Thermal error compensation for a high precision lathe

IWMF214, 9 th INTERNATIONAL WORKSHOP ON MICROFACTORIES OCTOBER 5-8, 214, HONOLULU, U.S.A. / 1 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: bkim@kimm.re.kr, TEL: +82-42-868-719, FAX: +82-42-868-718 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.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.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. 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 decades 1,2. The thermal error accounts for 4-7% of the total machining errors and it is the most significant factor influencing the machine tool accuracy 1. Thermally induced errors are quasistatic 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 distribution 3-5. 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. Moriwaki 6 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. Li 7 adopted an auto-regressive (AR) model to predict spindle thermal errors from spindle speed. Hong 8 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 3-4 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 239

D eviation arcsec D eviation m IWMF214, 9 th INTERNATIONAL WORKSHOP ON MICROFACTORIES OCTOBER 5-8, 214, HONOLULU, U.S.A. / 2 of the X-axis is 2 mm and that of the Z-axis 5 mm. The linear encoders for position feedback have a 1 nanometer resolution for the X- and Z-axes. The temperature of the machining room is maintained by an air conditioner with.6 C variations. Covering the lathe with metal panels reduces the ambient temperature change around the machine to be.3 C. The temperature of the oil from the hydrostatic oil bearing is controlled by an oil-cooler within.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 7 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, rpm, thus the lathe is ready to operate, the magnitude of the oscillation increases to be around 3-4 nanometers. than 5 nanometers and the yaw error less than.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..1. -.1.5. S traightness error Yaw ing error -.5 5 1 15 2 P osition m m Fig. 3 Straightness and yaw error profiles corrected by ACC 3. Linear Thermoelastic System Identification Fig. 1 Picture of a high precision lathe 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 2 mm over the tool holder. Spindle Tool holder 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 Z 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 X 24

IWMF214, 9 th INTERNATIONAL WORKSHOP ON MICROFACTORIES OCTOBER 5-8, 214, HONOLULU, U.S.A. / 3 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 7 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 (.52) has the most influence on the thermal drift, and the next is the table and the oil drain temperature (.48 and -.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..2.1 -.1 -.2 6 12 18 24 Spindle Housing Table.2.2.1 -.1 -.2 6 12 18 24 Ambient.4.2 -.2 -.4 6 12 18 24 -.1 -.2 6 12 18 24 Oil Drain.2 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.1.1 -.1 -.2 6 12 18 24 system identification methods were tested and evaluated. They were ARX(auto-regressive exogenous input), ARMAX(autoregressive 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.1 degrees to the ambient temperature and subtracted.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 statespace 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..4.2 Predictions with normal temperature data -.2 Measured Predic. by OE Predic. by SS -.4 3 6 9 12 15 18 Predictions with perturbed-mean temperature data.4.2 -.2 -.4 3 6 9 12 15 18 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) 241

Phase (deg) Amplitude IWMF214, 9 th INTERNATIONAL WORKSHOP ON MICROFACTORIES OCTOBER 5-8, 214, HONOLULU, U.S.A. / 4 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 (1). 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 statespace 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 = k sampling period. Our sampling period was 1 second. The matrixes A, B, C, D, K and an initial condition were A = [9.949e-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 = [, ], K = [1.79e-1, -7.878e-2] T, x() = [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. 1 1 1 1-1 1-2 1-3 1-6 1-5 1-4 1-3 1-2 1-1 -1-2 Ambient -> Thermal Drift Oil Drain -> Thermal Drift -3 1-6 1-5 1-4 1-3 1-2 1-1 Frequency (Hz) 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 readin 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 6 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..2.1 -.1 Compesation ON RMS = 45.2 nano -.2 6 12 18 24 Fig. 8 Experimental results of the thermal error compensation 5. Conclusions Compesation OFF RMS = 61.4 nano Compesation ON RMS = 51.5 nano Compesation OFF RMS = 65.9 nano Real-time Prediction w/o Compensation 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 242

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