*3.4. General Considerations*

The definition of multi-objective weights is usually accomplished as an empirical matter. Typically, a search process is run in order to find—after a lot of trials—values that fit some optimal criteria. This kind of methods is typically used when the parametric function is planned to be used many times. However, in the exploration context, this assumption or even the possibility of running trials are frequently out of the question. It is not possible to assume that all scenarios where the exploration will be conducted will be similar between each other and, for this reason, is neither possible to assume that the best *α* and *β* values can remain unchangeable.

Furthermore, when these procedures are followed, at the end of the training stage it is often tough to associate the resultant parameter values with real aspects of the problem (e.g., performance metrics like time, distances, energy, or even connectivity levels). This lack of understanding may, in turn, wrongly influence the fine tuning of such parameters without rerunning a portion of trials. Taking

those shortcomings into account, an analytic approach—through which the *α* and *β* values might be set independently of the scenarios—is explored.

### **4. Adaptive** *α***-Value Computation**

When a multi-robot exploration process is going to run under communication constrained conditions, choosing between *only exploring* or *exploring preserving connectivity level* is a crucial decision. The first choice would be suitable when connectivity is out of the question, or it is impossible for a robot to keep connected and explore at once. In such a case, connectivity does not play any role in the decision-making process. On the contrary, the second choice is suitable when it is necessary to interleave high-performance exploration (minimising the total exploration time) and acceptable connectivity level (avoiding robot isolation as much as possible).

To this end, the human operator is let to use his application field expertise in order to influence the robot decision—defining a criterion to balance the importance of both objectives—by merely setting a parameter before the exploration starts.

Therefore, since *α* and *β* parameters determine the behaviour of the robots concerning target selection, two questions come up: (i) How can the value of those parameters be defined in order to ensure the applicability of the human-operator criterion along the exploration process? (ii) Should these values be adapted during the exploration process?

Henceforth, the task selection framework and the human-operator criterion are formalised. Besides, several proofs to demonstrate the existence and correctness of an adaptive *α*-value that makes the robots behave following the criterion mentioned above are conducted.

#### *4.1. Task Selection Framework*

This process is always made iteratively from a list, comparing the currently best task against the rest, one by one. Therefore, without loss of generality, the most relevant aspects can be studied just analysing all the possible relations between an arbitrary pair of tasks. Regarding the distance to a specific robot location and the connectivity level (number of connections with the rest of the fleet), any task can be classified according to Table 1.


**Table 1.** Task classification.

Therefore, the meaning of these categories is straightforward: regarding the assignment of the fleet, *Co* means that the task location would offer to the robot at least the minimum level of connectivity (i.e., one connection to another fleet member); *NC* means the opposite; regarding the spatial distribution of tasks, *Cl* means that the task under consideration is closest to the robot than any other; *F* means that the task is furthest to the robot than any other task.

Moreover, let *Ri* a robot and *Tj* and *Tk* two tasks such that *class*(*T*) can belong to any class defined in Table 1. In any scenario, these tasks can be related to each other according to Table 2. Given that *Tj* and *Tk* are arbitrary tasks, the matrix can be considered symmetric. Thus, taking one of the triangular matrices is enough to study all possible cases.

From the lower triangular, it is possible to identify some cases where one task is better (regarding both path utility and connectivity utility) than the other. Such an example is the [Cl/Co;F/NC] where *Tj* is closer to the robot than *Tk*, and it is the only one that keeps the robot connected as well. Similarly, in the [Cl/NC;F/NC] case neither task can keep the robot connected, and in consequence, the closest task *Tj* results more convenient than *Tk*. Thus, in both previous cases, the criterion to choose a task is

clear: the closest task should be selected. However, in the other cases, it is not clear at all which task should be selected. In one case, [Cl/NC;F/Co], whichever selection implies either traversing longer distances or losing connectivity. In the other case, [Cl/Co;F/Co], selecting the closest task *Tj* ensures traversing the shortest path but could imply losing connectivity. By contrast, selecting the furthest task *Tk* would be acceptable only when the gain in connectivity oppose a more significant travelling effort.

**Table 2.** Possible cases when selecting from two tasks.

**Definition 1.** *The human operator threshold* HO-Threshold *expresses the human operator criterion through a distance that represents the extra effort made by robots that the human operator is willing to accept in order to maintain or enlarge the size of the robot communication network.*

In other words, the human operator criterion is determined by setting the distance threshold until which the targets that preserve or enlarge connectivity are preferred over the rest, for all robots.

For instance, in the [Cl/NC;F/Co] case the selection will be conditioned as follows: *Tk* will be selected if and only if the length of the shortest path between *Tk* and the robot location is less than or equal to *HO-Threshold*. *Tj* will be selected otherwise.

In order to make the influence of *HO-Threshold* clearer, an example scene is depicted in Figure 7. Note that all tasks are within the *HO-Threshold*, but only *T*<sup>3</sup> can enlarge the connectivity level of the robot *R*1. Thus, applying Definition 1 leads to the selection of task *T*<sup>3</sup> because it enables the robot *R*<sup>1</sup> to travel more distance to gain connectivity. On the contrary, whether the *HO-Threshold* ≤ 3, *T*<sup>3</sup> would be no longer preferred over the rest, and consequently the closest task *T*<sup>2</sup> would be selected instead.

Hence, in the presence of some specific conditions, it is expected that the application of the *HO* criterion can make the fleet more cohesive than following approaches that do not take communication constraints into account and less restrictive than the ones that do not permit disconnections or force re-connections as well.

Next, the proofs of correctness and existence of *α* (and *β*) values that implement the *HO* criterion are conducted regarding the cases present in the lower triangular of Table 2. The cases {[Cl/Co,F/NC];[Cl/NC,F/NC]} are considered first, while the remaining {[Cl/Co,F/Co];[F/Co,Cl/NC]} are considered afterwards.

**Figure 7.** Two robots are carrying out an exploration mission. The communication and sensory ranges are drawn around the robots with red and green dashed lines, respectively. It is assumed that *R*<sup>2</sup> has already chosen the task *T*<sup>4</sup> whereas *R*<sup>1</sup> is still selecting from *T*1, *T*2, and *T*3. Dotted lines are used to show the sight-line between *R*<sup>2</sup> and the tasks. The corresponding Euclidean distance is also shown. *HO-Threshold* is set to 6.

#### *4.2. [Cl/Co,F/NC] and [Cl/NC,F/NC] Cases*

In Figure 8a,b, two instances of these cases are depicted, respectively.

**Figure 8.** Two robots are carrying out an exploration mission. It is assumed that *R*<sup>2</sup> has already chosen the task *T*<sup>4</sup> whereas *R*<sup>1</sup> is still making its decision. The communication and sensory ranges are drawn around the robots with red and green dashed lines, respectively. Dotted lines are used to show the sight-line between *R*<sup>2</sup> and the tasks. The corresponding Euclidean distance is also shown. (**a**) [Cl/Co,F/NC] case: robot *R*<sup>1</sup> is selecting from targets *T*<sup>2</sup> that is the closest and keeps it connected and *T*<sup>3</sup> that is the furthest and cause a disconnection; (**b**) [Cl/NC,F/NC] case: robot *R*<sup>1</sup> is selecting from targets *T*1—the closest—and *T*2—the furthest—given that both targets cause a disconnection.

**Proposition 1.** *When Tj and Tk belong to* [*Cl*/*Co*, *F*/*NC*] *or* [*Cl*/*NC*, *F*/*NC*]*, the values of α and β do not make any difference in the selection process.*

**Proof.** This claim can be derived directly from the following facts:

• in the [Cl/Co,F/NC] case the furthest task *Tk* makes the robot disconnected, and then applying (7) to *Tj* and *Tk* leads to:

$$\Phi\_i(\mathfrak{a}, T\_k, \mathbb{R}) = \mathfrak{a} \cdot \Psi\_i(T\_k) \le \mathfrak{a} \cdot \Psi\_i(T\_j) + \mathfrak{f} \cdot \Omega\_i(T\_j, \mathbb{R}) = \Phi\_i(\mathfrak{a}, T\_j, \mathbb{R}), \forall \mathfrak{a} \tag{10}$$

$$\text{s.t.} \qquad \Omega\_i(T\_j, \mathbb{R}) > 0, \Omega\_i(T\_{k'}, \mathbb{R}) = 0$$

$$\Psi\_i(T\_k) \le \Psi\_i(T\_{\bar{j}})$$

• in the [Cl/NC,F/NC] case both tasks make the robot to be disconnected, and thus the Φ function value will depend only on the Ψ term:

$$\Phi\_i(a, T\_k, \mathbb{R}) = a \cdot \Psi\_i(T\_k) \le a \cdot \Psi\_i(T\_j) = \Phi\_i(a, T\_{j\prime}, \mathbb{R})\_\prime \quad \forall a \tag{11}$$

$$\text{s.t. } \quad \Omega\_i(T\_{j\prime}\mathbb{R}) = 0, \Omega\_i(T\_{k\prime}\mathbb{R}) = 0$$

$$\Psi\_i(T\_k) \le \Psi\_i(T\_{\bar{j}})$$

In conclusion, in any of these cases, the task selection is not affected by *α*.

### *4.3. [Cl/Co,F/Co] and [F/Co,Cl/NC] Cases*

In the [Cl/Co,F/Co] case both tasks offer the possibility to be connected. On the contrary, in the [F/Co,Cl/NC] case opposite objectives are present: one task is closer but disconnected while the other is connected but further. Thus, the latter case is taken to prove the existence of an *α*, that can respect any given *HO* criterion. The former case is finally used to corroborate the non-existence of any possible unwanted side effect caused by the achieved *α* expression.
