## 2 HGV Tracking with Modified Aerodynamic Model
### HGV Aerodynamic Model  
Assuming the earth is a homogeneous sphere with rotation [16] and sideslip angle is zero, the three-dimensional point-mass dynamics of HGV in semi-speed coordinate system are described by the following equations of motion [17]:  
\[\left\{ \begin{array}{l}\frac{dV}{dt}=-D-\text{gsin}\theta\\ \frac{d\rho}{dt}=\frac{L\text{cosy}}{V}+\left(\frac{V}{r}-\frac{g}{\rho}\right) \text{cos}\theta+2\omega_{e}\text{cos}\phi\text{sin}\sigma\\ \frac{d\sigma}{dt}=\frac{L\text{siny}}{V\text{cos}\theta}+\frac{V\text{cos} \text{sin}\sigma\text{tan}\phi}{r}-2\omega_{e}\left(\text{cos}\phi\text{tan} \theta\text{cos}\sigma-\text{sin}\phi\right)\\ \frac{d\sigma}{dt}=\frac{V\text{cos}\theta\text{cos}\sigma}{r\text{cos}\phi}\\ \frac{d\sigma}{dt}=V\text{sin}\theta\end{array} \right. \tag{3}\]  
where \(V,\theta,\sigma,r,\lambda\) and \(\phi\) are the state variables, which stands for speed, flight path angle, velocity azimuth angle measured from the North in a clockwise direction, radial distance from the center of the earth to the vehicle, longitude and latitude, respectively. Bank angle \(\
u\) and attack angle \(\alpha\) are the two unknown control variables, which will be estimated in the following sections for tracking HGV. \(g\) is gravity acceleration. \(t\) is the time. \(\omega_{e}\) is a coefficient related to the earth's rotation. \(L\) and \(D\) are lift acceleration and drag acceleration, respectively, which are defined as following:  
\[\left\{ \begin{array}{l}D=\frac{C_{D}d5_{M}}{m}\\ L=\frac{C_{L}d5_{M}}{m}\end{array} \right. \tag{4}\]  
where \(q=\frac{1}{2}\rho V^{2}\) is dynamic pressure. \(S_{M}\) is the HGV reference area. \(m\) is the HGV mass. \(\rho\) is mass density of air. HGV is designed based on the high-lift common aerodynamic vehicles (CAV-H) [18] so we can study the HGV aerodynamic model using the CAV-H aerodynamic parameters: \(m=908\) kg, \(S_{M}=~{}0.484\) m\({}^{2}\). Based on the U.S. Standard Atmosphere, 1976(USSA1976) [19], the mass density of air denoted as \(\rho\) and the sound speed \(a\) can be functions of height as  
\[\rho=1.225\times 10^{-0.1133h} \tag{5}\]  
\[a=f\left(h\right)=\left\{ \begin{array}{rl}-4.029h+340.3,h\leq 10\\ -7.978\times 10^{-4}h^{3}+0.1027h^{2}-2.873h+316.9,10<h\leq 50\\ -1.63h+412.6,h>50\end{array} \right. \tag{6}\]  
where \(h\) is height in kilometers. The drag and lift coefficient \(C_{D}\) and \(C_{L}\) are functions of attack angle \(\alpha\) and Mach number \(Ma\), which can be expressed as:  
\[\left\{ \begin{array}{l}C_{D}=C_{D0}+C_{D1}\alpha^{2}+C_{D2}e^{\left(C_{D3}Ma \right)}\\ C_{L}=C_{L0}+C_{L1}\alpha+C_{L2}e^{\left(C_{L3}Ma\right)}\end{array} \right. \tag{7}\]  
where \(C_{D0},C_{D1},C_{D2},C_{D3},C_{L0},C_{L1},C_{L2}\) and \(C_{L3}\) are some fixed numerical coefficients. Thus, we have almost established aerodynamic model. Furthermore, there are some important parameter constraints that must be considered, such as dynamic pressure \(q\), aerodynamic load \(n\) and heat flux \(\dot{Q}\). These constraints can be expressed as  
\[\left\{ \begin{array}{rl}q\leq q_{\max}\\ n=\frac{\sqrt{D^{2}+L^{2}}}{g}\leq n_{\max}\\ \dot{Q}=K\left(\frac{\rho}{\rho_{0}}\right)^{5}\left(\frac{\
u}{\
u_{e}}\right) ^{3.15}\leq\dot{Q}_{\max}\end{array} \right. \tag{8}\]where \(q_{\max},\ n_{\max}\) and \(\dot{Q}_{\max}\) are the maximum dynamic pressure, maximum aerodynamic load and maximum heat flux. \(\rho_{0}\) is the mass density of air at sea level. \(K\) is a coefficient associated with the head of HGV. When the bank angle \(\
u\) is not zero, the trajectory will maneuver laterally. Usually, in the HGV controlling, it is common to keep the attack angle fixed and keep the bank angle varying to make maneuvering laterally and skipping longitudinally trajectory. The bank angle is the main control variable and the attack angle is the auxiliary control variable for HGV controlling.