Stabilizing Dynamic Systems through Neural Network Learning: A Robust Approach

TL;DR

The paper presents a neural-network–based framework for learning autonomous dynamical systems from demonstrations with stability guarantees for both point-to-point and periodic motions. It uses a neural Lyapunov function $V(\boldsymbol{x})$ to enforce stability in point-to-point tasks and enforces transverse contraction to ensure convergence to a limit cycle for periodic tasks, avoiding the limitations of quadratic Lyapunov constraints typical of prior methods. The approach employs a two-network architecture: one learns a Lyapunov function and the other outputs velocity commands aligned with stability requirements, with projection-based decompositions $\boldsymbol{R}_1, \boldsymbol{R}_2$ (and $\boldsymbol{R}_3, \boldsymbol{R}_4$ for cycles). Demonstrations from LASA and real Franka robot data validate the method, showing improved reproduction accuracy and robust limit-cycle learning under disturbances, albeit with higher training time than conventional DMPs or GMMs. Overall, the method offers a principled, stability-aware pathway for robust LfD in both point-to-point and periodic motion regimes, with clear implications for reliable robotic manipulation and locomotion tasks.

Abstract

Point-to-point and periodic motions are ubiquitous in the world of robotics. To master these motions, Autonomous Dynamic System (DS) based algorithms are fundamental in the domain of Learning from Demonstration (LfD). However, these algorithms face the significant challenge of balancing precision in learning with the maintenance of system stability. This paper addresses this challenge by presenting a novel ADS algorithm that leverages neural network technology. The proposed algorithm is designed to distill essential knowledge from demonstration data, ensuring stability during the learning of both point-to-point and periodic motions. For point-to-point motions, a neural Lyapunov function is proposed to align with the provided demonstrations. In the case of periodic motions, the neural Lyapunov function is used with the transversal contraction to ensure that all generated motions converge to a stable limit cycle. The model utilizes a streamlined neural network architecture, adept at achieving dual objectives: optimizing learning accuracy while maintaining global stability. To thoroughly assess the efficacy of the proposed algorithm, rigorous evaluations are conducted using the LASA dataset and a manually designed dataset. These assessments were complemented by empirical validation through robotic experiments, providing robust evidence of the algorithm's performance

Figures (4)

• Figure 1: The simulation utilizing the proposed algorithm is depicted in this paper. Images with a black-yellow background display the learned vector fields. Within these visuals, the white dotted lines represent the original demonstration data, while the red solid lines depict the reproductions from identical initial points. The target points are denoted as "$\cdot$" in these illustrations.
• Figure 2: The proposed algorithm to learn a complex periodic trajectory, the white dotted lines represent the original demonstration data, while the red solid lines depict the reproductions from identical initial points.
• Figure 3: The proposed algorithm is used to learn a limit cycle from actual trajectory data collected by a robot. In the visual representation, the white dotted lines correspond to the original demonstration data, whereas the colored solid lines illustrate the algorithm's generated trajectories initiated from various random starting points.
• Figure 4: The proposed algorithm is designed to perform a massage task, leveraging the learned limit cycle derived from the actual trajectory data collected from a robot. Despite encountering various disturbances during the massage process, the robot is capable of maintaining task performance effectively.