Universal rigidity of bar frameworks in general position: a Euclidean distance matrix approach

TL;DR

This paper introduces a unified Euclidean distance matrix approach to analyze the universal rigidity of bar frameworks in general position, enabling the use of semi-definite programming to solve the problem.

Contribution

It presents a novel approach based on EDMs and Gram matrices for universal rigidity, with a focus on general position configurations, and surveys recent results and proofs.

Findings

Unified EDM-based approach for rigidity analysis

Application of semi-definite programming to rigidity problems

Results specific to configurations in general position

Abstract

A configuration p in r-dimensional Euclidean space is a finite collection of labeled points p^1,p^2,...,p^n in R^r that affinely span R^r. Each configuration p defines a Euclidean distance matrix D_p = (d_ij) = (||p^i-p^j||^2), where ||.|| denotes the Euclidean norm. A fundamental problem in distance geometry is to find out whether or not, a given proper subset of the entries of D_p suffices to uniquely determine the entire matrix D_p. This problem is known as the universal rigidity problem of bar frameworks. In this chapter, we present a unified approach for the universal rigidity of bar frameworks, based on Euclidean distance matrices (EDMs), or equivalently, on projected Gram matrices. This approach makes the universal rigidity problem amenable to semi-definite programming methodology. Using this approach, we survey some recently obtained results and their proofs, emphasizing the case where the points p^1,...,p^n are in general position.