Control of nonholonomic systems: from sub-Riemannian geometry to motion planning.

*(English)*Zbl 1309.93002
SpringerBriefs in Mathematics; BCAM SpringerBriefs. Cham: Springer; Bilbao: BCAM – Basque Center for Applied Mathematics (ISBN 978-3-319-08689-7/pbk; 978-3-319-08690-3/ebook). x, 104 p. (2014).

This monograph aims at presenting a new notion of approximation to the motion planning problem for non-holonomic systems, namely nonlinear control systems which depend linearly on the control.

The author thinks that the sub-Riemannian geometry plays for these systems the same role as Euclidean geometry does for linear systems. In fact, the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The book is divided into three chapters and two appendices.

In Chapter 1, starting with a discussion on linearization of control systems, the author presents the basic definitions and results on non-holonomic systems and the associated sub-Riemannian distances, different from the ones associated with sub-Riemannian manifolds. Then, the main result on controllability, namely the Chow-Rashevsky Theorem is stated and a rough estimates for sub-Riemannian distances is given.

Chapter 2 provides a detailed exposition of the notions of first-order approximation and of the basis of an infinitesimal calculus adapted to non-holonomic systems. To this aim a fundamental role is played by the concept of non-holonomic order of a function at a point. The approximations to the first-order are seen as nilpotent approximations, in the sense that the vector fields defining the system are approximated by vector fields that generate a nilpotent Lie algebra. These approximations are used to obtain estimates of the sub-Riemannian distance in terms of privileged coordinates. Then, a purely metric interpretation of this first-order theory is given and it is shown how the distance estimates allow to compute Hausdorff dimensions. As an application, these notions are shown to be useful in describing the tangent structure to a Carnot-Carathéodory space (the metric space defined by a sub-Riemannian distance). The chapter ends with the presentation of desingularization procedures that are necessary to recover uniformity in approximations and distance estimates. The main theoretical part about controllability and first-order theory is self-contained and contains the proof of all the results.

In Chapter 3 the author shows, in particular, how to apply the tools from sub-Riemannian geometry to give solutions to the motion planning problem namely, how to construct a control law steering a control system from a given state to another one. For specific classes of systems, in particular for nilpotent non-holonomic systems, the motion planning problem can be solved exactly. For a general non-holonomic system an exact solutions to the problem cannot be found. A solution is given through an algorithm which steers the system to an arbitrarily small neighbourhood of the goal. The construction of the algorithm is based on the notion of approximation and the concepts introduced in the previous chapters. An overview of the existing methods for non-holonomic motion planning concludes this chapter.

Appendix A contains some results on composition of flows in connection with the Campbell-Hausdorff formula and Appendix B contains some complements on different systems of privileged coordinates.

The author points out that, from the point of view of the sub-Riemannian geometry, this book is intended to be complementary to that of Ludo vie Rifford in the same collection. As a consequence, the subjects that are extensively talked about in the latter (for instance sub-Riemannian geodesics) are not discussed here.

The author thinks that the sub-Riemannian geometry plays for these systems the same role as Euclidean geometry does for linear systems. In fact, the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The book is divided into three chapters and two appendices.

In Chapter 1, starting with a discussion on linearization of control systems, the author presents the basic definitions and results on non-holonomic systems and the associated sub-Riemannian distances, different from the ones associated with sub-Riemannian manifolds. Then, the main result on controllability, namely the Chow-Rashevsky Theorem is stated and a rough estimates for sub-Riemannian distances is given.

Chapter 2 provides a detailed exposition of the notions of first-order approximation and of the basis of an infinitesimal calculus adapted to non-holonomic systems. To this aim a fundamental role is played by the concept of non-holonomic order of a function at a point. The approximations to the first-order are seen as nilpotent approximations, in the sense that the vector fields defining the system are approximated by vector fields that generate a nilpotent Lie algebra. These approximations are used to obtain estimates of the sub-Riemannian distance in terms of privileged coordinates. Then, a purely metric interpretation of this first-order theory is given and it is shown how the distance estimates allow to compute Hausdorff dimensions. As an application, these notions are shown to be useful in describing the tangent structure to a Carnot-Carathéodory space (the metric space defined by a sub-Riemannian distance). The chapter ends with the presentation of desingularization procedures that are necessary to recover uniformity in approximations and distance estimates. The main theoretical part about controllability and first-order theory is self-contained and contains the proof of all the results.

In Chapter 3 the author shows, in particular, how to apply the tools from sub-Riemannian geometry to give solutions to the motion planning problem namely, how to construct a control law steering a control system from a given state to another one. For specific classes of systems, in particular for nilpotent non-holonomic systems, the motion planning problem can be solved exactly. For a general non-holonomic system an exact solutions to the problem cannot be found. A solution is given through an algorithm which steers the system to an arbitrarily small neighbourhood of the goal. The construction of the algorithm is based on the notion of approximation and the concepts introduced in the previous chapters. An overview of the existing methods for non-holonomic motion planning concludes this chapter.

Appendix A contains some results on composition of flows in connection with the Campbell-Hausdorff formula and Appendix B contains some complements on different systems of privileged coordinates.

The author points out that, from the point of view of the sub-Riemannian geometry, this book is intended to be complementary to that of Ludo vie Rifford in the same collection. As a consequence, the subjects that are extensively talked about in the latter (for instance sub-Riemannian geodesics) are not discussed here.

Reviewer: Anna Maria Perdon (Ancona)

##### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93C25 | Control/observation systems in abstract spaces |

53C17 | Sub-Riemannian geometry |

93C85 | Automated systems (robots, etc.) in control theory |

53B21 | Methods of local Riemannian geometry |

93B27 | Geometric methods |

68T40 | Artificial intelligence for robotics |