Digital Twin-Based Geometry-Thermal Error Compensation System for Face Gear and Worm Grinding Machines

Abstract

This invention discloses a geometry-thermal error compensation system for a face gear worm grinder based on digital twins, comprising a geometry-thermal error mapping model, a thermal error prediction model, and a digital twin-driven geometry-thermal error compensation system. The geometry-thermal error mapping model is constructed based on the mapping relationship between geometry-thermal errors and tooth surface errors in the grinding machine's kinematic chain. The thermal error prediction model is constructed based on an improved gated LSTM network. The digital twin-driven geometry-thermal error compensation system includes a layered architecture of a sensing and control layer, an edge layer, and a cloud layer. The sensing and control layer is used to collect data in real time to provide thermal information data of the face gear worm grinder. The edge layer deploys the thermal error prediction model for real-time thermal error prediction and integrates it with the geometry-thermal error mapping model. The cloud layer trains the thermal error prediction model using historical thermal information data and sends the updated model parameters to the edge layer to synchronously update the thermal error prediction model.

Description

Technical Field

[0001] This invention belongs to the field of machining error compensation technology, specifically relating to a geometric-thermal error compensation system for a face gear and worm gear grinding machine based on digital twins. Background Technology

[0002] Face gear (FG) transmission systems are characterized by high interchangeability, strong load-bearing capacity, and efficient power distribution. They can realize motion and power transmission between intersecting or staggered shafts and are expected to replace bevel gears in military helicopters, hovercraft, and tanks. As the final machining process after roughing, worm grinding plays a decisive role in achieving the geometric accuracy of FG and directly affects the service performance of the transmission system. The face gear worm grinder (FGWGMT) is a key piece of equipment to ensure machining accuracy and efficiency, but its geometric and thermal errors (GTEs) can significantly reduce machining accuracy, impair FG geometric accuracy, and affect the overall performance of the transmission system. Therefore, adopting effective error suppression methods is crucial to improving machining accuracy.

[0003] Improving machine tool machining accuracy relies on effective error suppression, which is based on error modeling. Typically, the machine tool's motion axes are treated as rigid bodies, and a topological model of the machine tool's transmission chain is established using multibody system theory. Then, a mapping model between motion axis errors and machining errors is constructed through homogeneous coordinate transformation. This method can elucidate the error propagation mechanism within the transmission chain and reveal the specific impact of each error source on machining accuracy. Other scholars have proposed a method for suppressing machine tool thermal errors (TE).

[0004] Current research on geometric thermal error (GTE) modeling for FGWGMT is still in its early stages. Most existing models are limited to analyzing only a few error components. Machine tool thermal errors mainly originate from the temperature rise of key components, including bearings, servo motors, and ball screw pairs. During operation, heat is continuously dissipated through convection and radiation, gradually bringing the temperature gradient towards thermal equilibrium. However, if the rate of heat generation consistently exceeds the heat dissipation capacity, heat accumulation occurs, leading to significant thermal deformation. Traditional empirical models typically employ methods such as multiple linear regression, random forests, and support vector machines, using temperature-thermal error data as input and output variables for prediction, but their accuracy and robustness are unsatisfactory. To overcome these limitations, researchers have begun applying deep learning models such as Long Short-Term Memory (LSTM), Gated Recurrent Units (GRU), and Bilinear Temporal Convolutional Networks (BTCN) to capture dynamic thermal characteristics. Despite progress in various machine tool thermal error modeling approaches, systematic research on FGWGMT remains insufficient. Complex and ever-changing working conditions pose severe challenges to the prediction accuracy and robustness of models. Existing models are difficult to adapt to the robustness requirements under varying working conditions, and there is an urgent need to optimize the matching relationship between thermal data and model hyperparameters. At the same time, the LSTM network structure also needs to be improved.

[0005] Besides thermal error prediction, real-time performance is crucial for the effectiveness of error compensation systems. However, the computational complexity of deep learning models limits real-time performance. Digital twin technology offers an innovative solution, and research shows it can effectively improve the development efficiency of error compensation systems. Lightweight compensation systems built with digital twin support can significantly enhance real-time performance. When integrating thermal error prediction and compensation models into the system, careful design of the system architecture and functional modules at each level is necessary; this structured design is key to maintaining system response speed. The FG grinding process involves complex coupling effects of multiple error components, making error modeling and compensation for FG GMT particularly difficult. The unique kinematic chain structure of the machine tool further increases the complexity of grinding linkage, while the interactive propagation mechanism of geometric thermal errors within the transmission chain remains unclear. Current research on geometric thermal error compensation models is scarce, and the equivalent motion error calculated based on error sourcing has biases, resulting in poor error compensation effects for FG grinding machines and hindering the improvement of machining accuracy.

[0006] The interaction between two types of errors in FGWGMT makes compensation for geometric and thermal errors extremely challenging. The mapping relationship between tooth surface errors (TSEs) and geometric and thermal errors is not yet clear, and the quantitative impact of each error component on the tooth surface is difficult to determine. These errors are transmitted through the drive train, forming 24 positional errors in the tool / workpiece system, ultimately leading to tooth surface deviations. Furthermore, tracing the machining errors back to the servo axis remains a challenge. Thermal errors exhibit time-varying, nonlinear, and unsteady-state characteristics, making accurate modeling and prediction difficult, thus hindering the achievement of high-precision and reliable thermal error compensation and limiting the improvement of grinding accuracy. The thermal error prediction model must be able to handle these complex characteristics, while the compensation system must guarantee real-time performance; currently, the system response speed is still insufficient. The main challenges are as follows: (1) Existing research has not yet elucidated the mapping mechanism between tooth surface error and geometric thermal error. Geometric thermal error is often modeled independently, and the understanding of its interactive propagation law in the FGWGMT transmission chain is limited. This approach is difficult to fully reflect its combined influence on machining accuracy, and a complete geometric thermal error compensation model has not yet been established. Existing research ignores the non-rotational characteristics of worm gear grinding tools and only considers the tool tip spatial error. Although the tool / workpiece system pose error is taken into account, the non-rotational characteristics of the tool and the grinding meshing principle are not considered. Therefore, a more refined and comprehensive FGWGMT compensation model is needed.

[0007] (2) Although various machine tool thermal error modeling technologies have matured, there is still a lack of specialized research on FGWGMT. Due to the complex and variable operating conditions, traditional statistical models and data-driven models struggle to achieve both high accuracy and robustness. FGWGMT is affected by the coupling of multiple factors, which further reduces the accuracy and robustness of the models. Therefore, it is necessary to develop innovative thermal error prediction models that can capture the temporal characteristics of thermal information.

[0008] (3) Although there have been studies on the suppression of geometric-thermal error (GTE) in cylindrical gear grinding, there is still a lack of relevant studies on form grinding machines (FGWGMT). Existing suppression models mainly focus on tool space error and calculate 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 gear and the workpiece, resulting in poor GTE suppression effect. Summary of the Invention

[0009] In view of this, the purpose of the present invention is to provide a geometric-thermal error compensation system for a face gear worm grinding machine based on digital twins, which can simultaneously realize geometric and thermal error compensation to improve the machining accuracy of face gears and improve the real-time performance of error compensation.

[0010] To achieve the above objectives, the present invention provides the following technical solution: A geometric-thermal error compensation system for a face gear and worm gear grinding machine based on digital twins, comprising a layered architecture of a sensing and control layer, an edge layer, and a cloud layer; The sensing and control layer is used to collect data in real time to provide thermal information data of the face gear and worm grinding machine; The edge layer contains a digital twin model and a thermal error prediction model of a face gear and worm gear grinding machine. 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 the obtained thermal error prediction data is input into the digital twin model. The digital twin model integrates a geometry-thermal error mapping model and an error compensation model. The digital twin model performs real-time simulation based on the thermal error prediction data and the geometry-thermal error mapping model to obtain the tooth surface error, and calculates the compensation component through the error compensation model. The cloud layer trains the thermal error prediction model using historical thermal information data, and sends the updated model parameters to the edge layer to synchronously update the thermal error prediction model. The geometry-thermal error mapping model is constructed based on the mapping relationship between geometry-thermal errors and tooth surface errors in the grinding machine kinematic chain; and the method for constructing the error mapping model is as follows: Construct the theoretical positive kinematic transformation matrix from the grinding worm gear to the workpiece; Calculate the actual homogeneous coordinate transformation matrix from the grinding worm wheel to the workpiece under error conditions; Based on the difference between the theoretical transformation matrix and the actual transformation matrix, the tool / workpiece posture error model is derived. By combining the thermal contact transformation matrix and the positive kinematic matrix of the grinding process, a mapping relationship between tooth surface error and geometric-thermal error is established. The thermal error prediction model is constructed based on an improved gating structure LSTM network. The improved gating structure LSTM network introduces an attention mechanism and merges the input gate and forget gate into a joint gating mechanism. By adding residual connections and weight allocation, it enhances the short-term data representation capability while ensuring the long-term feature representation performance.

[0011] Furthermore, the tool / workpiece attitude error model is as follows: In the formula: and These are the position vector and the attitude vector, respectively. and These represent the position error vector and attitude error vector of the worm grinding wheel, respectively; , and express Element; , and express Element; This is the actual homogeneous coordinate transformation matrix from the grinding worm gear to the workpiece; This represents the theoretical positive kinematic transformation matrix from the grinding worm gear to the workpiece.

[0012] Furthermore, the equation for the formed grinding tooth surface, including errors, is expressed as: in: Represent the equation of the tooth surface containing errors; It is the normal vector; The set representing all errors; Let be the positive kinematics matrix containing the error, and its value is... , and The function; and For the new meshing conditions; The mapping relationship between tooth surface error and geometric-thermal error is as follows: in: , and Represents the position vector components of the grinding wheel; , and The vector components representing the grinding wheel; This is the position error vector; , and It is the position error component; This is the attitude error vector; , and This represents the attitude error components.

[0013] Furthermore, the principle of improving the gated structure of LSTM networks is as follows: In the formula: Indicates input; Indicates weighted input; For input gates; The output of the input gate; For output gate; This is the updated state; It is in a hidden state; , and As a weighted term; , and For bias terms; For weights.

[0014] Furthermore, within the cloud layer, the parameters of the thermal error prediction model are updated using a nonlinear chimpanzee optimization algorithm; the principle of the nonlinear chimpanzee optimization algorithm is as follows: The formula for updating the position of each chimpanzee is: in: and These represent the upper and lower bounds of the search space, respectively; rand is a random number. During the hunt, the distance between the chimpanzee and its prey and its location update letter for: in: and It is an interval Nonlinear random variables within; It is a random number; It is a chaotic factor; These are the conditions used to determine the location update mode; It is an interval Random variables within; It is a nonlinear convergence factor, the value of which varies with the number of iterations. The increase decreases linearly from 2 to 0; Indicates the maximum number of iterations; and: when This indicates that the chimpanzee is approaching its prey. ;when This indicates that the chimpanzees were forced to move away from their prey and disperse to a wider area to search; The optimal solution for the group is , , and This represents the position vectors corresponding to attacking, surrounding, driving away, and chasing the prey, respectively; the distances of these four positions from the prey. , , and From position vector , , and Joint decision: In the formula: , , and This represents the updated position vectors of the four types of chimpanzees; , , and These are the hindering factors influencing the four types of hunting behaviors; , , and It is the corresponding chaos factor; , , and For associated random vectors; The hunting behavior of chimpanzee populations is characterized by: Nonlinear convergence factor for: in: This represents the initial convergence factor.

[0015] Furthermore, the error compensation model includes a first-order error decoupling and source tracing model and a second-order error source tracing model.

[0016] Furthermore, the first-order error decoupling source tracing model determines six equivalent error decoupling quantities in the direction of motion: in: , , , and These are the equivalent error decoupling values ​​in the X, Y, Z, A, B, and C axis directions, respectively. , and These represent the positional errors of the tool / workpiece in the X / Y / Z directions, respectively. , and These represent the angular errors of the tool / workpiece in the X / Y / Z directions, respectively. These represent motion commands for the B, Y, A, Z, X, and C axes, respectively.

[0017] Furthermore, the second-order error source tracing model determines the actual compensation components: in: , and These are the actual compensation components for the X-axis, Z-axis, and C-axis, respectively. The helix angle; This indicates the relationship between the B-axis and the C-axis.

[0018] The beneficial effects of this invention are as follows: This invention presents a geometry-thermal error compensation system for face gear and worm gear grinding machines based on digital twins. First, it establishes the mapping relationship between total transmission error (TSE) and geared error (GTE) and constructs a GTE suppression model. Then, it proposes an LSTM network based on a gated structure to predict transmission error (TE). Finally, it designs a lightweight digital twin-driven geometry-thermal error compensation system that can simultaneously achieve geometry and thermal error compensation to improve the machining accuracy of face gears and enhance the real-time performance of error compensation.

[0019] The main innovations and technical effects of this invention are as follows: (1) The TSE-GTE correlation model of FGWGMT was established for the first time. The influence mechanism of the meshing relationship between the worm gear and the workpiece on TSE was revealed by the motion axis error transmission mapping. Finally, a comprehensive GTE suppression model covering multi-axis linkage compensation was formed.

[0020] (2) A prediction model based on an improved gating structure NChOA-LSTM is proposed: the correlation of information flow processing is enhanced by a novel gating mechanism that integrates the input gate and the forget gate, and the prediction accuracy reaches 97.62% by combining NChOA hyperparameter optimization, and it also has working condition robustness.

[0021] (3) Construct a lightweight digital twin drive system, integrating GTE suppression and TE prediction models. By adopting an edge-cloud architecture, the system execution time is reduced to 55.6% of that of the sensor control-cloud architecture, and the TSE on the left side is reduced from 21.8μm to 5.9μm, and the TSE on the right side is reduced from 22.9μm to 5.7μm. Attached Figure Description

[0022] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration: Figure 1 The kinematic chain for a face gear and worm gear grinding machine; Figure 2 This is a schematic diagram illustrating the impact of errors on the actual trajectory. Figure 3 For the distribution of error components on the left tooth surface (when hour); Figure 4 The variation law of grinding error with the translational error related to the position of the linear axis; Figure 5 The variation law of grinding error with linear axis position-related rotational error; Figure 6 For existing LSTM cell structures; Figure 7 It is an improved LSTM cell structure; Figure 8 A framework for digital twin modeling and GTE suppression applications; Figure 9 For building digital twin systems; Figure 10 Construction of FGWGMT twin data model; Figure 11 This is a schematic diagram of the system's operating principle. Figure 12 For geometric error measurement; Figure 13 Geometric errors of the X, Y, and Z axes; Figure 14 Location of temperature measuring points; Figure 15 X-axis thermal behavior data; Figure 16 Measurement of linear axis thermal behavior; Figure 17This is the thermal information data for the B-axis under operating condition #1; Figure 18 For working conditions The training results; Figure 19 For working conditions Test results; Figure 20 This is the grinding test site; Figure 21 For the tooth surface measurement grid and measurement point layout; Figure 22 For the tooth surface measuring point at Rotational projection on a plane; Figure 23 TSE measurement for FG gears; Figure 24 For the geometric accuracy of the tooth surface. Detailed Implementation

[0023] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0024] I. Geometric-Thermal Error Compensation System for Face Gear and Worm Grinding Machine Based on Digital Twin This embodiment presents a digital twin-based geometric-thermal error compensation system for face gear and worm gear grinding machines, comprising a layered architecture of a sensing and control layer, an edge layer, and a cloud layer.

[0025] In this embodiment, the sensing control layer is used to collect data in real time to provide thermal information data of the face gear and worm grinding machine; In this embodiment, a digital twin model and a thermal error prediction model of a face gear and worm gear grinding machine are deployed within 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 the obtained thermal error prediction data is input into the digital twin model. The digital twin model integrates a geometry-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 geometry-thermal error mapping model to obtain the tooth surface error, and calculates the compensation component through the error compensation model. In this embodiment, the cloud layer trains the thermal error prediction model using historical thermal information data, and sends the updated model parameters to the edge layer to synchronously update the thermal error prediction model. In this embodiment, the geometry-thermal error mapping model is constructed based on the mapping relationship between geometry-thermal errors and tooth surface errors in the grinding machine kinematic chain; and the method for constructing the error mapping model is as follows: Construct the theoretical positive kinematic transformation matrix from the grinding worm gear to the workpiece; Calculate the actual homogeneous coordinate transformation matrix from the grinding worm wheel to the workpiece under error conditions; Based on the difference between the theoretical transformation matrix and the actual transformation matrix, the tool / workpiece posture error model is derived. By combining the thermal contact transformation matrix and the positive kinematic matrix of the grinding process, a mapping relationship between tooth surface error and geometric-thermal error is established. In this embodiment, the thermal error prediction model is constructed based on an LSTM network with an improved gating structure. The improved gating structure LSTM network introduces an attention mechanism and merges the input gate and forget gate into a joint gating mechanism. By adding residual connections and weight allocation, the short-term data representation capability is enhanced while ensuring the performance of long-term feature representation.

[0026] 1.1 Geometric-Thermal Error Mapping Model (1) On the forward kinematic model Figure 1 The figure shows the forward kinematics model of FGWGM, with the mathematical expression of the motion transmission chain from the tool to the workpiece derived from the tool as the reference point.

[0027] The homogeneous coordinate transformation matrix (HCTM) between adjacent components is shown in Table 1 below.

[0028] Table 1 HCTM between adjacent components in: Represents the identity matrix; These represent motion commands for the B, Y, A, Z, X, and C axes, respectively.

[0029] The transmission sequence is obtained based on the transmission chain theory. The theoretical forward kinematic transformation matrix of FGWGM is derived.

[0030] A theoretical forward kinematics model is established by solving the HCTMs equations for each axis system and moving component from the tool to the workpiece. Its general expression is: In the formula to These are matrix elements; see Appendix A for specific values.

[0031] (2) Actual homogeneous coordinate transformation matrix according to Figure 2 , obtained department Relative to components Error propagation matrix: in: This represents the actual motion transformation matrix under error conditions; For components Positioning coordinate matrix in the reference coordinate system; Representative components The positioning coordinates; It is a component along directional movement distance The transformation matrix.

[0032] The transformation matrices of grinding-related motion axes are summarized in Table 2.

[0033] Table 2 Transformation matrices for each motion axis Based on the above error propagation principle, the thermal contact transformation matrix (HCTM) between the grinding wheel and the workpiece under actual working conditions is calculated as follows: In the formula: and Let represent the actual thermal contact transformation matrices from the grinding wheel to the B-axis, B-axis to the Y-axis, Y-axis to the A-axis, A-axis to the Z-axis, Z-axis to the X-axis, X-axis to the bed, C-axis to the bed, and workpiece to the C-axis, respectively. The actual forward kinematic model can then be expressed as: in: to These are matrix elements. And: (3) Tool / workpiece pose error model The tool / workpiece pose error model is derived from the difference in the calculation results: In the formula: and These are the position vector and the attitude vector, respectively. and These represent the position error vector and attitude error vector of the worm grinding wheel, respectively; , and express Element; , and express Element; This is the actual homogeneous coordinate transformation matrix from the grinding worm gear to the workpiece; This represents the theoretical positive kinematic transformation matrix from the grinding worm gear to the workpiece.

[0034] (4) Mapping relationship between tooth surface error and geometric-thermal error By combining the transformation matrix With the positive kinematics matrix of the grinding process Establish the relationship: Therefore, the theoretical motion control commands for the gear grinding process are determined: in: Represents the transmission ratio and is equal to In the actual grinding process, the Z-axis first moves to the specified height.

[0035] yes and The function of . Therefore, the tooth surface equation and its normal vector can be equivalently expressed as: The form grinding (FG) tooth surface equation incorporating these errors can be expressed as: in: Represent the equation of the tooth surface containing errors; It is the normal vector; The set representing all errors; Let be the positive kinematics matrix containing the error, and its value is... , and The function; and For the new meshing conditions.

[0036] The forward kinematic transformation matrix changes significantly. The mapping relationship between tooth surface errors (TSEs) and geometric-thermal errors (GTEs) is as follows: in: , and Represents the position vector components of the grinding wheel; , and The vector components representing the grinding wheel; This is the position error vector; , and It is the position error component; This is the attitude error vector; , and This represents the attitude error components.

[0037] The impact of one or more error terms on the TSE distribution can be evaluated by introducing them into the model. When the X-axis error term... Set as At that time, the distribution of the six TSE terms on the left tooth surface is as follows: Figure 3 As shown. For and The error is smaller at the root and larger at the tip, and gradually increases from the root to the tip along the tooth width direction. exist and The distribution pattern 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 and The error increases from the outer diameter of the tooth tip to the inner diameter of the tooth root. Although the pose error of the left tooth surface shows a similar distribution trend, their magnitudes differ.

[0038] Position-independent errors remain static, while position-dependent errors vary with position, resulting in dynamic characteristics for translational errors (TEs). This analysis reveals how linear axis errors affect TSEs and provides a basis for error compensation in flow fields (FGs). Based on the measured linear axis error distribution range, the translational error is set at... Inside, the rotational error is set at Within the arc. Each error component of the linear axis is treated as an independent variable. The spatial pose of 45 points on the tooth surface is calculated and compared with the ideal pose to obtain 45 sets of error values. Finally, the maximum value of translation error, rotation error and normal error is used as the evaluation index.

[0039] like Figure 4 and 5 As shown, tooth surface spatial error With normal error The correlation between them is thus revealed. These errors are transmitted through the machine tool drive train, and their values ​​change before reaching the gear face. Figure 4 This indicates that translational errors have a consistent impact on both tooth surface pose error and the normal error in the same direction. Translational errors in specific directions of the linear axis mainly affect the corresponding directional components of the tooth surface pose error, with the Y-axis and Z-axis translational errors having a particularly significant impact on the normal error. Figure 5 The results show that the effects of Y-axis and Z-axis rotation errors on tooth surface pose error and normal error are similar, which differs from the mechanism of X-axis rotation error. It is worth noting that only... and It has a significant impact on the normal error of the tooth surface, and and The impact on tooth surface pose and normal error is negligible. Although certain position-related error components in the linear axis continuously affect tooth surface pose and normal error, the motion commands for each axis differ during the actual grinding process, leading to changes in the position of the motion axis and thus causing variations in the magnitude of position-related errors. When considering the combined effect of all error components on tooth surface pose and normal error, it is difficult to directly identify the contribution of a single component. Furthermore, error components couple and accumulate during machining, and their impact on tooth surface accuracy is not a simple algebraic summation. This complexity makes error tracing and compensation difficult, and the multi-error interaction mechanism within the FGWGMT system is not yet clear. These factors collectively limit the effectiveness of error compensation strategies.

[0040] 1.2 Thermal Error Prediction Model (1) LSTM Neural Network Figure 6 The LSTM structure was demonstrated, which includes three types of gate units: input gates. Forgotten Gate and output gate Activation function Nonlinear mapping of input information to The interval is used to filter and update information; while the tanh activation function constrains the information within the range of [-1, 1], regulating the input and output flow in each cycle. This mechanism effectively prevents gradient vanishing or exploding, ensuring computational stability and fast convergence.

[0041] The working principle of LSTM is described as follows: In the formula: Indicates input; and Represent and The state of the LSTM cell at any given moment is used as a memory unit to retain historical information; yes LSTM node output at time; Indicates weight; For vector element-wise multiplication; It is a bias term; Both weights and biases are iteratively updated during model training. LSTM networks, through their chain-like recurrent structure and gating mechanism, can effectively extract short-term and long-term features from time-series data.

[0042] (2) Improved gating structure of LSTM network Thermal error (TE) is strongly correlated with short-term information and weakly correlated with long-term information. Therefore, optimizing the LSTM network cell structure reduces its dependence on long-term information while enhancing its ability to represent short-term information. The improved LSTM cell structure is shown below. Figure 7 As shown: First, an attention mechanism is introduced, merging the input gate and the forget gate into a new gating mechanism to strengthen the correlation of information flow and realize the synchronous screening of forgotten information and the updating of new information, making the process more efficient and orderly; Second, residual connections, which are widely used in deep learning, are added to improve convergence, training performance and generalization ability; The output gate also integrates an attention mechanism and establishes residual connections, which strengthens the short-term data representation ability while ensuring the performance of long-term feature representation through weight allocation.

[0043] The principle of the improved gating structure of LSTM networks is as follows: In the formula: Indicates weighted input; For input gates; The output of the input gate; For output gate; This is the updated state; It is in a hidden state; , and As a weighted term; , and For bias terms; For weights.

[0044] 1.3 Digital Twin-Driven Geometry-Thermal Error Compensation System Figure 8 The framework of a digital twin-driven geometry-thermal error compensation system was demonstrated. First, a geometric model was established based on measurable shape and size. Second, a multi-domain interactive system including mechanical, electrical control, and thermal systems was constructed to characterize the inherent properties and operating mechanism of the machine tool. Data was collected through sensing technology, and internal information and historical data were used as inputs to construct a TE model. Finally, the data was fed back to the actual machine tool to form a virtual-real closed loop.

[0045] 1.3.1 Digital Twin Modeling Methods (1) Geometric model construction Geometric modeling is the first step in creating a digital twin, supporting the parametric construction, assembly, and simulation of features. Fidelity and lightweight design are crucial for geometric model construction. The FGWGMT mainly consists of components such as the spindle, column, table, feed axes, and bed. Physical factors such as inertia, damping, and elastic deformation are not considered during modeling; each machine tool component is simplified to a rigid body using modeling software. The model accurately reflects the assembly relationships, reference points, and dependencies of the physical FGWGMT, maintaining structural consistency. Furthermore, to avoid latency caused by excessive memory usage during data transmission, the model employs a lightweight design, achieving high fidelity with minimal data volume.

[0046] (2) Physical model construction After the geometric model is constructed, physical modeling is based on this model, endowing it with inherent knowledge and mechanisms to describe the physical characteristics and constraints of the system. Since machine tools involve multiple fields such as mechanics, electrical engineering, hydraulics, and control, a multi-level precision modeling method is required. This object-oriented modeling method can accurately and objectively describe entities in different spatial dimensions or domains, and must fully consider the model's multi-level interaction capabilities in space. 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 FGWGMT multi-domain model is achieved.

[0047] A software platform supporting multi-domain modeling is employed to achieve interaction between mechanical, electrical, control, and thermal systems. These subsystems are integrated through energy conversion interfaces between models, ultimately forming a digital twin model of FGWGMT multi-domain interaction. Based on geometric models and multi-domain interactions, multi-dimensional models are linked, combined, and integrated to construct a comprehensive, high-fidelity virtual machine bed model (e.g., ...). Figure 9 (As shown). The next section will detail the data modeling method, as the digital twin model needs continuous updating and optimization through data modeling. To maintain the temporal consistency between the virtual and physical FGWGMT, a unified multi-domain modeling principle is adopted to study the interaction relationships of each subsystem, constructing mechanical, control, and electrical subsystems to form a digital twin model of FGWGMT multi-system interaction. 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.

[0048] (3) Twin data construction FGWGMT twin data model construction as follows Figure 10 As 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 handling, and feature extraction, static and dynamic data are loaded and mapped. Data fusion is performed using principles such as mean weighting and reliability priority. A data model of rotational speed, temperature field, TE, torque, coordinates, vibration, and ambient temperature is constructed in the digital space, ultimately providing services such as TE prediction, cluster analysis, thermal behavior monitoring, data mining, anomaly detection, and thermal information analysis.

[0049] 1.3.2 System Operating Principle Digital twin-driven geometry-thermal error compensation systems employ a layered architecture comprising a sensing and control layer, an edge layer, and a cloud layer, such as... Figure 11 As shown.

[0050] (1) Sensing and control layer The sensing and control layer acquires real-time data through temperature sensors (such as the DS18B20), vibration sensors (such as the PCB Piezotronics 356A16), and displacement sensors (such as the Renishaw LM10) to provide critical FGWGMT thermal information. Sensor data is captured by the acquisition card and processed... The converter performs post-processing, with the Raspberry Pi acting as a gateway for data transmission. Compensation components send data from the edge layer to the CNC system, where the servo controller adjusts servo axis motion accordingly. Real-time feedback from position sensors ensures the FGWGMT operates within set parameters to achieve TE compensation. A PLC (such as a Siemens S7-1200) manages machine tool operation, adjusting parameters based on edge layer feedback. Control software (such as Siemens TIA Portal) is used to program and design control logic, ensuring motion accuracy and reducing errors.

[0051] (2) Edge layer At the edge layer, sensor data is processed near the FGWGMT (five-axis grinding machine) to reduce latency. Devices such as the NVIDIA Jetson Nano handle local data processing, employing edge computing software like EdgeX Foundry for real-time analysis of thermal information and thermal error prediction. This allows the system to make rapid decisions without transmitting all data to the cloud, thus improving system response speed. This layer integrates a virtual mapping of the physical machine tool—a digital twin model—and uses tools such as Siemens NX or Dassault Systèmes SOLIDWORKS to perform real-time simulation of the FGWGMT. The digital twin model monitors the operating status, providing insights into machine tool performance. It also calculates compensation components through a compensation model and periodically updates the thermal error prediction model to reflect changes in thermal behavior, sending the prediction results to the error compensation model for execution.

[0052] (3) Clouds The cloud layer utilizes powerful platforms such as AWS or Microsoft Azure, providing scalable storage and computing resources. Data modeling and optimization are performed using data analysis tools such as Python, R, and MATLAB. Machine learning libraries such as TensorFlow and PyTorch are used to train an NChOA-LSTM model with an improved gate, using long-term historical thermal information data as input. Finally, the updated model parameters are distributed to the edge layer, synchronously updating the thermal error prediction model used in that layer.

[0053] 1.3.3 Training and updating of the thermal error prediction model In the cloud layer, the parameters of the thermal error prediction model are updated using the nonlinear chimpanzee optimization algorithm (NChOA); the principle of the nonlinear chimpanzee optimization algorithm is as follows.

[0054] The formula for updating the position of each chimpanzee is: in: and These represent the upper and lower bounds of the search space, respectively; rand is a random number. During the hunt, the distance between the chimpanzee and its prey and its location update letter for: in: and It is an interval Nonlinear random variables within; It is a random number; It is a chaotic factor; These are the conditions used to determine the location update mode; It is an interval Random variables within; It is a nonlinear convergence factor, the value of which varies with the number of iterations. The increase decreases linearly from 2 to 0; Indicates the maximum number of iterations; and: when This indicates that the chimpanzee is approaching its prey. ;when This indicates that the chimpanzees were forced to move away from their prey and disperse to a wider area to search; The optimal solution for the group is , , and This represents the position vectors corresponding to attacking, surrounding, driving away, and chasing the prey, respectively; the distances of these four positions from the prey. , , and From position vector , , and Joint decision: In the formula: , , and This represents the updated position vectors of the four types of chimpanzees; , , and These are the hindering factors influencing the four types of hunting behaviors; , , and It is the corresponding chaos factor; , , and For associated random vectors; The hunting behavior of chimpanzee populations is characterized by: when At one time, the chimpanzee is approaching its prey, preparing to attack; conversely, it is not. This indicates that they are searching for prey in a dispersed manner. The value of and convergence factor Closely related, this factor controls the algorithm's local and global search capabilities. However, the convergence factor... Linear decay is insufficient for complex nonlinear optimization tasks, easily leading to slow convergence and getting trapped in local optima. Therefore, NChOA introduces a nonlinear transformation strategy to enhance the convergence factor.

[0055] In the early stages of iteration, the convergence factor decays slowly to enhance global search capability, while in the later stages it decays rapidly to promote efficient local optimization. Therefore, NChOA exhibits superior optimization performance compared to linear ChOA. The nonlinear convergence factor proposed in this embodiment... for: in: This represents the initial convergence factor.

[0056] The nonlinear convergence factor enables the algorithm to adaptively adjust the iteration process: in the early stage, it focuses on global search to accelerate convergence, and when it is close to convergence, it shifts to local fine optimization to improve the optimization accuracy.

[0057] 1.3.4 Error Compensation Model The edge layer calculates the compensation component using an error compensation model. The error compensation model proposed in this embodiment includes a first-order error decoupling and source tracing model and a second-order error source tracing model.

[0058] (1) First-order error decoupling and source tracing model The differential motion of the machine tool's cumulative pose error can be expressed as: This relationship is derived by combining the separation of motion axis differential motion error and the decoupling analysis of motion direction equivalent error: In the above equation, the equality of the left and right matrices means that the elements at corresponding positions are equal. From this, twelve equations can be established, containing the unknowns of the six equivalent decoupling quantities of the motion axes, thereby determining the six equivalent error decoupling quantities in the motion direction: in: , , , and These are the equivalent error decoupling values ​​in the X, Y, Z, A, B, and C axis directions, respectively. , and These represent the positional errors of the tool / workpiece in the X / Y / Z directions, respectively. , and These represent the angular errors of the tool / workpiece in the X / Y / Z directions, respectively. These represent motion commands for the B, Y, A, Z, X, and C axes, respectively.

[0059] (2) Second-order error source tracing model Based on this envelope linkage relationship, the equivalent error traced back to the motion axis needs to undergo second-order tracing to determine the actual compensation component. The linkage ratio between the X-axis and Y-axis is: In the formula: Indicates the helix angle.

[0060] when Shaft position occurs When changes occur, it will cause The axis position changes synchronously: Therefore, the final compensation component for the X-axis is determined as follows: Similarly, the relationship between the B-axis and the C-axis is expressed as follows: In actual machining, the position of the B-axis The change will cause the C-axis position to change synchronously: The final compensation component for the C-axis is: Since the A-axis is always locked at zero position during movement, no A-axis compensation is required.

[0061] The shaft compensation component is: The compensation components for each motion axis were finally determined as follows: in: , and These are the actual compensation components for the X-axis, Z-axis, and C-axis, respectively. The helix angle; This indicates the relationship between the B-axis and the C-axis.

[0062] By discretizing the motion trajectory during the grinding process, compensation components at discrete positions are calculated. These components are then fitted to a smooth motion trajectory using spline curves, and interpolation methods are employed to determine the motion compensation amount at any point on the trajectory. Finally, preset values ​​are converted into pitch compensation amounts, enabling automatic adjustment of the motion axis positioning to achieve geometric thermal error (GTE) compensation.

[0063] II. Experimental Verification 2.1 Experimental Setup and Measurement Procedure This embodiment uses the FGWGMT (five-axis grinding machine) as the core experimental object, whose structure is optimized to meet the needs of precision grinding. The machine tool is equipped with a Siemens 840Dsl system and uses an electronic gearbox to synchronously achieve multi-axis control. The geometric error of the linear axes is measured using an XL-80 laser interferometer; the position of the reflector assembly during the measurement process is shown in [reference needed]. Figure 12 The measurement results are shown below. Figure 13 The thermal behavior of the linear axis of the machine tool is determined by... Figure 14 The equipment shown was used for measurement; the operating conditions are shown in Table 3; and the measurement data are shown in [Table 3]. Figure 15 B-axis thermal behavior is based on Figure 16 The measurement plan was implemented, and the results are shown below. Figure 17 .

[0064] Table 3 Different operating conditions 2.1.1 Validation of the thermal error model The NChOA parameters need to be determined before training the thermal error prediction model: and The upper and lower bounds are set to 10 and 500 respectively, the data dimension d is set to 3, the population size is 3, the maximum number of iterations is 4, and the optimization interval is [missing information]. The hyperparameters of the thermal error models used in the ablation and comparison experiments are shown in Table 4. The drastic changes in thermal error place high demands on the model capabilities; therefore, each model was trained using the training dataset. For example... Figure 18As shown in (a), the NChOA-LSTM with improved gates outperforms both the LSTM with improved gates and the standard LSTM, while the BP neural network performs the worst, verifying the effectiveness of the improved gates and NChOA—the LSTM with improved gates significantly outperforms the standard LSTM. Figure 18 (b) The residual variability of the BP network is higher than that of the other four types of models, and the residual variability of the standard LSTM is greater than that of the LSTM with improved gates and NChOA-LSTM. The high fitting accuracy of the LSTM series models stems from their powerful temporal modeling capabilities. Studies have shown that LSTM can effectively model error mechanisms, and the improved gates further enhance the temporal characteristics. The BP network, lacking memory, cannot store long-term thermal history information, highlighting the importance of temporal modeling for error prediction. Figure 18 For working conditions The training results.

[0065] Table 4 Model Hyperparameters The parameters are shown in Table 5. The NChOA-LSTM with improved gates has the highest fitting accuracy, outperforming both the LSTM with improved gates and the standard LSTM. Using ChOA to optimize the parameters ensures a good match between the LSTM parameters and the characteristics of the thermal information data, thereby improving prediction accuracy. Notably, the LSTM with improved gates surpasses the standard LSTM in accuracy, further demonstrating the effectiveness of the improved gates.

[0066] Table 5 Evaluation of Training Results 2.1.2 Predictive Performance according to Figure 19 As shown, the application of historical thermal information data is crucial. For TE prediction, we draw a similar conclusion: memory performance is far more important than parameter optimization. The evaluation parameters for each model are detailed in Table 6. The prediction accuracies of BP neural network, LSTM, improved gated LSTM, and NChOA improved gated LSTM are 93.85%, 94.74%, 96.35%, and 97.62%, respectively. After adopting the NChOA algorithm, the prediction performance of the improved gated LSTM improves from 96.35% to 97.62%. The prediction accuracy of the improved gated LSTM alone is 96.38%, lower than that of the NChOA improved gated LSTM. This significant improvement verifies the effectiveness of the NChOA algorithm. Comparing Tables 3 and 4, we can see that the improved gated LSTM and NChOA improved gated LSTM have the smallest accuracy decay under different operating conditions, while the accuracy decline of BP network and traditional LSTM is more significant, indicating that they have stronger robustness. The prediction accuracy of NChOA improved gated LSTM is slightly higher than that of the improved gated LSTM, therefore it is an ideal model for TE prediction.

[0067] Table 6 Evaluation of Prediction Results 2.3 Verification of face gear grinding The comparative experiment used the same worm gear grinding wheel tool, made of DA bonded grinding wheel material. The face gear (FG) control group consisted of gear sets with the same machining allowance, all machined on an FGWGMT machine tool. The gear material was AISI E9310 alloy steel; specific parameters are shown in Table 7.

[0068] Table 7 Parameters of Worm Grinding Wheels and Face Gears (FG) FG grinding experimental setup such as Figure 20 As shown. The linear axis moves rapidly during the idle phase and slowly during the grinding phase. The single feed rates for roughing, semi-finishing, and finishing grinding are 20. , .

[0069] FG gears differ significantly from cylindrical and bevel gears. The accuracy assessment standards for cylindrical and bevel gears are not applicable to FG gears. Therefore, existing standards for bevel gears are often used to assess the geometric accuracy of FG tooth surfaces. The TSE measurement of FG gears is mainly aimed at 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 Grid (e.g.) Figure 21 As shown), the grid is arranged along the tooth height direction. Shrinkage rate and 10% shrinkage rate in the tooth width direction are used for classification.

[0070] By using the rotational projection method, the measurement grid and measurement points are projected onto the plane of symmetry of two symmetrical tooth surfaces within the tooth groove of the tooth surface being measured. Above (e.g.) Figure 22 (As shown). and These represent the minimum and maximum values ​​of the measuring points on the tooth surface in the tooth height direction (corresponding to the root height and tip height); and For measuring points exist Xianghe To coordinates; , These are the four boundary points of the measurement grid. The coordinate values ​​are respectively and In the formula: and These represent the inner and outer radii of the FG gear being measured, respectively.

[0071] Interval along the tooth width direction Divide into equal parts, the first Measuring points at the quantile point coordinate Given by the following formula Points on the transition curve Xianghe The coordinates are determined by the following formula. Therefore, the upper and lower boundaries of the mesh in the tooth height direction are determined as follows: In the tooth height direction ( (axis) will be the interval Divide into equal parts, and finally get measuring point The coordinates are as and Follow these steps: First, perform GTE suppression treatment, then start the gear grinding program to complete the FG grinding with suppression. Measure the TSE values ​​of the gears with and without GTE suppression for comparison and evaluation. The measured tooth surface geometric accuracy of the machined FG gear is as follows: Figure 23 As shown. During measurement, theoretical measurement point data is imported into the measurement module. First, the coordinate measuring machine is calibrated and a positioning datum and measurement coordinate system are established. The probe measures each point according to the planned path, moving along the normal direction near the measurement points to capture machining errors. Based on the normal error data, a topographic error map of 45 measurement points on the tooth surface is generated.

[0072] The distribution of tooth surface normal error before and after compensation measures are as follows: Figure 24 As shown. The maximum normal error of the left tooth surface is from Down to The length of the error distribution interval was shortened from 25.7 μm to The maximum normal error of the right tooth surface is from Reduce to The length of the error distribution interval is determined by Reduced to .

[0073] 2.4 Compensation Efficiency According to the data in Table 8, the computation times of the compensation system using the improved gated NChOA-LSTM, improved gated LSTM, standard LSTM, and BP network as error prediction models are 126 seconds, 73 seconds, 184 seconds, and 105 seconds, respectively. Under a 5-minute processing cycle, all four models can complete error prediction within the compensation period. The compensation system using the improved gated LSTM as the control model has the shortest execution time, thanks to its elimination of the NChOA initial parameter optimization step, and the improved gated LSTM's convergence speed is superior to that of the standard LSTM.

[0074] Table 8. Efficiency of the compensation system under different prediction models The sampling frequency for temperature and error data was set to 0.01 Hz, and the total data volume collected over six months was 29.6 GB. In this study, a sensing and control edge cloud framework was designed to improve the system's execution efficiency. The sensing and control cloud framework was used as a comparison framework. Data was processed using both frameworks, and the processed data volume is shown in Table 9.

[0075] Table 9 Data volume processed by the thermal error compensation system The execution time and efficiency 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, validating its rationality. In the Sensing Control-Edge-Cloud framework, the edge layer is close to FGWGMT and carries the TE prediction and compensation models. These models do not require a large amount of computing resources. The HP Z240 Tower workstation cloud server located in the cloud layer provides sufficient computing power to meet these needs. For the Sensing Control-Cloud framework, although the edge layer time is zero, the data transmission and cloud layer time increase significantly. In addition, the increased workload in the cloud layer leads to an increase in runtime 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. Functions and tasks are well divided between different layers. The total execution time of both frameworks is less than five minutes. Furthermore, 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.

[0076] Table 10 Execution Time and Efficiency III. Conclusion This embodiment establishes the mapping relationship between geometric thermal error (GTE) and final thermally induced error (TSE), constructs a thermal error prediction model based on the nonlinear chimpanzee optimization algorithm-long short-term memory network (NChOA-LSTM), and develops a digital twin-driven GTE error suppression system. This system integrates prediction and compensation models, and through the application of the digital twin-driven GTE error suppression system, (1) For functional gradient worm grinding machine tools (FGWGMT), the mapping relationship between GTE and TSE was established, the influence of translation and rotation errors on TSE was studied, and a GTE compensation model was developed to analyze the mechanism of error on TSE.

[0077] (2) A thermal error prediction model was constructed based on NChOA-LSTM. In particular, an improved gated structure NChOA-LSTM was proposed to capture temporal features, and NChOA was used for hyperparameter multi-objective optimization. The prediction accuracy of the improved gated structure nonlinear chimpanzee optimization algorithm-LSTM model reached 97.62%.

[0078] (3) Develop a digital twin-driven GTE error control system to reduce the left-side TSE from 21.8. Down to TSE on the right Down to By adopting the sensor control-edge-cloud computing framework, the system execution time is reduced by 59.6% compared to the traditional sensor control-cloud framework.

[0079] The embodiments described above are merely preferred embodiments for 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 all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A geometric-thermal error compensation system for a face gear and worm gear grinding machine based on digital twins, characterized in that: A layered architecture including a sensing and control layer, an edge layer, and a cloud layer; The sensing and control layer is used to collect data in real time to provide thermal information data of the face gear and worm grinding machine; The edge layer contains a digital twin model and a thermal error prediction model of a face gear and worm gear grinding machine. 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 the obtained thermal error prediction data is input into the digital twin model. The digital twin model integrates a geometry-thermal error mapping model and an error compensation model. The digital twin model performs real-time simulation based on the thermal error prediction data and the geometry-thermal error mapping model to obtain the tooth surface error, and calculates the compensation component through the error compensation model. The cloud layer trains the thermal error prediction model using historical thermal information data, and sends the updated model parameters to the edge layer to synchronously update the thermal error prediction model. The geometry-thermal error mapping model is constructed based on the mapping relationship between geometry-thermal errors and tooth surface errors in the grinding machine kinematic chain; and the method for constructing the error mapping model is as follows: Construct the theoretical positive kinematic transformation matrix from the grinding worm gear to the workpiece; Calculate the actual homogeneous coordinate transformation matrix from the grinding worm wheel to the workpiece under error conditions; Based on the difference between the theoretical transformation matrix and the actual transformation matrix, the tool / workpiece posture error model is derived. By combining the thermal contact transformation matrix and the positive kinematic matrix of the grinding process, a mapping relationship between tooth surface error and geometric-thermal error is established. The thermal error prediction model is constructed based on an improved gating structure LSTM network. The improved gating structure LSTM network introduces an attention mechanism and merges the input gate and forget gate into a joint gating mechanism. By adding residual connections and weight allocation, it enhances the short-term data representation capability while ensuring the long-term feature representation performance.

2. The geometric-thermal error compensation system for face gear and worm gear grinding machine based on digital twin as described in claim 1, characterized in that: The tool / workpiece attitude error model is as follows: In the formula: and These are the position vector and the attitude vector, respectively. and These represent the position error vector and attitude error vector of the worm grinding wheel, respectively; , and express Element; , and express Element; This is the actual homogeneous coordinate transformation matrix from the grinding worm gear to the workpiece; This represents the theoretical positive kinematic transformation matrix from the grinding worm gear to the workpiece.

3. The geometric-thermal error compensation system for face gear and worm gear grinding machines based on digital twins according to claim 1, characterized in that: The equation for the profiled grinding tooth surface, including errors, is expressed as: in: Represent the equation of the tooth surface containing errors; It is the normal vector; The set representing all errors; Let be the positive kinematics matrix containing the error, and its value is... , and The function; and For the new meshing conditions; The mapping relationship between tooth surface error and geometric-thermal error is as follows: in: , and Represents the position vector components of the grinding wheel; , and The vector component representing the grinding wheel; This is the position error vector; , and It is the position error component; This is the attitude error vector; , and This represents the attitude error components.

4. The geometric-thermal error compensation system for face gear and worm gear grinding machines based on digital twins according to claim 1, characterized in that: The principle of the improved gating structure of LSTM networks is as follows: In the formula: Indicates input; Indicates weighted input; For input gates; The output of the input gate; For output gate; This is the updated state; It is in a hidden state; , and As a weighted term; , and For bias terms; For weights.

5. The geometric-thermal error compensation system for face gear and worm gear grinding machines based on digital twins according to claim 1, characterized in that: Within the cloud layer, the parameters of the thermal error prediction model are updated using a nonlinear chimpanzee optimization algorithm; the principle of the nonlinear chimpanzee optimization algorithm is as follows: The formula for updating the position of each chimpanzee is: in: and These represent the upper and lower bounds of the search space, respectively; rand is a random number. During the hunt, the distance between the chimpanzee and its prey and its location update letter for: in: and It is an interval Nonlinear random variables within; It is a random number; It is a chaotic factor; These are the conditions used to determine the location update mode; It is an interval Random variables within; It is a nonlinear convergence factor, the value of which varies with the number of iterations. The increase decreases linearly from 2 to 0; Indicates the maximum number of iterations; and: when This indicates that the chimpanzee is approaching its prey. ;when This indicates that the chimpanzees were forced to move away from their prey and disperse to a wider area to search; The optimal solution for the group is , , and This represents the position vectors corresponding to attacking, surrounding, driving away, and chasing the prey, respectively; the distances of these four positions from the prey. , , and From position vector , , and Joint decision: In the formula: , , and This represents the updated position vectors of the four types of chimpanzees; , , and These are the hindering factors influencing the four types of hunting behaviors; , , and It is the corresponding chaos factor; , , and For associated random vectors; The hunting behavior of chimpanzee populations is characterized by: Nonlinear convergence factor for: in: This represents the initial convergence factor.

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

7. The geometric-thermal error compensation system for face gear and worm gear grinding machines based on digital twins according to claim 6, characterized in that: The first-order error decoupling source tracing model determines six equivalent error decoupling quantities in the direction of motion: in: , , , and These are the equivalent error decoupling values ​​in the X, Y, Z, A, B, and C axis directions, respectively. , and These represent the positional errors of the tool / workpiece in the X / Y / Z directions, respectively. , and These represent the angular errors of the tool / workpiece in the X / Y / Z directions, respectively. These represent motion commands for the B, Y, A, Z, X, and C axes, respectively.

8. The geometric-thermal error compensation system for face gear and worm gear grinding machines based on digital twins according to claim 6, characterized in that: The second-order error source tracing model determines the actual compensation component: in: , and These are the actual compensation components for the X-axis, Z-axis, and C-axis, respectively. The helix angle; This indicates the relationship between the B-axis and the C-axis.

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