## 3 Error Analysis and Pointing Model
### Coordinates Definition and Transformation  
As shown in Figure 2, the error conversion matrix from the motion platform navigation system coordinate \(m\) to the geographic coordinate \(g\) is  
\[\hat{R}_{g}^{m}=Rot(Y,Rol+\Delta r)Rot(X,Pit+\Delta p)Rot(Z,Yaw+\Delta y) \tag{5}\]  
where \(Rol,Pit\), and \(Yaw\) are the roll, pitch, and azimuth angles respectively, serving as the real attitudes of the platform, and \(\Delta r,\Delta p,\Delta y\) are the corresponding measurement errors. The orientation-sensitive axis of the INS and the zero position of the base can be normally aligned by a right-angle prism mounted on the device, and the pitch and roll can be obtained from a leveling instrument, so it is reasonable to assume that the initial installation error matrix is known, here set to \(R_{b1}\). Moreover, supposing that the transformation matrix between the base coordinate and the platform coordinate is \(R_{b2}\) when the platform is in motion. Accordingly, the alignment error between \(b\) and \(m\) can be simply set as  
\[\hat{R}_{b}^{m}=R_{b1}R_{b2}=R_{b1}Rot(Y,\Delta w)Rot(X,\Delta x)Rot(Z,\Delta z) \tag{6}\]  
where \(\Delta w,\Delta x,\Delta z\) are angular vibration errors along each axis direction. The shafting error of the azimuth axis relative to the base, demonstrated in Figure 5, mainly comprises the geometric frame error, servo control system error, and encoder error. Several coordinates are involved, defined as follows: (1) The base coordinate \(b\), whose \(X_{B}\) and \(Y_{B}\) axes are in the mounting plane of the telescope and the moving platform, while the \(Z_{B}\) axis is perpendicular to that plane. (2) The azimuth base coordinate \(A0\), whose \(Y_{A0}\) axis  
Figure 4: Error source analysis. is parallel to the zero direction of the azimuth encoder and the \(Z_{A0}\) axis is perpendicular to the outer flange plane of the azimuth frame. (3) The azimuth gimbal coordinate \(A1\), which is attached to the azimuth frame and used to describe the azimuth axis movement. The transformation matrix of the azimuth axis can be expressed as  
\[ \begin{cases}\hat{R}_{B}^{A0}=Rot(Y,-\Delta\gamma_{0})Rot(X,\Delta\beta_{0})\\ \hat{R}_{A0}^{A1}=Rot(Y,-\Delta\gamma_{1})Rot(X,\Delta\beta_{1})Rot(Z,\hat{A}) \end{cases}  \tag{7}\]  
\[\hat{R}_{B}^{A0}= \begin{bmatrix}\cos(\Delta\gamma_{0})&0&\sin(\Delta\gamma_{0}) \\ 0&1&0\\ -\sin(\Delta\gamma_{0})&0&\cos(\Delta\gamma_{0})\end{bmatrix}\begin{bmatrix}1&0 &0\\ 0&\cos(\Delta\beta_{0})&\sin(\beta_{0})\\ 0&-\sin(\beta_{0})&\cos(\beta_{0})\end{bmatrix} \]  
\[Rot(Z,\hat{A})= \begin{bmatrix}\cos(\hat{A})&-\sin(\hat{A})&0\\ \sin(\hat{A})&\cos(\hat{A})&0\\ 0&0&1\end{bmatrix} \]  
where \((\Delta\gamma_{0},\beta_{0})\) denotes the perpendicularity error, \(\Delta\gamma_{1},\Delta\beta_{1}\) denotes the rotation error, \(\hat{A}=A_{b}+\Delta A=A_{b}+\Delta A_{s0}+\Delta A_{s1}+\Delta A_{s2}\), \(A_{b}\) denotes the desired azimuth, \(\Delta A_{s0}\) denotes the zero position error, \(\Delta A_{s1}\) denotes the tracking error, and \(\Delta A_{s2}\) denotes the encoder measurement error. Similar to the azimuth axis, Figure 6 illustrates the elevation axis shafting errors. The azimuth gimbal coordinate \(A1\), elevation base coordinate \(E0\), and elevation gimbal coordinate \(E1\) are involved. The transformation matrix from the azimuth gimbal \(A1\) to the elevation gimbal \(E1\) can be expressed as  
\[ \begin{cases}\hat{R}_{A1}^{E0}=Rot(Y,\Delta\gamma_{2})Rot(Z,\Delta\alpha_{2}) \\ \hat{R}_{E0}^{E1}=Rot(Y,-\Delta\gamma_{3})Rot(X,\hat{E})Rot(Z,\Delta\alpha_{3}) \end{cases}  \tag{8}\]  
where \(\hat{E}=E_{b}+\Delta E=E_{b}+\Delta E_{s0}+\Delta E_{s1}+\Delta E_{s2}\). Figure 7 portrays the optical axis perpendicularity error between the target coordinate \(L\) and the elevation coordinate \(E\). Due to the two-axis frame of the telescope, there is no relative movement of the optical axis with respect to the elevation axis, implying no rotation error.