## 3 Four-Dimensional Dynamic Cooperative Path Planning Algorithm
### Dynamic Priority RRT* Algorithm
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Center**} & **Radius** \\ \cline{2-4}
**Rader** & 1:(200, 80, 0) & 35 & \\  & 2:(100, 350, 5) & 35 & \\ \hline \multirow{4}{*}{**Missile**} & Center & Radius & Height \\ \cline{2-4}  & & (70, 170, 0) & 30 & 60 \\ \cline{2-4}  & \multirow{2}{*}{**Artillery**} & **Center** & **Radius** & **Height** \\ \cline{3-4}  & & 1:(300, 100, 0) & 30 & 40 \\  & & 2:(260, 180, 0) & 25 & 40 \\ \cline{3-4}  & \multirow{2}{*}{**No-fly Tower**} & **Center** & **Radius** & **Height** \\ \cline{3-4}  & & 1:(200, 290, 10) & 15 & 50 \\ \cline{3-4}  & & 2:(100, 275, 10) & 20 & 40 \\ \cline{3-4}  & \multirow{2}{*}{**Sudden moving**} & **Center** & \multirow{2}{*}{**Speed**} & **Moving direction** & **Appearing time** \\ \cline{3-4}  & & **(t = 4.7s)** & & Positive direction \\ \cline{3-4}  & \multirow{2}{*}{**threat**} & (126, 125, 0) & 3 m/s &
\begin{tabular}{c} **\#** \\ **of Y-axis** \\ \end{tabular} & t = 4.7 s \\ \cline{3-4}  & \multirow{2}{*}{**Sudden threats**} & **Center** & **Radius** & **Height** & **Appearing time** \\ \cline{3-4}  & & 1:(260, 205, 0) & 15 & 50 \\ \cline{3-4}  & & 2:(270, 215, 0) & 15 & 50 \\ \hline \multirow{4}{*}{**UAVs**} & **Start position** & **Goal position** & **Speed** \\ \cline{2-4}  & **UAV 1** & (20, 100, 40) & & \\ \cline{1-1} \cline{2-4}  & **UAV 2** & (20, 150, 40) & & \\ \cline{1-1} \cline{2-4}  & **UAV 3** & (20, 200, 40) & (380, 200, 50) & 15 m/s \\ \cline{1-1} \cline{2-4}  & **UAV 4** & (20, 250, 40) & & \\ \cline{1-1} \cline{2-4}  & **UAV 5** & (20, 300, 40) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Complicated Scenario. Figure 4: t = 4.5 s cooperative paths of UAVs, where **(a)** is the 3D view and **(b)** is the 2D view. Figure 5: t = 6 s cooperative paths of UAVs, where **(a)** is the 3D view and **(b)** is the 2D view. Figure 3: t = 3 s cooperative paths of UAVs, where **(a)** is the 3D view and **(b)** is the 2D view. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Expected 4D Path Length** & **Actual 4D Path Length** & **Deviation** \\ \hline UAV 1 & 243.36 & 244.85 & 0.61\% \\ UAV 2 & 226.49 & 226.37 & 0.05\% \\ UAV 3 & 244.42 & 244.42 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The deviation ratio between the actual path length obtained by the 4D cooperative path algorithm and \(l_{\text{co}}\). Figure 6: t = 7.59 s cooperative paths of UAVs, where (**a**) is the 3D view and (**b**) is the 2D view. Figure 7: 4D cooperative paths of UAVs, where (**a**) is the 3D view and (**b**) is the 2D view. Based on the above simulation results and analysis, the proposed algorithm is able to realize the planning of a 4D dynamic cooperative path of all UAVs in the Normal Scenario, thus proving the effectiveness of the proposed algorithm. ### Scenario 2: Complicated Scenario

In Scenario 2, the relevant parameters of the HAPF are set as follows: the attractive potential field coefficient is \(k_{att}\)= 8 and the repulsive potential field coefficient is \(k_{rep}\)= 30. The influence distance of obstacle threat is \(\rho_{0}=20\) m, the unit time step is \(t=0.01\) s, and the adjustment factor is \(n=0.7\). The maximum steering angle is \(\phi_{\max}=60^{\circ}\) and the maximum climbing/diving angle is \(\gamma_{\max}=35^{\circ}\). It can be seen in Table 2 that we have set up two kinds of sudden threats at different times. The first one is a mobile threat and the second one is a static threat group simulating a local minimum trap. Figures 8-10 show the pre-planned 4D coordination path and the satisfaction of 4D coordination constraints of the path when t = 0 s. Figure 11 shows the local collision avoidance planning path of some UAVs and the 4D cooperative replanning path of all UAVs after the first sudden moving threat. Figures 12 and 13 show the satisfaction of 4D cooperative constraints of the pre-planned path based on the current planning space.