*3.3. Connectivity Utility*

Connectivity utility computes, optimistically, the connectivity level present in a location at a certain moment. The *connectivity* utility function <sup>Ω</sup>*<sup>i</sup>* : *<sup>T</sup>* <sup>×</sup> *<sup>R</sup><sup>M</sup>* <sup>×</sup> *<sup>S</sup>m*×*<sup>n</sup>* <sup>→</sup> [0, 1] is defined as follows:

$$
\Omega\_{\rm i}(T\_{\rm f}, R, E\_{\rm known}) = \frac{\log\_2\left( (2^{\rho} - 1) \cdot \frac{|\mathcal{N}\_{\rm i}(E\_{\rm known}(t), t)|}{M - 1} + 1 \right)}{\rho} \tag{9}
$$

$$
\text{s.t.}$$

$$1 \le i \le M = |\mathbb{R}| , 1 \le j \le N = |T| $$

Given the current state of the fleet *R* and the current environment knowledge *Eknown*, the function Ω*<sup>i</sup>* estimates the connectivity utility obtained by a robot *Ri* in case of selecting the task *Tj*. Particularly, it is interesting to do so concerning the arrival time to *Tj*. The current fleet state refers to both the location of the assigned tasks in case of assigned robots and the robot positions otherwise. The parameter *ρ* works as a shaping factor that could be used to tune the relation between connectivity level and utility. Note that the utility is decreasing in the number of robots, and may favour the adoption of MANET compliant connectivity techniques. In such networks, messages travel from source to destination members in more than one hop, where intermediate nodes forward messages until the destination is reached. The shape and usefulness of the Ω function may be appreciated in Figure 6.

**Figure 6.** (**a**) Connectivity utility function shape; (**b**) A scene where the usefulness of the Ω function can be appreciated. Three robots (coloured dots) and several walls were arranged simulating an ongoing exploration process. Robots are surrounded with two lines of the same dot colour indicating sensory (dashed) and communication ranges (solid); (**c**) Shows the connectivity level map corresponding to (**b**). Therefore, as long as the will of another robot is to keep connected with the fleet, it would be able to take this perspective into account when deciding where going to.
