A Geometric Error Identification Method Based on a Ball Bar for Rotary Axes Independent of Position

Abstract

The present invention relates to a method for identifying geometric errors independent of the position of a rotating axis based on a ball bar. By installing the center of the tool ball at the intersection point of the AC axis and offsetting the workpiece ball in the XY direction, the single-axis movement of the two axes is controlled successively to achieve the switching of two measurement modes under one installation. Eight position-independent geometric errors of the rotating axis are identified through three installations. This method is easy to install, does not require the use of an extension rod, nor the identification of the axis of the rotating axis, improving the efficiency of error measurement. By inversely solving the coordinates of the workpiece ball through the inverse matrix, the initial coordinates of the tool ball and the workpiece ball under the RCS are established. A comprehensive model of the rod length including the installation error of the ball bar is established based on homogeneous coordinate transformation, and the preset values and the identified values are compared through simulation analysis. The results show that the residuals of the eight errors are very small, and the proposed identification method has high accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of geometric error identification, and particularly to a method for identifying geometric errors independent of the position of a rotating axis based on a ball bar. Background Art

[0002] Compared with three-axis machine tools, five-axis machine tools have better flexibility and can machine complex parts such as turbine blades and engine blocks. However, the addition of two rotating axes introduces more errors, affecting the accuracy of five-axis machine tools. Therefore, the measurement and identification of geometric errors of rotating axes are of great significance for improving the machining accuracy of five-axis machine tools. For the measurement of geometric errors of rotating axes, currently commonly used instruments include laser trackers, contact trigger probes, R-tests, ball bars, etc. Compared with other instruments, ball bars have the advantages of low cost, convenient installation, and stable measurement, and are widely used in the measurement of rotating axis errors.

[0003] For the measurement of geometric errors independent of the position of a rotating axis, currently commonly used methods include multi-axis linkage measurement and single-axis motion measurement. However, for the multi-axis linkage measurement method, since the geometric errors of the translational axes cannot be completely compensated, the measurement results will be affected by the translational axes, resulting in inaccurate final identification results. By controlling single-axis motion measurement, the influence of translational axis errors on the identification of rotating axis errors can be eliminated, the coupling problem of geometric errors between translational axes and rotating axes can be solved, and the error identification accuracy can be improved. At the same time, most of the existing identification methods using ball bars require more installation positions, increasing the installation time and measurement difficulty of the ball bar and affecting the simplicity of error measurement. Summary of the Invention

[0004] The purpose of the present invention is to at least solve one of the deficiencies of the prior art, and provide a method for identifying geometric errors independent of the position of a rotating axis based on a ball bar.

[0005] To achieve the above purpose, the present invention adopts the following technical solutions:

[0006] Specifically, a method for identifying geometric errors independent of the position of a rotating axis based on a ball bar is proposed, which is applied to a five-axis machine tool and includes the following:

[0007] Establish coordinate systems successively according to the workpiece end motion chain of the five-axis machine tool. The five-axis machine tool consists of three linear axes, namely the X-axis, Y-axis, and Z-axis, and two rotary axes, namely the A-axis and C-axis. The motion chain structure of the five-axis machine tool is divided into a tool chain R-Y-X-Z-T and a workpiece chain R-A-C-W. Since only the rotary axes are identified, the linear axes are not considered during the coordinate system establishment. The A-axis coordinate system ACS is established at the intersection of the axes of the A-axis and C-axis, and the C-axis coordinate system CCS is established at the center of the rotary table. To simplify the modeling, the workpiece coordinate system WCS is made to coincide with CCS, and the reference coordinate system RCS is made to coincide with ACS, and the Z-axis axes of WCS, CCS, RCS, and ACS are collinear;

[0008] Three installation modes are adopted for measurement. In the three installation modes, the tool ball B 1 is connected to the spindle through the tool cup, and the workpiece ball B 2 is fixed on the worktable through the tool cup and the magnetic suction seat;

[0009] Specifically,

[0010] In the first installation mode, control the movement of the A-axis or C-axis to measure 4 items of position errors respectively to obtain the first ball bar length model;

[0011] In the second installation mode, control the movement of the A-axis to measure 2 items of perpendicularity errors to obtain the second ball bar length model;

[0012] In the third installation mode, control the movement of the C-axis to measure 2 items of perpendicularity errors to obtain the third ball bar length model;

[0013] Based on the MATLAB function, combine the first ball bar length model, the second ball bar length model, and the third ball bar length model to fit and solve for 8 items of position-independent geometric errors of the rotary axes.

[0014] Furthermore, specifically, in the first installation mode, control the movement of the A-axis or C-axis to measure 4 items of position errors respectively to obtain the first ball bar length model, including,

[0015] The initial coordinates of the balls B 1 and B 2 in the reference coordinate system are:

[0016] (1)

[0017] (2),

[0018] where, e x 、 e y are respectively the tool ball B 1The installation error of the sphere center in the X and Y directions of the machine tool reference coordinate system, X L 、 Y L is the workpiece sphere B 2 The offset of the sphere center relative to the tool sphere B in the X and Y directions; 1 When only controlling the movement of the A axis, the workpiece sphere B

[0019] makes a circular motion around the A axis, and the actual transformation matrix from the A axis coordinate system to the reference coordinate system is: 2

[0020] (3)(3)

[0021] Calculate the initial coordinates of the workpiece sphere B 2 in the A axis coordinate system through inverse matrix transformation as:

[0022] (4)

[0023] After coordinate matrix transformation, the actual coordinates of B 2 in the reference coordinate system are:

[0024] (5)

[0025] And the tool sphere B 1 remains fixed during the measurement, so its coordinates in the reference coordinate system remain unchanged.

[0026] Substitute the coordinates of sphere B 1 and B 2 Calculate, simplify and ignore the high-order error small quantities to obtain the actual length of the ball bar as L A1 :

[0027] (6)

[0028] When only controlling the movement of the C axis, the workpiece sphere B 2 makes a circular motion around the C axis, and the actual transformation matrix from the C axis coordinate system to the A axis coordinate system is:

[0029] (7)

[0030] Calculate the initial coordinates of the workpiece sphere B 2 in the C axis coordinate system:

[0031] (8)

[0032] Then find the actual coordinates of B 2 in the reference coordinate system:

[0033] (9)

[0034] Finally, the actual length of the ball bar is obtained based on the coordinates of the two balls. L C1 It is:

[0035] (10).

[0036] Furthermore, specifically, in the second installation mode, controlling the movement of the A-axis measures the two perpendicularity errors to obtain the second ball bar length model, including

[0037] Only controlling the movement of the A-axis, due to the change in the installation position of the ball bar, the initial coordinates of ball B 1 and B 2 also change, as shown in Equations (11) and (12):

[0038] (11)

[0039] (12)

[0040] In the formula, l is the distance from the center of the tool ball B 1 to the axis of the C-axis, L is the nominal length of the ball bar.

[0041] Similar to the first installation mode, first calculate the actual coordinates of the workpiece ball B 2 in the reference coordinate system, and then obtain the actual length of the ball bar L A2 It is:

[0042] (13).

[0043] Furthermore, specifically, in the third installation mode, controlling the movement of the C-axis measures the two perpendicularity errors to obtain the third ball bar length model, including

[0044] Only controlling the movement of the C-axis, the initial coordinates of ball B 1 and B 2 are as follows:

[0045] (14)

[0046] (15)

[0047] In the formula, h is the distance from the center of the tool ball B 1 to the axis of the A-axis.

[0048] Similarly, the B 2The actual coordinates, and finally obtain the actual length of the ball bar L C3 are:

[0049] (16).

[0050] The beneficial effects of the present invention are:

[0051] The present invention provides a method for identifying geometric errors independent of the position of the rotating axis based on a ball bar. By installing the center of the tool ball at the intersection point of the AC axis, offsetting the workpiece ball in the XY direction, and controlling the single-axis movement of the two axes successively, the switching of two measurement modes is realized under one installation, and 8 geometric errors independent of the position of the rotating axis are identified through three installations. This method is easy to install, does not require the use of an extension rod, nor the identification of the axis of the rotating axis, improving the efficiency of error measurement. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] By describing in detail the embodiments shown in the accompanying drawings, the above and other features of the present disclosure will become more obvious. The same reference numerals in the drawings of the present disclosure represent the same or similar sampling monitoring points. Obviously, the drawings in the following description are only some embodiments of the present disclosure. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings. In the drawings:

[0053] Figure 1 Figures (a), (b), and (c) respectively show schematic diagrams of three installation modes involved in a method for identifying geometric errors independent of the position of the rotating axis based on a ball bar according to the present invention;

[0054] Figure 2 Figures (a) and (b) respectively show schematic diagrams of the machine tool structure and coordinate system of a five-axis machine tool to which a method for identifying geometric errors independent of the position of the rotating axis based on a ball bar according to the present invention is applied;

[0055] Figure 3 Figures (a) and (b) respectively show schematic diagrams of the definition principle of geometric errors independent of the position;

[0056] Figure 4 shows a schematic diagram of the installation error of the tool ball;

[0057] Figure 5 Figures (a), (b), and (c) respectively show schematic diagrams of three installation modes at the installation and measurement site in one embodiment;

[0058] Figure 6 Figures (a) and (b) respectively show curves of the ball bar length data measured twice in one embodiment;

[0059] Figure 7As shown in (a) and (b), the diagrams respectively show the length prediction errors of the ballbar in three installation modes in one embodiment. Detailed implementation mode

[0060] The following will clearly and completely describe the concept, specific structure and technical effects of the present invention in combination with embodiments and drawings to fully understand the purpose, solution and effects of the present invention. It should be noted that, without conflict, the embodiments in the present application and the features in the embodiments can be combined with each other. The same reference numerals used throughout the drawings indicate the same or similar parts.

[0061] This solution only measures the geometric errors independent of the position of the rotating axis, controls the movement of a single rotating axis during the measurement process, and reduces the influence of other axes on the error measurement. In Embodiment 1, the present invention proposes a method for identifying geometric errors independent of the position of the rotating axis based on a ballbar, which is applied to a five-axis machine tool and includes the following:

[0062] Establish coordinate systems in sequence according to the workpiece end motion chain of the five-axis machine tool. The structure of the five-axis machine tool is as Figure 2 shown in a. The five-axis machine tool consists of three linear axes, namely the X-axis, Y-axis, and Z-axis, and two rotating axes, namely the A-axis and C-axis. The motion chain structure of the five-axis machine tool is divided into a tool chain R-Y-X-Z-T and a workpiece chain R-A-C-W. Since only the rotating axes are identified, the linear axes are not considered during the coordinate system establishment process. As Figure 2 shown in b, the A-axis coordinate system ACS is established at the intersection of the axes of the A-axis and C-axis, and the C-axis coordinate system CCS is established at the center of the rotary table. To simplify the modeling, the workpiece coordinate system WCS is coincident with the CCS, the reference coordinate system RCS is coincident with the ACS, and the Z-axis axes of the WCS, CCS, RCS, and ACS are collinear; where the capital letter H + CS refers to the coordinate system where the capital letter H is located, and the value of H is R-Y-X-Z-T or R-A-C-W;

[0063] Three installation modes are adopted for measurement. In the three installation modes, the tool ball B 1 is connected to the spindle through a tool cup (installed at the intersection of the axes of the A and C axes), and the workpiece ball B 2 is fixed on the workbench through a tool cup and a magnetic suction seat;

[0064] Specifically,

[0065] In the first installation mode, control the movement of the A-axis or C-axis to measure 4 position errors respectively to obtain the first ballbar rod length model; as Figure 1 shown in a, install the tool ball center at the intersection of the axes of the A and C axes, and install the workpiece ball at a position 80 mm from the A-axis axis and 60 mm from the C-axis axis.

[0066] In the second installation mode, control the movement of the A axis to measure two perpendicularity errors to obtain the second ballbar length model; as Figure 1 shown in b, install the center of the tool ball on the A axis, 60 mm away from the other axis.

[0067] In the third installation mode, control the movement of the C axis to measure two perpendicularity errors to obtain the third ballbar length model; as Figure 1 shown in c, install the center of the tool ball on the C axis, 60 mm away from the other axis.

[0068] Based on the MATLAB function, combined with the first ballbar length model, the second ballbar length model, and the third ballbar length model, fit and solve for the 8 position-independent geometric errors of the rotating axis.

[0069] According to the ballbar length models in the three installation modes, combined with the ballbar length data in the actual measurement process, and measure the ballbar installation error through a digital display micrometer and a ballbar. Substitute these data into the length model, and use the MATLAB function to fit and solve for the 8 position-independent geometric errors of the rotating axis.

[0070] In this embodiment 1, by installing the center of the tool ball at the intersection of the A and C axis lines, offset the workpiece ball in the XY direction, control the single-axis movement of the two axes successively, and realize the switching of two measurement modes in one installation. Identify the 8 position-independent geometric errors of the rotating axis through three installations. This method is easy to install, does not require the use of an extension rod, nor does it require identifying the axis of the rotating axis, improving the efficiency of error measurement.

[0071] Considering that the position-independent geometric error of the rotating axis is caused by the deviation during the assembly of the machine tool components, it will cause errors in the position and angle of the axis, affecting the actual machining accuracy of the machine tool. The machine tool structure studied in this paper is as Figure 2 shown in a, its rotating axis is composed of the A axis and the C axis, with a total of 8 position-independent geometric errors. Each rotating axis has two position errors and two perpendicularity errors, as Figure 3 shown in (a) and (b). δ ya and δ za are the position deviation amounts of the A axis along the Y R and Z R directions in the machine tool reference coordinate system, S ya and S za are the angles by which the A axis rotates around Y R and Z RPerpendicularity error between two axes δ xc and δ yc are the position deviation amounts of the C-axis along the X A and Y A directions in the A-axis coordinate system, S xc and S yc is the perpendicularity error of the C-axis around the X A and Y A two axes.

[0072] According to the three installation modes, the initial coordinates of the tool ball B 1 and the workpiece ball B 2 including the installation error of the ballbar are established in sequence under the RCS. During the measurement, only the movement of a single rotating axis is controlled, and the coordinates of the tool ball do not change. The actual initial position of the workpiece ball on the measured axis is determined through the inverse transformation matrix. Then, the actual coordinates of the workpiece ball under the RCS are determined through the homogeneous transformation matrix, so that the actual coordinates of the ball B 1 and B 2 are both under the RCS, ensuring the consistency of the two ball coordinates and improving the identification accuracy of the model.

[0073] As a preferred embodiment of the present invention, specifically, in the first installation mode, the movement of the A-axis or C-axis is controlled to measure the four position errors respectively to obtain the first ballbar length model, including,

[0074] The initial coordinates of the ball B 1 and B 2 in the reference coordinate system are:

[0075] (1)

[0076] (2),

[0077] In the formula, e x , e y are respectively the installation errors of the center of the tool ball B 1 in the X and Y directions of the machine tool reference coordinate system, X L , Y L are the offsets of the center of the workpiece ball B 2 relative to the center of the tool ball B 1 in the X and Y directions; as Figure 4As shown, the axes of the tool cup and the tool shank deviate from the axis of the reference coordinate system (RCS) during installation, resulting in errors in the X and Y directions of the center of the tool ball. e x and e y The installation error of the workpiece ball has little effect on the change in rod length and can be ignored in the actual measurement test.

[0078] When only the movement of axis A is controlled, the workpiece ball B 2 makes a circular motion around axis A. The actual transformation matrix from the coordinate system of axis A to the reference coordinate system is:

[0079] (3)

[0080] Calculate the initial coordinates of the workpiece ball B 2 in the coordinate system of axis A through inverse matrix transformation:

[0081] (4)

[0082] After coordinate matrix transformation, the actual coordinates of B 2 in the reference coordinate system are:

[0083] (5)

[0084] The tool ball B 1 remains stationary during the measurement process, so its coordinates remain unchanged in the reference coordinate system.

[0085] Substitute the coordinates of ball B 1 and B 2 , calculate and simplify, and ignore the small high-order error terms to obtain the actual length of the ball bar L A1 as:

[0086] (6)

[0087] When only the movement of axis C is controlled, the workpiece ball B 2 makes a circular motion around axis C. The actual transformation matrix from the coordinate system of axis C to the coordinate system of axis A is:

[0088] (7)

[0089] Calculate the initial coordinates of the workpiece ball B 2 in the coordinate system of axis C:

[0090] (8)

[0091] Then find the actual coordinates of B 2 in the reference coordinate system:

[0092] (9)

[0093] Finally, the actual length of the ball bar is obtained based on the coordinates of the two balls. L C1 It is:

[0094] (10).

[0095] As a preferred embodiment of the present invention, specifically, in the second installation mode, controlling the movement of the A-axis to measure the two perpendicularity errors to obtain the second ball bar length model, including,

[0096] Only controlling the movement of the A-axis, due to the change in the installation position of the ball bar, the initial coordinates of ball B 1 and B 2 also change, as shown in equations (11) and (12):

[0097] (11)

[0098] (12)

[0099] In the formula, l is the distance from the center of the tool ball B 1 to the axis of the C-axis, L is the nominal length of the ball bar.

[0100] Similar to the first installation mode, first calculate the actual coordinates of the workpiece ball B 2 in the reference coordinate system, and then obtain the actual length of the ball bar L A2 It is:

[0101] (13).

[0102] As a preferred embodiment of the present invention, specifically, in the third installation mode, controlling the movement of the C-axis to measure the two perpendicularity errors to obtain the third ball bar length model, including,

[0103] Only controlling the movement of the C-axis, the initial coordinates of ball B 1 and B 2 are as follows:

[0104] (14)

[0105] (15)

[0106] In the formula, h is the distance from the center of the tool ball B 1 to the axis of the A-axis.

[0107] Similarly, the actual coordinates of B can be obtained, and finally the actual length of the ballbar is obtained. 2 L C3 is:

[0108] (16).

[0109] In this preferred embodiment, the establishment of the workpiece end coordinate system is simplified. At the same time, during the model establishment process, the influence of the ballbar installation error on the identification model is considered, and the installation error is eliminated as much as possible through actual measurement, simplifying the error identification model and improving the error identification accuracy.

[0110] When carrying out actual application,

[0111] 1 Simulation

[0112] To further verify the accuracy of the identification method, the method is verified by simulation here. First, 8 position-independent geometric errors of the A-axis and C-axis are generated. According to the simulation parameters in Table (1), the ballbar length data including the ballbar installation error is calculated through equations (6), (10), (13), and (16). Then, the MATLAB fitting function is used to identify the actual position-independent geometric errors. Finally, the preset values and the identified values are compared, and the residuals of each error are calculated.

[0113] Table 1 Simulation basic parameters

[0114]

[0115] Table 2 Simulation results

[0116]

[0117] The simulation results are shown in Table (2). It can be seen that the residuals of each error are very small. The simulation results show that the identification method has high accuracy and can accurately identify 8 position-independent geometric errors of the rotating axis.

[0118] 2 Identification experiment

[0119] In the measurement experiment, the nominal length of the ballbar is 100 mm, the movement range of the A-axis is from -15° to 70°, and the movement range of the C-axis is from 0° to 360°. After the installation error measurement is completed, the ballbar is installed according to the three installation modes shown in Figure 1 . After the installation is completed, the A-axis or C-axis is controlled to move uniaxially in each mode, and the ballbar length data is recorded every 5°. Each group of measurements is repeated twice to avoid the contingency of the measurement results. The actual measurement is shown in Figure 5 (a), (b), (c). In mode one and mode two, the A-axis movement is controlled, and in mode one and mode three, the C-axis movement is controlled.

[0120] After the measurement is completed, the two results of each group of measurements are curve-fitted as shown in Figure 6 Figures (a) and (b). It can be seen that the curves of the two measurements basically coincide, and the four groups of data results have high repeatability.

[0121] Take the average value of the two measurement data of the ball bar, substitute the angle and length data into the MATLAB function to identify 8 position-independent geometric errors. According to the identification results of the errors, the predicted values of the ball bar length under three installation modes are obtained, and then the difference between the predicted value and the actual measured value is calculated to obtain the predicted error of the rod length. The predicted error can reflect the accuracy of the error identification results from the side. The smaller its value, the closer the identified error value is to the true value. The predicted errors under three installation modes are shown in Figure 7 Figures (a) and (b).

[0122] The results show that the maximum value of the predicted error of the rod length in the four measurement experiments does not exceed 1 μm. Therefore, the proposed identification method has high precision and can effectively identify the position-independent geometric errors of the rotating axis.

[0123] Summary:

[0124] (1) In order to quickly and accurately identify the geometric errors of the rotating axis, a method for measuring and identifying the position-independent geometric errors of the rotating axis based on three installations of the ball bar is proposed. By installing the center of the tool ball at the intersection point of the AC axis, and offsetting the workpiece ball in the XY direction, and controlling the single-axis movement of the two axes successively, the switching of two measurement modes under one installation is realized, and 8 position-independent geometric errors of the rotating axis are identified through three installations. This method is easy to install, does not require the use of an extension rod, and does not require identifying the axis of the rotating axis, improving the efficiency of error measurement.

[0125] (2) By inversely solving the workpiece ball coordinates with the inverse matrix, the initial coordinates of the tool ball and the workpiece ball under the RCS are established. Based on homogeneous coordinate transformation, a comprehensive rod length model including the installation error of the ball bar is established, and the preset value and the identified value are compared through simulation analysis. The results show that the residuals of the 8 errors are very small, and the proposed identification method has high precision.

[0126] (3) In the identification experiment, the installation error is measured by a digital display micrometer and a ball bar, basically eliminating the influence of the tool end installation error. And the effectiveness of the identification method is verified by comparing the predicted errors of the ball bar length. The maximum value of the predicted error of the rod length in the four measurement experiments does not exceed 1 μm, and the proposed identification method can effectively identify the position-independent geometric errors of the rotating axis.

[0127] Although the description of the present invention has been quite detailed and several of the described embodiments have been particularly described, it is not intended to be limited to any of these details or embodiments or any particular embodiment, but rather should be regarded as providing a broad interpretation of these claims in light of the prior art by reference to the appended claims, so as to effectively cover the intended scope of the present invention. In addition, the present invention has been described above in terms of embodiments foreseeable by the inventors for the purpose of providing a useful description, and non-substantive modifications to the present invention that are not currently foreseeable may still represent equivalent modifications of the present invention.

[0128] As described above, these are only the preferred embodiments of the present invention. The present invention is not limited to the above-described embodiments. As long as the same means are used to achieve the technical effects of the present invention, they should fall within the protection scope of the present invention. Within the protection scope of the present invention, various different modifications and variations may be made to its technical solutions and / or embodiments.

Claims

1. A method for identifying position-independent geometric errors of a rotating axis based on a ballbar, characterized in that: Applicable to five-axis machine tools, including the following: The coordinate system is established in sequence according to the kinematic chain of the workpiece end of the five-axis machine tool. The five-axis machine tool consists of three linear axes, X-axis, Y-axis, and Z-axis, and two rotary axes, A-axis and C-axis. The kinematic chain structure of the five-axis machine tool is divided into a tool chain RYXZT and a workpiece chain RACW. Since only the rotary axis is identified, the linear axis is not considered in the process of establishing the coordinate system. The A-axis coordinate system ACS is established at the intersection of the A-axis and C-axis axes, and the C-axis coordinate system CCS is established at the center of the rotary table. To simplify the modeling, the workpiece coordinate system WCS is overlapped with the CCS, and the reference coordinate system RCS is overlapped with the ACS, and the Z-axis axes of the WCS, CCS, RCS, and ACS are collinear. Three installation modes are used for measurement. In the three installation modes, the tool ball B1 is connected to the spindle through the tool cup, and the workpiece ball B2 is fixed to the workbench through the tool cup and the magnetic seat; Specifically, In the first installation mode, the A-axis or C-axis movement is controlled to measure the four position errors respectively to obtain the first ballbar length model; In the second installation mode, the A-axis movement is controlled to measure two perpendicularity errors to obtain the second ballbar length model; In the third installation mode, the C-axis movement is controlled to measure two perpendicularity errors to obtain the third ballbar length model; Based on MATLAB function combined with the first ballbar length model, the second ballbar length model and the third ballbar length model, eight position-independent geometric errors of the rotation axis are solved; Specifically, in the first installation mode, the A-axis or C-axis movement is controlled to measure four position errors respectively to obtain a first ballbar length model, including: The initial coordinates of balls B1 and B2 in the reference coordinate system are: (1)， (2)， In the formula, e x , e y are the installation errors of the center of tool ball B1 in the X and Y directions of the machine tool reference coordinate system, X L , Y L is the offset of the center of the workpiece ball B2 relative to the center of the tool ball B1 in the X and Y directions; When only the A-axis is controlled to move, the workpiece ball B2 moves in a circle around the A-axis. The actual transformation matrix from the A-axis coordinate system to the reference coordinate system is: (3)， The initial coordinates of the workpiece ball B2 in the A-axis coordinate system are calculated by inverse matrix transformation: (4)， After coordinate matrix transformation, the actual coordinates of B2 in the reference coordinate system are: (5)， The tool ball B1 is fixed during the measurement process, so its coordinates remain unchanged in the reference coordinate system. Substituting the coordinates of balls B1 and B2, simplifying the calculation and neglecting small higher-order errors, we get the actual length of the ballbar: L A1 for: (6)， When only the C-axis is controlled, the workpiece ball B2 moves in a circle around the C-axis. The actual transformation matrix from the C-axis coordinate system to the A-axis coordinate system is: (7)， in Z AC represents the distance between ACS and CCS, δ ya and δ za The A-axis is along the machine tool reference coordinate system. Y R and Z R Position deviation in direction, S ya and S za The A axis is around Y R and Z R The verticality error of the two axes, δ xc and δ yc The C-axis is along the A-axis coordinate system X A and Y A Position deviation in direction, S xc and S yc The C-axis is around X A and Y A The verticality error of the two axes; Calculate the initial coordinates of the workpiece ball B2 in the C-axis coordinate system: (8)， Then find the actual coordinates of B2 in the reference coordinate system: (9)， Finally, the actual length of the ballbar is obtained according to the coordinates of the two balls. L C1 for: (10)； Specifically, in the second installation mode, the A-axis movement is controlled to measure two verticality errors to obtain a second ballbar length model, including: Only the A-axis motion is controlled. As the installation position of the ballbar changes, the initial coordinates of balls B1 and B2 also change, as shown in equations (11) and (12): (11)， (12)， In the formula, l is the distance from the center of tool ball B1 to the axis of C axis, L is the nominal length of the ballbar, Same as the first installation mode, first calculate the actual coordinates of the workpiece ball B2 in the reference coordinate system, and then get the actual length of the ballbar L A2 for: (13)； Specifically, in the third installation mode, the C-axis movement is controlled to measure two verticality errors to obtain a third ballbar length model, including: Only the C-axis is controlled, and the initial coordinates of balls B1 and B2 are as follows: (14)， (15)， In the formula, h is the distance from the center of tool ball B1 to the axis of A axis, Similarly, the actual coordinates of B2 can be calculated, and finally the actual length of the ballbar can be obtained. L C3 for: (16)。

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