Curve-Induced Dynamical Systems on Riemannian Manifolds and Lie Groups

2026-03-05

AI Summary

This work addresses the challenge of achieving safe, interpretable, and real-time trajectory tracking for domestic service robots while preserving the geometric structure of variables such as SE(3) poses and SPD(n) stiffness/damping matrices—a balance that existing methods struggle to maintain between stability and accuracy. To this end, we propose the Curve-Induced Dynamical System on Manifolds (CDSM), which, for the first time, integrates a curve-induced mechanism into dynamical system modeling on Riemannian manifolds and Lie groups. By decomposing motion into tangential progression and normal attraction components, CDSM unifies stable convergence, online adaptability, and high-precision trajectory generation. Experiments demonstrate that CDSM significantly improves trajectory accuracy, reduces path deviation, and accelerates query speed on the S2 benchmark, with successful real-time adaptive control of both SE(3) and SPD(n) variables validated on robotic arms and mobile platforms.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Riemannian manifolds
• Lie groups
• dynamical systems
• robotic behavior
• geometric structure

Innovation

• Curve-induced Dynamical Systems
• Riemannian Manifolds
• Lie Groups
• Geometric Control
• Real-time Motion Generation