4.3.2. [Cl/Co,F/Co] Case

This analysis is devoted to checking the applicability of the *α*HO when the conditions to achieve a good trade-off between path cost and connectivity level are less demanding than in the [F/Co,Cl/NC] case. In the [Cl/Co,F/Co] case, although one task is closer than the other, the differences in the positive connectivity level offered by them could make the furthest task more attractive than the closest. From that, considering the connectivity level offered by the closest, two cases may be identified: (i) When *Tj* offers a higher level of connectivity than *Tk*. In such a case, there is no doubt that independently of the *α*HO value, the selection would always favour the task *Tj* because it is the closest as well; (ii) On the contrary, when *Tk* offers a higher level of connectivity than *Tj*, the selection of *Tk* will depend on both how distant from robot it is and how much more connected would be the robot on *Tk* respect to *Tj*.

Finally, to show that the *α*HO value does not introduce any unwanted side effect on the task selection process when tasks belong to the [Cl/Co,F/Co] case, it is needed to prove that it neither contradicts the first case nor restricts the occurrence of the second case.

**Proposition 4.** *In the presence of two tasks subject to the [Cl/Co,F/Co] case conditions, if Tj is the closest and simultaneously the one which provides the highest level of connectivity, then the application of the αHO value will never result in the selection of Tk.*

**Proof.** By contradiction, it is assumed that under these conditions the selection could be in favour of *Tk*, implying that the following inequality holds:

$$\begin{aligned} \Phi\_i(\mathfrak{a}, T\_j, \mathbb{R}) &= \mathfrak{a} \cdot \Psi\_i(T\_j) + \mathfrak{f} \cdot \Omega\_i(T\_j, \mathbb{R}) \le \mathfrak{a} \cdot \Psi\_i(T\_k) + \mathfrak{f} \cdot \Omega\_i(T\_k, \mathbb{R}) = \Phi\_i(\mathfrak{a}, T\_k, \mathbb{R}) \\ \mathfrak{a} \cdot (\Psi\_i(T\_j) - \Psi\_i(T\_k)) + \mathfrak{f} \cdot (\Omega\_i(T\_j, \mathbb{R}) - \Omega\_i(T\_k, \mathbb{R})) &\le 0 \end{aligned} \tag{17}$$

Which implies that, independently of the *α*HO value, the terms (Ψ*i*(*Tj*) − Ψ*i*(*Tk*)) and (Ω*i*(*Tj*, *R*) − Ω*i*(*Tk*, *R*)) should not be positive simultaneously. Thus, either (Ψ*i*(*Tj*) ≤ Ψ*i*(*Tk*)) or (Ω*i*(*Tj*, *R*) ≤ Ω*i*(*Tk*, *R*)). However, this contradicts the hypothesis where *Tj* is stated as the closest and the one which simultaneously provides the highest level of connectivity, and accordingly the proposition has been demonstrated.

**Proposition 5.** *In the presence of two tasks subject to the [Cl/Co,F/Co] case conditions, if Tj is the closest and Tk the one which provides the highest level of connectivity, then the application of the αHO value will never be conclusive concerning the task selection.*

**Proof.** The relation between the utility of tasks is written as follows in (18):

$$\Phi\_i(a, T\_j, R) = a \cdot \Psi\_i(T\_j) + \oint \Omega\_i(T\_j, R) \lessgtr \, a \cdot \Psi\_i(T\_k) + \oint \Omega\_i(T\_k, R) = \Phi\_i(a, T\_k, R)$$

$$\begin{aligned} a \cdot (\Psi\_i(T\_j) - \Psi\_i(T\_k)) \lessgtr \, \not\, \rho \cdot (\Omega\_i(T\_k, R) - \Omega\_i(T\_j, R)) \\ a \cdot (\Psi\_i(T\_j) - \Psi\_i(T\_k)) \lessgtr \, (1 - a) \cdot (\Omega\_i(T\_k, R) - \Omega\_i(T\_j, R)) \\ a \lessgtr \, \frac{\Omega\_i(T\_k, R) - \Omega\_i(T\_j, R)}{(\Omega\_i(T\_k, R) - \Omega\_i(T\_j, R)) + (\Psi\_i(T\_j) - \Psi\_i(T\_k))} \end{aligned} \tag{18}$$

Substituting (Ω*i*(*Tk*, *R*) − Ω*i*(*Tj*, *R*)) = *x* and (Ψ*i*(*Tj*) − Ψ*i*(*Tk*)) = *u*, it is possible to state that in order to favour the selection of *Tj* the inequality (19) must be held, otherwise the (20):

$$
\alpha \ge \frac{\mathbf{x}}{\mathbf{u} + \mathbf{x}} \implies \alpha = \sup \left( \frac{\mathbf{x}}{\mathbf{u} + \mathbf{x}} \right) \tag{19}
$$

$$a \le \frac{\mathbf{x}}{\mathbf{u} + \mathbf{x}} \implies a = \inf \left( \frac{\mathbf{x}}{\mathbf{u} + \mathbf{x}} \right) \tag{20}$$
 
$$\text{s.t. } 0 \le \mathbf{x} \le 1$$

$$\begin{aligned} 0 \le \mu \le 1\\ \text{given that: } \Omega\_i(T\_k, \mathbb{R}) \ge \Omega\_i(T\_{\circ}, \mathbb{R})\\ \Psi\_i(T\_{\circ}) \ge \Psi\_i(T\_k) \end{aligned}$$

On this domain, the function *<sup>x</sup> <sup>u</sup>*+*<sup>x</sup>* presents an absolute maximum equal to 1 in the point (*x*, *u*)=(1, 0), and absolute minima equal to 0 along the line segment defined by (*x*, *u*)=(0, *u*). Assessing the *α*HO expression derived in (15) with (0, *u*) leads to (21) and (22), respectively:

$$0 = \frac{\Omega\_1}{1 + \Omega\_1 - \Psi\_i(T\_{\rm HO})} \quad \therefore \quad 0 = \Omega\_1 \tag{21}$$

$$\begin{aligned} 1 &= \frac{\Omega\_1}{1 + \Omega\_1 - \Psi\_i(T\_{\rm HO})}\\ 1 - \Psi\_i(T\_{\rm HO}) &= 0 \quad . \quad \Psi\_i(T\_{\rm HO}) = 1 \end{aligned} \tag{22}$$

From which, while the condition expressed in (21) is reached when |*R*| → ∞, the one expressed in (22) is reached when *HO-Threshold* tends to 0. The condition (21) is unreachable in practice implying that no *α*HO can make the task *Tk* always preferred over *Tj*. Conversely, the condition (22) is reachable if, and only if, the human operator deliberately does not want to care about connectivity. Otherwise, there is no positive *α*HO-value that can make the task *Tj* always preferred over *Tk*.

Consequently, when *α*HO ∈ (0..1] under the [Cl/Co,F/Co] conditions, it is not possible to hold a single preference over time.

#### *4.4. Considerations and Usefulness*

In order to establish the task selection criterion, the human operator only needs to choose the extra distance *HO-Threshold*—according to his expertise and knowledge—he is willing to ask the robots to travel in order to keep or enlarge the connectivity level of the fleet. Once the *HO-Threshold* is set, robots are capable of selecting tasks consistently with the *HO* criterion following the Equation (15). Furthermore, it is important to note that the *HO-Threshold* value does not change along the exploration but, as was pointed out, the *αHO* does, due to the dependency on the Ψ function. This explains the need for auto-adaptive capabilities concerning the multi-objective Φ function.

Additionally, it also worth noticing that setting *HO-Threshold*= ∞ it is a practical way to implement an event-based connectivity approach where the tasks that provide connectivity will always be preferred over the rest, no matter how close they are.

### **5. Task Allocation Scheme**

The allocation scheme is founded on two pillars: the coordination method and the task selection algorithm.

#### *5.1. Coordination Method*

In order to take advantage of the individual computing power of the robots, to avoid the single point of failure, and to deal better with the presence of real communication constraints during the exploration, a decentralised approach is followed. Typically, estimation of travelling costs and target benefits, as well as mapping and localisation, are the tasks chosen to be made locally by the robots. However, to achieve a cooperative behaviour, both the local map and localisation information must be shared among team members.

Additionally, the relation between |*T*| and |*R*| can result in two somewhat different behaviours: (i) If |*T*| < |*R*|, not all robots would be needed to reach all targets. Some robots may choose to keep quiet; (ii) When |*T*|≥|*R*| all robots would be needed in order to reach the maximum amount of targets at a time. When robots decide to explore, the task selection is made coordinately. Robots coordinate their actions implicitly, sharing specific information (such as locations, eventually already-done-selections, and local maps) and running the same selection algorithm. Thus, it is possible for the multi-robot system to compute a coordinated-tasks-to-robots distribution in a decentralised way [15,17,44].

To do so properly, the exchanging information time is carefully set up. The system is fully asynchronous, meaning that: (i) Robots do not wait for others; (ii) After selecting a task, the robots do exchange their selection in order to prevent future overlappings; (iii) Local maps and—by means of this —the sets of new available tasks are periodically exchanged, each time two conditions are met: (1) A waypoint of the planned path is reached; (2) New information has been gathered; (iv) Localisation data is exchanged at a higher rate than maps because its influence on the task selection algorithm is higher too.

While localisation data is exchanged periodically, the rest of data exchanging is triggered by events instead. These policies make the system more efficient and flexible because: (i) No data is transmitted when there is no new information to exchange; (ii) There is no need to set up any rate parameter when exploring different environments. The robot life-cycle algorithm is sketched in Algorithm 1.

