# Validation of Lumbar Compressive Force Simulation in Forward Flexion Condition
## 4 Discussion  
The accuracy of simulators that provide validation data of the safety standards of physical assistant robots must be high. We compared the performance of three types of biomechanical models with the data obtained from an invasive experiment involving ten healthy subjects. We hypothesized that a CF simulator can improve the precision of CF estimation over the models proposed by [PERSON] and [PERSON] under dynamic \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Slope** & \(R^{2}\) & **RMSE (N)** \\ \hline [PERSON] [16] & 1.070 & 0.658 & 259 \\ \hline [PERSON] [15] & 0.641 & 0.675 & 205 \\ \hline CF simulator & 1.012 & 0.683 & 269 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Linear regression analysis between the estimated compressive force (CF) and invasive measurements. Figure 7: Regression analysis between the estimated compressive force (CF) obtained from (**a**) the model by [PERSON] et al. (2009), (**b**) the CF simulator, and (**c**) the model by [PERSON] et al. (1997) with respect to the CF determined from invasive measurements; (**d**) Kruskal–Wallis test results: significance level (\(p\leq 0.001\)) among the distances between markers and \(y=x\) line estimated by the three models. conditions [15; 16]. Under forward flexion conditions, the results of each model were compared with those of invasive measurements [30; 31; 32; 33]. The CF was estimated under the flexion angle \(\theta\) of \(0^{\circ}\), \(10^{\circ}\), \(20^{\circ}\), and \(30^{\circ}\). A Wilcoxon test was performed to evaluate the difference between the CFs obtained with the three models and the invasive measurements at L5/S1 and \(0^{\circ}\) and \(30^{\circ}\). The angles of \(10^{\circ}\) and \(20^{\circ}\) were excluded because of the lack of invasive data at those angles. Among all body parameters, the body height and weight have the greatest influence on the CF estimation [9]. The results of the Mann-Whitney test imply that there is no significant difference in the height and weight between the subjects in the present study and those in the invasive experiments. Thus, we consider the two groups of subjects to be in the same population and the group difference in height and weight to be negligible small. As a result, the CF of each model significantly differed from the CF obtained from the invasive measurements. Furthermore, as shown in Figure 4, the CF discrepancies imply that the model by [PERSON] and the proposed CF simulator yield results that are more similar to those of the invasive measurements than those of the model by [PERSON]. The ratio of the increment of CF increasing with respect to flexion angle observed in ratio of the invasive experiment decreases from 49 to 11.5 N/deg as \(\theta\) increases from AR1 to AR3. Only the CF simulator and the model by [PERSON] et al. show a trend similar to that of the invasive measurements, while the ratio of the increment of compressive force CF with respect to flexion angle obtained by [PERSON]'s model increases from 35 to 37 N/deg from AR1 to AR2 and decreases from AR2 to AR3. Therefore, the CF simulator has a good correspondence with the invasive experiment data, in terms of both the trend and the absolute value. In Figure 4, the error of the CF simulator, compared with the experimental results, at \(30^{\circ}\) is 11.8%, which is much lower than the 16.8% and 20.6% errors of the models by [PERSON] and [PERSON], respectively. The differences between the CF simulator and the model by [PERSON] is probably due to the dynamic/static conditions: at \(0^{\circ}\), \(10^{\circ}\), \(20^{\circ}\), and \(30^{\circ}\), the CF estimated with [PERSON]'s model is 241 N (52%), 326 N (38%), 426 N (34%), and 450 N (29%) smaller than that estimated with the CF simulator, respectively. This result is in good agreement with the results of [PERSON], who reported an error range of 11-38% of dynamic estimation [17]. We can conclude that [PERSON]'s model performs better than [PERSON]'s one when comparing with invasive data. The possible reasons are as follows.