A method for measuring geometric errors of a rotary axis of a machine tool

Abstract

The present application relates to a kind of geometric error measurement methods of machine tool rotary shaft, standard ball is installed on the rotary shaft worktable of machine tool, probe is installed on machine tool spindle, comprising the following steps: rotary rotary shaft worktable, obtain the center coordinates of several groups of standard ball;Select the center coordinates of one group of standard ball, calculate the best fitting plane;According to the best fitting plane and the center coordinates of the rest group of standard ball, obtain position-independent error;Transformation matrix is constructed, and the coordinate system after eliminating position-independent error is obtained;Combine the center coordinates of several groups of standard ball, transformation matrix, coordinate system, obtain matrix equation;Solve matrix equation, obtain position-dependent error.This measurement method is based on numerical control machine tool rotary shaft geometric error model and considers the influence of position-independent error on the initial position of standard ball center position, calculates four position-independent errors and six position-dependent errors of rotary shaft step by step, realizes the comprehensive measurement of ten errors of rotary shaft.

Description

Technical Field

[0001] This invention belongs to the field of CNC machine tool technology, and relates to machine tool errors, particularly a method for measuring the geometric error of a machine tool rotary axis. Background Technology

[0002] CNC machine tools are a key technology in high-end manufacturing, especially for complex curved surfaces such as turbines and blades, which place higher demands on the machining accuracy of CNC machine tools. Compared with three-axis machine tools, five-axis machine tools have two more rotary axes, which, while providing greater flexibility, also introduce more errors and adversely affect the machining accuracy.

[0003] Existing methods for measuring the geometric error of rotating axes mainly include laser interferometers, ballbars, and R-TEST, which have the following problems:

[0004] 1. The measurement process of a laser interferometer requires operations such as light alignment, which are complex and cumbersome. Installation and debugging take a lot of time. Moreover, it can only measure the positioning error of the rotating axis of the machine tool, and the measurement results are not comprehensive enough. The cost of using the instrument is also high.

[0005] 2. Ballbar instruments have high requirements for machine tool space. Like the R-TEST measurement method, they will introduce installation errors, resulting in inaccurate identification results and high instrument usage costs. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the prior art and provide a simple and comprehensive method for measuring the geometric error of machine tool rotary axes.

[0007] To solve the above problems, the technical solution of the present invention is as follows:

[0008] A method for measuring the geometric error of a machine tool rotary axis, comprising the following steps: a standard ball is mounted on the machine tool's rotary axis worktable, and a probe is mounted on the machine tool spindle.

[0009] Rotate the rotary table to obtain the center coordinates of several standard spheres;

[0010] Select one set of standard sphere center coordinates and calculate the best-fit plane;

[0011] The position-independent error is obtained based on the best-fit plane and the center coordinates of the remaining standard spheres.

[0012] Construct the transformation matrix and obtain the coordinate system after eliminating position-independent errors;

[0013] By combining the center coordinates of several standard spheres, transformation matrices, and coordinate systems, the matrix equation is obtained.

[0014] Solve the matrix equation to obtain the position-related error.

[0015] In a further embodiment, rotating the rotary axis stage to obtain several sets of standard sphere center coordinates specifically includes:

[0016] Set several rotation angles for the rotary axis worktable and obtain the sphere center coordinates for each rotation angle;

[0017] By repeatedly changing the installation position of the standard ball, the coordinates of the ball's center at each rotation angle are obtained sequentially.

[0018] In a further embodiment, the center coordinates of one set of standard spheres are selected, and the best-fit plane is calculated, specifically including:

[0019] Construct a least-squares optimization function using the coordinates of the sphere's center as input:

[0020]

[0021] In the formula, F j (A j B j C j D j ) represents the least squares optimization function; i represents the number of rotation angles set on the rotary axis worktable; j represents the number of installation positions for the standard ball; A j B j C j D is the plane normal vector of the best-fit plane; j λ is the perpendicular distance from the best-fit plane to the origin; X(i,j), Y(i,j), and Z(i,j) are the X-axis, Y-axis, and Z-axis coordinates of the standard sphere, respectively; j These are Lagrange multipliers used to constrain the normalization of plane vectors;

[0022] Taking the partial derivatives of the plane normal vector, the perpendicular distance from the best-fit plane to the origin, and the Lagrange multipliers, and setting the partial derivatives to zero, and combining this with the least squares optimization function, we obtain the equation of the best-fit plane as follows:

[0023] A j X+B j Y = C j Z+D j =0;

[0024] In the formula, X, Y, and Z are calculated by substituting X(i,j), Y(i,j), and Z(i,j), respectively.

[0025] In a further embodiment, the position-independent error is obtained based on the best-fit plane and the center coordinates of the remaining standard spheres, specifically including:

[0026] Project the center of the standard sphere onto the best-fit plane to construct the equation of the axis of rotation;

[0027] The position-independent error is obtained based on the equation of the axis of rotation.

[0028] In a further embodiment, the center of the standard sphere is projected onto the best-fit plane to construct the equation of the rotation axis, specifically including:

[0029] By projecting the coordinates of each sphere's center onto the fitting plane using the equation of the best-fitting plane, the projection points of the sphere's center coordinates are obtained. The projection formula is as follows:

[0030]

[0031] In the formula, X′ (i，j) 、Y′ (i，j) Z′ (i，j) These are the coordinates of the sphere's center after projection onto the fitted plane;

[0032] During the rotation of the rotary axis worktable (1), the trajectory of the center of the standard ball (4) is a planar circle. An optimization function is constructed to fit the center of the planar circle, and the center is constrained to lie on the best-fit plane:

[0033]

[0034] L j =μ j (A j X 0，j +B j Y 0，j +C j Z 0，j +D j );

[0035] G j (X 0，j Y 0，j Z 0，j r j μ j ) = S j +L j ;

[0036] In the formula, s j The objective function is to minimize the error of a planar circle; X (0，j) Y (0，j) Z (0，j) L represents the coordinates of the center of the planar circle on the best-fit plane. j The constraint condition for the center of the planar circle to lie on the best-fit plane; μ j G is the Lagrange multiplier used to constrain the center of the planar circle to lie on the best-fit plane; j The Lagrangian function for jointly optimizing the objective and constraints; r j Let be the radius of the planar circle.

[0037] For X (0，j) Y (0，j) Z (0，j) r j μ j By taking the partial derivatives and setting them to zero, we can obtain the coordinates of the center of the planar circle on the best-fit plane.

[0038] Using the plane normal vector of the best-fit plane as the direction vector, and taking the coordinates of the center of the circular plane on the best-fit plane as fixed points, establish the equation of the axis of rotation:

[0039]

[0040] In a further embodiment, the position-independent error is obtained based on the equation of the rotation axis, specifically including:

[0041] The rotation axis error term is obtained from the plane normal vector:

[0042]

[0043]

[0044] In the formula, S xc，j S yc，j These represent the perpendicularity errors of the rotation axis to the X and Y axes measured at the j-th installation position, respectively.

[0045] The conditions for defining the first plane are as follows:

[0046] Z = 0;

[0047] Based on the equation of the axis of rotation, the intersection point of the axis of rotation and the first plane is solved to obtain the position error term:

[0048]

[0049] In the formula, O xc，j O yc，j These represent the positional errors of the standard ball relative to the X-axis and Y-axis at the j-th installation position, respectively.

[0050] Based on the coordinates of the centers of several standard spheres, several sets of perpendicularity errors and position errors were calculated. The arithmetic mean of these calculations was then performed to obtain the position-independent error.

[0051]

[0052] In the formula, S xc S yc O xc O yc The error is independent of position, and n is the number of sets of coordinates of the center of the standard sphere.

[0053] In a further embodiment, a transformation matrix is ​​constructed, and a coordinate system after eliminating position-independent errors is obtained, specifically including:

[0054] When the rotation angle of the rotary axis table is zero, the coordinate system of the rotary axis table is set as the machine tool coordinate system. The ideal transformation matrix from the working coordinate system to the machine tool coordinate system based on the standard sphere is:

[0055]

[0056] In the formula, C is the ideal transformation matrix; i This represents the rotation angle of the rotary axis worktable.

[0057] The actual transformation matrix from the working coordinate system to the machine tool coordinate system is:

[0058]

[0059]

[0060] In the formula, T is the actual transformation matrix; PIGE T is the position-independent error transformation matrix; PDGE Here is the position-related error transformation matrix; δ x (C i ), δ y (C i ), δ z (C i ), ε x (C i ), ε y (C i ), ε z (C i ) represents the position-related error parameter;

[0061] The initial coordinates of the standard sphere in the machine tool coordinate system are calculated using inverse matrix transformation:

[0062]

[0063] In the formula, P′ (1，j) P represents the initial coordinates of the standard sphere's center after error elimination at the j-th installation position; (1，j) The coordinates of the center of the standard sphere measured at the j-th installation position;

[0064] Based on the initial coordinates, the coordinates of the center of the standard sphere measured at different angles of the rotary table are as follows:

[0065]

[0066] In the formula, P(i，j) Let be the rotation angle of the i-th rotary axis worktable and the coordinates of the standard sphere center measured at the j-th installation position.

[0067] In a further embodiment, by combining the center coordinates of several sets of standard spheres, transformation matrices, and coordinate systems, a matrix equation is obtained, specifically including:

[0068] By combining the coordinates of the centers of several standard spheres, we obtain the matrix equation:

[0069] A 9×1 =B 9×6 ·E 6×1 ;

[0070]

[0071]

[0072] In the formula, A is the difference between the measured center coordinates of the standard spheres and the center coordinates of the standard spheres after error elimination, obtained in the step of rotating the rotary axis worktable to obtain several sets of center coordinates of the standard spheres; B is the error coefficient matrix; and E is the error vector related to the position to be identified at one of the rotation angles of the rotary axis worktable.

[0073] In a further embodiment, solving the matrix equation to obtain the position-related error specifically includes:

[0074] Solving the matrix equation using the least squares method:

[0075] E = (B T B) -1 B T A;

[0076] By combining the formula for the position-related error vector to be identified at one of the rotation angles of the rotary axis stage, the coordinates of the sphere center at each rotation angle of the rotary axis stage are calculated to obtain the position-related error.

[0077] In a further embodiment, the standard ball is mounted on the rotating shaft worktable via a magnetic base.

[0078] Compared with the prior art, the beneficial effects of the present invention are:

[0079] 1. This measurement method mounts a standard ball on the machine tool's rotary axis worktable using a magnetic base, facilitating installation, removal, and movement. A probe is installed on the machine tool spindle to measure the center position of the standard ball. Based on the geometric error model of the CNC machine tool's rotary axis and considering the influence of position-independent errors on the initial position of the standard ball's center, the four position-independent errors and six position-dependent errors of the rotary axis are calculated step-by-step based on the obtained center positions at different angles. This achieves comprehensive measurement of ten errors of the rotary axis, improving the accuracy and efficiency of error measurement while maintaining low cost. It can be applied to the measurement of geometric errors of rotary axes in five-axis machine tools with dual rotary table structures of different sizes.

[0080] 2. This measurement method combines standard sphere measurement point planning and data acquisition methods, and adopts an automated measurement strategy driven by CNC macro programs. By acquiring data from multiple positions and angles, it constructs the sphere center coordinate dataset required for error identification. This enables the measurement device and method to effectively identify four position-independent errors and six position-dependent errors, realizing comprehensive measurement of the geometric errors of the rotary axis of CNC machine tools, and providing a complete error data foundation for subsequent machine tool accuracy analysis.

[0081] 3. Compared with other error measurement equipment, this measurement method is lower in cost and simpler; at the same time, it is easy to integrate with CNC systems to realize the automated measurement of error items. This not only reduces the investment cost of error measurement equipment in traditional measurement methods, but also significantly shortens the error measurement and identification cycle, improves the efficiency of machine tool error detection, and is conducive to the rapid implementation of machine tool accuracy calibration.

[0082] 4. This measurement method fully considers the influence of position-independent errors on the initial measurement coordinates. Through step-by-step identification, it decouples position-independent errors and position-dependent errors, enabling efficient and accurate measurement of relevant error values. This provides a solid theoretical basis and reliable technical support for machine tool accuracy calibration and error compensation, and helps improve the machining accuracy of CNC machine tools. Attached Figure Description

[0083] Figure 1 This is one of the standard sphere schematic diagrams for a method of measuring the geometric error of a machine tool rotary axis;

[0084] Figure 2 Schematic diagram of a standard sphere for a method of measuring the geometric error of a machine tool rotary axis (Part 2);

[0085] Figure 3 This is a flowchart of a method for measuring the geometric error of a machine tool rotary axis.

[0086] In the diagram: 1. Rotary axis worktable; 2. Magnetic base; 3. Support rod; 4. Standard ball; 5. Probe; 6. Triggering device; 7. Probe holder; 8. Machine tool spindle; 9. Receiver; 10. Machine tool body. Detailed Implementation

[0087] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicating orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the invention. Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined with "first," "second," etc., may explicitly or implicitly include one or more of that feature. In the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0088] Example 1:

[0089] A method for measuring the geometric error of a machine tool rotary axis, such as... Figures 1 to 3 As shown, it includes the following steps:

[0090] S101. Rotate the rotary table 1 to obtain several sets of center coordinates of the standard sphere 4:

[0091] A probe measurement module is installed on a CNC machine tool. The probe measurement module includes a magnetic base 2, a support rod 3, a standard ball 4, a probe 5, a triggering device 6, a probe holder 7, and a receiver 9. The magnetic base 2, support rod 3, and standard ball 4 are sequentially fixed on the rotary table 1 of the machine tool, so that the standard ball 4 can be easily and quickly installed, removed, and repositioned by magnetic attraction. The probe holder 7, triggering device 6, and probe 5 are sequentially fixed on the machine tool spindle 8. The probe 5 is used to detect the center coordinates of the standard ball 4. The receiver 9 is fixed on the machine tool body 10. The probe 5, as the front-end sensing component of the measurement, causes the triggering device 6 to generate a trigger signal when it contacts the standard ball 4. The receiver 9 receives the trigger signal and transmits it to the CNC system. After receiving the signal, the CNC system issues an interrupt request, latches the measured center coordinates of the standard ball 4, records it in the storage location of the CNC system, and performs a program jump through the G31 instruction, thereby driving the probe to move in the opposite direction.

[0092] Write a macro program using the G31 command to set the rotary table 1 to eight rotation angles: 0°, 45°, ..., 270°, and 315°. Then, measure the center coordinates of the standard sphere 4 at each rotation angle. Figure 2 As shown, measurement points A, B, C, D, and E are set on the left, right, front, back, and top of the surface of standard sphere 4, respectively. First, measurements are taken at measurement points A, B, C, and D of standard sphere 4 in the XY plane. The coordinates of the center of standard sphere 4 in the XY plane are calculated by the average coordinates of measurement points A and B, and the average coordinates of measurement points C and D. Based on these center coordinates, a Z-axis measurement procedure is performed on measurement point E at the top of standard sphere 4. The measured value is then subtracted from the radius of standard sphere 4 to complete the measurement of the center coordinates of standard sphere 4 at one rotation angle. The rotary table 1 is then rotated to the next angle, and the machine tool translation axis is moved to the measurement position at that angle. The measurement steps are repeated to obtain the center coordinates of standard sphere 4 at each rotation angle, and the center coordinates are stored in a common variable.

[0093] After completing the measurement of the center coordinates of a set of standard spheres 4, the initial installation position of the standard spheres 4 is changed, and the measurement steps for the center coordinates of the standard spheres 4 at each rotation angle are repeated. To identify all errors of the rotation axis, at least three different initial installation positions are set, or at least three standard spheres 4 with different initial positions are installed on the rotation axis worktable 1, and the initial installation positions cannot be collinear. In this embodiment, three different initial installation positions are set, thereby obtaining a total of 24 center coordinates (X). (i，j) Y (i，j) Z (i，j) ), where (X) (i，j) Y (i，j) Z (i，j) () represents the coordinates of the center of the standard sphere at the i-th rotation angle and the j-th installation position.

[0094] S103. Select one set of coordinates of the center of standard sphere 4 and calculate the best-fit plane:

[0095] During the rotation of the rotary table 1, the trajectory of the center of the standard sphere 4 is a planar circle in space. For the coordinates of the center of the standard sphere 4 at the 8 angles in the j-th group, the best-fit plane is obtained using the least squares method. The center coordinates (X...) (i，j) Y (i，j) Z (i，j) Using ) as input, Lagrange multipliers are introduced to construct the least squares optimization function, as shown in Equation (1) and Equation (2):

[0096]

[0097] In the formula, F j (A j B j C j D j ) represents the least squares optimization function; i represents the number of rotation angles set on the rotary axis worktable; j represents the number of installation positions for the standard ball; A j B j C j D is the plane normal vector of the best-fit plane; j λ is the perpendicular distance from the best-fit plane to the origin; j is the Lagrange multiplier used to constrain the normalization of plane vectors.

[0098] For A j B j C j D j , λ j Taking the partial derivatives and setting them to zero, and combining them with the least squares optimization function, the equation of the best-fit plane is obtained as shown in formula (3):

[0099] A j X+B j Y+C j Z+D j =0 (3)

[0100] In the formula, X, Y, and Z are calculated by substituting X(i,j), Y(i,j), and Z(i,j), respectively.

[0101] S105. Based on the best-fit plane and the coordinates of the center of the remaining standard spheres 4, obtain the position-independent error:

[0102] The rotation axis error term is obtained through the plane normal vector as shown in formulas (4) and (5):

[0103]

[0104] In the formula, S xc，j S yc，j These represent the perpendicularity errors of the rotation axis to the X and Y axes, respectively, measured at the j-th installation position.

[0105] The projection points are obtained by projecting the coordinates of each sphere center onto the best-fit plane using formula (3), as shown in formula (6):

[0106]

[0107] In the formula, X′ (i，j) 、Y′ (i，j) Z′ (i，j) These are the coordinates of the sphere's center after projection onto the fitted plane;

[0108] Construct the center of the plane circle that fits the trajectory of the standard ball 4 using the optimization function, and constrain the center of the circle to lie on the best-fit plane, as shown in formulas (7), (8), and (9):

[0109]

[0110]

[0111] In the formula, S j The objective function is to minimize the error of a planar circle; X (0，j) Y (0，j) Z (0，j) L represents the coordinates of the center of the planar circle on the best-fit plane. j The constraint condition for the center of the planar circle to lie on the best-fit plane; μ j G is the Lagrange multiplier used to constrain the center of the planar circle to lie on the best-fit plane; j The Lagrangian function for jointly optimizing the objective and constraints; r j Let be the radius of the planar circle.

[0112] For X (0，j) Y (0，j) Z (0，j) r j μ j Find the partial derivatives and set them to zero to obtain the coordinates of the center of the planar circle on the best-fit plane;

[0113] Using the plane normal vector of the best-fit plane as the direction vector, and the center X of the fitted circle as the direction vector... (0，j) Y (0，j) Z (0，j) For a fixed point, establish the axis equation of the machine tool's rotating axis, as shown in formula (10):

[0114]

[0115] Let Z = 0, solve for the intersection of the axis equation and the Z = 0 plane, and obtain the position-independent error term, as shown in formulas (11) and (12):

[0116]

[0117] In the formula, O xc，j O yc，j These represent the positional errors of the rotation axis relative to the X and Y axes, respectively, measured at the j-th installation position.

[0118] The above calculations were performed on the center coordinate data of the three sets of standard spheres 4, and the three independent results were arithmetically averaged to obtain the final position-independent error parameter, as shown in formula (13):

[0119]

[0120] In the formula, S xc S yc O xc O yc The error is independent of position, and n is the number of sets of coordinates of the center of the standard sphere.

[0121] S107. Construct the transformation matrix and obtain the coordinate system after eliminating position-independent errors:

[0122] A kinematic model of the geometric error of the rotary axis is established using homogeneous coordinate transformation theory. When the rotation angle of the rotary axis worktable 1 is 0, the coordinate system of the rotary axis worktable 1 is set as the machine tool coordinate system. When the rotary axis worktable 1 is at other rotation angles, the ideal transformation matrix from the working coordinate system based on the standard sphere 4 to the machine tool coordinate system is shown in formula (14):

[0123]

[0124] In the formula, C is the ideal transformation matrix; i This represents the rotation angle of the rotary axis worktable.

[0125] Under the influence of geometric errors, the actual transformation matrix from the working coordinate system to the machine tool coordinate system is shown in formulas (15), (16), and (17):

[0126]

[0127] In the formula, T is the actual transformation matrix; PIGE T is the position-independent error transformation matrix; PDGE Here is the position-related error transformation matrix; δ x (C i ), δ y (C i ), δ z (C i ), ε x (C i ), ε y (C i ), ε z (C i ) represents the position-related error parameter;

[0128] In the actual measurement process, the center point of the standard sphere 4 measured at its initial position is affected by position-independent error. The initial coordinates of the standard sphere 4 in the reference coordinate system are calculated by inverse matrix transformation as shown in formula (18):

[0129]

[0130] In the formula, P′ (1,j) P represents the initial coordinates of the standard sphere's center after error elimination at the j-th installation position; (1，j) The coordinates of the center of the standard sphere measured at the j-th installation position;

[0131] Therefore, the coordinates of the center of the standard sphere 4 measured at different rotation angles of the rotary table 1 are shown in formula (19):

[0132]

[0133] In the formula, P (i，j) Let be the rotation angle of the i-th rotary axis worktable and the coordinates of the standard sphere center measured at the j-th installation position.

[0134] S109. Combining the coordinates of the center of several standard spheres 4, the transformation matrix, and the coordinate system, obtain the matrix equation:

[0135] The three sets of sphere center coordinates are calculated using the above steps. The three sets of data are then combined to form a matrix, as shown in formulas (20), (21), (22), and (23).

[0136] A 9×1 =B 9×6 ·E 6×1 (20)

[0137]

[0138] In the formula, A is the difference between the measured center coordinates of the standard spheres and the center coordinates of the standard spheres after error elimination, obtained in the step of rotating the rotary axis worktable to obtain several sets of center coordinates of the standard spheres; B is the error coefficient matrix; and E is the error vector related to the position to be identified at one of the rotation angles of the rotary axis worktable.

[0139] S111. Solve the matrix equation to obtain the position-related error:

[0140] Solving equations (20), (21), (22), and (23) using the least squares method yields equation (24):

[0141] E = (B T B) -1 B T A (24)

[0142] Combining formula (24), the coordinates of the center of the sphere at each rotation angle of the rotary axis worktable 1 are calculated to obtain the position-related error and realize the decoupling identification of the position-related error.

[0143] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A method for measuring the geometric error of a machine tool rotary axis, wherein a standard ball (4) is installed on the rotary axis worktable (1) of the machine tool and a probe (5) is installed on the machine tool spindle (8), characterized in that, Includes the following steps: Rotate the rotating axis worktable (1) to obtain the center coordinates of several sets of the standard spheres (4); Select one set of the center coordinates of the standard sphere (4) and calculate the best fitting plane; Using the coordinates of the sphere's center as input, construct a least-squares optimization function: ； ； In the formula, It is a least squares optimization function; The number of rotation angles set for the rotary axis worktable. This refers to the number of installation positions for the standard ball; , , The plane normal vector of the best-fit plane; The perpendicular distance from the best-fit plane to the origin of the coordinate system; , , These are the X, Y, and Z coordinates of a standard sphere, respectively. These are Lagrange multipliers used to constrain the normalization of plane vectors; Taking the partial derivatives of the plane normal vector, the perpendicular distance from the best-fit plane to the origin, and the Lagrange multipliers, and setting the partial derivatives to zero, and combining them with the least squares optimization function, the equation of the best-fit plane is obtained as follows: ； In the formula, , , Substitute into the calculation respectively , , ; Based on the best-fit plane and the center coordinates of the remaining standard spheres (4), the position-independent error is obtained; Project the center of the standard sphere (4) onto the best-fit plane to construct the axis equation of rotation; The center coordinates of each sphere are projected onto the fitting plane using the best-fit plane equation to obtain the projection point of the center coordinates. The projection formula is as follows: ； In the formula, , , These are the coordinates of the sphere's center after projection onto the fitted plane; During the rotation of the rotary axis worktable (1), the trajectory of the center of the standard ball (4) is a planar circle. An optimization function is constructed to fit the center of the planar circle, and the center is constrained to lie on the best-fit plane. ； ； ； In the formula, The objective function is to minimize the error of a planar circle. , , Let be the coordinates of the center of the planar circle on the best-fit plane; The constraint condition for the center of the planar circle to lie on the best-fit plane; Let Lagrange multipliers be used to constrain the center of a planar circle to lie on the best-fit plane. For the joint optimization of the objective and constraints of the Lagrangian function; The radius of the plane circle; right , , , , Find the partial derivatives and set them to zero to obtain the coordinates of the center of the planar circle on the best-fit plane; Using the plane normal vector of the best-fit plane as the direction vector, and taking the coordinates of the center of the circular plane on the best-fit plane as a fixed point, establish the equation of the axis of rotation: ； The position-independent error is obtained based on the equation of the axis of rotation. Construct the transformation matrix and obtain the coordinate system after eliminating the position-independent error; By combining the center coordinates of several sets of the standard spheres (4), the transformation matrix, and the coordinate system, the matrix equation is obtained; Solve the matrix equation to obtain the position-related error.

2. The method for measuring the geometric error of a machine tool rotary axis according to claim 1, characterized in that, Rotating the rotary axis worktable (1) to obtain several sets of center coordinates of the standard spheres (4) specifically includes: Set several rotation angles for the rotary axis worktable (1) and obtain the sphere center coordinates for each rotation angle; The installation position of the standard ball (4) is changed multiple times, and the center coordinates of the ball are obtained for each rotation angle in sequence.

3. The method for measuring the geometric error of a machine tool rotary axis according to claim 2, characterized in that, The position-independent error is obtained based on the equation of the rotation axis, specifically including: The rotation axis error term obtained from the plane normal vector is: ； ； In the formula, , The first The perpendicularity error between the rotation axis and the X-axis and Y-axis measured at each installation position; The conditions for defining the first plane are as follows: ； Based on the equation of the rotation axis, the intersection point of the rotation axis and the first plane is solved to obtain the position error term: ； ； In the formula, , The first The positional errors of the rotary axis relative to the X and Y axes measured at each installation position; Based on the coordinates of the centers of the aforementioned standard spheres, several sets of perpendicularity errors and position errors are calculated, and the arithmetic mean of these calculation results is taken to obtain the position-independent error. ； In the formula, , , , This is a position-independent error. The number of coordinate sets for the center of the standard sphere.

4. The method for measuring the geometric error of a machine tool rotary axis according to claim 3, characterized in that, Constructing the transformation matrix and obtaining the coordinate system after eliminating the position-independent error specifically includes: When the rotation angle of the rotary axis worktable (1) is zero, the coordinate system of the rotary axis worktable (1) is set as the machine tool coordinate system, and the ideal transformation matrix from the working coordinate system based on the standard sphere (4) to the machine tool coordinate system is: ； In the formula, It is an ideal transformation matrix; This represents the rotation angle of the rotary axis worktable. The actual transformation matrix from the working coordinate system to the machine tool coordinate system is: ； ； ； In the formula, This is the actual transformation matrix; This is the position-independent error transformation matrix; This is the position-related error transformation matrix; , , , , , The location-related error parameter; The initial coordinates of the standard sphere (4) in the machine tool coordinate system are calculated by inverse matrix transformation as follows: ； In the formula, For the first Initial coordinates of the standard sphere's center after error elimination at each installation position; For the first The coordinates of the center of the standard sphere measured at each installation location; Based on the initial coordinates, the coordinates of the center of the standard sphere (4) measured at different angles on the rotating axis worktable (1) are as follows: ； In the formula, For the first The rotation angle of the first rotary axis worktable, the first The coordinates of the center of the standard sphere measured at each installation location.

5. The method for measuring the geometric error of a machine tool rotary axis according to claim 4, characterized in that, By combining the center coordinates of several sets of the standard spheres (4), the transformation matrix, and the coordinate system, the matrix equation is obtained, specifically including: By combining the coordinates of the centers of several standard spheres (4), we obtain the matrix equation: ； ； ； ； In the formula, The difference between the measured center coordinates of the standard spheres and the center coordinates of the standard spheres after error elimination, obtained in the step of rotating the rotary axis worktable to obtain several sets of center coordinates of the standard spheres; This is the error coefficient matrix; This is the position-related error vector to be identified at one of the rotation angles of the rotary axis worktable.

6. The method for measuring the geometric error of a machine tool rotary axis according to claim 5, characterized in that, Solving the matrix equation to obtain the position-related error specifically includes: The matrix equation is solved using the least squares method: ； By combining the formula of the position-related error vector to be identified under one of the rotation angles of the rotary axis worktable, the sphere center coordinates under each rotation angle of the rotary axis worktable (1) are calculated to obtain the position-related error.

7. The method for measuring the geometric error of a machine tool rotary axis according to claim 6, characterized in that, The standard ball (4) is mounted on the rotating shaft worktable (1) via a magnetic base (2).

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