## Nonholonomic Mechanics and ControlNonholonomic Mechanics and Control develops the rich connections between control theory and geometric mechanics. Control theory is linked with a geometric view of classical mechanics in both its Lagrangian and Hamiltonian formulations and especially with the theory of nonholonomic mechanics (mechanical systems subject to motion constraints). Both controllability and optimal control are treated, including the Pontryagin maximum principle. In addition, the stability, control, and stabilization of mechanical systems are discussed. In particular, these items are considered for nonholonomic systems. The aim of the book is to provide a unified treatment of nonlinear control theory and constrained mechanical systems, incorporating material that has not yet made its way into texts and monographs. Detailed illustrations and exercises are included throughout the text. This book is intended for graduate and advance undergraduate students in mathematics, physics and engineering who wish to learn this subject and for researchers in the area who want to enhance their techniques. |

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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 |

439 | |

472 | |

### Common terms and phrases

### Popular passages

### References to this book

### References from web pages

Additional Bibliography: Nonholonomic Mechanics and Control References

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