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 $0<r \le \frac{1}{2}$, 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 $r> \frac{1}{2}$, 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.