## 3 Error Model of Laser Processing Platform for MCR Frequency Modulation
### Error Source Analysis of Laser Processing Platform for MCR Frequency Modulation
where L is a lower-numbered operator, i and j are symbols of a typical body, and Bj is called an n-th higher-numbered body of Bj, which satisfies the following:  
\[\rm{L^{n}(j)=L(L^{n-1}(j))} \tag{3}\]  
\[\rm{L^{0}(j)=j} \tag{4}\]  
\[\rm{L^{n}(0)=0} \tag{5}\]  
When Bj is an adjacent lower-numbered body of Bj,  
Figure 8:0  
According to the formulas above, the serial number of each low-numbered body array of multi-body system and the lower-numbered body array of the topological structure of the whole multi-body system can be obtained, and the results are shown in Figure 8:1  
Error Model of the Laser Processing Platform for MCR Frequency Modulation Based on Multi-Body System Theory  
According to the multi-body system theory, the error matrix \(T_{ij}\) between the typical bodies of the laser processing platform is as follows:  
\[T_{ij}=\left[ \begin{array}{ccccc}1&-\Delta\gamma_{ij}&\Delta\beta_{ij}& \Delta x_{ij}\\ \Delta\gamma_{ij}&1&-\Delta\alpha_{ij}&\Delta y_{ij}\\ -\Delta\beta_{ij}&\Delta\alpha_{ij}&1&\Delta z_{ij}\\ 0&0&0&1\end{array} \right] \tag{7}\]  
where \(\Delta\alpha_{ij}\) is the error between typical body \(i\) and \(j\) around the \(X\)-axis, \(\Delta x_{ij}\) is the error between typical body \(i\) and \(j\) between typical body \(i\) and \(j\) along the \(X\)-axis, \(\Delta\beta_{ij}\) is the error between typical body \(i\) and \(j\) around the \(Y\)-axis, \(\Delta y_{ij}\) is the error between typical body \(i\) and \(j\) along the \(Y\)-axis, \(\Delta\gamma_{ij}\) is the error between typical body \(i\) and \(j\) around the \(Z\)-axis, and \(\Delta z_{ij}\) is the error between typical body \(i\) and \(j\) along the \(Z\)-axis. In order to express this more clearly, subscripts p and s in the matrix represent static and motion errors, respectively. The homogeneous transformation matrix of the error model of the laser processing platform for MCR frequency modulation is established as follows:  
The gantry is fixed on the plate. The ideal static error \(T_{01p}\), ideal motion error \(T_{01s}\), actual static error \(\Delta T_{01p}\), and actual motion error \(\Delta T_{01s}\) of typical body 1 and typical body 0 are relatively small, which are denoted as \(I_{4\times 4}\). The frequency test probe is fixed on the gantry. The ideal static error \(T_{12p}\) and the actual static error \(\Delta T_{12p}\) of typical body 2 and typical body 1 are relatively small, which \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Typical Body j** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \(\rm{L^{0}(j)}\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \(\rm{L^{1}(j)}\) & 0 & 1 & 0 & 3 & 4 & 5 & 6 \\ \(\rm{L^{2}(j)}\) & 0 & 0 & 0 & 0 & 3 & 4 & 5 \\ \(\rm{L^{3}(j)}\) & 0 & 0 & 0 & 0 & 0 & 3 & 4 \\ \(\rm{L^{4}(j)}\) & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ \(\rm{L^{5}(j)}\) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \(\rm{L^{6}(j)}\) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \(\rm{L^{7}(j)}\) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: LBA of the laser processing platform for MCR frequency modulation. are denoted as \(I_{4\times 4}\). The homogeneous transformation matrices of ideal motion and actual motion are as follows:  
\[T_{12s}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&z_{12s}\\ 0&0&0&1\end{array} \right] \tag{8}\]  
\[\Delta T_{12s}=\left[ \begin{array}{cccc}1&0&\Delta\beta_{12s}&\Delta x_{12s} \\ \Delta\gamma_{12s}&1&-\Delta\alpha_{12s}&\Delta y_{12s}\\ -\Delta\beta_{12s}&\Delta\alpha_{12s}&1&\Delta z_{12s}\\ 0&0&0&1\end{array} \right] \tag{9}\]  
The \(X\)-axis linear motor is fixed on the plate, and the homogeneous transformation matrices of ideal motion and actual motion between typical body 3 and typical body 0 are \(T_{03s}\) and \(\Delta T_{03s}\). The \(Y\)-axis linear motor is installed on the \(X\)-axis linear motor, and the homogeneous transformation matrices of ideal motion, actual static, and actual motion between the typical body 5 and the typical body 4 are \(T_{45s}\), \(\Delta T_{45p}\), and \(\Delta T_{45s}\), respectively.