## 5 Results
### Estimated Parameters and Parameter Sensitivity  
The stiffness and damping coefficients estimated by GBA, GA, and HOM for 10 translational parameters: \(k_{21}\), \(k_{22}\), \(k_{31}\), \(k_{32}\) and \(k_{51}\) and \(c_{21}\), \(c_{22}\), \(c_{31}\), \(c_{32}\) and \(c_{51}\) and 8 torsional parameters: \(k_{1}^{T}\), \(k_{2}^{T}\), \(k_{3}^{T}\), \(k_{4}^{T}\) and \(c_{1}^{T}\), \(c_{2}^{T}\), \(c_{3}^{T}\), \(c_{4}^{T}\) at 0.5 and 2.0 m/s\({}^{2}\) rms forcing magnitudes are presented in Figures 12 and 13. The estimates were obtained by minimizing the squared error between the calculated and measured transmissibility for the head, trunk, thigh, and leg segments of the 7-DOF model. We analyzed the estimated parameters and compared the performance of each optimization method to gain insights into the key parameters that affect the appearance and magnitude of resonance frequencies and other modal characteristics of the model. While HOM produced nearly the same values for the estimated translational and torsional parameters at either 0.5 or 2.0 m/s\({}^{2}\) rms for all body segments, GBA and GA showed significant variances for all estimated parameters. We attribute the high randomness in estimated parameters by GA to the nature of its random initial population assignment and that of GBA to the limitations of guessing the appropriate initial values for the estimated parameters. To improve the results of estimated parameters, we performed a parameter sensitivity analysis by establishing a parameter sensitivity matrix, \(\mathbf{P}\) with elements [26]:  
\[\mathrm{P}_{ij}=\frac{\Delta R_{i}/R_{i}}{\Delta\psi_{j}/\psi_{j}}, \tag{27}\]  
where \(\psi=\{k_{21},k_{22},k_{31},k_{32},k_{51},c_{21},c_{22},c_{31},c_{32},c_{51},k _{1}^{T},k_{2}^{T},k_{3}^{T},k_{4}^{T},c_{1}^{T},c_{2}^{T},c_{3}^{T},c_{4}^{T}\}\); \(\psi_{j}\) and \(\Delta\psi_{j}\) represent the \(j\)th (\(j=1,2,\ldots,18\)) parameter at its nominal value and its value after a small perturbation; \(R_{i}\) represent the responses to be examined, in our case, the first resonance frequency (\(i=1\)), the second resonance frequency (\(i=2\)), the peak magnitude of resonance frequency (\(i=3\)), computed with the nominal value of each term in \(\psi\), extracted from the final optimization results that allow us to fit the model to experiment as presented in Figures 8 and 9; \(\Delta R_{i}\) represent the differences in the value of the above responses after perturbation. We set the perturbation factor, \(\alpha\) to 30% such that \(\Delta\psi_{j}=\alpha\psi_{j}\) and the new values of each term in \(\psi_{j}\), is \(\psi_{j}^{new}=\psi_{j}\pm\Delta\psi_{j}\) with other parameters kept at their nominal values. Then, new values of each examined response, \(R_{i}^{new}\) are computed with \(\psi_{j}^{new}\) such that \(\Delta R_{i}=\left|R_{i}^{new}-R_{i}\right|\). Figure 13: Torsional parameters (top: stiffness coefficients, bottom: damping coefficients). At \(\pm\)30%, the resultant parameter sensitivity matrix, normalized based on the maximum value in each row, is as follows:  
\[\mathbf{P}_{+30\%}= \begin{bmatrix}0.02&0.05&0.00\\ 0.00&0.00&0.00\\ \mathbf{1.00}&\mathbf{1.00}&0.09\\ 0.01&0.01&0.00\\ \mathbf{0.41}&\mathbf{0.31}&0.02\\ \mathbf{0.21}&0.05&\mathbf{0.91}\\ 0.01&0.00&0.00\\ 0.02&0.01&\mathbf{1.00}\\ 0.00&0.00&0.03\\ 0.09&0.00&\mathbf{0.32}\\ 0.00&0.00&0.00\\ 0.00&0.00&0.00\\ 0.00&0.00&0.00\\ 0.02&0.01&0.05\\ 0.05&0.01&0.04\\ 0.00&0.00&\mathbf{0.86}\\ 0.06&0.03&0.01\\ \end{bmatrix} ^{T}\]  
and \(\mathbf{P}_{-30\%}= \begin{bmatrix}0.05&0.08&0.00\\ 0.00&0.00&0.00\\ \mathbf{1.00}&\mathbf{1.00}&0.05\\ 0.01&0.00&0.00\\ 0.01&0.00&0.00\\ 0.00&0.00&0.00\\ 0.00&0.00&0.00\\ 0.00&0.00&0.00\\ 0.00&0.00&0.00\\ 0.01&0.01&0.03\\ 0.02&0.01&0.02\\ 0.00&0.00&\mathbf{0.78}\\ 0.08&0.05&0.01\\ \end{bmatrix} \)  
For each examined response, the sensitivity of each parameter was analyzed based on their score in the parameter sensitivity matrix. The greater the value, the more sensitive the parameter is to the considered response.