On-line grinding chatter detection based on minimum entropy deconvolution and autocorrelation function

On-line detection of chatter is one of the key techniques to avoid the harmful effects caused by chatter in grinding process. The key to chatter detection is to capture reliable chatter features and thresholds. To achieve this, it is important to make clear and extract the essential characteristics of the grinding chatter signal, which has not yet been well studied. In this paper, we are going to investigate the essential characteristics of the grinding chatter signal and propose a new approach for on-line detection of grinding chatter. The proposed approach for on-line detection of grinding chatter is based on minimum entropy deconvolution and autocorrelation function, in which the minimum entropy deconvolution is employed to deconvolve the effect of transmission path, and further to restore the essential characteristics of the chatter signals. To eliminate the interference of the non-periodic impulse signals in the measured vibration signals, an autocorrelation function is introduced. Kurtosis is employed to indicate chatter according to the changes of the processed signal. The validity of the proposed method is demonstrated through the measured vibration signals obtained from grinding processes and the presented chatter detection index is independent from the grinding conditions with excellent detection accuracy and permissible computational efficiency. This demonstrates the effectiveness of proposed method in on-line implementation.

Typically, the chatter encountered in machining is of the regenerative type, and it is the focus of the remainder of this paper [1]. In the grinding process, the occurrence of chatter can be particularly critical since it can detrimentally affect the form accuracy and surface finish of the ground workpieces [4][5][6]. On-line identification and isolation of the onset of chatter can prevent machining processes from the occurrence of these defects [7]. Given the chatter features are submerged by forced vibration and noise in the initial stage of chatter occurrence, the key point to realize early chatter identification is to extract the essential characteristics of the chatter signals. To a great extent, these demands rely on the measured signal type and feature extraction technique.
Over the years, many kinds of sensor signals have been used to monitor chatter, such as acceleration signal [8], force [6], sound [9], motor current [10][11][12], torque signal [9], and acoustic emission signal [13]. In [9], several sensors were compared to determine which signals are most sensitive to chatter onset by Kuljanic. The results indicate that force signals are more sensitive to chatter than other sensor signals since the time-varying cutting force is the root of chatter. However, it is difficult to apply the force transducer in industrial conditions since it is not compatible with the tool changer, which may reduce the system stiffness. However，the acceleration sensor is usually used for flutter detection as it is easy to install and higher the signal-to-noise ratio of the signal. In this paper, the acceleration signals of the machine tool are used for chatter detection. No matter what kind of sensors are chosen, the feature extraction technique is much more important, and it is the focus of the remainder of this paper. At present, many feature extraction methods have been developed for chatter detection, and these detection methods can be classified into two kinds. One is based on the changes in the characteristics of feature signals, which are sensitive to chatter occurrence, and the other is based on the changes in machining dynamics caused by the onset of chatter. For the former one, spectral analysis [1,14], wavelet [7,8,10], correlation analysis [12], ensemble empirical mode decomposition [15,16], and Hilbert-Huang transform [8], variational mode decomposition [17,18] and are introduced to identify chatter according to the change of timefrequency characteristics of the measured signals. The latter chatter detection principle uses coarsegrained entropy rates [19][20][21], coarse-grained information rate [6], summed non-stationary wavelet bispectrum feature [11] as the indicators which reflect the change of the dimensional dynamics during machining. A review of other methods for chatter detection is given in [22].
Although many feature extraction methods have been used for chatter detection, the chatter threshold selected of existing methods is empirical and not valid over a wide range of processing conditions. Therefore, these methods are mostly not suitable for on-line chatter detection.
To solve the problem that the chatter threshold cannot be easily determined. It is necessary to reveal the essential characteristics of the measured chatter signal. Chatter is a kind of self-excited vibration, and the main source of the self-excited chatter vibrations in grinding is also due to the regenerative effect [22][23][24]. The theory of regenerative chatter explains that the machine structure is excited in the range of its dominant natural frequencies, which are reflected in the form of waviness on the surface of the workpiece. When the next part of the grinding wheel is grinding on this surface, it leads to a renewed excitation of the machine structure. The process becomes unstable when the damping in the system is insufficient. In [23], the simulation result shows that grinding force is a periodic impact signal when grinding chatter occurred, and the period of grinding force is equal to the rotational period of the spindle. Besides, a lot of research efforts have indicated that chatter frequencies are closed to the mechanical natural frequencies of machine tools, and more than one dominant frequency can appear, each of which is approximately a multiplicity of the wheel rotational frequency; if machine operation is stable, these frequencies do not arise [22,24,25].
Consequently, according to the previous chatter studies, vibration mechanics, and digital signal processing (Fast Fourier Transform and convolution properties), we can conclude that the measured chatter signal should be a convolution result of the periodic impact forces and natural frequencies of machine tools. To better understand the above content, the production process of a simplified chatter signal can be described as in Fig.1. The remainder of the paper is organized as follows: Section 2 describes the theoretical background, Section 3 addresses the anti-interference capability of the MED method will be discussed. Section 4 presents the effectiveness of the proposed approach with validation results using various experimental data. Section 5 concludes this paper.

Theoretical background 2.1 Minimum entropy deconvolution (MED)
MED was firstly proposed for application in analyzing seismic recordings by Ralph Wiggins in 1978 [26]. And it has been evaluated for its effectiveness in extracting the hidden impulse signals from a mixture of response signals [27][28][29]. The basic idea of MED is to find an inverse filter that counteracts the effect of the transmission path [26,30]. It is designed to reduce the spread of the impulse response signal, and then obtain the signal which is closer to the original impulse signal. Fig.2 illustrates the basics of the MED method. The impulse signal ( ) passes through the structural filter ℎ whose output is mixed with noise ( ) to give the measured output ( ). This is a convolution process of the impulse signal and the resonance frequency band of the component.
The inverse filter produces output ( ), which has to be as close as possible to the original input ( ). This process eliminates the influence of the structure resonance. The input ( ) is unknown, so it is assumed to be as impulsive as possible. The inverse filter f is modeled as an FIR filter with L coefficients and we have where ( ) has to invert the system IRF ℎ such as: The delay is such that the inverse filter can be causal. It will displace the whole signal by but will not change the pulse spacing.
The inverse filter was implemented by maximizing the kurtosis of the output signal ( ) [19,24]. The kurtosis is taken as the normalized fourth-order moment given by: and the maximum kurtosis of ( ) can be obtained according to ( ) for which the derivative of the objective function is zero such as: Details of the MED method could be founded in [26,31].

Autocorrelation function
The autocorrelation function is an important diagnostic tool for analyzing time series in the time domain. The autocorrelation function [31], as defined by Equation (11), is the average product of the sequence x( ) with a time-shifted m, version of itself.
In this paper, unbiased autocorrelation is employed to extract periodic impulse response signals from a filtered signal contains a bigger non-periodic impact response signal. The unbiased autocorrelation function [31], as defined by Equation (12).
3.The anti-interference capability of the MED method The MED method has shown its effectiveness in fault diagnosis of gears and bearings [27]. But for on-line chatter detection, the robustness of the proposed method is very important. Here, the anti-interference capability of the MED method will be discussed.

Simulated signal
In the machining process, there are usually some non-periodic interference signals in the measured vibration signals. A simulated signal is generated according to the characteristics of the chatter signal, and the simulated signal contains a bigger non-periodic impact signal. The simulated signal can be expressed by Equation (13).
The simulated signal ( ) is composed of five terms. The first term represents a series of the periodic impulse response signals excited by chatter, it is a convolution result of the periodic impact signal and natural frequencies of machine tools, where is the amplitude of the periodic impulse signal and is the rotation period of the grinding wheel, is a resonance damping coefficient depending on the exciting structure, is the natural frequency of the exciting structure. The second term represents a bigger non-periodic impact response signal from unknown sources, it is a convolution result of the non-periodic impact signal and unknown natural frequencies. The third and fourth terms represent the fundamental frequency, is the fundamental frequency. The fifth terms denote the measurement noise. The parameters of simulated signals ( ) chosen in Table 1: In this simulation, the signal to noise ratio (SNR) of the periodic impulse response signal is −5.74 dB. The sampling frequency is 2048 Hz and the time length of data is 1.5 s. Produce process of the simulated signal is shown in Fig.2.   Fig.2(a) shows the convolution process weakens the impact characteristics of the simulated signal, and Fig.2(b) shows the fundamental frequency and white noise further weaken the impact characteristics of the simulated signal.

Analysis of the anti-interference capability of the MED
Based on the analysis above, parameters are all selected as recommended values of the MED method [27] in this paper, the length of the filter was selected at 30, the termination number of iterations was selected at 30, and termination threshold of iterations was selected at 0.01. Fig.3 depicts the simulated signal filtered with the MED method. It can be seen in Fig.3 that periodic impulse characteristics can be observed after the MED method. It demonstrates the MED method has the anti-interference ability in filtering chatter signal.

Extract periodic impulse characteristics based on unbiased autocorrelation
To eliminate the interference of the non-periodic impulse signals, the unbiased autocorrelation function can be used to extract periodic impulse characteristics from the measured signals. Figure   4 shows the unbiased autocorrelation of two signals.

4.The procedure of the chatter recognition method
The essential characteristics of the measured chatter signal have been discussed and conclude that the measured chatter signal is a convolution result of the periodic impact forces and the natural frequencies of machine tools. To reduce the spread of the convolution process, the MED is employed to deconvolve the effect of the transmission path, and restore the impact feature of the chatter signals.
The autocorrelation function is introduced to extract periodic impulse characteristics. Kurtosis is employed to indicate chatter.
Kurtosis is a measure of the heaviness of the tails in the distribution of the signal ( ). Outlier Step 1: Seek each raw vibration signal from the raw signal using a rectangular window function.
Step 2: Deconvolve the effect of the transmission path of each vibration signal using MED, and restore the essential characteristics of the chatter signals.
Step 3: Extract periodic impulse signals using unbiased autocorrelation, and the unbiased autocorrelation sequences should be selected to describe the characteristics of periodic impulse in a length T/5,t=T/5,...,2T/5.
Step 4: Calculate the kurtosis of each unbiased autocorrelation sequence ( ).
Step 5. Chatter recognition through comparing with threshold. Chatter state normally has a value of greater than 3, and a proper threshold should be selected for more accurate chatter detection according to the analysis results of experimental data in the next section.

Experimental setup
As shown in Fig.6, the experimental platform is a numerical control worm wheel gear grinding machine, the working principle of this machine is similar to the gear hobbing machine. The B axis is the grinding wheel spindle, and the C axis is the workpiece spindle. X-axis, Y-axis, and Z-axis are the linear feed axis. A one-dimensional acceleration sensor is installed on the top right side of the grinding wheel spindle housing, and Z-axis vibration can be measured. Vibration signals were sampled at 2560Hz by asynchronous data acquisition card and stored on a PC (Pentium(R) 2.93GHz and 2G RAM) for further analysis, which was carried out in the MathWorks MATLAB environment.
Accelerometer Data acquisition card Computer Fig.6 Experimental platform In the monitoring process, a cylindrical gear part profile is ground. The modulus of the gear is 3mm with 70 teeth, and the width of the gear is 45 mm. The main process parameters are shown in Table 1. The mechanical natural frequencies of the grinding carriage are measured by impact hammer testing, Fig.7 illustrates the natural frequencies of the grinding carriage, it can be noted that there are two resonance frequency bands.

Application Results
In this section, the proposed method is applied to the chatter detection of grinding processes under various conditions. Two practical cases are considered to validate the proposed method for chatter detection, and the parameters of the MED method were all selected in section 3.2.
(1) Case 1 In this case, the proposed method is applied to chatter detection during the coarse grinding processes. The aim is to verify the effectiveness of the proposed method. The grinding parameters selected in this study are summarized in Table 4. The measured signals were divided into four parts according to the grinding strokes. The time waveform and short-time Fourier transform(STFT) spectrograms of the measured signals during the coarse grinding processes are shown in Fig.8.  which includes four stages. From Fig.8 (a), it can be seen that the grinding process seems to be stable at each stage, and some bigger non-periodic impact interference signals can be observed at stage one and stage two. Besides, the details of Fig.8 (a) shows that there are no periodic impact components in the raw measured signal. Fig.8 (b) and (c) show the STFT spectrograms of the measured signal, color represents amplitude in the time-frequency spectrum, red and blue respectively represent maximum and minimum values. From Fig.8 (b) and (c), the following phenomena can be observed: more than one dominant frequency appears at stage two, stage three, and stage four, and each equally spaced frequency is equal to the rotating frequency, and these frequencies are in the range of two natural frequency bands. What's more, it can be seen that the amplitude of dominant frequency increases significantly at stage two, stage three, and stage four in Fig.8 (b)and (c). According to the characteristics of the chatter signal discussed in Section 1 and the observed phenomena in Fig.8 (b) and (c), it demonstrates that grinding chatter occurred at stage two, stage three, and stage four.
Furthermore, the proposed method was used to detect chatter. As shown in Fig.9,  Fig.9 (a) shows the unbiased autocorrelation of the MED filtered signal. The details of Fig.9(a) depicts that there is no impact component at stage one, and it demonstrates that the grinding process is stable. Besides, the details of Fig.9 (a) depicts that there are many obvious periodic impact components at stage two, stage three, and stage four, and it demonstrates that these grinding processes are unstable. Fig.9 (b) shows the kurtosis ( ) of the unbiased autocorrelation sequences, and the red dashed line represents the kurtosis is 3. From Fig.9 (b), it can be seen that the value variations are consistent with the chatter state. The value is about 3when the grinding process is stable, and the is greater than 3 when chatter occurred. It demonstrates that the proposed method can effectively detect chatter, and 5 can be easily chosen to be a proper threshold value in this process.
(2) Case 2 In this case, the proposed method is applied to chatter detection during the semi-finish and finish grinding processes. The aim is to verify the effectiveness of the proposed method. The grinding parameters selected in this study are summarized in Table 4. The measured signals were divided into two parts according to the grinding strokes. The time waveform and short-time Fourier transform(STFT) spectrograms of the measured signals during the semi-finish and finish grinding processes are shown in Fig.10.  Fig.10 (a) shows the time waveform of the measured vibration signal in the grinding process. Fig.10 (b) and (c) show the STFT spectrum of the measured signals, and the chatter characteristics similar to Fig.8 can also be observed. Fig.10 (b) and (c) demonstrate that grinding chatter occurred in the finish grinding process, and the semi-finish grinding processes are in a stable state.
In the following, the proposed method is applied. A window size of =1280 samples were used to truncate the raw vibration signal, and the calculation time of each sequence is about 0.20 seconds. The time waveforms and kurtosis ( ) of the preprocessed signal are shown in Fig.11.  Fig.11 (a) shows the unbiased autocorrelation of the MED filtered signal. The details of Fig.11 (a) depict that there are no impact components when the grinding process is stable, and the periodic impulse characteristics is quite significant when the grinding process is unstable. Fig.11 (b) shows the kurtosis( ) of the unbiased autocorrelation sequences, and the red dashed line represents the kurtosis is 3. The value variations are consistent with the chatter state. The value is about 3 when the grinding process is stable, and the is greater than 3 when chatter occurred. It demonstrates that the proposed method can effectively detect chatter, and 5 can be easily chosen to be a proper threshold value in this process.
From the results above, it can be shown that the proposed approach can effectively detect chatter at the onset of chatter, and a chatter threshold can be easily chosen according to the distribution characteristics of impulse signals.

Conclusions
In this paper, the essential characteristics of the measured chatter signal were discussed. And