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

TL;DR

This paper extends the understanding of optimal geodesic curvature constrained paths for Dubins vehicles on a sphere, identifying the types and conditions of optimal paths for larger turning radii.

Contribution

It generalizes previous results to larger turning radii, reducing candidate optimal paths and establishing conditions for their optimality.

Findings

Optimal paths are of type CG, CC, or degenerate for r ≤ √3/2.

Only one LG and one RG path can be optimal for a given final location.

Seven candidate paths are sufficient to describe optimal solutions for larger radii.

Abstract

In this paper, motion planning for a Dubins vehicle on a unit sphere to attain a desired final location is considered. The radius of the Dubins path on the sphere is lower bounded by $r$. In a previous study, this problem was addressed, wherein it was shown that the optimal path is of type $CG, CC,$ or a degenerate path of the same for $r \leq \frac{1}{2}.$ Here, $C = L, R$ denotes an arc of a tight left or right turn of minimum turning radius $r,$ and $G$ denotes an arc of a great circle. In this study, the candidate paths for the same problem are generalized to model vehicles with a larger turning radius. In particular, it is shown that the candidate optimal paths are of type $CG, CC,$ or a degenerate path of the same for $r \leq \frac{\sqrt{3}}{2}.$ Noting that at most two $LG$ paths and two $RG$ paths can exist for a given final location, this article further reduces the candidate optimal paths by showing that only one $LG$ and one $RG$ path can be optimal, yielding a total of seven candidate paths for $r \leq \frac{\sqrt{3}}{2}.$ Additional conditions for the optimality of $CC$ paths are also derived in this study.