Using Neural Networks to Predict Bending Angle of Sheet

7/30/2019 Using Neural Networks to Predict Bending Angle of Sheet

1/13

International Journal of Machine Tools & Manufacture 40 (2000) 11851197

Using neural networks to predict bending angle of sheetmetal formed by laser

P.J. Cheng, S.C. Lin *

Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC

Received 14 June 1999; received in revised form 1 October 1999; accepted 12 November 1999

Abstract

In this paper, three supervised neural networks are used to estimate bending angles formed by a laser.Inputs to these neural networks are known forming parameters such as spot diameter, scan speed, laserpower, and workpiece geometries including thickness and length of sheet metal workpiece. For comparison,regression models are also used to estimate bending angle. Verification experiments are then conducted toevaluate the performance of these models. It is shown that the radial basis function neural network modelis superior to other models in predicting bending angle. The volume energy model is better than the line

energy model in angle prediction. 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Laser forming; Neural network; Bending angle

1. Introduction

Sheet metal forming is a very efficient and highly developed process for high output productionsituations. However, the cost and time associated with the manufacturing, alteration or adjustmentof tooling are considerable and may be prohibitive in prototyping or one-off production. This is

an area where laser forming may offer a solution by providing a technique which enables formingto be carried out without the need of hard tooling.The mechanism of laser forming is considerably complex. Vollertsen and Geiger [1] and Kerm-

anidis et al. [2] used a finite element method and a finite difference method to estimate the bendingangle formed by a laser. In general, using these methods requires long computing time. Vollertsen[3], Vollertsen and Rodle [4], and Kao [5] proposed analytical models to predict bending angle.However, these analytical models are generally established based on over-simplified conditions.

* Corresponding author. Tel.: +886-3-5742-920; fax: +886-3-5722-840.

E-mail address: [email protected] (S.C. Lin).

0890-6955/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 8 9 0 - 6 9 5 5 ( 9 9 ) 0 0 1 1 1 - X


2/13

1186 P.J. Cheng, S.C. Lin / International Journal of Machine Tools & Manufacture 40 (2000) 11851197

Scully [6], Geiger [7], Geiger et al. [8], and Yau et al. [9] suggested that the bending angle couldbe expressed as a function of the line energy. The line energy is defined as the ratio between thelaser power and the scan speed. However, the change in bending angle may also result from someother factors. In order to predict bending angle accurately, parameters with significant effects onbending angle should all be taken into account.

Recently, the application of neural networks has attracted great interest. Superior learning, noisesuppression, and parallel computation abilities are major advantages of the neural network method[10]. In this paper, three supervised neural networks are used to estimate bending angles. In thebeginning of this paper, the structure and operation of these models are presented. Experimentsare then conducted and the data collected are used to train these neural networks. Additionalexperiments are conducted to evaluate the performance of these models. Finally, conclusionsare made.

2. Back propagation neural networks

In this paper, three supervised neural networks are used to estimate bending angles formed bya laser. The first one is a back propagation model using a hyperbolic tangent function as theactivation function (BPHTF); the second is a back propagation model using a logistic function asthe activation function (BPLF); and the last one is a radial basis function neural network (RBFN).

The back propagation type neural network is very popular, especially in the area of in-processmonitoring. This is because the structure and operation of the back propagation type neural net-works are simple. Therefore, the back propagation type neural network is adopted in this research.

The general back propagation network is depicted in Fig. 1. It is composed of many simpleprocessing neurons operating in parallel. The structure includes three layers: an input layer, a

Fig. 1. The general architecture of a neural network.


3/13

1187P.J. Cheng, S.C. Lin / International Journal of Machine Tools & Manufacture 40 (2000) 11851197

hidden layer, and an output layer. All neurons are fully connective between layers. The architec-ture of each individual neuron is shown in Fig. 2. The operation of the neural network modelscan be divided into two main phases: forward computing and backward learning.

2.1. Forward computing

The input patterns applied to the neurons of the first layer are just a stimulus to the network.On the other hand, there is no computation in the input layer. As depicted in Fig. 2, each neuronin the hidden layer determines a net input value based on all its input connections. The net inputis calculated by summing the input values multiplied by their corresponding weight. Once thenet input is calculated, it is converted to an activation value. The weight on the connection fromthe ith neuron in the forward layer to the jth neuron is indicated as wij.

The output value Yj of neuron j is computed by the following equations:

netjn

i0

wijxi (1)

Yjfact(netj) (2)

where netj is the linear combination of each of the xi values multiplied by wij; n is the numberof inputs to the jth neuron; fact is the activation function of neuron j.

In this paper, activation functions of back propagation used in the hidden and output layers arehyperbolic tangent function (BPHTF) and logistic function (BPLF) [11]. The value Yj is propa-gated through each upper layer until an output value of the output layer is generated.

2.2. Backward learning

The generated output of the network is compared to the desired output, and an error is computedfor each output neuron. The error vector E between desired values and the output value of thenetwork is defined as

Fig. 2. The architecture of an individual neuron for BPN.


4/13


E[e1,e2,,ej,,ek] (3)

ej

1

2(Y

j

Yj)

2

(4)

where Yj is the desired value of the jth output neuron; Yj is the output value of the jth output neu-ron.

Errors are then transmitted backward from the output layer to each neuron in the forward layer.The process repeats layer by layer. Connection weights are updated by each neuron to cause thenetwork to converge. This process is named the backward learning rule (or delta rule).

The delta rule is a data-adaptive technique for deriving a least-square-error solution. The inter-connection weights in the network are determined so that the error, E, is minimized. Weights areadjusted using the gradient information as

wijhE

wijawprev (5)

wnewwprevwij (6)

The learning rate, h, is a constant between 0 and 1. The momentum constant, a, is usually setto something between 0.1 and 1.

3. Radial basis function neural networks

The structure and operation of the radial basis function network (RBFN) [11] is quite similarto that of the back propagation neural network, but the RBFN has some additional advantagessuch as rapid learning, and less error. Therefore, the radial basis function network is also adoptedin this research.

The structure of the radial basis function network is the same as that of the back propagationneural network as depicted in Fig. 1. The architecture of each individual neuron for the radialbasis function network is shown in Fig. 3.There are no connection weights between the input layer and the hidden layer. The output of theith neuron in the hidden layer is

diRi(x)Ri(xui/s2i ) (7)

where x is a multidimensional input vector; ui is a vector with the same dimension as x; s2i is

the variance of the ith radial basis function; and Ri() is the ith radial basis function with a singlemaximum at the origin. Typically, Ri() is a Gaussian function and can be represented as:

Ri(x)exp xui2

2s2i (8)

The final output is the weighted sum of the output value associated with each neuro in the hid-den layer:


5/13


Fig. 3. The architecture of an individual neuron for RBFN.

YjH

i1

wijdiH

i1

wijRi(x) (9)

In the backward learning process, the values of ui, and si and interconnection weights between

the hidden layer and the output layer are determined so that the error, E, is minimized. Thesevalues are adjusted using the gradient information as

uihE

uiauprev (10)

sihE

sia sprev (11)

wnew(YtjYj)

1YtjY

j (12)


6/13


4. Experimental setup and experimental design

In order to establish models to predict bending angle of sheet metal formed during the laserforming process, a series of experiments are conducted. These experiments are conducted withthree levels of scan speeds, three levels of laser powers, three levels of length of workpiece, twolevels of thickness of workpiece, and two levels of the spot diameter of the laser beam. Parameterlevels used in these experiments are shown in Table 1. The workpiece material used in these testsis 304 stainless steel. For a higher absorption of the laser beam, specimens are coated with carbongraphite powder.

Experimental set-up for these tests is depicted in Fig. 4. These tests are carried out on a FANUCCO2 laser with a maximum output power of 1 kW. The CO2 laser operating in continuous wave(CW) mode with a TEM00 energy distribution is used. TEM stand for Transverse ElectroMagnetic[12]. TEM00 has a Gaussian shaped energy distribution and provides maximum energy intensity

in the middle of the laser beam. The bending angles of the specimens are measured using thelaser displacement sensor Keyence LB 1001.

5. Results and discussions

Statistical analysis is used to study the effects of laser forming parameters on bending angle andto identify factors that have significant effects on bending angle. ANOVA (analysis of variance) isemployed to quantify and verify the effects of these forming parameters on bending angle. Resultsof ANOVA are shown in Table 2. As shown in Table 2, it is found that the main effects ofpower, scan speed, spot diameter, thickness of workpiece, and length of workpiece on bendingangle are significant within the range studied. Interaction effects of these parameters are also sig-nificant.

5.1. Neural network approach

Factors that have significant effects on bending angle are used as inputs of neural networks.They are laser power, scan speed, spot diameter, thickness of workpiece, and length of workpiece.The only output is estimated bending angle. In order to prevent the saturation of the activationfunction, the data used to train neural networks are normalized before used.

In this paper, only one hidden layer is used in these neural networks. The learning rate and

Table 1Parameter levels used in training experiments

Scan velocity (mm/min) 1000, 3000, 5000Power of laser (W) 100, 300, 500Spot diameter of laser (mm) 0.5, 1.0Thickness of workpiece (mm) 1.0, 1.5Length of workpiece (mm) 30, 60, 90Width of workpiece (mm) 60


7/13


Fig. 4. Experimental setup.

momentum constant are adequately increased when the root mean square error (RMSE) of thisepoch is smaller than that of the previous epoch [13]. Conversely, the learning constants areadequately decreased when the RMSE of this epoch is larger than that of the previous epoch.

Fig. 5 shows the effect of the number of neurons in the hidden layer on the root mean square

error. The training epoch for each neural network is 5000. It is shown that the training error isminimized when 4, 7 and 10 neurons are used for BPHTF, BPLF, and RBFN, respectively. There-fore, these neural models with minimum errors are adopted for further study. Fig. 6 shows theconvergence of the error of these models during the learning process. It is shown that the RBFNis superior to the other two models in learning.

Bending angle estimated using these neural networks and those measured experimentally areshown in Fig. 7. As shown in Fig. 7, bending angles estimated using these models are scatteredaround the line, and the average error is 0.0136, 0.0233, and 0.0366 for RBFN, BPLF, andBPHTF, respectively. The maximum error is 0.0473, 0.0985, and 0.1556 for RBFN, BPLF, andBPHTF, respectively.


8/13


Table 2ANOVA results of experiment for bending angle

Factors Degree of freedom Sum of square Mean square F0a

Spot diameter 1 0.4684 0.4684 308.4*

Thickness 1 1.2666 1.2666 833.8*

Length 2 0.0627 0.0314 20.6*

Power 2 7.0355 3.5178 2315.8*

Scan speed 2 5.9447 2.9723 1956.7*

Spot diameterthickness 1 0.0766 0.0766 50.4*

Spot diameterlength 2 0.0090 0.0045 3.0*

Spot diameterspeed 2 0.0056 0.0028 1.9Spot diameterpower 2 0.2838 0.1419 93.4*

Thicknesspower 2 0.0008 0.0004 0.3Thicknessspeed 2 0.0419 0.2074 136.6*

Thicknesslength 2 0.0549 0.0275 18.1*

Lengthspeed 4 0.0118 0.0030 1.9Lengthpower 4 0.0170 0.0043 2.8*

Powerspeed 4 1.0922 0.2730 179.7*

Error 75 0.1139 0.0018Total 108 16.4855

a F0=(Mean squareparameter)/(mean squareerror).

Fig. 5. Effects of the number of neurons in the hidden layer on the root mean square error.

5.2. Regression model

Scully [6], Geiger [7], Geiger et al. [8], and Yau et al. [9] suggested that the bending anglecould be expressed as a function of the line energy. Fig. 8 shows the relationship between thebending angle and the line energy for the experimental data. The least square estimation is usedto determine the model for bending angle, a, as a function of line energy, p/v. The model isrepresented by the following equation:


9/13


Fig. 6. The RMSE of neural networks during 5000 epochs.

Fig. 7. Bending angle estimated using neural networks versus bending angle measured experimentally.

a0.00473.7556p

n2.5968

p

n

2

R20.8158 (13)

where R is the correlation coefficient, and the units for a, p, and n are degree, W, andmm/min, respectively.


10/13


Fig. 8. The relationship between the bending angle and the line energy for the experimental data.

It should be noticed that previous experimental results show that the bending angle is alsoaffected by some other factors such as spot diameter, thickness of workpiece, and length of work-piece. It is believed that these parameters with significant effects on bending angle should be takeninto account when establishing models to estimate the bending angle. Therefore, it is proposed tomodel bending angle as a function of volume energy instead of line energy. The volume energyis defined as the ratio of the laser power and the product of scan speed, spot diameter and thicknessof workpiece.

Fig. 9 shows the relationship between the bending angle and the volume energy for the experi-mental data. The least square estimation is used to determine the model for bending angle, a, asa function of volume energy, p/(vdt). The model is represented by the following equation:

Fig. 9. The relationship between the bending angle and the volume energy for the experimental data.


11/13


a0.01212.9116p

ndt1.4255

p

ndt 2 R20.9055 (14)

where the units for d and t are mm.By comparing Figs. 8 and 9, it is shown that the volume energy model is better than the line

energy model in estimating bending angles. The average error is around 0.0937 and 0.1367 forvolume energy and line energy, respectively. The maximum error is around 0.3542 and 0.4658for volume energy and line energy, respectively.

6. Verification experiments

In order to verify the feasibility of the proposed models, additional experiments are conducted.

The forming conditions for these experimental tests are all listed in Table 3. Table 4 lists bendingangles measured experimentally and those estimated via neural networks and regression models.The maximum error and average error for using neural networks and the regression model topredict bending angles are also listed Table 4. Fig. 10 shows bending angles estimated using theregression model and neural networks versus those measured experimentally. Examining theseresults, the following observations can be drawn.

1. The RBFN is superior to other models in predicting bending angle. The maximum error forthe RBFN is only 0.0687 while the error for BPLF, BPHTF, line energy model, and volumeenergy model is 0.3162, 0.3457, 0.4015, and 0.2722, respectively. The average error for theRBFN is only 0.0266 while that for BPLF, BPHTF, line energy model, and volume energy

model is 0.0996, 0.1010, 0.1608, and 0.1064, respectively.

Table 3Forming conditions for verification experiments

Test No. Diameter (mm) Thickness (mm) Length (mm) Power (W) Speed (mm/min)

1 0.5 1 30 200 20002 0.5 1 30 200 40003 0.5 1 30 400 20004 0.5 1 30 400 40005 0.5 1.5 30 200 2000

6 0.5 1.5 30 200 40007 0.5 1.5 30 400 20008 0.5 1.5 30 400 40009 1 1 30 200 200010 1 1 30 200 400011 1 1 30 400 200012 1 1 30 400 400013 1 1.5 30 200 200014 1 1.5 30 200 400015 1 1.5 30 400 200016 1 1.5 30 400 4000


12/13


Table 4Bending angle measured experimentally and those estimated via neural networks and regression models

Test No. Measured () RBFN () BPHTF () BPLF () Line energy () Volume energy()

1 0.514 0.5827 0.4986 0.4770 0.3449 0.53742 0.401 0.3525 0.2778 0.2536 0.1766 0.28903 0.728 0.7158 0.9451 0.9434 0.6425 0.94874 0.679 0.6840 0.5454 0.5379 0.3449 0.53745 0.361 0.3276 0.2790 0.2454 0.3449 0.37506 0.152 0.1595 0.0927 0.0977 0.1766 0.19997 0.415 0.3920 0.5655 0.4867 0.6425 0.68728 0.176 0.2267 0.2215 0.1957 0.3449 0.37509 0.297 0.3059 0.2972 0.2685 0.3449 0.289010 0.258 0.2185 0.2027 0.1714 0.1766 0.1541

11 0.384 0.3756 0.7297 0.7002 0.6425 0.537412 0.462 0.4742 0.4976 0.4709 0.3449 0.289013 0.198 0.1682 0.1947 0.1644 0.3449 0.199914 0.129 0.0895 0.0771 0.0795 0.1766 0.107615 0.241 0.2793 0.4444 0.4215 0.6425 0.375016 0.124 0.124 0.2179 0.2117 0.3449 0.1999Max. error 0.0687 0.3457 0.3162 0.4015 0.2722Average error 0.0266 0.1010 0.0996 0.1608 0.1064

Fig. 10. Bending angle estimated using neural network and regression models versus bending angle measured exper-imentally.

2. The average error for the volume energy model is about the same as that for BPLF and BPHTF,while the maximum error for the volume energy model is smaller than that for BPLF andBPHTF. The volume energy model is also proved to be better than the line energy model inbending angle prediction.


13/13


7. Conclusions

In this paper, three supervised neural networks are used to estimate bending angles formed bylaser. Inputs to these neural networks are known forming parameters such as spot diameter, scanspeed, laser power, and workpiece geometries including thickness and length of sheet metal work-piece. For comparison, regression models are also used to estimate bending angle. The perform-ance of these models is evaluated and the results compared experimentally. It is shown that theRBFN is superior to other models in predicting bending angle. The volume energy model is alsoproved to be better than the line energy model in angle prediction. The average error for thevolume energy model is about the same as that for BPLF and BPHTF, while the maximum errorfor the volume energy model is smaller than that for BPLF and BPHTF.

Acknowledgements

This work was supported by the National Science Council, Taiwan, Republic of China throughGrant No. NSC 87-2212-E-007-031.

References

[1] F. Vollertsen, M. Geiger, W.M. Li, FDM- and FEM-simulation of laser forming: a comparative study, in:Advanced Technology of Plasticity, Proceedings of the Fourth International Conference on Technology of Plas-ticity, 1993, pp. 17931798.

[2] Th.B. Kermanidis, A.K. Kyrsanidi, S.G. Pantelakis, Numerical simulation of the laser forming process in metallicsheet metals, in: Proceedings of the International Conference on Computer Methods and Experimental Measure-ments for Surface Treatment Effects, 1997, pp. 307316.

[3] F. Vollertsen, An analytical model for laser bending, Lasers in Engineering 2 (1994) 261276.[4] F. Vollertsen, M. Rodle, Model for the temperature gradient mechanism of laser bending, Laser Assisted Net

Shape Engineering Proceedings of the LANE 1 (1994) 371378.[5] M.T. Kao, Elementary study of laser sheet forming of single curvature, Master Thesis of Department of Power

Mechanical Engineering, Tsing Hua University, 1996.[6] K. Scully, Laser line heating, Journal of Ship Production 3 (4) (1987) 237246.[7] M. Geiger, F. Vollertsen, G. Deinzer, Flexible straightening of car body shells by laser forming, in: Sheet Metal

and Stamping Symping Symposium SAE Special Publication, no. 944 SAE, 1993, pp. 3744.[8] M. Geiger, H. Arnet, F. Vollertsen, Laser forming, Manufacturing Systems 24 (1) (1995) 4347.[9] C.L. Yau, K.C. Chan, W.B. Lee, Laser bending of leadframe materials, Journal of Materials Processing Technology

82 (1998) 117121.[10] L. Fausett, Fundamentals of Neural Networks: Architectures, Algorithms, and Applications, Prentice-Hall, Inc,

New York, 1994.[11] J.S.R. Jang, C.T. Sun, E. Mizutani, Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning

and Machine Intelligence, Prentice-Hall, Inc, New York, 1997.[12] G. Berkhahn, CO2 Laser Fabricating Machine ToolsTheir Numbers in Use and Practical Uses in Sheet Metal

Fabrication, Technical Paper MS89-552, Society of Manufacturing Engineers, 1989.[13] J.M. Zurada, Introduction to Artificial Neural Systems, West Publishing Co, 1992.

Documents

Using Neural Networks to Predict Bending Angle of Sheet