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Screw Theory and Lie Groups: the Two Faces of Robotics

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Screw Theory and Lie Groups: the Two Faces of Robotics SCREW THEORY AND LIE GROUPS: THE TWO FACES OF ROBOTICS By UDAI BASAVARAJ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1991 ACKNOWLEDGEMENTS The author acknowledges the Northeast Regional Data Center at the University of Florida for a grant under the Research Computing Initiative for Numerically Intensive Computing in Higher Education. The author would also like to thank his advisors, Dr. Joseph Duffy and Dr. Neil White. 11 TABLE OF CONTENTS ACKNOWLEDGEMENTS ii LIST OF TABLES v LIST OF FIGURES vi ABSTRACT viii 1 INTRODUCTION 1 1.1 Overview 1 1.2 The Fundamental Concept of Motion Capability 4 1.3 Summary of Chapters 5 2 PRELIMINARIES 8 3 THE EUCLIDEAN GROUP 20 3.1 Euclidean Space 20 3.2 Rigid Body Motion in Space and Time 25 3.3 The Euler Parameterization of the Euclidean Group 27 4 TANGENT BUNDLES 31 n 4.1 The Tangent Bundle of E” and R 31 4.2 Directional Derivatives as Operators Acting on Functions 36 4.3 The Tangent Bundle of Differentiable Manifold 41 5 THE TANGENT BUNDLE OF THE EUCLIDEAN GROUP 53 5.1 The Tangent Space at the Identity Element 53 5.2 The Right Tangential Mapping 57 5.3 The Left Tangential Mapping 57 6 THE ADJOINT MAP 76 6.1 The Composition of Mappings 75 6.2 Composition Interpreted as a Change in Frames 77 6.3 The Effect on the Fixed and Moving Generalised Axodes 79 111 6.4 The Physical Implications on the Fixed Generalised Axode 81 6.5 The Physical Implications on the Moving Generalised Axode 88 6.6 The Relationship Between the Generalised Axodes 95 7 THE EXISTENCE OF A SEMI-RIEMANNIAN METRIC 99 8 APPLICATIONS TO ROBOTICS 103 8.1 The Group Structure in Robotics 103 8.2 The Concept of Volume 104 8.3 Planar Motion 112 8.4 Redundant Planar Motion 121 8.5 Euclidean Motion 126 8.6 Application to a Typical Industrial Robot 129 8.7 Redundant Manipulators 138 9 CONCLUSIONS 144 APPENDICES A THE GENERAL ELECTRIC P60 ROBOT 146 B THE FLIGHT TELEROBOTIC SERVICER MANIPULATOR SYSTEM . 153 REFERENCES 180 BIOGRAPHICAL SKETCH 183 IV LIST OF TABLES 6.1 Interpretation of Changing Frames via Composition 79 6.2 Effect of a Change in Frames on the Generalised Axodes 82 8.1 Mechanism Parameters for the GE P60 Robot 133 8.2 Mechanism Parameters for the FTSMS Design A 141 8.3 Mechanism Parameters for the FTSMS Design B 141 8.4 Intermediate Values of Motion Capability for the FTSMS Designs . 143 v LIST OF FIGURES 2.1 Topologies on a Three Element Set 9 2.2 Compatible Co-ordinate Systems 11 2.3 Covering of the Sphere 12 2.4 Co-ordinate Chart on the Sphere 13 3.1 Definition of the Euler Angles 28 4.1 Differentiability on a Manifold 42 4.2 Differentiability Between Manifolds 43 4.3 Tangent Vector to a Curve 50 4.4 The Tangential Mapping Between Manifolds 52 5.1 Parameterization of a Rolling Cylinder 62 8.1 Semicircular Region in the Plane 105 8.2 Parameterization of the Surface of the Sphere 110 8.3 Tangent Plane to the Sphere HI 8.4 Definition of Planar Parameters 112 8.5 Planar RRR Robot 115 8.6 Physical Workspace of Planar RRR Robot 117 8.7 Planar PRP Robot H8 8.8 Physical Workspace of Planar PRP Robot 119 vi 8.9 Redundant Planar 4R Robot 122 8.10 Joint Definitions 130 8.11 Kinematic Model of the GE P60 Robot 133 8.12 GE P60 singularity sin d3 = 0 137 8.13 GE P60 singularity sin = 0 137 8.14 GE P60 singularity S5 sin 0432 + a34 cos 032 + a 23 cos 02 = 0 138 8.15 FTS Manipulator System Design A 139 8.16 FTS Manipulator System Design B 139 8.17 Kinematic Model of the FTSMS Design A 140 8.18 Kinematic Model of the FTSMS Design B 140 vii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SCREW THEORY AND LIE GROUPS: THE TWO FACES OF ROBOTICS By Udai Basavaraj December 1991 Chairman: Dr. Neil White Major Department: Mathematics The two parallel approaches of screw theory and Lie groups, used in the analysis of robot manipulators, are contrasted. The fundamental aspects of screw theory are shown to arise as a natural consequence of the group structure and the differential ge- ometric properties of the Euclidean group. The Lie algebra is shown to be equivalent to the space of twists encountered in screw theory. As an application of the theoretical framework provided by the semi-Riemannian structure of the Euclidean group, an invariant measure is defined and used to quantify the concept of motion capability of a robot manipulator. The measure is shown to arise from a symmetric bilinear form also known as the reciprocal product. This measure is shown to be applicable to manipulators of arbitrary design. In particular, it is shown to be independent of the parameterization of the manipulator and the type and number of joints in the manipulator. The singularities of the manipulator are also shown to be predicted by the measure. Finally, it is shown that, when applied to redundant manipulators, the measure quantifies the importance of each joint of the manipulator. viii CHAPTER 1 INTRODUCTION 1.1 Overview The pages that follow are an attempt to unify two parallel approaches that currently exist in the study of spatial kinematics and its application to robot manip- ulators. A robot manipulator can be considered to be an open chain of rigid bodies connected together by joints which define the relative freedom of motion of adjacent bodies in the chain. One approach to spatial kinematics uses screw theory which models the instanta- neous motion of a body in space as an infinitesimal rotation about some axis together with an infinitesimal translation along that axis. This is interpreted as an instanta- neous screw-like motion, called a twist, which is the fundamental geometric concept on which the theory is based. The other approach involves the use of Lie groups, and models the motion of a body in space as a curve in a six dimensional space consisting of all possible rotations and translations of an object. The engineer has traditionally used screw theory in the analysis of the subject. This approach was first comprehensively addressed by Ball [1]. This inspiring work is notable for its geometric insight and intuitive reasoning in the absence of a systematic representation which exists today via the use of line geometry, Plucker co-ordinates, and dual quaternions. In contrast, the mathematician working in the field of spatial kinematics has preferred the use of Lie groups, which is a combination of group theory and differential 1 2 geometry. This approach is partly due to training, but also due to the large body of supporting theory in these areas which can be used to one’s advantage. Indeed, the origins of differential geometry can be traced back well over a century to the works of Riemann. His preliminary papers, which are found translated by Clifford [5], laid the foundations for differential geometry and exhibit an appreciation of the concepts of a manifold and metric decades before these ideas were fully articulated and formalized. Today, the parallel approaches to spatial kinematics are typified by two compre- hensive works. The first is by Hunt [8] and contains an exhaustive study on systems of screws and the concept of reciprocity which describes the relationship between the constraints on, and freedoms of, a body. The second is by Karger [11]. This work views the Euclidean group, which can be considered to be the set of all possible rotations and translations of an object, as a six dimensional differentiable manifold endowed with a semi-Riemannian metric. This approach yields precisely the same results, concerning screw systems and reciprocity, and differs only in its formulation in terms of group theoretic and differential geometric structures such as group orbits and bilinear forms. Indeed, both approaches can be seen to be utilised by Bottema and Roth [3]. One advantage of screw theory is its ability in explaining a wide variety of con- cepts in spatial kinematics without the use of groups and differentiable manifolds. Screw theory has probably gained prominence due to the engineer’s tendency to choose a physically natural description of phenomena rather than look for hidden mathematical structures. Indeed, at first glance one finds little need of manifolds and semi-Riemannian metrics, preferring the appealing concept of an instantaneous screw-like motion instead. Unfortunately, this approach inherently precludes infor- mation about the finite motions from which the instantaneous motions are derived. 3 This results in obscuring the structure of the Euclidean group, and omits the rich mathematical framework contained therein. Very recently, a few works have begun to address this problem with applications of spatial kinematics to robot manipulators. The work by Rico [15] studies explicit representations of the Euclidean group and their application to manipulators. Also, Brockett [4] and Loncaric [12] apply the group structure to the problem of the control of manipulators. Finally Karger [9] [10], in two key papers, makes a direct application of some well known results in the areas of Lie groups and Riemannian geometry to manipulators. Unfortunately, these last two papers seem to have been overlooked in the field due to their highly mathematical nature. Karger’s papers illustrate, in an unambiguous way, the primary advantage in using Lie groups in the study of robot manipulators.
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