A unified analytical form-finding of truncated regular octahedral tensegrities

Symmetric configurations are preferred in various application scenarios of tensegrity and the representative examples include Z-based and rhombic truncated regular polyhedral (TRP) tensegrities. A key step in the design of such tensegrities is the determination of their self-equilibrated and stable configurations, known as form-finding, which have been extensively but individually investigated by many research groups. To unify the existing form-finding results of these two types of tensegrities, we propose here a novel hybrid type of the TRP tensegrities that can be readily converted into Z-based and rhombic. Based on this structural transformation, two unified form-finding models of the Z-based, rhombic, and hybrid truncated regular octahedral (TRO) tensegrities are established using the equilibrium/force-density matrix methods. A distribution coefficient for the force-densities of strings is defined for our models to directly yield the self-equilibrium and super-stability conditions for each type of TRO tensegrities, with no need for additional derivation. This work elucidates a connection between the Z-based, rhombic, and hybrid tensegrities in terms of form-finding, and may motivate the exploration of potential unification between more types of structures.