# Rapid Tracking Satellite Servo Control for Three-Axis Satcom-on-the-Move Antenna
## 2 Kinematic Modeling of Three-Axis SOTM Antenna
In order to make Figure 3 clear, \(d_{VL1}=0\) and \(d_{VL2}=0\) are set in the figure to represent the two groups of coincident coordinate systems. Coordinate systems 4 and 6 represent the roll and pitch axes, with \(Z_{4}\) and \(Z_{6}\) denoting the rotation axes of the roll and pitch axes, respectively. This paper refers to the above method as the NMDH method. Aiming at the global coordinate system configuration of the three-axis SOTM antenna mechanical structure shown in Figure 3, the D-H table of the three-axis SOTM antenna mechanical structure kinematic model is shown in Table 1.  
According to the link parameters shown in the D-H table of the three-axis SOTM antenna system, the transformation matrices between each link \({}_{i}^{i-1}T\) are obtained as follows:  
\[{}_{1}^{0}T= \begin{bmatrix}cos\theta_{1}^{*}&-sin\theta_{1}^{*}&0&0\\ sin\theta_{1}^{*}&cos\theta_{1}^{*}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{1}\]  
\[{}_{\frac{1}{2}}T= \begin{bmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&l_{1}\\ 0&0&0&1\end{bmatrix}  \tag{2}\]  
\[{}_{\frac{2}{3}}T= \begin{bmatrix}0&-1&0&0\\ cos\left(\pi/2+\beta\right)&0&-sin\left(\pi/2+\beta\right)&0\\ sin\left(\pi/2+\beta\right)&0&cos\left(\pi/2+\beta\right)&0\\ 0&0&0&1\end{bmatrix}  \tag{3}\] \begin{table}
\begin{tabular}{c c c c c} \hline \(i\) & \(\boldsymbol{\alpha_{i-1}}\) & \(\boldsymbol{a_{i-1}}\) & \(\boldsymbol{d_{i}}\) & \(\boldsymbol{\theta_{i}}\) \\ \hline
1 & 0 & 0 & 0 & \(\theta_{1}^{*}\) \\
2 & \(-\pi\) & 0 & \(-l_{1}\) & 0 \\
3 & \(\left(\pi/2\right)+\beta\) & 0 & 0 & \(\pi/2\) \\
4 & \(-\pi\) & \(-l_{2}\)sin \(\beta\) & \(l_{2}\)cos \(\beta\) & \(\theta_{2}^{*}+\pi\) \\
5 & \(-\pi\) & 0 & \(l_{3}\) & 0 \\
6 & \(-(\pi/2)\) & 0 & 0 & \(\theta_{3}^{*}\) \\ \hline \end{tabular}
\end{table} 
Table 1: D-H table of the three-axis SOTM antenna. \[ \begin{array}{c}{}_{4}^{3}T=\begin{bmatrix}cos\ (\theta_{2}^{*}+\pi)&-sin\ ( \theta_{2}^{*}+\pi)&0&-l_{2}sin\beta\\ -sin\ (\theta_{2}^{*}+\pi)&-cos\ (\theta_{2}^{*}+\pi)&0&0\\ 0&0&-1&-l_{2}cos\beta\\ 0&0&0&1\end{bmatrix}\end{array}  \tag{4}\]  
\[ \begin{array}{c}{}_{4}^{4}T=\begin{bmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&-l_{3}\\ 0&0&0&1\end{bmatrix}\end{array}  \tag{5}\]  
\[ \begin{array}{c}{}_{5}^{5}T=\begin{bmatrix}cos\theta_{3}^{*}&-sin\theta_{3}^{* }&0&0\\ 0&0&1&0\\ -sin\theta_{3}^{*}&-cos\theta_{3}^{*}&0&0\\ 0&0&0&1\end{bmatrix}\end{array}  \tag{6}\]  
The total transformation matrix \({}_{6}^{0}T\) between the reference coordinate system 0 and the end pitch axis coordinate system 6 in the three-axis antenna system can be obtained from the above equations, then the forward kinematics model is calculated as follows:  
Let \(\theta_{1}^{*}=\theta_{1}\), \(\theta_{2}^{*}+\pi=\theta_{2}\), \(\theta_{3}^{*}=\theta_{3}\), \(\frac{\pi}{2}+\beta=\varphi\). Then,  
\[ \begin{array}{c}{}_{6}^{0}T={}_{1}^{0}T{}_{1}^{2}T{}_{3}^{3}T{}_{4}^{3}T{}_{5 }^{5}T=\begin{bmatrix}r_{11}&r_{12}&r_{13}&p_{x}\\ r_{21}&r_{22}&r_{23}&p_{y}\\ r_{31}&r_{32}&r_{33}&p_{z}\\ 0&0&0&1\end{bmatrix}=\begin{bmatrix}P_{Aute}&p_{x}\\ \begin{bmatrix}P_{y}\\ \begin{bmatrix}P_{z}\\ 0\\ \end{bmatrix}\end{bmatrix}\\ \end{array}  \tag{7}\]  
where  
\[