Optimal Geodesic Curvature Constrained Dubins’ Paths on a Sphere

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, U max , 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, U max . 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.