## 3 Description of the Integrated Improved Algorithm
### Artificial Potential Field Algorithm  
The artificial potential field method treats the environment in which the unmanned aerial vehicle operates as an artificial force field. The target exerts an attractive force on the UAV, creating a gravitational potential field, while obstacles generate repulsive forces, forming a repulsive potential field. The UAV moves towards the target under the combined effects of attraction and repulsion forces.  
The gravitational potential field between the target point and the unmanned aerial vehicle is defined as:  
\[U_{att}(X)=\frac{1}{2}K_{att}(x-x_{G})^{2} \tag{28}\]  
In the equation, \(K_{att}\) is the position gain coefficient, \(x\) represents the position of the unmanned aerial vehicle, and \(x_{G}\) represents the position of the target point. The attraction force between the UAV and the target point is the negative gradient of \(U_{att}(X)\) and can be expressed as:  
\[F_{att}(X)=-grad[K_{att}]=-K_{att}(x-x_{G}) \tag{29}\]  
The repulsive potential field generated by obstacles on the unmanned aerial vehicle can be expressed as:\[U_{rep}(X)=\begin{cases}\frac{1}{2}K_{rep}\Big{(}\frac{1}{x-x_{0}}-\frac{1}{\rho_{0 }}\Big{)}^{2}&,x-x_{0}\leq\rho_{0}\\ 0&,x-x_{0}>\rho_{0}\end{cases} \tag{30}\]  
In the equation, \(K_{rep}\) is the repulsion gain coefficient, \(x-x_{0}\) represents the distance between the UAV and the obstacle, and \(\rho_{0}\) represents the influence distance of the obstacle. The repulsion force between the UAV and the obstacle is the negative gradient of \(F_{rep}(X)\) and can be expressed as:  
\[F_{rep}(X)=-grad\big{[}K_{rep}\big{]}=\begin{cases}K_{rep}\Big{(}\frac{1}{x-x _{0}}-\frac{1}{\rho_{0}}\Big{)}\Big{(}\frac{1}{(x-x_{0})^{2}}\Big{)}&,x-x_{0} \leq\rho_{0}\\ 0&,x-x_{0}>\rho_{0}\end{cases} \tag{31}\]  
Once the attraction force from the target point to the unmanned aerial vehicle and the repulsive forces from various obstacles on the UAV are determined, the resultant force acting on the UAV can be obtained.  
\[F_{all}=F_{att}+\sum_{i=1}^{n}F_{rep\_i} \tag{32}\]  
In a two-dimensional environment, due to the constraints of the task environment dimensions, the APF typically follows a more direct path to reach the target point. However, in a three-dimensional environment, with the increase in dimensions, obstacles may have larger volumes or heights, while the influence range of the force field is limited. When the UAV is far from the obstacles, the repulsive force may not be sufficient to completely avoid the obstacles, resulting in the occurrence of a path crossing through obstacles. Moreover, due to the complexity of the shape and distribution of the obstacles, traditional force field functions may not accurately describe the influence of obstacles, leading to more possible flight directions and paths during the planning process, which increases the number of local minima. Therefore, the UAV may become trapped in a local minimum due to the characteristics of the force field, unable to find a better path. Figure 3 shows the effect of the traditional APF algorithm for three-dimensional path planning. It can be clearly seen that during path planning with the APF, the route crosses over the peaks.