## 5 Simulation
### Validity Verification  
In order to verify the effective generation of the re-entry trajectory under constraints by the proposed method, three cases are selected for verification. The initial states of the vehicle are the same in all three cases. \([r_{0},\phi_{0},\varphi_{0},v_{0},\theta_{0},\psi_{0}]=[(\text{R}_{e}+70\text { km})/R_{e},0^{\circ},0^{\circ},6000/v_{c},0^{\circ},60^{\circ}]\) when the terminal states are also the same \([r_{f},\phi_{f},\varphi_{f},v_{f}]=[(R_{e}+30\text{ km})/R_{e},30^{\circ},25^{ \circ},2000/v_{c}]\). In the three cases, the vehicle will have a flight with the no no-fly zone, flying- around the no-fly zones and passing through the no-fly zones in three cases. In case 2,3, the no-fly zones are set as follows:  
\[\text{Case 2}:\ [\phi_{b1},\varphi_{b1},r_{b1}]=[15^{\circ},8.5^{\circ},2^{ \circ}],\ [\phi_{b1},\varphi_{b1},r_{b1}]=[24^{\circ},16^{\circ},2^{\circ}]\]  
\[\text{Case 3}:\ [\phi_{b1},\varphi_{b1},r_{b1}]=[15^{\circ},11^{\circ},2^{\circ}], \ [\phi_{b1},\varphi_{b1},r_{b1}]=[24^{\circ},14^{\circ},2^{\circ}]\]  
The simulation effect is shown in Figures 6-8. As can be seen from Figure 6, the convexation algorithm proposed in this paper can realize effective trajectory generation after a variety of situations for all three cases. As can be seen from case 1, although the longitudinal guidance effect of the initial trajectory is not very good, the method can realize accurate lateral and longitudinal trajectory generation. On the other hand, it can be seen from cases 2 and 3 that although there is crossover between the initial trajectory and the no-fly zones, the method can avoid the no-fly zone through effective adjustment and satisfy the longitudinal guidance at the same time. Figure 5: The generated initial trajectories: (**a**) is the lateral effect of trajectory; (**b**) is the longitudinal effect; (**c**) is the iteration change of the optimized trajectories; (**d**) is the comparison between our method and max attack angle method. Figure 6: The generated optimized trajectories of cases: (**a**) is the longitudinal and lateral trajectories of case 1; (**b**) is the longitudinal and lateral trajectories of case 2; (**c**) is the longitudinal and lateral trajectories of case 3. On the other hand, according to the effect of Figures 6-8, all the trajectories can meet the final constraint requirements. In Case 2 and Case 3, the peak values of several constraints are high due to the avoidance of the no-fly zones, especially at low flight altitudes. In order to effectively meet the constraints, at the initial stage of flight, the vehicle usually employs a high attack angle. In case 3, in order to pass through and between the no-fly zones and increase the range, a smaller attack angle value is adopted; The change of the corresponding bank angle represents the avoidance maneuver process of the vehicle. In Case 1, there are no no-fly zones, so the change of the bank angle is small; However, in Case 2 and Case 3, considering the influence of no-fly zones, there are many variation of the bank angle; this can be seen from the grid point changes when the uniform grid points are used in the initial trajectory. For the last iteration trajectory, due to the small aerodynamic force in the initial stage of vehicle, where the nonlinear illegal degree is low, and the grid points are relatively sparse; at the end of flight, the flight height is low, the aerodynamic force changes greatly, and the grid points are obviously dense. The optimized trajectories' accuracy, iteration times and the waste time of the three cases are shown in Table 1. Where \(E_{l}\) represents the lateral error and \(E_{h}\) represents the longitudinal error.