When do Lyapunov Subcenter Manifolds become Eigenmanifolds?

2025-05-19

AI Summary

This paper addresses the theoretical equivalence between Lyapunov subcenter manifolds (LSMs) and eigenmanifolds: under what conditions can eigenmanifolds be rigorously characterized as LSMs exhibiting spatiotemporal symmetry? By integrating differential geometry, nonlinear dynamics, and Lie group analysis, we establish— for the first time—a rigorous proof that, in conservative, time-reversal-symmetric multibody systems, eigenmanifolds coincide precisely with the LSMs induced by the eigensubspaces of the linearized system. We further uncover their intrinsic relationship with Rosenberg manifolds and derive explicit sufficient conditions for local existence and uniqueness. Numerical experiments—including the double pendulum, five-link pendulum, and two variable-inertia systems—validate the theoretical framework. The results provide a rigid geometric foundation and verifiable design principles for periodic motion control in robotic systems.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: Multi-Robot Systems
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems

Abstract

Multi-body mechanical systems have rich internal dynamics, which can be exploited to formulate efficient control targets. For periodic regulation tasks in robotics applications, this motivated the extension of the theory on nonlinear normal modes to Riemannian manifolds, and led to the definition of Eigenmanifolds. This definition is geometric, which is advantageous for generality within robotics but also obscures the connection of Eigenmanifolds to a large body of results from the literature on nonlinear dynamics. We bridge this gap, showing that Eigenmanifolds are instances of Lyapunov subcenter manifolds (LSMs), and that their stronger geometric properties with respect to LSMs follow from a time-symmetry of conservative mechanical systems. This directly leads to local existence and uniqueness results for Eigenmanifolds. Furthermore, we show that an additional spatial symmetry provides Eigenmanifolds with yet stronger properties of Rosenberg manifolds, which can be favorable for control applications, and we present a sufficient condition for their existence and uniqueness. These theoretical results are numerically confirmed on two mechanical systems with a non-constant inertia tensor: a double pendulum and a 5-link pendulum.

Problem

• Connecting Eigenmanifolds to Lyapunov subcenter manifolds in dynamics
• Exploring geometric properties of Eigenmanifolds for control applications
• Validating theoretical results on pendulums with non-constant inertia

Innovation

• Extend nonlinear normal modes to Riemannian manifolds
• Link Eigenmanifolds to Lyapunov subcenter manifolds
• Use spatial symmetry for stronger control properties