## 5 Verification by Simulations
As a result, the follower missiles converged to the same value as the leader missile. The MPC-based cooperative guidance algorithm optimizes the trajectory to meet the terminal impact angle constraints, based on the impact time of the cooperative missile group. From Figures 10 and 11, it can be observed that the MPC-based cooperative guidance law can make multiple missiles participating in cooperative operations hit the target with separately specified impact angles at a specified time. The miss distance meets the requirements and is less than that of the cooperative time guidance. Additionally, the strike effect is more accurate. Due to the limited adjustable time and angle of the missiles under different position conditions, the omni-directional saturation attack of the multiple missiles can be realized through reasonable state estimation, mission planning, and assignment of multiple terminal guidance missiles. The simulation results verified the effectiveness of this method in attacking the stationary and maneuvering targets. From Figures 7-10, it can be observed that the trajectory curve optimized by the MPC algorithm is smoother and easier to implement. By tracking the distance between the missile and the target, each follower missile achieved a cooperative impact time with the virtual leader missile. When hitting target 1, target 2, and target 3, the cooperative impact time of the four missiles was the same as the impact time of the virtual lead missile, which was 41.66 s, 41.28 s, and 41.82 s, respectively. However, the impact angle did not meet the requirements at this time, and the iterative optimization of the guidance instructions through the MPC algorithm achieved the impact angle constraints at the terminal. In the simulation example, the impact angle constrained by each follower missile and the terminal impact angle are given by Figures 12-14, when different guidance methods were used to attack an unfixed target. Figure 11: Projection of flight trajectory curve of cooperative attack on target 3 in the XOY plane. Figure 12: Velocity-inclination change curve of each follower missile during the cooperative attack on target 1 based on the MPC method: (**a**) Trajectory deflection curve; (**b**) Trajectory inclination curve. Figures 12-14 show the changing curves of the impact angle when attacking a stationary target, a uniformly moving target, and a maneuvering target by the MPC method. Using the MPC method can not only realize the cooperative time but can also satisfy the impact angle constraints. Both the time cooperative guidance and the MPC-based guidance can make four missiles hit the target at the same time, but the former could only constrain the impact time of four missiles, while the latter achieved a multi-directional cooperative attack with designated angles on the target. This verified the superior guidance performance of the MPC-based 3D cooperative guidance law with multiple constraints. Figure 15 shows the tracking curve of the missile-target distance in the time cooperative strategy with the designed parameters. Figure 16 shows the curve of the remaining time error with the time obtained by the MPC method. Tracking the missile-target distance of the leader missile achieves a cooperative impact time between the follower missiles and the leader missile, but it does not meet the impact angle constraints. Therefore, the MPC method was used to iteratively optimize the control commands to meet the impact angle constraints. Figure 14: Velocity-inclination change curve of each follower missiles during the cooperative attack on target 3 based on the MPC method: **(a)** Trajectory deflection curve; **(b)** Trajectory inclination curve. Figure 13: Velocity-inclination change curve of each follower missiles during cooperative attack on target 2 based on the MPC method: **(a)** Trajectory deflection curve; **(b)** Trajectory inclination curve. Figure 15: Tracking of the missile-target distance.