Optimal Planar Turns Under Acceleration Constraints

This paper considers the problem of finding optimal trajectories for a particle moving in a two-dimensional plane from a given initial position and velocity to a specified terminal heading under a magnitude constraint on the acceleration. The cost functional to be minimized is the integral over time of a general non-negative power of the particle´s speed. Special cases of such a cost functional include travel time and path length. Unlike previous work on related problems, variations in the magnitude of the velocity vector are allowed. Pontryagin´s maximum principle is used to show that the optimal trajectories possess a special property whereby the vector that divides the angle between the velocity and acceleration vectors in a specific ratio, which depends on the cost functional, is a constant. This property is used to obtain the optimal acceleration vector and the parametric equations of the corresponding optimal paths. Solutions of the time-optimal and the length-optimal problems are obtained as special cases