## 3 Error Model of Laser Processing Platform for MCR Frequency Modulation  
### Error Source Analysis of Laser Processing Platform for MCR Frequency Modulation  
The error of laser processing platform for MCR frequency modulation is mainly divided into external error and internal error. Figure 8 lists the common error sources. When the laser processing platform runs in the dust-free workshop, the influence of external error can be reduced by controlling environmental factors. The laser processing of MCR is non-contact processing and the thermal behavior has little impact on the laser processing platform. The contact area between the frequency test probe and MCR during processing is very small and the influence of elastic deformation error is very small. The part processing error, geometric error, and assembly error have a great impact on the laser processing platform, in which the geometric error is the main error source. Error Transfer Model of Laser Processing Platform for MCR Frequency Modulation Based on Multi-Body Topology Structure  
The error transfer circuit diagram of the laser processing platform for MCR frequency modulation is shown in Figure 9. The error is transmitted through the adjacent contact surface, and the error accumulation is mostly based on the main installation surface. In the error modeling, six errors on the main path of the error transfer should be mainly considered, which are the translational errors \(\Delta x_{ij}\), \(\Delta y_{ij}\), and \(\Delta z_{ij}\) and the rotational errors \(\Delta\alpha_{ij}\), \(\Delta\beta_{ij}\), and \(\Delta\gamma_{ij}\) along the X, Y, and Z axis, respectively. Figure 8: Error source of the laser processing platform for MCR frequency modulation. Figure 7: Positioning error of MCR. The lower-numbered body array (LBA) is used to describe the multi-body system topology structure. The typical bodies that produce direct errors in the laser processing platform for MCR frequency modulation are labeled, as shown in Figure 10.  
According to the constraint relationship between parts, the multi-body system can be divided into open-loop structure and closed-loop structure. The parts in the open-loop structure are only constrained by the front-end parts, and their error transmission direction and path are determined, while the parts in the closed-loop structure may be constrained by two or more front-end parts at the same time. The fully open-loop multi-body system cannot meet the needs of laser processing platform error analysis. According to the processing characteristics of the laser processing platform studied in this paper, the frequency test probe only contacts the MCR circuit during laser processing. The laser processing platform can be regarded as an open-loop multi-body system with specific constraints, and the LBA of the platform can be constructed. It is assumed that the bottom plate is the inertial body \(\mathrm{B_{0}}\), and the typical body 1 is selected as body \(\mathrm{B_{1}}\). Then, along the direction away from \(\mathrm{B_{1}}\), according to the natural growth sequence, each typical body is numbered in turn. \(\mathrm{B_{j}}\) is an arbitrary typical body in the system. The serial number of the n-th lower-numbered body of typical body \(\mathrm{B_{j}}\) is defined as follows:  
\[\mathrm{L^{n}(j)=i} \tag{2}\]  
Figure 10: Typical bodies of the laser processing platform for MCR frequency modulation. Figure 9: Error transmission line of the laser processing platform for MCR frequency modulation.