Obstacle distance for car-like robots

This paper shows how to compute the nonholonomic distance between a pointwise car-like robot and polygonal obstacles. Geometric constructions to compute the shortest paths from a configuration (given orientation and position in the plane of the robot) to a position (i.e., a configuration with unspecified final orientation) are first presented. The geometric structure of the reachable set (set of points in the plane reachable by paths of given length L) is then used to compute the shortest paths to straight-line segments. Obstacle distance is defined as the length of such shortest paths. The algorithms are developed for robots that can move both forward and backward (Reeds and Shepp´s car) or only forward (Dubins´ car). They are based on the convexity analysis of the reachable set