Properties of Lyapunov Subcenter Manifolds in Conservative Mechanical Systems

TL;DR

This paper analyzes Lyapunov subcenter manifolds in conservative mechanical systems, revealing their properties and potential for control applications, supported by theoretical proofs and numerical examples.

Contribution

It establishes conditions under which Lyapunov subcenter manifolds are Eigenmanifolds and explores their symmetry properties, with implications for control in robotics.

Findings

LSMs are Eigenmanifolds under mild non-resonance conditions

Existence of a unique generator for zero-velocity points

Stronger properties of Rosenberg manifolds due to spatial symmetry

Abstract

Multi-body mechanical systems have rich internal dynamics, whose solutions can be exploited as efficient control targets. Yet, solutions non-trivially depend on system parameters, obscuring feasible properties for use as target trajectories. For periodic regulation tasks in robotics applications, we investigate properties of nonlinear normal modes (NNMs) collected in Lyapunov subcenter manifolds (LSMs) of conservative mechanical systems. Using a time-symmetry of conservative mechanical systems, we show that mild non-resonance conditions guarantee LSMs to be Eigenmanifolds, in which NNMs are guaranteed to oscillate between two points of zero velocity. We also prove the existence of a unique generator, which is a connected, 1D manifold that collects these points of zero velocity for a given Eigenmanifold. Furthermore, we show that an additional spatial symmetry provides LSMs with yet stronger properties of Rosenberg manifolds. Here all brake trajectories pass through a unique equilibrium configuration, which can be favorable for control applications. These theoretical results are numerically confirmed on two mechanical systems: a double pendulum and a 5-link pendulum.