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Modification of a neural network utilizing hybrid filters for the compensation of thermal

International Journal of Machine Tools &Manufacture 47(2007)376–387

Modi?cation of a neural network utilizing hybrid ?lters for the

compensation of thermal deformation in machine tools

Yuan Kang a,?,Chuan-Wei Chang a ,Yuanruey Huang a ,Chuag-Liang Hsu b ,I-Fu Nieh b

Department of Mechanical Engineering,Chung Yuan Christian University,Chung Li,Taiwan 320,ROC

Chung Shan Institute of Science and Technology,Lung Tan,Taiwan 325,ROC

Received 28June 2004;received in revised form 15February 2006;accepted 13March 2006

Available online 22May 2006

Abstract

This study proposes a modi?ed method that combines feed-forward neural network (FNN)and hybrid ?lters to improve the accuracy and reduce computation times for the prediction of thermal deformation in a machine tool.The hybrid ?lter consists of the linear regression (LR),moving average (MA)and autoregression (AR).Their outputs serve as input of FNN,which are estimated by the static and dynamic relationships between the temperature distributions and thermal deformations.This modi?ed method enables the propagation accuracy between input and output layers of a static FNN to be improved and the learning time to be reduced.Furthermore,the modi?ed method is compared with other three ones,which are traditional ARMA,FNN,and FNN combined with LR by numerical analysis and practical experiments.In analysis,the error margins of various approaches are compared using a ?nite element model that is determined for the relationships between thermal deformation and temperature distribution.Also,practical experiments of these approaches for a grinding machine are realized to compare the deformation predications according to temperature measurements.r 2006Elsevier Ltd.All rights reserved.

Keywords:Thermal deformation;Feed-forward neural network;Hybrid ?lter;Finite element analysis;Machine tools

1.Introduction

During a cutting process,the thermal deformations induce a major part of error of working piece due to temperature rising and nonuniform distribution in machine tools [1].Postlethwaite et al.[2]discuss the methods used to reduce thermal deformation.The results show that the improvement of the ?nishing accuracy with error compen-sation by software method is ef?cient as it does not change the design of structure and mechanism.

The software method utilizes prediction model to determine the thermal deformations according to tempera-tures measured.The linear regression (LR)model as presented in [3,4]has been used to predict thermal deformations for turning lathe and horizontal machine center,respectively.Yang and Ni [5]used autoregressive moving average (ARMA)to predict thermal deformation for a horizontal machining center.However,in LR model,

time delay cannot be taken into consideration and ARMA model is linear;they have insuf?cient capability to describe nonlinear relationship between thermal deformations and temperature distribution.Thus,Hattori et al.[6]and Veldhuis and Elbestawi [7]proposed feed-forward neural network (FNN)with back-propagation algorithm to modify the error compensation method for a vertical milling machine and ?ve-axis machine,respectively.In [8,9],the prediction errors of FNN model and LR model for a three-axis horizontal machining center and CNC machine tool are compared,and the results show that FNN can yield more accurate results than other methods.

For the prediction model of thermal deformation,a large number of temperature sensors are necessary.However,the in?uence of sensor locations is more signi?cant.Vanherck et al.[10]and Mou et al.[11]estimated the numbers and the locations of sensors for the use of FNN in the prediction of thermal deformation in a CNC machine tool.Also node numbers effected the predicted results of FNN model.Accordingly,Lippmann [12]proposed a method to select the number of hidden layer nodes,which are

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determined by the numbers of input and output nodes. And,Bebis and Georgiopoulos[13]proposed how to illustrate that a small network is better than the bigger network for FNN model.

Since a smaller network needs less computation in training phase,the number of input nodes is reduced and the prediction accuracy of FNN can be improved as the training time being kept constant when node number is enough.

This study proposed a modi?ed model that is combined of a hybrid?lter and FNN.The hybrid?lter is composed of LR,MA and AR model.The vast measurements can be replaced by few inputs of FNN,which are outputs of the hybrid?lter and a bias.Thus,the accuracy of propagation between input and output layers can be increased and the training time can be reduced.The prediction accuracy of thermal deformations in an actual grinding machine is compared among ARMA,FNN,FNN with LR,and the proposed model,using?nite element method and experi-ment for feasibility and availability.

2.Modi?ed model for thermal deformation

The models used to predict the thermal deformations by measuring temperatures of a machine tool are static or dynamic neural network.For the dynamic model of FNN as shown in Fig.1,the total inputs of all sensors are sampling from time(nàJ)to time n and the outputs been

Fig.1.Dynamic FNN

model.

Fig.2.Modi?ed model of thermal deformation.

Y.Kang et al./International Journal of Machine Tools&Manufacture47(2007)376–387377

sampling from time (n àK )to time (n à1).Where J ,K and L denote the number of total sampling steps,of inputs,outputs,and total input sensors,respectively.For a static FNN model,the inputs are total measurements of all sensors at the same time.Therefore,the required number of inputs of the dynamic model is a J multiple of the input number of static model.

The modi?ed model proposed by this study is shown in Fig.2,which is a dynamic FNN,including a hybrid ?lter for inputs of static FNN.The hybrid ?lter consists of LR,MA and AR.LR and MA estimate thermal deformation

by using temperature measurements at the same time and the previous times within previous J steps,respectively.The determinations are expressed as ˉY

1en T?a 0tX L i ?0

a i T i en T

(1)

of LR model,and ˉY

j t1en T?b 0tX N à2j ?1X L i ?1

b ji T i en àj T

(2)

of MA modal,where T i (á)is the temperature measured by the i th thermal sensor at time denoted in parentheses.The coef?cients of a 0,a i ,b 0and b ji are estimation parameters,which can be determined by the least-square method.

The AR model estimates the relations between inputs and output of FNN by using N à1outputs of LR and MA models at the same time of the n th step and K sampling outputs of FNN to yield the output as expressed by ˉY

N en T?c 0tX K k ?1

c k ^Y

en àk TtX N à1j ?1

c j ˉY

j en T,(3)

where ^Y

en àk Tis the output of the FNN at sampling time of the (n àk )step,and ˉY

j en Tis the output of the LR and the MA model at sampling time of the n th step.Also,the coef?cients c 0,c k and c j can be determined by the least-square method.

The static FNN model is shown in Fig.3.The hybrid ?lter outputs are normalized by a constant C Y and served as inputs of FNN.The inputs and outputs of the hidden

Fig.3.Static FNN

model.

Fig.4.Original draft and ?nite element model for a practical machine tool.

Y.Kang et al./International Journal of Machine Tools &Manufacture 47(2007)376–387

378

layer are determined by

NET jenT?

X N

i?1

eW jienT~Y ienTTty,(4) where~Y ienT?C YˉY ienTand y is the bias of input layer,and

O jenT?feNET jenTT?tanhegáNET jenTT,(5) where g40,respectively.The displacement error of thermal deformation can be decomposed into these components due to coordinates as U x,U y,U z.

The determinations are decoupled using independent models.Then one of the error components is determined from the output of a static FNN model as expressed by

‘

enT?feNET‘enTT?tanhegáNET‘enTT;‘?x;y;z,

(6) where NET k(n)is determined by

NET‘enT?

X J

j?1

W‘jenTO jenT,(7)

where the connection weights W ji and W‘j between three layers are updated using the back propagation method.

In this modi?ed model,the variations of measured temperatures between two adjacent steps of sampling times are taken into consideration by the inputs of the LR and the MA models.Also,the static FNN outputs from the (nà1)th step to the(nàk)th step,which are determined from the outputs of LR model and MA model at the n th step,are all taken into consideration by inputs of AR model.

3.Determinations of error margins

3.1.Finite element analysis

The error margins of the prediction models can be determined using?nite element analysis(FEA)to simulate the temperature distribution and the thermal deformations of practical machine tools under the suppositions of regular heat?ux.However,the?nite element model must be simpli?ed by neglecting complex details and contact joints and using the same material.

The?nite element model for a practical machine tool as illustrating example of error margins is shown in Fig.4. The tetrahedral solid element with10nodes for one element is used.The material is assumed to be the same cast iron for all structures,which has elasticity coef?cient E?1.65?1011N/m2,Poisson’s ratio n?0:28,density r?7300kg=m3,speci?c heat448J/kg1C,and thermal expansion coef?cient10.2?10à6m/m1C,for its environ-ment the heat transfer coef?cient is79.4W/m1C and 12.5W/m1C for convection and radiation,respectively. The total number of all nodes in the meshed model is over 30,000.

Four heat?uxes induced by spindle and sideways of X,Y and Z axes are denoted by h s,h x,h y and h z,respectively.These heat?uxes are assumed to be uniformly distributed on all nodes of individual sources.Two data of heat ?uxes are assumed to be constant values of600and 1000W/m2for X-axis,200and150W/m2for Y-axis,400 and700W/m2for Z-axis,and800and1500W/m2for spindle.For both data of heat?uxes,the ambient temperature is assumed to be kept at constant221C.Other two typical data of heat?uxes are supposed as shown in Figs.5(a)and(b),which are regular and random changing. For both,the later ambient temperature as shown in Fig.5(c)is supposed by h a,which behaves according to the temperature changing from morning to noon and to evening during1day.

012345678

012345678 0

500

1000

250

500

150

300

500

1000

time (hour)

(

)

500

1000

400

800

300

600

500

1000

time (hour)

(

)

(

)

h x

h y

h z

h s

h x

h y

h z

h s

(a)

(b)

(c)

Fig. 5.Supposed variations of heat sources:(a)regular variations;

(b)random variations;(c)ambient temperature.

Y.Kang et al./International Journal of Machine Tools&Manufacture47(2007)376–387379

02468t

m e p e r a t u r e (°C )time (hour)

time (hour)

0246810

12t e m p e r a t u r e (°C )

(a)(b)

Fig.

6.Temperature distributions and variations at locations T1:

-202468101214t e m p e r a t u r e (°C )t e m p e r a

t u r e (°C )

time (hour)

(a)(b)

Fig.7.Temperature distributions and variations at locations T1:;T ;T

The commercial package ANSYS [14]is used for modeling,meshing,loading and solving in this model.The analysis results are illustrated by temperature varia-tions of four typical locations,shown in Fig.6for two constant heat ?uxes,and in Figs.7(a)and (b)for the regular and the random heat ?uxes,respectively.The temperature distributions at the same time are also shown by gray-level plots on the right side of these ?gures.For this time heat ?uxes into and off this model sustain 8h.The analysis results of both components in Y and Z directions for the relative displacement of the spindle end to the worktable center,which is located at center position of machine tools,are shown in Figs.8and 9,respectively.

3.2.Margin errors of prediction

The FEA results are used to determine the error margins of prediction models.Three traditional methods,including ARMA,FNN,and LR model with NN,and the modi?ed model are used to predict the error margins and compare each others.The temperatures form the (n à1)th step to the (n à3)th step of sample times are used for the inputs of

-202

t h e r m a l d e f o r m a t i o n U Y (μm )

time (hour)

Fig.8.Thermal deformation in Y direction for constant heat ?uxes

(the ?rst case:,th

t h e r m a l d e f o r m a t i o n U Y (μm )

time (hour)

Fig.9.Thermal deformation in Y direction for heat ?uxes belonging to variations (regular variations:;ra

error

(b)

Fig.10.Two phases of the determination of error margins in prediction model:(a)training phase;(b)prediction phase.

-4

-202468t h e r m a l d e f o r m a t i o n (μm )

time (hour)

Fig.11.Thermal deformation due to the second constant heat ?ux (FEM:

;A ;L ;m

ARMA model.The three layer FNN is used for the traditional FNN,LR model with NN,and the modi?ed model,simultaneously.

The supposed heat ?uxes as mentioned above are applied to ?nite element model;then temperature distributions and

relative displacements between spindle end and worktable center are obtained.These simulation results according to FEA are used as inputs and estimated outputs of prediction model.

In training phase,temperature distributions and thermal deformations are determined using ?nite element method.The prediction model as shown in Fig 10(a)is trained using these data,which belong to ?nite element model of a machine tool.In prediction phase,the thermal deforma-tions are determined for both ?nite element model and prediction model.As shown in Fig.10(b),the prediction errors are obtained by the comparison of the thermal deformations of ?nite element model.

In determinations of error margin,the ?rst constant heat ?ux and regular heat ?ux are used for training the prediction models.The second constant heat ?ux and the random heat ?ux both superposed by white noise are used as examples for the comparisons of prediction errors.Both the prediction results are shown in Figs.11and 12for constant and random heat ?uxes,respectively.The errors of these prediction results are listed in Table 1.When the noise contamination in temperature measurement from environment is taken into consideration,the white noises of 72%,75%and 710%levels are superposed in the analysis results of temperature distributions.For the contamination of white noises,the prediction errors are listed in Tables 2and 3for constant heat ?ux and random ?ux,respectively.

In these tables (from 1to 3),the ?rst values in each lattice are rms of error and the maximum errors are presented in parenthesis.

From the comparisons in error margins,the modi?ed model can yield more accurate results identically for constant and random ?uxes with or without noise contamination.4.Experiments 4.1.Rig and setup

A practical grinding machine with 1645?2140?2181mm of size speci?cation and 152?355?305mm of working space is used.The worktable motion in the X -axis is driven by hydraulic mechanism and controlled by a directional value.The worktable motions in Y -and Z -axis and spindle motion are controlled by AC servomotors,respectively.Thus,pre-diction and compensation of the thermal deformations are reduced into both motions in the Y -and Z -axis.

The measurement system in experiments for this study is shown in Fig.13(a).The thermal sensors and noncontact displacement sensors are used to measure the temperature and thermal deformations,respectively.The measurement data are dealt by the ampli?er circuit and the data acquisition (DAQ)of NI PCI 6024E.These data are analyzed using commercial software Labview with a CPU of AMD Duron 800,the ram of 384MB and a hard disk of 40GB in a PC.

-505101520t h e r m a l d e f o r m a t i o n (μm )time (hour)

Fig.12.Thermal deformation due to the random heat ?ux (FEM:

;A ;L ;m

Table 1

Error margins (m m)for constant ?ux and random ?ux Flux

Model ARMA

FNN LR/NN Modi?ed model Constant 0.63(0.70)0.91(1.08)0.50(0.62)0.35(0.44)Random

1.76(3.73)

1.24(3.28)

1.59(3.64)

1.17(1.97)

Table 2

Error margins for constant ?ux with contaminant of white noises Noise level (%)ARMA FNN LR/NN Modi?ed model 720.70(1.16)0.94(1.44)0.59(1.17)0.43(0.99)750.81(1.98) 1.13(2.76)0.92(2.11)0.74(1.95)710

1.56(3.33)

1.59(3.43)

1.85(4.05)

1.18(3.21)

Table 3

Error margins for random ?ux with contaminant of white noises Noise level (%)ARMA FNN LR/NN Modi?ed model 72 1.77(3.78) 1.30(3.34) 1.63(3.91) 1.18(2.24)75 1.81(3.98) 1.41(3.71) 1.68(4.58) 1.32(2.69)710

2.06(5.22)

1.78(5.27)

1.89(4.76)

1.69(3.44)

Y.Kang et al./International Journal of Machine Tools &Manufacture 47(2007)376–387

382

The temperature sensors used are AD590,which are J type with dimension+5.3?4mm.These embedded in the holes with dimension of+6?15mm on the structure of main components,as shown in Fig.13(b). There are28locations for the temperature measure-ment.In this study,the locations and number of temperature sensors as shown in Fig.13(b)are chosen by trial and error empirically;which locate at spindle shield, the column,and slideways of Y-and Z-axis,hydraulic oil tank,respectively,and10sensors are located nearby from this machine for the measurement of environment temperatures.

The displacement sensors are capacitance type with accuracy of0.1m m and mounted on two precise vises as shown in Fig.13(d),which are used to measure the rela-tive displacements the spindle end in directions of Y-and Z-axis.

4.2.Case study of error prediction and compensation

In the?rst and second experiments,the spindle runs at constant speed of3600rpm for8and7.5h,respectively, and sustainedly.The spindle end is?xed atà75mm from the mechanical origin.The worktable moves more in X-and Z-axis with constant speed of1000mm/min repeatedly from the mechanical origin toà360andà120mm, respectively.One minute is selected to be sampling time for temperature and deformation data.Within this time,

(b)

(c)

Fig.13.Experiment:(a)measurement system;(b)setup of thermal couple;(c)locations of temperature measurements;(d)displacement measurements of thermal deformation.

Y.Kang et al./International Journal of Machine Tools&Manufacture47(2007)376–387383

?ve measurements are executed continuously and the average of these ?ve measurements are used as sampling data for reducing random noises.

The third experiment sustains 4.5h,the heat ?uxes induced by spindle and worktable are interrupted in a repeated form of running continuously for 30min and resting continuously for 30min with ?ve times.

Some typical measurement data of temperatures for experiments 1–3are shown in Fig.14.The measurements of thermal deformations expressed by percentage are shown in Table 4.The measurement data of the ?rst experiment are used for training prediction model.The hidden layer of FNN has 30nodes.The normalization factor is selected to be 0.02.The hyperbolic tangent function is selected for FNN,since the function value is between +1and à1,the compensation values are limited between à50m m and +50m m.The initial weights of FNN are randomly generated between à0.5and +0.5.The prediction of this modi?ed model is built by learning

352

6t e m p e r a t u r e (°C )

t e m p e r a t u r e (°C )t e m p e r a t u r e (°C )

t e m p e r a t u r e (°C )

t e m p e r a t u r e (°C )t e m p e r a t u r e (°C )

time (hour)

01352

4678

01352

4678

time (hour)

352

time (hour)

0481216200

-11230

012302

4(a)(b)

(d)

(f)

(c)(e)

Fig.14.Measurement temperatures at typical locations (Exp 1sustains 8h:

Table 4

Distribution percentage (%)of thermal deformation (m m)Axis à15to à11à11to à7à7to à3à3to 11to 55to 9Exp.1Y 015.560.2524.2500Z 00046.528.7524.75Exp.2Y 42.7024.0515.9517.3000Z 9.7331.6219.4639.1900Exp.3

Y 00703000Z

0.3

39.46

58.5

1.74

Y.Kang et al./International Journal of Machine Tools &Manufacture 47(2007)376–387

384

10,000.The compensation equation of the modi?ed model is expressed as

^Y en T?f X 30j ?1

W kj K f X

3i ?1

W ji ~Y i ty ! ! !.(8)

For the ?rst experiment,the prediction results and error distributions in percentage of thermal deformations are shown in Fig.15and Table 5,respectively.

For the second and third experiments,prediction results and measurement data of thermal deformations are shown in Figs.16and 17,respectively.The comparison results are determined by subtracting prediction data from measure-ment data and listed in Table 6.And the percentages of prediction data are listed in Tables 7and 8,for the second and third experiment,respectively.

From the comparison results,the modi?ed model has less error than other models.The prediction data of thermal deformation are transferred into control coding to compensate the motion coding due to motion controller.The compensation system for errors of thermal deforma-tion is shown in Fig.18,which uses an external machine coordinate shift system (EMCSS).The EMCSS contains an optocoupler circuit and an input/output (I/O)card in the programmable logic controller (PLC).The optocouple circuit can isolate the noise in working environment from 8255card to the machine controller of machine tools.In compensation operation,the prediction data are trans-ferred to the I/O card from 8255card.

-8

-6

-4

-2

t h e r m a l d e f o r m a t i o n (μm )

time (hour)

(b)

(a)

-4-20246

t h e r m a l d e f o r m a t i o n (μm )

time (hour)

Fig.15.Thermal deformation of training results for the modi?ed model (experimental measurement:;p

)(

Table 5

Distribution percentage (%)of error (m m)0.3to –0.10.1to à0.1à0.1to à0.3à0.3to à0.5à0.5to à0.7Y -axis 1.324.55120.8 2.4Z -axis

6.8

62.8

29.9

0.5

-20

-15-10-50

5r e l a t i v e d i s p l a c e m e n t (μm )

r e l a t i v e d i s p l a c e m e n t (μm )

time (hour)

-20

-15-10-50

5(a)

(b)

Fig.16.Thermal deformation of the second experiment (experimental measurement:,a

;L ;a

5.Conclusion

A modi?ed model composed of hybrid ?lters and static FNN are proposed to improve the accuracy of the machine tools by error compensation.The comparison in error margins by using ?nite element model shows the modi-?ed method can yield better prediction results.Also,the proposed model can reduce the training time from

Table 6

Prediction error (m m)and training time (h)Model

Error Y -axis Z -axis Training time (h)

rms

max.rms max.ARMA Exp.2 5.07.4 4.97.90.02Exp.3 4.2 6.9 2.6 4.7FNN Exp.2 3.4 4.8 3.0 5.148Exp.3 3.9 6.1 2.2 4.0LR/NN Exp.2 3.5 5.1 2.6 3.94Exp.3 2.8 4.6 1.5 3.2Modi?ed model

Exp.2 2.4 3.9 1.9 3.58

Exp.

1.9

3.8

1.1

2.4

234

-9-6

-3

r e l a t i v e d i s p l a c e m e n t (μm )

time (hour)

(a)

2345

time (hour)

-30

6r e l a t i v e d i s p l a c e m e n t (μm )

(b)

Fig.17.Thermal deformation of the third experiment (experimental measurement:

;L ;a

Table 7

Error distribution (%)of prediction results for the second experiment (m m)

Model

Error 8to 4

4to 33to 22to 11to 00to –1à1to à2à2to à3à3to à4à4to à6Y -axis

ARMA 55.4 6.8 6.2 2.4 6.5 5.4710.300FNN 00.30.50.50.3 1.4 2.2 3.035.955.9LR/NN

37.521.6 5.7 4.9 2.2 6.5 4.6 6.510.50Modi?ed model 027.824.910.37.69.21010.200Z -axis

ARMA 23.528.412.4 5.9 4.9 5.9 3.8 4.610.50FNN 000.20.30.5 3.826.819.225.124.1LR/NN

00000 3.232.231.932.70Modi?ed model

14.9

53.5

23.2

8.4

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386

48to8h,compared with the traditional FNN.In the same training iteration,the modi?ed model can improve the accuracy of prediction results.

Acknowledgement

This study was supported by the National Science Council,the Republic of China,under grant number NSC91-2622-E033-002-CC3.

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Table8

Error distribution(%)of prediction results for the third experiment

(m m)Model Error

8to44to33to22to11to00toà1à1toà2à2toà3à3toà4à4toà7 Y-axis ARMA46.4218.77 5.129.5613.99 6.140000 FNN00000.349.9013.9910.2417.7547.78

LR/NN00009.9014.339.9017.7537.5410.58

Modi?ed model000011.2627.9921.5029.359.90

Z-axis ARMA0000 2.7324.9118.4317.7520.8215.36 FNN000029.6915.3615.3618.0921.500

LR/NN000031.0621.1619.1125.94 2.730

Modi?ed model0000.3439.5922.8726.9610.2400

machine Tools

https://www.wendangku.net/doc/a01317814.html,pensation system for thermal deformation.

Y.Kang et al./International Journal of Machine Tools&Manufacture47(2007)376–387387