Constructive Nonlinear Control of Underactuated Systems via Zero Dynamics Policies (2408.14749v1)

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.

• The paper introduces Zero Dynamics Policies (ZDPs) as a novel framework combining constructive analysis and learning-based approaches to systematically stabilize underactuated systems.
• It proves that ZDPs can be locally constructed by designing outputs whose zero dynamics are stable, ensuring full system stabilization.
• The framework integrates learning guided by optimal control to extend the region of attraction, demonstrated effectively on the cartpole problem.

Constructive Nonlinear Control of Underactuated Systems via Zero Dynamics Policies

Overview

The paper "Constructive Nonlinear Control of Underactuated Systems via Zero Dynamics Policies" by William Compton et al. addresses the complex challenge of stabilizing underactuated systems, which include systems such as legged robots and dexterous manipulators where not all degrees of freedom can be directly controlled. The authors introduce the notion of Zero Dynamics Policies (ZDPs), proposing a methodological framework to stabilize these systems by leveraging both constructive analysis and learning-based approaches.

Key Contributions

The authors make several noteworthy contributions:

1. Zero Dynamics Policies (ZDPs): The concept of ZDPs is formalized as a strategic mapping from unactuated to actuated coordinates, thus providing a systematic approach to stabilize the full system state. The approach is built on the recognition that underactuated systems can be decomposed into actuated and unactuated components, each described by zero dynamics when the output has been rendered zero.
2. Proof of Local Existence: The paper offers a proof that ZDPs can be locally constructed in a neighborhood of the origin for a broad class of locally controllable nonlinear systems. This constructive method involves designing outputs such that their zero dynamics are stable, ensuring the stabilization of the entire system.
3. Learning-Based Approach: Recognizing the limitations of linearization and local feedback control, the authors advocate the integration of machine learning techniques to extend the region of attraction for ZDPs. Optimal control informs the learning process, ensuring improved performance over conventional linear control methods like LQR.
4. Cartpole Demonstration: The methodology is validated using the canonical cartpole problem, where the application of ZDPs extends the region of attraction significantly beyond that of nominal LQR, showcasing improved stability and control performance even in the presence of nonlinear damping.

Technical Detail

The construction of ZDPs relies on the decomposition of the system into output (actuated) and zero (unactuated) dynamics. By choosing output coordinates that can be stabilized to zero, the authors show that this renders the system's zero dynamics invariant and stable. The paper leverages control Lyapunov functions (CLFs) and feedback linearization techniques for constructing stable outputs.

Importantly, the paper proposes a learning methodology that utilizes optimal control principles to define a loss function. This function guides the learning of ZDPs, extending their applicability across a broader state space. It suggests that, for underactuated systems, optimal control strategies that lead to invariant and stable zero dynamics manifolds can be expressed and fine-tuned as ZDPs.

Implications and Future Directions

The results have significant implications for the control of underactuated systems. Practically, this work enables system designers to achieve robust stabilization over larger domains compared to traditional methods. Theoretically, it provides an innovative mechanism to unify disparate methods of stabilization (such as Hybrid Zero Dynamics and feedback linearization) under a cohesive framework.

Future work may focus on expanding the discovery of manifold structures beyond local regions, exploring more sophisticated learning frameworks that integrate deeper reinforcement learning methods, and applying this approach to a wider array of robotic and mechatronic systems to realize its full potential in real-world applications.

In conclusion, this paper provides a seminal exploration into the construction and application of Zero Dynamics Policies for the control of underactuated systems, merging both classical control theory and modern learning techniques to overcome traditional barriers in stabilization.