6.4.2. Effectiveness Assessment

We start the analysis highlighting that all implemented approaches—and particularly all *AAMO* instances—can adequately explore all the environments presented above in Section 6.1.1. Coherently, all instances achieve a high level of *CR* when exploring the *Maze* environment, as can be appreciated in Figure 17.

**Figure 17.** *AAMO* Coverage ratio. Regardless of how different the *HO-Threshold* values are, in all cases, the *AAMO* approach can cover more than 99% of the terrain.

#### 6.4.3. *AAMO vs.* Baseline Comparison

Concerning *TT*, as was expected in multi-robot systems, all *AAMO* instances benefit from adding robots to the fleet. This result can be seen in Figure 18a. Nevertheless, compared to the baseline results all *AAMO* instances show performance degradation (see Figure 18b).

**Figure 18.** *AAMO* Total Exploration Time (TT) under non-ideal communication conditions and Degradation with respect to baseline results. (**a**) All *AAMO* instances show a decreasing trend of TT as the fleet size increase. The *Yamauchi* and *MinPos* approach results (coloured in purple and green, respectively) obtained under ideal communication conditions are placed together to make the comparison easier; (**b**) The degradation is expressed in terms of the difference between the TT achieved by each of the *AAMO* instances and the one achieved by the *MinPos* approach, for each fleet size.

The evidence indicates that the more efforts made in favour of connectivity (bigger HO-Threshold) the worst *TT*. In other words, not all *HO-Threshold* setup values produce the same level of performance

degradation. Since the degradation of TT performance could be very problematic in many application fields, this subject is carefully analysed.

At first, the PL indicator can help to initially explain why the fleet spends more time under *AAMO* approach than under the *MinPos* approach, to explore the same environment. In Figure 19a it is possible to observe the same behaviour as in the baseline (see Section 6.3): larger fleets imply bigger PL; while Figure 19b shows the difference between the corresponding total length of the paths traversed by fleets.

**Figure 19.** *AAMO* Path length (PL) under non-ideal communication conditions and Degradation with respect to baseline results. (**a**) An increasing trend of PL is shown by all *AAMO* instances as the fleet size increase. The *Yamauchi* and *MinPos* approach results (coloured in purple and green, respectively) obtained under ideal communication conditions are placed together to make the comparison easier; (**b**) The degradation is expressed in terms of the difference between the PL achieved by each of the *AAMO* instances and the one achieved by the *MinPos* approach, for each fleet size.

The similarity between Figures 18b and 19b is remarkable and could explain, to a large extent, the origin of TT degradation. Simply, under the *AAMO* approach, the robots are asked to invest some effort (translated as a distance using the HO-Threshold) in order to keep the fleet connected and hence it is logic to get a bigger PL as a result. Moreover, the tradeoff between path and connectivity utility discussed in Section 3.1 shows up through these results, reflecting that the price of connectivity maintenance is the inability to apply an optimal policy concerning path costs.

Nevertheless, there exists a small portion of the TT degradation that cannot be explained by the PL increasing. Therefore, the hypothesis assumed in the tractability analysis made at the end of Section 5.2 are compared here with the simulation results in order to add a complementary explanation on the TT degradation. Furthermore, this TT degradation shows a parabolic trend as the fleet size increase, reaching a maximum about three-sized fleets, independently of the *HO-Threshold* values. Thus, the analysis will be conducted observing what happens when the fleet size does change but the *HO-Threshold* does not (in order to explain the shape of the curve or the relative values), and the opposite conditions are imposed in order to explain the absolute values.

In any case, it is worth knowing that the *Task selection* algorithm is the most demanding software component in the software architecture of the robots. Hence, the overall performance of the multi-robot system is highly determined by the performance of this component. In turn—as was pointed out in Section 5.2—its performance is strongly influenced by the number of unassigned thresholded tasks *<sup>n</sup>* <sup>=</sup> <sup>|</sup>*T*HO<sup>|</sup> and the number of unassigned robots in a connected component *<sup>m</sup>* <sup>=</sup> <sup>|</sup>*Ru*<sup>|</sup> that are making a decision at the same time, in the following way: <sup>|</sup>*Ar<sup>n</sup> <sup>m</sup>*<sup>|</sup> <sup>=</sup> *<sup>n</sup>*! (*n*−*m*)! <sup>=</sup> <sup>Π</sup>*n*−<sup>1</sup> *<sup>m</sup>*=0(*<sup>n</sup>* <sup>−</sup> *<sup>m</sup>*) <sup>→</sup> *<sup>O</sup>*(*nm*). Therefore, the smaller <sup>|</sup>*T*HO<sup>|</sup> and <sup>|</sup>*Ru*<sup>|</sup> the faster the algorithm will run. Please recall that <sup>|</sup>*T*HO<sup>|</sup> is upper bounded by the amount of unassigned tasks <sup>|</sup>*Tu*|.

Firstly, from Figures <sup>20</sup> and 21, it is possible to examine how <sup>|</sup>*Ru*<sup>|</sup> and <sup>|</sup>*Tu*<sup>|</sup> change along explorations depending on the fleet size. In all cases, both values show well defined patterns that are easily identifiable. Concerning <sup>|</sup>*Ru*<sup>|</sup> (see Figure 20) it is possible to state that in all *AAMO* instances—working in a fully asynchronous modality—the probability of two or more robots simultaneously running a decision making process is negligible. Thus the majority of time either none robot is making a decision or at most one robot is evaluating the available tasks.

**Figure 20.** The maximum amount of unassigned robots <sup>|</sup>*Ru*<sup>|</sup> in any connected component over time under different sized multi-robot systems. All images concern instances of *AAMO* set with HO-Threshold = 15. Blue dots represent the <sup>|</sup>*Ru*<sup>|</sup> (on average) that are simultaneously deciding along the exploration.

Results obtained during simulations are summarised in Table 6 and show a behaviour that is consistent with this last statement independently of the fleet size. The low ratio of robot coincidences is remarkable (e.g., for 3-sized fleets, about 96% of the decision making moments have only one robot participating on them).


**Table 6.** *AAMO* Robot Coincidence on Decision Making moments.

In conclusion, in practice, the worsening of the TT performance is apparently only related to the incidence of the *HO-Threshold* on the <sup>|</sup>*T*HO<sup>|</sup> value. Next, this relation is carefully studied, and some answers are essayed.

The parabola described by the TT degradation values in Figure 18 suggests the presence of two factors impacting on this behaviour. One presses the trend upwards and the other in a counter sense. In the following, two particular factors are analysed: the fleet size and the bounded condition of the environment. (i) As the fleet size increase robots make progress faster, causing <sup>|</sup>*T*HO<sup>|</sup> to increase more quickly as well. When <sup>|</sup>*T*HO<sup>|</sup> rises, the task selection algorithm becomes slower, and thus the increase in the fleet size could explain the first increasing section of the trend; (ii) In bounded environments, the multi-robot exploration systems typically show two mobility patterns that characterise, in turn, two different exploration stages: (1) One is characterised by the dispersion of the fleet on the terrain. In such a stage, the new available tasks appear closer to each other, and its total amount <sup>|</sup>*Tu*<sup>|</sup> is upward; (2) On the contrary, the second exploration stage is characterised by the convergence of the fleet to the remaining unexplored zones starting when it is no longer possible to disperse the fleet until the end of

the exploration. In such a stage, the new available tasks generally appear further to each other and its total amount <sup>|</sup>*Tu*<sup>|</sup> is decreasing. Therefore, since the tasks *<sup>T</sup>*HO are the ones which are closer than a relative distance *HO-Threshold*, under the *AAMO* approach, it is statistically less demanding for the robots to select a task during the last exploration stage than in the initial one.

Additionally, either when the fleet size increase or the *HO-Threshold* decrease, the transition from the first to the second exploration stage is achieved faster. This fact can be corroborated in both Figures 21 and 22. For instance, concerning Figure 21, the 3-Robot system spends about 410 s to reach the end of the dispersion stage whereas the 5-Robot system and 8-Robot system spend about 320 s and 260 s, respectively. Likewise, from Figure 22, the AAMO:20 instance spends about 310 s to reach the end of the dispersion stage whereas the AAMO:15 and AAMO:10 spend about 260 s and 150 s, respectively.

**Figure 21.** Amount of unassigned tasks <sup>|</sup>*Tu*<sup>|</sup> over time for different sized multi-robot systems. All images concern instances of *AAMO* set with HO-Threshold = 15. The maximum <sup>|</sup>*Tu*<sup>|</sup> and the end of the dispersion stage are reached at the same time. Red dots represent the <sup>|</sup>*Tu*<sup>|</sup> considered by robots (on average) along the exploration.

**Figure 22.** Number of unassigned tasks <sup>|</sup>*Tu*<sup>|</sup> over time for different instances of the *AAMO* approach on 8-Robot systems. The maximum <sup>|</sup>*Tu*<sup>|</sup> and the end of the dispersion stage are reached at the same time. Red dots represent the <sup>|</sup>*Tu*<sup>|</sup> considered by robots (on average) along the exploration.

Hence, despite the fact the impact of the fleet size on the exploration stage transition appears to be higher than the one caused by the *HO-Threshold* value, both aspects contribute to reducing the task selection effort enabling robots to save time in the task allocation procedure anticipatedly.

In conclusion, when the *AAMO* is executed in bounded environments, the addition of robots and the decreasing of *HO-Threshold* can almost entirely mitigate the worsening in the total exploration time performance. Please note that the performance degradation of AAMO:10 instances is almost null for eight-sized fleets.

From these promising results, in the following, all *AAMO* instances are compared with the other approaches concerning non-ideal communication conditions.
