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

TL;DR

The paper addresses how to characterize optimal robotic trajectories under inertia, gravity, and drag beyond traditional geodesic planning. It develops a Riemannian variational calculus-based optimal-control framework where a general force is expressed as a function of configuration $q$ and velocity $\dot{q}$, and derives an optimal-spline equation via the Pontryagin Maximum Principle. Key contributions include identifying inertia-manifold curvature, potential-field curvature, and drag-induced path shortening, and validating the framework on a two-link arm and a UR5 to reveal a unified geometric view of motion. This geometry-informed approach provides insights for energy-efficient planning and can extend to more forces, constraints, and locomotion systems.

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.

Figures (4)

• Figure 1: (a) Optimal trajectories for a UR5 manipulator moving its first three joints. The black-$\times$-marked curve considers only inertia effects, where curvature-induced distortion directs acceleration to reduce inertia (e.g., folding the arm). The red-$\circ$-marked curve incorporates both inertia and gravity, influenced by the potential field curvature on the inertia manifold. The blue curve accounts for inertia and drag, where drag, modeled as joint viscous friction, results in an Euclidean drag metric, shortening the trajectory in configuration space. (b) The same optimal trajectories in task space, with the initial and final configurations of the UR5. Only the first three joints, starting from the base joint, are actuated in this work. The manipulator's appearance transitions from white to gray, representing its initial and final configurations.
• Figure 2: (a) A two-link manipulator and its optimal trajectories in task space, with and without gravity. Each link has the same mass ($m$) and length ($\ell$), and the joint angle of the $i$-th link is denoted as ${q}_i$. Gravity acts downward. (b) The same optimal trajectories in configuration space, where the potential field is shown as a filled contour. Tissot’s indicatrix (ellipses and cross lines) visualizes the inertia metric, indicating spatial contraction/expansion and the velocity needed for unit kinetic energy. (c) Optimal trajectories in a metric-stretched space, approximating the Riemannian manifold's geometry by scaling basis vectors for uniform kinetic energy representation ramasamy_geometry_2019.
• Figure 3: (a) Optimal trajectories of a two-link manipulator with and without joint friction. The black and blue curves represent the trajectories without and with friction, respectively. In configuration space, joint friction follows an Euclidean metric, resulting in circular-shape Tissot indicatrices. The trajectory with friction is closer to a straight path. (b) Optimal trajectories of the same manipulator with an end-effector drag force. The black and blue curves represent trajectories without and with drag. In task space, drag follows an Euclidean metric, yielding circular-shape Tissot indicatrices. The trajectory with drag is closer to a straight path.
• Figure 4: (a) Optimal trajectories for a UR5 moving its first three joints under the combinational effect of inertia, gravitation, and drag force. Tissot indicatrices visualize the metric space at given points of the trajectory. The red-colored curve represents an optimal trajectory under the actuation-based cometric, while the blue-colored curve corresponds to the Euclidean metric, equivalent to interpolation between initial and final conditions. The set of surfaces denotes a level set of a potential field. (b) An optimal trajectory of the UR5 manipulator under the actuation-based cometric in task space. (c) An optimal trajectory under the Euclidean metric or interpolation between initial and final conditions in task space. In (b) and (c), the manipulator transitions from white to gray, representing its initial and final configurations. (d) A time versus acceleration norm cost, defined as $\int_{t_0}^{t_f}\lVert u \rVert^2_{M}dt$.

Theorems & Definitions (1)

• Proposition 1