## 1 Introduction  
Large-aperture optical components are widely used in military defense, scientific research, and civil applications, including laser weapons, high-caliber satellites, airborne reconnaissance cameras, laser communications, gravitational wave detection programs in space, large optical infrared telescopes, and the \"Shen Guang\" facility. The \"Shen Guang\" facility is a high-performance, high-power neodymium glass laser device, in which the large-size diffractive optical elements (DOEs) required are used to achieve uniform illumination of the laser beam, which is a key component of the laser beam homogenization for inertial confinement fusion (ICF) systems. Therefore, the optical components used need extremely high precision, and the detection of optical components based on measuring machines can realize the detection of large field of view, high precision, and high stability, ensuring the quality and performance of optical components. The application of freeform surfaces in optical systems can enhance the system's degrees of freedom, reduce residual errors, and minimize the number of optical elements, thus simplifying the structure and improving image quality. Consequently, large-diameter complex surface elements--particularly freeform surface optical elements--and diffractive lens elementsare the future of ultra-high-performance optical systems [1]. From milling/grinding to polishing, the surface accuracy of the optical element falls between the coordinate measurement accuracy [2] and interferometric accuracy [3]. Neither method can quickly and accurately assess surface accuracy, resulting in an inefficiency in processing. Therefore, to meet the demand for modern large-caliber optical systems, innovations in high-precision, large-stroke, flexible measurement methods and key technologies are required. Geometric error is a key factor that restricts the measurement accuracy of measuring machines and compensating for these errors can effectively improve accuracy [4]. However, the compensation of geometric errors is complex due to various errors. Establishing error models is complicated, and the full compensation of all errors is costly. Therefore, analyzing the weights of the geometric errors and clarifying key error terms is crucial to improve the efficiency of the error compensation and reduce its cost. Many scholars have conducted in-depth research on geometric error weighting analyses, such as the Sobol method [5], extended Fourier amplitude sensitivity method [6], Morris method [7], reliability theory [8], multiplicative dimensional reduction method [9], and high-order moment standardization technology [10]. [PERSON] et al. [11] established comprehensive tool and tooth flank error models for a flush worm gear grinding machine and used the improved Sobol method to conduct a segmental sensitivity analysis to determine the critical geometric errors. Similarly, [PERSON] et al. [12] used a global quantitative sensitivity analysis method to identify the key geometric errors in a five-axis machine tool and evaluated the coupling effects between the geometric error components and different error sources at different indexing angles. [PERSON] et al. [13] also proposed two methods for evaluating axis error sensitivity to determine the critical axes. One method utilizes the error contribution weights of the axes and the other utilizes the axes' error sensitivity coefficients. [PERSON] et al. [14] developed geometric error and weighting models based on the multi-body theory, which relates the overall motion accuracy to the individual component accuracies. It also calculates the accuracies of all individual components from the geometric error weights derived from the proposed model. Additionally, [PERSON] et al. [15] designed a novel non-contact optical measurement system to measure geometric errors in six degrees of freedom of the rotary axes by compensating for the high weighting error.