## 4 Results and Discussion
### Optimization Results for Four Degrees of Freedom
Figure 12 compares the mean square error of \(\alpha\) (MSE \(\alpha\)) for different combinations of \(NP\times NG\) and different versions of DE investigated. As expected, in general, the MSE \(\alpha\)  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline  & & \multicolumn{2}{c}{**DE1**} & \multicolumn{2}{c}{**DE2**} & \multicolumn{2}{c}{**DE3**} \\ \hline \multirow{3}{*}{\(H_{0}/L_{0}\)} & \multirow{3}{*}{\(NP\times NG\)} & **Average** & **Standard** & **Average** & **Standard** & **Average** & **Standard** \\  & & **(\(\theta_{max}\))3m** & **Deviation** & **(\(\theta_{max}\))3m** & **Deviation** & **(\(\theta_{max}\))3m** & **Deviation** \\  & & **(\(\times\)10\({}^{-3}\))** & **(\(\times\)10\({}^{-4}\))** & **(\(\times\)10\({}^{-3}\))** & **(\(\times\)10\({}^{-4}\))** & **(\(\times\)10\({}^{-4}\))** & **(\(\times\)10\({}^{-3}\))** & **(\(\times\)10\({}^{-4}\))** \\ \hline \multirow{3}{*}{2.25} & 10 \(\times\) 50 & 32.05 & 44.37 & 33.25 & 48.44 & 31.32 & 40.51 \\  & 20 \(\times\) 100 & 28.30 & 3.99 & 30.09 & 34.23 & 29.42 & 31.38 \\  & 30 \(\times\) 100 & 28.39 & 12.86 & 29.75 & 33.39 & 28.30 & 6.20 \\  & 40 \(\times\decreased with the increase in \(NP\) and \(NG\) parameters. Despite that, in some conditions, the growth of \(NP\) and \(NG\) parameters does not influence the MSE \(\alpha\), e.g., for the interval 9.7 \(\leq H_{0}/L_{0}\leq\) 12.1 with the DE2 version, a significant decrease in MSE \(\alpha\) is not observed, as can be noticed in Figure 12a-d.  
For \(H_{0}/L_{0}\geq\) 14.6, the decrease in MSE \(\alpha\) is more accentuated for DE1 and DE3 versions, mainly for DE1 with \(NP\) = 20 and \(NG\) = 100. These two algorithm versions had more difficulty obtaining the optimal value of \(\alpha\) for \(H_{0}/L_{0}\) = 9.7, Figure 12a-c. This behavior indicates that, for this ratio of \(H_{0}/L_{0}\), there are more points of local minimal for the prediction of \((\alpha)_{20}\). For \(NP\) = 40 and \(NG\) = 112, Figure 12d, the algorithms DE1 and DE3 led to MSE \(\alpha\) tending to zero, except for \(H_{0}/L_{0}\) = 17.1, when DE2 and DE3 have a similar performance. The mean square error for \(\beta\) (MSE \(\beta\)) for each algorithm version in different \(NP\) and \(NG\) combinations is shown in Figure 13. For the first combination of \(NP\) and \(NG\) with \(NP\) = 10 and \(NG\) = 50, the three algorithm versions analyzed presents several errors for all values of \(H_{0}/L_{0}\), Figure 13a. With the increase of the \(NP\) and \(NG\) parameters, a strong reduction in the magnitude of MSE \(\beta\) was efficiently obtained, as can be seen in Figure 13b  
Figure 12: Effect of \(H_{0}/L_{0}\) over mean square error (MSE) of \(\alpha\) for different DE versions and combinations of \(NP\times NG\): (**a**) \(NP\) = 10, \(NG\) = 50; (**b**) \(NP\) = 20, \(NG\) = 100; (**c**) \(NP\) = 30, \(NG\) = 100; (**d**) \(NP\) = 40, \(NG\) = 112.  
d, mainly for versions DE1 and DE3, which obtained MSE \(\beta\) almost null in all ranges of \(H_{0}/L_{0}\) investigated. It is worth mentioning that, for \(NP=20\) and \(NG=100\), Figure 13b, the DE1 version still shows almost zero errors for all ratios of \(H_{0}/L_{0}\). Results of Figure 13 show that the DE2 version was not efficient in completely reducing the MSE \(\beta\) for different magnitudes of \(H_{0}/L_{0}\). Despite that, the performance was enormously superior to that noticed for the reduction of MSE \(\alpha\), which can be associated with the sensitivity of the ratio \(H_{0}/L_{0}\) over \((\alpha)_{20}\) and \((\beta)_{0}\). Figure 14 shows the mean square error for \(S_{1}/H_{0}\) (MSE \(S_{1}/H_{0}\)) as a function of the ratio \(H_{0}/L_{0}\) for different versions of DE and magnitudes of \(NP\) and \(NG\).