## 3 Problem Formulation  
The problem formulation assumes a wireless network composed of a set \(S\) of sensors positioned on the ground and a set \(U\) of UAVs positioned in the air. The sensors are low-power and the UAVs must come into close range and in line of sight to download the data from them. All communications are assumed to be orthogonal. The terrain is modelled using a 3D surface and the airspace includes no-fly zones that the UAVs must avoid. After being polled by UAV \(u\), sensor \(s\) transmits its data. Following the model presented in [2], the received Signal-to-Noise Ratio (SNR) \(\gamma_{s,u}\) by UAV \(u\), can be computed as follows. \[\gamma_{s,u}=\frac{\mathbf{p_{s}}\,d_{s,u}^{-\alpha}}{\sigma^{2}} \tag{1}\]  
where \(P_{s}\) is the power of the signal emitted by sensor \(s\), \(d_{s,u}\) is the distance between sensor \(s\) and UAV \(u\), \(\alpha>0\) is the path-loss exponent and \(\sigma^{2}\) is the noise power. The communication link is successful if \(\gamma_{s,u}\geq\gamma_{th}\), where \(\gamma\)th is the SNR threshold. Accordingly, a maximum communication range can be defined as follows:  
\[d_{s,u}\leq\ d_{th}=\left(\frac{\mathbf{p_{s}}}{\sigma^{2}\gamma_{th}}\right)^{ \frac{1}{\alpha}} \tag{2}\]  
The following notation shall be used:  
* \(T\) to represent the region of space under the surface of the terrain;
* \(NF\) to represent the region of space that corresponds to the no-fly zones;
* \(U=\left\{u_{i}:\ i=1\dots N_{u}\right\}\) to represent the set of UAVs;
* \(\mathbf{W_{u}}=\left\{\mathbf{W_{u,i}}\ |\ i=1\dots N_{u,W}\right\}\) as the set of waypoints that defines the trajectory of UAV \(u\);
* \(v_{u,i}=\) the average speed of UAV \(u\) between its waypoints \(\mathbf{W_{u,i}}\) and \(\mathbf{W_{u,i+1}}\);
* \(S=\left\{\mathbf{s}_{i}:\ i=1\dots N_{s}\right\}\) as the set of sensors, positioned on the ground at the coordinates \(\left\{\mathbf{s}_{i}\right\}\);
* \(D_{s}=\) the amount of data to be transferred by sensor \(\mathbf{s}\);
* \(r_{s}=\) the rate of transfer of the data by sensor \(\mathbf{s}\);
* \(DP=\left\{\mathbf{p}_{i}\ |\ i=1\ \dots\ N_{DP}\right\}\) as the set of download points, with the coordinates \(\left\{\mathbf{p}_{i}\right\}\);
* \(DP_{u}=\left\{\mathbf{p}_{u,i}\ |\ i=1\dots N_{u,DP}\right\}\) as the set of download points from which UAV \(u\) will gather data; and
* \(S_{\mathbf{p}}=\left\{\ \mathbf{s}\in S\ |\ d_{s,\mathbf{p}}\leq d_{th}\ \right\}\) the set of sensors that can be polled from download point \(\mathbf{p}\). In this work, it is assumed that all sensors transmit the same amount of data, at the same rate, have the same transmission range and that at any given download point \(\mathbf{p}_{u}\), UAV \(u\) successively gathers data from every sensor that is in range. Thus, the time required by UAV \(u\) to download all of its data depends on the number of sensors associated with the download point and the transfer rate. The download time and the travel time for UAV \(u\) in this mission, is therefore  
\[\mathcal{T}_{u,download}=\sum_{\mathbf{p}\ \in\ DR_{u}}\left\{\sum_{\mathbf{s}\in\ \mathbf{s}_{\mathbf{p}}}\frac{D_{\mathbf{s}}}{r_{s}}\right\} \tag{3}\]  
and  
\[\mathcal{T}_{u,travel}=\sum_{i=1\dots N_{u,u}}\frac{\parallel\mathbf{W_{u,i+1}}- \mathbf{W_{u,i}}\parallel}{v_{u,i}} \tag{4}\]The problem of wireless sensor data acquisition using a team of UAVs can now be expressed as that of minimizing the maximum time taken by any UAV to perform its mission, i.e., finding  
\[ \begin{array}{c}\min\\ S_{\textbf{{p}}}\text{ with }\textbf{{p}}\ \in\ DP\\ DP_{u}\text{ and }\textbf{{W}}_{u}\text{ with }u\ \in\ U\end{array}