Nonholonomic Mechanics and Control

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Springer Science & Business Media, Apr 8, 2003 - Language Arts & Disciplines - 483 pages
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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.
  

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Contents

Introduction
1
and NewtonEuler Balance
2
12 Hamiltons Principle
6
13 The LagrangedAlembert Principle
11
14 The Vertical Rolling Disk
15
15 The Falling Rolling Disk
21
16 The Knife Edge
23
17 The Chaplygin Sleigh
25
44 Averaging and Trajectory Planning
190
45 Stabilization
194
46 Hamiltonian and Lagrangian Control Systems
198
Nonholonomic Mechanics
207
51 Equations of Motion
211
52 The LagrangedAlembert Principle
216
53 Projected Connections and Newtons Law
221
54 Systems with Symmetry
223

18 The Heisenberg System
29
19 The Rigid Body
34
110 The Roller Racer
42
111 The Rattleback
43
112 The Toda Lattice
45
Mathematical Preliminaries
49
22 Differentiable Manifolds
62
23 Stability
71
24 Center Manifolds
74
25 Differential Forms
79
26 Lie Derivatives
86
27 Stokess Theorem Riemannian Manifolds Distributions
93
28 Lie Groups
97
29 Fiber Bundles and Connections
105
Basic Concepts in Geometric Mechanics
119
31 Symplectic and Poisson Manifolds and Hamiltonian Flows
120
32 Cotangent Bundles
123
33 Lagrangian Mechanics and Variational Principles
124
34 Mechanical Systems with External Forces
128
35 LiePoisson Brackets and the Rigid Body
130
36 The EulerPoincare Equations
134
37 Momentum Maps
136
38 Symplectic and Poisson Reduction
139
39 A Particle in a Magnetic Field
142
310 The Mechanical Connection
144
311 The LagrangePoincaré Equations
146
312 The EnergyMomentum Method
150
313 Coupled Planar Rigid Bodies
158
314 Phases and Holonomy the Planar Skater
167
An Introduction to Aspects of Geometric Control Theory
175
42 Controllability and Accessibility
177
43 Representation of System Trajectories
182
55 The Momentum Equation
228
56 Examples of the Nonholonomic Momentum Map
238
57 More General Nonholonomic Systems with Symmetries
248
58 The Poisson Geometry of Nonholonomic Systems
254
Control of Mechanical and Nonholonomic Systems
277
62 Stabilization of the Heisenberg System
284
63 Stabilization of a Generalized Heisenberg System
290
64 Controllability Accessibility and Stabilizability
301
65 Smooth Stabilization to a Manifold
303
66 Nonsmooth Stabilization
308
67 Nonholonomic Systems on Riemannian Manifolds
318
Optimal Control
328
72 Optimal Control and the Maximum Principle
336
73 Variational Nonholonomic Systems and Optimal Control
340
74 Kinematic SubRiemannian Optimal Control Problems
342
75 Optimal Control and a Particle in a Magnetic Field
353
76 Optimal Control of Mechanical Systems
359
Stability of Nonholonomic Systems
367
82 Overview
370
83 The Pure Transport Case
372
84 The Nonpure Transport Case
377
85 General Case the LyapunovMalkin Method
387
86 EulerPoincaréSuslov Equations
392
EnergyBased Methods for Stabilization of Controlled Lagrangian Systems
399
92 Feedback Design and Matching
402
93 Stabilization of a Class of Nonholonomic Systems
410
94 Averaging for Controlled Lagrangian Systems
415
95 Dynamic Nonholonomic Averaging
434
References
439
Index
472
Copyright

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Page 468 - Utkin, VI, 1978, Sliding Modes and Their Application in Variable Structure Systems, Moscow, MIR Utkin, VI, 1984, "Variable Structure Systems Present and Future,
Page 453 - A recursive technique for tracking control of nonholonomic systems in chained form.
Page 470 - On global representations of the solution of linear differential equations as a product of exponentials.

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