# A Collaborative UAV-WSN Network for Monitoring Large Areas
## 2 Materials and Methods
### Trajectory Planning for the Estimation of Sensor Positions  
As stated earlier, possible methods for the widespread dispersion of sensor nodes involve the use of air vehicles. Such a solution easily leads to an irregular distribution (characterized by a connected graph). Considering this aspect, it is mandatory to define a mechanism to obtain an estimate of the position of each sensor node placed on the ground. We start by defining a flight plan to ensure full coverage of the area of interest. Several specific parameters such as altitude of flying, UAV radio communication range, both for WSN nodes and UAV and others must be considered. We consider therefore a multi-obstacle environment a priori known. That is, we consider a collection of obstacles \(\mathrm{O}=\{\mathrm{O}_{\mathrm{i}}\}\subset\mathrm{R}^{n}\), as illustrated in the proof-of-concept example shown in Figure 2 (while the space dimension is \(n=3\) or even \(n=2\) if we assume the UAVs to fly at constant height, we keep \(n\) to highlight that the results hold for a generic space dimension). Within this environment we consider a collection of sensors \(\mathbb{S}=\{S_{j}\}\). Each sensor is characterized by its location \(z_{j}\
otin\mathbb{O}\) and communication radius \(r_{j}\), the former to be determined at a later step. In Figure 3, a grid of 38 sensors (blue dots) are randomly distributed through the feasible domain (the complement of the union of three restricted areas (gray areas). Assuming that the sensors are already deployed we require initial flights which cover the area of interest and, based on the information retrieved from the ground, provide the estimate \(\hat{z}_{j}\) of the true sensor location \(z_{j}\). For a UAV following trajectory \(r(t)\), the estimation of the \(j\)-th sensor position is then given as:  
\[\hat{z}_{j}\in\hat{Z}_{j}\triangleq\bigcap_{t:S_{j}\text{visible}}\{\,z:\, \big{|}|r(t)-z\big{|}\big{|}\leq r_{j}\,\}. \tag{1}\]  
Equation (1) simply states that the sensor's position has to lie within all the circles centered on the points of the UAV trajectory where signal from the \(j\)-th sensor is received. While the UAV trajectory \(r(t)\) can be arbitrary, it is much simpler to assume a regular trajectory: we consider a photogrammetry procedure in which the UAV covers the area with parallel lines separated by inter-distance \(\Delta\) and with measurements done along each line at distance \(\delta\) as illustrated in Figure 4.  
Figure 2: Hybrid UAV-WSN large scale monitoring functional diagram. Applying (1) for one sensor provides its location estimation (the blue region highlighted in the inset), as shown in Figure 5.  
Strictly speaking any pair \((\Delta,\delta)\) which checks \(\sqrt{\Delta^{2}+\delta^{2}}<\min_{j}r_{j}\) guarantees that no sensor is missed (it will be \"felt\" at least once during the flight). In practice, we require that \(\Delta\) and \(\delta\) are sufficiently small to obtain a fair estimate of the position. Later on, we require that the UAV spends enough time in the neighborhood of a cluster head. If its position is uncertain we can no longer be  
Figure 4: Photogrammetry trajectories for sensor location estimation. Figure 5: Sensor position estimation through a photogrammetry procedure (with highlight for the uncertainty radius). (**a**) estimation detail; (**b**) uncertainty range. Figure 3: Example of aerial vehicle flight path planning in a multi-obstacles environment. sure that communication is ensured. As illustrated in Figure 5b we consider the estimated position of the sensor and find the smallest circle (defined by its center \(c_{j}\) and radius \(r_{j}^{\prime}\)) which encloses it. The value of \(r_{j}^{\prime}\) ultimately depends on the the pair of parameters (\(\Delta\),\(\delta\)); thus, these have to be taken of appropriate values in order to obtain an acceptable value for the uncertainty radii \(r_{j}^{\prime}\).