Extremal trajectories for bounded velocity mobile robots

Previous work has presented the time optimal trajectories for three classes of nonholonomic mobile robots: steered cars that can only go forwards, steered cars that go forwards or backwards, and differential drives. Each of the vehicles is modelled as a rigid body in the plane with velocity and angular velocity controls. The systems are differentiated only by the bounds on the controls, but the optimal trajectories are qualitatively different for each system. We explore this difference by considering the effect that control bounds have on the extremal trajectories of bounded velocity vehicles, where the extremal trajectories are defined to be the set of trajectories that satisfy Pontryagin´s maximum principle, a necessary condition for optimality.