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 Network15.1 Introduction


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


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



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


(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
=

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



. 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
∂
∂
=

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
∆

()
=
≥<∨∧
()
()
=
≠
≠
≠
≠
,
,
,
,
∆
∆
∆
∆
2
1
ww
s

©2001 CRC Press LLC

Equation (15.9)
Set
Equation (15.10)
Then
Equation (15.11)

(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


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


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,

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
()
=
()
∞
+∞

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

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

(

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

ψ
ψ
–
–
,
ψψ
jk
d
j
tk
,
=
()
−
2
2
–
–
cftt
jk jk,,
=
() ()
∫
ψ
φφ
jk
d
tk
d
d

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
,



gn k
j
j
– 2
[]
cgnkd
jk
n
jk+
=
[]
∑
1
2
,,
–
dhnkd
jk
n
jk+
=
[]
∑
1
2
,,
–
cft htc ft

{}
=
()
⋅
{}
=
()
∑
∑
–
–
2
2
©2001 CRC Press LLC
Then Equation 15.27 can be written as follows:
Equation (15.29)
Clearly, DWT is only the approximation c
j-1
[f(t)] but not the detail signals d
j-1
[f(t)]; wavelet packet
transforms don’t omit the detail signals. Therefore, wavelet packet transforms is expressed as follows:
Equation (15.30)
Let Q
j
i
(t) be the i
th
packet on j
th

sensitive for small tool breakage when compared to force sensing and AE sensing; the system of current
measurement is only reliable in monitoring tool breakage at medium and heavy cuts [26].
This section presents on-line tool breakage detection of small diameter drills by sensing the AC
servomotor current [27]. The continuous wavelet transforms (CWT) were used to decompose the spindle
cft Hc ft
dft Gc ft
jj
jj
()
[]
=
()
[]
{}
()
[]
=
()
[]
{}
–
–
1
1
cft Hc ft Gd ft
dft Gc ft Hd ft
jj j
jj j
()
[]

i
j
i
0
1
21
1
2
1
()
=
()
()
=
()
()
=
()
–
–
–
©2001 CRC Press LLC
AC servomotor current signals and the discrete wavelet transforms (DWT) were used to decompose the
feed AC servomotor current signals in the time–frequency domain. The features of tool breakage were
extracted from the decomposed signals. Experimental results showed that the proposed monitoring
system could work in real time; in addition, it had a low sensitivity to changes in the cutting conditions
and a high detection rate for the breakage of small diameter drills [28].
15.4.1 Experimental Setup
The schematic diagram of the experimental setup is shown in Figure 15.7. Cutting tests were performed
on a Machine Center Makino-FNC74-A20. The four axles (spindle, X, Y, and Z) of the machine have

N/2
N/4
f(t)
2
1
2
2
2
j
N/2
j
Q
1
1
Q
2
1
Q
2
2
Q
2
3
Q
2
4
Q
1
2
©2001 CRC Press LLC

conditions Feed rate 30 mm/min
Without coolant
Workpiece 45# quench steel
©2001 CRC Press LLC
15.5.1 Experimental Setup and Results
Figure 15.16 shows a schematic diagram of the experimental setup. Cutting tests were performed on a
Machine Center Makino-FNC74-A20. The AC servomotor current signals of the Machine Center were
measured through Hall Current Sensor. The signals were first passed though low-pass filters (cut-off
frequency: 500 HZ), and then sent to a personal computer via an A/D converter. Table 15.2 shows the
experimental conditions.
During the experiments, both spindle and feed current amplitude changed because of the change of
tool wear, spindle speed, feed speed, and the depth of cut. The main conclusions are as follows:
1. Both spindle and feed current increase as tool wear increases; this is due to the increase of friction
between tool and workpiece. Moreover, current increases almost linearly as tool wear. In addition,
we found that tool wear had a more significant effect on feed current than spindle current.
2. Both spindle and feed current increase as the depth of cut increases. Moreover, feed current
increases almost linearly as the depth of cut increases, while spindle current increase is proportional
to the square of the depth of cut.
3. The current signal increases overall as the spindle speed increases, but current fluctuates at the
range of 20 to 30 m/min, see Figure 15.17. The reason for the change of current signals is complex;
the main influence factor is temperature, and the effect of temperature is small at the low speed,
but increases as spindle speed increases.
4. The current signal increases overall as the feed speed increases, and current fluctuates, see Figure
15.18. The reason for the change of current signal is complex; see the discussion in [32].
FIGURE 15.8 (Top): Live tool breakage spindle current signals, cutting speed 250 r/min, feed speed 30 mm/min,
drill diameter 2 mm and (Bottom): Live tool breakage feed current signals, cutting speed 250 r/min, feed speed
30 mm/min, drill diameter 2 mm. (Reprinted with permission of Springer-Verlag London, Ltd. From “Real-Time
Detection of the Breakage of Small Diameter Drills with Wavelet Transform,” Int. Adv. Manuf. Technol., 1999, 14,
539–543.)
5