Thermography-Based Investigation into Thermally Induced Positioning Errors of Feed Drives By Example of a Ball Screw

U. Heisel (1), G. Koscsák, T. Stehle

Institute for Machine Tools, University of Stuttgart, Germany

Abstract The following paper presents a method, based on thermographic measurements, to calculate the thermally induced positioning errors of feed drives in machine tools, here described with the example of a ball screw drive. Especially the effect of an uneven temperature distribution in the components on the total deformation is examined in this method. A transformation between the 3-D object space and the 2-D image space was created by means of digital image processing and photogrammetry as well as defined reference points on the structure of the feed drive with the surrounding machine components. The transformation enables the necessary identification of the defined structure points in the thermographic picture. With the help of the temperature information contained in the thermographic images, the respective thermally induced errors can then be calculated depending on the respective axial position by means of a temperature deformation model. Keywords: Temperature, Deformation, Thermography

1 INTRODUCTION For the extensive reduction or complete avoidance of thermally induced influences on the machining accu-racy of machine tools, it is necessary to exactly determine and analyse their thermal behaviour under real conditions [1-3]. According to ISO 230-3, the thermal behaviour is estab-lished by means of temperature profiles measured at se-lected points on the machine structure and by displace-ment measurements at the tool centre point (TCP), de-pending on various parameters such as e.g. main spindle speed or speed of feed drives as a function of a particular period of time. Usually the drift measurement is carried out only at one point within the work space of a machine in the direction of the linear motion axes x, y, and z. It can be taken as a simplified basis here that the thermally in-duced positioning and orientation deviations are equal at all other points of the work space. In fact the amounts of the positioning and orientation deviations can considera-bly differ from each other at different points of the work space, depending on the geometric conditions of the ma-chine and particularly in the feed drives of indirect posi-tioning systems. Hence the correlations between individ-ual temperature profiles at the machine structure and par-ticular relative displacements at the TCP can describe the real thermal behaviour of a machine only inadequately [4]. Compensation models which are based on such correla-tions are correspondingly inaccurate, determining individ-ual temperature profiles at selected points along the ma-chine structure as input quantities and generating position correction values for the feed drives as output quantities. In order to be able to realise an effective compensation in the whole work space, the temperature conditions of the components affected by thermal drift must be recorded in detail, and their proportionate effect on the positioning de-viations must be taken into consideration. Furthermore, the components affected by thermal drift are divided into parts generating motion and parts moved by them, which have to be examined separately from each other. Belonging to the components generating motion are the feed drives and, in a direct position measurement, the measuring systems as well. The thermally induced ex-pansions of these components mainly determine the posi-tion-dependent portion of the displacements.

The components moved by others include parts which have a steadying function. They participate only passively in the motion. This means that the thermoelastic deforma-tions of these components determine the position-independent portion of the displacements. However, they can also influence the position-dependent portion, de-pending on the particular design of the machine. Models for electromechanical feed drives, with which thermal processes can also be simulated, are known from literature [5, 6]. These have, however, not been applied yet in thermal drift compensation. The reason for this is that the thermodynamic modelling of a feed drive is com-plex and depends on numerous parameters. Not being taken into consideration here are the operating character-istics of the machine, which are changing in time, for ex-ample, due to wear of components, change of tribological properties or alterations of preload. Hence it has not been possible until now to realise a reli-able compensation of the positioning errors of electrome-chanical feed drives, which is based on temperature measurements. In the known compensation methods by means of temperature measurements, the temperature is usually measured at the bearings of the ball screw spindle and at the spindle nut. Other measurement points are of-ten difficult to reach with classic temperature measuring techniques, due to the accessibility and the motion of the components. 2 APPLICATION OF THERMAL IMAGING FOR DE-

TERMINING THE TEMPERATURE DISTRIBUTION OF A BALL SCREW

In order to be able to determine the position-dependent portion of thermally induced displacements, the tempera-ture profile of the components has to be detected along the feed drives by an indirect positioning and along the measuring system in a direct position measurement. Infra-red thermography (IR thermography) seems to be particu-larly suitable for this task, since it provides a high metro-logical information content as well as enables the pictorial representation and processing of the measuring results in a comparatively simple way. In addition, time-varying temperature distributions can be examined and depicted.

Annals of the CIRP Vol. 55/1/2006

2.1 Calculation model Figure 1 presents schematically the calculation model for a ball screw, which is described in the following. Input quantities are information gained by thermographic im-ages of the examined feed drive as well as the CAD data of the test stand.

Figure 1: Scheme of the calculation model

The real structure was depicted in a geometric-kinematic model. This model serves to establish the geometric and kinematic boundary conditions (degrees of freedom, posi-tion and orientation of the structure) as well as the points to be identified in the thermographic picture. The thermographic measurements were carried out with a camera of which the detector array has a resolution of 256 x 256 pixels. The temperatures determined with the software of the camera were converted into an ASCII file. Then they were imported into the calculation environment (MATLAB) in a 256 x 256 temperature matrix. In order to be able to assign the determined temperature values to the structure, a transformation had to be carried out, describing the connection between the world coordi-nate system of the real structure or its model and the im-age coordinate system in the thermographic picture or temperature matrix.

2.2 Transformation between world and image coor-dinates

The used transformation maps the 3-D object points onto the 2-D image plane:

P

WxPx ⋅= (1)

where are the image coordinates in pixel, and W are the world coordinates. The vectors of the coordinates are defined according to homogeneous notation. The matrix of the transformation (also called camera matrix) can always be calculated with the concatenation of three sub-transformations [7]:

x x

P

TP LLLP ⋅⋅= π (2)

where T is the coordinate transformation between the world coordinate system and the camera-fixed coordinate system, P is the central projection by the camera optic, and π is the conversion of the metric image coordinates into pixel coordinates.

L

LL

The matrix P has a size of 3 x 4 and contains information about the interior and exterior orientation of the thermo-graphic camera towards the test object. The six parame-ters of the exterior orientation describe the position and orientation of the camera-fixed coordinate system in the world coordinate system and therefore the transformation

. These parameters are: TL

• the coordinates , , and of the origin of the camera coordinate system,

KWx K

Wy KWz

• the cardan angles ω , ϕ , and κ around the corre-sponding coordinate axes.

The origin of the camera coordinate system is also the centre of projection. The parameters of the interior orientation determine the central projection P . Depending on the camera model used, there are the following parameters:

L

• the camera constant , i.e. the distance between the centre of projection and the image plane,

c

• the coordinates of the image's principal point , in the image plane,

HBx H

By

• the shear s between the axes of the image coordi-nate system.

The shear s can be neglected in modern cameras. The determination of the other three parameters is required for setting up the transformation between the object space and the image space. The parameters of the camera's interior orientation can be established by means of either the technical specifications by the manufacturer or an optical calibration. In optical calibration, the camera matrix is directly determined by means of reference or control points (points of which the coordinates are known in both the object space and the image space) and with the help of the collinearity equa-tions [8]. The wanted parameters of the interior orientation can be established by means of decomposing the camera matrix into an orthogonal matrix and an upper triangular matrix [9]. The camera used for the examinations de-scribed here was calibrated with the help of a coordinate measuring machine in three series of measurements with 40 control points each. The focus of the camera was fixed at the focus used in the measurements. The exterior orientation was also calculated with the help of control points. When the interior orientation is known, the six parameters can be calculated using least squares. Here the objective function is:

∑=

=⋅=m

iirf

1

2T )(21)()(

21)( ppRpRp (3)

where is the vector of unknown parameters, R is the vector of residuals, m is the number of control points, and i is the residual of the i th point. The residual can be calculated as follows:

p

r ir

iWiiiir xpPxxxp ⋅−=−= )(ˆ)( (4)

The control points for calculating the exterior orientation were placed on the test stand with red LEDs. Their posi-tions were arbitrarily selected. It is merely not allowed that they lie in one plane. Their coordinates in the object space were established by means of the available CAD data of the test stand. In the conducted experimental investiga-tions, the exterior orientation was calculated with the five set control points. To obtain a sufficient number of equa-tions, the residuals described in Equation 4 were calcu-lated coordinate-wisely. An image process chain was built up to identify the control points, put on the test stand by the LEDs, in the thermo-graphic picture. The separation of the image foreground from the background is carried out with threshold segmen-tation. The identification of the features has to be per-formed manually at the first measurement (e.g. when the test stand is cold), but it takes place automatically for the further measurements.

2.3 Experimental investigations Due to the built-in safety appliances and the dimensions of the slider, it was only possible to drive on the middle 600 mm long part of the thread shaft. The experiments were carried out on this section with four different motion cycles. In each of the tests V1 – V3, a 100 mm alternating motion was performed at different positions of the ball screw. In test V4 the 500 mm long path overlapped the ranges of the tests V1 – V3. After a particular number of performed motion cycles, thermographic photos were taken. In parallel to this, position measurements with a laser interferometer were carried out on the 500 mm long path of V4 with a step length of 50 mm.

Test V1 V2 V3 V4

startz 335 535 735 335

travell 100 100 100 500

Figure 2: Starting positions and travels of the conducted tests

In order to be able to detect the temperature profile of the ball screw completely, two thermographic pictures for every temperature measurement had to be saved. The first picture was taken when the slide with a load of 600 kg was at the starting point of the position measurement. The second photo was taken when the slide was at the end point of the position measurement.

Figure 3: Temperature matrices with the points for the

transformation as well as the points for the temperature measurement; measurement V1 in cold condition (top)

and after 4,000 cycles (below) The pictures were taken by the thermographic camera be-ing at a distance of approximately 7 m from the test stand. From this distance, it was possible to achieve a resolution of ca. 7 mm on the thread shaft. The temperature meas-urement points on the shaft were defined with a distance of 10 mm along the entire length. This means that there

were 131 temperature measurement points on the shaft. Such a resolution is almost impossible with a usual tem-perature measuring equipment. Figure 3 presents the temperature matrices generated out of the thermographic images. The yellow rhombuses show the identified mark-ings. The red crosses show the points of the ball screw which were selected for the temperature measurement. When the test stand is cold, the reflections of the objects in the surroundings are also visible (upper part of the ball screw) besides the markings put on the bed. The ball screw warms up due to the friction between the rolling elements and the thread shaft. Part of the heat of the mo-tor is conducted into the ball screw via the coupling and the fixed bearing (on the right-hand side of the photos). In the images, the reflections of the now warmed up compo-nents could be observed well too. Furthermore one of the measuring points had to be cho-sen, determining the transition between the two series of measuring data („slide at the starting point“ and „slide at the end point“). This was necessary because the slide with the nut covers a certain area of the spindle, depend-ing on its respective position. The transition point must be a point of the ball screw which is visible in both thermo-graphic pictures. Such a point can be determined by means of the difference between the thermographic im-ages. For establishing the temperature distribution on the points at the front of the ball screw (between fixed bearing and transition point), the measuring data were taken from those thermograms in which the slide is at the end point of the position measurement. The temperature values for the points at the back (between transition point and movable bearing) were taken from the measuring data in which the slide is at the starting point of the position measurement.

Figure 4: Determined temperature profile at the ball screw during the measurement V1 in cold condition (blue) and

after 4,000 cycles (red) Figure 4 presents the established temperature distribution on the ball screw in cold and warm condition. When the test stand is cold, a slight temperature increase can be observed on the side of the fixed bearing (left-hand side of the diagram). This may be attributed to the heat produced by the motor. The variations in temperature discernible on the side of the movable bearing can be explained by the change in the thermal emissivity ε . Due to multiple reflec-tions there is another thermal emissivity in the spiral groove than at the cylindrical outer surface of the shaft. The motion cycles were being conducted during the con-sidered measurement V1 between the positions of 335 mm and 435 mm. After 4,000 cycles the side of the fixed bearing is clearly warmed up. Near the bearing block the temperature of the shaft was approximately 29°C. Due to the friction between the rolling elements and the shaft, the highest temperature was reached with 45°C about the

middle of the load travel at position 470 mm. Being much more clearly visible in warm condition, the waviness of the temperature profile can also be explained by the thermal emissivity of the spiral groove, which differs from that of the outer surface.

2.4 Calculation of the thermally induced displace-ments

By means of the points defined in the object space, the thread shaft is divided into 10 mm long sections. The ex-pansion of the shaft caused by the temperature change of the shaft can be calculated in sections by means of the established temperature profiles. The displacements of the individual points can be calculated by cumulating the expansions of the sections:

( )∑=

−−

−+−+

⋅−⋅=∆n

i

iiiin

iitttt

zzz1

0011 2

)()(1α (5)

where n is the displacement of the nth point of the shaft,

z∆α is the linear expansion coefficient of the shaft,

i is the respective z-coordinate, i is the reference

temperature, and it is the current temperature of the ith point. In Figure 5 the displacements calculated with Equa-tion 5 are compared with the measured values. As the temperature profiles were determined for the entire length of the thread shaft, the calculation of the displacements could be carried out for the whole length of the shaft as well. The position measurements could, however, be con-ducted only between the positions 335 – 835 mm. To be able to compare the calculated displacements with the measured ones, the measured values were shifted to the calculated values along the axis of ordinates.

z t0

Figure 5: Calculated (red) and measured (blue) displace-ments of the ball screw in the measurements V1 – V4

The determined drift profiles show different characteris-tics, depending on the underlying load cycles. The profile of the calculated displacements shows very good agree-ments with the measured values. The greatest deviations can be detected in measurement V2 with 21 % and in V4 with 15 %. These deviations can be attributed to the inac-curacies of the temperature measurement. Therefore, a better accuracy can be achieved with a slighter tempera-ture change (measurements V1 and V3). The achieved results demonstrate the very good suitability of the presented procedure for determining thermally in-duced displacements on the basis of thermographic im-

ages. Moreover, the results show that, under the condition of a precise temperature measurement, the drift of ball screw drives can be estimated with a high accuracy, irre-spective of the underlying thermal load and the place of heat induction. 3 CONCLUSION AND PROSPECT By means of thermographic measurements, also uneven temperature distributions on the components of a machine tool, for example, along the shaft of a ball screw drive can be detected under real conditions. The aim of this is to calculate the thermally induced positioning error in differ-ent positions of a feed drive as precisely as possible. A temperature displacement model was created for this cal-culation. This model differs from correlative models known until now in that the temperature measurement are carried out by the processing of a planar measurement data in-stead of measuring the temperature at discrete points of the structure. In this paper it could be demonstrated that the displacements calculated with the described method show good agreements with the measured drift in the dif-ferent positions along a feed drive with ball screw. Minor deviations are the result of shortcomings in the thermo-graphic temperature measurement. Therefore, there is the possibility for improving this method by, among other things, a correction of radiometric parameters such as thermal emissivity or environmental reflections. This can be achieved, for example, by considering the thermal ra-diation beside the measured temperature and including them in the calculation. The present paper contains results from the research pro-ject HE 1656/76-1 and -2, funded by the German Re-search Foundation (DFG). REFERENCES [1] Jedrezejewski, J., Strauchold, S., 1988, Directions of

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