# Rapid Tracking Satellite Servo Control for Three-Axis Satcom-on-the-Move Antenna
## 2 Kinematic Modeling of Three-Axis SOTM Antenna
\begin{array}{l}r_{11}=\cos\theta_{3}(\sin\theta_{1}\cos\theta_{2}\cos \varphi-\cos\theta_{1}\sin\theta_{2})+\sin\theta_{1}\sin\theta_{3}\sin\varphi, \\ r_{21}=\cos\theta_{3}(\sin\theta_{1}\sin\theta_{2}-\cos\theta_{1}\cos\theta_{2 }\cos\varphi)-\cos\theta_{1}\sin\theta_{3}\sin\varphi,\\ r_{31}=-\cos\theta_{2}\sin\theta_{3}\cos\varphi-\sin\theta_{3}\cos\varphi, \\ r_{12}=-\sin\theta_{3}(\sin\theta_{1}\cos\theta_{2}\cos\varphi-\cos\theta_{1} \sin\theta_{2})+\sin\theta_{1}\cos\theta_{3}\sin\varphi,\\ r_{22}=-\sin\theta_{3}(-\cos\theta_{1}\cos\theta_{2}\cos\varphi+\sin\theta_{ 1}\sin\theta_{2})-\cos\theta_{1}\cos\theta_{3}\sin\varphi,\\ r_{32}=\cos\theta_{2}\sin\theta_{3}\sin\varphi-\cos\theta_{3}\cos\varphi, \\ r_{13}=\sin\theta_{1}\sin\theta_{2}\cos\varphi+\cos\theta_{1}\cos\theta_{2}, \\ r_{23}=-\cos\theta_{1}\sin\theta_{2}\cos\varphi-\sin\theta_{1}\cos\theta_{2}, \\ r_{33}=\sin\theta_{2}\sin\varphi,\\ p_{x}=-\sin\theta_{1}\sin\theta\cdot l_{3}-\sin\theta_{1}\cos\varphi\sin \beta\cdot l_{2}+\sin\theta_{1}\sin\varphi\cos\beta\cdot l_{2},\\ p_{y}=\cos\theta_{1}\sin\varphi\cdot l_{3}+\cos\theta_{1}\cos\varphi\sin \beta\cdot l_{2}-\cos\theta_{1}\sin\varphi\cos\beta\cdot l_{2},\\ p_{z}=\cos\varphi\cdot l_{3}+\sin\varphi\sin\beta\cdot l_{2}+\cos\varphi\cos \beta\cdot l_{2}+l_{1},\\ \end{array} \]  
\[ \begin{array}{c}{}_{Puint}R=\begin{bmatrix}r_{11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33}\end{bmatrix}\end{array} \]  
is the direction matrix pointing the antenna towards the target satellite. Next, the inverse kinematics solution of the three-axis SOTM antenna will be obtained, so as to obtain the real-time rotation angles of the azimuth axis, pitch axis and roll axis during the operation of the three-axis SOTM antenna. \( \begin{subarray}[]{c}{}_{Puint}^{Puint}R\\ \begin{subarray}[]{c}Ante\end{subarray}\end{subarray} \) in Equation (7) can be calculated by the following equation:  
\[ \begin{array}{c}{}_{Ante}^{Point}R={}_{1}^{0}R{}_{2}^{1}R{}_{3}^{2}R{}_{4}^{ 3}R{}_{5}^{4}R{}_{6}^{5}R\end{array}  \tag{8}\]  
where \({}_{i}^{i-1}R\) represents the rotation matrix from coordinate system \(i-1\) to coordinate system \(i\). The inverse kinematics solution of the three-axis SOTM antenna contains the following two cases:  
(1) During the process of pointing the antenna towards the satellite in SOTM, the entire hemisphere can be scanned using azimuth and pitch axes. When the pitch angle of the tracking antenna is not 90\({}^{\circ}\), the roll axis is fixed and \(\theta_{2}^{*}\) is set to 0. The inverse kinematic solution for the azimuth angle \(\theta_{1}^{*}\) is obtained as:  
\[\theta_{1}^{*}=\theta_{1}=\text{atan2}\Big{(}1,\pm\sqrt{{r_{13}}^{2}+{r_{23}}^{2}- 1}\Big{)}-\text{atan2}(r_{23},r_{13}) \tag{9}\]  
The inverse kinematic solution for the pitch angle \(\theta_{3}^{*}\) is:  
\[\theta_{3}^{*}=\theta_{3}=\text{atan2}(\sin\varphi,-\cos\varphi) \tag{10}\]  
or  
\[\theta_{3}^{*}=\theta_{3}=\text{atan2}(-\sin\varphi,\cos\varphi) \tag{11}\]  
Since the pitch angle must be between \([0^{\circ},\,90^{\circ}]\), therefore \(\theta_{3}^{*}\) should be within the range of \([0^{\circ},\,90^{\circ}]\). (2) When the pitch angle of the SOTM antenna reaches \(90^{\circ}\), the antenna is in a locked state, and changing the azimuth axis cannot change the system's orientation. In this case, the roll axis needs to be used, with the azimuth axis in a fixed state and \(\theta_{1}^{*}\) is taken as the angle at which the antenna was last operated between entering the locked state; at this point, \(\theta_{1}^{*}\) is known. The inverse kinematic solution for the roll angle \(\theta_{2}^{*}\) is obtained as:  
\[\left\{ \begin{array}{l}\sin\theta_{2}=\frac{r_{13}\sin\theta_{1}+r_{23}\cos \theta_{1}}{\cos^{2}\theta_{1}-\sin^{2}\theta_{1}}=a\\ \cos\theta_{2}=\frac{r_{13}\sin\theta_{1}+r_{23}\cos\theta_{1}}{\cos\sin^{2} \theta_{1}-\cos\varphi\sin^{2}\theta_{1}}=b\end{array} \right. \tag{12}\]  
Figure 3: Parameters and coordinate system configuration of three-axis SOTM antenna.