# Analytic Time Reentry Cooperative Guidance for Multi-Hypersonic Glide Vehicles
## 3 Analytical Time-Constrained Reentry Guidance
### Design of the Bank Angle Profile of the Gliding Phase by Considering Time Constraint  
After the transition point of the reentry process, the hypersonic reentry glider enters the quasi-equilibrium glide phase. The flight process not only needs to meet the drag-velocity (D-V) reentry corridor at all times, but also needs to meet the QEGC. In order to ensure that the aircraft meets the time constraints and accurately hits the target point, the remaining flight time and the remaining flight range can be estimated based on the QEGC [26]. Equation (7) can be transformed into:  
\[m=\frac{L\cos\sigma}{g-V^{2}/r} \tag{15}\]  
Combining Equations (1) and (15), the following can be obtained:  
\[\frac{dV}{dt}=-\frac{1}{r}\cdot\frac{D}{L}\cdot\frac{gr-V^{2}}{\cos\sigma} \tag{16}\]  
\[dt=-r\frac{L}{D}\cos\sigma\frac{dV}{gr-V^{2}} \tag{17}\]  
Since the flight altitude is much smaller than the radius of Earth, \(R_{e}/r\approx 1\), combining with Equation (17), the remaining time estimate can be obtained by integrating both sides at the same time:  
\[\int_{t}^{t_{f}}dt=\int_{V}^{V_{f}}-R_{e}\frac{L}{D}\cos\sigma\frac{dV}{g_{0} R_{e}-V^{2}} \tag{18}\]\[T_{g^{0}}=\frac{\sqrt{R_{e}}}{2\sqrt{g_{0}}}\frac{L}{D}\cos\sigma\ln\left(\frac{ \sqrt{g_{0}R_{e}}+V}{\sqrt{g_{0}R_{e}}-V}\sqrt{g_{0}R_{e}}-V_{f}}\right) \tag{19}\]  
If the flight time constraint and terminal speed are satisfied, then we have the following:  
\[T_{g^{0}}^{*}=\frac{\sqrt{R_{e}}}{2\sqrt{g_{0}}}\frac{L}{D}\cos\sigma\ln\left( \frac{\sqrt{g_{0}R_{e}}+V}{\sqrt{g_{0}R_{e}}-V}\frac{\sqrt{g_{0}R_{e}}-V_{f}^{ *}}{\sqrt{g_{0}R_{e}}+V_{f}^{*}}\right) \tag{20}\]  
and  
\[\left|\sigma_{g}\right|=\arccos\frac{T_{g^{0}}^{*}}{T_{g^{0}}^{0}} \tag{21}\]  
where \(T_{g^{0}}^{0}\) is the value of Equation (20) when the bank angle is 0. The range-to-go is the distance from the current point to the landing point, and the change rate of the range-to-go obtained from [23] is as follows:  
\[\frac{ds}{dt}=\frac{R_{e}}{r}V\cos\gamma \tag{22}\]  
Similarly, by bringing Equation (17) into Equation (22), we obtain as below:  
\[ds=-R_{e}\frac{L}{D}\cos\sigma\frac{VdV}{g_{0}R_{e}-V^{2}} \tag{23}\]  
By integrating both sides at the same time, the remaining range estimation can be obtained as follows:  
\[\int_{s}^{s_{f}}ds=\int_{V}^{V_{f}}-R_{e}\frac{L}{D}\cos\sigma\frac{VdV}{g_{0} R_{e}-V^{2}} \tag{24}\]  
\[\frac{dV}{dt}=-\frac{1}{r}\cdot\frac{D}{L}\cdot\frac{gr-V^{2}}{\cos\sigma} \tag{16}\]0  
The relationship between the time-to-go and the range-to-go obtained by Equations (19) and (25) is as follows:  
\[\frac{dV}{dt}=-\frac{1}{r}\cdot\frac{D}{L}\cdot\frac{gr-V^{2}}{\cos\sigma} \tag{16}\]1  
When the total flight time and the target point are known, and letting \(a=\sqrt{g_{0}R_{e}}\), Equation (24) can be transformed into that below:  
\[\frac{dV}{dt}=-\frac{1}{r}\cdot\frac{D}{L}\cdot\frac{gr-V^{2}}{\cos\sigma} \tag{16}\]2  
There is only one variable in the above equation, if the appropriate \(V_{f}\) is found, and the flight time constraint and the range constraint can be satisfied at the same time. Then Equation (27) can be transformed into the following:  
\[\frac{dV}{dt}=-\frac{1}{r}\cdot\frac{D}{L}\cdot\frac{gr-V^{2}}{\cos\sigma} \tag{16}\]3  
In this paper, the particle swarm optimization algorithm is used to solve Equation (28), and the required terminal velocity is recorded as \(\overline{V_{f}}\). Combined with Equation (21),the amplitude of the bank angle command that satisfies both the flight time constraint and range constraint is as follows:  
\[\frac{dV}{dt}=-\frac{1}{r}\cdot\frac{D}{L}\cdot\frac{gr-V^{2}}{\cos\sigma} \tag{16}\]4  
In summary, the glide phase profile can be obtained, and the time constraint and range constraint can be satisfied. After obtaining the bank angle amplitude, a comparison with the maximum bank angle amplitude is made, so that the analytical bank angle amplitude is limited to the allowable range of the control amount.