# Research on Head Pursuit Interception Strategy for Hypersonic Target
## 3 Design of Head Pursuit Interception Guidance Strategy
### Design of Terminal Guidance Law  
The normal overload required by traditional proportional guidance when intercepting the target is directly related to the speed of the missile at the interception point and the direction of the missile's attack, and the normal overload required fluctuates greatly when the end approaches the target. Therefore, in this paper, the terminal guidance strategy is designed based on the sliding mode control theory under angle constraints and overload constraints. (1) Longitudinal channel  
For the non-linear system described by Equation (14), the system state quantity is taken as  
\[X=\left[ \begin{array}{cc}R&\eta_{MH}&\eta_{TH}\end{array} \right]^{\mathsf{T}} \tag{41}\]  
By substituting into Equation (14), the following can be obtained:  
\[\dot{X}=\left[ \begin{array}{c}\dot{R}\\ \dot{\eta}_{MH}\\ \dot{\eta}_{TH}\end{array} \right]=f+\left[ \begin{array}{c}0\\ 1/V_{M}\\ 0\end{array} \right]u+\left[ \begin{array}{c}0\\ 0\\ 1/V_{T}\end{array} \right]w \tag{42}\]  
From Equation (42),  
\[f=\left[ \begin{array}{c}V_{T}\cos\eta_{TH}\cos\eta_{TL}+V_{M}\cos\eta_{MH} \cos\eta_{ML}\\ -\frac{V_{M}}{R}\sin\eta_{ML}\tan q_{H}\Big{(}\cos\eta_{MH}\sin\eta_{ML}- \frac{V_{T}}{V_{M}}\cos\eta_{TH}\sin\eta_{TL}\Big{)}-\frac{V_{M}}{R}\cos \eta_{ML}\Big{(}\sin\eta_{MH}-\frac{V_{T}}{V_{M}}\sin\eta_{TH}\Big{)}\\ -\frac{V_{M}}{R}\sin\eta_{TL}\tan q_{H}\Big{(}\cos\eta_{MH}\sin\eta_{ML}- \frac{V_{T}}{V_{M}}\cos\eta_{TH}\sin\eta_{TL}\Big{)}-\frac{V_{M}}{R}\cos\eta _{TL}\Big{(}\sin\eta_{MH}-\frac{V_{T}}{V_{M}}\sin\eta_{TH}\Big{)}\end{array} \right] \tag{43}\]  
The sliding mode surface is designed as follows:  
\[S_{H\_ter}=\eta_{MH}-c_{1}\eta_{TH} \tag{44}\]  
The reaching law is designed as follow:  
\[\dot{S}_{H\_ter}=-\frac{c_{1}N_{Y\_pre}}{V_{T}|S_{H\_ter}|}S_{H\_ter}-K_{1} \text{sgn}(S_{H\_ter}) \tag{45}\]  
where \(c_{1}\) and \(K_{1}\) are sliding mode guidance parameters, which are constants greater than 0. Define the Lyapunov function:  
\[V=\frac{1}{2}S_{H\_ter}^{2} \tag{46}\]Then,  
\[\dot{V} =S_{H\_te}\dot{S}_{H\_te\sigma} \tag{47}\] \[=S_{H\_te}\bigg{(}-\frac{c_{1}N_{Y\_pre}}{V_{T}|S_{H\_te}|}S_{H\_te} -K_{1}\text{sgn}(S_{H\_te\sigma})\bigg{)}\] \[=-K_{1}|S_{H\_te}|-\frac{c_{1}N_{Y\_pre}}{V_{T}|S_{H\_te}|}S_{H\_te \sigma}^{2}<0\]  
According to Equation (47), the designed sliding surface meets the Lyapunov stability criterion. By substituting Equations (44) and (45) into Equation (4), the terminal guidance law of the longitudinal channel can be obtained. \[N_{MY}=\frac{V_{M}}{g}\bigg{[}-(f_{2}-c_{1}f_{2})-K_{1}\text{sgn}(S_{H\_te})- \frac{c_{1}N_{Y\_pre}}{V_{T}}\text{sgn}(S_{H\_te\sigma})+g\cos\eta_{MH}+\frac{ c_{1}g}{V_{T}}N_{TY}\bigg{]} \tag{48}\]  
(2) Lateral channel  
Similarly to the longitudinal channel, the system state quantity is taken as  
\[X=\big{[} \begin{array}{cc}R&\eta_{ML}&\eta_{TL}\end{array} \big{]}^{T} \tag{49}\]  
By substituting into Equation (14), the following can be obtained. \[\dot{X}=\left[ \begin{array}{c}\dot{R}\\ \dot{\eta}_{ML}\\ \dot{\eta}_{TL}\end{array} \right]=h+\left[ \begin{array}{c}0\\ -1/(V_{M}\cos\eta_{MH})\\ 0\end{array} \right]u+\left[ \begin{array}{c}0\\ 0\\ -1/(V_{T}\cos\eta_{TH})\end{array} \right]w \tag{50}\]  
From Equation (50),  
\[h=\left[