## 4 Open Issues and Challenges
### Straight Path  
When the trajectory of the UAV is a straight path, the scenario can be further refined as: sensor is static (Figure 3), sensor and the UAV are move at the same direction (Figure 4), and in the opposite direction (Figure 5). The relative movement distance between sensor and the UAV is denoted by \(d_{r}\). UAV fly at a height \(h\) with constant velocity \(v\). The range of the sensor is denoted by \(r\). The original distance between UAV and node is \(d_{x}\). Then, we will discuss the calculation of the contact duration time as follows:  
* The node is static (Figure 3). In this case, the relative velocity between the node and the UAV is \(v\). The relative movement distance between them is \(d_{r}=A_{s}B_{s}\), i.e., \[d_{r}=2(r^{2}-h^{2}-d^{2})^{\frac{1}{2}}\,.\] (6) when the UAV is out of the range (Figure 3a). In Figure 3b, when the UAV is within the range of the node, \(d_{r}=D_{s}B_{s}\), i.e., \[d_{r}=(r^{2}-h^{2}-d^{2})^{\frac{1}{2}}+d_{x}\,.\] (7) The _contact duration_ between the UAV and the static node, \(T_{cd}\) can be given by, \[T_{cd}=\frac{d_{r}}{v}\,.\] (8) This is based on an assumption that the flying height of UAV is smaller than the range of the node (\(h<r\)) and the UAV is out of the range of the node in the beginning (\(d_{x}>r\)). The relative movement distance will be different if their topologies are changed (as illustrated in Figures 4 and 5). * The node and the UAV move at the same direction (Figure 4). The sensor velocity is \(v_{s}\). The contact duration time depends on the original deployment of the UAV if the sensor and the UAV have the same speeds (\(v_{s}=v\)). The relative velocity between the node and the UAV is 0. Let the moving time and distance of node are \(T\) and \(L\) respectively. The relative movement distance between the UAV and the sensor is \(d_{r}=0\) when the UAV is out of the range of the UAV (\(d_{x}>r\) in Figure 4a). \(d_{r}=L\) when the UAV is within the range of the node (\(d_{x}\leq r\) in Figure 4b). When \(v_{s}\
eq v\), similarly to static case, the relative distance between them can be given by, \[d_{r}=\left\{ \begin{array}{ll}0&v_{s}>v,d_{x}\geq r\\ A_{s}B_{s}&v_{s}<v,d_{x}\geq r\\ A_{s}D_{s}&v_{s}>v,d_{x}<r\\ D_{s}B_{s}&v_{s}<v,d_{x}<r\,.\end{array} \right.\] (9) Then, according to Equation (8) and Figure 4b, the \(T_{cd}\) between the UAV and the mobile sensor (moving at the same direction), can be given as in, \[T_{cd}=\left\{ \begin{array}{ll}T&v_{s}=v,d_{x}\leq r\\ 0&v_{s}=v,d_{x}>r\\ 0&v_{s}>v,d_{x}\geq r\\ \frac{A_{s}C_{s}}{p-v_{s}}&v_{s}<v,d_{x}\geq r\\ \frac{A_{s}C_{s}-d_{x}}{p-v_{s}}&v_{s}>v,d_{x}<r\\ \frac{A_{s}C_{s}+d_{x}}{p-v_{s}}&v_{s}<v,d_{x}<r\end{array} \right.\]* The node and the UAV move at opposite direction (as shown in Figure 5). Similarly to the front case, the relative movement distance depends on the dynamic topology. Hence, it can be given by, \[d_{r}=\left\{ \begin{array}{cc}A_{s}B_{s}&d_{x}\geq r\\ A_{s}D_{s}&d_{x}<r\,.\end{array} \right.\] Then, the \(T_{cd}\) between the UAV and the sensor (moving at the opposite direction) can be calculated by, \[T_{cd}=\left\{ \begin{array}{cc}\frac{2A_{s}C_{d}}{p+b_{s}}&d_{x}\geq r\\ \frac{A_{s}C_{s}-d_{x}}{p+b_{s}}&d_{x}<r\,.\end{array} \right.\] (11)  
As shown in Figures 3-5, \(T_{cd}\) achieves the maximum when the UAV flies on the top of the node. This conclusion is corroborated in Equations (8), (10) and (11). The longer the \(T_{cd}\), the higher opportunity for the sensor to communicate with the UAV. Figure 4: Simple scenario with one UAV and one Sensor (Same direction). Figure 5: Simple scenario with one UAV and one Sensor (opposite direction). Figure 3: Simple scenario with one UAV and one Sensor (Static). 2. Curve Path When the trajectory of the UAV is a curve path (Figure 6), the contact duration is obtained through a beacon based prediction mechanism [31]. The midline of the pre-defined path is \(y=f(x)\) (Figure 7). At \(t_{k}\), the UAV sends a beacon message to its coverage, and \(S\) receives it and successfully sends the join message to the UAV.