Time-Optimal Path Planning for a Kinematic Car with Variable Speed

This paper investigates the minimum-time path-planning problem for a kinematic car with variable speed and turn rate controls. The speed is strictly positive, ranging from a lower to an upper limit, and the turn rate limits are symmetric about zero. The minimum principle is used to characterize the extremal controls and, using additional geometric arguments, a finite and sufficient set of candidate optimal controls is derived. It is found that, in addition to straight and maximum rate turning segments at maximum-speed, minimum-time paths may include “cornering” turns at the minimum forward speed and the maximum turn rate. A procedure is proposed for solving the path synthesis problem of constructing the minimum-time path between two “oriented points” in the plane.