## 3 Materials and Method
### Design Phase
#### 3.1.1 Geometry Optimization: 2D FE Model
Both the nodes of the COMBIN14 element were applied as _direct attachment_ to the nodes of the connected parts. The elastic elements were joined together with two pretensioned M8 bolts in the physical MyFlex-\(\gamma\) that could be seen as a single bolt in the sagittal plane. In addition, the middle blade and the spring holder were joined with preloaded M6 screws. Bolt pretension was simulated as a normal force given to the _longitudinal springs body-to-body joints_, which corresponded to the standard bolt pretensions for M6 (7.4 kN, 8.8 class, frictional coefficient = 0.20) and M8 (13.7 kN, 8.8 class, frictional coefficient = 0.20). Connection links with hinge joints at both extremities were modeled with _body-to-body beam joints_ (BEAM3, ANSYS). BEAM3 is a 2D uniaxial element with tension, compression and bending capabilities and it has the translations in both directions \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Surface 1** & **Surface 2** & **Type** & **Formulation** & **Frict. Coeff.** & **Norm. Stiff. Fact.** \\ \hline AF top & UB bottom & bonded & augm.Lagrange & - & 1.00 \\ UB bottom & MB top & frictional & pure penalty & 0.20 & 0.01 \\ MB bottom & LB top & frictional & pure penalty & 0.20 & 0.01 \\ MB bottom & SH top & frictional & pure penalty & 0.20 & 0.01 \\ AF ankle & TC ankle & no separation & augm.Lagrange & - & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Contacts’ properties. See also Figure 7. AF = ankle frame; UB = upper blade; MB = middle blade; LB = lower blade; SH = spring holder; TC = tube connector. Figure 7: Joint and contact modeling in the 2D FE model. See also Table 4 for detailed information. of the \(x\)-\(y\) plane and the rotation around the z-direction at each node. In MyFlex-\(\gamma\), the link part, characterized by two hinge joints that connected the tube connector to the spring holder, was modeled by BEAM3. _Simulation conditions_: ESR feet elastic elements are subjected to high deflections. Therefore, the 2D FE model was simulated in a _nonlinear static structural analysis_ environment. _Simulation outputs_: The behavior of the foot can be evaluated in two different modalities by plotting the _reaction force_ at the fixed constraint against the platform _displacement_ in the \(y_{pf}\) direction (see Figure 4) or against the foot _rotation_. The _foot rotation_ is calculated by using two virtual markers (**G** and **M** in Figure 8). The positions of the virtual markers were chosen considering the marker-set protocol [21,25]. Since the shank was fixed, the **J** marker was considered as (0,0). The displacements of the virtual markers (given in the global \(x\)-\(y\) reference) are given as direct output of the simulation and they can be considered as functions of the platform displacement \(v_{pf}\) in the vertical direction \(y_{pf}\) of the same platform (see Figure 4). Therefore, the displacements of the G marker in the global \(x\)- and \(y\)-directions are  
\[u_{G}=u_{G}(y_{pf}) \tag{1}\]  
\[v_{G}=v_{G}(y_{pf}) \tag{2}\]  
The displacement of the M marker in the global \(x\)- and \(y\)-directions are  
\[u_{M}=u_{M}(y_{pf}) \tag{3}\]  
\[v_{M}=v_{M}(y_{pf}) \tag{4}\]  
If \((x_{\text{GO}};y_{\text{GO}})\) and \((x_{M0};y_{M0})\) are the initial positions of G and M markers, referred to J (0;0), the coordinates of G and M are:  
\[x_{G}(y_{pf})=x_{\text{GO}}+u_{G}(y_{pf}) \tag{5}\]  
\[y_{G}(y_{pf})=y_{\text{GO}}+v_{G}(y_{pf}) \tag{6}\]  
\[x_{M}(y_{pf})=x_{M0}+u_{M}(y_{pf}) \tag{7}\]  
Figure 7.0  
The foot rotation \(\Delta\alpha\) is calculated as the variation of the angle between the GM line and the horizontal direction (Figure 9). The initial angle \(\alpha_{0}\) between the GM line and the horizontal direction and the rotation \(\Delta\alpha\) of the foot are calculated as:  
Figure 7.1  
Figure 7.2  
Figure 7.3 The foot rotation \(\alpha\) is calculated as the variation of the angle between the shank axis and the **GM** line.