Pergamon Int. I. Mach. Tools Manufact. Vol. 37, No. 2, pp. 159-169, 1997

Copyright © 1996 Published by Elsevier Science D.d Printed in Great Britain. All rights reserved

0890--6955/97517.00 + .00

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F A S T C A L I B R A T I O N A N D M O D E L I N G 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 M A C H I N I N G

J.-S. CHEN'~

(Received 16 January 1996; in final form 16 April 1996)

Abstract---Calibration and modeling of thermally induced errors is a critical part of enhancing machine accuracy by software error compensation. In most applications, parametric thermal errors of a machine tool are calibrated and modeled individually by air-cutting experiments. Calibrating thermal errors individually is time-consuming and may neglect thermal interaction among thermal sources. The accuracy of the air-cutting model in real machining is also questionable. In this report, thermal errors of multiple machine axes in real cutting were calibrated simultaneously by a quick set-up measurement system consisting of on-machine probes and artifacts. Characteristics of thermal errors in real cutting under different cutting conditions, cutting paths and workl~ece materials were investigated. It was found that thermal errors in real machining were distinct from those in air cutting. Copyright © 1996 Published by Elsevier Science Lid

1. INTRODUCTION

Thermally induced error is recognized as one of the major contributors to the volumetric positioning inaccuracy of machine tools. For a multi-axis machine tool, the volumetric positioning accuracy is determined by an open kinematic chain consisting of the spindle axis, three linear axes and supporting machine structures. Thermal deformations of these machine components include dimensional elongation such as in the spindle and ballscrew, and angular distortions such as column bending and tool-axis inclination. Consequently, thermally induced volumetric positioning error is time-variant in the cutting time-span and spatial-variant in the machine working zone. Volumetric error calibration of a multi-axis machine can be done by functional measurement (i.e. workspace distortion measurement) where the overall volumetric positioning accuracy of the cutting edge in the working zone is measured [1--4], or by parametric measurement (i.e. error element measurement) where the error elements of each machine axis and the metrology framework are measured [5, 6].

The major problem with previous works [7-9] is that thermal errors are calibrated using air-cutting experiments (i.e. no load) as specified in the ANSI, ISO and BS standards [10- 12]. In order to simulate the cutting load under real-cutting conditions, the air-cutting experiments are often conducted with high spindle speeds and high feedrates which are not characteristic of real cutting. Although high-speed air cutting does increase the loading of drive motors, it has been found [13] that there are some heat sources generated in actual machining that cannot be addressed by high-speed air cutting, such as the cutting load- induced friction of the drive mechanisms. Hot chip and cutting coolant accumulated on the working table and machine base might be the additional thermal sources not taken into account by the air-cutting approach. Another problem with previous works is that parametric thermal errors are calibrated one at a time: Calibrating thermal errors individu- ally is not only time-consuming, but also may neglect thermal interaction among thermal sources, because the multiple axes of a machine are operated simultaneously in real-mach- ining conditions.

In this report, a measurement system which calibrates thermal errors of multiple machine axes simultaneously in real cutting is described. The thermal errors calibrated by this system include the thermal drifts of the cutting edge at a reference coordinate, thermal

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

159

160 J.-S. Chen

expansion of three linear axes, thermal bending of the column, and thermal inclination of the cutting tool axis. The spatial-variant volumetric thermal error at any cutting location in the machine working zone can be easily determined by error synthesis. Characteristics of thermal errors in real cutting using different cutting conditions, cutting paths and work- piece materials were studied. A multiple-output ANN model was used to model all cali- brated thermal error sources together. In other words, all thermal errors are predicted while machining using only one ANN model. The set of temperature sensors to be used in the model was screened in advance by a stepwise regression analysis. The prediction accuracy and robustness of the neural network model in the air-cutting and real-cutting experiments were also investigated.

2. FAST THERMAL ERROR CALIBRATION SYSTEM

Thermally induced positioning error of the cutting edge for a vertical machining center results from a combination of spindle growth, cantilever-arm bending, z-axis expansion, column bending, y-axis expansion and x-axis expansion (see Fig. 1). Consequently, the volumetric positioning error of the cutting edge is spatial-variant in the working zone and time-variant in the cutting time span. In our research, these thermal error sources were calibrated simultaneously by a quick set-up measurement system consisting of on-machine probes and artifacts. This system can be operated in real-cutting conditions where the spindle and the three linear axes are operated simultaneously, and the cutting load and coolant are applied. Steps in calibrating these thermal errors are discussed in the follow- ing paragraphs.

(1) Calibration of the thermal drift of a machine coordinate with a specified probe length. The thermal drift of a machine coordinate is calibrated by a spindle-mounted MP7 probe and a gauge block fixed on the working table. At the cold start, the coordinates [x0(0), yo(0), Zo(0)] of the gauge block were measured using the MP7 probe. Then the machine was operated continuously in order to heat it up and the coordinates of the gauge block were measured periodically. The thermal drift of the specified machine coordinate can be calculated as follows:

AP(n, Xo, Yo, Zo, MP7)= [xo(n)-xo(O), yo(n)-yo(O), zo(n)-zo(O)]. (1)

[Xo(n), yo(n), Zo(n)] are the measured coordinates of each measurement cycle. Because of the column bending, tool inclination and ballscrew expansion, AP(n, Xo, Yo, Zo, MP7) depends on the specified machine coordinates [Xo, Yo, Zo] as well as the tool length of the MP7 probe.

(2) Calibration of the linear-axis expansion. Thermal expansion of the horizontal linear axis is determined by the variation of the calibrated length of a quartz tube put in parallel with the axis. Quartz material was preferred, due to its very low thermal expansion coef- ficient. Thermal expansion of the vertical linear axis was calibrated using a granite height gauge. The temperature variation of the granite gauge during testing was monitored con-

z-axis

column :~:l

Fig. 1. Thermal error sources.

Fast calibration and modeling of thermally induced machine tool errors 161

tinuously in order to compensate for the dimensional deviation of granite gauge due to the thermal effect. Because the calibrated value depends on the nominal length L(0) of the gauge tube, it is better to present the thermal expansion as a gain value. Gain values of the thermal expansion in the x-, y- and z-directions can be determined as

K~(n) = (Lx(n)-Lx(O))lLx(O), (2)

= (Ly(n)-t,y(O))/Ly(O), (3)

Kt.z(n) = (Lz(n)-Lz(O))lLz(O). (4)

(3) Calibration of the column bending. As shown in Fig. 2, the thermal bending of the column is determined by the measured coordinates [xl(n), yl(n)] and [x2(n), y2(n)] of a gauge block at two z levels. Gain values of the column bending in the x- and y-directions can be calculated as

Ksx(n) = (x2(n)-x,(n)-x2(O) + xl(O))lL, (5)

Ksr(n) = (yz(n)--yl(n)-y2(O) + yl(O))/L. (6)

(4) Calibration of the tool-axis inclination. To calculate the inclination of the tool-axis, the thermal drift of the cutting edge at a specified machine coordinate must be measured using different tool lengths. This requirement was achieved by using a spindle-mounted MP7 probe, and a table-mounted MP4 probe (as shown in Fig. 3). Assume that the length difference between the MP7 and the test bar is L. The gain value of the tool-axis inclination in the x- and y-directions can be calculated as

Krx(n) = (xE(n)-xl(n)-x2(O) + xl(O))lL, (7)

1

x2(n), y2(n)

xl(n), yl(n),

Fig. 2. Calibration of column bending.

x2(n),

I

~ ~ - - - ~ xl(n),

Fig. 3. Calibration of the tool-axis inclination.

162 J.-S. Chen

Krr(n) = (y2(n)-yl(n)-y2(O) + yl(O))/L. (8)

After the thermal error sources are identified, the volumetric positioning error of any location in the working zone is then determined by error synthesis from these error sources. First, notice that the calibrated error AP(n, Xo, Yo, Zo, MP7) in Equation (1) is actually the volumetric positioning error of the specified machine coordinates [Xo, Yo, Zo] using a speci- fied cutting tool length of the MP7 probe. Therefore, when the machine is operated at coordinates [x, y, z] different to [Xo, Yo, Zo] and with a cutting tool length TL different from the MP7 probe, modification terms due to the linear-axis expansion, column bending and tool-axis inclination should be added to Equation (1). That is,

AP(n,x,y,z,TL) = AP(n,xo,Yo,Zo,MP7) + APlinear_axis(n,X-Xo,y-yo,Z-Zo ) (9) + APcolumn_bending(n,Z--Zo ) + APtool_inclination(n,TL-MP7 )

APlinear-axis=(n,x--Xo,y--yo,Z--Zo) is the added modification term for the thermal expansion of the linear axes, which depends on the distance between the current coordinates [x, y, z] and the reference coordinate [Xo, Yo, Zo]. That is,

(X-Xo)K~v(n)]

APli,e~-axis(n,X-Xo,Y-Yo,Z-Zo) = (Y--yo)KLr(n) ]. (lO)

(Z--Zo)KLz(n) J

APcotum-bending(n,Z--Zo) is the added modification term for the thermal column bending, which depends on the distance between the current z level and the reference Zo level. That is,

A Pcolum,_be,ding( n,Z-- Zo) = "(Z-Zo)Ksx(n) ]

(Z-Zo)~sz(n)J. (11)

APtool-i,clination(n, TL-MP7) is the added modification term for the thermal inclination of the cutting tool-axis, which depends on the length difference between the current tool length TL and the MP7 probe length. That is,

(TL-MP7)Krx(n)] APtoo,-incn,at,o,,(n,TL-MPT)= [(TL-MP~)KTr(n)J. (12)

3. CHARACTERIZATION OF THE THERMAL ERRORS

Some experiments were conducted to investigate the characteristics of thermal errors under air-cutting and real-cutting conditions.

3.1. Effect of the cutting conditions and coolant To study the characteristics of thermal errors under different cutting conditions, four

tests with the cutting conditions listed in Table 1 are compared. The thermal drift in the Table 1. List of cutting conditions for the air cutting and aluminum cutting

Depth of cut (mm) Spindle speed (rpm) Feedrate (mm/min) Cutting coolant

Low-speed air 0 1000 300 No cutting High-speed air 0 5000 3000 No cutting Aluminum finishing 0.5 3000 200 No Aluminum roughing 5 3000 200 Yes

Fast calibration and modeling of thermally induced machine tool errors 163

z-direction due to the spindle growth is generally recognized as the most important error source and is often correlated to the temperature rise of the spindle housing. We plot the experimental data in Fig. 4 using the temperature rise of the spindle as the independent variable. It shows that the relationship for each cutting condition is nonlinear. Also, the thermal drift could not be correlated to this single temperature sensor on the spindle hous- ing for all cutting conditions because there are multiple possibilities. For example, when AT is 6°C, the magnitude of the thermal drift could vary between 20 and 40 ttm, depending on the cutting conditions. This is because the thermal drift of the cutting tool tip arises from a combination of several thermal error sources, such as the spindle growth, cantilever arm bending, column bending and leadscrew expansion as shown in Fig. 1. For that reason, multiple temperature sensors from several machine components must be considered for an error model.

The thermal expansion of the x-axis is often correlated to the temperature rise of the motor in air cutting because the motor is considered as the major heat source. However, Figs 5 and 6 show that here again there is no such simple relationship for all cutting

o " o

Fig. 4. Thermal

5O

4O

3O

2O

l0

0 2 4 6 S l0

Temperature rise (°C)

drift in the z-direction versus temperature rise of the spindle.

4O 4

35 coolant applied

~ i 2

0 i 00 200 300

Time (min)

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

6 - -

s 2"

3 1

.

o 100 200 30O

Time (min)

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

164 J.-S. Chen

conditions. Also notice that the error magnitude in low feedrate roughing (200 mm/min) is almost three times that of air cutting using an extremely high feedrate (3000 mm/min), although the temperature rises of the motor were similar in magnitude in that two cases. Hot chip and cutting coolant accumulated on the working table and machine base had produced significant thermal errors. This can be proved by temperature measurement of the working table, where a very high temperature rise during aluminum roughing is found (see Fig. 7).

Figure 8 shows that the column bends very little in low spindle-speed air cutting and aluminum finishing, but bends backward significantly in high-speed air cutting. This is because the spindle unit generates much heat in high-speed air cutting. However, the column bends in the opposite direction during aluminum roughing. This indicates that the high-speed air cutting approach cannot accurately simulate the cases happened in real cutting. We 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 side and rear side of the column (see Figs 9, 10 and 11).

6

eJ

• ~. 4

2

. . . .

, ~ A ~ coolant applied

,)~L 3

, & , • 2

100 200 3O0

Time (rain)

Fig. 7. Temperature rise of the working table.

6 ' "

4 2

4

"80 100 200 300

Time (rain)

Fig. 8. Thermal bending of the column.

I==

"3, J l U

c o o . control electricity valve control box

'1 i1 I coo=,, coolant ~ = ~ r , pump "-~J_:mID

Fig. 9. Arrangement of the cutting coolant system.

Fast calibration and modeling of thermally induced machine tool errors 165

0.6

0.4

0.2

o

"~ -0.2

~- -0.4

-0.6

-0.80 l ~ 2~ 3O0 Time (min)

Fig. 10. Temperature gradient of the column (from front to rear).

c

Fig. 1 i.

1.6

!.2 4

0.8 coolant\app lied

0.4 O.

O2 " 0 IO0 2O0 30O Time (rain)

Temperature gradient of the column (from left to right).

3.2. Effect of the cutting paths The effect of cutting paths was investigated under the cutting conditions listed in Table

2. Because different directions of cutting paths produce different friction effects on the x- and y-axes, we arranged zig-zag and window cutting paths for the 2 mm depth of cut. Figs 12 and 13 show that the thermal expansion of the x- and y-axes strongly depends on the depth of cut and coolant condition, but are little affected by the change of cutting paths.

Table 2. List of the cutting conditions and cutting paths

Cutting path Depth of cut (mm) Spindle speed (rpm) Feedrate (ram/rain) Coolant

Zig-zag (y) 0.5 3000 200 No Zig-zag (y) 2 3000 200 Yes Zig-zag (x) 2 3000 200 Yes Window 2 3000 200 Yes Zig-zag (y) 5 3000 200 Yes

45 4 0 , . , . / - 5

,_, 3 5 ~ 30 2

"~ 25 20

I 0 1 5 o!

-5 ~' I(~0 200 300 Time (min)

Fig. 12. Thermal expansion of the x-axis in different cutting paths.

166 J.-S. Chen

..i

3{]

25

20

15

10

5

ol -5

0

Time (min)

300

Fig. 13. Thermal expansion of the y-axis in different cutting paths.

3.3. Effect of the workpiece materials As shown in Table 3, a direct comparison of the thermal errors among different work-

piece materials is difficult, because the cutting conditions are different. The objective of this investigation was to see whether any additional heat source appeared when cutting different materials. Figures 14 and 15 show examples which indicate that the character- istics of thermal errors in machining iron followed the same pattern found in aluminum

Table 3. List of the cutting conditions for different workpiece materials

Depth of cut (ram) Spindle speed (rpm) Feedrate (mm/min) Coolant

Aluminum finishing 0.5 3000 200 No Aluminum roughing 5 3000 200 Yes Iron finishing 0.5 1500 120 No Iron roughing 3 1500 120 Yes

45 40 35

g 3o 25

~ 2o

:~ zo 5

0

-5~ 300

4

1

3

1~0 2do Time (rain)

Fig. 14. Thermal expansion of the x-axis.

2

I 3

° I~ "

Time (rain)

Fig. 15. Thermal bending of the column.

Fast calibration and modeling of thermally induced machine tool errors 167

machining. The same phenomena were found when cutting carbon steel materials. In con- clusion, there was found no additional thermal source when cutting different workpiece materials.

4. MODELING AND PREDICTION OF THERMAL ERRORS

In this research, a three-layer artificial neural network (ANN) with a supervised backpro- pagadon training algorithm was used to map calibrated thermal errors to temperature measurements, A stepwise regression analysis with a computerized automatic search algor- ithm was used in advance to screen the temperature variables of the ANN inputs. One advantage of the ANN model is that multiple thermal errors can be easily modeled together. In other words, all thermal sources can be monitored while machining using only one ANN model with multiple output. However, this must be done carefully. In some cases a single-output ANN built for each thermal error may perform better than a multiple- output ANN built for all thermal errors. This is because the input variables for each thermal error can be independently and optimally screened, while for the multiple-output ANN all thermal errors must share a common set of temperature variables. To demonstrate the importance of screening temperature variables, air cutting-based ANN models were built and used to predict thermal errors in a new air cutting and an aluminum roughing. The new air cutting was actually an interpolation prediction, because the magnitude and pattern of the 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 very different from those used in the model esti- mation. Figure 16 shows one example where the multiple-output ANN performs well in interpolation prediction, but becomes poor in extrapolation prediction. On the other hand, the single-output ANN with optimally screened input variables performs well in both cases. However, it was also observed that if the temperature patterns were totally different from those used in the model estimation, such as in the thermal bending of the column, the optimally screened single-output ANN built from the air cutting tests also became unac- ceptable in real cutting application (see Fig. 17).

Finally, a hybrid ANN model with multiple-output was trained with the data collected

30

2O -4

o I0 "0

-8

m

measurement single output ANN ~ " ~ . ,~

oooo multiple output ANN S A ~k ~

" -- ~ - -ex t rapo la t ion I "3

Interpolation ~ o o o o o o o o o (air cutting) I (aluminium roughing)

| i i , 4 h o u r s 4 hours

Time

Fig. 16. Prediction of the thermal drift of the cutting edge in the x-direction.

i

"~ measurement ~,'N.. j,~ "~ = AAAA single output ANN v " ~ "~ -5 oooo multiple output ANN ~ • A

' ' xtrapolation W ~ lnterpolaUon ~ e (air cutting) I (aluminium mufhinl[)

-10 4 hours 4 hours Time

Fig. 17. Prediction of the thermal bending of the column.

168 J.-S. Chen

from the air cutting and real cutting experiments. Figures 18, 19, 20 and 21 show that in some cases the air cutting-based ANN model predicts errors in the opposite direction of the actual error in real cutting. The hybrid ANN model, however, gives satisfactory accu- racy under both air-cutting and real-cutting conditions.

5. CONCLUSIONS

This work has found that although the traditional high-speed air cutting test does increase the loading of the motor, the cutting load and cutting coolant in real cutting can produce significant thermal errors not addressed by the air-cutting approach. Because the tempera- ture pattems in real cutting are very different from those in air cutting, the air-cutting- based model may perform poorly in real-cutting applications. On the other hand, the hybrid model estimated from the air-cutting and real-cutting data can give satisfactory accuracy under both air-cutting and real-cutting conditions.

50

4O

i 30 '-" 20

l0

-8

measurement x ~ ( x X X x x x x air cutting model X " Y X'XXX X x

"0ooo hybrid mo~e~ '/ f ~ X X'' ~ , , /x, . ,oooooo . , . ~ r : ' d o x o ~ v_

Z": ] , / , , ,

£ - - - - "-- " ~ "--'-- - - - - ~ ' s t e e l cutting air cutting alummmm cutting iron cutting ~ " ~ 4 hours 4 hours 4 hours 4 hours

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

40

3O

=L 20

"o

• ~ I0

0

-10

4 i

2

~0 v

,g '~ -2

-4

measurement XXXX air cutting model _.O,~, 0o0o hybrid model ~ ~ )1 J ~ ~ "

f . ~ : ~ ~ o xxXX~ / o ~ V x x x ~ ,o ° x x~

- :

4 - air cutting "~a luminium c u t t i n g ' ~ iron cutting " a ~ s t e e l cutting .1~ 4 hours 4 hours 4 hours 4 hours

Fig. 19, Prediction of the thermal expansion of the x-axis.

• .XSo x

: P ~ , o ~ "XXx - ×Xxxxx~ xxx xxxxx ¢ s . . . . . - ~ : ~ - ~ , ~ % - - - - : ~ g q ~ - ~ -

• ./..,,, m: : rm2Lo, V oooo hybrid model

air cutting -~aluminium cutting-a~iron cutting "~h~steel cutting "~ 4 hours 4 hours 4 hours 4 hours

Fig. 20. Prediction of the thermal bending of the column.

Fast calibration and modeling of thermally induced machine tool errors 169

m e a s u r e m e n t . .

~¢ 5 xxxx air cutting model XXXXXXXX XxxX ~XXX XXxXXX oooo hybrid model

-10

air cutting "~aluminium cutting -~6" iron cutting "~'~-steel cutting " ~ 4 hours 4 hours 4 hours 4 hours

Fig. 21. Prediction of the thermal inclination of the cutting tool-axis.

Acknowledgements--Financial support from the National Science Council of Taiwan, R.O.C., project number NSC 83-0422-E-194-002, is gratefully acknowledged.

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