4 9 Texts in Applied Mathematics Editors J.E. Marsden L. Sirovich M. Golubitsky Advisors G. Iooss P. Holmes D. Barkley M. Dellnitz P. Newton Texts in Applied Mathematics I. Sirovich: Introduction to Applied Mathematics. 2. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed. 3. Hale/Ko~ak: Dynamics and Bifurcations. 4. Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed. 5. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Ordinary Differential Equations. 6. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed. 7. Perko: Differential Equations and Dynamical Systems, 3rd ed. 8. Seaborn: Hypergeometric Functions and Their Applications. 9. Pipkin: A Course on Integral Equations. I 0. Hoppensteadt/Peskin: Modeling and Simulation in Medicine and the Life Sciences, 2nd ed. II. Braun: Differential Equations and Their Applications, 4th ed. 12. Stoer!Bulirsch: Introduction to Numerical Analysis, 3rd ed. 13. Renardy/Rogers: An Introduction to Partial Differential Equations, 2nd ed. 14. Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and Applications. 15. Brenner/Scott: The Mathematical Theory of Finite Element Methods, 2nd ed. 16. Van de Velde: Concurrent Scientific Computing. 17. Marsden!Ratiu: Introduction to Mechanics and Symmetry, 2nd ed. 18. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Higher- Dimensional Systems. 19. Kaplan/Glass: Understanding Nonlinear Dynamics. 20. Holmes: Introduction to Perturbation Methods. 21. Curtain!Zwart: An Introduction to Infinite-Dimensional Linear Systems Theory. 22. Thomas: Numerical Partial Differential Equations: Finite Difference Methods. 23. Taylor: Partial Differential Equations: Basic Theory. 24. Merkin: Introduction to the Theory of Stability of Motion. 25. Naber: Topology, Geometry, and Gauge Fields: Foundations. 26. Po/derman/Willems: Introduction to Mathematical Systems Theory: A Behavioral Approach. 27. Reddy: Introductory Functional Analysis with Applications to Boundary-Value Problems and Finite Elements. 28. Gustafson/Wilcox: Analytical and Computational Methods of Advanced Engineering Mathematics. 29. Tveito/Winther: Introduction to Partial Differential Equations: A Computational Approach. 30. Gasquet/Witomski: Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets. (continued after index) Francesco Bullo Andrew D. Lewis Geometric Control of Mechanical Systems Modeling, Analysis, and Design for Simple Mechanical Control Systems With 102 Illustrations ~Springer Francesco Bullo Andrew D. Lewis Department of Mechanical & Department of Mathematics Environmental Engineering and Statistics University of California Queen's University at Santa Barbara Kingston, ON K7L 3N6 Santa Barbara, CA 93106-5070 Canada USA [email protected] [email protected] Series Editors J.E. Marsden L. Sirovich Control and Dynamical Systems, 107-81 Laboratory of Applied Mathematics California Institute of Technology Department of Biomathematical Sciences Pasadena, CA 91125 Mount Sinai School of Medicine USA New York, NY 10029-6574 [email protected] M. Golubitsky Department of Mathematics University of Houston Houston, TX 77204-34 76 USA Mathematics Subject Classification (2000): 93XX Library of Congress Cataloging-in-Publication Data Bullo, Francesco. Geometric control of mechanical systems : modeling, analysis, and design for simple mechanical control systems : monograph / Francesco Bullo and Andrew D. Lewis. p. cm. Includes bibliographical references and index. ISBN 978-1-4419-1968-7 (alk. paper) l. Automatic control. 2. Geometry, Differential. 1. Lewis, Andrew D. Il. Title. 1]213.87835 2004 629.8-dc22 2004052220 ISBN 978-1-4419-1968-7 ISBN 978-1-4899-7276-7 (eBook) DOI 10.1007/978-1-4899-7276-7 © 2005 Springer Science+Business Media New York Originally published by Springer Science+Business Media, Inc. in 2005 Softcover reprint oftbe hardcover 1st edition 2005 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with anv form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. (MP) 9 8 7 6 5 4 3 2 1 SPIN 10942574 springeronline.com To our families Series Preface Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in re search and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numeri cal and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe matical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich Houston, Texas M. Golubitsky Preface In the course of the past decade, a great deal of progress has been made in the theory and application of ideas in nonlinear control theory to mechanical systems. The areas of application of control theory for mechanical systems are diverse and challenging, and this constitutes an important factor for the inter est in these systems. Such areas of application include robotics and automa tion, autonomous vehicles in marine, aerospace, and other environments, flight control, problems in nuclear magnetic resonance, micro-electromechanical sys tems, and fluid mechanics. What is more, the areas of overlap between me chanics and control possess the sort of mathematical elegance that makes them appealing to study, independently of applications. This book is an outgrowth of our research efforts in the area of mechan ics and control, and is a development of separate courses offered by us at our respective institutions, at both the advanced undergraduate and graduate lev els. The book reflects our point of view that differential geometric thinking is useful, both in nonlinear control theory and in mechanics. Indeed, the pri mary emphasis of the book is the exploration of areas of overlap between mechanics and control theory for which differential geometric tools are useful. This area of overlap can be essentially characterized by whether one adopts a Hamiltonian or a Lagrangian view of mechanics. In the former, symplectic or Poisson geometry dominates. On the Lagrangian side, it is less clear what is the most useful geometric framework. We sidestep this to some extent by restricting our attention to the special class of mechanical systems known in the literature as "simple." 1 Simple mechanical systems are characterized by the fact that their Lagrangian is kinetic energy minus potential energy. While it is certainly true that simple mechanical systems are not completely general, it is also true that a very large number of applications, indeed the majority 1 This terminology appears to originate with Smale [1970]. As the reader might conclude after going through the material in our book, the word "simple" should not be taken to have its colloquial meaning. In some of the literature on mechanics, those systems that we call "simple" are called "natural." x Preface considered in the research literature, fall into the "simple" category. Simple mechanical systems also offer the advantage of providing the useful geometric structure of a Riemannian metric, with its attendant Levi-Civita affine con nection. It is this Riemannian geometric, or more generally affine differential geometric, point of view that dominates the presentation in this book. Intended audience This book is intended to serve as a reference book for graduate students and for researchers wishing to learn about the affine connection approach to control theory for mechanical systems. The book is also intended to be a textbook for graduate students and undergraduates with a suitable background. Let us say a few words about what a suitable background might be, and then a few words about which portion of the research audience we are aiming for. Prerequisites. Students who use this book as a text should have a back ground in analysis, linear algebra, and differential equations. An analysis course, rigorously covering such topics as continuity, differentiability, and con vergence, is essential. At Queen's University the book has been used as a text for students with a second-year course in analysis. Linear algebra beyond a first course is also essential. Students will be expected to have a knowledge of abstract vector spaces, linear maps, norms, and inner products. At Queen's a sufficient background is achieved with two courses in linear algebra. A basic course in differential equations may be sufficient, although an advanced course will be very useful. An undergraduate instructor teaching students not having seen this material can be sure to spend a significant amount of time in review using external sources. For graduate students in engineering, our experience is that they will either have had this material, or are capable of filling in the gaps as necessary. At the University of Illinois at Urbana-Champaign the course is offered as a first-year, second-semester graduate course. Students are required to take a prerequisite introductory graduate course on linear control theory. More will be said later in this preface on the use of the book as a text. A note to researchers. We are certainly aware that a great deal of valuable research is done on control theory for mechanical systems without the aid of the mathematical methodologies advocated by this book. We acknowledge that certain examples of mechanical control systems, and certain general prob lems in the control theory of mechanical systems, can be treated effectively without the aid of differential geometry. Furthermore, we are sympathetic to the fact that a significant effort is required in order for the reader to be come comfortable with the necessary mathematics. However, at some point the unity offered by a differential geometric treatment becomes advantageous and we feel that this is merely a necessary part of the subject, as we see it. It was our objective to write a book about a class of mechanical control systems, and not about specific examples. In our approach, examples provide motiva tion, and they provide a testing ground for general ideas. Nonetheless, we Preface xi hope that the effectiveness of our ideas, as applied to specific examples, will provide some impetus to the reader unsure of whether they should invest the effort in learning the necessary background. Additionally, one of the features of much of the literature on geometric mechanics and on geometric nonlinear control theory, is that it is written in a precise, mathematical style. We have chosen not to deny this mathematical sophistication. Rather, we have adopted the approach of trying to prepare the uninitiated reader, by providing, under one cover, the necessary background as well as the motivation for what is a demanding mathematical formalism. Objectives We state our objectives roughly in descending degree of generality. Broad aim. Broadly, our objectives were to write a book that could serve as a textbook for instructors with some knowledge of the subject area, and to write a book that would be a reference book, stating some of the important results in the field, and providing a guide to the literature for others. These objectives need not be at odds. Indeed, in terms of a graduate text, the objectives align quite nicely, since students can at the same time learn the basics of the subject, and get a glimpse at some advanced material. For example, at the University of Illinois at Urbana-Champaign this text has been used as an introduction to current research results on nonlinear control of mechanical systems. At the same time, the text has also been used for an advanced undergraduate course at Queen's University. In this capacity the book serves to illustrate to students the value of certain mathematical ideas in the physical sciences, and it serves as a spark to some students to pursue research in related fields. Control theory for mechanics, and mechanics for control theory. Another objective of the book is to provide a background in (parts of) ge ometric mechanics for researchers familiar with geometric nonlinear control, and to provide a background in (parts of) geometric nonlinear control for read ers familiar with geometric mechanics. This objective is served in two ways. For readers familiar with geometric nonlinear control theory, we describe in some detail the mathematical structure of the physical models we consider. We also clarify the connections between (some of) the tools of the nonlinear control theoretician (e.g., Lyapunov functions, Lie brackets) and (some of) the tools of the geometric mechanic (e.g., Riemannian metrics, nonholonomic constraints). For the reader with a background in geometric mechanics, we provide, in each chapter dealing with matters of control theory, a thorough overview of that area of control theory. It should be noted that the resulting coverage of control theory we provide is quite biased to serving our sometimes rather focused objectives. Therefore, a reader should not feel as if our treat ment of nonlinear control is even close to comprehensive. We refer the reader to the texts and monographs [Agrachev and Sachkov 2004, Bloch 2003, Isidori 1995, 1999, Jurdjevic 1997, Khalil 2001, Nijmeijer and van der Schaft 1990,