|This paper presents the time
optimal trajectories for differential drive vehicles in the unobstructed
plane. The wheel angular velocities are bounded, but may be discontinuous.
The paper proves the existence of optimal controls, derives the structure
of optimal trajectories, and develops an algorithm for producing a time
optimal trajectory between any two configurations. Every nontrivial optimal
trajectory is composed of straight segments alternating with turns about
the robot's center. Optimal trajectories may have as many as five actions,
but four actions are sufficient - for every optimal trajectory of five actions,
there is an equally fast trajectory with four actions.