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.

翻译：流匹配近期已成为模仿学习中一种强大的方法，能够实现可扩展、富有表现力的多模态运动策略。然而，将形式化的稳定性保证（这是确保机器人行为安全且可泛化的前提条件）融入这些生成模型，仍是一个重大挑战。虽然将机器人运动建模为动力系统能够引入基于稳定性的归纳偏置，但现有框架难以捕捉复杂机器人任务中固有的丰富动作分布。本文提出稳定流匹配动力系统（Stable Flow Matching Dynamical Systems, SFMDS），这是一个新颖的框架，它弥合了高容量生成模型与形式化李雅普诺夫稳定性保证之间的鸿沟。SFMDS通过流匹配参数化动力系统，同时将模型约束于一族稳定的解。我们提出了两种变体：基于惩罚项的软约束，以及直接嵌入模型架构的硬结构约束。我们还将这两种公式扩展到了李群。在基准数据集、仿真实验以及人形机器人上的实验表明，SFMDS能够在低维和高维状态空间中学习稳定、可扩展且多模态的动力系统，从而生成安全且富有表现力的机器人运动。