# Multi-Objective PSO with Variable Number of Dimensions for Space Robot Path Optimization
## 4 Results and Discussion
### Pathfinding Problems  
In this section, we will apply the VND algorithms to solve four artificial (labeled as \(\mathcal{P}1\)-\(\mathcal{P}4\)) and one real-life (\(\mathcal{P}5\)) pathfinding problem instances. The feasible dimensionalities for all the problems are \(\mathcal{N}=\{2,4,\ldots,20\}\). Maps of obstacles for the problems are shown in Figure 8. They were designed so that they can be solved with different numbers of TPs. See e.g., problem \(\mathcal{P}4\) where the Pareto front constitutes solutions leading to three different paths with one, two, or three TPs (see Figure 8). The real-life problem instance, \(\mathcal{P}5\), is then defined based on the photograph of the landing area of the famous Apollo \(11\) mission reaching the surface of the Moon [46], see Figures 8e,f. The problem instance, \(\mathcal{P}4\), is a relatively easy one. It is simple enough to perform an exhaustive search on it to reveal the true Pareto front, which is shown in Figure 9a. The corresponding paths are then shown in Figure 9b. The Pareto front is built from three solutions, with one (the longest one but with the least number of necessary turns), three (the shortest one but with the most number of necessary turns), and two (compromise) TPs. The first solution can be described as the easiest to control (purple curve in Figure 9b), while the third one (red curve in Figure 9b) is the fastest one. With the information about the trade-offs between objectives, the final path can be than selected. The convergence of the VNDMOPSO is compared against the VNDGDE3, VLMOPSO, and SCMOPSO algorithms. The quality of the found solutions is expressed in terms of the dHV metric defined in (9). Nevertheless, the true Pareto front is not known for the pathfinding problems. Therefore, \(\mathrm{HV}^{\mathcal{P},\mathcal{F}}\) is computed for a combined non-dominated set created from the union of all runs of the four algorithms (every algorithm was tested 100 times). In addition, the reference point necessary for the calculation of individual HV values was taken as the combination of the worst values in individual objective values for the combined non-dominated set. Figure 7: Median values of dHV metric versus parameter \(p_{\mathrm{g}}\). The median is selected from 100 independent runs of the VNDMOPSO algorithm for 21 benchmark problems with dimensional settings \(\mathcal{S}1\). The dHV results are shown in form of boxplots in Figure 10. The VNDMOPSO clearly outperforms the other three algorithms in all the pathfinding problem instances \(\mathcal{P}1\)-\(\mathcal{P}5\). This can be explained by a proper balance of the convergence speed and the robustness provided by optimal setting of the \(p_{\mathrm{g}}\) and \(p_{\mathrm{b}}\) parameters of VNDMOPSO. VLMOPSO also outperforms the VNDGDE3 algorithm. This is probably due to the fact that the PSO algorithm, due to its general approach for updating the position of agents, is generally more suitable for solving problems that seek the optimal locations of a set of components. The SCMOPSO achieves the worst results from the whole tested set of algorithms. This is probably caused by the fact that particles in SCMOPSO can learn only from particles within the same class and not from the whole swarm. Therefore, running SCMOPSO resembles more the situation when we would run a conventional MOPSO for each possible dimension separately but with a limited number of agents. Figure 8: Maps of obstacles for: (**a–d**) artificial problem instances \(\mathcal{P}1\)–\(\mathcal{P}4\), (**e**) real-life photograph of Apollo 11 landing area (46), and (**f**) instance \(\mathcal{P}5\) based on (**e**). Markers ‘\(\times\)’ show locations of the robot’s starting point, \(\mathbf{r}_{\mathcal{S}}\) (red color), and target point, \(\mathbf{r}_{\mathcal{T}}\) (blue).