Learning Riemannian Stable Dynamical Systems via Diffeomorphisms

TL;DR

This paper introduces a method to learn stable dynamical systems on Riemannian manifolds for robotic motion, ensuring stability and geometric consistency using neural manifold ODEs, outperforming Euclidean-based methods.

Contribution

The paper presents a novel approach to learning Riemannian stable dynamical systems with Lyapunov guarantees, incorporating geometric constraints via neural diffeomorphisms.

Findings

Successfully learned complex vector fields on Riemannian manifolds

Enforced stability guarantees on non-Euclidean spaces

Outperformed Euclidean methods in manipulation tasks

Abstract

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. This is challenging as the robot state does not evolve on a Euclidean space, and therefore the stability guarantees and vector field encoding need to account for the geometry arising from, for example, the orientation representation. To tackle this problem, we propose learning Riemannian stable dynamical systems (RSDS) from demonstrations, allowing us to account for different geometric constraints resulting from the dynamical system state representation. Our approach provides Lyapunov-stability guarantees on Riemannian manifolds that are enforced on the desired motion dynamics via diffeomorphisms built on neural manifold ODEs. We show that our Riemannian approach makes it possible to learn stable dynamical systems displaying complicated vector fields on both illustrative examples and real-world manipulation tasks, where Euclidean approximations fail.