Characterizing the universal rigidity of generic tensegrities

• Published in: Mathematical programming Vol. 197; no. 1; pp. 109 - 145
• Main Authors: Oba, Ryoshun, Tanigawa, Shin-ichi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2023; Springer Nature B.V
• Subjects: Cables; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Rigidity; Semidefinite programming; Stability; Struts; Symmetry; Tensegrity; Theoretical; Universal rigidity; Semidefinite programming; Super stability; 52C25 Rigidity and flexibility of structures (aspects of discrete geometry); 90C22 Semidefinite programming; Tensegrity; Graph rigidity; Strict complementarity
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-021-01730-2

A tensegrity is a structure made from cables, struts, and stiff bars. A d -dimensional tensegrity is universally rigid if it is rigid in any dimension d ′ with d ′ ≥ 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 representations of finite groups.