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

• URL: http://arxiv.org/abs/2510.05707v1
• Date: Tue, 07 Oct 2025 09:16:48 GMT
• Title: Stable Robot Motions on Manifolds: Learning Lyapunov-Constrained Neural Manifold ODEs
• Authors: David Boetius, Abdelrahman Abdelnaby, Ashok Kumar, Stefan Leue, Abdalla Swikir, Fares J. Abu-Dakka,
• Abstract summary: We propose a general framework for learning stable dynamical systems on Riemannian manifold.<n>Our method guarantees stability by projecting the neural vector field evolving on the manifold so that it strictly satisfies the Lyapunov stability criterion.<n>We demonstrate the performance, scalability, and practical applicability of our approach through extensive simulations and by learning robot motions in a real-world experiment.
• Score: 6.454329937275794
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• MSINO: Curvature-Aware Sobolev Optimization for Manifold Neural Networks [0.0]
  We introduce MSINO, a curvature aware training framework for neural networks defined on Riemannian manifold.<n>We derive geometry dependent constants that yield Descent Lemma with a manifold Sobolev smoothness constant.<n>MSINO provides training time guarantees that explicitly track curvature and transported Jacobians.
  arXiv  Detail & Related papers  (2026-02-26T12:27:00Z)
• ODELoRA: Training Low-Rank Adaptation by Solving Ordinary Differential Equations [54.886931928255564]
  Low-rank adaptation (LoRA) has emerged as a widely adopted parameter-efficient fine-tuning method in deep transfer learning.<n>We propose a novel continuous-time optimization dynamic for LoRA factor matrices in the form of an ordinary differential equation (ODE)<n>We show that ODELoRA achieves stable feature learning, a property that is crucial for training deep neural networks at different scales of problem dimensionality.
  arXiv  Detail & Related papers  (2026-02-07T10:19:36Z)
• Analytic and Variational Stability of Deep Learning Systems [0.0]
  We show that uniform boundedness of stability signatures is equivalent to the existence of a Lyapunov-type energy that dissipates along the learning flow.<n>In smooth regimes, the framework yields explicit stability exponents linking spectral norms, activation regularity, step sizes, and learning rates to contractivity of the learning dynamics.<n>The theory extends to non-smooth learning systems, including ReLU networks, proximal and projected updates, and subgradient flows.
  arXiv  Detail & Related papers  (2025-12-24T14:43:59Z)
• Merging Memory and Space: A State Space Neural Operator [8.378604588491394]
  State Space Neural Operator (SS-NO) is a compact architecture for learning solution operators of time-dependent partial differential equations.<n>We show that SS-NO achieves state-of-the-art performance across diverse PDE benchmarks.
  arXiv  Detail & Related papers  (2025-07-31T11:09:15Z)
• Generative System Dynamics in Recurrent Neural Networks [56.958984970518564]
  We investigate the continuous time dynamics of Recurrent Neural Networks (RNNs)<n>We show that skew-symmetric weight matrices are fundamental to enable stable limit cycles in both linear and nonlinear configurations.<n> Numerical simulations showcase how nonlinear activation functions not only maintain limit cycles, but also enhance the numerical stability of the system integration process.
  arXiv  Detail & Related papers  (2025-04-16T10:39:43Z)
• Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems [49.819436680336786]
  We propose an efficient transformed Gaussian process state-space model (ETGPSSM) for scalable and flexible modeling of high-dimensional, non-stationary dynamical systems.<n>Specifically, our ETGPSSM integrates a single shared GP with input-dependent normalizing flows, yielding an expressive implicit process prior that captures complex, non-stationary transition dynamics.<n>Our ETGPSSM outperforms existing GPSSMs and neural network-based SSMs in terms of computational efficiency and accuracy.
  arXiv  Detail & Related papers  (2025-03-24T03:19:45Z)
• Learning Controlled Stochastic Differential Equations [61.82896036131116]
  This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled differential equations with non-uniform diffusion. We provide strong theoretical guarantees, including finite-sample bounds for (L2), (Linfty), and risk metrics, with learning rates adaptive to coefficients' regularity. Our method is available as an open-source Python library.
  arXiv  Detail & Related papers  (2024-11-04T11:09:58Z)
• Distributionally Robust Model-based Reinforcement Learning with Large State Spaces [55.14361269378122]
  Three major challenges in reinforcement learning are the complex dynamical systems with large state spaces, the costly data acquisition processes, and the deviation of real-world dynamics from the training environment deployment. We study distributionally robust Markov decision processes with continuous state spaces under the widely used Kullback-Leibler, chi-square, and total variation uncertainty sets. We propose a model-based approach that utilizes Gaussian Processes and the maximum variance reduction algorithm to efficiently learn multi-output nominal transition dynamics.
  arXiv  Detail & Related papers  (2023-09-05T13:42:11Z)
• Learning Riemannian Stable Dynamical Systems via Diffeomorphisms [0.23204178451683263]
  Dexterous and autonomous robots should be capable of executing elaborated dynamical motions skillfully. Learning techniques may be leveraged to build models of such dynamic skills. To accomplish this, the learning model needs to encode a stable vector field that resembles the desired motion dynamics.
  arXiv  Detail & Related papers  (2022-11-06T16:28:45Z)
• ImitationFlow: Learning Deep Stable Stochastic Dynamic Systems by Normalizing Flows [29.310742141970394]
  We introduce ImitationFlow, a novel Deep generative model that allows learning complex globally stable, nonlinear dynamics. We show the effectiveness of our method with both standard datasets and a real robot experiment.
  arXiv  Detail & Related papers  (2020-10-25T14:49:46Z)
• Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems [74.80320120264459]
  We present an approach to learn such motions from a limited number of human demonstrations. The complex motions are encoded as rollouts of a stable dynamical system. The efficacy of this approach is demonstrated through validation on an established benchmark as well demonstrations collected on a real-world robotic system.
  arXiv  Detail & Related papers  (2020-05-27T03:51:57Z)
• Formal Synthesis of Lyapunov Neural Networks [61.79595926825511]
  We propose an automatic and formally sound method for synthesising Lyapunov functions. We employ a counterexample-guided approach where a numerical learner and a symbolic verifier interact to construct provably correct Lyapunov neural networks. Our method synthesises Lyapunov functions faster and over wider spatial domains than the alternatives, yet providing stronger or equal guarantees.
  arXiv  Detail & Related papers  (2020-03-19T17:21:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.