## 2 Research on UAV Post-Disaster Path Planning Problem
### Fitness Function  
During the research of UAV path planning using optimization algorithms, the quality of the planned path can be measured by setting a fitness function. In this paper, considering the actual flight conditions of the UAV and safety concerns, a fitness function is constructed by taking into account three aspects: path length, obstacle threat, and ground collision cost. A lower fitness value indicates a better generated path. 1. Cost of path length This study selects the minimization of path length as a criterion to determine the optimality of the path [35]. As UAVs are controlled by ground control stations, the flight path \(C_{i}\) of the UAV is represented as a list composed of \(n\) waypoints, where each waypoint corresponds to a path node in the path planning search map. Assuming there are \(i\) flight paths and \(j\) path nodes, and each node's coordinates can be represented as \(P_{ij}(x_{ij},y_{ij},z_{ij})\), the calculation of the cost \(T_{1}\) of path length can be expressed as: \[T_{1}(C_{i})=\sum_{j=1}^{n-1}\xrightarrow[P_{i,j}P_{i,j+1}]\] (4)
2. Navigation altitude cost The smoothness of the path is determined by the UAV's stable flight. A stable flight altitude not only enables the planning of smooth flight paths but also helps save fuel, alleviating the burden on the UAV control system to some extent [36]. Furthermore, in practical UAV applications, the flight altitude is often restricted between two given extremes. It is necessary to cruise at an appropriate altitude to meet requirements such as search operations and payload delivery. Assuming the minimum and maximum altitudes are \(h_{\min}\) and \(h_{\max}\), respectively, the cost \(T_{2}\) of flight altitude can be expressed as: \[T_{2}(C_{i})=\sum_{j=1}^{n}H_{ij}\] (5) \[H_{ij}= \begin{cases}\left(h_{\min}-h_{ij}\right),\ h_{ij}<h_{\min}\\ 0,\ h_{\min}\leq h_{ij}\leq h_{\max}\\ \left(h_{ij}-h_{\max}\right),\ h_{ij}>h_{\max}\end{cases} \] (6) In the equation, \(h\) represents the flight altitude relative to the ground. 3. Ground collision risk During the operation of UAVs, any issues such as explosions can affect the safety of the life and property of ground personnel. Therefore, it is necessary to use ground collision \begin{table}
\begin{tabular}{c c c c} \hline
**Number** & **Center** & **Height** & **Range** \\ \hline
1 & (20, 20) & 45 & \\
2 & (39, 29) & 45 & \\
3 & (30, 30) & 50 & \\
4 & (16, 40) & 55 & \\
5 & (61, 43) & 61 & (10, 10, 5) \\
6 & (81, 51) & 60 & \\
7 & (45, 55) & 60 & \\
8 & (36, 61) & 50 & \\
9 & (70, 70) & 60 & \\
10 & (88, 92) & 65 & \\ \hline \end{tabular}
\end{table} 
Table 2: Obstacle parameter settings. risk \(T_{3}\) to determine the impact of UAV accidents on ground personnel. The objective function is as follows:  
\[T_{3}=N_{r}D \tag{7}\]  
In the equation, \(N_{r}\) represents the number of injured personnel after the UAV collides with the ground, and \(D\) is the fatality rate resulting from the collision. For \(N_{r}\) and \(D\), they can be obtained using Equations (7), (8), (9) and (10) respectively. \[N_{r}=A_{r}\rho \tag{8}\]  
\[A_{r}=W_{UAV}\sqrt{L_{UAV}^{2}+H_{UAV}^{2}} \tag{9}\]  
In the equation, \(\rho\) represents the population density, and \(A_{r}\) represents the coverage area of the accident, which is determined by the UAV's wingspan \(W_{UAV}\), length \(L_{UAV}\), and height \(H_{UAV}\). This article evaluates the impact mortality rate d based on the logistic growth variable model proposed by [PERSON] [37].