Characterizing the universal rigidity of generic frameworks

TL;DR

This paper characterizes the universal rigidity of generic frameworks by linking it to the existence of a positive semi-definite stress matrix of maximal rank, providing a complete criterion for universal rigidity.

Contribution

It establishes that a generic framework's universal rigidity is equivalent to having a positive semi-definite stress matrix of maximal rank, extending previous results.

Findings

Universal rigidity characterized by positive semi-definite stress matrix

Maximal rank stress matrix is necessary and sufficient for generic frameworks

Extension to genericity of strict complementarity in semidefinite programming

Abstract

A framework is a graph and a map from its vertices to E^d (for some d). A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. We show that a generic universally rigid framework has a positive semi-definite stress matrix of maximal rank. Connelly showed that the existence of such a positive semi-definite stress matrix is sufficient for universal rigidity, so this provides a characterization of universal rigidity for generic frameworks. We also extend our argument to give a new result on the genericity of strict complementarity in semidefinite programming.