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

TL;DR

This paper develops a Riemannian geometric framework for robotic motion optimization, deriving equations that incorporate inertia, gravity, and drag effects to better understand and compute optimal trajectories.

Contribution

It introduces a novel optimal control equation based on Riemannian calculus that accounts for fundamental physical effects in trajectory planning.

Findings

Validated on a two-link manipulator and UR5 robot.

Identified three key effects shaping optimal trajectories.

Provides a unified geometric understanding beyond geodesic planning.

Abstract

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.