## 5 Experiments and Results
### Evaluation and Testing
We trained the method proposed by [PERSON] et al [12] on our simulation setup with the same robot kind we are using. Their method was implemented using the same setup, network architecture, and hyperparameters that they proposed in their paper. The training process took around 22 h. For the method proposed by [PERSON] et al. [13], we used the same policy that we trained for [PERSON] et al. 's [12] method as part of the hybrid control system similar to what [PERSON] et al. [13] did in their paper. To test and evaluate each method, we repeated the evaluations for 2 h. The goals' positions were changed every time each robot reached a unique goal or five episodes passed by since the last goal position change \(E_{g}=5\). The results are shown in Figure 5. It shows that both methods are on par with each other with a slight advantage of the method proposed by [PERSON] et al. [12] over both of the other methods in terms of the success rate. In terms of extra tie and extra distance, the method proposed by [PERSON] et al. [13] has an advantage over the other two methods. This advantage is attributed to the PID policy of the hybrid control system, which drives the robot directly toward the goal position if there is no obstacle nearby or the goal position is closer to the robot than the nearest obstacle. On the other hand, the \(TAI\) metric highlights the advantages of our method over the other two methods since our method is the only one that performs both MRTA and navigation from end-to-end. Our method takes input information about all the goals in the environment (\(N_{i}=N_{g}\) in Equation (14)); thus, the only factor affecting the value of the \(TAI\) metric in this case is the percentage of episodes ending with Category B collisions \(N_{CategoryB}\), as shown in Table 3. On the other hand, the \(TAI\) metric for both [PERSON] et al. 's [12] and [PERSON] et al. 's [13] methods is zero since both methods do not perform MRTA, and they only have input information about the preassigned goal position for each robot (\(N_{i}=1\)). Table 3 shows the statistics related to the number of episodes and the way they ended for each method. Both methods have a very low percentage of episodes ending with collisions, yet [PERSON] et al. 's [13] method has a slight advantage. We attribute our method's performance to our method's tendency to move the robots with an average speed that is higher than the average speed produced by the other methods--as shown in Figure 5--which can lead to the robots in our method having more collisions. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Metric** & **Our Method** & **[PERSON] et al.** & **[PERSON] et al.** \\ \hline Number of episodes & 268 & 402 & 425 \\ \% of episodes ending by collision & 8.955\% & 2.239\% & 2.118\% \\ \% of episodes ending by Category A collision & 1.522\% & 1.254\% & 2.118\% \\ \% of episodes ending by Category B collision & 7.433\% & 0.985\% & 0.0\% \\ \% of episodes ending by Category C collision & 0.0\% & 0.0\% & 0.0\% \\ \% of episodes ending by exhausting episode time & 0.746\% & 0.0\% & 0.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance metrics for both methods during a 2 h test. Table 3 also shows the percentage of the occurrence of each collision category during testing. It shows that all methods managed to avoid any collisions of Category C. Our method tends to have collisions of Category B about five times more than that of Category A, which indicates that the policies on both robots chose the same goal position to move towards. Samples of both categories of collisions are visualized in Figure 6.  
Figure 5: The performance results of each method during a 2 h test. Figure 6: The trajectories of the robots over three different episodes that ended by collisions during testing performed by each method. The trajectory of Robot 0 is shown in red, while the trajectory of Robot 1 is shown in green. The more opaque the color of the trajectory is, the more advanced in time it is.