Stable Robot Motions on Manifolds: Learning Lyapunov-Constrained Neural Manifold ODEs

2025-10-07

AI Summary

This work addresses the challenge of ensuring stability in motion planning and control for robots operating on Riemannian manifolds—such as the unit quaternion manifold (S^3) and the symmetric positive-definite (SPD) matrix manifold. We propose a Lyapunov-stable dynamical system learning framework based on neural ordinary differential equations (Neural ODEs). Our method explicitly projects the neural vector field onto the tangent space, enforcing Lyapunov stability conditions for arbitrary differentiable Lyapunov functions and manifold parameterizations. Crucially, it directly models and solves dynamics on intrinsic manifolds—including (S^3) and (mathbb{R}^3 imes S^3)—bypassing distortion-prone Euclidean embeddings. To our knowledge, this is the first approach to achieve pointwise asymptotic stability in learnable dynamical systems on Riemannian manifolds. We validate the framework on the Riemannian LASA dataset and real-robot experiments, demonstrating high accuracy, strong generalization across trajectories and manifolds, and real-time control capability—thereby significantly improving reliability and scalability of geometry-aware motion learning under geometric constraints.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: Learning & Optimization for ROB
• Humans and AI: Human-Aware Planning and Behavior Prediction

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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 imes 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.

Problem

• Learning stable robot motion from data on manifolds
• Extending stability guarantees to Riemannian manifold trajectories
• Ensuring Lyapunov stability for neural dynamical systems

Innovation

• Learning stable neural ODEs on manifolds
• Projecting vector fields for Lyapunov stability
• Evolving solutions directly on manifold constraints