Generalized prismatic tensegrity derived by dihedral symmetric lines

Classic prismatic tensegrity structures, characterized by dihedral symmetry with one orbit of nodes, are among the simplest and possibly the earliest spatial tensegrity structures invented. This paper introduces a generalized form of the prismatic tensegrity structures by converting a single-loop linkage into truss, in which the lines of joint axes rather than the nodes have dihedral symmetry. Since the vector space formed by the line coordinates of these joints has rank degeneracy one, the generated tensegrity structures are kinematically and statically indeterminate. These tensegrity structures are further proved to be prestress-stable, generally, for the total or partial parameter space of lines based on a second-order analysis of screws, and are called dihedral-line tensegrity structures in this paper. Specifically, this paper focuses on symmetric dihedral-line tensegrity structures, in which the nodes also have dihedral symmetry but in two orbits and members in seven orbits, and are called two-orbit dihedral-line tensegrity structures. It is found that there are at least N struts for the generated tensegrity with $D_N$ symmetry. And the classic prismatic tensegrity structures can be recovered from these dihedral-line tensegrity structures by removing certain zero-force members. Symmetric-adapted force density matrices are also provided as well as the relation to that of the classic prismatic tensegrity. Given $4N+6$ dimensional parameters inherent to these tensegrity structures, a rich variety of tensegrity structure family is presented.