## 5 Optimization Scheme
### Simulation  
Human gait is considered as a process of minimum mechanical energy. In this regard, researchers such as [PERSON] et. al. (2007) [PERSON] et al. (2007) determined normal gait patterns by optimization, starting from an upright position without walking. This approach was implemented in this study, for which the coefficients \(X_{0}\) correspond to the result of an optimization in which the human starts from a static position.  
Figure 4 presents the optimization scheme implemented in the HSI optimization, which starts by defining anthropic parameters and gait conditions. Then, the initial coefficients \(X_{0}\) (\(a_{0}^{(i)}\),\(a_{n}^{(i)}\) and \(b_{n}^{(i)}\)) used as the starting point of the optimization are defined. These initial coefficients are adjusted in successive iterations (\(X\)) by the optimization function to reduce the mechanical energy expenditure (**Step 1**).  
In **Step 2**, the \(X\) coefficients are evaluated using Equation (1), which determines the angular movements of each body segment (\(\theta_{i}\) and \(\varphi_{i}\)). In **Step 2.1**, the Ankle-Foot Rollover was calculated using the process detailed in Section 2.1.1. The kinematics of the human body are then calculated using the methodology presented in Section 2.1, obtaining the displacements (\(x_{i}\), \(y_{i}\), \(z_{i}\)) and accelerations (\(i_{i}\), \(y_{i}\), \(z_{i}\)) for each body segment.  
**Step 3** involves the calculation of gait kinetics starting from the previously calculated kinematics and using the inverse dynamics methodology, as described in Section 2.2, to get the ground reaction forces GRFs (\(F_{x}\), \(F_{y}\) and \(F_{z}\)). After, in **Step 4**, the bridge properties are defined and used to calculate the structural system (**Step 4.1**) as described in Section 3. Subsequently, taking as input (1) the structural system, (2) the kinematics (contact position), and (3) the kinetics (GRFs) of the pedestrian, the dynamic response of the coupled HSI structural system is calculated following the methodology presented in Section 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Joint**} & \multicolumn{2}{c}{**Flexion**} & \multicolumn{2}{c}{**Abduction**} \\ \cline{2-5}  & \(\theta_{min}^{(i)}\) [\({}^{\circ}\)] & \(\theta_{max}^{(i)}\) [\({}^{\circ}\)] & \(\varphi_{min}^{(i)}\) [\({}^{\circ}\)] & \(\varphi_{max}^{(i)}\) [\({}^{\circ}\)] \\ \hline Neck & 60 & 120 & 50 & 130 \\ Elbow & 45 & 135 & 50 & 180 \\ Wrist & 45 & 135 & 50 & 180 \\ Lumbosacral & 60 & 120 & 75 & 105 \\ Hip & 60 & 120 & 75 & 105 \\ Knee & 30 & 180 & 75 & 105 \\ Ankle & 30 & 135 & 40 & 120 \\ Metatarsal & 45 & 200 & 65 & 125 \\ Elbow * & 35 & 180 & - & - \\ Knee * & 55 & 180 & - & - \\ \hline \end{tabular}  
* Relative angular displacement at joints.  
\end{table}
Table 5: Limit range of absolute angular displacements at joints.  
The mechanical energy during the entire time interval was determined and checked in **Step 5** (Equation (5)) to determine if the predefined constraints were met. If the constraints are not met, the process is repeated by changing coefficient \(X\) (**Step 6**). However, if the constraints are met, the process proceeds to verify if the solution converges, and if convergence has not been achieved, the process iterates again from the definition of the \(X_{0}\) coefficients, adjusting them as necessary, and repeating the entire analysis. Finally, in **Step 7**, the iterative cycle continues until the constraints are met and convergence is achieved, at which point the optimization process is terminated, thus achieving an optimal solution for the system under analysis.