# Rapid Parametric Modeling and Robust Analysis for the Hypersonic Ascent Based on Gap Metrics
## 2 Model Description
### Force Estimation  
As soon as the configuration of the aerospace vehicle is identified, the following task is to acquire the forces for establishing the dynamic model. In this article, we divide the vehicle body into several surface elements, considered the triangle surface [22]. For each surface element in the face of freeflow, the oblique shock theory is used to estimate the enforced pressure coefficient \(C_{p}\), and it is expressed by [23]  
\[C_{p}=\frac{4}{\gamma_{c}+1}(\sin^{2}\beta_{s}-\frac{1}{Ma^{2}}) \tag{1}\]  
where \(\gamma_{c}\) denotes the specific heat ratio; \(Ma\) is the flight Mach number; and \(\beta_{s}\) is the shock wave angle, which is solved based on the flow turning angle \(\delta\) by the following equation:  
\[\tan\delta=2\cot\beta_{s}\cdot\left[\frac{Ma^{2}\sin^{2}\beta_{s}-1}{Ma^{2}( \gamma_{c}+\cos 2\beta_{s})+2}\right] \tag{2}\]  
Alternatively, if the surface element lies on the leeward side, the Prandtl-Meyer theory is adopted to estimate the pressure coefficient, which is calculated by [24]  
\[C_{p}=\frac{2}{\gamma_{c}Ma_{u}^{2}}\left[\left(\begin{array}{c}1+\frac{ \gamma_{c}-1}{2}Ma_{u}^{2}\\ 1+\frac{\gamma_{c}-1}{2}Ma_{d}^{2}\end{array}\right)^{\frac{\gamma_{c}}{\gamma _{c}-1}}-1\right] \tag{3}\]  
where the subscripts \(u\) and \(d\) denote the parameters before and after the shock wave, respectively, which are solved based on the flow turning angle by  
\[|\delta|=\
u(Ma_{d})-\
u(Ma_{u}) \tag{4}\]  
where \(\
u(Ma)\) is the Prandtl-Meyer function.  
Once the pressure coefficient is obtained for the surface element, the suffered pressure is computed by  
\[P_{l}=C_{pi}s_{a,i}+P_{\infty} \tag{5}\]  
where \(P_{\infty}\) denotes the static pressure, and \(s_{a,i}\) denotes the area of the \(i\)-th surface element. The aerodynamic forces and moments are yielded by superimposing these pressures for all surface elements, and they are expressed by [25]  
\[\left\{\begin{array}{l}F_{x}=-\sum P_{i}n_{xi}S_{i}\\ F_{y}=-\sum P_{i}n_{yi}S_{i}\\ F_{z}=-\sum P_{i}n_{zi}S_{i}\end{array}\right. \tag{6}\]  
Figure 1: Geometrical configuration of aerospace vehicle.  
\[\left\{\begin{array}{l}M_{x}=\sum P_{i}(r_{zi}u_{yi}-r_{yi}u_{zi})S_{i}\\ M_{y}=\sum P_{i}(r_{xi}u_{zi}-r_{zi}u_{xi})S_{i}\\ M_{z}=\sum P_{i}(r_{yi}u_{zi}-r_{xi}u_{yi})S_{i}\end{array}\right. \tag{7}\]  
where \(\mathbf{r}_{i}=r_{xi}e_{x}+r_{yi}e_{y}+r_{zi}e_{z}\) represents the distance vector from the vehicle centroid to the center of each surface element.  
Aside from aerodynamic forces, the thrust is calculated using the momentum theorem, and it is expressed by [26]  
\[F_{T}=\dot{m}_{a}[(1+f_{st})V_{5}-V_{1}]+(P_{5}-P_{1})A_{e}-(P_{2}-P_{1})A_{1} \tag{8}\]  
where \(\dot{m}_{a}\) denotes the air flow rate; \(\dot{\varphi}\) is the stoichiometric ratio; \(f_{st}\) represents the stoichiometric fuel-air ratio; \(V_{1}\) and \(P_{1}\) indicate the airflow parameters at the inlet; \(V_{5}\) and \(P_{5}\) denote the airflow parameters at the nozzle; \(A_{e}\) and \(A_{1}\) are the areas in relation to the nozzle exit plane and inlet plane, respectively; \(P_{2}\) represents the pressure of the compressed section.  
Furthermore, the pitching moment induced by the propulsive force is expressed by  
\[\mathbf{M}_{T}=\mathbf{r}_{TG}\times\mathbf{F}_{T} \tag{9}\]  
Similar to the aerodynamic coefficient, the thrust coefficient and propulsive moment coefficient are formulated by  
\[C_{T}=\frac{2F_{T}}{\rho_{\infty}V_{\infty}^{2}S_{\mathrm{cap}}} \tag{10}\]  
where \(S_{\mathrm{cap}}\) denotes the capture area of the engine and \(\rho_{\infty}\) indicates freeflow density.