## 4 UAV Path Planning with Cooperative Detection Using BLSSPIO
### Path-Planning Models  
* Models of the detection network:  
The detection network is designed to characterize communication quality and comprises a connectivity model and a communication reliability model. In this paper, the detection units represent the impact of different forbidden areas. Firstly, the concepts of the average degree, clustering coefficient, and path length are introduced to describe the connectivity model. The average degree, which represents the quantity of information interaction, can be determined via the following equation:  
\[\overline{d}=1/N_{net}\sum_{i=1}^{N_{net}}d^{i} \tag{22}\]  
where \(N_{net}\) is the detection units' number and \(d_{i}\) is the node degree of \(i\in\{1,2,\ldots,N_{net}\}\). The average clustering coefficient measures the redundancy of local information, and is given as follows:  
\[\overline{c}=1/N_{net}\sum_{i=1}^{N_{net}}c^{i} \tag{23}\]  
Figure 5: The coordination of UAVs path planning. where \(c^{i}=\frac{2\cdot c^{i}}{d^{i}\cdot(\vec{x}-1)}\), \(c^{i}\) is the edge number of unit \(i\). The path length indicates the maximum distance between the unit \(i\) and \(j\in\{1,2,\ldots,N_{net}\}\) in the network. It measures the transmission performance and efficiency between any two units, which is given by the following equation:  
\[\vec{l}^{i,j}=\max_{i\
eq j,i,j\in c^{i}}l^{i,j} \tag{24}\]  
where \(l^{i,j}\) represents the path length from the unit \(i\) to \(j\). Then, the connectivity between the two units is as follows:  
\[p^{i,j}=\vec{d}\cdot\varepsilon/\vec{l}^{i,j} \tag{25}\]  
In addition, the communication reliability model is formulated as an exponential function linked to the waypoint index:  
\[\xi=e^{(-\sigma\cdot\Delta)} \tag{26}\]  
where \(\sigma\in(0,1)\) and \(\Delta\) are the offset of the current waypoint index number and the formerly passed-through waypoint index, respectively. Therefore, the communication quality is determined by the following equation:  
\[q^{i,j}=p^{i,j}\cdot\xi \tag{27}\]  
* Detection modeling:  
Two types of detection units can communicate with each other to share information. For type I, the indicating probability of unit \(i\) at waypoint \(n\) of UAV \(m\) is calculated as follows:  
\[p^{i,m,n}_{A}=1-\prod_{i\
eq j,j=1}^{N_{pass}}\left(1-p^{i,m,n}_{r}\cdot q^{i, j}\right) \tag{28}\]  
where \(N_{pass}\) is the number of units passed. \(p^{i,m,n}_{r}\) is the detection probability of unit \(j\) at a waypoint \(n\) without the indicating probability, which is calculated as follows:  
\[p^{i,m,n}_{r}=1/\left(1+\left(r_{2}\cdot R^{4}_{j,m,n}\left/\sigma_{j,m,n} \right.\right)^{r_{1}}\right) \tag{29}\]  
where \(r_{1}\), \(r_{2}\) are the parameters of the detection unit, \(R_{j,m,n}\) is the distance between UAV \(m\) and detection unit \(j\) at the waypoint \(n\). \(\sigma_{j,m,n}\sim E(\mu)\) satisfies exponential distribution. The detection probability of all units of type I at waypoint \(n\) is expressed as follows:  
\[p^{m,n}_{detect}=1-\prod_{i=1}^{N_{tot}}\left(1-p^{i,m,n}_{SD}\right) \tag{30}\]  
where \(p^{i,m,n}_{SD}=1-\left(1-p^{i,m,n}_{r}\right)\cdot\left(1-p^{i,m,n}_{A}\right)\). Then, the detection cost of type I is calculated using the following equation:  
\[F^{m,n}_{D}=\left\{ \begin{array}{cc}10^{4}\cdot p^{m,n}_{detect}&\text{if }p^{m,n}_{detect}>p^{d}_{lim}\\ 0&\text{else}\end{array} \right. \tag{31}\]  
where \(p^{d}_{lim}\) is a detection threshold. For the type II, the maximum/minimum ranges are \(R^{k}_{max}\) and \(R^{k}_{min}\), then the detection possibility is calculated using the following equation:  
\[p^{k,m,n}_{t}=\left\{\exp\left\{-\left(\frac{2\left(R_{k,m,n}-\frac{R^{k}_{min}+R^ {k}_{max}}{2}\right)}{R^{k}_{max}-R^{k}_{min}}\right)^{2}\right\}\text{if}R^{k} _{min}\leq R_{k,m,n}\leq R^{k}_{max}\right. \tag{32}\] \[\left.0,\qquad\text{else}\right.\]  
where \(R_{k,m,n}\) is the distance between the UAV \(m\) and the center position of the unit \(k\) at the waypoint \(n\), \(R^{k}_{min}=1/2R^{k}_{max}\).