求解球面上车辆最优路径，6段内完成转向与移动。
Time-optimal Convexified Reeds-Shepp Paths on a Sphere
- 用庞特里亚金原理分析最优路径结构
- 在转向速率≥1时，路径最多6段，共23种类型
- 适用于卫星姿态控制与球面机器人
本文研究单位球面上凸化瑞德-谢普（CRS）车辆的时间最优路径规划问题，该车辆可正向与反向运动，速度大小受限于1，转向速率大小受限于给定常数。当转向速率约束不小于1时，通过庞特里亚金最大值原理和相图分析，证明从初始构型到目标构型的最优路径由至多六段构成，来自三种运动基元：紧转弯、大圆弧和原地转向。完整分类得到23种最优路径类型，并推导出闭式表达的各段角度。转向速率小于1的互补情况通过等价重构解决。该方法适用于欠驱动卫星姿态控制、球面滚动机器人及在球面或缓曲面上运行的移动机器人。求解与可视化源代码已公开于https://github.com/sixuli97/Optimal-Spherical-Convexified-Reeds-Shepp-Paths。
原文摘要 · Abstract (English)
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.
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