Time-optimal Convexified Reeds-Shepp Paths on a Sphere
Provides a theoretical and practical solution for time-optimal path planning on spheres, applicable to satellite attitude control and spherical robots, but is incremental as it extends known Reeds-Shepp results to a spherical domain.
This paper solves the time-optimal path planning problem for a convexified Reeds-Shepp vehicle on a sphere, showing that optimal paths consist of at most six segments from three motion primitives, with a complete classification of 23 optimal path types and closed-form segment angles.
This article studies the time-optimal path planning problem for a convexified Reeds-Shepp (CRS) vehicle on a unit sphere, capable of both forward and backward motion, with speed bounded in magnitude by 1 and turning rate bounded in magnitude by a given constant. For the case in which the turning-rate bound is at least 1, using Pontryagin's Maximum Principle and a phase-portrait analysis, we show that the optimal path connecting a given initial configuration to a desired terminal configuration consists of at most six segments drawn from three motion primitives: tight turns, great circular arcs, and turn-in-place motions. A complete classification yields a finite sufficient list of 23 optimal path types with closed-form segment angles derived. The complementary case in which the turning-rate bound is less than 1 is addressed via an equivalent reformulation. The proposed formulation is applicable to underactuated satellite attitude control, spherical rolling robots, and mobile robots operating on spherical or gently curved surfaces. The source code for solving the time-optimal path problem and visualization is publicly available at https://github.com/sixuli97/Optimal-Spherical-Convexified-Reeds-Shepp-Paths.