## 3 Error Model of Laser Processing Platform for MCR Frequency Modulation
### Error Source Analysis of Laser Processing Platform for MCR Frequency Modulation
The stage is installed on the \(Z\)-axis lifting motor, and the homogeneous transformation matrix of the actual static error between typical body 6 and typical body 5 is \(\Delta T_{56p}\). The MCR to be processed is mounted on the stage, and the homogeneous transformation matrix of the actual static error between typical body 7 and typical body 6 is \(\Delta T_{67p}\). According to the multi-body system theory, the change between the coordinate system of each typical body is represented by a \(4\times 4\) order homogeneous matrix with the parameter of Euler angle, and its expression is as follows:  
\[T_{ij}=T_{ijp}\Delta T_{ijp}T_{ijs}\Delta T_{ijs} \tag{10}\]  
where \(T_{ijp}\) is the ideal static error between typical body \(i\) and \(j\), \(T_{ijs}\) is the ideal motion error motion error between typical body \(i\) and \(j\), and \(\Delta T_{ijs}\) is the actual motion error between typical body \(i\) and \(j\). Point \(P_{t}\) is the homogeneous coordinate of the end of the frequency test probe in the typical body coordinate system, and point \(P_{w}\) is the homogeneous coordinate of the preset contact point of the MCR circuit to be processed in the typical body. The homogeneous coordinate in the generalized coordinate system \(O-x_{0}y_{0}z_{0}\) is calculated by homogeneous coordinate transformation. \[P_{t}=\left( \begin{array}{cccc}x_{t}&y_{t}&z_{t}&1\end{array} \right)^{T} \tag{11}\]  
\[P_{w}=\left( \begin{array}{cccc}x_{w}&y_{w}&z_{w}&1\end{array} \right)^{T} \tag{12}\]  
Through the LBA of the laser processing platform given in Table 2, it can be obtained that the homogeneous coordinate transformation matrix \(P_{t0}\) of the end contact point of the frequency test probe in the generalized coordinate system \(O-x_{0}y_{0}z_{0}\) is  
\[P_{t0}=T_{02}\cdot P_{t} \tag{13}\]  
\[T_{02}=T_{01}T_{12} \tag{14}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Positioning Error of Probe End** & **Mean Value (\(\mu\)m)** & **Standard Deviation (\(\mu\)m)** \\ \hline Ex & \(-0.0243\) & \(20.3\) \\ Ey & \(-0.0211\) & \(26.8\) \\ Ez & \(-0.0116\) & \(12.1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean value and standard deviation of the positioning error. The homogeneous coordinate transformation matrix \(P_{w0}\) of the preset contact point of the MCR circuit to be processed in the generalized coordinate system \(O-x_{0}y_{0}z_{0}\) is  
\[P_{w0}=T_{07}\cdot P_{w} \tag{15}\]  
\[T_{07}=T_{03}T_{34}T_{45}T_{56}T_{67} \tag{16}\]  
Finally, the matrix \(E\), the positioning error of laser processing platform for MCR frequency modulation, is as follows:  
\[E=P_{10}-P_{w0}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&z_{12s}\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}1&0&\Delta\beta_{12s}& \Delta x_{12s}\\ \Delta\gamma_{12s}&1&-\Delta\alpha_{12s}&\Delta y_{12s}\\ -\Delta\beta_{12s}&\Delta\alpha_{12s}&1&\Delta z_{12s}\\ 0&0&0&1\end{array} \right]\left( \begin{array}{c}x_{t}\\ y_{t}\\ z_{t}\\ 1\end{array} \right)-\] \[\left[ \begin{array}{cccc}1&0&0&x_{03s}\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}1&-\Delta\gamma_{03s}& \Delta\beta_{03s}&\Delta x_{03s}\\ \Delta\gamma_{03s}&1&-\Delta\alpha_{03s}&\Delta y_{03s}\\ -\Delta\beta_{03s}&\Delta\alpha_{03s}&1&\Delta z_{03s}\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}1&-\Delta\gamma_{34p}& \Delta\beta_{34p}&0\\ \Delta\gamma_{34p}&1&0&\Delta\beta_{34p}\\ -\Delta\beta_{34p}&0&1&\Delta z_{34p}\\ 0&0&0&1\end{array} \right] \tag{17}\] \[\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&y_{34s}\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}1&-\Delta\gamma_{34s}& \Delta\beta_{34s}&\Delta x_{34s}\\ \Delta\gamma_{34s}&1&-\Delta\alpha_{34s}&\Delta y_{34s}\\ -\Delta\beta_{34s}&\Delta\alpha_{34s}&1&\Delta z_{34s}\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{cccc}1&-\Delta\gamma_{45p}& \Delta\beta_{45p}&\Delta x_{45p}\\ \Delta\gamma_{45p}&1&-\Delta\alpha_{5p}&\Delta y_{45p}\\ 0&\Delta\alpha_{56p}&1&0\\ 0&0&0&1\end{array} \right]\] \[\left[