Nonholonomic motion planning versus controllability via the multibody car system example

A multibody car system is a non-nilpotent, non-regular, triangularizable and well-controllable system. One goal of the current paper is to prove this obscure assertion. But its main goal is to explain and enlighten what it means. Motion planning is an already old and classical problem in Robotics. A few years ago a new instance of this problem has appeared in the literature: motion planning for nonholonomic systems. While useful tools in motion planning come from Computer Science and Mathematics (Computational Geometry, Real Algebraic Geometry), nonholonomic motion planning needs some Control Theory and more Mathematics (Differential Geometry). First of all, this paper tries to give a computational reading of the tools from Differential Geometric Control Theory required by planning. Then it shows that the presence of obstacles in the real world of a real robot challenges Mathematics with some difficult questions which are topological in nature, and have been solved only recently, within the framework of Sub-Riemannian Geometry. This presentation is based upon a reading of works recently developed by (1) Murray and Sastry, (2) Lafferiere and Sussmann, and (3) Bellaiche, Jacobs and Laumond.

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