Motion planning for nonholonomic robots in a limited workspace

In this paper we pose the following two questions: Given two points, start and target, within a relatively small closed planar area W subset of R-2, each with a prescribed direction of motion (orientation) in it, and assuming a possibility of reversals of motion, (i) what is the shortest path of bounded curvature that connects the points and lies completely in WI (ii) what is the minimum number of motion reversals (path cusps) one needs to arrive at the target point with the prescribed orientation? Such questions appear in various applications with nonholonomic motion constraints, as for example in motion planning for driverless cars. The proposed approach solves both problems. It makes use of a tool dubbed the Reflective Unfolding operator which has a clear geometric interpretation and provides an interesting means for solving other trajectory design problems.