Optimal Planar Turns Under Acceleration Constraints

This technical note considers the problem of finding optimal trajectories for a particle moving in a 2-D 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 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 paths possess a special property whereby the vector that divides the velocity and acceleration vectors in a specific ratio, which depends on the cost functional, is a constant. This property is used to characterize the optimal acceleration vector and obtain the parametric equations of the corresponding optimal paths. Sufficiency conditions for optimality are used to show that trajectories satisfying the necessary conditions are actually optimal if the heading change required is sufficiently small. Solutions of the time-optimal and the length-optimal problems are obtained as special cases of the general problem.