Physical guided edge cloud collaborative thermal and geometric error real-time compensation system for precision manufacturing

Abstract

The invention discloses a physically guided edge cloud collaborative thermal and geometric error real-time suppression system for precision manufacturing, and belongs to the technical field of error control. The system comprises a sensing control layer, an edge computing layer and a cloud platform layer. The sensing control layer is used for collecting a multi-source physical signal of manufacturing equipment and issuing a compensation instruction; the edge calculation layer deploys a lightweight error prediction and compensation model to realize real-time reasoning and instruction generation; the cloud platform layer is responsible for data aggregation, model training and global optimization. The system realizes dynamic task allocation and load balancing through an edge-cloud cooperation mechanism, ensures real-time task low-delay processing through server request priority scheduling, and simplifies a complex model trained by the cloud into a lightweight version suitable for edge deployment through a model distillation technology. According to the method, the problem of real-time high-precision suppression of thermal errors and geometric errors in the precision manufacturing process is effectively solved, and the machining precision and the system response speed are remarkably improved.

Description

Technical Field

[0001] This invention belongs to the field of error control technology, specifically a physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing. Background Technology

[0002] Face gear (FG) transmission systems enable motion and power transmission between intersecting or staggered shafts, and hold promise as a replacement for bevel gears in equipment such as military helicopters, hovercraft, and tanks. As the final process after rough machining, worm grinding directly determines the geometric accuracy of the face gear, thus affecting the service performance of the transmission system. The face gear worm grinder (FGWGMT) is the core equipment ensuring machining accuracy, but its geometric-thermal errors (GTEs) significantly reduce machining accuracy, ultimately impairing the overall performance of the transmission system. Therefore, compensating for these errors through effective compensation methods is crucial for improving machining accuracy.

[0003] Improving machine tool machining accuracy relies on error compensation, which is fundamentally based on error modeling. Traditional methods treat machine tool motion axes as rigid bodies, establishing a transmission chain topology model based on multibody system theory, and constructing a mapping relationship between motion axis errors and machining errors through homogeneous coordinate transformation. This method reveals the error transmission mechanism within the transmission chain. However, the face gear grinding process involves multiple error components and complex coupling effects, making error modeling extremely challenging: the unique motion chain structure of the machine tool increases the complexity of the grinding process, and the interactive propagation mechanism of geometric-thermal errors within the transmission chain remains unclear. Current research on GTEs compensation models is scarce, leading to deviations in the equivalent motion error calculated based on error sourcing, making it difficult to achieve effective compensation for face gear grinding and hindering the improvement of machining accuracy.

[0004] Research on GTE modeling ensembles for FGWGMT is currently in its early exploratory stages. Most existing models focus only on a limited range of error components. In predicting thermal errors (TEs), empirical models typically employ methods such as multiple linear regression, random forests, and support vector machines. However, these methods are unsatisfactory in terms of prediction accuracy and robustness. To overcome these limitations, deep learning models such as Long Short-Term Memory (LSTM), Gated Recurrent Units (GRU), and Bilinear Temporal Convolutional Networks (BTCN) have been introduced to better capture dynamic temporal features. Given the spatiotemporal coupling characteristics of thermal errors, researchers have proposed Spatiotemporal Graph Convolutional Networks (STGCN). However, these methods still suffer from insufficient prediction accuracy and poor robustness, and the unclear physical meaning of the models limits their practical engineering applications.

[0005] Besides predicting thermal errors, ensuring real-time performance is crucial for the effectiveness of error compensation systems. However, the computational demands of deep learning models can impact real-time performance, as they typically require long inference times. Digital twin technology offers an innovative solution to this problem. Research shows that digital twins can significantly improve the development efficiency of error compensation systems. When integrating thermal error prediction and compensation models into a system, careful design of the framework and functional modules at each level is essential for achieving rapid response. However, existing compensation systems still face real-time challenges under varying operating conditions. In precision manufacturing environments with severe thermal deformation, achieving low-latency error prediction and compensation is crucial for maintaining machining accuracy and productivity. Traditional architectures relying on cloud processing or centralized computing are often constrained by high communication latency and insufficient response time, making it difficult to meet real-time requirements. Summary of the Invention

[0006] In view of this, the purpose of the present invention is to provide a physical-guided edge-cloud collaborative real-time compensation system for thermal and geometric errors in precision manufacturing, which can not only compensate for thermal and geometric errors in precision manufacturing, but also improve response speed and ensure high efficiency and real-time performance.

[0007] To achieve the above objectives, the present invention provides the following technical solution:

[0008] A physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing includes:

[0009] The sensing and control layer is used to collect multi-source physical signals of precision manufacturing equipment in real time during the processing, and output real-time compensation commands from the edge computing layer to the control system of the precision manufacturing equipment.

[0010] The edge computing layer is equipped with a lightweight error prediction model and an error compensation model, which are used to perform real-time inference based on the multi-source physical signals uploaded by the sensing and control layer, generate real-time compensation instructions for the precision manufacturing equipment, and send the instructions to the sensing and control layer.

[0011] The cloud platform layer is used to aggregate the data uploaded by the edge computing layer, perform model training and global optimization tasks, and update the optimized model parameters to the edge computing layer.

[0012] The system uses an edge-cloud collaborative mechanism for task allocation and load balancing, a server request priority scheduling mechanism to ensure low-latency processing of real-time tasks, and a model distillation technique to simplify the teacher model trained at the cloud platform layer into a student model suitable for edge deployment.

[0013] Furthermore, the edge-cloud collaboration mechanism achieves dynamic task allocation and load balancing in the following ways:

[0014] Real-time task T edge Distribute computationally intensive tasks T to the edge for processing. cloud Offloading to the cloud for processing, allocation is based on factors including available computing resources R at the edge. edge Cloud-based computing resources R cloud Edge load L edge and cloud load L cloud , is represented as:

[0015] T edge =f(T) real-time ,R edge ,L edge ),T cloud =f(T) compute-intensive ,R cloud ,L cloud )

[0016] Wherein: T real-time and T compute-intensive These represent real-time tasks that are highly sensitive to latency and computationally intensive tasks, respectively.

[0017] Edge load L edge It is dynamically determined by the following formula:

[0018]

[0019] The updated model M is periodically sent from the cloud. cloud Synchronize to edge model M edge ,Right now

[0020] M edge ←Sync(M cloud (frequency)

[0021] The frequency parameter controls how often the cloud model is synchronized to the edge.

[0022] Furthermore, the request priority scheduling mechanism is as follows: task requests are stored in a queue according to their priority.

[0023] T priority =PriorityQueue(T1,T2,…,T) n ,P1,P2,…,P n )

[0024] Wherein: T priority P represents a priority task queue; i For task T i Priority assignment;

[0025] Real-time error compensation tasks have the highest priority; for low-priority tasks, an exponential backoff algorithm is used to dynamically adjust the request frequency, and the time interval for the k-th request is:

[0026] Δt k =Δt0·2 k

[0027] Where: Δt k Δt0 represents the time interval of the k-th request; Δt0 is the initial request interval; and k is the number of request attempts.

[0028] Furthermore, the model distillation technique is implemented as follows: a high-precision physical constraint spatiotemporal model and a geometric-thermal error comprehensive compensation model are trained at the cloud platform layer to serve as the teacher model; a student model is trained by minimizing the distillation loss function, and the student model is deployed at the edge computing layer for real-time inference; the distillation loss function is expressed as:

[0029] Γ priority =λ·Γ student +(1-λ)·Γ teacher

[0030] Wherein: Γ andom and Γ teacher λ represents the loss values ​​of the student model and the teacher model, respectively; λ is a hyperparameter that controls the knowledge weight of the teacher model during the distillation process.

[0031] Furthermore, the lightweight error prediction model deployed in the edge computing layer is a physically constrained spatiotemporal model, including:

[0032] A multi-scale time encoder module is used to extract multi-scale temporal features from historical temperature sequences;

[0033] A physically constrained spatial graph convolutional network is used to construct a spatial graph based on sensor topology and temperature gradient, and embed thermal diffusion physics rules for spatial feature modeling.

[0034] The gated fusion unit dynamically weights and fuses the multi-scale temporal features with the spatial node embeddings.

[0035] A dynamic fusion layer is used to fuse the outputs of the multi-scale temporal encoder and the physical constraint spatial graph convolutional network to predict thermal errors.

[0036] The physical constraint spatiotemporal model introduces a physical residual loss term based on the heat conduction equation during training, and its loss function is:

[0037] Γ total =Γ MsE +λ1·Γ physics +λ2·Γ smooth

[0038]

[0039] Wherein: Γ MSE Γ represents the mean squared error between the predicted and actual values; λ1 and λ2 are the weighting coefficients for physical loss and smoothing loss; physics For physical residuals; Γ smooth This represents the smoothing loss term used to ensure continuity between spatially adjacent nodes and compensate for oscillations; α is the thermal diffusivity; T is the temperature field; and t is the time variable. is the Laplace operator in a three-dimensional Cartesian coordinate system.

[0040] Furthermore, the multi-scale temporal encoder module includes a multi-scale temporal encoder, a dynamic temporal reweighting module, a lightweight graph convolutional network, and a cross-scale feature fusion module;

[0041] The multi-scale timing encoder employs layered stacking of hollow GRU units with different expansion rates, based on the following principle:

[0042]

[0043] in: Represents large-scale temporal embedding; For small-scale temporal embedding; GRU L (T i ) and GRU S (T i () represent stacked hole GRU modules with large and small expansion coefficients, respectively; τ represents the historical temperature sequence, and τ is the length of the historical time window.

[0044] The dynamic temporal reweighting module adaptively adjusts the weights, and the reweighting feature is represented as follows:

[0045]

[0046] in: and They represent and The repetition feature; ⊙ indicates element-wise multiplication; g L and g s The learned temporal importance weights are represented as:

[0047]

[0048] Where: σ is the Sigmoid activation function; W L and W S b is a learnable weight matrix; L With b SFor bias terms;

[0049] The lightweight graph convolutional network is used to enhance local temporal correlations, focusing on local short-range thermal interactions between adjacent time steps, and is represented as:

[0050]

[0051] in: Temporal features encoded by GCN;

[0052] The cross-scale feature fusion module fuses multi-scale temporal features:

[0053]

[0054] in: This represents the fused multi-scale temporal features obtained through fusion. and They represent respectively to and Features obtained by performing attention-based cross-scale fusion and uniformly normalizing them to the same scale; α L α s and α GCN Indicates the scale attention coefficient; The fused features are obtained by feature fusion through attention-weighted matrix multiplication. In the feature Add input T above i The obtained features are used to enhance gradient flow; W g and b g These are learnable global gating parameters.

[0055] Furthermore, the physical constraint spatial graph convolutional network is used to construct a physically guided spatiotemporal graph based on sensor spatial coordinates and historical temperature sequences, and aggregates the adjacency matrix across time steps through a time decay matrix. Finally, graph convolution operations are performed to obtain spatially coupled node embeddings.

[0056] In the spatiotemporal graph, for time slice t, the adjacency matrix elements between nodes i and j Represented as:

[0057]

[0058] Where: x i and x j These are the spatial coordinates of nodes i and j, respectively; T i (t) and T j (t) For temperature readings; σ controls spatial decay rate; β controls thermal similarity sensitivity;

[0059] The time decay matrix is ​​represented as:

[0060]

[0061] Where: t and t ′ Representing different time indices; the method for aggregating the adjacency matrix of all time steps is as follows:

[0062]

[0063] Among them: A * For aggregated adjacency matrix; A *(t) Let be the adjacency matrix of time slice t;

[0064] Based on the aggregated adjacency matrix A * Perform graph convolution operations to propagate spatial features and obtain spatially coupled node embeddings:

[0065] H (l+1) =σ(A * H (l) W (l) )

[0066] Wherein: H (l) W is the feature matrix of the nodes in the l-th layer; (l) σ is the learnable transformation weight matrix; σ(·) is the activation function.

[0067] Furthermore, the input of the dynamic fusion layer is connected to the output of the multi-scale temporal encoder module and the physical constraint spatial graph convolutional network, used to fuse the input temporal features with spatial embedding and output thermal error prediction values; the dynamic fusion layer uses a gated fusion mechanism to achieve feature fusion, and the gated fusion mechanism is expressed as:

[0068] F fused =σ(W1W1)⊙F1+σ(W2·F2)⊙F2+F res

[0069] Where: F1 and F2 represent the feature vectors of the multi-scale temporal encoder module and the physically constrained spatial graph convolutional network, respectively; W1 and W2 are learnable weight matrices; σ represents the Sigmoid function that generates the gated weights; ⊙ is the Hadamard product; F res This represents a residual connection that enhances gradient flow.

[0070] Furthermore, the error compensation model is a unified geometric-thermal error compensation model. This model establishes a mapping relationship between machine tool motion axis errors and workpiece tooth surface errors through forward kinematics, and decouples errors through sensitivity analysis, ultimately generating compensation commands for each motion axis. The mapping relationship is determined by the following formula:

[0071]

[0072] Where: r xw r yw and r zw Indicates the position vector components of the grinding wheel; i w j w and k w The vector component representing the grinding wheel; E δf δ is the position error vector; xg δ yf and δ zg It is the position error component; E εf ε is the attitude error vector; xg ε yg and ε zg Indicates the attitude error components; and T GT These are the actual forward kinematics transformation matrix and the theoretical forward kinematics transformation matrix, respectively.

[0073] The geometric-thermal error unified compensation model quantifies the contribution of each axis error to the tooth surface error by establishing an error sensitivity Jacobian matrix, and decouples the errors accordingly.

[0074] δ TSE =J GTE ·δ axis

[0075] Among them: J GTE The Jacobian matrix representing the sensitivity between shaft error and tooth surface error; δ axis A vector representing the thermal deformation or geometric error of the machine tool shaft system.

[0076] Furthermore, the geometric-thermal error mitigation model considers the linkage between motion axes when generating compensation commands. For face gear and worm gear grinding machines, the final compensation amount for each axis is calculated using the following formula:

[0077]

[0078] Where: X C Z C and C C The compensation amounts for the X, Z, and C axes are respectively, e x e y e B and e C To trace back to the equivalent error of each motion axis through error decoupling; λ w For the gear helix angle; i pw This is the transmission ratio.

[0079] The beneficial effects of this invention are as follows:

[0080] This invention relates to a physical-guided edge-cloud collaborative real-time compensation system for thermal and geometric errors in precision manufacturing. By constructing a three-layer collaborative architecture of sensing and control layer, edge computing and cloud platform, and integrating three core mechanisms of edge-cloud collaboration, priority scheduling and model distillation, it achieves efficient real-time compensation for thermal and geometric errors in precision manufacturing, and has achieved the following significant technical effects.

[0081] (1) The edge-cloud collaborative architecture fundamentally optimizes system resource allocation and response capabilities. The edge layer is responsible for real-time inference and compensation, ensuring sub-second low-latency response, directly meeting the stringent timeliness requirements of precision machining; the cloud platform layer relies on powerful computing power for model training and optimization, ensuring the continuous evolution and high precision of the algorithm. The collaboration of the two overcomes the dual limitations of high latency in pure cloud solutions and limited computing power in pure edge solutions.

[0082] (2) By combining server request priority scheduling with model distillation, the reliable execution of critical tasks is achieved on the resource-constrained edge side. Priority scheduling ensures the transmission priority of error compensation instructions, while model distillation effectively preserves the prediction accuracy of the teacher model while greatly compressing the model complexity, making it possible to run complex physical information models on edge devices, thereby improving the accuracy of real-time prediction at the source.

[0083] (3) It provides the possibility for deploying advanced physical guidance models. Embedding physical constraints such as the law of heat conduction into neural networks for spatiotemporal modeling significantly improves the physical rationality and extrapolation reliability of thermal error prediction; while the unified geometry-thermal error compensation model achieves error decoupling and source tracing through forward kinematics and sensitivity analysis, realizing the coordinated and accurate compensation of multi-axis errors.

[0084] In summary, this invention solves the problems of real-time performance, accuracy, and scalability of error compensation in precision manufacturing, and has outstanding technical advantages and practical value. Attached Figure Description

[0085] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration:

[0086] Figure 1 This is an architecture diagram of the physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing according to the present invention.

[0087] Figure 2 This is a schematic diagram of the error compensation model; Figure 3 A schematic diagram illustrating the principle framework for digital twin modeling and geometric thermal error compensation applications.

[0088] Figure 4 A flowchart illustrating the construction method of a digital twin system; Figure 5 A schematic diagram illustrating the construction principle of a twin data model; Figure 6 Here is the overall architecture diagram of PSM; Figure 7 This is a diagram of the internal structure of the MTE; Figure 8 A high-fidelity internal structure diagram of PC-SGCN; Figure 9 A detailed flowchart for DFL;

[0089] Figure 10 This is a schematic diagram of a gated fusion unit; Figure 11 This is a schematic diagram illustrating the impact of thermal / geometric errors on the actual machining trajectory.

[0090] Figure 12 A graph showing the relationship between real-time latency and model inference time; Figure 13 A comparison of inference time between the original PSM and GEMM models and the distillation model; Figure 14 The results are the PSM training results under operating condition #1. Figure 15 The test results for PSM under operating condition #2; Figure 16 This is the experimental site for FG grinding and system verification. Figure 17 For FG tooth surface measurement grid and measurement point layout; Figure 18 This is the rotational projection of the tooth surface measurement point onto the xOz plane;

[0091] Figure 19 For FG tooth surface error measurement; Figure 20 Comparison of tooth surface geometric accuracy before and after GTE compensation; Figure 21 The average delay variation under different system loads; Figure 22 A comparison of inference time for the distillation model on different edge devices. Detailed Implementation

[0092] 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.

[0093] 1. Physically Guided Edge-Cloud Collaborative Real-Time Compensation System Architecture for Thermal and Geometric Errors

[0094] 1.1 System Architecture Overview

[0095] To achieve real-time adaptive compensation for geometric thermal error (GTE) in precision manufacturing, a method is proposed as follows: Figure 1The diagram illustrates a hierarchical edge-cloud collaborative architecture for a physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system used in precision manufacturing. Specifically, this embodiment of the physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing includes a sensing and control layer, an edge computing layer, and a cloud platform layer. Each layer coordinates and completes a specific subset of tasks through intelligent scheduling and bidirectional data exchange. Specifically, the sensing and control layer is used to collect multi-source physical signals from the precision manufacturing equipment during the processing in real time, and outputs real-time compensation instructions from the edge computing layer to the control system of the precision manufacturing equipment. The edge computing layer is equipped with a lightweight error prediction model and an error compensation model, which are used to perform real-time inference based on the multi-source physical signals uploaded by the sensing and control layer, generate real-time compensation instructions for the precision manufacturing equipment, and send the instructions to the sensing and control layer. The cloud platform layer is used to aggregate the data uploaded by the edge computing layer, and perform model training and global optimization tasks, updating the optimized model parameters to the edge computing layer. In this embodiment, the system uses an edge-cloud collaborative mechanism for task allocation and load balancing, a server request priority scheduling mechanism to ensure low-latency processing of real-time tasks, and a model distillation technique to simplify the teacher model trained by the cloud platform layer into a student model suitable for edge deployment.

[0096] (1) Sensing and Control Layer

[0097] The sensing and control layer interacts directly with the physical machine tool, acquiring multi-source data through embedded and external sensors (including temperature, displacement, and vibration signals) and transmitting it to the edge nodes via an industrial-grade gateway. Simultaneously, it receives real-time compensation commands from the edge layer, which are then applied to the CNC system via a PLC module, forming a complete sensing-control-execution closed loop. At the sensing and control layer, PT100 temperature sensors, eddy current vibration sensors (such as the PCBPiezotronics 356A16), and displacement sensors (such as the Renishaw LM10) are used to acquire data in real time. This data provides crucial information for the thermal analysis of the FGWGMT (Frequency Gear Grinding Machine). The sampling rate for displacement and temperature signals is 1kHz. Eighteen four-wire PT100 platinum resistance temperature sensors (accuracy ±0.1℃, 10kHz) are deployed in key areas of the worm gear grinding machine: the front and rear spindle bearing housings (applied with thermally conductive silicone grease and secured with polyimide tape) monitor frictional thermal elongation; the spindle motor housing monitors copper loss; the XYZ axis ball screw nuts and drive-end bearing housings characterize feed axis heat flow; symmetrically arranged on the bed-column joint surface and saddle guide rails detect thermal deformation; the workpiece fixture center quantifies tool-workpiece thermal drift; the stainless steel tee at the cooling circuit outlet assesses actual heat dissipation efficiency; and an environmental baseline is set 150mm from the exhaust port inside the machine housing for zero-drift compensation. All cables are connected to the Raspberry Pi edge gateway via oil-resistant cable connections, and sensor IDs are directly mapped to the machine coordinate data stream for subsequent physical modeling and real-time inference. Sensor data acquired by the acquisition card is converted from A / D and transmitted by the Raspberry Pi gateway. Compensation components send data from the edge layer to the CNC system, and the servo controller adjusts axis motion accordingly. The closed-loop sampling rate is 10kHz (one control cycle every 0.1ms), determined by the displacement / temperature acquisition card, synchronizing the entire sensing-edge-PLC loop. The PLC (such as Siemens S7-1200) manages machine tool operation, adjusting parameters based on edge-level feedback. Control software (such as Siemens TIAPortal) programs the control logic to ensure motion accuracy. Real-time feedback from the position sensor ensures the FGWGMT operates within the set parameters, achieving GTE compensation.

[0098] (2) Edge layer

[0099] The edge layer is designed for low-latency processing, deploying lightweight PSM and GEMM models to infer GTE in real time, generate compensation commands, and execute short-term predictions. A priority queuing mechanism ensures that critical operations (such as real-time compensation) are processed first. The edge layer processes sensor data near the FGWGMT (using devices such as NVIDIA Jetson Nano), analyzing thermal information and TE predictions in real time using software such as EdgeX Foundry, reducing cloud transmission latency. A digital twin model (virtual copy of the machine tool) is integrated into this layer, using Siemens NX or Dassault Systèmes for real-time simulation, monitoring operational status, and optimizing performance. The GEMM model calculates the compensation components, the TE prediction model continuously updates thermal behavior changes, and sends the prediction results to the error compensation model for execution.

[0100] (3) Cloud Platform Layer

[0101] The cloud provides high-performance computing capabilities for offline model training, long-term data analysis, and global optimization. It aggregates data from multiple edge deployment nodes, retrains or fine-tunes the base model, and periodically pushes the updated distillation model back to the edge layer. Furthermore, the cloud layer is responsible for maintaining system-wide synchronization, scalability, and knowledge sharing across devices and deployment nodes. The cloud layer utilizes robust platforms such as AWS or Microsoft Azure, providing scalable storage and computing resources. Data analysis tools such as Python, R, and MATLAB are used for data analysis and model optimization. Through machine learning libraries such as TensorFlow and PyTorch, historical thermal information data is used as input to train the PSM (parametric thermal model). Finally, the updated PSM parameters are transmitted to the edge layer, updating the TE (thermal error) prediction model used by that layer accordingly.

[0102] like Figure 2As shown, the system integrates edge-cloud collaboration, server request priority scheduling, and model distillation techniques to build a powerful real-time error compensation system. This error compensation framework integrates multiple components to achieve real-time GTE (Geometric Thermal Error) compensation in precision manufacturing. The system first continuously collects real-time data through streaming clients and sensors, transmitting it to the processing system via HTTP / WebSocket. Model update and error prediction requests are stored in a cache queue, with an exponential backoff clock mechanism managing the load and preventing bottlenecks. An expiration filter ensures that timed-out requests are discarded, and remaining requests are processed according to priority. The system uses a well-trained PSM (Predicted Physical Model) for accurate thermal error prediction and a GEMM (Generalized Matrix Multiplication) model to handle GTE. To optimize real-time performance, an edge-cloud collaboration strategy is implemented: real-time tasks are processed at the edge to achieve low latency, while heavy computational tasks are offloaded to the cloud. Simultaneously, model distillation technology simplifies the PSM and GEMM models, making them adaptable to edge devices with limited computing resources. This architecture ensures efficient real-time error compensation, maintaining high accuracy and responsiveness in dynamic manufacturing environments.

[0103] 1.1.1 Edge-Cloud Collaborative Optimization for Real-Time Performance

[0104] Edge-cloud collaboration features adaptive bidirectional characteristics. The cloud continuously optimizes the underlying model using aggregated data and distills it into resource-efficient sub-models for deployment at the edge, supporting low-latency inference. Simultaneously, edge nodes provide runtime feedback to the cloud, including compensating performance logs, fault alerts, and information on environmental changes. This feedback forms a learning loop, building a control system that combines edge intelligence with cloud scalability.

[0105] To achieve real-time GTE compensation, the system utilizes an edge-cloud collaborative mechanism: critical tasks are processed at the edge, while computationally intensive tasks are offloaded to the cloud. This hybrid edge-cloud architecture ensures low-latency processing and high computational efficiency, enabling dynamic task allocation and load balancing. Edge devices process real-time sensor data and perform rapid predictions to achieve immediate error compensation, while cloud computing handles heavy computational tasks such as deep learning model training and long-term system optimization. Specifically, in this embodiment, the edge-cloud collaborative mechanism achieves dynamic task allocation and load balancing in the following way: real-time task T... edge Distribute computationally intensive tasks T to the edge for processing. cloud Offloading to the cloud for processing, allocation is based on factors including available computing resources R at the edge. edge Cloud-based computing resources R cloud Edge load L edge and cloud load L cloud , is represented as:

[0106] T edge =f(T) real-time ,R edge, L edge),T cloud =f(T) compute-intensive ,R cloud ,L cloud )

[0107] Wherein: T real-time and T compute-intensive These represent real-time tasks that are highly sensitive to latency and computationally intensive tasks, respectively.

[0108] In this embodiment, T, R, and L are purely geometric operators (not physical measurement quantities): T represents a 4×4 homogeneous transformation matrix that maps the coordinate system from the tool coordinate system to the workpiece coordinate system; T and R contain 3×3 rotation submatrices (dimensionless), while L is a 3×1 translation vector (in μm). Edge end load L edge It is dynamically determined by the following formula:

[0109]

[0110] The updated model M is periodically sent from the cloud. cloud Synchronize to edge model M edge ,Right now:

[0111] M edge ←Sync(M cloud (frequency)

[0112] The frequency parameter controls how often the cloud model is synchronized to the edge.

[0113] 1.1.2 Low-latency service request priority scheduling

[0114] To ensure real-time performance and reduce latency, the system employs a service request priority scheduling mechanism. Tasks such as real-time error prediction are prioritized over non-urgent tasks, and critical tasks are always executed first. The dynamic request priority mechanism is as follows: real-time error compensation has the highest priority, and predictive error models are processed first; non-urgent tasks (such as model updates and long-term data processing) are placed in a low-priority queue for delayed processing. In this embodiment, the request priority scheduling mechanism is as follows: task requests are stored in a queue according to their priority.

[0115] T priority =PriorityQueue(T1,T2,…,T) n ,P1,P2,…,P n )

[0116] Wherein: T priority P represents a priority task queue; i For task T i Priority assignment.

[0117] Real-time error compensation tasks have the highest priority; to prevent system overload during periods of high load, an exponential backoff algorithm is used to dynamically adjust the request frequency for low-priority tasks, with the time interval for the k-th request being:

[0118] Δt k =Δt0·2 k

[0119] Where: Δt k Δt0 represents the time interval of the k-th request; Δt0 is the initial request interval; and k is the number of request attempts.

[0120] 1.1.3 Model Simplification and Distillation for Edge Device Deployment

[0121] To ensure real-time prediction efficiency at the edge, model distillation is employed to simplify large models without sacrificing accuracy. The teacher model is a large, complex model trained with high precision, while the student model is a lightweight model trained by mimicking the behavior of the teacher model. Model distillation is implemented as follows: a high-precision physical constraint spatiotemporal model and a geometric-thermal error compensation model are trained at the cloud platform layer to serve as the teacher model; the student model is trained by minimizing the distillation loss function and deployed at the edge computing layer for real-time inference; the distillation loss function is expressed as:

[0122] Γ priority =λ·Γ student +(1-λ)·Γ teacher

[0123] Wherein: Γ andom and Γ teacher λ and λ represent the loss values ​​of the student model and teacher model, respectively; λ is a hyperparameter controlling the knowledge weights of the teacher model during the distillation process. The student model is deployed on edge devices to achieve real-time performance with minimal computational cost.

[0124] 1.2 Digital Twin Modeling Methods

[0125] Figure 3 This study demonstrates a framework for digital twin modeling and geometric thermal error compensation. First, a geometric model is established based on measurable shape and dimensions. Then, a multi-domain interactive system encompassing mechanical, electrical control, and thermal systems is constructed to characterize the machine tool's inherent properties and operating mechanisms. Data is collected through sensing technology, and a thermal error model is built using internal information and historical data as input. Finally, feedback is applied to the actual machine tool, forming a closed-loop virtual-physical system.

[0126] 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 axis, and bed. Physical factors such as inertia, damping, and elastic deformation are not considered during modeling; the machine tool components are simplified to rigid bodies using modeling software. All machine tool sub-components in the current thermoelastic model are treated as rigid bodies. This simplification is acceptable for the worm gear grinding process studied because:

[0127] (i) The force gauge test during the machining process showed that the maximum tangential cutting force was <20N, and the resulting quasi-static offset of the tool center point was only 0.3-0.5μm - two orders of magnitude smaller than the 30-50μm thermal drift accumulated in a two-hour production cycle;

[0128] (ii) Modal analysis of the impact hammer showed that the first structural characteristic frequency was approximately 147Hz, which is much higher than the excitation bandwidth of the grinding force ≤15Hz. Therefore, the influence of elastic deformation on the subhertz error spectrum of our controller is negligible.

[0129] (iii) Preliminary force-temperature coupling simulations confirm that the volumetric error caused by force accounts for only <2% of the total positioning error budget. Therefore, thermal effects dominate the accuracy loss and are modeled in detail, while force-induced deformation is ignored in this preliminary implementation.

[0130] The model also accurately reflects the assembly relationships, reference points, and dependencies of the physical FGWGMT, maintaining structural consistency. To avoid latency issues caused by excessive memory consumption during data transmission, the model adopts a lightweight design, achieving high fidelity with minimal data volume. After the geometric model is constructed, physical modeling is based on this model to endow it with the inherent knowledge and mechanisms describing the physical characteristics and constraints of the system. Since machine tools involve multiple fields such as mechanics, electrical, hydraulics, and control systems, multi-level precision modeling is required. This method adopts an object-oriented modeling approach to accurately and objectively describe entities in different spatial dimensions or fields, and must fully consider the multi-level interaction capabilities of the model in space. The material properties, physical mechanisms, connection relationships, topological structures, functional component constraints, and driving mechanisms of each subsystem are defined through multidisciplinary knowledge to achieve coupling of the FGWGMT multi-domain model. With the help of a software platform that supports multi-domain modeling, interactions are generated 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 the geometric model and multi-domain interaction, multi-dimensional models are linked, combined, and integrated into a comprehensive high-fidelity virtual machine tool model (such as...). Figure 4(As shown). To maintain temporal consistency between virtual and physical FGWGMT, a unified multi-domain modeling principle is adopted to study the interaction relationships between subsystems, constructing mechanical, control, and electrical subsystems to form a digital twin model of FGWGMT multi-system interaction. These subsystems contain multidisciplinary components; although the physical forms of the models in each domain are different, they can be described by similar mathematical expressions.

[0131] The construction of the FGWGMT twin data model, as follows Figure 5 As shown, all temperature, displacement, and power trajectory data are derived from the physical sensors of the actual worm gear grinder, not from numerical models. Data on rotational speed, temperature field, thermal error (TE), torque, coordinates, vibration, and ambient temperature are collected and stored in a database. After statistical fitting, redundancy processing, anomaly handling, and feature extraction, static and dynamic data loading and mapping are performed. Data fusion is conducted using averaging, weighting, and reliability-priority principles. A data model for rotational speed, temperature field, TE, torque, coordinates, vibration, and ambient temperature is constructed in digital space, ultimately providing services such as TE prediction, cluster analysis, thermal behavior monitoring, data mining, anomaly detection, and thermal information analysis.

[0132] 2. Physically Constrained Spatiotemporal Model (PSM) for Thermal Error Prediction

[0133] To address these issues, a hybrid PSM that integrates domain physics knowledge into a deep neural architecture is proposed. The model comprises three key components: a multi-temporal encoder (MTE), a physically constrained spatial graph convolutional network (PC-SGCN), and a dynamic fusion layer (DFL). The MTE captures long and short temporal dependencies by expanding GRU branches and employs attention fusion and lightweight graph convolution enhancement to extract multi-scale temporal features from historical temperature sequences. The PC-SGCN constructs a spatial graph based on sensor topology and thermal gradients and embeds thermal diffusion physics rules for spatial feature modeling. The DFL integrates the outputs of the MTE and PC-SGCN through gated residual fusion to obtain a refined spatiotemporal representation for predicting thermal errors. Furthermore, a physically guided loss function is introduced to minimize the residuals of the partial differential equation (PDE) for heat conduction, explicitly maintaining consistency with Fourier's laws during model training.

[0134] 2.1 Governing equations for thermal field evolution

[0135] The governing partial differential equation (PDE) for the thermal field evolution considering the properties of isotropic materials is expressed as:

[0136]

[0137] Where: T is the temperature field; k is the thermal conductivity; Q is the internal volumetric heat generation rate; ρ is the material density; c p Specific heat capacity; It represents the time derivative of temperature, reflecting instantaneous thermal changes; t is a time variable.

[0138] For most processing steps, the internal volumetric heat source of the bulk material is negligible compared to boundary frictional heat. Therefore, setting Q≈0, the above equation simplifies to the classical transient heat conduction equation:

[0139]

[0140] in: denoted as thermal diffusivity.

[0141] To simulate the thermal field evolution with computational efficiency suitable for a digital twin framework, the above equation is discretized using the finite difference method. For time discretization, a forward Euler scheme is used, and for spatial discretization, a central difference scheme is employed. With uniform grid spacing Δv, Δy, Δz and time step Δt, the Laplacian operator in the three-dimensional Cartesian coordinate system is... The central difference approximation is:

[0142]

[0143] The forward difference approximation of the time derivative is:

[0144]

[0145] get:

[0146]

[0147] Temperature updated at each node Depends on its own preceding temperature The structure, including the temperatures of adjacent nodes, reflects strong local spatiotemporal characteristics during thermal evolution. The physical residual term is defined as:

[0148]

[0149] Thermal expansion of machine tool components under non-uniform temperature fields leads to thermal errors. The infinitesimal linear thermal strain tensor is expressed as:

[0150] ε thermal =β(T-T0)I

[0151] Where: β is the linear thermal expansion coefficient; T0 is the reference temperature (e.g., ambient or assembly temperature); I is the unit tensor. According to the small deformation theory, the relationship between the displacement field and thermal strain is:

[0152]

[0153] Solving the above equation yields the displacement field:

[0154] u i (x j,t)=β∫(T(x j ,t)-T0)dx j

[0155] The error is the result of the cumulative integral effect of the local temperature gradient. Different axes (x, y, z) produce different expansions according to the heat distribution, resulting in multi-axis positioning errors. The explicit time-stepping scheme of formula (11) imposes a stability constraint on Δt, namely the Courant-Friedrichs-Lewy (CFL) condition.

[0156]

[0157] When Δt is too large relative to the spatial resolution, it can lead to instability (oscillation or divergence). In practical applications, the temporal resolution must be strictly controlled according to the spatial grid size and material diffusivity.

[0158] 2.2 Design of Multi-Module Spatiotemporal Deep Networks for Physical Information Models

[0159] To accurately predict spatiotemporal thermal errors under operating conditions, this embodiment proposes a novel deep learning architecture called PSM. This network integrates three collaborative modules: the MTE module captures temperature change patterns at different time scales, taking into account both fast and slow thermodynamic processes; the PC-SGCN module models the spatial heat conduction and deformation interactions between adjacent nodes; and the DFL module adaptively fuses spatiotemporal features under physical constraints and outputs a thermal error prediction. The overall data flow is as follows: Figure 6 As shown. To ensure consistency between data-driven predictions and fundamental physical laws of thermal diffusion, a physical constraint residual loss function is introduced:

[0160] Γ total =Γ MSE +λ1·Γ physics +λ2·Γ smooth

[0161] Wherein: Γ MSE Γ represents the mean squared error between the predicted and actual values; λ1 and λ2 are the weighting coefficients for physical loss and smoothing loss; physics For physical residuals; Γ smooth This represents the smoothing loss term used to ensure continuity between spatially adjacent nodes and to compensate for oscillations.

[0162] 2.2.1 Multi-scale timing encoder (MTE) module

[0163] To effectively capture the dynamic characteristics of thermal fields at different time scales and enhance the richness of spatiotemporal modeling features, this embodiment proposes an MTE module. For example... Figure 7As shown, this module integrates three major branches: a large-scale dilated GRU encoder, a small-scale dilated GRU encoder, and a lightweight 1×3 graph convolutional network (GCN), achieving comprehensive temporal representation through cross-scale feature fusion.

[0164] (1) Multi-scale temporal encoder

[0165] The input of the multi-scale timing encoder is A layered stack of gated cyclic units (GRUs) with different expansion rates is employed, comprising two parallel hollow GRU branches: the L-scale hollow GRU uses a larger expansion coefficient to capture long-term heat accumulation, while the S-scale hollow GRU uses a smaller expansion coefficient to capture short-term heat accumulation. The principle is as follows:

[0166]

[0167] in: Represents large-scale temporal embedding; For small-scale temporal embedding; GRU L (T i ) and GRU S (T i () represent stacked hole GRU modules with large and small expansion coefficients, respectively; τ represents the historical temperature sequence, and τ represents the historical time step.

[0168] (2) Dynamic temporal reweighting based on Sigmoid gate

[0169] The weights of each branch output are adaptively adjusted via a Sigmoid gate:

[0170]

[0171] Where: σ is the Sigmoid activation function; W L and W S b is a learnable weight matrix; L With b S For bias terms; g L and g s These are the learned temporal importance weights. The reweighted features are calculated as follows:

[0172]

[0173] In the formula: ⊙ represents element-wise multiplication.

[0174] (3) Enhanced timing features of 1×3GCN

[0175] Lightweight graph convolutional networks are used to enhance local temporal correlations, focusing on local short-range thermal interactions between adjacent time steps, and are represented as follows:

[0176]

[0177] in: The temporal features encoded by GCN. This network focuses on local short-range thermal interactions between adjacent time steps.

[0178] (4) Cross-scale feature fusion

[0179] right and The heavily weighted features are fused across scales based on attention, and all features are uniformly normalized to the same scale:

[0180]

[0181] After average pooling, attention scores are calculated using Softmax activation for each normalized feature.

[0182]

[0183] Feature fusion is then performed using attention-weighted matrix multiplication.

[0184]

[0185] Add a shortcut connection to the original input on the fused features to enhance gradient flow:

[0186]

[0187] Finally, global sigmoid gating is applied for channel-level recalibration:

[0188]

[0189] in: This represents the fused multi-scale temporal features obtained through fusion. and They represent respectively to and Features obtained by performing attention-based cross-scale fusion and uniformly normalizing them to the same scale; α L α s and α GCN Indicates the scale attention coefficient; The fused features are obtained by feature fusion through attention-weighted matrix multiplication. In the feature Add input T above i The obtained features are used to enhance gradient flow; W g and b g These are learnable global gating parameters.

[0190] 2.2.2 Physically Constrained Spatial Graph Convolutional Network (PC-SGCN)

[0191] To accurately model spatial coupling behavior while maintaining physical consistency with thermal diffusion mechanisms, this embodiment proposes PC-SGCN. This network explicitly encodes spatial locality, thermodynamic consistency, and temporal decay dynamics into a graph structure (see...). Figure 8 The historical thermal feature sequence of node i is: The input sequence is divided into multiple time steps (e.g., t, t+1, t+2), forming parallel feature branches T. i (t) T i (t+1) and T i (t+2) T i (t) This represents a feature slice at time step t. Each temporal slice is enhanced using a sinusoidal position coding function to preserve temporal information.

[0192]

[0193] Where: t is the time index, i is the feature dimension index, and d is the embedding dimension. Node features T i (t) Generated after position encoding Construct a physical guidance space graph G for each time slice t. (t) =(v,ε (t) ), where v is the node set (sensor locations), ε (t) This represents the edge set capturing the spatial and thermodynamic relationships at time t. The original adjacency matrix is ​​calculated using the following formula:

[0194]

[0195] Where: x i and x j These are the spatial coordinates of nodes i and j, respectively, and T i (t) and T j (t) For temperature readings, σ controls the spatial decay rate, and β controls the thermal similarity sensitivity. To model the temporal decay of effects across different time steps, a time decay matrix is ​​introduced:

[0196]

[0197] Where: t and t ′ Representing different time indices, the adjusted adjacency matrix for all time steps is aggregated:

[0198]

[0199] This yields a composite adjacency matrix encoding spatial and temporal diffusion behavior. Based on the aggregated adjacency matrix A... * Performing graph convolution operations to propagate spatial features:

[0200] H (l+1) =σ(A * H (l) W (l) )

[0201] Wherein: H (l) W is the feature matrix of the nodes in the l-th layer; (l) σ is the learnable transformation weight matrix; σ(·) is the activation function (such as ReLU). After multi-layer graph convolution, the final output contains spatially coupled and thermodynamically consistent node embeddings.

[0202] 2.2.3 Dynamic Fusion Layer (DFL)

[0203] To effectively predict thermal errors, this embodiment designs a Dynamic Fusion Layer (DFL) to dynamically integrate temporal features and spatial interactions, achieving accurate prediction of thermal errors. For example... Figure 9 As shown, the Gated Recurrent Unit (GRU) processes the temporal features obtained from the MTE. Given the input spatiotemporal features, the GRU model is expressed as:

[0204] h GRU =GRU(x t ,h t-1 )

[0205] Where: x i h represents the input features at time step t; i-1 This represents the hidden state at time step t-1. This step captures the temporal dependencies of the thermal data and calculates sequence features characterizing the thermal evolution process. In parallel with GRU, a Temporal Convolutional Network (TCN) is used to extract long-range dependencies from the time-series data. The TCN uses causal convolutions to ensure that the model can only access current and historical information.

[0206] h TCN =TCN(x t )

[0207] TCN provides a complementary perspective to GRU, capturing complex temporal patterns that GRU may not have fully captured. The output features of GRU and TCN are concatenated to integrate the temporal information captured by both models. The concatenation operation is described as follows:

[0208] h concat =Concat(h GRU ,h TCN )

[0209] This concatenated feature vector integrates the sequence dependencies captured by GRU and the macroscopic temporal patterns captured by TCN. To reduce feature dimensionality, 1×1 convolutions are used for channel compression, improving model efficiency. The convolution output is expressed as:

[0210] h conv =Conv1×1(h concat )

[0211] A 1×1 convolutional layer is used as a channel compression operator. After convolution, the feature vectors are converted to nonlinearity through the ReLU activation function.

[0212] h relu =RELU(h conv )

[0213] The processed features are then fed into the feature processing module for final optimization to ensure they are suitable for thermal error prediction. This step is described as follows:

[0214] h final =FeatureProcessing(h relu )

[0215] Finally, optimize feature h final Used for predicting thermal errors:

[0216] Output = Regressor(h) final )

[0217] Where: Regressor is a linear layer or a fully connected layer, which outputs the predicted deformation.

[0218] 2.2.4 Gating Fusion Unit

[0219] Gated Fusion Unit (GFU) is a core component of PSM (Purely Magnetic Mixture System). Figure 10 The GFU dynamically adjusts and fuses spatiotemporal features from different sub-models through a gating mechanism. This unit applies Sigmoid activation to the input features of MTE and PC-SGCN to generate gating weights, and then performs feature weighting through Hadamard product. To preserve original information, residual connections are introduced to add the input features to the fused result. This design enables the GFU to dynamically optimize feature utilization and output fused features for hot error prediction, significantly improving the model's adaptability and accuracy in handling complex spatiotemporal data. In this embodiment, the gating fusion mechanism is expressed as follows:

[0220] F fused =σ(W1·F1)⊙F1+σ(W2·F2)⊙F2+F res

[0221] Where: F1 and F2 represent the feature vectors of the multi-scale temporal encoder module and the physically constrained spatial graph convolutional network, respectively; W1 and W2 are learnable weight matrices; σ represents the Sigmoid function that generates the gated weights; ⊙ is the Hadamard product; F res This represents a residual connection that enhances gradient flow.

[0222] 3. Geometric-Thermal Error Compensation Model (GEMM)

[0223] In multi-axis machine tools, thermally induced deformation and geometric errors jointly affect toolpath accuracy and surface quality. Existing models handle these error types in isolation and often rely on offline measurements, lacking real-time applicability and adaptability to dynamic process changes. To address this, a unified geometric-thermal error hybrid model (GEMM) was developed to simultaneously model the propagation of thermal / geometric errors in the kinematic chain of a five-axis simultaneous gear grinding machine (FGWGMT). This method includes forward kinematics with error embedding, error sensitivity decoupling, and second-order kinematic compensation. Error sensitivity decoupling, as a first-order error decomposition model, quantifies the contribution of each axis to the total spatial errors (TSEs), providing error tracing capabilities. Second-order kinematic compensation maps traceable errors to compensable motions by considering the correction mechanisms of inter-axis interactions and meshing constraints (such as XY and BC axis coupling). This embodiment first outlines the model framework for associating total spatial errors with geometric thermal errors (GTEs), and finally proposes a geometric thermal error compensation model for a five-axis simultaneous gear grinding machine.

[0224] 3.1 Mapping relationship between geometric thermal error and total spatial error

[0225] according to Figure 11 We obtain the error propagation matrix of component Q relative to component P.

[0226]

[0227] T pPQ =T x (p x )T y (p y )T z (p z )

[0228] T mQ =T x (X Q )

[0229] in: T represents the actual motion transformation matrix under error conditions; PPQ Here is the positioning coordinate matrix of component Q in the reference coordinate system; (p x ,p y ,p z() represents the positioning coordinates of component Q; C is the distance X that component Q moves along the x(i) direction. Q The transformation matrices of the grinding-related motion axes are summarized in Table 1.

[0230] Table 1 Transformation matrices for each motion axis

[0231]

[0232] Based on the above error propagation principle, the homogeneous coordinate transformation matrix (HCTM) from the grinding wheel to the workpiece under actual working conditions can be calculated as follows:

[0233]

[0234] in: and These represent the actual forward kinematics distances (HCTM) 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 kinematics model can be represented as:

[0235]

[0236] Where: b 11 To b 34 These represent matrix elements. The tool / workpiece pose error model is derived by calculating the differences in the obtained values.

[0237]

[0238] Here [a,b,c,1] T With [i,j,k,0] T δ represents the position vector and attitude vector, respectively; w and ε w These represent the position error vector and attitude error vector of the worm grinding wheel, respectively; δ xw δ yw and δ zw δ w element; ε xw ε yw With ε zw Then it is ε w The elements. This relationship is expressed by the transformation matrix T. GT With the positive kinematic matrix M of the grinding process gw Established through combination:

[0239]

[0240] Therefore, the theoretical motion control commands for the gear grinding process are determined:

[0241]

[0242] Where: i gw Indicates the transmission ratio and its value is N. w / N g In the actual grinding process, the Z-axis first moves to the specified height.

[0243] T GT It is l w and The function of . Therefore, the tooth surface equation and its normal vector can be equivalently expressed as:

[0244]

[0245] The FG tooth surface equation, which includes these errors, can be expressed as:

[0246]

[0247] in: Represent the error tooth surface equation; E is the normal vector; e The set representing all errors; It is the positive kinematics matrix containing errors, which is l w and E e. The function; and This constitutes a new meshing condition.

[0248] In this embodiment, the error compensation model is a unified geometric-thermal error compensation model. This model establishes a mapping relationship between machine tool motion axis errors and workpiece tooth surface errors through forward kinematics, and decouples errors through sensitivity analysis, ultimately generating compensation commands for each motion axis. The mapping relationship is determined by the following formula:

[0249]

[0250] Where: r xw r yw and r zw Indicates the position vector components of the grinding wheel; i w j w and k w The vector component representing the grinding wheel; E δf δ is the position error vector; xg δ yg and δ zg It is the position error component; E εf ε is the attitude error vector; xg ε yg and ε zg Indicates the attitude error components; and T GTThese are the actual forward kinematics transformation matrix and the theoretical forward kinematics transformation matrix, respectively.

[0251] The geometric-thermal error unified compensation model quantifies the contribution of each axis error to the tooth surface error by establishing an error sensitivity Jacobian matrix, and decouples the errors accordingly.

[0252] δ TSE =J GTE ·δ axis

[0253] Among them: J GTE The Jacobian matrix representing the sensitivity between shaft error and tooth surface error; δ axis A vector representing the thermal deformation or geometric error of the machine tool shaft system.

[0254] 3.2 Thermal Error Compensation Model

[0255] 3.2.1 First-order error decoupling and source tracing model

[0256] The differential motion of the machine tool's cumulative pose error can be expressed as:

[0257]

[0258] This relationship is derived by combining the separation of motion axis differential motion error and the decoupling analysis of equivalent motion direction error:

[0259] T GT Δ GT =T m-C Δ C1 T mX T mZ T mA T mY T mB +T m-C T mX Δ X1 T mZ T mA T mY T mB +T m-C T mX T mZ Δ Z1 T mA T mY T mB

[0260] +T m-C T mX T mZ T mA Δ A1 T mY T mB +T m-C T mX TmZ T mA T mY Δ Y1 T mB +T m-C T mX T mZ T mA T mY T mB Δ B1

[0261] In the above equation, the equality of the left and right matrices means that the elements at corresponding positions are equal. This yields a system of twelve equations, containing six unknowns with equivalent decoupling values ​​for the motion axes. Based on this, six equivalent error decoupling values ​​in the motion direction can be determined:

[0262] e x =u x cosB+u z sinB+Yv z cosB-Yv x sinB

[0263] 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

[0264] e z =-((u) x +Yv z )cosAsinB-(u z -Yv x cosAcosB-Xv z sinAcosB+Xv x sinAsinB) / cos 2 A

[0265] e A =v x cosB+v z sinB

[0266] e B =(v y cosA-v z sinAcosB+v x sinAsinB) / cosA

[0267] e C =(v z cosB-v x sinB) / cosA

[0268] 3.2.2 Second-order error source tracing model

[0269] 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:

[0270]

[0271] Where: λ w Indicates the helix angle.

[0272] When the Y-axis position changes by Δy, it will cause a synchronous change in the X-axis position:

[0273] Δx=-Δycotλ w

[0274] Therefore, the final compensation component of the X-axis is determined as follows:

[0275] X C =-(e x -e y cotλ w )

[0276] Similarly, the relationship between the B-axis and the C-axis is expressed as follows:

[0277]

[0278] In actual machining, a change in the position of the B-axis by Δb will cause a synchronous change in the position of the C-axis:

[0279] Δc=i gw Δb

[0280] The final compensation component of the C-axis:

[0281] C C =-(e C +i gw e B )

[0282] Since the A-axis is always locked at zero position during movement, no A-axis compensation is required. The Z-axis compensation component is:

[0283] Z C =-e C

[0284] Implement model-based shaft compensation command optimization:

[0285]

[0286] in: The measured TSE vector is represented by Δu; the compensation command vector for each axis is represented by Δu; and the regularization coefficient for overcompensation is represented by β. The compensation components for each motion axis are determined by the following formula:

[0287]

[0288] 4. Experimental verification

[0289] 4.1 Real-time performance

[0290] To verify the real-time performance of the thermal error compensation system, the experimental setup was designed to simulate actual working conditions. The system includes a CNCFGWGMT machine tool, integrating temperature and positioning sensors to monitor key components such as the spindle and workpiece. Real-time acquisition of sensor data such as temperature, displacement, and load drives the compensation model. The data is processed in real-time by edge devices to predict and compensate for thermal errors before being transmitted back to the machine tool control system for adjustment. Real-time performance is verified by measuring the latency and throughput at each level of the system, including calculating the time required for the system to process sensor data and implement compensation. The latency, from the start of sensor data acquisition to the implementation of compensation, must meet strict real-time requirements to ensure that compensation does not affect the machining process. The target latency is <0.2s to ensure effective real-time error compensation. Throughput refers to the system's ability to process multiple real-time data streams simultaneously without latency or bottlenecks. The system should have a throughput of >95%, meaning it can efficiently process massive amounts of sensor data while maintaining undiminished compensation performance.

[0291] 4.1.1 Comparison of real-time latency and model inference time

[0292] Further testing of the system performance under different operating conditions was conducted to evaluate its robustness and responsiveness: Real-time compensation performance under typical processing conditions without external pressure was tested; the system load was simulated by increasing the number of sensors or complex processing tasks to test performance retention under high computational demands; communication latency between edge devices and the cloud was simulated to verify the system's ability to compensate for network congestion and the continuity of real-time feedback. The performance of this compensation system was compared with that of a baseline system without real-time compensation functionality. Figure 12 As shown, both model inference time and real-time latency increase with increasing task load. The system optimization goal is to reduce inference time by using lightweight models (such as model distillation) and ensure that latency is kept within a reasonable range (e.g., <0.2 seconds).

[0293] Comparing the inference times of PSM and GEMM before and after distillation ( Figure 13The inference time of PSM was reduced from 0.5s in the original model to 0.1s in the distillation model; while that of GEMM was reduced from 0.7s in the original model to 0.2s in the distillation model. These results indicate that model distillation effectively improves inference efficiency, enabling edge devices to achieve faster real-time predictions.

[0294] 4.1.2 Throughput Comparison

[0295] Throughput comparison evaluates the system's ability to efficiently process real-time data, ensuring high throughput (>95%) even under high system load. By implementing a server request prioritization mechanism, critical tasks (such as real-time error compensation) are prioritized, giving higher priority to critical thermal error prediction, while tasks with lower timeliness requirements (such as model updates or background data processing) are queued for processing. Table 2 lists the throughput and latency performance indicators for different tasks in the system, focusing on real-time prediction, compensation, model updates, and low-priority tasks. The system is designed to maintain high throughput (>95%) and low latency (<0.2s) for real-time tasks, while model update tasks processed by the cloud, although with slightly lower throughput, still meet the overall performance requirement of >80%. Furthermore, the compensation time for real-time tasks is controlled within <1s, ensuring immediate execution of thermal error compensation, which is crucial for maintaining processing accuracy. This section verifies that the server request prioritization mechanism can achieve efficient task management under different load conditions, and that the system can still process real-time thermal error compensation tasks in a timely manner even when other computationally intensive tasks (such as model updates) are running concurrently.

[0296] Table 2 Throughput Comparison

[0297]

[0298] 4.1.3 Real-time performance of systems under different frameworks

[0299] The sampling frequency for temperature and error data was set to 1Hz, and the total amount of data collected over six months reached 29.6GB. This embodiment employs a sensor control-edge-cloud framework to improve system execution efficiency, and uses the sensor control-cloud framework as a control framework. The data processing volume after processing by these two frameworks is shown in Table 3.

[0300] Table 3. Data volume processed by the mitigation system (GB)

[0301]

[0302] Table 4 compares the throughput and real-time task processing efficiency of each component of the error compensation system. The edge layer is responsible for real-time thermal error prediction and compensation, exhibiting extremely high real-time task processing efficiency, with latency below 0.2s and throughput exceeding 95%, ensuring immediate response to thermal errors without delay. The cloud layer mainly handles heavy computational tasks such as model optimization and retraining, and large-scale data analysis. Although its real-time responsiveness is slightly lower than that of the edge layer, it still maintains more than 80% throughput, ensuring that complex model updates do not significantly hinder system operation. By implementing a server request priority allocation mechanism, the system ensures that real-time tasks always receive the highest processing priority. This mechanism can maintain the latency of error compensation tasks below 0.2s even under high system load. The model simplification technology based on knowledge distillation ensures that edge devices can quickly complete inference. The inference time of the distilled student model is less than 0.1s, which can quickly generate thermal error prediction results suitable for real-time compensation. The overall real-time compensation performance of the system is excellent, with a compensation accuracy exceeding 95%, accurately and promptly eliminating thermal errors and maintaining processing precision under dynamic operating conditions. In summary, the results in Table 4 show that, through the synergistic effect of edge-cloud collaboration, task priority management, and model distillation technology, the system in this embodiment achieves excellent throughput, low latency, and high compensation accuracy, meeting the stringent requirements of the precision manufacturing field for real-time applications.

[0303] Table 4 Throughput Comparison

[0304]

[0305] 4.2 System Effectiveness

[0306] This embodiment uses the FGWGMT (Precision Grinding Machine Tool) as an example, and all experimental data were collected from this machine tool. Its structure is specifically optimized to meet the requirements of FG (Precision Grinding), equipped with a Siemens 840Dsl system, and employs electronic gearbox synchronization technology to achieve multi-axis control. Linear axis geometric errors were measured using an XL-80 laser interferometer.

[0307] 4.2.1 Validation of the thermal error model

[0308] (1) Fitting and prediction performance

[0309] Drastic variations in temperature error (TE) place extremely high demands on model capabilities. In TE prediction models built based on the training dataset, such as... Figure 14 As shown in (a), PSM (Physically Constrained Model) significantly outperforms LSTM (Long Short-Term Memory) and STGCN (Spatiotemporal Graph Convolutional Network), while backpropagation (BP) performs the worst. The effectiveness of PSM is verified, and STGCN far surpasses LSTM. Figure 14(b) The results show that the residual volatility of the BP network is higher than that of the other three models, while the residual volatility of LSTM is greater than that of STGCN and PSM. The high fitting accuracy of LSTM stems from its excellent temporal modeling ability, while the superior performance of STGCN and PSM is attributed to their powerful spatiotemporal modeling capabilities. Research indicates that PSM can effectively characterize the error mechanism, and physical constraint modeling strengthens the spatiotemporal behavior representation. The BP network, lacking memory capacity, cannot store long-term thermal history information, highlighting the importance of spatiotemporal behavior modeling for error prediction.

[0310] The evaluation parameters are shown in Appendix Table 5. PSM showed higher fitting accuracy than other models, and both STGCN and PSM outperformed BP network and LSTM. By ensuring a high degree of fit between PSM and the characteristics of thermal information data, the model effectively improved prediction accuracy. Notably, PSM's accuracy surpassed that of STGCN, validating the effectiveness of the TE modeling method.

[0311] Appendix 5 evaluates the PSM training results. This table presents the performance evaluation of the PSM training phase, including mean absolute error (MAE), root mean square error (RMSE), and coefficient of determination (R²). 2 Key metrics such as [list of metrics] were tested. The results show that the model achieves low training error and high fitting accuracy, proving that it effectively learns the thermal / geometric error characteristics from the training data. This excellent training performance lays the foundation for subsequent testing and real-time deployment.

[0312] Table 5 Evaluation of PSM Training Results

[0313]

[0314] according to Figure 15 As shown, similar conclusions can be drawn regarding TE prediction, namely, the importance of spatiotemporal behavioral features. Detailed evaluation parameters for each model are shown in Table 6. The prediction accuracies of BP neural network, LSTM, STGCN, and PSM are 93.85%, 94.74%, 96.35%, and 97.62%, respectively. Comparing Tables 5 and 6, it can be seen that STGCN and PSM exhibit the smallest accuracy degradation under different operating conditions, while BP neural network and LSTM show a more significant decrease in accuracy. This indicates that STGCN and PSM are more robust than BP neural network and LSTM. Among them, PSM's prediction accuracy is slightly higher than STGCN; therefore, PSM is a suitable model for TE prediction.

[0315] Table 6 presents the evaluation results of PSM prediction. Table 6 summarizes the quantitative evaluation of PSM prediction performance under different operating conditions, with evaluation indicators including MAE, RMSE, and R... 2 Data shows that both models achieve low prediction errors and high fitting accuracy, indicating that they have excellent generalization ability and robustness under varying operating conditions.

[0316] Table 6 Evaluation of PSM Prediction Results

[0317]

[0318] (2) Ablation experiments and cross-validation

[0319] The K-fold cross-validation results of PSM (PropensityScoreMatching) were obtained. As shown in Table 7, 5-fold cross-validation was performed, and the results show that the model consistently achieves low prediction error (average MAE of 5.24 μm) and high generalization ability (R²). 2 -0.94) validated its robustness and accuracy in real-time thermal error prediction in the precision manufacturing field. Table 7 shows the results of the PSM model using 5-fold cross-validation, verified by MAE, RMSE, and R... 2 Three indicators are used to evaluate the model's performance.

[0320] Table 7. PSM Cross-Validation Results

[0321]

[0322]

[0323] Table 8 illustrates the impact of disabling individual components in the PSM ablation study. The study shows that disabling edge-cloud collaboration leads to increased latency and decreased throughput, while server request priority scheduling and model distillation improve overall performance. The complete system with all components enabled performs best, exhibiting low latency (0.20 seconds), high throughput (95%), and high error correction accuracy (96%). Table 8 presents the results of the error mitigation system ablation study, evaluating the impact of disabling key components on system latency, throughput, and error correction accuracy.

[0324] Table 8 Ablation Study Results of Error Mitigation System

[0325]

[0326] 4.2.2 Verification of end face gear grinding

[0327] The comparative experiments used the same type of worm gear grinding wheel, made with DA bonded wheel material. The FG control group consisted of gears with the same machining allowance, all machined on an FGWGMT machine tool. The gear material was AISIE9310 alloy steel, as detailed in Table 9. Table 9 lists the key geometric and process parameters of the worm gear grinding wheel and face gears used in the experimental setup, including dimensions, material properties, grinding wheel specifications, and machining conditions. These parameters form the basis for machining process analysis, error modeling, and subsequent verification of thermal / geometric error mitigation strategies.

[0328] Table 9 Parameters of Worm Grinding Wheel and Face Gear

[0329]

[0330] FG grinding experimental setup such as Figure 16 As shown. The linear shaft moves rapidly during the idle phase and slowly during the grinding phase. The single feed rates for roughing, semi-finishing, and finishing are 20 μm, 10 μm, and 5 μm, respectively. FG differs significantly from cylindrical and bevel gears. The accuracy evaluation standards for cylindrical and bevel gears are not applicable to FG. Therefore, existing standards for bevel gears are often referenced to evaluate the surface geometry of FG gears. TSE measurements for FG are primarily for the working tooth surface and require special methods and equipment to accommodate its unique characteristics. According to AGMA2009-B01-2001, the measurement area is defined as a 5×9 grid (…). Figure 17 The grid is divided according to a 5% shrinkage rate in the tooth height direction and a 10% shrinkage rate in the tooth width direction. Using a rotational projection method, the measurement grid and measurement points are projected onto the symmetry plane xOz of the tooth groove containing the measured tooth surface, as shown below. Figure 18 As shown.

[0331] z d1 and z d2 These represent the minimum and maximum values ​​of a point on the tooth surface along the tooth height direction (corresponding to the root height and the addendum), respectively; a i and z ijare Let G(i,j) be 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, with x-direction coordinates a,j, and j respectively. min and a max :

[0332]

[0333] Where Rmin and Rmax represent the inner and outer radii of the measured FG, respectively. Interval [a] min ,a max Divide the tooth width direction equally, and find the x-coordinate a of the measuring point at the i-th division. i It is given by the following formula:

[0334]

[0335] The x-coordinates and z-coordinates of a point on the transition curve are determined by the following formula:

[0336]

[0337] Therefore, the upper and lower boundaries of the mesh in the tooth height direction are determined as follows:

[0338]

[0339] In the tooth height direction (z-axis), the interval [z] i5 ,z i1Divide into equal parts. The final result is:

[0340]

[0341] The coordinates of the measurement point G(i,j) are denoted as a. i and z ij The specific steps are as follows: First, perform GTE (Geometric Thermal Error) compensation, then start the gear grinding program to complete the compensated FG (Fine Grinding) machining. Measure the gear tooth surface errors (TSEs) with and without GTE compensation, and compare and evaluate them. Figure 19 The following is the process of measuring the geometric accuracy of the tooth surface of the FG gear after machining: The theoretical measurement point data is imported into the measurement module. First, the coordinate measuring machine is calibrated and a positioning reference and measurement coordinate system are established. The probe measures point by point according to the planned path and moves along the normal direction near the measurement point to capture machining errors. Finally, a topology error map of 45 measurement points on the tooth surface is generated based on the normal error data. Figure 20 The distribution of tooth surface normal error with and without GTE compensation is shown. The maximum normal error on the left tooth surface decreased from 22.5 μm to 6.3 μm, and the maximum normal error on the right tooth surface decreased from 23.2 μm to 6.2 μm.

[0342] 4.3 System Reliability and Robustness

[0343] 4.3.1 Cross-validation and ablation studies

[0344] To validate the performance of PSM (Predictive State Model) and GEMM (Geometric Error Machine Model), K-fold cross-validation, incorporating sensor readings and machine tool data, was employed: the dataset was divided into K=5 subsets, with each subset used alternately as the validation set, and the remainder used for training. Evaluation metrics included MAE (Mean Absolute Error), RMSE (Root Mean Square Error), and R². 2 Table 10 shows that the average MAE and RMSE of both models indicate low prediction errors, and R... 2 A value close to 1 indicates a strong correlation between the predicted and actual values, confirming that the model has good generalization ability across different data subsets. Table 10 demonstrates the performance of the PSM model using K-fold cross-validation, employing MAE, RMSE, and R-squared. 2 The indicator evaluation uses 50% of the data. Each percentage point represents R. 2 The consistently high value indicates that the model has low prediction error and high generalization ability, which verifies the reliability of the proposed model in real-time thermal / geometric error prediction and compensation.

[0345] Table 10 PSM Cross-Validation Results

[0346]

[0347] In this ablation study, the impact of key system components such as edge cloud collaboration, server request priority scheduling, and model distillation on overall real-time error compensation performance was evaluated. This was accomplished by systematically disabling each component and measuring its impact on latency, throughput, and compensation accuracy. Specific experiments are as follows:

[0348] (1) Disable edge cloud collaboration: Edge devices perform all tasks without cloud support.

[0349] (2) Disable server request priority scheduling: High-latency tasks are not prioritized during task processing.

[0350] (3) Disable model distillation: Deploy the full model (non-distilled version) on edge devices.

[0351] As shown in Table 11, the ablation experiment results highlight the contributions of each component to system performance. Edge-cloud collaboration significantly improved real-time latency by optimizing task allocation between the edge and cloud, reducing latency from 0.45s to 0.20s. Server request priority scheduling improved throughput and ensured timely processing of critical tasks, maintaining a high throughput (≥91%). Model distillation significantly reduced model inference time on edge devices, achieving faster processing without sacrificing prediction performance, thus contributing to reduced latency and improved accuracy. Table 11 illustrates the impact of disabling key components in the system on the real-time performance of the thermal and geometric error compensation process. The evaluated components include edge-cloud collaboration, server request priority scheduling, and model distillation. The results show the changes in latency, throughput, and compensation accuracy when each component is disabled. The complete system with all components enabled exhibits the best performance in terms of latency reduction, throughput improvement, and compensation accuracy, validating the contribution of each component to the overall efficiency and effectiveness of the system's real-time error compensation.

[0352] Table 11 Ablation Study Results of Error Compensation System

[0353]

[0354] 4.3.2 Scalability Verification under Increasing System Load

[0355] To evaluate the system's scalability in a real-world environment with varying sensor density, the experiment involved progressively increasing the number of connected sensors while simultaneously increasing the data transmission rate. The average real-time latency was recorded under different system loads. The experimental configuration was as follows: initial sensor count of 5, increment step of 5 seconds per test, maximum sensor count of 30, and data acquisition frequency of 1 Hz. Figure 21As shown, when the number of sensors is less than 20, the system maintains an average delay of less than 0.2s; as the system load continues to increase, the delay gradually increases but remains within an acceptable range (less than 0.5s), even with 30 sensors.

[0356] 4.3.3 System robustness verification under abnormal conditions

[0357] To verify the system's robustness to network outages and harsh environments, two additional experiments were conducted: 1) The network connection between the edge device and the cloud server was artificially interrupted for 60 seconds during operation. The system's recovery capability was evaluated based on data transmission recovery ability and task restart latency. During the network outage, the edge device independently maintained its local prediction task, activated the data buffer, and no significant data loss occurred. After the network was restored, it automatically reconnected to the cloud and resumed synchronization within 5 seconds; 2) The system was tested in a high-temperature environment (maximum 20℃). A slight increase in latency was observed (average latency increase of approximately 2%), but it remained within the real-time threshold. Table 12 summarizes the system's performance under various abnormal conditions, including network outages, high-temperature exposure (50℃), and vibration stress. The results show that the system can recover from network outages within 5 seconds without loss of prediction accuracy, with only a slight decrease in accuracy observed, confirming the system's robustness and stability in harsh environments.

[0358] Table 12 System robustness verification under abnormal conditions

[0359]

[0360] 4.3.4 Comparison of inference time on different edge devices

[0361] To evaluate the system's adaptability to different hardware platforms, the inference time of the distillation model was measured on two typical edge devices (NVIDIA Jetson Nano and NVIDIA Xavier NX). The test subjects were the distillation PSM and distillation GEMM models, with a batch size of 1 (real-time single prediction). Figure 22 As shown, even on the lower-performance Jetson Nano platform, the inference time can still be controlled within 0.15s, meeting real-time requirements. Deployment on high-performance devices such as XavierNX further shortens the inference time, providing flexible options for different industrial scenarios. These supplementary experiments confirm that the system proposed in this embodiment not only maintains high real-time performance under increased load and exhibits strong robustness to network outages and environmental stress, but also demonstrates excellent adaptability among different edge devices.

[0362] 5. Conclusion

[0363] This embodiment proposes an innovative physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing, addressing several key challenges in the field. Through system architecture design, physical constraint spatiotemporal model development, and geometric-thermal error compensation model integration, it advances the technological frontier of error prediction and compensation. The main conclusions and contributions of this embodiment are as follows:

[0364] (1) The system architecture proposed in this embodiment successfully integrates edge computing and cloud computing, maintaining high computational efficiency while ensuring real-time performance. Through the edge-cloud collaborative mechanism, the system effectively balances low-latency tasks such as thermal error prediction / compensation with computationally intensive tasks such as model optimization / training, and further optimizes performance by employing server request priority scheduling and model distillation techniques. Real-time verification shows that the system can maintain a low-latency response (<0.2s) and high accuracy (>95%) under various operating conditions. Model distillation shortens the inference time, and the distilled model achieves faster predictions than the original model, ensuring efficient real-time thermal error compensation. The maximum normal error of the left tooth surface decreased from 22.5μm to 6.3μm, and the maximum normal error of the right tooth surface decreased from 23.2μm to 6.2μm.

[0365] (2) The physical constraint spatiotemporal model developed in this embodiment significantly improves the accuracy and efficiency of thermal error prediction by integrating physical principles and spatiotemporal data. Traditional thermal error models often ignore the dynamic and complex interaction between machine tool components and the thermal environment, leading to inaccurate predictions. By introducing physical constraints and considering spatiotemporal behavioral characteristics, the proposed model improves prediction accuracy while reducing computational complexity, achieving a prediction accuracy of 97.62%.

[0366] (3) The geometric-thermal error compensation model proposed in this embodiment achieves a significant breakthrough by integrating thermal error compensation and geometric error compensation into a unified framework. Unlike existing systems that handle these two types of errors separately, this model can collaboratively address both issues in real time, providing a more comprehensive solution.

[0367] In summary, this embodiment provides an innovative method for thermal / geometric error compensation in precision manufacturing. By addressing key real-time challenges and integrating advanced technologies such as digital twin systems, edge-cloud collaboration, and model distillation, it lays the foundation for developing a more efficient and accurate compensation system. The proposed system is not only scalable to adapt to different manufacturing conditions but also provides a complete solution for real-time error prediction and compensation, making it a practical tool in the field of precision manufacturing. The current framework focuses on the main thermal errors and does not consider deformation caused by cutting forces. While this assumption is applicable to low-load worm gear grinding, it may fail under heavy cutting conditions. Therefore, future work plans to integrate a low-order compliance kernel function based on spindle power signals and force gauge data to achieve synchronous compensation of thermal / mechanical deformation under high-load scenarios.

[0368] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A real-time compensation system for physical-guided edge-cloud coordinated thermal and geometric errors in precision manufacturing, characterized in that: include: The sensing and control layer is used to collect multi-source physical signals of precision manufacturing equipment in real time during the processing, and output real-time compensation commands from the edge computing layer to the control system of the precision manufacturing equipment. The edge computing layer is equipped with a lightweight error prediction model and an error compensation model, which are used to perform real-time inference based on the multi-source physical signals uploaded by the sensing and control layer, generate real-time compensation instructions for the precision manufacturing equipment, and send the instructions to the sensing and control layer. The cloud platform layer is used to aggregate the data uploaded by the edge computing layer, perform model training and global optimization tasks, and update the optimized model parameters to the edge computing layer. The system uses an edge-cloud collaborative mechanism for task allocation and load balancing, a server request priority scheduling mechanism to ensure low-latency processing of real-time tasks, and a model distillation technique to simplify the teacher model trained at the cloud platform layer into a student model suitable for edge deployment.

2. The physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing according to claim 1, characterized in that: The edge-cloud collaboration mechanism achieves dynamic task allocation and load balancing through the following methods: Real-time task T edge Distribute computationally intensive tasks T to the edge for processing. cloud Offloading to the cloud for processing, allocation is based on factors including available computing resources R at the edge. edge Cloud-based computing resources R cloud Edge load L edge and cloud load L cloud , is represented as: T edge =f(T real-time ,R edge ,L edge ),T cloud =f(T compute-intensive ,R cloud ,L cloud ) Wherein: T real-time and T compute-intensive These represent real-time tasks that are highly sensitive to latency and computationally intensive tasks, respectively. Edge load L edge It is dynamically determined by the following formula: The updated model M is periodically sent from the cloud. cloud Synchronize to edge model M edge ,Right now M edge ←Sync(M cloud ,frequency) The frequency parameter controls how often the cloud model is synchronized to the edge.

3. The physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing according to claim 1, characterized in that: The request priority scheduling mechanism is as follows: task requests are stored in a queue according to their priority. T priority =PriorityQueue(T1,T2,…,T n ,P1,P2,…,P n ) Wherein: T priority P represents a priority task queue; i For task T i Priority assignment; Real-time error compensation tasks have the highest priority; for low-priority tasks, an exponential backoff algorithm is used to dynamically adjust the request frequency, and the time interval for the k-th request is: Δt k =Δt0·2 k Where: Δt k Δt0 represents the time interval of the k-th request; Δt0 is the initial request interval; and k is the number of request attempts.

4. The physical-guided edge-cloud collaborative thermodynamic and geometric error real-time compensation system for precision manufacturing according to claim 1, characterized in that: The model distillation technique is implemented as follows: a high-precision physical constraint spatiotemporal model and a geometric-thermal error comprehensive compensation model are trained at the cloud platform layer to serve as the teacher model; a student model is trained by minimizing the distillation loss function, and the student model is deployed at the edge computing layer for real-time inference; the distillation loss function is expressed as: C priority =λ·Γ student +(1-λ)·C teacher Wherein: Γ andom and Γ teacher λ represents the loss values ​​of the student model and the teacher model, respectively; λ is a hyperparameter that controls the knowledge weight of the teacher model during the distillation process.

5. The real-time compensation system for physical-guided edge-cloud coordinated thermodynamic and geometric errors in precision manufacturing according to claim 1, characterized in that: The lightweight error prediction model deployed in the edge computing layer is a physically constrained spatiotemporal model, including: A multi-scale time encoder module is used to extract multi-scale temporal features from historical temperature sequences; A physically constrained spatial graph convolutional network is used to construct a spatial graph based on sensor topology and temperature gradient, and embed thermal diffusion physics rules for spatial feature modeling. The gated fusion unit dynamically weights and fuses the multi-scale temporal features with the spatial node embeddings. A dynamic fusion layer is used to fuse the outputs of the multi-scale temporal encoder and the physical constraint spatial graph convolutional network to predict thermal errors. The physical constraint spatiotemporal model introduces a physical residual loss term based on the heat conduction equation during training, and its loss function is: C total =C MSE +λ1·C physics +λ2·C smooth Wherein: Γ MSE Γ represents the mean squared error between the predicted and actual values; λ1 and λ2 are the weighting coefficients for physical loss and smoothing loss; physics For physical residuals; Γ smooth This represents the smoothing loss term used to ensure continuity between spatially adjacent nodes and compensate for oscillations; α is the thermal diffusivity; T is the temperature field; and t is the time variable. is the Laplace operator in a three-dimensional Cartesian coordinate system.

6. The physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing according to claim 5, characterized in that: The multi-scale temporal encoder module includes a multi-scale temporal encoder, a dynamic temporal reweighting module, a lightweight graph convolutional network, and a cross-scale feature fusion module. The multi-scale timing encoder employs layered stacking of hollow GRU units with different expansion rates, based on the following principle: in: Represents large-scale temporal embedding; For small-scale temporal embedding; GRU L (T i ) and GRU S (T i () represent stacked void GRU modules with large and small expansion coefficients, respectively; τ represents the historical temperature sequence, and τ is the length of the historical time window. The dynamic temporal reweighting module adaptively adjusts the weights, and the reweighting feature is represented as follows: in: and They represent and The repetition feature; ⊙ indicates element-wise multiplication; g L and g s The learned temporal importance weights are represented as: Where: σ is the Sigmoid activation function; W L and W S b is a learnable weight matrix; L With b s For bias terms; The lightweight graph convolutional network is used to enhance local temporal correlations, focusing on local short-range thermal interactions between adjacent time steps, and is represented as: in: Temporal features encoded by GCN; The cross-scale feature fusion module fuses multi-scale temporal features: in: This represents the fused multi-scale temporal features obtained through fusion. and They represent respectively to and Features obtained by performing attention-based cross-scale fusion and uniformly normalizing them to the same scale; α L α s and α GCN Indicates the scale attention coefficient; The fused features are obtained by feature fusion through attention-weighted matrix multiplication. In the feature Add input T above i The obtained features are used to enhance gradient flow; W g and b g These are learnable global gating parameters.

7. The physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing according to claim 5, characterized in that: The physically constrained spatial graph convolutional network is used to construct a physically guided spatiotemporal graph based on sensor spatial coordinates and historical temperature sequences, and aggregates the adjacency matrix across time steps through a time decay matrix. Finally, graph convolution operations are performed to obtain spatially coupled node embeddings. In the spatiotemporal graph, for time slice t, the adjacency matrix elements between nodes i and j Represented as: Where: x i and x j These are the spatial coordinates of nodes i and j, respectively; and For temperature readings; σ controls spatial decay rate; β controls thermal similarity sensitivity; The time decay matrix is ​​represented as: Where: t and t ′ Representing different time indices; the method for aggregating the adjacency matrix of all time steps is as follows: Among them: A * For aggregated adjacency matrix; A *(t) Let be the adjacency matrix of time slice t; Based on the aggregated adjacency matrix A * Perform graph convolution operations to propagate spatial features and obtain spatially coupled node embeddings: H (l+1) =σ(A * H (l) W (l) ) Wherein: H (l) W is the feature matrix of the nodes in the l-th layer; (l) σ is the learnable transformation weight matrix; σ(·) is the activation function.

8. The real-time compensation system for physical-guided edge-cloud coordinated thermodynamic and geometric errors in precision manufacturing according to claim 5, characterized in that: The input of the dynamic fusion layer is connected to the output of the multi-scale temporal encoder module and the physically constrained spatial graph convolutional network, and is used to fuse the input temporal features with spatial embeddings and output thermal error prediction values; the dynamic fusion layer uses a gated fusion mechanism to achieve feature fusion, and the gated fusion mechanism is expressed as follows: F fused =σ(W1·F1)⊙F1+σ(W2·F2)⊙F2+F res Where: F1 and F2 represent the feature vectors of the multi-scale temporal encoder module and the physically constrained spatial graph convolutional network, respectively; W1 and W2 are learnable weight matrices; σ represents the Sigmoid function that generates the gated weights; ⊙ is the Hadamard product; F res This represents a residual connection that enhances gradient flow.

9. The physical-guided edge-cloud collaborative real-time thermal and geometric error compensation system for precision manufacturing according to claim 1, characterized in that: The error compensation model is a unified geometric-thermal error compensation model. This model establishes a mapping relationship between machine tool motion axis errors and workpiece tooth surface errors through forward kinematics, and decouples errors through sensitivity analysis, ultimately generating compensation commands for each motion axis. The mapping relationship is determined by the following formula: Where: r xw r yw and r zw Indicates the position vector component of the grinding wheel; i w j w and k w The vector component representing the grinding wheel; E δf δ is the position error vector; xg δ yg and δ zg It is the position error component; E εf ε is the attitude error vector; xg ε yg and ε zg Indicates the attitude error components; and T GT These are the actual forward kinematics transformation matrix and the theoretical forward kinematics transformation matrix, respectively. The geometric-thermal error unified compensation model quantifies the contribution of each axis error to the tooth surface error by establishing an error sensitivity Jacobian matrix, and decouples the errors accordingly. d Tse =J GTE ·d axis Among them: J GTE The Jacobian matrix representing the sensitivity between shaft error and tooth surface error; δ axis A vector representing the thermal deformation or geometric error of the machine tool shaft system.

10. The real-time compensation system for physical-guided edge-cloud coordinated thermodynamic and geometric errors in precision manufacturing according to claim 1, characterized in that: The geometric-thermal error mitigation model considers the linkage between motion axes when generating compensation commands. For face gear and worm gear grinding machines, the final compensation amount for each axis is calculated using the following formula: Where: X C Z C and C C The compensation amounts for the X, Z, and C axes are respectively, e x e y e B and e C To trace back to the equivalent error of each motion axis through error decoupling; λ w i is the gear helix angle; pw This is the transmission ratio.

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