An adaptive-network-based fuzzy inference system for predicting

AN ADAPTIVE-NETWORK-BASED FUZZY INFERENCE SYSTEM FOR PREDICTING
SPRINGBACK OF U-BENDING
Bor-Tsuen Lin1 and Kun-Min Huang1,2
1 Institute
of Engineering Science and Technology, National Kaohsiung First University of Science and Technology,
Kaohsiung, Taiwan
2 Metal Industries Research and Development Center, Kaohsiung, Taiwan
E-mail: [email protected]; [email protected]
ICETI 2012-A1043_SGI
No. 13-CSME-20, E.I.C. Accession 3478
ABSTRACT
Springback will occur when the external force is removed after bending process in sheet metal forming.
This paper proposed an adaptive-network-based fuzzy inference system (ANFIS) model for prediction the
springback angle of the SPCC material after U-bending. Three parameters were selected as the main factors
of affecting the springback after bending, including the die clearance, the punch radius, and the die radius.
The training data were obtained from results of U-bending experiment. The training data with four different
membership functions – triangular, trapezoidal, bell, and Gaussian functions – were employed in the ANFIS
to construct a predictive model for the springback of the U-bending. After the comparison of the predicted
value with the checking data, the results show that the triangular membership function has the best accuracy,
which make it the best function to predict the springback angle of sheet metals after U-bending.
Keywords: U-bending; springback; adaptive-network-based fuzzy inference system; sheet metal forming.
SYSTÈME D’INFÉRENCE FLOUE ORGANISÉ EN RÉSEAU ADAPTIF POUR LA
PRÉDICTION DE RETOUR ÉLASTIQUE DANS LE CAS D’UN PLIAGE EN U
RÉSUMÉ
Le retour élastique se produira quand les forces externes sont retirées après le procédé de pliage dans le
formage d’une feuille de métal. Cet article propose un modèle d’inférence floue paramétré par apprentissage
neuronal (ANFIS) pour prédire l’angle de retour élastique d’un matériau SPCC après le pliage en U. Trois
paramètres ont été désignés comme facteurs principaux affectant le retour élastique après le pliage : le
dégagement de matrice, le rayon de pignon, le rayon de matrice. Les données d’apprentissage avec quatre
différentes fonctions d’appartenance – rectangulaire, trapézoïdale, de courbe en cloche, et les fonctions
gausiennes – ont été utilisées dans l’ANFIS pour construire un modèle de prévision pour le pliage en U.
Suite à la comparaison de la valeur prédite avec les données de vérification, les résultats démontrent que la
fonction d’appartenance triangulaire présente la plus grande exactitude ; ce qui en fait la meilleure fonction
pour prédire l’angle de retour élastique des métaux en feuille après pliage en U.
Mots-clés : pliage en U ; retour élastique ; système d’inférence floue organisé en réseau adaptif ; formage
de métal en feuille.
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335
1. INTRODUCTION
Bending is one of the most important processes in sheet metal forming to yield the desired shape and size.
When a sheet metal is taken out of the die after the bending process, spring-back will occur, which will
affect the precision and quality of products. Accordingly, minimizing the spring-back effect will avoid reprocessing and greatly reduce the time and cost of die development. Livatyali and Altan[1] conducted a set
of experiments and concluded that the following process parameters would heavily affect the springback in
straight flanging: the punch nose radius, the die corner radius, the punch-die clearance, pad force, and the
material type. In order to minimize the maximum bending load and springback, Bahloul et al. [2] proposed
three optimization procedures based on the response surface method by an experimental approach. Levy
[3] constructed an experimental equation to predict the degree of springback, and took the yield strength,
bending radius, die clearance, and the thickness of the sheet metal into consideration.
Ramesh et al. [4] focused on the prediction and optimization of cutting process parameters using ANN
and RSM methods for phosphor bronze damping material attached to the boring tool. Wang et al. [5]
presented an integrated manufacturing capability-oriented design and Taguchi’s method to tackle the high
resolution miniature camera/cell phone lens issues. The adaptive network fuzzy inference system (ANFIS)
was first introduced by Jang in 1993 [6]. Lin and Liu [7] adopted an ANFIS to predict the surface roughness
of the chemical-mechanical polishing processes. Sharma et al. [8] estimated the tool wear rate in turning
operations using ANFIS. Hayati et al. [9] developed ANFIS model for prediction of the heat transfer rate
of the wire-on-tube type heat exchanger. Yeh et al. [10] combined ANFIS, FEM and true strain method to
determine the anisotropic optimum blank in stretch flange process.
This paper presents the implementation of the ANFIS to predict the springback angle of the SPCC sheet
metal after U-bending. Three parameters were selected as the main factors of affecting the spring-back
(SB) after bending, including the die clearance (DC), the punch radius (PR), and the die radius (DR), were
experimented on U-bending. The training data of the ANFIS are obtained from experimental results. In
predicted model of ANFIS, four different types of membership functions (triangular function, trapezoidal
function, bell function, and Gaussian function) were adopted. The results from the four developed prediction
models were compared with the verification data to confirm the feasibility of this approach.
2. ANFIS APPROACH
ANFIS is employed to model the relationship between the features of bending parameters and vibration
signal with the measured values of springback. The architecture of ANFIS was first introduced by Jang in
1993 [6]. To apply the ANFIS, three inputs DC, PR, and DR, and a single output SB are considered in the
fuzzy inference system. The architecture of ANFIS used in this paper is shown in Fig. 1. The Sugeno fuzzy
model is utilized in fuzzification and defuzzification of the system. For a first-order Sugeno fuzzy model, a
typical rule set with 27 fuzzy if-then rules can be expressed as
Rule n: If (DC is Ai ) and (PR is B j ) and (DR is Ck ) then ( fi jk = pi jk DC + qi jk PR + ri jk DR + si jk ),
(1)
where n = i + ( j − 1) × 3 + (k − 1) × 9 (i, j, k = 1, 2, 3), Ai , B j and Ck (i, j, k = 1, 2, 3) are the linguistic terms
of the precondition part with membership functions µAi (DC), µB j (PR), and µCk (DR), pi jk , qi jk , ri jk and si jk
(i, j, k = 1, 2, 3) are the consequent parameters, and fi jk is the output variable.
The complete ANFIS consists of five layers, the fuzzy layer, production layer, normalization layer, defuzzy layer, and total output layer. Each layer involves several nodes, which are described by the node
function.
The node function in each layer, which performs the same operation, is detailed below:
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Fig. 1. The frameworks of ANFIS.
Layer 1 is the fuzzy layer, in which every node is an adaptive node with node function as
O1Ai = µAi (DC),
i = 1, 2, 3,
(2)
O1B j = µB j (PR),
j = 1, 2, 3,
(3)
OC1 k = µCk (DR),
k = 1, 2, 3,
(4)
where DC, PR and DR are the inputs of nodes µAi , µB j and µCk denote the membership functions of the
fuzzy set. Four different types of membership functions (triangular function, trapezoidal function, bell
function, and Gaussian function) were adopted in this paper. In each membership function, the parameters
can be treated as premise parameters.
Layer 2 is the production layer, in which every node is a fixed node with node function to multiply input
signals to serve as output signal:
O2i jk = µAi (DC) × µB j (PR) × µCk (DR) = wi jk ,
i, j, k = 1, 2, 3
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(5)
337
where wi jk represents the firing strength of a rule.
Layer 3 is the normalization layer, in which every node is a fixed node with node function to normalize
firing strength by calculating the ratio of this node firing strength to the sum of the firing strength:
O3i jk = w̄i jk =
wi jk
,
∑i ∑ j ∑k (wi jk )
i, j, k = 1, 2, 3,
(6)
where w̄i jk denotes the normalized firing strength.
Layer 4 is the de-fuzzy layer, in which every node is an adaptive node with node function to compute the
consequence of each fuzzy rule using the following formula:
O4i jk = w̄i jk fi jk = w̄i jk (pi jk (DC) + qi jk (PR) + ri jk (DC) + si jk ),
i, j, k = 1, 2, 3.
(7)
Layer 5 is the total output layer, in which the single node is a fixed node with node function to calculate the
overall output:
3
3
SB = O51 = ∑ ∑
4
∑ w̄i jk fi jk ,
i, j, k = 1, 2, 3,
(8)
i=1 j=1 k=1
where SB denotes the inferred output (i.e., the predicted springback angle) of the ANFIS.
3. EXPERIMENTS
3.1. Experimental Factors Planning
In this paper, the factors that affect the SB after U-bending are discussed. These factors include the DC,
PR, and DR, as shown in Fig. 2. First, the levels are set for each of the factors of the prediction model. For
the DC, the mean value of three chosen levels are decided that equals thickness 0.525mm of sheet metal
as the basis. Then, one value that is smaller than the basis and two values that are greater than the basis,
with an incremental value of 1/10 of the basis are chosen. The three levels for the DC are 0.475, 0.525, and
0.575mm. For the PR, PR = 1 mm as the basis with an incremental value of 2 mm are chosen. The four
levels for the PR are 1, 3, 5, and 7 mm. For the DR, DR = 2 mm as the basis with an incremental value
of 2 mm are chosen. The four levels for the DR are 2, 4, 6, and 8 mm. Therefore, there are 48 different
combinations, as shown in Table 1. The experimental result of each combination is used to construct the
model to predict the SB angle.
In addition, the levels are set for each of the factors of the checking model, where the value of each level
is the mean value of the corresponding values in the prediction model. Therefore, the two levels of the DC
are 0.5 and 0.55 mm. The three levels of the PR are 2, 4, and 6 mm. The three levels of the RD are 3, 5,
and 7 mm. There are a total of 18 different combinations, as shown in Table 2. The verification data of each
combination are used to verify the accuracy of the prediction model.
3.2. Experiments Planning
The CATIA software is used to design the experiment bending die, which include the punch, the die, the
pad and the other auxiliary components. Once the design was finished, those components were processed
and assembled, as shown in Fig. 3.
The following is a list of the experimental procedures:
1. According to the combination described in Tables 1 and 2, attach the punch and the die to the die seat.
2. Setup the experiment bending die to the press machine.
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Fig. 2. Definition of die parameters.
Fig. 3. Experimental bending die.
3. Adjust the die clearance to be in accordance with the value described in Tables 1 and 2.
4. Clean the experiment sheet, and give a number to it.
5. Place the experiment sheet on the experiment bending die, and start the bending experiment.
6. After the loading is removed, SB of the experiment sheet occurs.
7. Use the optical projector machine (Brand: Nikon, Model: V-12B) to measure the SB angle.
The experiment for each of the combinations shall be conducted five times. With the highest and lowest
values being discarded, the value of the degree of the SB is the mean values of the remaining three values.
Tables 1 and 2 show the experimental results.
3.3. Construction of the ANFIS System
The ANFIS theory is used to construct the model to predict the SB angle after bending. MATLAB 7.01
Fuzzy Logic Toolbox was used in our research. The experimental results of the SB angle for U-shape bending are used as the training data (48 groups, as shown in Table 1) for the prediction model. The training data
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Table 1. Comparison between the experimented and predicted springback under different membership functions.
Parameters
No.
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
T13
T14
T15
T16
T17
T18
T19
T20
T21
T22
T23
T24
T25
T26
T27
T28
T29
T30
T31
T32
T33
T34
T35
T36
T37
T38
T39
T40
T41
T42
T43
T44
T45
T46
T47
T48
340
DC (mm)
PR (mm)
DR (mm)
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.475
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.525
0.575
0.575
0.575
0.575
0.575
0.575
0.575
0.575
0.575
0.575
0.575
0.575
0.575
0.575
0.575
0.575
1
1
1
1
3
3
3
3
5
5
5
5
7
7
7
7
1
1
1
1
3
3
3
3
5
5
5
5
7
7
7
7
1
1
1
1
3
3
3
3
5
5
5
5
7
7
7
7
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
Experimented
SB (degree)
1.28
1.52
1.61
1.69
1.86
1.94
2.11
2.35
2.57
2.78
2.88
2.92
3.25
3.41
3.50
3.65
2.21
2.35
2.58
2.69
2.98
3.25
3.32
3.45
3.55
3.69
3.81
3.98
4.01
4.18
4.42
4.50
3.11
3.33
3.45
3.63
3.93
4.20
4.53
4.66
5.09
5.20
5.32
5.63
5.98
6.03
6.22
6.52
RMSE
Triangular
membership
function
1.29
1.50
1.63
1.68
1.85
1.97
2.12
2.32
2.58
2.72
2.86
2.99
3.25
3.42
3.53
3.61
2.20
2.38
2.55
2.70
3.00
3.20
3.33
3.43
3.53
3.72
3.87
3.99
4.01
4.20
4.36
4.50
3.12
3.31
3.47
3.62
3.95
4.24
4.48
4.67
5.03
5.17
5.38
5.63
6.02
6.03
6.20
6.52
0.0307
Predicted SB (degree)
Trapezoidal
Bell
membership membership
function
function
1.29
1.29
1.50
1.50
1.63
1.63
1.68
1.68
1.85
1.85
1.97
1.97
2.12
2.12
2.32
2.31
2.58
2.58
2.73
2.72
2.86
2.86
2.99
2.99
3.25
3.25
3.42
3.42
3.53
3.53
3.61
3.61
2.20
2.20
2.38
2.38
2.55
2.55
2.70
2.70
3.00
3.00
3.20
3.19
3.33
3.33
3.43
3.43
3.53
3.53
3.72
3.72
3.87
3.87
3.99
3.99
4.01
4.01
4.20
4.19
4.36
4.36
4.50
4.50
3.12
3.12
3.31
3.31
3.47
3.47
3.62
3.62
3.95
3.95
4.24
4.24
4.47
4.47
4.67
4.67
5.03
5.03
5.18
5.18
5.38
5.38
5.63
5.64
6.01
6.01
6.03
6.03
6.20
6.20
6.52
6.51
0.0308
0.0313
Gaussian
membership
function
1.29
1.50
1.63
1.68
1.85
1.97
2.12
2.32
2.58
2.73
2.86
2.99
3.25
3.42
3.53
3.61
2.20
2.38
2.55
2.70
3.00
3.19
3.33
3.43
3.53
3.72
3.87
3.99
4.01
4.19
4.36
4.50
3.12
3.31
3.47
3.62
3.95
4.24
4.47
4.67
5.03
5.18
5.38
5.64
6.01
6.03
6.20
6.51
0.0311
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Fig. 4. Fuzzy rule architecture of ANFIS model.
Table 2. Comparison between the experimented and predicted springback under different membership functions.
Parameters
No.
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
DC (mm)
PR (mm)
DR (mm)
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55
2
2
2
4
4
4
6
6
6
2
2
2
4
4
4
6
6
6
3
5
7
3
5
7
3
5
7
3
5
7
3
5
7
3
5
7
Experimented
SB (degree)
1.85
1.97
2.05
2.26
2.55
2.78
3.32
3.45
3.56
2.79
2.93
3.12
3.67
3.87
4.01
4.45
4.64
4.95
RMSE
Triangular
membership
function
2.09
2.25
2.38
2.67
2.82
2.95
3.33
3.51
3.61
2.92
3.11
3.25
3.61
3.80
3.94
4.32
4.47
4.65
0.2023
Predicted SB (degree)
Trapezoidal
Bell
membership membership
function
function
1.82
2.39
2.09
2.60
2.24
2.75
2.44
2.82
2.65
3.02
2.86
3.14
3.64
3.48
3.89
3.74
4.05
3.88
2.74
3.63
3.01
3.92
3.23
4.16
3.49
4.22
3.83
4.53
4.04
4.80
5.02
5.81
5.16
5.88
5.49
6.25
0.2968
0.7887
Gaussian
membership
function
2.10
2.27
2.39
2.54
2.71
2.78
2.96
3.23
3.32
3.58
3.86
4.06
4.30
4.58
4.83
5.77
5.86
6.18
0.7601
has three input parameters (the DC, PR and DR), and one output (the SB angle), as shown in Fig. 4. First the
training data are loaded into MATLAB Fuzzy Logic Toolbox, and a membership function is chosen. Different membership functions for each of the input parameters can also be chosen. The chosen membership
functions are triangular membership function, trapezoidal membership function, bell membership function,
and Gaussian membership function. After the model is trained using the hybrid- learning rule, the results
output by different membership functions were verified against the verification data (18 groups, as shown
in Table 2). The accuracy of each membership function is determined using the root-mean-square-error
(RMSE). The prediction model with the smallest RMSE is the best.
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Fig. 5. Initial (left) and final (right) triangular membership function of die clearance (DC).
Fig. 6. Initial (left) and final (right) triangular membership function of punch radius (PR).
Fig. 7. Initial (left) and final (right) triangular membership function of die radius (DR).
4. RESULTS AND DISCUSSIONS
4.1. Comparison of Prediction Models Using Different Membership Functions
Table 1 shows the RMSE for each of the result output by the aforementioned four different membership
functions. Comparing with the training values, the RMSE of the predicted springback degrees output by
triangular membership function, trapezoidal membership function, bell membership function and Gaussian
membership function are 0.0307, 0.0308, 0.0313 and 0.0311 respectively. The prediction model using
triangular membership function has the smallest RMSE.
Figures 5–7 show the comparison between the triangular membership function of the three input parameters before and after the training respectively. As shown in the figures, the triangular membership function
of the DC and PR differs greatly before and after the training, and the triangular membership functions of
the DR changed locally before and after the training.
Moreover, the 18 groups of verification data were fed into the prediction models those were constructed
using the triangular, trapezoidal, bell and Gaussian membership functions. The ERMS of the output are
0.2023, 0.2968, 0.7887 and 0.7601 respectively, as shown in Table 2. Therefore, the prediction model using
triangular membership function also has the smallest RMSE and the best predictive value.
4.2. Impact of the Bending Parameters on the SB
Figure 8 shows the impacts of the DC, PR, and DR on the SB for U-shape bending. The smaller the DC,
the smaller the SB angle, because the smaller clearance generated higher compressive stress in the bending
area of the sheet during the bending process can reduce larger force that brings the sheet back to its original
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Fig. 8. Contour surface for (a) DR and PR on the SB; (b) DC and DR on the SB; (c) PR and DC on the SB.
shape. A smaller PR can also reduce the SB angle, since sheets are subject to greater bending. The purpose
of the DR is to make the bending process easier. As shown in the figure, it has a very small impact on the
SB. Therefore, the SB angle is proportional to the DC, the PR, and the DR.
From Fig. 8a it can be seen that the slopes of PR are higher than the slopes of DR. In Fig. 8c the slopes of
PR are higher than that of DC. Therefore, PR has the most impact on the SB.
5. CONCLUSIONS
Our research used the ANFIS and the training data collected by the experiments to construct the predicted
model to forecast the springback angle for U-shape bending. The predicted model was verified by comparing
the predicted values with the checking data. Based on our research, the following conclusions are drawn:
• The model constructed using the ANFIS theory can effectively predict the springback angle of Ushape bending by using three bending parameters (die clearance, the punch radius, and the die radius).
• The ANFIS is used to train four different membership functions to determine the most suitable model.
The prediction model using triangular membership function has the smallest RMSE (0.0307) after the
training. The 18 groups of verification data were fed into the prediction model that was constructed
using the triangular membership function. The RMSE of the output is 0.2023, which is also the
smallest. The results show that the most suitable membership function is the triangular membership
function.
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ACKNOWLEDGEMENTS
This work was supported by Grant No. 99-2622-E-327-013-CC3 from the National Science Council of
the Taiwan. The authors would also like to thank the Metal Industries Research and Development Center,
Taiwan, for their financial support and useful suggestions.
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