An adaptive polishing and grinding method for aero-engine blades

Abstract

This invention discloses an adaptive polishing and grinding method for aero-engine blades, comprising: constructing an aero-engine blade polishing and grinding process database; performing collaborative preprocessing and deep feature extraction on multi-source data in the database; constructing an adaptive decision model based on a dual-path strategy of offline generation and online adjustment to generate and adjust process parameters in real time; and optimizing the process parameters offline based on measured quality feedback of the ground blades, and updating the process database and the adaptive decision model. This invention achieves adaptive decision-making and optimization control of process parameters.

Description

Technical Field

[0001] This invention relates to the field of deep learning technology, and in particular to an adaptive polishing and grinding method for aero-engine blades. Background Technology

[0002] As a core component of the power system, the surface quality of aero-engine blades directly determines the overall performance, service life, and reliability of the engine. Blades endure extremely high aerodynamic loads and thermal stresses under complex operating conditions, thus requiring extremely stringent precision and surface integrity. Traditional polishing and grinding processes heavily rely on operator experience, often determining process parameters through trial and error. This makes it difficult to guarantee consistent quality in mass production and cannot adapt to the geometric changes brought about by new blade models. The complex blade profiles and challenging material processing mean that even minor surface defects or contour deviations can lead to airflow loss, efficiency reduction, or even structural failure. Therefore, achieving precise, stable, and adaptive control of the blade polishing and grinding process is a crucial step in improving engine manufacturing capabilities and ensuring flight safety.

[0003] With the rapid development of intelligent manufacturing and industrial big data technologies, introducing multi-source sensing, deep learning, and adaptive control into the precision manufacturing field has become an important industry trend. Developing an intelligent polishing algorithm capable of autonomously sensing processing status, extracting process features in real time, and dynamically adjusting parameters not only helps to break through the reliance on manual experience in traditional processes but also enables online optimization and stable control of processing quality. By integrating visual, force control, and process timing data, this algorithm can deeply understand the complex dynamic relationships in the polishing process, maintaining process consistency even when facing material fluctuations, tool wear, or blade geometric differences, thereby driving the evolution of blade manufacturing towards intelligence and flexibility.

[0004] Overall, developing adaptive polishing and grinding technology for aero-engine blades is an inevitable requirement to meet the challenges of high-precision manufacturing and improve production efficiency and product quality.

[0005] The existing blade adaptive polishing and grinding methods are as follows: Chinese invention patent application number CN202511306773.7 relates to a constant-force grinding device and method based on gravity-elastic dynamic balance. This invention solves the problem of contact force fluctuations caused by blade profile deviations, actuator positioning errors, and grinding wheel wear, ensuring that the normal pressure remains stable at a set threshold during polishing. The device includes a constant-force coil spring, a motor, a base, a limiting block, a linear guide rail, a counterweight, a grinding wheel, and a basic frame. The vertical linear guide rail is mounted on the basic frame, and the base is slidably connected to the linear guide rail, allowing it to move up and down on the rail. The motor is fixed to the base. The upper end of the constant-force coil spring is fixed to the top of the basic frame, and the lower end is connected to the base. A limiting block is fixed to the linear guide rail below the base. The motor drives the grinding wheel to rotate. The counterweight is mounted on the motor or the base, and the counterweight requirement is calculated based on the target downward pressure. This invention improves contact force stability and extends the maintenance cycle: the constant-force floating structure requires no electrical maintenance.

[0006] Chinese invention patent application number CN202311477766.4 belongs to the field of grinding and polishing technology for end grooves of aero-engine blades. Specifically, it is a grinding and polishing device for end grooves of aero-engine blades. The device uses a machining positioning base plate to move in a plane on the machining positioning center, which drives the small end edge plate grinding positioning plate and the large end edge plate grinding positioning plate to move and grind and polish around the grinding equipment. The dual positioning hole design of the large end edge plate directional corner clamp and the small end edge plate grinding directional corner clamp enables the rapid and precise reversal of the aero-engine blades. This allows for precise automated control of the grinding and polishing of the end grooves of aero-engine blades, improves the consistency of grinding and polishing, and saves grinding and polishing costs.

[0007] However, the above method has the following limitations: (1) Insufficient intelligence and sensing capabilities in the process. Existing technologies mainly rely on precise mechanical structures or fixtures to achieve physical compensation and positioning, lacking the ability to sense and intelligently analyze the dynamic evolution of process parameters and real-time contact status during polishing and grinding.

[0008] (2) Limited adaptability to complex geometry and working conditions. The constant force control or special fixture scheme based on fixed mechanical principles is difficult to flexibly adapt to different models of blades with complex free-form surface features, as well as the disturbances caused by wear of the mold and slight changes in material properties during the processing. The process parameters are usually preset fixed values ​​and cannot be adjusted online.

[0009] (3) Lack of multi-source information fusion and collaborative decision-making. Existing methods fail to effectively integrate multimodal sensing information such as vision, force control, and process parameters. Process decisions are fragmented. For example, constant force control does not consider real-time feedback of surface quality, and the positioning device is not linked with process parameters, making it impossible to achieve collaborative optimization control oriented towards the final quality goal. Summary of the Invention

[0010] To address the technical problems existing in the prior art, this invention proposes an adaptive polishing and grinding method for aero-engine blades. By deploying a multi-source sensor array to collect process data in real time, and constructing a deep feature extraction network that integrates visual, temporal, and force control signals, the method ultimately achieves adaptive decision-making and optimized control of process parameters based on a dual-pathway strategy of offline generation and online adjustment.

[0011] To achieve the above objectives, the present invention provides an adaptive polishing and grinding method for aero-engine blades, comprising: Construct a database of polishing and grinding processes for aero-engine blades; Collaborative preprocessing and deep feature extraction are performed on the multi-source data in the database; Based on a dual-path strategy of offline generation and online adjustment, an adaptive decision-making model is constructed to generate and adjust process parameters in real time. Based on the measured quality feedback of the polished blades, the process parameters are optimized offline, and the process database and adaptive decision model are updated.

[0012] Preferably, the process database for polishing and grinding the aero-engine blades is constructed, including: Deploy a multi-source sensor array to collect real-time process parameter time-series data, blade surface visual data and force sensing time-series data during the grinding process, and collect the final quality index data of the blade offline, record the control parameter setting data corresponding to the grinding process, and synchronously import historical process archive data. The multi-source data is spatiotemporally aligned and fused, and the polishing quality status is marked according to the quality indicators. Based on the historical process archive data, the target values ​​of optimized process parameters are marked for the data samples to construct a structured process database. The multi-source sensor array includes process parameter sensors, vision sensors, force sensors, and quality inspection equipment.

[0013] Preferably, the spatiotemporal alignment and fusion processing of the multi-source data includes: A spatial phase compensation method based on blade kinematics is used to perform spatiotemporal alignment of the visual data of the blade surface and the temporal data of force sensing. Based on historical process archive data, an algorithm for labeling optimized process parameters is used to label the data samples with the target values ​​of optimized process parameters.

[0014] Preferably, the visual data of the blade surface and the temporal data of force sensing are spatiotemporally aligned, including: Based on the grinding path trajectory, spindle rotation angle and feed axis position, the time compensation required for the force sensor measurement point to move to the vision sensor shooting point is calculated in real time, and the vision data timestamp is compensated. The algorithm for labeling the target values ​​of optimized process parameters includes: For samples that have passed the grinding quality test, the average value of the actual process parameters will be marked as the target value. For non-compliant samples, the geometric feature-high-quality process parameter regression model trained based on gradient boosting decision tree is used to predict and label the target values ​​of optimized process parameters based on the blade geometric feature vector.

[0015] Preferably, collaborative preprocessing and deep feature extraction are performed on the multi-source data in the database, including: The visual data of the blade surface is preprocessed by denoising, enhancement and normalization, and then input into the surface defect perception convolutional neural network to extract the visual depth feature vector. The surface defect perception convolutional neural network includes a curvature-guided spatial attention mechanism to extract features of geometrically complex areas of the surface. The time series data of process parameters are preprocessed by outlier removal, missing point filling and standardization, and then input into the process dynamic evolution gated recurrent unit network to extract the process dynamic feature vector. The process dynamic evolution gated recurrent unit network includes a multi-scale temporal convolution module to capture process patterns at different time scales. The force sensing time series data is filtered, statistically characterized, and force tracking error is calculated. The processed results are then input into a steady-state contact force feature extraction network to extract the force-controlled steady-state feature vector.

[0016] Preferably, the curvature-guided spatial attention mechanism specifically comprises: A local curvature map is generated based on the computer-aided design model of the blade or the photometric stereo vision method. The curvature map is then used as a priori information channel and convolved with the convolutional feature map. After channel compression and activation processing, a spatial-channel composite attention weight map is generated, which is used to perform weighted fusion of the original feature map.

[0017] Preferably, based on a dual-pathway strategy of offline generation and online adjustment, an adaptive decision-making model is constructed to generate and adjust process parameters in real time, including: Based on the geometric information of the blade to be ground, an initial grinding path trajectory and initial process parameter combination are generated through similar case retrieval and parameter generation network. During the grinding process, a multimodal feature fusion adaptive controller receives real-time extracted visual depth feature vectors, process dynamic feature vectors, and force control steady-state feature vectors. Through feedforward compensation path and online feedback adjustment path, it calculates and outputs real-time spindle speed setpoints and real-time feed rate setpoints.

[0018] Preferably, the processing procedure of the similar case retrieval and parameter generation network is as follows: The current blade geometric feature vector is matched with the geometric feature vectors of historical samples to retrieve the top K samples with the highest similarity. Attention weights are calculated based on the polishing quality status and similarity scores of the K samples. The process parameter combinations of high-quality historical samples are then weighted and aggregated to generate initial process parameters.

[0019] Preferably, the multimodal feature fusion adaptive controller includes: Feedforward compensation path: Based on the Preston equation and real-time force control data, calculate the theoretical process parameter reference values, and predict the nonlinear deviation between the theoretical spindle speed and feed rate reference values ​​and the parameters required for actual high-quality process through the physical model residual compensation network; Online feedback adjustment pathway: Through a lightweight online update neural network, fine adjustment amounts are calculated based on real-time surface visual features and process-force control fusion features to optimize surface quality; The controller adds the initial process parameters, feedforward compensation, and online feedback adjustment to output the final real-time process parameter setpoints.

[0020] Preferably, based on the measured quality feedback of the polished blades, the process parameters are optimized offline, and the process database and adaptive decision model are updated, including: After the grinding of a single blade is completed, based on the deviation between the measured quality and the target value, the Bayesian process parameter optimizer is activated to perform offline parameter optimization, and the new data samples and optimized parameters are updated to the process database. The surface defect perception convolutional neural network, process dynamic evolution gated recurrent unit network, steady-state contact force feature extraction network, similar case retrieval and parameter generation network, and multimodal feature fusion adaptive controller are integrated and deployed in the blade polishing and grinding CNC system. The effectiveness of the algorithm in terms of grinding pass rate, batch blade surface quality consistency and debugging cycle is verified through comparative experiments. The Bayesian process parameter optimizer employs a Gaussian process surrogate model, and its kernel function is a composite kernel, including: Automatic correlation determination kernel function is used to characterize the interaction relationship between process parameters; The Matern kernel function is used to model the nonlinear relationship between process characteristics and quality deviations. Periodic kernel functions are used to characterize the effects of periodic fluctuations in a process.

[0021] Compared with the prior art, the present invention has the following advantages and technical effects: (1) Achieving intelligent perception and autonomous decision-making throughout the polishing process, significantly reducing reliance on human experience: Existing technologies mainly rely on precision mechanical structures for passive compensation or special fixtures for positioning, with fixed process parameters that cannot cope with dynamic changes during processing. This invention deploys a multi-source sensor array to acquire surface morphology, process dynamics, and contact force information online, and uses a deep neural network to automatically extract high-dimensional features. Furthermore, by integrating offline case retrieval and online multimodal feedback with an adaptive controller, process parameters are generated and adjusted in real time, enabling the system to autonomously adapt to changes such as blade geometry differences and grinding wheel wear. This transforms process decision-making from relying on manual trial and error to data-driven intelligent closed-loop optimization, significantly improving the consistency and reliability of the process.

[0022] (2) Breaking through the limitations of traditional single-factor control, achieving multimodal collaborative optimization control for final quality: Existing constant force control or dedicated tooling methods often only focus on a single physical quantity or a single link, with each control link being isolated and difficult to coordinate to ensure the final surface quality. This invention comprehensively characterizes the process state by constructing a deep feature extraction network of vision, time sequence, and force control. The designed multimodal feature fusion adaptive controller has a feedforward path based on process and force control feature prediction compensation to maintain process stability; the online feedback path incorporates real-time surface visual features to make fine adjustments guided by the target value of optimized process parameters.

[0023] (3) Enhancing the system's adaptability and robustness to complex surfaces and dynamic disturbances: Traditional methods based on fixed rules or simple feedback have limited adaptability to the complex and variable free-form surface profiles of aero-engine blades and the uncertain disturbances during processing. On the one hand, this invention retrieves historical high-quality process cases through geometric feature similarity matching to provide reliable initial parameters for unknown blades and shorten the debugging cycle; on the other hand, it dynamically adjusts parameters based on real-time multimodal sensing features through a lightweight online learning network, enabling the system to effectively cope with disturbances such as individual differences in blades and changes in local material properties. Attached Figure Description

[0024] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a flowchart of an adaptive polishing and grinding method for aero-engine blades according to an embodiment of the present invention; Figure 2 This is a flowchart illustrating the adaptive decision-making model processing based on offline generation and online adjustment, according to an embodiment of the present invention. Figure 3 This is a diagram showing the experimental results of multi-source data feature analysis according to an embodiment of the present invention; Figure 4The figure shows the experimental results of the performance verification of the adaptive model in this embodiment of the invention. Detailed Implementation

[0025] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0026] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0027] This embodiment proposes an adaptive polishing and grinding method for aero-engine blades, such as... Figure 1 ,include: Construct a database of polishing and grinding processes for aero-engine blades; Collaborative preprocessing and deep feature extraction are performed on the multi-source data in the database; Based on a dual-path strategy of offline generation and online adjustment, an adaptive decision-making model is constructed to generate and adjust process parameters in real time. Based on the measured quality feedback of the polished blades, the process parameters are optimized offline, and the process database and adaptive decision model are updated.

[0028] Furthermore, a database of polishing and grinding processes for the aero-engine blades is constructed, including: Deploy a multi-source sensor array to collect real-time process parameter time-series data, blade surface visual data and force sensing time-series data during the grinding process, and collect the final quality index data of the blade offline, record the control parameter setting data corresponding to the grinding process, and synchronously import historical process archive data. The multi-source data is spatiotemporally aligned and fused, and the polishing quality status is marked according to the quality indicators. Based on the historical process archive data, the target values ​​of optimized process parameters are marked for the data samples to construct a structured process database. The multi-source sensor array includes process parameter sensors, vision sensors, force sensors, and quality inspection equipment.

[0029] Specifically, firstly, a multi-source sensor array is deployed at key locations on the grinding head spindle, feed axis, and end of the grinding head to collect time-series data of process parameters, visual data of the blade surface, and time-series data of force sensing. At the end of the production line, the final quality index data of the blade is collected offline, and the grinding path trajectory file generated by offline programming is simultaneously recorded and imported into structured historical process archive data. Secondly, based on a unified network clock protocol, the multi-source data is aligned with basic timestamps, and a spatial phase compensation method based on blade kinematics is introduced to achieve precise spatiotemporal physical alignment of visual data and force control data. Thirdly, the offline measured final quality index data of the blade is associated with the full-process time-series data, and the data samples are labeled with grinding quality status according to enterprise standards and national standards. Finally, based on historical process archive data, an optimization target back-calculation algorithm is used to label the data samples with optimized process parameter target values, constructing a database of aero-engine blade polishing and grinding processes.

[0030] Multi-source data acquisition and sensor deployment: Multi-source sensor arrays are deployed on the main spindle, feed shaft, and key parts of the grinding head end of the aero-engine blade polishing and grinding equipment.

[0031] Process parameter sensors, specifically photoelectric encoders and Hall current sensors, are deployed on the grinding head spindle and feed axis to collect real-time process parameter timing data, including spindle speed, feed speed, and grinding head clamping force setpoint.

[0032] A vision sensor, specifically a high-resolution industrial area array camera, is deployed directly above the polishing area to collect visual data of the blade surface at fixed points during polishing intervals, i.e., grayscale images of the polished area.

[0033] A force sensor, specifically a torque sensor, is integrated between the end of the grinding head and the actuator to collect real-time force sensing timing data, i.e., normal contact force.

[0034] Quality inspection equipment, specifically a contact surface roughness meter and an optical coordinate measuring machine, is deployed at the end of the production line to collect the final quality index data of the blades offline, including the surface roughness value Ra and the profile accuracy deviation value.

[0035] All sensors are connected to the central industrial control computer via Ethernet or fieldbus, synchronously collecting and recording the control parameter settings data corresponding to each batch of grinding processes, including grinding path trajectory files generated by offline programming. At the same time, structured historical process archive data is imported from the enterprise's manufacturing execution system. The historical process archive data fields include blade model, historical process parameter combinations, historical blade 3D geometric information, and historical quality inspection report data.

[0036] Furthermore, the multi-source data undergoes spatiotemporal alignment and fusion processing, including: A spatial phase compensation method based on blade kinematics is used to perform spatiotemporal alignment of the visual data of the blade surface and the temporal data of force sensing. Based on historical process archive data, an algorithm for labeling optimized process parameters is used to label the data samples with the target values ​​of optimized process parameters.

[0037] Specifically, data alignment and process database construction: This step processes the acquired and imported multi-source data. First, based on a unified network clock protocol, the time-series data of process parameters, visual data of blade surfaces, and time-series data of force sensors are aligned with basic timestamps. To address the spatial mismatch in the acquired data caused by the different installation locations of the visual and force sensors, a spatial phase compensation method based on blade kinematics is introduced.

[0038] Specifically, based on the offline-programmed grinding path trajectory file, combined with the spindle rotation angle and feed axis position information, the time Δt required for the grinding head-blade contact point measured by the force sensor to move to the shooting point directly below the vision sensor at any given moment is calculated in real time. This compensation amount Δt is calculated using the movement speed of the contact point along the grinding trajectory and the spatial distance between the two acquisition points. Subsequently, the timestamps of the visual data are uniformly compensated by a compensation amount Δ, thereby achieving precise physical alignment in time and space between the force control data and the visual data reflecting the surface state of the same physical point. The final quality index data of the blade measured offline is linked to the corresponding full-process time-series data through the batch number and the workpiece's unique identification code.

[0039] Next, based on the manufacturing standards for high-pressure compressor blades for aero-engines and the national standards for geometric tolerances, the grinding quality status of each data sample was marked: When the surface roughness Ra ≤ 0.4 μm and the contour accuracy deviation ≤ 0.05 mm, it is marked as qualified; When the surface roughness value Ra > 0.4 μm, it is marked as roughness exceeding the tolerance; When the profile accuracy deviation is greater than 0.05mm, it is marked as profile deviation out of tolerance.

[0040] Then, based on historical process archive data, an algorithm for labeling optimized process parameters is used to label the data samples with the target values ​​of optimized process parameters.

[0041] The algorithm for optimizing the target value annotation of process parameters specifically executes the following steps: First, for each data sample in the database, if its own labeled grinding quality status is qualified, the average value of the time series data of the actual process parameters used during the stable grinding stage is directly marked as the target spindle speed and target feed rate of that sample. Second, for all samples whose grinding quality status is roughness or profile out of tolerance, the algorithm calls a pre-trained geometric feature-high-quality process parameter regression model.

[0042] The training process of the geometric feature-optimized process parameter regression model is as follows: A training set is constructed by selecting all samples with acceptable grinding quality from the database. From the historical 3D geometric information of these acceptable samples, a set of blade geometric feature vectors characterizing the blade profile geometry is extracted. Using the target spindle speed and target feed rate already labeled for these acceptable samples as regression targets, a gradient boosting decision tree regression model is trained. Then, for each unacceptable sample, its own blade geometric feature vector is extracted and input into the trained geometric feature-optimized process parameter regression model. The speed and feed rate values ​​predicted by the geometric feature-optimized process parameter regression model are used as the optimized process parameter target values ​​for that unacceptable sample. Finally, the aligned multi-source data, labeled grinding quality status, and optimized process parameter target values ​​are integrated to construct an aero-engine blade polishing and grinding process database.

[0043] Furthermore, collaborative preprocessing and deep feature extraction are performed on the multi-source data in the database, including: The visual data of the blade surface is preprocessed by denoising, enhancement and normalization, and then input into the surface defect perception convolutional neural network to extract the visual depth feature vector. The surface defect perception convolutional neural network includes a curvature-guided spatial attention mechanism to extract features of geometrically complex areas of the surface. The time series data of process parameters are preprocessed by outlier removal, missing point filling and standardization, and then input into the process dynamic evolution gated recurrent unit network to extract the process dynamic feature vector. The process dynamic evolution gated recurrent unit network includes a multi-scale temporal convolution module to capture process patterns at different time scales. The force sensing time series data is filtered, statistically characterized, and force tracking error is calculated. The processed results are then input into a steady-state contact force feature extraction network to extract the force-controlled steady-state feature vector.

[0044] Specifically, the multi-source data collaborative preprocessing and deep feature extraction process includes: First, the visual data of the blade surface is preprocessed by denoising, enhancement, and normalization, and then input into a surface defect perception convolutional neural network to extract visual depth feature vectors. Second, the time series data of process parameters is preprocessed by outlier removal, missing point filling, and standardization, and then input into a process dynamic evolution gated recurrent unit network to extract process dynamic feature vectors. Finally, the time series data of force sensing is filtered, statistically analyzed, and force tracking error is calculated, and the results are input into a steady-state contact force feature extraction network to extract force-controlled steady-state feature vectors.

[0045] Visual feature depth representation based on surface defect-aware convolutional neural network: This step preprocesses the visual data of the blade surface acquired by the vision sensor, specifically the grayscale image of the polished area. First, a Gaussian filter kernel is used for image denoising to eliminate high-frequency noise interference. Then, image enhancement is performed, using histogram equalization to improve surface texture contrast. Finally, all images are scaled to a uniform size and pixel values ​​are normalized. The preprocessed image data is then input into a specially designed surface defect-aware convolutional neural network.

[0046] The surface defect-aware convolutional neural network consists of four cascaded convolutional modules: The first convolutional module uses small-sized convolutional kernels to extract edge and primary texture features from the blade surface.

[0047] The second convolutional module innovatively introduces a curvature-guided spatial attention mechanism. This mechanism first calculates and generates a local curvature map corresponding to the current polishing area based on the computer-aided design model of the blade or images acquired from multiple angles using photometric stereo vision. This map encodes the geometric convexity and concavity of each point on the surface using pixel values. During forward propagation, this curvature map is downsampled to the same spatial size as the current feature map and used as an additional prior information channel for concatenation. Subsequently, the attention submodule processes the fused features through a two-layer fully connected network with a 16x channel compression ratio (i.e., first compressing the number of channels to 1 / 16 of the original number, then restoring it to the original number) to generate an initial weight vector with a spatial dimension of 1x1 and a channel dimension identical to the main feature map. This weight vector is then multiplied element-wise with the normalized curvature map, giving higher attention gain coefficients to high curvature regions. Finally, a final spatial-channel composite attention weight map is generated using a sigmoid activation function and then weighted and fused with the original main feature map. This mechanism enables the network to adaptively adjust the receptive field and focus of feature extraction based on the local geometric complexity of the blade surface, thereby enhancing its ability to capture texture and defect features in areas with drastic curvature changes.

[0048] The third convolutional module further integrates intermediate semantic features to capture the local texture and macroscopic morphological undulation patterns reflected by changes in surface light reflection.

[0049] The fourth convolutional module utilizes a larger receptive field to capture global surface integrity information.

[0050] Each convolutional module is followed by a batch normalization layer, a ReLU activation function, and a max pooling layer. At the end of the network, a global average pooling layer is used to aggregate spatial information and output a fixed-dimensional visual depth feature vector F_vis. The visual depth feature vector F_vis comprehensively encodes the macroscopic texture, light reflection characteristics, and visible defect distribution information of the blade surface captured by an industrial area scan camera and enhanced with curvature information.

[0051] Furthermore, based on the temporal feature extraction of the process dynamic evolution gated cyclic unit network, this step preprocesses the time-series data of process parameters collected by the process parameter sensors, specifically including: First, a sliding window method is used to calculate the mean and standard deviation of the data within the window. Data points exceeding three times the standard deviation of the mean are identified as outliers and removed. Then, missing points resulting from outlier removal or signal loss are filled using linear interpolation of data from preceding and following time points. Finally, the entire time series is Z-score standardized to eliminate the influence of dimensions. The processed data is then input into a dynamically evolving gated recurrent unit network.

[0052] The process dynamic evolution gated recurrent unit (GRU) network incorporates a process parameter sensing embedding layer at the input. This layer maps two physically different parameters—spindle speed and feed rate—to a unified high-dimensional feature space using independent learnable embedding matrices. The embedded sequence is then fed into a multi-scale temporal convolution module. This module uses three dilated one-dimensional convolutional layers with different dilation coefficients in parallel to capture process patterns at three timescales: short-term fluctuations, medium-term trends, and long-term cycles. The outputs of the three convolutional paths are concatenated along the channel dimension to form a feature rich in multi-scale temporal context information. This feature is then fed into the core gated recurrent unit (GRU) layer. The GRU layer controls the retention of historical information through its update gate and controls the combination of the current input and historical states through its reset gate, effectively modeling the dynamic evolution and dependencies of process parameters during the grinding process. The hidden state of the last time step of the process dynamic evolution gated recurrent unit network is output as the process dynamic feature vector F_proc, which characterizes the stability and changing trends of the process operation over a given period.

[0053] Furthermore, based on the force control signal characterization using a steady-state contact force feature extraction network, this step preprocesses the force sensing time-series data acquired by the force sensor, focusing primarily on the normal contact force, including: First, a first-order low-pass digital filter is used to filter out high-frequency noise caused by equipment vibration, while retaining the low-frequency component of the force signal that reflects the actual contact state.

[0054] Subsequently, within a continuous sliding time window, the statistical characteristics of the force signal are calculated: the average of all force values ​​within the window is calculated to obtain the force mean μ_F, reflecting the average clamping force level; the variance is calculated to obtain the force variance σ_F^2, reflecting the degree of force fluctuation; and the force fluctuation index I_F = σ_F / μ_F is calculated to quantify the relative stability of the force. Simultaneously, the difference between the real-time force value and the set value of the grinding head clamping force is calculated to obtain the force tracking error sequence, and its root mean square error is calculated.

[0055] The preprocessed original filtered force sequence fragment, the calculated statistical features (μ_F, σ_F^2, I_F), and the root mean square value of the force tracking error are concatenated into a comprehensive feature vector. This vector is then input into a steady-state contact force feature extraction network, which consists of a first fully connected layer with 64 neurons, a ReLU activation function, a second fully connected layer with 32 neurons, and a final Tanh activation function. Through this nonlinear transformation, multi-dimensional force information is fused and compressed into a low-dimensional, information-dense force-controlled steady-state feature vector F_force. The force-controlled steady-state feature vector F_force comprehensively characterizes the accuracy and stability of constant force control during the grinding process, as well as the contact state between the workpiece and the grinding head.

[0056] Furthermore, based on a dual-pathway strategy of offline generation and online adjustment, an adaptive decision-making model is constructed to generate and adjust process parameters in real time, including: Based on the geometric information of the blade to be ground, an initial grinding path trajectory and initial process parameter combination are generated through similar case retrieval and parameter generation network. During the grinding process, a multimodal feature fusion adaptive controller receives real-time extracted visual depth feature vectors, process dynamic feature vectors, and force control steady-state feature vectors. Through feedforward compensation path and online feedback adjustment path, it calculates and outputs real-time spindle speed setpoints and real-time feed rate setpoints.

[0057] Specifically, an adaptive decision-making model based on offline generation and online adjustment is constructed, and the model structure diagram is as follows: Figure 2 As shown, the process includes: The extracted features are input into an adaptive decision-making model based on offline generation and online adjustment. First, based on the computer-aided design model of the blade to be ground and the geometric information of the area to be ground, the initial grinding path trajectory and initial process parameter combination are automatically generated through similar case retrieval and parameter generation network. Second, during the grinding process, the multimodal feature fusion adaptive controller receives visual depth feature vectors, process dynamic feature vectors, and force control steady-state feature vectors. Finally, the controller integrates the outputs of the feedforward compensation path, which includes the physical model residual compensation network, and the online feedback adjustment path, which includes the online update neural network, to calculate and issue real-time spindle speed setpoints and real-time feed rate setpoints in real time.

[0058] The initial grinding path and process parameter combination are automatically generated, including: Based on the robot offline programming software platform, the computer-aided design model of the blade to be ground and the geometric information of the grinding area specified in the process plan are imported. A similar case search and parameter generation network is then initiated.

[0059] The similar case retrieval and parameter generation network first includes a feature encoder that encodes the geometric information of the current blade's grinding area into a current geometric feature vector. Secondly, it utilizes a pre-trained geometric feature similarity matching model. This model organizes input features in a concatenated manner, specifically by concatenating the current geometric feature vector to be queried with the geometric feature vectors of historical samples, and additionally calculating the absolute differences between the two in each feature dimension. This results in a comprehensive feature vector that integrates the original features and the differences, which serves as the model input. This geometric feature similarity matching model is a ranking or regression model based on a gradient boosting decision tree. During the training phase, for regression tasks, its loss function is defined as the mean squared error between the model's predicted similarity score and the true similarity label; for ranking tasks, a pair-based ranking loss is used, aiming to ensure that high-quality historical samples (with acceptable grinding quality) achieve significantly higher similarity scores compared to low-quality samples. Finally, the geometric feature similarity matching model takes the difference or composite feature of two geometric feature vectors as input and outputs a similarity score.

[0060] Using a geometric feature similarity matching model, the current geometric feature vector is compared one by one with the geometric feature vectors of all historical samples in the aero-engine blade polishing and grinding process database. The top K samples with the highest similarity scores are retrieved, and the corresponding historical process parameter combinations and grinding quality status are obtained for these samples. Next, the attention-weighted aggregation layer in the network calculates the attention weight for each historical sample based on the grinding quality status of these K samples and their geometric similarity score with the current blade, assigning higher weights to qualified samples. Finally, the historical process parameter combinations of these high-quality historical samples are aggregated through weighted summation to automatically generate an initial grinding path trajectory and initial process parameter combinations suitable for the current region of the current blade, including the initial spindle speed. With initial feed rate .

[0061] Furthermore, the multimodal feature fusion adaptive controller includes: Feedforward compensation path: Based on the Preston equation and real-time force control data, calculate the theoretical process parameter reference values, and predict the nonlinear deviation between the theoretical spindle speed and feed rate reference values ​​and the parameters required for actual high-quality process through the physical model residual compensation network; Online feedback adjustment pathway: Through a lightweight online update neural network, fine adjustment amounts are calculated based on real-time surface visual features and process-force control fusion features to optimize surface quality; The controller adds the initial process parameters, feedforward compensation, and online feedback adjustment to output the final real-time process parameter setpoints.

[0062] Specifically, during the polishing process, the multimodal feature fusion adaptive controller runs in real time. The multimodal feature fusion adaptive controller receives visual depth feature vectors. Process dynamic feature vector Sum of force-controlled steady-state eigenvectors .

[0063] The core of the multimodal feature fusion adaptive controller is a hybrid structure with dual pathways of feedforward compensation and online feedback adjustment. It innovatively combines a theoretical model based on the Preston equation with a residual neural network. The Preston equation establishes the theoretical relationship between material removal rate, relative velocity, and contact pressure, providing a physical basis for process parameter adjustment. In the feedforward compensation pathway, the process dynamic feature vectors are... With force-controlled steady-state eigenvectors The data is then stitched together to obtain the process-force control fusion features. The feedforward compensation path, based on the Preston equation and real-time force control data, calculates the theoretical spindle speed and feed rate baselines required to maintain a constant theoretical removal rate. The feedforward compensation path introduces a physical model residual compensation network, which takes the process-force control fusion features as input. Its learning objective is to predict the nonlinear deviation between the theoretical spindle speed and feed rate baselines and the parameters required for a high-quality process. The network outputs the pre-compensation amount for the spindle speed and feed rate. and This aims to proactively offset predictable disturbances not covered by the theoretical model, maintaining the stability of the constant force control loop. In the online feedback adjustment pathway, the visual depth feature vector representing the real-time surface state is used... The process-force control fusion features are input together into a lightweight, online update neural network.

[0064] The online update neural network is trained using the residuals between theoretical and actual parameters and the target values ​​of optimized process parameters labeled in the S1 database as supervision signals. It can dynamically calculate fine-tuning amounts aimed at optimizing the final surface quality based on real-time sensed surface morphology and process stability. and .

[0065] Ultimately, the real-time process parameter setpoints output by the controller are determined by the following formula: , ,in, This indicates the real-time spindle speed setting value. This represents the real-time feed rate setpoint. This output value is directly sent to the actuator, thereby achieving multi-objective collaborative optimization control that stabilizes contact force, adapts to surface changes, and pursues optimal surface quality.

[0066] Furthermore, based on the measured quality feedback of the polished blades, the process parameters are optimized offline, and the process database and adaptive decision model are updated, including: After the grinding of a single blade is completed, based on the deviation between the measured quality and the target value, the Bayesian process parameter optimizer is activated to perform offline parameter optimization, and the new data samples and optimized parameters are updated to the process database. The surface defect perception convolutional neural network, process dynamic evolution gated recurrent unit network, steady-state contact force feature extraction network, similar case retrieval and parameter generation network, and multimodal feature fusion adaptive controller are integrated and deployed in the blade polishing and grinding CNC system. The effectiveness of the algorithm in terms of grinding pass rate, batch blade surface quality consistency and debugging cycle is verified through comparative experiments. The Bayesian process parameter optimizer employs a Gaussian process surrogate model, and its kernel function is a composite kernel, including: Automatic correlation determination kernel function is used to characterize the interaction relationship between process parameters; The Matern kernel function is used to model the nonlinear relationship between process characteristics and quality deviations. Periodic kernel functions are used to characterize the effects of periodic fluctuations in a process.

[0067] Specifically, after the single blade grinding task is completed, the quality deviation is first calculated based on the measured surface roughness value and the measured profile accuracy deviation value. When the deviation exceeds the allowable tolerance threshold, the Bayesian process parameter optimizer is activated to perform offline parameter optimization, and the complete data tuple and the generated new process parameters are combined as new knowledge samples to update the aero-engine blade polishing and grinding process database. Secondly, the trained surface defect perception convolutional neural network, process dynamic evolution gated recurrent unit network, steady-state contact force feature extraction network, similar case retrieval and parameter generation network, and multimodal feature fusion adaptive controller are integrated and deployed on the blade polishing and grinding CNC system platform. Through the design of multiple rounds of comparative verification experiments, the effectiveness and engineering practicality of the algorithm in terms of single blade grinding pass rate, batch blade surface quality consistency, and process parameter debugging cycle are verified.

[0068] Furthermore, offline optimization and knowledge updating of process parameters based on measured feedback: After the grinding of a single blade is completed, the measured surface roughness value and measured profile accuracy deviation value of the blade are obtained by the quality inspection equipment. The system compares this measured result with the expected quality target corresponding to the target value of the optimized process parameters marked on the blade, and calculates the surface roughness quality deviation. Contour accuracy quality deviation If the surface roughness quality deviation Or contour accuracy quality deviation If the absolute value of any index exceeds its corresponding allowable tolerance threshold, the system will activate the Bayesian process parameter optimizer to optimize the parameters.

[0069] The construction and workflow of the Bayesian process parameter optimizer are as follows: First, the optimizer takes the actual process parameter sequence executed in this polishing task and the extracted visual depth feature vector. Process dynamic feature vector Force-controlled steady-state eigenvectors And the two calculated quality deviations ( , Together they form the initial observation dataset. The core of the optimizer is to construct a Gaussian process surrogate model. This model employs a composite kernel function specifically designed for the characteristics of blade grinding processes. This composite kernel consists of three parts: an autocorrelation determination kernel function, used to capture the complex interactions and importance between different process parameters; a Matern kernel function, used to model the visual depth feature vector. Process dynamic feature vector Force-controlled steady-state eigenvectors The nonlinear and non-stationary correlation between quality deviation and quality; a periodic kernel function to characterize the impact of inherent periodic fluctuations in processes such as spindle rotation on quality results.

[0070] The Gaussian process surrogate model takes process parameters and process characteristics as input and predicted quality deviations as output, thus establishing a probabilistic mapping relationship between the process and the quality outcome. Based on this surrogate model, the optimizer uses the expected improvement acquisition function to evaluate the value of unexplored points in the parameter space. The optimization process is an iterative loop: in each iteration, the acquisition function selects a new set of candidate process parameter combinations that are expected to minimize quality deviations; the system applies this combination to the simulation model; subsequently, this set of parameters and their corresponding predicted deviations are added to the observation dataset. In this process, the parameters are used to update the Gaussian process surrogate model. After multiple iterations, the optimizer will recommend a new combination of process parameters that minimizes the predicted quality deviation from all evaluated parameters. Regardless of whether the measured quality of this polishing task meets the standards, the complete data tuple corresponding to this task (including the process feature set, the actual sequence of process parameters used, the measured surface roughness value, and the measured profile accuracy deviation value) will be used as a new knowledge sample. Simultaneously, if the Bayesian process parameter optimizer is activated and generates a new combination of process parameters... This combination will also serve as a candidate for optimized process parameter target values ​​for similar geometric features. These new samples and optimized candidate parameters will be added to the constructed aero-engine blade polishing and grinding process database.

[0071] Furthermore, the system integration and intelligent polishing platform undergoes online verification, including: The trained surface defect perception convolutional neural network, process dynamic evolution gated recurrent unit network, steady-state contact force feature extraction network, similar case retrieval and parameter generation network, and multimodal feature fusion adaptive controller are integrated and deployed on the blade polishing and grinding CNC system platform. The blade polishing and grinding CNC system platform uses an industrial personal computer as the host computer and a real-time motion controller as the slave computer. During the actual grinding process, the host computer software receives and processes sensor data streams in real time, generating real-time spindle speed setpoints online. With real-time feed rate setpoint These settings are transmitted to the lower-level machine via a high-speed real-time Ethernet bus. The lower-level machine drives the servo motor and spindle motor to precisely execute the adaptive polishing process. Through multiple rounds of comparative verification experiments, the adaptive polishing process based on this algorithm and the traditional manual polishing process based on fixed parameters were run on the same equipment and the same batch of blades. The performance data of the two processes in terms of single-blade polishing pass rate, batch blade surface quality consistency, and process parameter debugging cycle for new blade models were recorded and analyzed. This comprehensively verifies the effectiveness and engineering applicability of this algorithm in improving process quality, stability, and adaptability.

[0072] To verify the effectiveness of the proposed aero-engine blade polishing process database construction and the adaptive decision-making model based on offline generation and online adjustment, this experiment uses a high-pressure compressor blade as the object, focusing on evaluating the performance of multi-source data feature extraction and adaptive control strategies. Experimental indicators include multi-source data feature discrimination, surface quality improvement effect under adaptive control, and process parameter optimization capability. Through comparative experiments, the distribution patterns of multi-source sensor data features are analyzed, and the superiority of adaptive polishing over traditional fixed-parameter polishing is verified.

[0073] First, based on the collected multi-source sensor data, the distribution pattern of process parameter time-series data in qualified and unqualified samples was analyzed. Ten qualified samples and ten samples with roughness exceeding tolerance were randomly selected from the process database, and the spindle speed time-series data of their stable grinding stage was extracted, with a time step of one hundred points. A line graph was used to visualize the change trend of spindle speed with time steps for qualified and unqualified samples. The horizontal axis represents the time step, and the vertical axis represents the spindle speed value in revolutions per minute. The two lines represent the mean sequence of qualified samples and the mean sequence of unqualified samples, respectively.

[0074] pass Figure 3 The line graph shows that the spindle speed sequence of the qualified samples remained relatively stable throughout the grinding stabilization phase, with small fluctuations and a mean value stable near the set value. In contrast, the spindle speed sequence of the unqualified samples exhibited significant fluctuations, especially between time steps 30 and 60, showing multiple abnormal peaks and troughs, deviating significantly from the set value. This difference indicates a close correlation between process parameter stability and grinding quality; the process parameters of the qualified samples were more precisely controlled, validating that multi-source data collaborative preprocessing and deep feature extraction can effectively capture the dynamic evolution of the process. Furthermore, the fluctuation characteristics of the time-series data provide crucial information for subsequent process parameter optimization, demonstrating that real-time monitoring and feature extraction based on sensor data are fundamental to improving grinding quality.

[0075] The experiment further evaluated the performance of the adaptive decision-making model in batch blade grinding. Twenty blades from the same batch were selected on the same polishing equipment and processed using both the adaptive grinding process based on this algorithm and the traditional manual grinding process with fixed parameters. The surface roughness Ra value of each blade after processing was recorded, and a line graph was plotted to compare the surface quality of the two methods. The horizontal axis represents the blade number, from one to twenty, and the vertical axis represents the surface roughness Ra value in micrometers. The two lines represent the Ra values ​​corresponding to the adaptive method and the traditional method, respectively.

[0076] pass Figure 4 The line graph shows that under the adaptive grinding process, the surface roughness Ra values ​​of all blades are below 0.4 micrometers, with small fluctuations, concentrated around 0.3 micrometers, exhibiting high consistency. In contrast, the Ra values ​​of the traditional fixed-parameter grinding process are more dispersed, with several blades having Ra values ​​exceeding 0.4 micrometers, resulting in a higher failure rate. Specifically, all blades using the adaptive method have acceptable Ra values, while six blades using the traditional method have Ra values ​​exceeding the tolerance. The results demonstrate that the adaptive decision model, by adjusting process parameters in real time, can effectively adapt to blade geometric differences and process disturbances, significantly improving surface quality consistency and pass rate. This verifies the engineering practicality of the multimodal feature fusion adaptive controller and online feedback adjustment path, providing a reliable solution for achieving stable and high-quality blade polishing.

[0077] In summary, this invention, through multi-source data feature analysis and adaptive decision model construction, achieves effective monitoring and optimization of the polishing and grinding process of aero-engine blades, significantly improving surface quality consistency and process adaptability, and verifying the effectiveness and practical value of the method in complex manufacturing scenarios.

[0078] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. An adaptive polishing and grinding method for aero-engine blades, characterized in that, include: Construct a database of polishing and grinding processes for aero-engine blades; Collaborative preprocessing and deep feature extraction are performed on the multi-source data in the database; Based on a dual-path strategy of offline generation and online adjustment, an adaptive decision-making model is constructed to generate and adjust process parameters in real time. Based on the measured quality feedback of the polished blades, the process parameters are optimized offline, and the process database and adaptive decision model are updated.

2. The adaptive polishing and grinding method for aero-engine blades according to claim 1, characterized in that, Constructing the aero-engine blade polishing and grinding process database includes: Deploy a multi-source sensor array to collect real-time process parameter time-series data, blade surface visual data and force sensing time-series data during the grinding process, and collect the final quality index data of the blade offline, record the control parameter setting data corresponding to the grinding process, and synchronously import historical process archive data. The multi-source data is spatiotemporally aligned and fused, and the polishing quality status is marked according to the quality indicators. Based on the historical process archive data, the target values ​​of optimized process parameters are marked for the data samples to construct a structured process database. The multi-source sensor array includes process parameter sensors, vision sensors, force sensors, and quality inspection equipment.

3. The adaptive polishing and grinding method for aero-engine blades according to claim 2, characterized in that, The spatiotemporal alignment and fusion processing of the multi-source data includes: A spatial phase compensation method based on blade kinematics is used to perform spatiotemporal alignment of the visual data of the blade surface and the temporal data of force sensing. Based on historical process archive data, an algorithm for labeling optimized process parameters is used to label the data samples with the target values ​​of optimized process parameters.

4. The adaptive polishing and grinding method for aero-engine blades according to claim 3, characterized in that, Spatiotemporal alignment of the visual data of the blade surface with the force sensing time-series data includes: Based on the grinding path trajectory, spindle rotation angle and feed axis position, the time compensation required for the force sensor measurement point to move to the vision sensor shooting point is calculated in real time, and the vision data timestamp is compensated. The algorithm for labeling the target values ​​of optimized process parameters includes: For samples that have passed the grinding quality test, the average value of the actual process parameters will be marked as the target value. For non-compliant samples, the geometric feature-high-quality process parameter regression model trained based on gradient boosting decision tree is used to predict and label the target values ​​of optimized process parameters based on the blade geometric feature vector.

5. The adaptive polishing and grinding method for aero-engine blades according to claim 1, characterized in that, Collaborative preprocessing and deep feature extraction of multi-source data in the database include: The visual data of the blade surface is preprocessed by denoising, enhancement and normalization, and then input into the surface defect perception convolutional neural network to extract the visual depth feature vector. The surface defect perception convolutional neural network includes a curvature-guided spatial attention mechanism to extract features of geometrically complex areas of the surface. The time series data of process parameters are preprocessed by outlier removal, missing point filling and standardization, and then input into the process dynamic evolution gated recurrent unit network to extract the process dynamic feature vector. The process dynamic evolution gated recurrent unit network includes a multi-scale temporal convolution module to capture process patterns at different time scales. The force sensing time series data is filtered, statistically characterized, and force tracking error is calculated. The processed results are then input into a steady-state contact force feature extraction network to extract the force-controlled steady-state feature vector.

6. The adaptive polishing and grinding method for aero-engine blades according to claim 5, characterized in that, The curvature-guided spatial attention mechanism is specifically as follows: A local curvature map is generated based on the computer-aided design model of the blade or the photometric stereo vision method. The curvature map is then used as a priori information channel and convolved with the convolutional feature map. After channel compression and activation processing, a spatial-channel composite attention weight map is generated, which is used to perform weighted fusion of the original feature map.

7. The adaptive polishing and grinding method for aero-engine blades according to claim 1, characterized in that, Based on a dual-pathway strategy of offline generation and online adjustment, an adaptive decision-making model is constructed to generate and adjust process parameters in real time, including: Based on the geometric information of the blade to be ground, an initial grinding path trajectory and initial process parameter combination are generated through similar case retrieval and parameter generation network. During the grinding process, a multimodal feature fusion adaptive controller receives real-time extracted visual depth feature vectors, process dynamic feature vectors, and force control steady-state feature vectors. Through feedforward compensation path and online feedback adjustment path, it calculates and outputs real-time spindle speed setpoints and real-time feed rate setpoints.

8. The adaptive polishing and grinding method for aero-engine blades according to claim 7, characterized in that, in, The processing procedure of the similar case retrieval and parameter generation network is as follows: The current blade geometric feature vector is matched with the geometric feature vectors of historical samples to retrieve the top K samples with the highest similarity. Attention weights are calculated based on the polishing quality status and similarity scores of the K samples. The process parameter combinations of high-quality historical samples are then weighted and aggregated to generate initial process parameters.

9. The adaptive polishing and grinding method for aero-engine blades according to claim 7, characterized in that, The multimodal feature fusion adaptive controller includes: Feedforward compensation path: Based on the Preston equation and real-time force control data, calculate the theoretical process parameter reference values, and predict the nonlinear deviation between the theoretical spindle speed and feed rate reference values ​​and the parameters required for actual high-quality process through the physical model residual compensation network; Online feedback adjustment pathway: Through a lightweight online update neural network, fine adjustment amounts are calculated based on real-time surface visual features and process-force control fusion features to optimize surface quality; The controller adds the initial process parameters, feedforward compensation, and online feedback adjustment to output the final real-time process parameter setpoints.

10. The adaptive polishing and grinding method for aero-engine blades according to claim 1, characterized in that, Based on the measured quality feedback of the polished blades, the process parameters are optimized offline, and the process database and adaptive decision model are updated, including: After the grinding of a single blade is completed, based on the deviation between the measured quality and the target value, the Bayesian process parameter optimizer is activated to perform offline parameter optimization, and the new data samples and optimized parameters are updated to the process database. The surface defect perception convolutional neural network, process dynamic evolution gated recurrent unit network, steady-state contact force feature extraction network, similar case retrieval and parameter generation network, and multimodal feature fusion adaptive controller are integrated and deployed in the blade polishing and grinding CNC system. The effectiveness of the algorithm in terms of grinding pass rate, batch blade surface quality consistency and debugging cycle is verified through comparative experiments. The Bayesian process parameter optimizer employs a Gaussian process surrogate model, and its kernel function is a composite kernel, including: Automatic correlation determination kernel function is used to characterize the interaction relationship between process parameters; The Matern kernel function is used to model the nonlinear relationship between process characteristics and quality deviations. Periodic kernel functions are used to characterize the effects of periodic fluctuations in a process.

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