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Safe and Stable Neural Network Dynamical Systems for Robot Motion Planning

Allen Emmanuel Binny, Mahathi Anand, Hugo T. M. Kussaba, Lingyun Chen, Shreenabh Agrawal, Fares J. Abu-Dakka, Abdalla Swikir

TL;DR

The paper tackles the problem of learning safe and stable robot motions from demonstrations in cluttered environments by proposing S$^2$-NNDS, an offline framework that jointly learns a neural dynamical system and neural Lyapunov and barrier certificates. It leverages split conformal prediction to provide probabilistic guarantees on the validity of the certificates, enabling safer deployment despite potential unsafe demonstrations. Empirical results on 2D LASA handwriting, 3D obstacle scenarios, and kinesthetic Franka Panda demonstrations show that S$^2$-NNDS can produce motions that closely follow demonstrations while respecting safety and stability, often outperforming SOS/bilinear baselines in non-convex obstacle configurations. Limitations include local rather than global stability guarantees, confinement to static obstacles, and sensitivity to hyperparameters, with future work aimed at extending guarantees and handling dynamic environments.

Abstract

Learning safe and stable robot motions from demonstrations remains a challenge, especially in complex, nonlinear tasks involving dynamic, obstacle-rich environments. In this paper, we propose Safe and Stable Neural Network Dynamical Systems S$^2$-NNDS, a learning-from-demonstration framework that simultaneously learns expressive neural dynamical systems alongside neural Lyapunov stability and barrier safety certificates. Unlike traditional approaches with restrictive polynomial parameterizations, S$^2$-NNDS leverages neural networks to capture complex robot motions providing probabilistic guarantees through split conformal prediction in learned certificates. Experimental results on various 2D and 3D datasets -- including LASA handwriting and demonstrations recorded kinesthetically from the Franka Emika Panda robot -- validate S$^2$-NNDS effectiveness in learning robust, safe, and stable motions from potentially unsafe demonstrations.

Safe and Stable Neural Network Dynamical Systems for Robot Motion Planning

TL;DR

The paper tackles the problem of learning safe and stable robot motions from demonstrations in cluttered environments by proposing S-NNDS, an offline framework that jointly learns a neural dynamical system and neural Lyapunov and barrier certificates. It leverages split conformal prediction to provide probabilistic guarantees on the validity of the certificates, enabling safer deployment despite potential unsafe demonstrations. Empirical results on 2D LASA handwriting, 3D obstacle scenarios, and kinesthetic Franka Panda demonstrations show that S-NNDS can produce motions that closely follow demonstrations while respecting safety and stability, often outperforming SOS/bilinear baselines in non-convex obstacle configurations. Limitations include local rather than global stability guarantees, confinement to static obstacles, and sensitivity to hyperparameters, with future work aimed at extending guarantees and handling dynamic environments.

Abstract

Learning safe and stable robot motions from demonstrations remains a challenge, especially in complex, nonlinear tasks involving dynamic, obstacle-rich environments. In this paper, we propose Safe and Stable Neural Network Dynamical Systems S-NNDS, a learning-from-demonstration framework that simultaneously learns expressive neural dynamical systems alongside neural Lyapunov stability and barrier safety certificates. Unlike traditional approaches with restrictive polynomial parameterizations, S-NNDS leverages neural networks to capture complex robot motions providing probabilistic guarantees through split conformal prediction in learned certificates. Experimental results on various 2D and 3D datasets -- including LASA handwriting and demonstrations recorded kinesthetically from the Franka Emika Panda robot -- validate S-NNDS effectiveness in learning robust, safe, and stable motions from potentially unsafe demonstrations.

Paper Structure

This paper contains 15 sections, 3 theorems, 12 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Related Work
  3. Contributions
  4. Preliminaries
  5. Notations
  6. DS for Motion Planning
  7. Proposed Methodology
  8. Neural Network Parameterizations
  9. Validation of Certificates via Conformal Prediction
  10. The S2-NNDS Algorithm
  11. Simulations and Experimental Results
  12. 2D and 3D Datasets
  13. Performance Analysis and Benchmark Comparisons
  14. Demonstrations from Franka Emika Robot
  15. Discussion

Key Result

Proposition 1

Consider the ds in eq:ds with an isolated equilibrium point $x^\star \in X$. A continuously differentiable real-valued function $V: X \rightarrow {\mathbb{R}}$ is a Lyapunov function, where $V(x^\star) = 0$, $\dot{V}(x^\star) = 0$, if for all $x \in X \setminus \{x^\star\}$ the following conditions The existence of a Lyapunov function guarantees that the system is locally asymptotically stable wi

Figures (7)

  • Figure 1: Overview of snnds Framework. Neural dynamics are first learned from expert demonstrations and iteratively refined to satisfy Lyapunov and barrier constraints using counterexamples. Verification with conformal prediction then provides formal statistical guarantees on safety and stability.
  • Figure 2: Neural ds generated by our proposed approach with obstacles for the LASA handwriting and robot demonstration datasets. Five demonstrations (blue) were used. The learned trajectories (pink) are simulated for two initial conditions within the initial set, and the robot path (brown) is obtained for another initial point. The region in green describes the safe set, while the arrows indicate the flow of the ds.
  • Figure 3: Neural ds generated by our proposed approach with obstacles for the 3D C-shaped motion. Ten demonstrations were used. The legend follows from Fig. \ref{['fig:lasa']}.
  • Figure 4: Drawing platform with Franka Panda Robot
  • Figure 5: Comparison of the learned trajectories and barrier functions obtained via snnds and ABC-DS, respectively. Barrier functions learned via our approach generally fit more tightly around the obstacles and offer less conservative results, as can be seen from the area computed in Table \ref{['tab:comparisons-area']}.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Proposition 1: Asymptotic Stability
  • Proposition 2: Safety
  • Theorem 3: Verification via Conformal Prediction