CN111046945B - Fault type and damage degree diagnosis method based on combined convolutional neural network - Google Patents
Fault type and damage degree diagnosis method based on combined convolutional neural network Download PDFInfo
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Abstract
The invention provides a fault type and damage degree diagnosis method based on a combined convolutional neural network, which comprises the following steps: s1, data acquisition and pretreatment; s2, constructing a one-dimensional convolutional neural network; s3, training a model; s4, adjusting super parameters and a network framework; s5, preparing a data set for diagnosing the fault type and the damage degree; s6, training each model respectively; s7, combining a plurality of convolution networks into a framework; s8, completing fault type identification and damage degree diagnosis. The method selects a one-dimensional convolutional neural network to extract the characteristics of the original vibration signals end to end; and meanwhile, the global maximum pooling layer is used for replacing the full-connection layer, so that training parameters are reduced, training speed is increased, and overfitting is prevented. Different models are trained by using one-dimensional original data with different severity degrees, so that the fault type can be identified, the purpose of fault damage degree classification can be achieved, and better effects than those of a single model can be achieved.
Description
Technical Field
The invention belongs to the field of deep learning and rotating machinery fault diagnosis, and relates to a mechanical fault type and damage degree diagnosis method based on a convolutional neural network.
Background
Today, modern industrial equipment is increasingly being developed towards large-scale, high-speed, fine-grained and automatic, and has been widely used in industries such as coal mine, petrochemical, electric power, etc., and monitoring the health condition of these equipment is becoming very complex. The large mechanical system can cause serious safety accidents once the large mechanical system fails, and huge economic loss and even casualties are caused. As one of the important components of the mechanical system, the bearing plays a role of a connecting rod or a gear shaft during the mechanical operation, and the damaged bearing can seriously affect the transmission and the transmission gear, thereby affecting the performance, the stability and the service life of the mechanical equipment. The fault positions of the bearing generally comprise an inner ring, an outer ring and rolling bodies, and the faults can be timely identified by using a fault diagnosis method, so that the safety performance of equipment is improved. However, early weak faults of equipment are often difficult to observe directly, and accidents are caused if the faults are not prevented in time, so that fault diagnosis for large-scale mechanical equipment is widely focused in the current society.
Conventional fault diagnosis methods can be divided into three categories: (1) analytical model-based methods such as parameter estimation, equivalent spatial and state estimation; (2) signal-based processing methods such as correlation analysis, spectrum analysis, wavelet analysis, etc.; (3) knowledge-based methods such as intelligent diagnostics, fuzzy reasoning, neural networks, etc. Among these methods, the analytical model-based method relies on expertise and requires a large amount of computation; the method based on signal processing has high requirements on professional knowledge, and cannot meet the requirements of industrial real-time monitoring; the third type of method has low requirements on the professional knowledge of technicians, and can realize real-time on-line monitoring of the working state of the industrial bearing.
With the development of machine learning, researchers also train various models of machine learning by using various indexes obtained through signal analysis as training samples (usually, the number of samples is small), so that the fault mode recognition accuracy is low. In recent years, with the advent of the big data age and the development of deep learning technology, an intelligent fault diagnosis method has been widely used. In particular, since 2016, deep learning has revolutionized practice, providing a useful tool for processing and analyzing large data, and data-driven mechanical fault diagnosis and health monitoring techniques have become increasingly popular.
Disclosure of Invention
Aiming at the fault diagnosis problem, the invention provides a mechanical fault type and damage degree diagnosis method based on deep learning. Because the vibration signal is a one-dimensional sequence, a one-dimensional convolutional neural network is selected to extract the characteristics of the original vibration signal end to end. By taking bearing faults as an example for explanation, the requirement of real-time on-line monitoring of the working state of the industrial bearing can be met, and the requirement of professional knowledge of technicians and equipment maintainers is low.
Aiming at the problems, the invention adopts a mechanical fault type and damage degree diagnosis method based on a convolutional neural network, which comprises the following steps:
s1, data acquisition and pretreatment: a sensor is used for collecting one-dimensional time sequence vibration signals of the mechanical equipment under different running states. Dividing the collected state signals into trainable samples, and performing overlapping sampling when the samples are insufficient so as to achieve the purpose of data enhancement; then constructing different data sets according to the fault diagnosis requirements; the data set is divided into training, verification and test samples and input into a one-dimensional convolutional neural network for training.
S2, constructing a one-dimensional convolutional neural network: the first layer of the one-dimensional convolutional neural network uses a convolutional kernel with the width of 8, the core of the convolutional neural network framework is a receptive field, and in order to enable the designed one-dimensional convolutional filter to learn characteristics irrelevant to displacement, the receptive field of the neurons of the last pooling layer in the network for an input signal is larger than the sampling point number of one rotation of a mechanical system. Furthermore, the present invention uses a global max-pooling layer after the convolutional layer instead of the fully-connected layer.
S3, training a model: the data set containing all fault types and damage degrees is input into a constructed one-dimensional convolutional neural network according to requirements, so that potential complex features in original vibration data are learned, and a multi-layer mapping relation from an original one-dimensional vibration signal to the fault type or the damage degree of the bearing is established.
S4, adjusting super parameters and a network framework: the super parameters and the network architecture of the one-dimensional convolutional neural network have influence on the model fault diagnosis and identification precision, and particularly, the method is used for comparing the test precision and the running time of the model aiming at the depth of the neural network, the width of the convolutional kernel, the global maximum pooling layer, the batch standardization (Batch Normalization, BN) layer and the filling (packing), thereby being beneficial to realizing the real-time fault diagnosis of mechanical equipment and achieving higher identification precision.
S5, preparing a data set for fault type and damage degree diagnosis, firstly forming the data containing all fault types and damage degrees into an integral data set, and then forming the data with different damage degrees under each fault type into a plurality of independent small data sets.
S6, training each model respectively: training a network that identifies only fault types using a data set that contains all fault types and severe damage; the data containing different damage levels for each fault type constitutes a plurality of independent data sets training a plurality of networks for identifying the damage levels of the faults. The network frameworks are obtained through learning to obtain different mapping relations, so that each framework has more pertinence, the precision of a single network framework is improved, and the overall higher precision can be realized.
S7, combining a plurality of convolution networks into a framework: the trained convolutional neural network is combined, and aiming at newly collected vibration data, the fault type can be identified through a pre-trained combined model, and then the fault damage degree is judged.
S8, completing fault type identification and damage degree diagnosis: and (3) completing automatic end-to-end feature extraction, high-precision fault type identification and damage degree diagnosis. Compared with the framework of a small convolution kernel used in the first layer and the traditional neural network framework of full connection used after the convolution layer, the method provided by the invention has higher accuracy and needs less training parameters. Meanwhile, compared with a single model, the combined model provided by the invention directly carries out fault type identification and damage degree diagnosis, and the model precision is greatly improved.
Preferably, the data acquisition and preprocessing in S1 includes the following steps:
s1.1, acquiring a large amount of one-dimensional time sequence vibration data under different operation conditions of mechanical equipment through a sensor to form a large data set for training a neural network.
S1.2, the sample input dimension is the premise of ensuring the diagnosis precision of the model, and when the sample input dimension is increased, the diagnosis precision is improved, but the running speed of the model is reduced, so that the sample dimension suitable for mechanical fault diagnosis is selected on the premise of ensuring the running speed of the model.
S1.3, the data enhancement method used in the invention is overlapped sampling, the length of a sample is assumed to be L, the offset is S, and if the data set has n data, the (n-L)/s+1 samples can be obtained. The invention uses overlapped sampling to divide the collected one-dimensional time sequence into required samples, and divides the signals under different running states into single samples to form different data sets.
S1.4, combining the plurality of data sets into one data set containing a plurality of different fault types and damage degrees.
Preferably, the step of constructing the one-dimensional convolutional neural network in S2 is as follows:
s2.1, according to the mechanical vibration signal, the first layer uses a convolution kernel of width 8, after which the convolution kernel uses a convolution kernel of width 3.
S2.2, aiming at the neuron of the last pooling layer, the receptive field of the input signal is required to be larger than the sampling point number of one rotation of the mechanical system. Let the receptive field of the neurons of the last pooling layer in the input signal be R (0) T is the number of points recorded by the accelerometer rotating around the bearing, L is the length of the input signal, and the receptive field R (0) Should satisfy T.ltoreq.R (0) L is less than or equal to L, and the specific calculation process is as follows:
receptive field R of neurons of the last pooling layer at the kth pooling layer (k) And receptive field R at the kth-1 pooling layer (k-1) The relation between the two is:
R (k-1) ＝S (k) (P (k) R (k) -1)+W (k) (1)
wherein S is (k) Is the step size, W, of the kth convolution kernel (k) Is the width of the kth convolution kernel, P (k) Is the number of downsampling points of the k layer.
When the layer number k is greater than 1, S (k) ＝1，W (k) ＝3，P (k) =2, therefore, formula (1) can be sorted as:
R (k-1) ＝2R (k) +2 (2)
when k is the lastR when a layer is pooled and a layer is n (n) =1, so the receptive field of the last pooled layer in the first pooled layer is:
R (1) ＝2 n-1 ×3-2 (3)
carry the above formula into formula R (k-1) ＝S (k) (P (k) R (k) -1)+W (k) Calculating the receptive field of the input signal in the last pooling layer as:
R (0) ＝S (k) (P (k) R (k) -1)+W (k) ＝2S (1) (2 n-1 ×3-2)+W (1) -S (1) ≈S (1) (2 n ×3-4) (4)
because T is less than or equal to R (0) L is less than or equal to T is less than or equal to S (1) (2 n X 3-4) is less than or equal to L, and the step length S is required (1) It should be possible to divide the signal length L.
S2.3, padding is carried out before each convolution, so that the feature graphs before and after the convolution are the same in size, and the purpose of fully extracting edge features is achieved.
S2.4, adding a BN layer after each convolution layer, wherein the purpose is to make the data mean value of the input network be 0 and the variance be 1, so that gradient propagation is facilitated, and a deeper network is constructed.
S2.5, realizing dimension reduction of the feature map by using global maximum pooling, reducing training parameters of a network, accelerating training speed and preventing overfitting.
S2.6, the optimization method uses root mean square transfer (Root Mean Square Prop, RMSProp) to solve the problems of convergence speed and local minimum point of small batch gradient descent.
S2.7, the invention combines model check point (ModelCheckPoint) and early termination (EarlyStopping) callback functions, when the monitoring target index is not changed any more in a set round, earyStopping termination model training is adopted, and simultaneously, the model can be continuously saved by the ModelCheckPoint in the training process so as to obtain an optimal model.
The steps for adjusting the super parameters and the network architecture described in the preferred S4 are as follows:
s4.1, adjusting the number of filters to avoid the condition of under fitting or over fitting of the model; when the filter types are small, the signal features cannot be sufficiently extracted, resulting in model under-fitting; too many filter types can result in overfitting.
S4.2, increasing the depth of the network, and measuring the change of training accuracy and running time until a proper network depth is found.
S4.3, firstly using a convolution kernel with the width of 3 and then using a convolution kernel with the width of 8 in the first layer of the model, and simultaneously examining the training accuracy of the model.
S4.4, after the last convolution layer of the network, a full connection layer is used first, then a global maximum pooling layer is used for replacing the full connection layer, and meanwhile training accuracy of the model is inspected.
S4.5, adding Padding before convolution, adding BN layer after convolution, and checking whether model training precision reaches the expectation.
Preferably, the combined convolutional neural network described in S7 includes the steps of:
s7.1, dividing a sample data set containing different fault states and different fault damage degrees into training, verifying and testing samples, training the one-dimensional convolution network for fault type identification in S2, and storing the trained model for fault type identification.
S7.2, dividing fault signals containing different damage degrees into a plurality of independent data sets, training the one-dimensional convolution networks in the S2, and storing the trained models for fault damage degree diagnosis.
And S7.3, combining the trained different models, identifying the fault type, and diagnosing the fault damage degree.
The invention has the advantages that: selecting a one-dimensional convolutional neural network to extract the characteristics of the original vibration signals end to end; and meanwhile, the global maximum pooling layer is used for replacing the full-connection layer, so that the method has the advantages of reducing network training parameters, accelerating training speed and preventing overfitting. In addition, the neural network structure and the self-adaptive optimization algorithm which are advanced and suitable for the mechanical vibration signals are used, so that higher accuracy is achieved. If the fault damage degree is different, one-dimensional original data with different severity degrees are used for respectively training different models, and the trained models are combined, so that the purposes of identifying the fault type and classifying the fault damage degree can be achieved, better effects than those of a single model are achieved, and staff can take corresponding measures according to the fault type and the severity degree, so that major accidents are avoided.
Drawings
Fig. 1 is a schematic diagram of a training process of a one-dimensional convolutional neural network 1 for fault type recognition according to an embodiment of the present invention.
FIG. 2 is a schematic diagram of a training process of a one-dimensional convolutional neural network 2-1 for inner ring fault damage level identification in accordance with one embodiment of the present invention.
FIG. 3 is a schematic diagram of a training process of a one-dimensional convolutional neural network 2-2 for outer ring fault damage level identification in accordance with one embodiment of the present invention.
FIG. 4 is a schematic diagram of a training process of a one-dimensional convolutional neural network 2-3 for identifying the failure damage level of rolling bodies according to an embodiment of the present invention.
FIG. 5 is a schematic diagram of a fault type identification and damage degree diagnosis process performed by combining a trained one-dimensional convolutional neural network 1, one-dimensional convolutional neural network 2-2 and one-dimensional convolutional neural network 2-3 according to an embodiment of the present invention.
FIG. 6 is a graph of model accuracy versus model accuracy using parallel training of large and small convolution kernels, respectively, 5 times in a first layer in accordance with one embodiment of the present invention.
FIG. 7 is a graph of model accuracy versus model accuracy after convolution using parallel training for 5 times with the fully connected layer and the global max pooling layer, respectively, in accordance with one embodiment of the present invention.
FIG. 8 is a graph of model accuracy versus model accuracy for one embodiment of the present invention, if Padding is performed before performing the convolution and if the BN layer is trained 5 times in parallel after the convolution.
Detailed Description
The invention is described in detail below with reference to the accompanying drawings.
Taking a bearing data set of Case Western Reserve University (CWRU) as an example to verify the effectiveness of the proposed method, the specific steps are as follows:
step one: data collected at 10 different operating conditions at 1 horsepower (hp) were selected as training, validation and test samples, with a sampling frequency of 12KHz. In the invention, the overlapping sampling is adopted, the overlapping rate is 0.8, the rotating speed of the motor is about 1772 Revolutions Per Minute (RPM), and the number of sampling points for one rotation of the bearing is about 400. The input dimension of the sample directly affects the diagnosis accuracy, and in particular, the input dimension is increased, so that the diagnosis accuracy is improved, but the model training speed is reduced, wherein the length of the training sample is 1024, namely, the number of sampling points which is larger than that of the sampling points for one rotation of the bearing is selected, and the purpose is to ensure high enough diagnosis accuracy and high running speed.
Data comprising health, inner ring failure, outer ring failure, and rolling element failure, and failure states comprising 3 different degrees of damage (failure sizes 0.007, 0.014, 0.021, respectively) were composed into data set 1. Table 1 lists the data used in this example, dividing the data into training, validation and test samples, in proportions of 70%, 15% respectively, training a one-dimensional convolutional neural network for fault type recognition.
Table 1 training dataset description
Step two: in this embodiment, the input signal length is 1024, the signal period T is approximately 400, and the convolution layer number is 5. Calculated from S2.2, S when the aforementioned requirement is satisfied (1) Only 8, the convolution width is not less than 3 steps, and the convolution kernel width is chosen to be 24 in this embodiment.
Step three: the data set 1 is firstly subjected to 10 classification training, so that the model directly learns 10 different complex mappings.
Step four: the width of the first convolution kernel in this embodiment is adjusted, and convolution kernels with a width of 3, a step length of 1, a width of 24, and a step length of 8 are used respectively, and the two models are trained in parallel for 5 times, so that the diagnostic accuracy is as shown in fig. 6, and it is seen that the wide convolution kernels help to improve the model accuracy.
After the last convolution layer, the full connection layer is replaced by the full connection layer or the global maximum pooling layer respectively, and the two models are trained for 5 times in parallel, and the diagnosis precision of the two models is shown in fig. 7. It can be seen that the two models have little influence on the diagnosis accuracy, however, the model parameters of the global maximum pooling layer are less, and the running speed is greatly improved.
The convolutional neural network is added with padding before a convolutional layer, the BN layer is added after the convolutional layer, the convolutional layer is compared with a framework which is not subjected to any treatment, the two models are trained 5 times respectively, and the diagnosis precision is shown in figure 8. The former can effectively improve the model diagnosis precision.
The one-dimensional convolutional network architecture constructed in accordance with the present invention is detailed in table 2.
TABLE 2 one-dimensional convolutional neural network structural parameters
Step five: dividing three fault types and corresponding damage degree data into different sample sets respectively: the inner ring fault data set 2, the outer ring fault data set 3 and the rolling bearing data set 4 are shown in table 3 in detail.
TABLE 3 training dataset description required for failure damage level identification
Step six: as shown in fig. 1, the data set 1 containing 10 operating states is first classified 4, i.e. the type of fault is first identified.
The training is then performed separately for the data set inner ring failure data set 2, outer ring failure data set 3 and rolling body failure data set 4 in order to identify the failure damage level.
Step seven: and (3) combining the four models trained in advance in the step (S6) aiming at the newly input vibration signals, and then carrying out fault type identification and damage degree diagnosis of the bearing, so that the diagnosis precision of the models can be greatly improved.
Step eight: the combination model provided by the invention is verified by using the data, and is compared with the one-dimensional convolutional neural network which is directly classified by 10, and model accuracy is obtained by training for 5 times in parallel and is shown in a table 4.
Table 4 model accuracy obtained by training the model 5 times in parallel.
Claims (5)
1. The fault type and damage degree diagnosis method based on the combined convolutional neural network is characterized by comprising the following steps of: the method comprises the following steps:
s1, data acquisition and pretreatment: collecting one-dimensional time sequence vibration signals of mechanical equipment in different running states by using a sensor; dividing the collected state signals into trainable samples, and performing overlapping sampling when the samples are insufficient so as to achieve the purpose of data enhancement; then constructing different data sets according to the fault diagnosis requirements; dividing a data set into training, verifying and testing samples and inputting the training, verifying and testing samples into a one-dimensional convolutional neural network for training;
s2, constructing a one-dimensional convolutional neural network, and using a global maximum pooling layer to replace a full-connection layer after a convolutional layer;
s3, training a model: inputting a data set containing all fault types and damage degrees into a constructed one-dimensional convolutional neural network according to requirements, and learning potential complex features in original vibration data and establishing a multi-layer mapping relation from an original one-dimensional vibration signal to the fault type or the damage degree of the bearing;
s4, adjusting super parameters and a network framework: comparing test accuracy and run time of the model for the depth of the neural network, the width of the convolution kernel, the global max pooling layer, the batch normalization layer Batch Normalization, BN and the filling Padding;
s5, preparing a data set for fault type and damage degree diagnosis, firstly forming the data containing all fault types and damage degrees into an integral data set, and then forming the data with different damage degrees under each fault type into a plurality of independent small data sets;
s6, training each model: training a network that identifies only fault types using a data set that contains all fault types and severe damage; the data containing different damage degrees under each fault type form a plurality of independent data sets for training a plurality of networks for identifying the damage degrees of the faults, and different mapping relations are obtained through learning;
s7, combining a plurality of convolution networks into a framework: combining the trained convolutional neural network, and aiming at newly collected vibration data, identifying the fault type through a pre-trained combined model, and judging the fault damage degree;
s8, completing fault type identification and damage degree diagnosis: and (3) completing automatic end-to-end feature extraction, high-precision fault type identification and damage degree diagnosis.
2. The method for diagnosing the fault type and the damage degree based on the combined convolutional neural network according to claim 1, wherein the method comprises the following steps of: the data acquisition and preprocessing in the step S1 comprises the following steps:
s1.1, acquiring a large amount of one-dimensional time sequence vibration data under different operation conditions of mechanical equipment through a sensor to form a large data set for training a neural network;
s1.2, selecting a sample dimension suitable for mechanical fault diagnosis on the premise of ensuring the running speed of the model;
s1.3, adopting a data enhancement method of overlapped sampling, assuming that the length of a sample is L and the offset is S, and if the data set has n data, obtaining (n-L)/s+1 samples; dividing the acquired one-dimensional time sequence into required samples by using overlapped sampling, dividing signals in different running states into single samples, and forming different data sets;
s1.4, combining the plurality of data sets into one data set containing a plurality of different fault types and damage degrees.
3. The method for diagnosing the fault type and the damage degree based on the combined convolutional neural network according to claim 1, wherein the method comprises the following steps of: the step of constructing the one-dimensional convolutional neural network in the step S2 is as follows:
s2.1, according to the mechanical vibration signal, a convolution kernel with the width of 8 is used for the first layer, and a convolution kernel with the width of 3 is used for the later convolution kernels;
s2.2, aiming at neurons of the last pooling layer, the receptive field of the input signal is larger than the sampling point number of one circle of rotation of the mechanical system; let the receptive field of the neurons of the last pooling layer in the input signal be R (0) T is the number of points recorded by the accelerometer rotating around the bearing, L is the length of the input signal, and the receptive field R (0) Satisfy T is less than or equal to R (0) L is less than or equal to L, and the specific calculation process is as follows:
receptive field R of neurons of the last pooling layer at the kth pooling layer (k) And receptive field R at the kth-1 pooling layer (k-1) The relation between the two is:
R (k-1) ＝S (k) (P (k) R (k) -1)+W (k) (1)
wherein S is (k) Is the step size, W, of the kth convolution kernel (k) Is the width of the kth convolution kernel, P (k) Is the number of downsampling points of the k layer;
when the layer number k is greater than 1, S (k) ＝1，W (k) ＝3，P (k) =2, therefore, formula (1) can be sorted as:
R (k-1) ＝2R (k) +2 (2)
when k is the last pooling layer n, R (n) =1, so the receptive field of the last pooled layer in the first pooled layer is:
R (1) ＝2 n-1 ×3-2 (3)
carry the above formula into formula R (k-1) ＝S (k) (P (k) R (k) -1)+W (k) Calculating the receptive field of the input signal in the last pooling layer as:
R (0) ＝S (k) (P (k) R (k) -1)+W (k) ＝2S (1) (2 n-1 ×3-2)+W (1) -S (1) ≈S (1) (2 n ×3-4) (4)
because T is less than or equal to R (0) L is less than or equal to T is less than or equal to S (1) (2 n X 3-4) is less than or equal to L, and the step length S is required (1) The signal length L can be divided;
s2.3, carrying out Padding before each convolution, so that the feature graphs before and after the convolution are the same in size, and fully extracting edge features;
s2.4, adding a BN layer after each convolution layer to enable the data mean value of an input network to be 0 and the variance to be 1, and constructing a deeper network;
s2.5, realizing dimension reduction of the feature map by using global maximum pooling, reducing training parameters of a network, accelerating training speed and preventing overfitting;
s2.6, optimizing Root Mean Square Prop and RMSProp by using a root mean square transmission method, and solving the problems of convergence speed and local minimum point of small batch gradient descent;
s2.7, combining model checkpoints with early termination of EarlyStopping callback functions, training by using an EarlyStopping termination model when the monitoring target index is not changed any more in a set round, and continuously storing the model in the training process by the model checkpoints so as to obtain an optimal model.
4. The method for diagnosing the fault type and the damage degree based on the combined convolutional neural network according to claim 1, wherein the method comprises the following steps of: the step of adjusting the super parameters and the network architecture in the step S4 is as follows:
s4.1, adjusting the number of filters to avoid the condition of under fitting or over fitting of the model;
s4.2, increasing the depth of the network, and measuring the change of training precision and running time until a proper network depth is found;
s4.3, firstly using a convolution kernel with the width of 3 in a first layer of the model, then using a convolution kernel with the width of 8, and simultaneously examining the training precision of the model;
s4.4, after the last convolution layer of the network, using a full connection layer firstly, and then using a global maximum pooling layer to replace the full connection layer, and simultaneously examining the training precision of the model;
s4.5, adding Padding before convolution, adding BN layer after convolution, and checking whether model training precision reaches the expectation.
5. The method for diagnosing the fault type and the damage degree based on the combined convolutional neural network according to claim 1, wherein the method comprises the following steps of: the combined convolutional neural network described in S7 includes the steps of:
s7.1, dividing a sample data set containing different fault states and different fault damage degrees into training, verifying and testing samples, and storing the one-dimensional convolution network model which is trained in the S2 and used for fault type identification;
s7.2, dividing fault signals containing different damage degrees into a plurality of independent data sets, training the one-dimensional convolution networks in the S2, and storing the trained models for fault damage degree diagnosis;
and S7.3, combining the trained different models, identifying the fault type, and diagnosing the fault damage degree.
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