error compensation

IWMF2014, 9th INTERNATIONAL WORKSHOP ON MICROFACTORIES OCTOBER 5-8, 2014, HONOLULU, U.S.A. / 1

1. Introduction

The machine tool accuracy directly affects on the

dimensional accuracy of the machined products. In response to

the increasing requirement of product quality, the nature of

thermally induced errors and geometric errors has been

investigated by many researchers over the past few decades1,2

.

The thermal error accounts for 40-70% of the total machining

errors and it is the most significant factor influencing the

machine tool accuracy1. Thermally induced errors are quasi-

static errors that change slowly in time and are closely related

to the machine tool structure. The commonly used empirical

thermal error model assumes that the thermal deformation of a

machine tool at a particular instant depends on the temperature

distribution in the machine tool at that particular time and

describes the relation between the thermal deformation and the

temperature distribution3-5

. These static error model

approaches often tend to be unreliable when working

conditions are different from the tested conditions.

To provide more robust models, researchers have taken into

consideration the dynamics of the thermoelastic process in a

machine tool. Moriwaki6 suggested using an empirically

determined transfer function model describing the process

parameters between spindle speed, ambient temperature, and

the thermal deformation in the machine tool. Li7 adopted an

auto-regressive (AR) model to predict spindle thermal errors

from spindle speed. Hong8 showed that the pseudo-hysteresis

effect is the main factor causing poor robustness of the static

error model and proposed to use a linear output error (OE)

model to predict the machine tool thermal deformation.

We present a thermal error compensation scheme in which

a state-space model is used for thermoelastic process modeling.

In our simulation, the state-space modeling approach showed

better robustness to capture the dynamic nature of thermal

deformation than other linear parametric models including AR

and OE models. Our machining process usually takes one or

two hours and during the machining a thermal drift appears

with roughly a period of 13-15 minutes and a magnitude of

300-400 nanometers.

2. Testbed Machine: a High Precision Lathe

The testbed high precision lathe has X- and Z-axes and

these axes are driven by linear motors and hydrostatic oil

bearing. The spindle in the testbed machine has a built-in

motor with air bearing. Air is supplied to the spindle for

cooling. A picture of the testbed is shown in Fig. 1. The stroke

Thermal error compensation for a high precision lathe

Byung-Sub Kim# and Jong-Kweon Park

Department of Ultra Precision Machines and Systems, Korea Institute of Machinery & Materials, Daejeon, South Korea # Corresponding Author / E-mail: [email protected], TEL: +82-42-868-7109, FAX: +82-42-868-7180

KEYWORDS : Thermal error model, Compensation, Thermoelastic process, Desktop 5-axis milling machine

High precision machines require very stable operational environment: temperature control and vibration isolation. Tight temperature control for machines usually demand high cost to operate air conditioners. Some of high precision machines require the ambient temperature changes to maintain within 0.1 degrees. In this paper, we present a thermal error compensation scheme and experimental results for improving machining accuracy of a high precision lathe. The testbed lathe has X- and Z-axes and they are driven by linear motors and hydrostatic oil bearing. Due to the temperature changes of the ambient air and supplied oil to the hydrostatic bearing, thermal deformation is generated and measured to be around 0.4 m. To identify the dynamic relations between the temperature changes and the thermal drift, a state-space model is used in which state variables are constructed from the input measured temperatures and the output thermal drift data. The identified model is implemented in a servo control loop and the predicted thermal error is compensated by subtracting the predicted thermal drift from the servo command. In our simulation, a thermal error of 97 nanometers RMS over 3 hours is reduced to 55 nanometers RMS. Experimental results show an average of 24% reduction in thermal drift and support the validity of our approach.

239


of the X-axis is 200 mm and that of the Z-axis 50 mm. The

linear encoders for position feedback have a 10 nanometer

resolution for the X- and Z-axes. The temperature of the

machining room is maintained by an air conditioner with 0.6

C variations. Covering the lathe with metal panels reduces

the ambient temperature change around the machine to be

0.3 C. The temperature of the oil from the hydrostatic oil

bearing is controlled by an oil-cooler within 0.13 degrees,

when the temperature is measured from the oil drain for the

accessibility reason. When the oil bearing and the spindle are

turned off, the distance between the spindle and the tool post

has a thermally induced oscillation of 70 nanometers reflecting

the ambient temperature changes. When the oil bearing and a

servo control are on and the spindle rotates at a speed of 1,000

rpm, thus the lathe is ready to operate, the magnitude of the

oscillation increases to be around 300-400 nanometers.

Fig. 1 Picture of a high precision lathe

Fig. 2 Measured error on a workpiece surface due to thermal

deformation and straightness error

After a diamond turning test, it was found that there was a

wavy form error on the machined surface induced by thermal

deformation and straightness error as shown in Fig. 2. Before

tackling the thermal error, we used two actively controlled

capillaries (ACC)9, which acted on one side of the X-axis, to

reduce the horizontal straightness and yaw errors of the X-axis.

The ACC has a piezo actuator to change the supplied oil

pressure to the hydrostatic bearing. The oil pressure change

along the X-axis offers two-degrees-of-freedom motion to the

X-axis stage. The input voltage values to the piezo actuators

were empirically determined so that the straightness and the

yaw errors of the X-axis could be reduced under a certain level.

The straightness error of the X-axis was corrected to be less

than 50 nanometers and the yaw error less than 0.15 arcsec as

shown in Fig. 3. The reason why the Z-axis was not used to

compensate for the horizontal straightness error of the X-axis,

was that the encoder resolution of the Z-axis was not small

enough and we could get finer dynamic response from ACC in

reduction of the straightness error. The next target for high

precision machining goes to the thermally induced error.

Fig. 3 Straightness and yaw error profiles corrected by ACC

3. Linear Thermoelastic System Identification

To obtain a linear thermal error model describing the

relation from the temperature inputs to the thermally induced

error output, four temperature sensors were attached to the

testbed lathe and the sensor values were measured at every 1

second. The temperature sensor locations are spindle housing,

table, oil drain, and ambient air around the machine. The red

dots in Fig. 4 show specific locations for temperature

measurement except the ambient temperature. A temperature

sensor for the ambient temperature is hung approximately 200

mm over the tool holder.

Fig. 4 Temperature sensor locations shown as red dots

Thermal drift and accompanying temperature sensor

values for 4 hours are shown in Fig. 5. The thermal drift is the

relative displacement change between the workpiece and the

-0 .1

0.0

0.1

0 50 100 150 200-0.5

0.0

0.5

Stra ightness error

De

via

tio

n

m

Yawing error

De

via

tio

n

arc

se

cPosition m m

X

Z

Spindle

Tool holder

240


tool holder, which is measured by a gap sensor, and all the

values in the plots are adjusted to have zero mean values. The

positive displacement value means that the workpiece and the

tool holder come closer in our experimental setup. The thermal

drift shown in Fig. 5 is 70 nanometers RMS. Notice the

symmetric configuration of the high precision lathe. Due to

symmetry, the relative displacement change between the

workpiece and the tool holder does not show noticeable

difference along the X-axis. In Fig. 5, we can see some pattern

in the thermal drift and the ambient temperature change.

Correlation coefficients indicate that the ambient temperature

(0.52) has the most influence on the thermal drift, and the next

is the table and the oil drain temperature (0.48 and -0.15,

respectively) in our experimental setup. The temperature of the

spindle housing appears to have least influence on the thermal

drift. It is because the spindle was continuously provided

cooling air through dedicated air tubes. The ambient

temperature around the lathe seemed to include the

characteristics of the table temperature. Thus, we chose the oil

drain temperature and the ambient temperature as the inputs to

our linear model and the thermal drift as the output. In our

simulation, adding the table temperature as one of the input

parameters to our thermoelastic model did not help improving

the accuracy of prediction for the thermal drift.

Fig. 5 Thermal drift (top) and accompanying temperature sensor

values (below four plots)

When a linear regression model was used, the predicted

thermal drift did not match well with the real data. Any

compensation based on the linear regression model seemed to

make the error bigger in our particular case. To consider the

dynamics of the thermoelastic process in the testbed, different

system identification methods were tested and evaluated. They

were ARX(auto-regressive exogenous input), ARMAX(auto-

regressive moving-average exogenous input), BJ(Box-

Jenkins), OE, and state-space models.

A state-space model did not always show best accuracy in

prediction of thermal drift among tested dynamic models, but

it showed more robust results than other models. For example,

an OE model and a state-space model, which were built based

on the data set depicted in Fig. 5, were applied to another data

set and their predictions were compared in Fig. 6. The solid

line shows the measured thermal drift of 97 nanometers RMS

over three hours. The top plot shows the predictions by the OE

model (dashed line) and the state-space model (dash-dot line)

with unperturbed temperature data. If the thermal drift is

compensated as much as predicted by the OE model, the

thermal error will be reduced to 87 nanometers RMS, and if

the state-model is used for compensation, the thermal error

will be 55 nanometers RMS in this particular simulation. We

can see that the both models can track the general shape of the

thermal drift. When a small constant bias is injected in the

temperature data, the simulation results are compared in the

bottom plot. We added 0.1 degrees to the ambient temperature

and subtracted 0.1 degrees from the oil drain temperature. The

state-space model seems to predict the thermal drift with

perturbed-mean temperature but the OE model goes in the

wrong direction. Similar phenomena could be seen with the

ARX and BJ models. Based on our observation, the state-

space model was selected as an appropriate thermal error

model for our experiments. When a thermoelastic model is

built, we remove the mean values from the input temperature

data before computation. Since it is an off-line computation

after collecting the temperature data, the accurate mean values

can be removed. But there is no guarantee that the mean

values in off-line computation will reappear as those in the

real-time on-line data. That is why the thermal error model

should be insensitive to the constant offset of the temperature

mean values.

Fig. 6 Comparisons of measured thermal drift and predictions by

a 2nd order OE and state-space models: with normal

temperature data (top) and with perturbed-mean temperature

data (bottom)

0 60 120 180 240-0.2

-0.1

0

0.1

0.2

Time (minutes)

Therm

al drift (

mic

ron)

0 60 120 180 240-0.2

-0.1

0

0.1

0.2Spindle Housing

Time (minutes)

Tem

p. change (

oC

)

0 60 120 180 240-0.2

-0.1

0

0.1

0.2Table

Time (minutes)

Tem

p. change (

oC

)

0 60 120 180 240-0.4

-0.2

0

0.2

0.4Ambient

Time (minutes)

Tem

p. change (

oC

)

0 60 120 180 240-0.2

-0.1

0

0.1

0.2Oil Drain

Time (minutes)

Tem

p. change (

oC

)

0 30 60 90 120 150 180-0.4

-0.2

0

0.2

0.4

Time (minutes)

Therm

al drift (

mic

ron)

Predictions with normal temperature data

Measured Predic. by OE Predic. by SS

0 30 60 90 120 150 180-0.4

-0.2

0

0.2

0.4

Time (minutes)

Therm

al drift (

mic

ron)

Predictions with perturbed-mean temperature data

241


State-space models are models that use state variables to

describe a system by a set of first-order difference equations,

rather than by one or more n’th-order difference equations.

State variables x(k) can be reconstructed from the measured

input-output data, but are not themselves measured during an

experiment (10). State-space models are not derived from

physical equations, so the states in a state-space model have

no direct physical meaning. More information on the state-

space system identification method can be found in (11).

Using Matlab

software, a second order state-space model

was obtained as follows,

x(k+1) = A x(k) + B u(k) + K e(k)

y(k) = C x(k) + D u(k) + e(k)

where x(k) is a state vector, u(k) is a temperature input

vector, u(k) = [Ambient temp.(k), Oil drain temp.(k)]T, e(k) is a

noise vector with Gaussian distribution, y(k) is a thermal error

drift output, k is a discrete-time step such that time t =

ksampling period. Our sampling period was 1 second. The

matrixes A, B, C, D, K and an initial condition were

A = [9.9490e-1, -1.7756e-2; 4.1983e-4, 9.9686e-1],

B = [6.4115e-4, -1.4311e-3; -6.5977e-6, -2.7177e-4],

C = [5.9741, -9.3924e-3],

D = [0, 0],

K = [1.7009e-1, -7.8078e-2]T,

x(0) = [1.3373e-2, -7.9594e-3]T.

Simulation result shown in Fig. 6 has carried out with no

noise and a zero initial condition. It may be interesting to see

the frequency response of the identified state-space model.

The Bode plots are shown in Fig. 7. It can’t be claimed that

the identified model exactly describes the real thermoelastic

process in our experimental setup, but there are common

dynamic characteristics captured by other dynamic models as

well as the state-space model. From the frequency responses,

we can say that the effect of the ambient temperature change

on the thermal drift is faster (small phase lag) and stronger

(large DC gain) than that of the oil drain temperature change.

Reflecting on the area they affect, it is not against our

common sense. From the step response test, a step change in

temperature takes about 21 (by ARX) to 33 minutes (by

state-space) until its effect fully appears in the thermal drift.

Fig. 7 Frequency responses of the identified state-space model

4. Experimental Results

In our experimental setup, the NC controller for the

testbed high precision lathe is a UMAC system from Delta

Tau Data Systems Inc. The servo control update rate is

approximately 2.26 kHz and the sampling rate for thermal

drift prediction is 1 Hz. Since there was a big difference

between these two rates, another DSP (Digital Signal

Processing) system was used to run the thermoelastic model

in real-time. The amount of thermal drift, which was

predicted from on-line temperature sensor values, was

returned to the NC system by the DSP system through an AD

(Analog-Digital) board. The NC system subtracted the read-

in thermal drift from the position command and made a

compensated actual position command for the Z-axis.

Experimental results are shown in Fig. 8. The thermal

error compensation had been alternatively on and off at every

60 minutes and the consecutive thermal drift was recorded for

four hours. The colored boxes in the figure enclose the

maximum and the minimum of the thermal drift in each period

for an easy comparison of peak-to-peak values. When the

compensation was off, the thermal drifts were 61.4 and 65.9

nanometers RMS from the second and the fourth period,

respectively. During the compensation on, they reduced to

45.2 (first period) and 51.5 (third period) nanometers RMS.

On average, the thermal error compensation obtained a 24%

reduction in thermal drift. The dashed line in the figure shows

the real-time prediction of the thermal drift without sending a

compensating signal to the NC system. We can see how

closely the identified state-space model predicts the thermal

drift in real-time and how big an error can be made by a small

phase mismatch.

Fig. 8 Experimental results of the thermal error compensation

5. Conclusions

In this paper, a state-space dynamic thermal error model is

presented for thermal error compensation in a high precision

lathe. Based on the system identification theory, a state-space

model is used for dynamic modeling and online prediction for

thermal drift. The thermally induced errors in the conventional

10-6

10-5

10-4

10-3

10-2

10-1

10-3

10-2

10-1

100

101

Am

plit

ude

10-6

10-5

10-4

10-3

10-2

10-1

-300

-200

-100

0

Phase (

deg)

Frequency (Hz)

Ambient -> Thermal Drift

Oil Drain -> Thermal Drift

0 60 120 180 240-0.2

-0.1

0

0.1

0.2

Time (minutes)

Therm

al drift (

mic

ron)

Compesation ON

RMS = 45.2 nano

Compesation ON

RMS = 51.5 nano

Compesation OFF

RMS = 61.4 nano

Compesation OFF

RMS = 65.9 nano

Real-time

Prediction w/o

Compensation

242


machine tools are usually more than scores of microns, but the

thermal error dealt in this paper is few hundred nanometers.

Thus, a linear thermal error model in our research should be

able to susceptibly follow any little temperature changes. The

effectiveness of state-space modeling approach is verified

through computer simulation and experiments. In our

simulation, a thermal error of 97 nanometers RMS over 3

hours is reduced to 55 nanometers RMS. Experimental results

show an average of 24% reduction in thermal drift and support

the validity of our approach.

REFERENCES

1. Ferreira, P. M. and Liu, C. R., “A method for estimating

and compensating quasi-static errors of machine tools,”

ASME Journal of Engineering for Industry, Vol. 115, No. 1,

pp. 149–159, 1993.

2. Donmez, M. A., Liu, C. R. and Barash, M. M., “General

Methodology for Machine Tool Accuracy Enhancement by

Error Compensation,” Precision Eng., Vol. 8, No. 4, pp.

187–196, 1986.

3. Yang, S., Yuan, J. and Ni, J., “The Improvement of

Thermal Error Modeling and Compensation on Machine

Tools by Neural Network,’’ Int. J. Machine Tools and

Manufacture, Vol. 36, No. 4, pp. 527–537, 1996.

4. Yang, J., Yuan, J. and Ni, J., “Thermal Error Mode

Analysis and Robust Modeling for Error Compensation on

a CNC Turning Center,” Int. J. Machine Tools and

Manufacture, Vol. 39, pp. 1367–1381, 1999.

5. Yang, J., Yuan, J. and Ni, J., “Thermal Error Mode

Analysis and Robust Modeling for Error Compensation on

a CNC Turning Center,” Int. J. Machine Tools and

Manufacture, Vol. 39, pp. 1367–1381, 1999.

6. Moriwaki, T. and Shamoto, E., “Analysis of Thermal

Deformation of an Ultraprecision Air Spindle System,”

CIRP Annals, Vol. 47, No. 1, pp. 315–319, 1998.

7. Li, S., Zhang, Y. and Zhang, G., “A Study of Pre-

compensation for Thermal Errors of NC Machine Tools,”

Int. J. Machine Tools and Manufacture, Vol. 37, No. 12, pp.

1715–1719, 1997.

8. Yang, H. and Ni, J., “Dynamic Modeling for Machine Tool

Thermal Error Compensation”, J. Manufacturing Science

and Engineering, Vol. 125, pp. 245-253, 2003.

9. Park, C. H., Oh, Y. J., Hwang, J. H., and Lee, D. W.,

“Motion error compensation method for hydrostatic tables

using actively controlled capillaries,” J. Mechanical

Science and Technology, Vol. 20, No. 1, pp. 51-58, 2006.

10. System Identification Toolbox, User’s Guide, MathWorks

Inc., 20013.

11. LJung, L. “System identification: Theory for the user,” Sec.

10.6, Prentice Hall Inc., 1999.

243

error compensation

Documents

Transcript of error compensation