## 4 Cluster Path Planning Method That Resembles a Starling Flock  
### Dubins Aircraft Path  
The automobile path, with a height component added, serves as the foundation for the Dubins aircraft path. According to [PERSON]'s method [44], the path of the Dubins aircraft is separated into three categories based on the altitude difference between the beginning point and the ending position, the length of the Dubins car's path, and the maximum output pitch angle (\(\tilde{\gamma}\)) of the aircraft [45]. Further incorporated are many circumstances, including low, medium, and high ascents. The Formula (12) yields the Dubins aircraft's minimum turning radius:  
\[R_{\min}=\frac{V^{2}}{g}\tan\tilde{\phi} \tag{12}\]  
\(\tilde{\phi}\) is the maximum tilt angle of the drone, \(V\) is the Aircraft airspeed, \(g\) is the Gravitational acceleration. When the altitude difference \(\Delta h=|x_{de}-x_{ds}|\) between the starting point and the ending point satisfies the following formula, it is a low climb situation:  
\[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]  
where \(L_{car}(R_{\min})\) is the minimum Dubins car path, and the right-hand side of the equation is the maximum altitude increment for flight climb at the limiting flight pitch angle \(\tilde{\gamma}\). \(\Delta h\) satisfies the Equation (14) for the medium climbing case:  
\[L_{car}(R_{\min})\tan\tilde{\gamma}|x_{de}-x_{ds}|\leqslant[L_{car}(R_{\min})+ 2\pi R_{\min}]\tan\tilde{\gamma} \tag{14}\]  
The high climb case applies when at a climb height even greater than the minimum path plus a section to complete the circling \(2\pi R_{\min}\) ascent:  
\[R_{\min}=[\frac{V^{2}}{g}+2\pi R_{\min}]\tan\tilde{\phi} \tag{15}\]  
The Dubins aircraft trajectories are produced for each of the three scenarios in the following three subsections:  
1. Low Climbing Dobins Path: The height increase between the start and end points can be achieved by increasing the flight path to satisfy the necessary pitch angle within the limitations, provided that the required climb height does not exceed the minimum path's limited climb height. The optimal route can be determined using the following equation: \[\gamma^{*}=\tan^{-1}(\frac{|x_{de}-x_{ds}|}{L_{car}(R_{\min})})\] (16) The formula below determines the length of the shortest flight path. \[L_{air}(R_{\min},\gamma^{*})=\frac{L_{car}(R_{\min})}{\cos\gamma^{*}}\] (17)The steep climb of the Dubins flight route is based on the minimum path of the car with the inclusion of the optimal flight pitch angle \(\gamma^{*}\), together with the pitch angles \(\psi_{s}\) for the start point attitude and the pitch angles \(\psi_{e}\) for the termination point attitude. A Dubins airplane path with a medium height difference of 75 m and a Dubins automobile route distance of 185 m are depicted in Figure 9.
2. Mid Climb Dobins Path: In the case of a medium climb, the climb height is both greater than the model's maximum height restriction at the minimum car path distance, is insufficient to add a climb to the initial path for a full climb spiral, and is also higher than the termination point without adjusting the proper pitch angle. This means that the pitch angle can reach the termination point under the model restriction, and only a section of an arc can be added to the path. In order to increase the path length in this paper so that the climb height with pitch angle \(\gamma=\pm\tilde{\gamma}\) is exactly \(X_{de}-X_{ds}\), an additional section of path \(\gamma=sign(x_{de}-x_{ds})\tilde{\gamma}\) must be inserted. As indicated in Figure 10a, the arc is added after the starting helix if the final height is greater than the starting height. In Figure 10b, the arc is inserted before the end helix if the final height is lower than the starting height. Figure 10: **(a)** Arc insertion method one. **(b)** Arc insertion method two. Figure 9: **(a)** Low-altitude climb Dobbins aircraft path. **(b)** Top view of the path.