# Research on Head Pursuit Interception Strategy for Hypersonic Target
## 3 Design of Head Pursuit Interception Guidance Strategy
### Design of Initial Guidance Law  
In the initial guidance phase, the combination of longitudinal and lateral channel control is used to make the interceptor climb to the predetermined height and direction. To ensure the stability of interceptor flight control loop in large airspace, the controllers of the longitudinal and lateral loop are designed based on the preset turning angle and improved proportional navigation guidance method in the initial phase. The interceptor needs to accelerate to a certain Mach number and meet the condition that the flight-path inclination angle of the interceptor is about 90 degrees at the end. That is, the interceptor turns at a certain angle at the end of the initial guidance, and thus the relationship of the interceptor and target does not need to be considered. The concept of a preset turning point is introduced, and the point is taken as the target in the initial guidance phase. The problem becomes that the interceptor intercepts the air-fixed target at a certain terminal ballistic angle.  
The preset turning point coordinates are assumed to be \(TP(X_{TP},Y_{TP},Z_{TP})\), the initial coordinates of the interceptor are \(M(X_{M0},Y_{M0},Z_{M0})\), and the initial coordinates of the target are \(T(X_{T0},Y_{T0},Z_{T0})\).  
Figure 6: Diagrammatic sketch of side force coefficient.  
The key to determining \(X_{TP}\) is to find its expression about the initial horizontal distance \(\Delta X\) of the missile and the target, that is, \(X_{TP}=f(\Delta X)\). According to the kinematic relationship, the following equation can be obtained:  
\[\Delta X_{\text{min}}<\Delta X-(X_{TP}+V_{T}t_{d})<\Delta X_{\text{max}} \tag{15}\]  
where \(\Delta X\) is the initial horizontal distance between interceptor and target; \(\Delta X_{\text{min}}\) is the minimum horizontal distance between the missile and the target at the end of the initial guidance, to ensure that the interceptor in the middle and terminal guidance phase is not exceeded by the target; and \(\Delta X_{\text{max}}\) is the maximum horizontal distance between the missile and the target at the end of the initial guidance, to ensure that the terminal guidance phase of the interceptor will not be too long.  
When \(\Delta X\) is certain and within the interception range, there is a feasible range corresponding to \(X_{TP}\). That is, as long as \(X_{TP}\in(X_{TP\text{min}},X_{TP\text{max}})\) is taken, the interceptor can hit the target.  
Through several groups of simulations, the corresponding feasible range of \(X_{TP}\) under different \(\Delta X\) and the optimal value \(X_{TPopt}\) of the corresponding \(X_{TP}\) are determined. Then, the expression of \(X_{TP}\) changing with \(\Delta X\) is fitted according to these data groups, which are shown in Table 2.  
By fitting the data in Table 2, the expression for \(X_{TP}\) can be obtained. Due to the small initial height difference and lateral distance difference between the target and the interceptor compared to the initial horizontal distance, the impact of changing \(Y_{TP}\) and \(Z_{TP}\) is minimal. Therefore, expressions for \(Y_{TP}\) and \(Z_{TP}\) can be designed within a reasonable range, all of which are listed as follows.  
\[\left\{\begin{array}{l}X_{TP}=-0.0001(X_{T0}-X_{M0})^{2}+0.0606(X_{T0}-X_{M0 })+21,385\\ Y_{TP}=Y_{T0}-Y_{M0}\\ Z_{TP}=Z_{m0}+\frac{1}{3}(Z_{T0}-Z_{M0})\end{array}\right. \tag{16}\]  
According to the three-dimensional proportional navigation guidance method, the longitudinal channel and lateral channel guidance laws of the initial guidance phase are designed as follows.  
(1) Longitudinal channel  
\[N_{MY}=\frac{k_{1}|\dot{R}|\dot{q}_{H}}{g} \tag{17}\]  
(2) Lateral channel  
\[N_{M\text{Z}}=\frac{k_{2}|\dot{R}|\dot{q}_{L}}{g} \tag{18}\]  
where \(k_{1}\) and \(k_{2}\) are proportional navigation guidance parameters, which are constants and typically range from 2 to 6.