Feedback Synthesis For Underactuated Systems Using Sequential Second-Order Needle Variations

TL;DR

This paper introduces a second-order needle variation control method for underactuated systems that leverages nonlinear controllability and Lie brackets, enabling control solutions where first-order methods fail, with demonstrated success in simulations.

Contribution

It develops a novel second-order feedback control synthesis based on needle variations, explicitly exploiting Lie brackets to improve control in nonlinear, underactuated systems.

Findings

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

Demonstrates convergence in underwater vehicle model with velocity field.

Shows effectiveness across different underactuated systems.

Abstract

This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actions---the second-order 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. Lastly, the underactuated dynamic underwater vehicle model demonstrates convergence even in the presence of a velocity field.