Face gear worm grinding machine error compensation method based on sensitivity analysis and multi-source error mapping model

Abstract

The invention discloses a face gear worm grinding machine error compensation method based on sensitivity analysis and a multi-source error mapping model, which comprises the following steps of: 1, constructing a multi-source error-tooth surface error mapping model based on a vector decomposition and truncation function method, the truncation function method is used for quantitatively describing propagation mechanisms of geometric errors and thermal errors in a machining chain and influences of the propagation mechanisms on tooth surface deviation, and high-order error terms are eliminated to improve calculation efficiency; 2, responding to the output of the mapping model, applying an improved Morris sensitivity analysis method, distributing a weighting coefficient higher than a geometric error component for a thermal error component, and calculating a weighting sensitivity coefficient so as to identify a key error component which has the greatest influence on the tooth surface precision; and 3, based on the key error component, a motion axis compensation value is generated through an error compensation model, and the compensation value is input into a machine tool control system to adjust the relative position of the worm grinding wheel and the workpiece, so that the tooth surface error is reduced.

Description

Technical Field

[0001] The present invention belongs to the technical field of gear grinding, and specifically relates to an error compensation method for a face gear worm grinding machine based on sensitivity analysis and a multi-source error mapping model. Background Art

[0002] Machine tool machining accuracy has become a critical factor in high-end equipment manufacturing. Face gear worm grinding machines (FGWGMTs) are widely used to machine face gears. Tooth surface geometric accuracy directly impacts gear meshing performance, load capacity, and system stability. However, in actual machining, multi-source errors (MSEs), including complex geometric errors and thermal errors (TEs), severely degrade tooth surface accuracy, posing a significant challenge to high-precision manufacturing. Therefore, effective control of multi-source errors in FGWGMTs is urgently needed.

[0003] Significant progress has been made in the theoretical modeling and practical application of multi-source error control. On the one hand, geometric error models based on homogeneous coordinate transformation (HCT) and multi-body system (MBS) theory have been developed to achieve systematic modeling and compensation in multi-axis systems. However, these approaches often generate high-order error (HOE) terms, resulting in computational complexity and poor real-time performance. Recent research has proposed vector decomposition and error truncation techniques to reduce the impact of these high-order terms, but computational efficiency in practical scenarios still needs to be improved. On the other hand, thermal errors are primarily caused by the heat generated by machine tool components during operation. Physical models (such as finite element analysis) and data-driven methods (such as regression models and artificial neural networks) are used to predict and compensate for thermal errors. However, these models are system-specific and require extensive experimental data for calibration. Furthermore, the nonlinear and time-varying nature of thermal errors limits their universal applicability. Understanding the transmission mechanisms of multi-source errors within the machining chain is crucial for designing effective compensation strategies. Previous studies have established error transmission models based on multi-body kinematics, revealing the cumulative impact of geometric and thermal errors on machining accuracy. However, the relative importance of different error components has often been overlooked. Sensitivity analysis methods such as Morris and Sobol are used to identify key error sources, but these methods are not yet fully applied to FGWGMT with high-dimensional errors. The combination of multi-source error mapping models and sensitivity analysis is expected to provide new ideas for error compensation, but the following key challenges still exist:

[0004] (1) Tooth surface errors have different sensitivities to geometric errors and thermal errors, but traditional compensation methods often treat all error components equally, resulting in low efficiency in error identification and compensation, and a lack of methods for systematically identifying and prioritizing key error sources.

[0005] (2) Currently, there are no reports on FGWGMT error compensation based on multi-source error mapping (MSE) models and sensitivity analysis. Traditional error compensation methods rely heavily on high-order error (HOE) modeling, which predicts and compensates for machining deviations by mathematically describing geometric errors and thermal deformation errors (TEs). Existing error models often introduce high-order terms, which complicates the calculation and reduces real-time applicability, especially for FGWGMTs with numerous error sources. The addition of HOE terms increases the computational complexity and reduces modeling efficiency.

[0006] (3) Insufficient knowledge of the mapping relationship between tooth surface errors and multi-source errors restricts the effectiveness of compensation strategies. The multi-source and nonlinear characteristics of error propagation make it difficult to identify key error components, thus limiting the effectiveness of targeted error compensation. Summary of the Invention

[0007] In view of this, the purpose of the present invention is to provide an error compensation method for face gear worm grinder based on sensitivity analysis and multi-source error mapping model, which can effectively reduce tooth surface error and computational complexity, and ensure accurate priority sorting of error components to achieve targeted compensation.

[0008] In order to achieve the above object, the present invention provides the following technical solutions:

[0009] An error compensation method for a face gear worm grinding machine based on sensitivity analysis and a multi-source error mapping model comprises the following steps:

[0010] Step 1: Based on vector decomposition and the truncation function method, a multi-source error-to-tooth surface error mapping model is constructed to quantitatively describe the propagation mechanism of geometric and thermal errors in the machining chain and their impact on tooth surface deviation. The truncation function method eliminates high-order error terms to improve computational efficiency.

[0011] Step 2: In response to the output of the mapping model, applying the improved Morris sensitivity analysis method to calculate the weighted sensitivity coefficient by assigning a higher weight coefficient to the thermal error component than to the geometric error component, so as to identify the key error component that has the greatest impact on the tooth surface accuracy;

[0012] Step 3: Based on the key error components, a motion axis compensation value is generated through an error compensation model, and the compensation value is input into the machine tool control system to adjust the relative position of the worm grinding wheel and the workpiece, thereby reducing the tooth surface error.

[0013] Furthermore, in step 1, based on the grinding kinematic chain of the face gear worm grinder, the ideal HCT matrix of the worm grinding wheel coordinate system WCS relative to the global coordinate system GCS is obtained:

[0014]

[0015] in: represents the ideal HCT matrix of WCS relative to GCS; Represents the ideal HCT matrix of the B-axis coordinate system BCS relative to the Y-axis coordinate system YCS; Represents the ideal HCT matrix of the Y-axis coordinate system YCS relative to the A-axis coordinate system ACS; Represents the ideal HCT matrix of the A-axis coordinate system ACS relative to the Z-axis coordinate system ZCS; Represents the ideal HCT matrix of the Z-axis coordinate system ZCS relative to the X-axis coordinate system XCS; Represents the ideal HCT matrix of the X-axis coordinate system XCS relative to the bed coordinate system RCS; Represents the ideal HCT matrix of the bed coordinate system RCS relative to the C-axis coordinate system;

[0016] Get the actual HCT matrix of the worm grinding wheel coordinate system WCS relative to the global coordinate system GCS:

[0017]

[0018] in: Represents the actual HCT matrix of WCS relative to GCS; Represents the actual HCT matrix of the B-axis coordinate system BCS relative to the Y-axis coordinate system YCS; Represents the actual HCT matrix of the Y-axis coordinate system YCS relative to the A-axis coordinate system ACS; Represents the actual HCT matrix of the A-axis coordinate system ACS relative to the Z-axis coordinate system ZCS; Represents the actual HCT matrix of the Z-axis coordinate system ZCS relative to the X-axis coordinate system XCS; Represents the actual HCT matrix of the X-axis coordinate system XCS relative to the bed coordinate system RCS; Represents the actual HCT matrix of the bed coordinate system RCS relative to the C-axis coordinate system; E BY(G) and E BY(T) are the HCT matrices of the position-dependent geometric error PDGEs and thermal error TEs of the B-axis coordinate system BCS relative to the Y-axis coordinate system YCS; E YA(G) is the HCT matrix of PDGEs of YCS relative to ACS; E AZ(G) is the HCT matrix of PDGEs in ACS relative to ZCS; E ZX(G) is the HCT matrix of PDGEs of ZCS relative to XCS; E XR(G) is the HCT matrix of PDGEs of XCS relative to RCS; E CR(G) and E CR(T) are the HCT matrices of the position-dependent geometric errors PDGEs and thermal errors TEs of CCS relative to RCS, respectively;

[0019] The mapping relationship between the position error and the angle error of the worm grinding wheel in the multi-source error-tooth surface error mapping model is obtained:

[0020]

[0021] Where: iw is the position error of the worm grinding wheel; εδ xw , δ yw and δ zw are the position errors of the worm grinding wheel along the X, Y and Z axes respectively; [x, y, z, 1] T is the theoretical position of the worm grinding wheel in the worm grinding wheel coordinate system WCS;

[0022] ε iw is the angular error of the worm grinding wheel; ε xw , ε yw and ε zw are the angular errors of the worm grinding wheel along the X, Y and Z axes respectively; [α, β, γ, 1] T It is the theoretical angle of the worm grinding wheel in the worm grinding wheel coordinate system WCS.

[0023] Furthermore, in step 1, the vector decomposition method separates the pose offset caused by geometric error and the additional pose offset caused by thermal error. The principle is:

[0024]

[0025] in: is the actual HCT matrix of object j relative to object i; is the ideal HCT matrix of object j relative to object i; E IJ(G) is the HCT matrix of the position-dependent geometric error PDGEs of object j relative to object i; E IJ(T) is the HCT matrix of the thermal error TEs of object j relative to object i.

[0026] Furthermore, in step 1, the truncation function method retains the first-order error term and discards the higher-order error term. The principle is:

[0027] M=M1·M2=(I+ω1+(ω1) 2 +Δ((ω1) 2 ))·(I+ω2+(ω2) 2 +Δ((ω2) 2 ))

[0028] ≈I+ω1+ω2+ω1·ω2

[0029] Where: M is the matrix obtained by multiplying the error matrices M1 and M2; 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 ) is a high-order error matrix.

[0030] Furthermore, the multi-source error-tooth surface error mapping model is expressed as:

[0031]

[0032] Where: g and ε g are the position error and angle error of the face gear tooth surface respectively; δ xg , δ yg and δ zg are the position errors of the face gear tooth surface along the X, Y and Z axes respectively; ε xg , ε yg and ε zg are the angular errors of the face gear tooth surface along the X, Y and Z axes respectively; and are the actual unit normal vectors of the global coordinate system GCS; and are the ideal unit normal vectors of the global coordinate system GCS respectively.

[0033] Furthermore, in step 2, the improved Morris sensitivity analysis method includes the following steps:

[0034] 21) Given the worm wheel parameters, face gear parameters, machine tool structure and machining motion configuration, calculate the theoretical and actual HCT matrices and all multi-source error MSE components from the global coordinate system GCS to the face gear coordinate system FCS, and use the Morris method to construct the random input matrix E * ;

[0035] 22) From the preset range [0, Δ i ,2Δ i ,…,1] randomly and uniformly sample the multi-source error MSE and construct the benchmark error vector X as the starting point of sensitivity analysis * ; Set the random permutation matrix P * Disrupt the order of error components to ensure random sampling, and use X * and P * Establish a random sampling matrix S representing the perturbation parameter set * ;

[0036] 23) S *Each row of the matrix is ​​substituted into the actual HCT matrix to calculate the gear tooth surface error component; for each error component, the basic effect is calculated based on the difference between the input parameter change and the output result; the corresponding weight is assigned to each error, and the basic effect EE is calculated. i (x i ) is updated to the weighted basic effect WEE i (x i );

[0037]

[0038] Where: F(·) is the multi-source error-tooth surface error mapping model; (x 1g ,x 2g ,…,x mg ,x It ,x 2t ,…,x nt ) are the geometric error and thermal error generated during the machining process, subscript g represents the geometric error, and subscript t represents the thermal error; σ 1g , σ i and σ nt The weighting coefficients assigned to specific error components;

[0039] 24) Loop through steps 22)-23) until the number of iterations reaches the set maximum number of iterations; after calculating the weighted basic effect for each iteration, calculate the average weighted basic effect of all iteration results, and finally calculate the weighted sensitivity coefficient wμ of each error parameter i and wσ i ;

[0040]

[0041] Where: wμ i and wσ i are the weighted mean and standard deviation respectively; μ i and σ i are the mean and standard deviation respectively; N is the maximum number of iterations;

[0042] 25) Compare the weighted sensitivity index of each error parameter to identify the sensitive error sources that significantly affect each tooth surface error component.

[0043] Furthermore, in step 3, the error compensation model is solved based on the Jacobian matrix, specifically including:

[0044] 31) defining the overall differential motion error E according to the differential motion error matrix;

[0045] 32) Define the Jacobian matrix J M Solve the axis error compensation component to satisfy the equation:

[0046] E=JM ·[δ X ,δ Y ,δ Z ,ε B ,ε C ] T

[0047] Where: x , ε y , ε z , δ x , δ y and δ z They represent the translation error component and angular error component of the worm grinding wheel respectively.

[0048] The beneficial effects of the present invention are:

[0049] Geometric and thermal errors, which significantly affect the machining accuracy of face gear worm grinders, are extremely difficult to compensate. A key challenge in ensuring machining accuracy lies in the lack of effective methods for identifying and compensating critical errors. Current geometric and thermal error compensation methods often rely on high-order error modeling, which is computationally expensive and lacks practicality in real-time scenarios. Furthermore, traditional compensation methods are limited by their inability to effectively track the most critical error components. To this end, the present invention proposes an error compensation method for face gear worm grinders based on sensitivity analysis and a multi-source error mapping model. For the first time, an innovative error compensation method is designed based on an efficient multi-source error-to-tooth surface error mapping model and an improved sensitivity analysis. This mapping model quantitatively describes the transmission mechanism of geometric and thermal errors in the machining chain and their impact on tooth surface deviations. An efficient multi-source error-to-tooth surface error mapping model is constructed using a vector decomposition method and a truncation function. Accurate and efficient compensation is achieved using an improved Morris sensitivity analysis method, effectively reducing modeling time, lowering the maximum tooth surface error after compensation, and significantly improving machining accuracy. The present invention fills a critical gap in error identification and compensation methods, providing a scalable solution for high-precision machining of face gear worm grinders.

[0050] Specifically, the main technical effects of the error compensation method for face gear worm grinding machine based on sensitivity analysis and multi-source error mapping model of the present invention are as follows:

[0051] (1) Multi-source error compensation is achieved based on the multi-source error-tooth surface error mapping model and improved sensitivity analysis, which can significantly reduce the tooth surface error and effectively improve the machining accuracy.

[0052] (2) A novel and efficient multi-source error mapping model is constructed to quantitatively describe the propagation mechanism of geometric and thermal errors in the machining chain. This model can establish a direct correlation between tooth surface errors and multi-source errors, providing a complete framework for error analysis and compensation. The computational challenges of high-order error terms are addressed through vector decomposition and truncation function methods, improving modeling efficiency while maintaining prediction accuracy.

[0053] (3) An improved Morris sensitivity analysis method is proposed to identify the key geometric and thermal error components that have the most significant impact on tooth surface accuracy. Compared with traditional sensitivity analysis techniques, this method can reduce computational complexity and ensure accurate prioritization of error components for targeted compensation. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] 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:

[0055] Figure 1 It is a forward kinematics model based on the FGWGMT structure;

[0056] Figure 2 is the PDGE distribution of the motion axis;

[0057] Figure 3 Schematic diagram of posture offset;

[0058] Figure 4 Compare the calculation time for the models;

[0059] Figure 5 Because of ε xb(t) The tooth surface error of the end gear caused by

[0060] Figure 6 Because of ε xx(g) The tooth surface error of the end gear caused by

[0061] Figure 7 Sensitive errors identified by the traditional Morris method;

[0062] Figure 8 To improve the sensitive errors identified by the Morris method;

[0063] Figure 9 It is the error compensation principle;

[0064] Figure 10 It is a machining process without compensation error;

[0065] Figure 11 For the measurement results. DETAILED DESCRIPTION

[0066] 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.

[0067] The present embodiment provides an error compensation method for a face gear worm grinding machine based on sensitivity analysis and a multi-source error mapping model, and includes the following steps.

[0068] Step 1: Based on vector decomposition and the truncation function method, a multi-source error-to-tooth surface error mapping model is constructed to quantitatively describe the propagation mechanism of geometric and thermal errors in the machining chain and their impact on tooth surface deviation. The truncation function method eliminates high-order error terms to improve computational efficiency.

[0069] 1.1 Multi-source error analysis

[0070] 1.1.1 Motion Transformation and Forward Kinematics Model

[0071] The transmission of multi-source errors is directly affected by the forward kinematics model, which determines the accumulation and amplification effect of errors. Figure 1 As shown in (a), the grinding motion chain consists of two sub-chains: the workpiece motion chain G (face gear) - C (C axis) - R (bed); the tool motion chain W (worm grinding wheel) - B (B axis) - Y (Y axis) - A (A axis) - Z (Z axis) - X (X axis) - R (bed). The final result is Figure 1 (b) The forward kinematic model WBYAZXRCG is shown.

[0072] Position-dependent geometric errors (PDGE) are errors associated with a specific position and vary as the machine tool moves from one location to another. For rotary axes, PDGE primarily include angular errors, roundness errors, and perpendicularity errors. Position-independent geometric errors (PIGE) are independent of the machine tool's specific location and motion path; these errors do not vary with the machine tool's motion. For rotary axes, PIGE primarily include concentricity and cylindricity errors. This article focuses on PDGE.

[0073] For the X-axis, there are six PDGEs, including δ x(g) , δ y(g) , δ z(g) , ε x(g) , ε y(g) and ε z(g) Ideally, there should be no geometric error in the X-axis, and the theoretical coordinate system is O X i X i Y i Z i However, according to Figure 2 (a) The presence of six PDGEs results in the central position shifting from O X i X i Y i Z i Change to O X a X a Y a Z a .

[0074] For the Y-axis, there are six PDGEs, including δ y(g) , δ y(g) , δ y(g) , ε xy(g) , ε yy(g) and ε xy(g) .

[0075] The Z axis also has six PDGEs, including δ x(g) , δ yz(g) , δ xz(g) , ε xz(g) , ε yz(g) and ε xz(g) .

[0076] The rotation axis also has six PDGEs, including δ xc(g) , δ yc(g) , δ zc(g) , ε xc(g) , ε yc(g) and ε zc(g) .

[0077] like Figure 2 As shown in (b), the theoretical coordinate system of the C axis is However, under the influence of six PDGEs, the central position shifted from becomes Similarly, when the A axis rotates to a specific angle A, six PDGEs are generated, namely δ xa(g) , δ ya(g) , δ za(g) , ε xa(g) , ε ya(g) and ε za(g) When the B axis rotates to a specific angle B, six PDGEs will be generated, including δ sb(g) , δ sb(g) , δ zb(g) , ε sb(g) , ε sb(g) and ε zb(g) In summary, there are 36 PDGEs in total.

[0078] Since the A-axis does not participate in the grinding process, its thermal errors (TEs) are not considered. The tool rotates along the workpiece surface, not along the spindle axis. Grinding forces act primarily on the workpiece surface in directions other than the spindle's rotation. This results in minimal thermal angular positioning errors associated with spindle rotation about its axis. Furthermore, thermal angular positioning errors have a minimal impact on grinding forces and grinding stability and can therefore be ignored in the modeling process.

[0079] 1.1.2 Mapping relationship between worm grinding wheel posture and geometric / thermal error

[0080] The coordinate system sequence is WCS→BCS→YCS→ACS→XCS→XCS→RCS→CCS. WCS can be combined with BCS to form a single coordinate system, and CCS can be combined with GCS to form a single coordinate system. When considering the motion from BCS to YCS, assuming that the kinematic pair is error-free under theoretical conditions, the HCT matrix of the B axis relative to the Y axis is expressed as:

[0081]

[0082] in: Indicates the rotation angle.

[0083] For the B axis, its PDGEs include δ xb(g) , δ b(g) , δ zb(g) , ε zb(g) , ε yb(g) and ε zb(g) TEs include δ xb(t) , δ yb(t) , δ zb(t) , ε zb(t) , ε yb(t) and ε zb(t) The HCT matrix of PDGEs of BCS relative to YCS is:

[0084]

[0085] The HCT matrix of TE of BCS relative to YCS is expressed as:

[0086]

[0087] Finally, the HCT matrix of BCS relative to YCS is obtained:

[0088]

[0089] The theoretical HCT matrix of YCS relative to ACS is:

[0090]

[0091] Where: Y represents the moving distance.

[0092] The geometric error HCT matrix of BCS relative to ACS is:

[0093]

[0094] After considering the geometric error, the actual HCT matrix of YCS relative to ACS is obtained:

[0095]

[0096] The theoretical HCT matrix of ACS relative to ZCS is:

[0097]

[0098] in: Indicates the rotation angle.

[0099]

[0100] After considering the geometric error, the actual HCT matrix of ACS relative to ZCS is obtained:

[0101]

[0102] The theoretical HCT matrix of ZCS relative to XCS is expressed as:

[0103]

[0104] Where: Z represents the moving distance. After considering the geometric error, the actual HCT matrix of ZCS relative to XCS is obtained:

[0105]

[0106] The actual HCT matrix of ZCS relative to XCS considering geometric errors is:

[0107]

[0108] Where: R o and represent the trajectory radius and azimuth angle respectively; α0 represents the static contact angle.

[0109] The theoretical HCT matrix of XCS relative to RCS is expressed as:

[0110]

[0111] After considering the geometric error, the actual HCT matrix of XCS relative to RCS has been obtained:

[0112]

[0113] The actual HCT matrix of XCS relative to RCS is:

[0114]

[0115] The theoretical HCT matrix from CCS to RCS can be expressed as:

[0116]

[0117] The HCT matrix of geometric error is:

[0118]

[0119] The HCT matrix of TE is:

[0120]

[0121] The actual HCT matrix from CCS to RCS is obtained as:

[0122]

[0123] The actual HCT matrix from RCS to CCS has been obtained:

[0124]

[0125] Specifically, based on the grinding kinematic chain of the face gear worm grinder, the ideal HCT matrix of the worm grinding wheel coordinate system WCS relative to the global coordinate system GCS is obtained:

[0126]

[0127] in: represents the ideal HCT matrix of WCS relative to GCS; Represents the ideal HCT matrix of the B-axis coordinate system BCS relative to the Y-axis coordinate system YCS; Represents the ideal HCT matrix of the Y-axis coordinate system YCS relative to the A-axis coordinate system ACS; Represents the ideal HCT matrix of the A-axis coordinate system ACS relative to the Z-axis coordinate system ZCS; Represents the ideal HCT matrix of the Z-axis coordinate system ZCS relative to the X-axis coordinate system XCS; Represents the ideal HCT matrix of the X-axis coordinate system XCS relative to the bed coordinate system RCS; Represents the ideal HCT matrix of the bed coordinate system RCS relative to the C-axis coordinate system.

[0128] Get the actual HCT matrix of the worm grinding wheel coordinate system WCS relative to the global coordinate system GCS:

[0129]

[0130] in: Represents the actual HCT matrix of WCS relative to GCS; Represents the actual HCT matrix of the B-axis coordinate system BCS relative to the Y-axis coordinate system YCS; Represents the actual HCT matrix of the Y-axis coordinate system YCS relative to the A-axis coordinate system ACS; Represents the actual HCT matrix of the A-axis coordinate system ACS relative to the Z-axis coordinate system ZCS; Represents the actual HCT matrix of the Z-axis coordinate system ZCS relative to the X-axis coordinate system XCS; Represents the actual HCT matrix of the X-axis coordinate system XCS relative to the bed coordinate system RCS; Represents the actual HCT matrix of the bed coordinate system RCS relative to the C-axis coordinate system; E BY(G) and E BY(T) are the HCT matrices of the position-dependent geometric error PDGEs and thermal error TEs of the B-axis coordinate system BCS relative to the Y-axis coordinate system YCS; E YA(G) is the HCT matrix of PDGEs of YCS relative to ACS; E AZ(G) is the HCT matrix of PDGEs in ACS relative to ZCS; E ZX(G) is the HCT matrix of PDGEs of ZCS relative to XCS; E xR(G) is the HCT matrix of PDGEs of XCS relative to RCS; E CR(G) and E cR(T) are the HCT matrices of the position-dependent geometric error PDGEs and thermal error TEs of CCS relative to RCS, respectively.

[0131] Using the traditional HCT principle to model will introduce high-order error terms. The actual HCT of BCS relative to YCS is:

[0132]

[0133] The theoretical position and angle of the worm grinding wheel (GWGW) in WCS are expressed as [x, y, z, 1] T and [α,β,γ,1] T The position error and angle error of GWGW are represented by δ iw and ε iw The mapping relationship between the position error and the angle error of the worm grinding wheel in the multi-source error-tooth surface error mapping model is obtained:

[0134]

[0135] Where: iw is the position error of the worm grinding wheel; δ xw , δ yw and δ zw are the position errors of the worm grinding wheel along the X, Y and Z axes respectively; [x, y, z, 1] T is the theoretical position of the worm grinding wheel in the worm grinding wheel coordinate system WCS; ε iw is the angular error of the worm grinding wheel; ε xw , ε yw and ε zw are the angular errors of the worm grinding wheel along the X, Y and Z axes respectively; [α, β, γ, 1]T It is the theoretical angle of the worm grinding wheel in the worm grinding wheel coordinate system WCS.

[0136] 1.1.3. Efficient Multi-Source Error Modeling without High-Order Terms Based on Vector Decomposition and Truncation Function Method

[0137] The complexity introduced by higher-order error terms is not only computationally challenging, but also significantly increases the sensitivity and complexity of the MSE model. Sensitivity analysis of these higher-order terms requires more detailed and time-consuming evaluations, which not only increases the difficulty of designing error compensation strategies but also complicates and costs the verification and implementation of error compensation schemes in practical applications. This embodiment proposes an efficient MSE modeling method without higher-order terms based on vector decomposition and truncation function methods. By addressing the challenges introduced by higher-order terms, the MSE model is simplified, thereby facilitating the design, verification, and application of error compensation strategies.

[0138] Under ideal conditions without errors, the HCT matrix of object j relative to object i can be expressed as:

[0139]

[0140] Among them: x, y and z are the motion of X, Y and Z axes respectively; α, β and γ are the motion of rotation axis respectively.

[0141] Under the error condition, the actual HCT matrix of object j relative to object i is:

[0142]

[0143] Where: Δx g , Δy g , Δz g , Δα g , Δβ g and Δγ g is the geometric error of X, Y, and Z axes; Δx t , Δy t , Δz t , Δα t , Δβ t and Δγ t is the thermal error of X, Y, and Z axes.

[0144] The HCT process under the combined effects of geometric errors and thermal errors (TEs) can be decomposed into two motions: one is the geometric error causing the theoretical coordinate system to shift in position, and the other is the thermal error causing the additional position shift. Figure 3As shown in (a), it can be understood that geometric error first causes an initial offset, and thermal error further deviates from the pose based on the geometric error displacement. Specifically, coordinate system X0Y0Z0O0 is first transformed to coordinate system X1Y1Z1O1 through the theoretical HCT. Then, due to the influence of geometric error, it deviates to coordinate system X2Y2Z2O2. Finally, due to the influence of thermal error, it further deviates to coordinate system X3Y3Z3O3. The vector values ​​and their corresponding HCT matrices are listed in Table 1.

[0145] Table 1 Vector values ​​and their corresponding HCT matrices

[0146]

[0147] Specifically, the mutual relationship is expressed as:

[0148]

[0149] like Figure 3 As shown in (b), After translation, it intersects with point O1 to form The coordinate system transformation is also expressed as E U(T) ,and The coordinate system transformation is expressed as and and then It can be expressed as:

[0150]

[0151] The following relationship is obtained:

[0152]

[0153] Since the error value caused by actual machining is small, the vector decomposition method separates the pose offset caused by geometric error and the additional pose offset caused by thermal error. The principle is:

[0154]

[0155] in: is the actual HCT matrix of object j relative to object i; is the ideal HCT matrix of object j relative to object i; E IJ(G) is the HCT matrix of the position-dependent geometric error PDGEs of object j relative to object i; E IJ(T) is the HCT matrix of the thermal error TEs of object j relative to object i.

[0156] The above equations complete the matrix decomposition, eliminate a large number of high-order error (HOE) terms, and reduce the complexity of the MSE model. To verify the correctness of the proposed method, take the B axis as an example and solve have to:

[0157]

[0158] Remove After removing the high-order error terms in , it can be observed that the two matrices are completely consistent, which proves the correctness of the proposed vector decomposition method. When the vector decomposition method is extended to the dual-motion axis coordinate system transformation, the model matrix established by the traditional HTM method is:

[0159]

[0160] The modeling matrix based on the vector decomposition method is:

[0161]

[0162] The final HTM matrix of the machine tool kinematic chain (based on the vector decomposition method) taking into account the errors is:

[0163]

[0164] As can be seen from the above equation, the number of matrices involved is still quite large, and matrix multiplication produces high-order error terms, making it difficult to solve the error model. Therefore, the truncated series method is introduced. This method can be applied to the multiplication of multiple error matrices. By reducing the high-order error terms and retaining the low-order terms with significant influence, the overall effect is reduced.

[0165] The following is the definition of the truncated series method in error modeling. Assume that there are two error matrices M1 and M2, whose Taylor series expansion is:

[0166] M1=I+ω1+(ω1) 2 +Δ((ω1) 2 )

[0167] M2=I+ω2+(ω2) 2 +Δ((ω2) 2 )

[0168] 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 ) is the high-order error (HOE) matrix.

[0169] Before matrix multiplication, each error matrix needs to be truncated to retain only specific order terms. In this embodiment, only the first-order error matrix is ​​retained, and then the error matrix multiplication operation is performed:

[0170] M=M1·M2=(I+ω1+(ω1) 2 +Δ((ω1)2 ))·(I+ω2+(ω2) 2 +Δ((ω2) 2 ))

[0171] ≈I+ω1+ω2+ω1·ω2

[0172] For the error matrix obtained after the multiplication operation, the truncated series method is used to discard the high-order error terms, retaining only the low-order terms of ω1, ω2, and ω1·ω2 to reduce the impact of high-order errors. In this equation, the continuous multiplication of a single geometric error matrix or TE matrix is ​​defined as a first-order error term, and the continuous multiplication of two error matrices is defined as a second-order error term. After matrix decomposition using the vector decomposition method, the highest-order error term is the fourth-order error term, and the final MSE model is:

[0173]

[0174] During machining, low-order error terms are typically small, while high-order error terms are closer to zero and much smaller than low-order terms. These terms have minimal impact on the MSE model but are still involved in the calculation. When multiple error matrices are multiplied, the number of high-order error terms increases dramatically. Traditional HTM methods introduce a large number of high-order error terms when constructing MSE models, resulting in a sharp increase in the complexity of model solution. High-order error terms not only increase model complexity and expand the scale of model equations, but also increase computational complexity and solution difficulty, often requiring more computing resources and time. In addition, high-order error terms have strict requirements on numerical stability and computational accuracy, requiring the support of high-precision and massive data. These factors collectively result in MSE models containing high-order error terms facing enormous challenges and computational costs when constructing and solving them. Therefore, it is necessary to balance model accuracy requirements with computational costs and select appropriate modeling methods and solution strategies.

[0175] The calculation time of MSE modeling based on the vector decomposition method combined with the truncation function method was compared with that of the traditional modeling method based on the HCT principle. The results showed that the proposed modeling method only takes 19 seconds, which is significantly shorter than the 7.1 minutes required by the traditional method ( Figure 4 ). The entire modeling process includes calculation time and high-order error term elimination time. The traditional HCT principle MSE model also has the problem of too many high-order error terms, which requires manual elimination of high-order error terms. This process is very likely to produce elimination errors, which seriously reduces the accuracy of the model. The efficient MSE model without high-order error terms based on the vector decomposition method and the truncation function method can quickly solve the accurate MSE model, and ensure the model time and accuracy without the need to manually eliminate high-order error terms. The efficient MSE model based on vector decomposition and truncation function proposed in this embodiment greatly reduces high-order error terms, circumvents the high-order error terms introduced by matrix multiplication in traditional modeling through a modular modeling method, reduces the complexity of the MSE model and the demand for computing resources, and is more suitable for error modeling.

[0176] 2.2 Multi-source error-tooth surface error mapping model

[0177] Based on the theoretical HCT matrix of WCS relative to GCS, the ideal tooth surface coordinates and unit normal vector in GCS are expressed as:

[0178]

[0179] The actual tooth surface coordinates and unit normal vector in GCS are:

[0180]

[0181] The multi-source error-tooth surface error mapping model is expressed as:

[0182]

[0183] Where: g and ε g are the position error and angle error of the face gear tooth surface respectively; δ xg , δ yg and δ zg are the position errors of the face gear tooth surface along the X, Y and Z axes respectively; ε xg , ε yg and ε zg are the angular errors of the face gear tooth surface along the X, Y and Z axes respectively; and are the actual unit normal vectors of the global coordinate system GCS; and are the ideal unit normal vectors of the global coordinate system GCS respectively.

[0184] Figure 5 Shows that the error ε xb(t) The six tooth surface errors caused by this error have different degrees of influence on each tooth surface error component: the influence on the three position errors comes from δ yz , which makes δ yz The relevant tooth surface errors show a distribution pattern that decreases from the tooth top to the tooth root and increases from the inner diameter to the outer diameter. This distribution is consistent with δ xy The distribution of the related tooth surface errors is consistent, but different from δ xy The distribution of the related tooth surface errors is opposite - where ε yy This will cause the tooth surface error to show a reverse change trend.

[0185] Figure 6 Shows ε xx(g) The six tooth surface errors caused by this error are related to the position error δ yg The impact is the most significant, but the δ yg Error distribution and ε ψ(t) .δ ygThe error distribution is opposite: the former increases from the top to the root of the tooth and decreases from the inner diameter to the outer diameter. g and δ g The error is consistent with δ pg The error is contradictory. g With ε g The distribution is uniform, and ε g With ε gg The error and ε gg The errors are inversely related. By comparison, it can be seen that the effects of different errors on the six tooth surface error components are significantly different. Therefore, the MSE components that have a significant impact on specific tooth surface error components should be analyzed in detail to provide a theoretical basis for the sensitivity error tracing method.

[0186] Step 2: In response to the output of the mapping model, an improved Morris sensitivity analysis method is applied to calculate weighted sensitivity coefficients by assigning higher weighting coefficients to thermal error components than to geometric error components to identify the key error components that have the greatest impact on tooth surface accuracy.

[0187] Local sensitivity analysis only considers parameter variations within a local range, ignoring their impact across the entire parameter space. This makes it limited when dealing with nonlinear or complex systems. In contrast, global sensitivity analysis encompasses parameter variations across the entire domain, comprehensively considering the overall impact of parameter changes on system output under different operating conditions. This method can examine both the effects of individual parameters and the interactions between multiple parameters and their combined impact on system output.

[0188] The Morris method is a global sensitivity analysis technology based on first-order and second-order effects. By introducing random perturbations of parameters and observing the changes in output responses, it evaluates the contribution of each parameter to the system output. At the same time, it can identify key parameters and sensitive factors that have a significant impact on tooth surface accuracy.

[0189] Tooth surface accuracy can be affected by multiple error components, and these effects may be coupled. Furthermore, system responses often exhibit nonlinear characteristics. Therefore, an error transmission model can be established using machine system errors (MSEs) as input variables and the six tooth surface errors as output variables:

[0190] E=F(G)

[0191] Where: E represents the tooth surface error, E=[δ x ,δ y ,δ z ,ε x ,ε y ,ε z ] T , G represents each error component, G=[x1,x2,…,x n ] Tis the total number of error terms.

[0192] The Morris method uses basic effects as the basic unit for evaluating the sensitivity of input parameters. The basic effects are calculated by changing the input parameter values ​​and observing the changes in the output: an initial value is set for each input parameter, and the parameter is adjusted within a preset step size or range (other parameters remain unchanged). The parameter sensitivity is evaluated by comparing the model output under different parameter values. When this method is applied to the error propagation model, it is assumed that the error component changes between two adjacent iterations, and the resulting output variation is observed. If the error component x i The change in is recorded as Δ, then the basic effect of Δ can be expressed as:

[0193]

[0194] Assuming that all basic effects follow a specific distribution, their mean μ can be calculated i and standard deviation σ i . The mean reflects the sensitivity of the error component to the tooth surface error, and the standard deviation characterizes the intensity of the interaction between the error component and other components. In order to reduce the influence of random errors, it is usually necessary to repeat the calculation process independently multiple times, and take the average value of the sensitivity index to improve the reliability of the results. Thermal deformation error (TE) accounts for a large proportion of machining errors, and its impact on machining accuracy is significantly greater than that of geometric error. When using the Morris method for sensitivity analysis, it is more reasonable to use a weighted error coefficient instead of the original error value: different error components have different effects on machining accuracy, and this difference can be reflected by assigning different weights. For example, the weight of TE's impact on machining accuracy is usually higher than that of geometric error, so it occupies a larger proportion in the total error.

[0195] This embodiment proposes an improved Morris method that introduces weighted error coefficients into the original Morris algorithm. This improved scheme ensures that the significant impact of TE (geometric transfer error) is accurately reflected in the total error by highlighting the importance of different error components. The improved method can provide a more realistic evaluation, which helps to develop effective error compensation strategies and optimize machine tool performance. The geometric errors and TE (x 1g ,x 2g ,…,x mg ,x It ,x 2t ,…,x nt ) is used as the input parameter of the improved Morris model, where the subscript g represents the geometric error and the subscript t represents TE. The tooth surface error component E = [δ x ,δ y ,δ z ,ε x ,ε y ,ε z ] T As the output target, the calculation includes μi and σ i The sensitivity coefficients of each error component, including , are used to identify the key sensitive error components that significantly affect the tooth surface accuracy. Specifically, the improved Morris sensitivity analysis method includes the following steps.

[0196] 21) Given the worm wheel parameters, face gear parameters, machine tool structure and machining motion configuration, calculate the theoretical and actual HCT matrices and all multi-source error MSE components from the global coordinate system GCS to the face gear coordinate system FCS, and use the Morris method to construct the random input matrix E * .

[0197] 22) From the preset range [0, Δ i ,2Δ i ,…,1] randomly and uniformly sample the multi-source error MSE and construct the benchmark error vector X as the starting point of sensitivity analysis * . Set the random permutation matrix P * Disrupt the order of error components to ensure sampling randomness and set the random permutation matrix P * Disrupt the order of error components to ensure random sampling, and use X * and P * Establish a random sampling matrix S representing the perturbation parameter set * .

[0198] 23) S * Each row of the matrix (representing a set of error values) is substituted into the actual HCT matrix to calculate the gear tooth surface error component; for each error component, the basic effect is calculated based on the difference between the input parameter change and the output result; in order to reflect the differentiated impact of different errors, the corresponding weight is assigned to each error, and the basic effect EE is calculated. i (x i ) is updated to the weighted basic effect WEE i (x i ).

[0199] The updated weighted basic effect calculation formula is:

[0200]

[0201] Where: F(·) is the multi-source error-tooth surface error mapping model; (x 1g ,x 2g ,…,x mg ,x It ,x 2t ,…,x nt ) are the geometric error and thermal error generated during the machining process, subscript g represents the geometric error, and subscript t represents the thermal error; σ 1g , σ i and σ ntThe weighting factor assigned to a specific error component.

[0202] 24) Loop through steps 22)-23) until the number of iterations reaches the set maximum number of iterations; after calculating the weighted basic effect for each iteration, calculate the average weighted basic effect of all iteration results, and finally calculate the weighted sensitivity coefficient wμ of each error parameter i and wσ i In this embodiment, the maximum number of iterations is 50.

[0203]

[0204] Where: wμ i and wσ i are the weighted mean and standard deviation respectively; μ i and σ i are the mean and standard deviation respectively; N is the maximum number of iterations;

[0205] 25) Compare the weighted sensitivity index of each error parameter to identify the sensitive error sources that significantly affect each tooth surface error component.

[0206] This method can trace the key contributing factors of each tooth surface error variation and provide an operational basis for error compensation and performance optimization. All error identifiers are input into the traditional Morris model to obtain the sensitive error components corresponding to the six tooth surface error components, such as Figure 7 As shown. The sensitivity error corresponding to each tooth surface error component can be tracked. Mean μ i Higher error components should be given priority. Larger standard deviation σ i This indicates that this error component has strong interaction with other components, which means that the error fluctuates significantly between different sampling points. The sum of the sensitivity coefficients of the 46 error components is 1. i By analyzing its distribution, the error component that has the greatest impact on each tooth surface error can be identified, thereby focusing on the key error sources that lead to the overall tooth surface error. Figure 7 The TE component of the rotation axis contributes minimally, while the geometric error component contributes significantly. This distribution differs from the actual geometric error and TE distributions. Further analysis shows that angular error is more easily coupled with other errors or motion parameters through the error transformation matrix, and therefore has a more significant impact on the tooth surface error component than position error.

[0207] Then the 46 error components are input into the improved Morris model, and the original sensitivity coefficient μ i and σ i Corrected to the error weighted sensitivity coefficient wμ i and wσ i ( Figure 8Table 1 compares the sensitivity error results of the six tooth surface error components identified by the traditional and improved Morris methods. A total of 25 sensitive errors were tracked. The error weighting coefficient of the rotation axis TE was increased to reflect the actual error distribution. Following the same analysis steps, wμ was selected i The error with the largest value is regarded as the sensitive error. Based on the sensitivity analysis process of the improved Morris method, the sensitive error identification results are compared with those of the traditional method, and the 46 error components are ranked (see Table 2 for error identifiers and definitions).

[0208] Table 2 Error numbers

[0209]

[0210] Table 3 lists the weighted sensitive error identification results for the six tooth surface error components, ultimately identifying 18 sensitive error components. Comparing the identification results using the traditional and improved Morris methods, the number of sensitive errors has decreased. Furthermore, the contribution of the rotary axis TE components has increased, more closely matching the actual error distribution. The rotary TE components of the A, B, and C axes require particular compensation. This is because TE is a dynamic error—the B and C axes rotate at high speeds and generate significant heat. Over time, TE accumulates and degrades the tooth surface error.

[0211] Table 3 Sensitive error and tooth surface error components corresponding to weighted sensitive error terms

[0212]

[0213] Step 3: Based on the key error components, a motion axis compensation value is generated through an error compensation model, and the compensation value is input into the machine tool control system to adjust the relative position of the worm grinding wheel and the workpiece, thereby reducing the tooth surface error.

[0214] Specifically, the error compensation model is solved based on the Jacobian matrix, which includes:

[0215] 31) Define the overall differential motion error E based on the differential motion error matrix.

[0216] The differential motion errors are listed in Table 4.

[0217] Table 4 Differential motion error matrix

[0218]

[0219]

[0220] The overall differential motion error E is calculated as:

[0221]

[0222] Where:B and ε C Represents the angular errors of the B-axis and C-axis respectively; δ x , δ y and δ z They correspond to the displacement errors of the X, Y, and Z axes respectively.

[0223] 32) Define the Jacobian matrix J M Solve for the axis error compensation components:

[0224]

[0225] Satisfies the equation:

[0226] E=J M ·[δ X ,δ Y ,δ z ,ε B ,ε C ] T

[0227] Where: x , ε y , ε z , δ x , δ y and δ z They represent the translation error component and angular error component of the worm grinding wheel respectively.

[0228] When the matrix J M When the matrix is ​​non-square and the dimension is 6×5, it is not possible to directly invert it. To facilitate the solution, multiply both sides of the equation by the matrix J M The transposed matrix J M T Perform indirect calculations:

[0229] J M T E=(J M T ·J M )·[δ X ,δ Y ,δ Z ,ε A ,ε C ] T

[0230] Therefore, the error compensation component is expressed as:

[0231]

[0232] In order to make the error compensation term equivalent to the error, the overall differential motion error E is redefined as:

[0233] E=[ε x,ε y ,ε z ,δ x ,δ y ,δ z ] T

[0234] Where: x , ε y , ε z , δ x , δ y and δ z represent the translation error component and angular error component of GWGW respectively.

[0235] δ X , δ Y , δ Z , ε B and ε C The calculation formula is:

[0236]

[0237] The differential motion error caused by the compensation error should offset the posture error caused by the actual error, so the following relationship is established.

[0238] [ε x ,ε y ,ε z ,δ x ,δ y ,δ z ]=-[ε xw ,ε yw ,ε zw ,δ xw ,δ yw ,δ zw ]

[0239] The following relationship is obtained:

[0240]

[0241] This embodiment carries out error compensation experiments. Figure 9 This paper demonstrates the error compensation principle of a face gear machining motion control system. The ideal command position generated by the interpolator is sent as an input reference value to the servo system driving the motion axis. However, due to geometric and thermal errors, the actual command position may deviate from the ideal value. After the position information is fed back to the computer, a temperature sensor and acquisition system acquire temperature and geometric error data in real time. The predicted error is then processed and analyzed using a geometric / thermal error model. The calculated error is input into the error compensation model to generate a compensation value, which is then sent to a programmable logic controller (PLC) to adjust the command position to offset the error. The corrected position is then re-input into the servo system, achieving precise motion and reducing the impact of errors during machining.

[0242] The speed of GWGW is set to 3000r / min, and the linear axis feed rate is set to 3mm / min. Figure 10 As shown in the figure, the cutting depth is 0.2 mm.

[0243] After the end gear is ground, the normal error of the tooth surface is measured using a Leitz PMM-C 12.10.7 coordinate measuring machine (CMM). Figure 11 The results of tooth surface normal error measurements with and without error compensation are presented. Without compensation, the maximum normal error on the left tooth surface is 4μm, the minimum error is -22.3μm, and the total absolute error reaches 26.3μm. The maximum error on the right tooth surface is 14.3μm, the minimum error is -23.9μm, and the total error is 38.2μm. After error compensation, the maximum error on the left tooth surface is reduced to 1.1μm, the minimum error is -7.7μm, and the total error is 8.8μm, a decrease of 17.5μm. The maximum error on the right tooth surface is reduced to 6.8μm, the minimum error is -7.2μm, and the total error is 14μm, an improvement of 24.2μm. These results demonstrate a significant improvement in tooth surface geometric accuracy, verifying that the proposed MSE compensation method can effectively improve FGWGMT machining accuracy.

[0244] This embodiment aims to solve the complex geometric error and thermal error problems that affect machining accuracy, and proposes a new error compensation method for face gear worm grinding machines. For the first time, an MSE-tooth surface error mapping model is established to reveal the transmission mechanism of geometric error and thermal error in face gear machining, and quantify its influence on tooth surface deviation. An efficient multi-source error model is proposed based on vector decomposition and truncation function method, which improves computational efficiency without sacrificing accuracy by eliminating high-order terms. In addition, an improved Morris sensitivity analysis method is proposed to accurately identify the key geometric error and thermal error components that have the most significant impact on machining accuracy, and achieve targeted and efficient compensation. The method of this embodiment has been experimentally verified on FGWGMT, and the main conclusions are as follows:

[0245] (1) An efficient multi-source error-tooth surface error mapping model was established, revealing the geometric / thermal error transmission mechanism in the machining chain and quantitatively describing its impact on tooth surface error. This model reduced the modeling time from 7.1 minutes to 19 seconds.

[0246] (2) The improved Morris sensitivity analysis method proposed successfully identified the key geometric / thermal error components that have the most significant impact on tooth surface errors, laying the foundation for accurate compensation.

[0247] (3) The error compensation model is applied to compensate for the key sensitive errors, achieving a maximum reduction of 24.2μm in tooth surface error.

[0248] This method is universally applicable and provides a new theoretical framework and technical path for error compensation in other complex multi-axis machine tools. Future research will focus on refining the error mapping model, exploring real-time compensation methods for high-dimensional nonlinear errors, and verifying its applicability in other high-precision machining scenarios.

[0249] 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 face gear worm grinding machine error compensation method based on sensitivity analysis and multi-source error mapping model, characterized by: The steps include: Step 1: Based on vector decomposition and the truncation function method, a multi-source error-to-tooth surface error mapping model is constructed to quantitatively describe the propagation mechanism of geometric and thermal errors in the machining chain and their impact on tooth surface deviation. The truncation function method eliminates high-order error terms to improve computational efficiency. Step 2: In response to the output of the mapping model, applying the improved Morris sensitivity analysis method to calculate the weighted sensitivity coefficient by assigning a higher weight coefficient to the thermal error component than to the geometric error component, so as to identify the key error component that has the greatest impact on the tooth surface accuracy; Step 3: Based on the key error components, a motion axis compensation value is generated through an error compensation model, and the compensation value is input into the machine tool control system to adjust the relative position of the worm grinding wheel and the workpiece, thereby reducing the tooth surface error.

2. The error compensation method for a face gear worm grinding machine based on sensitivity analysis and multi-source error mapping model according to claim 1, characterized in that: In step 1, based on the grinding kinematic chain of the face gear worm grinder, the ideal HCT matrix of the worm grinding wheel coordinate system WCS relative to the global coordinate system GCS is obtained: in: represents the ideal HCT matrix of WCS relative to GCS; Represents the ideal HCT matrix of the B-axis coordinate system BCS relative to the Y-axis coordinate system YCS; Represents the ideal HCT matrix of the Y-axis coordinate system YCS relative to the A-axis coordinate system ACS; Represents the ideal HCT matrix of the A-axis coordinate system ACS relative to the Z-axis coordinate system ZCS; Represents the ideal HCT matrix of the Z-axis coordinate system ZCS relative to the X-axis coordinate system XCS; Represents the ideal HCT matrix of the X-axis coordinate system XCS relative to the bed coordinate system RCS; Represents the ideal HCT matrix of the bed coordinate system RCS relative to the C-axis coordinate system; Get the actual HCT matrix of the worm grinding wheel coordinate system WCS relative to the global coordinate system GCS: in: Represents the actual HCT matrix of WCS relative to GCS; Represents the actual HCT matrix of the B-axis coordinate system BCS relative to the Y-axis coordinate system YCS; Represents the actual HCT matrix of the Y-axis coordinate system YCS relative to the A-axis coordinate system ACS; Represents the actual HCT matrix of the A-axis coordinate system ACS relative to the Z-axis coordinate system ZCS; Represents the actual HCT matrix of the Z-axis coordinate system ZCS relative to the X-axis coordinate system XCS; Represents the actual HCT matrix of the X-axis coordinate system XCS relative to the bed coordinate system RCS; Represents the actual HCT matrix of the bed coordinate system RCS relative to the C-axis coordinate system; E BY(G) and E BY(T) are the HCT matrices of the position-dependent geometric error PDGEs and thermal error TEs of the B-axis coordinate system BCS relative to the Y-axis coordinate system YCS; E YA(G) is the HCT matrix of PDGEs of YCS relative to ACS; E Az(G) is the HCT matrix of PDGEs in ACS relative to ZCS; E ZX(G) is the HCT matrix of PDGEs of ZCS relative to XCS; E XR(G) is the HCT matrix of PDGEs of XCS relative to RCS; E cR(G) and E CR(T) are the HCT matrices of the position-dependent geometric errors PDGEs and thermal errors TEs of CCS relative to RCS, respectively; The mapping relationship between the position error and the angle error of the worm grinding wheel in the multi-source error-tooth surface error mapping model is obtained: Where: iw is the position error of the worm grinding wheel; δ xw , δ yw and δ zw are the position errors of the worm grinding wheel along the X, Y and Z axes respectively; [x, y, z, 1] T is the theoretical position of the worm grinding wheel in the worm grinding wheel coordinate system WCS; ε iw is the angular error of the worm grinding wheel; ε xw , ε yw and ε zw are the angular errors of the worm grinding wheel along the X, Y and Z axes respectively; [α, β, γ, 1] T It is the theoretical angle of the worm grinding wheel in the worm grinding wheel coordinate system WCS.

3. The error compensation method for a face gear worm grinding machine based on sensitivity analysis and multi-source error mapping model according to claim 1, characterized in that: In step 1, the vector decomposition method separates the pose offset caused by geometric error from the additional pose offset caused by thermal error. The principle is: in: is the actual HCT matrix of object j relative to object i; is the ideal HCT matrix of object j relative to object i; E IJ(G) is the HCT matrix of the position-dependent geometric error PDGEs of object j relative to object i; E IJ(T) is the HCT matrix of the thermal error TEs of object j relative to object i.

4. The error compensation method for a face gear worm grinding machine based on sensitivity analysis and multi-source error mapping model according to claim 1, characterized in that: In step 1, the truncation function method retains the first-order error term and discards the higher-order error term. The principle is: M=M1·M2=(I+ω1+(ω1) 2 +Δ((ω1) 2 ))·(I+ω2+(ω2) 2 +Δ((ω2) 2 )) ≈I+ω1+ω2+ω1·ω2 Where: M is the matrix obtained by multiplying the error matrices M1 and M2; 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 ) is a high-order error matrix.

5. The error compensation method for a face gear worm grinding machine based on sensitivity analysis and multi-source error mapping model according to claim 1, characterized in that: The multi-source error-tooth surface error mapping model is expressed as: Where: g and ε g are the position error and angle error of the face gear tooth surface respectively; δ xg , δ yg and δ zg are the position errors of the face gear tooth surface along the X, Y and Z axes respectively; ε xg , ε yg and ε zg are the angular errors of the face gear tooth surface along the X, Y and Z axes respectively; and are the actual unit normal vectors of the global coordinate system GCS; and are the ideal unit normal vectors of the global coordinate system GCS respectively.

6. The error compensation method for a face gear worm grinding machine based on sensitivity analysis and multi-source error mapping model according to claim 1, characterized in that: In step 2, the improved Morris sensitivity analysis method includes the following steps: 21) Given the worm wheel parameters, face gear parameters, machine tool structure and machining motion configuration, calculate the theoretical and actual HCT matrices and all multi-source error MSE components from the global coordinate system GCS to the face gear coordinate system FCS, and use the Morris method to construct the random input matrix E * ; 22) From the preset range [0, Δ i ,2Δ i ,…,1] randomly and uniformly sample the multi-source error MSE and construct the benchmark error vector X as the starting point of sensitivity analysis * ; Set the random permutation matrix P * Disrupt the order of error components to ensure random sampling, and use X * and P * Establish a random sampling matrix S representing the perturbation parameter set * ; 23) S * Each row of the matrix is ​​substituted into the actual HCT matrix to calculate the face gear tooth surface error component; For each error component, the basic effect is calculated based on the difference between the input parameter change and the output result; Assign corresponding weights to each error and divide the basic effect EE i (x i ) is updated to the weighted basic effect WEE i (x i ); Where: F(·) is the multi-source error-tooth surface error mapping model; (x 1g ,x 2g ,…,x mg ,x It ,x 2t ,…,x nt ) are the geometric error and thermal error generated during the machining process, subscript g represents the geometric error, and subscript t represents the thermal error; σ 1g , σ i and σ nt The weighting coefficients assigned to specific error components; 24) Loop through steps 22)-23) until the number of iterations reaches the set maximum number of iterations; after calculating the weighted basic effect for each iteration, calculate the average weighted basic effect of all iteration results, and finally calculate the weighted sensitivity coefficient wμ of each error parameter i and wσ i ; Where: wμ i and wσ i are the weighted mean and standard deviation respectively; μ i and σ i are the mean and standard deviation respectively; N is the maximum number of iterations; 25) Compare the weighted sensitivity index of each error parameter to identify the sensitive error sources that significantly affect each tooth surface error component.

7. The error compensation method for a face gear worm grinding machine based on sensitivity analysis and multi-source error mapping model according to claim 1, characterized in that: In the step 3, the error compensation model is solved based on the Jacobian matrix, specifically including: 31) defining the overall differential motion error E according to the differential motion error matrix; 32) Define the Jacobian matrix J M Solve the axis error compensation component to satisfy the equation: E=J M ·[d X ,d Y ,d Z ,he B ,he C ] T Where: ε x , ε y , ε z , δ x , δ y and δ z They represent the translation error component and angular error component of the worm grinding wheel respectively.

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