Optimal Geodesic Curvature Constrained Dubins' Paths on a Sphere

Abstract

In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, Umax of geodesic curvature of its path; this constrains the object to change the heading at the fastest rate only when traveling on a tight smaller circular arc of radius r < 1, where r depends on the bound, Umax. We show in this article that if 0 < r ≤ 1 2 , the shortest path between any two configurations of the rigid body on the sphere consists of a concatenation of at most three circular arcs. Specifically, if C is the smaller circular arc and G is the great circular arc, then the optimal path can only be CCC, CGC, CC, CG, GC, C or G. If r > 1 2 , while paths of the above type may cease to exist depending on the boundary conditions and the value of r, optimal paths may be concatenations of more than three circular arcs. Mathematical Formulation ModelConsider the trajectory of a rigid massless robot (modeled as a short pointed stick) on a unit sphere as shown by the green spherical curve in Fig. 1.Let s denote arc length (traversed by the robot), and X(s) be a C 1 -smooth spherical curve, representing the position of the robot (geometric center of the stick) on the sphere. Let T(s) := dX ds represent the direction