## 2 Variation in Jamming Power Reception Due to the Attitude Change
The angular location \(\theta_{I}\) and \(\phi_{I}\) for the UAV in reference to the local coordinate system of the jammer antenna \(\left(\hat{\mathbf{u}}_{I},\,\hat{\upsilon}_{I},\,\hat{\upsilon}_{I}\right)\) can be determined in a similar manner. However, if a UAV attitude change occurs, the radiation pattern of the UAV antenna changes in reference to the global coordinate system. The attitude change can be described as three successive rotations [16; 17]. The attitude change is modeled by the following equation:  
\[\left[ \begin{array}{c}\hat{\mathbf{u}}_{A}^{\prime}\\ \hat{\mathbf{\upsilon}}_{A}^{\prime}\\ \hat{\mathbf{\upsilon}}_{A}^{\prime}\end{array} \right]=R(\hat{\mathbf{n}}_{Y},\,\theta _{Y})R(\hat{\mathbf{n}}_{P},\,\theta_{P})R(\hat{\mathbf{n}}_{R},\,\theta_{R})\left[ \begin{array}{c}\hat{\mathbf{u}}_{A}\\ \hat{\mathbf{\upsilon}}_{A}\\ \hat{\upsilon}_{A}\end{array} \right], \tag{10}\]  
where \(\hat{\mathbf{u}}_{A}^{\prime},\hat{\mathbf{\upsilon}}_{A}^{\prime}\), and \(\hat{\mathbf{\upsilon}}_{A}^{\prime}\) are the unit vectors representing the local coordinate system of the UAV antenna affected by the attitude change; \(R(\hat{\mathbf{n}}_{R},\,\theta_{R})\), \(R(\hat{\mathbf{n}}_{P},\,\theta_{P})\), and \(R(\hat{\mathbf{n}}_{Y},\,\theta_{Y})\) are, respectively, the rotation transformation matrix of roll, pitch, and yaw; \(\hat{\mathbf{n}}_{R}\), \(\hat{\mathbf{n}}_{P}\), and \(\hat{\mathbf{n}}_{Y}\) are, respectively, the rotation axes of roll, pitch, and yaw; \(\theta_{R},\,\theta_{P}\), and \(\theta_{Y}\) are, respectively, the rotation angles of roll, pitch, and yaw. Equation (8) is for the case where the antenna attitude change is in the order of roll, pitch, and yaw. If the rotation order changes, the equation can be changed. The rotation transformation matrix \(R(\hat{\mathbf{n}}_{R},\,\theta_{R})\), for example, is defined as:  
\[R(\hat{\mathbf{n}}_{R},\,\theta_{R})=\left[ \begin{array}{rl}{n_{R,x}}^{2}(1-\cos \theta_{R})+\cos\theta_{R}&{n_{R,x}}{n_{R,y}}(1-\cos\theta_{R})-{n_{R,z}}\sin \theta_{R}&{n_{R,x}}{n_{R,z}}(1-\cos\theta_{R})+{n_{R,y}}\sin\theta_{R}\\ {n_{R,x}}{n_{R,y}}(1-\cos\theta_{R})+{n_{R,z}}\sin\theta_{R}&{n_{R,y}}^{2}(1- \cos\theta_{R})+\cos\theta_{R}&{n_{R,y}}{n_{R,z}}(1-\cos\theta_{R})-{n_{R,x}} \sin\theta_{R}\\ {n_{R,x}}{n_{R,z}}(1-\cos\theta_{R})-{n_{R,y}}\sin\theta_{R}&{n_{R,y}}{n_{R,z }}(1-\cos\theta_{R})+{n_{R,x}}\sin\theta_{R}&{n_{R,z}}^{2}(1-\cos\theta_{R})+ \cos\theta_{R}\end{array} \right], \tag{11}\]  
where \(\hat{\mathbf{n}}_{R}={n_{R,x}}\hat{\mathbf{x}}+{n_{R,y}}\hat{\mathbf{y}}+{n_{R,z}}\hat\mathbf{{z}}\). The angular location \(\theta_{A}^{\prime}\) and \(\phi_{A}^{\prime}\) for the jammer in reference to \((\hat{\mathbf{u}}_{A}^{\prime},\,\hat{\mathbf{\upsilon}}_{A}^{\prime},\,\hat{\mathbf{ \upsilon}}_{A}^{\prime})\) can be determined by the following equation:  
\[\theta_{A}^{\prime}=\cos^{-1}\left(\frac{\vec{r}_{A}\cdot\hat{\mathbf{\upsilon} }_{A}^{\prime}}{|\vec{r}_{A}||\hat{\mathbf{\upsilon}}_{A}^{\prime}|}\right) \tag{12}\]  
\[\phi_{A}^{\prime}=\left\{ \begin{array}{rl}&+\cos^{-1}\left(\frac{\vec{r}_{A,uv}\cdot\hat{\mathbf{\upsilon}}_{A}^{\prime}}{|\vec{r}_{A,uv}||\hat{u}_{A}^{ \prime}|}\right)&\text{if }\vec{r}_{A}\cdot\hat{\mathbf{\upsilon}}_{A}^{\prime}\geq 0\\ &-\cos^{-1}\left(\frac{\vec{r}_{A,uv}\cdot\hat{u}_{A}^{\prime}}{|\vec{r}_{A,uv}|| \hat{u}_{A}^{\prime}|}\right)&\text{if }\vec{r}_{A}\cdot\hat{\mathbf{\upsilon}}_{A}^{\prime}<0\end{array} \right., \tag{13}\]  
where \(\vec{r}_{A,uv}^{\prime}(=\overline{R}_{A}-(\overline{R}_{A}\cdot\hat{\mathbf{ \upsilon}}_{A}^{\prime})\hat{\mathbf{\upsilon}}_{A}^{\prime})\) is the projection of \(\overline{r}_{A}\) on the plane containing \(\hat{\mathbf{u}}_{A}^{\prime}\) and \(\hat{\mathbf{\upsilon}}_{A}^{\prime}\). The jamming power received by the UAV receiver affected by the attitude change \(P_{A}^{\prime}\) can be calculated by following equation:  
\[