## 4 Comparison and Analysis
### Comparison and Analysis of Numerical  
The first example [19] is used to test the fitting process of this method using the IRMO algorithm. The catenary trajectory parameters and five-segment trajectory parameters are shown in Table 1.  
To determine the influence of different algorithm parameters on the fitting results, the value of \(R^{2}\) is used to manifest the similarity of fit. As shown in Table 2, the difference in the value of \(R^{2}\) is quite minor and is greater than 0.998 in each case. The overlap of the fitted trajectory images for series 1, 3, and 5 is relatively high (Figure 6). Moreover, the time resource of each case increases as the calculation becomes more complex. The optimal fitting result shows a lower \(SOS\) value searched by the IRMO algorithm (in Table 3), but more time costing. Compared with the results of the genetic algorithm (GA), the IRMO algorithm can give more accurate calculations with an acceptable extra time costing. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Series** & _dx_ **[m]** & \(N\) & **Generation** & _SOS_ & **R\({}^{2}\)** & **Time [ms]** \\ \hline
1 & 50 & 100 & 100 & 10.25 & 0.9991 & 765 \\
2 & 50 & 50 & 100 & 21.01 & 0.9981 & 487 \\
3 & 5 & 50 & 100 & 70.68 & 0.9991 & 532 \\
4 & 5 & 50 & 250 & 57.27 & 0.9993 & 1275 \\
5 & 1 & 50 & 250 & 287.22 & 0.9995 & 1680 \\ \hline \hline \end{tabular} Note: R\({}^{2}\) = \(1-\) SSE/SSR, R\({}^{2}\)\(\in\)(0,1); the closer the value of R\({}^{2}\) is to 1, the better the goodness of fit.  
\end{table} 
Table 2: Comparison of goodness of fit for different algorithm parameters. Figure 6: Solutions of fittings: Series 1, 3, and 5. During the 100 iterations, the global optimal \(SOS\) tends to a stable convergence when approaching the approximate 50 th generation (shown in Figure 7). The optimal result can be improved when the total number of iterations is increased. As shown in Table 4, the optimization result of \(L_{3}\) in this study is largely similar to that of the GA [19]. The other parameter optimization results have \(\theta_{1}\), \(\theta_{2}\), and \(R_{2}\) that are all smaller than those of the GA, while \(R_{2}\) is larger. The difference between the optimization results is attributed to the mutual influence and restriction of each parameter. Taking Series 4 in Table 3. as an example, the \(SOS\) is calculated 20 times consecutively (Figure 8), and the 20 trajectories are shown in Figure 9. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(\mathbf{L_{1}}\) **[m]** & \(\mathbf{\theta_{1}}\) **[\({}^{\circ}\)]** & \(\mathbf{R_{2}}\) **[m]** & \(\mathbf{\theta_{2}}\) **[\({}^{\circ}\)]** & \(\mathbf{R_{4}}\) **[m]** & \(\mathbf{L_{3}}\) **[m]** \\ \hline GA [12] & — & \(-9.3\) & 2473.96 & 7.44 & 2378.51 & 100.78 \\ IRMO (this study) & — & \(-8.83\) & 2678.16 & 6.49 & 2133.25 & 100.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of parameter optimization results. Figure 7: The convergence process of searching for the optimal SOS: Series 3 in Table 2.  
The calculations completed by the IRMO showed great stability and accuracy, with the standard deviation of the \(SOS\) being 3.61244, consistent with the trajectory images. In Figure 9, different colors are used to distinguish each segment of the five-segment trajectory. The fitting results of each segment are highly similar as well. The parameter collocation of the optimal fitting trajectory cannot be identical here. As the fitting similarity is higher, the mutual influences and restrictions of each parameter on the others will be enlarged causing different results. To achieve a high similarity of fitting, the results of the parameters, especially \(L_{1}\), \(R_{2}\), and \(R_{4}\), fluctuate in a small range after 20 calculations (Figures 10-12), while the fitting results are not much different (Figure 9).