# A Spatial 4-DOF Laser Collimation Measurement System
## 3 Error Analysis  
### Analysis of Installation Error (Abbe Error) Using the Homogeneous Transformation Matrix Method  
1. The homogeneous transformation matrix method was used for the installation error (Abbe error) analysis. The central coordinate point OOO1 of O-X1Y1Z1 should be perpendicular to the O0 \(\times\) O7O and O1X1Y1 planes, and the axes of the moving-end coordinate system and the fixed-end coordinate system O0-X0Y0Z0 should be parallel to one another. However, as Figure 10 illustrates, there will be a misalignment between the fixed-end coordinate system O0-X0Y0Z0 and the moving-end coordinate system O-X1Y1Z1. This misalignment can be measured using six indicators and is referred to as an installation error (Abe error). That is, the position error is \(\Delta_{X}\), \(\Delta_{Y}\), and \(\Delta_{Z}\) in the three directions of the XYZ axis of the two coordinate systems; the Angle error is \(\alpha\), \(\beta\), and \(\gamma\) in the three directions of the XYZ axis. This misaligned state can be represented by the homogeneous coordinate matrix \(H\). The homogeneous matrix \(H\) contains all six of these indicators, which are \(\Delta_{X}\), \(\Delta_{Y}\), \(\Delta_{Z}\), \(\alpha\), \(\beta\) and \(\gamma\). \[H= \begin{bmatrix}1&\alpha&-\beta&\Delta_{X}\\ -\alpha&1&\gamma&\Delta_{Y}\\ \beta&-\gamma&1&\Delta_{Z}\\ 0&0&0&1\end{bmatrix}  \tag{21}\]  
1. Establishment of a straightness error model under the homogeneous transformation matrix method. The straightness error, which does not consider the installation error, can be represented by matrix \(P_{CMOS1_{0}}\):  
\[P_{CMOS1_{1}}=HP_{CMOS1_{0}}= \begin{bmatrix}1&\alpha&-\beta&\Delta_{X}\\ -\alpha&1&\gamma&\Delta_{Y}\\ \beta&-\gamma&1&\Delta_{Z}\\ 0&0&0&1\end{bmatrix} [2\delta_{X}&2\delta_{Y}&0&1]^{T}= \begin{bmatrix}2\delta_ {X}+2\alpha\delta_{Y}+\Delta_{X}\\ 2\delta_{Y}-2\alpha\delta_{Y}+\Delta_{Y}\\ 2\beta\delta_{X}-2\gamma\delta_{Y}+\Delta_{Z}\\ 1\end{bmatrix}  \tag{22}\]  
Therefore, the straightness error in the x direction is \(2\delta_{X}+2\alpha\delta_{Y}+\Delta_{X}\), and the straightness error in the y direction is \(2\delta_{Y}-2\alpha\delta_{X}+\Delta_{Y}\)  
Figure 10: CMOS installation error model. i.e., equality:  
\[\delta^{\prime}_{X}=\delta_{X}+\alpha\delta_{Y}+\frac{\Delta_{X}}{2} \tag{23}\]  
\[\delta^{\prime}_{Y}=\delta_{Y}-\alpha\delta_{X}+\frac{\Delta_{Y}}{2} \tag{24}\]  
\(\frac{\Delta_{X}}{2}/\frac{\Delta_{Y}}{2}\) represents the translation errors due to CMOS1 installation, which have the same effect on each measuring point. This effect can be eliminated using differential operations between the measuring point and the initial point. \(\alpha\delta_{X}/\alpha\delta_{Y}\) is the straightness distance crosstalk between two straightness errors; \(\alpha\) can be determined via calibration. \(\delta_{X}/\delta_{Y}\) is the original ideal straightness distance in the X and Y directions. * Establishment of the error model of pitch angle and deflection angle under the homogeneous transformation matrix method. The direction vector of the laser beam in the O0-X0Y0Z0 coordinate system and the direction vector after being reflected by plane mirror M can be represented as follows:  
\[\overrightarrow{n}= \begin{bmatrix}0&0&1\end{bmatrix} \overrightarrow{n_{1}}= \begin{bmatrix}2\alpha&2\beta&-1\end{bmatrix}  \tag{25}\]  
As shown in Figure 11, according to the ABCD matrix, in the coordinate system O0-X0Y0Z0, in the \(X\)-axis direction, the laser beam matrix that considers the distance to the planar convex lens can be expressed as \(L\)\(M_{X}\):  
\[M_{X}= \begin{bmatrix}1&L\\ 0&1\end{bmatrix}\begin{bmatrix}0\\ 2\alpha\end{bmatrix} = \begin{bmatrix}2L\alpha\\ 2\alpha\end{bmatrix}  \tag{26}\]  
In the coordinate system O0-X0Y0Z0, in the \(Y\)-axis direction, the laser beam matrix that considers the distance to the planar convex lens can be expressed as \(L\)\(M_{Y}\):  
\[M_{Y}= \begin{bmatrix}1&L\\ 0&1\end{bmatrix}\begin{bmatrix}0\\ 2\beta\end{bmatrix} = \begin{bmatrix}2L\beta\\ 2\beta\end{bmatrix}