## 2 Problem Formulation
### Communication Model  
This model aims to support the connectivity awareness ability of robots needed to deal with disconnection situations during the exploration. Given the position of their teammates and obstacles, robots can estimate the connectivity degree of a specific location considering some of the communication constraints that are widely present in real scenarios, mainly indoor (e.g., office-like and buildings).  
The signal strength function (\(\Gamma_{i}\) represents a slight adaptation of the signal strength function presented in [38]) \(\Gamma_{i}:\mathbb{N}\times S^{m\times n}\times\mathbb{R}\rightarrow\mathbb{R}\) is defined as follows:  
\[\begin{split}\Gamma_{i}(j,E_{known}(t),t)&=\Gamma_{i }^{0}-d_{Att}(i,j,t)-w_{Att}(i,j,E_{known}(t),t)\\ \Gamma_{i}^{0}&=10\cdot D_{af}\cdot log_{10}(c_{i}/r_ {i})\\ d_{Att}(i,j,t)&=10\cdot D_{af}\cdot log_{10}(d_{i} (j,t)/r_{i})\\ d_{i}(j,t)&=\left\|X_{i}(t),X_{j}(t)\right\|_{2}\\ w_{Att}(i,j,E_{known}(t),t)&=\begin{cases}w_{i}(j,E _{known}(t),t)\cdot W_{af}&\text{if }w_{i}(j,E_{known}(t),t)<C\\ C\cdot W_{af}&\text{otherwise}\end{cases}\end{split} \tag{1}\]  
where, \(d_{Att}\) and \(w_{Att}\) stand for _distance attenuation_ and _wall attenuation_ terms, respectively. In addition, \(d_{i}(j,t)\) represents the Euclidean distance between two robot locations at time \(t\): typically the transmitter(\(X_{i}(t)\)) and receiver (\(X_{j}(t)\)), \(w_{i}(j,E_{known}(t),t)\) represents the number of walls (robots cannot distinguish between different kind of rigid obstacles, but the term _wall_ is used for simplicity and in order to be consistent with the underlying proposal) present in the known region between transmitter and receiver locations at time \(t\), \(D_{af}\) represents a distance attenuation factor, and \(W_{af}\) represents a wall attenuation factor. Finally, \(C\) represents the maximum number of walls up to which the \(W_{af}\) factor causes a significant effect in function \(\Gamma_{i}\). When \(w_{i}(j,E_{known}(t),t)\geq C\), the distance attenuation effect dominates. Finally, note that in [38] the independent term \(\Gamma_{i}^{0}\) is suggested to be either derived empirically or obtained directly from the wireless network device specification. Nevertheless, in this work the model is adapted in order to become independent from specific deployments (communication devices), deriving the \(\Gamma_{i}^{0}\) value so that the signal strength \(\Gamma_{i}(j,E_{known}(t),t)=0\) when \(d_{i}(j,t)=c_{i}\).  
In Figure 2 the shape of the function \(\Gamma_{i}\), as well as the attenuation effects caused by both distances and walls, are plotted.  
Unfortunately, due to uncertain and incomplete knowledge, the \(\Gamma_{i}\) function only can either confirm the absence of connectivity or deliver an optimistic estimation of connectivity level instead. Although this model represents a valuable improvement in relation to others (e.g., the classic disk or line of sight models [20]), for the sake of simplicity other impairments also common in communication (e.g., bandwidth, information losses, fading, and multi-path propagation phenomenon [39; 40]) are not considered in this work.