# Multi-Directional Shape Change Analysis of Biotensegrity Model Mimicking Human Spine Curvature
## 3 Shape Change Strategies for Spine Biotensegrity Models
### Optimization by Sequential Quadratic Programming  
The shape change of the spine biotensegrity model is achieved via the changeable length of cables which is enforced in numerical analysis with the forced elongation of cables. The shape change strategy is planned for a set of identified monitored nodes to advance from their initial position to target positions. This advancement to target position represented by a set of prescribed coordinates of the monitored nodes is achieved by minimizing the following objective function:  
\[\min f(\vec{x})=\left|{}^{t}\vec{x}^{\prime}-\vec{\vec{x}}\right| \tag{13}\]  
where \(\vec{\vec{x}}\) is the prescribed target coordinates and \({}^{t}\vec{x}^{\prime}={}^{t-1}\vec{x}^{\prime}+{}^{t}\dot{\vec{x}}^{\prime}\) is the current coordinates for all of the specified monitored nodes at current step \(t\) during the shape change analysis.  
The Sequential Quadratic Programming (SQP) method is used to solve the nonlinear optimization problem as shown in Equation (13). The quadratic programming sub-problem can be written as follows:  
\[\begin{array}{l}\min_{x\in R^{2}}\!\!f(\vec{x})=\mathbf{g}^{T}{}^{t}\dot{ \mathbf{i}}+\frac{1}{2}{}^{t}\dot{\mathbf{i}}^{T}\mathbf{H}{}^{t}\dot{\mathbf{i }}\\ \text{subject to }\mathbf{A}_{2}{}^{t}\dot{\mathbf{i}}\geq\ \mathbf{b}_{2}\\ \text{where}\\ \mathbf{g}=\mathbf{B}_{2}{}^{T}\!\!\left({}^{t-1}\vec{x}^{\prime}-\vec{\vec{x }}\right)\\ \mathbf{H}=\mathbf{B}_{2}{}^{T}\!\!\mathbf{B}_{2}\end{array} \tag{14}\]  
\({}^{t}\dot{\mathbf{i}}\) is a \(m_{\textit{cf}}\times 1\) vector of incremental forced elongation which corresponds to the optimization variable, and \(\mathbf{H}\) is a positive-definite approximation of the Hessian matrix of the Lagrangian function. \(m_{\textit{cf}}\) is the number of cables excluding the undeformed cables. Matrix \(\mathbf{A}_{2}\) and vector \(\vec{b}_{2}\) are the inequality constraints corresponding to the limitation in allowable axial forces and forced elongation, respectively, as follows:  
\[\mathbf{A}_{2}=\left\{\begin{array}{c}\mathbf{D}_{2}\\ \ldots\\ -\mathbf{D}_{2}\\ \ldots\\ \mathbf{I}_{mcf}\\ \ldots\\ -\mathbf{I}_{mcf}\end{array}\right\};\mathbf{b}_{2}=\left\{\begin{array}{c} \mathbf{n}_{L}-{}^{t-1}\mathbf{n}\\ \ldots\\ -\big{(}\mathbf{n}_{U}-{}^{t-1}\mathbf{n}\big{)}\\ \ldots\\ -d_{cl}\\ \ldots\\ -d_{cl}\end{array}\right\} \tag{15}\]  
where \(\mathbf{A}_{2}\) is a \((2m+2m_{\textit{cf}})\times m_{\textit{cf}}\) matrix and \(\mathbf{b}_{2}\) is a \((2m+2m_{\textit{cf}})\times 1\) vector. \(\mathbf{n}_{L},{}^{t-1}\mathbf{n},\mathbf{n}_{U}\) are vectors of the lower limit of axial force, axial force at step \((t-1)\) and upper limit of axial force, respectively. \(d_{cl}\) (with \(d_{cl}>0\)) is the limit for length imposed on the magnitude of forced elongation \({}^{t}\dot{\mathbf{i}}\).  
For elastic material properties, the lower and upper axial force limits for cables and struts are defined as:  
\[0\leq n_{c}\leq\sigma_{c}A_{c} \tag{16}\]\[\max\bigg{\{}-\frac{\pi^{2}E_{s}I_{s}}{l_{s}{}^{2}},-\sigma_{s}A_{s}\bigg{\}}\leq n _{s}\leq\sigma_{s}A_{s} \tag{17}\]  
where \(n_{c}\), \(\sigma_{c}\) and \(A_{c}\) are axial forces, the yield stress and the cross-sectional area for cable elements, respectively; \(n_{s}\), \(E_{s}\), \(I_{s}\), \(L_{s}\), \(\sigma_{s}\) and \(A_{s}\) are axial forces, Young modulus, the moment of inertia, the element current length, the yield stress and the cross sectional area for strut elements, respectively. In this study, all strut elements have a circular cross-section.  
The obtained converged \({}^{t}\dot{1}\) is then used in each incremental step of shape change analysis to determine the updated incremental nodal coordinates in Equation (11) and the incremental axial forces in Equation (12). Figure 4 shows the proposed algorithm for the shape change analysis of tensegrity.