Closed-form Generation of Paths for Motion Planning of a Convexified Reeds–Shepp Vehicle on a Sphere

Abstract

This paper presents an analytical derivation of the path generation process for time-optimal convexified Reeds–Shepp paths on a sphere. Specifically, a sufficient list of 23 optimal path types for this problem was derived in \cite{li2025time}; we focus here on the explicit construction of candidate paths using inverse kinematics. Given an initial configuration, a desired terminal configuration, and a specified maximum turning rate $U_{max}$, closed-form expressions for the segment angles corresponding to each path type in the sufficient list are derived. These expressions enable efficient and accurate generation of feasible time-optimal paths on the sphere. The results are used for the implementation at https://github.com/sixuli97/Optimal-Spherical-Convexified-Reeds-Shepp-Paths.