Modular representation and control of floppy networks

TL;DR

This paper introduces a geometric and algebraic framework to identify and control floppy, under-constrained networks, enabling better understanding and manipulation of their hierarchical and modular properties for applications in robotics and elastic materials.

Contribution

It combines geometric rigidity and algebraic sparsity to analyze floppy networks, providing new tools for their characterization and control beyond critical points.

Findings

Framework effectively identifies zero-energy floppy modes.

Application to robotic motion primitives demonstrates control capabilities.

Physical experiments validate predictions of network responses.

Abstract

Geometric graph models of systems as diverse as proteins, robots, and mechanical structures from DNA assemblies to architected materials point towards a unified way to represent and control them in space and time. While much work has been done in the context of characterizing the behavior of these networks close to critical points associated with bond and rigidity percolation, isostaticity, etc., much less is known about floppy, under-constrained networks that are far more common in nature and technology. Here we combine geometric rigidity and algebraic sparsity to provide a framework for identifying the zero-energy floppy modes via a representation that illuminates the underlying hierarchy and modularity of the network, and thence the control of its nestedness and locality. Our framework allows us to demonstrate a range of applications of this approach that include robotic reaching tasks with motion primitives, and predicting the linear and nonlinear response of elastic networks based solely on infinitesimal rigidity and sparsity, which we test using physical experiments. Our approach is thus likely to be of use broadly in dissecting the geometrical properties of floppy networks using algebraic sparsity to optimize their function and performance.