# Dynamic UAV Inspection Boosted by Vehicle Collaboration Under Harsh Conditions in the IoT Realm
## 3 Problem Formulation
### Initial Static Task Model
#### 3.1.2 UAV Flight Route Planning Model  
In the multi-UAV-vehicle cooperative data acquisition problem studied in this paper, after the vehicle transports the multi-UAVs to a designated docking anchor point, the UAVs will disperse and take off to their respective mission points for data acquisition. The single-trip flight route planning problem for each UAV is essentially a closed-loop Traveling Salesman Problem (TSP). Specifically, the UAVs take off from the vehicle anchor point and return to the vehicle anchor point after visiting each task point, either to recharge for the next trip or dock and wait for the vehicle to travel to the next anchor point. The distance between any two mission points \(p_{i}\) and \(p_{j}\) is denoted as \(d_{p_{i},p_{j}}\). The time for UAV \(u_{k}\) to fly from task node \(p_{i}\) to \(p_{j}\) is defined as \(t_{p_{i},p_{j}}^{u_{k}}\), \(t_{p_{i},p_{j}}^{u_{k}}\) is calculated as follows:  
\[t_{p_{i},p_{j}}^{u_{k}}=\frac{d_{p_{i},p_{j}}}{v_{uav}} \tag{4}\]  
In a single-trip flight of a single UAV \(u_{k}\), it is assumed that the set of data collection task points it needs to visit is \(Q=\{q_{1},q_{2},\ldots,q_{s}\}\), where \(Q\subseteq P\). The anchor point for takeoff is denoted as \(a_{q}\), and, based on this set of task points, the flight route of UAV \(u_{k}\) is generated. A particular single-trip flight route is denoted as \(r_{i}=\{a_{q},q_{1},q_{2},\ldots,q_{n},a_{q}\}\), where \(a_{q}\in A_{c}\) and \((q_{1},\ldots,q_{n}\in Q)\). The route \(r_{i}\) indicates that the UAV takes off from the anchor point \(a_{q}\), sequentially visits the mission points \(q_{1}\) to \(q_{n}\), and, finally, returns to the anchor point \(a_{q}\). Let us denote the number of UAVs as \(M_{\text{uav}}\). For all \(M_{\text{uav}}\) UAVs, we compile their flight routes into a set of UAV flight routes \(R_{\text{uav}}=\{r_{1},r_{2},\ldots,r_{n_{r}}\}\). Therefore, the set of global routes in the multi-UAV-vehicle coordination problem is \(R=\{r_{0},r_{1},r_{2},\ldots,r_{n_{r}}\}\). To determine if UAV \(u_{k}\) flies from task node \(q_{i}\) to \(q_{j}\) in flight route \(r_{i}\), we use the auxiliary decision variable \(y_{r_{i}}(q_{i},q_{j})\) to represent it. If this is the case, then \(y_{r_{i}}(q_{i},q_{j})=1\); otherwise, \(y_{r_{i}}(q_{i},q_{j})=0\). Accordingly, the length \(l_{r_{i}}\) of the flight route \(r_{i}\) can be calculated by Equation (5). The time \(t_{a_{q_{i}}}^{u_{k}}\) for UAV \(u_{k}\) to make a single trip at the anchor point \(a_{q}\) following the flight route \(r_{i}\) is calculated by Equation (6). \[l_{r_{i}}=\sum_{q_{i}\in Q}\sum_{q_{j}\in Q}y_{r_{i}}(q_{i},q_{j})d_{q_{i},q_{ j}} \tag{5}\]  
\[t_{a_{q_{i}}}^{u_{k}}=\sum_{i=0}^{n-1}t_{q_{i},q_{i+1}}^{u_{k}}+t_{q_{n},q_{0} }^{u_{k}}=\frac{l_{r_{i}}}{v_{uav}} \tag{6}\]  
Due to the limited range and power of the UAV, only a few mission data points can be accessed for execution in a single flight. Therefore, UAV \(u_{k}\) may have multiple trips at the anchor point \(a_{q}\). Based on the UAV's data collection task points, the center console will run the assigned algorithm to calculate the UAV route and the number of flight trips and then send the corresponding commands. Alternatively, a binary variable \(\text{z}(u_{k},r_{i})\) is used to indicate whether route \(r_{i}\) is assigned to UAV \(u_{k}\). Specifically, if the route is assigned, then \(\text{z}(u_{k},r_{i})=1\); otherwise, \(\text{z}(u_{k},r_{i})=0\). In the previous subsection, the variable \(x(a_{i},r_{0})\) was used to indicate whether the candidate anchor point \(a_{i}\) was selected as a stopping anchor point and included in the route \(r_{0}\). In this subsection, the auxiliary decision variable \(x(a_{q_{t}},r_{i})\) is used to indicate whether the docking anchor point \(a_{q}\) is included in the single-trip route \(r_{i}\) of a particular UAV. If it is included, then \(x(a_{q_{t}},r_{i})=1\); otherwise, \(x(a_{q_{t}},r_{i})=0\).