Geometric control and numerical aspects of nonholonomic systems.

*(English)*Zbl 1009.70001
Lecture Notes in Mathematics. 1793. Berlin: Springer. xv, 219 p. (2002).

The theory of nonholonomic systems may be regarded as a rather particular case of classical Lagrange problem of the calculus of variations. Here the action functionals \({\mathcal I}:C^2(q_0,q_1,[a, b])\to \mathbb{R}\) given by first-order Lagrangians \({\mathcal I}(c)= \int^b_a L(\dot c(t)) dt\) are investigated on the class \(C^2\) of twice differentiable curves \(c\) with fixed end points in a manifold \(Q\), \(C^2(q_0, q_2,[a, b])= \{c:[a, b]\to Q\), \(c(a)= q_0\), \(c(b)= q_1\}\), under the assumption that the tangent \(\dot c(t)\) lies in a given submanifold \(M\) of the tangent space \(TQ\). The attention is (as a rule) restricted to regular Lagrangians \(L: TQ\to\mathbb{R}\) of the mechanical type \(L= T-V\), where \(T\) is the kinetic energy identified with a Riemannian metric \(g\) on \(Q\), and \(V: Q\to \mathbb{R}\) is a potential energy. There are many geometric structures that can be intrinsically related to this problem. The book provides a very nice and transparent overview both of the general theory and particular examples and applications.

Contents: the first chapter presents the literature review (a large number of references and historical comments thorough all book should be pointed out especially). The second chapter surveys the necesary tools: tensor fields and differential forms, distributions and codistributions, Lie group actions, connections, Riemannian metrics, symplectic structures, Poisson manifolds and tangent bundle geometry. Variational principles together with symplectic and affine connection reformulations are presented in the third chapter, whereas the symmetries of nonholonomic systems are thoroughly classified in the fourth chapter. The fifth chapter is devoted to Chaplygin systems where the configuration space \(Q\) is a principal \(G\)-bundle \(\pi: Q\to Q/G\), the constraint submanifold \(M\subset TQ\) is identified with the horizontal distribution of a principal connection, and both the kinetic energy \(T\) and potential energy \(V\) are \(G\)-invariant (the existence of invariant measure is studied and in general rejected). Three concluding chapters are the most instructive: they deal with constraints of variable rank, with integration theory including the discrete Lagrange-d’Alembert principle, and with controllability.

The following examples considered in the book are worth mentioning: a rolling disk, a ball on rotating table (consisting of rough and smooth pieces), a snakeboard model, a plate with knife edge, several classes of moving robots, and many others.

Contents: the first chapter presents the literature review (a large number of references and historical comments thorough all book should be pointed out especially). The second chapter surveys the necesary tools: tensor fields and differential forms, distributions and codistributions, Lie group actions, connections, Riemannian metrics, symplectic structures, Poisson manifolds and tangent bundle geometry. Variational principles together with symplectic and affine connection reformulations are presented in the third chapter, whereas the symmetries of nonholonomic systems are thoroughly classified in the fourth chapter. The fifth chapter is devoted to Chaplygin systems where the configuration space \(Q\) is a principal \(G\)-bundle \(\pi: Q\to Q/G\), the constraint submanifold \(M\subset TQ\) is identified with the horizontal distribution of a principal connection, and both the kinetic energy \(T\) and potential energy \(V\) are \(G\)-invariant (the existence of invariant measure is studied and in general rejected). Three concluding chapters are the most instructive: they deal with constraints of variable rank, with integration theory including the discrete Lagrange-d’Alembert principle, and with controllability.

The following examples considered in the book are worth mentioning: a rolling disk, a ball on rotating table (consisting of rough and smooth pieces), a snakeboard model, a plate with knife edge, several classes of moving robots, and many others.

Reviewer: Jan Chrastina (Brno)

##### MSC:

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

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

70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |

70Q05 | Control of mechanical systems |

93B29 | Differential-geometric methods in systems theory (MSC2000) |

49S05 | Variational principles of physics |