# A Collaborative UAV-WSN Network for Monitoring Large Areas
## 3 Results
### Simulations
#### 3.1.1 Sensor Location Estimation and Trajectory Planning  
To highlight the procedure for sensors' location estimation, in Figure 8 we illustrate the estimation results for a choice of parameters \(\Delta=100\) m, \(\delta=50\) m and \(r_{j}=150\) m, \(\forall j\). Recalling the construction of the uncertain region from Section 2.2 we proceed as follows: at each point (denoted by red \"\(x\)\" symbols in Figure 8) along the trajectory where measurements are taken, the UAV gathers information from the sensors and, using its current position and the communication radius \(r_{j}=150\) m provides an uncertain region \(\hat{Z}_{j}\) for each of the sensors as in relation (1). As can be seen in Figure 9, the real sensor position (denoted by blue-filled circles), lies inside the yellow-filled uncertain regions (1). These, in turn, are over-approximated, as shown in optimization procedure (2), through circles (the red-filled shapes) characterized by centers \(c_{j}\) (hollow blue circles) and radii \(r_{j}\)\({}^{\prime}\) as illustrated earlier in Figure 5.  
**Remark 5**. : _Note that the assumption of a known communication radius leads to a bounded uncertainty: while the actual position of the sensor is unknown it is, nonetheless, guaranteed to lie within the bounds of uncertain region (1)._  
For this particular example we observe that the uncertainty radii \(r_{j}^{\prime}\) stay in the interval \([10.51\text{ m},\,61.97\text{ m}]\). Since the worst estimation, \(r_{j}^{\prime}=61.97\text{ m}\) is significantly less than the communication radius \(r_{j}=150\text{ m}\) we consider the identification procedure to be precise enough. Note as well that the worse uncertainties are for sensors near the obstacles. This happens because we assumed that no measurements are taken above the interdicted regions which means that there are fewer possibilities to assess the sensors' position. Note for further use that the guaranteed communication region is given by the circle centered in \(c_{j}\) with radius \(\hat{r}_{j}=r_{j}-r^{\prime}_{j}\). This allows later to consider the sensors as if they are centered in \(c_{j}\) with radius \(\hat{r}_{j}\) from the point of view of the subsequent trajectory generation procedures. The \"worst\" or \"best\" estimates are obtained a posteriori: once the UAV has done its initial passing, we proceed to bound the actual sensors' position through their associated uncertain regions (1) and to subsequently over-approximate these by circles of radii \(r_{j}^{\prime}\). If the approximation is unsatisfactorily (e.g., \(r_{j}^{\prime}\) is comparable in size with the communication radius \(r_{j}\)) we need to repeat the procedure with modified parameters (smaller inter-line \(\Delta\), and intra-line spacings \(\delta\)). Figure 10 shows various trajectories passing through or near the cluster heads. These trajectories are obtained as per the procedure highlighted in Section 2.4. Briefly: we assume that each trajectory is a weighted sum of basis functions (in our case, the B-splines defined in (4)-(6)) whose weights are the result of a constrained optimization procedure (the cost is the total trajectory length and the constraints are taken from (7)-(18)). We enumerate them in increasing order of complexity:(i) the dashed blue curve describes a trajectory which passes through the cluster heads (large black dots) ignoring the obstacles; (ii) the dotted green curve shows the slightly more complex case where the trajectory is required to pass through the neighbourhood of the cluster head (i.e., to be inside the communication radius--semi-transparent red regions), still ignoring the obstacles; (iii) the magenta solid curve shows a trajectory which ensures both the sufficient communication time constraint and the obstacle avoidance constraint. Figure 9: Estimation of the sensor location in both exact and over-approximated (by circles).