Face gear worm grinding machine geometric-thermal error compensation system based on digital twinning

Abstract

The invention discloses a face gear worm grinding machine geometric-thermal error compensation system based on digital twinning. The face gear worm grinding machine geometric-thermal error compensation system comprises a geometric-thermal error mapping model, a thermal error prediction model and a digital twinning driving geometric-thermal error compensation system. The geometric-thermal error mapping model is constructed based on a mapping relation between geometric-thermal errors and tooth surface errors in a grinding machine kinematic chain; the thermal error prediction model is constructed based on an LSTM network of an improved gating structure; the digital twin drive geometric-thermal error compensation system comprises a layered architecture of a sensing control layer, an edge layer and a cloud layer. The sensing control layer is used for collecting data in real time so as to provide heat information data of the face gear worm grinding machine. An edge layer deploys a thermal error prediction model to carry out real-time thermal error prediction, and the thermal error prediction model is integrated with a geometric-thermal error mapping model; and the cloud layer trains a thermal error prediction model through historical thermal information data, and issues updated model parameters to the edge layer to synchronously update the thermal error prediction model.

Description

Technical Field

[0001] The present invention belongs to the technical field of machining error compensation, and particularly relates to a geometric-thermal error compensation system for face gear worm grinding machines based on digital twin. Background Art

[0002] The face gear (FG) drive system is characterized by high interchangeability, strong load-bearing capacity, and efficient power distribution, and can achieve motion and power transmission between intersecting or staggered axes, and is expected to replace bevel gears in military helicopters, hovercrafts, and tanks. As the final machining process after rough machining, the worm grinding process plays a decisive role in achieving the geometric accuracy of FG, directly affecting the service performance of the drive system. The face gear worm grinding machine (FGWGMT) is a key equipment to ensure machining accuracy and efficiency, but its geometric and thermal errors (GTEs) will significantly reduce machining accuracy, damage the geometric accuracy of FG, and affect the overall performance of the drive system. Therefore, adopting effective error suppression methods is crucial for improving machining accuracy.

[0003] Improving the machining accuracy of machine tools depends on effective error suppression, and its foundation lies in error modeling. Usually, the machine tool moving axes are regarded as rigid bodies, and the topological model of the machine tool transmission chain is established by applying the multi-body system theory, and then the mapping model between the moving axis error and the machining error is constructed through homogeneous coordinate transformation. This method can clarify the error transfer mechanism within the transmission chain and reveal the specific influence of each error source on machining accuracy. Some scholars have also proposed methods for suppressing the thermal error (TE) of machine tools.

[0004] At present, the research on the geometric thermal error (GTE) modeling of FGWGMT is still in the preliminary exploration stage. Most of the existing models are limited to the analysis of a few error components. The thermal error of machine tools mainly originates from the temperature rise of key components, including bearings, servo motors, and ball screw pairs. During operation, heat continuously dissipates through convection and radiation, making the temperature gradient gradually tend to the thermal equilibrium state. However, if the heat generation rate often exceeds the heat dissipation capacity, it will lead to heat accumulation and cause significant thermal deformation. Traditional empirical models usually use methods such as multiple linear regression, random forest, and support vector machine, and use temperature-thermal error data as input and output variables for prediction, but their accuracy and robustness are not satisfactory. To break through these limitations, researchers have begun to apply deep learning models such as long short-term memory network (LSTM), gated recurrent unit (GRU), and bilinear temporal convolutional network (BTCN) to capture dynamic thermal characteristics. Although progress has been made in the modeling of various machine tool thermal errors, the systematic research on FGWGMT is still insufficient. The complex and changeable working conditions pose severe challenges to the prediction accuracy and robustness of the model, and the existing models are difficult to meet the robustness requirements under variable working conditions. It is urgent to optimize the matching relationship between thermal data and model hyperparameters, and at the same time, the LSTM network structure also needs to be improved.

[0005] In addition to thermal error prediction, real-time guarantee is crucial for the effectiveness of the error compensation system. However, the computational complexity of deep learning models restricts real-time performance. Digital twin technology provides an innovative solution for this. Research shows that it can effectively improve the development efficiency of the error compensation system. The lightweight compensation system constructed with the support of digital twins can significantly enhance real-time performance. When integrating the thermal error prediction and compensation model into the system, it is necessary to carefully design the system architecture and functional modules at each level. This structured design is the key to maintaining the system response speed. The FG grinding process involves complex coupling effects of multiple error components, making the error modeling and compensation of FGWGMT particularly difficult. The unique kinematic chain structure of the machine tool further increases the complexity of grinding linkage, and the interaction propagation mechanism of geometric thermal errors in the transmission chain is not yet clear. Currently, there is a lack of research on geometric thermal error compensation models, and there are deviations in the equivalent motion errors based on error tracing calculations, resulting in poor error compensation effects for FG grinders and restricting the improvement of machining accuracy.

[0006] The interaction between two types of errors in FGWGMT makes the compensation of geometric and thermal errors extremely challenging. The mapping relationship between tooth surface errors (TSEs) and geometric thermal errors is not yet clear, and it is difficult to determine the quantitative impact of each error component on the tooth surface. These errors are transmitted through the transmission chain, forming 24 position errors in the tool / workpiece system, ultimately leading to tooth surface deviation. In addition, the reverse tracing of machining errors to the servo axis is still a difficult problem. Thermal errors have the characteristics of time-varying, non-linear, and non-steady state, making it difficult to accurately model and predict, resulting in difficult realization of high-precision and reliable thermal error compensation and restricting the improvement of grinding accuracy. The thermal error prediction model must be able to handle these complex characteristics, and at the same time, the compensation system needs to ensure real-time performance. Currently, the system response speed is still insufficient. The main challenges are as follows:

[0007] (1) Existing research has not clarified the mapping mechanism between tooth surface errors and geometric thermal errors. Geometric thermal errors are mostly modeled independently, and there is limited understanding of their interaction propagation law in the FGWGMT transmission chain. This treatment method is difficult to comprehensively reflect their combined impact on machining accuracy, and a complete geometric thermal error compensation model has not been established yet. Existing research ignores the non-rotary characteristics of the worm wheel grinding tool and only considers the spatial error of the tool tip. Although the pose error of the tool / workpiece system is taken into account, the non-rotary characteristics of the tool and the grinding meshing principle are not considered. Therefore, a more refined and comprehensive FGWGMT compensation model needs to be established.

[0008] (2) Although various machine tool thermal error modeling technologies have become mature, there is still insufficient specific research on FGWGMT. Due to the complex and changeable working conditions, it is difficult for traditional statistical models and data-driven models to balance high precision and strong robustness. FGWGMT is affected by multiple factor couplings, further reducing the model accuracy and robustness. Therefore, it is necessary to develop an innovative thermal error prediction model that can capture the temporal characteristics of thermal information.

[0009] (3) Although there have been studies on the suppression of geometric-thermal errors (GTE) in cylindrical gear grinding, the related research on form grinding worm gear machines (FGWGMT) is still scarce. Existing suppression models mainly focus on the spatial error of the tool, calculating the compensation amount through single-axis motion, but do not consider the multi-axis linkage relationship in FGWGMT and the meshing effect between the grinding worm wheel and the workpiece, resulting in poor GTE suppression effect. Summary of the Invention

[0010] In view of this, the purpose of the present invention is to provide a geometric-thermal error compensation system for face gear worm grinding machines based on digital twin, which can simultaneously achieve geometric and thermal error compensation to improve the machining accuracy of face gears and can improve the real-time performance of error compensation.

[0011] To achieve the above purpose, the present invention provides the following technical solutions:

[0012] A geometric-thermal error compensation system for face gear worm grinding machines based on digital twin, including a hierarchical architecture of a sensing control layer, an edge layer, and a cloud layer;

[0013] The sensing control layer is used to collect data in real time to provide thermal information data of the face gear worm grinding machine;

[0014] The digital twin model and the thermal error prediction model of the face gear worm grinding machine are deployed inside the edge layer; the thermal error prediction model performs real-time thermal error prediction according to the thermal information data collected by the sensing control layer, and inputs the obtained thermal error prediction data into the digital twin model; the digital twin model integrates a geometric-thermal error mapping model and an error compensation model, and the digital twin model performs real-time simulation based on the thermal error prediction data and the geometric-thermal error mapping model to obtain tooth surface error, and calculates the compensation component through the error compensation model;

[0015] The cloud layer trains the thermal error prediction model through historical thermal information data, and sends the updated model parameters to the edge layer to synchronously update the thermal error prediction model;

[0016] The geometric-thermal error mapping model is constructed based on the mapping relationship between geometric-thermal errors and tooth surface errors in the grinding machine motion chain; and the construction method of the error mapping model is:

[0017] Construct the theoretical forward kinematic transformation matrix from the grinding worm wheel to the workpiece;

[0018] Calculate the actual homogeneous coordinate transformation matrix from the grinding worm wheel to the workpiece under error conditions;

[0019] Derive the tool / workpiece attitude error model according to the difference between the theoretical transformation matrix and the actual transformation matrix;

[0020] Combining the thermal contact transformation matrix and the forward kinematics matrix of the grinding process, a mapping relationship between the tooth surface error and the geometric-thermal error is established;

[0021] The thermal error prediction model is constructed based on the LSTM network with an improved gating structure. The LSTM network with the improved gating structure introduces an attention mechanism and combines the input gate and the forget gate into a joint gating mechanism. By adding residual connections and weight allocation, while ensuring the long-term feature representation performance, the short-term data representation ability is strengthened.

[0022] Furthermore, the tool / workpiece attitude error model is:

[0023]

[0024] In the formula: [a, b, c, 1] T and [i, j, k, 0] T are the position vector and the attitude vector respectively; δ w and ε w represent the position error vector and the attitude error vector of the worm grinding wheel respectively; δ xw , δ yw and δ zw represent the elements of δ w ; ε xw , ε yw and ε zw represent the elements of ε w ; is the actual homogeneous coordinate transformation matrix from the grinding worm wheel to the workpiece; T mGT is the theoretical forward kinematics transformation matrix from the grinding worm wheel to the workpiece.

[0025] Furthermore, the formed grinding tooth surface equation containing errors is expressed as:

[0026]

[0027] Where: represents the tooth surface equation with errors; is the normal vector; E e represents the set of all errors; is the forward kinematics matrix with errors, and its value is l w and E e function; and are the new meshing conditions;

[0028] The mapping relationship between the tooth surface error and the geometric-thermal error is:

[0029]

[0030] Where: rxw , r yw and r zw represent the position vector components of the grinding wheel; i w , j w and k w represent the vector components of the grinding wheel; E δf is the position error vector; δ xg , δ yg and δ zg are the position error components; E ef is the attitude error vector; ε xg , ε yg and ε zg represent the attitude error components.

[0031] Furthermore, the principle of the improved gated structure LSTM network is as follows:

[0032]

[0033] i t = σ(W i [h t-1 , x t + b i )

[0034]

[0035] m t = σ(W m [h t-1 , x t + b m )

[0036]

[0037] In the formula: represents the weighted input; i i is the input gate; is the output of the input gate; m i is the output gate; c t is the updated state; h t is the hidden state; W i , W c and W m are the weight terms; b i , b c and b m are the bias terms; W is the weight value.

[0038] Furthermore, in the cloud layer, the parameters of the thermal error prediction model are updated using the non - linear chimp optimization algorithm; the principle of the non - linear chimp optimization algorithm is as follows:

[0039] The position update formula for each chimpanzee is:

[0040] X i = rand×(U b - L b ) + L b

[0041] Where: U b and L b represent the upper and lower bounds of the search space respectively; rand is a random number;

[0042] During the hunting process, the distance D between the chimpanzee and the prey and its position update function X chimp are as follows:

[0043] D = |hX prey (t) - mX chimp (t)|

[0044] X chimp (t + 1) = X chimp (t) - a×D, (μ < 0.5)

[0045] X chimp (t + 1) = m, (μ ≥ 0.5)

[0046] a = 2f×r1 - f

[0047] h = 2×r2

[0048] Where: r1 and r2 are non - linear random variables within the interval [0, 1]; h is a random number; m is a chaos factor; μ is a condition for judging the position update mode; a is a random variable within the interval [-f, f]; f is a non - linear convergence factor, whose value linearly decreases from 2 to 0 as the iteration number t increases; t max represents the maximum number of iterations; and: when |a| < 1, it indicates that the chimpanzee is approaching the prey position X prey ; when |a| > 1, it means that the chimpanzee is forced to move away from the prey and disperse to a wider area for searching;

[0049] The global optimal solution is represented by X Att , X Bar , X Cha and X Dri , corresponding to the position vectors of attacking, surrounding, driving away, and chasing the prey respectively; the distances D Att , D Bar , D cha and D Dri from these four positions to the prey are jointly determined by the position vectors X Att , X Bar , X Cha and X Dri :

[0050]

[0051] In the formula: X1, X2, X3, and X4 represent the updated position vectors of four types of chimpanzees; h1, h2, h3, and h4 are the obstacle factors affecting four types of hunting behaviors; m1, m2, m3, and m4 are the corresponding chaotic factors; a1, a2, a3, and a4 are associated random vectors;

[0052] The hunting behavior of the chimpanzee population is shown as:

[0053]

[0054] The non-linear convergence factor f is:

[0055]

[0056] Where: f0 represents the initial convergence factor.

[0057] Furthermore, the error compensation model includes a first-order error decoupling and tracing model and a second-order error tracing model.

[0058] Furthermore, the first-order error decoupling and tracing model determines six equivalent error decoupling quantities in the movement direction:

[0059] e x = u x cosB + u z sinB + Yv z cosB - Yv x sinB

[0060] e y = (u y cos 2 A - Xv z cosB + Xv x sinB - (u z - Yv x )cosAsinAcosB + (u x + Yv z )cosAsinAsinB) / cos 2 A

[0061] e z = -((u x + Yv z )cosAsinB - (u z - Yv x )cosAcosB - Xv z sinAcosB + Xv x sinAsinB) / cos 2 A

[0062] e A = v x cosB + v z sinB

[0063] e B = (v y cosA - v z sinAcosB + v x sinAsinB) / cosA

[0064] e C = (v z cosB - v x sinB) / cosA

[0065] Where: e x 、e y 、e z 、e A 、e B and e C are the equivalent error decoupling amounts in the directions of the X-axis, Y-axis, Z-axis, A-axis, B-axis, and C-axis respectively; u x 、u y and u z are the position errors of the tool / workpiece in the X / Y / Z directions respectively; v x 、v y and v z are the angular errors of the tool / workpiece in the X / Y / Z directions respectively; B, Y, A, Z, X, and C represent the motion commands of the B, Y, A, Z, X, and C axes respectively.

[0066] Furthermore, the second-order error tracing model determines the actual compensation components:

[0067]

[0068] Where: X C 、Z C and C C are the actual compensation components of the X-axis, Z-axis, and C-axis respectively; λ w is the helix angle; i gw represents the correlation between the B-axis and the C-axis.

[0069] The beneficial effects of the present invention are as follows:

[0070] The geometric-thermal error compensation system for face gear worm grinding machines based on digital twins first establishes the mapping relationship between the total transmission error (TSE) and GTE, constructs a GTE suppression model; then proposes an LSTM network based on an improved gating structure for predicting the transmission error (TE); finally designs a lightweight digital twin-driven geometric-thermal error compensation system; which can simultaneously achieve geometric and thermal error compensation to improve the machining accuracy of face gears and can improve the real-time performance of error compensation.

[0071] The main innovations and technical effects of the present invention are as follows:

[0072] (1) For the first time, establish the TSE-GTE correlation model of FGWGMT, reveal the influence mechanism of the meshing relationship between the worm wheel and the workpiece on TSE through the error transfer mapping of the motion axis, and finally form a GTE comprehensive suppression model covering multi-axis linkage compensation.

[0073] (2) Propose an NChOA-LSTM prediction model based on an improved gating structure: enhance the relevance of information flow processing through a new gating mechanism that fuses the input gate and the forget gate, and combine NChOA hyperparameter optimization to make the prediction accuracy reach 97.62%, and have working condition robustness.

[0074] (3) Construct a lightweight digital twin-driven system, integrating the GTE suppression and TE prediction models. Adopt an edge-cloud architecture to reduce the system execution time to 55.6% of the sensing control-cloud architecture, and achieve a reduction of the left TSE from 21.8μm to 5.9μm and the right TSE from 22.9μm to 5.7μm. Description of the Drawings

[0075] In order to make the objectives, technical solutions, and beneficial effects of the present invention clearer, the present invention provides the following drawings for illustration:

[0076] Figure 1 Is the kinematic chain of the face gear worm grinding machine; Figure 2 Is a schematic diagram of the influence of errors on the actual trajectory;

[0077] Figure 3 Is the distribution of the left tooth surface error component (when δ x (X)=50μm);

[0078] Figure 4 Is the variation law of the grinding error with the translational error related to the position of the linear axis;

[0079] Figure 5 Is the variation law of the grinding error with the rotational error related to the position of the linear axis;

[0080] Figure 6 Is the existing LSTM cell structure; Figure 7 Is the improved LSTM cell structure;

[0081] Figure 8 For digital twin modeling and GTE suppression application framework; Figure 9 For the construction of digital twin system;

[0082] Figure 10 For the construction of FGWGMT twin data model; Figure 11 For the system operation schematic diagram;

[0083] Figure 12 For geometric error measurement; Figure 13 For geometric errors of X, Y, and Z axes; Figure 14 For the layout position of temperature measurement points;

[0084] Figure 15 For X-axis thermal behavior data; Figure 16 For linear axis thermal behavior measurement; Figure 17 For B-axis thermal information data under working condition #1;

[0085] Figure 18 For the training results of working condition 1#; Figure 19 For the test results of working condition 2#;

[0086] Figure 20 For the grinding experiment site; Figure 21 For the tooth surface measurement grid and the arrangement of measurement points;

[0087] Figure 22 For the rotational projection of tooth surface measurement points on the xOz plane;

[0088] Figure 23 For FG gear TSE measurement; Figure 24 For the tooth surface geometric accuracy. Specific implementation manner

[0089] The following further illustrates the present invention in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and be able to implement it, but the exemplified embodiments do not limit the present invention.

[0090] I. Geometric-thermal error compensation system for face gear worm grinding machine based on digital twin

[0091] The geometric-thermal error compensation system for face gear worm grinding machine based on digital twin in this embodiment, a geometric-thermal error compensation system for face gear worm grinding machine based on digital twin, includes a hierarchical architecture of a sensing control layer, an edge layer, and a cloud layer.

[0092] In this embodiment, the sensing control layer is used to collect data in real time to provide thermal information data of the face gear worm grinding machine;

[0093] In this embodiment, a digital twin model and a thermal error prediction model of a face gear worm grinding machine are deployed inside the edge layer; the thermal error prediction model performs real-time thermal error prediction based on the thermal information data collected by the sensing and control layer, and inputs the obtained thermal error prediction data into the digital twin model; a geometric-thermal error mapping model and an error compensation model are integrated in the digital twin model, and the digital twin model performs real-time simulation based on the thermal error prediction data and the geometric-thermal error mapping model to obtain tooth surface error, and calculates a compensation component through the error compensation model.

[0094] In this embodiment, the cloud trains the thermal error prediction model with historical thermal information data, and sends the updated model parameters to the edge layer to synchronously update the thermal error prediction model.

[0095] In this embodiment, the geometric-thermal error mapping model is constructed based on the mapping relationship between geometric-thermal error and tooth surface error in the grinding machine kinematic chain; and the construction method of the error mapping model is as follows:

[0096] Construct the theoretical forward kinematic transformation matrix from the grinding worm to the workpiece.

[0097] Calculate the actual homogeneous coordinate transformation matrix from the grinding worm to the workpiece under error conditions.

[0098] Derive the tool / workpiece pose error model according to the difference between the theoretical transformation matrix and the actual transformation matrix.

[0099] Combine the thermal contact transformation matrix and the forward kinematic matrix of the grinding process to establish the mapping relationship between tooth surface error and geometric-thermal error.

[0100] In this embodiment, the thermal error prediction model is constructed based on an LSTM network with an improved gating structure. The LSTM network with the improved gating structure introduces an attention mechanism and combines the input gate and the forget gate into a joint gating mechanism. By adding residual connections and weight allocation, while ensuring the long-term feature representation performance, the short-term data representation ability is strengthened.

[0101] 1.1 Geometric-thermal error mapping model

[0102] (1) Theoretical forward kinematic model

[0103] Figure 1 The FGWGM forward kinematic model is shown, and the mathematical expression of the motion transfer chain from the tool as the reference point to the workpiece is derived.

[0104]

[0105] T GT =(T mRG ) -1 TmRT

[0106] The homogeneous coordinate transformation matrix (HCTM) between adjacent components is shown in Table 1 below.

[0107] Table 1 HCTM between adjacent components

[0108]

[0109] Where: I 4×4 represents the identity matrix; B, Y, A, Z, X, and C respectively represent the motion commands of the B, Y, A, Z, X, and C axes.

[0110] According to the transmission chain theory, the transmission sequence T→B→Y→A→Z→X→C→G is obtained, and the theoretical forward kinematic transformation matrix of FGWGM is derived.

[0111] By solving the HCTMs equations of each axis system and moving components from the tool to the workpiece, a theoretical forward kinematic model is established. Its general expression is:

[0112]

[0113] Where a 11 to a 34 are matrix elements, and the specific values are shown in Appendix A.

[0114]

[0115] (2) Actual homogeneous coordinate transformation matrix

[0116] According to Figure 2 , the error transfer matrix of part Q relative to part P is obtained:

[0117]

[0118] T pPQ = T x (p x )T y (p y )T z (p z )

[0119] T mQ = T x (X Q )

[0120] Where: represents the actual motion transformation matrix under error conditions; T PPQ is the positioning coordinate matrix of part Q in the reference coordinate system; (p x , p y , p z) represent the positioning coordinates of component Q; C is the distance X that component Q moves along the x(i) direction Q of the transformation matrix.

[0121] The transformation matrices of the grinding-related motion axes are summarized in Table 2.

[0122] Table 2 Transformation Matrices of Each Motion Axis

[0123]

[0124] Based on the above error transfer principle, the actual heat contact transformation matrix (HCTM) from the grinding wheel to the workpiece under actual working conditions is calculated as:

[0125]

[0126] In the formula: and respectively represent the actual heat contact transformation matrices from the grinding wheel to the B axis, from the B axis to the Y axis, from the Y axis to the A axis, from the A axis to the Z axis, from the Z axis to the X axis, from the X axis to the bed, from the C axis to the bed, and from the workpiece to the C axis. Then the actual forward kinematic model can be expressed as:

[0127]

[0128] Where: b 11 to b 34 are matrix elements. And:

[0129]

[0130]

[0131] (3) Tool / workpiece pose error model

[0132] Derive the tool / workpiece pose error model through the difference in the calculation results:

[0133]

[0134] In the formula: [a, b, c, 1] T and [i, j, k, 0] T are the position vector and the attitude vector respectively; δ w and ε w respectively represent the position error vector and the attitude error vector of the worm grinding wheel; δ xw , δ yw and δ zw represent the elements of δ w ; ε xw , ε yw and ε zw represent the elements of ε w ; is the actual homogeneous coordinate transformation matrix from the grinding worm to the workpiece; T mGT is the theoretical forward kinematics transformation matrix from the grinding worm to the workpiece.

[0135] (4) Mapping relationship between tooth surface error and geometric-thermal error

[0136] By combining the transformation matrix T GT with the forward kinematics matrix M of the grinding process gw this relationship is established:

[0137]

[0138] Thus, the theoretical motion control commands for the gear grinding process are determined:

[0139]

[0140] where: i gw represents the transmission ratio and is equal to N w / N g . During the actual grinding process, the Z-axis first moves to the specified height.

[0141] T GT is a function of l w and . Therefore, the tooth surface equation and its normal vector can be equivalently expressed as:

[0142]

[0143] The form grinding (FG) tooth surface equation containing these errors can be expressed as:

[0144]

[0145] where: represents the tooth surface equation with errors; is the normal vector; E e represents the set of all errors; is the forward kinematics matrix with errors, and its value is l w and E e is a function of; and are the new meshing conditions.

[0146] The forward kinematics transformation matrix changes significantly. The mapping relationship between tooth surface errors (TSEs) and geometric-thermal errors (GTEs) is:

[0147]

[0148] where: r xw 、r ywand r zw represent the position vector components of the grinding wheel; i w , j w and k w represent the vector components of the grinding wheel; E δf is the position error vector; δ xg , δ yg and δ zg are the position error components; E ef is the attitude error vector; ε xg , ε yg and ε zg represent the attitude error components.

[0149] By introducing one or more error terms into the model, its impact on the TSE distribution can be evaluated. When the X-axis error term δ x (X) is set to 50 μm, the distribution of the six TSEs on the left tooth surface is as Figure 3 shown. For δ yg and δ zg , the error is smaller at the tooth root and larger at the tooth tip, showing a gradually increasing trend from the root to the top along the tooth width direction. δ x (X) The distribution pattern on δ yg and δ zg shows that the error is smaller in the middle of the tooth height and gradually increases towards both sides along the tooth height direction. For ε xg and ε zg , the error increases from the outer diameter of the tooth tip to the inner diameter of the tooth root. Although the pose errors on the left tooth surface show a similar distribution trend, there are differences in their magnitudes.

[0150] The errors independent of position remain static, while the position-dependent errors change with position, and the TEs exhibit dynamic characteristics. This analysis reveals how the linear axis errors affect the TSEs and provides a basis for error compensation in FGs. According to the measured distribution range of the linear axis errors, the translational error is set within [0, 0.05] mm, and the rotational error is set within [0, 0.0002] radians. Treating each error component of the linear axis as an independent variable, calculate the spatial poses of 45 points on the tooth surface and compare them with the ideal poses to obtain 45 sets of error values. Finally, take the maximum value of the translational error, rotational error, and normal error as the evaluation index.

[0151] As Figure 4 and 5 shown, the spatial errors on the tooth surface (δ xg , δ yg , δ zg , ε xg , ε xg , ε xy ) and the normal error (Δ f) The correlation relationship between them is presented. These errors are transmitted through the machine tool transmission chain, and their values will change before reaching the tooth surface. Figure 4 It shows that the translational error has a consistent effect on the tooth surface pose error and the co-directional normal error. The translational error in a specific direction of the linear axis mainly affects the corresponding direction component of the tooth surface pose error, and the translational errors in the Y and Z directions have a particularly significant impact on the normal error. Figure 5 It shows that the influence laws of the rotational errors of the Y-axis and Z-axis on the tooth surface pose error and the normal error are similar, which is different from the action mechanism of the rotational error of the X-axis. It should be noted that only ε x (Y) and ε x (Z) have a significant impact on the tooth surface normal error, while ε y (Y) and ε y (Z) have a negligible impact on the tooth surface pose and normal errors. Although some position-related error components in the linear axis will continuously affect the tooth surface pose and normal errors, there are differences in the motion commands of each axis during the actual grinding process, resulting in changes in the positions of the motion axes and further causing differences in the magnitudes of the position-related errors. When comprehensively considering the combined effects of all error components on the tooth surface pose and normal errors, it is difficult to directly identify the contribution degree of a single component. In addition, the error components will be coupled and accumulated during the processing, and their impact on the tooth surface accuracy is not a simple algebraic superposition. This complexity makes error tracing and compensation difficult, and the multi-error interaction mechanism within the FGWGMT system is not yet clear. These factors jointly restrict the effectiveness of the error compensation strategy.

[0152] 1.2. Thermal error prediction model

[0153] (1) LSTM neural network

[0154] Figure 6 It shows the LSTM structure, which includes three types of gating units: input gate (i l ), forget gate (f l ), and output gate (o t ). The activation function σ non-linearly maps the input information to the interval [0, 1] to achieve information screening and updating; while the tanh activation function constrains the information within the range [-1, 1] to regulate the input and output streams of each cycle. This mechanism effectively prevents the vanishing or explosion of gradients, ensuring computational stability and rapid convergence.

[0155] The working principle of LSTM is described as follows:

[0156]

[0157] In the formula: x t represents the input; c t-1 and c tRepresent the LSTM cell states at times t-1 and t respectively, and retain historical information as memory units; h t is the output of the LSTM node at time t; W i represents the weight; is the element-wise multiplication of vectors; b i is the bias term; i = 1 to 4. The weights and biases are iteratively updated during model training. The LSTM network can effectively extract short-term and long-term features in time series data through its chain-like recurrent structure and gating mechanism.

[0158] (2) LSTM network with improved gating structure

[0159] The thermal error (TE) is strongly correlated with short-term information and weakly correlated with long-term information. Therefore, the structure of the LSTM network unit is optimized to reduce the dependence on long-term information while enhancing the short-term information representation ability. The improved LSTM unit structure is as Figure 7 shown: First, an attention mechanism is introduced, and the input gate and forget gate are merged into a new gating mechanism to strengthen the relevance of the information flow, synchronize the screening of forgotten information and the update of new information, and make the process more efficient and orderly; Second, a residual connection widely used in deep learning is added to improve the convergence, training performance, and generalization ability; The output gate also integrates the attention mechanism and establishes a residual connection. By weight allocation, while ensuring the long-term feature representation performance, the short-term data representation ability is strengthened.

[0160] The principle of the LSTM network with improved gating structure is:

[0161]

[0162] i t = σ(W i [h t-1 , x t + b i )

[0163]

[0164] m t = σ(W m [h t-1 , x t + b m )

[0165]

[0166] In the formula: represents the weighted input; i i is the input gate; is the output of the input gate; m i is the output gate; c t is the updated state; ht is in the hidden state; W i , W c and W m are weight terms; b i , b c and b m are bias terms; W is the weight value.

[0167] 1.3. Digital Twin - Driven Geometry - Thermal Error Compensation System

[0168] Figure 8 The framework of the digital twin - driven geometry - thermal error compensation system is shown. First, a geometric model is established based on the measurable shape dimensions. Secondly, a multi - domain interaction system including mechanical, electrical control, and thermal systems is constructed to characterize the inherent properties and operating mechanisms of the machine tool. Data is collected through sensing technology, and an TE model is constructed using internal information and historical data as inputs. Finally, it is fed back to the actual machine tool to form a virtual - physical closed - loop.

[0169] 1.3.1 Digital Twin Modeling Method

[0170] (1) Geometric Model Construction

[0171] The geometric model is the first step in creating a digital twin. It supports parametric construction, assembly, and simulation of features. Fidelity and lightweight design are crucial for geometric model construction. FGWGMT mainly consists of components such as the spindle, column, worktable, feed axis, and bed. Physical factors such as inertia, damping, and elastic deformation are not considered during the modeling process. Each machine tool component is simplified as a rigid body through modeling software. The model also accurately reflects the assembly relationship, reference points, and dependencies of the physical FGWGMT, maintaining structural consistency. In addition, to avoid delays caused by excessive memory occupancy during data transmission, the model adopts a lightweight design to achieve high fidelity with the minimum amount of data.

[0172] (2) Physical Model Construction

[0173] After completing the geometric model construction, physical modeling endows the model with internal knowledge and mechanisms to describe the physical characteristics and constraints of the system. Since the machine tool involves multiple fields such as mechanical, electrical, hydraulic, and control, a multi - level precision modeling method needs to be adopted. This object - oriented modeling method can accurately and objectively describe entities in different spatial dimensions or fields, and the multi - level interaction ability of the model in space must be fully considered. By defining the material properties, physical mechanisms, connection relationships, topological structures, functional component constraints, and driving mechanisms of each subsystem through multidisciplinary knowledge, the coupling of the multi - domain models of FGWGMT is realized.

[0174] Adopt a software platform that supports multi-domain modeling to achieve the interaction between mechanical, electrical, control, and thermal systems. These subsystems are integrated through the energy conversion interface between models, and finally form a digital twin model with multi-domain interaction of FGWGMT. Based on the geometric model and multi-domain interaction, link, combine, and integrate multi-dimensional models to construct a comprehensive and high-fidelity virtual machine tool model (such as Figure 9 as shown). The data modeling method will be described in detail in the next section, because the digital twin model needs to be continuously updated and optimized through the data model. To maintain the temporal consistency between the virtual and physical FGWGMT, adopt a unified multi-domain modeling principle to study the interaction relationship of each subsystem, construct mechanical, control, and electrical subsystems, and form a digital twin model with multi-system interaction of FGWGMT. These subsystems integrate multidisciplinary components. Although the physical forms of the models in each domain are different, they can be described by similar mathematical expressions.

[0175] (3) Construction of twin data

[0176] The construction of the FGWGMT twin data model is as Figure 10 shown. Data such as rotational speed, temperature field, thermal error (TE), torque, coordinates, vibration, and ambient temperature are collected and stored in the database. After statistical fitting, redundancy processing, anomaly processing, and feature extraction, static and dynamic data loading mapping is performed, and data fusion is carried out according to principles such as mean weighting and reliability priority. Construct data models of rotational speed, temperature field, TE, torque, coordinates, vibration, and ambient temperature in the digital space, and finally provide services such as TE prediction, clustering analysis, thermal behavior monitoring, data mining, anomaly detection, and thermal information analysis.

[0177] 1.3.2. System operation principle

[0178] The digital twin-driven geometric-thermal error compensation system includes a hierarchical architecture of a sensing control layer, an edge layer, and a cloud layer, as Figure 11 shown.

[0179] (1) Sensing control layer

[0180] The sensing and control layer collects real-time data through temperature sensors (such as DS18B20), vibration sensors (such as PCB Piezotronics 356A16), and displacement sensors (such as Renishaw LM10), etc., providing key hot state information of FGWGMT. The sensor data is captured by a data acquisition card, processed after being converted by an A / D converter, and the Raspberry Pi is used as a gateway to transmit the data. The compensation components are sent from the edge layer to the CNC system, and the servo controller adjusts the servo axis movement accordingly. The real-time feedback of the position sensor ensures that FGWGMT operates within the set parameters to achieve TE compensation. The PLC (such as Siemens S7-1200) manages the machine tool operation, adjusts parameters according to the feedback from the edge layer, and the control software (such as Siemens TIA Portal) is used for programming and designing the control logic to ensure motion accuracy and reduce errors.

[0181] (2) Edge layer

[0182] In the edge layer, the sensor data is processed near the FGWGMT (five-axis grinding machine) to reduce latency. Devices such as NVIDIA Jetson Nano are responsible for local data processing, and edge computing software such as EdgeX Foundry is used to perform real-time analysis and thermal error prediction on the thermal information, enabling the system to make quick decisions without transmitting all data to the cloud, thus improving the system response speed. This layer integrates the virtual mapping of the physical machine tool - the digital twin model, and tools such as Siemens NX or Dassault Systemes SOLIDWORKS are used to perform real-time simulation on FGWGMT. The digital twin model monitors the operating state, provides insights into the machine tool performance, calculates the compensation components through the compensation model at the same time, and regularly updates the thermal error prediction model to reflect changes in thermal behavior, and sends the prediction results to the error compensation model for execution.

[0183] (3) Cloud layer

[0184] The cloud layer uses powerful platforms such as AWS or Microsoft Azure to provide scalable storage and computing resources. Data modeling and optimization are carried out through data analysis tools such as Python, R, and MATLAB, and machine learning libraries such as TensorFlow and PyTorch are used to train the NChOA-LSTM model with improved gates using long-term historical thermal information data as input. Finally, the updated model parameters are sent down to the edge layer to synchronously update the thermal error prediction model used in this layer.

[0185] 1.3.3. Training and updating of the thermal error prediction model

[0186] In the cloud layer, the parameters of the thermal error prediction model are updated using the Nonlinear Chimp Optimization Algorithm (NChOA); the principle of the Nonlinear Chimp Optimization Algorithm is as follows. The position update formula for each chimpanzee is:

[0187] X i = rand × (U b - L b ) + L b

[0188] Where: U b and L b represent the upper and lower bounds of the search space respectively; rand is a random number;

[0189] During the hunting process, the distance D between the chimpanzee and the prey and its position update function X chimp are as follows:

[0190] D = |hX prey (t) - mX chimp (t)|

[0191] X chimp (t + 1) = X chimp (t) - a × D, (μ < 0.5)

[0192] X chimp (t + 1) = m, (μ ≥ 0.5)

[0193] a = 2f × r1 - f

[0194] h = 2 × r2

[0195] Where: r1 and r2 are non - linear random variables within the interval [0, 1]; h is a random number; m is a chaos factor; μ is a condition for judging the position update mode; a is a random variable within the interval [-f, f]; f is a non - linear convergence factor, whose value linearly decreases from 2 to 0 as the iteration number t increases; t max represents the maximum number of iterations; and: when |a| < 1, it indicates that the chimpanzee is approaching the prey position X prey ; when |a| > 1, it means that the chimpanzee is forced to move away from the prey and disperse to a wider area for searching;

[0196] The global optimal solution is represented by X Att , X Bar , X Cga and X Dri , corresponding to the position vectors for attacking, surrounding, driving away, and chasing the prey respectively; the distances D Att , D Bar , D cha and D Dri from these four positions to the prey are jointly determined by the position vectors X Att , X Bar , X Cha and X Dri :

[0197]

[0198] Where: X1, X2, X3, and X4 represent the updated position vectors of four types of chimpanzees; h1, h2, h3, and h4 are the hindrance factors affecting four types of hunting behaviors; m1, m2, m3, and m4 are the corresponding chaotic factors; a1, a2, a3, and a4 are the associated random vectors;

[0199] The hunting behavior of the chimpanzee population is shown as:

[0200]

[0201] When a < 1, the chimpanzee is approaching the prey to prepare for an attack; conversely, a > 1 indicates that it is dispersing to search for prey. The value of a is closely related to the convergence factor f, which controls the local and global search capabilities of the algorithm. However, the linear decay of the convergence factor f is difficult to cope with complex non-linear optimization tasks and is prone to slow convergence speed and falling into local optima. Therefore, NChOA introduces a non-linear transformation strategy to enhance the convergence factor.

[0202] In the initial stage of iteration, the convergence factor decays slowly to enhance the global search ability, and in the later stage, it decays rapidly to promote efficient local optimization. Therefore, NChOA has better optimization performance than linear ChOA. The non-linear convergence factor f proposed in this embodiment is:

[0203]

[0204] Where: f0 represents the initial convergence factor.

[0205] The non-linear convergence factor enables the algorithm to adaptively adjust the iteration process: focusing on global search to accelerate convergence in the early stage and turning to local fine optimization near convergence to improve the optimization accuracy.

[0206] 1.3.4. Error Compensation Model

[0207] The edge layer calculates the compensation component through the error compensation model. The error compensation model proposed in this embodiment includes a first-order error decoupling and traceability model and a second-order error traceability model.

[0208] (1) First-order Error Decoupling and Traceability Model

[0209] The differential motion of the cumulative pose error of the machine tool can be expressed as:

[0210]

[0211] This relational expression is derived by combining the differential motion error separation of the motion axis and the equivalent error decoupling analysis of the motion direction:

[0212] T Gt Δ GT =T m-C ΔC1 T mX T mZ T mA T mY T mB

[0213] +T m-C T mX Δ X1 T mZ T mA T mY T mB

[0214] +T m-C T mX T mZ Δ Z1 T mA T mY T mB

[0215] +T m-C T mX T mZ T mA Δ A1 T mY T mB

[0216] +T m-C T mX T mZ T mA T mY Δ Y1 T mB

[0217] +T m-C T mX T mZ T mA T mY T mB Δ B1

[0218] In the above formula, the equality of the left and right matrices means that the elements in the corresponding positions are equal. From this, twelve equations can be established, including six unknowns of the equivalent decoupling quantities of the six motion axes, so as to determine the six equivalent error decoupling quantities in the motion direction:

[0219] e x =u x cosB+u z sinB+Yv z cosB-Yv x sinB

[0220] e y =(u y cos 2 A-Xvz cosB + Xv x sinB - (u z -Yv x )cosAsinAcosB + (u x +Yv z )cosAsinAsinB) / cos 2 A

[0221] e z = -((u x +Yv z )cosAsinB - (u z -Yv x )cosAcosB - Xv z sinAcosB + Xv x sinAsinB) / cos 2 A)

[0222] e A =v x cosB + v z sinB

[0223] e B =(v y cosA - v z sinAcosB + v x sinAsinB) / cosA

[0224] e C =(v z cosB - v x sinB) / cosA

[0225] Among them: e x 、e y 、e z 、e A 、e B and e C are the equivalent error decoupling amounts in the directions of the X-axis, Y-axis, Z-axis, A-axis, B-axis, and C-axis respectively; u x 、u y and u z are the position errors of the tool / workpiece in the X / Y / Z directions respectively; v x 、v y and v z are the angular errors of the tool / workpiece in the X / Y / Z directions respectively; B, Y, A, Z, X, and C represent the motion commands of the B, Y, A, Z, X, and C axes respectively.

[0226] (2) Second-order error traceability model

[0227] Based on this envelope linkage relationship, the equivalent error of the motion axis needs to be traced secondarily to determine the actual compensation component. The linkage ratio between the X-axis and the Y-axis is:

[0228]

[0229] In the formula: λ w represents the helix angle.

[0230] When the position of the Y-axis changes by Δy, it will cause a synchronous change in the position of the X-axis:

[0231] Δx = -Δy cot λ w

[0232] Therefore, the final compensation component of the X-axis is determined as:

[0233] X C = -(e x - e y cot λ w )

[0234] Similarly, the correlation relationship between the B-axis and the C-axis is expressed as:

[0235]

[0236] During actual machining, a change in the position Δb of the B-axis will cause a synchronous change in the position of the C-axis:

[0237] Δc = i gw Δb

[0238] The final compensation component of the C-axis is:

[0239] C C = -(e C + i gw e B )

[0240] Since the A-axis is always locked at the zero position during movement, there is no need to compensate the A-axis.

[0241] The compensation component of the Z-axis is:

[0242] Z C = -e C

[0243] The compensation components of each motion axis are finally determined as

[0244]

[0245] Among them: X C 、Z C and C C are the actual compensation components of the X-axis, Z-axis, and C-axis respectively; λ w is the helix angle; igw Represents the correlation between the B-axis and the C-axis.

[0246] By discretizing the motion trajectory in the grinding process, the compensation components at discrete positions are calculated. Subsequently, these components are fitted into a smooth motion trajectory using a spline curve, and the interpolation method is used to determine the motion compensation amount at any point on the trajectory. Finally, the preset value is converted into the pitch compensation amount to realize the automatic adjustment of the motion axis positioning for the purpose of geometric thermal error (GTE) compensation.

[0247] II. Experimental Verification

[0248] 2.1. Experimental Setup and Measurement Process

[0249] In this embodiment, the FGWGMT (five-axis grinding machine) is used as the core experimental object, and its structure is optimized to meet the requirements of precision grinding. The machine tool is equipped with a Siemens 840Dsl system and uses an electronic gearbox for synchronization to achieve multi-axis control. The geometric errors of the linear axes are measured by an XL-80 laser interferometer. The position of the mirror assembly during the measurement is shown in Figure 12 , and the measurement results are shown in Figure 13 . The thermal behavior of the linear axes of the machine tool is measured by the equipment shown in Figure 14 . The working conditions are shown in Table 3, and the measurement data are shown in Figure 15 . The thermal behavior of the B-axis is measured according to the Figure 16 scheme, and the results are shown in Figure 17 .

[0250] Table 3 Different Working Conditions

[0251]

[0252] 2.1.1. Verification of the Thermal Error Model

[0253] Before training the thermal error prediction model, the NChOA parameters L b and U b are set with the upper and lower bounds of 10 and 500 respectively, the data dimension d is set to 3, the population size is 3, the maximum number of iterations is 4 times, and the optimization interval is [10, 500]. The hyperparameters of the thermal error model used in the ablation experiment and the comparative experiment are shown in Table 4. The drastic change of the thermal error poses high requirements on the model's ability, and each model is trained using the training dataset. As shown in Figure 18 (a), the performance of the NChOA-LSTM with an improved gate is better than that of the LSTM with an improved gate and the standard LSTM, and the BP neural network performs the worst, verifying the effectiveness of the improved gate and NChOA - the performance of the LSTM with an improved gate is significantly better than that of the standard LSTM. Figure 18(b) It shows that the residual fluctuations of the BP network are higher than those of the other four types of models, and the residual fluctuations of the standard LSTM are greater than those of the LSTM with improved gates and NChOA-LSTM. The high fitting accuracy of the LSTM series models stems from their strong time series modeling ability. Research shows that LSTM can effectively model the error mechanism, and the improved gates further strengthen the time series characteristics. The BP network lacking memory function cannot store long-term thermal history information, highlighting the importance of time series modeling for error prediction. Figure 18 It is the training result of operating condition 1#.

[0254] Table 4 Model hyperparameters

[0255]

[0256] The parameters are shown in Table 5. The NChOA-LSTM with improved gates has the highest fitting accuracy, and both it and the LSTM with improved gates are superior to the BP network and the standard LSTM. Using ChOA to optimize the parameters can ensure a good match between the LSTM parameters and the characteristics of the thermal information data, thus improving the prediction accuracy. It is worth noting that the accuracy of the LSTM with improved gates exceeds that of the standard LSTM, once again proving the effectiveness of the improved gates.

[0257] Table 5 Training result evaluation

[0258]

[0259] 2.1.2. Prediction performance

[0260] According to Figure 19 As shown, the application of thermal information data in the historical state is crucial. For TE prediction, we draw a similar conclusion: the memory performance is far more important than parameter optimization. The evaluation parameters of each model are shown in detail in Table 6. The prediction accuracies of the BP neural network, LSTM, improved gated LSTM, and NChOA improved gated LSTM are 93.85%, 94.74%, 96.35%, and 97.62% respectively. After using the NChOA algorithm, the prediction performance of the improved gated LSTM is improved from 96.35% to 97.62%. The separate prediction accuracy of the improved gated LSTM is 96.38%, which is lower than that of the NChOA improved gated LSTM. This significant improvement verifies the effectiveness of the NChOA algorithm. By comparing Table 3 and Table 4, it can be found that the accuracy decay of the improved gated LSTM and the NChOA improved gated LSTM is the smallest under different operating conditions, while the accuracy decline of the BP network and the traditional LSTM is more obvious, indicating that they have stronger robustness. The prediction accuracy of the NChOA improved gated LSTM is slightly higher than that of the improved gated LSTM, so it is an ideal model for TE prediction.

[0261] Table 6 Prediction result evaluation

[0262]

[0263]

[0264] 2.3, Face Gear Grinding Verification

[0265] In the comparative experiment, the same worm grinding wheel tool was used, and the material was a vitrified bond grinding wheel material. The control group of the face gear (FG) was a gear set with the same machining allowance reserved, and all were machined on the FGWGMT machine. The gear material was AISI E9310 alloy steel, and the specific parameters are shown in Table 7.

[0266] Table 7 Parameters of Worm Grinding Wheel and Face Gear (FG)

[0267]

[0268] The FG grinding experimental device is as Figure 20 shown. The linear axis moves rapidly during the idle stroke stage and slowly during the grinding stage. The single feed rates for rough grinding, semi-finishing grinding, and finish grinding are 20μm, 10μm, and 5μm respectively.

[0269] There are significant differences between FG gears and cylindrical gears and bevel gears. The accuracy evaluation criteria for cylindrical gears and bevel gears are not applicable to FG gears. Therefore, the existing standards for bevel gears are often referred to evaluate the geometric accuracy of the FG tooth surface. The TSE measurement of FG gears mainly targets the working tooth surface, and special methods and equipment are required to adapt to its unique characteristics. According to the AGMA 2009-B01-2001

[29] standard, the measurement area is defined as a 5×9 grid (as Figure 21 shown), and this grid is divided according to a 5% shrinkage rate in the tooth height direction and a 10% shrinkage rate in the tooth width direction.

[0270] Through the rotation projection method, the measurement grid and measurement points are projected onto the symmetric plane xOz of two symmetric tooth surfaces in the tooth groove where the measured tooth surface is located (as Figure 22 shown). Among them, z d1 and z d2 respectively represent the minimum and maximum values of the tooth surface measurement points in the tooth height direction (corresponding to the tooth root height and tooth tip height); a i and z ijare are the coordinates of the measurement point G(i,j) in the x and z directions; M, N, J, and K are the four boundary points of the measurement grid, and their x-direction coordinate values are a min and a max

[0271]

[0272] In the formula: Rmin and Rmax respectively represent the inner radius and outer radius of the measured FG gear.

[0273] Along the tooth width direction, the interval [amin , a max equally divided, the x - coordinate a of the measurement point at the i - th th percentile is given by the following formula i as follows

[0274]

[0275] The x - direction and z - direction coordinates of the points on the transition curve are determined by the following formula

[0276]

[0277] Thus, the upper and lower boundaries of the grid in the tooth height direction are

[0278]

[0279] In the tooth height direction (z - axis), the interval [z i5 , z i1 is equally divided, and finally

[0280]

[0281] The coordinates of the measurement point G(i, j) are denoted as a i and z ij . Operate according to the following steps: First, perform GTE suppression processing, and then start the gear grinding program to complete FG grinding with suppression. Measure and compare the TSE values of the gears with and without GTE suppression respectively. The measured tooth surface geometric accuracy of the FG gear after machining is as Figure 23 shown. When measuring, import the theoretical measurement point data into the measurement module. First, calibrate the coordinate measuring machine and establish the positioning reference and measurement coordinate system. The probe measures each point according to the planned path and moves along the normal direction near the measurement point to capture the machining error. Generate a topography error map of 45 measurement points on the tooth surface according to the normal error data.

[0282] The distribution of the normal error of the tooth surface before and after taking compensation measures is as Figure 24 shown. The maximum normal error of the left - hand tooth surface drops from 21.8 μm to 5.9 μm, and the length of the error distribution interval shortens from 25.7 μm to 6.6 μm; the maximum normal error of the right - hand tooth surface decreases from 22.9 μm to 5.7 μm, and the length of the error distribution interval reduces from 30.7 μm to 9.7 μm.

[0283] 2.4. Compensation efficiency

[0284] According to the data in Table 8, when using the improved gated NChOA-LSTM, improved gated LSTM, standard LSTM, and BP network as error prediction models, the calculation times of the compensation system are 126 seconds, 73 seconds, 184 seconds, and 105 seconds respectively. At a 5-minute processing beat, all four models can complete error prediction within the compensation period. Among them, the compensation system using the improved gated LSTM as the control model has the shortest execution time, which benefits from omitting the initial parameter optimization step of NChOA, and the convergence speed of the improved gated LSTM is better than that of the standard LSTM.

[0285] Table 8 Compensation system efficiency under different prediction models

[0286]

[0287] The sampling frequency of temperature and error data is set to 0.01 Hz, and the total data volume collected within half a year is 29.6 GB. In this study, a sensing and control edge-cloud framework was designed to improve the execution efficiency of the system. The sensing and control cloud framework was used as a comparison framework. The two frameworks were used to process data, and the processed data volumes are shown in Table 9.

[0288] Table 9 Data volume processed by the thermal error compensation system

[0289]

[0290] The execution times and efficiencies of the two system frameworks are shown in Table 10. The execution time of the sensing control-edge-cloud framework is 232 seconds, and that of the sensing control-cloud framework is 389 seconds. The proposed system framework requires the least time, verifying its rationality. The edge layer in the sensing control-edge-cloud framework is close to the FGWGMT and carries the TE prediction and compensation models. These models do not require a large amount of computing resources. The cloud server HP Z240 Tower workstation located in the cloud layer provides sufficient computing power to meet these needs. For the sensing control-cloud framework, although the time of the edge layer is zero, the data transmission and cloud layer times increase significantly. In addition, the increase in the cloud layer workload results in the running time increasing from 43 seconds to 228 seconds. For the proposed sensing control-edge-cloud framework, the total execution time is 59.6% of that of the sensing control-cloud framework. Therefore, the designed compensation system framework is reasonable. The functions and tasks are well divided among different layers. The total execution times of both frameworks are less than five minutes. In addition, the performance of the sensing control-edge-cloud framework is faster than that of the sensing control-cloud framework, confirming that the former design is more efficient.

[0291] Table 10 Execution time and efficiency

[0292]

[0293] III. Conclusion

[0294] In this embodiment, the mapping relationship between geometric thermal error (GTE) and final thermal induced error (TSE) is established. A thermal error prediction model is constructed based on the non-linear chimp optimization algorithm-long short-term memory network (NChOA-LSTM), and a digital twin-driven GTE error suppression system is developed. This system integrates prediction and compensation models. By applying the digital twin-driven GTE error suppression system,

[0295] (1) For the functionally graded worm grinding machine tool (FGWGMT), the mapping relationship between GTE and TSE is established, the influence law of translational and rotational errors on TSE is studied, and a GTE compensation model is developed to analyze the action mechanism of errors on TSE.

[0296] (2) A thermal error prediction model is constructed based on NChOA-LSTM. In particular, NCohA-LSTM with an improved gating structure is proposed to capture time series features, and NCohA is used for multi-objective optimization of hyperparameters. The prediction accuracy of the non-linear chimp optimization algorithm-LSTM model with the improved gating structure reaches 97.62%.

[0297] (3) A digital twin-driven GTE error control system is developed, which reduces the left TSE from 21.8 μm to 5.9 μm and the right TSE from 22.9 μm to 5.7 μm. After adopting the sensing control-edge-cloud computing framework, the system execution time is shortened to 59.6% compared with the traditional sensing control-cloud framework.

[0298] The above-described embodiments are only preferred embodiments given to fully illustrate the present invention, and the protection scope of the present invention is not limited thereto. Equivalent substitutions or transformations made by those skilled in the art on the basis of the present invention are all within the protection scope of the present invention. The protection scope of the present invention shall be subject to the claims.

Claims

1. A geometric-thermal error compensation system for face gear worm grinders based on digital twin, characterized in that: A hierarchical architecture including a sensing control layer, an edge layer, and a cloud layer; The sensing control layer is used to collect data in real time to provide thermal information data of the face gear worm grinding machine; A digital twin model and a thermal error prediction model of the face gear worm grinding machine are deployed inside the edge layer; the thermal error prediction model performs real-time thermal error prediction based on the thermal information data collected by the sensing control layer, and inputs the obtained thermal error prediction data into the digital twin model; a geometric-thermal error mapping model and an error compensation model are integrated in the digital twin model, and the digital twin model performs real-time simulation based on the thermal error prediction data and the geometric-thermal error mapping model to obtain tooth surface error, and calculates the compensation component through the error compensation model; The cloud layer trains the thermal error prediction model with historical thermal information data, and sends the updated model parameters to the edge layer to synchronously update the thermal error prediction model; The geometric-thermal error mapping model is constructed based on the mapping relationship between geometric-thermal error and tooth surface error in the grinding machine motion chain; and the construction method of the error mapping model is: Construct a theoretical forward kinematic transformation matrix from the grinding worm to the workpiece; Calculate the actual homogeneous coordinate transformation matrix from the grinding worm to the workpiece under error conditions; Derive a tool / workpiece attitude error model based on the difference between the theoretical transformation matrix and the actual transformation matrix; Combine the thermal contact transformation matrix and the forward kinematic matrix of the grinding process to establish the mapping relationship between tooth surface error and geometric-thermal error; The thermal error prediction model is constructed based on an LSTM network with an improved gating structure. The LSTM network with the improved gating structure introduces an attention mechanism and combines the input gate and the forget gate into a joint gating mechanism. By adding residual connections and weight allocation, while ensuring the long-term feature representation performance, it strengthens the short-term data representation ability.

2. The geometric-thermal error compensation system of a face gear worm grinding machine based on digital twin according to claim 1, characterized in that: The tool / workpiece attitude error model is: Where: [a, b, c, 1] T and [i, j, k, 0] T are the position vector and the attitude vector respectively; δ w and ε w represent the position error vector and the attitude error vector of the worm grinding wheel respectively; δ xw , δ yw and δ zw denote the elements of δ w ; ε xw , ε yw and ε zw denote the elements of ε w ; is the actual homogeneous coordinate transformation matrix from the grinding worm wheel to the workpiece; T mGT is the theoretical forward kinematic transformation matrix from the grinding worm wheel to the workpiece.

3. The geometric-thermal error compensation system for face gear worm grinding machines based on digital twins according to claim 1, characterized in that: The formed grinding tooth surface equation containing errors is expressed as: Wherein: represents the error-containing tooth surface equation; is the normal vector; E e represents the set of all errors; is the forward kinematics matrix with error, and its value is l w and E e function; and are the new meshing conditions; The mapping relationship between tooth surface error and geometric-thermal error is: Where: r xw 、r yw and r zw represent the position vector components of the grinding wheel; i w 、j w and k w represent the vector components of the grinding wheel; E δf is the position error vector; δ xg 、δ yg and δ zg are the position error components; E ef is the attitude error vector; ε xg 、ε yg and ε zg represent the attitude error components.

4. The geometric-thermal error compensation system for face gear worm grinding machine based on digital twin according to claim 1, characterized in that: The principle of the LSTM network with the improved gating structure is: i t = σ(W i [h t-1 , x t + b i ) m t = σ(W m [b t-1 , x t + b m ) Wherein: represents the weighted input; i i is the input gate; is the output of the input gate; m i is the output gate; c t is the updated state; h t is the hidden state; W i 、W c and W m are weight terms; b i 、b c and b m are bias terms; W is the weight value.

5. The geometric-thermal error compensation system of the face gear worm grinding machine based on digital twin according to claim 1, characterized in that: In the cloud layer, a non-linear chimpanzee optimization algorithm is used to update the parameters of the thermal error prediction model; the principle of the non-linear chimpanzee optimization algorithm is: The position update formula for each chimpanzee is: X i = rand × (U b - L b ) + L b Where: U b and L b represent the upper and lower bounds of the search space respectively; rand is a random number; During the hunting process, the distance D between the chimpanzee and the prey and its position update function X chimp are as follows: D = |hX prey (t) - mX chimp (t)| X chimp (t + 1)=X chimp (t)-a×D, (μ < 0.5) X chimp (t + 1) = m, (μ ≥ 0.5) a = 2f × r1 - f h＝2×r2 Where: r1 and r2 are non-linear random variables within the interval [0, 1]; h is a random number; m is a chaotic factor; μ is a condition for judging the position update mode; a is a random variable within the interval [-f, f]; f is a non-linear convergence factor, whose value linearly decreases from 2 to 0 as the iteration number t increases; t max represents the maximum number of iterations; and: when |a| < 1, it indicates that the chimpanzee is approaching the prey position X prey ; when |a| > 1, it means that the chimpanzee is forced to move away from the prey and disperse to a wider area for searching; The group optimal solution is represented by X Att , X Bar , X Cha and X Dri , corresponding to the position vectors of attacking, surrounding, driving away, and chasing prey respectively; the distances D Att , D Bar , D cha and D Dri are jointly determined by the position vectors X Att , X Bar , X Cha and X Dri : Where: X1, X2, X3, and X4 represent the position vectors of the four types of chimpanzees after update; h1, h2, h3, and h4 are the obstacle factors affecting the four types of hunting behaviors; m1, m2, m3, and m4 are the corresponding chaotic factors; a1, a2, a3, and a4 are associated random vectors; The hunting behavior of the chimpanzee population is: The non-linear convergence factor f is: Where: f0 represents the initial convergence factor.

6. The geometric-thermal error compensation system of the face gear worm grinding machine based on digital twin according to claim 1, characterized in that: The error compensation model includes a first-order error decoupling and tracing model and a second-order error tracing model.

7. The geometric-thermal error compensation system for face gear worm grinding machine based on digital twin according to claim 6, wherein: The first-order error decoupling and tracing model determines six equivalent error decoupling quantities in the motion direction: e x = u x cos B + u z sin B + Yv z cos B - Yv x sin B e y = (u y cos 2 A - Xv z cosB + Xv x sinB - (u z -Yv x )cosAsinAcosB + (u x +Yv z )cosAsinAsinB) / cos 2 A e z = -((y x + Yv z ) cos A sin B - (u z - Yv x ) cos A cos B - Xv z sin A cos B + Xv x sin A sin B) / cos 2 A e A = v x cos B + v z sin B e B = (v y cosA - v z sinAcosB + v x sinAsinB) / cosA e C = (v z cos B - v x sin B) / cos A where: e x , e y , e z , e A , e B and e C are the equivalent error decoupling amounts in the directions of the X-axis, Y-axis, Z-axis, A-axis, B-axis, and C-axis respectively; u x , u y and u z are the position errors of the tool / workpiece in the X / Y / Z directions respectively; v x , v y and v z are the angular errors of the tool / workpiece in the X / Y / Z directions respectively; B, Y, A, Z, X, and C represent the motion commands of the B, Y, A, Z, X, and C axes respectively.

8. The geometric-thermal error compensation system of a face gear worm grinding machine based on digital twin according to claim 6, characterized in that: The second-order error tracing model determines the actual compensation component: Wherein: X C , Z C and C C are the actual compensation components of the X-axis, Z-axis and C-axis respectively; λ w is the helix angle; i gw represents the correlation between the B-axis and the C-axis.

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