Geometric error measuring, modeling, and compensation for CNC machine tools: A review

Kinematic accuracy is a key reason which influence workpiece's geometric error precision on traditional working process of precisely CNC（Computerized Numerical Control）P3G（polygon profile with 3 lobes） grinding machine. A systematic geometric error model has been presented for CNC P3G grinding machine， proposed multi-body system theory integrate with the structure of CNC P3G grinding machine tools, researched on the machine's space geometric errors. By means of separate geometric errors from the machine tools, build geometric mathematical error model. Then, identify 21 error parameters through method of 9 lines, analysis and calculate the total space geometric errors of the workpiece and wheel. Finally, formed a parameter-list and applied software error compensational technique ， achieved real-time control to the motions of workpiece and wheel. Experimental results shown that the geometrical error modeling technique is accurate and efficient, and the precision of CNC P3G grinding machine is highly raised 70%.

The article is devoted to increasing the volumetric accuracy of multi-axis machine tools. Based on the model of measurement and calculation of the volumetric geometric error of machine tools presented by the authors, a method has been developed which allows to signifi cantly improve the accuracy of the movement of the operating unit of a three-axis metal-cutting machine tool to a given point. The astatic law and the calculation of the coordinate correction according to the differential geometric model of the volumetric error are used to control the movement of the operating unit. The developed method is implemented programmatically and allows to perform the correction in the control program represented in G-code. As a result of experiments on software correction of volumetric errors according to the developed method, conducted at the State Engineering Centre of the Moscow State University of Technology “STANKIN” on a threeaxis CNC machine tool, a signifi cant reduction (up to 90 %) of geometric volumetric errors was shown. The developed method can be used to improve the accuracy of metal-cutting machine tools by creating postprocessors that calculate corrections to the coordinates set in the control system based on the results of interferometric measurements.Keywords: volumetric accuracy, interference measurements, three-axis machine tools, machine tool accuracy, geometric errors, astatic control law, mathematical modelling, differential geometry

Modeling of geometric errors of linear guideway and their influence on joint kinematic error in machine tools

This paper proposes a recognition methodology for key geometric errors using the feature extraction method and accuracy retentivity analysis and presents the approach of optimization compensation of the geometric error of a multiaxis machine tool. The universal kinematics relations of the multiaxis machine tool are first modelled mathematically based on screw theory. Then, the retentivity of geometric accuracy with respect to the geometric error is defined based on the mapping between the constitutive geometric errors and the time domain. The results show that the variation in the spatial error vector is nonlinear while considering the operation time of the machine tool and the position of the motion axes. Based on this aspect, key factors are extracted that simultaneously consider the correlation, similarity, and sensitivity of the geometric error terms, and the results reveal that the effect of the position‐independent geometric errors (PIGEs) on the error vectors of the position and orientation is greater than that of the position‐dependent geometric errors (PDGEs) of the linear and rotary axes. Then, the fruit fly optimization algorithm (FOA) is adopted to determine the compensation values through multiobjective tradeoffs between accuracy retentivity and fluctuation in the geometric errors. Finally, an experiment on a four‐axis horizontal boring machine tool is used to validate the effectiveness of the proposed approach. The experimental results show that the variations in the precision of each test piece are lower than 25.0%, and the maximum variance in the detection indexes between the finished test pieces is 0.002 mm when the optimized parameters are used for error compensation. This method not only recognizes the key geometric errors but also compensates for the geometric error of the machine tool based on the accuracy retentivity analysis results. The results show that the proposed methodology can effectively enhance the machining accuracy.

Inevitable machine motion errors change the cutting tool trajectory and degrade the machined surface quality. Compared to the mature error measurement technologies developed for traditional precision CNC machine tools, the increasing use of ultra-precision machine tools (UPMTs) has shown some distinctive characteristics in error modelling, measurement, and compensation. This paper attempts to summarise state-of-the-art research in the calibration of geometric errors of UPMTs. A general routine for a UPMT error calibration is proposed in this literature review. Various error modelling methods, instruments, and measurement methods applicable to the geometric error measurement of both the linear and rotary axes are discussed using typical case studies. With respect to these achievements, there is a real concern regarding the reproducibility of measurement sensors used for the calibration of UPMTs and it remains challenging to decompose the volumetric motion error of UPMTs. Owing to the high flexibility in practice, trial cutting and a sensitivity analysis-based error measurement and compensation provide a promising solution to achieve a fast UPMT calibration.

A three-point method based on sequential multilateration is applied to measuring the geometric error components (GECs) of three-axis machine tools (MTs). To meet the accuracy requirement of geometric error mapping, a sequential multilateration scheme is developed for high-accuracy point measurement by introducing four additional targets into the measuring system, and the uncertainty of point measurement is verified by simulation. Three independent targets fixed on the MT’s spindle assembly move along with each axis step by step and their coordinates at each step can be determined by the distance data acquired by laser tracker at all steps based on sequential multilateration. Then the volumetric errors of the three target points can be obtained by comparing the actual coordinates and the corresponding desired coordinates, and nine equations can be established by substituting volumetric errors into the error model of linear axis, so that the six GECs of each axis can be obtained by solving these equations. The three squareness errors can be determined by computing the angles between the average lines of the three axes which are achieved by linear curve fitting. Experiments are conducted to measure these 21 GECs, and the volumetric errors in the three-axis MT’s workspace, which are determined by these measured GECs based on the error model of three-axis MT, are compensated. Finally, the positioning errors of the MT with compensation and without compensation are evaluated by laser interferometer, respectively, the experimental results of which demonstrate that the positioning errors are significantly reduced by the error compensation.

The simultaneous measurement and identification of position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs) of dual rotary axes in five-axis machine tools remain a significant challenge in precision machining. To address this issue, this study develops a geometric error measurement system for the dual rotary axes of a five-axis machine tool by integrating a scanning probe, a block, and a calibration sphere. Unlike conventional touch-trigger probes, the scanning probe acquires continuous measurement signals along the surfaces of the block and the calibration sphere, enabling the extraction of richer geometric features and providing sufficient data for the simultaneous analysis of multiple geometric error components. The proposed measurement system is implemented on a five-axis machine tool, Uni5X-400, manufactured by FALCON Machine Tools Co., Ltd., to measure the geometric information of the calibration sphere and the block and thereby identify 20 PIGEs and PDGEs associated with the dual rotary axes. A kinematic model of the measurement system is established using homogeneous coordinate transformation matrices. By applying forward and inverse kinematics, the spatial positions of the calibration sphere and block are derived, forming an explicit relationship between volumetric errors and geometric errors. The resulting error identification equations are solved using the least squares method. Experimental results demonstrate that the proposed approach can efficiently and accurately identify the geometric errors of the dual rotary axes and maintain robust reliability under real-world operating conditions, making it suitable for practical calibration and accuracy improvement of five-axis machine tools.

The relative position and orientation deviation between grinding wheel and workpiece caused by geometric errors affect the machining accuracy of five-axis CNC gear profile grinding machine tools directly. Therefore, geometric error modeling and compensation are presented according to homogeneous transformation and differential motion matrix based on the multi-body system theory for the accuracy enhancement of the machine tools. Firstly, the open kinematic chain and the ideal homogeneous transformation matrix from workpiece to grinding wheel are established according to the topological structure of the CFXZAY-type, five-axis gear profile grinding machine tools and the basic homogeneous transformation matrix between coordinate frames. Secondly, the homogenous transformation matrices of linear pairs and rotary pairs with geometric errors are calculated, and the relationships of the error propagation from workpiece to grinding wheel are acquired as well on the basis of 37 geometric error components analysis. The geometric error models including position and orientation errors of grinding wheel in workpiece coordinate system are obtained with matrix multiplication by using small-angle approximation and ignoring the second-order and high-order error terms. Then, the Jacobian is obtained by using transforming differential motion matrix of each motion axis to compensate the integrated error components of the grinding wheel, which can make the compensation effective and convenient with using the corresponding homogeneous transformation matrix. Finally, error measurement, error compensation, and machining experiments are carried out on a five-axis CNC gear profile grinding machine tool SKMC-1200W/10 to verify the applicability and effectiveness of the proposed error modeling, error compensation, and research approach.

Geometric error is the major part of the inaccuracy of NC machine tools. This paper presents an integrated geometric error measuring, modeling and compensation method for NC machine tools. 21 geometric errors are anlysised. Taking three-axis machine tool as example, the positioning error is measured by a laser interferometer. A new method is proposed using MATLAB to fit the curve measured. The efficient methodology is developed and described for the establishment of the geometric error model. Finally, the compensation mechanism is executed based on the model. The experimental results show that the geometric accuracy of the machine tool is significantly improved by the error compensation.

Geometric error compensation is a widely used and effective approach for enhancing the accuracy of Coordinate Measuring Machines (CMMs) and Computer Numerical Control (CNC) machines. The research in this field primarily focuses on the challenging task of developing geometric error modelling techniques. Currently, there are several well-established methods for modelling geometric errors. However, there is scarce availability of methods that can validate the established geometric error models. Therefore, this article proposes an analytical validation method for geometric error models based on the error transformation equation. The principle of the method is elaborately explained, and its correctness is verified through specific model verification example. The method proposed in this article enables rapid and accurate model correction and iteration, effectively reducing the economic and time costs of model validation in the current geometric error compensation process. Therefore, the method provides guidance for standardising the modelling process and could be widely applied to various structural forms of measurement and machining systems.

The purpose of this study was to improve the accuracy of three-axis CNC Milling Vertical engines with a general approach by using mathematical modeling methods of machine tool geometric errors. The inaccuracy of CNC machines can be caused by geometric errors that are an important factor during the manufacturing process and during the assembly phase, and are factors for being able to build machines with high-accuracy. To improve the accuracy of the three-axis vertical milling machine, by knowing geometric errors and identifying the error position parameters in the machine tool by arranging the mathematical modeling. The geometric error in the machine tool consists of twenty-one error parameters consisting of nine linear error parameters, nine angle error parameters and three perpendicular error parameters. The mathematical modeling approach of geometric error with the calculated alignment error and angle error in the supporting components of the machine motion is linear guide way and linear motion. The purpose of using this mathematical modeling approach is the identification of geometric errors that can be helpful as reference during the design, assembly and maintenance stages to improve the accuracy of CNC machines. Mathematically modeling geometric errors in CNC machine tools can illustrate the relationship between alignment error, position and angle on a linear guide way of three-axis vertical milling machines.

The volumetric accuracy of machine tools is significantly impacted by thermal errors, making the evaluation of volumetric accuracy a significant challenge for the manufacturing industry. Several technical solutions have been implemented, both at the industrial and research level, to address this issue. However, considering thermal errors in the volumetric error characterization adds complexity to the process and remains an open topic of research in the field.In this study, a novel volumetric error characterization technique based on simulated Thermal Modal Analysis is proposed for a three-axis machine tool. This approach involves analyzing the impact of thermal modes of the machine tool's structural elements on the volumetric error, providing a characterization that incorporates the physical meaning of the structure. This simulated characterization will serve as a foundation for future experiments that combine it with other techniques to achieve a comprehensive understanding of volumetric errors.

Total differential methods based universal post processing algorithm considering geometric error for multi-axis NC machine tool

Virtual manufacturing systems can provide useful means for products to be manufactured without the need of physical testing on the shop floor. As a result, the time and cost of part production can be decreased. There are different error sources in machine tools such as tool deflection, geometrical deviations of moving axis and thermal distortions of machine tool structures. Some of these errors can be decreased by controlling the machining process and environmental parameters. However other errors like tool deflection and geometrical errors which have a big portion of the total error, need more attention. This paper presents a virtual machining system in order to enforce dimensional, geometrical and tool deflection errors in three-axis milling operations. The system receives 21 dimensional and geometrical errors of a machine tool and machining codes of a specific part as input. The output of the system is the modified codes which will produce actual machined part in the virtual environment.

To address the problems of the low accuracy of geometric error identification and incomplete identification results of the linear axis detection of computer numerical control (CNC) machine tools, a new 21-item geometric error identification method based on double ball-bar measurement was proposed. The model between the double ball-bar reading and the geometric error term in each plane was obtained according to the three-plane arc trajectory measurement. The mathematical model of geometric error components of CNC machine tools is established, and the error fitting coefficients are solved through the beetle antennae search particle swarm optimization (BAS–PSO) algorithm, in which 21 geometric errors, including roll angle errors, were identified. Experiments were performed to compare the optimization effect of the BAS–PSO and PSO and BAS and genetic particle swarm optimization (GA–PSO) algorithms. Experimental results show that the PSO algorithm is trapped in the local optimum, and the BAS–PSO is superior to the other three algorithms in terms of convergence speed and stability, has higher identification accuracy, has better optimization performance, and is suitable for identifying the geometric error coefficient of CNC machine tools. The accuracy and validity of the identification results are verified by the comparison with the results of the individual geometric errors detected through laser interferometer experiments. The identification accuracy of the double ball-bar is below 2.7 µm. The proposed identification method is inexpensive, has a short processing time, is easy to operate, and possesses a reference value for the identification and compensation of the linear axes of machine tools.