## 3 Antenna Pointing Direction and Its Blockage by UAVs  
### Target Commands of Antenna Driving Angle  
In the proposed system, two-axis gimbals with directional antennas are attached to the underside of the fixed-wing UAV to establish a wireless link between a fixed-wing UAV for relay and one ground station. The gimbals point the antenna's main beams toward the ground station by driving the azimuth and elevation angles. The target antenna's pointing direction is calculated by using the turning position and attitude of the fixed-wing UAV and the position of the ground station, as shown in Figures 4 and 5. In order to use the azimuth and elevation angles for the control system as target commands, those angles must be expressed in the body-fixed coordinates of the UAV. Those angles are calculated using the directional vector \(\underline{a}\) from the UAV to the ground station, which is expressed in \(X_{\rm E}-Y_{\rm E}-Z_{\rm E}\) coordinates, as well as transforming it to \(\underline{a}_{\rm B}\), which is expressed in \(X_{\rm B}-Y_{\rm B}-Z_{\rm B}\) coordinates. Detailed operations are shown in Equation  
Figure 4: Position of the fixed-wing UAV and ground station. Figure 5: Driving angle of the two-axis gimbal. (1)-(6) for the elevation angle target command \(\theta_{\rm GC}\) and in Equation (8)-(11) for the azimuth target command \(\psi_{\rm GC}\)  
The vector in the direction from the fixed-wing UAV to the ground station is expressed, as shown in Equation (1), by using positions for the UAV and the station expressed in \(X_{\rm E}-Y_{\rm E}-Z_{\rm E}\) coordinates. Position data for the UAV are measured by using the GNSS system. \[\underline{a}=\underline{QO_{\rm E}}= \begin{bmatrix}-R\cos\psi_{\rm P}\\ -R\sin\psi_{\rm P}-D\\ -H\end{bmatrix}  \tag{1}\]  
The transformation matrix between the ground-fixed coordinate system and the fixed-wing UAV's body frame coordinate system is defined as shown in Equation (2). \[ \begin{bmatrix}X_{\rm B}\\ Y_{\rm B}\\ Z_{\rm B}\end{bmatrix} = \begin{bmatrix}\sin\psi_{\rm P}&-\cos\psi_{\rm P}&0\\ -\cos\psi_{\rm P}\cos\phi&-\sin\psi_{\rm P}\cos\phi&-\sin\phi\\ \cos\psi_{\rm P}\sin\phi&\sin\psi_{\rm P}\sin\phi&-\cos\phi\end{bmatrix} \begin{bmatrix}X_{\rm E}\\ Y_{\rm E}\\ Z_{\rm E}\end{bmatrix}  \tag{2}\]  
Using the transformation matrix in Equation (2), the vector in Equation (1) is ex-pressed in the body frame coordinate system as in Equation (3). \[ \begin{split}\underline{a_{\rm B}}&=\begin{bmatrix}\sin\psi_{ \rm P}&-\cos\psi_{\rm P}&0\\ -\cos\psi_{\rm P}\cos\phi&-\sin\psi_{\rm P}\cos\phi&-\sin\phi\\ \cos\psi_{\rm P}\sin\phi&\sin\psi_{\rm P}\sin\phi&-\cos\phi\end{bmatrix} \underline{a}\\ &=\begin{bmatrix}D\cos\psi_{\rm P}\\ H\sin\phi+(R+D\sin\psi_{\rm P})\cos\phi\\ H\cos\phi-(R+D\sin\psi_{\rm P})\sin\phi\end{bmatrix}\end{split}  \tag{3}\]  
By using the unit vector \(\underline{n_{\rm B}}\) in the Z-axis direction of the body frame coordinate system expressed in Equation (4), the inner product with \(\underline{a_{\rm B}}\) and \(\underline{n_{\rm B}}\) is calculated as shown in Equation (5). \[\underline{n_{\rm B}}= \begin{bmatrix}0\\ 0\\ 1\end{bmatrix}  \tag{4}\]  
\[\underline{a_{\rm B}}\cdot\underline{n_{\rm B}}=|\underline{a_{\rm B}}|\cos \left(\frac{\pi}{2}-\theta_{\rm GC}\right) \tag{5}\]  
Defining parameters \(\alpha\) and \(\beta\), as shown in Equation (6), and by using the turning radius \(R\), the horizontal distance \(D\), and flight altitude \(H\) of the UAV, the elevation angle target command, \(\theta_{\rm GC}\), is derived as shown in Equation (7).