# A Noncontact Method for Calibrating the Angle and Position of the Composite Module Array
## 3 Calibration Method
### Influence of Array Placement Error on Calibration
#### 3.2.1 Influence of Array Placement Error on Angle Calibration
Tables 1-3 show the angle calibration errors calculated according to the Equations (15) and (16) when \(\gamma\) is 180\({}^{\prime\prime}\), 360\({}^{\prime\prime}\), 720\({}^{\prime\prime}\), \(\theta_{xi}\) and \(\theta_{yi}\) (to simplify the calculation, \(\theta_{xi}=\theta_{yi}\)) are 50\({}^{\prime\prime}\), 100\({}^{\prime\prime}\), 150\({}^{\prime\prime}\), 200\({}^{\prime\prime}\), 300\({}^{\prime\prime}\), 500\({}^{\prime\prime}\), 800\({}^{\prime\prime}\), respectively. 2. The placement error angle of the composite module array around the X-axis is \(\beta\). At present, the plane normal vector of each composite module is  
\[\stackrel{{\rightarrow}}{{n_{1}}}=\left( \begin{array}{ccc}1&0&0 \\ 0&\cos\beta&-\sin\beta\\ 0&\sin\beta&\cos\beta\end{array} \right)\left( \begin{array}{c}0\\ 0\\ 1\end{array} \right)=\left( \begin{array}{c}0\\ -\sin\beta\\ \cos\beta\end{array} \right) \tag{17}\]  
\[\stackrel{{\rightarrow}}{{n_{i}}}=\left( \begin{array}{ccc}1&0&0 \\ 0&\cos\beta&-\sin\beta\\ 0&\sin\beta&\cos\beta\end{array} \right)\left( \begin{array}{c}\cos\theta_{ xi}\sin\theta_{yi}\\ \sin\theta_{xi}\cos\theta_{yi}\\ \cos\theta_{xi}\cos\theta_{yi}\end{array} \right)=\left( \begin{array}{c} \cos\theta_{xi}\sin\theta_{yi}\\ \cos\theta\sin\theta_{xi}\cos\theta_{yi}-\sin\beta\cos\theta_{xi}\cos\theta_{ yi}\\ \sin\theta\sin\theta_{xi}\cos\theta_{yi}+\cos\beta\cos\theta_{xi}\cos\theta_{yi} \end{array} \right) \tag{18}\] \begin{table}
\begin{tabular}{c c c} \hline \(\gamma\)/arcsec & \(\theta_{xi}\)=\(\theta_{yi}\)/arcsec & \(\Delta_{x}\)=\(\Delta_{y}\)/arcsec \\ \hline
180 & 50 & 0.044 \\
180 & 100 & 0.087 \\
180 & 150 & 0.131 \\
180 & 200 & 0.174 \\
180 & 300 & 0.262 \\
180 & 500 & 0.436 \\
180 & 800 & 0.698 \\ \hline \end{tabular}
\end{table} 
Table 1: Angle calibration error when \(\gamma\) is 180\({}^{\prime\prime}\). \begin{table}
\begin{tabular}{c c c} \hline \(\gamma\)/arcsec & \(\theta_{xi}\)=\(\theta_{yi}\)/arcsec & \(\Delta_{x}\)=\(\Delta_{y}\)/arcsec \\ \hline
360 & 50 & 0.087 \\
360 & 100 & 0.174 \\
360 & 150 & 0.262 \\
360 & 200 & 0.349 \\
360 & 300 & 0.523 \\
360 & 500 & 0.872 \\
360 & 800 & 1.395 \\ \hline \end{tabular}
\end{table} 
Table 2: Angle calibration error when \(\gamma\) is 360\({}^{\prime\prime}\). \begin{table}
\begin{tabular}{c c c} \hline \(\gamma\)/arcsec & \(\theta_{xi}\)=\(\theta_{yi}\)/arcsec & \(\Delta_{x}\)=\(\Delta_{y}\)/arcsec \\ \hline
720 & 50 & 0.174 \\
720 & 100 & 0.348 \\
720 & 150 & 0.523 \\
720 & 200 & 0.697 \\
720 & 300 & 1.045 \\
720 & 500 & 1.742 \\
720 & 800 & 2.788 \\ \hline \end{tabular}
\end{table} 
Table 3: Angle calibration error when \(\gamma\) is 720\({}^{\prime\prime}\). Similar to the above steps, the projection vectors \(\overrightarrow{n_{1x}}\), \(\overrightarrow{n_{1y}}\), \(\overrightarrow{n_{ix}}\), \(\overrightarrow{n_{iy}}\), of \(\overrightarrow{n_{1}}\) and \(\overrightarrow{n_{i}}\) on the YOZ and XOZ planes can be calculated. Finally, the cosine values of the relative angles around the X-axis and Y-axis between the plane normal vectors of each composite module are  
\[\cos\theta_{xi^{\prime}}=\cos\left(\overrightarrow{n_{1x}},\overrightarrow{n_ {ix}}\right)=\cos\theta_{xi} \tag{19}\]  
\[\cos\theta_{yi^{\prime}}=\cos\left(\overrightarrow{n_{1y}}^{\rightarrow}, \overrightarrow{n_{iy}}\right)=\frac{\cos(\theta_{xi}-\beta)\cos\theta_{yi}}{ \sqrt{\cos^{2}(\theta_{xi}-\beta)\cos^{2}\theta_{yi}+\cos^{2}\theta_{xi}\cos^ {2}\theta_{yi}}} \tag{20}\]  
According to the Equations (19) and (20), the relative angle (\(\theta_{xi^{\prime}}\), \(\theta_{yi^{\prime}}\)) can be solved when the placement error angle is \(\beta\).