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AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures
A Subseries of Proceedings of Symposia in Applied Mathematics
Volume 41 ROBOTICS Edited by R. W. Brockett {Louisville, Kentucky, January 1990)
Volume 40 MATRIX THEORY AND APPLICATIONS
Edited by Charles R. Johnson {Phoenix, Arizona, January 1989)
Volume 39 CHAOS AND FRACTALS: THE MATHEMATICS BEHIND THE COMPUTER GRAPHICS Edited by Robert L. Devaney and Linda Keen {Providence, Rhode Island, August 1988)
Volume 38 COMPUTATIONAL COMPLEXITY THEORY
Edited by Juris Hartmanis {Atlanta, Georgia, January 1988)
Volume 37 MOMENTS IN MATHEMATICS
Edited by Henry J. Landau {San Antonio, Texas, January 1987)
Volume 36 APPROXIMATION THEORY
Edited by Carl de Boor {New Orleans, Louisiana, January 1986)
Volume 35 ACTUARIAL MATHEMATICS
Edited by Harry H. Panjer {Laramie, Wyoming, August 1985)
Volume 34 MATHEMATICS OF INFORMATION PROCESSING
Edited by Michael Anshel and William Gewirtz {Louisville, Kentucky, January 1984)
Volume 33 FAIR ALLOCATION
Edited by H. Peyton Young {Anaheim, California, January 1985)
Volume 32 ENVIRONMENTAL AND NATURAL RESOURCE MATHEMATICS
Edited by R. W. McKelvey {Eugene, Oregon, August 1984)
Volume 31 COMPUTER COMMUNICATIONS
Edited by B. Gopinath {Denver, Colorado, January 1983)
Volume 30 POPULATION BIOLOGY
Edited by Simon A. Levin {Albany, New York, August 1983)
Volume 29 APPLIED CRYPTOLOGY, CRYPTOGRAPHIC PROTOCOLS, AND COMPUTER SECURITY MODELS By R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J Lipton {San Francisco, California, January 1981)
Volume 28 STATISTICAL DATA ANALYSIS
Edited by R. Gnanadesikan {Toronto, Ontario, August 1982)
Volume 27 COMPUTED TOMOGRAPHY
Edited by L. A. Shepp {Cincinnati, Ohio, January 1982)
Volume 26 THE MATHEMATICS OF NETWORKS
Edited by S. A. Burr {Pittsburgh, Pennsylvania, August 1981)
Volume 25 OPERATIONS RESEARCH: MATHEMATICS AND MODELS
Edited by S. I. Gass {Duluth, Minnesota, August 1979)
Volume 24 GAME THEORY AND ITS APPLICATIONS
Edited by W. F. Lucas {Biloxi, Mississippi, January 1979)
Volume 23 MODERN STATISTICS: METHODS AND APPLICATIONS
Edited by R. V. Hogg {San Antonio, Texas, January 1980)
Volume 22 NUMERICAL ANALYSIS Edited by G. H. Golub and J. Oliger {Atlanta, Georgia, January 1978)
http://dx.doi.org/10.1090/psapm/041
PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
Volume 21 MATHEMATICAL ASPECTS OF PRODUCTION AND DISTRIBUTION OF ENERGY Edited by P. D. Lax (San Antonio, Texas, January 1976)
Volume 20 THE INFLUENCE OF COMPUTING ON MATHEMATICAL RESEARCH AND EDUCATION Edited by J. P LaSalle {University of Montana, August 1973)
Volume 19 MATHEMATICAL ASPECTS OF COMPUTER SCIENCE
Edited by J. T. Schwartz (New York City, April 1966)
Volume 18 MAGNETO-FLUID AND PLASMA DYNAMICS
Edited by H. Grad (New York City, April 1965)
Volume 17 APPLICATIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN MATHEMATICAL PHYSICS Edited by R. Finn (New York City, April 1964)
Volume 16 STOCHASTIC PROCESSES IN MATHEMATICAL PHYSICS AND ENGINEERING Edited by R. Bellman (New York City, April 1963)
Volume 15 EXPERIMENTAL ARITHMETIC, HIGH SPEED COMPUTING, AND MATHEMATICS Edited by N. C Metropolis, A. H. Taub, J. Todd, and C B. Tompkins (Atlantic City and Chicago, April 1962)
Volume 14 MATHEMATICAL PROBLEMS IN THE BIOLOGICAL SCIENCES
Edited by R. Bellman (New York City, April 1961)
Volume 13 HYDRODYNAMIC INSTABILITY
Edited by R. Bellman, G. Birkhoff and C C Lin (New York City, April I960)
Volume 12 STRUCTURE OF LANGUAGE AND ITS MATHEMATICAL ASPECTS
Edited by R. Jakobson (New York City, April I960)
Volume 11 NUCLEAR REACTOR THEORY
Edited by G. Birkhoff and E. P. Wigner (New York City, April 1959)
Volume 10 COMBINATORIAL ANALYSIS
Edited by R. Bellman and M. Hall, Jr. (New York University, April 1957)
Volume 9 ORBIT THEORY
Edited by G. Birkhoff and R. E. Langer (Columbia University, April 1958)
Volume 8 CALCULUS OF VARIATIONS AND ITS APPLICATIONS
Edited by L. M. Graves (University of Chicago, April 1956)
Volume 7 APPLIED PROBABILITY
Edited by L. A. MacColl (Polytechnic Institute of Brooklyn, April 1955)
Volume 6 NUMERICAL ANALYSIS
Edited by J. H. Curtiss (Santa Monica City College, August 1953)
Volume 5 WAVE MOTION AND VIBRATION THEORY
Edited by A. E. Heins (Carnegie Institute of Technology, June 1952)
Volume 4 FLUID DYNAMICS
Edited by M. H. Martin (University of Maryland, June 1951)
Volume 3 ELASTICITY
Edited by R. V. Churchill (University of Michigan, June 1949)
Volume 2 ELECTROMAGNETIC THEORY
Edited by A. H. Taub (Massachusetts Institute of Technology, July 1948)
Volume 1 NON-LINEAR PROBLEMS IN MECHANICS OF CONTINUA Edited by E. Reissner (Brown University, August 1947)
AMS SHORT COURSE LECTURE NOTES Introductor y Surve y Lecture s
publishe d as a subserie s o f
Proceeding s o f Symposi a in Applie d Mathematic s
PROCEEDING S O F SYMPOSI A IN APPLIE D MATHEMATIC S
Volum e 41
Robotic s J. Baillieu l an d D . P. Marti n
R. W . Brocket t Bruc e R. Donal d
Richar d M . Murra y an d S. Shanka r Sastr y Madhusuda n Raghava n
America n Mathematica l Societ y Providence , Rhod e Islan d
LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE
ROBOTICS HELD IN LOUISVILLE, KENTUCKY
JANUARY 16-17, 1990
The AMS Short Course Series is sponsored by the Society's Committee on Employment and Educational Policy (CEEP). The series is under the direction of the Short Course Advisory Subcommittee of CEEP.
Library of Congress Cataloging-in-Publication Data Robotics/R. W. Brockett, editor; J. Baillieul...[et al.].
p. cm. — (Proceedings of symposia in applied mathematics, ISSN 0160-7634; v. 41) ISBN 0-8218-0163-5 (acid-free paper) 1. Robotics—Mathematics. I. Brockett, Roger W. II. Baillieul, J. (John) III. American
Mathematical Society. IV. Series. TJ211.M3675 1990 90-1220 629.8,92/0151—dc20 CIP
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1980 Mathematics Subject Classification. (1985 Revision). Primary 53A17, 68-02, 68G05, 70B15, 93-02.
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Table of Contents
Preface ix
Some Mathematical Aspects of Robotics R. W. BROCKETT 1
Manipulator Kinematics MADHUSUDAN RAGHAVAN 21
Resolution of Kinematic Redundancy J. BAILLIEUL AND D. P. MARTIN 49
Grasping and Manipulation using Multifingered Robot Hands RICHARD M. MURRAY AND S. SHANKAR SASTRY 91
Planning and Executing Robot Assembly Strategies in the Presence of Uncertainty BRUCE R. DONALD 129
Formal Languages for Motion Description and Map Making R. W. BROCKETT 181
Index 195
vii
Preface
The emergence of the field of robotics has provided the occasion to analyze, and to attempt to replicate, the patterns of movement required to accomplish useful tasks. On the whole, this has been a sobering experience. Just as the ever-closer examination of the physical world occasionally reveals inadequacies in our vocabulary and mathematics, roboticists have found that it is quite awkward to give precise, succinct descriptions of effective movements using the syntax and semantics in common use. Perhaps it has always proved easier to demonstrate than to describe, but in any case, mankind has reached its present state without the benefit of a particularly expressive means for discussing movement. Yet, this is what is needed if we are to convey our wishes to general-purpose robots capable of doing what we ask them to do. In this volume we focus on some of the ways mathematics can be used to address problems in this area.
Because robotics is a broad field, it can be examined with profit from many points of view. The perspectives afforded by computer science, electrical engineering, mechanical engineering, psychology, and neuroscience all yield important insights. Even so, there are pervasive common threads, such as the description of spatial relations and their time evolution. One often finds that ideas from three-dimensional geometry and kinematics are not far from the center of the stage. The concept of a kinematic chain is basic to robotic manipulation, and these objects show up in considerable variety in practical applications. It is, therefore, particularly pleasant to observe that a very natural description of kinematic chains is afforded by a product of one-parameter Lie groups. It turns out that a key step in the design of controllers for industrial robots is equivalent to finding an algorithm for converting between coordinates of the first and second type for the group of rigid motions in three dimensions. We mention a second, perhaps unexpected, mathematical fact related to the manipulation of objects. In considering the application of grasping forces to objects, systems of inequalities play a central role because fingers can only push against, and not pull, objects. In fact, the study of grasping involves convex analysis, models of friction and details of the interface between the hand and the object that go considerably beyond simple mechanics. It happens that many tasks, including, but not limited to, grasping, can be done in more than one way. This may happen because the robot has more than the minimal number of degrees of freedom or because the task description is ambiguous. Disposing of such problems is usually called resolution of redundancy, and in some cases it leads to nonlocal questions in geometry.
From the point of view of the programmer, high-level languages are more efficient than low-level ones. Likewise, in directing the motion of a robot the programmer would like to say as little as possible about the means, preferring to focus attention
ix
x PREFACE
on the ends. In order to make this possible, it is necessary to incorporate automatic path-planning algorithms in the software and to write compilers that are capable of converting high-level directives into the motor control programs needed to execute motion segments. This means that motion-planning algorithms are an important part of any high-level programming environment and that proving correctness of the motion programs produced by such compilers is an issue.
Computing power is far less expensive now then it has been in the past, and there now exists an effective collection of software tools together with a large group of scientists who know how to use them. The emergence of robotics as a practical activity is one consequence of these developments. The field is immature, and at this stage unification is more easily expressed in terms of the goals than methods. The work presented here demonstrates again the effectiveness of mathematics. Even so, many difficult problems remain.
Roger Brockett
Index
adjoint action, 13 algorithmic singularities, 64, 67, 86 algorithms for EDR strategies, 172 avoidable singularities, 59
backprojections, 138, 154
Chasles, 4 compliance, 1 compliance control, 15 compliant motion, 133, 136, 167 computed torque, 111 conditional expectation, 193 conditioned reflex, 189 constrained manipulatability index, 64, 67 contact coordinates, 98 control uncertainty, 135
damper equation, 144, 152 descent programming, 188 dialytical elimination, 22, 23, 28, 30, 31, 34,
40 dynamic model, 135
error detection and recovery, 134, 139, 143, 162, 168
etage, 117 euclidean graph, 7 Euler angles, 5 Euler's theorem, 4 extended Jacobian. 66 extended Jacobian technique, 50, 65
finger kinematics, 102 finger repositioning, 114, 124 finite state language, 182 First Entry Set, 156 first etage controllable, 119 fixed contact, 94 fixed object, 185 forces, 13 formal motion language, 182 forward kinematics problem, 21, 22, 23, 25, 26 forward projection, 146, 161, 169 friction cone, 95, 103, 112
(/-equivalence, 52 general six-revolute-jointed manipulator, 21 generalized configuration space, 164 generalized inverse methods, 60 geometrical theory of planning, 133 grasp map, 96 grasping control problem, 110
harmonic maps, 15 Hodge star, 7 homogeneous transformation, 51 homotopy classes, 70 homotopy continuation, 50 homotopy continuation methods, 68
initialized automaton, 181 internal forces, 96, 109, 111, 112 internal motion, 110 internal motions, 113 inverse kinematics, 11 inverse kinematics problem, 11, 21, 22, 23, 25,
26, 27, 34, 39, 40
joint subgroup, 50
Killing form, 5 kinematic chains, 1, 8 kinematic machine, 182 kinematic singularities, 57 kinematically redundant, 53 kinematically redundant robot arms, 49 kinematically redundant robotic mechanisms,
49 kinematics, 21, 22, 40 kinematics mapping, 56 Klein form, 7
language A/1, 184 Lie algebra, 3 Lie bracket, 3 Lie group, 3 linking conditions, 174 local methods, 65 lower pair joints, 50
manipulable grasp, 103
195
196
manipulatability index, 61, 67 manipulator, 21, 22, 23, 24, 27, 28, 31, 32, 39,
40 matrix exponentials, 51 maximal backprojection, 156 maximal tree, 191 model error, 136, 162 motion strategy, 133 movable rigid objects, 185
navigation, 189
operational space, 51, 56 orthogonal matrices, 2
pathwise optimization, 50 pathwise resolution of kinematic redundancy,
68 PD control, 112 planar mechanism, 16 planning, 189 polynomial systems, 28 posture, 189 potential method, 186 prehensile, 111 prehensile grasp, 103, 112 preimages, 142, 144, 152, 173 probabilistic strategies, 143 proprioceptive feedback, 189 push-forward, 173 pushing, 37
INDEX
rationalization, 191 Riemannian metric, 5 rigid motion, 92 robot, 21 robot hand dynamics, 107 rolling contact kinematics, 98
screw motions, 3 second etage controllable, 121 singular perturbation, 72 stable grasp, 103 strictly triangular systems, 122 structure theorems for elimination, 21, 33
task symmetries, 52 termination predicate, 145, 161, 167 torques, 13 twist, 93, 94
unavoidable singularities, 59
volume measure, 6
work, 14 wrists, 10
