## 5 Verification by Simulations  
Let M1, M2, M3, and M4 denote four missiles participating in cooperative operations. The target is in the horizontal plane, and its initial position is at the origin of the coordinates. Table 1 lists the initial parameters of the proposed virtual leader missile (denoted as M) and follower missiles. Let us consider a case where a stationary target, a straight moving target, and a snake-like maneuvering target is attacked (specific movement of the target is shown in Table 2). The control coefficients of the augmented proportional guidance method adopted by the five missiles are \(K_{01}=K_{i1}=10\), \(K_{02}=5\); the parameters are \(c_{1}=0.7\), \(c_{2}=0.9\); the bandwidth of the fast dynamics sub-system is \(K_{\psi}=5\); and the overload limit of the missile is 8 g. The impact time during the attack on a fixed target is shown in Table 3. According to the above design strategy, under the premise of adopting the augmented proportional guidance method, the impact time of the leader missile is greater than that of each follower missile. Therefore, the use of this scheme can provide a sufficient flight time adjustment margin for the follower missiles. In the adjustable range, in order to enhance the ability of attack and damage on moving targets in a horizontal plane, the ideal impact angles (\(\theta^{*}_{w},\psi^{*}_{vm}\)) of M1, M2, M3, and M4 are designated as \((50^{\circ},-20^{\circ})\), \((44^{\circ},-34^{\circ})\), \((-30^{\circ},0^{\circ})\), and \((35^{\circ},-20^{\circ})\), respectively. The ideal impact time \(t^{*}\) is determined based on the flight time of the leader missile attacking different targets. The simulation results of cooperative attack by missiles on targets 1, 2, and 3 are shown in Figures 7-11. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Name** & \multicolumn{3}{c}{**Initial Position**} & \multicolumn{2}{c}{**Velocity**} & **Ballistic Angle** & **Ballistic Drift Angle** \\ \hline
**Variable** & \(x\) & \(y\) & \(z\) & **Vm** & \(\theta\) & \(\psi\) \\ \hline
**Unit** & **m** & **m** & **m** & **m/s** & (\({}^{\circ}\)) & (\({}^{\circ}\)) \\ \hline Virtual Pilot Missile & 4000 & 1800 & 3000 & 133 & 45 & 45 \\  & M1 & \(-3000\) & 1500 & \(-3800\) & 155 & 45 & 45 \\ Follow & M2 & \(-3500\) & 1200 & \(-2500\) & 140 & 40 & 40 \\ Missile & M3 & 3500 & 800 & 2000 & 125 & 10 & 30 \\  & M4 & \(-3500\) & 1600 & 2500 & 140 & \(-15\) & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial parameters of the virtual leader and the follower missiles. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & \multicolumn{3}{c}{**Target Motion State**} \\ \hline Target 1 & \multicolumn{3}{c}{Static target} \\ Target 2 & \(v_{t}=10\) m/s & \(a_{t}=0\) m/s\({}^{2}\) & \(\psi_{vt}=45^{\circ}\) \\ Target 3 & \(v_{t}=10\) m/s & \(y_{t}=60\sin(x_{t}/100)\) & \(K_{01}=K_{i1}=10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial parameters of the virtual leader missile and follower missiles. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Target Type**} & \multicolumn{3}{c}{**Virtual Pilot Missile**} & \multicolumn{3}{c}{**Follow Missile**} \\ \cline{2-7}  & **Unit** & **M** & **M1** & **M2** & **M3** & **M4** \\ \hline Target 1 & s & 41.66 & 33.92 & 32.85 & 33.25 & 32.93 \\ Target 2 & s & 39.48 & 41.36 & 36.58 & 33.39 & 36.54 \\ Target 3 & s & 39.82 & 39.95 & 34.67 & 34.01 & 35.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Impact time of APN guidance. Figure 8: Flight trajectory curve of the cooperative attack on target 2. Figure 10: Projection of flight trajectory curve of cooperative attack on target 3 in the XOZ plane. Figure 7: Flight trajectory curve of the cooperative attack on stationary target 1. Figure 9: Flight trajectory curve of the cooperative attack on target 3. As shown in Figures 7-9, the cooperative strategy based on the tracking of the missile-target distance could well constrain the missile-target distance during the flight of multiple projectiles.