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Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning

Abstract : Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular 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 aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems. The notes are divided into three chapters and two appendices. In Chapter 1 we introduce the basic definitions on nonholonomic systems and sub-Riemannian geometry, and give the main result on controllability, namely the Chow-Rashevsky Theorem. Chapter 2 provides a detailed exposition of the notions of first-order approximation, including nonholonomic orders, privileged coordinates, nilpotent approximations, and distance estimates such as the Ball-Box Theorem. As an application we show how these notions allow us to describe 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. Chapter 3 is devoted to the motion planning problem for nonholonomic systems. We show in particular how to apply the tools from sub-Riemannian geometry in order to give solutions to this problem, first in the case where the system is nilpotent, and then in the general case. An overview of the existing methods for nonholonomic motion planning concludes this chapter. Finally, we present some results on composition of flows in connection with the Campbell-Hausdorff formula in Appendix A, and some complements on the different systems of privileged coordinates in Appendix B.
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Frédéric Jean. Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning. Springer International Publishing, 2014, SpringerBriefs in Mathematics, 978-3-319-08689-7. ⟨10.1007/978-3-319-08690-3⟩. ⟨hal-01137580⟩

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