Generation of Paths for Motion Planning for a Dubins Vehicle on Sphere

TL;DR

This work provides analytic closed-form expressions for candidate optimal spherical Dubins paths of a vehicle constrained to move on a sphere, leveraging the Sabban frame and Euler–Rodriguez rotations. By expressing path segments with rotation matrices $R_L$, $R_G$, and $R_R$, it reduces the problem to solving for arc angles $\phi_1$, $\phi_2$, and $\phi_3$ via cosine and sine relations, including a comprehensive set of special and degenerate cases (e.g., $\phi_2=0$ or $\pi$, $r=1/\sqrt{2}$, and $\cos\phi_2$ special values). The paper provides closed-form procedures for a wide family of path types (LGL, RGR, LGR, RGL, LRL, RLR, and their variants with $\pi$-turns and multiple turns), together with systematic handling of degeneracies and reflections to enable practical computation. The results culminate in a complete analytic toolkit for spherical Dubins-path planning, with an implementation available on GitHub. This advances real-time motion planning for spherical geometry constraints in robotics and aerial systems.

Abstract

In this article, the candidate optimal paths for a Dubins vehicle on a sphere are analytically derived. In particular, the arc angles for segments in $CGC$, $CCC$, $CCCC$, and $CCCCC$ paths, which have previously been shown to be optimal depending on the turning radius $r$ of the vehicle by Kumar \textit{et al.}, are analytically derived. The derived expressions are used for the implementation provided in https://github.com/DeepakPrakashKumar/Motion-planning-on-sphere.