## 2 Description of Cooperative 3D Terminal Guidance for Multiple Missiles  
The relative motion relationship between the missile and the target is shown in Figure 1. Herein, M and T represent the missile and the target, respectively. The ground inertial coordinate system is expressed in the \(A\)-\(XYZ\) plane. This is the relative distance between the missile and the target, is the line-of-sight elevation angle and the line-of-sight azimuth angle, represents the trajectory inclination angle and the trajectory deflection angle of the missile, and represents the trajectory inclination angle and the trajectory deflection angle of the target. Unless otherwise specified, all angles were positive counterclockwise.  
According to the relative geometric relationship between the missile and the target in Figure 1, the relative motion can be obtained as:  
\[\left\{\begin{array}{l}\frac{\mathrm{d}\mathbf{L}}{dt}=\frac{\mathrm{d}}{ \mathrm{d}t}(\dot{r}i_{L})=V_{T}\dot{t}_{T}-V_{M}\dot{t}_{M}=\dot{r}i_{L}+ \mathbf{\Omega}_{L}\times\dot{r}i_{L}\\ \mathbf{A}_{T}=A_{y}\dot{t}_{T}+A_{z}k_{T}=\mathbf{\Omega}_{L}\times\mathbf{V} _{T}+\mathbf{\Omega}_{T}\times\mathbf{V}_{T}\\ \mathbf{\Omega}_{M}=A_{ym}\dot{m}_{M}+A_{zm}k_{M}=\mathbf{\Omega}_{L}\times V _{M}+\mathbf{\Omega}_{M}\times V_{M}\\ \mathbf{\Omega}_{L}=\dot{\psi}_{L}\sin\theta_{L}i_{L}-\dot{\theta}i_{L}+\dot{ \psi}_{L}\cos\theta_{L}k_{L}=\dot{\lambda}_{x}i_{L}+\dot{\lambda}_{y}i_{L}+ \dot{\lambda}_{z}k_{L}\\ \mathbf{\Omega}_{M}=\dot{\psi}_{m}\sin\theta_{m}i_{M}-\theta_{m}\dot{y}_{M}+ \dot{\psi}_{m}\cos\theta_{m}k_{M}\\ \mathbf{\Omega}_{T}=\dot{\psi}_{t}\sin\theta_{t}i_{T}-\dot{\theta}i_{T}+\dot{ \psi}_{t}\cos\theta_{t}k_{T}\end{array}\right. \tag{1}\]  
where \(\mathbf{V}_{T}\) and \(\mathbf{V}_{M}\) are the velocity vectors of the target and the missile, respectively; \(A_{zm}\) and \(A_{ym}\) are the normal accelerations of the turning plane and the dive plane of the missile, respectively; \(A_{zt}\) and \(A_{yt}\) are the normal accelerations of the target turning plane and the dive plane, respectively; \(\mathbf{\Omega}_{L}\) is the rotation angular velocity vector of the line-of-sight coordinate system; \(\mathbf{\Omega}_{T}\) is the rotation angular velocity vector of the target relative to the line of sight coordinate system; and \(\mathbf{\Omega}_{M}\) is the rotation angular velocity vector of the missile relative to the line of sight coordinate system.  
From Equation (1), the following differential equations can be obtained:  
\[\left\{\begin{array}{l}\dot{r}=V_{m}\cdot(\rho\cos\theta_{t}\cos\psi_{t}- \cos\theta_{m}\cos\psi_{m})\\ r\dot{\lambda}_{y}=V_{m}\cdot(\sin\theta_{m}-\rho\sin\theta_{t})\\ r\dot{\lambda}_{z}=V_{m}\cdot(\rho\cos\theta_{t}\sin\psi_{t}-\cos\theta_{m} \sin\psi_{m})\\ \dot{\theta}_{m}=\frac{A_{ym}}{V_{m}}+\frac{1}{r}V_{m}\tan\lambda_{y}\sin\psi _{m}(\rho\cos\theta_{t}\sin\psi_{t}-\cos\theta_{m}\sin\psi_{m})-\frac{1}{r}V_ {m}\cos\psi_{m}(\rho\sin\theta_{t}-\sin\theta_{m})\\ \dot{\psi}_{m}=\frac{V_{m}}{V_{m}\cos\theta_{m}}-\frac{1}{r\cos\theta_{m}}V_ {m}\sin\theta_{m}\cos\psi_{m}\times\tan\lambda_{y}(\rho\cos\theta_{t}\sin \psi_{t}-\cos\theta_{m}\sin\psi_{m})\\ -\frac{1}{r\cos\theta_{m}}V_{m}\sin\theta_{m}\sin\psi_{m}(\rho\sin\theta_{t}- \sin