Time-optimal Convexified Reeds-Shepp Paths on a Sphere

TL;DR

This paper advances time-optimal path planning for a spherical convexified Reeds-Shepp vehicle on the unit sphere under a maximum turning rate $U_{ ext{max}}$ and unit speed. By applying Pontryagin's Maximum Principle and phase-portrait analysis to the spherical CRS dynamics in the Sabban frame, it classifies optimal trajectories into three regimes and derives a finite sufficient set of 23 path types for $U_{ ext{max}}\ge1$, composed of segments $C$, $G$, and $T$ with a tight-turn radius $r=\frac{1}{\sqrt{1+U_{ ext{max}}^2}}$; closed-form angles are obtained for all paths. The authors prove non-optimality and redundancy for many candidate types and provide a complete generator to compute feasible, time-optimal paths, along with open-source code for solving the problem and visualization. The results enable efficient attitude control and spherical-robot motion planning on spheres or locally spherical terrains, with potential impact on planetary exploration and surveillance tasks.

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.

Figures (22)

• Figure 1: Configurations on a sphere
• Figure 2: Illustration of $u_g$ and $v$
• Figure 3: Schematic plot of the spherical rolling robot
• Figure 4: Repositioning of the shutter
• Figure 5: Approximating local uneven terrain with a spherical patch

Theorems & Definitions (57)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• proof
• Remark 4
• Lemma 2
• proof
• Remark 5
• Lemma 3