# 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
Then the angle calibration error can be calculated as:  
\[\Delta_{x}=0 \tag{21}\]  
\[\Delta_{y}=\theta_{yi^{\prime}}-\theta_{yi} \tag{22}\]  
Tables 4-6 show the angle calibration errors calculated according to the Equations (21) and (22) when \(\beta\) is 180\({}^{\prime\prime}\), 360\({}^{\prime\prime}\), 720\({}^{\prime\prime}\), \(\theta_{xi}\) and \(\theta_{yi}\) (to simplify the calculation, \(\  
At present, the plane normal vector of each composite module is  
\[\overrightarrow{n_{1}}=\left( \begin{array}{ccc}\cos\alpha&0&\sin\alpha\\ 0&1&0\\ -\sin\alpha&0&\cos\alpha\end{array} \right)\left( \begin{array}{c}0\\ 0\\ 1\end{array} \right)=\left( \begin{array}{c}\sin\alpha\\ 0\\ \cos\alpha\end{array} \right) \tag{23}\]  
\[\overrightarrow{n_{i}}=\left( \begin{array}{ccc}\cos\alpha&0&\sin\alpha\\ 0&1&0\\ -\sin\alpha&0&\cos\alpha\end{array} \right)\left( \begin{array}{c}\cos\theta_ {xi}\sin\theta_{yi}\\ \sin\alpha_{xi}\cos\theta_{yi}\\ \cos\theta_{xi}\cos\theta_{yi}\end{array} \right)=\left( \begin{array}{c} \cos\alpha\cos\theta_{xi}\sin\theta_{yi}+\sin\alpha\cos\theta_{xi}\cos\theta_ {yi}\\ \sin\theta_{xi}\cos\theta_{yi}\end{array} \right) \tag{24}\]  
Similarly, the projection vectors \(n_{1x}^{\rightarrow}\), \(n_{1y}^{\rightarrow}\), \(n_{ix}^{\rightarrow}\), \(n_{iy}^{\rightarrow}\), \(n_{iy}^{\rightarrow}\) 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  
\[\cos\theta_{xi^{{}^{\prime}}}=\cos\left( \begin{array}{c}n_{1x}^{\rightarrow} \end{array} ,n_{ix}^{\rightarrow}\right)=\frac{\cos\left(\theta_{yi}+\alpha \right)\cos\theta_{xi}}{\sqrt{\cos^{2}(\theta_{yi}+\alpha)\cos^{2}\theta_{xi}+ \sin^{2}\theta_{xi}\cos^{2}\theta_{yi}}} \tag{25}\]  
\[\cos\theta_{yi^{{}^{\prime}}}=\cos\left( \begin{array}{c}\stackrel{{ \rightarrow}}{{n_{1y}}^{\rightarrow}},n_{iy}^{\rightarrow}\end{array}  \right)=\cos\theta_{yi} \tag{26}\]  
According to the Equations (25) and (26), the relative angle \((\theta_{xi^{{}^{\prime}}},\ \theta_{yi^{{}^{\prime}}})\) can be solved when the placement error angle is \(\alpha\). Therefore, the angle calibration error can be obtained. \[\Delta_{x}=\theta_{xi^{{}^{\prime}}}-\theta_{xi} \tag{27}\]  
\[\Delta_{y}=0 \tag{28}\]  
Tables 7-9 show the angle calibration errors calculated according to the Equations (27) and (28) when \(\alpha\) 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. From Tables 1-9, note that when the placement error angle \((\gamma,\beta,\alpha)\) of the composite module array is fixed, the larger the relative angles \((\theta_{xi},\theta_{yi})\) between the composite module give larger angle calibration errors. Meanwhile, when the relative angles between the composite module are fixed, the calibration angle error introduced by the array placement error angle \(\gamma\) is the largest. However, the calibration angle errors introduced by \(\beta\) and \(\alpha\) are both less than \(6\times 10^{-3}\) arcsec, which has little effect on the calibration results and can be ignored. In order to ensure the accuracy of the calibration, it is essential to ensure that the 2-D relative angle between the plane normal vectors of each composite module is not more than 300\({}^{\prime\prime}\) in the calibration state, and the placement error angle of the array around the Z-axis of the machine tool should be controlled within \(\pm 180^{\prime\prime}\). Calculation of the uncertainty of the relative angle calibration data between the composite module at this time is within \(\pm 0.262^{\prime\prime}\), and the calibration data can be considered to be credible.