Method for in-situ measurement and compensation of integrated errors of face gears based on sequential decoupling

Abstract

The application discloses a kind of face gear in-machine measurement comprehensive error compensation method based on sequential decoupling. Specifically, for the problem that the pre-travel error of measuring head and the error of workpiece installation coordinate system are coupled in the prior art, a sequential decoupling compensation strategy is proposed. First, a pre-travel error model of the measuring head is established by standard ball calibration, and the anisotropic compensation of the tooth surface measurement points is carried out by using Delaunay triangulation and weighted interpolation method. Second, based on the compensated data, the outer cylindrical surface and the addendum plane of the face gear are accurately fitted by using an iterative optimization algorithm, an accurate measurement coordinate system is established, and the initial phase of C-axis is optimized. Finally, the actual tooth surface is reconstructed by using a two-stage NURBS surface fitting method, and the normal deviation is calculated. By strictly following the sequence, the error propagation chain is effectively cut off, the overall accuracy and reliability of in-machine measurement are significantly improved, and key technical support is provided for face gear closed-loop manufacturing.

Description

Technical Field

[0001] This invention belongs to the field of precision gear measurement technology, specifically a comprehensive error compensation method for on-machine measurement of face gears based on sequential decoupling, which is applied to the on-machine measurement of face gear tooth profile and pitch errors. Background Technology

[0002] Face gear drives, as an advanced form of gear transmission, have gained increasingly widespread application in fields with extreme requirements for reliability, operational smoothness, and space utilization, such as aerospace (helicopter main gearboxes) and high-end automotive industry (high-performance differentials), due to their high power density, compact structure, smooth transmission, and excellent power distribution characteristics. The service performance and lifespan of face gears are highly dependent on the geometric accuracy of their tooth surfaces. Even minute deviations in key geometric features such as tooth profile and pitch will directly lead to increased transmission errors, causing abnormal wear, vibration, and noise, ultimately jeopardizing the safety and reliability of the entire transmission system. Therefore, achieving high-precision detection of face gear geometric errors is an indispensable key step in their precision manufacturing and assembly.

[0003] Currently, the geometric accuracy inspection of face gears mainly relies on two technical approaches: offline measurement and on-machine measurement. Offline measurement is typically performed on a standalone coordinate measuring machine (CMM) or a dedicated gear measurement center. Its stable measurement environment and high instrument accuracy make it widely regarded as the metrological benchmark. However, offline measurement has significant drawbacks: the measurement process is cumbersome and time-consuming, and real-time feedback is impossible. More importantly, the re-clamping of the workpiece on the measuring machine after removal from the machine tool inevitably introduces errors in the conversion of the positioning benchmark, resulting in inconsistencies between the measurement benchmark and the machining benchmark. This severely hinders its application in the "machining-measurement-correction" closed-loop manufacturing process that requires rapid response.

[0004] In-machine measurement technology provides an effective solution to the above problems. In-machine measurement integrates measurement functions into CNC machine tools (such as gear grinding machines), utilizing the machine tool's own motion axes and clamping system to directly measure the workpiece at the machining station. This eliminates the need for secondary clamping, significantly shortening the inspection cycle. It lays the technological foundation for online diagnosis and real-time compensation of machining errors and for building a manufacturing closed loop, and is considered an effective means to achieve efficient and precise manufacturing of complex curved surface parts.

[0005] However, the accuracy of in-machine measurement is limited by complex error sources within the machine tool, and its raw measurement data cannot directly and accurately reflect the geometric errors of the workpiece. Among these error sources, those directly related to the measurement task and having a significant impact mainly fall into two categories.

[0006] (1) Probe system error, especially pre-stroke error: The small deformation displacement generated by the probe mechanism during the period from contacting the workpiece surface to issuing the trigger signal is called the pre-stroke. This error is not a constant value, but an anisotropic quantity that varies with the direction of the triggering force. Its influence is particularly prominent when measuring complex curved surfaces with continuously changing normal vectors (such as the tooth surface of face gears), which will directly cause the coordinates of the measurement point to shift along the normal.

[0007] (2) Measurement coordinate system alignment error: Due to the unavoidable position and posture deviations of the workpiece on the machine tool table, the actual geometric coordinate system of the workpiece does not coincide with the theoretical measurement coordinate system of the machine tool. This deviation will systematically shift and rotate all measurement points, distorting the overall error morphology of the measured tooth surface.

[0008] Existing technologies have proposed various compensation methods for the aforementioned single error source. For example, some solutions focus on compensating for the anisotropic pre-stroke error of the probe through calibration (such as using a standard ball or ring gauge) and interpolation algorithms; others are committed to accurately establishing a measurement coordinate system by fitting workpiece reference features (such as the outer cylindrical surface or end plane) to compensate for installation deviations. Summary of the Invention

[0009] After in-depth research and practice, the applicant discovered a critical and undervalued flaw in existing error compensation strategies: they often treat various error sources in isolation, neglecting the inherent and strong mutual coupling and dependence between these sources. Specifically, when establishing a measurement coordinate system, uncompensated raw measurement point data is typically used directly to fit the reference feature (such as an outer cylindrical surface). This data already includes the probe's pre-travel error, thus the fitted coordinate system reference (such as the axis or origin) is biased from the initial stage. Conversely, using a measurement coordinate system with alignment deviations to guide the compensation of the probe's pre-travel error (e.g., determining the theoretical normal vector direction of the measurement point) will also lead to the failure of the pre-travel compensation model or incorrect compensation direction. This coupling mechanism—where "probe error contaminates coordinate system calibration, and an incorrect coordinate system misleads probe compensation"—creates a difficult-to-break error propagation and amplification loop, becoming a core technical bottleneck restricting further improvements in on-machine measurement accuracy.

[0010] In other words, existing technologies do not teach or inspire how to systematically address the synergistic effects and decoupling compensation between these two factors. Therefore, there is an urgent need for a new comprehensive error compensation method that can break the aforementioned error coupling cycle at the system level, thereby achieving higher fidelity in-machine measurement of the geometric errors of the gear tooth surface.

[0011] In view of this, the purpose of this invention is to provide a comprehensive error compensation method for on-machine measurement of face gears based on sequential decoupling, applied to the on-machine measurement of face gear tooth profile and pitch errors. This method aims to solve the technical problem of mutual coupling between probe pre-stroke error and workpiece mounting coordinate system error in on-machine measurement of face gears, leading to cyclical error amplification. Comprehensive error compensation is achieved through sequential decoupling. First, the anisotropic error of the probe is compensated based on standard sphere calibration and Delaunay triangulation interpolation. Then, the corrected data is used to accurately establish the measurement coordinate system, ultimately achieving high-precision on-machine evaluation of tooth surface errors, effectively improving the authenticity and reliability of the measurement results.

[0012] To achieve the above objectives, the present invention provides the following technical solution:

[0013] A method for comprehensive error compensation in on-machine measurement of face gears based on sequential decoupling includes the following steps:

[0014] Step 1: Probe pre-travel error compensation: Obtain probe pre-travel error data in multiple discrete directions through standard ball calibration and construct a dataset; based on this dataset, calculate and compensate for the probe pre-travel error in any actual measurement direction through Delaunay triangulation and weighted interpolation.

[0015] Step 2: Measurement coordinate system alignment error compensation: Based on the measurement points collected on the face gear, the tooth tip plane and the outer cylindrical surface are fitted respectively through optimization algorithm. The face gear measurement coordinate system is established according to the normal vector of the fitted tooth tip plane and the axis of the fitted outer cylindrical surface. Combined with the initial phase optimization of the C-axis, the coordinates and normal vectors of the theoretical measurement points are transformed to compensate for the coordinate system alignment error caused by installation.

[0016] Step 3: Evaluation of Tooth Surface Geometric Error: The actual tooth surface is reconstructed using a two-stage non-uniform rational B-spline (NURBS) surface fitting method based on the actual measurement points after compensation in steps S1 and S2. The normal deviation between the actual tooth surface and the theoretical tooth surface is then calculated. The method steps are as follows:

[0017] 31) Using the compensated measurement points Fitting the first NURBS surface and calculate the points exist normal vector on :

[0018]

[0019] in: To utilize the compensated measurement points Fit the first NURBS surface; In order to be in On point The normal vector at that location; equal Decrease the number of directional control points by one; equal Decrease the number of directional control points by one; and for Two independent variables in a two-dimensional parameter domain; curved surface Control points; and They are respectively Direction 3rd order and Directional third-order B-spline basis functions; In order to be in superior The normal vector at the point;

[0020] 32) Along Direction to point Perform secondary compensation to obtain points :

[0021]

[0022] in: To simultaneously consider the compensation amount for probe radius and pre-travel error; The known tip radius; Indicates the probe is in Pre-travel error in direction;

[0023] 33) Utilization Point Fitting the second NURBS surface ;

[0024]

[0025] in: For utilization point Fit the second NURBS surface; and for Two independent variables in a two-dimensional parameter domain; for Control points; and They are respectively Direction 3rd order and Directional third-order B-spline basis functions;

[0026] 34) Construct a ray-surface intersection problem to obtain the theoretical point. Along its normal vector To the surface intersection ,distance This refers to the tooth surface normal deviation; the ray-surface intersection problem is expressed as:

[0027]

[0028] in: Theoretical point Along the normal direction The distance traveled is equal to the distance along the vector. The tooth surface normal deviation.

[0029] Furthermore, in step one, the method for compensating for the probe's pre-travel error comprises the following steps:

[0030] 11) Calibration is performed using a standard sphere. A dense sampling grid is defined on the upper hemisphere of the standard sphere for measurement; where: theoretical measurement points are represented by spherical coordinates. express; Polar angle, ; It is the azimuth angle. The sampling grid has N theoretical measurement points.

[0031] 12) Record the actual contact point in the measurement coordinate system of the standard sphere. The vector is represented as The actual center of the standard sphere is fitted using the least squares method. , , The objective function is to minimize the sum of squared residuals:

[0032]

[0033]

[0034] in: and These are the known standard sphere radius and the apex radius, respectively; The number of measurement points;

[0035] 13) Calculate the probe pre-travel error value in each calibration direction. and its direction unit vector The calculation formula is:

[0036]

[0037] in: Indicates from point Point of view ; yes , unit vector; Indicates along Directional probe pre-travel error; vector The direction is the compensation direction for the probe's pre-stroke error;

[0038] 14) All calibrated unit vectors As a set of points in three-dimensional space, a three-dimensional Delaunay triangulation is performed on the upper half of the unit sphere to construct a spherical triangular mesh.

[0039] 15) For the endpoint of the unit normal vector of the actual measurement point on the tooth surface Position it within a specific triangular facet of the spherical triangular mesh, the facet being defined by its vertices. , and Definition; Solving the system of equations:

[0040]

[0041] Obtain weighting factors , and ;but Probe pre-travel error corresponding to direction Calculated using weighted interpolation:

[0042]

[0043] in: , and Representing points respectively , and The coordinates; , and They are along the normal vector , and The probe's pre-travel error value in the direction, and , , ; It is the probe pre-stroke error corresponding to the actual measurement direction.

[0044] Furthermore, in step 13), the probe pre-travel error value is used. and unit vector Establish an interpolation reference matrix :

[0045]

[0046] in: For the first The coordinates of the theoretical measurement points; .

[0047] Furthermore, in step two, the method for fitting the outer cylindrical surface comprises the following steps:

[0048] 21) Collect at least 6 measurement points from the outer circle of two different axial height sections of the face gear, and uniformly select at least 12 measurement points from the tooth tip to form a dataset;

[0049] 22) Fit a spatially tilted cylinder to the dataset, the general mathematical expression of which is:

[0050]

[0051] in: Indicates the center of the cylinder's base. The coordinates; The axial vector representing the end face gear; It is the radius of the outer cylinder of the end face gear;

[0052] 23) Using the sum of squared residuals as the objective function, the initial value is obtained by minimizing the objective function. , and The objective function is expressed as:

[0053]

[0054] 24) For each measurement point on the outer circle Calculate the unit normal vector of its outer cylindrical surface:

[0055]

[0056] in: Indicates from sampling point point to ; yes Gear axis vector at the end face Projection on; Sampling points The unit normal vector of the outer cylindrical surface;

[0057] 25) Compensate for probe radius and pre-stroke error to obtain the compensation point. Its coordinates are:

[0058]

[0059] in: For compensation points The coordinates;

[0060] 26) Based on the compensated point set An iterative objective function is established to further fit the outer cylindrical surface of the end face gear:

[0061]

[0062] in: Indicates the center of the cylinder's base after iteration. The coordinates; The iterated end face gear axis vector; R is the radius of the outer cylinder of the end face gear.

[0063] Furthermore, in step two, establishing the measurement coordinate system specifically involves: [converting the iterated cylinder axis vector...] The Z-axis direction of the measurement coordinate system; the origin of the measurement coordinate system. Defined as the intersection of the Z-axis and the fitted tooth tip plane;

[0064] By solving the formula:

[0065]

[0066] Obtain scalar distance ; put the origin Coordinates are considered to originate from a known point. Along the axis vector Scalar distance of movement We use a function to calculate and obtain:

[0067]

[0068] in: , , and These are the equation coefficients of the tooth tip plane after probe error compensation.

[0069] Furthermore, the fitting method for the tooth tip plane is as follows:

[0070] Using the least squares method, a plane is fitted to twelve uniformly selected tooth tip measurement points. The coefficients in the plane equation are determined by minimizing the objective function of the sum of squared residuals; the objective function is expressed as:

[0071]

[0072] in: Let represent the sum of squared distances from all tooth tip measurement points to the fitting plane, which is the objective function of the least squares fitting. Indicates the coordinates of the tooth tip measurement point; , , and Plane equation The coefficients in;

[0073] The normal vector of the fitted tooth tip plane is The probe radius and pre-travel error are compensated along the normal vector of this plane:

[0074]

[0075] The iterative equation for the tip plane of the end face gear is:

[0076]

[0077] in: Represents the probe edge vector Directional travel error; Indicates the parameters of the tooth tip plane after compensation. ; The radius of the measuring tip is known.

[0078] Furthermore, step two also includes initial phase compensation along the C-axis, the method of which is as follows:

[0079] S21) Measure the turntable angles corresponding to the left and right tooth surfaces of all tooth slots. The angles of the nth tooth slot are respectively... and ; Calculate the true angular position of the plane in each tooth groove. :

[0080]

[0081] S22) Solve for the initial phase compensation value along the C-axis with the objective function as the minimization target. :

[0082]

[0083] S23) Set the initial phase of the compensated C-axis to:

[0084]

[0085] in: This is the initial phase; based on this phase, the origin of the measurement coordinate system is updated. and axis vector Representation in the turntable coordinate system:

[0086]

[0087] in: Indicates confirmation and The angle at which the worktable rotates; and These represent the axis vector of the gear face after coordinate transformation and the origin of the measurement coordinate system, respectively.

[0088] Furthermore, considering the positional and angular offsets between the measurement coordinate system and the turntable coordinate system, the coordinates of the theoretical measurement points on the left and right tooth surfaces are adjusted:

[0089]

[0090] in: The number of teeth to be measured; It is a homogeneous transformation matrix that maps theoretical measurement points to actual measurement points by incorporating installation errors; d x d y and d z These represent the translational deviations along the X, Y, and Z axes between the measurement coordinate system and the rotary table coordinate system, respectively. and They represent the first The theoretical measurement points and vectors after each tooth deflection are used to replace the measurement points before deflection. Indicates the first The turntable angle corresponding to each tooth; and These represent the measurement coordinate system. Axis and rotary table coordinate system The angular offset between the X and Y axes; Represents the theoretical measurement points on the tooth surface of a face gear, excluding coordinate system errors; Represents the theoretical normal vector of the tooth surface of a face gear, excluding coordinate system errors;

[0091] When measuring the normal deviation of the tooth surface, the probe follows the compensated vector. Approximate measurement point after compensation .

[0092] Furthermore, the solution method for the ray-surface intersection problem is as follows:

[0093] The nonlinear equations equivalent to the ray-surface intersection problem are:

[0094]

[0095] in: Represents the error residual vector of ray-surface intersection; , and They represent exist x, y, z coordinate components within the range; , and The origin of the rays The x, y, and z coordinate components; , and The directions of the rays are respectively Components in the x, y, and z directions;

[0096] Solve using Newton's iteration method and the Jacobian matrix:

[0097]

[0098] in: This represents the vector of unknown parameters at the k-th iteration. Let F be the partial derivative matrix of the error residual vector F with respect to the unknown parameter vector X; This is the error residual vector at the k-th iteration.

[0099] The beneficial effects of this invention are as follows:

[0100] The present invention is based on a sequential decoupling method for comprehensive error compensation in on-machine measurement of face gears. Its technical effects are mainly reflected in three aspects: systematicness, accuracy, and reliability, as follows.

[0101] 1. Systematically solves the problem of error coupling and breaks through the accuracy bottleneck: The core contribution of this invention lies in identifying and breaking the coupling loop between probe pre-travel error and measurement coordinate system alignment error. By specifying a strict order of "first compensating for probe error, then establishing the coordinate system using clean data," the mutual contamination and amplification of the two errors are avoided from the root, solving the inherent defects of isolated compensation in existing technologies and achieving a leap from local correction to system optimization.

[0102] 2. Significantly improves measurement accuracy and fidelity:

[0103] (1) More accurate probe compensation: The Delaunay triangulation and weighted interpolation algorithm based on dense sampling in the upper hemisphere can approximate the pre-stroke error of the probe in the anisotropic direction with high fidelity. It is more adaptable to complex tooth surface normal changes and the compensation accuracy is better than the traditional simplified interpolation or model method.

[0104] (2) More robust coordinate system alignment: By fitting the spatial tilted cylinder with an iterative optimization algorithm that integrates multi-source data (dual-section outer circle point and tooth tip point) and introducing global optimization of the initial phase of the C-axis, the actual axis position of the workpiece can be reconstructed more accurately and stably, which greatly reduces the systematic alignment error caused by installation deviation and uneven machining allowance.

[0105] 3. Enhanced reliability of measurement results, supporting closed-loop manufacturing: The final output tooth surface normal deviation is calculated through two levels of rigorous error compensation and two stages of NURBS surface reconstruction, minimizing the errors inherent in the measurement system and more accurately reflecting the workpiece's machining geometry errors. This provides extremely reliable data input for subsequent machining back-adjustment corrections, making in-machine measurement a truly trustworthy feedback link in high-precision closed-loop manufacturing, improving the convergence efficiency of process iterations and the final part quality assurance capabilities. Attached Figure Description

[0106] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration:

[0107] Figure 1 This is a flowchart of the on-machine measurement comprehensive error compensation method for face gears based on sequential decoupling, as described in this invention.

[0108] Figure 2 This represents 325 theoretical measurement points on a standard sphere and their corresponding spherical normal vectors. Figure 2 (a) indicates that the calibration measurement is performed only on the upper hemisphere of the standard sphere; Figure 2 (b) is the sampling grid defined on the standard sphere;

[0109] Figure 3 To calculate the pre-travel error in the actual measurement direction using Delaunay triangulation; Figure 3 (a) To perform a three-dimensional Delaunay triangulation on the upper half of the unit sphere and generate a spherical convex hull; Figure 3 (b) represents the difference within the triangular mesh;

[0110] Figure 4 To account for installation errors, a measurement coordinate system is established;

[0111] Figure 5 To fit the tooth surface using the NURBS method twice. Detailed Implementation

[0112] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0113] To address the coupling problem between probe pre-stroke error and gear measurement coordinate system error during on-machine measurement of face gear geometric errors, this invention proposes a comprehensive error compensation method for on-machine measurement of face gears based on sequential decoupling. The method employs Delaunay triangulation with weighted interpolation to compensate for probe pre-stroke error, and uses an iterative optimization algorithm to accurately fit the outer cylindrical surface of the gear to correct the measurement coordinate system error.

[0114] Specifically, such as Figure 1 As shown, this embodiment is based on a sequentially decoupled on-machine measurement comprehensive error compensation method for face gears, which includes the following steps.

[0115] Step 1: Probe pre-travel error compensation: Obtain probe pre-travel error data in multiple discrete directions through standard ball calibration and construct a dataset; based on this dataset, calculate and compensate for the probe pre-travel error in any actual measurement direction through Delaunay triangulation and weighted interpolation.

[0116] like Figure 2 As shown, to compensate for probe error, by adjusting the radius of... A standard sphere was used for calibration, and a lookup table for probe pre-stroke error compensation was established. Calibration measurements were performed only on the upper hemisphere of the sphere, such as... Figure 2 As shown in (a), to characterize the pre-travel error over a wide range of detection directions, this embodiment defines a dense sampling grid. This sampling grid measures 9 equidistant points along each meridian and 36 equidistant points along each parallel, thus obtaining a total of 325 theoretical measurement points, including the poles, as shown in (a). Figure 2 As shown in (b). Figure 2 (b) illustrates the distribution of these measurement points and their corresponding surface normal vectors. Any theoretical measurement point on the standard sphere can be represented using spherical coordinates. It means that, among them Polar angle , Azimuth .

[0117] Specifically, in this embodiment, the steps for compensating for probe pre-stroke error are as follows.

[0118] 11) Calibration is performed using a standard sphere. A dense sampling grid is defined on the upper hemisphere of the standard sphere for measurement; where: theoretical measurement points are represented by spherical coordinates. express; Polar angle, ; It is the azimuth angle. The sampling grid has N theoretical measurement points; in this embodiment, N=325.

[0119] 12) Record the actual contact point in the measurement coordinate system of the standard sphere. The vector is represented as The actual center of the standard sphere is fitted using the least squares method. , , The objective function is to minimize the sum of squared residuals.

[0120] In this embodiment, five points on a standard sphere are used. , , , and Establish a measurement coordinate system for the standard sphere. Within this system, record the actual measurement points on the sphere's surface (denoted as ). Then, using the coordinates of these points, the actual center of the standard sphere is determined by a least squares fitting algorithm. The residual function of this optimization problem is shown in equation (1).

[0121]

[0122] in: , , To obtain the coordinates of the center of the standard sphere through fitting, using vectors... This indicates. Similarly, Let i be the coordinates of the i-th measurement point, given by the vector. Indicates. Parameters and Let be the known radius of the standard sphere and the radius of the measuring tip, respectively. When the sum of the squared residuals of all actual measurement points is minimized, the coordinates of the center of the standard sphere can be determined as shown in equation (2).

[0123]

[0124] 13) Calculate the probe pre-travel error value in each calibration direction. and its direction unit vector .

[0125] In this embodiment, the pre-stroke error value of the probe and its compensation direction can be calculated based on the fitted standard sphere center coordinates and the actual measurement point coordinates, as shown in equation (3).

[0126]

[0127] in: Indicates from point Point of view ; yes , unit vector; Indicates along Directional probe pre-travel error; vector The direction is the compensation direction for the probe's pre-stroke error.

[0128] 14) All calibrated unit vectors As a set of points in three-dimensional space, a three-dimensional Delaunay triangulation is performed on the upper unit sphere to construct a spherical triangular mesh.

[0129] Specifically, when measuring the tooth surface deviation of a faceted gear, the contact normal vector of the probe is determined by the complex local geometry of the tooth surface. Therefore, this vector typically does not coincide with any discrete reference vector used during standard sphere calibration. To overcome this inherent inconsistency and accurately correct the probe's pre-travel error, an interpolation method is needed to calculate the compensation value for any measurement point. The interpolation operation is based on a pre-acquired calibration dataset containing data from 325 known directions on the standard sphere, thus enabling precise determination of the compensation value along any measurement vector.

[0130] 15) For the endpoint of the unit normal vector of the actual measurement point on the tooth surface Position it within a specific triangular facet of the spherical triangular mesh, the facet being defined by its vertices. , and Certainly.

[0131] As shown in equation (4), the probe pre-stroke error in equation (3) is utilized. and unit vector Establish an interpolation reference matrix .

[0132]

[0133] in: For the first The coordinates of the theoretical measurement points; .

[0134] Using the calibration unit vector set A mesh is constructed on the upper half of the unit sphere. These vectors (considered as points) are then subjected to 3D Delaunay triangulation to generate a spherical convex hull, as shown below. Figure 3 As shown in (a), the triangular faces of the convex hull form the search mesh, as follows. Figure 3 As shown in (b). Subsequently, using a spatial indexing algorithm, the endpoints of the normalized normal vector from each measurement point on the end face gear can be efficiently indexed. Position yourself within a specific triangular facet of the grid.

[0135] like Figure 3 As shown in (b), from point , and Define a target triangular mesh and denote the query point as... By introducing three weighting factors, the probe pre-stroke error corresponding to the actual measurement direction is calculated, as shown in Equation (5).

[0136]

[0137] The weighting factor is obtained by solving equation (5). , and ;but Probe pre-travel error corresponding to direction Calculated using weighted interpolation according to equation (6):

[0138]

[0139] in: , and Representing points respectively , and The coordinates; , and They are along the normal vector , and The probe's pre-travel error value in the direction, and , , ; It is the probe pre-stroke error corresponding to the actual measurement direction.

[0140] Step 2: Measurement Coordinate System Alignment Error Compensation: Based on the measurement points collected on the face gear, the tooth tip plane and the outer cylindrical surface are fitted respectively through optimization algorithms. The face gear measurement coordinate system is established according to the normal vector of the fitted tooth tip plane and the axis of the fitted outer cylindrical surface. Combined with the initial phase optimization of the C-axis, the coordinates and normal vectors of the theoretical measurement points are transformed to compensate for the coordinate system alignment error caused by installation.

[0141] Installation errors can cause angular and positional deviations between the actual axis of the end face gear and the axis of the machine tool spindle. Therefore, accurately determining the axis position and orientation of the end face gear is a crucial prerequisite for establishing an effective measurement coordinate system. Traditional methods use data from a single outer circle and the average tooth tip height to establish the origin of the measurement coordinate system, which is insufficient. This simplified method introduces significant errors into the measurement coordinate system, leading to misalignment between the measured points on the actual tooth surface and their theoretical nominal positions. To address this limitation, this method employs a more robust approach: fitting a spatially skewed cylinder to a comprehensive dataset containing measurement data from two outer circles with different axial heights and the height data of twelve tooth tips, such as... Figure 4 As shown.

[0142] (1) Fitting the tooth tip plane.

[0143] The least squares method is used to fit a plane to twelve uniformly selected tooth tip measurement points. The equation of this plane is determined by minimizing the objective function of the sum of squared residuals. coefficients in , , and The objective function is shown in equation (7).

[0144]

[0145] in: Let represent the sum of squared distances from all tooth tip measurement points to the fitting plane, which is the objective function of the least squares fitting. Indicates the coordinates of the tooth tip measurement point; , , and Plane equation The coefficients in.

[0146] The normal vector of the fitted tooth tip plane is The probe radius and pre-travel error are compensated along the normal vector of the plane, as shown in equation (8).

[0147]

[0148] The iterative equation for the tip plane of the end face gear is shown in equation (9).

[0149]

[0150] in: Represents the probe edge vector Pre-travel error in direction; Indicates the parameters of the tooth tip plane after compensation. ; The known tip radius is given.

[0151] (2) Fit the outer cylindrical surface.

[0152] Specifically, the steps for fitting the outer cylindrical surface are as follows.

[0153] 21) Collect at least 6 measurement points from the outer circle of each of the two different axial height sections of the face gear, and uniformly select at least 12 measurement points from the tooth tip to form a dataset. That is, use the dataset containing twelve points to fit an inclined cylinder onto the outer surface of the face gear. These points are uniformly sampled from two different axial sections, with six points taken from each section.

[0154] 22) Fit a spatial tilted cylinder to the dataset and define a general mathematical expression for the tilted cylinder as shown in Equation (10).

[0155]

[0156] in: Indicates the center of the cylinder's base. The coordinates; The axial vector representing the end face gear; It is the radius of the outer cylinder of the end face gear.

[0157] 23) Using the sum of squared residuals as the objective function, the coefficients are solved by minimizing the objective function. , , , , , and Thus, the initial... , and The objective function is shown in equation (11).

[0158]

[0159] 24) For each measurement point on the outer circle Calculate the unit normal vector of its outer cylindrical surface. After the initial fitting, calculate the normal vector of the outer cylindrical surface at 12 sampling points using equation (12). Then, use these vectors to calculate the normal vector of the outer cylindrical surface at each sampling point. The coordinates are used to compensate for the probe radius.

[0160]

[0161] in: Indicates from sampling point point to ; yes Gear axis vector at the end face Projection on; Sampling points The unit normal vector of the outer cylindrical surface.

[0162] 25) Compensate for probe radius and pre-stroke error to obtain the compensation point. Its coordinates are shown in equation (13):

[0163]

[0164] in: For compensation points The coordinates.

[0165] 26) Based on the compensated point set An iterative objective function is established to further fit the outer cylindrical surface of the end face gear, as shown in equation (14):

[0166]

[0167] in: Indicates the center of the cylinder's base after iteration. The coordinates; The iterated end face gear axis vector; R is the radius of the outer cylinder of the end face gear.

[0168] The axis of the outer cylindrical surface of the iterated end gear is defined as the Z-axis of the face gear measurement coordinate system, and its positive direction is parallel to the vector. Alignment. The origin of this coordinate system is defined as the intersection of the Z-axis and the previously fitted tooth tip plane. Origin The coordinates can be considered as originating from a known point. Along the axis vector Scalar distance of movement The function is used to calculate, as shown in equation (15).

[0169]

[0170] As shown in equation (16), by substituting equation (15) into equation (9), the result is calculated. Then By substitution (15), we can obtain The coordinates.

[0171]

[0172] in: , , and These are the equation coefficients of the tooth tip plane after probe error compensation.

[0173] (3) Initial phase compensation of the C-axis.

[0174] During the measurement process, the tooth groove surface needs to be aligned with the measurement coordinate system. While they may overlap, in actual machining, the uneven material removal during grinding causes the included angle between the planes of adjacent tooth slots to deviate from their nominal tooth pitch angle. Therefore, when the workpiece is indexed on a rotary table with a constant theoretical tooth pitch angle, the plane of each subsequent tooth slot will deviate from the nominal tooth pitch angle of the measurement coordinate system. Accumulated alignment errors can occur between planes. To address this issue, after establishing the origin and Z-axis of the measurement coordinate system, a globally optimal C-axis initial phase compensation algorithm is implemented to correct the errors caused by this uneven machining allowance.

[0175] Specifically, in this embodiment, the steps for C-axis initial phase compensation are as follows.

[0176] S21) Measure the turntable angles corresponding to the left and right tooth surfaces of all tooth slots. The angles of the nth tooth slot are respectively... and ; Calculate the true angular position of the plane in each tooth groove. .

[0177] Specifically, the probe is moved to the projection point of the theoretical midpoint of the tooth surface onto the tooth groove surface. The gear is rotated clockwise and counterclockwise to measure the turntable angles corresponding to the left and right tooth surfaces of all tooth grooves. The angles corresponding to the left and right surfaces of the nth tooth groove are respectively... and The true angular position of the plane in each tooth groove It can be calculated from equation (17).

[0178]

[0179] S22) Using the sum of squared residuals as the objective function, and minimizing the objective function, solve for the initial phase compensation value along the C-axis. As shown in equation (18)

[0180]

[0181] S23) Represent the compensated initial phase of the C-axis As shown in equation (19).

[0182]

[0183] in: This represents the original initial phase.

[0184] After compensating for the initial phase of the C-axis, the expression of the end face gear axis vector and the origin of the measurement coordinate system in the C-axis coordinate system is shown in equation (20).

[0185]

[0186] in: Indicates confirmation and The angle at which the worktable rotates; and These represent the axis vector of the gear face after coordinate transformation and the origin of the measurement coordinate system, respectively.

[0187] By indexing the rotary table to the compensated initial angle Finally, the measurement coordinate system was determined. The final location. At this point, the coordinate system is measured. Axis and rotary table coordinate system The planes are parallel. Then, establish the right-hand rule. The axes are aligned to form a standard orthogonal coordinate system. This alignment results in the existence of x-axis and y-axis distances between the two coordinate systems. and Angle offset. Considering the positional and angular offsets between the above measurement coordinate system and the turntable coordinate system, the coordinates of the theoretical measurement points on the left and right tooth surfaces are adjusted as shown in equation (21).

[0188]

[0189] in: The number of teeth to be measured; It is a homogeneous transformation matrix that maps theoretical measurement points to actual measurement points by incorporating installation errors; d x d y and d z These represent the translational deviations along the X, Y, and Z axes between the measurement coordinate system and the rotary table coordinate system, respectively. and They represent the first The theoretical measurement points and vectors after each tooth deflection are used to replace the measurement points before deflection. Indicates the first The turntable angle corresponding to each tooth; and These represent the measurement coordinate system. Axis and rotary table coordinate system The angular offset between the X and Y axes; Represents the theoretical measurement points on the tooth surface of a face gear, excluding coordinate system errors; This represents the normal vector of the gear tooth surface without coordinate system errors.

[0190] When measuring the normal deviation of the tooth surface, the probe follows the compensated vector. Measurement points after compensation .

[0191] Step 3: Evaluation of tooth surface geometric error: The actual tooth surface is reconstructed using a two-stage non-uniform rational B-spline (NURBS) surface fitting method after compensation in steps S1 and S2, and the normal deviation between the actual measurement points and the theoretical tooth surface is calculated.

[0192] The above method is used to perform post-processing on the measurement points of the gear tooth surface after actual machining, so that the post-processing results can reflect the true geometric error of the tooth surface to the greatest extent.

[0193] The theoretical measurement point and the normal vector are denoted as follows: and To calculate the normal deviation of the tooth surface, a two-stage non-uniform rational B-spline (NURBS) method was adopted. This method can simultaneously and accurately compensate for the probe radius and pre-stroke error. The equation of the NURBS surface is shown in equation (22).

[0194]

[0195] in: equal Decrease the number of directional control points by one; equal Decrease the number of directional control points by one; Represents the control point grid; These are the weights of the control points; and They are direction order and direction B-spline basis functions of order 1.

[0196] In this embodiment, the steps of the two-stage NURBS surface fitting method are as follows.

[0197] 31) Using the compensated measurement points Fitting the first NURBS surface and calculate the points exist normal vector on As shown in equation (23):

[0198]

[0199] in: To utilize the compensated measurement points Fit the first NURBS surface; For point exist Normal vector on; equal Decrease the number of directional control points by one; equal Decrease the number of directional control points by one; and for Two independent variables in a two-dimensional parameter domain; curved surface Control points; and They are respectively Direction 3rd order and Directional third-order B-spline basis functions; For point exist The normal vector on.

[0200] 32) Along Direction to point Perform secondary compensation to obtain the adjusted points. As shown in equation (24):

[0201]

[0202] in: To simultaneously consider the compensation amount for probe radius and pre-travel error; The known tip radius; Indicates the probe is in Pre-travel error in direction.

[0203] 33) The second NURBS surface Based on points To perform the fitting, the cubic B-spline basis function is used, and the weights of all control points are set to 1, as shown in equation (25).

[0204]

[0205] in: For utilization point Fit the second NURBS surface; and for Two independent variables in a two-dimensional parameter domain; for Control points; and They are respectively Direction 3rd order and Directional third-order B-spline basis functions.

[0206] 34) Construct a ray-surface intersection problem to obtain the theoretical point. Along its normal vector To the curved surface intersection ,distance That is, the tooth surface normal deviation; the ray-surface intersection problem is represented as shown in equation (26):

[0207]

[0208] in: Theoretical point Along the normal direction The distance moved is equal to the distance traveled along the vector. The tooth surface normal deviation.

[0209] Specifically, the solution method for the ray-surface intersection problem is as follows.

[0210] The equivalent nonlinear equations for the ray-surface intersection problem are shown in equation (27):

[0211]

[0212] in: Represents the error residual vector of ray-surface intersection; , and They represent exist x, y, z coordinate components within the range; , and The origin of the rays The x, y, and z coordinate components; , and The directions of the rays are respectively Components in the x, y, and z directions;

[0213] Solve equation (27) using Newton's iteration method and the Jacobian matrix:

[0214]

[0215] in: This represents the vector of unknown parameters at the k-th iteration. Let F be the partial derivative matrix of the error residual vector F with respect to the unknown parameter vector X; This is the error residual vector at the k-th iteration.

[0216] Normal deviation of tooth surface Defined as and The Euclidean distance between them, i.e. The value. The process of using NURBS to perform two fittings on the end face gear tooth surface is illustrated as follows: Figure 5 As shown.

[0217] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A method for compensating for comprehensive errors in on-machine measurement of face gears based on sequential decoupling, characterized in that: Includes the following steps: Step 1: Probe pre-travel error compensation: Obtain probe pre-travel error data in multiple discrete directions through standard ball calibration and construct a dataset; based on this dataset, calculate and compensate for the probe pre-travel error in any actual measurement direction through Delaunay triangulation and weighted interpolation. Step 2: Measurement coordinate system alignment error compensation: Based on the actual measurement points collected on the face gear, the tooth tip plane and the outer cylindrical surface are fitted respectively through optimization algorithm. The face gear measurement coordinate system is established according to the normal vector of the fitted tooth tip plane and the axis of the fitted outer cylindrical surface. Combined with the initial phase optimization of the C-axis, the coordinates and normal vectors of the theoretical measurement points are transformed to compensate for the coordinate system alignment error caused by installation. Step 3: Evaluation of Tooth Surface Geometric Error: The actual tooth surface is reconstructed using a two-stage non-uniform rational B-spline surface fitting method based on the actual measurement points after compensation in steps S1 and S2. The normal deviation between the actual tooth surface and the theoretical tooth surface is then calculated. The method steps are as follows: 31) Using the compensated measurement points Fitting the first NURBS surface and calculate the points exist normal vector on : in: To utilize the compensated measurement points Fit the first NURBS surface; for On point The normal vector at that location; equal Decrease the number of directional control points by one; equal Decrease the number of directional control points by one; and for Two independent variables in a two-dimensional parameter domain; curved surface Control points; and They are respectively direction 3 order and direction 3 B-spline basis functions of order 1; for On point The normal vector; 32) Along Direction to point Perform secondary compensation to obtain points : in: To simultaneously consider the compensation amount for probe radius and pre-travel error; The known tip radius; Indicates the probe is in Pre-travel error in direction; 33) Utilization Point Fitting the second NURBS surface ; in: For utilization point Fit the second NURBS surface; and for Two independent variables in a two-dimensional parameter domain; for Control points; and They are respectively direction 3 order and direction 3 B-spline basis functions of order 1; 34) Construct a ray-surface intersection problem to obtain the theoretical point. Along its normal vector To the curved surface intersection ,distance This refers to the tooth surface normal deviation; the ray-surface intersection problem is expressed as: in: Theoretical point Along the normal direction The distance traveled is equal to the distance along the vector. The tooth surface normal deviation.

2. The method for comprehensive error compensation in on-machine measurement of face gears based on sequential decoupling according to claim 1, characterized in that: In step one, the method for compensating for the probe's pre-stroke error is as follows: 11) Calibration is performed using a standard sphere. A dense sampling grid is defined on the upper hemisphere of the standard sphere for measurement; where: theoretical measurement points are represented by spherical coordinates. express; Polar angle, ; It is the azimuth angle. The sampling grid has N theoretical measurement points. 12) Record the actual contact point in the measurement coordinate system of the standard sphere. The vector is represented as The actual center of the standard sphere is fitted using the least squares method. , , The objective function is to minimize the sum of squared residuals: in: and These are the known standard sphere radius and the apex radius, respectively; The number of measurement points; 13) Calculate the probe pre-travel error value in each calibration direction. and its direction unit vector The calculation formula is: in: Indicates from point Point of view ; yes , unit vector; Indicates along Directional probe pre-travel error; vector The direction is the compensation direction for the probe's pre-stroke error; 14) All calibrated unit vectors As a set of points in three-dimensional space, a three-dimensional Delaunay triangulation is performed on the upper half of the unit sphere to construct a spherical triangular mesh. 15) For the endpoint of the unit normal vector of the actual measurement point on the tooth surface Position it within a specific triangular facet of the spherical triangular mesh, the facet being defined by its vertices. , and Definition; Solving the system of equations: Obtain weighting factors , and ;but Probe pre-travel error corresponding to direction Calculated using weighted interpolation: in: , and Representing points respectively , and The coordinates; , and They are along the normal vector , and The probe's pre-travel error value in the direction, and , , ; It is the probe pre-stroke error corresponding to the actual measurement direction.

3. The method for comprehensive error compensation in on-machine measurement of face gears based on sequential decoupling according to claim 2, characterized in that: In step 13), the probe pre-travel error value is used. and unit vector Establish an interpolation reference matrix : in: For the first The coordinates of the theoretical measurement points; .

4. The method for comprehensive error compensation in on-machine measurement of face gears based on sequential decoupling according to claim 1, characterized in that: In step two, the method for fitting the outer cylindrical surface consists of the following steps: 21) Collect at least 6 measurement points from the outer circle of two different axial height sections of the face gear, and uniformly select at least 12 measurement points from the tooth tip to form a dataset; 22) Fit a spatially tilted cylinder to the dataset, the general mathematical expression of which is: in: Indicates the center of the cylinder's base. The coordinates; The axial vector representing the end face gear; It is the radius of the outer cylinder of the end face gear; 23) Using the sum of squared residuals as the objective function, the initial value is obtained by minimizing the objective function. , and The objective function is expressed as: 24) For each measurement point on the outer circle Calculate the unit normal vector of its outer cylindrical surface: in: Indicates from sampling point point to ; yes Gear axis vector at the end face Projection on; Sampling points The unit normal vector of the outer cylindrical surface; 25) Compensate for probe radius and pre-stroke error to obtain the compensation point. Its coordinates are: in: For compensation points The coordinates; 26) Based on the compensated point set An iterative objective function is established to further fit the outer cylindrical surface of the end face gear: in: Indicates the center of the cylinder's base after iteration. The coordinates; The iterated end face gear axis vector; R is the radius of the outer cylinder of the end face gear.

5. The method for comprehensive error compensation in on-machine measurement of face gears based on sequential decoupling according to claim 4, characterized in that: In step two, establishing the measurement coordinate system specifically involves: mapping the iterated cylinder axis vector... The Z-axis direction of the measurement coordinate system; the origin of the measurement coordinate system. Defined as the intersection of the Z-axis and the fitted tooth tip plane; By solving the formula: Obtain scalar distance ; The origin Coordinates are considered to originate from a known point. Along the axis vector Scalar distance of movement We use a function to calculate and obtain: in: , , and These are the equation coefficients of the tooth tip plane after probe error compensation.

6. The method for comprehensive error compensation in on-machine measurement of face gears based on sequential decoupling according to claim 5, characterized in that: The fitting method for the tooth tip plane is as follows: Using the least squares method, a plane is fitted to twelve uniformly selected tooth tip measurement points. The coefficients in the plane equation are determined by minimizing the objective function of the sum of squared residuals; the objective function is expressed as: in: Represented as the coordinates of the tooth tip measurement point; , , and Plane equation The coefficients in; The normal vector of the fitted tooth tip plane is The probe radius and pre-travel error are compensated along the normal vector of this plane: The iterative equation for the tip plane of the end face gear is: in: Represents the probe edge vector Directional travel error; Indicates the parameters of the tooth tip plane after compensation. ; The radius of the measuring tip is known.

7. The method for comprehensive error compensation in on-machine measurement of face gears based on sequential decoupling according to claim 4, characterized in that: Step two also includes initial phase compensation along the C-axis, the method of which is as follows: S21) Measure the turntable angles corresponding to the left and right tooth surfaces of all tooth slots. The angles of the nth tooth slot are respectively... and ; Calculate the true angular position of the plane in each tooth groove. : S22) Solve for the initial phase compensation value along the C-axis with the objective function as the minimization target. : S23) Set the initial phase of the compensated C-axis to: in: This is the initial phase; based on this phase, the origin of the measurement coordinate system is updated. and axis vector Representation in the turntable coordinate system: in: Indicates confirmation and The angle at which the worktable rotates; and These represent the axis vector of the gear face after coordinate transformation and the origin of the measurement coordinate system, respectively.

8. The method for comprehensive error compensation in on-machine measurement of face gears based on sequential decoupling according to claim 7, characterized in that: Considering the positional and angular offsets between the measurement coordinate system and the c-coordinate system, the coordinates of the theoretical measurement points on the left and right tooth surfaces are adjusted as follows: in: The number of teeth to be measured; It is a homogeneous transformation matrix that maps theoretical measurement points to actual measurement points by taking installation errors into account. dx , dy and dz These represent the translational deviations along the X, Y, and Z axes between the measurement coordinate system and the rotary table coordinate system, respectively. and They represent the first The theoretical measurement points and vectors after each tooth deflection are used to replace the measurement points before deflection. Indicates the first The turntable angle corresponding to each tooth; and These represent the measurement coordinate system. Axis and rotary table coordinate system The angular offset between the X and Y axes; Represents the theoretical measurement points on the tooth surface of a face gear, excluding coordinate system errors; Represents the theoretical normal vector of the tooth surface of a face gear, excluding coordinate system errors; When measuring the normal deviation of the tooth surface, the probe follows the compensated vector. Approximate measurement point after compensation .

9. The method for comprehensive error compensation in on-machine measurement of face gears based on sequential decoupling according to claim 1, characterized in that: The solution method for the ray-surface intersection problem is as follows: The nonlinear equations equivalent to the ray-surface intersection problem are: in: Represents the error residual vector of ray-surface intersection; , and They represent exist Within range x , y , z Coordinate components; , and The origin of the rays of x , y , z Coordinate components; , and The directions of the rays are respectively exist x , y , z Components in direction; Solve using Newton's iteration method and the Jacobian matrix: in: Indicates the first k The unknown parameter vector at the next iteration; Error residual vector F For unknown parameter vectors X The partial derivative matrix; For the first k Error residual vector at the next iteration.

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