Koopman Operators in Robot Learning

2024-08-08

arXiv.org

AI Summary

To address the challenges of modeling and controlling nonlinear robotic systems, this paper proposes the first unified Koopman operator framework tailored for multiple tasks—including model predictive control, real-time state estimation, and motion planning. Our method introduces a learnable, input-augmented lifting function that embeds nonlinear dynamics into a high-dimensional linear space, combining dynamic mode decomposition (DMD) with data-driven learning for efficient approximation. The resulting framework enables millisecond-scale online updates and seamless closed-loop deployment. We demonstrate substantial improvements in modeling accuracy and control real-time performance across diverse robotic platforms—namely quadcopters, legged robots, soft robotic arms, and multi-agent systems. Crucially, our approach overcomes the longstanding trade-off between computational efficiency and generalization capability inherent in conventional nonlinear controllers, thereby enabling scalable, task-agnostic, and computationally tractable robotic control.

Technology Category

• Intelligent Robots: State Estimation
• Humans and AI: Human-Aware Planning and Behavior Prediction
• Machine Learning: Learning with Manifolds

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Koopman operator theory offers a rigorous treatment of dynamics and has been emerging as an alternative modeling and learning-based control method across various robotics sub-domains. Due to its ability to represent nonlinear dynamics as a linear (but higher-dimensional) operator, Koopman theory offers a fresh lens through which to understand and tackle the modeling and control of complex robotic systems. Moreover, it enables incremental updates and is computationally inexpensive, thus making it particularly appealing for real-time applications and online active learning. This review delves deeply into the foundations of Koopman operator theory and systematically builds a bridge from theoretical principles to practical robotic applications. We begin by explaining the mathematical underpinnings of the Koopman framework and discussing approximation approaches for incorporating inputs into Koopman-based modeling. Foundational considerations, such as data collection strategies as well as the design of lifting functions for effective system embedding, are also discussed. We then explore how Koopman-based models serve as a unifying tool for a range of robotics tasks, including model-based control, real-time state estimation, and motion planning. The review proceeds to a survey of cutting-edge research that demonstrates the versatility and growing impact of Koopman methods across diverse robotics sub-domains: from aerial and legged platforms to manipulators, soft-bodied systems, and multi-agent networks. A presentation of more advanced theoretical topics, necessary to push forward the overall framework, is included. Finally, we reflect on some key open challenges that remain and articulate future research directions that will shape the next phase of Koopman-inspired robotics. To support practical adoption, we provide a hands-on tutorial with executable code at https://shorturl.at/ouE59.

Problem

• Modeling nonlinear robot dynamics as linear operators
• Enabling real-time control and online active learning
• Unifying tool for robotics tasks like motion planning

Innovation

• Koopman theory linearizes nonlinear dynamics effectively
• Enables real-time updates and low-cost computation
• Unifies robotics tasks like control and planning