Regularizing Numerical Extremals Along Singular Arcs: A Lie-Theoretic Approach

TL;DR

This paper introduces a Lie-theoretic method to improve numerical solutions of time-optimal control problems with singular arcs, combining explicit formulas with software tools to enhance accuracy and understanding of optimal trajectories.

Contribution

It presents a novel approach that integrates explicit Lie-theoretic formulas into numerical optimal control methods to better handle singular solutions in mechanical systems.

Findings

Enhanced control solutions by combining theory and numerical algorithms.

Reduced artifacts in numerical solutions of singular arcs.

Applied method successfully to a robotic arm example.

Abstract

Numerical ``direct'' approaches to time-optimal control often fail to find solutions that are singular in the sense of the Pontryagin Maximum Principle, performing better when searching for saturated (bang-bang) solutions. In previous work by one of the authors, singular solutions were shown to exist for the time-optimal control problem for fully actuated mechanical systems under hard torque constraints. Explicit formulas, based on a Lie theoretic analysis of the problem, were given for singular segments of trajectories, but the global structure of solutions remains unknown. In this work, we review the aforementioned framework, and show how to effectively combine these formulas with the use of general-purpose optimal control software packages. By using the explicit formula given by the theory in the intervals where the numerical solution enters a singular arc, we not only obtain an algebraic expression for the control in that interval but we are also able to remove artifacts present in the numerical solution. In this way, the best features of numerical algorithms and theory complement each other and provide a better picture of the global optimal structure. We illustrate the technique on a two degree of freedom robotic arm example, using two distinct optimal control numerical software packages running on different programming languages.