# Research on Head Pursuit Interception Strategy for Hypersonic Target
## 2 Model of Interceptor
### Model of Head Pursuit Interception  
The study of guidance trajectory is based on the laws of classical mechanics. In the preliminary design phase of a guidance system, kinematics analysis methods are generally used in order to simplify the research. The relative motion equation describes the relative motion of the interceptor and target. The establishment of the relative motion equation is the basis of a guidance trajectory kinematics analysis method. In the interception process controlled by a direct driving force, it is assumed that the maneuvering acceleration vectors of the interceptor and the target are perpendicular to their velocity vectors in order to simplify the analysis process of the problem, that is, the maneuvering acceleration vectors only change their velocity directions without changing the size of the velocity scalars. The \(x\)-direction of this coordinate system coincides with the velocity vector of the missile center of mass and has the same direction. The \(y\)-direction is perpendicular to the velocity vector and is in the vertical plane containing the \(y\)-axis. The \(z\)-direction is determined by the right-hand rule. The ballistic (trajectory) coordinate system is a dynamic coordinate system fixed to the velocity vector \(\mathbf{V}\), with the origin fixed to the missile body and taken at the center of mass of the missile. The coordinate system moves with the motion of missiles and is often used to study the motion characteristics of missile centroids. Figure 5 shows the relative motion of the missile and the target. \(T\) and \(M\) represent the target and interceptor, respectively. \(OXYZ\) represents the inertial coordinate system, and \(OX_{l}Y_{l}Z_{l}\) represents the line-of-sight coordinate system. \(R\) is the range of them. \(V_{T}\) and \(V_{M}\) represent velocity vectors, and \(A_{T}\) and \(A_{M}\) represent acceleration vectors. \(q_{H}\) and \(q_{L}\) represent the pitch angle and yaw angle of the line-of-sight. \(\eta_{TH}\) and \(\eta_{TL}\) represent the angle between \(V_{T}\) and the line-of-sight, which are defined as the attitude angle of the target. \(\eta_{MH}\) and \(\eta_{ML}\) represent the angle between \(V_{M}\) and the line-of-sight, which are defined as the attitude angle of the interceptor. In the interception process studied in this paper, the target performs a maneuvering flight. According to the classification of the maneuver trajectory, typical terminal maneuver modes include the jump-dive maneuver, snake maneuver, pendulum maneuver, and spiral maneuver. The hypersonic target studied in this paper starts the snake maneuver at a certain time, and the maximum maneuver overload is 5 g.  
The kinematic equations of the maneuvering target are established in the inertial frame as follows. \[\left\{ \begin{array}{l}\dot{X}_{T}=V_{T}\cos\varphi_{T}\cos\psi_{T}\\ \dot{Y}_{T}=V_{T}\sin\varphi_{T}\\ \dot{Z}_{T}=-V_{T}\sin\varphi_{T}\cos\psi_{T}\\ \end{array} \right. \tag{1}\]  
Considering the ballistic frame, the following expressions can be obtained. \[\left\{ \begin{array}{l}\dot{\varphi}_{T}=\frac{N_{T}\chi}{V_{T}}\\ \dot{\psi}_{T}=-\frac{N_{T}\chi}{V_{T}\cos\varphi_{T}}\\ \end{array} \right. \tag{2}\]  
Figure 4: Side force coefficient. The kinematic equations of the interceptor are established in the inertial frame as follows. \[\left\{ \begin{array}{l}\dot{X}_{M}=V_{M}\cos\varphi_{M}\cos\psi_{M}\\ \dot{Y}_{M}=V_{M}\sin\varphi_{M}\\ \dot{Z}_{M}=-V_{M}\sin\varphi_{M}\cos\psi_{M}\\ \end{array} \right. \tag{3}\]  
Considering the ballistic frame, the following expressions can be obtained. \[\left\{ \begin{array}{l}m\dot{V}_{M}=P\cos\alpha\cos\beta-X-mg\sin\varphi_{M} \\ mV_{M}\phi_{M}=P\sin\alpha+Y-mg\cos\varphi_{M}\\ mV_{M}\psi_{M}=P\cos\alpha\sin\beta-Z\\ \dot{\varphi}_{M}=\frac{(N_{M\gamma}-\cos\varphi_{M})\beta}{V_{M}}\\ \dot{\psi}_{M}=-\frac{N_{MZ}\beta}{V_{M}\cos\varphi_{M}}\\ \end{array} \right.