# A Noncontact Method for Calibrating the Angle and Position of the Composite Module Array
## 3 Calibration Method
### Calibration of Composite Module Array
#### 3.1.2 Position Calibration  
Generally, the geometric center of the rotating paraboloid is used to measure the position between the composite modules. Due to the shape characteristics of the rotating paraboloid, it is difficult to directly obtain the geometric center coordinates of the rotating paraboloid to calibrate the relative position. Therefore, the calibration of the relative position of each composite module is achieved by introducing the definition of feature point. Under the condition that the curvature of the rotating parabola is continuous, there is a point on the surface where the tangent plane at that point is parallel to the composite module plane. The point is defined as the feature point corresponding to the composite module as shown in Figure 3. The reflected ray at the feature point is parallel to the reflected ray on the plane, and the coordinates of the center spot at the two places are the same on the angle sensor. Based on these properties, the relative position between the feature points is used to evaluate the relative 2-D position information of the composite module  
The composite module array is placed along the X direction of the machine, and the machine tool spindle drives the angle sensor to move along the X direction. As shown in Figure 4, \(N_{1}\), \(N_{i}\) are the measuring points projected by the angle sensor onto two rotating paraboloids. If \(M_{1},\ M_{i}\) are the feature points of two rotating paraboloids, then the relative positions between the two feature points are:  
\[\left\{ \begin{array}{l}X_{M_{i}}=\left(X_{N_{1}}-X_{M_{1}}\right)+l_{X_{N_{ 1}}}X_{N_{i}}+\left(X_{M_{i}}-X_{N_{i}}\right)\\ Y_{M_{i}}=\left(\ Y_{N_{1}}-Y_{M_{1}}\ \right)+l_{Y_{N_{1}}}Y_{N_{i}}+\left(\ Y_{M_{i}}-Y_{N_{ i}}\right)\end{array} \right. \tag{2}\]  
where parameters (\(X_{M_{i}}\), \(Y_{M_{i}}\)) and ( \(X_{M_{1}}\), \(Y_{M_{1}}\)) are the coordinates of the feature points. Parameters (\(X_{N_{i}}\), \(Y_{N_{i}}\)) and ( \(X_{N_{1}}\), \(Y_{N_{1}}\)) are the coordinates of the measurement points. Parameters \(l_{X_{N_{1}}X_{N_{i}}}\) and \(l_{Y_{N_{1}}Y_{N_{i}}}\) are the displacement between the \(N_{1}\) and \(N_{i}\) detected by the laser interferometer. Figure 4: Calibration of the position. Figure 3: Feature point diagram. According to the principle of displacement measurement, it can be transformed as follows:  
\[\left\{ \begin{array}{l}X_{N_{1}}-X_{M_{1}}=q\Big{(}x_{N_{1}}-x_{M_{1}}\Big{)}\\ Y_{N_{1}}-Y_{M_{1}}=q\Big{(}y_{N_{1}}-y_{M_{1}}\Big{)}\end{array} \right. \tag{3}\]  
\[\left\{ \begin{array}{l}X_{M_{i}}-X_{N_{i}}=q\Big{(}x_{M_{i}}-x_{N_{i}}\Big{)} \\ Y_{M_{i}}-Y_{N_{i}}=q\Big{(}y_{M_{i}}-y_{N_{i}}\Big{)}\end{array} \right. \tag{4}\]  
where parameter \(q\) is the displacement coefficient. Parameters (\(x_{N_{1}}\), \(y_{N_{1}}\)) and (\(x_{N_{i}}\), \(y_{N_{i}}\)) are the spot center coordinates of the measurement points on the angle sensor. Parameters (\(x_{M_{1}}\), \(y_{M_{1}}\)) and (\(x_{M_{i}}\), \(y_{M_{i}}\)) are the spot center coordinates of the feature points on the angle sensor. Based on the characteristics of the feature points, parameters (\(x_{M_{1}}\), \(y_{M_{1}}\)) and (\(x_{M_{i}}\), \(y_{M_{i}}\)) are the spot center coordinates obtained when the angle sensor measures the planes. The relative positions of the feature points of each composite module can be expressed as:  
\[\left\{ \begin{array}{l}X_{M_{i}}=q\Big{(}x_{N_{1}}-x_{M_{1}}\Big{)}+l_{X_{N_{ 1}}}x_{N_{i}}+q\Big{(}x_{M_{i}}-x_{N_{i}}\Big{)}\\ Y_{M_{i}}=q\Big{(}y_{N_{1}}-y_{M_{1}}\Big{)}+l_{Y_{N_{1}}}Y_{N_{i}}+q\Big{(}y_ {M_{i}}-y_{N_{i}}\Big{)}\end{array} \right. \tag{5}\]  
Considering the relative angle between the composite modules and the angle error of the machine tool spindle, the calibration formula of the relative position is:  
\[\left\{