Characterizing the Universal Rigidity of Generic Tensegrities

TL;DR

This paper extends the characterization of universal rigidity in tensegrities, showing that generic universally rigid tensegrities are super stable and that this holds even with symmetric configurations, using advanced algebraic techniques.

Contribution

It proves that generic universally rigid tensegrities are super stable and extends this characterization to symmetric tensegrities using representation theory.

Findings

Universal rigidity implies super stability in generic tensegrities.

The characterization holds for symmetric tensegrities modulo symmetry.

Block-diagonalization and representation theory are key tools in the proof.

Abstract

A tensegrity is a structure made from cables, struts and stiff bars. A $d$-dimensional tensegirty is universally rigid if it is rigid in any dimension $d'$ with $d'\geq d$. The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representation of finite groups.