NONHOLONOMIC MOTION PLANNING

THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE

ROBOTICS: VISION, MANIPULATION AND SENSORS

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NONHOLONOMIC MOTION PLANNING

Edited by:

Zexiang Li Dept. of Electrical & Electronic Engineering

Hong Kong University of Science & Technology Clear Water Bay

Kowloon, Hong Kong

J.F. Canny Dept. of Electrical Engineering & Computer Sciences

University of California at Berkeley Berkeley, CA 94720

.... " SPRINGER SCIENCE+BUSINESS MEDIA, LLC

Library oC Congress Cataloging-in-Publication Data

Nonholonomic motion planning / edited by Zexiang Li, J.F. Canny. p. cm. -- (The Kluwer international series in engineering and

computer science. Robotics) Inc1udes bibliographical references and index.

ISBN 978-1-4613-6392-7 ISBN 978-1-4615-3176-0 (eBook) DOI 10.1007/978-1-4615-3176-0 1. Robots--Motion. 1. Li, Zexiang, 1961- . II. Canny, John. III. Series. TJ211.4.N66 1992 629. 8' 92--dc20 92-27560

CIP

Copyright © 1993 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 1993 Softcover reprint of the hardcover 1 st edition 1993

Ali rights reserved. No part of this publication may be reproduced, stored in a retrieval system ar transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+ Business Media, LLC.

Printed on acid-free paper.

TABLE OF CONTENTS

Introduction .................................. Vll

1. Non-holonomic Kinematics and the Role of Elliptic Functions in Constructive Controllability - R.W. Brockett, Liyi Dai ....... 1

2. Steering Nonholonomic Control Systems Using Sinusoids -Richard M. Murray, S. Shankar Sastry ................... 23

3. Smooth Time-Periodic Feedback Solutions for Nonholonomic Motion Planning - L. Gurvits, Z.X. Li ................... 53

4. Lie Bracket Extensions and Averaging: The Single-Bracket Case - Hector J. Sussmann, Wensheng Liu ................ 109

5. Singularities and Topological Aspects in Nonholonomic Motion Planning - Jean-Paul Laumond ................... 149

6. Motion Planning for Nonholonomic Dynamic Systems-Mahmut Reyhanoglu, N. Harris McClamroch, Anthony M. Bloch .. 201

7. A Differential Geometric Approach to Motion Planning-Gerardo Lafferriere, Hector J. Sussmann .................. 235

8. Planning Smooth Paths for Mobile Robots-Paul Jacobs, John Canny ............................ 271

9. Nonholonomic Control and Gauge Theory-Richard Montgomery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

10. Optimal Nonholonomic Motion Planning for a Falling Cat-C. Fernandes, L. Gurvits, Z.X. Li ..................... 379

11. Nonholonomic Behavior in Free-floating Space Manipulators and its Utilization - Evangelos G. Papadopoulos ............. 423

Index ....................................... 447

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constraint limits only the freedom of motion. The case of parallel parking subject to rolling contact constraints serves as a perfect il­lustrative example. First, if cars were able to move sideways, then a careless mistake can cause a severe car accident. Second, given adequate knowledge about the constraint, one can actively explore or maneuver the constraint to move from one configuration to an­other. The motions of a falling cat serve as another illustrative example.

Research in traditional motion planning problem or the so­called Piano Movers' Problem has a long history, but research in Nonholonomic Motion Planning is very recent, and has been driven by studies of the following problems.

1. Mobile robots navigating in a cluttered environment. The kinematics of the driving mechanisms of robot carts results in constraints on the instantaneous velocities that they can achieve. For instance, a cart with two forward drive wheels and two back wheels cannot move sideways. This was first pointed out by Laumond in the context of motion planning for a mobile robot at Toulouse, France ([LS89]).

2. Multifingered hands rolling on the surface of a grasped ob­ject. If the object is twirled through a cyclic motion which returns it to its initial position and orientation, and the fin­gers roll without slipping on the surface of the object, then they would not return to their initial positions. This fea­ture can be used to plan the regrasp of a poorly grasped object or to choose a good initial grasp. This application of nonholonomic motion planning was first pointed out by Li ([LC90] and [LCS89]). A formulation of nonholonomic mo­tion planning in the framework of a nonlinear control system was given in ([LC90l, [LCS89] and [MS90]).

3. Space Robots. Unanchored robots in space are difficult to control with either thrusters or internal motors since they conserve total angular momentum. In fact the motion of astronauts on space walks is of this ilk, so that the planning of the strategy required to reorient an astronaut is a problem

ix

of this nature (also gymnasts and divers are good examples of this kind of motion planning). Application of nonholonomic motion planning in studying these problems can be found in ([VDS7], [NM90), [LM90], [Kri90), [BMR90] and [WS91)).

4. Ultrasonic Motors. Based on the traveling wave method and nonholonomic constraints, Panasonic Inc. developed a class of ultrasonic motors whose torque-to-mass ratio could be orders of magnitude higher than that of traditional motors. Operating principles and controls of the ultrasonic motors were studied in [BroSS].

The link made between nonholonomic motion planning and constructive nonlinear control has sparked a series of research ac­tivities in this problem: (1) more applications of nonholonomic motion planning are being found in planning/control of systems such as redundant robots, space based robots and even a one­legged hopper; (2) classical and modern tools in nonlinear and optimal control, starting with the works of Hermann in 1963 and continuing with the works of Brockett, Krener, Sussmann and others (e.g., see [BroS1], [SusS3), [HK77] and [HH70)) are being applied to this problem; (3) differential geometric methods and classical mechanics are being utilized to characterize solutions of nonholonomic motion planning ([Mon90)); (4) new algorithms and approaches are being developed for computing collision-free paths of nonholonomic motion planning.

This book grew out of the Workshop on Nonholonomic Motion Planning that took place at the 1991 IEEE International Confer­ence on Robotics and Automation. It consists of a collection of papers representing new development in this area. Contributors of the book include robotics engineers, nonlinear control experts, differential geometers and applied mathematicians.

The chapters of this book may be arranged into three groups:

x

1 Controllability

One of the key mathematical tools needed to study nonholonomic motion planning is nonlinear controllability. While nonholonomic constraints usually appear as differential form constraints, they can be easily dualized to give steering problems in the form of control systems. There is a long history of activity in control the­ory and optimal control for steering problems on ~n as well as Lie Groups and other manifolds beginning with the work of Hermann in 1963 and continuing with the work of several researchers in­cluding Brockett, Krener, Sussmann and others([Bro81]' [Sus83], [HK77], [HH70] for example). In the present collection we have four papers in the area of controllability:

1. "Nonholonomic Kinematics and the Role of Elliptic Func­tions in Constructive Controllability", by Brockett and Dai. In this paper the authors continue with and extend the ear­lier work of Brockett which showed that the optimal con­trollaws needed for steering drift free control systems with control Lie algebra being the Heisenberg algebra were sinu­soids at integrally related frequencies. In the current paper, the authors show that for certain other control Lie algebras, which are nilpotent but involve two layers of brackets, the optimal steering inputs are elliptic functions. They apply the theory to the regrasping of a ball grasped between two parallel plates.

2. "Steering Nonholonomic Control Systems Using Sinusoids", by Murray and Sastry. In this paper the authors extend the results of Brockett alluded to above to constructively steer several classes of nonholonomic systems using sinusoids and some Fourier analysis even when their control Lie algebras are not Heisenberg algebras. The classes of systems steerable are referred to as "chained form systems" .

3. "Smooth Time Periodic Feedback Solutions for Nonholo­nomic Motion Planning", by Gurvits and Li. In this paper, the authors use averaging techniques to compute asymptotic

xi

behaviors of nonholonomic systems under application of a class of highly oscillatory inputs and present an algorithm for computing time-periodic feedback solutions for nonholo­nomic motion planning. Analytic solutions of a class of non­holonomic systems using Fourier analysis are also discussed.

4. "Lie Bracket Extensions and Averaging: the Single Bracket Case", by Sussmann and Liu. Here Sussmann and Liu show how highly oscillatory high amplitude sinusoids may be used to generate Lie bracket motion in a drift free control system. Their method is general (though only the "Single-Bracket" case is discussed here) and enables the generation of inputs required to steer the system along the basis directions of a Phillip Hall basis for the control Lie algebra. Consequently, algorithms for steering the control system along all possible directions are given.

2 Motion Planning for Mobile Robots

In recent years there has been a great deal of activity in the gener­ation of efficient motion planning algorithms for robots. A tremen­dous variety of algorithms for particular motion planning problems have appeared, both heuristic and guaranteed. Most of this work has focussed on the "Piano Movers' Problem" and generalizations, where the obstacle positions are known, and dynamical constraints are not considered. In this section on motion planning the papers are focussed on problems with non-holonomic velocity constraints as well as constraints on the generalized coordinates. These lead to the most difficult planning problems, since it is non-trivial to find a path even in the absence of obstacles. There are three papers in this area:

1. "Singularity and Topological Aspects in Nonholonomic Mo­tion Planning", by Laumond. In this survey paper, Lau­mond gives a comprehensive discussion of the aggregation

xii

of tools from controllability and the sub-Riemannian geom­etry to solve the problem of steering a mobile robot with n trailers in a field of obstacles.

2. "Motion Planning for Nonholonomic Dynamic Systems", by Reyhanoglu, McClamroch and Bloch. The authors develop a class of nonholonomic systems called Caplygin systems, whose constraints are cyclic in the variables. Methods for steering such systems are discussed and applied to a knife edge moving on a planar surface or a planar multi-body sys­tem.

3. "A Differential Geometric Approach to Motion Planning", by Lafferriere and Sussmann. The authors develop methods for steering drift free systems using the basic definition of the Lie Bracket and feedback nilpotentization of the associated control Lie algebra.

4. "Planning Smooth Paths for Mobile Robots", by Jacobs and Canny. The authors define a set of canonical trajectories that satisfy non-holonomic constraints and present a graph search algorithm to compute approximate paths for mobile robots.

3 Falling Cats, Space Robots and Gauge Theory

There are numerous connections to be made between symplectic geometry techniques for the study of holonomies in mechanics, gauge theory and control. In this section, using the back drop of examples drawn from space robots and falling cats reorienting themselves, we have three papers which aim to make these con­nections:

1. "Nonholonomic Control and Gauge Theory" by Montgomery. The author provides a dictionary between gauge theory and

xiii

control and shows how it is useful to study the control of deformable bodies (cats, gymnasts, astronauts). It also con­tains some useful results on stabilization.

2. "Optimal Nonholonomic Motion Planning for a Falling Cat", by Fernandes, Gurvits and Li. Here the authors work out the details of how a falling cat might change her orientation in mid-air to land on her feet. First, they modeled a falling­cat as two rigid bodies coupled by a universal or a ball-in­socket joints. Then, they applied Ritz approximation theory in functional analysis to reduce the motion planning prob­lem into a finite dimensional optimization problem. Finally, they developed a Basis Algorithm to solve the optimization problem.

3. "Nonholonomic Behavior in Free-floating Space Manipula­tors and its Utilization", by Papadopoulos. The author dis­cusses the reorientation of a spacecraft using internal mo­tions of an on-board manipulator.

We invite the reader to plunge into this exciting fusion which is a representative but not complete sample of the fantastic amount of new literature in this field. Nonholonomies are around us ev­erywhere: for example, walking and swimming are cyclic motions which produce a forward drift!

Finally, we would like to thank all contributors whose efforts have made this workshop and book possible. We would also like to thank J. Burdick of California Institute of Technology and J. Wen of Rensselaer Polytechnic Institute for co-organizing the workshop.

S.S. Sastry, Berkeley, California Z.X. Li, New York, New York

xiv

References

[BMR90] A. Bloch, N.H. McClamroch, and M. Reyhanoglu. Con­trollability and stabilizability properties of a nonholo­nomic control system. In Conf. on Decision and Con­trol, pages 1312-1314, 1990.

[Bro81] R. Brockett. Control theory and singular riemannian geometry. In P. Hilton and G. Young, editors, New Di­rections in Applied Mathematics, pages 11-27. Springer­Verlag, 1981.

[Bro88] R. W. Brockett. On the rectification of vibratory mo­tion. In Proceedings of Microactuators and Micromech­anism, Salt Lake City, Utah, 1988.

[HH70] G.W. Haynes and H. Hermes. Nonlinear controllability via lie theory. SIAM J. Control, 8(4):450-460, 1970.

[HK77] R. Hermann and A.J. Krener. Nonlinear controllability and observability. IEEE Trans. on Automatic Control, 22:728-740, 1977.

[Kri90] P.S. Krishnaprasad. Geometric phases, and optimal re­configuration for multibody systems. Technical report, System research center, University of Maryland at Col­lege Park, 1990.

[LC90] Z. Li and J. Canny. Motion of two rigid bodies with rolling constraint. IEEE Trans. on Robotics and A u­tomation, RA2-06:62-72, 1990.

[LCS89] Z.X. Li, J. F. Canny, and S.S. Sastry. On motion plan­ning for dextrous manipulation. In IEEE International conference on robotics and automation, 1989.

[LM90] Z. Li and R. Montgomery. Dynamics and optimal con­trol of a legged robot in flight phase. In Proceedings of IEEE Int. Conf. on Robotics and Automation, 1990.

xv

[LS89] J.P. Laumond and T. Simeon. Motion planning for a two degrees of freedom mobile robot with towing. Tech­nical report, LAAS/CNRS, Report 89148, Toulouse, France, 1989.

[Mon90] R. Montgomery. Isoholonomic problems and some ap­plications. Communication in Mathematical Physics, pages 128:565-592, 1990.

[MS90] R. Murray and S. Sastry. Grasping and manipulation using multifingered robot hands. In R.W. Brockett, edi­tor, Robotics: Proceedings of Symposia in Applied Math­ematics, volume 41, pages 91-128. American Mathemat­ical Society, 1990.

[NM90] Y. Nakamura and R. Mukherjee. Nonholonomic path planning of space robotics via bidirectional approach. In Proceedings of IEEE Int. Conf. on Robotics and A u­tomation, 1990.

[Sus83] H. J. Sussmann. Lie brackets, real analyticity and geo­metric control. In R. Millman R.W. Brockett and H.J. Sussman, editors, Differential Geometric Control The­ory. Birkhauser, Boston, 1983.

[VD87] Z. Vafa and S. Dubowsky. On the dynamics of manipu­lators in space using the virtual manipulator approach. In Proceedings of IEEE Int. Conf. on Robotics and A u­tomation, 1987.

[WS91] G. Walsh and S. Sastry. On reorienting linked rigid bod­ies using internal motions. In IEEE Conf. on Decision and Control, 1991.