Nonholonomic Mechanics and Control

1. Front Matter

   Pages i-xix

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2. Introduction

   • A. M. Bloch

   Pages 1-48

3. Mathematical Preliminaries

   • A. M. Bloch

   Pages 49-118

4. Basic Concepts in Geometric Mechanics

   • A. M. Bloch

   Pages 119-174

5. An Introduction to Aspects of Geometric Control Theory

   • A. M. Bloch

   Pages 175-205

6. Nonholonomic Mechanics

   • A. M. Bloch

   Pages 207-276

7. Control of Mechanical and Nonholonomic Systems

   • A. M. Bloch

   Pages 277-327

8. Optimal Control

   • A. M. Bloch

   Pages 329-366

9. Stability of Nonholonomic Systems

   • A. M. Bloch

   Pages 367-397

10. Energy-Based Methods for Stabilization of Controlled Lagrangian Systems

    • A. M. Bloch

    Pages 399-437

11. Back Matter

    Pages 439-484

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Our goal in this book is to explore some of the connections between control theory and geometric mechanics; that is, we link control theory with a g- metric view of classical mechanics in both its Lagrangian and Hamiltonian formulations and in particular with the theory of mechanical systems s- ject to motion constraints. This synthesis of topics is appropriate, since there is a particularly rich connection between mechanics and nonlinear control theory. While an introduction to many important aspects of the mechanics of nonholonomically constrained systems may be found in such sources as the monograph of Neimark and Fufaev [1972], the geometric view as well as the control theory of such systems remains largely sc- tered through various research journals. Our aim is to provide a uni?ed treatment of nonlinear control theory and constrained mechanical systems that will incorporate material that has not yet made its way into texts and monographs. Mechanicshastraditionallydescribedthebehavioroffreeandinteracting particles and bodies, the interaction being described by potential forces. It encompasses the Lagrangian and Hamiltonian pictures and in its modern form relies heavily on the tools of di?erential geometry (see, for example, Abraham and Marsden [1978]and Arnold [1989]). From our own point of view,ourpapersBloch,Krishnaprasad,Marsden,andMurray[1996],Bloch and Crouch [1995], and Baillieul [1998] have been particularly in?uential in the formulations presented in this book. Control Theory and Nonholonomic Systems. Control theory is the theory of prescribing motion for dynamical systems rather than describing vi Preface their observed behavior.

• Optimal control
• control
• control theory
• mechanics
• nonlinear control
• stability
• stabilization

From the reviews:

"As aptly formulated in its preface, Nonholonomic Mechanics and Control links control theory with a geometric view of classical mechanics in both its Lagrangian and Hamiltonian formulations and in particular, with the theory of mechanical systems subject to kinematic motion constraints. The analysis and control of mechanical systems has been an active research area over the last several decades. This book aims to present some of this material, often scattered throughout the literature, in a cohesive manner. … Although, unavoidably, the opening chapters provide only a ‘crash course’ at some points, the material has been written with much care. In fact, in many cases, the clarity of the presentation is unmatched elsewhere in the literature. … This book is a welcome addition to the existing literature and will certainly become a standard reference. … It is to be expected that Bloch’s book will be a continuing source of inspiration for further research in this area. … In summary, this is a delightful book that will be valuable for both the control community and researchers working on the geometric theory of mechanical systems. With its extensive illustrations and exercises, this book is eminently suited for a graduate course. The author should be congratulated for such an admirable job." (A.J. van der Schaft, IEEE Control Systems Magazine, June 2005)

 

"… A well-written and comprehensive reference that can be used as a graduate-level textbook, complete with exercises. The book also gives a nice history of the development of the methods covered, and it is an excellent resource for references for further reading. … The book’s background material in differential geometry, mechanics, and geometric control gives the book a timeless feel, while the extensive coverage of recent research results conveys the vitality of the field. This book will make an excellent reference for students andresearchers in differential geometric control. It will also make a fine textbook for engineering graduate students with a background in differential geometry. For graduate students with a more traditional engineering mathematics background, portions of this book can be coupled with supplementary reading and example-based introduction of the relevant mathematical concepts." (IEEE Transactions on Automatic Control)

"Nonholonomic Mechanics and Control links control theory with a geometric view of classical mechanics … . This book is a welcome addition to the existing literature and will certainly become a standard reference. … Bloch’s book will be a continuing source of inspiration for further research in this area. … this is a delightful book that will be valuable for both the control community and researchers … . is eminently suited for a graduate course. The author should be congratulated for such an admirable job." (A.J. van der Schaft, IEEE Control System Magazine, June, 2005)

"The book … aims to explore connections between control theory and geometric mechanics. … This book can be considered as a mixture between a research monograph and a textbook, as a lot of exercises … are given. … contains an exhaustive list of contemporary references. … After reading this book the reader will be convinced that the aim of the book … is successfully reached. … it is a nice and useful learning tool for scientists and engineers from academia and industry." (Clementina Mladenova, Zentralblatt MATH, Vol. 1045 (20), 2004)

"The aim of this book is ‘to provide a unified treatment of nonlinear control theory and constrained mechanical systems that will incorporate material that has not yet made its way into texts and monographs’. … the authors of the present book have contributed a great deal to the way in which nonholonomic systems are currently understood … . The book constitutes an accurate reflection of this work,and covers a broad variety of topics and problems concerning nonholonomic systems and control." (Jorge Cortes, Mathematical Reviews, 2004e)

"The focus of this book is mechanics, controllability, and feedback stabilization of mechanical control systems … . The result is a well-written and comprehensive reference … . The book’s background material … gives the book a timeless feel, while the extensive coverage of recent research results conveys the vitality of the field. This book will make an excellent reference for students and researchers in differential geometric control. It will also make a fine textbook for engineering graduate students … ." (Kevin M. Lynch, IEEE Transactions on Automatic Control, Vol. 49 (5), May, 2004)

• Deparment of Mathematics, University of Michigan, Ann Arbor, USA

  A. M. Bloch

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