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

TL;DR

This work tackles the challenge of learning stable robot motions on Riemannian manifolds by introducing sNMODE, a neural manifold ODE framework that enforces stability via a Lyapunov-based projection. It jointly learns a base vector field and a neural Lyapunov function on the manifold, formulating $f(x)=g(x)-\nabla_{x}^{\mathcal{M}}V(x) \frac{\text{ReLU}(\mathcal{L}_{g}V(x)+\alpha V(x))}{\|\nabla_{x}^{\mathcal{M}}V(x)\|^2}$ to guarantee ${\mathcal{L}_{f}V}(x)\le -\alpha V(x)$ and hence exponential stability. The paper also identifies and corrects a critical equilibrium-point issue in prior Euclidean NODE stability work by enforcing $g(x_e)=0$, and it introduces a three-stage training strategy to efficiently learn stable dynamics from demonstrations. Extensive experiments on Riemannian LASA datasets, multi-manipulator tasks, and a real-world Panda robot demonstrate improved stability, scalability, and applicability over existing methods such as RSDS and SDS-RM, highlighting the practical impact of stable, manifold-aware learning for robotics.

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 \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.

Figures (6)

• Figure 1: Learning Stable Motions on Riemannian Manifolds using sNMODE. A user demonstrates position-orientation motions (Left), which lie on a Riemannian manifold (Centre); sNMODE learns a stable vector field that enables the robot to autonomously perform the motion (Right).
• Figure 2: Our sNMODE Three-Stage Training Strategy. The NMODEs and the Lyapunov function are trained on the human demonstrations .
• Figure 3: Numerical solutions of a vector field stabilised using (a) ManekKolter2019 and (b) our approach. In this figure, $f(x_e) \neq 0$ when using ManekKolter2019 as described in \ref{['sec:main-stability-issue']}. This leads to numerical instability in ManekKolter2019, which is addressed by our correction. In this figure, $\mathcal{M} = \mathbb{R}^2$ and $V(x) = d_{\mathcal{M}}(x, x_e)$.
• Figure 4: Unit Quaternion LASA Datasets. In our improved dataset, the LASA shapes are visibly distorted by the manifold geometry, while the original dataset maintains the Euclidean geometry. The unit quaternions are displayed in an axis-angle representation. The stable NMODEs solutions follow the demonstrations from both datasets.
• Figure 5: Multi-Manipulator Results. We train stable NMODEs to jointly perform $2$--$9$ tasks controlling multiple manipulators simultaneously. "Train Time" is the duration of training, "Loss" is the root mean squared distance between the demonstrations and the learned solutions, and "Dimension" is the dimension of the manifold. While the loss is independent of problem size, the runtime of our approach scales linearly with the problem size. The task illustrations are taken from AuddyEtAl2023Continual.

Theorems & Definitions (5)

• Example 1
• Definition 1: Quasi-Global Stability
• Theorem 1: Lyapunov Function
• Theorem 2
• proof