## 1 Introduction  
A micro crystal resonator (MCR) is a kind of micro electronic component that can provide stable clock frequency. It has the advantages of high frequency performance, small package size, high stability, and low power consumption. MCR is widely used in small electronic products to generate stable clock signals and provide reference signals for communication systems. The working frequency of MCR studied in this paper is 32.768 KHz. As 32,768 is equal to the 15 th power of 2, the 32.768 KHz MCR can easily divide the frequency to produce a 1 s clock frequency, so 32.768 KHz MCR is widely used in various industrial control instruments. As a key process of MCR manufacturing, frequency modulation directly determines the performance and life of the product. The laser processing platform is the main equipment for the MCR frequency modulation. It mainly completes the preliminary frequency modulation process of the resonator to make its frequency reach the target. The positioning error and frequency modulation characteristics of laser processing platform directly affect the quality and efficiency of frequency modulation processing. At present, the research on laser processing for MCR frequency modulation mainly focuses on microstructure error modeling, microstructure error analysis, and laser micromachining technology. The research on the modeling method of microstructure error mostly focuses on machine tools. [PERSON] et al. [1] used the triangular geometry method for geometric error modeling of mechanical systems with a simple structure. [PERSON] et al. [2] utilized the space vector method to establish the error model of a machine tool to analyze the influence of geometric error. [PERSON] et al. [3] established the machining center model based on the homogeneous coordinate transformation matrix of multi-body system theory and analyzed the error variables. [PERSON] [4] established the geometric error model of a large five-axis machining center based on the virtual rigid body method, used least square theory to calculate the error value of any position, and used the recursive software compensation method to compensate. [PERSON] [5] combined differential motion with the D-H model and proposed a new method to analyze the error transfer motion of each axis. [PERSON] [6] proposed a new modeling method that introduced a stream of variation into multi-axis system theory of the geometric error model. [PERSON] [7] established a machine tool error compensation model based on spinor theory to perform geometric error identification and structure analysis of the rotating shaft. Microstructure error analysis methods mainly include the experimental simulation method, statistical tolerance method, Monte Carlo simulation method, and range analysis method. [PERSON] et al. [8] analyzed the machining error of orthogonal three-axis machine tool with Monte Carlo simulation, and obtained the probability distribution characteristics of arbitrary spatial points and motion paths for error compensation and machining process optimization. Considering the influence of tolerance on the error of the statically indeterminate mechanism, [PERSON] et al. [9] analyzed the error distribution by the Monte Carlo method and statistical tolerance method. [PERSON] et al. [10] used the kernel density estimation method to improve the Monte Carlo simulation method, which effectively reduced the number of random samples while improving reliability. [PERSON] et al. [11] explored the relationship between minimizing tolerance and cost by combining the Monte Carlo method and discrete optimization method for the problem of tolerance allocation of nonlinear constraints. In order to evaluate the influence of tolerance on gear quality, [PERSON] [12] performed Monte Carlo simulation based on tooth contact analysis and tolerance model. [PERSON] [13] studied the mathematical formulation of tolerance analysis.