Learning Lyapunov-Stable Polynomial Dynamical Systems through Imitation

TL;DR

This paper introduces a novel imitation learning approach that models globally stable polynomial dynamical systems with Lyapunov functions, improving safety, accuracy, and efficiency in complex robotic motion planning tasks.

Contribution

It proposes jointly learning polynomial dynamical systems and Lyapunov functions for stable, accurate imitation learning in robotics, addressing limitations of prior methods.

Findings

Demonstrates high sample efficiency in simulation and real-world tests.

Achieves accurate reproduction of complex trajectories.

Maintains stability under perturbations.

Abstract

Imitation learning is a paradigm to address complex motion planning problems by learning a policy to imitate an expert's behavior. However, relying solely on the expert's data might lead to unsafe actions when the robot deviates from the demonstrated trajectories. Stability guarantees have previously been provided utilizing nonlinear dynamical systems, acting as high-level motion planners, in conjunction with the Lyapunov stability theorem. Yet, these methods are prone to inaccurate policies, high computational cost, sample inefficiency, or quasi stability when replicating complex and highly nonlinear trajectories. To mitigate this problem, we present an approach for learning a globally stable nonlinear dynamical system as a motion planning policy. We model the nonlinear dynamical system as a parametric polynomial and learn the polynomial's coefficients jointly with a Lyapunov candidate. To showcase its success, we compare our method against the state of the art in simulation and conduct real-world experiments with the Kinova Gen3 Lite manipulator arm. Our experiments demonstrate the sample efficiency and reproduction accuracy of our method for various expert trajectories, while remaining stable in the face of perturbations.