Optimal Spherical Geodesic Curvature Constrained Paths

Abstract

Path planning for vehicles is an essential study that must be undertaken to make good use of resources such as fuel (which is always a limited resource) to ascertain that a vehicle/robot completes its mission efficiently. The current study deals with the path planning of a Dubins’ vehicle on a sphere. A Dubins’ vehicle is one that moves only forwards, with a constant speed and with a minimum turning radius constraint; and is named after L.E. Dubins due to his seminal work [1] on the nature of optimal curves in the plane. The result being that optimal paths must be of the following types only: CSC, CCC, SC, CS, CC, or C can be optimal. This study aims to understand the nature of optimality of the Dubins’ type paths on a sphere. The main tools employed are Pontryagin’s Minimum Principle and the Sabban frame (same setup as in Monroy-Pérez’s work [2]). The final result obtained as a result of analytical study and corroboration with numerical computation is that Dubins’ type paths are optimal on the sphere for r in the interval (0,(1/sqrt(2))] on account of the CCCC type path being non-optimal in the same interval.