Nonholonomic Mechanics and Control

The focus of this book is mechanics, controllability, and feedback stabilization of mechanical control systems subject to nonintegrable constraints on the velocity variables. Example systems include vehicles with wheels that roll without slipping and space systems with conserved angular momentum.While the mechanics of nonholonomic systems is “classical,” the control of such systems has received increasing attention in the last 30 years. The purpose of this book is to bring together in one place results of this recent work in a differential geometric framework. The result is 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 content of the book is primarily influenced by the work of the author, as well as the work of J. Baillieul, R. Brockett, F. Bullo, P. Crouch, P. S. Krishnaprasad, N. Leonard, A. Lewis, J. Marsden, R. Murray, J. Ostrowski, T. Ratiu, and D. Zenkov. This book requires a fairly high level of mathematical sophistication from the reader, beyond the typical mathematical training of an engineering student. Sections are devoted to concise development of mathematical preliminaries in an effort to make the book as self-contained as possible, but the treatment is likely to be inaccessible to readers without some background in differential geometry. Readers will be unable to skip to topics in later chapters of the book without first mastering the mathematical background and notation. The chapters on mechanics (Chapters 3 and 5) contain material that can be found in part in books on geometric mechanics [1], [6], while some material on control and stabilization (Chapters 4 and 6–9) can be found in books on nonlinear and geometric control theory [3]–[5], [7], [8]. This book is unique in that the inquiry is focused on nonholonomic mechanical systems (though many of the methods introduced can be applied to other systems). This allows a deeper and more comprehensive treatment of this subject, using both Lagrangian and Hamiltonian formulations, with special emphasis on systems possessing symmetries in the equations of motion. Many recent results appear here for the first time in a book. A forthcoming monograph by Bullo and Lewis [2] also uses a differential geometric framework to study the control of “simple” mechanical systems, with or without nonholonomic constraints. As this book deals with both mechanics and control, a recurring theme emphasizes the difference between “dynamic nonholonomic” and “variational nonholonomic” (or “vakonomic”) equations, which are occasionally confused in the literature. The former are the dynamic equations of motion for a nonholonomic system, while the latter are satisfied by optimal controls for the constrained system. The dynamic nonholonomic equations arise from taking variations (e.g., Hamilton’s principle) and then applying the constraints to project to motions satisfying the nonholonomic constraints. The variational nonholonomic equations arise from applying the constraints before taking variations. A Brief Summary of the Book: Chapter 1 quickly introduces Hamiltonian and Lagrangian formulations of the equations of motion