Neural Contractive Dynamical Systems

TL;DR

This work tackles stability in learned robotic dynamics by introducing Neural Contractive Dynamical Systems (NCDS), a neural architecture that guarantees contraction through a Jacobian network yielding a negative definite symmetric part for all parameters. To handle high-dimensional data, it couples NCDS with an injective-decoder variational autoencoder, enabling latent contractive dynamics that decode to contractive, data-space behavior. The approach is extended to full-pose motions on the Lie group $\mathrm{SO}(3)$ via careful orientation parameterization and the first-cover diffeomorphism, and obstacle avoidance is achieved with a contraction-preserving modulation in data space. Empirically, NCDS demonstrates strong performance and scalable stability on LASA trajectory benchmarks and real robotic tasks, including 7-DoF Panda joint-space motions and full-pose end-effector dynamics, while maintaining safe obstacle avoidance and robust extrapolation. Overall, the proposed framework provides a unified, provably stable and scalable pathway for learning complex robot dynamics from demonstrations.

Abstract

Stability guarantees are crucial when ensuring a fully autonomous robot does not take undesirable or potentially harmful actions. Unfortunately, global stability guarantees are hard to provide in dynamical systems learned from data, especially when the learned dynamics are governed by neural networks. We propose a novel methodology to learn neural contractive dynamical systems, where our neural architecture ensures contraction, and hence, global stability. To efficiently scale the method to high-dimensional dynamical systems, we develop a variant of the variational autoencoder that learns dynamics in a low-dimensional latent representation space while retaining contractive stability after decoding. We further extend our approach to learning contractive systems on the Lie group of rotations to account for full-pose end-effector dynamic motions. The result is the first highly flexible learning architecture that provides contractive stability guarantees with capability to perform obstacle avoidance. Empirically, we demonstrate that our approach encodes the desired dynamics more accurately than the current state-of-the-art, which provides less strong stability guarantees.

Figures (22)

• Figure 1: Robot motion executed via a neural contractive dynamical system (NCDS).
• Figure 2: The learned vector field (grey) and demonstrations (black). Yellow and green trajectories show path integrals starting from demonstration starting points and random points, respectively.
• Figure 3: Aspects of the Lie group $\mathcal{SO}(3)$.
• Figure 4: Architecture overview: a single iteration of NCDS simultaneously generating position and orientation dynamics. (A) VAE (pink box): The encoder processes the concatenated position-orientation data ${\bm{p}}_t$, yielding a resulting vector that is subsequently divided into two components: the latent code ${\bm{z}}$ (yellow squares) and the surplus (gray squares). The $\operatorname{Unpad}$ function in Equation \ref{['eq:encoding_process']} removes the unused segment. The unpadded latent code ${\bm{z}}$ is fed to the contraction module and simultaneously padded with zeros (white squares) before being passed to the injective decoder. (B) Contraction (blue box): The Jacobian network output, given the latent codes, is reshaped into a square matrix and transformed into a negative definite matrix using Equation \ref{['eq:negative_def_Jacobian']}. The numerical integral solver then computes the latent velocity $\dot{{\bm{z}}}$. Later, using Eq. \ref{['eq:velocity_map']}, $\dot{{\bm{z}}}$ is mapped to the input-space velocity via the decoder's Jacobian $\bm{J}{\mu_{\bm{\xi}}}$.
• Figure 5: Visualization of the LASA-2D dataset: Gray contours represent the learned vector field, black trajectories depict demonstrations, and orange/green trajectories illustrate path integrals starting from the initial points of the demonstrations and plot corners. The magenta circles indicate the initial points of the path integrals.

Theorems & Definitions (2)

• Definition 1: Contraction stability Lohmiller1998ContractionAnalysis
• Theorem 1: Contraction invariance under diffeomorphisms Manchester17:ControlContractionMetric