Riemannian Variational Calculus: Optimal Trajectories Under Inertia, Gravity, and Drag Effects

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.