Nonholonomic mechanics and control. With the collaboration of J. Baillieul, P. Crouch, and J. Marsden. With scientific input from P. S. Krishnaprasad, R. M. Murray, and D. Zenkov.

*(English)*Zbl 1045.70001
Interdisciplinary Applied Mathematics 24. New York, NY: Springer (ISBN 0-387-95535-6/hbk). xix, 483 p. (2003).

The book under review aims to explore connections between control theory and geometric mechanics. The authors link control theory with geometrical methods of classical mechanics (both Lagrangian and Hamiltonian formulations), especially with the theory of mechanical systems subject to motion constraints. This link is quite reasonable because there is a very close connection between mechanics and nonlinear control theory which is nicely presented in diagrams inside the book and depicted at the very beginning. This book can be considered as a mixture between a research monograph and a textbook, as a lot of exercises, especially in the first chapter, are given. The book is organized in nine chapters and contains an exhaustive list of contemporary references.

Chapter I “Introduction” is devoted to basic ideas of mechanics, nonholonomic mechanics, control, and optimal control and a lot of classical illustrative examples are given. Chapter II “Mathematical preliminaries” covers a fairly wide topics from mathematics that are used in the next chapters within the area of differential manifolds, Lie groups and Lie algebras. Chapter III “Basic concepts in geometric mechanics” summarizes and, where necessary, develops the basic concepts in geometric mechanics exploring the geometric view of mechanics. It is stressed that its two branches, i.e. Hamiltonian and Lagrangian mechanics, are able to serve as two starting points in mechanics. The Lagrangian side of mechanics focuses on variational principles for its basic formulation, while the Hamiltonian one focuses on geometric structures called symplectic (Poisson) structures.

Chapter IV “An introduction to aspects of geometric control theory” touches a few aspects that are important for the major topics, and the advances made by Jurdjevic, Isidori, Nijmeijer and van der Schaft, Sontag and Brockett, Sussman and etc. are explained at appropriate points.

Chapter V “Nonholonomic mechanics” is a basic chapter on nonholonomic mechanics and discusses the basic geometric approach following the papers of Bloch, Krishnaprasad, Marsden and Murray. Chapter VI “Control of mechanical and nonholonomic systems” treats various aspects of control and stabilization of nonholonomic systems both for kinematic and dynamic systems. Chapter VII “Optimal control” begins with discussing the relationships between variational nonholonomic control systems and the classical Lagrange problem with the optimal control. A selection of techniques and results in optimal control theory are given that are actually optimization problems for mechanical systems including nonholonomic systems.

Chapter VIII “Stability of nonholonomic systems” discusses an energy-momentum-based approach to the stability of nonholonomic systems based on the thesis work of Zenkov and related works by the author and Marsden. In chapter IX “Energy-based methods for stabilization of controlled Lagrangian systems” the authors briefly discuss two recent energy-based methods for stabilizing second-order nonlinear systems and their application to nonholonomic systems. The first is the method of controlled Lagrangians (or Hamiltonians) and “matching”. The second one is a geometric approach to averaging of second-order systems that arise as models of controlled superactuated (or underactuated) mechanical systems.

After reading this book the reader will be convinced that the aim of the book – to integrate the intimately related disciplines which have evolved to a great extent almost independently of each other and gave rise to two separate research communities – is successfully reached. The book covers almost all research achievements in this field. It subsumes the recent scientific advances in the theory of nonholonomic mechanical systems, and it is a nice and useful learning tool for scientists and engineers from academia and industry. It is also worthy of note that the author together with his collaborators presents one of the best teams in the area of the modern nonholonomic mechanics and control theory.

Chapter I “Introduction” is devoted to basic ideas of mechanics, nonholonomic mechanics, control, and optimal control and a lot of classical illustrative examples are given. Chapter II “Mathematical preliminaries” covers a fairly wide topics from mathematics that are used in the next chapters within the area of differential manifolds, Lie groups and Lie algebras. Chapter III “Basic concepts in geometric mechanics” summarizes and, where necessary, develops the basic concepts in geometric mechanics exploring the geometric view of mechanics. It is stressed that its two branches, i.e. Hamiltonian and Lagrangian mechanics, are able to serve as two starting points in mechanics. The Lagrangian side of mechanics focuses on variational principles for its basic formulation, while the Hamiltonian one focuses on geometric structures called symplectic (Poisson) structures.

Chapter IV “An introduction to aspects of geometric control theory” touches a few aspects that are important for the major topics, and the advances made by Jurdjevic, Isidori, Nijmeijer and van der Schaft, Sontag and Brockett, Sussman and etc. are explained at appropriate points.

Chapter V “Nonholonomic mechanics” is a basic chapter on nonholonomic mechanics and discusses the basic geometric approach following the papers of Bloch, Krishnaprasad, Marsden and Murray. Chapter VI “Control of mechanical and nonholonomic systems” treats various aspects of control and stabilization of nonholonomic systems both for kinematic and dynamic systems. Chapter VII “Optimal control” begins with discussing the relationships between variational nonholonomic control systems and the classical Lagrange problem with the optimal control. A selection of techniques and results in optimal control theory are given that are actually optimization problems for mechanical systems including nonholonomic systems.

Chapter VIII “Stability of nonholonomic systems” discusses an energy-momentum-based approach to the stability of nonholonomic systems based on the thesis work of Zenkov and related works by the author and Marsden. In chapter IX “Energy-based methods for stabilization of controlled Lagrangian systems” the authors briefly discuss two recent energy-based methods for stabilizing second-order nonlinear systems and their application to nonholonomic systems. The first is the method of controlled Lagrangians (or Hamiltonians) and “matching”. The second one is a geometric approach to averaging of second-order systems that arise as models of controlled superactuated (or underactuated) mechanical systems.

After reading this book the reader will be convinced that the aim of the book – to integrate the intimately related disciplines which have evolved to a great extent almost independently of each other and gave rise to two separate research communities – is successfully reached. The book covers almost all research achievements in this field. It subsumes the recent scientific advances in the theory of nonholonomic mechanical systems, and it is a nice and useful learning tool for scientists and engineers from academia and industry. It is also worthy of note that the author together with his collaborators presents one of the best teams in the area of the modern nonholonomic mechanics and control theory.

Reviewer: Clementina Mladenova (Sofia)

##### MSC:

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

70F25 | Nonholonomic systems related to the dynamics of a system of particles |

93B27 | Geometric methods |

93D15 | Stabilization of systems by feedback |

70Q05 | Control of mechanical systems |