## 2 Mechanism behind Thermal-Induced Measurement Errors
\begin{array}{c}0\\ 0\\ v^{\prime}\end{array} \right]+\textbf{T}^{{}^{\prime}} \tag{4}\]  
\(\textbf{R}^{*}\) and \(\textbf{T}^{{}^{\prime}}\) are the rotation matrix and translation vector from the coordinate system \(O_{s^{\prime}}\)-\(X_{s^{\prime}}Y_{s^{\prime}}Z_{s^{\prime}}\) to \(O_{s}\)-\(X_{s}\)-\(Y_{s^{\prime}}\)-\(Z_{s^{\prime}}\), respectively, expressed as:  
\[\textbf{R}^{{}^{\prime}}=\left[\begin{array}{ccc}\cos(A^{CCD}_{Z})&\sin(A^ {CCD  
and  
\[\mathbf{T}^{\prime}=\left[ \begin{array}{c}-L_{2}\sin(A_{Z}^{C})-L_{1}\sin(A_{Y}^ {C})\\ -\frac{L_{2}\sin(A_{Z}^{C})\sin(\frac{A_{Z}^{C}}{2})}{\cos(\frac{A_{Z}^{C}}{2}) }+\frac{\sqrt{\left(L_{1}\right)^{2}+\left(L_{2}\right)^{2}}\sin(A_{X}^{C}) \cos(\frac{\pi}{2}-\arctan(\frac{L_{1}}{L_{2}})+\frac{A_{X}^{C}}{2})}{\cos( \frac{A_{Z}^{C}}{2})}-t_{Y}^{C}\\ \frac{\sqrt{\left(L_{1}\right)^{2}+\left(L_{2}\right)^{2}}\sin(A_{X}^{C})\sin( \frac{\pi}{2}-\arctan(\frac{L_{1}}{L_{2}})+\frac{A_{X}^{C}}{2})}{\cos(\frac{A _{Z}^{C}}{2})}+\frac{L_{1}\sin(A_{Y}^{C})\sin(\frac{A_{Z}^{C}}{2})}{\cos( \frac{A_{Z}^{C}}{2})}-\delta_{Z}^{C}-\delta_{Z}^{m}-\delta_{Z}^{l}\\ \end{array} \right] \tag{8}\]  
\(v^{\prime}\) is expressed as:  
\[v^{\prime}=v+\delta_{Z}^{m}+\delta_{Z}^{l} \tag{9}\]  
\(dx^{\prime}\) and \(dy^{\prime}\) are the physical sizes of one pixel after the thermal behaviours occurred and are expressed as:  
\[ \begin{array}{c}dx^{\prime}=\frac{\delta_{X}^{CCD}+l_{x}}{l_{x}}dx\\ dy^{\prime}=\frac{\delta_{Y}^{CCD}+l_{y}}{l_{y}}dy\end{array}  \tag{10}\]  
where \(l_{x}\) and \(l_{y}\) are the initial physical lengths of the CCD focal plane along two directions. Combined with Equations (1) and (2), the image error model based on the thermal behaviours was established. Next, the image error generated by each thermal behaviour was simulated, and the forms of image errors are shown in Figure 3. Table 1 lists the corresponding error form for each thermal behaviour. Figure 4 shows the pixel drift caused by temperature variation in the investigated camera system under real experimental conditions. It can be seen that the actual image error is a combination of the previously mentioned error forms, while the influence degree of each error form is different. The previously mentioned thermal behaviours were studied experimentally and the image errors caused by them were analysed. This was done to eliminate the thermal behaviours that have little influence on image error and simplify the image error model discussed above. Figure 2: Imaging light path before and after thermal behaviours occur, wherein \(L_{I}\) and \(L_{2}\) are the \(Z\)-directional and \(Y\)-directional distance between the rotation center of the camera case and optical center, respectively, and the \(X\)-directional distance is assumed to be zero. The rotation center of the CCD focal plane is assumed to be the principal point. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Components**} & \multicolumn{2}{c}{**Type of Thermal**} & \multirow{2}{*}{**Thermal Behaviours**} & \multirow{2}{*}{**Type of Image Error**} \\ \cline{3-3} \cline{5-5}  & & \(A_{X}^{c}\) & I \\ \cline{3-5} \multirow{3}{*}{Case} & Rotation & \(A_{Y}^{c}\) & III \\ \cline{3-5}  & & \(A_{Z}^{c}\) & VI \\ \cline{3-5}  & Translation & \(t_{Y}^{c}\) & II \\ \cline{3-5}  & Deformation & \(\delta_{Z}^{c}\) & V \\ \hline \multirow{5}{*}{CCD focal plane} & \multirow{3}{*}{Translation} & \(A_{X}^{CCD}\) & VII \\ \cline{3-5}  & & \(A_{Y}^{CCD}\) & VII \\ \cline{3-5}  & & \(A_{Z}^{CCD}\) & VI \\ \cline{3-5}  & & \(A_{X}^{CCD}\) & IV \\ \cline{3-5}  & Translation & \(t_{Y}^{CCD}\) & II \\ \cline{3-5}  & & \(t_{Z}^{CCD}\) & V \\ \cline{3-5}  & Deformation & \(\delta_{X}^{CCD}\) & IX \\ \cline{3-5}  & & \(\delta_{Y}^{CCD}\) & X \\ \hline Mount & Deformation & \(\delta_{Z}^{m}\) & V \\ \hline Lens & Deformation & \(\delta_{Z}^{l}\) & V \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Error forms corresponding to each thermal behaviour. Figure 3: Different forms of image errors induced by thermal behaviours.