# Multi-Objective PSO with Variable Number of Dimensions for Space Robot Path Optimization
## 4 Results and Discussion
### Benchmark Problems
The results of the [PERSON]'s test comparing VNDMOPSO with VNDGDE3 and VNDMOPSO with VLMOPSO are summarized in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \multicolumn{2}{c}{**Settings \(\mathcal{S}\)1**} & \multicolumn{2}{c}{**Settings \(\mathcal{S}\)2**} & \multicolumn{2}{c}{**Settings \(\mathcal{S}\)3**} \\ \hline \hline
**VNDMOPSO vs.** & **VNDGDE3** & **VLMOPSO** & **VNDGDE3** & **VLMOPSO** & **VNDGDE3** & **VLMOPSO** \\ \hline VNDMODTIZ1 & – & + & – & + & – & = \\ VNDMODTIZ2 & + & – & + & – & + & – \\ VNDMODTIZ3 & – & + & – & – & – & = \\ VNDMODTIZ4 & – & – & – & – & – & – \\ VNDMODTIZ5 & + & + & + & – & + & + \\ VNDMODTIZ6 & = & + & + & + & + & – \\ VNDMODTIZ7 & – & – & – & – & – & = \\ VNDMOLI1 & – & + & – & + & – & + \\ VNDMOLZ1 & – & + & + & + & + & + \\ VNDMOLZ2 & – & – & – & + & – & + \\ VNDMOLZ3 & – & + & + & + & + & + \\ VNDMOLZ4 & – & + & + & + & = & + \\ VNDMOLZ5 & = & + & + & + & + & + \\ VNDMOLZ7 & = & + & + & = & + & + \\ VNDMOLZ8 & – & = & + & + & + & + \\ VNDMOLZ9 & – & – & – & + & – & + \\ VNDMOZDT1 & + & – & – & – & – & – \\ VNDMOZDT2 & – & – & – & – & – & – \\ VNDMOZDT3 & + & + & – & – & – & – \\ VNDMOZDT4 & + & – & + & – & + & + \\ VNDMOZDT6 & – & – & – & – & + & – \\ \hline
**Overall** & 5 / 3 / 13 & 9 / 2 / 10 & 10 / 0 / 11 & 10 / 2 / 9 & 10 / 1 / 10 & 11 / 3 / 7 \\ \hline \end{tabular}
\end{table} 
Table 4: Results of [PERSON]’s test (at significance level \(\alpha=0.05\)) when comparing the VNDMOPSO algorithm with VNDGDE3 and VLMOPSO for different dimensionality settings: ‘+’ denotes that the VNDMOPSO algorithm is significantly better, ‘\(-\)’ denotes that the the second algorithm is significantly better, ‘\(-\)’ denotes that the difference is not significant. The last row summarizes the number of the test results for the whole benchmark set: VNDMOPSO is better/difference not significant/VNDMOPSO is worse. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Settings** & **Feasible Dimensions**, \(\mathcal{N}\) & **Optimal Dimensions**, \(\mathcal{D}\)(**opt**) \\ \hline \hline \(\mathcal{S}\)1 & \(3,4,\ldots,12\) & \(3,4,5\) \\ \(\mathcal{S}\)2 & \(3,4,\ldots,22\) & \(7,8,9\) \\ \(\mathcal{S}\)3 & \(3,4,\ldots,52\) & \(10,11,12\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dimensionality settings for the comparative study between VNDMOPSO and reference algorithms VLMOPSO and VNDGDE3. Figure 4: Results of dHV metric for algorithms VNDMOPSO, VLMOPSO, and VNDGDE3 for dimensionality settings \(\mathcal{S}2\): \(N_{\text{d}}\in\{2,3,\ldots,22\}\). Figure 3: Results of dHV metric for algorithms VNDMOPSO, VLMOPSO, and VNDGDE3 for dimensionality settings \(\mathcal{S}1\): \(N_{\text{d}}\in\{2,3,\ldots,12\}\). When comparing VNDMOPSO with VNDGDE3, we can see that the performance of VNDMOPSO is inferior for the settings with lower dimensions (\(\mathcal{S}1\): \(N_{\text{max}}=12\)) whereas the VNDGDE3 is significantly better for 13 problems while the VNDMOPSO is better for 5 problems only. The results become more comparable for the more-dimensional settings, \(\mathcal{S}2\) and \(\mathcal{S}3\). For them, VNDMOPSO significantly outperforms VNDGDE3 for at least 10 problems. Next, it is clearly visible that VNDMOPSO achieves at least comparable results to VLOPSO for settings \(\mathcal{S}1\) and outperforms VLMOPSO for more-dimensional settings, \(\mathcal{S}2\) and \(\mathcal{S}3\). Most importantly, the VNDMOPSO algorithm outperforms the other two algorithms on problems from the LZ benchmark suite (please refer to supplementary material S1, Equations (S22)-(S39)). The nature of these problems is very similar to searching for the correct location of some points in an \(N\)-dimensional space, which is similar to the main subject of this paper--searching for the locations of the turning points of a space robot. Therefore, VNDMOPSO can be seen as the best candidate to solve the SPRR.