## 3 Mathematical Description  
The plane truss finite element of Young modulus \(E_{e}\), cross-sectional area \(A_{e}\) and lengths \(L_{e}\) (Figure 1a) is used to model the components of tensegrity structures. In global coordinate system (\(x\), \(y\)) a truss element is described respectively by the compatibility matrix \(\mathbf{B}_{e}\big{(}\in\mathbb{R}^{1\times 4}\big{)}\) and by the geometric stiffness matrix \(\mathbf{K}_{Ge}\big{(}\in\mathbb{R}^{4\times 4}\big{)}\), taking into account the impact of the self-stress state \(S_{e}\):\[\mathbf{B}_{e}=\left[ \begin{array}{ccc}-c&-s&c&s\end{array} \right]\text{, }\mathbf{K}_{Ge}=\frac{S_{e}}{L_{e}}\left[ \begin{array}{cc} \mathbf{I}_{e}&-\mathbf{I}_{e}\\ -\mathbf{I}_{e}&\mathbf{I}_{e}\end{array} \right]\text{; }\mathbf{I}_{e}=\left[ \begin{array}{cc}s^{2}&-cs\\ -cs&c^{2}\end{array} \right]\text{,} \tag{1}\]  
where: \(c=\frac{x_{j}-x_{l}}{L_{e}}\), \(s=\frac{y_{j}-y_{l}}{L_{e}}\). The qualitative analysis focuses on \(n\)-membered \(\left(e=1,2,\ldots,n\right)\) two-dimensional trusses, described by elasticity matrix \(\mathbf{E}(\in\mathbb{R}^{n\times n})\):  
\[\mathbf{E}=diag\left[ \begin{array}{ccc}\frac{E_{1}A_{1}}{L_{1}}&\frac{E_{2 }A_{2}}{L_{2}}&\ldots&\frac{E_{n}A_{n}}{L_{n}}\end{array} \right] \tag{2}\]  
with \(m\) degrees of freedom \(\mathbf{q}\big{(}\in\mathbb{R}^{m\times 1}\big{)}\):  
\[\mathbf{q}=\left[ \begin{array}{cccc}q_{1}&q_{2}&\ldots&q_{m}\end{array}  \right]^{T} \tag{3}\]  
The compatibility matrix \(\mathbf{B}\big{(}\in\mathbb{R}^{n\times m}\big{)}\) and the geometric stiffness matrix \(\mathbf{K}_{G}(\in\mathbb{R}^{m\times m})\) for trusses are determined using the finite element formalism [49; 50; 51]:  
\[\mathbf{B}=\left[ \begin{array}{c}\mathbf{B}_{1}\mathbf{C}_{1}\\ \mathbf{B}_{2}\mathbf{C}_{2}\\ \ldots\\ \mathbf{B}_{n}\mathbf{C}_{n}\end{array} \right]\text{, }\mathbf{K}_{G}=\sum_{i=1}^{n} \mathbf{C}_{e}^{T}\mathbf{K}_{G_{e}}\mathbf{C}_{e}\text{,} \tag{4}\]  
where \(\mathbf{C}_{e}\big{(}\in\mathbb{R}^{4\times m}\big{)}\) is a Boolean matrix. It is assumed that the number of global degrees of freedom of each element \(q_{i}\) (\(i=1\), \(2\), \(3\), \(4\)) corresponds to the number of global nodes of elements \(n_{1}\), \(n_{2}\) as shown in Figure 1b. Consequently, the non-zero elements of \(\mathbf{C}_{e}\) for each element can be expressed as \(C_{iq_{i}}=1\). Equilibrium equations for trusses can be written in a stress form:  
\[\mathbf{B}^{T}\mathbf{S}=\mathbf{P}\text{,} \tag{5}\]  
where \(\mathbf{S}\big{(}\in\mathbb{R}^{n\times 1}\big{)}\) is the normal force vector and \(\mathbf{P}\big{(}\in\mathbb{R}^{m\times 1}\big{)}\) is the load vector, or via nodal displacements:  
\[\mathbf{K}_{L}\mathbf{q}=\mathbf{P}\text{,} \tag{6}\]  
where \(\mathbf{K}_{L}\big{(}\in\mathbb{R}^{m\times m}\big{)}\) is the linear stiffness matrix:  
\[\mathbf{K}_{L}=\mathbf{B}^{T}\mathbf{E}\mathbf{B}\text{.} \tag{7}\]  
To take into account the non-linearity of the equilibrium equations and the influence of the prestress (self-stress) in the structure, the geometric stiffness matrix (4)\({}_{2}\) is introduced. Then the system of displacement equations (6) takes the form:  
\[(\mathbf{K}_{L}+\mathbf{K}_{G})\mathbf{q}=\mathbf{P}\text{.} \tag{8}\]  
A mathematical description of tensegrity-like pin-jointed structures can be found in [51; 52; 53; 54; 55; 56]. Figure 1: (a) Plane truss finite element, (b) global degrees of freedom of element \(e\). The qualitative analysis of the set of equations for the structure can be done through singular value decomposition [57; 58; 59; 60; 61; 62; 63] of the compatibility matrix \(\mathbf{B}\)[64; 65; 66]. The singular value decomposition of matrix \(\mathbf{B}\) is a factorization in the form:  
\[\mathbf{B}=\mathbf{Y}\mathbf{N}\mathbf{X}^{\mathsf{T}}, \tag{9}\]  
where \(\mathbf{Y}(\in\mathbb{R}^{n\times n})\) and \(\mathbf{X}(\in\mathbb{R}^{m\times m})\) are the orthogonal matrices and \(\mathbf{N}(\in\mathbb{R}^{n\times m})\) is rectangular diagonal matrix.