Abstract: This paper analyzes in detail the causes of geometric errors in CNC machine tools, summarizes the methods for compensating systematic errors, and elaborates on the application scenarios of various error compensation methods, laying the foundation for further soft upgrades in machine tool accuracy. Keywords: CNC machine tool; geometric error; error compensation Introduction There are two methods to improve machine tool accuracy. One is to eliminate possible error sources by improving the level of part design, manufacturing, and assembly, called error prevention. This method is mainly constrained by the accuracy of the machining mother machine, and the increased machining costs due to improved part quality limit its application. The other method is error compensation, which usually involves modifying the machine tool's machining instructions to compensate for errors, achieving an ideal motion trajectory and realizing a soft upgrade in machine tool accuracy. Studies show that geometric errors and temperature-induced errors account for about 70% of the total machine tool error, among which geometric errors are relatively stable and easy to compensate. Compensation for geometric errors in CNC machine tools can improve the processing level of the entire machinery industry, and is of great significance for promoting scientific and technological progress, enhancing China's national defense capabilities, and ultimately greatly strengthening China's comprehensive national power. 1. Causes of Geometric Errors It is generally believed that geometric errors in CNC machine tools are caused by the following factors: 1.1 Original manufacturing errors of the machine tool refer to the machine tool motion errors caused by the geometric shape, surface quality, and positional errors between the working surfaces of the various components of the machine tool. This is the main cause of geometric errors in CNC machine tools. 1.2 Control system errors of the machine tool include servo errors (contour following errors) of the machine tool axis system and CNC interpolation algorithm errors. 1.3 Thermal deformation errors are caused by the structural thermal deformation of the machine tool due to internal heat sources and environmental thermal disturbances. 1.4 Errors caused by deformation of the process system due to cutting loads include errors caused by deformation of the machine tool, cutting tools, workpiece, and fixture. This error is also known as "tool deflection," which causes shape distortion of the machined parts, especially when machining thin-walled workpieces or using slender cutting tools. 1.5 Vibration error of machine tool During cutting, due to the flexibility of the process and the variability of the operation, the operating state of CNC machine tool is more likely to fall into the unstable area, thus arousing strong chatter. This leads to the deterioration of the surface quality and geometric shape error of the workpiece. 1.6 The test error of the detection system includes the following aspects: (1) The error of the measurement sensor feedback system itself caused by the manufacturing error of the measurement sensor and its installation error on the machine tool; (2) The error of the measurement sensor caused by the error of machine tool parts and mechanism and the deformation during use. 1.7 External interference error Random error caused by changes in environment and operating conditions. 1.8 Other errors such as errors caused by programming and operation errors. The above errors can be classified into two categories according to their characteristics and nature: namely, systematic error and random error. The systematic error of CNC machine tool is the inherent error of the machine tool itself and is repeatable. The geometric error of CNC machine tool is its main component and is also repeatable. Using this characteristic, it can be "measured offline", and the "offline detection - open-loop compensation" technique can be used to correct and compensate it, so as to reduce it and achieve the purpose of strengthening the machine tool accuracy. Random error has randomness, and the "online detection - closed-loop compensation" method must be used to eliminate the influence of random error on the machining accuracy of the machine tool. This method has strict requirements on measuring instruments and measuring environment, and is difficult to promote. 2 Geometric error compensation technology According to different types of errors, error compensation can be divided into two categories. Random error compensation requires "online measurement", and the error detection device is directly installed on the machine tool. While the machine tool is working, the error value at the corresponding position is measured in real time, and the machining command is corrected in real time using this error value. Random error compensation has no requirements on the error nature of the machine tool and can compensate for the random error and systematic error of the machine tool at the same time. However, it requires a complete set of high-precision measuring devices and other related equipment, which is too costly and not economically efficient. Reference [4] carried out online measurement and compensation of temperature, but failed to achieve practical application. Systematic error compensation involves pre-testing the machine tool with appropriate instruments, specifically obtaining the error values ​​of the machine tool's workspace command positions through "offline measurement," and using these values ​​as functions of the machine tool coordinates. During machine operation, the corresponding error values ​​are retrieved based on the coordinates of the machining points for correction. This requires high machine tool stability to ensure the determinism of machine tool errors, facilitating correction. The accuracy of the compensated machine tool depends on its repeatability and environmental conditions. Under normal circumstances, the repeatability of CNC machine tools is far higher than their spatial comprehensive error; therefore, systematic error compensation can effectively improve the accuracy of the machine tool, and even improve its accuracy level. To date, there are many methods for systematic error compensation both domestically and internationally, which can be categorized as follows: 2.1 Single-item error synthesis compensation method: This compensation method is based on the error synthesis formula. First, the original individual error values ​​of the machine tool are measured directly. Then, the error components at the compensation points are calculated using the error synthesis formula, thereby achieving error compensation for the machine tool. Leete was the first to measure the position error of a coordinate measuring machine (CMM). He used trigonometric relationships to derive the representation method of the machine tool's coordinate axis errors, without considering the influence of rotation angles. Professor Hocken was one of the earliest to perform error compensation. For a CMM model Moore5-Z (1), he measured the errors of a large number of points in the workspace within 16 hours, taking into account the influence of temperature and identifying the error model parameters using the least squares method. Since the position signal of the machine tool motion was obtained directly from the laser interferometer, the influence of angle and straightness errors was considered, and relatively satisfactory results were obtained. In 1985, G. Zhang successfully performed error compensation on a CMM. The flatness error of the worktable was measured. Except for the slightly larger value at the edge of the worktable, the others did not exceed 1 μm, verifying the reliability of the rigid body assumption. The 21 errors measured using a laser interferometer and a level were synthesized through linear coordinate transformation and error compensation was implemented. Measurement experiments on the XY plane show that before compensation, 20% of all measurement points had errors greater than 20 μm. After compensation, no more than 20% of the points had errors greater than 2 μm, demonstrating an accuracy improvement of nearly 10 times. Besides error compensation for coordinate measuring machines, research on error compensation for CNC machine tools has also yielded certain results. In 1977, Professor Schultschik used vector diagrams to analyze the errors of various machine tool components and their impact on geometric accuracy, laying the foundation for further research on machine tool geometric errors. Ferreira and his collaborators also studied this method, deriving a general model for machine tool geometric errors and contributing to the single-item error synthesis compensation method. J. Nietal further applied this method to online error compensation, obtaining relatively ideal results. Nietal established a 32-item error model, of which the 11 redundant items are related to temperature and machine tool origin error parameters. Compensation experiments on horizontal machining centers showed an accuracy improvement of 10 times. Eung-SukLeaetal used almost the same measurement method as G. Zhang to measure 21 errors of the Bridgeport three-axis milling machine. He derived an error model using the error synthesis method, and the compensated results were verified using a laser interferometer and a Renishaw DBB system, demonstrating improved machine tool accuracy. 2.2 Direct Error Compensation Method: This method requires precise measurement of the machine tool's spatial vector error. Higher compensation accuracy requires more measurement precision and a greater number of measurement points. However, knowing the error at any point in the measurement space in detail is impossible. Interpolation is used to obtain the error components at the compensation points for error correction. This method requires establishing an absolute measurement coordinate system consistent with the compensation process. In 1981, Dufour and Groppetti measured the errors at machine tool workspace points under different load and temperature conditions, constructing an error vector matrix to obtain machine tool error information. This error matrix was then stored in a computer for error compensation. Similar studies include those by ACOkaforetal, who measured the relative errors of multiple points on a standard reference within the machine tool's workspace, using the first point as a reference, and then converted them into absolute coordinate errors. Error compensation was then performed using interpolation, resulting in a 2-4 fold improvement in accuracy. Hooman, using a three-dimensional linear dynamic range measurement device (LVTDS), obtained the errors of 27 points in the machine tool space (resolution 0.25 μm, repeatability 1 μm) and performed similar work. Further considering the influence of temperature, measurements were taken every 1.2 hours for a total of 8 measurements, and the error compensation results were corrected for temperature coefficients. The drawback of this method is the large workload and the large amount of data to store. Currently, there is no completely suitable instrument, which limits the further application and development of this method. 2.3 Relative Error Decomposition and Synthesis Compensation Method Most error measurement methods only obtain the relative comprehensive error, from which the individual machine tool errors can be decomposed. Further utilizing error synthesis methods for machine tool error compensation is feasible. Currently, some progress has been made in this area both domestically and internationally. In 2000, Professor Jun Ni's doctoral student Chen Guiquan at the University of Michigan conducted an experiment, using a ballbar (TBB) to measure the geometric errors of a three-axis CNC machine tool at different temperatures. He established a rapid temperature prediction and error compensation model and performed error compensation. Christopher used a laser ballbar (LBB) to obtain machine tool error information within 30 minutes, established an error model, and evaluated the error compensation results five times over a nine-month period. The results showed that software error compensation can improve machine tool accuracy and maintain it over a relatively long period. Error synthesis requires measuring the original errors of each axis of the machine tool. A relatively mature measurement method is the laser interferometer, which offers high measurement accuracy. However, error measurement using a dual-frequency laser interferometer is time-consuming and requires a high level of operator skill. More importantly, it has high requirements for the error measurement environment and is often used for coordinate measuring machine (CMM) inspection, making it unsuitable for production site operation. The relative error decomposition and synthesis compensation method is relatively simple, obtaining data for the entire circumference in a single measurement, and can simultaneously meet the requirements for machine tool accuracy inspection and evaluation. Currently, there are many error decomposition methods. However, due to the varying conditions of machine tools, it is difficult to find a suitable universal mathematical model for error decomposition. Furthermore, original error terms that have the same impact on measurement results cannot be decomposed, hindering widespread application. Direct error compensation methods generally use standard parts as a reference to obtain spatial vector errors for direct compensation, eliminating intermediate steps and more closely reflecting the practical situation of machine tools. However, obtaining a large amount of information requires different standard parts, which is difficult to achieve, thus limiting the compensation accuracy. In China, many research institutions and universities have also conducted research on machine tool error compensation in recent years. In 1986, the Beijing Machine Tool Research Institute conducted research on machine tool thermal error compensation and coordinate measuring machine compensation. In 1997, Li Shuhe et al. of Tianjin University conducted modeling and thermal error compensation research on machine tool errors. In 1998, Liu Youwu et al. of Tianjin University established an error model for machine tools using a multibody system and presented 22-line, 14-line, and 9-line laser interferometer measurement methods for geometric errors. In 1999, they also conducted a comprehensive study on error compensation for CNC machine tools, achieving encouraging results. In 1998, Yang Jianguo of Shanghai Jiao Tong University conducted research on thermal error compensation for lathes. From 1996 to 2000, with the support of the National Natural Science Foundation of China and the National 863 Program, Huazhong University of Science and Technology carried out research on geometric error compensation for CNC machine tools and intelligent adaptive control based on online cutting force identification, achieving some results. In summary, error measurement is key to error compensation for CNC machine tools, and error models are fundamental. Through error compensation, the accuracy of machine tools can be effectively improved, contributing to the advancement of my country's manufacturing industry. References [1] Ni Jun. A review and prospect of the research on error compensation of CNC machine tools [J]. China Mechanical Engineering, 1997, 8 (1): 29-32. [2] Ramesh R, Mannan MA, Poo A N. Error compensation in machine tools — a review part I: geometric, cutting-force induced and fixture-dependent errors. 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