Tool wear monitoring based on wavelet packet coefficient and hidden Markov model*

Ying QIU，Feng-yun XIE

School of Mechanical and Electronical Engineering，East China Jiaotong University，Nanchang 330013，China

Tool wear monitoring is crucial in order to prevent tool failures during the automation machining process.However，the on-line tool wear monitoring is not an easy task due to the complexity of the process.For many years，lots of scholars have studied tool wear monitoring by various methods.There are important contributions presented for condition monitoring，for instance，on-line tool monitoring by using Artificial intelligence was presented by Vallejo［1］，a method of state recognitions based on wavelet and hidden Markov model(HMM)was presented by Xie［2］.On-line condition monitoring based on empirical mode decomposition and neural network was proposed by Xie［3］.A prediction tool wear in machining processes based on ANN was proposed by Haber et al［4］.A new hybrid technique for cutting tool wear monitoring，which fuses a physical process model with an artificial neural networks(ANN)model，is proposed for turning by Sick［5］.However，ANN in tool wear monitoring requires a lot of empirical data for the learning algorithm.Otherwise，it will reduce the recognition rate of the tool wear.

In this paper，an approach based on wavelet packet coefficient and HMM for tool wear monitoring is proposed.In order to monitor the tool wear states in machining process，the dynamometer is used for data acquisition.The wavelet packet decomposition is adopted for data processing.The root mean square(RMS)of the wavelet packet coefficients at different scales are taken as the feature observations vector.The HMM is used to recognize the states of tool wear.The results show that the proposed method has a relatively high recognition rate.

1.Introduction

1.1.Wavelet packet analysis

Wavelet packet decomposes the lower as well as the higher frequency bands and leads to a balanced binary tree structure.Wavelet Packet could be defined as:

Where，hl-2kand gl-2kare called as orthogonal mirror filter，the function series W(2-jt-k)is called as orthogonal wavelet packet.

Wavelet packet function is defined as

Where，N is the set of positive integers and Z is the set of integers;n is the oscillation parameter;j and k are the frequency localization parameters and the time localization parameter，respectively.

The first two wavelet packet functions are defined as:

Where，φ(t)is the scaling function，and Ψ(t)is the wavelet function.h1and g1are the low-pass and high-pass filters.The function 2-j/2Wn(2-jt-k)is localized both in time and frequency.Each of them is a function of unit energy，with a scale of 2j，centered at 2jk， and with an oscillation parameter n， i.e.，.

The basic wavelet function Ψ(t)is defined as:

Where，a，b ∈L2(R)(square-integrable space)，a≠0.Parameter a is called as scale parameter，which is related to the frequency.Parameter b is called as position parameter，which determines the time-domain orspace-domain information in the transformed results.

The diagram of this algorithm is shown in Figure 1，where，A and D are the wavelet packet coefficients［6］.

Figure 1.Wavelet packet decomposition tree at level 3

When sampling frequency 2fsis adopted，the different frequency bands range by three layers of wavelet packet decomposition could be shown in Table 1.

The decomposition coefficients of a signal f(t)into Wavelet Packet are computed by applying the low-pass and high-pass filters iteratively.The decomposition coefficients are defined as:

Table 1.Different frequency bands range

1.2.Hidden Markov model

HMM is an extension of Markov chains.Unlike Markov chains，HMM is doubly stochastic process，i.e.，not only is the transition from one state to another state stochastic，but the output symbol generated at each state is also stochastic.Thus the model can only be observed through another set of stochastic processes.The actual sequences of states are not directly observable but are“hidden”from observer.A HMM are illustrated in Figure 2.

Figure 2.Hidden Markov model

The hidden states{qt}can not be directly observed，but could be inferred indirectly by the observation sequence O=(o1，o2，…，oT).A HMM has the following elements.The N possible values of the hidden states are S={S1，S2，…，SN}.The state variable at time t is denoted as qt.The M possible distinct observation symbols per state are V={v1，v2，…，vM}.The parameters of an HMM are usually denoted as λ=A，B，{}π.A=(aij)N×Nis the state transition probability matrix，where，aij=P(qt+1=Sj|qt=Si)(1≤i，j≤N)is the probability of transition from the state i to the state j.B=(bj(k))N×Mis the observation probability matrix，where，bj(k)=P(ot=vk|qt=Sj)(1≤j≤N，1≤k≤M).πi=P(q1=Si)(1≤i≤N)is the initial state probability distribution.The engineering problem may be solved by the HMM basic algorithms，i.e.，the evaluation，decoding，and training algorithms［7］.

2.Experiment and feature extraction

The experimental setup used in this study is illustrated in Figure 3.The cutting tests are conducted on five-axis machining center Mikron UCP800 Duro.The thrust force is measured by a Kistler 9253823 dynamometer.The force signals are amplified by Kistler multichannel charge amplifier 5070 and simultaneously recorded by NI PXIe-1802 data recorder with 5 kHz sampling frequency.The collected signals are displayed by Cathode ray tube CRT.The workpiece is continuously processed under different processing conditions until the obvious cutting tool wear is observed.

Figure 3.Experimental setup for cutting processing

The tool wear states are classified into three categories:the initial processing status of the tool is named as sharp state(pattern 1)，the wear processing status of the tool is named as wear state(pattern 3)，and the status between sharp state and wear state is named as slight wear state(pattern 2)［8］.

The real-time cutting processing signals under different cutting tool condition are shown in Figure 4.Signal I represents the sharp cutting tool condition.Signal II represents the slight wear cutting tool condition.Signal III represents the wear cutting tool condition.By using the fast Fourier transforms(FFT)processing，the time domain signals are shown in Figure 5.We can see that the time-frequency amplitude is different significantly for these three wear states.

Figure 4.Dynamometer signals

Figure 5.The chart of frequency spectrum

A four-level wavelet packet decomposition is used in this paper.The root mean square(RMS)of the wavelet coefficients at different scales is shown in Figure 6.It could be found that RMS results are significantly different for these three states.The RMS of the wavelet coefficients at different scales are taken as the feature observations vector.

Figure 6.The RMS of wavelet coefficient in three wear states

3.Tool wear monitoring

Flow chart of the tool states recognition based on HMM is shown in Figure 7.It is composed of the wavelet-based feature extraction and the RMS of the wavelet coefficients for HMM input.Each HMM pattern is trained by the RMS from post treatment，and the test sample is recognized by the HMM based classification method.As shown in Table 1，21 test samples are recognized.The same recognition procedure based on the BP neural network and the recognition results are presented in Table 2.

Table 2.Pattern classification results of the tool wear

Figure 7.Flow chart of the tool states recognition

As shown in Table 2，most samples have been recognized correctly and the accuracy rate of HMM is 20/21=95%，the accuracy rate of HMM is 19/21=90%.The results show that the HMM-based classification hasa higherrecognition rate than that of ANN.

4.Conclusion

Tool wear monitoring in machining process is very important for mechanical manufacturing process.In this paper，an approach based on wavelet packet coefficient and HMM for tool wear monitoring is proposed.Wavelet packet decomposition is used for signal processing.The RMS of the wavelet coefficients is adopted for the input of HMM.According to HMM-based recognition method，tool wear states are recognized.In future works，uncertainty in processing should be regarded in modeling and signal acquisition.

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