a study of thermally induced machine tool errors in real cutting conditions

~ P e r g a m o n Int. J. Mach. Tools Manufact. Vol. 36, No. 12, pp. 1401-1411, 1996

Copyright © 1996 Published by EL*,evier Science Ltd Printed in Great Britain. All fillh~ reserved

08gO-.6955/96515.00 + .00

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A S T U D Y O F T H E R M A L L Y I N D U C E D M A C H I N E T O O L E R R O R S IN R E A L C U T T I N G C O N D I T I O N S

i

JENQ-SHYONG CHEN~"

(Original received 27 June 1995 i

Abstract--A therm,'d error model for software error compensation i s usually established from air cutting experiments. The aca:uracy of the air cutting model in real machining applications is often questioned. In this research the characteristics of thermal errors in real cutting conditior~s are studied. An analysis based on a regression error model and an artificial neural network-based error model are studied and compared for free air cutting and real i~iiiiMg- conditions. The result shows that the prediction accuracy of the air cutting model is unacceptable in real cutting applications. This is because the cutting 10ad and the cutting coolant applications produce significant ~termal effects not taken into account by the air cutti[Ig approach. In some cases the predicted result Of an air cutting model may go in the opposite direction to the actual error in real cutting. On the other hand, the hybrid model derived from air cutting and real cutting data!gives a satisfactory prediction for real cutting conditions. Copyright © 1996 Published by Elsevier Science Ltd

i

1. INTRODUCTION

The effectiveness of software error compensation for thermally induced machine tool errors relies on the prediction accuracy of the dynamic ~hermal errors produced during machining. On-line prediction of thermal error is usually Achieved through pre-established models which correlate machine thermal errors to temperature measurements of some critical points on the machine structures. The major difficulty is finding a model which can give satisfactory prediction accuracy for different cutting conditions. This problem stems from the fiict that the temperature response of thermal errors can vary widely under different cutting conditions.

A FEM-based analytical model by heat transfer and themaoelasticity principles has been proposed [ 1, 2], but the prediction accuracy is lowered by an insufficient knowledge of machine boundm'y conditions [3]. On the other hand empirical-based error models using the curve-fitting of experimental measurements, such as regression analysis [4-8] and neural networks [9-11], have been demonstrated to predict ithermal errors with satisfactory prediction accuracy in many applications. The major problem of previous works is that the developed thermal error models are all based on air cutting conditions which generate the increase in temperature of the machine tool. AdditiOnally, in order to emulate the cutting load under real cutting conditions, the air cutting tests are often conducted at high spindle speeds mad high feedrates, which are unrealistic real cutting conditions. Although the high-speed air cutting does increase the output loading of the drive motors, a recent study [12] has found that there are some heat sources generated in actual machining not explored by high-speed air cutting, such as the increased friction of the drive mechanisms due to the cutting load. Hot chips and applied coolant which accumulate on the working table and machine base might be the additional thermal Sources not found in air cutting. The objective of this research is to compare the prediction accuracy of the thermal error model built using the air cutting and real cutting conditions. A multi-variant regression analysis (MRA)-rbased error model and an artificial neural network (ANN)-based error model are studied.

tDepartment of Mechanical Engineering, National Chung Cheng University, Chia-Yi, 621 Taiwan, Republic of China.

m se-sv~ 1401

1402 Jenq-Shyong Chen

2. CHARACTERIZATION OF THERMAL ERRORS

For the vertical machining center studied in this research, thermal error sources were classified into the thermal drift of the cutting tool tip, column bending, leadscrew expansion and tool-axis inclination. The temperature response of these thermal errors can vary widely, depending on the cutting conditions. The issue is further complicated by the nonlinear thermoelastic behavior of machine joints, whose heat flow characteristics nonlinearly interact with the structural deflection of the machine [13]. Therefore characterizing thermal errors in different cutting conditions is a necessity. Calibrating the thermal errors individu- ally not only becomes a time-consuming work, but also may neglect the thermal interaction between thermal sources. In this research, a quick setup and multiple-error measurement system consisting of on-machine probes and artifacts was developed to calibrate these thermal errors at the same time in one setup.

The thermal drift of the cutting tool tip was calibrated using a spindle-mounted probe (called a MP7) to periodically check the coordinate of a gage block mounted on the working table. The deviation of the measured coordinates from the values obtained at cold- start was then regarded as the thermal drift of the cutting edge. Thermal expansions of the horizontal linear axes were calibrated by probing the lengths of quartz tubes aligned to the machine axes. Quartz was used for its dimensional stability. Thermal expansion of the vertical linear axis was calibrated using a granite height gage. The temperature vari- ation of the granite gage during testing was monitored continuously in order to compensate for the dimensional deviation of this gage. The granite gage was also used to measure the thermal bending of the column. To calibrate the inclination of the tool-axis, it was neces- sary to measure the thermal drifts of the tool tip with different tool lengths while main- taining the same slide positions. This was achieved by the combination of a spindle- mounted probe and a table-mounted probe, called a MP4.

To demonstrate the problem of thermal error modeling, four tests using the cutting conditions listed in Table 1 are compared. The spindle thermal growth is recognized as the major error source and is usually correlated with a temperature rise of the spindle housing. However, Figs 1 and 2 show that no such simple relationship exists for the four cutting conditions. We plot the experimental data in Fig. 3 using the temperature rise of

Table 1. A list of the cutting conditions

Type Depth of cut Spindle speed Feedrate Cutting coolant (mm) (rpm) (mm min-i)

High speed air 0 5000 3000 No cutting Aluminum air 0 3000 200 No cutting Aluminum 0.5 3000 200 No finishing Aluminum 5 3000 200 Yes roughing

30 .~ 25

20

0 Time (min)

~0

Fig. 1. Thermal drift of the cutting edge in the z-direction.

Thermally Induced Machine Tool Errors in Real Cutting Conditions 1403

, - , 1 0 / . ~ 41

T i m e (min) Fig. 2. Temperature rise of the spindle housing.

. [ 20

n n I

O 0 2 4 6 i ~ 10 Tempera tu re Rise ( 'C)

Fig. 3. Thermal drift in the z-direction versus tempe~are rise of the spindle.

the spindle as the independent variable. This shows that the relationship is nonlinear for each cutting condition. Also, the thermal drift could no i be correlated to this single temperature sensor because there are multiple possibilities depending on the particular cutting condition. For example, when AT is 6°C, the magnitude of the thermal drift could vary between 20 and 40/xm, depending on the spindle speed and temperature history. In sum- mary, these results show the complicated nature of the thermal errors under different cut- ring conditions. This is because the thermal drift of the cutting tool tip is the combination of several thermal error sources, such as spindle growth, column bending, leadscrew expansion and tool-axis inclination, as shown in Fig. 4. For this reason, an empirical model with multiple-input variables including the temperature isensors from several components is required to model the thermal error. [

The thermal expansion of the x-axis, often correlated to the temperature rise of the motor, is generally considered to be the major heat source for this error in air cutting.

z-axis ~ m

Fig. 4. Thermal error sources.


However, Figs 5-7 show that there is no such simple relationship for all cutting conditions. This implies that the heat from the motor is not the only factor contributing to the thermal expansion of a linear axis. There are some heat sources in real cutting not explored by air cutting, such as the increased friction due to the cutting load, hot chips and applied coolant. Figures 8 and 9 demonstrate that it is also impossible to correlate the thermal expansion error to the temperature rise of the ballscrew. The same characteristic also occurs in the y-axis. However, the thermal expansion of the z-axis is affected little by the cutting depth and coolant. This may indicate that the hot chips and applied coolant which accumulate on the work table and machine base are the reasons for the exaggerated thermal expansions of the x- and y-axes in real cutting conditions. Figure 10 shows that the column bends very little in low spindle speed air cutting and aluminum finishing, but bends back- ward significantly in high-speed air cutting. However, the column bends in the opposite direction in aluminum roughing. This indicates that the high-speed air cutting approach cannot give an accurate picture in real cutting conditions. It is found that the heat generated from the coolant tube and control valve box attached at the left side of the column produces significant temperature rises on the left and rear sides of the column.

45

V

,~ 30 "~ 25 .~ 20

15 lO

G

c a ntap Time (min)

Fig. 5. Thermal expansion of the x-axis.

)(3

6

0

"--" 4

'~ 3

"~ 2

~* 1

-1'0 100 200 300 Time (min)

Fig. 6. Temperature rise of the x-axis motor.

~ 4 0

f

" 20 ,A.~ A'

10

Temperature (*C)

Fig. 7. Thermal expansion of the x-axis versus the motor temperature rise.


10

"7 ,, 3

4.

O0 100 ~ v " 2 i ~ v = - v 300 Time (min)

Fig. 8. Temperature rise of the x-axis ballscrew.

:t I I I I

i

'= 20 2 a'/(

I0 i

0 ' ' :' 0 2 4 6 8 10

Temperature ('C)

Fig. 9. Thermal expansion of the x-axis versus the motor temperature rise.

i 6 4

a. 2 • = 0

-2 ° 4 t~

2 - 6

"80 .1130 20d !' Time (rain)

' 00 i

Fig. 10. Thermal expansion of the z-axis.

3. MULTI-VARIANT REGRESSION ANALYSIS (MRA)

In general, a multi-variant regression model can be Presented as

8 = {AI}{AT] + higher order terms ' (1)

where {Aa} is the coefficients vector of regression model and {AT} is the temperature rise vector of machine structures I

Although higher-order terms, including the interactio n terms, can fit the error to very fight residuals, it was found that the resulting models were usually not very robust against conditions that differ from those used in the model estimation [8]. Therefore a relatively simple model of linear combinations of the significant t~mperature sensors is sometimes preferred. One of the most difficult problems in regression analysis is the selection of the set of independent variables which are to be used in the model. It is important not to neglect any key variables because this could damage the power of the model to predict


new observations. However, a regression analysis with a large number of independent variables may overfit the experimental data, including the measurement noise. It is reported that selecting sensors based on their correlation with the thermal errors may be difficult [8]. Since all temperature rises originate in a limited number of heat sources, they show a high degree of dependence. These strong linearly dependent input variables may produce one problem known as multi-collinearity, where a serious round-off problem in estimating the model coefficients using a least-squares method may occur.

In this research, independent variables were initially screened out based on experience and statistical correlation analysis. However, typically the number of independent variables that remained after this initial screening was still large. The stepwise regression analysis with a computerized automatic search algorithm was then used for a second screening. This search method develops a sequence of regression models, which at each step adds or deletes a new variable using the F-test or other statistical criteria. The screened regression model may also be more robust than the unscreened model. To demonstrate this, two air cutting models with and without screening were used to predict thermal errors obtained from new air cutting and aluminum roughing proceedures. The air cutting prediction was actually an interpolation prediction, because the magnitude and the pattern of temperature rises were similar to those used in the model estimation. The aluminum roughing, on the other hand, was an extrapolation prediction, because the magnitude and pattern of the temperature rises were outside the range of those used in the model estimation. Figure 11 shows one example, in which the unscreened regression model performs well in interpolation prediction, but becomes poor in extrapolation prediction. On the other hand, the screened regression model performs well in both cases. However, it was also noted that the prediction accuracy of the screened regression model was unacceptable in some cases where the temperature rise patterns were totally different from those used in the model estimation, such as the thermal bending of the column (see Fig. 12).

Finally, a hybrid stepwise MRA model was built from a data set collected from air

5 0 1 measuaementm . I

I "a

V 4 hours 4 hours

Time

Fig. 11. Prediction of the thermal drift of the cutting edge in the z-direction.

101 measurement I '~ [ ^^A^ unscreened M~A

, •

• ~- "5F interpolation '! .......... __~/~k ~ - l O / ~ i - ; ~terp°l~i°n v v '

, (aluminum roughimz)

4 hours 4 hours Time

Fig. 12. Prediction of the thermal bending of the column.


cutting and real cutting data. Figures 13-16 shows that the air cutting MRA model predicted errors well in air cutting, but became poor in reali cutting. For some cases in real cutting the air cutting model made the serious mistake of proceeding in the opposite direction of the actual error. The hybrid MRA model, however, gives satisfactory accuracy in both air and real cutting conditions.

4. ARTIFICIAL NEURAL NETWORKS

In this research, a three-layer feed-forward artificial neural network (ANN)-based model was used to nonlinearly map calibrated thermal items toltempemture measurements (see Fig. 17). The activation function of each artificial neuron ~s the "sigmoid" function which can perform automatic gain control on the input signal. The nonlinear mapping between the thermal errors and temperature measurements are represented in the form of connected weights between neurons through a supervised backpropogation training algorithm. The

m m I

50 measurement

30

.~ 20

lO

0

xxxx air c.uning.~odei

r f m m I m ~ ~ m m ~ m ~

air cutting iuum cutting iron cutting steel cutting 4 ;hours 4 hours 4 hours 4 hours


40 , ~ I - - m e - - r e = n , ~ l

3oi- x ~ al, cu~.gmode, ×x x x ~ x × l , . , / c~oo hybrid model X . .X x [

• ~ 1

4 hours 4 hours 4 hours: 4 hours

Fig. 14. Prediction of the thermal expansion of the y:axis.

4 ° t _ m . - - n , ~ 3 0 xxxx aircutting model [ cooo hybrid model .X . x ,X X [

¢ X Xi X o

-1 . , I

4 hours 4 hours 4 hours 4 hours

Fig. 15. Prediction of the thermal bending of the column,

1408

8

6

"~ 4 3 .~.'4

2 ca

0

Jenq-Shyong Chen

air cutting 4hours

- - m e a s u r e m e n t xxxx air cutting model oooo hybrid model

~-uminum cutting'~irpn cutting .h. 4 hours 4 hours

. d

steel cutting 4 hours

Fig. 16. Prediction of the thermal inclination of the cutting tool axis.

A ' l { - ' " O ~ ~ ~ ~ ~ d ~ C ) ' - " 61

hidden 6 m ATn input ouput Fig. 17. Three-layer feedforward ANN.

initial weights of the network were randomized between -0 .3 and 0.3. A learning rate (i.e. updating the rate for the weights at every iteration) of 0.2 was chosen. All training pairs were normalized between 0.1 and 0.9. The normalization of input and output variables was done by normalizing each input channel separately because it had the advantage of fully utilizing the signal of every channel. The weights of the ANN were updated with each pattern passed.

One advantage of the ANN model over the MRA model is that multiple thermal errors can be easily modeled together. In other words, all of the thermal sources were modeled using only one ANN model with multiple inputs and outputs. However, care must be taken when doing this. For the extrapolation prediction, the single output model for each error may give a better performance, because the input variables of each error model can be independently screened to an optimal case. The multiple output model, on the other hand, needs a common set of input variables for each error. Therefore in some cases (as shown in Fig. 18), the single output model performs more robustly than the multiple output model. However, as with the stepwise regression model, if the temperature patterns are

0 I ..... measuremerlt ~ f

"-'E 20 AAAA single output ANN / ~ _ ~ =L oooo multiple ouTut A N N / ~ A / ~

o 10

.= 0

ca -1G interpolation ! e (air cutting) !

-2G , (aluminum roughing)


Fig. 18. Prediction of the thermal drift of the cutting edge in the x-direction.


totally different from those used in model estimation, ithc prediction accuracy of the screened ANN model also becomes unacceptable (see Fig. 19).

Finally, a hybrid multiple input and output ANN model was also built. The air cutting ANN model predicted errors well in air cutting conditions, but became unacceptable in real cutting conditions (see Figs 20-23). The hybrid ANN model, however, was sufficiently accurate in both air and real cutting conditions.

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5. CONCLUSIONS

In this research a multi-variant regression analysis (MRA)-based error model and an artificial neural network (ANN),based error model built from air and real cutting conditions are studied. The MRA model has the advantage in that the physical meaning of the model can be easily interpreted and the sensitive gain value of each coefficient is available. One advantage of the regression analysis is its robustness ag~finst conditions that differ from those used in tbe model estimation can be increased by careful selection of the input variables using the stepwise regression analysis. The ANN model has the advantage that

51 , [ I ' ,~ I

"~ "5~°°°°.. ~ ple°utp~itANN ~ h A

-10 l n )


Fig. 19. Prediction of the thermal bending of, the column.

--~ /measurel~nt I' 1

,.'rou .gm e, xxXX xxXX X x

",~ ~x x

]"=air cutting - ~ u t ~ u m cutting'~-iron cutting "~s tee l cutting 4 hours 4 hours 4 hours 4 hours


~, 4 0 [ - - measurement ~30~ xxxx air cutting model ,.~ __1 ¢xx3o hybrid model .~*' ! ,~q. .f

. , o t . . . . . .

4 hours 4 hours 4 hotWs 4 hours

Fig. 21. Prediction of the thermal expansion of the x-axis.


4. +_~o x

~ ' 2 " " '

,~ 0 ~ ~ ] ' , ~ ~ . ~ x Xx x,.xr, x x x~xY Xx x xx~ x xx,+m, xx x..X, x x-x ..+-°,+I . . . . . ~ -4 =l ~. measurement el ~ J ~ V" " ' cn xxxx air cutting mod

-61 oooo, h~'brid model

8 mg ~ron cutung stee= cutUng 4 hours 4 hours 4 hours 4 hours

Fig. 22. Prediction of the thermal bending of the column.

o "o 0 L ° * , ,d

=0 -5 ¢1 -10

• X X o ; • ° ° ° o ~ o

measurement X " X ' X [ I

XXXooo~X ahi;bUtdfimngodeltTelXxxxxxXX XxxxX X x XxXX~ I air cutting n cutting "~s l ee l cutting ' '1 4 hours 4 hours 4 hours 4 hours

Fig. 23. Prediction of the thermal inclination of the cutting tool axis.

very complex models can be automatically learned. However, it is difficult to interpret the physical meaning from the ANN model. The ANN model also has the advantage that multiple thermal errors can be easily modeled together. In other words, all of the thermal sources were modeled using only one ANN model with multiple input and outputs.

Because the cutting load and applied cutting coolant in real cutting conditions can produce significant thermal effects not explored by the air cutting test, patterns of temperature rises in real cutting are very different from those generated in air cutting. Consequently, the air cutting model performs well in air cutting, but becomes unacceptable in real cutting applications. In some cases the air cutting model produces an output in the direction opposite to the actual error. On the other hand, the hybrid model estimated from the air cutting and real cutting data gives satisfactory accuracy in both air and real cutting conditions.

REFERENCES

[1] R. Venugopal and M. Barash, Thermal effects on the accuracy of numerically controlled machine tool. Ann. CIRP 35(1), 255-258 (1986).

[2] M. Weck and L. Zangs, Computing the thermal behavior of machine tools using the finite element method- possibilities and limitations, 16th MATADOR Conf., Vol. 16, pp. 185-194 (1975).

[3] J. Bryan, International status of thermal error research. Ann. CIRP 39(2), 645--656 (1990). [4] J.S. Chen, J.X. Yuan, J. Ni and S.M. Wu, Real-time compensation of time-variant volumetric error on a

machining center. ASME Trans. J. Engng Ind. 115, 472-479 (1993). [5] M.A. Donmez et al., A general methodology for machine tool accuracy enhancement by error compensation,

Precision Engng 8(4), 187-195 (1986). [6] K.C. Fan, J.F. Lin and S.S. Lu, Measurement and compensation of thermal error on a machining center,

29th MATADOR Conf., England, April, pp. 261-268 (1992). [7] B.R. Hardwick, Improving the accuracy of CNC machine tools using software error compensation for

thermally induced errors, 29th MATADOR Conf. England, April, pp. 269-276 (1992). [8] J.A. Soons, H.A. Spaan and P.H. Schellekens, Thermal error models for software compensation of machine

tools, Proc. 9th Ann. Meet. American Society for Precision Engineering, October, pp. 69-75 (1994). [9] J.S. Chen, J.X. Yuan, J. Ni and S.M. Wu, Thermal error modeling for volumetric error compensation, ASME

Winter Ann. Meet. Sensors and Signal Processing for Manufacturing, PED-Vol. 55, pp. 113-125 (1992). [10] Y. Hatamura, T. Nagao, M. Mitsuishi, K. Kato, S. Taguchi and T. Okumura, Development of an intelligent

Thermally Induced Machine Tool Errors in Real CuRing Conditions 1411

machining center incorporating active compensation for thermal idistortion. Ann. CIRP 42(1), 549-552 (1993).

[11] N. Srinivasa and J.C. Ziegert, Real-time learning of thermal errors in machine tools using a fuzzy logic based neural nel~vork. Manufact. Sci. Engng ASME, 64, 235-240 (1993).

[12] K.C. Fan and K.Y. Hung, Error compensation of spindle expansion by cutting model on a machining center, 31st MATADOR Conf., England, April, pp. 269-275 (1995).

[13] M.H. ARia and L. Kops, Computer simulation of nonlinear therml3elastic behavior of a joint in machine tool structure arid its effect on thermal deformation. ASME Trans. J. Engng Ind. 101, 355-361 (1979).

a study of thermally induced machine tool errors in real cutting conditions

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