A Geometric Algorithm to Compute Time-Optimal Trajectories for a Bidirectional Steered Robot

This paper addresses the problem of determining time-optimal trajectories, between two specified configurations, for a nonholonomic bidirectional steered robot. It presents an original geometric reasoning that is grounded on Pontryagin´s maximum principle, which provides analytical solutions of this problem in a visually clear way and allows for an effective algorithm to compute the exact optimal trajectories between two arbitrarily specified configurations. The proposed geometric reasoning is based on the analysis of the switching functions of the optimal controller and the definition of a switching vector from which it is able to determine a unit vector rotating along a unit circle of an appropriate coordinate system. It is shown that simple geometric rules are sufficient to determine all possible rotations of this unit vector, from which the time-optimal trajectories can be uniquely determined. The proposed algorithm, which is based on this geometric reasoning, is guaranteed to be complete and has a low computational cost. Moreover, the proposed geometric representation provides an interesting insight into the structure of this class of nonholonomic systems, thereby offering a model for further studies.