## 2 Problem Analysis and Formulation
### The Aiming Model of BVR Attack Fire Control  
Consider the aiming geometry diagram of the UCAV in the attack guidance state in the BVR air combat as shown in Figure 4.  
The flight trajectory pointing to the ideal aiming point \(P\) from the current position of the UCAV is the ideal attack guidance trajectory in the state of complete aiming. It means that the UCAV flies a time of \(T_{1}\) along the correct direction at the current velocity \(v_{u}\), launches the BVR AAM at the point \(O\), and then the AAM hits the point \(P\) after flying a distance of \(R\) along the ideal straight line trajectory. At the same time, the target has just reached the point \(P\) after flying at the velocity \(v_{t}\) for a time of \(T\). However, in the actual attack guidance process, there is always a deviation between the actual velocity vector \(\mathbf{v}_{u}\) of the UCAV and the ideal attack guidance trajectory, that is, the deviation vector \(\mathbf{\Delta L}\) shown in Figure 4. Suppose the actual velocity vectors of the target is \(\mathbf{v}_{t}\), and \(\mathbf{D}\) is the distance vector pointing to the target from the current position of the UCAV. The UCAV launches the AAM at the point \(O^{\prime}\) after flying a time of \(T_{1}\) along the direction of \(\mathbf{v}_{u}\). If the AAM still flies along the ideal distance vector \(\mathbf{R}\), then the hit point is \(P^{\prime}\). The vector pointing from \(P^{\prime}\) to \(P\) is \(\mathbf{\Delta L}\). The angle between \(\mathbf{v}_{u}\) and the ideal attack guidance trajectory is the aiming error angle \(\delta\), and the lead angle of the UCAV is \(\eta\), where \(\eta=\sqrt{\eta_{y}^{2}+\eta_{z}^{2}}\). Besides, under the assumptions in this paper, \(\delta\) and \(\eta\) correspond to the angles in Figure 2.  
Based on this, the aiming vector equation can be obtained as  
\[\mathbf{v}_{u}T_{1}+\mathbf{R}+\mathbf{\Delta L}=\mathbf{D}+\mathbf{v}_{t}T \tag{3}\]  
where \(T=T_{1}+R/v_{m}\). \(R\) and \(v_{m}\) are the ideal attack range and the corresponding average velocity along the ideal straight line trajectory of the AAM, which are both preset according to the missile performance. To facilitate the projection decomposition calculation of the deviation vector, \(\mathbf{\Delta L}\) is represented as  
\[\mathbf{\Delta L}=\mathbf{D}+\mathbf{v}_{t}T-(\mathbf{R}+\mathbf{v}_{u}(T-R/v_{m})) \tag{4}\]  
The projection of \(\mathbf{\Delta L}\) on the trajectory coordinate system of the UCAV is decomposed to obtain three deviation components on the corresponding axes, \(\Delta L_{x}\), \(\Delta L_{y}\), and \(\Delta L_{z}\), that is  
\[\left[ \begin{array}{c}\Delta L_{x}\\ \Delta L_{y}\\ \Delta L_{z}\end{array} \right]= \tag{5}\]  
Figure 4: Schematic diagram of the aiming geometry of the UCAV in the attack guidance state in the BVR air combat. where \([D_{x},D_{y},D_{z}]^{\rm T}\) and \([v_{tx},v_{yy},v_{tz}]^{\rm T}\) are the projections of \(\mathbf{D}\) and \(\mathbf{v}_{t}\) on the three axes of the geographic coordinate system, respectively. \(\mathbf{T}_{z}(\theta_{u})\mathbf{T}_{y}(\phi_{u})\) is the transformation matrix from the geographic coordinate system to the trajectory coordinate system of the UCAV, which is given by  
\[\mathbf{T}_{z}(\theta_{u})\mathbf{T}_{y}(\phi_{u})=\left[ \begin{array}{ccc}\cos\theta _{u}\cos\phi_{u}&\sin\theta_{u}&\cos\theta_{u}\sin\phi_{u}\\ -\sin\theta_{u}\cos\phi_{u}&\cos\theta_{u}&-\sin\theta_{u}\sin\phi_{u}\\ -\sin\phi_{u}&0&\cos\phi_{u}\end{array} \right] \tag{6}\]  
By changing the attack guidance time \(T_{1}\) of the UCAV, it can always make \(\Delta L_{x}=0\), thus the full attack time \(T\) can be obtained as  
\[T=\frac{R(1-v_{u}/v_{m})-D_{x}\cos\theta_{u}\cos\phi_{u}-D_{y}\sin\theta_{u}- D_{z}\cos\theta_{u}\sin\phi_{u}}{v_{tx}\cos\theta_{u}\cos\phi_{u}+v_{ty}\sin \theta_{u}+v_{tz}\cos\theta_{u}\sin\phi_{u}-v_{u}} \tag{7}\]  
Therefore, \(\Delta L_{y}\) and \(\Delta L_{z}\) can be obtained by combining Equations (5) and (7).