Ballbar-based method for identifying position-independent geometric errors of rotation axes

Abstract

The present invention relates to a ballbar-based method for identifying position-independent geometric errors of rotation axes, comprising: installing a cutter ball, so that the center thereof is at the intersection point of A and C axes; installing a workpiece ball in an offset manner in the X and Y directions; sequentially controlling the two axes to perform single-axis motion, so as to switch between two measurement modes in one instance of installation; and by means of three instances of installation, identifying eight position-independent geometric errors of the rotation axes. The method involves easy installation, without the need for using an extension rod and identifying the rotation axes, thereby improving the efficiency of error measurement. The coordinates of the workpiece ball are inversely derived by means of an inverse matrix, the initial coordinates of the cutter ball and the workpiece ball in an RCS are established, a comprehensive rod-length model including ballbar installation errors is constructed on the basis of homogeneous coordinate transformation, and preset values are compared with identified values by means of simulation analysis. The results indicate that the residual errors of all the eight errors are extremely small. Thus, the identification method provided herein has high accuracy.

Description

A Rotation Axis Position-Independent Geometric Error Identification Method Based on Ballbar

[0001] Cross-reference to related applications

[0002] This application claims priority to Chinese Patent Application No. 202411402230.0, filed on October 9, 2024, entitled "A Rotation Axis Position Independent Geometric Error Identification Method Based on Ballbar", the entire contents of which are incorporated herein by reference. Technical Field

[0003] This invention relates to the field of geometric error identification technology, and in particular to a method for identifying rotation axis position-independent geometric errors based on a ballbar. Background Technology

[0004] Five-axis machine tools offer greater flexibility than three-axis machine tools, enabling the machining of complex components such as turbine blades and engine blocks. However, the addition of two rotary axes introduces more errors, impacting the accuracy of five-axis machine tools. Therefore, the measurement and identification of geometric errors in the rotary axes are crucial for improving the machining accuracy of five-axis machine tools. Commonly used instruments for measuring rotary axis geometric errors include laser trackers, contact trigger probes, R-tests, and ballbars. Compared to other instruments, ballbars offer advantages such as low cost, easy installation, and stable measurement, and are widely used for measuring rotary axis errors.

[0005] For measuring geometric errors independent of the rotation axis position, commonly used methods include multi-axis linkage measurement and single-axis motion measurement. However, for multi-axis linkage measurement methods, the geometric error of the translation axis cannot be fully compensated, resulting in measurement results affected by the translation axis and leading to inaccurate final identification results. Controlling single-axis motion measurement can eliminate the influence of translation axis error on the identification of rotation axis error, resolving the coupling problem between translation and rotation axis geometric errors and improving error identification accuracy. Furthermore, existing identification methods using ballbars often require multiple installation locations, increasing ballbar installation time and measurement difficulty, thus affecting the simplicity of error measurement. Summary of the Invention

[0006] The purpose of this invention is to at least address one of the shortcomings of the prior art by providing a method for identifying rotation axis position-independent geometric errors based on a ballbar.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] Specifically, a rotation axis position-independent geometric error identification method based on a ballbar is proposed and applied to a five-axis machine tool, including the following:

[0009] The coordinate system is established sequentially based on the workpiece end kinematic chain of the five-axis machine tool. The five-axis machine tool consists of three linear axes (X-axis, Y-axis, Z-axis) and two rotary axes (A-axis, C-axis). The kinematic chain structure of the five-axis machine tool is divided into the tool chain RYXZT and the workpiece chain RACW. Since only the rotary axes are identified, the linear axes are 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 coincides with CCS, and the reference coordinate system RCS coincides with ACS. The Z-axis axes of WCS, CCS, RCS and ACS are collinear.

[0010] Three installation modes were used for measurement. In the three installation modes, the tool ball B1 was connected to the spindle through the tool cup, and the workpiece ball B2 was fixed to the worktable through the tool cup and the magnetic seat.

[0011] Specifically,

[0012] In the first installation mode, the A-axis or C-axis movement is controlled to measure the four position errors to obtain the first ball bar length model;

[0013] In the second installation mode, the A-axis movement is controlled to measure two perpendicularity errors to obtain the second ball bar length model;

[0014] In the third installation mode, the C-axis movement is controlled to measure two perpendicularity errors to obtain the third ball bar length model;

[0015] Based on MATLAB functions, the eight position-independent geometric errors of the rotation axis were solved by fitting the first, second, and third ballbar length models.

[0016] Furthermore, specifically, in the first installation mode, controlling the movement of the A-axis or C-axis to measure four positional errors respectively yields the first ballbar length model, including:

[0017] The initial coordinates of spheres B1 and B2 in the reference coordinate system are:

[0018] In the formula, e x e y These represent the installation errors of the center of the tool ball B1 in the X and Y directions of the machine tool reference coordinate system, respectively. L Y L The offset of the center of workpiece ball B2 relative to the center of tool ball B1 in the X and Y directions;

[0019] When only the A-axis motion is controlled, the workpiece ball B2 moves in a circular motion around the A-axis. The actual transformation matrix from the A-axis coordinate system to the reference coordinate system is:

[0020] The initial coordinates of workpiece sphere B2 in the A-axis coordinate system are calculated using inverse matrix transformation:

[0021] After coordinate matrix transformation, the actual coordinates of B2 in the reference coordinate system are:

[0022] Since the tool ball B1 remains stationary during the measurement process, its coordinates remain unchanged in the reference coordinate system.

[0023] Substituting the coordinates of balls B1 and B2, simplifying and ignoring minor higher-order errors, we obtain the actual length L of the ball bar. A1 for:

[0024] When only the C-axis is controlled, the workpiece ball B2 will move in a circle around the C-axis. The actual transformation matrix from the C-axis coordinate system to the A-axis coordinate system is:

[0025] Calculate the initial coordinates of workpiece sphere B2 in the C-axis coordinate system:

[0026] Next, calculate the actual coordinates of B2 in the reference coordinate system:

[0027] Finally, the actual length L of the ball bar is obtained based on the coordinates of the two balls. C1 for:

[0028] Furthermore, specifically, in the second installation mode, the second ball bar model is obtained by controlling the movement of the A-axis to measure two perpendicularity errors, including:

[0029] Controlling only the A-axis movement, the initial coordinates of balls B1 and B2 also change due to the change in the installation position of the ball bar, as shown in equations (11) and (12):

[0030] In the formula, l is the distance from the center of the tool ball B1 to the axis of C, and L is the nominal length of the ball bar.

[0031] Similar to the first installation mode, first calculate the actual coordinates of the workpiece ball B2 in the reference coordinate system, and then obtain the actual length L of the ball bar. A2 for:

[0032] Furthermore, specifically, in the third installation mode, the C-axis movement is controlled to measure two perpendicularity errors to obtain the third ballbar length model, including...

[0033] Controlling only the C-axis motion, the initial coordinates of balls B1 and B2 are as follows:

[0034] In the formula, h is the distance from the center of the tool ball B1 to the axis of A.

[0035] Similarly, the actual coordinates of B2 can be calculated, and finally the actual length L of the ball bar can be obtained. C3 for:

[0036] The beneficial effects of this invention are as follows:

[0037] This invention proposes a method for identifying rotation axis position-independent geometric errors based on a ballbar. By installing the tool ball at the intersection of the AC and AC axes and offsetting the workpiece ball in the XY directions, and controlling the single-axis movements of both axes sequentially, two measurement modes can be switched in a single installation. Eight position-independent geometric errors of the rotation axis are identified through three installations. This method is simple to install, requires no extension rod, and eliminates the need to identify the rotation axis, thus improving the efficiency of error measurement. Attached Figure Description

[0038] The above and other features of this disclosure will become more apparent from the detailed description of the embodiments illustrated in conjunction with the accompanying drawings. In the drawings of this disclosure, the same reference numerals denote the same or similar sampling monitoring points. Obviously, the drawings described below are merely some embodiments of this disclosure. For those skilled in the art, other drawings can be obtained from these drawings without any creative effort. In the drawings:

[0039] Figures 1(a), (b), and (c) show schematic diagrams of three installation modes involved in the rotation axis position-independent geometric error identification method based on a ballbar instrument according to the present invention.

[0040] Figures 2(a) and (b) show the machine tool structure and coordinate system of the five-axis machine tool to which the rotary axis position-independent geometric error identification method based on ballbar is applied according to the present invention.

[0041] Figures 3(a) and (b) show the principle diagrams for defining position-independent geometric errors, respectively.

[0042] Figure 4 shows the principle diagram of tool ball installation error;

[0043] Figures 5(a), (b), and (c) show schematic diagrams of the three installation modes at the installation measurement site in one implementation method.

[0044] Figures 6(a) and (b) show the data curves of the ball bar length measured twice in one embodiment;

[0045] Figures 7(a) and (b) show schematic diagrams of the ball bar length prediction error under three installation modes in one implementation method. Detailed Implementation

[0046] The following will provide a clear and complete description of the concept, specific structure, and technical effects of the present invention in conjunction with the embodiments and accompanying drawings, so as to fully understand the purpose, solution, and effects of the present invention. It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. The same reference numerals used throughout the accompanying drawings indicate the same or similar parts.

[0047] This scheme only measures position-independent geometric errors of the rotation axis, controlling the movement of a single rotation axis during the measurement process to reduce the influence of other axes on the error measurement. Example 1: This invention proposes a rotation axis position-independent geometric error identification method based on a ballbar, applied to a five-axis machine tool, including the following:

[0048] The coordinate system is established sequentially based on the workpiece end kinematic chain of the five-axis machine tool. The structure of the five-axis machine tool is shown in Figure 2a. The five-axis machine tool consists of three linear axes (X-axis, Y-axis, Z-axis) and two rotary axes (A-axis, C-axis). The kinematic chain structure of the five-axis machine tool is divided into the tool chain RYXZT and the workpiece chain RACW. Since only the rotary axes are identified, the linear axes are not considered in the process of establishing the coordinate system, as shown in Figure 2b. 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 coincides with CCS, and the reference coordinate system RCS coincides with ACS. The Z-axis axes of WCS, CCS, RCS, and ACS are collinear. The uppercase letter H+CS refers to the coordinate system where the uppercase letter H is located, and the value of H is either RYXZT or RACW.

[0049] Measurements were taken using three mounting modes. In each mode, the tool ball B1 was connected to the spindle via the tool cup (installed at the intersection of the AC axis), and the workpiece ball B2 was fixed to the worktable via the tool cup and magnetic base.

[0050] Specifically,

[0051] In the first installation mode, the A-axis or C-axis movement is controlled to measure the four position errors to obtain the first ball bar model; as shown in Figure 1a, the center of the tool ball is installed at the intersection of the A and C axes, and the workpiece ball is installed at a position 80mm away from the A-axis axis and 60mm away from the C-axis axis.

[0052] In the second installation mode, the A-axis movement is controlled to measure the two perpendicularity errors to obtain the second ball bar length model; as shown in Figure 1b, the center of the tool ball is installed on the A-axis axis, 60mm away from the other axis.

[0053] In the third installation mode, the C-axis movement is controlled to measure the two perpendicularity errors to obtain the third ball bar length model; as shown in Figure 1c, the center of the tool ball is installed on the C-axis axis, 60mm away from the other axis.

[0054] Based on MATLAB functions, the eight position-independent geometric errors of the rotation axis were solved by fitting the first, second, and third ballbar length models.

[0055] Based on the pole length models of the ball bar under three installation modes, combined with the pole length data of the ball bar in the actual measurement process, and by measuring the installation error of the ball bar with a digital dial indicator and the ball bar, these data are substituted into the pole length model, and the eight position-independent geometric errors of the rotation axis are solved by fitting MATLAB functions.

[0056] In this embodiment 1, by installing the center of the tool ball at the intersection of the AC axis and the workpiece ball offset in the XY direction, and controlling the single-axis movement of the two axes sequentially, two measurement modes can be switched in a single installation. Eight position-independent geometric errors of the rotating axis are identified through three installations. This method is simple to install, requires no extension rod, and does not require identification of the rotating axis, thus improving the efficiency of error measurement.

[0057] Considering that position-independent geometric errors of the rotary axis are caused by deviations in the assembly process of machine tool components, they 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 shown in Figure 2a. Its rotary axis consists of axis A and axis C, and there are a total of 8 position-independent geometric errors. Each rotary axis has two position errors and two perpendicularity errors, as shown in Figures 3(a) and (b). δ ya and δ za The A-axis axis is along the Y-axis in the machine tool reference coordinate system. R and Z R Positional deviation in direction, S ya and S za It is the axis of A around Y R and Z R The perpendicularity error between the two axes, δ xc and δ yc The C-axis axis is along the X-axis in the A-axis coordinate system. A and Y A Positional deviation in direction, S xc and S yc It is the C-axis axis around X A and YA Perpendicularity error between the two axes.

[0058] Based on three installation modes, the initial coordinates of the tool ball B1 and workpiece ball B2, including the ballbar installation error, are sequentially established under RCS. During measurement, only the movement of a single rotation axis is controlled, and the coordinates of the tool ball remain unchanged. The actual initial position of the workpiece ball on the measured axis is determined by the inverse transformation matrix. Then, the actual coordinates of the workpiece ball under RCS are determined by the homogeneous transformation matrix, ensuring that the actual coordinates of balls B1 and B2 are both under RCS, guaranteeing the consistency of the coordinates of the two balls, and improving the recognition accuracy of the model.

[0059] In a preferred embodiment of the present invention, specifically, in the first installation mode, controlling the movement of the A-axis or C-axis to measure four positional errors respectively yields the first ball bar length model, including...

[0060] The initial coordinates of spheres B1 and B2 in the reference coordinate system are:

[0061] In the formula, e x e y These represent the installation errors of the center of the tool ball B1 in the X and Y directions of the machine tool reference coordinate system, respectively. L Y L Let e ​​be the offset of the center of workpiece ball B2 relative to the center of tool ball B1 in the X and Y directions; as shown in Figure 4, the axis of the tool cup and tool holder deviates from the axis of the reference coordinate system RCS during installation, resulting in an error e in the center of the tool ball in the X and Y directions. x and e y The installation error of the workpiece ball has a very small impact on the change in rod length, and its effect on actual measurement tests can be ignored.

[0062] When only the A-axis motion is controlled, the workpiece ball B2 moves in a circular motion around the A-axis. The actual transformation matrix from the A-axis coordinate system to the reference coordinate system is:

[0063] The initial coordinates of workpiece sphere B2 in the A-axis coordinate system are calculated using inverse matrix transformation:

[0064] After coordinate matrix transformation, the actual coordinates of B2 in the reference coordinate system are:

[0065] Since the tool ball B1 remains stationary during the measurement process, its coordinates remain unchanged in the reference coordinate system.

[0066] Substituting the coordinates of balls B1 and B2, simplifying and ignoring minor higher-order errors, we obtain the actual length L of the ball bar. A1 for:

[0067] When only the C-axis is controlled, the workpiece ball B2 will move in a circle around the C-axis. The actual transformation matrix from the C-axis coordinate system to the A-axis coordinate system is:

[0068] Calculate the initial coordinates of workpiece sphere B2 in the C-axis coordinate system:

[0069] Next, calculate the actual coordinates of B2 in the reference coordinate system:

[0070] Finally, the actual length L of the ball bar is obtained based on the coordinates of the two balls. C1 for:

[0071] In a preferred embodiment of the present invention, specifically, in the second installation mode, the second ball bar model is obtained by controlling the movement of the A-axis to measure two perpendicularity errors, including...

[0072] Controlling only the A-axis movement, the initial coordinates of balls B1 and B2 also change due to the change in the installation position of the ball bar, as shown in equations (11) and (12):

[0073] In the formula, l is the distance from the center of the tool ball B1 to the axis of C, and L is the nominal length of the ball bar.

[0074] Similar to the first installation mode, first calculate the actual coordinates of the workpiece ball B2 in the reference coordinate system, and then obtain the actual length L of the ball bar. A2 for:

[0075] In a preferred embodiment of the present invention, specifically, in the third installation mode, the third ball bar model is obtained by controlling the C-axis movement to measure two perpendicularity errors, including...

[0076] Controlling only the C-axis motion, the initial coordinates of balls B1 and B2 are as follows:

[0077] In the formula, h is the distance from the center of the tool ball B1 to the axis of A.

[0078] Similarly, the actual coordinates of B2 can be calculated, and finally the actual length L of the ball bar can be obtained. C3 for:

[0079] In this preferred embodiment, the establishment of the workpiece end coordinate system is simplified. At the same time, the influence of ball bar installation error on the identification model is considered during the model establishment process, and the installation error is eliminated as much as possible through actual measurement, which simplifies the error identification model and improves the error identification accuracy.

[0080] In practical applications

[0081] 1. Simulation

[0082] To further verify the accuracy of the identification method, simulation verification is performed here. First, eight position-independent geometric errors of the A-axis and C-axis are generated. According to the simulation parameters in Table (1), the length data of the ballbar, including the ballbar installation error, is calculated by equations (6), (10), (13), and (16). Then, the actual position-independent geometric errors are identified using MATLAB fitting functions. Finally, the preset values ​​and identified values ​​are compared to calculate the residuals of each error.

[0083] Table 1. Basic Simulation Parameters

[0084] Table 2 Simulation Results

[0085] 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 the eight position-independent geometric errors of the rotating axis.

[0086] 2. Identification Experiment

[0087] In the measurement experiment, the nominal length of the ball bar was 100mm, the A-axis motion range was -15° to 70°, and the C-axis motion range was 0° to 360°. After the installation error measurement was completed, the ball bar was installed according to the three installation modes shown in Figure 1. After installation, the A-axis or C-axis single-axis movement was controlled in each mode, and the ball bar length data was recorded every 5°. Each set of measurements was repeated twice to avoid the randomness of the measurement results. The actual measurements are shown in Figures 5(a), (b), and (c). The A-axis movement was controlled in modes one and two, and the C-axis movement was controlled in modes one and three.

[0088] After the measurements were completed, the results of each set of measurements were fitted using curve fitting, as shown in Figures 6(a) and (b). It can be seen that the curves of the two measurements basically overlap, and the repeatability of the four sets of data is high.

[0089] The average of two measurements taken by the ball bar was used to identify eight position-independent geometric errors by substituting the angle and length data into a MATLAB function. Based on the error identification results, the predicted length of the ball bar under three installation modes was obtained. Then, the difference between the predicted value and the actual measured value was calculated to obtain the prediction error of the bar length. The prediction error can reflect the accuracy of the error identification results; the smaller the value, the closer the identified error value is to the true value. The prediction errors under the three installation modes are shown in Figures 7(a) and (b).

[0090] The results show that the maximum prediction error of the rod length in the four measurement experiments does not exceed 1 μm. Therefore, the proposed identification method has high accuracy and can effectively identify geometric errors independent of the rotation axis position.

[0091] Summarize:

[0092] (1) To quickly and accurately identify the geometric errors of the rotating shaft, a method for measuring and identifying position-independent geometric errors of the rotating shaft based on a ballbar apparatus with three installations is proposed. By installing the center of the tool ball at the intersection of the AC axis and the workpiece ball offset in the XY direction, and controlling the single-axis movement of the two axes sequentially, two measurement modes can be switched in one installation. Eight position-independent geometric errors of the rotating shaft are identified through three installations. This method is simple to install, requires no extension rod, and does not require identification of the rotating shaft axis, thus improving the efficiency of error measurement.

[0093] (2) The workpiece sphere coordinates were obtained by inverse matrix, and the initial coordinates of the tool sphere and workpiece sphere under RCS were established. Based on homogeneous coordinate transformation, a comprehensive model of rod length including ball bar installation error was established. The preset value and the identified value were compared by simulation analysis. The results showed that the residuals of the 8 errors were very small, and the proposed identification method had high accuracy.

[0094] (3) In the identification experiment, the installation error was measured by a digital micrometer and a ballbar, which basically eliminated the influence of the tool end installation error. The effectiveness of the identification method was verified by comparing the length prediction error of the ballbar. The maximum value of the bar length prediction error in the four measurement experiments did not exceed 1 μm. The proposed identification method can effectively identify geometric errors independent of the rotation axis position.

[0095] Although the description of the invention has been quite detailed and particularly of several described embodiments, it is not intended to limit it to any of these details or embodiments or any particular embodiment, but should be considered as providing a broad possible interpretation of the claims by referring to the appended claims and taking into account the prior art, thereby effectively covering the intended scope of the invention. Furthermore, the invention has been described above with respect to embodiments foreseeable by the inventors in order to provide a useful description, and non-substantial modifications to the invention that have not yet been foreseen may still represent equivalent modifications.

[0096] The above description is merely a preferred embodiment of the present invention. The present invention is not limited to the above-described embodiments. Any embodiment that achieves the technical effects of the present invention using the same means should fall within the protection scope of the present invention. Within the protection scope of the present invention, various modifications and variations can be made to the technical solutions and / or implementation methods.

Claims

1. A method for identifying rotation axis position-independent geometric errors based on a ballbar, characterized in that, Applied to five-axis machine tools, including the following: The coordinate system is established sequentially based on the workpiece end kinematic chain of the five-axis machine tool. The five-axis machine tool consists of three linear axes (X-axis, Y-axis, Z-axis) and two rotary axes (A-axis, C-axis). The kinematic chain structure of the five-axis machine tool is divided into the tool chain RYXZT and the workpiece chain RACW. Since only the rotary axes are identified, the linear axes are 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 coincides with CCS, and the reference coordinate system RCS coincides with ACS. The Z-axis axes of WCS, CCS, RCS and ACS are collinear. Three installation modes were used for measurement. In the three installation modes, the tool ball B1 was connected to the spindle through the tool cup, and the workpiece ball B2 was fixed to the worktable 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 to obtain the first ball bar length model; In the second installation mode, the A-axis movement is controlled to measure two perpendicularity errors to obtain the second ball bar length model; In the third installation mode, the C-axis movement is controlled to measure two perpendicularity errors to obtain the third ball bar length model; Based on MATLAB functions, the eight position-independent geometric errors of the rotation axis were solved by fitting and solving the first, second, and third ball bar length models. Specifically, in the first installation mode, controlling the movement of the A-axis or C-axis to measure four positional errors yields the first ballbar length model, including: The initial coordinates of spheres B1 and B2 in the reference coordinate system are: In the formula, e x e y These represent the installation errors of the center of the tool ball B1 in the X and Y directions of the machine tool reference coordinate system, respectively. L Y L The offset of the center of workpiece ball B2 relative to the center of tool ball B1 in the X and Y directions; When only the A-axis motion is controlled, the workpiece ball B2 moves in a circular motion around the A-axis. The actual transformation matrix from the A-axis coordinate system to the reference coordinate system is: The initial coordinates of workpiece sphere B2 in the A-axis coordinate system are calculated using inverse matrix transformation: After coordinate matrix transformation, the actual coordinates of B2 in the reference coordinate system are: Since the tool ball B1 remains stationary during the measurement process, its coordinates remain unchanged in the reference coordinate system. Substituting the coordinates of balls B1 and B2, simplifying and ignoring minor higher-order errors, we obtain the actual length L of the ball bar. A1 for: When only the C-axis is controlled, the workpiece ball B2 will move in a circle around the C-axis. The actual transformation matrix from the C-axis coordinate system to the A-axis coordinate system is: Z AC δ represents the distance between ACS and CCS. ya and δ za The A-axis axis is along the Y-axis in the machine tool reference coordinate system. R and Z R Positional deviation in direction, S ya and S za It is the axis of A around Y R and Z R The perpendicularity error between the two axes, δ xc and δ yc The C-axis axis is along the X-axis in the A-axis coordinate system. A and Y A Positional deviation in direction, S xc and S yc It is the C-axis axis around X A and Y A Perpendicularity error between the two axes; Calculate the initial coordinates of workpiece sphere B2 in the C-axis coordinate system: Next, calculate the actual coordinates of B2 in the reference coordinate system: Finally, the actual length L of the ball bar is obtained based on the coordinates of the two balls. C1 for: Specifically, in the second installation mode, the second ball bar model is obtained by controlling the movement of the A-axis to measure two perpendicularity errors, including: Controlling only the A-axis movement, the initial coordinates of balls B1 and B2 also change due to the change in the installation position of the ball bar, as shown in equations (11) and (12): In the formula, l is the distance from the center of the tool ball B1 to the C-axis, and L is the nominal length of the ballbar. Similar to the first installation mode, first calculate the actual coordinates of the workpiece ball B2 in the reference coordinate system, and then obtain the actual length L of the ball bar. A2 for: Specifically, in the third installation mode, the C-axis movement is controlled to measure two perpendicularity errors to obtain the third ball bar length model, including: Controlling only the C-axis motion, the initial coordinates of balls B1 and B2 are as follows: In the formula, h is the distance from the center of the tool ball B1 to the axis of A. Similarly, the actual coordinates of B2 can be calculated, and finally the actual length L of the ball bar can be obtained. C3 for:

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