Geometric Optimal Control of Mechanical Systems with Gravitational and Resistive Force

2024-10-12

arXiv.org

AI Summary

This work addresses the common omission of fundamental physical constraints—such as inertia, gravity, and viscous drag—in robot motion optimization. We propose a unified optimal control framework grounded in differential geometry. Methodologically, we model viscous drag as a Riemannian metric on the configuration manifold, thereby unifying kinetic energy and gravitational potential fields, and derive geometric optimal control equations incorporating curvature effects. Indirect optimal control is solved via Lagrangian mechanics and manifold-based variational calculus. Experiments on a two-link planar manipulator and a UR5 robot demonstrate that the proposed model substantially alters optimal trajectory shape and energy distribution, enhancing both physical realizability and energy efficiency. Our core contribution is a novel geometric modeling paradigm that synergistically integrates drag, curvature, and potential fields—establishing a new theoretical foundation for physics-informed robotic trajectory optimization.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Optimal control plays a crucial role in numerous mechanical and robotic applications. Broadly, optimal control methods are divided into direct methods (which optimize trajectories directly via discretization) and indirect methods (which transform optimality conditions into equations that guarantee optimal trajectories). While direct methods could mask geometric insights into system dynamics due to discretization, indirect methods offer a deeper understanding of the system's geometry. In this paper, we propose a geometric framework for understanding optimal control in mechanical systems, focusing on the combined effects of inertia, drag, and gravitational forces. By modeling mechanical systems as configuration manifolds equipped with kinetic and drag metrics, alongside a potential field, we explore how these factors influence trajectory optimization. We derive optimal control equations incorporating these effects and apply them to two-link and UR5 robotic manipulators, demonstrating how manifold curvature and resistive forces shape optimal trajectories. This work offers a comprehensive geometric approach to optimal control, with broad applications to robotic systems.

Problem

• Derives optimal control equation for general forces.
• Identifies effects of inertia, gravity, and drag.
• Validates framework on robotic manipulators for optimal trajectories.

Innovation

• Riemannian geometry for motion optimization
• Optimal control equation for general forces
• Unified geometric framework for trajectories