## 3 Mixed-Strategy Harris Hawk Optimization
### Gaussian Random Walk Strategy
\[ppc(t)=\frac{\sum_{i=1}^{N}X_{ij}(t)}{dim} \tag{26}\]3

where \(\varepsilon\) represents the fuel consumption per unit flight distance, measuring the fuel needed for a UAV to fly a special distance. \(Tf\) is the total fuel carried by the UAV, determining the maximum distance it can fly without refueling. The entire flight path is divided into \(n\) segments, and \(L_{j}\) denotes the length of the jth flight leg. Considering that the probability of the airframe being affected by low temperature is directly related to the flight altitude, while the risk of terrain threat is inversely proportional to the flight altitude, the maximum flight altitude, hmax, and the minimum flight altitude, hmin, are set to ensure UAV safety. Assuming the current flight altitude is \(h_{j}\), flying altitude cost function can be expressed as follows. \[f_{j}= \begin{cases}\frac{h_{j}-h_{min}}{h_{max}-h_{min}}&h_{min}<h_{j}<h_{max} \\ \infty&\text{others}\end{cases}  \tag{32}\]

\[CF_{altitude}=\frac{\sum_{j=1}^{n}f_{j}}{n} \tag{33}\]

Taking into account factors such as environmental terrain, extreme weather, and military operations, the cost associated with flight risk becomes a crucial consideration in the aviation domain. There are \(k\) flying risk points. The entire flight path is divided into n segments, each of which is divided into m small segments. The flying risk cost function associated with flying in segment \(L_{j}\) can be defined as in (34). The total risk cost of UAV trajectory planning can be expressed as in (35). \[f_{nj,r}=\frac{\sum_{i=1}^{m}p_{r}\times d_{r,i}}{m} \tag{34}\]

\[CF_{risk}=\frac{\sum_{j=1}^{n}\sum_{r=1}^{k}f_{nj,r}}{k\times n} \tag{35}\]

where \(n_{j}\) is the jth segment of the trajectory \(L_{j}\) in the trajectory planning, \(p_{r}\) is the probability of UAV damage at the \(r\)th risk point, and \(d_{r,i}\) is the distance from the rth risk point to the \(i\)th sub-segment of the trajectory segment \(L_{j}\). When a UAV is flying, it is necessary to comprehensively consider the cost of flight distance, the cost of flight altitude, and the cost of flight risk as three key factors. Each cost function has different impacts on the flight trajectory. In order to effectively balance these factors, weight coefficients can be set for them, denoted as \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) such that \(\omega_{1}+\omega_{2}+\omega_{3}=1\). The objective function for UAV path planning can be expressed as follows. \[\text{CF}=\omega_{1}\times\text{CF}_{\text{distance}}+\omega_{2}\times\text{ CF}_{\text{altitude}}+\omega_{3}\times\text{CF}_{\text{risk}} \tag{36}\]

#### 5.1.2 UAV Path Planning Simulation and Analysis

To verify the effectiveness of MSHHO, a simulation of the three-dimensional flight trajectory of a UAV was conducted in Matlab. The simulation space was 150 km \(\times\) 100 km \(\times\) 3 km, with starting coordinates at (10, 90) and ending coordinates at (130, 10). The position information of risk areas is shown in Table 6 and marked in blue in the simulation space, as seen in Figure 11. The population size \(N\) was set to 30, and the maximum number of iterations \(T\) was set to 500. HHO and MSHHO algorithms were separately used to solve the three-dimensional path planning problem for UAVs. The two-dimensional flight trajectory curves of HHO and MSHHO are shown in Figure 12.

The comparison of HHO and MSHHO in Figure 12 clearly shows that the HHO algorithm plans a longer flight distance, consuming more fuel, bringing the route closer to risk areas, and posing potential risks. Moreover, the flight curve is not smooth, as there are obvious turns. In contrast, MSHHO achieves a shorter flight distance and smoother trajectory while avoiding risk areas, demonstrating more reasonable and safer flight paths. Additionally, MSHHO allows UAVs to fly closer to mountains, reducing risks associated with high-altitude low temperature and hypoxia.