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

arXiv: https://arxiv.org/abs/2510.05707 · HTML: https://mlsanity.com/paper/2510.05707/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 41 references that are in the map.
- First posted on arXiv 7 Oct 2025.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Reachability Analysis of Deep Neural Networks with Provable Guarantees](https://mlsanity.com/paper/1805.02242/) (2018)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [On The Verification of Neural ODEs with Stochastic Guarantees](https://mlsanity.com/paper/2012.08863/) (2020)
- [Multiple shooting for training neural differential equations on time series](https://mlsanity.com/paper/2109.06786/) (2021)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints](https://mlsanity.com/paper/2306.09739/) (2023)
- [Continual Learning from Demonstration of Robotics Skills](https://mlsanity.com/paper/2202.06843/) (2022)
- [Learning Stable Robotic Skills on Riemannian Manifolds](https://mlsanity.com/paper/2208.13267/) (2022)
- [Learning Complex Motion Plans using Neural ODEs with Safety and Stability Guarantees](https://mlsanity.com/paper/2308.00186/) (2023)
- [Learning Deep Dynamical Systems using Stable Neural ODEs](https://mlsanity.com/paper/2404.10622/) (2024)
- [Scalable and Efficient Continual Learning from Demonstration via a Hypernetwork-generated Stable Dynamics Model](https://mlsanity.com/paper/2311.03600/) (2023)
- [Riemannian Motion Policy Fusion through Learnable Lyapunov Function Reshaping](https://mlsanity.com/paper/1910.02646/) (2019)
- [Learning Deep Robotic Skills on Riemannian manifolds](https://mlsanity.com/paper/2210.15244/) (2022)
- [FxTS-Net: Fixed-Time Stable Learning Framework for Neural ODEs](https://mlsanity.com/paper/2411.09118/) (2024)

## Abstract (arXiv)

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 (S3) and symmetric positive-definite matrix manifolds, as well as robotic motions evolving on ℝ3 × S3. We demonstrate the performance, scalability, and practical applicability of our approach through extensive simulations and by learning robot motions in a real-world experiment.