## 3 Search Strategies for Spine Biotensegrity Models
### Equilibrium Equations of Spine Biotensegripty Model
#### 3.2.2 Equilibrium Equations: Formulation  
As the spine biotensegripty model consists of struts and cables that are pin and connected, the set of equilibrium equations adopted in this study is derived based on the basic formulations of pin-jointed structures. Such formulation for tensegripty structures can also be found in [32; 33]. The spine biotensegripty model in this study consists of \(m\) elements, \(n_{o}\) nodes, \(n_{c}\) supports and \(n_{u}=3(n_{o}\)-\(n_{c})\) unconstrained degree of freedoms. Consider an element \(k\) of the spine biotensegripty model connecting node \(i\) and \(j\) (\(i<j\)) in a Cartesian coordinate system \(O\)-\(XYZ\) as shown in Figure 8. The vectors of nodal coordinates of node \(i\) and \(j\) are denoted as \(\vec{\mathrm{x}}_{i}\) and \(\vec{\mathrm{x}}_{j}\), respectively. The length of element \(k\) is evaluated using the following expression:  
\[L_{T}=\sqrt{\left(x_{j}-x_{i}\right)^{T}\left(x_{j}-x_{i}\right)} \tag{1}\]  
The vector of directional cosines \(\lambda\) of an element \(k\) is defined as follows:  
\[\lambda=\frac{\left(x_{j}-x_{i}\right)}{L_{T}} \tag{2}\]  
Figure 8: A three-dimensional pinned jointed spine biotensegripty model element with the nodal forces. Figure 8 also shows the axial force \(n\) of element \(k\) for the spine biotensegrity model, and the components of the nodal force at \(i\) and \(j\), \(f_{i}\) and \(f_{j}\), respectively (where \(f_{i}=\) - \(f_{j}\)), which are in equilibrium with \(n\). Considering equilibriums at all nodes, the following equation can be obtained:  
\[\mathbf{F=Bn} \tag{3}\]  
where \(\mathbf{B}\) is a \(n_{u}\times m\) matrix consisting of the vector of directional cosines \(\boldsymbol{\lambda}\) of all elements in the spine biotensegrity model with respect to \(x\), \(y\), z axes, \(\mathbf{F}\) is a vector of nodal forces with size \(n_{u}\) and \(\mathbf{n}\) is a vector of axial forces for all elements with size \(m\). By making use of the Moore-Penrose generalized inverse, the necessary and sufficient condition for the existence of solution for Equation (3) can be expressed using the following equation:  
\[\mathbf{F=BB^{+}F} \tag{4}\]  
which can be rewritten in the following form  
\[[\mathbf{I}_{n_{u}}-\mathbf{BB^{+}}]\mathbf{F=}0 \tag{5}\]  
where \(\mathbf{I}_{n_{u}}\) is an identity matrix of order \(n_{u}\) and \(\mathbf{B^{+}}\) is Moore-Penrose generalized inverse of matrix \(\mathbf{B}\). Singular value decomposition method is used in this study to compute the Moore-Penrose generalized inverse of the matrix \(\mathbf{B}\) with rank deficiency (i.e., rank \(<n_{u}\)). For the case of self-equilibrated state without external forces, \(\mathbf{F=0}\). Under such condition, the solution for the vector of axial forces \(\mathbf{n}\) for the spine biotensegrity model is given by the following equation:  
\[\mathbf{n=}[\mathbf{I}_{m}-\mathbf{B^{+}B}]\mathbf{\beta} \tag{6}\]  
where \(\mathbf{\beta}\) is a vector of arbitrary coefficient of size \(m\). Denoting matrix \([\mathbf{I}_{m}-\mathbf{B^{+}B}]\) in Equation (6) as matrix \(\mathbf{G}\) of size \(m\times m\) and applying eigenvector basis decomposition method, the real symmetric matrix \(\mathbf{G}\) can be expressed as follows:  
\[\mathbf{G=\Phi\Lambda\Phi^{T}} \tag{7}\]  
where \(\Lambda\) is a diagonal matrix whose diagonal elements \(\Lambda i\) (\(i=1,2,\ldots,m\)) correspond to eigenvalues of matrix \(\mathbf{G}\); \(\Phi\) is an orthogonal matrix whose \(i\)th column \(\Phi\)\({}_{i}\) (\(i=1,2,\ldots,m\)) corresponds to eigenvector associated with eigenvalue \(\Lambda\)i. Since matrix \(\mathbf{G}\) is an orthogonal projection matrix, the eigenvalues \(\Lambda_{i}\) returns the value of either 0 or 1. The eigenvectors \(\Phi_{i}\) correspond to vectors of independent self-equilibrium stress modes for biotensegrity models considered in this study. It is noted that the number of vectors of independent self-equilibrium stress modes is equal to the number of non-zero eigenvalues of matrix \(\mathbf{G}\) or rank \([\mathbf{G}]\).