## 6 Results and Discussion  
As described before, two ANN models have been considered in this study. The first model is an AAN-BP model whose weights and biases are determined by a backpropagation algorithm (BP) i.e., the Levenberg-Marquardt algorithm (LMA), and the second one is the ANN-PSO model, which is tuned by the PSO algorithm. The same architecture of 10 neurons in a single hidden layer has been \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Number of Particles**} & \multirow{2}{*}{**Max Iteration**} & \multicolumn{4}{c}{**Train**} & \multicolumn{2}{c}{**Test**} \\ \cline{3-8}  & & \(r\) & \(R^{2}\) & \(RMSE\) (\(\mathbf{k}\mathbf{m}\)) & \(r\) & \(R^{2}\) & \(RMSE\) (\(\mathbf{k}\mathbf{N}\)) \\ \hline
25 & 1000 & 0.965 & 0.931 & 14.237 & 0.968 & 0.938 & 14.030 \\
35 & 1000 & 0.971 & 0.944 & 13.394 & 0.969 & 0.939 & 13.818 \\
45 & 1000 & 0.972 & 0.945 & 13.074 & 0.970 & 0.941 & 13.129 \\
65 & 1000 & 0.968 & 0.937 & 13.823 & 0.963 & 0.927 & 14.751 \\
85 & 1000 & 0.966 & 0.933 & 14.026 & 0.967 & 0.935 & 14.441 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The number of particles impact on the results (output = load). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Number of Particles**} & \multirow{2}{*}{**Max Iteration**} & \multicolumn{4}{c}{**Train**} & \multicolumn{2}{c}{**Test**} \\ \cline{3-8}  & & \(r\) & \(R^{2}\) & \(RMSE\) (\(\mathbf{mm}\)) & \(r\) & \(R^{2}\) & \(RMSE\) (\(\mathbf{mm}\)) \\ \hline
25 & 1000 & 0.893 & 0.797 & 1.836 & 0.773 & 0.597 & 2.667 \\
35 & 1000 & 0.887 & 0.787 & 1.882 & 0.784 & 0.615 & 2.618 \\
45 & 1000 & 0.901 & 0.811 & 1.785 & 0.824 & 0.680 & 2.311 \\
65 & 1000 & 0.912 & 0.831 & 1.685 & 0.791 & 0.625 & 2.538 \\
85 & 1000 & 0.892 & 0.795 & 1.850 & 0.801 & 0.641 & 2.492 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The number of particles impact on the results (output = slip). Figure 7: Impact of the swarm size on \(RMSE\) value: (**a**) Output = load; (**b**) Output = slip. adopted in both of the models. Most (70%) of the inputs were randomly assigned to the training phase, and the remaining 30% were assigned to the testing phase. Figure 8 illustrates the predicted and measured slip by ANN-BP and ANN-PSO models in scatter diagrams. Figure 8a shows the training phase of the ANN-BP model, whose performance parameters are \(r=0.889\), \(R^{2}=0.791\), and \(RMSE=1.863\) (mm). Figure 8b depicts the testing phase of the ANN-BP model with performance parameters of \(r=0.804\), \(R^{2}=0.646\), and \(RMSE=2.569\) (mm). As can be realized, the ANN-BP model has been able to show acceptable performance in the prediction of the slip, since the values of \(r\) and \(R^{2}\) are near to 1, and the \(RMSE\) value is almost low in the slip range of 0 to 16 mm. Figure 8c represents the training phase of the ANN-PSO model. Performance indices of the ANN-PSO model in this phase are \(r=0.908\), \(R^{2}=0.824\), and \(RMSE=1.708\) (mm). As can be seen, some improvement in the performance of the ANN has been resulted in this phase by using the PSO algorithm such that the \(r\) and \(R^{2}\) values have increased, and the \(RMSE\) value has decreased. Figure 8d shows the testing phase of the ANN-PSO model with performance indices of \(r=0.865\), \(R^{2}=0.750\), and \(RMSE=2.069\) (mm). The testing phase has also improved with respect to that of the ANN-BP model. Figure 8: Comparison of the predicted and measured slip: (**a**) Training phase of the ANN-BP model (**b**) Testing phase of the ANN-BP model (**c**) Training phase of the ANN-PSO model (**d**) Testing phase of the ANN-PSO model. BP: backpropagation, PSO: particle swarm optimization. Figure 9 shows the capability of the models in the testing phase to predict each of the measured values of the test samples. As can be realized, both of the models are capable of predicting most of the test samples closely. However, the better performance of the ANN-PSO model is almost clear.