submitted to Florida Institute of Technology in partial satisfaction ... the moving frame to each of the n desired locations, sum these distances, and then to employ ...
Approximate Motion Synthesis of Robotic Mechanical Systems
by Venkatesh Venkataramanujam B.E. (Mechanical Engineering.) University of Mumbai, 2002
A THESIS submitted to Florida Institute of Technology in partial satisfaction of the requirements for the degree of
MASTER OF SCIENCE in Mechanical Engineering
Melbourne, Florida May, 2007
©2007 VENKATESH VENKATARAMANUJAM ALL RIGHTS RESERVED
The author grants permission to make single copies
We the undersigned committee hereby recommend that the attached document be accepted as fulfilling in part the requirements for the degree of Master of Science in Mechanical Engineering. “Approximate Motion Synthesis of Robotic Mechanical Systems” by Venkatesh Venkataramanujam
Pierre M. Larochelle, Ph.D., P.E. Professor, Mechanical and Aerospace Engineering Department
Bo Yang, Ph.D. Assistant Professor, Mechanical and Aerospace Engineering Department
Hamid K. Rassoul, Ph.D. Associate Dean, College of Sciences Professor, Physics and Space Sciences Department Founding Director, Geospace Physics Laboratory
Pei-feng Hsu, Ph.D. Professor, Mechanical and Aerospace Engineering Department Department Head
Abstract Approximate Motion Synthesis of Robotic Mechanical Systems by Venkatesh Venkataramanujam Professor Pierre M. Larochelle, Chair This is an endeavor to investigate the approximate motion synthesis of open and closed chain mechanisms. This thesis presents a novel methodology for approximate motion synthesis of planar and spherical mechanisms. The suggested methodology uses homogenous transformations to represent planar and spherical locations, the homogenous transforms in the planar case can be approximated by elements in SO(3) using the Polar Decomposition based distance metric. The forward kinematics of the moving frame is represented in terms of the mechanisms dimensional synthesis variables. The locations are then compared using a bi-invariant distance metric in SO(N) which calculates the distance between two locations of a rigid body. Our approach is to utilize the metric discussed above to determine the distance from the moving frame to each of the n desired locations, sum these distances, and then to employ nonlinear optimization techniques to vary the dimensional synthesis parameters such that the total distance to the desired locations is minimized. The result is a methodology for performing dimensional synthesis of planar and spherical mechanisms for approximate motion generation. Software tools that facilitate the iii
synthesis are critical to moving this theory out of the laboratory into industrial practice. A M AT LAB T M implementation of the proposed methodologies allows designers with little or no knowledge of the synthesis techniques to interactively design and explore solutions for open and closed chain mechanisms for motion generation. Kinematic routines to generate and animate the resulting optimal chains have been developed.
iv
Contents Abstract
iii
List of Figures
viii
List of Tables
xi
Table of Symbols
xiv
Acknowledgements
xv
1 Introduction 1.1
1.2
1
Background and Motivation . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Kinematic Chains 2.1
2.2
6
Planar Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.1
Planar Displacements . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.2
Planar Serial Chains . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.3
Planar RR Dyad . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.1.4
Planar 4R Mechanism . . . . . . . . . . . . . . . . . . . . . .
11
Spherical Mechanisms
. . . . . . . . . . . . . . . . . . . . . . . . . .
v
20
2.3
2.2.1
Spherical Displacements . . . . . . . . . . . . . . . . . . . . .
20
2.2.2
Spherical Serial Chains . . . . . . . . . . . . . . . . . . . . . .
22
2.2.3
Spherical RR Dyad . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.4
Spherical 4R Mechanism . . . . . . . . . . . . . . . . . . . . .
25
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3 Metric
34
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2
The PD Based Projection Metric . . . . . . . . . . . . . . . . . . . .
35
3.3
Dubrulle’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.4
Computation of Characteristic Length . . . . . . . . . . . . . . . . .
39
3.5
Distance Between Elements in SO(N) . . . . . . . . . . . . . . . . . .
40
3.6
Finite Sets of Locations
. . . . . . . . . . . . . . . . . . . . . . . . .
40
3.7
Principal Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.7.1
Determination of the Principal Frame . . . . . . . . . . . . . .
43
3.8
Summary of the Technique . . . . . . . . . . . . . . . . . . . . . . . .
44
3.9
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4 Approximate Motion Synthesis
46
4.1
Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.2
Inner Optimization Routine . . . . . . . . . . . . . . . . . . . . . . .
48
4.3
Outer Optimization Routine . . . . . . . . . . . . . . . . . . . . . . .
51
4.4
Constraints on Optimization . . . . . . . . . . . . . . . . . . . . . . .
51
4.4.1
Constraints in Planar Case . . . . . . . . . . . . . . . . . . . .
53
4.4.2
Constraints in Spherical Case . . . . . . . . . . . . . . . . . .
54
4.5
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
vi
5 Planar Design Case Studies
58
5.1
Planar - Case Study 1 . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.2
Planar - Case Study 2 . . . . . . . . . . . . . . . . . . . . . . . . . .
63
5.3
Planar - Case Study 3 . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6 Spherical Design Case Studies
7
74
6.1
Case Study 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
6.2
Case Study 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
6.3
Case Study 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
6.4
Case Study 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
M AT LAB T M Simulation
91
7.1
The Computer Graphics Module . . . . . . . . . . . . . . . . . . . . .
91
7.2
The Graphical User Interface . . . . . . . . . . . . . . . . . . . . . .
92
7.2.1
Design Window . . . . . . . . . . . . . . . . . . . . . . . . . .
93
7.2.2
Input Window . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
7.2.3
Graphics Controls . . . . . . . . . . . . . . . . . . . . . . . . .
95
7.2.4
The Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
8 Conclusion
98
8.1
Summary of Research . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
8.2
Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
A Euler Parameters
101
B Matlab Source Codes
103
B.1 Graphical User Interface Code Listing . . . . . . . . . . . . . . . . . . 103 B.2 Optimization Code Listing . . . . . . . . . . . . . . . . . . . . . . . . 162 B.3 Simulation Code Listing . . . . . . . . . . . . . . . . . . . . . . . . . 168
vii
Bibliography
171
viii
List of Figures 2.1
Revolute Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Prismatic Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Planar Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.4
Planar Serial Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.5
Forward Kinematics of a Planar RR Dyad . . . . . . . . . . . . . . .
10
2.6
Planar 4R Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.7
Planar 4R Mechanism Parameters . . . . . . . . . . . . . . . . . . . .
15
2.8
Input θ Range Cases . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.9
Planar 4R Mechanism Forward Kinematics . . . . . . . . . . . . . . .
19
2.10 Spherical Displacement . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.11 Spherical Serial Chain . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.12 Spherical RR Dyad . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.13 Spherical 4R Mechanism . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.14 Spherical 4R Mechanism Parameters . . . . . . . . . . . . . . . . . .
27
2.15 Forward Kinematics of a Spherical 4R Mechanism . . . . . . . . . . .
32
3.1
Planar Case: SE(2) ⇒ SO(3) . . . . . . . . . . . . . . . . . . . . . .
36
3.2
Spatial Case: SE(3) ⇒ SO(4)(Figure adapted from Larochelle) . . . .
36
3.3
General Case: SE(N-1) ⇒ SO(N) . . . . . . . . . . . . . . . . . . . .
37
3.4
Four Possible Orientations for the Principal Frame . . . . . . . . . . .
44
ix
4.1
Local and Global Minima . . . . . . . . . . . . . . . . . . . . . . . .
48
4.2
Objective Function Surface . . . . . . . . . . . . . . . . . . . . . . . .
49
4.3
Flow Chart for the Inner Optimization Loop . . . . . . . . . . . . . .
50
4.4
Flow Chart for the Outer Optimization Loop . . . . . . . . . . . . . .
52
4.5
Joint Space and Task Space . . . . . . . . . . . . . . . . . . . . . . .
54
4.6
Constraints on Optimization for Spherical Mechanisms . . . . . . . .
55
4.7
Simulation of an Optimized Mechanism . . . . . . . . . . . . . . . . .
56
5.1
11 Planar Locations with Respect to the Fixed Frame . . . . . . . . .
60
5.2
11 Planar Locations with Respect to the Principal Frame . . . . . . .
61
5.3
11 Scaled Planar Locations . . . . . . . . . . . . . . . . . . . . . . . .
62
5.4
Planar RR Dyad Optimized for 10 Rigid Body Locations . . . . . . .
64
5.5
The 10 Rigid Body Locations . . . . . . . . . . . . . . . . . . . . . .
66
5.6
The 10 Rigid Body Locations with Respect to the Principal Frame . .
67
5.7
Optimized Planar RR Dyad for 10 Rigid Body Locations . . . . . . .
69
5.8
Optimized Planar 4R Mechanism . . . . . . . . . . . . . . . . . . . .
72
5.9
Optimized Planar 4R Mechanism nearest the Sixth Location . . . . .
73
6.1
The 10 Desired Rigid Body Orientations . . . . . . . . . . . . . . . .
76
6.2
Optimal RR Dyad for 10 Orientations . . . . . . . . . . . . . . . . . .
78
6.3
Optimal RR Dyad at Seventh Orientation . . . . . . . . . . . . . . .
79
6.4
The 4 Rigid Body Orientations . . . . . . . . . . . . . . . . . . . . .
82
6.5
Optimum Spherical 4R Mechanism for 4 Orientations . . . . . . . . .
83
6.6
Spherical 4R Mechanism at the Fourth Orientation . . . . . . . . . .
84
6.7
Optimal RR Dyad for 4 Desired Orientations . . . . . . . . . . . . . .
86
6.8
The 5 Desired Orientations . . . . . . . . . . . . . . . . . . . . . . . .
87
6.9
The Optimal RR Dyad . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6.10 The Optimal RR Dyad at the First Orientation . . . . . . . . . . . .
90
x
7.1
GUI for Spherical Mechanism Synthesis. . . . . . . . . . . . . . . . .
92
7.2
Input Window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
7.3
Task Specification Window. . . . . . . . . . . . . . . . . . . . . . . .
95
7.4
Input Window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
7.5
Optimized RR Dyad. . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
xi
List of Tables 2.1
Planar 4R Linkage Types . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
Classification of the Input Link of the Planar 4R Mechanism . . . . .
17
2.3
Spherical 4R Linkage Types . . . . . . . . . . . . . . . . . . . . . . .
29
2.4
Classification of the Input Link of the Spherical 4R Mechanism . . . .
30
5.1
Planar Locations Case Study 1 - 11 Rigid Body Locations . . . . . .
59
5.2
Planar RR Dyad Joint Parameters and Distances for 11 Locations . .
63
5.3
Planar Locations Case Study 2 - 10 Rigid Body Locations . . . . . .
65
5.4
Planar RR Dyad Joint Parameters and Distances for 10 Locations . .
68
5.5
Optimized Planar 4R Mechanism for 10 Locations . . . . . . . . . . .
70
5.6
Planar 4R Mechanism Joint Parameters and Distances for 10 Locations 71
6.1
Spherical Orientations Case Study 1 - 10 Orientations . . . . . . . . .
75
6.2
Optimized RR Dyad for 10 Orientations . . . . . . . . . . . . . . . .
77
6.3
Joint Parameters and Distances for 10 Desired Orientations . . . . . .
77
6.4
Case Study 2 - 4 orientations . . . . . . . . . . . . . . . . . . . . . .
79
6.5
Optimized Spherical 4R Mechanism for 4 Orientations
. . . . . . . .
81
6.6
4R Joint Parameters and Distances for 4 Desired Orientations . . . .
81
6.7
Case Study 3 - 4 Orientations . . . . . . . . . . . . . . . . . . . . . .
84
6.8
RR Joint Parameters and Distances for 4 Orientations . . . . . . . .
85
6.9
Spherical Orientations Case Study 4 - 5 Orientations . . . . . . . . .
85
xii
6.10 Case Study 4 - 5 Orientations . . . . . . . . . . . . . . . . . . . . . .
87
6.11 RR Dyad Joint Parameters and Distances for 5 Orientations . . . . .
89
xiii
Table of Symbols a column vector [X]
brackets denote a matrix
ksk norm of a vector or matrix a · b dot product of a and b a × b cross product of a and b F AR
denotes the linear transformation from F to A
PD denotes Polar Decomposition PF denotes Principal Frame
xiv
Acknowledgements The dream begins with a teacher who believes in you, who tugs and pushes and leads you to the next plateau, sometimes poking you with a sharp stick called “truth”. ∼ Dan Rather. This work might not have been possible without the support and encouragement of Dr. Larochelle. An engineer in the truest sense of the word, he tackles every issue at hand with a sensibility and ease that still “stumps” me. To quote Newton “If I have seen a little further it is by standing on the shoulders of Giants”, if I haven’t seen further I can certainly see things much clearer. Thank you for condoning our eccentricities: Living in “India Time Zone”, showing up late for meetings, our prodigious talent for messing up the lab and taking a smoke break 5 minutes before flight departure. Many thanks to Dr. Marzocca for encouraging me to look ahead and helping me make a difficult decision. A lot of what I have achieved, I owe to you. It would not have been possible without your understanding and cooperation. I am indebted to Deshmukh Sir, Dange Sir, Sangram Sir, Miss Nina Francis, Suma Menon and all the teachers who have influenced my education. This is a debt that I cannot repay only pass on. To my lovely little sister Bharti for thinking I am superhuman and that nothing is beyond me, though I doubt she would ever admit to that. To Prasanna for being a sobering influence and a great friend, if I need advise I know whom to turn to. To my father for constantly stressing the importance of education and making me see xv
the big picture and my mother for giving me the courage to do what I believed in. What I am and hope to be I owe to their love and sacrifice. To Pratima, Rama and my lovely nieces Akhila and Alekhia for the great times in Boston. Big up to the Clarkson University gang Ganesh, Kaushik, Jeff, Kiran, Mahesh, Sameer, Elaine, Dustin I have had some of the best days of my life with you and it will be a cherished memory for years to come. To the the CU Golden Knights for their awesome games and showing that size does not matter... Brown, Harvard and Ohio State bear testimony to the fact and the crew at Cheel Center for all the cookies, pizzas and the ohh so delicious pasta. To my roommates Alex and Vivek for condoning my faults and being great friends. Alex thank you for sharing my joys and sorrows and being with me in times good or bad. My warmest wishes to you and I am sure you have a great future ahead of you.
All I can say is Forward...Drink! ; P.S. The bet on the car still stays.
To Maheshbhai, Janakbhai, Dev, Neera, Bela, Magan Uncle and the whole Patel consortium for making Melbourne a really sunny place for me. I am thankful to have had the company of such beautiful people. To Rizwanbhai and Heena didi for the awesome biryani, great chai, lively banter and great times. To Saurabh, Jackie and Deepti for the great times we have had together. I only wish they never did end. I wish you all nothing but the very best in whatever you might choose. To the RASSL gang who have been a profound influence on me these years, Danny and John with their infinite wisdom and patience. Jason, A.J and Saurabh with their enthusiasm and willingness to believe that nothing is impossible. To Vicki and Arlene for doing more than they had to to make my life easier. The Department is richer and livelier because of you pretty ladies. To Messrs. Gilmore, Waters, Plant, Norah Jones, Morrison, Clapton, Satriani,
xvi
Springsteen for being great company on long nights. I would give my life to play like any of you.. In retrospect you probably did. To the M∗A∗S∗H fraternity .. Holy Hemostat .. I am almost done. To the Zephyr gang at L.T.C.E for the great times during my senior year. I will always cherish the fond memories of the cricket, basketball and the drinking binges that I had with you guys. The Pahadi gang at KK, Ricki, Nonie, Khaddu, Thakur, Rahul,Jakku, Rocky, Krishna, Mattoo and the Raina brothers. To Anna and Anni for the endless cups of “chai” and “vada-pav” and love. To my very large family for supporting me all the way, i have been fortunate to come this far because of your prayers and wishes. To Meena aunty and Laxmi aunty, Kala’s Mom, Ramasubramanian uncle for all the help, advise and encouragement and love. There was no way I could have given up with a team like you rooting for me. To my friends in Nerul, Dinesh, Chakya, Shreedhar, Kiran, Sandeep, Nitin, Manjinder, Chakya, Ashok for all the great times. To the Madras gang for the whopper time in 2004... Lots of xoxo for the Temptress for all the support that she has given me all these years. You have spurred me to be the best that I can be, I would not have made it this far without your love and affection. Here’s wishing all your trials and tribulations become mine.
xvii
Dedication
To my Mother and Father for holding my hand for a while and my heart forever. If I had been with them all this while, as I wished, and they deserved, this thesis would not have been studied or written. Whatever its quality, it is a poor substitute.
xviii
Chapter 1 Introduction 1.1
Background and Motivation
Mechanisms are mechanical devices whose purpose is to transfer motion and/or forces from an input to an output in a prescribed manner. Kinematic synthesis is the process of designing mechanisms to accomplish a desired task. Dimensional synthesis is the part of kinematic synthesis which determines the dimensions of the mechanisms structural parameters. There are three common types of problems in dimensional synthesis: function generation, path generation and motion generation. Function generation mechanisms produce a desired relationship between the mechanism’s input and output. In path generation mechanisms, a particular point of the mechanism follows a specified path. In motion generation mechanisms, also referred to as rigidbody guidance mechanisms a rigid body that is part of the mechanism attains a sequence of desired locations, where the location of a rigid body prescribes both its position and orientation. The work presented studies the synthesis of mechanisms for rigid body guidance, Suh and Radcliffe [44]. Rigid body guidance is also referred to as motion generation, (Sandor and Erdman [11]) and finite-position synthesis, (Roth [41]). 1
In dimensional synthesis there are two approaches: exact motion synthesis and approximate or optimal synthesis. Exact motion synthesis implies that the endeffector frame (most frequently attached to the coupler in a four-bar linkage and the floating link in an RR chain) passes through a certain number of desired (exact) locations. Exact motion synthesis is however restricted by the number of locations it can be used for. It is seen that exact motion synthesis becomes progressively difficult and nonlinear and no solutions exist for a large number of locations (n ≥ 5). Approximate motion synthesis of mechanisms is the process of determining the best possible mechanism to meet the specified requirements, and is most often used in dimensional synthesis. Dimensional Synthesis is the determination of elements of the given mechanism (lengths, angles, coordinates) necessary for the desired motion. The objective function may contain various restrictions, such as: restriction on the ratio of link lengths, prevention of negative lengths of the links, restrictions on the transmission angles, location of the fixed pivots etc. Many applications in robotics depend on the measurement of distances between coordinate frames. In approximate motion synthesis the distance between a number of desired locations and the moving frame of a mechanism defines the optimization criterion of the mechanism synthesis routine. The motion generation task is specified as a set of finitely separated locations for the moving frame.
The number of
positions determines whether the problem is underconstrained, exactly constrained or overconstrained. Depending the nature of the problem the approach to the solution varies. In the first case, there are infinite solutions and the optimization criteria may be applied to determine an optimal mechanism. In the second case, there exist a finite number of solutions depending on the type of mechanism investigated. In the third case, the moving frame (end-effector) cannot attain all the locations and optimization procedures may be be used to solve for the mechanism that minimizes
2
the overall distance from the moving frame of the mechanism to the desired locations. As the number of desired locations increases, it becomes increasingly difficult to design a mechanism by traditional synthesis methods. The third situation forms the classic case where approximate motion synthesis might be effectively used for rigid body guidance. Spatial and spherical mechanisms are designed using algebraic calculations and the design process is free from inaccuracies inherent in graphical construction methods. The process though efficient is cumbersome and time consuming.
Moreover a
prototype of the mechanism has to be constructed in order to determine the feasibility of the mechanism for the desired task. This process is repeated until a suitable mechanism is found. The advent of computers has simplified the design process and reduced the time required. Computer aided design softwares can do all the calculations and computer graphic simulations allow the user to to visualize the entire mechanism and investigate its motion without building a prototype. It is essential that the mechanism designer be able to visualize the entire problem and computer graphics can be an effective tool for providing this necessary visualization of the problem to the designer.
1.1.1
Objectives
The primary objective of this thesis is the design of planar and spherical, open and closed chain mechanisms for approximate motion synthesis. A secondary objective is to create a graphical user interface (GUI) to facilitate task specification and the design of planar and spherical mechanisms using this methodology.
3
1.2
Thesis Outline
The thesis proceeds as follows, Chapter 2 introduces the background required for the approximate motion synthesis of planar and spherical RR and 4R chains. It begins with a discussion of the planar mechanisms covering their background and notation. Analysis of the planar RR and 4R mechanisms is carried out and the forward kinematics equations are derived. The theories of planar mechanisms are then applied to spherical mechanisms in the later part of the chapter. In chapter 3 we proceed to discuss the polar decomposition (PD) based distance metric. We also discuss how the metric may be applied to finite position synthesis by transforming the elements in SE(N-1) (Special Euclidean Group) to SO(N) (Special Orthogonal Group). We also discuss the Frobenius norm which is used for finding the distance between the elements in SO(N). In chapter 4, the optimization methods pertaining to the design of planar and spherical mechanisms are enumerated. We discuss the bounds on the optimization as applicable to both the cases. Flowcharts describing the Outer and Inner optimization routines are shown. In chapter 5, studies in the planar case are shown. Planar RR and 4R chains are synthesized using the PD based distance metric and the errors to each of the desired locations and the overall error is shown. In chapter 6, we enumerate a few examples showing the applicability of the theory to the synthesis of RR and 4R spherical chains. The Frobenius norm is used in these examples to find the distance between the moving frame and the desired orientations. To demonstrate how the principles discussed in Chapters 2 and 4 can be implemented for the synthesis of planar and spherical mechanisms, a M AT LAB T M GUI for designing a Spherical RR dyad is presented in chapter 7. Finally, chapter 8
4
presents the conclusions of this research and makes recommendations for future work.
5
