Let the Dynamics Flow: Stable Flow Matching Dynamical Systems
For robot imitation learning, this work provides a method to combine high-capacity generative models with formal stability guarantees, addressing a key safety requirement.
SFMDS integrates flow matching with Lyapunov stability guarantees for imitation learning, enabling stable, scalable, and multimodal robot motion policies. It achieves stable behavior on benchmark datasets, simulations, and a humanoid robot.
Flow matching has recently emerged as a powerful approach for imitation learning, enabling scalable, expressive, and multimodal motion policies. However, incorporating formal stability guarantees into these generative models, a prerequisite to ensure safe and generalizable robot behaviors, remains a significant challenge. While modeling robot motions as dynamical systems allows for such stability-based inductive biases, existing frameworks struggle to capture the rich action distributions inherent in complex robotic tasks. This paper introduces Stable Flow Matching Dynamical Systems (SFMDS), a novel framework that bridges the gap between high-capacity generative modeling and formal Lyapunov stability guarantees. SFMDS parametrizes dynamical systems via flow matching while simultaneously constraining the model to a family of stable solutions. We propose two variants: a soft constraint based on a penalty term, and a hard structural constraint embedded directly in the model architecture. We further extend both formulations to Lie groups. Experiments on benchmark datasets, in simulation, and on a humanoid robot show that SFMDS learns stable, scalable, and multimodal dynamical systems in low- and high-dimensional state spaces, enabling safe and expressive robot motion generation.