## 4 Cluster Path Planning Method That Resembles a Starling Flock
### Dubins Aircraft Path
The intermediate arc is parameterized as shown in Figure 11 to demonstrate how the Dubins path can be obtained for modest climb heights, where the expression for \(z_{i}\) is given by Equation:  
\[z_{i}=c_{s}+R(\varphi)(z_{s}-c_{s}) \tag{18}\]  
On the basis of the discovered position \((z_{i},\psi_{s}+\varphi)\), a typical Dubins aircraft path is then plotted to the endpoint. The path is a distance from:  
\[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]0  
The pitch angle \(\varphi^{*}\) is determined using the binary search algorithm, which produces:  
\[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]1  
The following equation yields the corresponding Dubins aircraft path distance:  
\[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]2  
The parameters required for modeling the planning of the Dubins aircraft path for the medium climb case are shown in \[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]3 The intermediate parameters introduced are \(c_{i},\psi_{i},\lambda_{i},w_{i},q_{i}\), so the parameters of the Dubins aircraft path can be defined as:  
\[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]4  
A Dubins airplane path with a medium altitude variation of 175 m and a Dubins automobile route distance of 185 m are depicted in \[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]5  
\[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]6 Illustration of the criteria for the medium-height climbing Dobbins path. 3. High Climbing Dobins Path: States within the model limitations cannot achieve height increments for starting or ending locations with significant height variations relative to the car path. To achieve the height increase, the Dubins car path can be extended by adding a specified number of spiral revolutions and increasing the turning radius at the beginning or end of the path. The minimum number of spiral revolutions required, if any, is the lowest integer \(k\) that satisfies the following conditions: \[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]7 (23) The formula for \(k\) is given by \[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]8 (24) The formula \(\lfloor x\rfloor\) represents a function that rounds \(x\) down to the nearest integer and then increases the radius of the beginning and ending spiral to \(R^{*}\), achieving: \[|x_{de}-x_{ds}|\leqslant L_{car}(R_{\min})\tan\tilde{\gamma} \tag{13}\]9 (25) To create a query in \(R^{*}\) that answers Equation (25), use a tiling search. The resulting path, which is the ideal Dobins plane path, has the following formula for path length: \[L_{car}(R_{\min})\tan\tilde{\gamma}|x_{de}-x_{ds}|\leqslant[L_{car}(R_{\min})+ 2\pi R_{\min}]\tan\tilde{\gamma} \tag{14}\]0 (26) The path of the automobile is 185 m, the height difference is 400 m, and Figure 13 shows the course of the high-climbing Dubins aircraft. The minimum radius \(R_{\min}\) is swapped out for \(R^{*}\) when defining the high-rise Dubins aircraft path, which otherwise shares the same specifications as the Dubins automobile path. Increase the optimal flight path angle \(\pm\bar{\gamma}\), the starting point flight pitch angle \(\psi_{S}\), the terminal point flight pitch angle \(\psi_{e}\), and the quantity of spiral spin \(k\) following the initial spiral. This section explains how Dubins vehicle routes can be used for aircraft. Planning outcomes for the UAV path can be improved by switching the RRT algorithm's path generation approach to the Dubins aircraft path. \[L_{car}(R_{\min})\tan\tilde{\gamma}|x_{de}-x_{ds}|\leqslant[L_{car}(R_{\min})+ 2\pi R_{\min}]\tan\tilde{\gamma} \tag{14}\]1 (**a**) Mid-altitude climbing Dobins aircraft path. (**b**) Top view of the path.