This paper presents a unifying framework for the form-finding and topology-finding of tensegrity structures. The novel computational framework is based on rank-constrained linear matrix inequalities. For form-finding, given the topology (i.e., member connectivities), the determination of the member force densities is formulated into a linear matrix inequality (LMI) problem with a constraint on the rank of the force density matrix. The positive semi-definiteness and rank deficiency condition of the force density matrix are well managed by the rank-constrained LMI-based formulation. A Newton-like algorithm is employed to solve the rank-constrained LMI problem. Two methods, named direct method and indirect method, are proposed to determine the nodal coordinates once the force densities have been obtained. For topology-finding, given the geometry (i.e., nodal coordinates), the determination of the topology is also formulated into an LMI problem with a constraint on the rank of the tangent stiffness matrix. Numerical examples demonstrate that different types of form-finding problems (such as tensegrity structures with single and with multiple self-stress states, symmetric and irregular tensegrity structures) can be uniformly and efficiently solved by the proposed approach. Furthermore, three well-known tensegrity structures are reproduced to verify the effectiveness of the proposed formulation on the topology-finding of tensegrity structures.

本文为张力结构的找形和拓扑查找提供了一个统一的框架。新颖的计算框架基于秩约束线性矩阵不等式。对于找形，给定拓扑结构（即，构件连接性），将构件力密度的确定公式化为线性矩阵不等式（LMI）问题，其中力密度矩阵的等级受到约束。力密度矩阵的正半定性和秩不足条件可以通过基于秩的LMI公式很好地管理。采用类牛顿算法来解决等级受限的LMI问题。两种方法，分别称为直接方法和间接方法一旦获得力密度，建议用来确定节点坐标。对于拓扑查找，在给定几何形状（即节点坐标）的情况下，将拓扑的确定也公式化为对切线刚度矩阵的秩有约束的LMI问题。数值算例表明，所提出的方法可以统一有效地解决不同类型的找形问题（例如具有单个和多个自应力状态的张紧结构，对称和不规则的张紧结构）。此外，复制了三个众所周知的张力结构，以验证所提出的公式对张力结构的拓扑查找的有效性。