# Analytic Time Reentry Cooperative Guidance for Multi-Hypersonic Glide Vehicles
## 3 Analytical Time-Constrained Reentry Guidance
### Parametric Design of the Initial Reentry Phase Bank Angle Profile  
The reentry guidance of a hypersonic vehicle can be divided into longitudinal guidance and lateral guidance. Longitudinal guidance can design the bank angle amplitude, and lateral guidance can determine the bank angle symbol through the heading angle deviation corridor to ensure that the vehicle points to the target point.  
When designing the longitudinal trajectory, the reentry process can be divided into the initial reentry phase and the quasi-equilibrium glide phase. The initial reentry phase is to ensure the state of the reentry vehicle smoothly enters the drag acceleration-velocity reentry corridor, and meet the QEGC (Equation (4)) to prepare for the glide phase. The drag acceleration-velocity reentry corridor can be obtained by the path constraints of heat flux, dynamic pressure, overload, and the QEGC. The drag acceleration is recorded as \(a_{D}\), the specific constraints of which are as follows:  
\[\left\{\begin{array}{l}a_{D}\leq\frac{\partial_{\rm max}S_{\rm r}}{2K_{0}m} \cdot\frac{C_{D}\rho^{\alpha 5}}{V^{1.15}}\\ a_{D}\leq\frac{\alpha_{\rm max}S_{\rm r}}{m}\cdot C_{\rm D}\\ a_{D}\leq\frac{N_{\rm max}S_{0}}{\sqrt{(L/D)^{2}+1}}\\ a_{D}\geq\frac{D}{L}\cdot(g-\frac{V^{2}}{r})\end{array}\right.. \tag{10}\]  
During the initial reentry phase, the atmosphere is thin, the path constraint is small, and the aerodynamic control force is limited. Therefore, in order to simplify the calculations, the bank angle of this flight process can be set to a constant value, \(\sigma_{0}\), until it reaches the conversion point of the quasi-equilibrium glide phase. The bank angle symbol at the initial time of reentry can be determined by:  
\[sign(\sigma_{0})=-sign(\Delta\psi_{0}). \tag{11}\]  
where \(\Delta\psi_{0}\) is the deviation between the heading angle and the line-of-sight direction to the target point, \(\Delta\psi_{0}=\psi_{0}-\psi_{los}\), \(\psi_{0}\) is the heading angle at the reentry time, and \(\psi_{los}\) is the line-of-sight azimuth of the target point. Based on the AOA profile given in Section 2.1, the bank angle and state value at the initial moment, together with the state quantity and the control quantity in the initial reentry phase can be obtained by continuous integration until the aircraft enters the reentry corridor and satisfies the QEGC. That is, the transition point from the initial reentry phase to the quasi-equilibrium glide phase is obtained, and the transition condition is as follows:  
\[\left|\frac{dr}{dV}-(\frac{dr}{dV})_{\text{QEGC}}\right|<\delta_{1} \tag{12}\]  
where \(\delta_{1}\) is a small constant and can be derived from Equations (1) and (7) as below:  
\[\left\{\begin{array}{l}\frac{dr}{dV}=\frac{V\sin\gamma}{-D/m-g\sin\gamma}\\ \left(\frac{dr}{dV}\right)_{\text{QEGC}}\doteq\frac{C_{1}S_{7}\cos\sigma_{0} PV+2 mV/r}{mV^{2}/r^{2}+\rho V^{2}C_{1}S_{7}\cos\sigma_{c}/2h_{s}}\end{array}\right. \tag{13}\]  
In order to ensure that the control quantity is continuous during the conversion from the initial reentry phase to the quasi-equilibrium glide phase, the bank angle should also meet the following:  
\[\left|\sigma_{0}-\sigma_{\text{QEGC}}\right|<\delta_{2} \tag{14}\]  
where \(\sigma_{\text{QEGC}}\) is the size of the bank angle obtained from the QEGC, and \(\delta_{2}\) is a small constant value. Through continuous iteration and judgment, the state and control quantity of the conversion point can be obtained.