## 5 Experiments and Results
### Evaluation and Testing
The environment spans the interval [\(-\)2, 2] for both the x and y axes. The black squares represent the starting positions of the robots; the blue circles represent the goal positions, and the “\(\times\)” shape represents the collisions’ locations. On the other hand, [PERSON] et al. 's [12] method tends to have slightly more collisions of Category A than that of Category B. Most of the Category A collisions occur when the goal position is near the obstacle wall, which means that the policy drives the robot to approach such goals at an angle that exposes it to collision. Alternatively, the Category B collisions occur when the other robot reaches a goal position that is in the middle between the ego-robot and its assigned goal position, as shown in Figure 6, which indicates that the policy failed to avoid a stationary robot in front of it. As for [PERSON] et al. 's [13] method, the only instance when a collision occurred was when a certain goal configuration appeared during testing. This collision instance is Category A, as shown in Figure 6. This collision happens when a goal position is spawned behind Robot 1 and the conservative safe policy of [PERSON] et al. 's [13] hybrid control system is the policy controlling the robot. Figure 7 shows the distribution of collisions within the environment. The figure reinforces the analysis of both Table 3 and Figure 6 as it shows that for our method, the collisions are concentrated in the middle of the environment and such collisions are Category B. On the other hand, for [PERSON] et al. 's [12] method, most collisions are near the edges of the environment, and such collisions are Category A. For [PERSON] et al. 's [13] method, the collisions only occur for Robot 1 at the same position, and this collision is Category A. In Figure 8, we visualize three different successful episodes from testing for each method. the visualization shows the trajectory of each robot color coded. It is clear form the graph that the methods of [PERSON] et al. [12] and [PERSON] et al. [13] tend to move the robots in more straight trajectories, while our method tends to move in trajectories that are more irregular and sometimes overshoots the goal positions. Figure 7: The distribution of collisions in the environment done by each robot controlled by each method during testing. The environment spans the interval [\(-\)2, 2] for both the x and y axes. The black squares represent the starting positions of the robots. Figure 9 shows the distribution of goals reached by each robot throughout the environment during testing for our method. It shows that Robot 0, which is spawned near the upper right corner of the environment, tends to reach the goals positioned in the environment's upper half, while Robot 1, which is spawned in the lower left corner, tends to reach the goals positioned in the environment's lower half. The past two observations indicate that the task allocation strategy learned during training is to split the environment into two halves, and each robot attempts to reach the goals available in its half. To analyze the reward function for our method that was introduced in Section 4.1.3 and to evaluate the effects of the values of the constants for the reward function that are shown in Table 2, Figures 10 and 11 visualize the reward function during a successful episode and an episode that ended with a collision, both episodes by our method. For the successful episode in Figure 10, the reward \(r^{d}\)--described in Equation (7)--which is used to incentivize the ego-robot to go towards a goal position, has a positive value for both robots as each of them goes towards the goal position and has a large terminal value once each robot reaches a unique goal position. An exception to that happened at the beginning of the episode for Robot 1 since the policy moved it away from the goal position for the first 10 time steps. As for the other reward functions, since there were no collisions, therefore the reward \(r^{col}\)--described in Equation (10)--had a zero value for both robots.