## 3 Antenna Pointing Direction and Its Blockage by UAVs
### Target Commands of Antenna Driving Angle
\[\alpha=\frac{R}{H},\beta=\frac{D}{H} \tag{6}\]  
\[\theta_{\rm GC}=\sin^{-1}\frac{\underline{a_{\rm B}}\cdot\underline{n_{\rm B} }}{|\underline{a_{\rm B}}|}=\sin^{-1}\frac{\cos\phi+(\alpha+\beta\sin\psi_{\rm P })\sin\phi}{\sqrt{\alpha^{2}+\beta^{2}+2\alpha\beta\sin\psi_{\rm P}+1}} \tag{7}\]  
In the same way, in order to obtain the azimuth angle target command, the inner product between the directional vector \(\underline{a_{\rm B,xy}}\) shown in Equation (8), which is projected to \(X_{\rm B}-Y_{\rm B}\) plane of the UAV, and the unit vector of the X-axis of the UAV, shown in Equation (9), is calculated as Equation (10). \[\underline{a_{\rm B,xy}}= \begin{bmatrix}D\cos\psi_{\rm P}\\ H\sin\phi+(R+D\sin\psi_{\rm P})\cos\phi\\ 0\end{bmatrix}  \tag{8}\]\[\underline{X_{\text{B}}}= \begin{bmatrix}1\\ 0\\ 0\end{bmatrix}  \tag{9}\]  
\[\underline{a_{\text{B,xy}}}\cdot\underline{X_{\text{B}}}=\Big{|}\underline{a_{ \text{B,xy}}}\Big{|}\cos\psi_{\text{GC}} \tag{10}\]  
In this case, the outer product between \(\underline{X_{\text{B}}}\) and \(a_{\text{B,xy}}\) is used to judge the sign of the azimuth angle, as shown in Equation (11). The outer product vector, \(\underline{X_{\text{B}}}\times\underline{a_{\text{B,xy}}}\), is shown in Equation (12). \[\underline{X_{\text{B}}}\times\underline{a_{\text{B,xy}}}= \begin{bmatrix}0\\ 0\\ Z_{\text{op}}\end{bmatrix}  \tag{11}\]  
\[Z_{\text{op}}=H\!\sin\phi+(R+D\!\sin\psi_{\text{P}})\cos\phi \tag{12}\]  
If the \(Z_{\text{op}}\) is bigger than or equal zero, the outer product vector will be upward, and the two-axis gimbal will be as shown in Figure 6. The azimuth target command, \(\psi_{\text{GC}}\), is derived, as shown in Equation (13). \[\psi_{\text{GC}}=\cos^{-1}\frac{a_{\text{B,XY}}\cdot\underline{X_{\text{B}}}} {\Big{|}\underline{a_{\text{B,XY}}}\Big{|}}=\cos^{-1}\frac{-\beta\!\cos\psi_{ \text{P}}}{\sqrt{\left(\beta\!\cos\psi_{\text{P}}\right)^{2}+\left\{\sin\phi- \left(\alpha+\beta\!\sin\psi_{\text{P}}\right)\cos\phi\right\}^{2}}} \tag{13}\]  
If the \(Z_{\text{op}}\) is smaller than zero, the outer product vector will be downward, and the two-axis gimbal will be as shown in Figure 7. Since the angle formed by the outer product is between \(0^{\circ}\) and \(180^{\circ}\), the azimuth target command, \(\psi_{\text{GC}}\), is derived, as in Equation (14), to represent an angle between \(0^{\circ}\) and \(360^{\circ}\). Figure 6: Two-axis gimbal when the \(Z_{\text{op}}\) is bigger than or equal to zero. Figure 7: Two-axis gimbal when the \(Z_{\text{op}}\) is smaller than zero. \[\psi_{\rm GC} = 2\pi-\cos^{-1}\frac{a_{\rm B,XY}\cdot X_{\rm B}}{\left|\frac{a_{\rm B,XY}\cdot X_{\rm B}}{\left|a_{\rm B,XY}\right|}\right.}\] \[= 2\pi-\cos^{-1}\frac{-\beta\cos\psi_{\rm P}}{\sqrt{(\beta\cos\psi_ {\rm P})^{2}+\{\sin\phi-(\alpha+\beta\sin\psi_{\rm P})\cos\phi\}^{2}}}\]  
As a result of above operation, the elevation angle target command, \(\theta_{\rm GC}\), is derived in Equation (7), whereas the azimuth target command is \(\psi_{\rm GC}\), which is derived in Equations (13) and (14). The \(\theta_{\rm GC}\) is the angle measured from \(X_{\rm B}-Y_{\rm B}\) plane of the UAV, whereas the \(\psi_{\rm GC}\) is the angle measured from the X_B-axis of the UAV.