C type storage element, like capacitors or springs, the flow is the rate of change of the
generalized displacement that is the conserved quantity and in the I type storage element, like
inductors and masses, the role of effort and flow is just interchanged. In other words, both types
of storage elements transform the energy reversibly.
The irreversible transformation of energy to heat pumped to the environment, such as electrical
resistors or mechanical friction, is often modeled as loss or waste of energy. In bond graphs it is
represented by an R element. Table 2.1 demonstrates the element analogies in various energy
domains[61,63].
2.2.1.2 Double-Port Elements (Transformers and Gyrators)
The most basic components of bond graphs are double-port elements. In these elements, power is
conserved that means the instantaneous power at one port equals the instantaneous power at the
other port. It can be demonstrated that because of the energy conservation, independent of the
domains, only two types of ideal double-port elements can exist. On the one hand, transformers,
denoted by TF, transduce effort at one port to the effort at the other port by a parameter called
transformer ratio (N) that is a matrix in general,
If N is a function of time, the element should have a signal port in addition to the power ports.
This kind of transformer is named modulated transformer and represented by MTF.
∫= dtep rr ; (2.33)
∫= dtfqrr . (2.34)
1221 fNfeNerrrr
=⎯⎯⎯⎯ →⎯= onconservatienergy . (2.35)
22
Transformers are utilized in the same energy domain, e.g. gearboxes and pulleys, or between
different domains such as electromotors and winches.
On the other hand, gyrators (GY) relate the flow at one port to the effort at the other port by a
factor labeled (H) that in general, can be a matrix,
The gyrator is called modulated gyrator (MGY) when H is variable. Gyrators are mostly
transducers representing domain transformation such as DC-motors, pumps and turbines [64].
2.2.1.3 Multi-Port Elements (Junctions)
In order to distribute power between the subsystems distributing elements are required. These
components are denoted by junctions. For a physical system, interchanging ports must have no
influence on the constitutive equations of junctions. This port symmetry and power continuity
properties result in two kinds of junctions called 0- (zero) and 1- (one) junctions. In 0-junction
the amount of effort remains constant in all ports, thus, the power continuity equation turns into:
where n is the number of ports. An example of 0-junctioin is parallel connection in electrical
circuits.
On the other hand, 1-junction maintains flows the same at all ports. Series connections in
electronics can be a clear instance of this kind of junctions. Considering power continuity in an
ideal 1-junction, the constraint equation can be derived as follows [64]:
2.2.2 Causality In the bond graphs representation at each power port of an element both power conjugates can be
considered as incoming or outgoing signals toward or from the component. However, at the
1221 fHefHerrrr
=⎯⎯⎯⎯ →⎯= onconservatienergy . (2.36)
;...
,...
0fff
eee
n21
n21
=+++
===rrr
rrr
(2.37)
....
,...0eee
fff
n21
n21
=+++===rrr
rrr
(2.38)
23
computational level one of the power variables must be chosen as the input and the other one as
the output of the port that is calculated based on the constitutive equations. The decision of
assigning causality is denoted by the causal stroke in the bond graphs notation. Causal stroke is
a perpendicular line at one end of the power bond that indicates the direction of the effort signal
whether it comes or goes toward or from the element. The result is now a causal bond graph that
can be interpreted as bi-directional signal flow. Figure 2.2 shows the different admissible
causality assignments and their corresponding block diagrams for all of the bond graphs
elements. However, this decision cannot be completely deliberately. Depending on the kind of
equation of the element, the element ports can impose constraints on the connected bonds. There
are four different constraints that should be treated in the causality analysis of the bond graphs.
2.2.2.1 Fixed Causality
Fixed causality occurs when the equations only allow one of the power variables to be the
outgoing or incoming signal. For instance, for flow or effort sources (sinks) the output (input) is
the known flow or effort signal.
2.2.2.2 Constrained Causality
At TF (MTF), GY (MGY), 0- and 1-junctions, there exist equations between power variables of
different ports. Therefore, the causality of a particular port imposes the causality of the other
ports. In a transformer element one of the ports has effort-out causality and the other port has
flow-out causality. Nevertheless, at a gyrator both ports has either effort-out or flow-out
causality. At a 0-junction where all the efforts are the same exactly one port must bring in the
effort and all other ports bring in the flow. At a 1-junction the role of flow and effort is changed
and exactly one port should bring in the flow and the rest bring in the effort.
2.2.2.3 Preferred Causality
At the storage elements, the causality indicates whether the integration or differentiation with
respect to time should be operated in order to determine the output signal. Since both are
calculated numerically during the simulation, because of less computational error in the
integration case, it is preferred against differentiation. Also when the input contains a step
function, the output signal becomes infinity using differentiation. Therefore, in assigning
causality it is always preferred to have integration instead of differentiation wherever possible.
24
2.2.2.4 Indifferent Causality
For all other bond graph elements that no constraint can be considered in assigning the causality,
like linear elements, where it does not matter which power conjugate is the output, the direction
of the input-output signals is assigned deliberately. However, other elements might impose a
preferred direction to assigning the causality in indifferent causality case.
25
Chapter 3: Linguistic Mechatronics
The engineering products have reached a level of sophistication that the traditional approaches of
subsystem partitioning for their design may not provide effective solutions. The decoupled
design methodology for the development of multidisciplinary systems often undermines
interconnections between various subsystems throughout the design process, resulting in more
iterations and less desirable outcomes. The alternative is Mechatronics as a synergistic approach
to the design, development and manufacturing of complex engineering systems, products and
processes. The emphasis is on the physical integration and information communication amongst
various subsystems in a collaborative manner.
The premise of mechatronics is to provide a common language to fill in the communication gap
between different engineering disciplines and to devise a means for helping them to collaborate
towards a common goal [65]. The necessity of communication and collaboration in mechatronics
implies that it must be closely linked to concurrent engineering in managing and conducting the
design process. The challenge, however, is that consequently a large number of multidisciplinary
objective and constraint functions must be taken into account simultaneously along with a great
number of design variables. If the formal optimization approach is followed, the multi-objective
constrained optimization problem with large number of variables is quite difficult to solve. Thus,
a practical multiattribute concurrent design method is required for mechatronics. Furthermore,
design of multidisciplinary systems involves many subjective notions, in addition to physical
features, that can hardly be captured by pure mathematical formulations. Both customers and
designers need to communicate beyond the equations to convey design requirements and
specifications. This necessity becomes even more critical in a multidisciplinary collaboration as
mechatronics mandates. Hence, there is a need for a communication means in mechatronics that
can convey qualitative and subjective notions that are used frequently in human interactions, in
addition to holistic criteria that finalize the design process based on objective performances in
the real physical world.
This chapter outlines a systematic conceptual design methodology, called Linguistic
Mechatronics (LM), for mechatronic systems, that emphasizes on the designer’s satisfaction,
instead of pure performance optimization, and brings the linguistic aspects of communication
26
into the design process. Furthermore, LM formalizes subjective notions of design by redefining
the ultimate goal of design based on the qualitative notions of wish and must satisfactions using
fuzzy logic. This methodology reduces the complicated multi-objective constrained optimization
problem to a single-objective unconstrained optimization utilizing fuzzy-logic operators to
aggregate satisfactions corresponding to the design attributes. Linguistic Mechatronics also
formalizes the effect of designer’s subjective attitude in the design process by employing
parameterized aggregation operators. The corresponding parameters, called attitude parameters,
are adjusted based on the reality of the system performance during the design process. Hence,
not only does the suggested approach ease the way of communication between designers of
different backgrounds and clients, but it also makes the multiattribute design solution more
practical and permits numerous design variables with different natures to be considered,
concurrently.
However, subjective aspects of design solely represent designer/client’s preferences. A holistic
criterion must finalize the design process based on objective performances of the design in the
real world. A proficient modeling scheme is capable of offering appropriate criteria. Since
mechatronic systems are multidisciplinary and sophisticated, in order to model them a unique
prototyping strategy is required. An alternative method is the hybrid utilization of bond graphs
and block diagrams.
A notable attempt to take into account imprecision in design is presented in [38], namely Method
of Imprecision (MoI). The approach defines a set of designer’s preferences for design variables
and performance parameters to model the imprecision in design. It determines and maximizes the
global performance under one of the two conservative or aggressive design tradeoff strategies,
and uses fuzzy-logic operators for tradeoff in the design space. In addition to subjective aspects
of design, MoI takes into account all design variables and performances concurrently and
modifies a multi-objective constrained optimization to achieve a single-objective problem.
Nonetheless, MoI does not distinguish between the constraints and goals in the aggregation
process and simply offers two extreme attitudes that are not justified with any objective
performance criterion. In this chapter, an alternative framework is presented for including
subjective notions and simplifying the design optimization, while addressing the above-
mentioned deficiencies through (a) dividing the design attributes into two inherently-different
classes, namely wish and must attributes; and (b) aggregating satisfactions using parametric
27
fuzzy-logic operators so that the designer’s attitude can be adjusted based on an objective
performance criterion.
3.1 Design Problem in LM Framework A design problem consists of two sets: design variables jX=X (j=1,2,…,n) and design
attributes iA=A (i=1,2,…,N). Design variables are to be configured to satisfy the design
requirements assigned for design attributes, subject to the design availability jD=D
(j=1,2,…,n). Each design attribute stands for a design function providing a functional mapping
ii AF →X: that relates a state of design configuration X to the attribute Ai, i.e., )(Xii FA =
(i=1,2,…,N). These functional mappings can be of any form, such as closed-form equations,
heuristic rules, or set of experimental or simulated data.
Given a set of design variables and a set of design attributes along with a database that conveys
the relationship between them, the process of Linguistic Mechatronics is performed in two
phases: (a) primary phase in which proper intervals for the design variables are identified subject
to design availability, and (b) secondary phase in which design variables are specified in their
intervals in order to maximize an overall design satisfaction based on the design requirements
and designer’s preferences. Thus, the secondary phase involves a single-objective optimization,
yet it is critically dependant on the initial values of a large number of design variables. The
primary phase makes the optimization more efficient by providing proper intervals for the design
variables from where the initial values are selected. The overall satisfaction is an aggregation of
satisfactions for all design attributes. The satisfaction level depends on the designer’s attitude
that is modeled by fuzzy aggregation parameters. However, different designers may not have a
consensus of opinion on satisfaction. Therefore, the system performance must be checked over a
holistic supercriterion to capture the objective aspects of design considerations in terms of
physical performance. Designer’s attitude is adjusted through iterations over both primary and
secondary phases to achieve the enhanced system performance. Therefore, this methodology
incorporates features of both human subjectivity (i.e., designer’s intent) and physical objectivity
(i.e., performance characteristic) in multidisciplinary system engineering.
28
Definition 1 (Satisfaction): A mapping μ such that ],[: 10→Yμ for each member of Y is called
satisfaction, where Y is a set of available design variables or design attributes based on design
requirements. The grade one corresponds to the ideal case or the most satisfactory situation. On
the other hand, the grade zero means the worst case or the least satisfactory design variable or
attribute.
Satisfaction on a design attribute )(XiAia μ≡ indicates the achievement level of the
corresponding design requirement based on the designer’s preferences. The satisfaction for a
design variable )(XjXjx μ≡ indicates the availability of the design variable. In the conceptual
phase, design requirements are usually subjective concepts that implies the costumer’s needs.
These requirements are naturally divided into demands and desires. A designer would use
engineering specifications to relate design requirements to a proper set of design attributes.
Therefore, in LM the design attributes are divided into two subsets, labeled must and wish design
attributes.
Definition 2 (Must design attributes): A design attribute is called must if it refers to costumer’s
demand, i.e., the achievement of its associated design requirement is mandatory with no room for
compromise. These attributes form a set coined M.
Definition 3 (Wish design attributes): A design attribute is called wish if it refers to costumer’s
desire, i.e., its associated design requirement permits room for compromise and it should be
achieved as much as possible. These attributes form a set coined W.
Therefore,
The satisfaction specified for wish attributes, iW (i=1,2,…,NW) is )()( XXiWiw μ≡ and the
satisfaction specified for must attributes, iM (i=1,2,…,NM) is )()( XXiMim μ≡ . For each design
attribute Ai (corresponding to either Mi or Wi), there is a predefined mapping to the satisfaction ai
(mi or wi), i.e., ],[:),( A∈∈∀ iii AN1iaA . Consequently, fuzzy set theory can be applied for
defining satisfactions through fuzzy membership functions and also for aggregating the
satisfactions using fuzzy-logic operators.
AWMWM =∪=∩ ,φ . (3.1)
29
Remark: ][][ )()()()( 2i
1i
2i
1i aaAA ≥⇔f for monotonically non-decreasing satisfaction. More
specifically, if 1ai ≠•)( then ][][ )()()()( 2i
1i
2i
1i aaAA >⇔f and if 1ai =•)( then
][ )()( 2i
1i AA f ][ )()( 2
i1
i aa =⇔ , where f denotes loosely superior and f represents strictly
superior. It means that the better the performance characteristic the higher the satisfaction will
be.
Definition 4 (Overall satisfaction): For a specific set of design variables X, overall satisfaction
is the aggregation of all wish and must satisfactions, as a global measure of design achievement.
3.2 Calculation of Overall Satisfaction Must and wish design attributes have inherently-different characteristics. Hence, appropriate
aggregation strategies must be applied for aggregating the satisfactions of each subset.
3.2.1 Aggregation of Must Design Attributes
Axiom 1: Given must design attributes, ],,[:),( M∈∈∀ iMii MN1imM , and considering
component availability, ],,[:),( D∈=∀ jjj Dn1jxD , the overall must satisfaction is the
aggregation of all must satisfactions using a class of t-norm operators.
Must attributes correspond to those design requirements that are to be satisfied with no room of
negotiation, and linguistically it means that all design requirements associated with must
attributes have to be fulfilled simultaneously. Therefore, for aggregating the satisfactions of must
attributes an AND logical connective is suitable. Considering satisfactions as fuzzy membership
degrees, the AND connective can be interpreted through the family of t-norm operators [58].
Thus, the overall must satisfaction is quantified using the p-parameterized class of t-norm
operators, i.e.,
where NM is the number of must attributes and n is the number of design variables. The
parametric t-norm operator T(p) is defined based on Equations (2.10) and (2.11) in the Chapter 2.
)();,...,,,,...,,()( )()( 0pxxxmmmT n21N21pp
M>=XMμ ; (3.2)
30
Parameter p can be adjusted to control the fashion of aggregation. Changing the value of p makes
it possible to obtain different tradeoff strategies. The larger the p, the more pessimistic
(conservative) designer’s attitude to a design will be, and vice versa.
3.2.2 Aggregation of Wish Design Attributes Definition 5 (Cooperative wish attributes): A subset of wish design attributes is called
cooperative if the satisfactions corresponding to these attributes all vary in the same direction
when changing design variables.
Therefore, wish attributes can be divided into two cooperative subsets:
(a) Positive-differential wish attributes ( +W ): In this subset the total differential of satisfaction
for each wish attribute (with respect to design variables) is non-negative.
where +wN is the number of positive-differential wish attributes. This subset includes all attributes
that tend to reach higher satisfaction when all design variables have infinitesimal increments.
(b) Negative-differential wish attributes ( −W ): In this subset the total differential of satisfaction
for each wish attribute (with respect to design variables) is negative.
where −wN is the number of negative-differential wish attributes. This subset includes all
attributes that tend to reach lower satisfaction when all design variables have infinitesimal
increment.
Since in each subset all wish attributes are cooperative, their corresponding design requirements
can all be fulfilled simultaneously in a linguistic sense. Hence, according to Axiom 1, similar to
must satisfactions, a q-parameterized class of t-norm operators is suitable for aggregating
satisfactions in either subsets of wish attributes.
],[ )(:),( +∈∀≥=+
WXXW N1i0dww ii ; (3.3)
],[ )(:),( -WXXW N1i0dww ii ∈∀<=− ; (3.4)
WWWWW =∪=∩ −+−+ ,φ . (3.5)
31
where ±W demonstrates either subsets of positive- or negative-differential wish attributes and
±WN denotes the number of their members.
Axiom 2: Given the satisfactions corresponding to positive- and negative-differential wish
attributes, )()( Xq
W +μ and )()( Xq
W −μ , the overall wish satisfaction can be calculated using an α-
parameterized generalized mean operator.
The two subsets of wish attributes cannot be satisfied simultaneously as their design
requirements compete with each other. Therefore, some compromise is necessary for aggregating
their satisfactions, and the class of generalized mean operators, i.e. Equation (2.16), reflects the
averaging and compensatory nature of their aggregation.
The class of generalized mean operators expressed in (3.7) tends to be monotonically increasing
with respect to α and varies between Min operator for −∞→α and Max operator for +∞→α .
This offers a variety of aggregation strategies from conservative to aggressive, respectively. The
overall wish satisfaction is governed by two parameters q and α, representing subjective tradeoff
strategies. They can be adjusted appropriately to control the fashion of aggregation. The larger
the α or the smaller the q, the more optimistic (aggressive) one’s attitude to a design will be, and
vice versa.
3.2.3 Aggregation of Overall Wish and Must Satisfactions Axiom 3: The overall satisfaction is quantified by aggregating the overall must and wish
satisfactions )()( XMpμ and )(),( XW
αμ q with the p-parameterized class of t-norm operators, i.e.,
)();,...,,()( )()( 0qwwwTW
N21qq >=
±± X
Wμ ; (3.6)
( ) ( )( ) )( .)()()( )()(),( +∞<<−∞⎥⎦⎤
⎢⎣⎡ += −+ αμμμ
αααα
1qqq
21 XXX
WWW . (3.7)
)()).(),(()( ),()()(),,( 0pT qppqp >= XXX WMαα μμμ . (3.8)
32
The aggregation of all wish satisfactions can be considered as one must attribute, i.e., it has to be
fulfilled to some extent with other must attributes with no compromise. Otherwise, if the overall
wish satisfaction becomes zero it means that none of the wish attributes is satisfied, which is
unacceptable in design. Therefore, the same aggregation parameter, p, that was used for must
attributes should be used for aggregating the overall wish and must satisfactions. In (3.8), three
parameters p, q and α, called attitude parameters, govern the overall satisfaction.
3.3 Primary Phase of LM Once the overall satisfaction is calculated, in order to obtain the best design, this index should be
maximized. The optimization schemes are critically dependent on the initial values and their
search spaces. Therefore, to enhance the optimization performance, suitable ranges of design
variables are first found in the primary phase of LM. In linguistic term, primary phase of LM
methodology provides a sketch of the final product and illustrates the decision-making
environment by defining some ranges of possible solutions. For this purpose, the mechatronic
system is represented by a fuzzy-logic model. This model consists of a set of fuzzy IF-THEN
rules that relates the ranges of design variables as fuzzy sets to the overall satisfaction; i.e.,
IF X1 is B11 AND…AND Xn is B1n THEN μ is D1 ALSO … (3.9) ALSO IF X1 is Br1 AND…AND Xn is Brn THEN μ is Dr
where μ is the overall satisfaction and Blj and Dl (j=1,2,…,n and l=1,2,…,r) are fuzzy sets on Xj
and μ, respectively, that can be associated with linguistic labels.
The fuzzy rule-base is generated from the available data obtained from simulations, experimental
prototypes, previous designs or etc., using fuzzy clustering as detailed in Chapter 2. The
consequent fuzzy sets, Dl, can be further defuzzified by means of Center of Area (CoA)
defuzzification method [57] to crisply express the level of overall satisfaction corresponding to
each rule.
]),[( ;)(
)(
],[
],[* r1ldd
dd
10 l
10 l
l ∈∀=∫∫
μμ
μμμμ (3.10)
33
where dl(μ) is the membership function of fuzzy set Dl. The rule with the maximum *lμ is
selected, and the set of its antecedents represents the appropriate intervals for the design
variables. These suitable intervals are denoted as ),...,,( n21jC j ==C and their fuzzy
membership functions are labeled as ),...,,( )( n21jXc jj = . Finally, these fuzzy sets are
defuzzified using CoA defuzzification method to introduce the set of initial values
),...,,( n21jX 0j ==0X for design variables in the secondary phase optimization process.
3.4 Secondary Phase of LM In the secondary phase, LM employs regular optimization methods to perform a single-objective
unconstrained maximization of the overall satisfaction. The point-by-point search is done within
the suitable intervals of design variables obtained from the primary phase. Therefore, the locally
unique solution Xs is obtained through
It can be shown that the answer is locally pareto-optimal (see Appendix A). As indicated in
Equation (3.12), various attitude parameters, p, q and α, result in different optimum design
values for maximizing overall satisfaction. Consequently, a set of satisfactory design alternatives
(Cs) is generated based on subjective considerations, including designer’s attitude and
preferences for design attributes.
3.5 Performance Supercriterion From the set of optimally satisfactory solutions (Cs) the best design needs to be selected based on
a proper criterion. In the previous design stages, decision making was critically biased by the
designer’s preferences (satisfaction membership functions) and attitude (aggregation
parameters). Therefore, the outcomes must be checked against a supercriterion that is defined
based on physical system performance. Indeed, such a supercriterion is used to adjust the
designer’s attitude based on the reality of system performance. A suitable supercriterion for
]),[( .)(
)(n1j
dXXc
dXXcXX
j
j
C jjj
C jjjj
0j ∈∀=∫∫
(3.11)
))(),((max)( ),()()(),,( XXX WMCXsαα μμμ qppqp T
∈= (3.12)
34
multidisciplinary systems should take into account interconnections between all subsystems and
consider the system holistically, as the synergistic approach of mechatronics necessitates. A
proficient system modeling strategy can offer the proper supercriterion.
A mechatronic system is a modern engineering system with controllable motion behaviour. It
consists of a complex combination of several subsystems. Generally, the subsystems can be
divided into three categories, namely: Generalized Executive Movement, Sensing and Testing
and Information Processing and Control. Flow of energy, material or information links these
three generic subsystems.
The generalized executive subsystem is the integration between driver elements and executive
mechanisms. The inputs of the drivers are various forms of energy and the outputs are various
motions that match with mechanisms. Subsequently, the mechanisms receive and create motions.
In other words, generalized executive mechanisms are performing tasks that the mechatronic
system is expected to do. Energy flows through out this subsystem.
Sensing and testing subsystem examines the information needed by generalized executive
subsystem, such as displacement, velocity, force, temperature of executive components and
environmental parameters. Ultimately, it transmits these state parameters to information
processing and control subsystem. Sensing and testing subsystem includes sensors and
microcomputers. Input to this subsystem is usually in the form of energy, and output is
information flow.
Information processing and control subsystem is composed of microprocessors and various
controllers. Input to this subsystem is information transmitted by sensing and testing module and
output is processed information needed to control generalized executive mechanisms [66].
Although in the generalized executive movement subsystem there exist several types of elements
and subsystems from different physical disciplines, the universal concept of energy and energy
exchange is common to all of them. Therefore, an energy-based model can deem all subsystems
together with their interconnections, and introduce generic notions that are proper for
mechatronics. A successful attempt in this direction is the conception of bond graphs in the early
60’s [60]. Bond graphs are domain-independent graphical descriptions of dynamic behaviour of
physical systems. In this modeling strategy, which is detailed in Chapter 2, all components are
35
recognized by the energy they supply or absorb, store or dissipate, and reversibly or irreversibly
transform. In [61,62] bond graphs are utilized to model mechatronic systems, and this provides
an efficient means to define holistic criteria for mechatronics.
In addition, information processing and control subsystem also plays a significant role in design.
Controller parameters can be traced to MTF and MGY parameters. Since bond graphs are
converted into block diagrams to be processed, they are completely compatible with block
diagrams which are the most popular technique to simulate a control system. Thus, information
processing and control subsystem of mechatronic systems can be represented by means of block
diagrams. Nonetheless, to attach a control system to bond graphs, transient subsystems are
required. Sensors are the subsystems that are capable of transmitting energies into information
signals. Testing and sensing subsystem, as the transient subsystem between the two other
subsystems, is out of the design scope of this research. Therefore, it can be simply represented by
signals going out of bond graphs and entering block diagrams during modeling process. A
combination of bond graphs and block diagrams as two powerful simulating means leads to an
alternative modeling tool for mechatronic systems in the concurrent design process that is able to
offer suitable performance supercriteria.
As an example, the supplied energy from the source elements can be used to define an
appropriate supercriterion, considering a predefined task for a mechatronic system. According to
the principle of conservation of energy, the total amount of supplied energy (S) is equal to the
sum of the requested work from the system, denoted by effective work (E), and the energy stored
or dissipated in the system, labeled as cost function (y).
In the proposed modeling approach (see Chapter 2 ), sources of energy are easily distinguishable
by Se and Sf with the bonds coming out of them. Therefore according to Equation (2.32) the
supplied power at ith energy source, iS& , can be calculated from the bond graphs model at each
instant.
)()( XX yES += . (3.13)
);().;();( XXX tftetS iii
rr& = . (3.14)
36
Hence, the supplied energy would be,
In Se and Sf, the effort and the flow remain unchanged, respectively. Therefore, by employing the
Equations (2.33) and (2.34), Equation (3.15) can be further simplified for each case and the total
supplied energy is calculated.
where Nf and Ne are the number of flow and effort sources, correspondingly. Given that the
desired effective work is constant, by minimizing supplied energy or cost function, depending on
the application, with respect to the attitude parameters the best design can be achieved in the set
of optimally satisfied solutions (Cs).
The energy criterion is appropriate for the systems where energy consumption is the main matter
of concern. Alternatively, for systems where response time is a crucial factor the rate of energy
transmission through the system, or agility, can be used for defining the performance
supercriterion. Thus, the supercriterion would be to minimize the time that the system needs to
reach a steady state as the result of a unit step change of supplied energy. A system reaches the
steady state when the rate of its dynamic energy, K& , becomes zero. Dynamic energy is
equivalent to the kinetic energy of masses in mechanical systems or the energy stored in
inductors in electrical systems. Masses and inductors resist the change of velocity and current,
respectively. In terms of bond graphs modeling, both velocity and current are considered as flow.
dttSSft
0ii ∫= );()( XX & . (3.15)
)(.);(.);(.)( XXXX ii
t
0
t
0iiiii qedttfedttfeSe
f f rrrrrr=== ∫ ∫ (3.16)
)(.);(.);(.)( XXXX ii
t
0
t
0iiiii pfdttefdttefSf
f f rrrrrr=== ∫ ∫ (3.17)
∑∑==
+=⇒ef N
1iii
N
1iii qepfS )(.)(.)( XXX rrrr
(3.18)
),,;(min)( αqpSS sCX
* XXss∈
= . (3.19)
37
Consequently, dynamic energy is defined as the energy stored in the elements of system that
inherently resist the change of flow.
where NI is the total number of I elements.
Therefore, Given a unit step change of supplied energy, the response time, denoted by T(X), is
the time instant after which the rate of change of dynamic energy remains below a small
threshold, ε.
As a design supercriterion, when the response time reaches its minimum value with respect to
attitude parameters the best design is attained in the set of optimally-satisfactory solutions (Cs).
Other factors can be also considered as holistic performance criteria of design, such as, structural
considerations, cost, precision, etc. The complete flowchart of LM methodology is presented in
Figure 3.1.
Note: It can be shown that the concepts behind the above supercriteria are analogous to the first
and second laws of thermodynamics (see Appendix B).
∑=
=IN
1iii tftetK );().;();( XXXrr& , (3.20a)
∑∑ ∫==
==II N
1iii
i
N
1i
t
0ii
i
tpteI1dete
I1tK );().;();().;(),( XXXXX rrrr& ττ ; (3.20b)
),(:)( ε<>∀= XX tKtttInfT 00& (3.21)
),,;(min)( αqpTT sCX
* XXss∈
= (3.22)
38
Chapter 4: Application, Simulation Results
In this chapter a design platform consisting of LM methodology, a simulation code based on
recursive Lagrange-Euler inverse dynamics and bond graphs model of a generic serial link
manipulator is employed to concurrently design a 5 D.O.F manipulator. In this design problem
all kinematic, dynamic and control parameters are considered simultaneously to achieve the best
configuration that can accomplish the main goal. The ultimate mission is defined as following a
predefined trajectory, satisfactorily. Consequently, the considered design variables and attributes
are introduced in the LM framework in the first section. Subsequently, a fuzzy modeling strategy,
as it was detailed in Chapter 2, is used in the primary phase of LM to find the initial
configuration of the secondary phase. The secondary phase of LM uses an appropriate single-
objective optimization algorithm in order to maximize the overall design satisfaction altering
design variables. Finally the bond graphs model of the generic 5 D.O.F manipulator is detailed
and utilized in the supercriterion section for defining a holistic performance criterion based on
energy consumption of the system and adjust the designer’s attitude. The entire simulation has
been executed on a computer platform and the design results are presented and discussed in the
last section.
4.1 Design Problem The first step in any design problem is introducing the design variables and attributes that are
considered in the design process in order to accomplish the main goal of design that is, in this
case, satisfactorily following a predefined pick and place trajectory. A simulation package
consisting of forward and inverse kinematics and inverse dynamics based on Lagrange-Euler
recursive algorithm is developed in MATLAB® to evaluate the design attributes during the
design process. The complete design architecture including the design process and simulation
section is shown in Figure 4.1.
4.1.1 Design Variables All kinematic, dynamic and control parameters of a generic 5 D.O.F robot manipulator with five
rotary joints are considered as design parameters. Kinematic characteristics of the robot are
defined based on standard Denavit-Hartenberg convention. Therefore, length (li) offset (di) and
39
twist (αi) are deemed as the kinematic design variables of the ith link. In order to take into
account dynamic parameters of the manipulator, each link is modeled with an L-shaped circular
cylinder along link length and offset. The radius of corresponding cylinder (ri), as a design
variable, specifies dynamic parameters of the ith link knowing the material and thus link density.
A PI position controller with velocity feedback and feedforward is considered for each joint of
the robot. Hence, the control design parameters for the ith joint include proportional (Pi), integral
(Ii), velocity feedback (Kvfb,i) and velocity feedforward (Kvff,i) gain. Consequently, the design
problem deals with forty design variables in total to identify the most satisfactory kinematic,
dynamic and control configuration of the robot manipulator.
4.1.2 Design Attributes Based on Linguistic Mechatronics, design attributes are divided into must and wish attributes and
design availabilities are considered as must attributes because of their similar nature.
4.1.2.1 Must Design Attributes
Must attributes are the design features that are corresponding to the costumers demands and have
to be satisfied with no room for compromise. In this research the following are taken into
account as must design attributes:
4.1.2.1.1 Design Availabilities
Each design variable has an acceptable range of values, considering its physical nature and
manufacturing constraints. They are taken into account by the following inequality expression.
where kX is the kth design variable, n is the number of design variables and minkX and max
kX are
the minimum and maximum values for kX , respectively.
4.1.2.1.2 Joint Constraint
Each joint module has restrictions due to physical properties of the actuator, mechanical
configuration of the manipulator and its location in the working environment. In this case study,
only the restrictions on the ranges of joint movements are considered as the following inequality,
maxminkkk XXX ≤≤ (k=1,2,…,n.); (4.1)
40
where jiθ is the angle between link i and i-1 at jth working point, ndof and N are the number of
joints and working points, and miniθ and max
iθ are the minimum and maximum angles allowed for j
iθ respectively.
4.1.2.1.3 Torque Constraint
Each joint module can handle a maximum amount of torque ( maxiτ ) that must not be approached
while a manipulator performs a task. This torque is usually corresponding to the stall torque of
the ith joint actuator. Therefore,
where j
iτmax is the ith joint maximum absolute value of the torque between jth and (j-1)th
working points.
4.1.2.1.4 Maximum Reachability
Reachability of a robot manipulator is defined as its ability to move its joints and links in free
space so that the end-effector reaches a target point. The farthest point that the manipulator can
reach is the maximum reachability of the robot (R) and because of environmental constraints it
should not exceed a certain number (Rmax). It means,
4.1.2.2 Wish Design Attributes
The must attributes are mostly constraints that are imposed by environment or physical
properties of the manipulator. However, the main mission of the robot is reflected in the wish
attributes. In this case study, this mission is defined as satisfactorily performing a predefined
pick and place trajectory. Along this trajectory seven working points are selected at which design
attributes are evaluated. The wish design attributes are listed as following;
maxmini
jii θθθ ≤≤ (i=1,2,…,ndof. j=1,2,…,N.); (4.2)
maxmax i
j
i ττ ≤ (i=1,2,…,ndof. j=1,2,…,N.); (4.3)
maxRR ≤ . (4.4)
41
4.1.2.2.1 End-Effector Error
The typical ultimate task for a robot manipulator is to follow a predefined trajectory. This
trajectory is defined by the position and orientation of the manipulator end-effector at a certain
number of working points. Therefore, the error that has been measured at the working points is
an appropriate wish attribute to minimize. If jΔ and jδ are the maximum permitted errors for the
end-effector position and orientation, respectively, at jth working point, then the end-effector
error can be defined as:
where jxΔ , jyΔ and jzΔ are the position errors in x, y and z directions, and jxδ , jyδ and jzδ
are the orientation errors about x, y and z directions at the jth working point. Note that orientation
errors are assumed sufficiently small so that the overall orientation error can be considered as a
vector. Also, for a 5 D.O.F manipulator only yaw and roll angles of the end-effector were
considered. A maximum of 1.5 mm for the translational error and 6º for the orientation error are
assigned for this design problem.
4.1.2.2.2 Manipulability
The manipulability index is used for checking the manipulator singularity at the working points.
This measure can be expressed as [67]:
where )( j0Jcond is the condition number of the jacobian matrix of the end-effector with respect
to the base frame at jth working point. At the singular points this index approaches to infinity and
its minimum value is one. Therefore, this wish attribute is satisfied when manipulability index is
close enough to one.
∑= ⎟⎟
⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ +++
++=
N
1j j
2j
2j
2j
j
2j
2j
2j zyxzyx
N1E
δ
δδδ
Δ
ΔΔΔ; (4.5)
∑=
=N
1jj
0JcondN1M )( ; (4.6)
42
4.1.2.2.3 Structural Length Index
A desirable manipulator is the one with a larger workspace using the least amount of material.
Structural length index summarizes this tradeoff in a simple formulation as:
where
and V is the workspace volume that can be numerically calculated based on a method detailed in
[68]. This ratio roughly encapsulates the relative amount of structure required to generate a given
workspace. Thus, a good design would be a manipulator with a small length sum (L) and a larger
V, which results in a small QL.
4.1.2.2.4 Total Required Torque
The torque generated by the joint motor is proportional to the electric current, and subsequently
the energy consumed at the joint. Therefore, another wish attribute that is considered in this
design is the total required torque at the joints. At each working point the total required torque
can be expressed as:
where jiτ is the torque of joint i at jth working point.
4.1.3 Assigning Satisfactions Once the design variables and attributes are specified, satisfactions are defined over the ranges of
their values. As it is noticeable, the must attributes should occasionally satisfy inequalities while
wish attributes should be as satisfactory as possible. One of the significant merits of
implementing LM to a design problem is that since this methodology employs fuzzy set theory,
the restrict binary behaviour of inequalities and optimization problems can be turned into flexible
3L
VLQ = ; (4.7)
∑=
+=ndof
1iii dlL )( ; (4.8)
∑=
=ndof
1i
ji
jT ττ ; (4.9)
43
and fuzzy one redefining their notions. This brings the subjective aspects of design into the
scope, nevertheless, simplifies the design process. One of the popular fuzzy membership
functions that can be used to demonstrate satisfaction is the trapezoidal membership function.
This function possesses four parameters (four corners of a trapezoid) that the designer should
decide about to specify the range in which the satisfaction is one and the slopes of the side
portions. This decision making is based on the design requirements and the designer’s
preferences. In other words, the trapezoidal parameters reflect how conservative or aggressive
the designer is in interpreting the design attributes. This membership function is easy to
comprehend and sufficiently flexible for our purpose. Therefore, in this case study, the
trapezoidal membership functions have been employed in order to define satisfactions over the
ranges of the design variables and attributes. These trapezoids are depicted in Figure 4.2. The
first and last points of a must satisfaction function are the minimum and maximum values of the
corresponding inequality, respectively. The middle points are picked in a manner that the
definition of the inequality is neither too fuzzy nor too crisp and it obeys the design
requirements. For a wish satisfaction function, the last point is the maximum allowed value of
the attribute and as it decreases the corresponding satisfaction approaches to one. The middle
point is selected based on designer’s consensus of the notion of minimum. All minimum and
maximum values of design variables and attributes are listed in Table 4.1.
4.2 Primary Phase of LM The primary phase of LM, as it was discussed in the third chapter, attempts to sketch the final
design by a set of ranges for design variables and introduce the initial configuration to start the
optimization problem in the secondary phase. In order to achieve this goal, the system is
modeled by a fuzzy modeling scheme that consists of several IF-THEN rules describing the
behavior of the system. The ranges of design variables as fuzzy sets along with the overall
satisfaction index form the antecedents and consequent of each rule, respectively. The required
dataset for the modeling process is generated by executing the simulation portion of the design
platform in a search process for 1730 times. In the procedure of fuzzy-logic modeling the first
step is to generate the rule base; therefore, a Fuzzy C-Means (FCM) clustering method is used to
cluster the output data that is the overall satisfaction corresponding to each configuration. As it
was discussed in the Chapter 2, the number of clusters (c), which is the number of rules, and the
weighting exponent (m) should be determined in a priori. An iterative algorithm based on scs and
44
sT is employed to determine the optimum m and c. First, the reliable domain for m that is the
middle range of sT is found in Figure 4.3. Subsequently, by minimizing the scs for a value of m in
this domain the best corresponding c is calculated (Figure 4.4). Then the resultant value of c and
m are checked in the Figure 4.3 whether they lie in the reliable range or not. This procedure
iterates until it converges to the optimum m and c. In this case study, the optimum values of
c=20 and m=1.5 are finally chosen. Therefore, the output variable of dataset, which is overall
satisfaction, is clustered employing these parameters.
Subsequently, in order to enhance the efficiency of the fuzzy-logic modeling process, significant
design variables are identified using non-significance measure, jπ . This index is calculate based
on the ranges of antecedents in which membership values are close to one that correspond to the
output membership function equals to one. Table 4.2 shows the calculated values of jπ for all
input variables. The design variables with the measure of non-significance of less than 1E-30 are
deemed in the model. Hence, according to Table 4.3, 22 design variables are selected as
considerably significant inputs of the fuzzy-logic model. Therefore, the output membership
functions are projected to a 22 dimensional input space to define the input membership functions
for the rules. This projection is conducted using line fuzzy clustering discussed in Chapter 2.
Hence, now, a rule base consisting of 20 rules with 22 antecedents and one consequent is
formed.
In order to be able to calculate the model output corresponding to a set of input variables, the
next step is choosing appropriate fuzzy connectives and fuzzy inference engine. Fuzzy algebraic
product and sum class of operators is used for AND and OR connectives and fuzzy Takagi-
Sugeno-Kang (TSK) inference mechanism, detailed in Chapter 2, is employed, accordingly.
Ultimately, the parameters of input membership functions and output coefficients must be tuned
based on the available database utilizing an algorithm detailed in [59]. In this algorithm an error
function that is basically the mean square error of the existing data points is tried to be
minimized varying the input and output parameters. In this attempt the performance of the fuzzy-
logic model has improved by 50%.
Once the fuzzy model is tuned, the consequent of each rule is averaged over the entire dataset
employing Equation (4.10) in order to crisply identify the satisfaction level corresponding to the
range of design variables in each rule.
45
where liμ (l=1,2,…,r, i=1,2,…,N) is the overall satisfaction corresponding to the ith data point in
lth rule, N is the number of data points in the existing database, blj (j=1,2,…,n) is the TSK
consequent coefficient corresponding to the jth design variable in the lth rule, xij is the jth design
variable in the ith data point and l*μ is the average of overall satisfactions in the lth rule.
The rule with the maximum l*μ is picked and the set of its antecedents is defuzzified based on
CoA defuzzification method (Equation (3.11)) to determine the initial values
),...,,( 00j n21iX ==0X for the significant design variables where n0 is the number of them.
The numbering of significant design variables and the antecedents and consequent parameters of
the most satisfactory rule are shown in Tables 4.4 and 4.5, respectively. The antecedents
membership functions of the most satisfactory rule are also depicted in Figure 4.5. The initial
values corresponding to the non-significant variables do not influence the secondary phase
intensively; hence, they are picked from the configuration that the data generation process started
with. Ultimately, the initial configuration for the secondary phase is stated in Table 4.6.
4.3 Secondary Phase of LM This phase of design is basically optimizing the overall design satisfaction using the initial
design variables calculated in the primary phase. A function called fminsearch in MATLAB®
optimization toolbox is utilized to solve this optimization problem. This function uses a
derivative-free search algorithm based on simplex method that is suitable for handling
discontinuity, sharp corners and noise in the objective function, which is the case in this research.
Nevertheless, the optimum value is a function of the attitude parameters, p, q and α. Therefore,
this optimization process forms a set of satisfactory design alternatives (CS) based on subjective
considerations of design, including designer’s attitude in aggregation and preferences in
assigning attributes satisfactions processes.
∑∑==
+++==N
1in
i1
i1l0l
N
1il
il xbxbb
N1
N1 )...( ln
* μμ ; (4.10)
46
4.4 Performance Supercriterion Ultimately, the best design configuration from the optimally satisfactory set of design
alternatives (CS) should be selected based on a holistic supercriterion. Since the previous steps of
design were biased by designer’s preferences and attitude, this measure ought to be an objective
criterion based on physical system performances. The energy supercriterion defined in Chapter 3
based on bond graph model of the system is utilized for this purpose.
Therefore, a generic 5 D.O.F robot manipulator consisting of joint modules, controllers and
mechanical elements is modeled based on hybrid utilization of bond graphs and block diagrams.
In order to demonstrate the bond graph model of the mechanical arm in Figure 4.6 vector bond
graphs are employed since they are more compact. This bond graph model basically represents
the recursive inverse dynamic equations of a generic serial link manipulator expressed in
Equations (4.11a) to (4.12b).
These equations include a forward loop of angular and linear velocities, i.e., Equations (4.11a-c)
and backward loop of forces and moments, i.e., Equations (4.12a,b) with boundary conditions of
zero angular and linear velocities at the base and zero force and moment at the end effector.
where iiω is the angular velocity of link i in frame i, and
iCi v and i
i v are the linear velocities of
the center of mass and frame origin of the link i expressed in frame i, respectively. 1ii R − is the
rotation matrix between frame i and i-1, i1i1i z θ−
− is the angle between link i-1 and i about the
joint axis and ii r and
iCi r are the distance between frame i and i-1 and center of mass position of
the ith link measured in the ith frame, both expressed in frame i, respectively. The following
dynamic equations reflect flow of effort in the bond graph representation of serial link
manipulators..
)( i1i1i
1i1i
1ii
ii zR θωω −
−−
−− += ; (4.11a)
ii
Ci
ii
1i1i
1ii
Ci
iirrvRv ω)~~( +−= −
−
− ; (4.11b)
ii
ii
1i1i
1i1
ii rvRv ω~+= −
−
− ; (4.11c)
47
where 1ii1i1i f −−− , and 1ii
1i1i T −
−− , are the force and moment acting from the link i to the link i-1 at the
joint and expressed in the (i-1)th frame, mi is the mass of link i, iC
i I is the moment of inertia of
link i about its center of mass and gi is the gravitational acceleration both expressed in frame i.
The bond graph representation of the electrical motors, shown in Figure 4.7, consists of two main
domains, i.e., electrical and mechanical. A gyrator element, using the torque coefficient of the
motor (imK ) as the gyrator ratio, relates these two physical domains. The circuit schematic of the
modeled motor is demonstrated in Figure 4.8. The electrical part includes a voltage source, a
motor driver, which is modeled by a gain intensifying the voltage, and a simple RL electrical
circuit. Using Kirchhoff’s circuit law the equation based on which the bond graph model is
generated can be expressed as:
where Ii, iau ,
imθ ,iml and
imr are the circuit current, source voltage after the motor driver, angular
displacement of the actuator shaft, the inductor coefficient and the total resistance of the circuit,
respectively, all of which correspond to the ith joint module.
In the mechanical portion a transmission mechanism with iη as transmission ratio and inertia of
the rotary motor (imj ) are deemed; therefore, the equation of motion would be:
where iθ is the angular displacement and iτ is the torque corresponding to the ith joint module.
All of the used constant coefficients in the motor modeling are listed in the Table 4.7.
))(( ,, iCi
ii
ii1iii
T1i
i1ii
1i1i vm
dtdgmfRf +−= +−−
−− ; (4.12a)
))~~()(~( ,,,, 1iii
1iCi
ii
ii
Ci
i1iiiC
ii1i
ii
T1i
i1ii
1i1i frrI
dtdfrTRT
iii −−++−−−− +++−= ω ; (4.12b)
)(iiii
i
mmimam
i KIrul1
dtdI θ−−= ; (4.13)
)( iiimm
ii IKjdt
di
i
τηηθ
−= ; (4.14)
48
Since bond graphs should be converted into block diagrams in order to be processed, they are
completely compatible with block diagrams which are the most popular technique to simulate a
control system. Hence, block diagram of the joint position controllers, demonstrated in Figure
4.9, are jointly utilized with the bond graphs. The block diagrams of the entire system generated
and processed in MATLAB® Simulink are shown in the Figures 4.10-4.12.
Once the model is ready, the sources of power should be recognized in order to calculate the total
energy consumption of the manipulator while it follows a trajectory. The power is flowed to the
system through the electric sources of energy in the electric motors. Therefore, by monitoring the
voltage and current of the motors at the sources, the instantaneous power and accordingly the
energy consumption of each joint module are determined, using a method of time integration.
Ultimately, the total energy consumption of the system as the supercriterion is calculated by:
where Ei is the voltage at the ith voltage source, which is constant in time, tf is the final time of
the simulation, ndof is the number of joints and XS is an optimally satisfactory solution of design
determined in the secondary phase of LM.
By minimizing this criterion over CS the best design is achieved:
4.5 Results and Discussion The detailed design architecture was executed and the final results of the 5 D.O.F manipulator
design is presented in Table 4.8. According to these results, not all of the design variables have
been changed to the required precision and interestingly, the minimum variations can be
observed in the non-significant design variables identified in the primary phase of LM (see Table
4.3). The link radii represent the dynamic characteristics of the robot. The third link radius has
been changed the most amongst other radii by almost 5%. In terms of kinematic parameters the
length of second and third links have been increased and decreased by almost 1% and 0.5%,
respectively. On the other hand, amongst link offsets, only the first link shows a considerable
∑ ∫=
=ndof
1i
t
0ii
f
dtIEq;pEnergy ),,( αSX ; (4.15)
)),,(min()( αq;pEnergyEnergySS CX
S* XX
∈= . (4.16)
49
adjustment. Nevertheless, none of the twist variables has been significantly changed. This
indicates that the fuzzy-logic model of the robot in the primary phase of LM was sufficiently
accurate to identify the initial configuration. Considering these modifications, the masses of the
first three links have been adjusted by -1.3%, -1.8% and +9%, respectively. Hence, it is seen that
a trivial variations of the kinematic parameters and link radii can considerably affect the dynamic
characteristics of the manipulator.
The block diagrams of the bond graph model of the robot were developed in MATLAB®
Simulink. This model used ODE solvers of Simulink® to simulate the system. Since the model
was so stiff, executing the simulation with completely off control gains would result in
divergence. Therefore, the controller must have been tuned manually with the initial
configuration. Hence, from the controller point of view, control gains have been slightly
modified by 0.3-1.5% in order to enhance the objectives of the design problem.
Perhaps the most crucial wish design attribute is the end-effector error, E. From Table 4.8, a
significant improvement in E is achieved. The final value of this attribute is almost 3.25 times
smaller than the initial value. Considering the designer’s preferences, the wish satisfaction
corresponding to this attribute has reached to 0.417 from the initial value of zero that means an
unsatisfactory case. An overview of other wish satisfactions demonstrates that all of them have
been improved to some extend. This indicates that in terms of designer’s preferences the system
performance has enhanced. Furthermore, the overall must satisfaction has also been increased.
This means that the system was even improved considering the constraint and availability
inequalities; hence, it went farther from the boundaries. Therefore, the final design would be
safer and more fault tolerable.
Nonetheless, all such candidates must checked against a real world objective supercriterion in
order to adjust the designer’s attitude in the aggregation process. The energy supercriterion
discussed in Chapter 3 was used to finalize the design process. Ultimately, the configuration with
the minimum energy consumption was picked as the final design. According to Table 4.8, the
energy consumption has been decreased by almost 6%. Comparing the achieved designer’s
attitude to the initial parameters reflects that the must aggregation parameter, p, has been reduced
by 5% that means the designer was initially slightly conservative in aggregating must attributes.
This indicates that instead of centralizing the overall must satisfaction around the minimum
50
attribute, the designer should give more weight to other must satisfactions, as well. In terms of
wish satisfaction aggregation, the value of α did not change, that is, the designer was able to
appropriately compromise between the two cooperative wish attribute subsets that are competing
with each other. Nonetheless, the other wish aggregation parameter corresponding to combining
the satisfaction of each wish attributes subset (q) has been adjusted by 13% increase. This
indicates that the initial attitude of the designer in this aggregation was too aggressive and the
designer should not try to enhance all cooperative wish design attributes at once and he/she
should more focus on improving the minimum attribute. In other words, overalls, in wish
satisfactions aggregation, the designer was too optimistic and he/she has to more concentrate on
the minimum attribute.
In this chapter a design architecture for synthesis of robot manipulators was proposed based on
alternative methodology of LM and Lagrange-Euler recursive inverse dynamics of a generic
manipulator. In the primary phase of LM first, a database was generated executing the simulation
portion of the architecture. The output data points were clustered employing FCM and after
identifying the significant input variables the output clusters were mapped to the input space in
order to define the input membership functions. Finally, connectives and inference mechanism
are selected and the generated fuzzy-logic model of the system was tuned. Subsequently, the
consequents of the rules were averaged employing the input data points and the most satisfactory
rule was picked in order to calculated a suitable initial configuration for the secondary phase of
LM. Secondary phase maximized the overall satisfaction changing the design variables;
nonetheless, the result is a function of attitude parameters that were defined in aggregation
process of satisfactions. Ultimately, the energy consumption of the system as a holistic
supercriterion finalized the design process. It was minimized varying the attitude parameters to
adjust the designer’s attitude in the aggregation process. In order to calculate the energy
consumption , the bond graph model of a generic 5 D.O.F manipulator in Simulink® was used.
51
Chapter 5: Application, Experimental Results
In this chapter a Robotic Hardware-In-the-Loop Simulation (RHILS) along with LM
methodology is utilized to concurrently redesign an industrial manipulator called, CRS CataLyst-
5. The RHILS platform was developed and validated in the Space Mechatronics Laboratory using
the mentioned robot components, and it is applied to evaluate the design attributes during a
design process [55]. The present RHILS involves physical joints and a control unit of a real robot
in order to reduce the complexity of simulation and to take into account phenomena that are
difficult to model mathematically. Therefore, it would provide the designer with a rapidly
changeable and sufficiently accurate design platform. The discussion about the architecture of
the entire design process, employing RHILS, along with the hardware and software
implementation is presented in the following section. The specific design problem of redesigning
CRS CataLyst-5 in the framework of Linguistic Mechatronics is formulated in the section 2.
Finally, the design results are demonstrated and the design platform is validated.
5.1 Design Architecture Including RHILS The design architecture detailed here provides a modular and generic test-bed for analysis and
concurrent synthesis of serial-link robot manipulators. The designer can change kinematic,
dynamic, and control parameters in order to enhance the performance of the system. The
platform consists of two parallel workstations, namely Host and Target, and physical
components of a robot manipulator, i.e. joint modules and controller unit. For each joint module
a load emulator is employed to apply simulated dynamic loads during the real-time execution.
The collection of load emulators, joint modules and control system is called Hardware
Emulation block. The Host and Target workstations and Hardware Emulation block are depicted
in Figure 5.1.
5.1.1 Host Workstation This block is the link between the platform and designer. All preferences and options of design
are set in the Host computer, where the main code that governs the design process is executed.
The options consist of initial kinematic, dynamic and control parameters, the predefined
trajectory of the end-effector, gravity conditions, payloads, and the simulation duration. This
52
block communicates with the controller to load control parameters such as proportional, integral,
derivative, feedback and feedforward gains through an FTP connection, and sends the command
signals to the trajectory planner using Python® software. The Host workstation also transfers data
via a TCP/IP connection in order to load kinematic and dynamic parameters and inverse
dynamics model of a design candidate to the Target workstation, and gathers positions and
torque saved on the Target workstation using MATLAB® xPC Target® toolbox.
5.1.2 Target Workstation This block is a barebones PC running the xPC Target®
real time kernel. On this workstation a
torque controller for load emulators and an inverse dynamics model of the manipulator, built in
Simulink® and compiled through Real-Time Workshop®, are executed. In the model torque
signals are calculated based on the manipulator configuration and joints position, velocity and
acceleration. Target workstation includes several interface cards in order to communicate with
the joint modules and load emulators. Joint positions and the torque simulated and sensed at the
modules are easily displayed on a monitor attached to this PC using Simulink® scopes. In order
to connect the Target workstation to the hardware components a data acquisition board and a
RS232 port are utilized
5.1.3 Hardware Emulation All physical hardware pieces that remain unchanged in the design process form Hardware
Emulation block. In this research the first three joint modules of an industrial manipulator, called
CRS CataLyst-5, their corresponding load emulators, and the CRS DM Master Controller unit are
implemented. Each joint module consists of a stepper motor, an encoder mounted on the motor
shaft, a harmonic drive as a transmission mechanism, and the driver unit. The module interfaces
with both controller and Target workstation in order to receive control signals via motor driver
and send joint positions back to the Target workstation. The load emulators are coupled directly
to the joint modules in order to apply computed loads. These torque signals represent the arm’s
dynamics and weight and payload effects that must be reflected on each joint actuator to have a
genuine simulation of the real system. Since the applied torque should be followed accurately, a
servo torque controller is designed and calibrated for each load emulator module. A reaction
torque sensor is installed between the load emulator case (stator) and its mounting fixture to
measure the feedback signal. The load emulator module sends and receives torque signals to and
53
from the Target PC. The controller unit includes a trajectory planner and a typical
feedback/feedforward controller for each physical joint module. The schematic of the controller
is depicted in Figure 5.2. The trajectory planner generates instantaneous desired position signals
with a frequency of 1 KHz based on the input of the controller. Joint trajectories can be broken
down into three sections: first, accelerating to the maximum speed with the nominal acceleration
of the joint module, second, constant speed motion and finally, decelerating to the final position
with the nominal acceleration.
5.2 Design Problem As it is stated in the last section, hardware components of CRS CataLyst-5 have been installed in
the mentioned RHILS platform. Hence, a valuable practice would be concurrently redesigning
this industrial manipulator, starting from the existing configuration. In this section the LM
methodology is implemented for redesigning kinematic, dynamic and control parameters of the
CRS CataLyst-5 manipulator. The RHIL platform is also utilized to evaluate the following design
attributes. Figure 5.3 shows the real CRS CataLyst-5 manipulator against the RHILS platform
developed using the manipulator components.
5.2.1 Design Variables CRS CataLyst-5 is a five degree-of-freedom industrial manipulator consisting of five rotary
joints. The first three joint modules, called waist, shoulder and elbow, respectively, have a
crucial role in the manipulator performance, and they are physically included in RHILS platform.
The last two joints are for roll and pitch motions of the manipulator wrist, and they are modeled
in computer. Kinematic characteristics of the manipulator are defined based on standard Denavit-
Hartenberg convention. Therefore, length (li), offset (di) and twist (αi) are deemed as kinematic
design variables of the ith link. In order to take into account dynamic parameters of the robot,
similar to the previous chapter, each link is considered as an L-shaped circular cylinder along
link length and offset. The radius of such cylinder (ri), as a design variable, specifies dynamic
parameters of the ith link knowing the material and thus link density. The CRS DM Master
Controller unit generates control signals for each joint consisting of proportional (Pi) and
integral (Ii) gains along with gains for feedback velocity ( ifbKv , ) and acceleration ( ifbKa , ) and
also feedforward velocity ( iffKv , ) and acceleration ( iffKa , ). Since the last two joints are small at
54
the tip of the manipulator with smaller moments of inertia than that those of the other joints, their
control gains are not considered in the design. Consequently, the design problem deals with
thirty-eight design variables in total to identify the most desirable kinematic, dynamic, and
control configuration of the manipulator.
5.2.2 Design Attributes Based on Linguistic Mechatronics, design attributes are divided into must and wish attributes. In
this case study, the same design attributes that were detailed in Chapter 4 are utilized.
Therefore, must design attributes can be listed as:
(a) design availabilities,
(b) joint restriction,
(c) torque restriction, and,
(d) maximum reachability
and the considered wish design attributes are:
(a) trajectory error,
(b) manipulability,
(c) structural length index, and,
(d) total required torque.
Once the design attributes are specified, their satisfactions are defined based on designer’s
preferences. In Figure 5.4 trapezoidal fuzzy membership functions are employed in order to
demonstrate attribute satisfactions for redesigning CRS CataLyst-5 manipulator. Table 5.1
specifies the range of design variables and attributes. Subsequently, the corresponding
satisfactions for any possible configuration are aggregated to determine the overall design
satisfaction.
55
However, it should be mentioned that since this design problem starts with an existing
manipulator configuration and the design platform is sufficiently accurate, more restrict
parameters than those in the simulation problem (Chapter 4) are chosen for defining the wish
satisfactions. This indicates smaller middle points; hence steeper trapezoid sides. Also, the
maximum allowed translational end-effector error has been decreased to 1 mm.
5.3 Design Process The LM design process can be divided into three steps. As it was discussed before, primary phase
selects a set of proper intervals for design variables based on which the initial configuration is
determined for proceeding optimization. However, in this case study, since a predesigned
manipulator is the matter of concern, the optimization can be safely started from the existing
configuration and this step would be skipped.
Secondary phase searches for the design variables that maximize the overall design satisfaction.
In this phase, in order to calculate the design attributes RHILS platform is utilized to simulate the
candidate configuration of 5 D.O.F robot manipulator while it follows the predefined pick and
place trajectory. In this procedure, first the Denavit-Hartenberg table and dynamic parameters of
the design candidate are determined based on the kinematic parameters and the links radii. They
are loaded onto the Target workstation as the parameters of the inverse dynamic model of the
manipulator. Afterward, the control gains are placed on the controller via an FTP connection. On
the Host computer an inverse kinematic code is executed in order to transform the end-effector
trajectory to the joint trajectories utilizing the kinematic parameters of the candidate. Once the
angles are specified, the corresponding control signals are sent to the controller from the Host
workstation using Python® software and simultaneously, while the real joint modules are moving
the joint torques calculated in Target PC are applied on them by means of a set of load
emulators. Then the corresponding position and torque signals are saved on the Target
workstation for further computations. Subsequently, the design availability, maximum
reachability, manipulability and structural length index attributes are calculated using the
kinematic parameters and the joint restriction, torque restriction and total torque required design
attributes are determined based on the saved signals. In addition, a forward dynamic code is
executed to compute the actual end-effector positions at the working points in order to evaluate
the end-effector error. Finally, the corresponding satisfactions are identified depending on the
56
designer’s preferences, that are trapezoid corners in defining satisfactions, and they are
aggregated using the attitude parameters. This phase of design involves a single-objective
optimization of the overall satisfaction over a large number of design variables. A function in
optimization toolbox of MATLAB® called, fminsearch, has been employed. This function uses a
derivative-free search algorithm based on simplex method that is suitable for handling
discontinuity, sharp corners and noise in the objective function, which is the case in this research.
This real-time process takes almost 1 minute for evaluating each configuration. Therefore, in
order to maximize the overall satisfaction the entire design architecture must run for at least one
day to find an optimally satisfactory solution for design.
Hence, by altering the designer’s attitude parameters (p, q and α) the secondary phase generates a
set of optimally satisfactory solutions for design. Nonetheless, the physical performance of the
system should also be checked against an objective supercriterion in order to adjust the
designer’s attitude. The total energy consumption of the manipulator, calculated by (5.1), is
employed to define an appropriate performance supercriterion. This function is determined by
the stored position and torque signals of the design candidates in the last step.
where XS is the set of design variables corresponding to a satisfactory design candidate.
Ultimately, by minimizing this criterion over optimally satisfactory solutions set (CS), the best
design (X*) is achieved.
Since it was not possible to continuously run the RHILS platform for more than a day, the
optimization of the supercriterion was performed manually. Therefore, the entire design platform
was executed for almost 2 weeks in order to generate 11 solutions for the secondary phase of LM
varying the attitude parameters. Finally, the solution with the least energy consumption was
picked as the best design.
∑ ∫=
=ndof
1iii
Ni
1i
dq;pEnergyθ
θ
θτα ),,( SX ; (5.1)
)),,(min()( αq;pEnergyEnergySS CX
S* XX
∈= . (5.2)
57
5.4 Results and Discussion The final results of the manipulator re-design are shown in Table 5.2. The initial values of design
variables were based on the current configuration of CRS CataLyst-5. The final design solution
shows notable modifications in some of the design variables. With respect to the manipulator
dynamics, i.e., link radii, the third link radius has been decreased by almost 10%. Furthermore,
the length of link 3 has also changed by 0.7%. These modifications result in 17.5% reduction of
the link mass. In addition, all other kinematic and dynamic parameters have been adjusted in
order to enhance the manipulator performance in accomplishing the final mission. For example
the link radii of the first and second link have been changed by almost 0.1% and 0.7%,
respectively. The length of link 2 and the first link offset have also been correspondingly altered
by 0.1% and 0.4%. Nevertheless, twist angles have remained almost unchanged. Therefore, in
terms of dynamic and kinematic design, the third link has been modified considerably.
In addition, since the controller of the existing manipulator was initially tuned, all control gains
have slightly modified by an average of 0.8% in order to enhance the design attributes.
Nonetheless, even this small change in the control parameters affected the end-effector position
and orientation error significantly that is observable in the design results.
From Table 5.2 a considerable improvement is made in the end-effector error, E. The final error
is approximately 78 times less than its initial value. The total must satisfaction has improved,
which indicates that the system is farther from the boundary points of design and it is more
reliable and safe. An increase in the level of satisfaction for all other wish attributes can be
observed from Table 2, as well. Therefore, based on the designer’s preferences, all the
considered attributes have been enhanced.
Nevertheless, all such design candidates were checked against an objective supercriterion, which
is the total consumed energy, through altering attitude parameters. Ultimately, the configuration
with the minimum energy consumption was picked as the final design. The energy consumption
was improved by 10%. Adjustment of attitude parameters during the design process indicates
that the initial designer’s attitude in aggregating must satisfactions was appropriate. That is, the
value of p did not change through the attitude adjustment. However, in aggregating wish
satisfactions the designer was originally too conservative, and the values of q and α were
58
decreased and increased by almost 50% and 140%, respectively, through the attitude adjustment.
This implies that instead of focusing on the worst wish attribute, the designer should equally
stress all wish design attributes in order to improve the energy consumption of the system.
Therefore, overalls, the designer should have been more aggressive (optimistic) in the design of
CRS CataLyst-5.
The RHILS platform discussed in this chapter provided a real-time and concurrent design
platform for robot manipulators. Implementing physical components reduced the simulation
complexity, yet included the phenomena that are difficult to model. Using analytical model of
the parts that need to be designed in concurrence with the hardware components has made the
platform suitable for rapid design alterations. In addition, Linguistic Mechatronics as an
alternative approach for synthesis of multidisciplinary systems was employed. This methodology
not only simplified the optimization complexities of concurrent design, but also brought in, in a
formal way, the subjective notions of design. The presented design architecture was used to
redesign an industrial manipulator, namely CRS CataLyst-5, whose components were utilized in
development of RHILS setup. This design problem consisted of thirty-eight design variables
including kinematic, dynamic and control parameters that were specified concurrently. The same
attributes used in the Chapter 4 were employed.
59
Chapter 6: Conclusions and Future Work
Since the reconfigurable robotic systems are multidisciplinary with a number of complex
electromechanical modules, the architecture and mechatronic design of these engineering devices
require an alternative design framework. Therefore, the main goals of this research were defined
as: (i) proposing a practical design methodology for synthesis of multidisciplinary systems and
(ii) taking advantage of RHILS platform in order to reduce the simulation complexities. These
goals were attained by formalizing Linguistic Mechatronics and utilizing an available RHILS to
solve a synthesis problem. Therefore, this final chapter is devoted to the conclusions of this
research and the discussions on potential future works on this topic.
6.1 Conclusions An alternative conceptual design methodology for concurrent synthesis of multidisciplinary and
complex engineering systems, such as reconfigurable robots, namely Linguistic Mechatronics
(LM), was developed. This design approach argues that the ultimate goal of design is not pure
objective optimization, and by considering the subjective aspects of design not only can the
design process be simplified but also the communication between different disciplines is
enhanced. LM employs fuzzy membership functions in order to define the notion of satisfaction
for design attributes that reflects the designer’s preferences in interpreting the design
requirements. In this approach, the design attributes are intuitively divided into two inherently
different subsets, called must and wish attributes, that correspond to the customer’s demands and
desires or optimization constraint and objective functions. Subsequently, in order to transform
the multi-objective constrained design problem to a single-objective unconstrained one an index,
coined as overall design satisfaction, is calculated combining all the attribute satisfactions using
parameterized fuzzy operators. These parameters continuously quantify the designer’s attitude in
aggregating satisfactions between two extremes, i.e., being aggressive (optimistic) and being
conservative (pessimistic). Nonetheless, this methodology not only considers the subjectivity in
design, but also brings the physical objectivity into the scope by defining holistic supercriteria
that can deem the interconnection between different subsystems in different physical domains
that plays a vital role in the synthesis of mechatronic systems. Therefore, an alternative
prototyping strategy based on the notion of energy and energy exchange for generalized
60
executive movement subsystems and signal transmission for information processing and control
subsystems that can consider the coupling between all components of a complex system was
proposed. This modeling technique was used to define appropriate design supercriteria in order
to pick the final design from the set of optimally satisfactory design candidates. This
consideration of the real world in design would also modify the subjective parameters in the
suggested design process, i.e., attitude parameters. Therefore, not only does LM provide the final
configuration based on the combination of subjective and objective notions of design, but it also
adjusts the designers attitude in treating a design problem through utilizing a holistic
supercriterion.
Subsequently, this promising design framework was employed to designing a 5 D.O.F
manipulator in order to illustrate the notions behind the LM methodology. This problem
consisted of forty design variables with different natures, including kinematic, dynamic and
control parameters. This is a large number of variables for traditional multi-objective
optimization algorithms. Eight typically used design attributes were considered. These attributes
are divided into four must and four wish design attributes. According to the final results, the must
attributes, demonstrating the environmental and physical constraints, slightly improved that
means a safer and more fault tolerable system. On the other hand, the wish design attributes,
specially the end-effector error function, expressing the final goals of the design, considerably
enhanced during the LM process. Therefore, according to the designer’s preferences the final
solution was more satisfactory. All of the optimally satisfactory design candidates were checked
against a performance supercriterion, i.e., energy consumption, to adjust the designer’s attitude
in aggregating the satisfactions. The final results showed almost 10% reduction of the energy
consumption during attitude parameters adjustment process.
Ultimately a Robotic Hardware-In-the-Loop Simulation (RHILS) was added to the design
platform in order to perform the attribute evaluation process of LM. This platform reduces the
simulation complexity and takes into account the exigently determinable phenomena, such as
energy loss in the form of friction and heat or flexibility in the subsystems, e.g. transmission
mechanisms, by implementing physical components of a real robot. The RHILS discussed in this
thesis provided a real-time and concurrent design platform for robot manipulators. Using
analytical model of the parts that need to be designed in concurrence with the hardware
components has made the platform suitable for rapid design alterations. The presented design
61
architecture was used to redesign an industrial manipulator, namely CRS CataLyst-5, whose
components were utilized in development of RHILS setup. This design problem consisted of
thirty-eight design variables including kinematic, dynamic and control parameters that were
specified concurrently. The same attributes used in the previous chapter were employed. The
final results demonstrated a considerable improvement in wish design attributes and particularly
in the defined end-effector error function, E, that the corresponding error for the final design was
78 times smaller than the initial value of the current commercial configuration. Finally, the
designer’s attitude parameters were adjusted employing energy consumption as a supercriterion
and the consequences were discussed in detail. It revealed that the designer was originally too
conservative in aggregating wish attributes.
6.2 Future work The proposed design platform showed a promising performance, and revealed some of its
potentials in the synthesis process. Two performance supercriteria were suggested by LM
methodology, but the second criterion that checks the agility of the system was never
investigated in a case study because of the thesis time limit. Furthermore, in the case studies,
since the number and type of joints were selected a priori, the structural length index and
manipulability design attributes did not change considerably in the design process. Therefore,
either the employed attributes may be changed or the number of degrees of freedom and types of
joints should be deemed as design variables. In addition, in the supercriterion section, it was
attempted to find the criteria that are appropriate for the synergy notion of design and can
consider the interconnections between the subsystems. The offered supercriteria were based on
the basic laws of physics and sufficiently holistic for this purpose. Nonetheless, the author
always tried to relate the state of satisfaction and synergy of the system through a unique
quantity such as entropy of design. Therefore, an investigation can be launched to
mathematically define a physically meaningful function for this criterion.
Another valuable practice is applying LM to more complicated realistic design case studies and
investigating whether this methodology works for mechatronic systems other than robot
manipulators. Furthermore, this flexible and generic design methodology can be employed to
synthesis of other types of processes and products, as well. Hence the author even suggests this
linguistic design framework for other applications.
62
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67
Figures
Initialize Simulation and Hardware
Read from Input Device
Evaluation Simulation Models
Write to OutputDevices
Delay Until Time to Start Next Frame
Integrate State Variables
Shut Down Simulation End of Run?
Yes
No
(a)
Black-Box (HILS)
Optimization
QualityCandidates
(c)
(b)
Sensor
Input Signals Operator Commands
Actuator
Cobntrol Signals Operator Display
Embedded System
Real Time Simulation
s
Figure 1.1 (a) HILS software flowchart, (b) HILS Block, (c) Real-time optimization with HILS architecture
68
BA
BA
Power Ports
(a)
(b)
ef
er
fr
Figure 2.1 (a) scalar bond, (b) vector bond connecting two power ports of components A and B
R
I
I
R
C
C er
fr
er
fr
er
fr
er
fr
er
fr
er
fr
TF 1 2
TF 1 2
GY 1 2
GY 1 2
0 1 2
3
1 1 2
3
R fr
er
R er
fr
qr
∫
C-1 er
fr
I-1
∫ er pr
fr
dtd
I pr
er
fr
C qr
dtd
er
fr
2er 1er
1fr
N 2fr
N-11er 2er
1fr
2fr
H 2fr
1er
1fr
2er
H-1 1er
2fr
2er
1fr
∑
1er 2er
3er 2fr
1fr
3fr
––
∑
1fr
2fr
3er
2er 1er
3fr
– –
Figure 2.2 Block diagrams for all possible causality assignments of bond graphs elements [69]
69
Calculate overall satisfaction μ(p,q,α)(X)
Maximize μ(p,q,α)(X)
Change X
Construct bond graphs model of the system
Minimizing Supercriterion
over Cs
Record
)*
(),*
(],***
[,*
XSXqpX μα or )*
( XT
Change ],,[ αqp
Converged
Converged
NO
NO
YES
YES
Construct fuzzy linguistic rule base
Database (X,A)
Select the rule with maximum defuzzified consequent
Calculate overall satisfaction μ(p,q,α)(X) for database
]0,0,0[ αqp
Obtain the suitable ranges of design variables and initial values
]0,0,0[ αqp X0 C
Calculate S(X)
Choose a supercriterion
S(X)
Primary Phase of LM
Secondary Phase of LM
Performance Supercriterion
Calculate T(X)
T(X)
Figure 3.1 Linguistic Mechatronics flowchart
70
Calculating Design
Attributes
Calculating Attributes’
satisfactions
Bond Graphs Model of the Manipulator
Maximize Overall
Satisfaction
Minimize Performance
Supercriterion
Calculating Overall
Satisfaction
Design Process
Interface
1- Database 2- Predefined
Trajectory 3- Design Attributes 4- Designer’s
Preferences 5- Initial Designer’s
Attitude Parameters
Final Design (X*) YES
NO Change X
YES Database
NO
Change p, q and α
Output
Trajectory Planner
Joint 1 Joint 2
Joint 3
Joint 1 Joint 2
Joint 3 Joint 4
Joint 5
Joint 1 Joint 2
Joint 3 Joint 4
Joint 5
Joint 1 Joint 2
Joint 3 Joint 4
Joint 5 Lagrange-
Euler Recursive Algorithm
Inverse Dynamics Position Controller Inverse kinematicsForward Kinematics
T ΘΘΘ &&& ,,
Joint 3 Joint 3
θθθ &&&,,
θθθ &&&,,
θθθ &&&,,
θθθ &&&,,
θθθ &&&,,
T
T
T
T
T
Simulation
( ΘΘΘ &&& ,, , T)
Modeling the System by Fuzzy TSK
Choosing the Most
Satisfactory Rule
Defuzzifying the
Antecedents X0
Calculating Energy
Consumption
Secondary Phase of LM Performance Supercriterion
Primary Phase of LM
Calculating Design
Attributes
Calculating Attributes’
satisfactions
Calculating Overall
Satisfaction
Figure 4.1 Design Architecture including simulation
71
Figure 4.3 Trace of total scatter matrix for identifying m
c=2c=3
c=20
Reliable
Domain sT
m
c
Figure 4.2 Satisfactions on design variables and attributes
0.75 Ymax
li,
72
m=1.4
m=1.5
m=2.1
m
c
scs
Figure 4.4 Specification of c
73
Figure 4.5 Antecedents membership functions for the most satisfactory rule
74
MTF 1nn R −
1 I: nJ MTF nC
n r~ 0
1 I: mn
Se: gm nn
MTF )~~( n
nC
n rrn+
1n1n
−− ω
1nn1n1n T −
−− ,
MTF 1nn R −
1
Flow sensor
∫
MOTOR
nθ
Trajectory Planner
Controller
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
Nn
2n
1n
N
2
1
t
tt
θ
θθ
MM,
dnθ
Gain nθ& nτ
1n1n v −−
1nn1n1n f −−− ,
nCn v
nθ& d
nθ&
0 0
MTF 1ii R −
1 I: iJ MTF iC
i r~
0
1 I: mi
Se: gm ii
MTF )~~( i
iC
i rri+
1i1i
−− ω 1ii
1i1i T −
−− ,
MTF 1ii R −
1
Flow sensor
∫
MOTOR
iθ
Trajectory Planner
Controller
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
Ni
2i
1i
N
2
1
t
tt
θ
θθ
MM,
diθ
Gain iθ& iτ
1i1i v −− 1ii
1i1i f −−− ,
iCi v
iθ& d
iθ&
0
iiω
i1iiiT ,+
ii v
i1iii f ,+
0
MTF 01 R
1 I: 1J MTF 1C
1 r~ 0
1 I: m1
Se: gm 11
MTF )~~( 1
1C
1 rr1+
Sf:0
1
Flow sensor
∫
MOTOR
1θ
Trajectory Planner
Controller
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
N1
21
11
N
2
1
t
tt
θ
θθ
MM,
d1θ
Gain 1θ&
1τ
1C1 v
1θ& d
1θ&
0
11ω
1211T ,
11 v
1211 f ,
0
Sf: 0
… ………
Se: 0 Se:0
Figure 4.6 Bond graph representation of a serial link manipulator
75
+ - Ei
Driver
imr iml
imK
ua
I
+
imj iη
Figure 4.8 Circuit schematic of an electric motor
Figure 4.7 Bond graph representation of an electric motor
MTF Se: Ei 1 GY 1
I:imj I:
iml
R:imr
imK
Gain
MSe: iτ iτ iθ&
Driver
Electrical Domain Mechanical Domain
ua MTF
iη
diθ
diθ&
∑
iP
SIi
∑
iffKv ,
Gain
ifbKv ,iθ&
iθ
+ - -+
++
Figure 4.9 Block diagram model of the controller
76
Figure 4.10 Simulink® model of a 5 D.O.F manipulator based on bond graphs
77
Figure 4.12 Simulink® model of an electric motor based on bond graphs
Figure 4.11 Simulink® model of a link based on bond graphs
78
Trajectory Planner
Joint 1 Joint 2
Joint 3
Motor Driver
Motor Driver
Motor Driver
Motor Driver
Motor Driver
Motor Driver
Joint 4 Joint 5
Trajectory Planner
Joint 1 Joint 2
Joint 3 Joint 4
Joint 5
Joint 1 Joint 2
Joint 3 Joint 4
Joint 5
Joint 1 Joint 2
Joint 3 Joint 4
Joint 5
Lagrange-Euler
Recursive Algorithm
Joint 1 Joint 2
Joint 3
Calculating Design
Attributes
Calculating Attributes’
satisfactions
Calculating Performance
Supercriterion
Maximize Overall
Satisfaction
Minimize Performance
Supercriterion
Calculating Overall
Satisfaction
Inverse Dynamics Torque
Controller Position Controller Inverse kinematics
Design Process
Interface
Host Workstation
Target Workstation Hardware Emulation
1- Initial guess (X0) 2- Predefined
Trajectory 3- Design Attributes 4- Designer’s
Preferences 5- Initial Designer’s
Attitude Parameters
Final Design (X*) YES
NO
Change X
YES X0
NO
Change p, q and α
Forward Kinematics
θθθ &&&,,
θθθ &&&,,
θθθ &&&,,
T
T
T
θθθ &&&,,
θθθ &&&,,
T
T
T ΘΘΘ &&& ,,
Output
Figure 5.1 The design architecture including RHILS
diθ ∑
iP
SIi
∑
iffKv ,
Joint Module
ifbKv ,iθ&
iθ+ - -+
++
S
S iffKa ,
+Encoder
diθ&
diθ&&
S
S ifbKv ,
-iθ&&
Figure 5.2 The schematic of CRS DM Master Controller
ifbKa ,
79
(a) (b)
Figure 5.3 (a) CRS CataLyst-5 robot (b) RHILS platform
Figure 5.4 Satisfactions on design variables and attributes
li,
80
Tables
Table 2.1 Bond graphs elements in various energy domains
Energy DomainEffort (e)
Flow (f)
Generalized Momentum (p= )
Generalized Displacement (q= )
C I R
Translational Mechanics
Force Velocity Momentum Displacement Spring Inertia Damper
Rotational Mechanics
Torque Angular velocity Angular
momentum Angle Rotational spring Moment of inertia
Rotational damper
Electronic Domain
Voltage Current Linkage flux Charge Capacitor Inductor Resistance
Magnetic Domain
Magnetomotive Force
Magnetic flux Rate
- Magnetic flux Magnetic capacitor
- Reluctance
Hydraulic Domain
Total pressure Volume flow rate Pressure
momentum Volume Reservoir Fluid inertia Flow resistance
Thermodynamic Temperature Entropy flow
rate - Entropy Heat capacitor - Heat resistance
Chemical Domain
Chemical potential
Molar flow - Molar mass - - -
SS2m
pP
S2m
pP
S &
∫ fdt∫edt
Table 4.1 Design variables and attributes ranges
i= 1 i= 2 i= 3 i= 4 i= 5[0,0.2] [0,0.2] [0,0.2] [0,0.2] [0,0.2][0,0.5] [0,0.5] [0,0.5] [0,0.5] [0,0.5][0,0.5] [0,0.5] [0,0.5] [0,0.5] [0,0.5]
[-180,180] [-180,180] [-180,180] [-180,180] [-180,180] [-180,180] [-110,10] [-100,70] [-110,110] [-180,180]
[0,5.5] [0,16.2] [0,5.5] [0,4.8] [0,2.4]R (m )
EM
All Control Gains
[-∞,+∞]
[0,16.5]
[0,0.87][0,2]
[0,1.6][1,24]
)(oiθ
)(md i
)(oiα
).(max mNiτ
LQ).( mNTτ
)(mri
)(ml i
81
Dynamic Parameter
Link 1 6.722E-02 7.249E+00 1.284E-03 4.700E-03 5.928E-01 1.976E+01 2.475E-15 1.003E-11Link 2 1.179E-01 3.242E-02 7.072E+10 4.084E-01 2.884E-02 8.415E+00 1.704E-04 4.220E-01Link 3 8.756E+00 7.552E-02 2.324E-01 2.432E-02 1.563E+00 1.000E+01 1.763E-03 2.155E-02Link 4 3.794E+01 1.562E+01 1.084E-11 1.109E+00 6.466E+02 1.878E+00 9.882E+01 7.536E+00Link 5 1.434E-02 1.240E+17 3.960E+02 2.007E-01 5.248E-01 3.576E-01 1.117E+01 1.053E+01
Kinematic Parameters Control Gainsil id iα iP iI ifbKv , iffKv ,ir
)( 30j 10×π
Table 4.2 Measure of non-significance for design variables
Table 4.3 Significant design variables
Dynamic Parameter
Link 1Link 2Link 3Link 4Link 5
Kinematic Parameters Control Gainsil id iα ir iP iI ifbKv , iffKv ,
×
××
× ×× ×
×××× ×
× × × ×× ×
×
i 1 2 3 5 1 2 5 2 3 1 3 4 1 2 5 5 1 2 3 1 2 3Xj (j= ) 1 2 3 4 6 7 12 8 10 5 9 11 13 16 21 22 14 17 19 15 18 20
ir il id iP iI ifbKv , iffKv ,iαTable 4.4 Numbering of significant design variables
Consequent Parameters
Xj a b c d bljj =0 -5.648E+01j =1 6.531E-02 6.561E-02 6.561E-02 6.607E-02 -1.190E+02j =2 2.768E-02 2.776E-02 2.776E-02 2.836E-02 -2.615E+02j =3 2.404E-02 2.406E-02 2.410E-02 2.458E-02 -2.839E+02j =4 1.000E-02 1.004E-02 1.025E-02 1.051E-02 1.145E+00j =5 -1.628E+00 -1.568E+00 -1.555E+00 -1.544E+00 -7.784E+01j =6 -9.279E-07 3.108E-06 3.370E-05 1.278E-04 -1.401E+01j =7 2.539E-01 2.539E-01 2.542E-01 2.577E-01 -2.013E+02j =8 -9.284E-05 -3.091E-05 4.355E-06 1.567E-04 -5.487E+02j =9 -7.111E-05 -4.823E-07 4.885E-05 9.680E-05 2.006E+01j =10 -3.075E-05 2.584E-06 4.820E-05 2.292E-04 1.692E+00j =11 -1.623E+00 -1.571E+00 -1.548E+00 -1.492E+00 0.000E+00j =12 9.976E-11 1.000E-10 1.001E-10 1.024E-10 -1.801E-01j =13 1.993E+01 1.999E+01 2.100E+01 2.100E+01 -3.449E-02j =14 3.998E+01 4.035E+01 4.156E+01 4.251E+01 -5.757E-02j =15 4.278E+01 4.449E+01 4.504E+01 4.549E+01 -8.479E-02j =16 2.200E+01 2.200E+01 2.215E+01 2.275E+01 -1.799E-02j =17 3.742E+01 3.859E+01 4.140E+01 4.271E+01 -1.270E-01j =18 4.740E+01 4.753E+01 4.800E+01 4.972E+01 -1.183E-01j =19 2.298E+01 2.362E+01 2.470E+01 2.503E+01 -7.598E-02j =20 3.256E+01 3.291E+01 3.352E+01 3.413E+01 -2.534E-01j =21 9.878E+00 9.939E+00 1.009E+01 1.028E+01 -2.339E+01j =22 9.863E-02 9.937E-02 1.010E-01 1.031E-01 5.139E+01
Antecedents Parameters
0.7469*μ
Table 4.5 Antecedents and consequent parameters of the most satisfactory rule
82
(V) ( ) (mH) (g.cm2)Link 1 4 3.3 3 0.2587 68 1/72Link 2 3.6 1.8 2.5 0.4414 300 1/72Link 3 3.6 1.8 2.5 0.4414 300 1/72Link 4 4 3.3 3 0.2587 68 1/19.6Link 5 4 3.3 3 0.2587 68 1/9.8
iEimr
imKimj iηΩ
imlimr
Table 4.7 Motors coefficients used in the simulation
j= 1 j= 2 j= 3 j= 4 j= 5 j= 6 j= 7Initial 2.1948 19.5192 1.3049 14.0631 12.1214 13.0851 12.1373 12.1434 13.1062 12.1474Final 0.6757 18.7397 1.2982 13.3135 11.3882 12.308 11.4063 11.4128 12.3297 11.4165
Wish Design Attributes
ME (N .m )LQ
jTτ
i= 1 i= 2 i= 3 i= 4 i= 5 i= 1 i= 2 i= 3 i= 4 i= 5Initial 65.6 27.9 24.2 10.0 10.0 0.0 255.2 254.0 0.0 0.0Final 65.9 28.0 23.0 10.1 10.2 0.0 257.9 255.1 0.0 0.0
i= 1 i= 2 i= 3 i= 4 i= 5 i= 1 i= 2 i= 3 i= 4 i= 5Initial 254.0 0.0 0.0 0.0 0.0 -90.4 0.0 0.0 -89.3 0.0Final 255.1 0.0 0.0 0.0 0.0 -90.6 0.0 0.0 -89.5 0.0
i=1 i=2 i=3 i=4 i=5 i=1 i=2 i=3 i=4 i=5Initial 20.48 22.26 13.00 12.00 10.05 0.100 0.100 0.150 0.200 0.101Final 20.73 22.35 13.07 12.04 10.08 0.100 0.101 0.152 0.201 0.101
i=1 i=2 i=3 i=4 i=5 i=1 i=2 i=3 i=4 i=5Initial 41.11 39.67 24.08 23.65 22.40 44.38 48.25 33.29 25.00 23.00Final 40.55 39.68 24.12 23.71 22.52 45.04 48.39 33.37 25.07 23.08
InitialFinal [9.56,1.69,0.50] 7.8049
8.2850[p,q,α]
[10.00,1.50,0.50]Energy (J )
)(oiα
iP iI
)(mmri)(mmli
)(mmdi
ifbKv , iffKv ,
Table 4.8 Design results
j= 1 j= 2 j= 3 j= 4 j= 5 j= 6 j= 7 μInitial 0.000 0.738 0.747 0.591 1.000 0.828 1.000 1.000 0.823 1.000 0.418 0.245Final 0.417 0.754 0.877 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.592 0.572
Overall Satisfaction
Wish Satisfactions Overall must Satisfaction
Eμ Mμ LQμj
Tτμ
Mμ
Dynamic Parameter
Link 1 4.555E-05 2.540E-01 -1.577E+00 6.566E-02 2.048E+01 1.000E-01 4.111E+01 4.438E+01Link 2 2.552E-01 2.082E-05 0.000E+00 2.793E-02 2.226E+01 1.000E-01 3.967E+01 4.825E+01Link 3 2.540E-01 7.093E-05 1.869E-05 2.423E-02 1.300E+01 1.500E-01 2.408E+01 3.329E+01Link 4 0.000E+00 0.000E+00 -1.558E+00 1.000E-02 1.200E+01 2.000E-01 2.365E+01 2.500E+01Link 5 0.000E+00 0.000E+00 0.000E+00 1.021E-02 1.005E+01 1.006E-01 2.240E+01 2.300E+01
Kinematic Parameters Control Gainsil id iα iP iI ifbKv , iffKv ,ir0jX
Table 4.6 Initial values of design variables
83
Table 5.2 Design Results
j= 1 j= 2 j= 3 j= 4 j= 5 j= 6 μInitial 0.000 0.606 0.455 0.838 0.593 0.838 0.838 0.609 0.609 0.250Final 1.000 0.620 0.626 1.000 0.896 1.000 1.000 0.896 1.000 0.607
Overall Satisfaction
Wish Satisfactions
Eμ Mμ LQμj
Tτμ
i= 1 i= 2 i= 3 i= 4 i= 5 i= 1 i= 2 i= 3 i= 4 i= 5Initial 65.6 27.7 24.1 10.0 10.0 0.0 254.0 254.0 0.0 0.0Final 65.7 28.0 21.8 10.0 10.0 0.0 253.6 255.9 0.0 0.0
i= 1 i= 2 i= 3 i= 4 i= 5 i= 1 i= 2 i= 3 i= 4 i= 5Initial 254.0 0.0 0.0 0.0 0.0 -90.0 0.0 0.0 -90.0 0.0Final 255.0 0.0 0.0 0.0 0.0 -90.8 0.0 0.0 -90.7 0.0
[p,q,α ]
i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3Initial 18.32 20.00 12.00 0.07325 0.05000 0.10000 40.7 40.0 20.0 [10,1.5,0.5]Final 18.46 20.16 12.10 0.07381 0.05039 0.10077 41.0 40.3 20.2 [10,0.7,1.2]
Energy (J )
i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3Initial 43.41 100.00 80.00 59.010 40.000 30.000 3473.0 100.0 120.0 6.2549Final 43.76 100.80 80.62 59.471 40.311 30.230 3483.6 100.8 120.9 5.6307
)(oiα
iP iI
)(mmri)(mmli
)(mmdi
ifbKv ,
iffKv , iffKa ,ifbKa ,
j= 1 j= 2 j= 3 j= 4 j= 5 j= 6Initial 1.4787 20.7223 1.3091 9.3557 10.2754 9.3561 9.3561 10.2172 10.2172Final 0.0189 19.4923 1.3025 8.3071 9.1391 8.3071 8.3071 9.1394 8.3071
Wish Design Attributes (N .m )ME LQ
jTτ
Table 5.1 Design variables and attributes ranges
i= 1 i= 2 i= 3 i= 4 i= 5[0,0.2] [0,0.2] [0,0.2] [0,0.2] [0,0.2][0,0.5] [0,0.5] [0,0.5] [0,0.5] [0,0.5][0,0.5] [0,0.5] [0,0.5] [0,0.5] [0,0.5]
[-180,180] [-180,180] [-180,180] [-180,180] [-180,180] [-180,180] [-110,0] [-90.6,35] [-110,110] [-180,180]
[0,13.8] [0,13.8] [0,13.8] [0,4.8] [0,2.4]R (m )
EM
All Control Gains
[-∞,+∞]
[0,12.5]
[0,0.87][0,2]
[0,1.6][1,24]
)(oiθ
)(mai
)(md i
)(oiα
).(max mNiτ
LQ).( mNTτ
)(mri
il
84
Appendix A (Pareto-optimality of the overall satisfaction optimization)
Assume that Xs is not locally pareto-optimal. Then CX 1 ∈∃ such that
and
Thus, according to the Remark, there must exist
to Equation (A.1) and
or
to Equation (A.2). Hence, according to monotonicity of t-norm operators, the following must be held
or
as the result of Equation (A.4a) or (A.4b) when aggregating the design attributes. Obviously, Equation
(A.5a) contradicts that Xs is a locally optimal solution as the premise, and so does Equation (A.5b)
providing that Xs is a locally unique solution. This leads to the conclusion that Xs is locally pareto-
optimal.
],[),()( N1iFF ii ∈∀s1 XX f (A.1)
],[).()( N1iFF 0ii 00∈∃s1 XX f (A.2)
],[),()( N1iaa ii ∈∀≥ s1 XX (A.3)
],[),()( N1iaa 0ii 00∈∃> s1 XX (A.4a)
],[,)()( N1i1aa 0ii 00∈∃== s1 XX (A.4b)
)()( )()(sM1M XX pp μμ >⇒ , (A.5a)
)()( )()(sM1M XX pp μμ ≥⇒ ; (A.5b)
85
Appendix B (First and second laws of thermodynamics from a different
perspective) As discussed in the thesis, first performance supercriterion was defined based on conservation of
energy. In another approach it is analogous to first law of thermodynamics. This law simply
states that the change of internal energy of a control mass is equal to the heat transferred to the
control mass and the work done by the control mass during a process.
Figure B.1 Energy representation of a system
For the system shown in Figure B.1., first law can be formed as follows:
where K is internal energy of the system and K0 is energy of the system before energy exchange.
Input energy (work) to the system from environment is ei, output work to the environment is
denoted by eo and Q is the dissipated heat transmitted to the environment.
The components of the first supercriterion will be:
Suppose a system without any input and output energy. Subsequently, first law of
thermodynamics can be reformulated as:
0io KeKQe =−++ ; (B.1)
yKKQandEeSe 0oi =−+== ,, . (B.2)
System
K
Isolated control volume
Environment ei
eo
Q
86
First law of thermodynamics states conservation of energy in a cycle, however, it places no
restriction on the direction heat flow and work. Second law of thermodynamics asserts that a
cycle is possible when ever the inequality of Clausius is satisfied. Therefore, based on the second
law and definition of entropy, total entropy (s) of an isolated system always increases until it
reaches equilibrium. Entropy is proportional to the heat transferred between the components of
the isolated system. For the control volume in Figure B.1, it is proportional to Q.
Differentiating Equation (B.3) and substituting Equation (B.4) in that will result in:
Equilibrium state is once entropy reaches its maximum or in other words the differentiation of
entropy is zero which is equivalent to 0=K& , according to Equation (B.5). Therefore, the time
takes to reach equilibrium can be considered as response time of the system and can be
calculated based on rate of dynamic energy of the system as the second supercriterion presents.
0KKQ =+ . (B.3)
0sTQ0TQs ≥=⇒≥= &&&
& . (B.4)
0KsT =+ && ; (B.5)
0K0s ≤⇒≥ && . (B.6)
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