Curve-Induced Dynamical Systems on Riemannian Manifolds and Lie Groups

TL;DR

This paper introduces a real-time framework for creating stable and accurate dynamical systems directly on Riemannian manifolds and Lie groups, enhancing robotic motion control in complex geometric spaces.

Contribution

It proposes Curve-induced Dynamical Systems on Smooth Manifolds (CDSM), a novel method that constructs dynamical systems based on a nominal curve on manifolds, with stability analysis and practical validation.

Findings

Improved trajectory accuracy on S2 benchmark

Reduced path deviation compared to existing methods

Faster trajectory generation and query times

Abstract

Deploying robots in household environments requires safe, adaptable, and interpretable behaviors that respect the geometric structure of tasks. Often represented on Lie groups and Riemannian manifolds, this includes poses on SE(3) or symmetric positive definite matrices encoding stiffness or damping matrices. In this context, dynamical system-based approaches offer a natural framework for generating such behavior, providing stability and convergence while remaining responsive to changes in the environment. We introduce Curve-induced Dynamical systems on Smooth Manifolds (CDSM), a real-time framework for constructing dynamical systems directly on Riemannian manifolds and Lie groups. The proposed approach constructs a nominal curve on the manifold, and generates a dynamical system which combines a tangential component that drives motion along the curve and a normal component that attracts the state toward the curve. We provide a stability analysis of the resulting dynamical system and validate the method quantitatively. On an S2 benchmark, CDSM demonstrates improved trajectory accuracy, reduced path deviation, and faster generation and query times compared to state-of-the-art methods. Finally, we demonstrate the practical applicability of the framework on both a robotic manipulator, where poses on SE(3) and damping matrices on SPD(n) are adapted online, and a mobile manipulator.