## 4 Optimization Methods
### Gradient-Based Algorithm (GBA)  
Gradient-based algorithms (GBAs) are commonly used in human model parameter estimation due to their speed and ease of implementation. However, they have some limitations, such as being highly sensitive to the initial values of the unknown parameters. This limitation becomes even more significant in a study like ours where there are no established data on the values of stiffness and damping coefficients that can accurately model the viscoelastic properties of the skin and seat subject to dynamic excitations.  
In this study, we execute GBA using the 'fmincon' MATLAB optimization function to minimize the sum of squared errors between measured and calculated transmissibility in the 7-DOF model. We initially set computation parameters, including a maximum iteration of 50, a step tolerance of \(1\times 10^{-15}\), and a function tolerance of \(1\times 10^{-12}\), under the bounds specified in Equation (25). However, this resulted in a poor fitting of the transmissibility and high objective function value at final solutions. The objective function value at final solutions (_FVAL_) is a real number returned when the algorithm converges. Typically, in a minimization problem, the lower the _FVAL_, the higher the chances that the obtained solution is close to the optimum solution of the desired problem [28].  
To improve the fitting results by GBA, we continuously adjusted the computation parameters using trial-and-error techniques. Finally, we achieved much better fits, with an _FVAL_ of 0.0513 obtained at the final solutions of GBA using an average of 100 iterations, a  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \begin{tabular}{c} **Mass** \\ **(kg)** \\ \end{tabular} & \multicolumn{2}{c}{\begin{tabular}{c} **Moment of** \\ **Inertia (kg\_m2)** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Length** \\ **(m)** \\ \end{tabular} } & \multicolumn{2}{c}{**Thickness (m)**} & \multicolumn{2}{c}{\
\begin{tabular}{c} **Initial Angle** \\ **(deg.)** \\ \end{tabular} } & \multicolumn{2}{c}{
\begin{tabular}{c} **Spring Ends** \\ **Distance (m)** \\ \end{tabular} } \\ \hline \(m_{1}\) & 5.66 & \(J_{1}\) & 0.037 & \(L_{1}\) & 0.263 & & & \(\theta_{1}\) & 100 & \(d_{1}\) & 0.100 \\ \(m_{2}\) & 37.96 & \(J_{2}\) & 2.050 & \(L_{2}\) & 0.599 & \(T_{2}\) & 0.224 & \(\theta_{2}\) & 111 & \(d_{2}\) & 0.479 \\ \(m_{3}\) & 18.74 & \(J_{3}\) & 0.324 & \(L_{3}\) & 0.495 & \(T_{3}\) & 0.156 & \(\theta_{3}\) & 12 & \(d_{3}\) & 0.088 \\ \(m_{4}\) & 6.08 & \(J_{4}\) & 0.061 & \(L_{4}\) & 0.368 & & & \(\theta_{4}\) & 306 & \(d_{4}\) & 0.459 \\ \(m_{5}\) & 1.56 & \(J_{5}\) & 0.005 & \(L_{5}\) & 0.248 & \(T_{5}\) & 0.076 & \(\theta_{5}\) & 10 & \(d_{5}\) & 0.124 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Identified human and geometric parameters.  
maximum function evaluation of 2700, and within the computation time of 1300 s. The difficulties in finding the appropriate initial values demonstrate the challenges associated with GBA in estimating human parameters and highlight the importance of carefully selecting and adjusting the computation parameters to obtain accurate and reliable solutions.