Lecture Notes in Mathematics Editors: J.–M Morel, Cachan F Takens, Groningen B Teissier, Paris 1793 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Jorge Cort´es Monforte Geometric, Control and Numerical Aspects of Nonholonomic Systems 13 Author Jorge Cort´es Monforte Systems, Signals and Control Department Faculty of Mathematical Sciences University of Twente P.O Box 217 7500 AE Enschede Netherlands e-mail: j.cortesmonforte@math.utwente.nl http://www.math.utwente.nl/ssb/cortesmonforte.htm Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Cortés Monforte, Jorge: Geometric, control and numeric aspects of nonholonomic systems / Jorge Cortés Monforte - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in mathematics ; 1793) ISBN 3-540-44154-9 Cover illustration by Mar´ıa Cort´es Monforte Mathematics Subject Classification (2000): 70F25, 70G45, 37J15, 70Q05, 93B05, 93B29 ISSN 0075-8434 ISBN 3-540-44154-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: Camera-ready TEX output by the author SPIN: 10884692 41/3142/ du - 543210 - Printed on acid-free paper A mi padre, mi madre, Ima y la Kuka Preface Nonholonomic systems are a widespread topic in several scientific and commercial domains, including robotics, locomotion and space exploration This book sheds new light on this interdisciplinary character through the investigation of a variety of aspects coming from different disciplines Nonholonomic systems are a special family of the broader class of mechanical systems Traditionally, the study of mechanical systems has been carried out from two points of view On the one hand, the area of Classical Mechanics focuses on more theoretically oriented problems such as the role of dynamics, the analysis of symmetry and related subjects (reduction, phases, relative equilibria), integrability, etc On the other hand, the discipline of Nonlinear Control Theory tries to answer more practically oriented questions such as which points can be reached by the system (accessibility and controllability), how to reach them (motion and trajectory planning), how to find motions that spend the least amount of time or energy (optimal control), how to pursue a desired trajectory (trajectory tracking), how to enforce stable behaviors (point and set stabilization), Of course, both viewpoints are complementary and mutually interact For instance, a deeper knowledge of the role of the dynamics can lead to an improvement of the motion capabilities of a given mechanism; or the study of forces and actuators can very well help in the design of less costly devices It is the main aim of this book to illustrate the idea that a better understanding of the geometric structures of mechanical systems (specifically to our interests, nonholonomic systems) unveils new and unknown aspects of them, and helps both analysis and design to solve standing problems and identify new challenges In this way, separate areas of research such as Mechanics, Differential Geometry, Numerical Analysis or Control Theory are brought together in this (intended to be) interdisciplinary study of nonholonomic systems Chapter presents an introduction to the book In Chapter we review the necessary background material from Differential Geometry, with a special emphasis on Lie groups, principal connections, Riemannian geometry and symplectic geometry Chapter gives a brief account of variational principles in Mechanics, paying special attention to the derivation of the non- VIII Preface holonomic equations of motion through the Lagrange-d’Alembert principle It also presents various geometric intrinsic formulations of the equations as well as several examples of nonholonomic systems The following three chapters focus on the geometric aspects of nonholonomic systems Chapter presents the geometric theory of the reduction and reconstruction of nonholonomic systems with symmetry At this point, we pay special attention to the so-called nonholonomic bracket, which plays a parallel role to that of the Poisson bracket for Hamiltonian systems The results stated in this chapter are the building block for the discussion in Chapter 5, where the integrability issue is examined for the class of nonholonomic Chaplygin systems Chapter deals with nonholonomic systems whose constraints may vary from point to point This turns out in the coexistence of two types of dynamics, the (already known) continuous one, plus a (new) discrete dynamics The domain of actuation and the behavior of the latter one are carefully analyzed Based on recent developments on the geometric integration of Lagrangian and Hamiltonian systems, Chapter deals with the numerical study of nonholonomic systems We introduce a whole new family of numerical integrators called nonholonomic integrators Their geometric properties are thoroughly explored and their performance is shown on several examples Finally, Chapter is devoted to the control of nonholonomic systems After exposing concepts such as configuration accessibility, configuration controllability and kinematic controllability, we present known and new results on these and other topics such as series expansion and dissipation I am most grateful to many people from whom I have learnt not only Geometric Mechanics, but also perseverance and commitment with quality research I am honored by having had them as my fellow travelers in the development of the research contained in this book Among all of them, I particularly would like to thank Manuel de Le´ on, Frans Cantrijn, Jim Ostrowski, Francesco Bullo, Alberto Ibort, Andrew Lewis and David Mart´ın for many fruitful and amusing conversations I am also indebted to my family for their encouragement and continued faith in me Finally, and most of all, I would like to thank Sonia Mart´ınez for the combination of enriching discussions, support and care which have been the ground on which to build this work Enschede, July 2002 Jorge Cort´es Monforte Table of Contents Preface VII Introduction 1.1 Literature review 1.2 Contents Basic geometric tools 2.1 Manifolds and tensor calculus 2.2 Generalized distributions and codistributions 2.3 Lie groups and group actions 2.4 Principal connections 2.5 Riemannian geometry 2.5.1 Metric connections 2.6 Symplectic manifolds 2.7 Symplectic and Hamiltonian actions 2.8 Almost-Poisson manifolds 2.8.1 Almost-Poisson reduction 2.9 The geometry of the tangent bundle 13 13 17 18 23 24 26 29 30 32 33 34 Nonholonomic systems 3.1 Variational principles in Mechanics 3.1.1 Hamilton’s principle 3.1.2 Symplectic formulation 3.2 Introducing constraints 3.2.1 The rolling disk 3.2.2 A homogeneous ball on a rotating table 3.2.3 The Snakeboard 3.2.4 A variation of Benenti’s example 3.3 The Lagrange-d’Alembert principle 39 39 39 42 43 45 47 49 50 51 ... )) = G(∇Z X, Y ) + G(X, ∇Z Y ) , X(G(Z, Y )) = G(∇X Z, Y ) + G(Z, ∇X Y ) , Y (G(X, Z)) = G(∇Y X, Z) + G(X, ∇Y Z) , for all X, Y, Z ∈ X(Q) Now Z(G(X, Y )) + X(G(Z, Y )) − Y (G(X, Z)) = G(∇X Z... better understanding of the geometric structures of mechanical systems (specifically to our interests, nonholonomic systems) unveils new and unknown aspects of them, and helps both analysis and design... then d(α|U ) = (dα)|U , where α ∈ Ω k (V ) Let f : Q −→ N be a smooth mapping and ω ∈ Ω k (N ) Define the pullback f ∗ ω of ω by f as f ∗ ω(q)(v1 , , vk ) = ω(f (q))(Tq f (v1 ), , Tq f (vk