Nonholonomic Mechanics and Control

Abstract

1R5. Nonholonomic Mechanics and Control. - AM Bloch (Dept of Math, Univ of Michigan, Ann Arbor MI 48109-1109). Springer-Verlag, New York. 2003. 483 pp. ISBN 0-387-95535-6. $69.95.Reviewed by B Brogliato (INRA, 655 Ave De L’Europe, Saint Ismier, 38334, France).This mathematically oriented book is dedicated to the modeling and control of a class of nonlinear mechanical systems, namely mechanical systems subject to nonholonomic (or non integrable) bilateral constraints. It is known that the control of such nonlinear systems requires specific tools, as they may not be stabilizable with continuous feedbacks, as a consequence of Brockett’s necessary conditions. Such problems have received considerable attention in the applied mathematics, mechanics, and systems and control communities, for many years. Therefore, this is a topic of major importance. The audience will mainly consist of graduate students, researchers, either working in this field or with a sufficiently advanced mathematical background (especially differential geometry tools). The book starts with an introductory chapter with many examples treated in detail. The next two chapters are devoted to presenting the mathematical tools needed to study such systems. This material will certainly be hard to follow for those who have no acquaintance in differential geometry and its applications to mechanics. However, in view of the literature on the subject, understanding the basics from geometry seems to be mandatory for someone who wants to understand the control of such mechanical systems. The fourth chapter is dedicated to nonlinear control theory, still with an emphasis on geometry. Controllability, stability and stabilization are reviewed, and the chapter ends with Hamiltonian and Lagrangian control systems. The fifth and sixth chapters are devoted to nonholonomic systems, their dynamics and their control. Stabilization techniques are explained, like time-varying and nonsmooth controllers. Here one can regret that the bibliography omits C Samson, one of the first contributors to the field and who introduced time-varying controllers for the control of chained systems in the paper Velocity and torque feedback control of a nonholonomic cart, in Advanced Robot Control, Proceedings of the International Workshop on Nonlinear and Adaptive Control: Issues in Robotics, Grenoble, France, November 21-23, 1990, Springer Verlag LNCIS 162. Samson’s paper Control of chained systems: Application to path following and time-varying point stabilization, IEEE Trans on Automatic Control, Volume 40, pp 64-77, 1995, would have also been welcome in the bibliography. Chapter 7 deals with optimal control and variational principles. The eighth chapter concerns stability analysis with energy arguments (generalization of the Lejeune-Dirichlet theorem on stability of fixed points). In a logical way the last chapter is devoted to energy-based control and stabilization. There are many other books that cover the topic of nonholonomic mechanical systems control in one or two chapters (for instance Sastry’s book in the same Springer’s series, or the book Theory of Robot Control, Springer CCE Series, 1996). But Nonholonomic Mechanics and Control is entirely dedicated to this topic and covers the aspects of modeling, analysis and control. It clearly belongs to the realm of geometry-oriented works in mechanics or control. The book is well organized, contains many examples and exercises, and can be recommended to all researchers working in the field (applied mathematics, mechanics or control).