Constructive Nonlinear Control of Underactuated Systems via Zero Dynamics Policies

TL;DR

This paper introduces Zero Dynamics Policies (ZDPs) for stabilizing underactuated systems by leveraging system decomposition, providing a constructive and learning-based method to enhance stability and performance beyond traditional control approaches.

Contribution

It formalizes ZDPs as a novel control paradigm, offering a constructive and learning-based approach to stabilize underactuated systems with larger regions of attraction.

Findings

ZDPs exist near the origin for underactuated systems.

Combining output stabilization with ZDPs ensures full system stability.

Learning-based ZDPs improve stability regions and performance over LQR.

Abstract

Stabilizing underactuated systems is an inherently challenging control task due to fundamental limitations on how the control input affects the unactuated dynamics. Decomposing the system into actuated (output) and unactuated (zero) coordinates provides useful insight as to how input enters the system dynamics. In this work, we leverage the structure of this decomposition to formalize the idea of Zero Dynamics Policies (ZDPs) -- a mapping from the unactuated coordinates to desired actuated coordinates. Specifically, we show that a ZDP exists in a neighborhood of the origin, and prove that combining output stabilization with a ZDP results in stability of the full system state. We detail a constructive method of obtaining ZDPs in a neighborhood of the origin, and propose a learning-based approach which leverages optimal control to obtain ZDPs with much larger regions of attraction. We demonstrate that such a paradigm can be used to stabilize the canonical underactuated system of the cartpole, and showcase an improvement over the nominal performance of LQR.