Optimal Geodesic Curvature Constrained Dubins' Path on Sphere with Free Terminal Orientation

TL;DR

This paper derives optimal vehicle paths on a sphere with curvature constraints, showing that for certain parameters, the best paths are limited to at most seven specific types involving circular and great arcs.

Contribution

It provides a complete characterization of optimal geodesic curvature constrained paths on a sphere with free terminal orientation, extending Dubins' path theory.

Findings

Optimal paths are limited to at most seven types for certain radii.

Paths include combinations of circular and great arcs.

The results are derived using Pontryagin's Minimum Principle.

Abstract

In this paper, motion planning for a vehicle moving on a unit sphere with unit speed is considered, wherein the desired terminal location is fixed, but the terminal orientation is free. The motion of the vehicle is modeled to be constrained by a maximum geodesic curvature $U_{max},$ which controls the rate of change of heading of the vehicle such that the maximum heading change occurs when the vehicle travels on a tight circular arc of radius $r = \frac{1}{\sqrt{1 + U_{max}^2}}$. Using Pontryagin's Minimum Principle, the main result of this paper shows that for $r \leq \frac{1}{2}$, the optimal path connecting a given initial configuration and a final location on the sphere belongs to a set of at most seven paths. The candidate paths are of type $CG, CC,$ and degenerate paths of the same, where $C \in \{L, R\}$ denotes a tight left or right turn, respectively, and $G$ denotes a great circular arc.