# A Bioinspired Humanoid Foot Mechanism
## 3 Results  
### Simulation and Results  
The behavior of the humanoid foot was checked through a nonlinear, dynamic study using FEA. Iterative schemes to solve Equation (1) through the Newton-Raphson method (NR), already integrated in the software, were used for each node (i). \[[M]^{t+\Delta t}\{U^{\prime\prime}\}^{(i)}+[C]^{t+\Delta t}\{U^{\prime}\}^{(i)} +\ ^{t+\Delta t}[K]^{(i)\ t+\Delta t}[\Delta U]^{(i)}\ =\ ^{t+\Delta t}\{R\}-\ ^{t+\Delta t}\{F\}^{(i-1)} \tag{1}\]  
where:  
\(M\) = Mass matrix of the system;  
\(C\) = Damping matrix of the system;  
\(\bullet\) \(\  
entire assembly. The temperature loads were included, setting a constant temperature of 298 Kelvin. The mean angular displacement of the ankle acquired from the subject was used as the input motion for the nonlinear dynamic simulation. The definition of the time curve for the angle displacement application was set to simulate the mean step duration for 0.63 s, using a force control technique. The input given to the simulation was extracted by the gait analysis data, and it corresponds to the average gait of Figure 3. The input motion was applied to the ankle joint axis, which is highlighted in blue in Figure 7.  
The humanoid foot is composed of four bodies, three of which are in ABS rigid plastic (Table 2), and the middle of which is set to thermoplastic polyurethane (TPU) for his flexible mechanical properties, simulating the muscle part. Another key component is the spring--shown in Figure 8--in the forefoot, which passively improves the ability of the foot to adapt to the ground; the spring was modelled with a 2 mm diameter, five coils, and an 8 mm length, and alloy steel was chosen as material in order to simulate a commercial part. After setting all of the required parameters, a mesh was generated with the parameters in Table 3 to perform the FEA. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **ABS** & **TPU** & **Alloy Steel** \\ \hline Yield strength & 4.50 \(\times\) 10\({}^{7}\) N/m\({}^{2}\) & 5.41 \(\times\) 10\({}^{7}\) N/m\({}^{2}\) & 2.76 \(\times\) 10\({}^{7}\) N/m\({}^{2}\) \\ Tensile strength & 7.30 \(\times\) 10\({}^{7}\) N/m\({}^{2}\) & 7.79 \(\times\) 10\({}^{7}\) N/m\({}^{2}\) & 6.89 \(\times\) 10\({}^{7}\) N/m\({}^{2}\) \\ Elastic modulus & 3.00 \(\times\) 10\({}^{9}\) N/m\({}^{2}\) & 1.48 \(\times\) 10\({}^{7}\) N/m\({}^{2}\) & 6.90 \(\times\) 10\({}^{10}\) N/m\({}^{2}\) \\ Poisson’s ratio & 0.35 & 0.55 & 0.33 \\ Mass density & 1200.00 kg/m\({}^{3}\) & 1217.00 kg/m\({}^{3}\) & 2700.00 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties of the materials in the FEA. Figure 8: Detailed view of the forefoot torsional spring. Figure 7: Ankle joint axis of the CAD model, on which the input motion is applied. Figure 9 shows the FEM nonlinear dynamic analysis stress results computed by the use of the Von Mises formulation. A maximum value of 2.57 \(\times\) 10\({}^{6}\) N/m\({}^{2}\) is observed in the upper part of the rigid midfoot, next to the connection to the compliant body. The maximum value is significantly smaller than the yield limit of the material, thus validating the proposed humanoid foot for an average walking gait. The behavior of the impact of the foot on the ground is obtained by applying the mean angle displacement taken from the analyzed subjects (see Section 2). The results can be observed in the step-by-step effect of the load on the different bodies of the humanoid foot, as is shown in the linear displacement distribution reported in Figure 10:  
* Starting from the rigid body of the hindfoot, the angular displacement is applied to the ankle joint. The hindfoot is connected to the flexible body of the midfoot, which starts to deform. * When the angular displacement causes the foot to hit the ground, the midfoot is characterized by a stress concentration in the upper part, as per Figure 9.
* The arc bottom part of the foot is loaded with a maximum tension of 1.57 \(\times\) 10\({}^{6}\) N/m\({}^{2}\), behaving as the human foot muscle when stepping.