用几何方法统一分析机器人运动中的惯性、重力与阻力影响
Riemannian Variational Calculus: Optimal Trajectories Under Inertia, Gravity, and Drag Effects
- 基于黎曼几何推导出包含三类力的最优控制方程
- 发现惯性流形曲率、势场曲率和阻力缩短效应三类关键影响
- 在双连杆与UR5机械臂上验证，适合机器人轨迹优化研究者
机器人运动优化常聚焦特定任务，忽视基本运动规律。基于黎曼几何与变分法（常作为最优控制的间接方法），我们推导出将一般力表示为构型与速度函数的最优控制方程，揭示了惯性、重力与阻力如何塑造最优轨迹。分析识别出三类关键效应：(i) 惯性流形的曲率效应，(ii) 势场的曲率效应，(iii) 阻力引起的缩短效应。在双连杆机械臂与UR5机器人上验证了该方法，展示了超越测地线规划的统一几何框架。
原文摘要 · Abstract (English)
Robotic motion optimization often focuses on task-specific solutions, overlooking fundamental motion principles. Building on Riemannian geometry and the calculus of variations (often appearing as indirect methods of optimal control), we derive an optimal control equation that expresses general forces as functions of configuration and velocity, revealing how inertia, gravity, and drag shape optimal trajectories. Our analysis identifies three key effects: (i) curvature effects of inertia manifold, (ii) curvature effects of potential field, and (iii) shortening effects from resistive force. We validate our approach on a two-link manipulator and a UR5, demonstrating a unified geometric framework for understanding optimal trajectories beyond geodesic-based planning.
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