## 3 Modeling and Simulations of the Assembly Error Effect on the Stability Accuracy
### Modeling and Simulations of the General Kinematics and Compensation Principle  
According to kinematics analysis of homogeneous coordinate transformation [19], the rotation transformation matrixes from \(\{b\}\) to \(\{a\}\) and from \(\{a\}\) to \(\{p\}\) can be expressed as Equation (3), where \(c\theta_{a}\) means \(cos\theta_{a}\), \(s\theta_{a}\) means \(sin\theta_{a}\), and likewise hereinafter for simplicity. \[R_{b}^{a}=\left[ \begin{array}{ccc}c\theta_{a}&s\theta_{a}&0\\ -s\theta_{a}&c\theta_{a}&0\\ 0&0&1\end{array} \right]\text{, }R_{a}^{p}=\left[ \begin{array}{ccc}c \theta_{p}&0&-s\theta_{p}\\ 0&1&0\\ s\theta_{p}&0&c\theta_{p}\end{array} \right]\text{, }R_{a}^{p}=\left[ \begin{array}{ccc}c \theta_{p}&0&-s\theta_{p}\\ 0&1&0\\ s\theta_{p}&0&c\theta_{p}\end{array} \right]\text{ } \tag{3}\]  
The rotation of the azimuth in \(\{i\}\) can be decomposed into that of the azimuth in \(\{b\}\) and that of the system base in \(\{i\}\), which is expressed as Equation (4), where \(\overrightarrow{\omega}_{ba}=\left[0,0,\theta a\right]\), representing the angular velocity vector of the azimuth in \(\{b\}\). \[\overrightarrow{\omega}_{ia}=R_{b}^{a}\overrightarrow{\omega}_{ib}+ \overrightarrow{\omega}_{ba}\text{, } \tag{4}\]  
Similarly, the rotation of the pitch in \(\{i\}\) can be decomposed into the pitch rotation in \(\{a\}\) and the azimuth rotation in \(\{i\}\), which is expressed as Equation (5), where \(\overrightarrow{\omega}_{ap}=\left[0,\dot{\theta}_{p},0\right]\), representing the angular velocity vector of the pitch in \(\{a\}\). \[\overrightarrow{\omega}_{ip}=R_{a}^{p}\overrightarrow{\omega}_{ia}+ \overrightarrow{\omega}_{ap}\text{, } \tag{5}\]  
By introducing Equation (3) and Equation (4) into Equation (5) and decomposing as needed, Equation (6) is obtained, where \(\overrightarrow{\omega}_{ib}^{p}=\left[a_{ibx}^{p},\omega_{iby}^{p},\omega_{ ibz}^{p}\right]^{T}\), representing the projection of the system base angular velocity relative to the inertia coordinate in \(\{p\}\), and \(\overset{\rightarrow}{\omega}_{bp}^{p}=\left[\omega_{bpx^{\prime}}^{p},\omega_{bpy^ {\prime}}^{p},\omega_{bpz}^{p}\right]^{T}\), representing the projection of the pitch angular velocity relative to the system base in \(\{p\}\). \[\overset{\rightarrow}{\omega}_{ip}=\left[ \begin{array}{c}\omega_{ipx}\\ \omega_{ipy}\\ \omega_{ipz}\end{array} \right]=\left[ \begin{array}{c}\omega_{ipx}^{p}\\ \omega_{ipy}^{p}\\ \omega_{ipz}^{p}\end{array} \right]+\left[ \begin{array}{c}\omega_{bpx}^{p}\\ \omega_{bpy}^{p}\\ \omega_{bpz}^{p}\end{array} \right]=\left[ \begin{array}{c}\omega_{ibx}c \theta_{a}c\theta_{p}+\omega_{iby}s\theta_{a}c\theta_{p}-\omega_{ibz}s\theta_{ p}\\ -\omega_{ibx}s\theta_{a}+\omega_{iby}c\theta_{a}\\ \omega_{ibx}c\theta_{a}s\theta_{p}+\omega_{iby}s\theta_{a}s\theta_{p}+\omega_{ ibz}c\theta_{p}\end{array} \right]+\left[ \begin{array}{c}\dot{\theta}_{a}s\theta_{p}\\ \dot{\theta}_{p}\\ \dot{\theta}_{a}c\theta_{p}\end{array} \right], \tag{6}\]  
As mentioned, axial rotation movements of y- and z-axes of the inertial coordinate coupled to the LOS must be compensated. Those are realized by using the motor-driven azimuth and pitch frames to make \(\omega_{ipy}=0\) and \(\omega_{ipz}=0\). In that case, the compensation angular velocities \(\dot{\theta}_{a}\) and \(\dot{\theta}_{p}\) can be inferred from Equation (6), which is expressed as Equation (7), where \(tg\theta_{p}\) represents \(tan\theta_{p}\). \[\left\{ \begin{array}{c}\dot{\theta}_{a}=-\frac{\omega_{ibz}^{p}}{c\theta_{p} }=-\left(\omega_{ibx}c\theta_{a}tg\theta_{p}+\omega_{iby}s\theta_{a}tg\theta_{ p}+\omega_{ibz}\right)\\ \dot{\theta}_{p}=-\omega_{iby}^{p}=-\omega_{iby}c\theta_{a}+\omega_{ibx}s\theta _{a}\end{array} \right., \tag{7}\]  
Practically, the azimuth and pitch gyroscopes are assembled on the pitch frame to detect the real-time value of \(\omega_{ibz}^{p}\) and \(\omega_{iby}^{p}\). The sensitive velocities of the two gyroscopes are denoted as \(\omega_{gyro\_z}=\omega_{ibz}^{p}\) and \(\omega_{gyro\_y}=\omega_{iby}^{p}\).