Accuracy Improvement for CNC System using Wavelet-Neural
Networks
Shashidhara H.L
Suneel T. S.
Vikram M. Gadre
Elect. Engg. Dept.
Mech. Engg. Dept.
Elect. Engg. Dept.
'
S. S. Pande
Mech. Engg. Dept.
Indian Institute of *TechnologyBombay, Powai, Mumbai - 400 076, INDIA
Abshct- Wavelet neural networks are investigated for learning a multidimensional inputoutput complex nonlinear function. The CNC
turning process is modeled using Wavelet neural
networks. The error on the component is different from the desired dimensions because of the
dynamics of the machining system. The error,
ifpredicted, apriori can be used for compensating the same thus improving the accuracy in the
part. In this work, wavelet neural networks are
employed to predict the e r r o r given the process
conditions as the input. Simulation studies are
carried out to arrive for the selection of a suitable
wav6let function for the CNC turning system in
pacticdar.
I
Introduction
The exigencies of modern manufacturing have spawned
several CAD/CAM/CIM [l]tools which have helped in
automating several tasks in the manufacturing cycle.
In this scenario CNC technology assumes a special significance as it provides a means of rapid conversion of
design models into physical prototypes. The accuracy
and utilization of CNC machine tools is dictated to a
large extent by the availability of an error free optimum
CNC program(code)[2]. CNC code generation is primarily based on ideal part geometry, disregarding tool/work
deflections, static and dynamic compliance and inherent
errors present in the machine tool[3]. Accuracy of parts
produced on the CNC machine tool can be enhanced
if one has an U priori knowledge about the pattern of
these errors on the machine tool and the factors causing
them[4].
A need therefore exists to intelligently capture this
functional relationship ,and use it further in a predictive manner for automatic correction of ideal CNC code.
The present research work is an attempt to predict the
error on the component using wavelet neural networks.
0-7803-5812-0/00/$10.0092000IEEE
At times it is difficult or impossible to model real life
functions mathematically. Artificial neural networks are
used in such situations since they are very promising
as function approximation tools. In recent years there
is a considerable advance in Artificial Neural Networks
(ANN) for universal function approximation, function
learning and classification. Despite its wide applications ANNs are beset with limitations such as determining the number of neurons, their interconnections,
choice of proper learning algorithm and poor convergence. Stochastic gradient algorithms are often used
during training and the aim of this.is to get to the minimum of the errors. But in doing so it gets stuck in
the local minima. More over the approximation class is
nonlinear in the adjustable parameters.
Of late, Wavelet Neural Networks (WNN) are used as
an alternative to ANNs for function learning and function approximation[5][6][7]since they overcome some of
the limitations of ANN mentioned above. WNNs can
also be used to learn a class of functions, where each
function in the class differs from the others in a minor
way and all these functions essentially correspond to a
single entity[8].
I1 Wavelets
Wavelets are mathematical functions that split the data
into various frequency bands and analyze each component with a resolution matched to its scale. In wavelet
analysis, scale plays a significant role. Wavelet algorithms process data at different resolutions or scales. In
order to capture gross features the signal is viewed with
a larger window and for capturing fine features of the
signal, a small window is selected. Thereby both the forest as well as the trees are seen. Thus wavelet analysis
provides both time and frequency localizations of the
signal and is better suited to the task of function learning than t he traditional Fourier transform, particularly
when the signal has sharp discontinuities and spikes.
The sine wave basis functions used in Fourier analysis
are non-local and stretch out to infinity and, therefore
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,
3
they are not well suited in approximating sharp spikes.
In wavelet analysis the approximating functions are contained in a finite domain and thus are well suited for
approximating signals with sharp discontinuities[9].
Advances in research and interaction of different fields
have led to various new wavelet applications in function
learning and function approximation. The wavelet networks proposed by Benviniste[5] overcome the problem
of poor convergence and provide a scheme for defining
the structure and determining the number of wavelons
required in the wavelet neural network[lO]. In the
work[8], the ability of a wavelet neural network has been.
successfully demonstrated to learn a class of functions
as against learning a single function. These encouraging
results were the basis for exploring further the abilities
of a wavelet neural network in learning a more complex
and dynamic system. In the present work, wavelet neural networks are used to model a CNC turning system
which is a sufficiently complex, dynamic and difficult
system to model[ll]. The motivation for modeling such
a system is to improve the accuracy on parts produced
during machining. The modeled system will be able to
predict the error on the component which can be used
for compensating the process to improve the part accuracy. This has been reported elsewhere[l2]. The selection of wavelet functions, number of wavelons etc.,
in modeling the network is important to arrive at an
optimum network design which can further be used for
the prediction of profile error in CNC machining. Thus
it is important to investigate the effect of variation of
wavelet functions, number of wavelons etc., on the performance of wavelet networks. In this paper different
wavelet functions are used to model the wavelet network
and their suitability for error prediction is investigated.
This paper is organized as follows. The CNC turning
system modeling and its training is discussed in section
111. Simulation studies of wavelet network using different wavelet functions is presented in the subsequent
section. Conclusions are drawn in the last section.
cessive memory requirements[l3] and complexity of designing a multidimensional system. The higher dimensional problem is thus reduced to a simple and easily implementable, combinations of one-dimensional networks.'
The details of wavelet neural network are shown in the
figure 2 . The weights ( k i j , W i j ) and other parameters
(translation and dilation) are adapted in the process of
training in order that the cost function is minimized.
The stochastic gradient descent technique is used for
training the wavelet neural network. The training algorithm used is similar to the back propagation algorithm for multilayer perceptrons in an artificial neural
network. The network is made to learn the relationship
between the input and output by feeding the data samples. The adaptation algorithm is aimed at minimizing
the cost function. The translations, dilations, weights of
the wavelet neural network and the weights of the linear combiner are adapted as per the following equations.
Results of the simulation studies are discussed below.
The gradients with respect to various network parameters are:
dC
-
ar: = ei
where $I(P)=
I11
and
System modeling
For modeling the CNC turning system with the help of
a wavelet neural network, the multidimensional inputs
are fed to a linear combiner as shown in Figure 1. These
outputs of the linear combiner are then pocessed with
the help of combinations of single dimensional wavelet
neural networks. In the application viz., CNC machining system, there are 5-inputs (Cutting speed, feed rate,
L/D ratio, nominal dimension and tool position) and
one-output(error on the component). This is a multidimensional system and any implementation using multidimensional wavelets become impracticable due to ex-
IV
Simulation studies
In the proposed model two-one dimensional wavelet neural networks were used. The learning rate parameter, -y
was taken to be 0.001 and the number of iterations used
was 10000. Initialization of various adaptable parameters viz., weights, mean etc was carried out as given
in [5]. The stochastic gradient algorithm was used for
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Figure 1: Architecture of the system model for CNC turning system
Figure 2: Structure of the wavelet neural network
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training the wavelet neural network. 900 training samples were used to train the wavelet neural network.
Different Wavelet functions used in the simulation
studies are as under:
0 Gaussian first order derivative
f(x) = -xexp-+
0
(7)
Gaussian second order derivative
Sinc function
sin(x)
X
(9)
Number of wavelons (combination of translation, dilation and wavelet function) were varied for all the simulation studies.
Table I shows the variation of network prediction error
with number of wavelons for Gaussiar? first order derivative as the wavelet function. The wavelet network has
two linear combiners and both of these wavelet networks
has the same number of wavelons in the simulations conducted.
Figure 3 shows the variation of prediction error with
number of wavelons for the wavelet function chosen.
As it is seen from the figure 3 the prediction error
reduces with the increase in number of wavelons. But
increasing number of wavelons beyond a certain limit is
seen to have no significant effect on the over all performance of the network. This is evident from the table I1
that variation of wavelons from 20 to 25 and 30 has a
very little influence on the overall network performance
both in training and testing phases. Thus 20 wavelons
were selected for further simulation studies.
Table I1 shows the effect of changing wavelet function
on the network performance. The prediction error for
both training and testing error are presented in table
11. Number of Wavelons were varied from 1 to 20, as it
was seen in the case of first order Gaussian derivative
that increasing number of wavelons beyond 20 did not
improve the network performance (Table 11). The plots
are shown in Figure 4.
It is seen from the table that the performance of the
network improved marginally with 2nd order Gaussian
wavelet function. Optimum results were obtained for 10
wavelons. 'Figure 4 shows the variat,ion of training and
testing errors with number of wavelons.
Simulation studies with Sinc wavelet function was also
carried out. This however did not improve the network
performance significantly as the prediction error (training) was observed to be 211% and testing error observed
was 210% with 20 wavelons used. 'Thus a detailed simulation study was not carried out for SiIic wavelet function.
TABLE I
E F F E C T O F VARIATION OF WAVELONS ON NETWORK
PERFORMANCE
Number of
wavelons
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
25
30
Raining error
% (epr)
67.52
65.90
64.61
64.44
63.47
63.21
58.40
47.84
63.02
62.99
63.30
62.78
62.52
63.32
61.37
58.50
62.20
58.60
62.06
62.00
61.92
61.91
Testing error
% (epr)
37.44
34.59
32.46
28.65
30.76
28.32
22.26
27.81
27.45
27.45
23.28
21.84
22.29
20.81
20.40
22.56
24.69
22.62
18.09
17.70
16.44
16.09
TABLE I1
EFFECT OF VARIATION OF WAVELONS ON NETWORK
PERFORMANCE
-
Number of Training error
wavelons 1
2
64.01
3
63.64
4
63.57
5
62.66
6
62.25
7
61.53
8
75.23
61.42
9
11
61.29
12
61.17
13
61.62
14
63.48
52.56
15
17
52.56
66.10
18
66.29
19
Testing error
37.20
37.54
37.05
26.00
24.43
22.12
19.81
18.75
19.14
18.99
19.87
19.85
19.85
27.93
19.97
39.31
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I
I
70
10
-
I
80
I
I
e
J
~
I
I
I
15
I
Number of wavelons
10
I
I
e
*
*
c
I
I
I
I
I
.
....
.
.
a
.
. . . . . . . ...... ....
..
. . . . , . . . . . . . . . . .. .
., . . . . . .
-
Testing
.d
40L
c0
.l
.U
L
i?2
302
20 I
I
I
I
I
I
I
I
10
12
14
16
18
I
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20
Thus Gaussian derivatives as the wavelet function can
be recommended for the problem in hand. The wavelet
function chosen here can be used for other function mapping purposes as the CNC turning system modeling is
considered as sufficiently complex and non-linear.
The wavelet network can further be used as a tool
to predict the error and use the error for correcting the
CNC code. This revised CNC code was further used
to machine a component. Error on the component was
reduced to a significantly. This is reported elsewhere
P21.
V
Conclusions
Simulation studies were conducted for designing an optimum wavelet network. CNC turning system was
modeled and simulated using various wavelet functions.
Gaussian wavelet function performed better than any
other wavelet function and thus can be recommended
to be used for most of the function approximation problems. The trained network can then be used to predict
the error likely to be produced on the component and
use it to revise the CNC code. The revised CNC code
can then be used to machine the components thus providing a means to improve the product accuracy.
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