## 1 Introduction  
In the processes of curved surface milling of complex parts and thin-walled parts, the machining accuracy directly affects the production quality of the parts [1]. The machining accuracy of machine tools is affected by various factors such as the dimensional accuracy of machine components, motion accuracy, deformations, and distortions generated by cutting force and cutting heat [2; 3; 4]. Due to the uncertainty of machine error, it is very difficult but valuable to identify and quantify its error source. Error budget provides a method to determine the source of machining error and predict the repeatable and unrepeatable errors of a machine [5]. The main error sources of machine tools can be divided into geometric, kinematic, thermal, rigidity, and radial errors of cutting tools [6; 7]. Among them, geometric and thermal errors account for most of the total errors of machine tools. At present, the geometric error of rotary axis is difficult to determine by direct measurement, and the measurement standards are not yet unified. Moreover, there are shortcomings and deficiencies in the identification model. Therefore, the efficient and accurate detection of the geometric error of the five-axis machine tool rotating axis is of particular importance. The geometrical error of machine tools is caused by imperfections of components and the imperfect assembly of parts, such as the straightness errors of the guideways, joint misalignments, angular offset and rotary axes separation errors [8; 9]. The geometric error modeling of CNC machine toolsis based on the mathematical relationship between the geometric errors and machining errors of each component. At present, many researchers are studying geometric error modeling methods of machine tools. For example, [PERSON] and [PERSON] [10] used the Denavit-Hartenberg (D-H) method to study the influence of relative position error between the coordinate axes of moving parts in five-axis machine tool kinematics on the comprehensive error of tool tip point. These studies are applicable to machine tool error estimation and accuracy optimization. [PERSON] et al. [11] proposed that the angular and displacement errors of the moving parts of the machine were independent from each other, and the model of the geometric and thermal errors of the five-axis machine tool was established. [PERSON] and [PERSON] [12] proposed a new method, which used matrix summation to establish a geometric error model for five-axis machine tools. This method decomposed the kinematic equation into six motion components, which were easy to understand and had a clear physical meaning. [PERSON] et al., [13] developed a methodology for the estimation of the tool tip position accuracy of five-axis milling centers based on the D-H formulation. [PERSON] et al., [14] proposed a method for modeling and compensating the geometric error of large grinders based on the D-H method. [PERSON] et al. [15] presented a method to estimate the geometrical accuracy of a multi-axis machines by using a homogenous matrix. [PERSON] [16] considered the geometric meaning of perpendicularity error and improved the accuracy of five-axis machine tool geometric error by improving the D-H modeling method. However, the applicability of the model required further verification. [PERSON] et al. [17] classified the geometric error of rotating axis into position dependent geometric errors (PDGEs) and position independent geometric errors (PIGEs). The two types of geometric error models were represented as polynomial equations with first-order continuity and n-order polynomial equations with constant terms, but the installation error of the measuring instrument was high, which affected the accuracy of the model. [PERSON] et al. [18] used the theory of rotational motion to establish a model for the comprehensive error of five-axis machine tools with 41 errors.