Face gear worm grinding wheel dressing error compensation method based on multi-source error modeling and sensitivity analysis

Abstract

The invention discloses a face gear worm grinding wheel dressing error compensation method based on multi-source error modeling and sensitivity analysis, which is characterized by comprising the following steps of: step 1, on the basis of a constructed worm grinding wheel tooth surface error model, carrying out multi-source error modeling; sequentially adopting a dynamic truncation strategy based on sensitivity, a high-order correction term approximate nonlinear error term and a vector orthogonalization method to obtain a multi-source error model of the face gear worm grinding wheel dressing process; 2, inputting the output of the multi-source error model into a sensitivity analysis model based on a generative adversarial network and a gradient elevator, wherein the generative adversarial network generates simulation data consistent with actual measurement error distribution; the gradient elevator calculates the sensitivity coefficient of the error term based on the analog data; when the sensitivity coefficient exceeds a threshold value, marking as a key error term; and step 3, inputting an error compensation model based on the key error term, calculating a compensation component through an error decoupling module, and generating a compensation instruction for the motion axis to realize error compensation.

Description

Technical Field

[0001] The present invention belongs to the technical field of gear dressing, and specifically provides a face gear worm grinding wheel dressing error compensation method based on multi-source error modeling and sensitivity analysis. Background Art

[0002] Face gears play a key role in modern mechanical transmission systems due to their efficient and precise transmission of motion and power. They are widely used in aerospace, automotive, and industrial sectors, where reliability and precision are paramount. The unique geometry of face gears enables compact design and high load-bearing capacity, making them suitable for demanding operating conditions. However, the manufacturing process of face gears is complex and requires precise control to ensure surface quality and dimensional accuracy. Before grinding face gears, the worm grinding wheel must be dressed with a diamond roller. Proper dressing ensures that the grinding wheel maintains accurate geometry and is crucial for achieving high-precision, high-quality face gear manufacturing.

[0003] The accuracy of the worm grinding wheel dressing process (WGWDP) directly affects the final grinding accuracy of the face gear. Any machining deviation will directly affect the performance and service life of the gear. The accuracy of this dressing process is directly affected by multi-source errors caused by geometric and thermal effects. These errors are directly transmitted to the workpiece during the grinding process, resulting in dimensional misalignment of the face gear. These errors are often interactive and nonlinear, which can significantly affect the geometric accuracy of the machined face gear. Effective error control ensures that the grinding wheel maintains the target geometry, minimizes cumulative errors, and improves the consistency of the machining process. Therefore, error compensation in the WGWDP is crucial to ensuring the accuracy and quality of face gear grinding.

[0004] Machine tool errors (especially geometric and thermal errors) often exhibit highly complex correlation characteristics due to their multi-source nature. These errors involve multiple physical mechanisms and may exhibit nonlinear, high-dimensional, and irregular characteristics. Multi-source error modeling is the key to achieving error compensation. The existence of multi-source errors such as geometric, thermal, and dynamic errors complicates error modeling and compensation. Traditional methods such as Taylor series expansion have been used to model error sources. These methods are effective for linear systems, but have difficulty handling highly nonlinear error interactions. To improve computational efficiency, simplified models such as the truncation function method have been introduced by discarding high-order terms, but these methods often sacrifice modeling accuracy in complex scenarios. The reason is that the traditional truncation function method discards high-order terms to simplify the error expression. Although computationally efficient, it has difficulty coping with the complexity of multi-source errors, especially in scenarios with strong nonlinear interactions.

[0005] In addition to multi-source error modeling, sensitivity analysis is essential for error compensation. This method plays an important role in simplifying the model by identifying key error components. Traditional sensitivity analysis methods, such as the Morris method and the Sobol method, can analyze the impact of individual error sources and perform well in low-dimensional systems with limited variable interactions. However, when dealing with the complex interactions of multi-source errors within machine tools, quantifying the impact of different errors on the overall system behavior remains a challenge. Traditional methods struggle to accurately quantify the interactions between error components, especially for systems with nonlinear relationships. Most sensitivity-based methods rely on fixed thresholds to identify key variables, potentially overlooking dynamic changes in system behavior. Machine learning-based sensitivity analysis and generative models offer new approaches to addressing these limitations. Generative adversarial networks (GANs) can generate samples from the same distribution as the original data without the need for explicit labels. In sensitivity analysis, GANs can simulate error combinations and their impact on system performance, making them particularly suitable for generating high-dimensional, complex, and sparse data. Gradient boosting machines (GBMs), as an ensemble learning method, optimize system prediction performance by iteratively constructing decision trees. They are effective in error sensitivity analysis for nonlinear relationships and high-dimensional data, identifying the impact of key error sources. Despite progress in this area, the following challenges remain:

[0006] (1) Existing models often sacrifice accuracy for computational efficiency, especially when discarding high-order terms or simplifying interactions. Balancing these two aspects remains a major challenge, especially for real-time applications.

[0007] (2) As the number of error sources increases, traditional sensitivity analysis and modeling methods face the problem of exponential growth in computational costs, and there is currently a lack of effective strategies for dealing with high-dimensional, strongly interactive systems.

[0008] (3) Many error sources exhibit nonlinear behaviors and dynamic interactions that are difficult to model using linear or static methods. Integrating these characteristics without introducing excessive computational complexity is a core challenge. Summary of the Invention

[0009] In view of this, the purpose of the present invention is to provide a face gear worm grinding wheel dressing error compensation method based on multi-source error modeling and sensitivity analysis, which integrates the sensitivity-based dynamic truncation strategy, high-order correction terms and vector orthogonalization technology, and combines sensitivity analysis at the same time, which can shorten the calculation time and improve the characterization accuracy.

[0010] In order to achieve the above object, the present invention provides the following technical solutions:

[0011] A method for compensating face gear worm grinding wheel dressing errors based on multi-source error modeling and sensitivity analysis includes the following steps:

[0012] Step 1: Based on the forward kinematics matrix of the face gear worm wheel dressing process, a worm wheel tooth surface error model is constructed. A sensitivity-based dynamic truncation strategy is used to dynamically determine whether to retain the error interaction terms, a high-order correction term is used to approximate the nonlinear error terms, and a vector orthogonalization method is used to process the tooth surface error model. Finally, a multi-source error model of the face gear worm wheel dressing process is constructed.

[0013] Step 2: Input the output of the multi-source error model into a sensitivity analysis model based on a generative adversarial network and a gradient boosting machine, wherein: the generative adversarial network generates simulated data consistent with the measured error distribution; the gradient boosting machine calculates the sensitivity coefficient of the error term based on the simulated data; when the sensitivity coefficient exceeds a threshold, it is marked as a critical error term;

[0014] Step 3: Based on the key error terms identified, the error compensation model is input, the compensation components are calculated through the error decoupling module, and compensation instructions for the motion axis are generated to achieve error compensation.

[0015] Furthermore, in step 1, the ideal forward kinematics matrix of the face gear worm grinding wheel dressing process is:

[0016]

[0017] in: and They represent the ideal HCT matrices from X-axis to B2-axis, B2-axis to C2-axis, C2-axis to Z2-axis, Z2-axis to F-axis, F-axis to X-axis, X-axis to Z-axis, Z-axis to A-axis, A-axis to Y-axis, Y-axis to B-axis, and B-axis to W-axis respectively;

[0018] The actual forward kinematics matrix of the face gear worm grinding wheel dressing process is:

[0019]

[0020] in: and represent the actual HCT matrices from X-axis to B2-axis, B2-axis to C2-axis, C2-axis to Z2-axis, Z2-axis to F-axis, F-axis to X-axis, X-axis to Z-axis, Z-axis to A-axis, A-axis to Y-axis, Y-axis to B-axis, and B-axis to W-axis, respectively, and

[0021] Furthermore, the worm grinding wheel tooth surface error model is expressed as:

[0022]

[0023] Where: w is the position error matrix; δ x , δ y and δ zRespectively represent the position error components; ε w is the angle error matrix; ε x , ε y and ε z Represent the angular error components respectively; and are the actual position vector and the ideal position vector respectively; and are the actual direction vector and the ideal direction vector respectively; is the initial angular position or reference angular position of the worm; θ x is the rotation angle along the X axis; E is the error vector; and:

[0024]

[0025] in: is the ideal forward kinematics matrix of the face gear worm grinding wheel dressing process; r x (θ x ) and n x (θ x ) are the position vector and the normal vector of a point on the worm tooth surface respectively; is the ideal relative velocity at the contact point;

[0026]

[0027] in: is the actual forward kinematics matrix of the face gear worm grinding wheel dressing process; r x (θ x ,E) and n x (θ x ,E) are the position vector of a point on the worm tooth surface and the normal vector of a point on the worm tooth surface respectively; is the actual relative velocity at the contact point.

[0028] Furthermore, the dynamic truncation strategy based on sensitivity is:

[0029]

[0030] Where: M is the matrix M A and M B The matrix obtained by multiplication operation; M A and M B are obtained by Taylor series expansion of error matrices A and B respectively; I is the identity matrix; ω1 and ω2 are first-order error matrices; ω1·ω2 is the interaction term; S AB is the sensitivity coefficient; ε is the preset threshold; and:

[0031]

[0032] Where: E is the error vector;

[0033] The method of using high-order correction terms to approximate nonlinear error terms is:

[0034] When S AB When ≥ε, it indicates that the interaction term ω1·ω2 has a significant effect and should be retained, and the high-order term and Using nonlinear functions to approximate it, we get:

[0035] M=M A ·M B ≈I+ω1+ω2+(1+α+β)ω1·ω2

[0036] Where: α and β represent the nonlinear interference strength during multi-rotation coupling;

[0037] The coefficients of α and β are determined by least squares fitting:

[0038]

[0039] Where: E i measured is the measured value of the i-th sample point; E i predicted is the predicted value of the i-th sample point; N is the total number of samples;

[0040] For the error matrix E = [ω1,ω2,ω1·ω2] T Implement vector orthogonalization processing and use the Gram-Schmidt vector orthogonalization method to obtain the orthogonalized error model:

[0041] E oth =[v1,v2,v3]

[0042] v1=ω1

[0043]

[0044] The multi-source error model of the face gear worm grinding wheel dressing process is constructed as follows:

[0045] M=M A ·M B ≈I+ω1+ω2+γω1·ω2

[0046]

[0047] Where: γ is the error coupling weight coefficient.

[0048] Furthermore, in step 2, the principle of the sensitivity analysis model is:

[0049] The generator generates simulation data that conforms to the actual error distribution from the random noise vector z. The optimization objective function of the generator is:

[0050]

[0051] Where: p z (z) is the random noise vector distribution; G(z) is the generated error data; D(G(z)) represents the discriminator’s probability estimate of the authenticity of G(z);

[0052] The loss function of the discriminator is defined as:

[0053]

[0054] Where: p data (x) is the true data distribution; D(x) represents the probability that x is true; D(G(z)) represents the probability that G(z) is true;

[0055] The loss function of the gradient boosting machine is based on minimizing the residual between the predicted value and the true value. The loss function of each iteration is:

[0056]

[0057] Where: y i and f(x i ) represent the true value and the predicted value respectively; n is the total number of samples;

[0058] The total sensitivity of all error components is:

[0059]

[0060] Where: S j is the total sensitivity of the error component; e j is the jth error term.

[0061] Furthermore, in step 3, the error compensation is achieved by error decoupling, and the error compensation component is:

[0062] [δ Y ,δ Z ,ε A ,ε B ,ε C2 ] T =(J M T ·J M ) -1 ·J M T ·E

[0063] Where: E is the differential motion error; J M is the Jacobian matrix; δy and δ z Respectively represent the position error components; ε w is the angle error matrix; ε A , ε B and ε C2 Indicates the angle error compensation amount.

[0064] The beneficial effects of the present invention are:

[0065] The present invention is based on a face gear worm grinding wheel dressing error compensation method based on multi-source error modeling and sensitivity analysis. Aiming at the limitations of the traditional truncation function method and sensitivity analysis technology in multi-source error modeling and compensation in the worm grinding and dressing process, an improved truncation function method is proposed. This method integrates a dynamic truncation strategy based on sensitivity, high-order correction terms and vector orthogonalization technology, which can effectively shorten the calculation time while improving the error characterization accuracy; a high-precision sensitivity analysis model constructed based on generative adversarial networks (GAN) and gradient boosting machines (GBM) is used to optimize the sensitivity terms. Finally, error compensation is achieved through the error compensation model, which significantly reduces the normal error, and its compensation accuracy and effect are better than traditional methods. Specifically, the present invention makes the following key contributions to the field of multi-source error modeling and compensation in precision machining:

[0066] (1) An improved truncation function method is proposed to address the shortcomings of traditional methods in capturing the nonlinear and interactive characteristics of multi-source errors. By introducing sensitivity-based dynamic truncation, high-order correction terms, and vector orthogonalization, the error characterization capability is enhanced while maintaining computational efficiency.

[0067] (2) Develop a new sensitivity analysis model based on GAN and GBM. This model reduces the sensitivity terms from 19 to 18 in high-dimensional strongly interactive systems, improving computational efficiency and multi-source error analysis accuracy.

[0068] (3) An efficient multi-source error model and a high-precision sensitivity analysis model are integrated into the error compensation framework. This framework performs well in compensating the normal error of face gear tooth surfaces, surpassing traditional compensation methods in both accuracy and effectiveness. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] In order to make the purpose, technical solutions and beneficial effects of the present invention more clear, the present invention provides the following drawings for illustration:

[0070] Figure 1 Schematic diagram of the error compensation method for face gear worm grinding wheel dressing based on multi-source error modeling and sensitivity analysis of the present invention;

[0071] Figure 2 It is the structural diagram of the grinding machine;

[0072] Figure 3 For the comparison of computational efficiency;

[0073] Figure 4 is the influence of geometric error;

[0074] Figure 5 is the directional sensitivity factor obtained by the Sobol method;

[0075] Figure 6 is the directional sensitivity factor obtained by the method of the present invention;

[0076] Figure 7 is the tooth surface error. DETAILED DESCRIPTION

[0077] 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 the present invention and implement it. However, the embodiments are not intended to limit the present invention.

[0078] like Figure 1 As shown, a face gear worm grinding wheel dressing error compensation method based on multi-source error modeling and sensitivity analysis in this embodiment includes the following steps.

[0079] Step 1: Based on the forward kinematics matrix of the face gear worm grinding wheel dressing process, a worm grinding wheel tooth surface error model is constructed. A sensitivity-based dynamic truncation strategy is used to dynamically decide whether to retain the error interaction terms, a high-order correction term is used to approximate the nonlinear error terms, and a vector orthogonalization method is used to process the tooth surface error model. Finally, a multi-source error model of the face gear worm grinding wheel dressing process is constructed.

[0080] 1.1 Theoretical kinematic model and practical kinematic model

[0081] like Figure 2 As shown in Figure 1, during the entire trimming process, the Z, Y, A, and B axes are in motion, the X, Z2, and C2 axes remain stationary, and the B2 axis rotates at high speed. Because the B and B2 axes are high-precision motorized spindles, only their thermal errors are considered, ignoring geometric errors. A total of 33 error items are listed in Table 1.

[0082] Table 1 List of error items in the trimming process

[0083]

[0084] The theoretical forward kinematics matrix of the face gear worm grinding wheel dressing process is expressed as:

[0085]

[0086] in: and They represent the ideal HCT matrices from X-axis to B2-axis, B2-axis to C2-axis, C2-axis to Z2-axis, Z2-axis to F-axis, F-axis to X-axis, X-axis to Z-axis, Z-axis to A-axis, A-axis to Y-axis, Y-axis to B-axis, and B-axis to W-axis, respectively; α, b, λ, y, and z represent the A-axis rotation angle, B-axis rotation angle, worm wheel helix angle, X-axis moving distance, and Z-axis moving distance, respectively.

[0087] The actual forward kinematics matrix of the face gear worm grinding wheel dressing process is:

[0088]

[0089] in: and represent the actual HCT matrices from X-axis to B2-axis, B2-axis to C2-axis, C2-axis to Z2-axis, Z2-axis to F-axis, F-axis to X-axis, X-axis to Z-axis, Z-axis to A-axis, A-axis to Y-axis, Y-axis to B-axis, and B-axis to W-axis, respectively, and

[0090] The ideal and actual HCT matrices are detailed in Appendix A, Table A1.

[0091] The above matrix calculation is extremely cumbersome. After ignoring the second-order and higher-order terms, the final calculation result is:

[0092]

[0093] matrix The elements are shown in Appendix B.

[0094] 1.2 Multi-source error and worm grinding wheel tooth surface error mapping model

[0095] 1.2.1 Worm Grinding Wheel Tooth Surface Error Model

[0096] The coordinate transformation process is:

[0097]

[0098] in: and are the ideal position vector and direction vector respectively; r x (θ x ) and n x (θ x ) are the position vector and the normal vector of a point on the worm tooth surface respectively; is the initial angular position or reference angular position of the worm; θ x is the rotation angle along the X axis.

[0099] The meshing equation is:

[0100]

[0101] in: is the ideal relative velocity at the contact point.

[0102] After substituting the HCT matrix, the coordinate transformation and meshing equations of the modification process under actual working conditions are:

[0103]

[0104] in: and are the actual position vector and direction vector respectively; is the actual relative velocity at the contact point; r x (θ x ,E) and n x (θ x ,E) are the position vector of a point on the worm tooth surface and the normal vector of a point on the worm tooth surface respectively; E is the error vector.

[0105] The formula for obtaining discrete points is:

[0106]

[0107] Where: r wli and n wli are position vector and direction vector respectively; r wl and n wl are the position vector and direction vector of the point set respectively. The error mapping model of the worm grinding wheel tooth surface is obtained as follows:

[0108]

[0109] Right now:

[0110] E w =[δ xw ,δ yw ,δ zw ,ε xw ,ε yw ,ε zw ] T

[0111] Where: w is the position error matrix; δ x , δ y and δ z Respectively represent the position error components; ε w is the angle error matrix; ε x , ε y and ε z Represent the angular error components respectively; and are the actual position vector and the ideal position vector respectively; and are the actual direction vector and the ideal direction vector respectively; is the initial angular position or reference angular position of the worm; θ x is the rotation angle along the X axis; E is the error vector.

[0112] 1.2.2 Efficient Multi-Source Error Modeling Based on the Improved Truncation Function Method

[0113] The dynamic truncation strategy based on sensitivity is as follows: In the traditional truncation function method, it is assumed that there are two error matrices A and B, whose Taylor series expansions are:

[0114] M A =I+ω1+(ω1) 2 +Δ((ω1) 2 )

[0115] M B =I+ω2+(ω2) 2 +Δ((ω2) 2 )

[0116] Where: I is the identity matrix; ω1 and ω2 are first-order error matrices; (ω1) 2 and (ω2) 2 is the second-order error matrix; Δ((ω1) 2 ) and Δ((ω2) 2 ) are higher-order terms. Before the matrix multiplication operation, each error matrix series is truncated, retaining only those up to a specific order. In this embodiment, only the first-order error matrix is ​​retained, and then the error matrix multiplication operation is performed.

[0117] M=M A ·M B ≈I+ω1+ω2+ω1·ω2

[0118] In complex scenarios such as strongly nonlinear interaction errors, discarded high-order terms can significantly affect system behavior. Existing methods assume that low-order terms are sufficient to capture the primary interactions between error components. However, in complex multi-source errors, low-order terms may not fully characterize the underlying interactions between error terms. To dynamically determine whether to retain the interaction term ω1·ω2 and its correction, a truncation decision is made based on the sensitivity of the error components to the overall error, yielding:

[0119]

[0120] Where: M is the matrix M A and M B The matrix obtained by multiplication operation; M A and M Bare obtained by Taylor series expansion of error matrices A and B respectively; I is the identity matrix; ω1 and ω2 are first-order error matrices; ω1·ω2 is the interaction term; S AB is the sensitivity coefficient of the interaction term ω1·ω2, which is used to characterize the contribution of the interaction term to the overall error; ε is the preset threshold used to determine whether to retain the interaction term. And:

[0121]

[0122] Where: E is the error vector.

[0123] The method of using high-order correction terms to approximate nonlinear error terms is:

[0124] When S AB When ≥ε, it indicates that the interaction term ω1·ω2 has a significant effect and should be retained. The original model replaces the high-order term and Completely discarded, this embodiment will be the high-order terms and A nonlinear function is used to approximate it, retaining key features while maintaining computational complexity. We obtain:

[0125]

[0126] The optimized model expression is:

[0127] M=M A ·M B ≈I+ω1+ω2+ω1·ω2+αω1·ω2+βω1·ω2

[0128] After combining like terms, we get:

[0129] M=M A ·M B ≈I+ω1+ω2+(1+α+β)ω1·ω2

[0130] Where: α and β are the nonlinear interference strengths during multi-rotation coupling.

[0131] The coefficients of α and β are determined by least squares fitting:

[0132]

[0133] Where: E i measured is the measured value of the i-th sample point; E i predicted is the predicted value of the i-th sample point; N is the total number of samples.

[0134] In order to reduce the interaction effect between error components (such as ω1 and ω2), the error matrix is ​​orthogonalized. The error matrix expression is:

[0135] E=[ω1,ω2,ω1·ω2] T

[0136] Using the Gram-Schmidt vector orthogonalization method, first retain ω1:

[0137] v1=ω1

[0138] Then ω2 is orthogonalized:

[0139]

[0140] Finally, orthogonalize ω1ω2:

[0141]

[0142] The orthogonalized error model is:

[0143] E oth =[v1,v2,v3]

[0144] The final error model becomes:

[0145] E=w T E oth

[0146] Where: w = [w1, w2, w3] represents the weight vector, which can be dynamically adjusted based on sensitivity analysis.

[0147] The multi-source error model of the face gear worm grinding wheel dressing process is constructed as follows:

[0148] M=M A ·M B ≈I+ω1+ω2+γω1·ω2

[0149]

[0150] Where: γ is the error coupling weight coefficient.

[0151] Computation time comparison Figure 3As shown in the figure, the traditional truncation function method takes only 15 seconds at its fastest because it omits all high-order term calculations. This model takes 25 seconds, which increases the computational time by 66.7%, but achieves a balance between efficiency and accuracy by introducing nonlinear corrections and a dynamic truncation strategy. In comparison, the traditional HCT method takes 482 seconds, 19.3 times longer than this model, due to the retention of high-order terms and complex calculations. Although the multi-source error model based on the traditional truncation function method is the most efficient, this model achieves a better balance between computational cost and modeling accuracy, making it more suitable for practical engineering applications.

[0152] In order to verify the validity of the multi-source error and worm grinding wheel tooth surface error mapping model, the geometric error e15(ε zA ) as an example to analyze its effect on δ yf and ε xf The specific impact of the pose error component. Let e15(ε zA ) has an average value of 0.0076 radians, and the remaining error terms are zero. Figure 4 As shown in (a), e15(ε zA ) caused by the pose error component δ yf The distribution is in a "saddle shape", and the error decreases from the middle of the grinding wheel to the end faces on both sides. Figure 4 (b) shows e15(ε zA ) for the right-handed worm grinding wheel tooth surface posture error component ε xF The influence of the tooth height changes approximately linearly, with the maximum value at the tooth root reaching -9.646×10 -4 millimeters, gradually increasing towards the tooth top, and the overall error magnitude is relatively large.

[0153] Step 2: Input the output of the multi-source error model into a sensitivity analysis model based on a generative adversarial network and a gradient boosting machine, wherein: the generative adversarial network generates simulated data consistent with the measured error distribution; the gradient boosting machine calculates the sensitivity coefficient of the error term based on the simulated data; when the sensitivity coefficient exceeds a threshold, it is marked as a critical error term.

[0154] 2.1 GAN and GBM Fusion Method

[0155] Sensitivity analysis is performed by combining GAN with GBM: the generator (G) generates simulated data that conforms to the actual error distribution from the random noise vector z, which is used for subsequent sensitivity analysis to simulate the impact of various error sources. The optimization objective function of the generator is:

[0156]

[0157] Where: p z (z) is the random noise vector distribution; G(z) is the generated error data; D(G(z)) represents the discriminator's probability estimate of the authenticity of G(z).

[0158] The loss function of the discriminator (D) is defined as:

[0159]

[0160] Where: p data (x) is the true data distribution; D(x) represents the probability that x is true; D(G(z)) represents the probability that G(z) is true.

[0161] The loss function of the Gradient Boosting Machine (GBM) is based on minimizing the residual between the predicted value and the true value. The most commonly used loss function in regression tasks is the mean square error (MSE). The loss function for each iteration is:

[0162]

[0163] Where: y i and f(x i ) represent the true value and the predicted value respectively; n is the total number of samples.

[0164] The total sensitivity of all error components is:

[0165]

[0166] Where: S j is the total sensitivity of the error component; e j is the jth error term.

[0167] 2.2 Recognition Results

[0168] The sampling matrix is ​​generated in the range of [0,1]. According to the measurement results of geometric error and multi-source error, the range of translation error and thermal expansion error is set to [0,50]μm, and the range of rotation error, thermal tilt and thermal pitch angle error is set to [0,0.03] degrees. Figure 5 As shown, in this embodiment, the traditional Sobol method is used, and errors with a sensitivity coefficient exceeding 0.05 are determined as critical errors, and the motion axes containing these critical errors are considered to be critical axes. For the traditional Sobol method:

[0169] Worm grinding wheel tooth surface posture error component δ x The key error term is e5(ε yz )、e9(T yZ )、e15(ε zA )、e24(ε zγ )、e26(T zγ ) and e29(T zB ), the sensitive parts are A-axis, Z-axis, Y-axis and B-axis;

[0170] Worm grinding wheel tooth surface posture error component δ yThe key error term is e4(ε xZ )、e8(T xZ )、e13(ε xA )、e22(ε xY )、e27(T xY )and The sensitive components are the Z axis, A axis, Y axis and B axis;

[0171] Worm grinding wheel tooth surface posture error component δ z The key error terms include e4(ε xZ )、e5(ε yZ )、e8(T x )、e9(T y )、e13(ε xA )、e12(ε xA ) and e23(ε yY ), the sensitive parts are Z axis, Y axis and A axis;

[0172] Rotation error component ε x The key error term is e8(T xZ )、e14(ε yA )、e15(ε zA )、e22(ε xY )、e23(ε yY )、e29(T zB ) and e33(T xB2 ), sensitive components involve Z axis, A axis, Y axis and B2 axis;

[0173] Rotation error component ε y The key error term is e4(ε xZ )、e8(T * )、e13(ε xA )、e22(ε xY )、e27(T xY ) and e33(T * ), corresponding to the Y axis and the B2 axis;

[0174] Rotation error component ε z The key error terms include e6(ε zZ )、e8(T xZ )、e15(ε zA )、e22(ε XY )、e24(ε zY )、e30(T′ xB )、e32(T′ zB2 ) and e33(T′ xB2 ), the sensitive parts cover A-axis, Z-axis, Y-axis, B-axis and B2-axis.

[0175] like Figure 5 As shown in Figure 2, the proposed model is also used to perform sensitivity analysis.

[0176] Worm grinding wheel tooth surface posture error component δ x The key error term is e9(T yZ )、e15(ε zA )、e24(ε zY )、e26(T zγ ) and e29(T zB ), the sensitive components are A-axis, Z-axis, Y-axis and B-axis.

[0177] Worm grinding wheel tooth surface posture error component δ y The key error term is e4(ε xZ )、e8(T xZ )、e13(ε xA )、e27(T xY )and Sensitive components involve Z, Y, A, and B axes;

[0178] Worm grinding wheel tooth surface posture error component δ z The key error term is e4(ε xZ )、e5(ε yZ )、e8(T xZ )、e9(T yZ ) and e13(ε x4 ), the sensitive components are the Z axis and the A axis.

[0179] Rotation error component ε x The key error term contains e8(T xX )、e14(ε yA )、e22(ε xY )、e29(T zB ) and e33(T xB2 ),

[0180] Sensitive components cover A, Z, Y, and B2 axes;

[0181] Rotation error component ε y The key error term is e8(T xZ )、e22(ε xY )、e27(T xY ) and e33(T xB2 ), the sensitive components include Z, Y, and B2 axes;

[0182] Rotation error component ε z The key error term is e6(ε zZ )、e8(T xZ )、e15(ε zA )、e24(εzY )、e30(T xB )、e32(T zB ) and e33(T xB2 ), the sensitive components involve Z, A, Y, B and B2 axes.

[0183] Analysis shows that the errors in the Z, Y, A, B and B2 axes have the most significant impact on the worm gear tooth surface accuracy, and the error compensation strategy should give priority to these axes.

[0184] The key error terms and sensitive axis series of the dressing process are shown in Table 3. The key error terms of the worm grinding wheel dressing process obtained by the proposed method include e4(ε xZ )、e5(ε yZ )、e6(ε zZ )、e8(T xZ )、e9(T yZ )、e13(ε xA )、e14(ε yA )、e15(ε zA )、e22(ε xY )、e24(ε zY )、e26(T zY )、e27(T xY )、e28(T 5 ) yB )、e29(T zB )、e30(T xB )、e32(T zB2 ) and e33(T xB ), a total of 17 items, the sensitive axes are Z axis, Y axis, A axis, B axis and B2 axis.

[0185] The key error term of worm gear dressing obtained by Sobol method is: e4(ε xZ )、e5(ε yZ )、e6(ε zZ )、e8(T xZ )、e9(T yZ )、e13(ε xA )、e14(ε yA )、e15(ε zA )、e22(ε xY )、e23(ε yY 0, e24(ε zY )、e26(T zY )、e27(T xY ), e29(T zB )、e30(T xB )、e32(T zB ) and e33(TxB2 ), a total of 18 items, the sensitive axis system is the same.

[0186] Table 3 Key errors and key axes of the trimming process

[0187]

[0188]

[0189] Due to the lack of effective measurement methods, including e8(T xZ )、,e9(T yZ ) and e27(T xY ) cannot be directly measured and will be treated as zero in subsequent research. By integrating a generative adversarial network (GAN) with a gradient boosting machine (GBM), the impact of machine tool errors (such as geometric and thermal errors) on sensitivity analysis models can be effectively simulated and analyzed. The GAN is used to generate high-dimensional, complex error data, while the GBM is responsible for fitting and optimizing the sensitivity analysis model and calculating the sensitivity of the error sources. This innovative sensitivity analysis method can more accurately identify key error sources, providing strong support for machine tool error compensation. Compared with traditional methods (such as the Sobol method), the GAN-GBM combination demonstrates significant advantages in handling high-dimensional, nonlinear, multi-source errors and error interactions. It is particularly suitable for scenarios with dynamic error source changes, nonlinear correlations, global sensitivity analysis, and massive data processing. The Sobol method is suitable for scenarios with simple error modeling, especially when the error sources exhibit linear or low-order interactions. However, its performance and computational efficiency significantly decline when analyzing complex, multi-source system errors. Therefore, the GAN-GBM combination offers superior flexibility, accuracy, and adaptability in modeling complex, dynamic, and high-dimensional errors, and overall outperforms the traditional Sobol method.

[0190] Step 3: Based on the key error terms identified, the error compensation model is input, the compensation components are calculated through the error decoupling module, and compensation instructions for the motion axis are generated to achieve error compensation.

[0191] 3.1 Error Decoupling

[0192] The expression is:

[0193]

[0194] This results in the following relationship:

[0195]

[0196] The error matrix can be expressed as:

[0197]

[0198] The matrix elements are shown in Appendix C.

[0199] 3.2 Error Compensation Model

[0200] The differential motion error is expressed as:

[0201]

[0202] Therefore, the differential motion error E of the dressing wheel is:

[0203]

[0204] Here J M represents the Jacobian matrix of the worm wheel grinding process, and J M The calculation formula is:

[0205]

[0206] The compensation components are:

[0207] [δ Y ,δ Z ,ε A ,ε B ,ε C2 ] T =(J M T ·J M ) -1 ·J M T ·E (36)

[0208] Assume E = [δ xx ,δ yx ,δ zx ,ε xx ,ε yx ,ε zx ] T , where δ xx ,δ yx ,δ zx ,ε xx ,ε yx and ε zx is the comprehensive error component of the dressing wheel in its own coordinate system, the compensation component can be obtained:

[0209] ε A =ε Xx cosλ+ε Yx sinλ

[0210]

[0211] δ Y =[y(εA sinλsinα-e C2 cosλcosα)-δ Xx -e Yx (z+ysinα)-yε Zx cosλcosα] / (sinλcosα)

[0212] d Z =d Zx +y(e Zx cosλcosα+ε Yx sinλcosα-ε A cosα)-δ Yx Sinai

[0213] e B =e A sinαsinb

[0214] -[e χx (sinλcosb+cosλsinαsinb)-ε χx (cosλcosb-sinλsinasinb)] / (cosαcosb)

[0215] Y axis、Z axis、A axis、B axis、C2 axis's compensation fraction is

[0216] e A =-(η x cosλ+η y sinλ)

[0217] e C2 =[(-η x cosλ-η y sinλ)sinλcosα+η y sinα+η z cosλcosα] / (cosλcosα)

[0218] d Y =[y(e A sinλsinα-e C2 cosλcosα)+p x +n y (z+ysinα)-yη z cosλcosα] / (sinλcosα)

[0219] d Z =-p z +y(-η z cosλcosα-η ysinλcosα-ε A cosα)+p y sinα

[0220] ε B =ε A sinαsinb

[0221] -[-η x (sinλcosb+cosλsinαsinb)+η y (cosλcosb-sinλsinαsinb)] / (cosαcosb)

[0222] 3.3 Error Compensation Principle

[0223] Error compensation principle Figure 1 As shown in the figure, the control system boasts powerful multi-axis motion control capabilities, supports a wide range of related software, and offers comprehensive functionality. Its comprehensive self-protection features facilitate operator convenience. A multi-source error model for the worm wheel dressing process has been embedded in the compensation system.

[0224] After the key geometric error compensation table is imported into the worm grinding wheel dressing process (WGWDP) machining program, the predicted thermal error compensation value is loaded into the temperature compensation module before the dressing operation begins. This embodiment conducts three compensation verifications: the first one does not use any error compensation; the second one is based on the proposed multi-source error modeling and sensitivity analysis for compensation; the third one is based on the traditional multi-source error modeling and Sobol method for compensation. The compensation results of the face gear machined with the dressed worm grinding wheel are shown in the figure below. Figure 7 As shown in the figure, the proposed error compensation strategy significantly reduces tooth surface errors, reducing the maximum normal errors of the left and right gear tooth surfaces from 22.4μm to 12.9μm and from 28.7μm to 11.4μm, respectively. In comparison, the traditional compensation method only reduced them to 17.6μm and 14.8μm, respectively. The compensation effect of this embodiment is significantly better than that of traditional methods, providing a practical solution for precision machining applications.

[0225] in conclusion

[0226] This embodiment implements thermal error compensation in face gear worm wheel dressing based on novel multi-source error modeling and sensitivity analysis. The improved truncation function method proposed in this embodiment is used for multi-source error modeling and compensation in the worm wheel dressing process. This method combines sensitivity-based dynamic truncation, high-order correction terms, and vector orthogonalization to overcome the limitations of traditional methods. Furthermore, a sensitivity analysis model was established based on a generative adversarial network and a gradient boosting machine. Experimental verification shows that the tooth surface error can be significantly reduced while maintaining computational efficiency. The main conclusions are as follows:

[0227] (1) An efficient multi-source error modeling method is proposed based on the improved truncation function method. Compared with the 15 seconds required by the traditional truncation function method and the 482 seconds required by the traditional homogeneous coordinate transformation method, the new model only takes 25 seconds to achieve a better balance between computational cost and modeling accuracy.

[0228] (2) A high-precision sensitivity analysis model was constructed based on a generative adversarial network and a gradient boosting mechanism, which reduced the number of sensitivity error terms from 18 to 17, improving computational efficiency and accuracy.

[0229] (3) The proposed error compensation strategy significantly reduces the tooth surface error. The maximum normal errors of the left and right gear tooth surfaces are reduced from 22.4 μm to 12.9 μm and 28.7 μm to 11.4 μm, respectively, while the traditional compensation method only reduces them to 17.6 μm and 14.8 μm, respectively.

[0230] These results indicate that the method proposed in this embodiment achieves a practical balance between accuracy and efficiency, and provides an optimization solution for high-precision machining applications.

[0231] Appendix A

[0232] Table A1 Ideal HCT matrix and actual HCT matrix

[0233]

[0234]

[0235] Where: x, y and z represent the movement distance of X, Y and Z axes respectively; a and b are the rotation angles of A and B axes respectively.

[0236] Appendix B

[0237]

[0238]

[0239] Appendix C

[0240]

[0241]

[0242] The above embodiments are merely preferred embodiments for the purpose of fully illustrating 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 within the scope of protection of the present invention. The scope of protection of the present invention shall be subject to the claims.

Claims

1. A method for compensating face gear worm grinding wheel dressing errors based on multi-source error modeling and sensitivity analysis, characterized by: The steps include: Step 1: Based on the forward kinematics matrix of the face gear worm wheel dressing process, a worm wheel tooth surface error model is constructed. A sensitivity-based dynamic truncation strategy is used to dynamically determine whether to retain the error interaction terms, a high-order correction term is used to approximate the nonlinear error terms, and a vector orthogonalization method is used to process the tooth surface error model. Finally, a multi-source error model of the face gear worm wheel dressing process is constructed. Step 2: Input the output of the multi-source error model into a sensitivity analysis model based on a generative adversarial network and a gradient boosting machine, wherein: the generative adversarial network generates simulated data consistent with the measured error distribution; the gradient boosting machine calculates the sensitivity coefficient of the error term based on the simulated data; when the sensitivity coefficient exceeds a threshold, it is marked as a critical error term; Step 3: Based on the key error terms identified, the error compensation model is input, the compensation components are calculated through the error decoupling module, and compensation instructions for the motion axis are generated to achieve error compensation.

2. The method for compensating for face gear worm grinding wheel dressing errors based on multi-source error modeling and sensitivity analysis according to claim 1, characterized in that: In step 1, the ideal forward kinematics matrix of the face gear worm grinding wheel dressing process is: in: and They represent the ideal HCT matrices from X-axis to B2-axis, B2-axis to C2-axis, C2-axis to Z2-axis, Z2-axis to F-axis, F-axis to X-axis, X-axis to Z-axis, Z-axis to A-axis, A-axis to Y-axis, Y-axis to B-axis, and B-axis to W-axis respectively; The actual forward kinematics matrix of the face gear worm grinding wheel dressing process is: in: and represent the actual HCT matrices from X-axis to B2-axis, B2-axis to C2-axis, C2-axis to Z2-axis, Z2-axis to F-axis, F-axis to X-axis, X-axis to Z-axis, Z-axis to A-axis, A-axis to Y-axis, Y-axis to B-axis, and B-axis to W-axis, respectively, and 3. The method for compensating for face gear worm grinding wheel dressing errors based on multi-source error modeling and sensitivity analysis according to claim 2, characterized in that: The worm grinding wheel tooth surface error model is expressed as: Where: w is the position error matrix; δ x , δ y and δ z Respectively represent the position error components; ε w is the angle error matrix; ε x , ε y and ε z Represent the angular error components respectively; and are the actual position vector and the ideal position vector respectively; and are the actual direction vector and the ideal direction vector respectively; is the initial angular position or reference angular position of the worm; θ x is the rotation angle along the X axis; E is the error vector; and: in: is the ideal forward kinematics matrix of the face gear worm grinding wheel dressing process; r x (θ x ) and n x (θ x ) are the position vector and the normal vector of a point on the worm tooth surface respectively; is the ideal relative velocity at the contact point; in: is the actual forward kinematics matrix of the face gear worm grinding wheel dressing process; r x (θ x ,E) and n x (θ x ,E) are the position vector of a point on the worm tooth surface and the normal vector of a point on the worm tooth surface respectively; is the actual relative velocity at the contact point.

4. The method for compensating face gear worm grinding wheel dressing errors based on multi-source error modeling and sensitivity analysis according to claim 1, characterized in that: The dynamic truncation strategy based on sensitivity is: Where: M is the matrix M A and M B The matrix obtained by multiplication operation; M A and M B are obtained by Taylor series expansion of error matrices A and B respectively; I is the identity matrix; ω1 and ω2 are first-order error matrices; ω1·ω2 is the interaction term; S AB is the sensitivity coefficient; ε is the preset threshold; and: Where: E is the error vector; The method of using high-order correction terms to approximate nonlinear error terms is: When S AB When ≥ε, it indicates that the interaction term ω1·ω2 has a significant effect and should be retained, and the high-order term and Using nonlinear functions to approximate it, we get: M=M A ·M B ≈I+ω1+ω2+(1+α+β)ω1·ω2 Where: α and β represent the nonlinear interference strength during multi-rotation coupling; The coefficients of α and β are determined by least squares fitting: Where: E i measured is the measured value of the i-th sample point; E i predicted is the predicted value of the i-th sample point; N is the total number of samples; For the error matrix E = [ω1,ω2,ω1·ω2] T Implement vector orthogonalization processing and use the Gram-Schmidt vector orthogonalization method to obtain the orthogonalized error model: E oth =[v1,v2,v3] v1=ω1 The multi-source error model of the face gear worm grinding wheel dressing process is constructed as follows: M=M A ·M B ≈I+ω1+ω2+γω1·ω2 Where: γ is the error coupling weight coefficient.

5. The method for compensating face gear worm grinding wheel dressing errors based on multi-source error modeling and sensitivity analysis according to claim 1, characterized in that: In step 2, the principle of the sensitivity analysis model is: The generator generates simulation data that conforms to the actual error distribution from the random noise vector z. The optimization objective function of the generator is: Where: p z (z) is the random noise vector distribution; G(z) is the generated error data; D(G(z)) represents the discriminator’s probability estimate of the authenticity of G(z); The loss function of the discriminator is defined as: Where: p data (x) is the true data distribution; D(x) represents the probability that x is true; D(G(z)) represents the probability that G(z) is true; The loss function of the gradient boosting machine is based on minimizing the residual between the predicted value and the true value. The loss function of each iteration is: Where: y i and f(x i ) represent the true value and the predicted value respectively; n is the total number of samples; The total sensitivity of all error components is: Where: S j is the total sensitivity of the error component; e j is the jth error term.

6. The method for compensating face gear worm grinding wheel dressing errors based on multi-source error modeling and sensitivity analysis according to claim 1, characterized in that: In step 3, the error compensation is achieved by error decoupling, and the error compensation component is: [d Y ,d Z ,he A ,he B ,he C2 ] T =(J M T ·J M ) -1 ·J M T ·E Where: E is the differential motion error; J M is the Jacobian matrix; δ y and δ z Respectively represent the position error components; ε w is the angle error matrix; ε A , ε B and ε C2 Indicates the angle error compensation amount.

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