A method for predicting the remaining life of grinding wheel based on physical information neural network

Abstract

The present invention discloses a method for predicting the remaining service life of a grinding wheel for a grinding machine based on a physical information neural network. The method comprises measuring and recording sensor data from n channels of the grinding wheel and converting the data into digital signals via an acquisition card; analyzing the sensor data; and if the amount of sensor data exceeds a first preset threshold, directly converting the digital signals into a feature matrix. If the amount of sensor data is less than a second preset threshold, extracting time domain and frequency domain features from the data signals to form a feature matrix, which is then normalized to obtain a normalized feature matrix. The normalized feature matrix is ​​used as an input matrix for training a neural network model to obtain a trained neural network model. The trained neural network model is then validated using the normalized feature matrix obtained by processing the equipment data to be predicted as an input matrix to obtain a predicted value for the remaining service life of the industrial equipment. This method improves the final prediction accuracy.

Description

Technical Field

[0001] The present invention belongs to the technical field of equipment health management, and in particular relates to a method for predicting the remaining life of a grinding wheel of a grinding machine based on a physical information neural network. Background Art

[0002] Prognostic health management (PHM) has long been a key research topic for industrial equipment, with remaining useful life prediction being a key focus. Accurate useful life prediction can help ensure that mission-critical equipment never ceases service before a failure occurs, reducing economic losses due to equipment reaching the end of its useful life and alleviating worker safety risks. Currently, mainstream prediction methods can be categorized into three categories: physics-based modeling, data-driven methods, and hybrid approaches that combine the two. Physics-based methods use prior physical knowledge to establish a degradation model for equipment. However, with the increasing sophistication of industrial equipment and complexity of manufacturing processes, establishing an accurate physical model to predict the remaining useful life of equipment is becoming increasingly challenging. Data-driven methods, on the other hand, often have an accuracy that is positively correlated with the complexity of the model, and their prediction results lack transparency and interpretability. Therefore, there is an urgent need to develop a hybrid approach that combines the advantages of both approaches to predict the remaining useful life of equipment. Summary of the Invention

[0003] In response to the above technical problems, the present invention provides a method for predicting the remaining life of a grinding wheel of a grinding machine based on a physical information neural network.

[0004] The technical solution adopted by the present invention to solve the technical problem is:

[0005] A method for predicting the remaining life of a grinding wheel of a grinding machine based on a physical information neural network, the method comprising the following steps:

[0006] S100: Measure and record the sensor data of the grinding wheel n channel and convert it into a digital signal through the acquisition card. The number of operations completed by the current device or the unit time is used as the current service life of the device;

[0007] S200: Analyze the obtained n-channel sensor data. If the amount of sensor data is greater than a first preset threshold, directly form a feature matrix from the digital signal. If the amount of sensor data is less than a second preset threshold, extract time domain and frequency domain features from the data signal to form a feature matrix. Normalize the feature matrix to obtain a normalized feature matrix.

[0008] S300: In the model training phase, the normalized feature matrix is ​​formed into an input matrix to train the Mamba network model under the physical information neural network framework to obtain a trained neural network model;

[0009] S400: In the model testing phase, the normalized feature matrix obtained after processing the equipment data to be predicted is used as the input matrix to verify the trained neural network model to obtain the remaining useful life prediction value of the industrial equipment.

[0010] Preferably, in S200, if the amount of sensor data is greater than a first preset amount threshold, directly forming a feature matrix from the digital signal, and normalizing the feature matrix to obtain a normalized feature matrix, including:

[0011] S210: The signal collected during one use process is regarded as a time unit. There are N use processes in total. The feature data extracted during each use process is m in total. The feature matrix X is formed by splicing them in chronological order. N*m , reshape the feature matrix dimension to X N*1*m ;

[0012] S220: Calculate the average value and standard deviation of each row of feature data according to the feature matrix, and normalize the feature matrix according to the average value and standard deviation of each row of feature data, specifically:

[0013]

[0014] in is the normalized feature data, y is the original sensor data, mean(y(:,j,:)) is the mean value of each row of feature data under the corresponding channel, std(y(:,j,:)) is the standard deviation of each row of feature data under the corresponding channel, i=1,…,N, j=1, k=1,…,m;

[0015] S230: Reconstruct the normalized feature matrix and reshape the dimension to X N*m .

[0016] Preferably, in S200, if the amount of sensor data is less than the second preset amount threshold, extracting time domain and frequency domain features from the data signal to form a feature matrix, and normalizing the feature matrix to obtain a normalized feature matrix includes:

[0017] S240: Take the signal collected during one use process as a time unit, a total of N use processes, and m feature data extracted during each use process, and splice them in time order to form a feature matrix X N*m , perform time domain and frequency domain feature extraction, re-extract time domain and frequency domain features for each original feature data, a total of a, and re-splice to form a feature matrix X N*m*a ;

[0018] S250: Calculate the average value and standard deviation of each row of feature data according to the feature matrix, and normalize the feature matrix according to the average value and standard deviation of each row of feature data, specifically:

[0019]

[0020] in is the normalized feature data, x is the extracted time domain feature and frequency domain feature data, mean(x(:,j,:)) is the mean value of each row of feature data under the corresponding channel, std(x(:,j,:)) is the standard deviation of each row of feature data under the corresponding channel, i=1,…,N, j=1,…,m, k=1,…,a;

[0021] S260: Reconstruct the normalized feature matrix and reshape the dimension to X N*ma .

[0022] Preferably, the time domain features extracted from the original sensor signal include: absolute mean, peak value, root mean square value, root amplitude, skewness value, kurtosis value, waveform factor, pulse factor, skewness factor, peak factor, margin factor, kurtosis factor, and the frequency domain features include: center of gravity frequency, mean square frequency, root mean square frequency and frequency variance.

[0023] Preferably, S300 includes:

[0024] S310: Divide the number of operations according to the set operation time, obtain the current service life as a label, and form the label into an output matrix. The specific form is:

[0025]

[0026] S320: The feature matrix X extracted previously N*ma or X N*m Input the neural network model for training;

[0027] S330: After one training process is completed, the prediction matrix is ​​obtained The root mean square error value is obtained according to the prediction results and the root mean square error RMSE evaluation function, specifically:

[0028]

[0029] S340: Using the back propagation principle, the neural network parameters are corrected to improve the network fitting ability. After the training cycle is completed, the network model training is completed.

[0030] Preferably, S320 includes:

[0031] S321: The feature matrix is ​​input into the neural network model, passes through the Mamba layer, and the state output from the Mamba network layer is added to the linear layer together with the original input matrix after the attention mechanism. After forward propagation, the hidden state is obtained;

[0032] S322: The hidden state and time are input into the multi-sensor, and the prediction result is output after passing through the fully connected layer. The first RMSE value is calculated between the prediction result and the true label;

[0033] S323: The partial derivative of the hidden state with respect to the output is obtained through the multi-layer perceptron, which is then input into the Mamba layer and the attention mechanism and then into the linear layer together with the hidden state. After forward propagation, the current state can be obtained. The current state plus the partial derivative of the output with respect to time is added to the value of 0 to calculate the second RMSE. The second RMSE value is added to the first RMSE calculated previously to form the loss function.

[0034] Preferably, S321 specifically includes:

[0035] The feature matrix is ​​input into the neural network model and first enters the Mamba layer. In the Mamba layer, the input data first passes through two linear layers. After passing through the linear layer one, the input data undergoes a convolution operation and passes through the activation function represented by σ. The formula is:

[0036]

[0037] After passing through the activation function, it passes through the state-space model, where it is multiplied by the output of the second linear layer and the activation function, and then passes through another linear layer and is output from the Mamba network. The state output from the Mamba network layer passes through the attention mechanism and is added to the linear layer together with the original input matrix at the beginning. After forward propagation, the hidden state is obtained.

[0038] Preferably, S400 includes:

[0039] S410: Perform an operation on the equipment for which the remaining life prediction is required, extract and transform the original sensor data, and obtain a normalized feature matrix;

[0040] S420: Input the normalized feature matrix into the trained neural network model to predict the remaining service life of the current grinding wheel.

[0041] The above-mentioned method for predicting the remaining service life of grinding wheels based on physical information neural networks adds a Mamba layer to the traditional neural network model to better extract potential features in the original data, making up for the insufficient ability of the neural network to utilize features in the hidden state, improving the final prediction accuracy, and realizing the remaining service life prediction of high-precision grinding wheels. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 This is a flow chart of a method for predicting the remaining life of a grinding wheel of a grinding machine based on a physical information neural network in one embodiment of the present invention;

[0043] Figure 2 A schematic diagram of a physical information neural network framework combined with Mamba provided in one embodiment of the present invention;

[0044] Figure 3 A diagram showing the principle structure of a Mamba network according to an embodiment of the present invention;

[0045] Figure 4 A schematic diagram of prediction results of a physical information neural network framework combined with a Mamba network in one embodiment of the present invention;

[0046] Figure 5 This is a schematic diagram of the prediction results of the physical information neural network framework without combining the Mamba network in one embodiment of the present invention. DETAILED DESCRIPTION

[0047] In order to enable those skilled in the art to better understand the technical solution of the present invention, the present invention is further described in detail below with reference to the accompanying drawings.

[0048] In one embodiment, Figure 1 As shown, a method for predicting the remaining life of a grinding wheel of a grinding machine based on a physical information neural network comprises the following steps:

[0049] S100: Measure and record sensor data from the n-channel grinding wheel of a grinding machine and convert it into digital signals via an acquisition card. The number of operations completed by the current device or the unit time is used as the current service life of the device. Furthermore, the sensor data includes vibration signals, force signals, temperature signals, etc., and the specific sensor data is determined by the specific device. In this embodiment, S100 extracts vibration signals from three channels of the high-precision grinding wheel spindle (X, Y, and Z) along the three axes. The data acquisition card collects 4096 data points every 1.6 seconds, with each 1.6 second period considered as one operation. For the training set, the operation time data is measured 234 times, for the validation set, the operation time data is measured 40 times, and for the test set, the operation time data is measured 32 times.

[0050] S200: Analyze the obtained n-channel sensor data. If the amount of sensor data is greater than a first preset number threshold, directly form a feature matrix from the digital signal. If the amount of sensor data is less than a second preset number threshold, extract time domain and frequency domain features from the data signal to form a feature matrix, and normalize the feature matrix to obtain a normalized feature matrix. The second preset number threshold is less than the first preset number threshold.

[0051] S300: In the model training phase, the normalized feature matrix is ​​formed into an input matrix to train the Mamba network model under the physical information neural network framework to obtain a trained neural network model;

[0052] S400: In the model testing phase, the normalized feature matrix obtained after processing the equipment data to be predicted is used as the input matrix to verify the trained neural network model to obtain the remaining useful life prediction value of the industrial equipment.

[0053] Compared to other neural network-based prediction methods, the aforementioned method for predicting the remaining life of grinding wheels for a grinding machine, based on physical information neural networks, better captures hidden correlations between feature data, improving life prediction accuracy. By leveraging sensor signals such as vibration and force, the method achieves the remaining useful life prediction of high-precision grinding wheels.

[0054] In one embodiment, if the amount of sensor data in S200 is greater than a first preset threshold, directly forming a feature matrix from the digital signal, and normalizing the feature matrix to obtain a normalized feature matrix includes:

[0055] S210: The signal collected during one use process is regarded as a time unit. There are N use processes in total. The feature data extracted during each use process is m in total. The feature matrix X is formed by splicing them in chronological order. N*m , reshape the feature matrix dimension to X N*1*m ;

[0056] S220: Calculate the average value and standard deviation of each row of feature data according to the feature matrix, and normalize the feature matrix according to the average value and standard deviation of each row of feature data, specifically:

[0057]

[0058] in is the normalized feature data, y is the original sensor data, mean(y(:,j,:)) is the mean value of each row of feature data under the corresponding channel, std(y(:,j,:)) is the standard deviation of each row of feature data under the corresponding channel, i=1,…,N, j=1, k=1,…,m;

[0059] S230: Reconstruct the normalized feature matrix and reshape the dimension to X N*m .

[0060] In one embodiment, if the amount of sensor data is less than a second preset threshold in S200, time domain and frequency domain features are extracted from the data signal to form a feature matrix, and the feature matrix is ​​normalized to obtain a normalized feature matrix, including:

[0061] S240: Take the signal collected during one use process as a time unit, a total of N use processes, and m feature data extracted during each use process, and splice them in time order to form a feature matrix X N*m , perform time domain and frequency domain feature extraction, re-extract time domain and frequency domain features for each original feature data, a total of a, and re-splice to form a feature matrix X N*m*a ;

[0062] S250: Calculate the average value and standard deviation of each row of feature data according to the feature matrix, and normalize the feature matrix according to the average value and standard deviation of each row of feature data, specifically:

[0063]

[0064] in is the normalized feature data, x is the extracted time domain feature and frequency domain feature data, mean(x(:,j,:)) is the mean value of each row of feature data under the corresponding channel, std(x(:,j,:)) is the standard deviation of each row of feature data under the corresponding channel, i=1,…,N, j=1,…,m, k=1,…,a;

[0065] S260: Reconstruct the normalized feature matrix and reshape the dimension to X N*ma .

[0066] In one embodiment, the time domain features extracted from the original sensor signal include: absolute mean, peak, root mean square value, root amplitude, skewness value, kurtosis value, waveform factor, pulse factor, skewness factor, peak factor, margin factor, and kurtosis factor; the frequency domain features include: center of gravity frequency, mean square frequency, root mean square frequency, and frequency variance.

[0067] Specifically, for the three channel data, 12 time domain features (absolute mean, peak value, root mean square value, root square amplitude, skewness value, kurtosis value, shape factor, pulse factor, skewness factor, peak factor, margin factor, and kurtosis factor) are extracted for each channel at each operation time, and 4 frequency domain features (center of gravity frequency, mean square frequency, root mean square frequency, and frequency variance) are extracted, for a total of 16 features. The feature matrix X is formed. 234*3*16 , after normalizing the feature matrix, adjust the matrix dimension to X 234*48 .

[0068] In one embodiment, S300 includes:

[0069] S310: Divide the number of operations according to the set operation time, obtain the current service life as a label, and form the label into an output matrix. The specific form is:

[0070]

[0071] S320: The feature matrix X extracted previously N*ma or X N*m Input the neural network model for training;

[0072] S330: After one training process is completed, the prediction matrix is ​​obtained The root mean square error value is obtained according to the prediction results and the root mean square error RMSE evaluation function, specifically:

[0073]

[0074] S340: Using the back propagation principle, the neural network parameters are corrected to improve the network fitting ability. After the training cycle is completed, the network model training is completed.

[0075] Specifically, the neural network model of the present invention adds a Mamba network layer and an attention mechanism under the framework of the physical information neural network. After the feature matrix is ​​input into the network, the spatial dimension features are extracted through the attention mechanism, and then the time dimension features are extracted through the Mamba network layer. The intermediate state obtained through these two layers of network models is then input into the subsequent network to complete the network training.

[0076] Specifically, according to the settings of the sensor and acquisition card, 4096 data points are collected every 1.6 seconds, so 1.6 seconds is used as the operation time, and every 1.6 seconds is regarded as one operation. The number of operations in the training set is 234, and the corresponding service life labels are 1-234. The number of operations in the validation set is 40, and the corresponding service life labels are 1-40. The number of operations in the validation set is 32, and the corresponding service life labels are 1-32.

[0077] In one embodiment, Figure 2 and Figure 3 As shown, S320 includes:

[0078] S321: The feature matrix is ​​input into the neural network model, passes through the Mamba layer, and the state output from the Mamba network layer is added to the linear layer together with the original input matrix after the attention mechanism. After forward propagation, the hidden state is obtained;

[0079] S322: The hidden state and time are input into the multi-sensor, and the prediction result is output after passing through the fully connected layer. The first RMSE value is calculated between the prediction result and the true label. The time refers to the time sequence number of the corresponding usage process, that is, the time of the first use is 1, the time of the second use is 2, and so on.

[0080] S323: The partial derivative of the hidden state with respect to the output is obtained through the multi-layer perceptron, which is then input into the Mamba layer and the attention mechanism and then into the linear layer together with the hidden state. After forward propagation, the current state can be obtained. The current state plus the partial derivative of the output with respect to time is added to the value of 0 to calculate the second RMSE. The second RMSE value is added to the first RMSE calculated previously to form the loss function.

[0081] In one embodiment, S321 specifically includes:

[0082] The feature matrix is ​​input into the neural network model and first enters the Mamba layer. In the Mamba layer, the input data first passes through two linear layers. After passing through the linear layer one, the input data undergoes a convolution operation and passes through the activation function represented by σ. The formula is:

[0083]

[0084] After passing through the activation function, it passes through the state-space model, where it is multiplied by the output of the second linear layer and the activation function, and then passes through another linear layer and is output from the Mamba network. The state output from the Mamba network layer passes through the attention mechanism and is added to the linear layer together with the original input matrix at the beginning. After forward propagation, the hidden state is obtained.

[0085] In one embodiment, S400 includes:

[0086] S410: Perform an operation on the equipment for which the remaining life prediction is required, extract and transform the original sensor data, and obtain a normalized feature matrix;

[0087] S420: Input the normalized feature matrix into the trained neural network model to predict the remaining service life of the current grinding wheel.

[0088] Specifically, the original data of the prediction set to be predicted is converted into an input feature matrix with a dimension of X 32 *48 , the feature matrix is ​​input into the trained neural network model, and the network finally outputs the prediction result.

[0089] In a detailed embodiment, the present invention collects X, Y, and Z three-axis vibration data for the grinding wheel spindle on a high-precision grinding machine, thereby predicting the remaining service life of the grinding wheel, with the grinding wheel speed being 1500 rpm. During the grinding process, a CMS-One TES001T three-axis vibration sensor and a CMS-One DAQ acquisition card are used to collect three channels of digital signals from the grinding wheel spindle X, Y, and Z axes at a sampling rate of 2.56kHz. 4096 data points are collected every 1.6 seconds, with a 1.6 second cycle, and each acquisition cycle is considered to be one grinding operation. Figure 4 This is a schematic diagram of the prediction results of the physical information neural network framework combined with the Mamba network. Figure 5 This is a diagram showing the prediction results using the physical information neural network framework without the Mamba network. It can be seen that the overall prediction results are closer to the true value and more accurate when the Mamba network is added as proposed in this patent. Table 1 shows the root mean square error (RMS) of the prediction results under different scenarios. This invention adds the Mamba network to the physical information network framework based on conventional methods.

[0090] Table 1 Prediction results of different methods

[0091]

[0092] Compared to other life prediction methods, the proposed method for predicting the life of high-precision grinding wheels incorporates a Mamba network layer within the neural network model architecture. This method better captures hidden information between feature data and improves life prediction accuracy. Using the grinding wheel spindle vibration signal, the remaining service life of high-precision grinding wheels can be predicted.

[0093] The above is a detailed introduction to the method for predicting the remaining life of a grinding wheel of a grinding machine based on a physical information neural network provided by the present invention. This article uses specific examples to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the core idea of ​​the present invention. It should be pointed out that for ordinary technicians in this technical field, without departing from the principles of the present invention, several improvements and modifications can be made to the present invention, and these improvements and modifications also fall within the scope of protection of the claims of the present invention.

Claims

1. A method for predicting the remaining life of a grinding wheel of a grinding machine based on a physical information neural network, characterized in that: The method comprises the following steps: S100: Measure and record the sensor data of the grinding wheel n channel and convert it into a digital signal through the acquisition card. The number of operations completed by the current device or the unit time is used as the current service life of the device; S200: Analyze the obtained n-channel sensor data. If the amount of sensor data is greater than a first preset threshold, directly form a feature matrix from the digital signal. If the amount of sensor data is less than a second preset threshold, extract time domain and frequency domain features from the data signal to form a feature matrix. Normalize the feature matrix to obtain a normalized feature matrix. S300: In the model training phase, the normalized feature matrix is ​​formed into an input matrix to train the Mamba network model under the physical information neural network framework to obtain a trained neural network model. In S300, the normalized feature matrix is ​​input into the neural network model for training, including: S321: The feature matrix is ​​input into the neural network model, passes through the Mamba layer, and the state output from the Mamba network layer is added to the linear layer together with the original input matrix after the attention mechanism. After forward propagation, the hidden state is obtained; S322: The hidden state and time are input into the multi-sensor, and the prediction result is output after passing through the fully connected layer. The first RMSE value is calculated between the prediction result and the true label; S323: Obtain the partial derivative of the hidden state with respect to the output through the multi-layer perceptron, input it into the Mamba layer and the attention mechanism, and then input it into the linear layer together with the hidden state. After forward propagation, the current state can be obtained. The current state is added to the partial derivative of the output with respect to time, and the second RMSE is calculated with 0. The second RMSE value is added to the first RMSE calculated previously to form the loss function; S400: In the model testing phase, the normalized feature matrix obtained after processing the equipment data to be predicted is used as the input matrix to verify the trained neural network model to obtain the remaining useful life prediction value of the industrial equipment.

2. The method according to claim 1, characterized in that In S200, if the amount of sensor data is greater than the first preset amount threshold, directly forming a feature matrix from the digital signal, and normalizing the feature matrix to obtain a normalized feature matrix, including: S210: The signal collected during one use process is regarded as a time unit. There are N use processes in total. The feature data extracted during each use process is m in total. The feature matrix X is formed by splicing them in chronological order. N*m , reshape the feature matrix dimension to X N*1*m ; S220: Calculate the average value and standard deviation of each row of feature data according to the feature matrix, and normalize the feature matrix according to the average value and standard deviation of each row of feature data, specifically: in is the normalized feature data, y is the original sensor data, mean(y(:,j,:)) is the mean value of each row of feature data under the corresponding channel, std(y(:,j,:)) is the standard deviation of each row of feature data under the corresponding channel, i=1,…,N, j=1, k=1,…,m; S230: Reconstruct the normalized feature matrix and reshape the dimension to X N*m .

3. The method according to claim 1, characterized in that In S200, if the amount of sensor data is less than the second preset amount threshold, time domain and frequency domain features are extracted from the data signal to form a feature matrix, and the feature matrix is ​​normalized to obtain a normalized feature matrix, including: S240: Take the signal collected during one use process as a time unit, a total of N use processes, and m feature data extracted during each use process, and splice them in time order to form a feature matrix X N*m , perform time domain and frequency domain feature extraction, re-extract time domain and frequency domain features for each original feature data, a total of a, and re-splice to form a feature matrix X N*m*a ; S250: Calculate the average value and standard deviation of each row of feature data according to the feature matrix, and normalize the feature matrix according to the average value and standard deviation of each row of feature data, specifically: in is the normalized feature data, x is the extracted time domain feature and frequency domain feature data, mean(x(:,j,:)) is the mean value of each row of feature data under the corresponding channel, std(x(:,j,:)) is the standard deviation of each row of feature data under the corresponding channel, i=1,…,N, j=1,…,m, k=1,…,a; S260: Reconstruct the normalized feature matrix and reshape the dimension to X N*ma .

4. The method according to claim 3, characterized in that The time domain features extracted from the original sensor signal include: absolute mean, peak value, root mean square value, root mean square amplitude, skewness value, kurtosis value, waveform factor, pulse factor, skewness factor, peak factor, margin factor, and kurtosis factor. The frequency domain features include: center of gravity frequency, mean square frequency, root mean square frequency, and frequency variance.

5. The method according to claim 4, characterized in that S300 includes: S310: Divide the number of operations according to the set operation time, obtain the current service life as a label, and form the label into an output matrix. The specific form is: S320: The feature matrix X extracted previously N*ma or X N*m Input the neural network model for training; S330: After one training process is completed, the prediction matrix is ​​obtained The root mean square error value is obtained according to the prediction results and the root mean square error RMSE evaluation function, specifically: S340: Using the back propagation principle, the neural network parameters are corrected to improve the network fitting ability. After the training cycle is completed, the network model training is completed.

6. The method according to claim 5, characterized in that S321 specifically: The feature matrix is ​​input into the neural network model and first enters the Mamba layer. In the Mamba layer, the input data first passes through two linear layers. After passing through the linear layer one, the input data undergoes a convolution operation and passes through the activation function represented by σ. The formula is: After passing through the activation function, it passes through the state-space model, where it is multiplied by the output of the second linear layer and the activation function, and then passes through another linear layer and is output from the Mamba network. The state output from the Mamba network layer passes through the attention mechanism and is added to the linear layer together with the original input matrix at the beginning. After forward propagation, the hidden state is obtained.

7. The method according to claim 6, characterized in that S400 includes: S410: Perform an operation on the equipment for which remaining life prediction is required, extract and transform the original sensor data, and obtain a normalized feature matrix; S420: Input the normalized feature matrix into the trained neural network model to predict the remaining service life of the current grinding wheel.

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