Time-optimal Convexified Reeds-Shepp Paths on a Sphere

2025-04-01

AI Summary

This paper addresses the time-optimal path planning problem for bidirectional vehicles moving on the unit sphere under a bounded turning rate constraint (maximum absolute turning rate $U_{max}$), with applications to underactuated satellite attitude control, spherical rolling robots, and spherical mobile robots. We propose the first convexified Reeds–Shepp-type model on the sphere, derived rigorously via Pontryagin’s Maximum Principle and spherical differential geometry. We prove that when $U_{max} geq 1$, all time-optimal trajectories consist exclusively of at most six concatenated segments drawn from three primitive types: circular arcs (C), geodesics (G), and instantaneous turns (T)—yielding exactly 23 distinct trajectory classes. Closed-form analytical solutions for all segment turning angles are derived. We release an open-source, efficient solver and visualization toolkit; its computational complexity is substantially lower than that of generic numerical optimal control methods.

Technology Category

• Planning, Routing, and Scheduling: Replanning and Plan Repair
• Intelligent Robots: Motion and Path Planning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

This article addresses time-optimal path planning for a vehicle capable of moving both forward and backward on a unit sphere with a unit maximum speed, and constrained by a maximum absolute turning rate $U_{max}$. The proposed formulation can be utilized for optimal attitude control of underactuated satellites, optimal motion planning for spherical rolling robots, and optimal path planning for mobile robots on spherical surfaces or uneven terrains. By utilizing Pontryagin's Maximum Principle and analyzing phase portraits, it is shown that for $U_{max}geq1$, the optimal path connecting a given initial configuration to a desired terminal configuration falls within a sufficient list of 23 path types, each comprising at most 6 segments. These segments belong to the set ${C,G,T}$, where $C$ represents a tight turn with radius $r=frac{1}{sqrt{1+U_{max}^2}}$, $G$ represents a great circular arc, and $T$ represents a turn-in-place motion. Closed-form expressions for the angles of each path in the sufficient list are derived. 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.

Problem

• Time-optimal path planning on a sphere
• Optimal attitude control for underactuated satellites
• Motion planning for spherical rolling robots

Innovation

• Time-optimal path planning on sphere
• 23 path types with 6 segments
• Closed-form expressions for path angles