# Mdpj _sciences_
## 3 Method  
### Aircraft Model and Missile Model  
In maneuver decision-making, the aircraft model adopts normal overload, tangential overload and roll angle as control parameters. To simplify the complexity of the problem, the angles of attack and angle of side slip are regarded as 0, and the ground coordinate system is treated as the inertial system. Meanwhile, the effects of the rotation of the earth are overlooked. The kinematic and dynamic model is shown as follows [45]:  
\[\left\{\begin{array}{l}\dot{x}=v\cos\gamma\cos\psi\\ \dot{y}=v\cos\gamma\sin\psi\\ \dot{z}=v\sin\gamma\\ \dot{v}=g(n_{x}-\sin\gamma)\\ \dot{\gamma}=\frac{g}{v\cos\gamma}n_{z}\sin\mu\\ n_{x}\in[-0.5,1.5]\\ n_{z}\in[-3,9]\\ \mu\in[-\pi,\pi]\end{array}\right. \tag{1}\]  
where \(x\), \(y\), and \(z\) indicate the positions of the aircraft in the inertial coordinate system; \(\gamma\) is the pitch angle, \(\psi\) is the yaw angle, \(v\) is the velocity, and \(g\) is the acceleration of gravity. Roll angle \(\mu\), tangential overload \(n_{x}\), and normal overload \(n_{z}\) are control parameters. The action space consists of \(\mu\), \(n_{x}\), and \(n_{z}\), which is continuous, as shown in (1). This means that the control parameters are selected from the three intervals instead of using space discretization as in previous methods. The kinematic model of the missile is [46]:  
\[\left\{\begin{array}{l}\dot{x}_{m}=v_{m}\cos\gamma_{m}\cos\psi_{m}\\ \dot{y}_{m}=v_{m}\cos\gamma_{m}\sin\psi_{m}\\ \dot{z}_{m}=v_{m}\sin\gamma_{m}\end{array}\right. \tag{2}\]  
where \(x_{m}\), \(y_{m}\), and \(z_{m}\) indicate the positions of the missile in the inertial coordinate system; \(v_{m}\) is the velocity, \(\gamma_{m}\) is the pitch angle, and \(\psi_{m}\) is the yaw angle. The dynamic model of the missile is:  
\[\left\{\begin{array}{l}\dot{v}_{m}=\frac{(P_{m}-Q_{m})g}{G_{m}}-g\sin\gamma_ {m}\\ \dot{\psi}_{m}=\frac{m_{m}g}{v_{m}\cos\gamma_{m}}\\ \dot{\gamma}_{m}=\frac{m_{m}g}{v_{m}}-\frac{g}{v_{m}}\cos\gamma_{m}\end{array}\right.\]  
where \(t_{w}=8.0\) s, \(\rho=0.607\), \(S_{m}=0.0324\), \(C_{Dm}=0.9\). It is assumed that the guidance coefficient of proportional guidance law is \(K\) in the control planes. The two overloads in the yaw and pitch directions are defined as:  
\[\left\{\begin{array}{l}n_{mc}=K\cdot\frac{\tau_{m}\cos\gamma_{L}}{g}\big{[} \dot{\beta}+\tan\epsilon\tan(\epsilon+\beta)\dot{\epsilon}\big{]}\\ n_{mh}=\frac{\tau_{m}}{g}\frac{K}{\cos(\epsilon+\beta)}\dot{\epsilon}\\ \end{array}\right. \tag{7}\]  
\[\left\{\begin{array}{l}\beta=\arctan(r_{y}/r_{x})\\ \varepsilon=\arctan(r_{z}/\sqrt{r_{x}^{2}+r_{y}^{2}})\\ \end{array}\right. \tag{8}\]  
\[\left\{\begin{array}{l}\dot{\beta}=(\dot{r}_{y}r_{x}-r_{y}r_{x})/(r_{x}^{2}+ r_{y}^{2})\\ \dot{\epsilon}=\frac{(r_{x}^{2}+r_{y}^{2})\dot{r}_{z}-r_{z}(r_{x}r_{x}+\dot{r} _{y}r_{y})}{R^{2}\sqrt{r_{x}^{2}+r_{y}^{2}}}\\ \end{array}\right. \tag{9}\]  
where \(\beta\) and \(\varepsilon\) are the yaw angle and pitch angle of line-of-sight, respectively. The line-of-sight vector is the distance vector \(r\), where \(r_{x}=x_{t}-x_{m}\), \(r_{y}=y_{t}-y_{m}\), \(r_{z}=z_{t}-z_{m}\) and \(R=\|r\|=\sqrt{r_{x}^{2}+r_{y}^{2}+r_{z}^{2}}\).  
The maximum overload of the missile is 40. When the minimum distance between the missile and the target is less than 12 m, the target is regarded as being hit; when the missile flight time exceeds 120 s, and it still fails to hit the target, the target is regarded as missed; during the midcourse guidance stage, the target is regarded as missed when its azimuth relative to the aircraft exceeds 85\({}^{\circ}\); during the final guidance stage, the target is regarded as missed when its azimuth relative to missile axis exceeds 70\({}^{\circ}\).