## 4 UAV Path Planning with Cooperative Detection Using BLSSPIO
### Path-Planning Models
Combined with the detection probability of the type I, the comprehensive probability is  
\[p^{m,n}_{K}=p^{m,n}_{t}\cdot p^{m,n}_{detect} \tag{33}\]  
where \(p^{m,n}_{t}=1-\prod\limits_{k=1}^{N_{ministic}}\left(1-p^{k,m,n}_{t}\right)\). Then, the detection cost of type II is calculated as follows:  
\[F^{m,n}_{K}=\left\{ \begin{matrix}10^{4}\cdot p^{m,n}_{K},&\text{if}\ p^{m,n}_{ K}>p^{k}_{lim}\\ 0&\text{else}\end{matrix} \right. \tag{34}\]  
Finally, the total costs of detection units are given as follows:  
\[F^{m}_{Threat}=\sum\limits_{n=1}^{N}\left(F^{m,n}_{D}+F^{m,n}_{K}\right) \tag{35}\]  
* Mountains cost:  
The mountains are represented obstacles, which can be simplified as the trapezoidal cylinders. The mountain cost of UAV \(m\) is determined as follows:  
\[F^{m,n}_{o}=\left\{ \begin{matrix}10^{4}\cdot\frac{1}{|\mathbf{P}^{\text{in},T} -\mathbf{P}^{\text{e}}|},&\text{if}\ h<h^{o}\ \text{and}\ \|\mathbf{P}^{m,n}-\mathbf{P}^{\text{e}}\|\leq \left(1-\frac{h}{P^{\text{e}}}\right)\cdot R^{o}_{a}+\frac{h}{P^{\text{e}}}R ^{o}_{b}\\ 0&\text{else}\end{matrix} \right. \tag{36}\]  
where \(\mathbf{P}^{o}\) denotes the two-dimensional global position and \(h^{o}\) represents the mountain's height. \(R^{o}_{a}\) and \(R^{o}_{b}\) (\(R^{o}_{a}>R^{o}_{b}\)) are the bottom and top radius of the mountain surface, respectively. Then, the mountain cost is calculated using the following equation:  
\[F^{m}_{mount}=\sum\limits_{o=1}^{N_{mount}}\sum\limits_{n=1}^{N}\left(F^{m,n} _{o}\right) \tag{37}\]  
* Coordination costs:  
The coordination cost ensures that the UAVs can communicate with each other without internal collision. Given that the safety distance is \(d_{safe}\) and communication range is \(r_{commun}\), suppose that the neighbor of the UAV \(m\) is labeled \(\widetilde{m}\in\left\{1,2,\ldots,M,\widetilde{m}\
eq m\right\}\). Then, the coordination cost is calculated as follows:  
\[F^{m}_{cour}=\left\{ \begin{matrix}0,&\text{if}\ d_{safe}\leq\left\|p^{m,n}-p ^{\widetilde{m},n}\right\|\leq r_{commun}\\ &10^{5},\text{else}\end{matrix} \right. \tag{38}\]In summary, the total costs combined with the path length can be described as shown below:  
\[F_{cost}^{m}=F_{Tluent}^{m}+F_{coor}^{m}+F_{path}^{m} \tag{39}\]  
where \(F_{path}^{m}\) is the length of the candidate path. * Flight constraints:  
The velocity and time stamp are influenced by flight constraints, such as the maximal voyage \(L_{max}\), the minimal step length \(S_{min}\), the airspeed range \([V_{min},V_{max}]\), and the minimum turning radius \(R_{min}\). The straight-line distance between the starting and the ending points is \(L_{m}\), then the time stamps satisfy the following equation:  
\[T_{step}=\cap_{m=1}^{M}[T_{min}^{m},T_{max}^{m}] \tag{40}\]  
\[T_{min}^{m}=\max\biggl{\{}\frac{S_{min}}{V_{max}},\frac{L^{m}}{V_{max}\cdot N} \biggr{\}}T_{max}^{m}=\min\biggl{\{}\frac{L_{max}}{V_{min}\cdot N^{\prime}}, \frac{L^{m}}{V_{min}\cdot N},\frac{T_{max}}{N}\biggr{\}} \tag{41}\]  
where \(T_{max}\) is the longest endurance time and \(\cap\) denotes the intersection. The common arrival time is given by the following equation:  
\[T_{end}=N\cdot T_{step}+T_{start} \tag{42}\]  
The maximum yaw angle is expressed as follows:  
\[\psi_{max}=\arcsin\frac{V_{min}\cdot T_{step}}{2\cdot R_{min}} \tag{43}\]  
In the local frame \(X_{m}O_{m}Y_{m}\), we have \(\overline{V}_{x}^{m,n}=L^{m}/\left(N\cdot T_{step}\right)\), and then the BSLSPIO is used to generate the optimal solution \(\overline{V}_{y}^{m,n}\in\left[-\overline{V}_{x}^{m,n}\cdot\tan\psi_{\max}, \overline{V}_{x}^{m,n}\cdot\tan\psi_{\max}\right]\).