Generalizing Dubins Curves: Minimum-time Sequences of Body-fixed Rotations and Translations in the Plane

In this paper we present the minimum-time sequences of rotations and translations that connect two configurations of a rigid body in the plane. The configuration of the body is its position and orientation, given by (x , y, θ ) coordinates, and the rotations and translations are velocities (x , y, θ ) that are constant in the frame of the robot. There are no obstacles in the plane. We completely describe the structure of the fastest trajectories, and present a polynomial-time algorithm that, given a set of rotation and translation controls, enumerates a finite set of structures of optimal trajectories. These trajectories are a generalization of the well-known Dubins and Reeds—Shepp curves, which describe the shortest paths for steered cars in the plane.