## 3 Error Analysis and Pointing Model
### Linear Model Construction  
In summary, the actual positioning equation of the observed target from the geographic coordinate \(g\) to the target coordinate \(L\) can be obtained based on Formulas (5)-(9). \[\hat{G}=\hat{R}_{m}^{g}\hat{R}_{B}^{m}\hat{R}_{A0}^{B}\hat{R}_{A1}^{A0}\hat{R}_{ E0}^{A1}\hat{R}_{E1}^{E0}\hat{R}_{L}^{E1}\hat{L} \tag{10}\]  
Accordingly, in practice, the pointing error model of the MPEOT is  
\[ \begin{cases}B_{1}=\hat{G}\hat{R}_{m}^{g}\hat{R}_{b}^{m}\\ B_{2}=\hat{R}_{A0}^{b}\hat{R}_{A1}^{A0}\hat{R}_{E1}^{A1}\hat{R}_{L}^{E0}\hat{R}_ {L}^{E1}\hat{L}\end{cases}  \tag{11}\]  
where \(\hat{G}=[X_{g},Y_{g},Z_{g}]^{T},\hat{L}=[0,1,0]^{T},B_{1}=[X_{b1},Y_{b1},Z_{b1} ]^{T},B_{2}=[X_{b2},Y_{b2},Z_{b2}]^{T}\). To sequentially expand Equation (11), using the following simplification principle, each error factor \(\Delta\) is a small quantity, i.e., \(\cos\Delta=1,\sin\Delta=0\) while ignoring the higher order error term. \[ \begin{cases}X_{b1}=k_{x}+m_{y}\Delta y+m_{p}\Delta p+m_{r}\Delta r+m_{z}\Delta z +m_{w}\Delta w\\ Y_{b1}=k_{y}+n_{y}\Delta y+n_{p}\Delta p+n_{z}\Delta z+n_{x}\Delta x\\ Z_{b1}=k_{z}+h_{y}\Delta y+h_{p}\Delta p+h_{r}\Delta r+h_{x}\Delta x+h_{w} \Delta w\end{cases}  \tag{12}\]  
Figure 6: Shafting errors of the elevation axis. Figure 7: Perpendicularity error of optical axis. \[ \begin{cases}X_{k2}=\Delta_{y0}\sin(E_{b})-(\Delta\beta_{4}+\Delta E-\Delta\beta_{ 1})\sin(A_{b})\sin(E_{b})+\Delta\alpha_{4}\cos(A_{b})\\ \quad+\cos(E_{b})\sin(A_{b})+(\Delta A+\Delta\alpha_{2}+\Delta\alpha_{3})\cos( A_{b})\cos(E_{b})\\ \quad+(\Delta\gamma_{1}-\Delta\gamma_{2})\sin(A_{b})\sin(E_{b})\\ Y_{k2}=\Delta\beta_{0}\sin(E_{b})-\Delta\alpha_{4}\sin(A_{b})-(\Delta\beta_{4}+ \Delta E-\Delta\beta_{1})\cos(A_{b})\sin(E_{b})\\ \quad+\cos(A_{b})\cos(E_{b})-(\Delta A+\Delta\alpha_{2}+\Delta\alpha_{3})\cos (E_{b})\sin(A_{b})\\ \quad-(\Delta\gamma_{1}-\Delta\gamma_{2})\sin(A_{b})\sin(E_{b})\\ Z_{k2}=(\Delta\beta_{1}-\Delta E)\cos(E_{b})-\sin(E_{b})-\Delta\beta_{4}\cos (E_{b})\\ \quad+\Delta\beta_{0}\cos(A_{b})\cos(E_{b})+\Delta\gamma_{0}\cos(E_{b})\sin( A_{b})\end{cases}  \tag{13}\]  
In addition, the conversion of Cartesian coordinates into pointing angles is given by  
\[ \begin{cases}\tan(A_{b}+\delta A_{1})=\hat{X}_{b1}/\hat{Y}_{b1}\\ \sin(E_{b}+\delta E_{1})=\hat{Z}_{b1}\end{cases}\begin{cases}\tan(A_{b}+\delta A _{2})=\hat{X}_{b2}/\hat{Y}_{b2}\\ \sin(E_{b}+\delta E_{2})=\hat{Z}_{b2}\end{cases}  \tag{14}\]  
Substituting Equations (12) and (13) into Equation (14) and expanding, a linear model of the pointing error, named the full parameter model (FPM), is obtained as  
\[ \begin{bmatrix}\delta A\\ \delta E\end{bmatrix} = \begin{bmatrix}\delta A_{1}+\delta A_{2}\\ \delta E_{1}+\delta E_{2}\end{bmatrix} =HX+ \begin{bmatrix}\epsilon_{A}\\ \epsilon_{E}\end{bmatrix}  \tag{15}\]  
\[H^{\mathcal{T}}= \begin{bmatrix}\Delta\beta_{0}&\Delta\beta_{0}\\ \Delta\gamma_{0}&\Delta\gamma_{0}\\ (\Delta\gamma_{1}-\Delta\gamma_{2})&\Delta\gamma\\ \Delta\alpha_{4}&\Delta p\\ \Delta\gamma&\Delta r\\ \Delta p&\Delta x\\ \Delta r&\Delta w\\ \Delta z&\Delta\beta_{1}-\Delta E-\Delta\beta_{4}-2\tan(E_{b})+k_{22}\\ \Delta x&0\\ \Delta w&0\\ \Delta A+\Delta\alpha_{2}+\Delta\alpha_{3}+k_{A2}&0\end{bmatrix} ,X= \begin{bmatrix} p_{1}&q_{1}\\ p_{2}&q_{2}\\ p_{3}&q_{3}\\ p_{4}&q_{4}\\ p_{5}&q_{5}\\ p_{6}&q_{6}\\ p_{7}&q_{7}\\ p_{8}&1\\ p_{9}&0\\ p_{10}&0\\ 1&0\end{bmatrix} \]  
\[