Shortest paths synthesis for a car-like robot

This paper deals with the complete characterization of the shortest paths for a car-like robot. Previous works have shown that the search for a shortest path may be limited to a simple family of trajectories. Our work completes this study by providing a way to select inside this family an optimal path to link any two configurations. We combine the necessary conditions given by Pontryagin's maximum principle with a geometric reasoning. This approach enables us to complete the local information with a global analysis of different wave fronts. We construct a partition of the configuration space in regions where the same kind of path is optimal to reach the origin. In other words, we determine a shortest path synthesis by providing, at each point, an optimal control law to steer the robot to the origin.