## Nonholonomic Mechanics and ControlOur 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 |

439 | |

472 | |

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### Common terms and phrases

action asymptotically stable Bloch bracket bundle Chaplygin Chapter condition conﬁguration space connection consider constant control system coordinates corresponding curve deﬁned Deﬁnition denote derivative diﬀerent diﬀerential equations discussed dynamics Ehresmann connection equations of motion equilibrium Euler-Lagrange equations example feedback ﬁber ﬁnd ﬁrst ﬁxed ﬂow function given Hamiltonian Heisenberg system holonomic horizontal inertia input integral invariant kinematic kinetic energy Krishnaprasad Lagrange multipliers Lagrangian Lie algebra Lie group linear Lyapunov manifold Marsden matrix mechanical systems metric momentum equation momentum map nonholonomic equations nonholonomic systems nonlinear obtain one-form optimal control orbit parameters phase space Poisson Poisson manifold potential Ratiu rattleback reduced Riemannian rigid body rolling disk rotating satisﬁes Schaft smooth stability submanifold symmetry symplectic tangent bundle tangent space Theorem theory tion Toda lattice trajectories variables variational vector ﬁeld velocity vertical zero

### Popular passages

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.