Robustness of rank minimization heuristics for form-finding of tensegrity structures. (1st July 2022) - Tkachuk, Anton
Highlights: Form finding of tensegrity uses rank minimization of the force density matrix. The approach requires conversion of the structure to a self-stressed version. Log-det heuristic is more robust than the trace heuristic for the form finding. Several examples where the trace heuristic fails are found and well-documented. The robustness of the form-finding is influenced by the topology and constraints. Abstract: The form-finding process for tensegrity structures with a given topology seeks a distribution of force densities that satisfies stability and equilibrium conditions. A rank minimization problem regarding the force density matrix can yield feasible distributions of force densities. A direct computation of the matrix rank is difficult and is generally avoided in practice. In a recent contribution by Wang [Engineering Structures 2021;227 :111419], a use the trace heuristic for form-finding is proposed. The trace heuristic does not guarantee recovery of the minimal rank. Insufficient rank deficiency leads to a degenerated geometry of the tensegrity. In this contribution, the approach is extended to log-det heuristic and a comparison between both methods is carried out focusing on robustness of the form-finding procedure for tensegrities. Robustness of the form-finding procedure is understood here as the ability to recover at least one feasible tensegrity design. Several examples show that the robustness is mainly influenced by the topology of the structure, constraints on force densities and the admissible set for force densities. Furthermore, the proposed extension with log-det heuristic outperforms the original method in terms of robustness, however it increases the computational cost.