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´
ECOLE CENTRALE DE NANTES UNIVERSIT`
A DEGLI STUDI DI GENOVA
MASTER ERASMUS MUNDUS
EMARO+ “European Master in Advanced Robotics”
2019 / 2020
Master Thesis Report
Presented by
Durgesh Haribhau Salunkhe
On August 27, 2020
Optimal kinematic design of a robot mechanism for
otological surgery
Jury
President: Philippe Wenger Research Director, LS2N, Nantes
Evaluators: Philippe Wenger Research Director, LS2N, Nantes
Marcello Sanguineti Associate Professor, University of Genova
Olivier Kermorgant Associate Professor, ´
Ecole Centrale de Nantes
Supervisors: Damien Chablat Research Director, LS2N, Nantes
(EMARO) Marcello Sanguineti Associate Professor, University of Genova
Guillaume Michel Surgeon, CHU Nantes
Shivesh Kumar Senior Researcher, DFKI-RIC, Bremen
Laboratory: Laboratoire des Sciences du Num´erique de Nantes LS2N
Abstract
The use of endoscope in otological surgery provides many benefits in terms of visibility
and access to the operating region but needs to be handled by the surgeon at all time
making such surgeries cumbersome. A mechanism that can act as an assistant to the
surgeon for manipulating endoscope can have a huge positive impact on the efficiency of
the surgeon and the surgeries.
This thesis explains the technical requirements, constraints and optimization process
for proposing a mechanism for such application. The importance of analysis and synthesis
of the kinematic properties of the structure for an optimized output is highlighted in the
study. The author presents a novel implementation of an optimization algorithm for
the design optimization. Various approaches researched in the past for the optimization
are presented and their advantages and disadvantages are pointed out too. The effect
of parametrization, different constraints as well as rewarding strategies are discussed to
stress the complexities in the optimization of closed loop kinematic chains. The thesis
also presents approaches used to consider the constraints for the passive joint limits as
well as avoiding internal collision in the mechanism. A novel methodology is used for
a faster and efficient global search of the optimization space. The work concludes by
presenting the optimised result as well as by discussing open questions regarding the
future of mechanism design.
1
Acknowledgements
Many people have played an important role during my journey of completing the pre-
sented work and I am indebted to each and everyone for the same. Having said that, I
may not be able to do justice to all of them through the following words and I ask for
forgiveness from those whom I may miss to mention.
I am deeply grateful to Prof. Damien Chablat, who has been a constant guide and a
companion during my Master thesis from the very beginning. The knowledge gained
as well as the experience of working under his guidance has had a great impact on me
and my decisions for the future. I extend the same intensity of gratitude towards Prof.
Marcello Sanguineti who has continuously motivated me and guided me in the unknown
territory of the optimization theory. The progress achieved in the thesis would have been
simply impossible without the enthusiasm he showed at every stage of the thesis. I thank
Dr. Guillaume Michel, for his strong support and friendly attitude because of which it
was quite easy for me to involve in the thesis with better understanding. Thanks for all
the free trips to the operation room. It will be unfair to thank Dr. Shivesh Kumar only
for the thesis work. He has been a great guide and a friend for more than 3 years and I
am indebted by the confidence he has shown in my work throughout these years. I am
thankful to Elise Olivier for her insights with an alternative perspective on the thesis.
Her guidance has helped improve the work to be more practical and she has been the per-
fect link between robotics theory and the user experience. Also, thanks for the history of
Nantes and your help with French language. Thanks to Prof. Philippe Bordure, Fran¸cois
Pasquier and Sarah Normant for the extraordinary hospitality in the operating room at
the CHU. Your efforts to go at lengths to demonstrate the different scenarios while using
endoscope have been very helpful in highlighting the importance of the presented work.
The people I will mention next are the most special ones. Aai ,Baba and Didi . I
find the art of expressing oneself through words very inadequate to tell you how grateful,
indebted and humbled I am to have you in my life. Starting from the journey of life
to taking small steps of my own in different fields of life, you have only supported me.
Thanks to Baba, who introduced me to the world of Mathematics and I was saved from
mundane learning at the school. Because of you, Mathematics was never a subject for
me, it became a topic of curiosity and appreciation and I am very proud of that.
I wish to express my special thanks to Sanket with whom I have shared so many thoughts
and have pondered upon ideas that ended up shaping my overall personality. The ex-
change of ideas and skills have helped me improve my experience of life. Thank you,
Yuri, for without you none of the experiences would be the same. Your disciplined way
of living has motivated me to transform myself and has given enough strength to perse-
vere. Thanks to my friends back in India, Vipul and Ankit, whom none can replace. We
have shared each other’s pain and sorrows while rejoiced each others success and I’ll be
there for you like I’ve been there before. Thanks to Anirvan, who has been inspirational
with his undeterred focus on research and with whom I had the good fortune to work
with. Thanks to Francesco for the times spent together, we have shared so much over
this 2 years and I stay grateful for everything. I know that you will reach greater heights
learning your way to the top. Thanks to Pablo, with whom I have debated on various
topics and have pondered over different aspects of life. Thanks to my Italian friends in
2
France, Angie, Nic, Mario, Klav, Erri and Andrea, and also Sunny for the fun we had
in Nantes, I wish we had more time to celebrate. Thanks to Mbarre, Orianna, Francesca
and Ricardo for an amazing time in Sicilia and Nantes, you guys will never be forgotten.
Thanks to Judith and Esra, my friends from Erasmus, for your cheerful company. Thanks
to Rohit, your friendship has played an important role in reassessing my perspective of
life. These people have helped me during various stages of my thesis and life in overall,
and I hope that I had at least some if not equal influence on their life.
THANK YOU
3
Notations
Dconfiguration space
κconditioning number
RDWddesired Regular Dextrous Workspace
vemanipulability ellipsoid volume
Ffeasible set
κ−1
gglobal quality index
qiinput vector
JJacobian matrix
Ooptimisation space
Koutput space
xgeneralized output vector
κ−1quality index
SE (3) special Euclidean group of dimension 3
SO(3) special orthogonal group of dimension 3
4
Abbreviations
CHU Centre Hospitalier Universitaire
COR centre of rotation
DE Differential Evolution
DGS direct geometric solution
DH Denavit Hartenberg
dof degrees of freedom
EE end-effector
GA Genetic Algorithm
IGS inverse geometric solution
MOEA/D Multi-Objective Evolutionary Algorithm based on Decomposition
MOO Multi Objective Optimization
NM Nelder-Mead
NSGA-II Non-dominated Sorting Genetic Algorithm
OR Operating Room
PKM Parallel Kinematic Manipulator
PSO Particle Swarm Optimization
RDW Regular Dextrous Workspace
RR Revolute-Revolute
SPM Spherical Parallel Mechanism
SS Spherical-Spherical
US Universal-Spherical
5
List of Figures
1.1 The comparison of the view and proximity to operation while using an
endoscope and a microscope. . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 The comparison of the number of instruments possible to use simultane-
ously while using an endoscope and a microscope. . . . . . . . . . . . . . 11
1.3 The serial architecture (Robotol) used for endoscope manipulation . . . . 12
2.1 The issue in current scenario . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 The details of the 2 degrees of freedom (dof) agile eye with an offset . . . 15
2.4 The architecture with redundant passive leg . . . . . . . . . . . . . . . . 16
3.1 The patented proposed mechanism for the presented work [1] . . . . . . . 18
3.2 The 3 variations of the proposed architecture . . . . . . . . . . . . . . . . 19
3.3 The general parameterization of the 2UPS-1U and 2PUS-1U variation . . 20
3.4 The initial frames for the joints in a leg . . . . . . . . . . . . . . . . . . . 21
3.5 The 4 intersection points for showing the possible inverse geometic solutions 23
3.6 Actuation singularity in 2PUS-1U variation . . . . . . . . . . . . . . . . 24
3.7 The workspace divided by 4 apsects. We can use only one of the aspects
for a t-connected feasible workspace. . . . . . . . . . . . . . . . . . . . . 24
3.8 The implemented spherical joint as well as the ball-socket joint . . . . . . 25
3.9 The different travel paths of an axis in tilt-torsion representation. . . . . 26
3.10 The different poses of the mechanism in Euler representation . . . . . . . 27
3.11 The polar representation of the azimuth-tilt angles . . . . . . . . . . . . 28
3.12 The visualization for endoscope motion in azimuth-tilt representation . . 28
3.13 The comparison of the desired workspace in Euler-angles (blue) represen-
tation and the azimuth-tilt (red) representation. . . . . . . . . . . . . . . 29
3.14 The renderings used for size comparison in the phase 1 questionnaire . . 30
3.15 The renderings used for size comparison in the phase 2 questionnaire . . 30
4.1 The models of the ear and the sinus presented in [2]. . . . . . . . . . . . 33
4.2 The change in plot while expressing the same constraint in different spaces 35
4.3 The geometric interpretation with the assumption of k˙qk ≤ 1....... 36
4.4 The geometric interpretation with the independent constraints . . . . . . 36
4.5 The comparison between the conditioning number and the velocity ampli-
ficationfactor ................................. 38
4.6 Illustration of the joint limit distance . . . . . . . . . . . . . . . . . . . . 38
4.7 Summary for the different objective functions . . . . . . . . . . . . . . . 39
4.8 The aspects of the Kspace in different cases . . . . . . . . . . . . . . . . 40
4.9 The industrial prismatic joints and the relation between ρmin and ρmax . 40
6
4.10 The varying search brackets for range of actuators. . . . . . . . . . . . . 41
4.11 Comparison of feasible workspace (white space) within the RDWdfor dif-
ferentsearchbrackets. ............................ 41
4.12 The different definitions of the same architecture . . . . . . . . . . . . . 42
4.13 The range of optimizing variables . . . . . . . . . . . . . . . . . . . . . . 43
4.14 Summary for different constraints in the optimization process . . . . . . 43
4.15 An example of mapping with 2 optimization variables . . . . . . . . . . . 44
4.16 The premature convergence while using a simplex of less than (n+1) points
in n-dimensional O-space. .......................... 44
4.17 An example of an operation on a simplex in 2-dimensional O....... 46
4.18 An example of the travel path of optimization in Nelder-Mead algorithm 48
4.19 The actuator considered for the mechanism and the discretized points for
collision calculation (right) . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.20 An illustration for binary reward of the feasible workspace . . . . . . . . 49
4.21 An illustration for biased reward of the feasible workspace . . . . . . . . 50
4.22 The minimum quality constraint . . . . . . . . . . . . . . . . . . . . . . . 50
4.23 The evaluations for infinite penalty . . . . . . . . . . . . . . . . . . . . . 51
4.24 The evaluations for smooth penalty . . . . . . . . . . . . . . . . . . . . . 54
4.25 The flowchart for single start of the implemented optimization methodology 55
4.26 Comparison between random and low-discrepancy samplings of the 2-
dimensional unit cube from [3] . . . . . . . . . . . . . . . . . . . . . . . . 57
4.27 The rough and fine turning in lathes . . . . . . . . . . . . . . . . . . . . 59
4.28 The flowchart for the complete implemented optimization methodology . 59
5.1 The singularities occurring in the 2PUS-1U variation . . . . . . . . . . . 61
5.2 The different priorities scored by the surgeons . . . . . . . . . . . . . . . 62
5.3 2UPS-1U mechanism corresponding to a single point in the 13 dimension O63
5.5 An example of the output space with constraints violation (right) and the
heat map for the quality index (left) . . . . . . . . . . . . . . . . . . . . 64
5.4 An example of a schematic plot of the 2UPS-1U mechanism . . . . . . . 64
5.6 Results for one of the best local optima acquired while maximizing the
globalqualityindex.............................. 65
5.7 Results for one of the best local optima acquired while maximizing the
jointlimitnorm................................ 66
5.8 Results for the implemented biggest actuator range . . . . . . . . . . . . 67
5.9 Results for the implemented best actuator range . . . . . . . . . . . . . . 68
5.10 Results for the curious case of collision avoidance . . . . . . . . . . . . . 70
5.11 Results for the optimized architecture using 4 parameters and 13 parameters 70
5.12 The valid feasible space and the heat map for the quality index for opti-
mized mechanism using 4 parameters . . . . . . . . . . . . . . . . . . . . 71
5.13 The valid feasible space and the heat map for the quality index for opti-
mized mechanism using 13 parameters . . . . . . . . . . . . . . . . . . . 71
5.14 The small subset of the evaluation surface while implementing the binary
rewarding strategy and biased rewarding strategy . . . . . . . . . . . . . 72
5.15 The small subset of the evaluation in top view to observe the penalty regions 72
5.16 Valid feasible space and heat map for the quality index with coarse search 73
5.17 Valid feasible space and heat map for the quality index after fine search . 74
7
Contents
1 Introduction 10
2 State of the art 13
2.1 State of the art: design choices . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Choice of mechanism - First things first . . . . . . . . . . . . . . . 13
2.1.2 The Orthoglide wrist . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 2 dof agile eye with an offset . . . . . . . . . . . . . . . . . . . . . 15
2.1.4 2 dof orientation mechanism with passive leg . . . . . . . . . . . . 16
2.2 State of the art: optimization . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Design choices 18
3.1 Proposed mechanism and its variations . . . . . . . . . . . . . . . . . . . 18
3.2 Inversegeometry ............................... 19
3.2.1 Inverse geometry: 2UPS-1U variation . . . . . . . . . . . . . . . . 19
3.2.2 Inverse geometry: 2PUS-1U variation . . . . . . . . . . . . . . . . 22
3.3 Singularityanalysis.............................. 23
3.4 Choosingjoints ................................ 24
3.4.1 Spherical joint limits . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Workspace representation . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5.1 Eulerangles.............................. 27
3.5.2 Azimuth-tilt angles . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 User-centricchoices.............................. 29
4 Mechanism optimization 32
4.1 Elements of the optimization . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Objectivefunction .............................. 32
4.2.1 Multiple objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.2 Multiple objectives : Workspace of the manipulator . . . . . . . . 34
4.2.3 Multiple objectives : Quality of the manipulator . . . . . . . . . . 34
4.2.4 Measuring quality via the manipulability ellipsoid . . . . . . . . . 34
4.2.5 Measuring quality via the conditioning number . . . . . . . . . . 35
4.2.6 Measuring quality via the velocity amplification factor . . . . . . 36
4.2.7 Multiple objectives : Joint limits norm . . . . . . . . . . . . . . . 37
4.2.8 Summary of the used objectives . . . . . . . . . . . . . . . . . . . 38
4.3 Constraints .................................. 39
4.3.1 Feasible actuator range . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Optimizingvariables ............................. 40
8
4.5 Local search algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5.1 The Nelder-Mead (NM) Algorithm . . . . . . . . . . . . . . . . . 44
4.5.2 Implementation............................ 46
4.5.3 Rewarding strategies . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5.4 Implementing best actuator range . . . . . . . . . . . . . . . . . . 51
4.5.5 Pros and cons of the Nelder-Mead (NM)-algorithm . . . . . . . . 51
4.6 Global search algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6.1 Initial simplexes for multi-start . . . . . . . . . . . . . . . . . . . 56
4.7 Coarse and fine searches . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.8 Summary ................................... 60
5 Results and discussions 61
5.1 Results from design choices . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.1 Different joint placements . . . . . . . . . . . . . . . . . . . . . . 61
5.1.2 User-centric choice . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Results from optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 How to read the results . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.2 Different objective functions . . . . . . . . . . . . . . . . . . . . . 64
5.2.3 Effect of change of constraints . . . . . . . . . . . . . . . . . . . . 65
5.2.4 Effect of parameterization . . . . . . . . . . . . . . . . . . . . . . 69
5.2.5 Different rewarding strategies . . . . . . . . . . . . . . . . . . . . 71
5.2.6 Coarse and fine search . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.7 Computational time . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 Conclusions and future works 75
Bibliography 77
9
Chapter 1
Introduction
The recent health crisis of 2020 due to the Covid-19 virus made us realise the high im-
portance of our health system. The situation took all the humanity by surprise and the
soldiers in this crisis, our doctors, were completely exhausted and found themselves in-
adequately equipped. Many important lessons were learnt at the various stages of the
crisis and the following report mainly addresses about the need of an efficient state of
art in every possible area of healthcare system. The medical field has made enormous
progress in the past century leading to a higher life expectancy. The growing age of the
population also resulted in the rise of medical emergencies and surgeries. It is estimated
that by 2030, 5000 surgeries will be conducted per 100,000 population [4]. In order to
cope up with the rise of surgeries, it is important that we explore different technologies
that can be combined to make surgeries faster yet safer and more reliable. Surgical robots
play an important role in revolutionizing the conventional surgery procedures and is re-
searched for more than three decades. The first Food and Drug Administration (FDA)
approved endoscopic robotic surgery system was commissioned in 1990 by the AESOP
system [5]. Several works have been reported earlier for minimally invasive surgeries by
robots [6, 7, 8] with great success.
The concept of human robot collaboration to improve the efficiency of a task by
combining the accuracy of a robot and intuition of a human has become popular in the
recent years. The implementation of robots in inner ear surgery [9] and middle ear surgery
[10, 11] were discussed in detail in [12]. Some of them provide a complete solution for
robotic surgery while some can be used as an assistant for the surgeon. The robots in
surgery can be used to replace tasks of the surgeon that are mandatory but demand no
human expertise. One of the applications in this area is the use of endoscope in otological
surgeries.
There are many advantages of using an endoscope instead of the conventionally used
microscope in otological surgeries [13]. Some of them include better view with easy
access to the operating area and proximity to the patient while operating (Fig. 1.1).
Endoscopic surgeries lead to less bone sacrifice and thus improves the recovery period of
the patient. Though agile in application, endoscopic surgeries pose different challenges to
the surgeon. The confrontation interviews conducted by Elise Olivier under this project
suggests that due to certain limitations of endoscope, surgeons prefer to operate with
microscope which is comfortable to use. Currently, the surgeon can only use one tool
at a time in endoscopic surgeries in contrast to two tools with the use of a microscope
as illustrated in Fig. 1.2. This makes the endoscopic surgeries cumbersome as the
10
(a) Surgeon observing ∼25
cm away from the ear
(b) View of the operating
area in a microscope
(c) View of the operating
area in an endoscope
Figure 1.1: The comparison of the view and proximity to operation while using an endo-
scope and a microscope.. Courtesy: Dr. Guillaume Michel, CHU Nantes
(a) The use of 1 hand to hold endoscope
limits the number of instruments
(b) The surgeon can use 2 instruments
while using a microscope
Figure 1.2: The comparison of the number of instruments possible to use simultaneously
while using an endoscope and a microscope.. Courtesy: Dr. Guillaume Michel, CHU
Nantes
surgeon has to switch between tools to operate and manage bleeding in the ear. The
incapability of using both hands for the surgery leads to frustration and fatigue of the
surgeon. The use of a robot arm to manipulate the endoscope as needed can improve the
performance of the otological surgeries remarkably. Using assistive systems can result in
drastic reduction in the operating time and positively affect both, the surgeon as well as
the patient. Previously, robot mechanisms with a serial architecture were proposed as
a solution for the endoscopic surgery [10] and one of such implementation is illustrated
in Fig. 1.3. The clinical report regarding the same has been published recently [14].
The serial manipulators have larger workspace in contrast to its parallel counterpart and
are relatively easier to design and analyse. That been said, parallel manipulators are
generally known for their stiffer structure and are kinematically more robust. This can
be attributed to the reason that the error in joints is cumulative in serial mechanisms. It
is easier to have a fixed centre of rotation (COR) in a parallel mechanism by the virtue of
its architecture and the joint selection. These inherent advantages of parallel mechanisms
gives them a considerable edge for the application of the endoscope manipulation.
In mechanism design, the optimisation methodology plays an important role and is
governed by the definition of the objective function and constraints implemented. In
11
Figure 1.3: The serial architecture (Robotol) used for endoscope manipulation
Source: Collin Medical, France
the past, several design optimization methods were proposed in the field of mechanism.
Some of them utilise the mathematical formulation of the objective function in order to
implement gradient descent method [15]. Where such luxury is not present, numerical
approaches and evolutionary algorithms are extensively implemented in the past. The
most widely submitted work in this area uses the Genetic Algorithm (GA) [16, 17, 18]
while there exists literature with other evolutionary algorithms [19]. A recent work imple-
mented co-optimzation strategy which presents the efficient search of implicitly expressed
optimization space [20].
In this report, we present the kinematic comparison of different parallel mechanisms
that can be used to manipulate an endoscope. Works presented in [21] have been con-
sidered for comparison along with a novel 2 dof rotational parallel mechanism [1] and its
variant. One of the main objectives for the thesis is to present new design optimization
methodology that can adapt to constraints involving internal collisions along with the
physical joint limits and physical stroke of the actuator. We propose a fast local search
algorithm, i.e. Nelder-Mead algorithm, coupled with a global search mechanism. This
method allows to achieve a global optimum faster even for mechanisms that have com-
putationally expensive objective functions. The overall output of the work is a general
algorithm for design optimization that is flexible with the definition of the objective func-
tion as well as is modular and adaptive to any constraints.
The report is organised in the following way: Chapter 3 defines the task and its con-
straints clarifying the surgeons’ requirement for the design. It also explains the choice
of architecture, joint type, joint placement, and their effect on the workspace. The
three Spherical Parallel Mechanism (SPM) are also detailed in this section. Chapter 4
is the most important part of the report where the authors detail upon the optimization
methodology along with the choice of the objective function and constraint implementa-
tion. The results of the thesis work are presented in Chapter 5 followed by the conclusions
and pointers to future work.
12
Chapter 2
State of the art
In this chapter, a detailed survey of the previously reported work is presented. The
problem statement is discussed in more details to explain the required degrees of freedom
and the nature of mobility of the mechanism. Different mechanisms of such mobility and
their synthesis presented in the past are discussed in the following sections. The imple-
mentations of these mechanisms in different fields are provided to highlight the ubiquity
of parallel mechanisms and therefore the importance of their design optimization. Later,
we discuss the state of the art for the optimization procedures. Different evolutionary
and novel algorithms that have been used successfully implemented in recent years are
presented in depth. The advantages and disadvantages of these methods are discussed
allowing the reader to appreciate the need of new methodology for the same.
2.1 State of the art: design choices
In otological surgeries, ossiculoplasty, stapedotomy, tympanoplasty and myringoplasty
can be carried out with better efficiency if performed with endoscope. On the other hand,
all the sinus surgeries make use of endoscope for its advantages. In order to operate on
some parts during these surgeries, it is of prime importance that the surgeons can use
both-hands simultaneously. Without a robotic manipulator, an assistant surgeon has to
perform the task of manipulating endoscope which is not only unergonomic and hard
to synchronize but also very less productive use of his presence in the Operating Room
(OR). Fig. 2.1 is an ideal illustration from the OR of the problem statement we are trying
to resolve.
2.1.1 Choice of mechanism - First things first
The final objective of the project is to have a complete robotic solution as an assistant
for the surgeons to handle an endoscope. There has been an active dialogue between the
surgeons of Centre Hospitalier Universitaire (CHU) Nantes and the authors in order to
decouple the problem and translate it into the language of mechanisms. This allowed the
authors to formalize the expectations of the proposed solution which should have:
1. A possibility to locate the system in the appropriate location in OR.
2. The ability to manipulate the endoscope with a fixed center of rotation (cor)
3. The ability to have a translation along the direction of the endoscope.
13
Figure 2.1: The issue in current scenario.The assistant surgeon needs to hold the endo-
scope to operate on certain part. Courtesy: Dr. Guillaume Michel, CHU Nantes
As the importance of rotational movement is higher than the translational movement,
it is a wise choice to decouple them. This allows us to use actuators of different size,
speed and accuracy according to the need of the movement. Such ideology has been
implemented in the past and is known as the macro/micro manipulators [22]. It is also
useful to have decoupled rotation and translation motion for analyzing the kinematic
properties of the mechanism and we can dare to use several tools without fearing the
entanglement of the dimensions of the Jacobian matrix. The rotation of an endoscope
about its own axis is not very useful and thus the essential dof of the mechanism used
to manipulate an endoscope are 2-rotations and 1-translation. The foundational study
and analysis of a suitable mechanism was performed prior to the start of the presented
work by Dr. Guillaume Michel, Prof. Philippe Bordure and Prof. Damien Chablat. As
the translational movement is considered to be decoupled, the presented work focuses on
choosing a 2-dof orientation mechanism. The spherical parallel mechanisms have been
extensively researched in the past [23, 24, 25] and remain a topic of interest to current
researchers too [26]. A lot of theory and research is available from the previous decade
regarding parallel mechanisms with 2 orientation dof. The following subsections present
different implementations of the same.
2.1.2 The Orthoglide wrist
The Orthoglide wrist shown in fig. 2.2b is a mechanism similar to a 2 dof agile eye [27]
mechanism. The wrist was introduced as the decoupled orientation mechanism for the
translational Orthoglide [28] mechanism. The center of rotation of the wrist is at the
intersection of the two motors as shown in the fig. 2.2b. The detailed geometric and
kinematic analysis of this manipulator has been mentioned in [25] and [27]. The number
of inverse geometric solution (IGS) depends on the architecture of the mechanism. For
example, the 2 dof agile eye mechanism proposed by [27] has 4 IGS while the [25] has only
one IGS. This result is attributed to the joint arrangements of the mechanisms as well
as the joint limits of the actuators and collision avoidance between the components. The
number of direct geometric solution (DGS)s for [27] are 2 while the [25] has a unique DGS.
The DGS and IGS were analyzed by using Denavit Hartenberg (DH) parameterization.
The type I and type II singularities of these mechanisms are elaborated by detailing the
conditions in which the Jacobian matrix, Aand Bmatrix mentioned in [29], loses a rank.
14
The advantage of this system is the robust construction and the kinematic precision due
to the closed loop. One of the drawbacks of this system is that the end-effector (EE)
has to be mounted in such a way that the point of interest of the EE coincides with the
COR.
(a) The schematic of orthoglide wrist (b) Orthoglide wrist as proposed in [25]
2.1.3 2 dof agile eye with an offset
The 2 dof agile eye mechanism with an offset was discussed in [24]. This mechanism is
same in construction as the 2 dof agile eye but the COR is displaced by using a paral-
lelogram joint as shown in fig. 2.3. The maximum joint displacement of the actuators
is 60 degrees as shown in fig. 2.3b. The number of geometric solutions of this mecha-
(a) The schematic of the mechanism (b) The mechanism in 600pitch
Figure 2.3: The details of the 2 dof agile eye with an offset. Courtesy: Prof. Damien
Chablat, LS2N
nism changes once the constraints related to the joint limits and internal collisions are
added. The advantage of this mechanism is that the COR can be displaced according
to the requirement. This helps in locating the actuators on ground and manipulating
the endoscope remotely. This leverage also leads to a more complex kinematic analysis
as the properties of the parallelogram structure also affect the overall performance. The
mechanical design of such mechanisms is of prime importance as they have long cantilever
beams and the load (weight of the endoscope) is applied at the extreme end which can
lead to bending issues.
15
Figure 2.4: The architecture with redundant passive leg
2.1.4 2 dof orientation mechanism with passive leg
Another implementation is done by introducing a motion constraint leg to achieve the
necessary degrees of freedom. In such architectures (with 2-dof), 2 legs with 6 degrees of
freedom are used and a passive redundant leg is introduced which governs the degrees of
freedom as well as the nature of mobility of the overall mechanism. The idea is illustrated
in Fig. 2.4. The passive leg with the universal joint, i.e. leg 3 in Fig. 2.4 defines the
center of rotation as there are no constraint wrenches applicable in the 6-dof legs, leg1
and leg2. The mechanism is actuated with a prismatic joint present in each active leg.
The type synthesis of such class of mechanism was first presented in [30]. A procedure
regarding the choice of joints while designing mechanisms with such architectures is also
detailed in the same.
A parallel mechanism with architecture 2UPS-1U is previously proposed for bone
milling operations [31]. This work presents the workspace analysis with some assumptions
in the architecture. This architecture has been widely proposed in past research for
various applications [32, 33, 34, 35, 36]. Other variant of the architecture is the 2PUS-
1U which has been implemented in the Valkyrie humanoid [37]. A recent variation of
the 2UPS-1U is the 2SPRR-1U [21] which highlights the advantage of using a Revolute-
Revolute joint as a replacement of the Universal joint in the active legs.
2.2 State of the art: optimization
Different algorithms have been suggested previously to optimize the mechanism design.
In problems where the objective function and the constraints are well formulated, al-
gorithms that exploit the derivative of the objective functions are most successful [15].
Unfortunately, in mechanism design, it is common to encounter a non-smooth objective
function with non-linear constraints. In such cases, the derivative-free algorithms are
16
implemented. We are particularly interested in this type of optimization and the reasons
for the same will be discussed in Chapter 4.
The search for global optimum in the mechanism design is a challenging topic and
different nature-inspired evolutionary algorithms are implemented during such endeav-
ors. The algorithms used for single objective are Differential Evolution (DE) [38] and
Genetic Algorithms (GA) [39] while most widely used algorithm for the Multi Objec-
tive Optimization (MOO) is the Non-dominated Sorting Genetic Algorithm (NSGA-II)
[16, 18, 40, 41] in which the theory of genetic evolution is implemented. It treats a point
in optimisation space (O) as a chromosome and evaluates the fitness value (reward strat-
egy) of the population (several chromosomes spread in O) in one iteration. Then the
operations such as selection, crossover and mutation take place to generate a set of new
population and the process continues. The advantage of this methodology is that it per-
forms a global search of the optimization space. Other evolutionary algorithms that have
been implemented are the Particle Swarm Optimization (PSO) [19] and Multi-Objective
Evolutionary Algorithm based on Decomposition (MOEA/D) and are claimed to be su-
perior than NSGA-II in [42]. Nonetheless, all the above mechanisms are computationally
expensive as the efficiency of the algorithm highly depends on the population size. Also,
only a guess of the required chromosomes is provided for an efficient global search of
the optimization space. This dependency makes them vulnerable in case of demanding
objective functions and also limits the quantity and nature of the constraints that can be
implemented.
A recent work in mechanism design optimization is the co-optimization with the mo-
tion trajectories [20]. In this work, the design parameters and the motion equations are
represented implicitly and efficient algorithms are used to traverse the implicitly defined
manifold. This type of methodology can provide better insights on the global optimum
of the optimization space. The work also highlights the importance of joint placement of
prismatic actuators in the optimization process. Keeping in focus the computational cost
of optimizing a mechanism, different local search methods can be implemented. To avoid
the solution converging in local area, different methodologies can be used to combine
local optimization methodologies with global searches [43, 44, 45, 46]. Such algorithms
have been proved to be utilizing the advantages of the individual methods.
Most of the literature presented above focuses primarily upon the problem formulation
and use the existing methodology as an optimization tool. The author believes that
diving deeper into the implementation of the optimization algorithm can provide better
flexibility and capability to handle different constraints efficiently.
17
Chapter 3
Design choices
In this chapter, the authors discusses in detail the mechanisms used for comparison along
with the motivation behind choosing them. The initial sections present different varia-
tions, their geometric solutions as well as the nature of their singularity curves. In next
sections, the type of joints chosen and the changes occurring in passive limits because of
them is discussed. Later, the importance of workspace representation is highlighted. The
chapter concludes by detailing the procedure used to decide on the important parame-
ters of the design optimization. The aim of this chapter is to familiarise the reader with
the background behind the mechanism chosen and subsequently the components for the
same.
3.1 Proposed mechanism and its variations
Based on the requirements and cross-functional discussions, a novel mechanism was in-
troduced to cater to the current scenario. A 2-dof parallel mechanism with a passive leg
was used to achieve the 2 rotational degrees of freedom while a parallelogram joint was
introduced for a remote center of rotation [1]. The required linear motion in endoscope
is achieved by a rack and pinion arrangement as shown in Fig. 3.1c.
(a) Side view of the mechanism in
home position
(b) Mechanism with the remoter
center of rotation
(c) Decoupled lin-
ear motion
Figure 3.1: The patented proposed mechanism for the presented work [1]
The proposed architecture can be implemented in several ways. Depending on the
choice of the actuators, rotary or linear, the kinematic properties such as singularities,
number of inverse geometric solutions and joint limits change dramatically. The presented
work limits the scope to only prismatic joint as actuators to keep the analysis compara-
tively simple. Even while using a prismatic joint, a designer has to choose between the
18
Figure 3.2: The 3 variations of the proposed architecture
different joint orders, type of joints and their placements. For example, the three varia-
tions occurring from different order and placement of joints for the patented idea using
prismatic joint actuator are shown in Fig. 3.2. The 2UPS-1U variation has a prismatic
joint linking the passive universal and spherical joint of a leg while the 2PUS-1U varia-
tion has a fixed orientation for the prismatic joint. A major consequence of this is the
possibility of using bigger or heavier actuators for the 2PUS-1U variation. In this case,
we do not have to worry about the inertia of the prismatic joint as well as the collision
between the two actuators. The 2UPS-1U variation has a better kinematic performance
compared to a 2PUS-1U architecture and allows us to have a larger workspace satisfying
all the necessary constraints. These qualities of different variations motivates the need of
a comparative analysis for a conclusive proposal of the architecture.
3.2 Inverse geometry
The general parameterization of the 2UPS-1U variation can be done with the help of 13
parameters as shown in Fig. 3.3a while the 2PUS-1U variation also requires 13 parame-
ters, it considers the orientation of the prismatic joint and assumes that the z-coordinate
of the base of the actuator as zero. If we use the Euler angles (α, β) to represent the
output space as detailed further in Section 3.5.1, the orientation of the end-effector can
be given as:
R=RαRβ=
cos(β) 0 sin(β)
sin(α)sin(β)cos(α)−cos(β)sin(α)
−cos(α)sin(β)sin(α)cos(α)cos(β)
(3.1)
3.2.1 Inverse geometry: 2UPS-1U variation
If the base frame of the manipulator is considered to be at the bottom point where all
the three legs meet as shown in Fig. 3.3a, then the coordinates of the 2 universal joints
19
(a) Parameters in 2UPS-1U (b) Parameters in 2PUS-1U
Figure 3.3: The general parameterization of the 2UPS-1U and 2PUS-1U variation
with respect to the origin are represented as:
u11 =
a1cos(φ1)
a1sin(φ1)
h1
,u21 =
a2cos(φ2)
a2sin(φ2)
h2
(3.2)
The universal joint of the third passive leg, u3, is given by [0,0, t]Tand the spherical
joints of the mechanisms are represented in the frame of the universal joint u3as:
s12 =
b1cos(ψ1)
b1sin(ψ1)
h3
,s22 =
b2cos(ψ2)
b2sin(ψ2)
h4
(3.3)
Here, a1and a2are the lengths of the links that join the base with the universal joints
and b1,b2are the lengths of the links joining the end-effector with the spherical joint as
shown in Fig. 3.3a. φ1,2are the angles deciding the orientation of the universal joints
with respect to the x-axis while ψ1,2are the angles deciding the placement of the spherical
joints with the respect to the x-axis of the chosen end-effector frame. The transformation
matrix corresponding to the end-effector frame at a certain pose (α, β) is:
0Te=R t
01,t= [0,0, t]T(3.4)
0s12 =0Tes12
0s22 =0Tes22 (3.5)
As we have all the joints in the origin frame, the distance between the universal joint as
well as the spherical joint will give us the required actuator length.
ρ1=
0s12 −u11
=f(α, β) (3.6)
20
ρ2=
0s22 −u21
=g(α, β) (3.7)
differentiating (3.6) and (3.7) with respect to time gives us:
˙ρ1
˙ρ2=
∂f
∂α
∂f
∂β
∂g
∂α
∂g
∂β
˙α
˙
β(3.8)
We define the Jacobian matrix (J) as the mapping between the output velocities of the
mechanism, ( ˙x), and the input joint velocities, (˙q), and is given by:
Jx˙x =Jq˙q
˙x =J−1
xJq˙q
J=J−1
xJq(3.9)
The matrix in (3.8), is the inverse mapping of the Jacobian matrix and is denoted as J−1.
This matrix is of particular importance for singularity analysis as well as the evaluation
of the quality of motion and will be presented in next sections. The joint orientations
Figure 3.4: The initial frames for the joints in a leg
are the next important thing to consider as the joint limits are one of the important
constraint in the optimization. We choose the tilt-torsion representation for calculating
the passive joint values and the motivation for the same has been discussed in Section
3.4.1. The spherical joints were aligned such that in the initial position all the joints are
in their default position with no rotation as shown in Fig. 3.4. The initial frame for the
universal joint can be calculated by:
zu11 =
0s12 −u11
k0s12 −u11k,yu11 =−u11 ×zu11
ku11k,xu11 =yu11 ×zu11 (3.10)
21
While the initial frame for the spherical joint can be calculated by:
zs12 =−s12
k−s12k,ys12 =−t×zs12
ktk,xs12 =ys12 ×zs12 (3.11)
The azimuth angle and the tilt angle for the universal joint is calculated by equivalent
axis representation for the z-axis of current and initial frame of the universal joint. θau
is the azimuth angle while θtu is the tilt angle for the universal joint.
ω=u11zinitial ×u11zcurrent
θau =atan2(ω(2), ω(1))
θtu =acos(u11zcurrent ·u11 zinitial) (3.12)
The rotation matrix after the tilt operation for the universal joint can be given by Euler-
Rodrigues’ formula. ˜ωis the anti-symmetric matrix notation of the vector ω
Rtu =e˜ω θtu
=I+ ˜ω sin(θtu) + ˜ω2(1 −cos(θtu)) (3.13)
Rts =
cos(θtu) + ω2
x(1 −cos(θtu)) ωxωy(1 −cos(θtu)) −ωzsin(θtu )ωysin(θtu) + ωxωz(1 −cos(θtu))
ωzsin(θtu) + ωxωy(1 −cos(θtu)) cos(θtu ) + ω2
y(1 −cos(θtu)) ωzωy(1 −cos(θtu)) −ωxsin(θtu )
ωxωz(1 −cos(θtu)) −ωysin(θtu)ωxsin(θtu ) + ωyωz(1 −cos(θtu)) cos(θtu) + ω2
z(1 −cos(θtu))
(3.14)
The same method is used to calculate the azimuth and the tilt angles for the spherical
joint. The torsion angle is calculated by using the Rts (refer (3.14)) matrix and comparing
the initial y-axis of the spherical joint’s frame with the current orientation.
s12ycurrent =Rts R(z, θtorsion)s12yinitial
3.2.2 Inverse geometry: 2PUS-1U variation
The 2PUS-1U variation can be defined with 13 parameters as shown in Fig. 3.3b. The
origin frame is considered be at the junction point of all three legs. The base of the
actuators in the origin frame is given by:
c1= [c1cos(φ1), c1sin(φ1),0],c2= [c2cos(φ2), c2sin(φ2),0] (3.15)
The line of actuator, l1and l2, along which the prismatic joint will act can be given as:
l1=c1+λ1v1
where, v1= [(c1−cos(θ1)) cos(φ1),(c1−cos(θ1)) sin(φ1), sin(θ1)]T(3.16)
λ1:constant
l2=c2+λ2v2
where, v2= [(c2−cos(θ2)) cos(φ2),(c2−cos(θ2)) sin(φ2), sin(θ2)]T(3.17)
λ2:constant
The co-ordinates of the spherical joint of the leg in the end-effectors frame is represented
as:
s12 =
b1cos(ψ1)
b1sin(ψ1)
h1
,s22 =
b2cos(ψ2)
b2sin(ψ2)
h2
(3.18)
22
Figure 3.5: The 4 intersection points for showing the possible inverse geometic solutions
The universal joint lies at one of the intersection points between the sphere centered at
s12 and s22 and the actuators line of action, l1and l2. We have 4 Inverse geometic solution
as shown in Fig. 3.5 and we always check for the 1-4 solution’s feasibility. Once we have
the universal joint’s position in the origin’s frame, we can calculate the actuator lengths,
ρ1and ρ2as:
ρ1=ku11 −c1k=f(α, β) (3.19)
ρ2=ku21 −c2k=g(α, β) (3.20)
The Jacobian matrix is derived by differentiating (3.19) and (3.20) with respect to time
similar to (3.8).
3.3 Singularity analysis
The singularity curves were analyzed for all the three variations shown in Fig. 3.2. The
condition for input-output singularity in the proposed mechanism is calculated from the
following condition:
det(J) = 0 (3.21)
As we are using a motion constraint generator, we do not have a case of constraint singu-
larities. The passive universal joint will always have 4 reciprocals and so the mechanism
will never have an instantaneous dof greater than 2. The redundancy added with pas-
sive leg avoids cases of actuation singularity in the 2UPS-1U variation with basic care in
modeling. But the 2PUS-1U is susceptible to the actuation singularity if not modeled
correctly and the parameters have to be chosen carefully. One such condition is shown in
Fig. 3.6. In this case, the actuation wrench for the active legs degenerate thus losing the
control of actuators on the end-effector. In the case shown below, the end-effector will
23
have 2 dof, rotation about the axes in the plane of the reciprocal screws shown in red in
Fig. 3.6.
Figure 3.6: Actuation singularity in 2PUS-1U variation. The reciprocals to the passive
joints are also reciprocal to the actuator
The singularity curve divides the workspace into aspects which are defined as t-
connected regions or the singularity free areas of the output space. It is therefore inferred
that we can have a feasible trajectory between two points in the output space if they
belong to the same aspect. The Fig. 3.7 shows a case for 2UPS-1U variation where the
singularity curve divides the workspace in 4 different singularity-free regions. The opti-
mization problem is related to choosing the parameters such that the singularity curves
lead to a large enough desired aspect.
Figure 3.7: The workspace divided by 4 apsects. We can use only one of the aspects for
a t-connected feasible workspace.
3.4 Choosing joints
As discussed in previous sections, the revolute joints or prismatic joints can be used as an
actuator. Apart from the active joints, the passive joints in the mechanism can be chosen
in various ways too. In the presented mechanism, we have one universal joint and one
spherical joint in a leg. Interestingly, a Spherical-Spherical (SS) joint pair can be used to
replace the Universal-Spherical (US) joint pair. This adds redundancy in the leg without
affecting the resultant dof of the mechanism and has been implemented in various previ-
ous researches [47]. Also, the universal joint can be replaced with a Revolute-Revolute
24
(RR) joint and has been reported to have better kinematic performance [21].
This creates several challenges, especially while calculating the passive joint limits.
The question that needs to be addressed is, ‘How can I define passive limits if there is
no order in the rotation?’ For example, if we use 3 revolute joints as a spherical joint,
Figure 3.8: The implemented spherical joint as well as the ball-socket joint
then we know the exact order of rotation as well as the limit on the individual axis. The
assumption of order fails while implementing a ball-socket joint but we do not have to
worry as we have a constant limit in any direction. But, it is not straightforward to im-
plement limits when we have spherical joints that are not in the ball-socket arrangement.
The spherical joints in both the arrangements are shown in Fig. 3.8. In the joint shown
in left, the joint limits depends on the axis about which the joint is being rotated. In
other words, if we represent the orientation of the joint with tilt-torsion [48], then the
passive joint limits is a function expressed in terms of the azimuth angle. Because of this
reason, the orientation of the spherical joint is also an important parameter to consider,
but more importantly, ‘What do we mean by passive joint limits?’
3.4.1 Spherical joint limits
The special orthogonal group of dimension 3 (SO(3)) is a non-euclidean space defining
all the orientations in the 3d-space and is the subspace of the special Euclidean group
of dimension 3 (SE(3)). In translational space (SE(3)\SO(3)), we have established a
single definition of distance, path and the shortest distance between two points but it is
quite hard to define the same uniquely in the SO(3). The shortest distance between two
orientations is solely dependent on the representation chosen. So, the shortest distance
between two orientations will differ with the XZX convention and the YZY convention.
It is not only Euler angles but also the tilt-torsion representation and the quaternions
which have their own definition of shortest distance and the path travelled to reach from
one orientation to another. So, if the path is representation-dependent, then aren’t the
limits dependent on the representation too?
For example, in Fig. 3.8, if we represent the orientation in XZX-convention and a
particular orientation is {250,320,450}. The very same orientation will have different
values in XYX-orientation and thus it is not valid to implement limits on individual axis.
25
In tilt-torsion representation, this can be countered in case of zero torsion. The reason
being, we know the azimuth angle and the tilt-angle and they are unique for an orien-
tation. So, we can still implement the joint limits while using the spherical joint as a
replacement for the universal joint. When we have non-zero torsion angle, there are more
challenges. The path assumed to be travelled in the zero-torsion case is shown in Fig.
3.9a. It can be seen that the limit is dependent on the orientation of the azimuth axis, so
there is a range of directions that have no limits and can rotate continuously. As shown
(a) Spherical rotation without torsion (b) Spherical rotation with torsion
Figure 3.9: The different travel paths of an axis in tilt-torsion representation.
in Fig. 3.9b, we have reached the final path by a combination of two different paths
that represent the tilt-rotation and the torsion-rotation. It is important to note that this
occurs solely because of the representation in the tilt-torsion convention. We would have
3 individual paths if we would have used the Euler angles. So, the natural question to be
asked is : ‘Can we use this path followed by the axis to reach from an initial point to a
final orientation for implementing joint limits?’
A solution can be to use tilt-torsion representation in the zero-torsion case as we have
an unique path connecting two orientations. In case of torsion, we can check limits for
the final orientation as well as the intermediate point as shown in Fig. 3.9a. In this case,
we will have to consider two axis as we are rotating about an axis in a plane, for example
the xy-plane, for tilt-angle and later rotate about the axis perpendicular to the plane,
the z-axis, for the torsion angle. So, if the path respects the joint limits, we know that
there is at least one way to reach the final orientation but if the path does not respect
the passive limits, in such case we can not tell the truth for sure. Thus, the limits by
tilt-torsion representation only implies the truth about the passive joint limits.
3.5 Workspace representation
In this section, the discussion from the above section has been continued but for a different
application. The representation of the workspace is crucial while considering optimiza-
tion variables and also to visualize the feasible workspace. The different possibilities to
represent the workspace and their pros and cons are discussed in this section.
26
3.5.1 Euler angles
Figure 3.10: The different poses of the mechanism in Euler representation
As we are proposing a 2-dof orientation mechanism, we can use Euler angles without
the issue of gimbal lock. The workspace is represented in two angles, αand β, that
represent an ordered rotation about x-axis and y-axis respectively. The value of αcan
be directly linked with the roll motion of the mechanism but βis the pitch movement
in the rolled-orientation. The feasible workspace can be represented as a rectangle with
independent limits on αand β. In fig.3.10, different end-effector poses are shown to
highlight the interpretation of individual angles in the Euler-angle representation. In this
representation a point in the output space can be represented as the combination of the
ordered rotation by αand βrespectively.
R(α, β) = RαRβ=
cos(β) 0 sin(β)
sin(α)sin(β)cos(α)−cos(β)sin(α)
−cos(α)sin(β)sin(α)cos(α)cos(β)
(3.22)
3.5.2 Azimuth-tilt angles
We can use the the tilt-torsion representation and treat the workspace as a zero-torsion
case [48]. The advantage of such representation is an easier visualization of the feasible
workspace and thus the the desired workspace can be easily analyzed as shown in Fig.
3.12. A big disadvantage that was found while analyzing the mechanism with this rep-
resentation is that we encounter a singularity at the default position (α= 0, β= 0) of
the manipulator. This can be attributed to a reason that when the tilt-angle is zero, the
azimuth angle has infinite solutions for the same representation. A further study into
this will be carried out to investigate the root of the issue and mitigate the same. The
azimuth angle in the tilt-torsion representation gives the direction in which the endoscope
is going to tilt and the tilt-angle gives the measure of amount of the inclination of the
endoscope. In this representation a point in the output space can be represented as the
rotation of a vector about axis ωby θtas stated in (3.23)
27
Figure 3.11: The polar representation of the azimuth-tilt angles
Rt=e˜ω θt
=I+ ˜ω sin(θt) + ˜ω2(1 −cos(θt)) (3.23)
The expansion of the same is given in (3.14). The azimuth angle is solely derived from
the components of vector ωand is given as:
θa=atan2(ωy, ωx) (3.24)
Figure 3.12: The visualization for endoscope motion in azimuth-tilt representation
28
Figure 3.13: The comparison of the desired workspace in Euler-angles (blue) representa-
tion and the azimuth-tilt (red) representation.
In Fig. 3.11, the workspace has been presented in polar co-ordinates. It is very easy to
visualize and understand the workspace in this representation. Fig. 3.12 shows the motion
of the endoscope when the workspace is represented in the azimuth-tilt representation.
The area covered in Fig. 3.11 is nothing but the base of the cone in Fig. 3.12. The height
of the cone is just a representation of the length of endoscope but it has no meaning in
the orientation workspace.
3.6 User-centric choices
In this section, we discuss the different choices considered while proposing an optimum
design. Taking into account the application in our case, it is of prime importance that
the feedback from surgeons is analyzed in order to tweak the requirements and solutions
related to them.
We created a questionnaire in order to understand the requirements and expectations
from the mechanism by the surgeons. The questionnaire was designed in two stages. In
the first stage, few preliminary questions were asked that could relate the answers to a
desired speed of the actuator as well as their accuracy. A 3D CAD-model was designed
in order to compare the size of the mechanism with respect to the workspace of the ear
as well as the sinus as shown in Fig. 3.14. To familiarise the surgeons with the speed of
the mechanism, we prepared simulations of the movement and asked the surgeons to rate
them as fast, slow or adequate. The questionnaire also presented an option to prioritize
between four requirements; (i) speed of the mechanism, (ii) size of the mechanism, (iii)
ease of operation and (iv) multiple operation capacity. We were able to elicit interesting
conclusions and the results have been presented in Fig. 5.2.
It was helpful to get a rough idea of what surgeons perceived when they were pre-
sented with the idea of a robotic assistance. In the first phase, the number of participants
were 9 and no information regarding their level of expertise was taken into account.
29
Figure 3.14: The renderings used for size comparison in the phase 1 questionnaire
Figure 3.15: The renderings used for size comparison in the phase 2 questionnaire
30
To get better insights, we designed a questionnaire with better understanding of the
surgeons perspective. As we learned that the surgeons prioritized the ease of use over
other parameters, we implemented the System Usability Scale (SUS), the most widely
used standardized questionnaire for the assessment of perceived usability and learnabil-
ity [49]. The information regarding the expertise and years of practise with and without
endoscope was also collected. This was important as for a technology to be accepted in
the environment, it is important to measure the comfort of adapting to such mechanisms.
It also allowed us to have weighted feedback in order to design a mechanism for future
operations. The complete environment was created on the CAD model for better percep-
tion of the size of the mechanism. The rendered image for the same has been shown in
Fig. 3.15
This questionnaire was also shared with a larger group of surgeons from various regions
of France in order to get a conclusive idea on the speed, size and accuracy required for
proposing a solution. The feedbacks from these questions provides a strong foundation
for the optimization problem where the optimized solution completely depends on the
constraints which are governed by the requirements from the surgeons.
31
Chapter 4
Mechanism optimization
Before we plunge into the core essence of the optimization methodology, it is important
that we understand the motivation behind it. In mechanism design the various choices
that have to be made are:
1. The architecture of the manipulator: the variants shown in Chapter 3
2. Type of joints: different combinations of joints to achieve the same dof
3. Pose of the joints: where to place and how to place a particular joint’s frame?
Making a particular choice is non-trivial, especially because of its effect on the workspace,
kinematic solutions and size of the mechanism. Another interesting challenge is that
the same architecture can be used to perform different tasks with either kinematic or
dynamic constraints and thus have to be optimized accordingly. This chapter presents
the optimization methodology adapted for the parallel mechanisms to be used for the
otological surgery.
4.1 Elements of the optimization
To solve any optimization problem, we should ask ourselves the following questions:
•What is the goal?
•What are the constraints?
•What can we change to reach the goal?
The answer to the above questions form the elements of the optimization problem. The
importance of each element is that they affect the final goal, the computational cost as
well as the algorithm suitable for the problem. The following sections elaborate on these
elements allowing us to further understand the choice of the algorithm.
4.2 Objective function
The choice of the objective function of a mechanism completely depends on the appli-
cation of the manipulator. Some use cases require a good kinematic behaviour, for e.g:
camera manipulation, while some may demand a good dynamic characteristic, for e.g:
32
machining tasks. In this thesis, we are primarily concerned about the kinematic charac-
teristics of the manipulator as our application is similar to that of a camera manipulation.
The most important property of a manipulator is its feasible workspace and so max-
imizing workspace is a natural choice of the objective function. A detailed analysis of
the volume available for endoscope manipulation in otological as well as sinus surgery
is presented in [2]. The work presents the study of the petrous bone and the paranasal
sinuses of several patients of different age (2 to 95 years) and sex. This establishes a firm
ground for the requirement of the orientation ranges in the endoscope manipulator. Fig.
(a) Different measurements of the ear (b) Different measurements of the sinus
Figure 4.1: The models of the ear and the sinus presented in [2].. The sinus model
suggests that we need a 90 degree travel of the endoscope (±45 deg)
4.1 illustrates the models of the ear and sinus and the endoscope manipulation in the
same. It was observed that the otological surgery has a small workspace and the range
required is low. In the case of sinus surgery, the workspace is larger as the partition in
the nose is sometimes removed. Also, as the centre of rotation of the manipulator can
be changed, there are positions where the endoscope has an orientation travel of π/2
radians.
4.2.1 Multiple objectives
When we are designing a manipulator with kinematic characteristics in mind, it is im-
portant that we evaluate the quality of the motion performed. There are several quality
indices presented in the past and a brief comparison between widely used indices is pre-
sented in this section. Apart from the quality of the motion, we also concentrate on the
passive joint limits of the mechanism. This is the reason that the presented work also
considered implementing a reward strategy related to keeping the joints as far as possi-
ble from their limits. The feasible workspace and the global quality of the manipulator
are directly related in our case and thus can be implemented together with appropriate
weights. The two objective functions, the quality index and the passive joint norm, ap-
pear to be contrasting in their nature and thus a compromise has to be made in order to
choose a mechanism performing well in both the objective functions.
33
4.2.2 Multiple objectives : Workspace of the manipulator
The presented work considers a Regular Dextrous Workspace (RDW) that is a square in
output space represented by the Euler angles. It is thus implied that the roll and pitch
movements of the manipulator are of equal importance for the endoscope manipulation.
As we were interested in exploring the possibility of implementing the mechanism for
multi-purpose surgeries, the required workspace is not treated as a constraint rather it
tries to achieve a workspace larger than required (±1 radian). This allowed us to also
observe the effect of different constraints if the manipulator is to be used in different
application.
The concept of safe working zone for parallel manipulators has been introduced in
[50] where a feasible workspace is free of singularities, internal link collisions as well as
respects passive joint limits. The presented work considers only the collision of actuating
prismatic joints as the rest of the links can be redesigned to counter the resulting collision
issues, if any. The context of feasible set (F) in this literature relates to the set of all
points in the in the discretized output spaceoutput space (K) that are:
1. Non-singular
2. Respect passive joint limits
3. There is no collision between the actuators at this configuration
The desired Regular Dextrous Workspace (RDWd) in the context of the report is:
RDWd:span{(α, β)|α, β ∈[−1,1]}
4.2.3 Multiple objectives : Quality of the manipulator
Different quality indices such as the manipulability ellipsoid volume (ve) [51] and the con-
ditioning number (κ) [52] were used previously for quantifying the quality of the motion
of a manipulator. Later, different quality indices were introduced in order to mitigate the
shortcomings of these indices. One such parameter is the velocity amplification factor
[53]. This parameter has a physical interpretation and directly represents the efficiency
of motion amplification in the actuators to that of the end-effectors.
4.2.4 Measuring quality via the manipulability ellipsoid
The manipulability ellipsoid measures the volume of the reachable workspace at a par-
ticular configuration of a manipulator when the l2-norm of the input vector (qi) is less
than 1 and is calculated as shown in (4.2). This ellipsoid gives the idea of the reachable
workspace but provides no information on the ease of direction of travel. In some cases,
the ellipsoid is a sphere and thus the manipulator can travel in every direction with equal
agility. This configuration is called the isotropic configuration and is highly desirable
for uniform performance of the manipulator. The manipulability ellipsoid quantifies a
sphere and an ellipsoid of same volume as equally good but we know that the isotropic
configuration is preferable. The linear mapping between the qiand generalized output
vector (x) is given by the Jas:
˙x =J ˙qi(4.1)
34
ve=pdet(JJT) (4.2)
Figure 4.2: The change in plot while expressing the same constraint in different spaces
4.2.5 Measuring quality via the conditioning number
To counter the shortcomings of the manipulability ellipsoid, the conditioning number (κ)
was introduced in [52] to quantify the quality of motion. The condition number is defined
as the value of the asymptotic worst-case relative change in output for a relative change
in input and is used to measure how sensitive a function is to changes in the input. It
is geometrically interpreted as the eccentricity of the ellipsoid giving information about
the ease of travel in a direction relative to the others. When the κis equal to 1, we have
a sphere and it is the isotropic configuration. The conditioning number (κ) can also be
interpreted as an index that tells us how far the current configuration is from the isotropic
configuration. The value of κranges from 1 to ∞and so its inverse is used as a quality
index (κ−1) for bounded values and is calculated in (4.3), where σis the singular value
of the jacobian matrix, J.
κ−1=σmin
σmax
, κ−1∈[0,1] (4.3)
Both the quality indices, the manipulability ellipsoid and the conditioning number,
suffer from dimensional inhomogeneity of the Jacobian matrix and are not suitable for
manipulators with both translational and rotational movements [54]. This is an impor-
tant issue to consider while implementing the proposed optimization methodology for a
general manipulator. The thesis was focused more on the optimization algorithm and its
implementation to a specific application. As the manipulator proposed in this work has
only 2 rotational dof, we decided to use the inverse of conditioning number as the quality
index. As we wanted a global quality index (κ−1
g), the mean of summation of the values
of κ−1over the discretized workspace point was used.
κ−1
g:=
W
P
1
κ−1
W, W : total workspace points (4.4)
35
4.2.6 Measuring quality via the velocity amplification factor
The velocity amplification factor (vaf) was introduced in [53] and has been used in mea-
suring the quality of motion in the past [55, 28]. The vaf allows us to analyze the relation
between the actuator speeds with the velocities of end-effector at a certain configuration.
The vaf can be bounded within desirable range according to the chosen actuators. One of
the main advantages of vaf over the previously mention quality indices is that the bounds
are very practical in nature and the performance of the manipulator can be analyzed with
physical interpretation. As we plan to use off the shelf actuators, it is very important
to have realistic bounds on the velocity of the actuators. Velocity amplification factor
helps us to measure the quality of the end-effector motion with respect to the speeds of
individual actuators.
In previous works, the vaf is calculated by assuming the l2-norm of the qito be
less than 1. It allows us to get a square matrix JJTand we can further analyse this
matrix with different mathematical tools. This choice of input vector is strange as we
make the input velocity of a particular joint dependent on the other input joints [56]. A
better condition would be to implement an independent constraints on the actuated joint
velocities and thus implementing the ∞-norm.
Figure 4.3: The geometric interpretation with the assumption of k˙qk ≤ 1
Figure 4.4: The geometric interpretation with the independent constraints
Let k1and k2be the maximum velocity of the individual actuators. The formulation
36
of the vaf is given as:
|˙ρ| ≤ k1
k2
|J−1˙
x| ≤ k1
k2(4.5)
|J11 ˙α+J12 ˙
β| ≤ k1(4.6)
|J21 ˙α+J22 ˙
β| ≤ k2(4.7)
We obtain four lines from (4.6, 4.7) which form two origin-centered rectangles as shown
in Fig. 4.4. The intersection of these two rectangles (the pink area in Fig. 4.4) is the
valid region in the output space that respects the limits of the actuator velocities. We
calculate the radius of the incircle as well as the circumcircle of this area to obtain the
minimum as well as the maximum amplification of the velocities. These distances are
used as bounded constraints and the quality of the manipulator is the ratio of the inradius
to the circumradius.
As we are using a bounded constraint on the inradius as well as the circumradius, it is
important to understand their properties. It was observed that the inradius as well as the
circumradius amplifies or shrinks upon scaling of the dimensions, i.e. if the manipulator
is scaled to have double size, the velocity amplification increases too. This can be a
problem as neither the workspace nor the quality of motion should change upon scaling
for manipulators that have pure rotational dof. This problem occurs if we use constant
limits for k1and k2as the bigger actuators have more stroke length but we are limiting
their maximum travel speed equal to the actuator with half of its stroke length. In our
implementation, we are choosing LA22 actuators from LINAK as shown in Fig. 4.19.
The maximum speed of this actuator is 37 mm/sec while the stroke length is 200 mm.
We use the same ratio for our optimization problem. If the scale of the manipulator is
such that the stroke of the actuator is smm, then the limits k1and k2are assigned as:
k1=k2=k=s37
200 (4.8)
The comparison of the velocity amplification factor with the conditioning number is
presented in Fig. 4.5. It is interesting to observe that the singular regions are not exactly
reflected in vaf. This is attributed to the reason that we derive the inverse jacobian
matrix, J−1, and do not synthesize and are not concerned with its eigen values or the
determinant. This means that even when the matrix is not invertible, there exists a stable
one way mapping between the input and output spaces.
4.2.7 Multiple objectives : Joint limits norm
After we have a good desirable workspace with acceptable quality, we can also try to keep
the joints away from the limits. This objective function was implemented to explore the
areas where simple joints with limited motion can be used. It is not straightforward to
define the distance of a joint orientation from its limits for a spherical joint. We have
proposed the projection of the rotated vector onto a plane to define the distance from
its limits as shown in Fig. 4.6. It is not the accurate presentation of the distance from
limits but a detailed analysis would require time and more mathematical analysis into
37
(a) Heatmap for the conditioning number
of chosen parameters
(b) Heatmap for the VAF of chosen param-
eters
Figure 4.5: The comparison between the conditioning number and the velocity amplifi-
cation factor
orientation space and defining the ’essence’ of travel, distance and shortest distance in
orientation. As we are implementing the same spherical joint, the distances were not
weighted and summation of the distances was used as the joint limit norm at a particular
configuration (qpi). The summation of qpi over the workspace is used as an evaluation
while maximizing the distance from the joint limits.
kqpik:=
4
X
n=1
(llimit −lcurrent )n, n :no.ofjoints (4.9)
kqpk:=
W
X
i=1 kqpik(4.10)
Figure 4.6: Illustration of the joint limit distance
4.2.8 Summary of the used objectives
So, to summarise the objective functions implemented in the presented methodology, we
are looking to:
1. Maximize the feasible set in the desired regular dextrous workspace
38
2. Maximize the quality of motion in the feasible workspace.
3. Maximize the distance of the passive joints from their respective limits
Mathematically denoting,
Objective functions :max F\RDWd(4.11)
max κ−1
g(4.12)
max (kqpk) (4.13)
Figure 4.7: Summary for the different objective functions
4.3 Constraints
Parallel Kinematic Manipulators (PKMs) have 2 distinct features from a serial chain as
they have:
1. Passive joints whose orientation can be calculated but not controlled explicitly.
2. Multiple legs - serial chains connecting the end-effector with the base.
These two points are of great importance as they affect the workspace of the manipulator.
So, we take note that the passive joint limits and avoiding internal collisions among
different legs of the Parallel Kinematic Manipulator (PKM) are two important constraints
to be implemented in our optimization problem. The Kspace of the mechanism is
separated by the singularity curves thus resulting into several connected regions also
called as the aspects [57]. As we cannot travel from one aspect to another, it is important
that the desired RDW (RDWd) lies in a single aspect. Fig. 4.8a illustrates a valid set of
parameters while Fig. 4.8b corresponds to a non-useful architecture for our application.
If the box in the Fig. 4.8 consists of multiple colors, then it is evident that there are
more than one aspect and thus we cannot travel to every configuration in the RDW.
4.3.1 Feasible actuator range
Another important constraint while designing is of the actuator which is represented
as the active joint ranges. This work specifically focuses on the range of the prismatic
joint to be chosen for maximizing the points in FTRDWd. Generally, a prismatic joint
is expressed as a constraint with a certain minimum and maximum range and with a
constraint on the ratio of the lengths in completely acuated state and its default length:
min ≤ρ≤max (4.14)
39
(a) RDW in one connected region (aspect),
a valid set of parameters
(b) RDW is intersected by the singularity
curves, an invalid set of parameters
Figure 4.8: The aspects of the Kspace in different cases
Figure 4.9: The industrial prismatic joints and the relation between ρmin and ρmax
Source: Hanpose linear actuator HPV5 SFU1204, www.pngegg.com/en/png-mckkp
ρmax ≤stroke ×ρmin, stroke ∈[1,2] (4.15)
Eq. 4.15 comes from the physical build of general prismatic joints. If the unextended
length of the actuator is ρmin, then it is not practical for common prismatic joints to
extend beyond their original length (ρmax <2.ρmin) as explained in Fig. 4.9. The novelty
in expression of the actuator range in our work is that we do not have a static value as
a limit as mentioned in (4.14), i.e, we express the constraint only in terms of the stroke
ratio expressed in (4.15). This allows us to choose the best actuator ranges to maximize
the feasible workspace without putting any constraint on the minimum or maximum size
of the prismatic joint. This is best illustrated in the Fig. 4.10 and Fig. 4.11 and the
implementation is detailed further in Section 4.5.4
4.4 Optimizing variables
Optimizing variables are the parameters that are changed in order to optimize the objec-
tive function. In a mechanism, the position or the orientation or both (pose) of the joints
in the kinematic chain are the design parameters that are to be optimized. The number
of variables form the dimension of the optimization problem which governs the size of
the search space (also denoted as O) and so the computational capacity. In deciding the
optimizing variables, we have to consider the following options:
40
(a) Different search brackets within the ac-
tuator space (I)
(b) Comparison of the size of the search
brackets showing their variable nature
Figure 4.10: The varying search brackets for range of actuators.. The blue dots correspond
to the pair of lengths of actuators for a configuration in RDW
(a) Feasible workspace
(white) when red bracket in
Fig. 4.10a is implemented
(b) Feasible workspace
(white) when black bracket
in Fig. 4.10a is implemented
(c) Feasible workspace
(white), magenta bracket in
fig.4.10a is implemented
Figure 4.11: Comparison of feasible workspace (white space) within the RDWdfor dif-
ferent search brackets.. The red and blue part represent the violation due to actuator
lengths of leg1 and leg2 respectively.
•Use a general approach, we have 3 variables x, y, z for each joint (Fig. 4.12a).
•Use human intuition to fix certain variables in order to reduce the search space.
Each of the above-mentioned options have some advantages and disadvantages. If
we use a general approach to optimize the position of the joints, then we can produce
some results that are counter-intuitive but efficient. This comes at a cost of dimensional
blowup as there are many joints in a parallel mechanism. For e.g, in the mechanism
architecture proposed in this work, we have two legs with 6 dof and a leg with 2 dof.
In a case where we use a combination of revolute joints to form a universal joint and
a spherical joint, we can have as much as 14 joints (6 dof ×2 legs + 2 dof ×1 leg).
This results into an optimization space of dimension 42 which is a very large space given
the nature of constraints and the time required to compute the objective function for
a particular configuration. In the other option, we can seriously reduce the dimension
for Oby using human intuition and trying to bring careful symmetry (avoiding obvious
singularities) in the mechanism. For example, in the 2UPS-1U variant of the proposed
architecture, we can reduce the optimization space to dimension 4 as shown in Fig. 4.12b.
41
The choice of fixing some variables comes at a cost that we may miss some configu-
rations that are better performing but it has several advantages. Thanks to the human
intuition, we use links of same length and try to optimize the mechanism by assuming
the leg1 and leg2 are identical from the manufacturing and assembling point of view. So,
as every designer seeks in every project he does, we can certainly explore the possibility
of a compromise between these 2 choices.
(a) Mechanism with 13 optimization vari-
ables shown in red (b) Mechanism reduced to dimension-4 by
human intuition (variables in red)
Figure 4.12: The different definitions of the same architecture . varying the number of
parameters. In 13 (left) parameters, we do not fix any variable for the position of the joints
while in 4 parameters (right), we assume equal lengths, symmetry and perpendicular legs.
42
Figure 4.13: The range of optimizing variables. The lower and upper joints of each leg
can be place in their respective cylindrical enclosures shown in blue and green.
Fig. 4.13 shows the space in which the universal joints and the spherical joints can lie
in. The arc angle of the span is 200 degrees with a minimum and maximum range. The
height of the span decides the range of the z-coordinate of the joints. It is to be noted
that the blue span is with respect to the ground while the green cylindrical arc is with
respect to the universal joint of the third passive leg. When the dimension is reduced to
4 as shown in Fig. 4.12b, the 2 cylindrical arc get reduced to 4 squares in 2 perpendicular
planes.
Figure 4.14: Summary for different constraints in the optimization process
4.5 Local search algorithm
The Nelder-Mead algorithm is a derivative-free optimization algorithm proposed by John
Nelder and Roger Mead [58] in 1965. It is also called the downhill-simplex algorithm as
it uses simplexes to search the space locally. In this section, we present the algorithm for
asingle start which searches in the local vicinity of the initial simplex. Later, we discuss
the implementation of the algorithm in mechanism optimization with different rewarding
43
strategies and detail the method for extracting the best actuator ranges from the solu-
tion. We conclude the section with a summary of the algorithm with its implementation
highlighting few strengths and weaknesses of the same.
4.5.1 The Nelder-Mead (NM) Algorithm
For a n-dimensional O, we require a simplex of at least n+1 points in Oto avoid prema-
ture convergence. This can be explained with a simple graphics for 2-dimensional Oas
show in Fig. 4.16.
Figure 4.15: An example of mapping with 2 optimization variables
(a) The traversing in O-space with 2 point
simplex, we can explore the points on the
line only
(b) The traversing in O-space with 3 point
simplex, we can travel in both directions
exploring the complete space
Figure 4.16: The premature convergence while using a simplex of less than (n+1) points
in n-dimensional O-space.
In our implementation of the algorithm, we start with a sorted simplex of n+1 points
(v0,v1, ...vn) such that the objective function evaluated of the ith vertex has a value
44
better than or equal to that of (i+ 1)th vertex. A mean point (vm) is calculated by
excluding the worst point.
vm:=
n−1
P
i=0
vi
n(4.16)
The optimization algorithm then compares the mean point and searches for better points
by geometrical operations termed as (i) reflection, (ii) expansion, (iii) contraction and
(iv) shrinkage. These operations can be explained as follows:
•Reflection:
vreflect := vm+r(vm−vn), r := reflection coefficient (r > 0) (4.17)
•Expansion:
vexpand := vm+e(vreflect −vm), e := expansion coefficient (e > 1) (4.18)
•Outside contraction:
voc := vm+k(vm−vn), k := contraction coefficient (0 < k < r) (4.19)
•Inside contraction:
vic := vm−k(vm−vn), k := contraction coefficient (4.20)
•Shrinkage:
∀i∈[1, n]vi=s .vi, s := shrinkage factor (0 <s<1) (4.21)
The new point (vnew) introduced in the simplex depends on the evaluation of the vreflect ,
vexpand,voc and vic. The operation is continued till the stopping criteria is reached. The
simplex stops if it shrinks below a certain value 1and the evaluations of every vertex of
the shrunk simplex vary by maximum threshold 2. The stopping criteria is presented in
Algorithm 1 and the complete procedure for one start of the NM-algorithm is given in
Algorithm 2. An example of the operations in a 2-dimension Ois illustrated in Fig. 4.17
to present the geometrical nature of search of the Oin NM-algorithm. In Fig. 4.18, an
example of the points explored during an optimization process is graphically represented.
45
Figure 4.17: An example of an operation on a simplex in 2-dimensional O
Algorithm 1: Stopping criteria for the NM algorithm
Result: Boolean for stopping condition
1sorted simplex {v0,v1,v2, ..., vn−1,vn};
2evaluations {e0, e1, e2, ..., en−1, en};
3maximum iteration max iter iteration count iter;
4lij =kvj−vik;
5eij =|ei-ej|;
6size = max(lij);
7eval = max(eij);
8if size ≤1&& eval ≤2then
9stop = 1
10 else
11 stop = 0
12 end
13 if iter ≥max iter then
14 stop = 1
15 else
16 stop = 0
17 end
4.5.2 Implementation
We will see an example for optimization of 2UPS-1U mechanism with 4 variables to opti-
mize as shown in Fig. 4.12b and also mention the method to optimize other mechanisms
with the algorithm. As discussed in Section 4.2, we want to calculate an evaluation for a
mechanism that circumpasses information related to the feasible workspace, FTRDWd,
along with the global quality, κ−1
g, or the passive joint limit norm kqpk. As we know
that the κ−1
gand kqpkare conflicting objectives, we design different rewarding functions
for each of them.
4.5.3 Rewarding strategies
Rewarding strategies refer to the different methods implemented to calculate the eval-
uation for a particular objective function. For example, to optimize the quality of the
46
Algorithm 2: Single start of the Nelder-Mead optimization algorithm
Result: Local minimum evaluation and the optimized parameters
1initial sorted simplex {v0,v1,v2, ..., vn−1,vn};
2evaluations {e0, e1, e2, ..., en−1, en};
3while stop == 0 do
4calculate vm,vreflect and eref lect;
5if (en< eref lect < e0)then
6vn=vreflect;
7else if (e0< ereflect)then
8if (eref lect < eexpand)then
9vn=vexpand;
10 else
11 vn=vreflect;
12 end
13 else if (en< ereflect < en−1)then
14 if (eoc > eref lect)then
15 vn=voc;
16 else
17 ∀i∈[1, n]vi=s.vi;
18 end
19 else if (ereflect > en)then
20 if (eic > eref lect)then
21 vn=vic;
22 else
23 ∀i∈[1, n]vi=s.vi;
24 end
25 sort the simplex;
26 if v0new >v0then
27 iter = 0
28 else
29 iter = iter + 1
30 end
31 Update stop from Algorithm 1
32 end
33 return v0, e0
47
Figure 4.18: An example of the travel path of optimization in Nelder-Mead algorithm
constraint, we can bias the reward towards the center. This results in mechanisms having
very good motion in and around the center of the desired Regular Dextrous Workspace
(RDWd). Different strategies can be used to evaluate the objective leading to desirable
mechanism parameters. We discretize the RDWdby dividing the range of each degree of
freedom into 201 points (2/201 radians division) resulting into equispaced 40401 points
to calculate the necessary evaluation at every configuration of the workspace.
For a given set of optimizing variables [a, a0, h, t] from Fig. 4.12b, we want to know
how good is the mechanism. We solve for the IGS of the mechanism at each configuration,
(α, β)∈RDWd, and get the complete configuration space (D). This provides us with
all the passive joint values and the actuated length for both legs. The same function is
also used to derive Jand its determinant. As we are also checking for collision between
actuators, a function has been written to accommodate the collision check. This function
is denoted as f(v) in Algorithm 3. The actuators considered for the mechanism has a
wider bore diameter at the bottom and is smaller at the top as shown in Fig. 4.19.
So, the actuator was divided in 5 points and the distance between these discretized
points was calculated. Knowing that the inner piston is smaller than the outer radius for
the actuator, we also reduced the constraint for the distance between points that belong
to the inner piston. For example, the distance used as a collision constraint for bottom
4 points in Fig. 4.19 is different than that for the 5th point as it lies on the inner piston
and thus is smaller in size. This adjustment in collision constraint though seemingly
small, has a huge impact on the evaluation depending on how the rewarding strategies
are defined.
Rewarding strategies : Only feasible workspace
As we explore the possibility of having a large workspace with the proposed mechanism,
it was decided to first see the results for maximizing the feasible workspace only. This is
done by the binary reward strategy. Each configuration is awarded either 1 or 0 depending
48
Figure 4.19: The actuator considered for the mechanism and the discretized points for
collision calculation (right)
upon the point respecting the passive joint limits and the actuator limits only as shown in
Fig. 4.20. The constraints for non-singular points and collision are treated more strictly.
If any configuration in the RDWdis singular or does not respect the collision constraint,
then the evaluation is given a very large penalty. This makes sure that no matter how big
the feasible workspace is, it is disqualified as a valid solution if it contains any singular
curve or colliding configurations. We can also bias the reward to have better kinematic
performances in and around the center of the feasible workspace. One of such ideas is
illustrated in Fig. 4.21
Figure 4.20: An illustration for binary reward of the feasible workspace
Rewarding strategies: minimum quality
As we are using a globality quality to quantify the ability of the manipulator to move
along any direction, it is understandable that we want to maximize the same. We want to
have a manipulator which has the capability to move in any direction with equal agility
in any configuration of the feasible workspace. This is a goal not easy to reach, in fact,
we already know that the workspace is mainly bound by the singularity curves and as
we go near singular configurations, the Jtends to lose at least one rank. As the global
49
Figure 4.21: An illustration for biased reward of the feasible workspace. The feasible
points near the center of the feasible workspace is given higher weightage
Figure 4.22: The minimum quality constraint. The left figure is the heatmap for the
quality index of a particular architecture of the mechanism. The right figure is a zoomed
view. The dashed white box in right image shows the area with a minimum quality
constraint.
quality, κ−1or the velocity amplification factor, depends on the singular values of J, it
becomes harder to have dextrous mobility near singular boundaries as shown in (4.2)
and (4.3). So, a constraint is necessary to place a minimum value of the quality index
in a particular workspace that will be used very often. This also allows us a buffer to
controllably stop the manipulator if it shoots away from this prescribed working area. We
bound the the quality with acceptable ranges in the 64% of the RDWd. So, the quality
of the manipulator is treated as a constraint in this box and rewarded with either binary
or biased reward outside the inner box. The concept is graphically represented in Fig.
4.22.
Rewarding strategies : Maximize quality or joint limit norm
In order to combine another objective function with the feasible workspace, we propose to
change the binary reward strategy to κ−1
gor kqpk. If we want to maximize the quality, then
the summation of inverse of conditioning number, κ−1at each configuration is treated as
the reward for the mechanism with given parameters. Otherwise, the joint limit norm
as detailed in Section 4.2 is used as an evaluation. In the joint limit, the weighted norm
can also be implemented if joints with different limits are implemented. In our case, we
50
are proposing to implement the same spherical joint for all cases and thus implemented
a non-weighted Manhattan (l1) norm.
4.5.4 Implementing best actuator range
We have to choose a physically possible range of actuators to assure that the points
deemed as feasible (mathematically) also respect the actuator lengths and their physical
limits. The constraints for actuators are shown in (4.15) and illustrated in Fig. 4.9 and
4.10. To implement the same, we recheck the recorded valid points in the RDWdfor
the different brackets of the actuator range. When we are maximizing the global quality
or the joint limit norm, we also ensure that the chosen bracket (refer to Fig. 4.10a)
has at least 90% of the RDWd. For example, if a particular bracket is chosen that has
higher global quality (meaning that there is a particular zone that is super good), but if
that bracket does not have at least 90% of the discretized 40401 points, then we ignore
the quality and choose the bracket with maximum valid points as detailed in Algorithm 4.
As the search is geometric, different rewarding functions were tested to check the
efficiency of the algorithm. The choice of rewarding function affects the convergence of
the algorithm. For example, the implementation in the next Algorithm 3 penalises the
mechanism with negative infinite value if even one point is near singularity. It is defined
as the the infinite penalty in the context of this report. It is hard to distinguish between
how close or far we are from a region free of singularity with such rewarding function.
Later, we change the penalty function to take into account this issue. If a singular point
is detected in RDWd, we count the number of points near singularity and completely
deny any reward for the feasible points in this case. Both these examples are illustrated
in Fig. 4.23 and Fig. 4.24.
Figure 4.23: The evaluations for infinite penalty. The different colors represent singularity
boundaries for different parameters. The red box is the RDWd. The colored dot on the
right side represent the evaluation corresponding to their colored singularity curve.
4.5.5 Pros and cons of the NM-algorithm
The Nelder-Mead algorithm is quite straightforward to model the optimization problem
for mechanism design. This advantage allows us to have a general methodology for op-
51
Algorithm 3: Implemented rewarding function for workspace only
Result: evaluation e for a given parameter set v
1input →v;
2for α=−1:0.01 : 1 do
3for β=−1:0.01 : 1 do
4f(v) := function that solves IGS, collision distance and κ−1;
5[det(J), passive joints, ρ1,ρ2,κ−1, collision distance] = f(v);
6if |det(J)| ≤ 3then
7e = -∞;
8break;
9else
10 reward = 1
11 end
12 if kqpk ≥ limits then
13 reward = 0
14 else
15 reward = 1
16 end
17 if collision distance ≥limits then
18 reward = 0
19 else
20 reward = 1
21 end
22 if reward == 1 then
23 e = e + reward;
24 valid points[i] = [ρ1,ρ2, reward]
25 ρvec[i] = [ρ1,ρ2];
26 end
27 end
28 rho range = [maximum(ρvec), minimum(ρvec)];
29 Checking for feasible set of the actuators;
30 if stroke ×rho range[1] ≥rho range[2] then
31 for rho bracket = ρmin : steps : ρmax do
32 for n = 1 : length(valid points) do
33 e = 0;
34 if ρ1,ρ2≤rho bracket || ρ1,ρ2≥stroke ×rho bracket then
35 feasible points[j] = valid points[n];
36 j = j+1;
37 e = e + valid points[n, 3];
38 end
39 eval vector[k]= [e, ρmin];
40 k = k + 1;
41 end
42 e = maximum(eval vector, 1);
43 rho range = ρmincorresponding to e;
44 return e
52
Algorithm 4: Implementation for choosing best actuator range considering
workspace and quality
Result: evaluation for a given parameter set v
1input →v;
2get the matrix of valid points from Algorithm 3 rho range = [maximum(ρvec),
minimum(ρvec)];
3Checking for feasible set of the actuators;
4if stroke ×rho range[1] ≥rho range[2] then
5for rho lower = ρmin : steps : ρmax do
6for n = 1 : length(valid points) do
7e = 0;
8if ρ1,ρ2≤rho lower || ρ1,ρ2≥stroke ×rho lower then
9feasible points[j] = valid points[n];
10 j = j+1;
11 e = e + valid points[n, 3];
12 end
13 eval vector[k]= [e, ρmin, j];
14 k = k + 1;
15 end
16 [e1,index1] = maximum(eval vector, 1);
17 [e2,index2] = maximum(eval vector, 3);
18 if eval vector[index1][3] ≤0.9 ×40401 then
19 e = eval vector[index2][1];
20 ρmin =eval vector[index2][2];
21 else
22 e = eval vector[index1][1];
23 ρmin =eval vector[index1][2];
24 end
25 rho range = [ρmin, stroke.ρmin ];
26 return e
timizing any parallel mechanism. As it is a derivative-free algorithm, we can introduce
complex objective functions that are hard to formalise. An example is the quality index,
κ−1
g, defined in Section 4.3. Also, as NM-algorithm is a local search algorithm, it returns
a stationary point (no proof for convergence) in a considerably low time compared to the
currently implemented global optimization methodologies. This makes it possible for the
designer to structure an objective function that is computationally expensive. Also, the
constraints can be constructed modularly allowing us to experiment different constraints
at any stage of the development. Another important advantage of the Nelder-Mead al-
gorithm relevant to the mechanism design is its geometric search method. The basis
of optimization space in NM-algorithm are the optimization variables themselves. It is
logical to use this method as the next best design parameters are chosen as a result of
the combination of parameters of previous simplex rather than using complex methods
to represent a mechanism in the optimization space which may not have geometrical
explanation for choosing the next best proposal, e.g: chromosomes in the GA. We can
also tune the exploring parameters, the reflection, expansion, contraction and shrinkage
coefficients, with human intuition and some prior knowledge about the importance of
53
Figure 4.24: The evaluations for smooth penalty. The different colors represent singularity
boundaries for different parameters. The red box is the RDWd. The colored dot on the
right side represent the evaluation corresponding to their colored singularity curve.
different parameters.
Though suitable for our application, there are certain disadvantages of using this al-
gorithm too. Under some hypotheses, the NM algorithm has proof of convergence up to
dimension 2 [59] and has no proof for convergence beyond 2-dimensional optimization.
If not implemented correctly, it gets into a collapsing simplex patterns thus converging
onto a non-stationary solution [60]. The convergence highly depends on the initial size
of the simplex and the effect of the coefficients have been discussed in [61]. Despite
these shortcomings, the NM-algorithm is useful in our case as the aim is not finding
the absolute optimized design parameter but to satisfy all constraints and then get an
acceptable quality of performance and has been implemented in such areas with great
success [45, 46]. Different convergent variants have also been proposed to get around the
premature convergence [62] allowing the algorithm to explore extra points in case of near
collapse.
To get better results, we complement the local search of the NM algorithm with few
variants and with a multi-start technique for a global search of the optimisation space,
as discussed in the next section.
54
mented a multi-start Nelder Mead algorithm with low discrepancy points for exploring
a global optimization space. In this method, we execute the NM algorithm with differ-
ent initial simplexes. It is very important that we have a uniformly distributed initial
simplexes over the optimization space in order to explore the maximum area of the opti-
mization space.
4.6.1 Initial simplexes for multi-start
An easily implementable way to obtain a sampling set OM⊂ O is Monte Carlo sampling
with a uniform distribution (see, e.g., [64]), i.e., random sampling. Unfortunately, it
is known [65] that the resulting points have the tendency to form clusters, particularly
in high-dimensional contexts, which undermine the uniformity of the discretization. A
better choice consists in having the Mpoints of the discretization OMof Ospread “well-
uniformly”. In particular, it is desirable the points to be close enough to one another,
without leaving space regions undersampled. To this end, as done in [66, 3, 67, 68], one
can use certain deterministic sampling techniques. The discussion of the properties of
such techniques contained in the remaining of this section is from [66].
Dispersion and discrepancy
Let OM⊂ O be a set of Msample points oi,i= 1, . . . , M and define the dispersion
[65] of OMas
θ(OM) := max
o∈O min
˜o∈OMko−˜ok(4.22)
It is clear from the definition that dispersion quantifies “how uniformly” the points are
spread in the space: it is a measure of the uniformity of the distribution of the points of
OMin O. Roughly speaking, a small value of θ(OM) guarantees that the points of OM
are spread on O“in a uniform way,” i.e., without leaving regions of the space “under-
sampled,” and by letting the points “close enough” to one another. The best dispersion
properties, in the sense described above, belong to special deterministic sequences called
low-discrepancy sequence, commonly employed in the fields of number-theoretic meth-
ods, statistics, and quasi-Monte Carlo integration. To introduce them, one needs the
concept of discrepancy [65, 69] to measure the uniformity of a set of points in a com-
pact domain. In the following we discuss results that are referred to the n-dimensional
unit cube In:= [0,1]n, but other compact sets can be considered by performing suitable
transformations.
Let OMbe a set of Mpoints in In. Define Pas the family of all subintervals I
of the form ×n
i=1[o1,i, o2,i], where o1,i, o2,i ∈[0,1], o1,i < o2,i, and let c(I, OM) be the
counting function, which counts the number of points of OMin I(i.e., c(I, OM) is the
number of points of OMthat belong to I). The discrepancy of OMis defined as [69]
D(OM) := sup
I∈P
c(I, OM)
M−λ(I)
,
where λ(I) is the Lebesgue measure of I.
Discrepancy is strictly related to dispersion. In particular, it can be shown [69] that, for
any set of Mpoints on In, the inequalities
(bnM)−1/n ≤θ(OM)≤√nD(OM)1/n
56
hold, where bnis the volume of the n-dimensional unit ball. Then, a “low-discrepancy
set” is also a “low-dispersion set”.
Low-discrepancy sequences
Sequences such that their discrepancy satisfies
D({OM})≤O(M−1(log M)n) (4.23)
are called low-discrepancy sequences. Their construction varies from method to method.
In [69], a common framework for the construction certain sequences, called (t, n)-sequences,
that satisfy deterministically the condition (4.23), was presented. Other examples are the
good lattice points sequence the Halton sequence, and the Hammersley sequence [69, 65].
The use of such low-discrepancy sequences is at the basis of the so-called quasi-Monte
Carlo methods [69], originally introduced as an efficient alternative to classic Monte Carlo
methods for the numerical computation of integrals.
To sum up: on the basis of the discussion above, an effective procedure to gener-
ate uniformly scattered deterministic sets of points consists in taking finite portions of
low-discrepancy sequences. They attain deterministically a rate of convergence for the
dispersion of order
OM−1/2n≤θ(SM)≤O√2nM−1/2n
Fig. 4.26 shows the comparison between a sampling of the 2-dimensional unit cube by a
sequence of 500 points i.i.d. according to the uniform distribution and by a sampling of
the same cube obtained via a low-discrepancy sequence (in this case, the Sobol sequence
[70]). It can be clearly seen how the space is better covered by the second sequence,
as well as how the largest empty spaces among the points appear in the first sampling
scheme.
Figure 4.26: Comparison between random and low-discrepancy samplings of the 2-
dimensional unit cube from [3]
57
4.7 Coarse and fine searches
In a normal execution of the Nelder-Mead algorithm, the iteration stops either when the
simplex has shrunk to a desirable size with near to same evaluations or if we have encoun-
tered the same best point for preset allowable maximum iterations, refer to Algorithm
1. In an attempt to decrease the time for local convergence allowing us to explore more
initial simplexes, we adapt a methodology inspired from the practise of rough and fine
turning in lathe machines. In general, when we want to remove the excess stock from the
workpiece as rapidly as possible, we increase the feed rate and do not focus on the finish
of the work. Later, when we are close to desired dimensions, the feed is decreased and
now the focus is shifted on the finishing of the work.
In our case, the idea is to initialise simplexes in multi-start and perform a coarse search
for the local optima. Later, we collect the local optima from all the used simplexes and
then implement a stricter stopping criteria on few selected local optima allowing them to
converge to a stationary point with finer quality. Fundamentally, we are discarding the
local optima that do not promise a good evaluation even after a longer search bringing
down the implementation time considerably. Also, as we already have an optimized vertex
as an initial simplex, we can build the rest of the vertices as per our choice thus controlling
the size of initial simplex. One of such implementation is detailed in Algorithm 5, where
the condition of incrementing the iteration is changed. We implement a condition that the
new evaluation found is better than the previous only if it exceeds the previous evaluation
by 1%.
Algorithm 5: An example of coarse local search criteria
Result: The iteration count, refer to Algorithm 2
1input : Previous and current best evaluations {v0,v0new,};
2For coarse search;
3max iter = 3 n;
4margin = 1.01 ... (suggesting 1% increment);
5For fine search;
6max iter = 10 n, margin = 1;
7if v0new > margin v0then
8iter = 0
9else
10 iter = iter + 1
11 end
12 if iter >max iter then
13 return v0new;
14 break;
15 return iter
58
Figure 4.27: The rough and fine turning in lathes. The motivation for implementing a
faster and efficient global search
Figure 4.28: The flowchart for the complete implemented optimization methodology
59
Algorithm 6: An example of implemented multi-start optimization
Result: The fine optimized result of the mechanism
1Assuming we have ‘m’ starts for a ‘n’ dimensional optimization problem;
2We need m.(n+1) valid n-dimensional points from the sobol set generated;
3later, we choose ‘k’ local optima for further fine search;
4for start = 1:m do
5Initial simplex = {v(m−1).(n+1)...vmn+m−1};
6Implement Single start NM-algorithm from Algorithm 2 with coarse search in
Algorithm 5;
7save local optima and evaluation;
8end
9for fine start = 1:k do
10 Generate n simplexes around the chosen local optima;
11 end
12 return Best point
4.8 Summary
In this chapter we present different objective functions that can be used for an optimized
result. The consequent sections discuss the different constraints such as passive joint lim-
its, actuator feasibility, internal collision as well as the non-singular condition. We also
discuss the methodology to choose the best actuator stroke for maximizing the feasible
workspace. Later, the various rewarding strategies are presented and their relevance in
certain situations is also discussed.
The existing Nelder-Mead algorithm is detailed to highlight the geometric nature of
search of the optimization space to motivate its relevance in the design optimization. It
is followed by the algorithms with implementation for different constraints and rewarding
strategies. In the last section, we present a novel implementation for changing nature
of search within the Nelder-Mead algorithm for faster and efficient global search of the
optimization space. Fig. 4.25 shows the flowchart for the implemented optimization
methodology. The figure also illustrates the different choices available and various deci-
sions needed to make for a complete definition of the optimization problem. In Fig. 4.28,
the complete process is illustrated. In this figure, the multi-start simplexes as well as the
order of coarse search and fine search in the single start Nelder-Mead algorithm for the
final optimized parameter is shown.
60
Chapter 5
Results and discussions
In this chapter, the author presents the different results obtained during the thesis work.
The results are arranged in the same manner as the topics are introduced in the report.
Along with the results, few interesting facts and observations are discussed to provide
the reader a clear flow of the choices made for the optimization of the mechanism design.
5.1 Results from design choices
In this section, the results regarding the design choices are presented. Different issues
such as the type of joint, the orientation of joints and their effect on the workspace are
analyzed. Later, we also present the important effect of orientation representation and
the problems posed in their transformation from one representation to another.
5.1.1 Different joint placements
We studied 3 variations of the 2UPS-1U type as illustrated in Fig. 3.2. After different
starts and kinematic analysis of the kinematic curves, it was concluded that the 2PUS-1U
with the prismatic joint in horizontal direction performed the worst.
(a) The singularity in 2PUS-1U hori-
zontal architecture
(b) The actuation singularity in
2PUS-1U vertical architecture
Figure 5.1: The singularities occurring in the 2PUS-1U variation
61
This can be reasoned with the reciprocal screw theory itself. The screw that is recip-
rocal to the universal joint as well as the spherical joint is a force passing through both
the center of the joints. If this screw is also reciprocal to the actuated prismatic joints,
then the manipulator loses its mobility as the constraint wrenches has dimension 6. The
reason why this architecture performs poor is that there are many such instances when
the above-mentioned case occurs and the manipulator is as good as a rigid structure.
The condition is illustrated in the Fig. 5.1a. Fig. 5.1b is an actuation singularity of
a special nature, there exists no actuation wrench in the active leg and this singularity
cannot be changed by altering the actuator joints. The effect of such singularity is that
the actuators cannot control the instantaneous motion of the end-effector.
5.1.2 User-centric choice
We collect the information from 9 surgeons of various expertise working on otologic as
well as sinus surgery at CHU Nantes. The feedback on speed and size seemed to be
coherent and everyone preferred the actuators with maximum speed. To analyze better
the importance of the features, the surgeons were asked to prioritize few parameters. It
was concluded that most of the surgeons expected a robotic assistant that is easy to use.
The next important parameter is the size of the mechanism while speed and multiple
operation capacity were not considered as high priority by the surgeons.
Figure 5.2: The different priorities scored by the surgeons
Fig. 5.2 shows the result of the priorities filled by the surgeons. It was thus concluded
that we will be focusing on compact size of the mechanism and choose actuators based
62
on the speed/size ratio. The low priority of multiple operation capacity also helped us in
concentrating on the minimum quality discussed in Section 4.5.3. The reason being that
the assembly of the architecture can be different for otological and sinus surgery such
that most common positions are the default positions of the mechanism (α= 0, β= 0).
5.2 Results from optimization
In this section, we present the results from the optimization process of the mechanism.
The effect of time on the choice of parameters as well as the choice of the objective
function is also discussed in the following subsections.
5.2.1 How to read the results
This subsection is aimed at making the reader comfortable with the interpretation of
the data presented. When the mechanism was parameterized with 13 parameters for
the 2UPS-1U case, the following array was used to define a point in the corresponding
optimization space, Oand corresponds to the mechanism shown in Fig. 5.3.
[a1, φ1, h1, b1, ψ1, h3, a2, φ2, h2, b2, ψ2, h4, t]
The bounds for these parameters are :
a1, a2∈[0.25,1.5], φ1, φ2, ψ1, ψ2∈[−1000,1000], b1, b2∈[0.25,2], h1, h4∈[−0.1,0.1]
and h2, h3∈[−0.5,0.5]. The optimized point was then visualized with a schematic
Figure 5.3: 2UPS-1U mechanism corresponding to a single point in the 13 dimension O
diagram. The red dotted line is the actuator of first leg while the black dotted line is the
actuator of the second leg. All the schematics are shown in the default position of α= 0
and β= 0. An example plot code with parameters [1, -1, 0, 1, -0.8, 0.5, 0.9, 0.2, 0.5, 1,
1, -0.2, 3.5] is illustrated in Fig. 5.4.
63
Figure 5.5: An example of the output space with constraints violation (right) and the
heat map for the quality index (left)
Figure 5.4: An example of a schematic plot of the 2UPS-1U mechanism
The evaluation of these parameters is done by checking for the constraints imple-
mented in the optimization problem. The complete output space is visualized to un-
derstand the area of violations and the constraints violated. Along with the constraint
violations, it is also helpful to visualize the local quality of the manipulator. It allows us
to graphical understand how good the manipulator will behave in the desired (frequently
used) workspace. For these reasons, we visualize the local quality as a heat map where the
minimum value (darkest) corresponds to the singular regions while the brightest regions
correspond to isotropic configuration.
5.2.2 Different objective functions
The two different objective functions used are the global conditioning number and the
joint limit norm. Till now, only single objective optimization has been carried out and
both the objective functions were implemented individually. The results obtained from
the same are very interesting. Fig. 5.6 shows the feasible workspace and the heat map
for the quality index for corresponding parameters. Fig. 5.6 is the result of maximizing
the conditioning number along with maximizing the feasible workspace. The black box
of ±1 radian is the RDWdand the white space in the plot is the feasible workspace.
64
Fig. 5.7 shows the feasible workspace and the heat map for the quality index for
the parameters for the results of maximizing the passive joint limit norm along with
maximizing the feasible workspace. It is interesting to note that the conditioning number
for the optimized result is very poor suggesting that the joint limit norm and the kinematic
performance index are contrasting objective functions.
Figure 5.6: Results for one of the best local optima acquired while maximizing the global
quality index. Parameters = [0.4584, -0.4439, -0.0846, 0.7423, -0.4097, 0.0183, 0.6314,
1.5885, -0.0307, 0.7247, 1.1574, 0.0124, 3.6], actuator range = [2.9864, 4.3548]
5.2.3 Effect of change of constraints
Best feasible actuator range
The method in which we can implement the Nelder-Mead algorithm allows us to calculate
the best actuator range for maximizing the feasible workspace. To illustrate the advantage
of the same, we initiate two instances of proposed Nelder-Mead algorithm with identical
initial simplex. The results obtained while using the biggest actuator range and the best
actuator range are illustrated in Fig. 5.8 and Fig. 5.9 In Fig. 5.8, we implemented
a constraint such that the largest possible actuator range is chosen for evaluation. Fig.
5.9 shows the results when the optimized actuator range is chosen for evaluation. This
example highlights the fact that the largest actuator range is not always the best actuator
range. As both the optimization methods were initiated with identical simplexes, we can
65
Figure 5.7: Results for one of the best local optima acquired while maximizing the
joint limit norm. Even though it has large feasible workspace, it performs kinemati-
cally poor, Parameters = [0.281024, -1.606316, 0.080873, 0.644999, -1.687512, -0.011566,
0.380352, -1.183068, -0.022100, 0.843396, -1.565658, 0.011168, 3.971212], actuator range
= [3.289266, 4.712601]
66
Figure 5.8: Results for the implemented biggest actuator range. Parameters = [1.056069,
-1.610967, 0.035785, 1.568429, -1.745329, 0.219276, 0.250000, 1.501642, -0.011370,
1.530870, -0.091670, 0.219896, 2.390018], actuator range = [2.5850, 3.8775]
67
Figure 5.9: Results for the implemented best actuator range. Parameters = [0.994852,
1.540909, 0.071788, 0.829097, 0.151599, 0.156304, 1.096556, -0.251495, 0.030696,
0.624417, 1.725550, -0.052983, 2.383970], actuator range = [2.2087, 3.3131]
68
also conclude that the way the constraints are implemented have a great impact on the
results achieved. The variable actuator range also allows us to choose the mechanism
with best kinematic performance too.
Collision constraint
Another important constraint to be considered is the collision constraint. The modeling
of the actuator as a cylinder or some other shape contributes majorly to the evaluation
of the mechanism. One such curious case was found when we applied a design that was
supposed to be valid according to the human intuition. The joint limits were respected
for a large workspace and the configuration was suitable for the global kinematic quality
too as shown in Fig. 5.10. Still, the algorithm never proposed a solution towards the
expected zone.
With manual intervention and forcing the algorithm to go near the zone, it was
observed that the actuators were colliding in a small area of the desired workspace.
To counter this problem, a tapered shape was implemented as discussed in Section 4.5.3.
This improved the result with great success and many such invalid cases were corrected
by the same. This is also an example to show that the efficiency of the algorithm highly
depends on the modeling of the constraints as well as the rewarding strategies.
5.2.4 Effect of parameterization
As discussed in the Chapter 4, the manipulator can be parameterized with 13 variables for
a general geometry or else we can implement a general architecture using human intuition
to reduce the dimension. The following results explain how close the human intuition
is to the optimized result for particular rewarding strategies. When we were optimizing
the mechanism by rewarding the quality of the manipulator inversely proportional to the
distance of the point (α,β) from (0, 0), the general parameterization lead to an optimized
result with parameters very close the the humanly chosen 4-parameterization.
The optimized results with the 4 parameters is: [0.357173, 0, 0, 0.438437, 0, 0.105441,
0.357173, 1.5708, 0, 0.438437, 1.5708, 0.105441, 3.723585] with actuator range of [3.411769,
4.154748].
The optimized results with the 13 parameters is: [0.4584, -0.4439, -0.0846, 0.7423,
-0.4097, 0.0183, 0.6314, 1.5885, -0.0307, 0.7247, 1.1574, 0.0124, 3.6] with actuator range
of [2.9864, 4.3548]
69
Figure 5.10: Results for the curious case of collision avoidance. The heat map and feasible
workspace highlights the high kinematic performance of the manipulator at this particular
constraint, Parameters = [1, 0, 0, 1, 0, 0, 1, 1.5708, 0, 1, 1.5708, 0, 4]
Figure 5.11: Results for the optimized architecture using 4 parameters and 13 parameters
70
Figure 5.12: The valid feasible space and the heat map for the quality index for optimized
mechanism using 4 parameters
Figure 5.13: The valid feasible space and the heat map for the quality index for optimized
mechanism using 13 parameters
5.2.5 Different rewarding strategies
While optimising the mechanism for a particular objective, we implemented various dif-
ferent rewarding strategies. The modeling of the rewarding strategies play an important
role in the optimized architecture. Fig. 5.14 shows the evaluation surface only with
respect to 2 of the optimizing variables. It is impossible to visualize the complete opti-
mization surface as we have 13 optimizing variables and the following figures are just the
tip of the iceberg. But, certain conclusions can be drawn from the same. Especially, how
the mechanism is penalized equally, irrespective of the rewarding strategy and also how
does the area near the penalty region is affected with change of the rewarding strategy
as illustrated in Fig. 5.15.
71
Figure 5.14: The small subset of the evaluation surface while implementing the binary
rewarding strategy and biased rewarding strategy
Figure 5.15: The small subset of the evaluation in top view to observe the penalty regions
5.2.6 Coarse and fine search
The proposed strategy of coarse and refined search for a faster and efficient global scan
of the optimization space has been explained in Section 4.7. The results of the same
for a single example has been shown in Fig. 5.17 to present the clear advantage of the
proposed search mechanism.
72
Figure 5.16: Valid feasible space and heat map for the quality index with coarse search.
Parameters = [0.824612, 1.911219, 0.082068, 1.061443, -0.323155, 0.010461, 1.136735,
1.911284, 0.039281, 0.667857, 0.805770, 0.217043, 3.265619]
73
Figure 5.17: Valid feasible space and heat map for the quality index after fine search.
The discretization of the workspace is also done with finer intervals, Parameters
= [0.802151, 1.251624, 0.060514, 0.849475, -0.438371, -0.083466, 1.367378, 1.702786,
0.009458, 0.878488, 1.072529, 0.079628, 3.328403]
5.2.7 Computational time
This is important in order to understand the importance of reducing the optimization
dimension as well as the effect of different constraints on the overall computational time.
The following table presents the number of iterations as well as the time for each iteration
while optimizing with 4-parameters and 13-parameters. Also, as the discretization as well
as the search criteria are stricter in fine search, it was observed that finer searches are
computationally very expensive as expected. This corroborates our proposal of a coarse
search allowing us to choose few valid simplexes in order to initiate a much deeper search
for a better optimum point.
Parameters type iterations(min-max) avg. time (sec/iteration)
4-parameters coarse 12 - 42 64
4-parameters fine 25-98 84
13-parameters coarse 34-138 286
13-parameters fine 58-152 410
74
Chapter 6
Conclusions and future works
The thesis work presents a detailed explanation of the current scenario and the existing
shortcomings in the use of an endoscope in otological surgeries. The requirements of the
surgeons for the same are then translated into technical requirements for better analysis.
The report then states the analysis and comparison of possible architectures that can be
used for the same. The state of the art on such mechanisms is presented and the novel
implemented architecture is detailed. It was observed that adding a motion generator leg
in the mechanism helps avoid constraint singularities and thus promise better workspace
qualities.
The main objective of the work is to optimize the proposed mechanism. Several ob-
jective functions and constraints are highlighted and their effect on the choice of the
optimization algorithm is discussed. The global kinematic quality implemented with in-
dependent constraints promises a result with physical interpretation allowing us to choose
proper actuators for the mechanism. A novel implementation of a fast local search al-
gorithm is presented coupled with a global search mechanism to combine the speed and
efficiency of the search of complete optimization space. The algorithm searches for an
optimized point in a geometric travel across the optimization space which is very relevant
for mechanism design. The simplicity of modeling the algorithm allows one to implement
the algorithm for general mechanism design optimization. This algorithm works in two
phases, the coarse search and the fine search, and the advantages of such methodology
is objectively presented in the report. Later, several rewarding strategies are mentioned
which further corroborates the optimized parameters for certain objective functions. The
report also touches upon the effect of parameterization of the mechanism and how human
intuition can help reduce the optimization time remarkably. In the presented variation
of the mechanism, we optimize the mechanism for a general 13 dimensional optimization
space and later note that the dimension can be reduced to 4 parameters with human
knowledge of the singularities and isotropic configurations of the mechanism.
Several results are shown to highlight the importance of each parameter, constraint
and rewarding strategy used. The report also showed the importance of careful modeling
of such strategies and their effect on the optimized parameters. A short comparison of
computational time has been done to get a relative idea of the optimization with different
parameters.
The further work of the thesis is to implement multi-objective optimisation using
Pareto front as detailed in [71, 72]. The author also plans to create an open library for
the implemented algorithm so that any parallel mechanisms can be optimized using the
75
presented work.
Human factors is a the study of how humans react and interact with equipment, ser-
vices and their environment : sometimes bad design is funny but in medicine it can be
harmful! To understand how human interacts with the proposed mechanism, we need to
make observations, questionnaires and interviews, and also need to test the same on the
physical prototype. Even though our questionnaire helped us prevent the big mistakes,
it is not enough to prevent smaller mistakes, so we need a prototype that the surgeons
can test. So, another important future task would be to develop a prototype for a mock
use by surgeons.
A virtual simulator can be produced to visualize the effect of dimension and scaling of
the mechanism on the motion of the manipulator.Another aspect in focus is the control
of the manipulator. It is important that the control environment is intuitive as well as
easy to use. A study on the virtual simulation with the control interface can be done to
understand the perspective of the surgeons on the use of the endoscope.
76
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