Learning stable dynamical systems from data is crucial for safe and reliable robot motion planning and control. However, extending stability guarantees to trajectories defined on Riemannian manifolds poses significant challenges due to the manifold's geometric constraints. To address this, we propose a general framework for learning stable dynamical systems on Riemannian manifolds using neural ordinary differential equations. Our method guarantees stability by projecting the neural vector field evolving on the manifold so that it strictly satisfies the Lyapunov stability criterion, ensuring stability at every system state. By leveraging a flexible neural parameterisation for both the base vector field and the Lyapunov function, our framework can accurately represent complex trajectories while respecting manifold constraints by evolving solutions directly on the manifold. We provide an efficient training strategy for applying our framework and demonstrate its utility by solving Riemannian LASA datasets on the unit quaternion (S^3) and symmetric positive-definite matrix manifolds, as well as robotic motions evolving on \mathbb{R}^3 \times S^3. We demonstrate the performance, scalability, and practical applicability of our approach through extensive simulations and by learning robot motions in a real-world experiment.

翻译：从数据中学习稳定的动态系统对于安全可靠的机器人运动规划与控制至关重要。然而，由于流形的几何约束，将稳定性保证扩展到黎曼流形上定义的轨迹面临重大挑战。为解决此问题，我们提出了一个使用神经常微分方程学习黎曼流形上稳定动态系统的通用框架。我们的方法通过将演化在流形上的神经向量场进行投影，使其严格满足Lyapunov稳定性准则，从而保证稳定性，确保系统在任意状态下的稳定。通过为基向量场和Lyapunov函数采用灵活的神经参数化，我们的框架能够精确表示复杂轨迹，同时通过直接在流形上演化解来满足流形约束。我们提供了应用本框架的高效训练策略，并通过求解单位四元数（S^3）和对称正定矩阵流形上的黎曼LASA数据集，以及在\mathbb{R}^3 \times S^3上演化的机器人运动，证明了其实用性。我们通过大量仿真和在真实世界实验中学习机器人运动，展示了所提方法的性能、可扩展性和实际适用性。