## 2 Modeling of Hypersonic Target Motion Characteristics  
### Motion Equation of Hypersonic Target  
In the process of establishing the motion equation of hypersonic targets, the following coordinate systems need to be considered (Figure 1):  
1. Earth-centered, Earth-fixed (ECEF) coordinate system: The origin \(O\) is located at the Earth's center, \(OX\) points to the zero degrees longitude in the equatorial plane, the \(OZ\) axis points to the north pole, and the \(OY\) axis is perpendicular to the \(OX\) and \(OZ\) axes in the equatorial plane. The unit vectors on the three \(O-XYZ\) axes are expressed as \(i\), \(j\), and \(k\). 2. Longitude, latitude, altitude (LLA) coordinate system: This is the geodetic coordinate system. The coordinate origin is located at the geo-center \(O\), and the longitude \(\mu\), latitude \(\lambda\), and altitude \(h\) of the target location. 3. Aircraft position coordinate system: The origin \(O\) is located at the center of the Earth. The \(Ox\) axis is the connecting line from the center of the Earth to the hypersonic target, and it is positive upward. The \(Oy\) axis is perpendicular to the axis and is in the equatorial plane. The due east direction is positive, and the \(Oz\) axis is determined by the other two axes and the right-hand law. The \(O-xyz\) coordinate system can be obtained by combining the \(O-XYZ\) coordinate system with the latitude and longitude of the hypersonic target position. 4. East-north-up (ENU): This is the radar coordinate system. The radar position \(R\) is the origin of the coordinate system, and the \(RE\), \(RN\), and \(RU\) axes point to the east, north, and sky directions perpendicular to each other. It can be used to represent the pitch angle \(a\) and azimuth angle \(e\) of the hypersonic target. Assuming the Earth is a homogeneous sphere with rotation [19], the aerodynamic force and gravity of the hypersonic target are converted into the same coordinate system through the flight heading angle, aerodynamic angle, attitude angle, and other important angles to obtain the six-degree-of-freedom motion equation of the hypersonic target. Due to the complex structure of the equation, it is simplified when analyzing the scene, and the target trajectory model is obtained as follows [20]:  
\[ \begin{cases}\dot{r}=V\sin\gamma\\ \dot{\mu}=\frac{V\cos\gamma\cos\xi}{r\cos\lambda}\\ \dot{\lambda}=\frac{V\cos\gamma\sin\xi}{r\cos\lambda}\\ \dot{V}=-\frac{D}{m}-g\sin\gamma+w_{e}^{2}r\cos\lambda(\sin\gamma\cos\lambda- \cos\gamma\sin\lambda\sin\xi)\\ \dot{\gamma}=\frac{L\cos\sigma}{mV}+\left(\frac{V}{r}-\frac{\xi}{r}\right) \cos\gamma+2w_{e}\cos\lambda\cos\xi\\ +\frac{w_{e}^{2}r}{V}\cos\lambda(\cos\gamma\cos\lambda+\sin\gamma\sin\lambda \sin\xi)\\ \dot{\xi}=\frac{L\sin\sigma}{mV\cos\gamma}-\frac{V}{r}\cos\gamma\cos\xi\tan \lambda+\\ 2w_{e}(\tan\gamma\cos\lambda\sin\xi-\sin\lambda)-\frac{w_{e}^{2}r}{V\cos\gamma }\sin\lambda\cos\xi\end{cases}  \tag{1}\]  
where \(r\) is the distance between the center of mass of the target aircraft and the geo-center, \(\mu\) is the geocentric longitude, and \(\lambda\) is the geocentric latitude. These three parameters describe the location of the target. \(V\) is the target velocity, \(\gamma\) is the trajectory inclination angle, and \(\xi\)  
Figure 1: Coordinate system relationship. \(O\) is geo-center, \(O-XYZ\) is the geocentric earth fixed coordinate system, and \(O-xyz\) is the aircraft position coordinate system. is the trajectory deflection angle. These three parameters describe the target velocity. \(D\) and, \(L\) are the aerodynamic resistance and aerodynamic lift, \(\sigma\) is the inclination angle, \(w_{e}\) is the value of the Earth's rotation angle velocity, and \(g\) is the local gravity acceleration. In the main movement process of the hypersonic target, the complex aerodynamic force is the main factor to be considered. The common flight aerodynamic lift \(L\) and drag \(D\) are as follows:  
\[ \begin{cases}L=C_{L}qS\\ D=C_{D}qS\\ q=0.5\rho V^{2}\end{cases}