On affine motions and universal rigidity of tensegrity frameworks

TL;DR

This paper extends the conditions for universal rigidity from bar frameworks to tensegrity frameworks and relaxes the general position assumption to a weaker affine spanning condition.

Contribution

It generalizes a known rigidity result to tensegrity frameworks and introduces a weaker assumption on the configuration for universal rigidity.

Findings

Extended rigidity conditions to tensegrity frameworks.

Replaced general position assumption with affine spanning condition.

Provided theoretical proof of the generalized result.

Abstract

Recently, Alfakih and Ye [Lin. Algebra Appl. 438:31--36, 2013] proved that if an $r$-dimensional bar framework $(G,p)$ on $n \geq r+2$ nodes in general position in $\R^r$ admits a positive semidefinite stress matrix with rank $n-r-1$, then $(G,p)$ is universally rigid. In this paper, we generalize this result in two directions. First, we extend this result to tensegrity frameworks. Second, we replace the general position assumption by the weaker assumption that in configuration $p$, each point and its neighbors in $G$ affinely span $\R^r$.