## 3 D-V Profile Method Preliminary
### Profile Tracking Guidance Law  
As illustrated in Figure 1, there are two major boundary constraints for the D-V profile design. On one hand, the upper black, thick, solid line is the consequence of the maximum heating rate, dynamic pressure, and load factor constraints combined as a D-V profile upper boundary constraint. That is a hard constraint, and to avoid serious safety problems including excessive thermal ablation of the vehicle, structural damage, and control failure, the D-V profile can only be designed to travel below this boundary. On the other hand, the lower black, thick, solid line, also described as the lower boundary, which is constrained by the quasi-equilibrium gliding condition, is a soft constraint that the vehicle is free to ignore and along which a greater range in the gliding phase can be obtained. The upper and lower boundaries are combined to give a range of profile designs known as flight corridors. The D-V profile is designed as a three-fold line in the drag acceleration corridor indicated by the constraints. The red line represents the profile to be designed. As the design profile moves upwards, it means \(D_{1}\) increases, and the range of the vehicle will gradually decrease. In contrast, as the design profile moves downwards, the range of the vehicle will gradually increase. When the D-V profile is designed in the form of a three-fold line, as in Figure 1, a quadratic (or sub-quadratic) curve of the designed D-V profile is uniformly expressed as follows:  
\[D=C_{1}+C_{2}V+C_{3}V^{2} \tag{7}\]  
The whole corresponding range \(R^{*}\) for this designed D-V profile is:  
\[R^{*}=R_{1}+R_{2}+R_{3} \tag{8}\]\[R_{1} =\frac{V_{1}-V_{0}}{D_{0}-D_{1}}\left(V_{1}-V_{0}-\frac{D_{1}V_{0}-D_ {0}V_{1}}{D_{0}-D_{1}}\ln\frac{D_{1}}{D_{0}}\right)\] \[R_{2} =\frac{V_{1}^{2}-V_{2}^{2}}{2D_{1}} \tag{9}\] \[R_{3} =\frac{V_{f}-V_{2}}{D_{1}-D_{f}}\left(V_{f}-V_{2}-\frac{D_{f}V_{2} -D_{1}V_{f}}{D_{1}-D_{f}}\ln\frac{D_{f}}{D_{1}}\right)\]  
Once the reference D-V profile is determined, the standard lift-to-drag ratio of the vehicle along that designed profile can be found [26; 27], denoted as \((L\cos v/D)_{0}\). \[\left(\frac{L\cos v}{D}\right)_{0}=-\frac{4h_{s}C_{1}}{V^{2}}-\frac{h_{s}C_{2} }{V}-\frac{1}{D}\left(\frac{V^{2}}{r}-g\right) \tag{10}\]  
The vehicle will also be guided to the predetermined target by a longitudinal trajectory tracking controller, and the required lift-to-drag ratio \(\left(L\cos\sigma/D\right)_{c}\) and its small increment \(\delta(L\cos\sigma/D)\) will be determined by:  
\[\left(\frac{L\cos v}{D}\right)_{c}=\left(\frac{L\cos v}{D}\right)_{0}+\delta \left(\frac{L\cos v}{D}\right) \tag{11}\]  
To maintain the true profile of the vehicle as closely as possible to the reference profile and to provide a smooth transition of the guidance process, the small increment of the required lift-to-drag ratio \(\delta(L\cos\sigma/D)\) is designed as follows:  
\[\delta\left(\frac{L\cos v}{D}\right)=f_{1}\delta D+f_{2}\delta h \tag{12}\]  
The details about how to calculate the value of \(f_{1}\) and \(f_{2}\) have been explained in the literature [10]. Thus, the required lift-to-drag ratio is given by:  
\[\left(\frac{L\cos v}{D}\right)_{c}=\left(\frac{L\cos v}{D}\right)_{0}+f_{1} \delta D+f_{2}\delta\dot{h} \tag{13}\]  
Figure 1: The D-V designed profile schematic. \[\left\{ \begin{array}{l}f_{1}=\frac{h_{\xi}}{D_{0}}\left[\begin{array}{c} \omega^{2}+\frac{1}{h_{\xi}}\left(\frac{V_{0}^{2}}{V_{0}}-g\right)+3\dot{D}_{0} \bigg{(}\frac{1}{V_{0}}-\frac{\dot{D}_{0}}{D_{0}^{2}}\bigg{)}-\frac{4D_{0}^{2} }{V_{0}^{2}}\\ \quad+\frac{2\dot{D}_{0}}{D_{0}}-\bigg{(}2\xi\omega-\frac{3D_{0}}{V_{0}}+\frac {2\dot{D}_{0}}{D_{0}}\bigg{)}\bigg{(}\frac{4D_{0}}{V_{0}}+\frac{\dot{h}_{0}}{h_ {\kappa}}\bigg{)}\end{array}\right],\\ f_{2}=-\frac{1}{D_{0}}\bigg{(}2\xi\omega-\frac{3D_{0}}{V_{0}}+\frac{2\dot{ D}_{0}}{D_{0}}\bigg{)},\\ \delta D=D-D_{0},\\ \delta\dot{h}=v\theta+h_{\rm s}\bigg{(}\frac{2D_{0}}{v}+\frac{\dot{D}_{0}}{D _{0}}-\frac{\dot{C}_{D_{0}}}{C_{D_{0}}}\bigg{)}.\end{array} \right.