# 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  
The plane normal vector relationship of the composite module is shown in Figure 6. Assuming that the plane normal vector \(\overrightarrow{n_{1}}\) of the reference composite module when the array is placed along the X direction of the machine tool is \(\overrightarrow{n_{1}}=(0,0,1)^{T}\), then the 2-D angle between the composite module \(i\) and the reference composite module is \((\theta_{xi},\theta_{yi})\). The plane normal vector \(\overrightarrow{n_{i}}\) of the composite module \(i\) is then:  
\[\overrightarrow{n_{i}}=\left(\cos\theta_{xi}\sin\theta_{yi},\sin\theta_{xi} \cos\theta_{yi},\cos\theta_{xi}\cos\theta_{yi}\right)^{T} \tag{7}\]  
Figure 5: Array placement error. The influence of different array placement error angles is discussed below. 1. The placement error angle of the composite module array around the Z-axis is \(\gamma\). Therefore, the plane normal vector of each composite module is: \[\vec{n_{1}}=\left( \begin{array}{ccc}\cos\gamma&-\sin\gamma&0\\ \sin\gamma&\cos\gamma&0\\ 0&0&1\end{array} \right)\left( \begin{array}{c}0\\ 0\\ 1\end{array} \right)=\left( \begin{array}{c}0\\ 0\\ 1\end{array} \right)\] (8) \[\vec{n_{i}}=\left( \begin{array}{ccc}\cos\gamma&-\sin\gamma&0\\ \sin\gamma&\cos\gamma&0\\ 0&0&1\end{array} \right)\left( \begin{array}{c}\cos\theta_{xi}\sin\theta_{yi} \\ \sin\theta_{xi}\cos\theta_{yi}\\ \cos\theta_{xi}\cos\theta_{yi}\end{array} \right)=\left( \begin{array}{c} \cos\gamma\cos\theta_{xi}\sin\theta_{yi}-\sin\gamma\sin\theta_{xi}\cos\theta_ {yi}\\ \sin\gamma\cos\theta_{xi}\sin\theta_{yi}+\cos\gamma\sin\theta_{xi}\cos\theta_ {yi}\\ \cos\theta_{xi}\cos\theta_{yi}\end{array} \right)\] (9)  
Furthermore, the projection vectors \(\vec{n_{1x}}\), \(\vec{n_{1y}}\) of \(\vec{n_{1}}\) on the YOZ and XOZ planes can be calculated as follows:  
\[\vec{n_{1x}}=\vec{n_{1y}}=(0,0,1)^{T} \tag{10}\]  
Similarly, the projection vectors \(\vec{n_{1x}}\), \(\vec{n_{iy}}\) of \(\vec{n_{i}}\) on the YOZ and XOZ planes are  
\[\vec{n_{1x}}=\left( \begin{array}{c}0\\ \sin\gamma\cos\theta_{xi}\sin\theta_{yi}+\cos\gamma\sin\theta_{xi}\cos\theta_ {yi}\\ \cos\theta_{xi}\cos\theta_{yi}\end{array} \right) \tag{11}\]  
\[\vec{n_{iy}}=\left( \begin{array}{c}\cos\gamma\cos\theta_{xi}\sin\theta_{yi} -\sin\gamma\sin\theta_{xi}\cos\theta_{yi}\\ 0\\ \cos\theta_{xi}\cos\theta_{yi}\end{array} \right) \tag{12}\]  
At this time, the cosine values of the relative angles around the X-axis and Y-axis between the plane normal vectors of each composite module are  
\[\cos\theta_{xi^{\prime}}=\cos\left( \begin{array}{c}\vec{n_{1x}},\vec{n_{ix} }\\ \end{array} \right)=\frac{\cos\theta_{xi}\cos\theta_{yi}}{\sqrt{\cos^{2}\theta_ {xi}\cos^{2}\theta_{yi}+\left(\cos\theta_{yi}\sin\theta_{xi}\cos\gamma+\cos \theta_{xi}\sin\theta_{yi}\sin\gamma\right)^{2}}} \tag{13}\]  
\[\cos\theta_{yi^{\prime}}=\cos\left\langle\vec{n_{1y}},\vec{n_{iy}}\right\rangle =\frac{\cos\theta_{xi}\cos\theta_{yi}}{\sqrt{\cos^{2}\theta_{xi}\cos^{2}\theta_ {yi}+\left(\cos\theta_{xi}\sin\theta_{yi}\cos\gamma+\cos\theta_{yi}\sin \theta_{xi}\sin\gamma\right)^{2}}} \tag{14}\]  
where parameters (\(\theta_{xi^{\prime}}\), \(\theta_{yi^{\prime}}\)) are the relative 2-D angle of each composite module when the array placement error is \(\gamma\). According to the Equations (13) and (14), the relative angle (\(\theta_{xi^{\prime}}\), \(\theta_{yi^{\prime}}\)) can be solved when the placement error angle is \(\gamma\). Figure 6: Schematic diagram of the plane normal vector. At this time, the angle calibration error introduced can be calculated as:  
\[\Delta_{x}=\theta_{xi^{\prime}}-\theta_{xi} \tag{15}\]  
\[\Delta_{y}=\theta_{yi^{\prime}}-\theta_{yi} \tag{16}\]  
where parameter \(\Delta_{x}\) is the angle calibration error around the X-axis. Parameter \(\Delta_{y}\) is the angle calibration error around the Y-axis.