Form-finding of tensegrity structures via rank minimization of force density matrix

This study proposes a general computational framework for the form-finding of tensegrity structures. The procedure is divided into two stages in which the member force densities and nodal coordinates are obtained respectively. In the first stage, the determination of force densities is transformed into a rank minimization problem regarding the force density matrix and then formulated into a semi-definite programming. The unilaterality condition of member forces and the positive semi-definiteness condition of force density matrix are incorporated as constraints. In the second stage, the determination of nodal coordinates is formulated into a constrained nonlinear programming model. The nodal positions and member lengths are assigned as constraints and auxiliary variables are introduced to homogenize the member lengths. The proposed formulation bypasses the number of self-stress states in a tensegrity structure thus applies to both tensegrity structures with single and multiple self-stress states. Different from existing studies that are based on the minimum required rank deficiency condition of force density matrix, the proposed method can handle tensegrity structures that have a force density matrix with rank deficiency greater than the required minimum number. Several examples are presented to verify the effectiveness of the proposed method on different types of tensegrity structures.