4.3.1. [F/Co,Cl/NC] case.

Based on the human-operator criterion (set by a threshold value) we want an *α*-value that makes, following the scenario depicted in Figure 9, *T*<sup>3</sup> preferred over *T*<sup>2</sup> if and only if *T*<sup>3</sup> belongs to the circular area defined by the *HO-Threshold*.

**Figure 9.** [F/Co,Cl/NC] case.

Next, the existence of such an *α* parameter will be demonstrated, and its value will be derived as well.

**Proposition 2.** *When Tj and Tk belong to [F/Co,Cl/NC] is always possible to find an α-value that satisfies the following inequality:*

$$\Phi\_i(a, T\_j, \mathbb{R}) = a \cdot \Psi\_i(T\_j) + \beta \cdot \Omega\_i(T\_j, \mathbb{R}) \ge a \cdot \Psi\_i(T\_k) = \Phi\_i(a, T\_k, \mathbb{R}) \tag{12}$$
 
$$\text{s.t.} \qquad \Omega\_i(T\_j, \mathbb{R}) > 0, \Omega\_i(T\_k, \mathbb{R}) = 0$$
 
$$\Psi\_i(T\_j) \le \Psi\_i(T\_k)$$

**Proof.** Let Ω<sup>1</sup> the utility assigned to the fact of being connected with only one teammate. Then, from (9), it is possible to state that: if *Tj* belongs to any [∗/*Co*] class, Ω<sup>1</sup> ≤ Ω*i*(*Tj*, *R*), ∀(*i*, *j*) over time. Moreover, if the number of robots does not change, it is also possible to state that Ω<sup>1</sup> remains invariant over time. Applying this result into (12) leads to the inequality presented next in (13):

$$\Phi\_i(\mathfrak{a}, T\_j, \mathbb{R}) \ge \mathfrak{a} \cdot \Psi\_i(T\_j) + \beta \cdot \Omega\_1 \ge \mathfrak{a} \cdot \Psi\_i(T\_k) = \Phi\_i(\mathfrak{a}, T\_k, \mathbb{R})$$

$$\mathfrak{a} \cdot \Psi\_i(T\_j) + (1 - \mathfrak{a}) \cdot \Omega\_1 \ge \mathfrak{a} \cdot \Psi\_i(T\_k)$$

$$\mathfrak{a} \cdot (\Psi\_i(T\_j) - \Omega\_1) + \Omega\_1 \ge \mathfrak{a} \cdot \Psi\_i(T\_k) \tag{13}$$

$$\frac{\Omega\_1}{\Psi\_i(T\_k) - \Psi\_i(T\_j) + \Omega\_1} \ge \mathfrak{a}$$

Besides, substituting Ω<sup>1</sup> = *x* and Ψ*i*(*Tk*) − Ψ*i*(*Tj*) = *u*, equation (13) may be rewritten as follows:

$$\alpha \le \frac{\mathbf{x}}{u + \mathbf{x}} \implies a \le \inf \left( \frac{\mathbf{x}}{u + \mathbf{x}} \right) \tag{14}$$
 
$$\text{s.t. } 0 < c \le \mathbf{x} \le 1$$
 
$$0 \le u \le 1$$
 
$$\text{given that: } \Omega\_1 = c$$
 
$$\Psi\_i(T\_k) \ge \Psi\_i(T\_j)$$

From (14) is possible to claim the existence of an *α*-value that obey any *HO-Threshold* if and only if the function *<sup>x</sup> <sup>u</sup>*+*<sup>x</sup>* presents an absolute minimum on the domain:

$$D = \{ (\mathfrak{x}, \mathfrak{u}) \mid 0 < \mathfrak{c} \le \mathfrak{x} \le 1, 0 \le \mathfrak{u} \le 1 \}.$$

This fact can be stated employing *Weierstrass* theorem (a function *f* has an absolute extreme if it is continuous and its domain is compact). Besides, the minimum point might be calculated analysing both: (i) the relative extrema and (ii) the points lying on the border of *D*. Following this procedure, it is possible to find the absolute extreme of the function *<sup>x</sup> <sup>u</sup>*+*<sup>x</sup>* in (*x*, *u*)=(*c*, 1).

Moreover, it is remarkable that this extreme represents a place where the most demanding conditions are reached: task *Tj* presents the lowest positive connectivity utility, and the distance between both tasks is the largest. Hence, the existence of a positive value *<sup>α</sup>* <sup>≤</sup> <sup>Ω</sup><sup>1</sup> <sup>1</sup>+Ω<sup>1</sup> (regardless of how demanding can be the distance relation between tasks) that might alter the task selection in favour of connectivity has been demonstrated.

Nevertheless, in (13) *α* is independent of the *HO-Threshold*. Consequently, its direct application would result in a strictly connectivity-guided exploration, where tasks that offer connectivity are always preferred over the rest no matter how far they are. Therefore, to relate it with an *HO-Threshold* the value of the term Ψ*i*(*Tj*) in (13) must be substituted by the utility of being *HO-Threshold* far from the robot, say Ψ*i*(*T*HO). Next, the value of the term Ψ*i*(*Tk*) is substituted by 1 since Ψ*i*(*Tk*) = 1 represents the necessary condition to reach the extreme coordinate *u* = 1 that arose from (14). Finally, the expression for an *HO-Threshold* dependent *α*, say *α*HO, is expressed in (15) as follows:

$$\alpha\_{\rm HO} = \frac{\Omega\_1}{1 - \Psi\_i(T\_{\rm HO}) + \Omega\_1} \tag{15}$$

**Proposition 3.** *The applicability of the αHO referred to in* (15) *causes any task within the threshold scope that also offers any positive connectivity level to be favoured over the rest of the tasks that do not offer any connectivity level, regardless of how close to the robot they are.*

**Proof.** Φ*i*(*α*, *Tj*, *R*) ≥ Φ*i*(*α*, *Tk*, *R*) is imposed to any tasks (*Tj*, *Tk*) that respect the [F/Co,Cl/NC] conditions:

$$\Phi\_i(\boldsymbol{\alpha}, \boldsymbol{T}\_j, \boldsymbol{\mathsf{R}}) = \boldsymbol{\alpha} \cdot \boldsymbol{\Psi}\_i(\boldsymbol{T}\_j) + \boldsymbol{\beta} \cdot \boldsymbol{\Omega}\_i(\boldsymbol{T}\_j, \boldsymbol{\mathsf{R}}) \geq \boldsymbol{\alpha} \cdot \boldsymbol{\Psi}\_i(\boldsymbol{T}\_k) = \boldsymbol{\Phi}\_i(\boldsymbol{\alpha}, \boldsymbol{T}\_k, \boldsymbol{\mathsf{R}})$$

$$\boldsymbol{\alpha} \cdot \boldsymbol{\Psi}\_i(\boldsymbol{T}\_j) + (1 - \boldsymbol{\alpha}) \cdot \boldsymbol{\Omega}\_i(\boldsymbol{T}\_j, \boldsymbol{\mathsf{R}}) \geq \boldsymbol{\alpha} \cdot \boldsymbol{\Psi}\_i(\boldsymbol{T}\_k)$$

$$\boldsymbol{\alpha} \cdot (\boldsymbol{\Psi}\_i(\boldsymbol{T}\_j) - \boldsymbol{\Omega}\_i(\boldsymbol{T}\_j, \boldsymbol{\mathsf{R}})) + \boldsymbol{\Omega}\_i(\boldsymbol{T}\_j, \boldsymbol{\mathsf{R}}) \geq \boldsymbol{\alpha} \cdot \boldsymbol{\Psi}\_i(\boldsymbol{T}\_k)$$

$$\frac{\boldsymbol{\Omega}\_i(\boldsymbol{T}\_j, \boldsymbol{\mathsf{R}})}{\boldsymbol{\Psi}\_i(\boldsymbol{T}\_k) - \boldsymbol{\Psi}\_i(\boldsymbol{T}\_j) + \boldsymbol{\Omega}\_i(\boldsymbol{T}\_j, \boldsymbol{\mathsf{R}})} \geq \boldsymbol{\alpha}$$

Then, applying (15) leads to (16):

$$\frac{\Omega\_i(T\_j, R)}{\Psi\_i(T\_k) - \Psi\_i(T\_j) + \Omega\_i(T\_{j'}, R)} \ge \frac{\Omega\_1}{1 + \Omega\_1 - \Psi\_i(T\_{\rm HO})}$$

$$\Psi\_i(T\_j) \ge \frac{\Omega\_i(T\_j, R)}{\Omega\_1} \cdot (\Psi\_i(T\_{\rm HO}) - 1) + \Psi\_i(T\_k) \tag{16}$$

$$\Psi\_i(T\_j) = \sup \left( \frac{\Omega\_i(T\_{j'}, R)}{\Omega\_1} \cdot (\Psi\_i(T\_{\rm HO}) - 1) + \Psi\_i(T\_k) \right)$$

Since *<sup>i</sup>*) <sup>Ω</sup><sup>1</sup> is constant, *ii*) <sup>Ω</sup>*i*(*Tj*,*R*) <sup>Ω</sup><sup>1</sup> ≥ 1, and *iii*) (Ψ*i*(*T*HO) − <sup>1</sup>) ≤ 0, it is possible to conclude that:


$$\begin{array}{ll} \text{(a)} & \Psi\_i(T\_{\text{HO}}) = 1\\ \text{(b)} & 0 \le \Psi\_i(T\_{\text{HO}}) < 1, \,\Omega\_i(T\_{j'}R) = \Omega\_1 \text{ and } \Psi\_i(T\_k) = 1. \end{array}$$

Please note that (a) is out of the proposition conditions. Instead, (16) can be rewritten imposing (b), leading to:

$$\begin{aligned} \Psi\_i(T\_{\vec{\}}) &\geq \frac{\Omega\_1}{\Omega\_1} \cdot (\Psi\_i(T\_{\text{HO}}) - 1) + 1), \\ \Psi\_i(T\_{\vec{\}}) &\geq \Psi\_i(T\_{\text{HO}}) \end{aligned}$$

which is true if, and only if, Δ*i*(*Tj*) ≤ *HO-Threshold*, which is indeed what the human operator would like to get from his criterion application to tasks within the *HO-Threshold*. Hence, following (15) under the [F/Co,Cl/NC] conditions it is always possible to compute an *α*HO-value that makes the robots behave following the human-operator criterion.

Likewise, it is important to highlight that the *α*HO-value needs to be calculated every time a robot is ready to make a decision. This need for adaptation arises from Ψ*i*(*T*HO), which is not constant. Its value depends on the relation between the *HO-Threshold* and the relative distance to the current furthest task. That way, the robots can autonomously adapt the weights of the *task* utility function according to the changing conditions of the environment in order to be always consistent with the human-operator criterion.
