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Li, Xiaoli "Fuzzy Neural Network and Wavelet for Tool Condition Monitoring" Computational Intelligence in Manufacturing Handbook Edited by Jun Wang et al Boca Raton: CRC Press LLC,2001 ©2001 CRC Press LLC 15 Fuzzy Neural Network and Wavelet for Tool Condition Monitoring 15.1 Introduction 15.2 Fuzzy Neural Network 15.3 Wavelet Transforms 15.4 Tool Breakage Monitoring with Wavelet Transforms 15.5 Identification of Tool Wear States Using Fuzzy Methods 15.6 Tool Wear Monitoring with Wavelet Transforms and Fuzzy Neural Network 15.1 Introduction To reduce operating costs and improve product quality are two objectives for the modern manufacturing industries, so most manufacturing systems are fast converting to fully automated environments such as computer integrated manufacturing (CIM) and flexible manufacturing systems (FMS). However, many manufacturing processes involve some aspects of metal cutting operations. The most crucial and deter- mining factor to successful maximization of the manufacturing processes in any typical metal cutting process is tool condition. It would seem be logical to propose that tool condition monitoring (TCM) will inevitably become an automated feature of such manufacturing environments. Due to failure, cutting tools adversely affect the surface finish of the workpiece and damage machine tools; serious failure of cutting tools may possibly endanger the operator’s safety. Therefore, it is very necessary to develop tool condition monitoring systems that would alert the operator to the states of cutting tools, thereby avoiding undesirable consequences [1]. Initial TCM systems focused mainly on the development of mathematical models of the cutting process, which were dependent upon large amounts of experimental data. Due to the complexity of the metal cutting process, an accurate model for wear and breakage prediction of cutting tools cannot be obtained, so that many researchers resort to sensor integration methods for replacing model methods. These results in a series of problems such as signal processing, feature extraction, and pattern recognition. To overcome the difficulty of these problems, computational intelligence (fuzzy systems, neural networks, wavelet transforms, genetic algorithms, etc.) has been applied in some TCM systems in recent years. The TCM systems based on computational intelligence, such as wavelet transforms [2], fuzzy inference [3–5], fuzzy neural networks [6–9], etc., have been established, in which all forms of tool condition can be monitored. Fuzzy systems and neural networks are complementary technologies in the design of intelligent systems. Neural networks are essentially low-level computational structures and algorithms that offer good per- formance in dealing with sensory data, while fuzzy systems often deal with issues such as reasoning on Xiaoli Li Harbin Institute of Technology ©2001 CRC Press LLC a higher lever than neural networks. However, since fuzzy systems do not have much learning capability, it is difficult for a human operator to tune the fuzzy rules and membership functions from the training data set. Also, because the internal layers of neural networks are always opaque to the user, the mapping rules in the network are not visible so that it is difficult to understand; furthermore, the convergence (learning time) is usually very slow or not guaranteed. Thus, it is very necessary to reap the benefits of both fuzzy systems and neural networks by combining them in a new integrated system, called a fuzzy neural network (FNN). FNN had been widely used in the TCM [10–12]. Spectral analysis and time series analysis are the most common signal processing methods in TCM. These methods have a good solution in the frequency domain but a very bad solution in the time domain, so that they lose some useful information during signal processing. In general, they are recommended only for processing stability stochastic signals. Recently, wavelet transforms (WT) have been proposed as a significant new tool in signal analysis and processing [13, 14]. They have been used to analyze some signals for tool breakage monitoring [15, 16]. WT has a good solution in the time–frequency domain so that it can extract more information in the time domain at different frequency bands from any signals [17]. Tool condition monitoring can be divided into the two types: tool breakage and tool wear. This chapter addresses how to apply the fuzzy neural network and wavelet transforms to TCM. First, the fuzzy neural network and the wavelet transforms are respectively introduced. Second, the continuous wavelet trans- forms (CWT) and discrete wavelet transforms (DWT) are used to decompose the spindle AC servomotor current signal and the feed AC servomotor current signal in the time–frequency domain, respectively. Real-time tool breakage detection of small-diameter drills is presented by using motor current decom- posed. Third, analyzing the effects of tool wear as well as cutting parameters on the current signals, the models of the relationship between the current signals and the cutting parameters are established, and the fuzzy classification method is effectively used to detect tool wear states. Finally, wavelet packet transforms are applied to decompose AE signals into different frequency bands in the time domain; the root means square (RMS) values extracted from the decomposed signals of each frequency band are referred to as the features of tool wear. The fuzzy neural network is presented to describe the relationship between the tool wear conditions and the monitoring features. 15.2 Fuzzy Neural Network 15.2.1 Combination of Fuzzy System and Neural Network Fuzzy system (FS) and neural networks (NN) are powerful tools for controlling the complex systems operating under a known or unknown environment. Fuzzy systems can easily be used to express approx- imate knowledge and to quickly implement a reaction, but have difficulty in executing learning processes [18]. Neural networks have strong learning abilities but are weak at expressing rule-based knowledge. Although the fuzzy system and neural networks possess remarkable properties when they are employed individually, there are great advantages to using them synergistically, resulting in what are generally referred to as neuro-fuzzy approaches [19]. Neural networks are organized in layers, each consisting of neurons or processing elements that are interconnected. The neurons or perceptions compute a weight sum of their inputs, generating an output. The connections between the neurons have weighted numerical inputs associated with them. There are a number of learning methods to train neural nets, but the backpropagation (BP) paradigm has emerged as the most popular training mechanism. The BP method works by measuring the difference between the system output and the observed output value. The values being calculated at the output layer are propagated to the previous layers and used for adjusting the connection weights. But there are potential drawbacks: (i) no clear guidelines on how to design neural nets; (ii) accuracy of results relies heavily on the size of the training set; (iii) the logic behind the estimate is hard to convey to the user; (iv) long learning time; (v) local convergence. In order to overcome its drawbacks, some hybrid models of neural network and fuzzy system are presented. There are many possible combinations of the two systems, but the four combinations shown in Figure 15.1 have been widely applied to actual systems [20]. ©2001 CRC Press LLC Figure 15.1(a) shows the case where one piece of equipment uses the two systems for different purposes without mutual cooperation. The model in Figure 15.1(b) shows NN used to optimize the parameters of FS by minimizing the error between the output of FS and the given specification. Figure 15.1(c) shows a model where the output of FS is corrected by the output of NN to increase the precision of the final system output. Figure 15.1(d) shows a cascade combination of FS and NN where the output of the FS or NN becomes the input of another NN or FS. The models in Figures 15.1(b) and 15.1(c) are referred to as a combination model with net learning and a combination model with equal structure, respectively. These are shwon in greater detail in Figure 15.2. Figure 15.2(a) shows that the total system is controlled by means of fuzzy system, but the membership of the fuzzy system is produced and adjusted by the learning power of the neural network. The model in Figure 15.2(b) shows that the fuzzy system can be controlled by the neural network; the inference processing of the fuzzy system is responded to by the neural network. 15.2.2 Fuzzy Neural Network In this chapter, a new neural network with fuzzy inference is presented. Let X and Y be two sets in [0,1] with the training input data ( x 1 , x 2 , . . . , x n ) and the desired output value ( y 1 , y 2 , . . . , y m ), respectively. The set of the corresponding elements of the weight matrix is ( w 11 , w 12 , . . . , w nm ). Based on the fuzzy inference, the definition is given as follows: Equation (15.1) and y j = max(min( x i , w ij )) ( i = 1, 2, … , n ; j = 1, 2, . . . , m) Equation (15.2) The fuzzy neural network topology is shown in Figure 15.3. Basically, the idea of backpropagation (BP) is used to find the errors of node outputs in each layer. Without any loss of generality, the detailed learning processes of a single layer for clarity are derived as follows. The derivation can easily be extended to the multiple-output case. The goal of the proposed learning algorithm is to minimize a least-squares error function: Equation (15.3) FIGURE 15.1 Combination type of neural network and fuzzy system. (Reprinted with permission of Springer-Verlag London, Ltd. From “Hybrid Learning for Tool Wear Monitoring,” Int. J. Adv. Manuf. Technol. , 2000, 16, 303–307.) FS FS FS FS NN NN NN NN (a) (c) (d) (b) YXW= o ETO jj = () –/ 2 2 ©2001 CRC Press LLC FIGURE 15.2 Combination model with (a) net learning, and (b) equal structure. (Reprinted with permission of Springer-Verlag London, Ltd. From “Hybrid Learning for Tool Wear Monitoring,” Int. J. Adv. Manuf. Technol., 2000, 16, 303–307.) FIGURE 15.3 FNN net topology. (Reprinted with permission of Chapman & Hall, Ltd. From “On-line Tool Condition Monitoring System with Wavelet Fuzzy Neural Network,” Journal of Intelligent Manufacturing , 1997, 8, 271–276.) NN input output (a) FS if then output input (b) y m x n x 2 x 1 y 2 y 1 W nm W 1m W 1l ©2001 CRC Press LLC where O j = max(min( x i , w ij )), T j is desired output values, O j is the actual values, the least-squares error between them is E . The general parameter learning rule used is as follows: Equation (15.4) where Equation (15.5) Set Equation (15.6) Define when , otherwise when otherwise a 2 = x s Assuming Equation (15.7) According to fuzzy min–max and smooth derivative ideas, a fuzzy ruler is constructed as follows: Equation (15.8) and ∂ ∂ = ∂ ∂ ⋅ ∂ ∂ E w E O O w ij j j ij ∂ ∂ = ∂∨ ∧ () () ∂∧ () ∂∧ () ∂ O w xw xw xw w j ij iij ssj ssj sj , , , a xw xw xw xw xw a xw w iij ssj ssj is iij ssj ssj sj 1 2 = ∂∨ ∧ () () ∂∧ () = ∂∨ ∧ () ∨∧ () ()       ∂∧ () = ∂∧ () ∂ ≠ , , ,, , , ∧ () ≥∨ ∧ () () = ≠ xw xw a ssj is iij ,,, 1 1 axw ssj1 =∧ () , ; xwa ssj ≥=, 2 1, ∂ ∂ = O w j sj ∆ if and then if and then if and then if and then xw x xw x xw x xw x xw w xw xw w xw ssj s is iij s ssj s is iij s ssj sj is iij ssj sj is iij <≥∨∧ () () = <<∨∧ () () = ≥≥∨∧ () () = ≥<∨∧ () () = ≠ ≠ ≠ ≠ , , , , ∆ ∆ ∆ ∆ 2 1 ww s ©2001 CRC Press LLC Equation (15.9) Set Equation (15.10) Then Equation (15.11) the changes for the weight will be obtained from a δ -rule with expression Equation (15.12) where µ is learning rates , µ ∈ [0,1]. To test the fuzzy neural network (FNN), it is compared with the BP neural networks (BPNN) [22]. Under the same conditions (training sample, networks structure (5 × 5), learning rate (0.8), convergence error (0.0001)), the training iteration of FNN is 7, but that of BPNN is 427. Figure 15.4 shows each training process. FIGURE 15.4 (Top): Training process BPNN and (Bottom): FNN. (Reprinted with permission of Chapman & Hall, Ltd. From “On-line Tool Condition Monitoring System with Wavelet Fuzzy Neural Network,” Journal of Intelligent Manufacturing , 1997, 8, 271–276.) ∂ ∂ = () E O TO i jj –– δ j j E O = ∂ ∂ ∂ ∂ = E w ij j δ∆ ∆∆ w ij j = µδ 0.06 0.04 0.02 0 50 100 150 200 250 300 350 400 450 0 Iteration Error 0 0 1 2345678 1.2 0.9 0.6 0.3 Error Iteration ©2001 CRC Press LLC 15.3 Wavelet Transforms 15.3.1 Wavelet Transforms (WT) An energy limited signal f ( t ) can be decomposed by its Fourier transforms F ( w ), namely Equation (15.13) where Equation (15.14) f ( t ) and F ( w ) are called a pair of Fourier transforms. Equation 15.13 implies that f ( t ) signal can be decomposed into a family in which harmonics e iwt and the weighting coefficient F ( w ) represent the amplitudes of the harmonics in f ( t ). F ( w ) is independent of time; it represents the frequency composition of a random process that is assumed to be stationary so that its statistics do not change with time. However, many random processes are essentially nonstationary signals such as vibration, acoustic emis- sion, sound, and so on. If we calculate the frequency composition of nonstationay signals in the usual way, the results are the frequency composition averaged over the duration of the signal, which can’t adequately describe the characteristics of the transient signals in the lower frequency. In general, a short-time Fourier transform (STFT) method is used to deal with nonstationary signals. STFT has a short data window centered at time (see Figure 15.5). Spectral coefficients are calculated for this short length of data, and the window is moved to a new position and repeatedly calculated. Assuming an energy limited signal, f (t) can be decomposed by STFT, namely Equation (15.15) where g ( t − t 0 ) is called window function . If the length of the window is represented by time duration T , its frequency bandwidth is approximately 1/ T . Use of a short data window means that the bandwidth of each spectral coefficient is on the order 1/ T , namely its frequency band is wide. A feature of the STFT is that all spectral estimates have the same bandwidth. Clearly, STFT cannot obtain a high resolution in both the time and the frequency domains. FIGURE 15.5 An illustration of the STFT. (Reprinted with permission of Elsevier Science, Ltd. From “Tool Wear Monitoring with Wavelet Packet Transform-Fuzzy Clustering Method,” Wear, 1998, 219(2), 145–154.) f(t) t t 0 g(t-t 0 ) ft Fwe dt iwt () = () ∞ +∞ ∫ 1 2 π – Fw f te dt iwt () = () ∞ +∞ ∫ – – Gwt f t gt t e dt R iwt ,– – 00 () = () ( ) ∫ ©2001 CRC Press LLC Wavelet transforms involve a fundamentally different approach. Instead of seeking to break down a signal into its harmonics, which are global functions that go on forever, the signals are broken down into a series of local basis functions called wavelets . Each wavelet is located at a different position on the time axis and is local in the sense that it decays to zero when sufficiently far from its center. At the finest scale, wavelets may be very long. Any particular local features of signals can be identified from the scale and position of the wavelets. The structure of nonstationary signals can be analyzed in this way, with local features represented by a close-packet wavelet of short length. Given a time varying signal f ( t ), wavelet transforms (WT) consist of computing a coefficient that is the inner product of the signal and a family of wavelets. In the continuous wavelet transforms (CWT), the wavelet corresponding to scale a and time location b is Equation (15.16) where a and b are the dilation and translation parameters, respectively. The continuous wavelet transform is defined as follows: Equation (15.17) where “*” denotes the complex conjugation. With respect to w f ( a,b ), the signals f ( t ) can be decomposed into Equation (15.18) where c ψ is a constant depending on the base function. Similar to the Fourier transforms, w f ( a,b ) and f ( t ) constitute a pair of wavelet transforms. Equation 15.17 implies that WT can be considered as f ( t ) signal decomposition. Compared with the STFT, the WT has a time-frequency function that describes the information of f ( t ) in various time windows and frequency bands. When a = 2 j , b = k 2 j , j, k ∈ Z, the wavelet is in this case Equation (15.19) The discrete wavelet transform (DWT) is defined as follows: Equation (15.20) where c j,k is defined as the wavelet coefficient, it may be considered as a time–frequency map of the original signal f(t). Multi-resolution analysis is used in discrete scaling function: Equation (15.21) Set Equation (15.22) ψψ ab a b a , – = () 1 1 ab Ra,,∈≠0 wab xt tdt fab , , * () = () () ∫ ψ ft c w a b dadb f a b a () = () () +∞ ∞ +∞ ∫∫ 1 0 1 1 ψ ψ – – , ψψ jk d j tk , = () − 2 2 – – cftt jk jk,, = () () ∫ ψ φφ jk d tk d d , =     − 2 2 2 2 – dfttdt jk jk, = () () ∫ φ , * ©2001 CRC Press LLC where d j,k is called the scaling coefficient, and is the sampled version of original signals. When j = 0, it is the sampled version of the original signals. Wavelet coefficients c j,k (j = 1, 2, . . . , J ) and scaling coefficients d j,k are given by Equation (15.23) and Equation (15.24) where x[n] are discrete-time signals, is the analysis discrete wavelets, and the discrete equivalents to , are called scaling sequence. At each resolution j > 0, the scaling coefficients and the wavelet coefficients can be written as follows: Equation (15.25) Equation (15.26) In fact, the structure of computations in DWT is exactly an octave-band filter [23]. The terms g and h can be considered as high-pass and low-pass filters derived from the analysis wavelet ψ (t) and the scaling function φ (t), respectively. 15.3.2 Wavelet Packet Transforms Wavelet packets are particular linear combinations of wavelets. They form bases that retain many of the orthogonality, smoothness, and location properties of their parent wavelets. The coefficients in the linear combinations are computed by a factored or recursive algorithm, with the result that expansions in wavelet packet bases have low computational complexity. The discrete wavelet transforms can be rewritten as follows: Equation (15.27) Set Equation (15.28) cxnhnk jk n j j , = [] [] ∑ – 2 dxngnk jk n j j , = [] [] ∑ – 2 hn k j j – 2 [] 222 2–/ – – jjj tk ψ ()     gn k j j – 2 [] cgnkd jk n jk+ = [] ∑ 1 2 ,, – dhnkd jk n jk+ = [] ∑ 1 2 ,, – cft htc ft dft gtc ft cft ft jj jj () [] = () () [] () [] = () () [] () [] = () * * – – 1 1 0 Hhkt Ggkt k k ⋅ {} = () ⋅ {} = () ∑ ∑ – – 2 2 . so most manufacturing systems are fast converting to fully automated environments such as computer integrated manufacturing (CIM) and flexible manufacturing. 1 axw ssj1 =∧ () , ; xwa ssj ≥=, 2 1, ∂ ∂ = O w j sj ∆ if and then if and then if and then if and then xw x xw x xw x xw x xw w xw xw w xw ssj s is iij