Grinding Forces Prediction Based Upon Experimental

Design and Neural Network Models

Ridha Amamou

1

, Nabil Ben Fredj

1

, Farhat Fnaiech

2

1

Laboratoire de Mécanique Matériaux et Procédés, Ecole Supérieure des

Sciences et Techniques de Tunis,

2

Centre de Recherche en Productique, Ecole Supérieure des Sciences et

Techniques de Tunis,

5 Av. Taha Hussein, Montfleury, 1008 Tunis, Tunisia

Abstract. The results presented are related to the prediction of the specific

grinding force components. The main problems associated with the prediction

capability of empirical models developed using the design of experiment (DOE)

method are given. In this study an approach suggesting the combination of

DOE method and artificial neural network (ANN) is developed. The inputs of

the developed ANNs were selected among the factors and interaction between

factors of the DOE depending on their significance at different confidence

levels expressed by the value of α%. Results have shown particularly, the

existence of a critical input set which improves the learning ability of the

constructed ANNs. The built ANNs using these critical sets have shown low

deviation from the training data and an acceptable deviation from the testing

data. A high prediction accuracy of these ANNs was tested between models

constructed using the developed approach and models developed by previous

investigations.

1 Introduction

Because of the importance of the grinding forces regarding to the process outputs

including wheel wear and surface integrity, many attempts were made to model its

normal and tangential specific components. However, analyses of the obtained

models have shown that the theoretical modeling [1] exhibits shortcomings from a

quantitative aspect because the permanent changes on shapes and on density of the

cutting edges cannot be clearly taken into account by these models. This suggests the

use of simplifying hypotheses affecting the reliability of these models and limiting

their employment to off-line prediction tasks. In contrast, empirical models, such as

the regression analysis model [2, 3, 4], the fuzzy logic model [5, 6] and the neural

network model [7-11] have, generally, shown satisfactory prediction accuracy,

particularly useful for the on-line response evaluation and control. In many cases, data

from design of experiment (DoE) were used to establish the regression models or to

develop the fuzzy rule sets or to train the neural networks. Indeed, the DoE method

offers the possibility to study the effects of several factors at one time and to

investigate the inter-relationships between these factors, while conducting a relatively

limited number of experiments [12].

Amamou R., Ben Fredj N. and Fnaiech F. (2005).

Grinding Forces Prediction Based Upon Experimental Design and Neural Network Models.

In Proceedings of the 1st International Workshop on Artiﬁcial Neural Networks and Intelligent Information Processing, pages 122-131

DOI: 10.5220/0001194101220131

Copyright

c

SciTePress

However, this method has to be used with awareness, especially when it is applied for

response minimization tasks. In fact, it was remarked that in many cases, the

regression analysis models established by using the DoE were unable to predict an

appropriate minimal response value [13].

This constitutes serious limitation of this method, particularly, for on-line process

control, where the predicted optimal values have to be determined with high accuracy

as they are continuously compared to targets to maintain the desired level of the

outputs.

These inconveniences give rise to the need to develop improved methods with

enhanced prediction capability. In this paper, feed forward neural networks using the

Bayesian regularization were developed. The goal was to train ANNs to include the

most important factors and interactions between factors affecting the surface

roughness in order to make accurate and consistent predictions for new combinations

of values for these factors. This was made by considering the extrapolation beyond

the training data. The developed ANNs were trained using an experimental data set of

a 48 runs factorial design and the best set of variables inputs, the number of neurons

and the ANNs structures selection criteria were discussed. The performance of the

developed ANNs on predicting the specific cutting force components F’

n

and F’

t

within the range of the factors levels fixed by the factorial design was compared to the

statistical multiple regression models obtained directly using the design of experiment

method. Here F’

n

and F’

t

are respectively the normal and the tangential specific

components of the grinding force.

2 ANN approach

Even though several learning methods have been developed [7-11, 14-19], the back

propagation method has been proven to be successful in applications related to

surface integrity prediction [7, 8, 10, 18]. However, the effect of the neural network

inputs selection was not elucidated sufficiently. Indeed, even though it is known that

the selected inputs of the ANNs is an important parameter controlling the outputs

prediction accuracy [8, 16 19], previous studies which have used the DoE data to train

the ANNs, have also used the independent variables of the DoE as inputs of the

developed ANNs. On the other hand the major problem encountered in the use of

ANNs is over fitting [20]. A neural network can predict correctly the trained data set

but it is unable to generalize for other input data. Consequently, the training error

function E is modified to include not just the sum of square errors E

a

but also the sum

of squares of the network weights and biases E

w

. This approach is called the weight

decay regularization [20]. This modification forces the network to have smaller

weights and biases and decrease the tendency of a model to over fit the training data.

The modified error function to be minimized is:

wa

EEE )1(

θθ

−+=

(1)

where θ is the weight decay parameter. The difficulty with regularization is in

assigning an optimum value of θ. If the selected weight decay parameter is too large,

so that overfitting may accrues. On the other hand if it is too small the network will

not adequately fit the training data. Finding the optimum value for the weight decay

123

parameter that is appropriate for the training data is therefore an important task. In

this investigation, the weight decay parameter was determined by the Bayesian

framework. In this case all weight and biases of the network are assumed to be

random variables with Gaussian distribution. The weight decay parameters are related

to the unknown variances associated with these distributions.

In this study, a 48 runs DoE rotatable central composite design was developed. The

data of this experimental design were utilized to train a one hidden layer back

propagation neural network. In fact, previous investigations have proven that this

architecture is enough for the majority of applications [7-11, 16-19].

Fig. 1. Algorithm for ANN Training and selection

As for the inputs selection, it was discussed based on the statistical significance of the

independent variables or interactions between these variables obtained from the

quadratic regression models developed using the data of the 48 runs DoE according to

the algorithm shown in figure 1. The point in this algorithm is that the ANN will be

also trained for learning the factors interactions effects. In fact, the effects of

interaction between factors can be in some cases, more significant than the effects of

the factors. M. Thomas et al [3] have shown in their study related to the prediction of

the surface roughness generated by the cutting process that the effect of the

interaction between the feed rate f and the depth of cut (a) is more significant than the

effect of the individual factors. On the other hand, as good generalization of the

developed ANNs requires that their inputs contain sufficient information pertaining to

the output, so that an accurate mathematical function relating the outputs to the inputs

Desi

g

n an ex

p

eriment

Establish the re

g

ression model

Classify factors and interaction

in an increasing order of the p_value

and corresponding te α%

Train the ANN using inputs

corresponding to α=

α

ι

%

and number of

neurons n=n

j

Save results and

p

erformances

n=n

j

+1

α

ι

=

α

ι

+ increment

Select the most efficient ANN

124

can be established [21], using the interactions between factors as additional inputs for

the ANNs improves this accuracy. Moreover, as the Multilayer perceptron (MLP)

architectures are good at ignoring both redundant and irrelevant inputs [19], non-

pertaining interactions, which are used as inputs for the developed ANNs will not be

considered automatically. Nevertheless, as the training time of the developed ANNs is

widely affected by the number of inputs, therefore, it is important to distinguish the

significant interactions and to use them as additional inputs. This fact is well

considered by the algorithm of figure 1.

The number of epochs was set to 200 and the ANN performances were evaluated by

the training error MSE, training time (s), percentage of deviation from training data,

percentage of deviation from testing data, and the values of the optimal outputs and

the corresponding inputs combinations.

3 Experimental Set-up and design

All the grinding tests were realized in down cut plunge surface grinding mode using a

Teknoscuola RT600 grinding machine. Grinding wheels were dressed using a single

point diamond dresser with a constant gross feed (0.2 mm/rev). The workpieces

dimension was 100

L

x30

H

x15

W

mm. The grinding force components were measured

using a piezo-electric transducer based type dynamometer (kistler 9257A). Three

workpiece materials having different structures and mechanical properties were

selected, 42CrMo4, 90MnCrV8 and X160CrMoV12. Chemical composition and

hardness of these materials are given in table 1. The selection was made based on the

wide industrial application of these materials.

Table 1. Chemical composition of the used material

material C Si Mn Cr Mo P S Hardness HR

C

42CrMo4 0.41 0.28 0.77 1.02 0.25 0.02 0.03 36

90MnCrV8 0.9 _ 2.0 0.4 0.7 _ _ 60

X160CrMoV12 1.55 _ _ 12.0 0.7 _ _ 63

Concerning the grinding parameters, table speed (v

w

), depth of cut (a), grinding wheel

grain mesh size (#), dressing depth (a

d

) and the number of passes (N

p

) were selected.

The selected values of the process parameters, given in table 2, cover conditions

related to both coarse and fine grinding. For experiments a 48 runs DoE rotatable

central composite design was selected and experiments were conducted in a random

order. Three factorial experimental designs using the L

27

[3

5

] standard table were used

for testing the prediction performances of the regression models and the different

ANN structures selected in this study. Here, the material was not considered as an

independent factor and the 27 testing experiments were repeated for each material

type.

125

Table 2. Process parameters and values

Factor Level

1 2 3

Material

v

w

(m/min)

a (mm)

N

p

a

d

(mm)

#

42CrMo4

1

0.050

2

0.010

46

90MnCrV8

5.5

0.100

11

0.020

60

X160CrMoV12

10

0.150

20

0.030

80

4 Results and discussions

4.1 Statistical results

Table 3 summarizes the statistical performances of the regression models developed

for F’

n

and F’

t

. This table shows, particularly, an important average percentage of

deviation from the testing data calculated for the three kinds of materials. Moreover,

negative optimal values of F’

n

and F’

t

are calculated. It can be notice that the

regression model of F’

n

and F’

t

established using the DoE presents low capability to

predict the optimal output value rather than the corresponding factors level. Indeed,

the combinations corresponding to the optimal values for F’

n

and F’

t

are in good

correlation with the results of previous studies [22,23]. This constitutes a serious

limitation of the prediction performances of the regression models established in this

study.

Table 3. Summary of the statistical performances of the DoE multiple regression models.

Results Model for F’

n

Model for F’

t

Percentage

deviation of

the training

data

51,58% 44,95%

Percentage

deviation of

the testing

data:

• 42CrMo4

• 90MnCrV8

• X160CrMo

V12

24,1%

28,01%

28,032%

23,31%

37,05%

35,62%

V

w

a N

p

a

d

# F’

n min

V

w

a N

p

a

d

#

F’

t

min

1 50 2 30 46 -1,273 1 50 2 30 46 -

0,74

4

1 50 2 28 46 -1,341 1 50 2 30 46 -

0,55

8

Grinding

conditions for

the minimal

value:

• 42CrMo4

• 90MnCrV8

• X160CrMo

V12

1 50 2 30 46 -0,045 1 50 2 10 46 0,06

8

126

4.2 ANN results

4.2.1 6-n-1 structure

At first, the common method for establishing ANN’s architecture was used. This was

realized by selecting the same inputs for the developed ANN as those selected for the

DoE. Hence, a 6-n-1 structure was constructed; with n is the number of neurons in the

hidden layer. The value of n was varied from 2 to 40 and the average

MSE error of

the 39 constructed artificial neural networks was 0.023 in the case of F’

n

and 0.0037

in the case of F’

t

. When considering the structure 6-5-1 for which the lowest training

error MSE (MSE=1.04e-03) and deviations from testing data were computed, the

calculated deviations from the training data were 0.84% for F’

n

and 0.47% for F’

t

.

However, the percentages of deviations from the testing data for F’

n

were 26.82% for

42CrMo4, 22.71% for 90MnCrV8 and 23.17% for X160CrMoV12. Concerning the

specific tangential force F’

t

, the calculated deviations from the testing data were

19.42% for 42CrMo4, 20.87 for 90MnCrV8 and 23.34% for X160CrMoV12. These

high deviations express poor generalization capability of the ANN structures with 6

inputs. Therefore, more training data are needed to improve the prediction efficiency

of this neural network architecture when extrapolation beyond the training data is

considered.

4.2.2 x-n-1 structure

For improving the prediction capability of the ANNs, the number of inputs was

varied. Therefore, instead of selecting the same inputs as the DoE, the factors and the

second order interactions between factors were selected as inputs. Figure 2, show the

relation between the averages

MSE

error for the 39 artificial neural networks

structures constructed using the significant factors and interactions at different

confidence levels for F’

n

et F’

t

. A clear improvement of the learning capability of the

constructed artificial neural networks can be seen from this figure as an important

reduction of the average

MSE

error could be realized. Moreover, these figures put in

evidence the existence of a threshold value of α% for which no significant learning

improvement can be realized by increasing the number of inputs. Here a threshold

value about 50% was observed for F’

n

and F’

t

.

On the other hand, as in this work we are particularly interested on the grinding force

minimization, extrapolation beyond the training data have to be considered. Hence,

calculation of a global error (E

g

) expressing the deviation of the predicted values

using the developed neural networks from the testing data sets is required to valid the

results of this investigation.

This error is composed of two terms: the bias which measures the extent to which the

average predicted output, over all testing data sets, of the network function differs

from the experimental values and the variance which measures the extent to which

the network function is sensitive to particular choice of data set [18]. As in this

investigation 27 testing experiments were conducted for each material (n=81) and 39

different networks (m=39) were trained for each input set, the expressions of the bias

and the variance can be written in the following forms [20]:

127

{}

{}

)5(])()((

1

[

1

var

)4(])()([

1

)(

)3()(

1

)(

)2(var)(

1

2

1

1

2

exp

2

1

2

∑∑

∑

∑

==

=

=

−=

−=

=

+=

m

i

k

predicted

kpredicted i

n

k

n

k

kerimentalk

predicted

m

i

predicted i

predicted

g

XFXF

mn

iance

XFXF

n

bias

XF

m

XF

iancebiasE

Here F represents F’

n

or F’

t

depending on the considered output and X and X

k

are the

input sets.

0

0.0005

0.001

0.0015

0.002

0.0025

0 0.2 0.4 0.6 0.8 1

Average errors (mse)of the ANNs

for F'n

confidence level

α

%

(a)

1 10

-5

1.5 10

-5

2 10

-5

2.5 10

-5

3 10

-5

3.5 10

-5

4 10

-5

00.20.40.60.81

Average error (mse) of the ANNs

for F't

Confidence level

α

%

(b)

Fig. 2. Relation between the average

MSE

errors of the 39 ANN structures and α%: (a) F’

n

and (b) F’

t

Figure 3 gives the relation between the average error E

g

for F’

n

and F’

t

respectively at

different confidence levels. It can be seen from these figures that for the testing data

sets selected in this investigation, the minimum average error E

g

occurs at α around

50% for the specific normal component F’

n

and 40% for F’

t

. However, the input set

which have to be selected for the developed ANNs is the set that offers,

simultaneously, high leaning performance of the training data characterised by low

MSE and good generalisation characterized by low value of E

g

. Therefore, inputs sets

corresponding to α=50% have to be retained for the ANNs related to F’

n

and F’

t

.

Therefore, the corresponding inputs are used to train the artificial neural networks and

the best structure offering the lowest deviation from the training and testing data were

16-18-1 for F’

n

and 19-11-1 for F’

t

. The full performances of these structures are

given in table 4. This table shows particularly positive values for the predicted

minimal cutting force parameters F’

n

and F’

t

. Here 0% and 0.028% deviations from

the training data were calculated respectively for F’

n

and F’

t

. The percentages of

deviations from the testing data for F’

n

were 7.6% for 42CrMo4, 8.55% for

90MnCrV8 and 7.46% for X160CrMoV12. These deviations are clearly lower than

those calculated using the 6-5-1 structure. Concerning the specific tangential force F’

t

,

the calculated deviations from the testing data were 7.32% for 42CrMo4, 8.58% for

90MnCrV8 and 7.11% for X160CrMoV12.According to these findings, it can be

concluded that a factor or an interaction between factors, which is statically not

significant in the case of the DoE, can be a significant input for the ANN.

128

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

00.20.40.60.81

Average error Eg for F'n

confidence level

α

%

(a)

0.15

0.2

0.25

0.3

0.35

0.4

00.20.40.60.81

Average error Eg for F't

confidence level

α

%

(b)

Fig. 3. (a) Relation between the confidence level α% and the average error E

g

: (a) F’

n

and (b)

F’

t

Table 4. Summary of the selected ANN performances

5 Conclusions

In this paper an approach combining the application of the design of experiment

(DoE) and the neural network methods was developed to establish accurate models

for specific grinding forces prediction. This approach uses data of the DoE to train

artificial neural network for which the input set is composed of significant factors and

interaction between the factors of the DoE. The significance was evaluated based on

Results Model for F’

n

(16-18-1) Model for F’

t

(19-11-1)

Number of inputs 16 19

Hidden nodes 18 11

Mean Square Error 1.956e-05 5.58e-12

Sum Square Error 8.804e-04 2.51e-10

Running time* (s) 18 34

Training cycles 200 200

Percentage deviation

of the training data

Percentage deviation

of the testing data

for:

• 42CrMo4

• 90MnCrV8

• X160CrMoV12

0%

7.6%

8.55%

7.46%

0.028%

7.32%

8.58%

7.11%

V

w

a N

p

a

d

# F'

n mini

V

w

a N

p

a

d

# F'

t mini

1 50 2 30 46 0,0757 1 50 2 30 46 0,0551

1 50 2 30 46 0,1371 1 50 2 20 46 0,0753

Grinding conditions

for the minimal value

• 42CrMo4

• 90MnCrV8

• X160CrMoV12

1 50 6 10 46 0,3956 1 50 2 10 46 0,0889

129

the Fisher test at different values of the confidence levels α%. When this level

increases, the number of significant factors and interactions increased, and thus the

number of inputs of the ANN increases. Average learning

MSE which express the

average error between the training data and the corresponding predicted values and

the generalization error

Eg

which express the deviation from the testing data have

shown the existence of a critical set of inputs offering the highest prediction capability

of the developed ANNs. By using this approach, substantial improvements of the

prediction capability of the ANNs could be realized comparatively with the prediction

ability of the quadratic models developed using the DoE. On the other hand the

developed ANNs have shown better capability comparatively with the commonly

used structures, which use the DoE factors as inputs.

It was also remarked that the developed ANNs present higher sensibility to the input

variations than the DOE as they can distinguish between particular phenomena

occurring at low and high work speeds. At last, problems related to the minimal

negative predicted value, calculated by using models established with DOE method

could be solved as ANNs respects the output sign.

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