Source free unsupervised domain adaptation for electro-mechanical actuator fault diagnosis

Jianyu WANG, Heng ZHANG, Qiang MIAO

College of Electrical Engineering, Sichuan University, Chengdu 610065, China

KEYWORDSData privacy;Electro-mechanical actuator;Pseudo-label clustering;Nearest centroid filtering;Unsupervised domain adaptation

AbstractA common necessity for prior unsupervised domain adaptation methods that can improve the domain adaptation in unlabeled target domain dataset is access to source domain dataset and target domain dataset simultaneously.However, data privacy makes it not always possible to access source domain dataset and target domain dataset in actual industrial equipment simultaneously,especially for aviation component like Electro-Mechanical Actuator(EMA)whose dataset are often not shareable due to the data copyright and confidentiality.To address this problem,this paper proposes a source free unsupervised domain adaptation framework for EMA fault diagnosis.The proposed framework is a combination of feature network and classifier.Firstly,source domain datasets are only applied to train a source model.Secondly,the well-trained source model is transferred to target domain and classifier is frozen based on source domain hypothesis.Thirdly,nearest centroid filtering is introduced to filter the reliable pseudo labels for unlabeled target domain dataset, and finally, supervised learning and pseudo label clustering are applied to fine-tune the transferred model.In comparison with several traditional unsupervised domain adaptation methods,case studies based on low-and high-frequency monitoring signals on EMA indicate the effectiveness of the proposed method.

1.Introduction

Electro-Mechanical Actuators(EMAs)have drawn wide attentions in modern aviation aircrafts like A-380 and Boeing 787.1,2Mechanical component in EMA is directly driven by electrical motor, and thereby, it can relieve complex structure and avoid oil leakage compared with traditional integrated actuator package and electro-hydraulic actuator.3To avoid catastrophic accidents caused by unexpected failures, health monitoring system is one of the most important systems in airplane, especially for the monitoring of EMA due to its critical role in aileron and landing gear.4As one of the key approaches in health monitoring system, fault diagnosis can identify failures based on monitoring signal.

In recent years,deep learning draws wide attention in actuator fault diagnosis because of its ability to automatically mine features from original signal and map them into different failure labels.It is easy to generalize it directly to different actuators, no longer labor intensive in developing observer and extracting features in traditional model- and feature-based methods,5respectively.Until now,many popular deep learning models such as deep auto-encoder,6Convolution Neural Network (CNN),7long-short term memory network,8and etc.,have been validated to be more effective than traditional feature-based methods in actuator fault diagnosis.However,a common limitation for those methods deployed in previous studies is that they only perform well in testing dataset with the same distribution as that in training dataset, while their performance will decrease a lot when facing a testing dataset under new working condition.Actually,EMA is usually driven by different commands with varying amplitudes and frequencies, and also, works in complicated loading conditions and various velocity environments.9Hence, all of those variable working conditions bring a data distribution discrepancy challenge for the application of aforementioned methods.

To cope with the distribution discrepancy challenge, transfer learning is introduced to enhance the model generalization ability in testing dataset under new working condition.10The key idea of transfer learning model is to take advantages of knowledge in source domain(i.e.,training dataset in old working condition)to serve for target domain(i.e.,testing dataset in new working condition) so as to reduce distribution discrepancy between these two domains and improve domain adaptation ability of transferred model in target domain dataset.11The domain adaptation task in transfer learning can be divided into two categories,12i.e., supervised domain adaptation and unsupervised domain adaptation, depending on whether there are labels in target domain dataset.For supervised domain adaptation, instance-weight samples13and fine-tuning network14are usually applied to achieve domain adaptation in target domain dataset.The former, by enhancing the weight of similar samples in source domain, can implement the data augmentation for target domain samples.The latter aims to fine-tune the parameters of model trained by source domain dataset, and then, accommodate it adapt well in target domain.Labeled target domain samples are required for these two efforts.However, it is generally labor-intensive and timeconsuming to label those samples, and even impossible to acquire the labels for them in actual industrial equipment.

In comparison with supervised domain adaptation, unsupervised domain adaptation (i.e., without labels in target domain dataset)is more realistic for mechanical fault diagnosis on actual industrial equipment.There are two popular unsupervised domain adaptation strategies,15i.e.,feature alignment and adversarial learning.The feature alignment introduces an additional distribution metric criterion in deep learning model to reduce the distribution discrepancy between two domains.For example,Lu et al.16utilized the Maximum Mean Discrepancy (MMD) to reduce the distribution distance in high dimensional space of deep learning network, which could address the distribution discrepancy in bearing and gear fault diagnosis, respectively.The correlation alignment loss can align the second order statistics of two domains, and then, it was introduced to validate its feasibility in bearing fault diagnosis under variable working conditions.17Subsequently,some improved distribution metric criterions like multi-kernel MMD, joint MMD, and so on, are also explored in different efforts.15,18,19In virtue of adversarial learning in generative adversarial network, the generator in unsupervised domain adaptation can create indistinguishable features for two domains.20Chen et al.21conducted the Domain Adversarial Network (DAN) in bearing and gear fault diagnosis, which indicated that it can perform well in unseen working conditions.Since only inter-domain clustering is concerned in DAN, further, Han et al.22combined the triplet loss with adversarial learning to strengthen the intra-domain clustering for fault diagnosis of rotating machinery.In addition, more variants inspired by DAN are also investigated in different efforts.23,24Those efforts have made advanced progress in addressing the distribution discrepancy challenge.However,the access to source domain dataset for feature alignment or adversarial learning approach is inevitable.In other words,source domain dataset and target domain dataset need to be involved in the model training process simultaneously.Once source domain dataset are absent, these two unsupervised domain adaptation approaches may not work properly.

Monitoring dataset in some industrial equipment are unavailable for researchers due to some laws and regulation constraints on data.Hence, a noteworthy phenomenon in actual equipment is that data privacy or potential conflict of interest from data are concerned for many owners.Many owners just want researchers to utilize them in their own interests but not share them,which is due to the fact that data copyright and confidentiality are of the utmost importance for different equipment, even for the data sampled from the same equipment but with different working conditions.Therefore, the data privacy of source domain dataset will greatly impede the application of traditional unsupervised domain adaptation methods in industrial equipment.In recent years, federated learning draws wide attention in protecting the data privacy.One idea in federated learning is to upload different private data to the central server and utilize encryption techniques to protect them.25However, data leakage risk still remains when central server is attacked or improperly managed.The other idea is to train different sub-models for different local dataset, and then, parameters from those models are aggregated to a global model to make it adapt well in different local dataset.26However, labeled samples are necessary to train those sub-models.As the challenge mentioned in supervised domain adaptation, the labels collection in actual industrial equipment is time-consuming and labor-intensive, or even impossible.

Hence, how to take advantages of knowledge in source domain dataset and protect their data privacy while still improving the domain adaptation for unlabeled target domain dataset is concerned in this paper.In other words, source domain dataset and target domain dataset would not been involved in model training process together.Similar to the fine-tuning network in supervised domain adaptation, Source Free Unsupervised Domain Adaptation(SFUDA)27can transfer the source model and fine-tune it to adapt well in unlabeled target domain dataset.Actually, the transferred deep learning model is a black box that is not interpretable for original dataset,28so it can protect the data privacy for original dataset while storing its knowledge in well-trained model.Until now,some researchers have noticed the SFUDA challenge in semantic segmentation29and object detection.30The key idea of fine-tuning model in target domain is to explore the reliable and valuable target samples for supervised learning.Therefore,pseudo labels will be estimated by the transferred model, and then, two available approaches,27i.e., pseudo label clustering and pseudo label filtering, are applied to enhance the domain adaptation based on supervised learning.Liang et al.31proposed a nearest classifier to minimize the centroid shift and improve clustering effect for pseudo labels.Further, information maximization and supervised pseudo labeling are introduced in target domain to improve the pseudo label clustering of transferred model.32For pseudo label filtering,a source domain hypothesis assumes that the pseudo labels from similar target domain dataset are close to the actual labels trained in source domain, and then, the similar samples will be applied to guide the training process of transferred model.A valuable phenomenon founded by Kim et al.33is that the target samples with low self-entropy will have higher reliability pseudo labels.Thereby, those samples with low-entropy will be filtered to improve the domain adaptation ability.In addition, contrastive learning is applied to give different weights for different target samples, and those samples with high confidence will be equipped with heavy weights.34Despite the advances of SFUDA aforementioned, there are still relatively limited researches focusing on the mechanical fault diagnosis to protect data privacy of source domain and improve the domain adaptation in unlabeled target domain, especially for the one-dimensional time series monitoring signal.For the monitoring dataset sampled from aviation airplane, it is usually critical to monitor the state of key component and their data privacy can not be ignored.

For the EMA, dozens of sensors with different sampling frequencies are applied to monitor its state.Different time series signals from different sensors may reflect different failure information.EMA is usually driven by command with different amplitudes,and also,works in variable working conditions like various loadings and speeds.The distribution discrepancy challenge in EMA fault diagnosis is inevitable.In consideration of its special application scenarios like in aileron or landing gear of aircraft, labeling new failure samples is a difficult task.Hence, the dataset collected by different users sampling from different working conditions also have data copyright and confidentiality,especially for those valuable labeled source domain dataset.Based on those preliminary hypothesis, the source domain dataset and target domain dataset can not meet each other in model training process.The source domain dataset from one working condition are labeled while their data privacy need to be protected.The target domain dataset from the other working condition are unlabeled while improving domain adaptation of transferred model is concerned.

It is still not clear whether SFUDA method can adapt well in domain adaptation task of EMA,because different methods may adapt well in different dataset and some traditional domain adaptation methods may bring a negative transfer learning phenomenon.Hence, a comparison between the SFUDA method and several traditional domain adaptation methods has been explored in two case studies of EMA fault diagnosis.The traditional domain adaptation methods need to access the source domain and target domain simultaneously,while the SFUDA framework is applied to address the unsupervised domain adaptation challenge and protect the data privacy of source domain.The model in the proposed framework is a combination of two networks, i.e., feature network and classifier.The proposed model follows the source domain hypothesis,32i.e., predicted encoding pseudo labels (one-hot encodings) from target domain dataset are similar to those of source domain dataset based on frozen classifier.Source domain dataset are only applied to train the source model firstly.Then, the well-trained source model will be transferred to target domain but the parameters in classifier are not allowed to be fine-tuned.Further, pseudo label filtering(guided by nearest centroid filtering) is applied to filter the noise pseudo labels, and then, supervised learning (guided by pseudo labels) and pseudo label clustering (guided by mutual information maximizing) are applied to fine-tune the feature network.Finally,the fine-tuned model can be applied in unlabeled testing target domain dataset.The main contributions of this paper are summarized as follows:

(1) Benefitted from two separate training process of the proposed framework in source domain dataset and target domain dataset respectively,the data privacy of source domain dataset can be protected and the generalization ability of transferred model can be improved by several fine-tuning strategies in unlabeled target domain dataset.

(2) Multiple monitoring signals sampled by different sampling frequencies but for the same failures are applied to illustrate the effectiveness of the proposed method in low- and high-frequency case studies.

(3) This paper gives a comparison about the proposed method and traditional unsupervised domain adaptation methods in unsupervised domain adaptation task of EMA fault diagnosis, which can provide a benchmark for future studies.

The rest of this paper are organized as follows.The details of proposed method are introduced in Section 2.Experiment descriptions including EMA dataset and implementation details are introduced in Section 3.Case studies are explored in Section 4.Conclusion is summarized in Section 5.

2.Proposed method

Although many unsupervised domain adaptation methods have been applied in mechanical fault diagnosis, the access to source domain dataset in domain adaptation stage is contrary to the data privacy.Hence, a SFUDA framework is applied to protect the data privacy of source domain dataset while improving the unsupervised domain adaptation in target domain dataset for EMA fault diagnosis.Section 2.1 illustrates the overview of the proposed method that can be divided into source model generation and target domain adaptation.In Section 2.2, the details of source model generation are introduced.In Section 2.3, the details of target model adaptation are introduced.In Section 2.4, model parameters and flow chart of the proposed method are illustrated.

2.1.Overview of proposed method

where Llossis a loss function.However, due to the noise pseudo labels existing in Eq.(1), the domain adaptation optimization of the model FTmay be impeded.

Following the basic idea in SFUDA, the proposed model FTis a combination of two networks including feature network fE（?;θE）and classifier fC（?;θC）, as shown in Fig.1.Feature network is a one-dimensional convolutional network that can mine the robust features from original time series signal.Classifier is a combination of fully connected layers and softmax function, which is applied to predict labels.Two stages are included in the proposed method,i.e.,source model generation and target domain adaptation.In source model generation, only source domain dataset with their labels are applied to train those two sub-networks.The source domain hypothesis used in this paper assumes that the pseudo labels(i.e., one-hot encodings) from target domain dataset are similar to those of source domain dataset based on the well-trained classifier.35In target domain adaptation, the well-trained model is transferred to target domain and classifier would be fixed based on source domain hypothesis.Later, the proposed method utilizes the nearest centroid filtering to filter the noise pseudo labels.The supervised learning through Eq.(2) and pseudo label clustering through mutual information maximization, and then, are applied to fine-tune the feature network.Finally,the fine-tuned network FTcan be applied to diagnosis the monitoring signal in EMA.

2.2.Source model generation

Pooling layers usually follow the convolutional layers,which can down sample the features and help to reduce the number of parameters in model.Two types of pooling layers,i.e., max pooling layer and average pooling layer, are usually adopted to acquire maximum value and mean value, respectively.With the help of pooling layer, the activation features in （l+1）-th layer can be described as follows:

The fully connected layers transform multi-kernel features into one-dimensional features.Then, a softmax function is applied to predict the class labels based on the onedimensional features.The CNN learns to make class prediction by minimizing the cross-entropy loss Lsrcbetween the predicted and true labels.

Fig.1 Framework of the proposed unsupervised domain adaptation model with source free dataset.

where α is a smoothing parameter and is set to 0.1 in this paper; K stands for the number of classes.

2.3.Target domain adaptation

The key idea of SFUDA in this paper is to fine-tune the welltrained source model based on unlabeled target domain dataset.Benefitted from the blank box characteristic of deep learning model, the data privacy of source domain dataset can be protected by transferring only source model to target domain.The source domain hypothesis assumes that the pseudo labels predicted by classifier in transferred model are similar to the labels in source domain dataset.Hence, the classifier is fixed in target domain adaptation process while feature network is fine-tuned by pseudo labels predicted by classifier.To filter the noise pseudo labels,pseudo label filtering(i.e.,nearest centroid filtering) is applied to filter the reliable pseudo labels for supervised learning and pseudo label clustering.Here, the pseudo label clustering based on mutual information maximization is introduced firstly,and then,the necessity of pseudo label filtering is further explained.Finally, the network finetuning process in target domain is concluded.

2.3.1.Pseudo label clustering

For traditional domain adaptation methods, it is easy to measure the distribution discrepancy between two domains when source domain dataset are not absent in domain adaptation stages.However, the source domain dataset are unavailable in target domain adaptation stage due to the data privacy.Based on the source domain hypothesis, the ideal outputs of target datasets would be similar to the one-hot encoding vectors predicted in source domain while belong to different classes.To reinforce the above hypothesis,mutual information maximization38is introduced to make the pseudo one-hot encoding vectors more individually decisive and globally fair.

The general idea of mutual information maximization is to retain as much information of input samples in labels as possible.Hence, the mutual information L（c;x ）between the input samples x and the labels c can be described as follows:

For the SFUDA, since the classifier is fixed, a good domain adaptation model for target domain should extract robust features from the feature network, and then, make the pseudo labels from classifier more individually certain and follow an even distribution between the classes.Based on the Eq.(10), thereby, maximizing mutual information LMIin the target domain of the proposed method can be illustrated as follows:

where FT=fTE（?;θE）+fTC（?;θC）; Ext?Xt[?] is an average operation.

2.3.2.Pseudo label filtering

The mutual information maximization can be applied to strengthen the pseudo label clustering,however,some samples are still matched to a wrong clustering center due to the effect of noise pseudo labels.Hence, this phenomenon will misguide the fine-tuning process, and even bring a negative effect on domain adaptation model.To alleviate such negative effect of noise pseudo labels, the nearest centroid filtering is applied to filter the pseudo labels before maximizing the mutual information.

2.3.3.Network fine-tuning

Based on such double nearest centroid filtering mechanisms from Eq.(12) to (15), the pseudo labels are reliable enough to relieve the noise labels in them, which is beneficial to strengthen the pseudo label clustering effect.Furthermore,those filtering pseudo labels will also be applied for the supervised learning in target domain dataset40.

2.4.Model parameters and flow chart

The architecture and parameters of the proposed model includes feature network and classifier, as shown in Table 1.The feature network includes four convolutional layers (Convolution 1–4), four batch normalization layers (Batchnorm 1–4), two pooling layers (Maxpooling and Averagepooling)and one fully connected layer.Three fully connected layers and softmax function are included in classifier.Parametric Rectified Linear Unit (PReLU) is regarded as the activation function in two networks, and two dropout strategies are applied to relieve the over-fitting effect in classifier.

The main idea of the proposed model has been introduced from Section 2.1 to Section 2.3, the detailed flow chat of the proposed model is illustrated as follows:

3.Experiment descriptions

In this section,experiment descriptions including EMA dataset and implementation details are introduced.The EMA dataset based on two different monitoring signals with different sampling frequencies is introduced in Section 3.1.The implementation details of the proposed method and several comparison methods are illustrated in Section 3.2.

3.1.Electro-mechanical actuator dataset

The NASA flyable EMA testbed41is shown in Fig.2 (a).In Fig.2 (b), there are three actuators installed in the test bed.Actuator X is applied to simulate different failures.Actuator Y is treated as a comparison without injecting any failures.Actuator Z is applied to simulate the external load and apply force to the first two actuators.Three failures, i.e., ballscrew jam,sensor fault and spall fault,and normal state are regarded as four labels in this paper.Since unsupervised domain adaptation task is considered in this paper, eight working conditions considering different commands, amplitudes,frequencies, loads and velocities, are considered in this paper.Here, they are divided into two groups according to the sinusoidal and trapezoidal commands, as shown in Table 2.

Two types of monitoring signals, i.e., low sampling frequency (100 Hz) signals and high sampling frequency(20 kHz) signals, are applied to monitor the state of actuator,which may reflect different failure information.Since there is big sampling frequency difference between them, it is difficult to combine them directly.Hence, two case studies are conducted in this paper,respectively.For low sampling frequency signals, four types of signals, i.e., motor voltage, motor current, motor temperature, and nut temperature, are regarded as the low-frequency case study.For high sampling frequencysignals,acceleration signals from nut accelerometer in x,y and z directions are regarded as the high-frequency case study.

Table 1 Parameters of the proposed method.

Fig.2 FLEA of NASA.

Table 2 Experiment dataset under variable working conditions.

3.2.Implementation details

The input size of the low- and high-frequency case studies in the proposed model are different.For low-frequency signal,a sliding window with size 256 is applied to capture points from four signals.Therefore, the size of each sample under low-frequency is a 1×1024 vector reshaped by 4×256 points.For high-frequency signals, a sliding window with a size of 1024 is applied to capture points from three signals in turns.Therefore, the size of each sample under high-frequency is a 1×3072 vector reshaped by 3×1024 points.In each working condition,there are total 600 samples with 150 samples in each label.50 % samples in source domain and target domain are set as the training data,while the other 50%samples in target domain are regarded as testing data.12 cross-domain adaptation tasks within the same command are constructed in following experiment.The unsupervised domain adaptation task based on low-frequency signal is defined by A, while the task based on high-frequency signal is defined by B.For instance,12 domain adaptation of task A under sinusoidal command will be grouped as A1-A2, A1-A3, A1-A4, A2-A1, A2-A3,A2-A4, A3-A1, A3-A2, A3-A4, A4-A1, A4-A2, and A4-A3.The balancing-factor λ1and λ2in Eq.(16) are set to 1 and 0.1 before exploring its influence in Section 4.2 and Section 4.3,respectively.Adam optimizer with learning rate 0.01 is adopted.Batch size and iterative epoch are 64 and 30, respectively.The average classification accuracy of ten repeated experiment is considered as an evaluation criterion in all case studies.In addition, the basic CNN and other six traditional feature alignment and adversarial learning models are regarded as the comparison methods in this paper.It needs to be pointed out that DeepCoral, DeepMMD, DAN and DAN + MMD only consider reducing the global interdomain distance between two domains, while DANT-S and DANT-ST add the triplet loss to help to reduce the intraclass domain distance based on DAN.

(1)CNN:It is a baseline network trained by source domain dataset.

(2)DeepCoral17:Correlation alignment is applied to reduce second-order statistics distance of two domains in feature network.

(3)DeepMMD16:MMD is applied to reduce mean distance of Hilbert space of two domains in feature network.

(4) DAN20: It is a domain adversarial network using Kullback-Leibler divergence in discriminator.

(5) DAN + MMD23: It is a combination of DAN and MMD.

(6)DANT-S22:It is a combination of DAN and triplet loss,however, only source domain samples appear in triplet loss.

(7) DANT-ST24: Based on the DANT-S method, target domain samples with pseudo labels are further applied in triplet loss.

4.Case study

In this section, the effectiveness of the proposed method is investigated on several case studies.In Section 4.1,the comparison results of traditional unsupervised domain adaptation methods and the proposed method are conducted on lowfrequency and high-frequency case studies, respectively.In Section 4.2, the influence of supervised learning in target domain adaptation is discussed.In Section 4.3, the influence of pseudo label clustering in target domain adaptation is further discussed.In Section 4.4, the negative transfer phenomenon of the proposed method in high-frequency case study under trapezoidal command is discussed.

4.1.Performance on unsupervised domain adaptation

Two case studies are designed for low-and high-frequency signals, respectively.For each case study, 12 domain adaptation tasks are further explored under sinusoidal and trapezoidal command, respectively.

4.1.1.Low-frequency case study

The comparison results of eight methods under sinusoidal command are shown in Table 3 and Fig.3.Among the 12 domain adaptation tasks, the proposed method obtains the best average accuracy 92.45 %.Except for the DANT-S,DANT-ST and the proposed method, negative transfer phenomenon exists in most domain adaptation tasks for the other four domain adaptation methods compared to CNN, especially for DeepMMD.Therefore, the average accuracy of DeepCoral,DeepMMD,DAN,and DAN+MMD are lower than that average accuracy 87.22%in basic CNN.With introducing the triplet loss to enhance intra-domain clustering based on pseudo labels, DANT-S and DANT-ST achieve better average accuracy than CNN, which indicates that intraclass domain clustering is beneficial to improve the domain adaptation.However, it is still necessary to acquire the source domain dataset and make them participate in unsupervised domain adaptation for those traditional domain adaptation methods.For the proposed method, only source model is transferred to target domain, which can protect the data privacy of source domain.Even though the source domain dataset are not available for target domain, the proposed method obtains the better performance than DANT-S and DANTST in most domain adaptation tasks.

The comparison results of eight methods under trapezoidal command are shown in Table 4 and Fig.4.The average accuracy drops a lot among eight methods indicate that the domain adaptation tasks under trapezoidal command are more challenging than that under sinusoidal command.Only DANTST and the proposed method obtain the better average accuracy than CNN.However, severe negative transfer occurring in other five domain adaptation methods (DeepCoral,DeepMMD, DAN, DAN + MMD, and DANT-S) result in their low average accuracy, which may be due to the fact that only reducing the global inter-domain distance is considered in these efforts.In other words, although the source domain dataset are involved in domain adaptation training process,the improvement of those methods are still limited,which indicate that the EMA dataset bring a challenge for traditional domain adaptation methods.The proposed method only transfers the source model,and then,fine-tunes the transferred model to make it adapt well in target domain.It obtains the best accuracy in most domain adaptation tasks and also obtains the best average accuracy 86.67 %.For the lowfrequency monitoring signal, to conclude, the proposed method can achieve the stable performance and obtain the best average accuracy in domain adaptation tasks under sinusoidal command and trapezoidal command, respectively.

4.1.2.High-frequency case study

The comparison results of eight methods under sinusoidal command are shown in Table 5 and Fig.5.The CNN in high-frequency case study obtains the better average accuracy than that in Table 3 of low-frequency case study, which illustrates that different monitoring signals contain the different failure information.Different from the good performance of DANT-ST in Table 3 and Table 4, the negative transfer phenomenon exists in DANT-ST in this experiment.Only DAN + MMD and the proposed method obtain the better average accuracy than CNN.The DAN + MMD obtainsthe best accuracy in B2-B1, B2-B4 and B3-B4 tasks, however,the difference of the results between the proposed method and DAN+MMD under these three tasks are very small.In addition,the CNN obtains the best accuracy in five tasks, i.e., B1-B2,B2-B3,B3-B2,B4-B2,and B4-B3,which indicates there are great similarities between source domain dataset and target domain dataset.Hence, negative transfer learning occurs in all seven domain adaptation methods among those five similar domain adaptation tasks.However, the proposed method still achieves positive domain adaptation results under other seven domain adaptation tasks and obtains the best average accuracy (90.61 %) among those comparison methods.

Table 3 Unsupervised domain adaptation accuracy of low-frequency signals under sinusoidal command.

Fig.3 Unsupervised domain adaptation accuracy histogram of Table 3.

Table 4 Unsupervised domain adaptation accuracy of low-frequency signals under trapezoidal command.

Fig.4 Unsupervised domain adaptation accuracy histogram of Table 4.

Table 5 Unsupervised domain adaptation accuracy of high-frequency signals under sinusoidal command.

Fig.5 Unsupervised domain accuracy histogram of Table 5.

The comparison results of eight methods under trapezoidal command are shown in Table 6 and Fig.6.Like the same inference from Table 4 and Fig.4, the domain adaptation tasks of high-frequency monitoring signals under trapezoidal command are also challenging.The CNN in high-frequency case study obtains the lower average accuracy than that in Table 4 and Table 5.Only DANT-S and DANT-ST obtain the better average accuracy than that in CNN.However, the proposed method does not obtains the impressive performance like in above three case studies.It gets better accuracy than CNN under B1-B4, B3-B1, B3-B2 and B3-B4 tasks, but the performance on other eight tasks not drop a lot compared to Deep-Coral, DeepMMD, DAN, and DAN + MMD.Further, the possible reason for its unsatisfactory performance under this case study will be discussed in Section 4.4.For the highfrequency monitoring signals, to conclude, there are different failures information between high sampling frequency and low sampling frequency signals, and also, the proposed method obtains the best average accuracy in domain adaptation tasks under sinusoidal command.

4.2.Influence of supervised learning

Since the supervised learning based on pseudo labels is applied to fine-tune the network in Eq.(16), the influence of the supervised learning is explored in this section.Hence, supervisedlearning in Eq.(16)by letting the λ1is equal to 0.The comparison results based on two types of monitoring signal are shown in Table 7 and Table 8, respectively.In addition, to make the comparison results more vividly,the sector diagram of Table 7 and Table 8 are shown in Fig.7 and Fig.8, respectively.For low-frequency case study, supervised learning is beneficial to improve the accuracy in almost all tasks, except for the A1-A4 and A3-A4 under trapezoidal command.The improvement of average accuracy in sinusoidal command and trapezoidal command are about 6 % and 3 %, respectively.For highfrequency case study, supervised learning is still beneficial for most domain adaptation tasks, except for the B2-B1, B2-B3 and B2-B4 task in trapezoidal command.The improvement of average accuracy in sinusoidal command and trapezoidal command are about 4 % and 7 %, respectively.In a word,when supervised learning based on pseudo labels is absent in target domain, negative transfer phenomenon is inevitable in most domain adaptation tasks.

Table 6 Unsupervised domain adaptation accuracy of high-frequency signals under trapezoidal command.

Fig.6 Unsupervised domain accuracy histogram of Table 6.

4.3.Influence of pseudo label clustering

Since the pseudo label clustering based on mutual information maximization is applied to enhance the clustering effects in Eq.(16), the influence of the pseudo label clustering is further discussed in this section.Hence,λ2is set to different values,from nonparticipation to a big weight in Eq.(16)with 0,0.1,0.3,0.5,0.7,and 1.0,respectively.The comparison results based on two case studies are shown in Table 9 and Table 10,respectively.In addition, the histograms of those two tables are drawn in Fig.9 and Fig.10,respectively.For low-frequency case study,when λ2is set to 0,the average accuracy drops about 1%than that in previous experiments with default value 0.1,which indicates that the pseudo label clustering is still necessary to improve the domain adaptation ability.However, when λ2is set to a big value 1, the accuracy of all domain adaptation tasks drops a lot, as shown in Table 9 and Fig.9.Hence, the proposed method is sensitive to select λ2and low values like 0.1 or 0.3 are beneficial to improve the domain adaptation ability for most tasks.For high-frequency case study, the application of pseudo label clustering still improves the domain adaptation ability in most tasks, especially for the experiment under trapezoidal command.It is preferred to select the low values for λ2, for instance, the performance of 0.1 and 0.3 are also more stable in Table 10 and Fig.10.The big value 1 for λ2brings a negative transfer effect on domain adaptation task.In a word, it is necessary to take the pseudo label clustering into account of the proposed method and λ2with a low weight value is recommended in Eq.(16).

4.4.Discussion

The proposed method obtains the best average accuracy from Table 3 to Table 5, however, the negative transfer phenomenon occurs in most domain adaptation tasks in Table 6.The potential reason for this phenomenon is discussed in thissection.The generalization ability of the well-trained model is critical to its performance on testing dataset,and also,a robust well-trained model also performs well in training dataset.Hence, the performance of training dataset under source model and target model are discussed.The accuracy of source domain training dataset (S_train) and target domain training dataset(T_train)are explored in Table 11 and Fig.11.For S_-train, the first three experiments (Table 3 to Table 5 corre-sponding to Sinusoidal low-frequency, Trapezoidal lowfrequency and Sinusoidal high-frequency, respectively) obtain the average accuracy 99.72 %, 98.92 %, and 94.52 %, respectively.The performance of last experiment (Table 6 corresponding to Trapezoidal high-frequency) drops a lot, with only average accuracy 88.64%in source domain training data-set.Later,these source models are fine-tuned by target training dataset, however, the performance in target domain training dataset will not exceed that in source domain training dataset,which are 92.55 %, 87.16 %, 91.09 %, and 75.65 % for four experiments, respectively.Hence, the generalization ability of source model is critical to deciding the performance on target domain dataset.That is because the source domain hypothesis assumes that classifier in the well-trained source model is fixed in target domain.Hence, the poor performance of source model results in the poor domain adaptation ability in highfrequency case study under trapezoidal command.

Table 7 Influence of supervised learning in low-frequency case study.

Table 8 Influence of supervised learning in high-frequency case study.

Fig.7 Unsupervised domain accuracy sector diagram of Table 7.

Fig.8 Unsupervised domain accuracy sector diagram of Table 8.

Table 9 Influence of pseudo label clustering in low-frequency case study.

Table 10 Influence of pseudo label clustering in high-frequency case study.

Fig.9 Unsupervised domain accuracy histogram of Table 9.

Fig.10 Unsupervised domain accuracy histogram of Table 10.

Table 11 Accuracy of training dataset in source domain and target domain.

Fig.11 Accuracy histogram of Table 11.

5.Conclusions

In this paper, a source free unsupervised domain adaptation framework for EMA fault diagnosis is proposed, which can improve the domain adaptation ability in unlabeled target dataset while source domain dataset are no longer needed to participate in domain adaptation process due to their data privacy.(A)Under the distribution discrepancy challenge of variable working conditions of EMA, two case studies based on monitoring signal with different sampling frequencies but for the same failures are applied to show the effectiveness of the proposed method.(B) Further, the influence of supervised learning and pseudo label clustering are explored in different experiment, respectively, which indicate they are beneficial to improve the unsupervised domain adaptation.(C)The possible negative transfer reasons for unsupervised domain adaptation accuracy of high-frequency signals under trapezoidal command are discussed to indicate that the generalization ability of source model is critical to the proposed method.(D) Only reducing the global inter-domain distance between two domains may be not a good choice for EMA fault diagnosis.These comparison results can also provide a benchmark for future studies on unsupervised domain adaptation for EMA fault diagnosis.(E) In the future, how to fuse the multiple monitoring signals with different sampling frequencies, search optimal hyper-parameters for those methods and explore more effectiveness methods are noteworthy research hotspots in domain adaptation task of EMA fault diagnosis.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (No.52075349), the Aeronautical Science Foundation of China (No.201905019001), and the China Scholarship Council (No.202106240078).