# Research on Head Pursuit Interception Strategy for Hypersonic Target
## 2 Model of Interceptor
### Model of Head Pursuit Interception
\tag{4}\]  
where \(P\) represents the thrust of the interceptor engine; \(X\), \(Y\), and \(Z\) represent the aerodynamic drag, lift, and side force of the interceptor, respectively; \(\alpha\) and \(\beta\) represent the angle of attack and the angle of side-slip of the interceptor; \(\varphi\) represents the flight-path inclination angle, which is the angle between the velocity vector and the horizontal plane; \(\psi\) represents the azimuth angle, which is the angle between the velocity vector and the vertical plane; \(\varphi_{T}\) and \(\psi_{T}\) represent the flight-pathinclination angle and azimuth angle of the target, respectively; \(\varphi_{M}\) and \(\psi_{M}\) represent the trajectory inclination angle and azimuth angle of the interceptor, respectively; \(N_{TY}\) and \(N_{TZ}\) represent the normal overload of the target in the longitudinal and lateral channels, with sizes of \(A_{TY}/g\) and \(A_{TZ}/g\), respectively; and \(N_{MY}\) and \(N_{MZ}\) represent the normal overload of the interceptor in the longitudinal and lateral channels, with sizes of \(A_{MY}/g\) and \(A_{MZ}/g\), respectively. The following matrices are derived through the coordinate system transformation by Equations (6)-(9). \[\mathbf{T}_{L}=\left[ \begin{array}{ccc}\cos q_{H}\cdot\cos q_{L}&\sin q_{H} &-\cos q_{H}\cdot\sin q_{L}\\ -\sin q_{H}\cdot\cos q_{L}&\cos q_{H}&\sin q_{H}\cdot\sin q_{L}\\ \sin q_{L}&0&\cos q_{L}\\ \end{array} \right] \tag{5}\]  
\[\mathbf{T}_{T}=\left[ \begin{array}{ccc}\cos\varphi_{T}\cdot\cos\psi_{T}& \sin\varphi_{T}&-\cos\varphi_{T}\cdot\sin\psi_{T}\\ -\sin\varphi_{T}\cdot\cos\varphi_{T}&\cos\varphi_{T}&\sin\varphi_{T}\cdot\sin \psi_{T}\\ \sin\psi_{T}&0&\cos\psi_{T}\\ \end{array} \right] \tag{6}\]  
\[\left[ \begin{array}{c}V_{TLX}\\ V_{TLY}\\ V_{TLZ}\\ \end{array} \right]=\mathbf{T}_{L}\cdot\mathbf{T}_{T}^{-1}\cdot\left[ \begin{array} []{c}V_{T}\\ 0\\ 0\\ \end{array} \right] \tag{7}\]  
\[\mathbf{T}_{M}=\left[ \begin{array}{ccc}\cos\varphi_{M}\cdot\cos\psi_{M}& \sin\varphi_{M}&-\cos\varphi_{M}\cdot\sin\psi_{M}\\ -\sin\varphi_{M}\cdot\cos\psi_{M}&\cos\varphi_{M}&\sin\varphi_{M}\cdot\sin \psi_{M}\\ \sin\psi_{M}&0&\cos\psi_{M}\\ \end{array} \right] \tag{8}\]  
\[\left[ \begin{array}{c}V_{MLX}\\ V_{MLY}\\ V_{MLZ}\\ \end{array} \right]=\mathbf{T}_{L}\cdot\mathbf{T}_{M}^{-1}\cdot\left[ \begin{array} []{c}V_{M}\\ 0\\ 0\\ \end{array} \right] \tag{9}\]  
where \(\mathbf{T}_{T}\) and \(\mathbf{T}_{M}\) represent the transformation matrix from the inertial or earth coordinate system to the ballistic coordinate system of the target and interceptor, respectively. \(\mathbf{T}_{L}\) represents the transformation matrix from the inertial coordinate system to the line-of-sight coordinate system. Thus, the representation of the relative motion of the interceptor and target can be deduced. \[\left\{ \begin{array}{l}\dot{R}=V_{TLX}-V_{TMX}\\ \dot{q}_{H}=\frac{V_{TLY}-V_{TMX}}{K\cos{q_{H}}}\\ \dot{q}_{L}=\frac{V_{TLZ}-V_{TMX}}{K\cos{q_{H}}}\\ \end{array} \right. \tag{10}\]  
The requirement of the head pursuit interception mode is to place the interceptor in front of the target, bring the interceptor close to the predicted flight trajectory of the target, and make them move in the same direction. The Mach number of the interceptor should be less than that of the target. The coefficient \(K\) is taken as follows:  
\[K=\frac{V_{M}}{V_{T}}<1 \tag{11}\]  
To intercept a target successfully, the range is limited not only to 0 at the intercept point but also to keep the direction of motion identically. This leads to the following:  
\[\lim_{R\to 0}\eta_{MH}=0\&\&\lim_{R\to 0}\eta_{ML}=0 \tag{12}\]  
\[\lim_{R\to 0}\eta_{TH}=0\&\&\lim_{R\to 0}\eta_{TL}=0 \tag{13}\]