## 3 Decision Mechanism for Collaborative Search
### Path-Planning Algorithm  
As shown in Figure 5, to balance the long-term benefits of UAV search, each decision made by the UAV determines two waypoints, which are located in two concentric annular regions centered on the UAV with different radii. The grids marked in red and blue  
Figure 4: Calculation of yaw cost. represent the candidate grids for the first and second waypoints, respectively, denoted here by the sets \(\mathbf{G}_{wp1}\) and \(\mathbf{G}_{wp2}\). The elements of these sets satisfy the following conditions:  
\[\left\{ \begin{aligned} R_{wp1}^{\text{min}}&\leq D_{wp1 }^{i}\leq R_{wp1}^{\text{max}}\\ R_{wp2}^{\text{min}}&\leq D_{wp2}^{i}\leq R_{wp2}^{ \text{max}}\end{aligned} \right. \tag{16}\]  
where \(D_{wp1}^{i}\) and \(D_{wp2}^{i}\) represent the distances from the centers of the elements in \(\mathbf{G}_{wp1}\) and \(\mathbf{G}_{wp2}\) to the current position of the UAV, respectively. For ease of subsequent analysis, \(R_{wp1}^{\text{min}}\) and \(R_{wp1}^{\text{max}}\) are expressed in units of grid width \(\Delta L\), and the following convention is established: if \(n\geq 2\) and \(n\in\mathbb{Z}\), \(\Delta n\in(0,1)\), \(R_{wp1}^{\text{min}}=(n-\Delta n)\cdot\Delta L\), \(R_{wp1}^{\text{max}}=(n+\Delta n)\cdot\Delta L\), and \(D_{wpj}^{i}\) satisfies Equation (16), then the fuzzy definition \(D_{wpj}^{i}=n\cdot\Delta L\) is considered valid. Under this definition, the candidate grids for the first waypoint are distributed as a single layer of grids around the UAV, and \(n\) represents the multiple of the grid width corresponding to the distance from the UAV. By using the index values \(g_{1,i}^{t}\) and \(g_{2,i}^{t}\) of \(\mathbf{G}_{wp1}\) and \(\mathbf{G}_{wp2}\) as the coordinates of particle \(i\) at time \(t\) in the PSO algorithm, denoted as \(\mathbf{x}_{i}^{t}=\left(g_{1,i}^{t},g_{2,i}^{t}\right)\), the solution space can be represented as:  
\[\mathbf{S}=\left\{\left(g_{1,i}^{t},g_{2,i}^{t}\right)\left|g_{1,i}^{t}\in\left[1, N\left(\mathbf{G}_{1}^{wp}\right)\right]\cap\mathbb{N},g_{2,i}^{t}\in\left[1,N \left(\mathbf{G}_{2}^{wp}\right)\right]\cap\mathbb{N}\right.\right\}, \tag{17}\]  
where \(\mathbb{N}\) represents the set of natural numbers. By using the index values of elements in set \(\mathbf{S}\) as the particle coordinates in the PSO algorithm and integrating the path grid determination algorithm, the optimal planned path can be obtained via the optimization of waypoint selection. The PSO algorithm is expressed as:  
\[v_{i}^{t+1}=w\cdot v_{i}^{t}+c_{1}\cdot r_{1}\cdot\left(x_{i}^{best}-x_{i}^{ t}\right)+c_{2}\cdot r_{2}\cdot\left(x_{g}^{best}-x_{i}^{t}\right), \tag{18}\]  
\[x_{i}^{t+1}=x_{i}^{t}+v_{i}^{t+1}, \tag{19}\]  
where \(v_{i}^{t}\) and \(x_{i}^{t}\) represent the velocity and position of particle \(i\) at time \(t\), respectively. The parameter \(w\) denotes the inertia weight, \(c_{1}\) and \(c_{2}\) are the learning factors for the individual historical best solution and the global historical best solution, respectively, and \(r_{1}\) and  
Figure 5: Schematic diagram of the selection range for two waypoints. are random numbers. Additionally, \(x_{i}^{best}\) and \(x_{g}^{best}\) correspond to the individual historical best solution and the global historical best solution, respectively. The fitness function of all particles should be expressed as:  
\[F_{i}=1-\frac{1}{4}\left(3 Cost_{path}+Cost_{turn}\right). \tag{20}\]  
Based on Equations (18-20), the UAV can complete a waypoint-planning decision. At this point, we have transformed the path-planning problem into a multi-objective optimization problem suitable for the PSO algorithm. In this paper, the number of particles is 25, the number of iterations is 15, and the learning factor and inertia weight are both set to 0.5. We stop iterating when the relative change in fitness function is less than 5%. Each UAV initiates intelligence exchange and situation information updates with other UAVs only after reaching its first waypoint, followed by path replanning.