Feedback Synthesis for Controllable Underactuated Systems using Sequential Second Order Actions

TL;DR

This paper introduces a nonlinear feedback control synthesis method for underactuated systems using second-order needle variations, exploiting system controllability and Lie brackets to improve control performance.

Contribution

It develops a novel control synthesis approach based on second-order needle variations that explicitly leverages nonlinear controllability via Lie brackets.

Findings

Method finds control solutions when first-order analysis is singular.

Demonstrates superior convergence compared to first-order needle variation methods.

Effective even with velocity fields in underwater vehicle models.

Abstract

This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actions---the needle variations of optimal control---as the basis for choosing each control response to the current state. A second result of the paper is that the method provably exploits the nonlinear controllability of a system by virtue of an explicit dependence of the second-order needle variation on the Lie bracket between vector fields. As a result, each control decision necessarily decreases the objective when the system is nonlinearly controllable using first-order Lie brackets. Simulation results using a differential drive cart, an underactuated kinematic vehicle in three dimensions, and an underactuated dynamic model of an underwater vehicle demonstrate that the method finds control solutions when the first-order analysis is singular. Moreover, the simulated examples demonstrate superior convergence when compared to synthesis based on first-order needle variations. Lastly, the underactuated dynamic underwater vehicle model demonstrates the convergence even in the presence of a velocity field.