# Multi-Directional Shape Change Analysis of Biotensegrity Model Mimicking Human Spine Curvature
## 3 Shape Change Strategies for Spine Biotensegrity Models
### Formulation of Incremental Equilibrium Equations  
This section shows the formulation of incremental equilibrium equations for spine biotensegrity models with \(m\) elements, \(n\) nodes, \(n_{c}\) constraints and \(n_{u}\left(3n-n_{c}\right)\) unconstrained degree of freedoms. The formulation is generally based on the static equilibrium equation, which could be expressed as follows:  
\[\mathbf{F}=\mathbf{B}\mathbf{n} \tag{1}\]  
where \(\mathbf{B}\) is a \(n_{u}\times m\) matrix of directional cosines, \(\mathbf{F}\) is a vector of nodal forces with size \(n_{u}\) and \(\mathbf{n}\) is a vector of axial forces with size \(m\). The vector of directional cosine \(\lambda\) for a member of a biotensegrity model at incremental step \(t\) could be written as follows:  
Figure 3: Self-equilibrated configuration for spine biotensegrity models (**a**) SB1, (**b**) SB2 and (**c**) SB3 [5]. \[{}^{t}\mathbf{\lambda}=\frac{\left({}^{t-1}\mathbf{x}_{j}-{}^{t-1}\mathbf{x}_{i}\right)}{{}^{t -1}I+{}^{t-1}u} \tag{2}\]  
where \(l\), \(u\), \(x_{i}\) and \(x_{j}\) are the initial geometrical inputs, particularly the length after forced elongation, elastic elongation, nodal coordinate at nodes \(i\) and \(j\), obtained from the previous step \(t-1\). The incremental equilibrium equation for the entire structure could be obtained by the differentiation of Equation (1) as follows:  
\[{}^{t}\dot{\textbf{F}}={}^{t}\dot{\textbf{B}}~{}^{t}\textbf{n}+{}^{t}\textbf{B }~{}^{t}\dot{\textbf{n}} \tag{3}\]  
Contents of equilibrium equations at the member level which are used in the summation process to obtain Equation (3) are shown as follows:  
\[\left\{ \begin{array}{c}{}^{t}\dot{\textbf{f}}_{i}\\ {}^{t}\dot{\textbf{f}}_{j}\end{array} \right\}=\left[ \begin{array}{c}-{}^{t} \dot{\lambda}\\ {}^{t}\dot{\lambda}\end{array} \right]^{t}\textbf{n}+\left[ \begin{array}{c}-{} ^{t}\lambda\\ {}^{t}\lambda\end{array} \right]^{t}\dot{\textbf{n}} \tag{4}\]  
where the \(\textbf{f}_{i}\), \(\textbf{f}_{j}\) are vectors of nodal force at end \(i\) and \(j\) of the member. The first term on the right-hand side of Equation (4) could be expressed as follows:  
\[\left[ \begin{array}{c}-{}^{t}\dot{\lambda}\\ {}^{t}\dot{\lambda}\end{array} \right]^{t}\textbf{n}={}^{t}\textbf{k}_{g}\left\{ \begin{array}{c}{}^{t}\mathbf{x}_{i}\\ {}^{t}\mathbf{x}_{j}\end{array} \right\} \tag{5}\]  
where \({}^{t}\textbf{k}_{g}\) is the 6 \(\times\) 6 geometrical stiffness matrix of which the contents are given as follows:  
\[{}^{t}\textbf{k}_{g}=\frac{{}^{t-1}n}{{}^{t-1}I+{}^{t-1}u}\left[ \begin{array} []{cc}\left[\textbf{I}_{3}-{}^{t}\mathbf{\lambda}^{t}\mathbf{\lambda}^{\text{T}}\right] &-\left[\textbf{I}_{3}-{}^{t}\mathbf{\lambda}^{t}\mathbf{\lambda}^{\text{T}}\right] \\ -\left[\textbf{I}_{3}-{}^{t}\mathbf{\lambda}^{t}\mathbf{\lambda}^{\text{T}}\right]& \left[\textbf{I}_{3}-{}^{t}\mathbf{\lambda}^{t}\mathbf{\lambda}^{\text{T}}\right] \end{array} \right] \tag{6}\]  
where \(\textbf{I}_{3}\) is a \(3\times 3\) identity matrix. The second term on the right-hand side of Equation (4) could be expressed as  
\[\left\{ \begin{array}{c}{}^{-t}\lambda\\ {}^{t}\lambda\end{array} \right\}^{t}\dot{\textbf{n}}={}^{t}\textbf{k}_{e}\! \left\{ \begin{array}{c}{}^{t}\mathbf{x}_{i}\\ {}^{t}\mathbf{x}_{j}\end{array} \right\}-\left\{ \begin{array}{c}{}^{-t}\lambda\\ {}^{t}\lambda\end{array} \right\}\frac{EA}{{}^{t-1}I}\!\left(1+\frac{{}^{t-1}u}{ {}^{t-1}I}\right)^{t}\!\dot{\textbf{j}} \tag{7}\]  
where \({}^{t}\textbf{k}_{e}\) is the 6 \(\times\) 6 linear stiffness matrix which is given by the following expression:  
\[{}^{t}\textbf{k}_{e}=\frac{EA}{{}^{t-1}I}\!\left[ \begin{array}{cc}{}^{t} \mathbf{\lambda}^{t}\mathbf{\lambda}^{\text{T}}&-{}^{t}\mathbf{\lambda}^{t}\mathbf{\lambda}^{ \text{T}}\\ -{}^{t}\mathbf{\lambda}^{\text{T}}&{}^{t}\mathbf{\lambda}^{\text{T}}\end{array}