Shaping Energy Exchange with Gyroscopic Interconnections: a geometric approach

TL;DR

This paper introduces a geometric framework to analyze energy exchange in gyroscopic interconnected systems, using invariant tori and resonance conditions to inform energy control strategies.

Contribution

It develops a novel inscribed-radius metric and an efficient method to analyze subsystem energy exchange without simulation, enhancing energy shaping design.

Findings

Low-order resonances can limit energy depletion due to phase-locking.

High-order resonances allow for better energy recovery and bounds.

The framework enables explicit design of energy absorption and containment strategies.

Abstract

Gyroscopic interconnections enable redistribution of energy among degrees of freedom while preserving passivity and total energy, and they play a central role in controlled Lagrangian methods and IDA-PBC. Yet their quantitative effect on transient energy exchange and subsystem performance is not well characterised. We study a conservative mechanical system with constant skew-symmetric velocity coupling. Its dynamics are integrable and evolve on invariant two-tori, whose projections onto subsystem phase planes provide geometric description of energy exchange. When the ratio of normal-mode frequencies is rational, these projections become closed resonant Lissajous curves, enabling structured analysis of subsystem trajectories. To quantify subsystem behaviour, we introduce the inscribed-radius metric: the radius of the largest origin-centred circle contained in a projected trajectory. This gives a lower bound on attainable subsystem energy and acts as an internal performance measure. We derive resonance conditions and develop an efficient method to compute or certify the inscribed radius without time-domain simulation. Our results show that low-order resonances can strongly restrict energy depletion through phase-locking, whereas high-order resonances recover conservative bounds. These insights lead to an explicit interconnection-shaping design framework for both energy absorption and containment control strategies, while taking responsiveness into account.