## 2 Variation in Jamming Power Reception Due to the Attitude Change
The unit vectors (i.e., \(\hat{\mathbf{r}}\), \(\hat{\mathbf{\theta}}\), and \(\hat{\mathbf{\phi}}\)) are calculated by the following equation:  
\[\left[ \begin{array}{c}\hat{\mathbf{p}}\\ \hat{\mathbf{\theta}}\\ \hat{\mathbf{\phi}}\end{array} \right]=\left[ \begin{array}{ccc}\sin\theta\cos \phi&\sin\theta\sin\phi&\cos\theta\\ \cos\theta\cos\phi&\cos\theta\sin\phi&-\sin\theta\\ -\sin\phi&\cos\phi&0\end{array} \right]\left[ \begin{array}{c}\hat{\mathbf{u}}\\ \hat{\mathbf{v}}\\ \hat{\mathbf{w}}\end{array} \right], \tag{5}\]  
where \(\hat{\mathbf{u}}\), \(\hat{\mathbf{v}}\), and \(\hat{\mathbf{w}}\) are, respectively, the unit vectors along the \(u-\), \(v-\), and \(w-\) axis representing the local-coordinate system of the antenna. The unit vectors (i.e., \(\hat{\mathbf{u}}\), \(\hat{\mathbf{v}}\), and \(\hat{\mathbf{w}}\)) are expressed as:  
\[\left[ \begin{array}{c}\hat{\mathbf{u}}\\ \hat{\mathbf{v}}\\ \hat{\mathbf{w}}\end{array} \right]=\left[ \begin{array}{ccc}\hat{\mathbf{u}}\cdot\hat {\mathbf{x}}&\hat{\mathbf{u}}\cdot\hat{\mathbf{y}}&\hat{\mathbf{u}}\cdot\hat{\mathbf{z}}\\ \hat{\mathbf{v}}\cdot\hat{\mathbf{x}}&\hat{\mathbf{v}}\cdot\hat{\mathbf{y}}&\hat{\mathbf{v}}\cdot \hat{\mathbf{z}}\\ \hat{\mathbf{v}}\cdot\hat{\mathbf{x}}&\hat{\mathbf{w}}\cdot\hat{\mathbf{y}}&\hat{\mathbf{w}}\cdot \hat{\mathbf{z}}\\ \end{array} \right]\left[ \begin{array}{c}\hat{\mathbf{x}}\\ \hat{\mathbf{y}}\\ \hat{\mathbf{z}}\end{array} \right]. \tag{6}\]  
The electric field components of the antenna for the \(x\), \(y\), and \(z\) components in the far field, \(E^{0}_{x}(\theta,\ \phi)\), \(E^{0}_{y}(\theta,\ \phi)\), and \(E^{0}_{z}(\theta,\ \phi)\), can be calculated by the following equation:  
\[\left[ \begin{array}{ccc}E^{0}_{x}(\theta,\ \phi)&E^{0}_{y}(\theta,\ \phi)&E^{0}_{z}(\theta,\ \phi)\\ \end{array} \right] \tag{7}\] \[=\left[ \begin{array}{ccc}0&E^{0}_{y}(\theta,\ \phi)&E^{0}_{\phi}(\theta,\ \phi)\\ \end{array} \right]\left[ \begin{array}{ccc}\sin\theta\cos\phi&\sin\theta \sin\phi&\cos\theta\\ \cos\theta\cos\phi&\cos\theta\sin\phi&-\sin\theta\\ -\sin\phi&\cos\phi&0\end{array} \right]\left[ \begin{array}{ccc}\hat{\mathbf{u} }\cdot\hat{\mathbf{x}}&\hat{\mathbf{u}}\cdot\hat{\mathbf{y}}&\hat{\mathbf{u}}\cdot\hat{\mathbf{z} }\\ \hat{\mathbf{v}}\cdot\hat{\mathbf{x}}&\hat{\mathbf{v}}\cdot\hat{\mathbf{y}}&\hat{\mathbf{v}}\cdot \hat{\mathbf{z}}\\ \hat{\mathbf{w}}\cdot\hat{\mathbf{x}}&\hat{\mathbf{w}}\cdot\hat{\mathbf{y}}&\hat{\mathbf{w}}\cdot \hat{\mathbf{z}}\\ \end{array} \right].\]  
\(G(\theta,\ \phi)\), \(\hat{\mathbf{\rho}}(\theta,\ \phi)\), and \(\overline{E}^{0}(\theta,\ \phi)\) are varied as the attitude of the antenna changes in a given direction in reference to the global coordinate system. To determine the parameters, the angular locations \(\theta\) and \(\phi\) need to be determined first. For the local coordinate system of the UAV antenna (\(\hat{\mathbf{u}}_{A},\ \hat{\mathbf{v}}_{A},\ \hat{\mathbf{w}}_{A}\)) and the vector from the UAV to the jammer \(\overline{\mathbf{r}}_{A}\) in Figure 2, the angular location for the jammer \(\theta_{A}\) and \(\phi_{A}\) is determined as follows:  
\[\theta_{A}=\cos^{-1}\!\left(\frac{\overline{\mathbf{r}}_{A}\cdot\hat{\mathbf{w}}_{A} }{\|\overline{\mathbf{r}}_{A}\|\hat{\mathbf{w}}_{A}\|}\right) \tag{8}\]  
Figure 1: Geometry of the UAV jammed by ground jammers considering the effect of the change in UAV attitude on the UAV jamming link. \[\phi_{A}=\left\{ \begin{array}{rl}&+\cos^{-1}\left(\frac{\vec{r}_{A,uv}\cdot\hat{ \psi}_{A}}{|\vec{r}_{A,uv}||\hat{u}_{A}|}\right)&\text{if }\vec{r}_{A}\cdot\hat{\psi}_{A}\geq 0\\ &-\cos^{-1}\left(\frac{\vec{r}_{A,uv}\cdot\hat{u}_{A}}{|\vec{r}_{A,uv}||\hat{u} _{A}|}\right)&\text{if }\vec{r}_{A}\cdot\hat{\psi}_{A}<0\end{array} \right., \tag{9}\]  
where \(\vec{r}_{A,uv}(=\vec{r}_{A}-(\vec{r}_{A}\cdot\hat{\psi}_{A})\hat{\psi}_{A})\) is the projection of \(\vec{r}_{A}\) on the plane containing \(\hat{\mathbf{u}}_{A}\) and \(\hat{\psi}_{A}\).