Safe and Stable Neural Network Dynamical Systems for Robot Motion Planning

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 in various 2D and 3D datasets – including LASA handwriting and demonstrations recorded kinesthetically from the Franka Emika Panda robot – validate the effectiveness of S$^2$-NNDS in learning robust, safe, and stable motions from potentially unsafe demonstrations. The source code, supplementary material and experiment videos can be accessed via https://github.com/allemmbinn/S2NNDS

💡 Research Summary

The paper introduces Safe and Stable Neural Network Dynamical Systems (S²‑NNDS), a novel learning‑from‑demonstration (LfD) framework that simultaneously learns a robot’s motion dynamics and formal certificates of stability and safety. Traditional LfD methods typically fit a dynamical system (DS) to demonstrated trajectories but ignore obstacle avoidance, leading to unsafe motions. Existing safety‑or‑stability approaches often rely on polynomial parameterizations and Sum‑of‑Squares (SOS) optimization, which are limited to low‑degree polynomials, suffer from numerical ill‑conditioning, and cannot capture complex, non‑algebraic obstacle shapes.

S²‑NNDS overcomes these limitations by representing the vector field f(x), a Lyapunov function V(x), and a barrier function B(x) as fully connected neural networks (NNs). The dynamics network fθ is constrained to satisfy fθ(0)=0 by zeroing all biases, ensuring the equilibrium at the origin. Vθ′ is also bias‑free to guarantee Vθ′(0)=0, while Bθ″ has no such restriction. Differentiable activation functions (e.g., tanh, ELU) are used to keep V and B continuously differentiable, a prerequisite for the Lyapunov and barrier conditions.

Training proceeds with three coupled loss terms: (1) an MSE loss L_MSE that forces fθ to reproduce the demonstrated velocities; (2) a Lyapunov loss L_lyap that penalizes violations of V(x)>0 and ∇V·f<0 using a leaky‑ReLU hinge formulation; and (3) a barrier loss L_bar that enforces B(x)≤0 on the initial set, B(x)>0 on the unsafe set, and ∇B·f≤0 in a narrow safety band. Hyper‑parameters λ and small tolerances δ control the relative importance of each term and provide numerical robustness.

Because these losses are evaluated only on sampled points, they cannot guarantee that the certificates hold over the entire continuous workspace. To bridge this gap, the authors employ split conformal prediction. The dataset is split into a training set (used to optimize the NNs) and a calibration set (used to estimate the empirical distribution of certificate violations). For a chosen confidence level (e.g., 95 %), a quantile of the calibration residuals yields a threshold that, when applied to unseen states, provides a Provably Approximately Correct (PAC) guarantee that the Lyapunov and barrier conditions are satisfied with the specified probability. This statistical validation is performed offline, so it does not affect real‑time control.

Experimental validation covers three domains. First, the LASA handwriting dataset demonstrates that S²‑NNDS can reproduce intricate 2‑D vector fields more accurately than the polynomial SOS‑based ABC‑DS method. Second, synthetic 2‑D environments with multiple irregular static obstacles show that the learned DS respects obstacle boundaries and converges to the goal without collisions. Third, real‑world kinesthetic demonstrations on a Franka Emika Panda robot in 3‑D space confirm that the method works with high‑dimensional state spaces and physical hardware. Notably, the authors deliberately include “unsafe demonstrations” that intersect obstacles; S²‑NNDS automatically corrects these portions during training, resulting in a final DS that never violates safety constraints.

Performance-wise, all training and conformal validation are performed offline, eliminating any additional computational load in the robot’s control loop. The approach therefore respects typical robot control frequencies (≥1 kHz). The neural‑network parameterization offers expressive power to capture highly nonlinear motions and arbitrary obstacle geometries, while the conformal prediction step supplies a quantifiable statistical safety guarantee.

Limitations include the reliance on sufficiently dense sampling of the workspace for loss evaluation, the relatively shallow network architectures used for the certificates, and the current focus on static obstacles only. Extending the framework to dynamic obstacles, online adaptation, or deeper certificate networks constitutes promising future work.

In summary, S²‑NNDS presents a compelling integration of neural dynamical systems, learned Lyapunov and barrier certificates, and split conformal prediction, delivering a practical, offline‑trained solution that guarantees both stability and safety for robot motion planning from demonstrations.