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Fault diagnosis of stamping process based on empirical mode decomposition and learning vector quantization
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International Journal of Machine Tools & Manufacture 47 (2007) 2298–2306 www.elsevier.com/locate/ijmactool
Fault diagnosis of stamping process based on empirical mode decomposition and learning vector quantization A.M. Bassiunya, Xiaoli Lib,, R. Duc a
Department of Mechanical Engineering, Faculty of Engineering, Helwan University, Helwan-Cairo, Egypt Center for Networking Control and Bioinformatics, Yanshan University, Qinhuangdao 066004, PR China c Institute of Precision Engineering, The Chinese University of Hong Kong, Hong Kong
b
Received 7 February 2007; received in revised form 25 May 2007; accepted 2 June 2007 Available online 10 July 2007
Abstract Sheet metal stamping process is widely used in industry due to its high accuracy and productivity. However, monitoring the process is a difficult task since the monitoring signals are typically non-stationary transient signals. In this paper, empirical mode decomposition (EMD) is applied to extract the main features of the strain signals. First, the signal is decomposed by EMD into intrinsic mode functions (IMF). Then the signal energy and the Hilbert marginal spectrum, which reflects the working condition and the fault pattern of the process, are computed. Finally, to identify the faulty conditions of process, the learning vector quantization (LVQ) network is used as a classifier with the Hilbert marginal spectrum as the input vectors. The performance of this method is tested by 107 experiments derived from different conditions in the sheet metal stamping process. The artificially created defects can be detected with a success rate of 96.3%. The method seems to be useful to monitor a sheet metal stamping process in practice. r 2007 Elsevier Ltd. All rights reserved. Keywords: Stamping process; Empirical mode decomposition (EMD); Hilbert marginal spectrum; Learning vector quantization (LVQ)
1. Introduction Sheet metal stamping is an important process in modern manufacturing. Everyday, millions of parts are made by stamping. In automotive industry, for example, nearly all body panels are made by stamping. The stamping process, however, is very complicated and involves many factors such as the elastic and plastic deformation of sheet metals, the design, manufacturing and lubrication of the dies, static and dynamic behavior of the press, etc. [1]. Some of these factors are rather difficult to control. Moreover, a slight change in one of these factors, for example, the change of the material property, may cause defects even if the other process parameters remain unchanged [2]. Therefore, monitoring and diagnosis of the stamping process is of great practical importance. Quick and correct fault
Corresponding author.
E-mail addresses: [email protected] (A.M. Bassiuny), [email protected] (X. Li), [email protected] (R. Du). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.06.006
diagnosis helps not only to avoid product quality problems but also to protect the stamping press and the dies. Monitoring the stamping process, however, is rather difficult. This is because the stamping process is a highly non-linear transient process, and hence the simple threshold checking and FFT analysis just do not work [3]. In the past decade, many research works have been carried out on the analysis and monitoring of stamping processes. Generally, various sensors, such as force sensor, strain sensor and acceleration sensor can be used for monitoring this process. The use of acceleration signal offers a number of advantages [4]: easy to acquire, resistance to various environment hazards and it contains rich information about the process dynamics. The main limitation, however, is its sensitivity. Owing to the low signal and noise ratio, it is difficult to extract the features of the signal that are only related to the malfunctions of the stamping process. The strain signal, also referred to as the tonnage signal as it is proportional to the stamping force, is the most commonly used monitoring signal. In [1], the principal component analysis was used to extract features from the
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tonnage signal, followed by hierarchical classification and cross-validation. In [3], a hidden Markov model (HMM) based fault detection method is developed. The method uses a number of autoregressive (AR) models to model the monitoring signal in different time periods and uses the residues as the features. HMM was then applied for classification. AR-based feature extraction, however, results in small improvement. To improve the performance, the authors suggest finding more effective features and developing more efficient training algorithm. Research efforts have been devoted to fully utilizing the waveform features in the whole cycle of press tonnage signals to improve the diagnosis performance. For example, Jin and Shi [5] developed a feature preserving data compression method using wavelets. Zhou et al. [6] described a statistical process control monitoring system that integrates the statistical process control technique and the Haar wavelet transform. They applied this algorithm to diagnose the faults in die cushion. Ge et al. [7] applied support vector machine (SVM). In spite of its improvement, several samples are still misdiagnosed. They attribute this to the similarity of the time-domain signals under different conditions. Li and Du [8] proposed to use a latent process model, by which the important features were extracted, and then use generative topographic mapping for classification. The performance is slightly better. Based on the discussions above, it is seen that feature extraction is the key element for monitoring the stamping process. In this paper the main innovation is to extract features from the strain signal using the Hilbert–Huang transform (HHT). The empirical mode decomposition (EMD) is used to decompose the signal onto the intrinsic mode functions (IMF). The significant instantaneous amplitude and instantaneous frequency are then obtained for each IMF and a marginal spectrum is created to generate a feature vector. Finally, the learning vector quantization (LVQ) network is applied for fault classification. The rest of the paper is organized as follows. Section 2 is a brief introduction of the theory as well as the proposed fault diagnosis method based on HHT and LVQ is given. In Section 3, the results of applying the proposed fault diagnosis method to the stamping process is discussed and analyzed. The conclusion of this paper is given in Section 4.
2. The method 2.1. Empirical mode decomposition Empirical mode decomposition (EMD) was originally proposed by Huang et al. [9] to allow for the subsequent application of the Hilbert transform to a data set. EMD has been used to decompose a signal into a set of completely data-adaptive basis functions called intrinsic mode functions (IMFs) [10–12]. Given a data x(t), it is
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decomposed into a number of IMFs, ci [13]: xðtÞ ¼
n X
ci þ r n ,
(1)
i¼1
where, rn is the residue after n numbers of IMFs are extracted. IMFs are simple oscillatory functions with varying amplitude and frequency. It must have the following properties [14]: (a) the number of extrema and the number of zero crossings may differ by no more than one, and (b) the local average is zero (the local average is defined as the average of the maximum and minimum envelopes). A sifting process is used to obtain the separate components IMFs, ci, i ¼ 1,2,y,n. The details of the shifting process can be found in [14]. The computation of IMFs is as below: Step 1: Find the local extrema of x(t). Step 2: Find the maximum envelope ex(t) of x(t) by passing a natural cubic spline through the local maxima. Similarly find the minimum envelope em(t) with the local minima. Step 3: Compute an approximation to the local average, mðtÞ ¼ ½ex ðtÞ þ em ðtÞ=2. Step 4: Find the proto-mode function zi ðtÞ ¼ xðtÞ mðtÞ. Step 5: Check whether zi(t) is an IMF. If zi(t) is not an IMF, go to Step 2; If zi(t) is an IMF then set: c1 ðtÞ ¼ zi ðtÞ. As shown above, the name, sifting, indicates the process of removing the lowest frequency information until only the highest frequency remains. The sifting procedure performed on x(t) can then be performed on the residual x1(t) ¼ x(t)c1(t) to obtain x2(t) and c2(t). This procedure is repeated until all IMFs are found. 2.2. Hilbert marginal spectrum Having obtained all the IMFs, the Hilbert transform can be applied to each component and the instantaneous frequency can be computed. It is known that with the Hilbert transform, the data, x(t), can be expressed in the following form [14]: R n X X ðtÞ ¼ aj ðtÞei oj ðtÞ dt (2) j¼1
This expression gives both the instantaneous amplitude a(t) and the instantaneous frequency o(t) of each component as functions of time. Eq. (2) also enables us to visualize the amplitude and the instantaneous frequency as functions of time in a three-dimensional plot. This frequency–time distribution of the amplitude is designated as the Hilbert spectrum H(o,t). Moreover, the Hilbert marginal spectrum, h(o), is defined as [13]: Z T hðoÞ ¼ Hðo; tÞ dt. (3) 0
The marginal spectrum offers a measure of the total amplitude (or energy) contributed from each frequency
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component. It represents the cumulated amplitude over the entire spectrum in a probabilistic sense. 2.3. Learning vector quantization LVQ network is a supervised version of vector quantization. As shown in Fig. 1, the LVQ network has three layers: the input layer, the competitive layer of the hidden layer, and the output layer [15]. The neurons in the competitive layer are divided into n groups. Each group has the same number of neurons and corresponds to an output layer neuron. Classification information is stored in the weight matrix connecting between the input layer neurons and the competitive layer neurons. LVQ forms a quantized approximation of the distribution of an input data set using a finite number of reference vectors [16]. The LVQ algorithm belongs to a class of signal approximation methods that model the probability density function f(x), of some stochastic variable X 2 Rn , using a finite set of codebook vectors, mi 2 Rn , i ¼ 1,2,y,k, where the subscript i represents the hypothesis index in detection techniques. Once a set of codebook vectors are determined, the approximation of X implies finding the codebook mc closest to X in the input space for a given distance metric (typically the Lp space, with p ¼ 1,2,y,N). The determination of mc is achieved by the following decision process [16,17]:
An optimal selection of mi minimizes the average expected square of the quantization error, defined below: Z (5) E ¼ jjx mc jj2 f ðxÞ dx. The codebook vectors are recursively updated by minimizing Eq. (5) as follows: 8 mc ðkÞ þ aðkÞ½X ðkÞ mc ðkÞ; > > > > < X ; mc 2 S c ; c 2 ½1; k; mc ðk þ 1Þ ¼ mc ðkÞ aðkÞ½X ðkÞ mc ðkÞ; > > > > : mc 2 S c ; X 2 S¯ c 2 ½1; k; mi ðk þ 1Þ ¼ mi ðkÞ;
iac,
(6)
(7)
where, a, 0pa(t)p1, is a learning rate, S¯ c is the complement set of Sc. In this paper the adjustment of the learning rate follows the Kohonen rule [18], aðk þ 1Þ ¼
aðkÞ . 1 þ xaðkÞ
(8)
Here, x ¼ 1 if the classification is correct and x ¼ 1 if the pattern is misclassified. An online version of the LVQ algorithm is described in [19]. 2.4. The proposed fault diagnosis scheme-
jjx mc jj ¼ mini fjjx mi jjg. That is c ¼ arg mini fjjx mi jjg.
(4)
In summary, the proposed fault diagnosis method consists of the following steps: Step 1: Collect samples under different conditions. For diagnosis of the stamping process, three circumstances are investigated; these are normal operation, mis-feed, and material too thick. Step 2: Decompose the signals into IMFs and the residual. Only IMFs that are correlated with the data are used to extract the feature. Step 3: Calculate the Hilbert transform of each IMF. Step 4: Calculate the instantaneous frequency and instantaneous amplitude of the IMF components. Step 5: Calculate the Hilbert marginal spectrum to construct the feature vector. Step 6: Train the LVQ-network. For diagnosis of the stamping process, there are three classes: Class 0: normal condition, Class 1: mis-feed, and Class 2: too thick material. Step 7: Test samples to identify the fault class of the stamping process. 3. Experiments and results 3.1. Experimental setup
Fig. 1. Topology of the LVQ network.
A large number of experiments were carried out on a Cframe mechanical press (manufacturer: SEYI; model: SN125) shown in Fig. 2(a). The press is a 25-ton mechanical
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Fig. 2. (a) Experimental setup and (b) work part.
Fig. 3. Typical strain signal during normal blanking.
press with an automatic feeder. The signals are acquired from a Kistler piezoelectric strain sensor (model: 9232A). The sampling frequency is 1 kHz. The part studied in this paper is a mounting bracket of a desktop computer shown in Fig. 2(b). The operation is simple blanking and the workpiece material is mild steel. Based on empirical calculation, the estimated punch force is about 12 ton. 3.2. Results and analysis A typical strain signal is shown in Fig. 3. The process can be classified into three main stages. First, the workpiece undergoes elastic deformation for about 0.35 s. Then, the plastic deformation begins and continues till the develop-
Fig. 4. Strain signals of normal condition, mis-feed, and thickness increase.
ment of micro cracks and the part begins to break off. The last part of the signal corresponds to the vibration of the anvil. During the experiments, two types of faults are artificially created: mis-feed and too thick material. The mis-feed sometimes occurs in practice where the sheet is more or less fed into a die. This fault is derived from the feed system. The fault of thickness is derived from the sheet used, for instance, a sheet of the thickness of 1.2 mm is used in place of one of 1.1 mm. Fig. 4 shows the strain signals representing the normal condition and the two faults. It can be seen from Fig. 4 that
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Fig. 5. Intrinsic mode functions by applying the sifting process to a strain signal of normal operation using: (a) mean prediction, (b) pattern prediction and (c) the corresponding reconstruction error.
the differences between the three signals are very small and cannot be observed that makes monitoring of such cases are rather difficult. One of the main drawbacks of the EMD is the problem of end effect. It pertains to the difficulty in estimation of the bottom and top envelopes of a signal near the beginning or end of the signal. The envelopes are typically created using cubic spline interpolation, but at the endpoints there are not enough data to perform a cubic spline. Kizhner et al. [20] proposed three methods to solve this problem. The first one, pattern prediction, is based on statistical analyses of the extremas at the borders of the data set and estimates the distances and amplitudes of the predicted extrema based on those results. The second copies the extrema points closest to the data borders. The last method uses the mean of the last three extrema closest to the edge of the data. All these algorithms have their respective advantages and disadvantages. Fig. 5 shows the results of the sifting process applied to a typical strain signal acquired during normal operation. The results of decomposition using the mean prediction are
shown in Fig. 5(a) and that of using pattern prediction is shown in Fig. 5(b). The first line is the original signal. The rest of the time series is the intrinsic mode functions, ck, k ¼ 1,2,3,4, by applying the algorithms described above. The last time series corresponds to the residue. When the cks are added, the original signal is recovered. Four IMFs, in addition to the residue, with different time scales characterizes the signal in different resolution ratio. The top component represents the shortest time scale whereas the last one represents the largest time scale. This implies that transitory events in the time domain may be located using the finest scale. It should be noted that the effect of the sifting is filtering the data in a band-pass form. The results shown in Fig. 5 indicate that the original signal is not contained in one IMF alone. As EMD is a dyadic filter, any delta type of signal will be decomposed into many components. In our case, the original signal should be the sum of residue and two more components, c3 and c4 using any one of the two methods. Examination of the two decomposition methods reveals the original strain
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Fig. 6. Intrinsic mode functions by applying the sifting process to strain signals: (a) mis-feed and (b) too thick material.
Table 1 The correlation between the original signal and the IMFs components at different states Signal
IMF1 IMF2 IMF3 IMF4 Res
Normal
Mis-feed
Too-thick material
1
0.9988
0.9971
0.0318 0.1291 0.2555 0.8055 0.6854
0.0006 0.1683 0.5829 0.8081 0.6664
0.0275 0.1894 0.7513 0.7686 0.6154
signal can be reconstructed by adding the IMF components. The numerical difference between the sum of the all IMFs and the original data as shown in Fig. 5(c) is of the order of 1016 V in both cases. Fig. 6 shows the IMFs result from decomposing strain signals in the case of mis-feed and in the case of too thick material. Comparing these results to that shown in Fig. 5(a), it is seen that the last two IMFs and the residue are almost the same in the case of normal operation and misfeed while a slight change can be observed in the fourth component (c4) in the case of too thick material. To identify components which are strongly correlated with the original signal, Peng et al. [21] suggested the use of the correlation between the original signal and the IMFs, ci(t). A physically meaningful IMF will have significant
correlation with the raw data [22]. Define the correlation coefficient as follows: R xðtÞci ðtÞ dt , (9) RC i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R R x2 ðtÞ dt c2i ðtÞ dt where RCiA[0,1] is the correlation coefficient between the IMF component and the original signal. Table 1 shows correlation coefficients between the healthy data set and IMFs components of the three data sets. The results in the table show that the correlation coefficients increase with the mode number and attain a maximum value for mode number four for the three cases. The last mode and the residue are strongly correlated with the original healthy signal with correlation coefficients of 0.81 and 0.68, respectively. The results reveal that the original signal is not contained in one IMF. Only those IMFs that are correlated with the data are physically meaningful and should be included in the analysis. The original signal can be reconstructed from the IMF components starting from the longest to the shortest periods as shown in Fig. 7(a)–(d). The numerical difference between the sum of the all IMFs and the original data are given in Fig. 1(e). The magnitude of the error is only of the order of 1016 V. Fig. 8 shows the results of reconstructing the strain signals using only the last two modes and the residue where the signals can be recognized in comparison with the results shown in Fig. 4.
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When a fault occurs, it is expected that the energy distribution of the signal change. Examining this change leads us to compute the energy of the signal based on EMD
decomposition. The energy of each IMF is computed by the expression: E k ðtÞ ¼ 12a2k ðtÞ.
(10)
Moreover, the total signal energy as a function of time is the sum of the energies of all IMFs giving EðtÞ ¼
n X
E k ðtÞ,
(11)
k¼1
Fig. 7. (a)–(d) Reconstruction of the original signal from the decomposed components and (e) the difference between the data and the reconstructed signal from all IMF.
Fig. 8. Strain signal reconstructed using c4+c5+Res.
where n is the number of sifts. In our experiments, it was found that the energy index of the signal under the normal conditions (1308.5) is bigger than that of the other two faulty conditions, which were 597.9 and 947.7, respectively. This may be due to the fact that when a fault occurs, a number of corresponding resonance frequency components are exhibited. As a result, the energy index is reduced because the energy distributes mainly in the resonance frequency band. The obtained IMFs are then used to compute the Hilbert spectrum and hence the marginal spectrum for the three types of strain signal as shown in Fig. 9. The marginal Hilbert spectrum, h(o), has a different meaning than that of the Fourier spectrum. In the Fourier spectrum, the existence of energy at a frequency o means a component of a sine or a cosine wave persisted through the time span of the data. In Hilbert spectrum, the existence of energy at the frequency o means only that, in the whole time span of data, there is a higher likelihood for such a frequency component to have appeared locally. Hence, the Hilbert marginal spectrum gives a more precise spectrum, i.e. without energy spreading. The main frequency of the signal can be computed based on the marginal spectrum. Fig. 9(b) shows the Hilbert marginal spectrum of a signal in the case of mis-feed and Fig. 9(c) shows the Hilbert marginal spectrum of a signal in the case of material too thick. If h(o) is the marginal spectrum, then om is called the main frequency of x(t) if hðom ÞXhðoÞ 8o [22]. This index can be used for condition monitoring. For the example above, the main frequencies
Fig. 9. The Hilbert marginal spectrum of the process under different conditions.
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Fig. 10. Hilbert marginal spectrum of strain signals during different stamping operations.
for the normal condition and the two faults condition are 2, 4.3, and 4.78, respectively.
3.3. Classification In this paper a total of 107 data sets collected during stamping process are used for classification. The data represent three groups: 1—normal, 2—mis-feed and 3—too thick material. The fault diagnosis method described in Section 2 is applied where the energy indices defined in Eqs. (10) and (11) are computed for monitoring purposes. For fault classification, the marginal spectrums of each data set are computed with a frequency resolution of 200 and are used for feature extraction. Fig. 10 shows the Hilbert marginal spectrum for some of the data set representing the three groups of signals. The LVQ neural network is developed using MATLABs neural network toolbox [23]. The calculated Hilbert marginal spectrum is used to construct the feature vector, which is used as input for the LVQ network. During training, 53 data sets are used among which 30 were from normal condition, 12 were from mis-feed condition and the other 11 were from too-thick material conditions. The length of each data set is 500 samples. The primary factors, which controlled the behavior of the LVQ network, are the number of hidden units, the learning rate and the training time. Two different schemes for initial placement of the hidden units were tried: random placement and placement of all units at the mean value of the training set. The networks seemed to stabilize to the same solution regardless of the initialization of the hidden layer weights. The number of hidden units had a greater effect on the stabilization of network. Many tests are carried out starting with three hidden neurons for each class and computing the percentage of classifier success in each case. The best LVQ performance for this application was achieved with 24 hidden units trained at a learning rate of 0.001. The network is stabilized after 427 epochs as shown in Fig. 11. The classification is 100% successfully performed during training. Reduction of the learning rate
Fig. 11. Training performance of the LVQ network.
to 0.01 reduces the training time considerably but on the expense of the classification accuracy. For checking purposes, 54 data sets are used among of which 30 were from normal condition, 12 were from misfeed condition and the other 12 were from too-thick material conditions. The results show that only 3.7% of the data are misclassified. One sample of normal condition is classified as too thick and a mis-feed sample is classified as normal. 4. Conclusions In this paper, a new approach to detect and classify faults for sheet metal stamping process is presented. Based on the discussions above, following conclusions can be made: 1. The features of the monitoring signal can be found based on the empirical mode decomposition (EMD) and the Hilbert transform. The former decompose the
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3.
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monitoring signal into a number of intrinsic mode functions (IMFs) and a residue. Moreover, based on the IMFs, an energy index can be derived and used as monitoring index. Hilbert marginal spectrum provides another set of monitoring features. Experimental results indicate the energy index of the signal under the normal conditions is different from that of two faulty conditions. Thus the energy index of the strain signals can be used to monitoring the sheet metal stamping process. Taking the marginal spectrum as the input vector for the learning vector quantization (LVQ) network, the fault patterns of the stamping process can be effectively identified. Experimental results indicated that the best LVQ performance is achieved with 24 hidden units trained with 53 input vector presentations at a learning rate of 0.001. Using the proposed method, the stamping process can be diagnosed at a success rate of 100% for the training data and 96.7% for checking data. Diagnosing the faults in the stamping process is a difficult task as the monitoring signals (strain) are nearly inseparable. The successes of the presented method indicate its potential. It is expected that the new method can also be used to monitor various processes, in which the monitoring signal is transient and nonlinear.
Acknowledgments The authors would like to thank Dr. M. Ge for providing some of the experimental data. This work was partly supported by the National Natural Science Foundation of China (60575012). References [1] J. Jin, J. Shi, Diagnostic feature extraction from stamping tonnage signals based on design of experiments, Transactions of ASME, Journal of Manufacturing Science and Engineering 122 (2) (2000) 360–369. [2] C. Garcia, Artificial intelligence applied to automatic supervision, diagnosis and control in sheet metal stamping processes, Journal of Materials Processing Technology 164–165 (2005) 1351–1357. [3] M. Ge, R. Du, Y. Xu, Hidden Markov model based fault diagnosis for stamping processes, Mechanical Systems and Signal Processing 18 (2004) 391–408. [4] G. Zhang, M. Ge, H. Tong, Y. Xu, R. Du, Bispectral analysis for online monitoring of stamping operation, Engineering Applications of Artificial Intelligence 15 (2002) 97–104. [5] J. Jin, J. Shi, Automatic feature extraction of waveform signals for inprocess diagnostic performance improvement, Journal of Intelligent Manufacturing 12 (3) (2001) 257–268.
[6] S. Zhou, B. Sun, J. Shi, An SPC monitoring system for cycle-based waveform signals using Haar transform, IEEE Transactions on Automation Science and Engineering 3 (1) (2006) 60–72. [7] M. Ge, R. Du, G. Zhang, Y. Xu, Fault diagnosis using support vector machine with an application in sheet metal stamping operations, Mechanical Systems and Signal Processing 18 (2004) 143–159. [8] X. Li, R. Du, Condition monitoring using a latent process model with an application to sheet metal stamping processes, Transactions of the ASME, Journal of Manufacturing Science and Engineering 127 (2005) 376–385. [9] N. Huang, et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis, Proceedings of Royal Society of London A 454 (1971) 903–995 March 1998. [10] Y. Yu, Y. Dejie, C. Junsheng, A roller bearing fault diagnosis method based on EMD energy entropy and ANN, Journal of Sound and Vibration 294 (2006) 269–277. [11] A. Bassiuny, X. Li, Flute breakage detection during end milling using Hilbert–Huang transform and smoothed nonlinear energy operator, International Journal of Machine Tools and Manufacture 47 (2007) 1011–1020. [12] J.N. Yang, Y. Lei, S. Lin, N. Huang, Hilbert–Huang based approach for structural damage detection, Journal of Engineering Mechanics 130 (2004) 85–95. [13] Z. Wu, N.E. Huang, Ensemble empirical mode decomposition: a noise-assisted data analysis method, Centre for Ocean–Land–Atmosphere Studies, Technical Report no. 173, 2004. [14] N. Huang, M. Wu, W. Qu, S. Long, S. Shen, J. Zhan, Applications of Hilbert–Huang transform to non-stationary financial time series analysis, Applied Stochastic Models in Business and Industry 19 (361) (2003) 246–268. [15] M.-T. Vakil-Baghmisheh, N. Pavesic, Premature clustering phenomenon and new training algorithms for LVQ, Pattern Recognition (2003) 1901–1912. [16] T. Kohonen, Self-Organizing Maps, third ed., Springer, Heidelberg, Germany, 2001. [17] C. Borgelt, D. Girimonte, G. Acciani, Learning vector quantization: cluster size and cluster number, Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS), Vancouver, Canada, 2004, IEEE Press, Piscataway, NJ, USA, 2004. [18] S. Seo, K. Obermayer, Soft learning vector quantization, Neural Computation 15 (2003). [19] S. Bharitkar, D. Filev, An online learning vector quantization algorithm, Proceedings of the 6th IEEE International Symposium on Signal Processing and Application (ISSPA), Kuala Lumpur, August 2001. [20] S. Kizhner, P. Flatley, N. Huang, K. Blank, E. Conwell, D. Smith, On the Hilbert–Huang transform data processing system development, MAPLD International Conference, Ronald Reagan Building and International Trade Center, Washington, DC, September 9–11, 2003. [21] Z. Peng, F. Chu, P. Tse, Detection of the rubbing-caused impacts for rotor–stator fault diagnosis using reassigned scalogram, Mechanical Systems and Signal Processing 19 (2) (2005) 391–409. [22] R. Jha, D. Raju, A. Sen, Analysis of tokamak data using a novel Hilbert transform based technique, IPR/RR-356/2006, 2006. Available from: /http://www.ipr.res.in/S. [23] Neural Network Toolbox for use with MATLAB (R14) (2001), MathWorks, Inc.
International Journal of Machine Tools & Manufacture 47 (2007) 2298–2306 www.elsevier.com/locate/ijmactool
Fault diagnosis of stamping process based on empirical mode decomposition and learning vector quantization A.M. Bassiunya, Xiaoli Lib,, R. Duc a
Department of Mechanical Engineering, Faculty of Engineering, Helwan University, Helwan-Cairo, Egypt Center for Networking Control and Bioinformatics, Yanshan University, Qinhuangdao 066004, PR China c Institute of Precision Engineering, The Chinese University of Hong Kong, Hong Kong
b
Received 7 February 2007; received in revised form 25 May 2007; accepted 2 June 2007 Available online 10 July 2007
Abstract Sheet metal stamping process is widely used in industry due to its high accuracy and productivity. However, monitoring the process is a difficult task since the monitoring signals are typically non-stationary transient signals. In this paper, empirical mode decomposition (EMD) is applied to extract the main features of the strain signals. First, the signal is decomposed by EMD into intrinsic mode functions (IMF). Then the signal energy and the Hilbert marginal spectrum, which reflects the working condition and the fault pattern of the process, are computed. Finally, to identify the faulty conditions of process, the learning vector quantization (LVQ) network is used as a classifier with the Hilbert marginal spectrum as the input vectors. The performance of this method is tested by 107 experiments derived from different conditions in the sheet metal stamping process. The artificially created defects can be detected with a success rate of 96.3%. The method seems to be useful to monitor a sheet metal stamping process in practice. r 2007 Elsevier Ltd. All rights reserved. Keywords: Stamping process; Empirical mode decomposition (EMD); Hilbert marginal spectrum; Learning vector quantization (LVQ)
1. Introduction Sheet metal stamping is an important process in modern manufacturing. Everyday, millions of parts are made by stamping. In automotive industry, for example, nearly all body panels are made by stamping. The stamping process, however, is very complicated and involves many factors such as the elastic and plastic deformation of sheet metals, the design, manufacturing and lubrication of the dies, static and dynamic behavior of the press, etc. [1]. Some of these factors are rather difficult to control. Moreover, a slight change in one of these factors, for example, the change of the material property, may cause defects even if the other process parameters remain unchanged [2]. Therefore, monitoring and diagnosis of the stamping process is of great practical importance. Quick and correct fault
Corresponding author.
E-mail addresses: [email protected] (A.M. Bassiuny), [email protected] (X. Li), [email protected] (R. Du). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.06.006
diagnosis helps not only to avoid product quality problems but also to protect the stamping press and the dies. Monitoring the stamping process, however, is rather difficult. This is because the stamping process is a highly non-linear transient process, and hence the simple threshold checking and FFT analysis just do not work [3]. In the past decade, many research works have been carried out on the analysis and monitoring of stamping processes. Generally, various sensors, such as force sensor, strain sensor and acceleration sensor can be used for monitoring this process. The use of acceleration signal offers a number of advantages [4]: easy to acquire, resistance to various environment hazards and it contains rich information about the process dynamics. The main limitation, however, is its sensitivity. Owing to the low signal and noise ratio, it is difficult to extract the features of the signal that are only related to the malfunctions of the stamping process. The strain signal, also referred to as the tonnage signal as it is proportional to the stamping force, is the most commonly used monitoring signal. In [1], the principal component analysis was used to extract features from the
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tonnage signal, followed by hierarchical classification and cross-validation. In [3], a hidden Markov model (HMM) based fault detection method is developed. The method uses a number of autoregressive (AR) models to model the monitoring signal in different time periods and uses the residues as the features. HMM was then applied for classification. AR-based feature extraction, however, results in small improvement. To improve the performance, the authors suggest finding more effective features and developing more efficient training algorithm. Research efforts have been devoted to fully utilizing the waveform features in the whole cycle of press tonnage signals to improve the diagnosis performance. For example, Jin and Shi [5] developed a feature preserving data compression method using wavelets. Zhou et al. [6] described a statistical process control monitoring system that integrates the statistical process control technique and the Haar wavelet transform. They applied this algorithm to diagnose the faults in die cushion. Ge et al. [7] applied support vector machine (SVM). In spite of its improvement, several samples are still misdiagnosed. They attribute this to the similarity of the time-domain signals under different conditions. Li and Du [8] proposed to use a latent process model, by which the important features were extracted, and then use generative topographic mapping for classification. The performance is slightly better. Based on the discussions above, it is seen that feature extraction is the key element for monitoring the stamping process. In this paper the main innovation is to extract features from the strain signal using the Hilbert–Huang transform (HHT). The empirical mode decomposition (EMD) is used to decompose the signal onto the intrinsic mode functions (IMF). The significant instantaneous amplitude and instantaneous frequency are then obtained for each IMF and a marginal spectrum is created to generate a feature vector. Finally, the learning vector quantization (LVQ) network is applied for fault classification. The rest of the paper is organized as follows. Section 2 is a brief introduction of the theory as well as the proposed fault diagnosis method based on HHT and LVQ is given. In Section 3, the results of applying the proposed fault diagnosis method to the stamping process is discussed and analyzed. The conclusion of this paper is given in Section 4.
2. The method 2.1. Empirical mode decomposition Empirical mode decomposition (EMD) was originally proposed by Huang et al. [9] to allow for the subsequent application of the Hilbert transform to a data set. EMD has been used to decompose a signal into a set of completely data-adaptive basis functions called intrinsic mode functions (IMFs) [10–12]. Given a data x(t), it is
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decomposed into a number of IMFs, ci [13]: xðtÞ ¼
n X
ci þ r n ,
(1)
i¼1
where, rn is the residue after n numbers of IMFs are extracted. IMFs are simple oscillatory functions with varying amplitude and frequency. It must have the following properties [14]: (a) the number of extrema and the number of zero crossings may differ by no more than one, and (b) the local average is zero (the local average is defined as the average of the maximum and minimum envelopes). A sifting process is used to obtain the separate components IMFs, ci, i ¼ 1,2,y,n. The details of the shifting process can be found in [14]. The computation of IMFs is as below: Step 1: Find the local extrema of x(t). Step 2: Find the maximum envelope ex(t) of x(t) by passing a natural cubic spline through the local maxima. Similarly find the minimum envelope em(t) with the local minima. Step 3: Compute an approximation to the local average, mðtÞ ¼ ½ex ðtÞ þ em ðtÞ=2. Step 4: Find the proto-mode function zi ðtÞ ¼ xðtÞ mðtÞ. Step 5: Check whether zi(t) is an IMF. If zi(t) is not an IMF, go to Step 2; If zi(t) is an IMF then set: c1 ðtÞ ¼ zi ðtÞ. As shown above, the name, sifting, indicates the process of removing the lowest frequency information until only the highest frequency remains. The sifting procedure performed on x(t) can then be performed on the residual x1(t) ¼ x(t)c1(t) to obtain x2(t) and c2(t). This procedure is repeated until all IMFs are found. 2.2. Hilbert marginal spectrum Having obtained all the IMFs, the Hilbert transform can be applied to each component and the instantaneous frequency can be computed. It is known that with the Hilbert transform, the data, x(t), can be expressed in the following form [14]: R n X X ðtÞ ¼ aj ðtÞei oj ðtÞ dt (2) j¼1
This expression gives both the instantaneous amplitude a(t) and the instantaneous frequency o(t) of each component as functions of time. Eq. (2) also enables us to visualize the amplitude and the instantaneous frequency as functions of time in a three-dimensional plot. This frequency–time distribution of the amplitude is designated as the Hilbert spectrum H(o,t). Moreover, the Hilbert marginal spectrum, h(o), is defined as [13]: Z T hðoÞ ¼ Hðo; tÞ dt. (3) 0
The marginal spectrum offers a measure of the total amplitude (or energy) contributed from each frequency
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component. It represents the cumulated amplitude over the entire spectrum in a probabilistic sense. 2.3. Learning vector quantization LVQ network is a supervised version of vector quantization. As shown in Fig. 1, the LVQ network has three layers: the input layer, the competitive layer of the hidden layer, and the output layer [15]. The neurons in the competitive layer are divided into n groups. Each group has the same number of neurons and corresponds to an output layer neuron. Classification information is stored in the weight matrix connecting between the input layer neurons and the competitive layer neurons. LVQ forms a quantized approximation of the distribution of an input data set using a finite number of reference vectors [16]. The LVQ algorithm belongs to a class of signal approximation methods that model the probability density function f(x), of some stochastic variable X 2 Rn , using a finite set of codebook vectors, mi 2 Rn , i ¼ 1,2,y,k, where the subscript i represents the hypothesis index in detection techniques. Once a set of codebook vectors are determined, the approximation of X implies finding the codebook mc closest to X in the input space for a given distance metric (typically the Lp space, with p ¼ 1,2,y,N). The determination of mc is achieved by the following decision process [16,17]:
An optimal selection of mi minimizes the average expected square of the quantization error, defined below: Z (5) E ¼ jjx mc jj2 f ðxÞ dx. The codebook vectors are recursively updated by minimizing Eq. (5) as follows: 8 mc ðkÞ þ aðkÞ½X ðkÞ mc ðkÞ; > > > > < X ; mc 2 S c ; c 2 ½1; k; mc ðk þ 1Þ ¼ mc ðkÞ aðkÞ½X ðkÞ mc ðkÞ; > > > > : mc 2 S c ; X 2 S¯ c 2 ½1; k; mi ðk þ 1Þ ¼ mi ðkÞ;
iac,
(6)
(7)
where, a, 0pa(t)p1, is a learning rate, S¯ c is the complement set of Sc. In this paper the adjustment of the learning rate follows the Kohonen rule [18], aðk þ 1Þ ¼
aðkÞ . 1 þ xaðkÞ
(8)
Here, x ¼ 1 if the classification is correct and x ¼ 1 if the pattern is misclassified. An online version of the LVQ algorithm is described in [19]. 2.4. The proposed fault diagnosis scheme-
jjx mc jj ¼ mini fjjx mi jjg. That is c ¼ arg mini fjjx mi jjg.
(4)
In summary, the proposed fault diagnosis method consists of the following steps: Step 1: Collect samples under different conditions. For diagnosis of the stamping process, three circumstances are investigated; these are normal operation, mis-feed, and material too thick. Step 2: Decompose the signals into IMFs and the residual. Only IMFs that are correlated with the data are used to extract the feature. Step 3: Calculate the Hilbert transform of each IMF. Step 4: Calculate the instantaneous frequency and instantaneous amplitude of the IMF components. Step 5: Calculate the Hilbert marginal spectrum to construct the feature vector. Step 6: Train the LVQ-network. For diagnosis of the stamping process, there are three classes: Class 0: normal condition, Class 1: mis-feed, and Class 2: too thick material. Step 7: Test samples to identify the fault class of the stamping process. 3. Experiments and results 3.1. Experimental setup
Fig. 1. Topology of the LVQ network.
A large number of experiments were carried out on a Cframe mechanical press (manufacturer: SEYI; model: SN125) shown in Fig. 2(a). The press is a 25-ton mechanical
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Fig. 2. (a) Experimental setup and (b) work part.
Fig. 3. Typical strain signal during normal blanking.
press with an automatic feeder. The signals are acquired from a Kistler piezoelectric strain sensor (model: 9232A). The sampling frequency is 1 kHz. The part studied in this paper is a mounting bracket of a desktop computer shown in Fig. 2(b). The operation is simple blanking and the workpiece material is mild steel. Based on empirical calculation, the estimated punch force is about 12 ton. 3.2. Results and analysis A typical strain signal is shown in Fig. 3. The process can be classified into three main stages. First, the workpiece undergoes elastic deformation for about 0.35 s. Then, the plastic deformation begins and continues till the develop-
Fig. 4. Strain signals of normal condition, mis-feed, and thickness increase.
ment of micro cracks and the part begins to break off. The last part of the signal corresponds to the vibration of the anvil. During the experiments, two types of faults are artificially created: mis-feed and too thick material. The mis-feed sometimes occurs in practice where the sheet is more or less fed into a die. This fault is derived from the feed system. The fault of thickness is derived from the sheet used, for instance, a sheet of the thickness of 1.2 mm is used in place of one of 1.1 mm. Fig. 4 shows the strain signals representing the normal condition and the two faults. It can be seen from Fig. 4 that
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Fig. 5. Intrinsic mode functions by applying the sifting process to a strain signal of normal operation using: (a) mean prediction, (b) pattern prediction and (c) the corresponding reconstruction error.
the differences between the three signals are very small and cannot be observed that makes monitoring of such cases are rather difficult. One of the main drawbacks of the EMD is the problem of end effect. It pertains to the difficulty in estimation of the bottom and top envelopes of a signal near the beginning or end of the signal. The envelopes are typically created using cubic spline interpolation, but at the endpoints there are not enough data to perform a cubic spline. Kizhner et al. [20] proposed three methods to solve this problem. The first one, pattern prediction, is based on statistical analyses of the extremas at the borders of the data set and estimates the distances and amplitudes of the predicted extrema based on those results. The second copies the extrema points closest to the data borders. The last method uses the mean of the last three extrema closest to the edge of the data. All these algorithms have their respective advantages and disadvantages. Fig. 5 shows the results of the sifting process applied to a typical strain signal acquired during normal operation. The results of decomposition using the mean prediction are
shown in Fig. 5(a) and that of using pattern prediction is shown in Fig. 5(b). The first line is the original signal. The rest of the time series is the intrinsic mode functions, ck, k ¼ 1,2,3,4, by applying the algorithms described above. The last time series corresponds to the residue. When the cks are added, the original signal is recovered. Four IMFs, in addition to the residue, with different time scales characterizes the signal in different resolution ratio. The top component represents the shortest time scale whereas the last one represents the largest time scale. This implies that transitory events in the time domain may be located using the finest scale. It should be noted that the effect of the sifting is filtering the data in a band-pass form. The results shown in Fig. 5 indicate that the original signal is not contained in one IMF alone. As EMD is a dyadic filter, any delta type of signal will be decomposed into many components. In our case, the original signal should be the sum of residue and two more components, c3 and c4 using any one of the two methods. Examination of the two decomposition methods reveals the original strain
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Fig. 6. Intrinsic mode functions by applying the sifting process to strain signals: (a) mis-feed and (b) too thick material.
Table 1 The correlation between the original signal and the IMFs components at different states Signal
IMF1 IMF2 IMF3 IMF4 Res
Normal
Mis-feed
Too-thick material
1
0.9988
0.9971
0.0318 0.1291 0.2555 0.8055 0.6854
0.0006 0.1683 0.5829 0.8081 0.6664
0.0275 0.1894 0.7513 0.7686 0.6154
signal can be reconstructed by adding the IMF components. The numerical difference between the sum of the all IMFs and the original data as shown in Fig. 5(c) is of the order of 1016 V in both cases. Fig. 6 shows the IMFs result from decomposing strain signals in the case of mis-feed and in the case of too thick material. Comparing these results to that shown in Fig. 5(a), it is seen that the last two IMFs and the residue are almost the same in the case of normal operation and misfeed while a slight change can be observed in the fourth component (c4) in the case of too thick material. To identify components which are strongly correlated with the original signal, Peng et al. [21] suggested the use of the correlation between the original signal and the IMFs, ci(t). A physically meaningful IMF will have significant
correlation with the raw data [22]. Define the correlation coefficient as follows: R xðtÞci ðtÞ dt , (9) RC i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R R x2 ðtÞ dt c2i ðtÞ dt where RCiA[0,1] is the correlation coefficient between the IMF component and the original signal. Table 1 shows correlation coefficients between the healthy data set and IMFs components of the three data sets. The results in the table show that the correlation coefficients increase with the mode number and attain a maximum value for mode number four for the three cases. The last mode and the residue are strongly correlated with the original healthy signal with correlation coefficients of 0.81 and 0.68, respectively. The results reveal that the original signal is not contained in one IMF. Only those IMFs that are correlated with the data are physically meaningful and should be included in the analysis. The original signal can be reconstructed from the IMF components starting from the longest to the shortest periods as shown in Fig. 7(a)–(d). The numerical difference between the sum of the all IMFs and the original data are given in Fig. 1(e). The magnitude of the error is only of the order of 1016 V. Fig. 8 shows the results of reconstructing the strain signals using only the last two modes and the residue where the signals can be recognized in comparison with the results shown in Fig. 4.
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When a fault occurs, it is expected that the energy distribution of the signal change. Examining this change leads us to compute the energy of the signal based on EMD
decomposition. The energy of each IMF is computed by the expression: E k ðtÞ ¼ 12a2k ðtÞ.
(10)
Moreover, the total signal energy as a function of time is the sum of the energies of all IMFs giving EðtÞ ¼
n X
E k ðtÞ,
(11)
k¼1
Fig. 7. (a)–(d) Reconstruction of the original signal from the decomposed components and (e) the difference between the data and the reconstructed signal from all IMF.
Fig. 8. Strain signal reconstructed using c4+c5+Res.
where n is the number of sifts. In our experiments, it was found that the energy index of the signal under the normal conditions (1308.5) is bigger than that of the other two faulty conditions, which were 597.9 and 947.7, respectively. This may be due to the fact that when a fault occurs, a number of corresponding resonance frequency components are exhibited. As a result, the energy index is reduced because the energy distributes mainly in the resonance frequency band. The obtained IMFs are then used to compute the Hilbert spectrum and hence the marginal spectrum for the three types of strain signal as shown in Fig. 9. The marginal Hilbert spectrum, h(o), has a different meaning than that of the Fourier spectrum. In the Fourier spectrum, the existence of energy at a frequency o means a component of a sine or a cosine wave persisted through the time span of the data. In Hilbert spectrum, the existence of energy at the frequency o means only that, in the whole time span of data, there is a higher likelihood for such a frequency component to have appeared locally. Hence, the Hilbert marginal spectrum gives a more precise spectrum, i.e. without energy spreading. The main frequency of the signal can be computed based on the marginal spectrum. Fig. 9(b) shows the Hilbert marginal spectrum of a signal in the case of mis-feed and Fig. 9(c) shows the Hilbert marginal spectrum of a signal in the case of material too thick. If h(o) is the marginal spectrum, then om is called the main frequency of x(t) if hðom ÞXhðoÞ 8o [22]. This index can be used for condition monitoring. For the example above, the main frequencies
Fig. 9. The Hilbert marginal spectrum of the process under different conditions.
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Fig. 10. Hilbert marginal spectrum of strain signals during different stamping operations.
for the normal condition and the two faults condition are 2, 4.3, and 4.78, respectively.
3.3. Classification In this paper a total of 107 data sets collected during stamping process are used for classification. The data represent three groups: 1—normal, 2—mis-feed and 3—too thick material. The fault diagnosis method described in Section 2 is applied where the energy indices defined in Eqs. (10) and (11) are computed for monitoring purposes. For fault classification, the marginal spectrums of each data set are computed with a frequency resolution of 200 and are used for feature extraction. Fig. 10 shows the Hilbert marginal spectrum for some of the data set representing the three groups of signals. The LVQ neural network is developed using MATLABs neural network toolbox [23]. The calculated Hilbert marginal spectrum is used to construct the feature vector, which is used as input for the LVQ network. During training, 53 data sets are used among which 30 were from normal condition, 12 were from mis-feed condition and the other 11 were from too-thick material conditions. The length of each data set is 500 samples. The primary factors, which controlled the behavior of the LVQ network, are the number of hidden units, the learning rate and the training time. Two different schemes for initial placement of the hidden units were tried: random placement and placement of all units at the mean value of the training set. The networks seemed to stabilize to the same solution regardless of the initialization of the hidden layer weights. The number of hidden units had a greater effect on the stabilization of network. Many tests are carried out starting with three hidden neurons for each class and computing the percentage of classifier success in each case. The best LVQ performance for this application was achieved with 24 hidden units trained at a learning rate of 0.001. The network is stabilized after 427 epochs as shown in Fig. 11. The classification is 100% successfully performed during training. Reduction of the learning rate
Fig. 11. Training performance of the LVQ network.
to 0.01 reduces the training time considerably but on the expense of the classification accuracy. For checking purposes, 54 data sets are used among of which 30 were from normal condition, 12 were from misfeed condition and the other 12 were from too-thick material conditions. The results show that only 3.7% of the data are misclassified. One sample of normal condition is classified as too thick and a mis-feed sample is classified as normal. 4. Conclusions In this paper, a new approach to detect and classify faults for sheet metal stamping process is presented. Based on the discussions above, following conclusions can be made: 1. The features of the monitoring signal can be found based on the empirical mode decomposition (EMD) and the Hilbert transform. The former decompose the
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2.
3.
4.
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monitoring signal into a number of intrinsic mode functions (IMFs) and a residue. Moreover, based on the IMFs, an energy index can be derived and used as monitoring index. Hilbert marginal spectrum provides another set of monitoring features. Experimental results indicate the energy index of the signal under the normal conditions is different from that of two faulty conditions. Thus the energy index of the strain signals can be used to monitoring the sheet metal stamping process. Taking the marginal spectrum as the input vector for the learning vector quantization (LVQ) network, the fault patterns of the stamping process can be effectively identified. Experimental results indicated that the best LVQ performance is achieved with 24 hidden units trained with 53 input vector presentations at a learning rate of 0.001. Using the proposed method, the stamping process can be diagnosed at a success rate of 100% for the training data and 96.7% for checking data. Diagnosing the faults in the stamping process is a difficult task as the monitoring signals (strain) are nearly inseparable. The successes of the presented method indicate its potential. It is expected that the new method can also be used to monitor various processes, in which the monitoring signal is transient and nonlinear.
Acknowledgments The authors would like to thank Dr. M. Ge for providing some of the experimental data. This work was partly supported by the National Natural Science Foundation of China (60575012). References [1] J. Jin, J. Shi, Diagnostic feature extraction from stamping tonnage signals based on design of experiments, Transactions of ASME, Journal of Manufacturing Science and Engineering 122 (2) (2000) 360–369. [2] C. Garcia, Artificial intelligence applied to automatic supervision, diagnosis and control in sheet metal stamping processes, Journal of Materials Processing Technology 164–165 (2005) 1351–1357. [3] M. Ge, R. Du, Y. Xu, Hidden Markov model based fault diagnosis for stamping processes, Mechanical Systems and Signal Processing 18 (2004) 391–408. [4] G. Zhang, M. Ge, H. Tong, Y. Xu, R. Du, Bispectral analysis for online monitoring of stamping operation, Engineering Applications of Artificial Intelligence 15 (2002) 97–104. [5] J. Jin, J. Shi, Automatic feature extraction of waveform signals for inprocess diagnostic performance improvement, Journal of Intelligent Manufacturing 12 (3) (2001) 257–268.
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