Learning a Stable Dynamic System with a Lyapunov Energy Function for Demonstratives Using Neural Networks

TL;DR

Addresses the stability-accuracy trade-off in learning from demonstrations for autonomous dynamical systems by learning a Lyapunov energy function from demonstration data. Proposes a neural-network framework with a residual y-design and a three-network architecture that yields $V(\boldsymbol{x})=\tfrac{1}{2}\boldsymbol{y}^{\mathrm{T}}\boldsymbol{y}$ and a stabilizing $\dot{\boldsymbol{y}}$ to guarantee convergence. Demonstrates improved reproduction accuracy on LASA handwriting trajectories and successful real-robot validation on a Franka Emika, outperforming a CLF-DM baseline by approximately $15.6\%$ in SEA and $13.2\%$ in $V_{rmse}$; notes training-time costs and potential smoothness limitations due to activation choices, suggesting avenues for efficiency and generalization improvements.

Abstract

Autonomous Dynamic System (DS)-based algorithms hold a pivotal and foundational role in the field of Learning from Demonstration (LfD). Nevertheless, they confront the formidable challenge of striking a delicate balance between achieving precision in learning and ensuring the overall stability of the system. In response to this substantial challenge, this paper introduces a novel DS algorithm rooted in neural network technology. This algorithm not only possesses the capability to extract critical insights from demonstration data but also demonstrates the capacity to learn a candidate Lyapunov energy function that is consistent with the provided data. The model presented in this paper employs a straightforward neural network architecture that excels in fulfilling a dual objective: optimizing accuracy while simultaneously preserving global stability. To comprehensively evaluate the effectiveness of the proposed algorithm, rigorous assessments are conducted using the LASA dataset, further reinforced by empirical validation through a robotic experiment.

Figures (5)

• Figure 1: The overall structure of the proposed algorithm for learning a DS while simultaneously learning a Lyapunov energy function.
• Figure 2: 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. Moreover, the transformed reproduction trajectories (aligned with $\boldsymbol{y}$ as illustrated in Fig. \ref{['fig1']}) are showcased against a white background. Within this context, solid points indicate the starting positions for these trajectories.
• Figure 3: Four simulation scenarios of learning from one demonstration by using the proposed algorithm are depicted.
• Figure 4: Four simulation scenarios of learning from one demonstration with modeling the $\dot{\boldsymbol{y}}=-\boldsymbol{y}$ are depicted.
• Figure 5: The real-world experiments involving the generation of multiple trajectories using the "Multi-Models-1" dataset from LASA. These experiments employ the proposed algorithm for trajectory learning.