Nonholonomic interpolation for kinematic problems, entropy and complexity

Abstract

Here we present the main lines of a theory we developed in a series of previous papers, about the motion planning problem in robotics. We illustrate the theory with a few academic examples. Our theory, although at its starting point, looks promising even from the constructive point of view. It does not mean that we have precise general algorithms, but the theory contains this potentiality. The robot is given under the guise of a set of linear kinematic constraints (a distribution). The cost is specified by a riemannian metric on the distribution. Given a non-admissible path for the robot, i.e. a path that does not satisfy the kinematic constraints), our theory allows to evaluate precisely and constructively the metric complexity and the entropy of the problem. This estimation of metric complexity provides methods for approximation of nonadmissible paths by admissible ones, while the estimation of entropy provides methods for interpolation of the nonadmissible path by admissible. © 2008 Springer-Verlag Berlin Heidelberg.