Table Of ContentNONHOLONOMIC
MOTION PLANNING
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
ROBOTICS: VISION, MANIPULATION AND SENSORS
Consulting Editor: Takeo Kanade
KINEMATIC MODELING, IDENTIFICATION AND CONTROL OF ROBOT
MANIPULATORS, H.W. Stone
ISBN: 0-89838-237-8
OBJECT RECOGNITION USING VISION AND TOUCH, P. Allen
ISBN: 0-89838-245-9
INTEGRATION, COORDINATION AND CONTROL OF MULTI-SENSOR ROBOT
SYSTEMS, H.F. Durrant-Whyte
ISBN: 0-89838-247-5
BAYESIAN MODEUNG OF UNCERTAINTY IN LOW-LEVEL VISION, R. Szeliski
ISBN 0-7923-9039-3
VISION AND NA VIGATION: THE CMU NA VLAB, C. Thorpe (editor)
ISBN 0-7923-9068-7
TASK-DIRECTED SENSOR FUSION AND PLANNING: A Computational Approach,
G. D. Hager
ISBN: 0-7923-9108-X
COMPUTER ANALYSIS OF VISUAL TEXTURES, F. Tomita and S. Tsuji
ISBN: 0-7923-9114-4
DATA FUSION FOR SENSORY INFORMATION PROCESSING SYSTEMS, J. Clark
and A Yuille
ISBN: 0-7923-9120-9
PARALLEL ARCHITECTURES AND PARALLEL ALGORITHMS FOR INTEGRATED
VISION SYSTEMS, AN. Choudhary, J. H. Patel
ISBN: 0-7923-9078-4
ROBOT MOTION PLANNING, J. Latombe
ISBN: 0-7923-9129-2
DYNAMIC ANALYSIS OF ROBOT MANIPULATORS: A Carlesian Tensor Approach,
C.A Balafoutis, R.V. Patel
ISBN: 07923-9145-4
PERTURBATION TECHNIQUES FOR FLEXIBLE MANIPULATORS: A Fraser and
R. W. Daniel
ISBN: 0-7923-9162-4
COMPUTER AIDED MECHANICAL ASSEMBLY PLANNING: L. Homen de Mello and
S.Lee
ISBN: 0-7923-9205-1
INTELLIGENT ROBOTIC SYSTEMS FOR SPACE EXPLORATION: Alan A Desrochers
ISBN: 0-7923-9197-7
MEASUREMENT OF IMAGE VELOCITY: David J. Fleet
ISBN: 0-7923-9198-5
DIRECTED SONAR SENSING FOR MOBILE ROBOT NAVIGATION: John J. Leonard,
Hugh F. Durrant-Whyte
ISBN: 0-7923-9242-6
A GENERAL MODEL OF LEGGED LOCOMOTION ON NATURAL TERRAIN: David
J. Manko
ISBN: 0-7923-9247-7
INTELLIGENT ROBOTIC SYSTEMS: THEORY, DESIGN AND APPLICATIONS K.
Valavanis, G. Saridis
ISBN: 0-7923-9250-7
QUALITATIVE MOTION UNDERSTANDING: W. Burger, B. Bhanu
ISBN: 0-7923-9251-5
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 1s t 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
viii
constraint limits only the freedom of motion. The case of parallel
parking subject to rolling contact constraints serves as a perfect il
lustrative 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 an
other. The motions of a falling cat serve as another illustrative
example.
Research in traditional motion planning problem or the so
called 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 ob
ject. If the object is twirled through a cyclic motion which
returns it to its initial position and orientation, and the fin
gers roll without slipping on the surface of the object, then
they would not return to their initial positions. This fea
ture 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 mo
tion 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 ac
tivities 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 one
legged 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 Confer
ence 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 the
ory 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 in
cluding 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 Func
tions in Constructive Controllability", by Brockett and Dai.
In this paper the authors continue with and extend the ear
lier work of Brockett which showed that the optimal con
trollaws needed for steering drift free control systems with
control Lie algebra being the Heisenberg algebra were sinu
soids 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 Nonholo
nomic 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 nonholo
nomic motion planning. Analytic solutions of a class of non
holonomic 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 gener
ation of efficient motion planning algorithms for robots. A tremen
dous 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 Mo
tion Planning", by Laumond. In this survey paper, Lau
mond gives a comprehensive discussion of the aggregation
xii
of tools from controllability and the sub-Riemannian geom
etry 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 sys
tem.
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 con
nections:
1. "Nonholonomic Control and Gauge Theory" by Montgomery.
The author provides a dictionary between gauge theory and
