Extremal trajectories for bounded velocity differential drive robots

This paper applies Pontryagin's maximum principle to the time optimal control of differential drive mobile robots with velocity bounds. The maximum principle gives necessary conditions for time optimality. Extremal trajectories are those which satisfy these conditions, and are thus a superset of the time optimal trajectories. This paper derives a compact geometrical structure for extremal trajectories and shows that extremal trajectories are always composed of rotations about the robot center and straight line motions. Further necessary conditions are obtained.