Optimal trajectories for symmetric turns

Abstract

The problem of determining minimal time trajectories in a plane constrained by an upper bound on the magnitude of the acceleration vector is reexamined. In the previous work [Am. J. Phys. 49(7), 685–688 (1981)], a stationary solution of a functional, applied over curves in two-dimensional velocity space, was used to find explicit expressions for what was claimed to be a minimum turn time trajectory. In this paper, this work is furthered by a formal demonstration that the turn time associated with this trajectory is indeed lower than that corresponding to any other smooth trajectory. Supporting evidence for this claim is provided by numerical procedures, which are developed to allow comparisons between the turn times of competing trajectories across a range of parameter values of the turn width, the initial speed, and the magnitude of the acceleration vector.