Home / Accuracy of Feed Axes (Part One)

Accuracy of Feed Axes (Part One)

The positioning error values of the feed axes play a critical role in the total error budget of a machine tool.


The accuracy of modern machine tools is measured with an increasing number of new and revised inspection and acceptance tests. Where years ago purely geometric acceptance tests predominated, today’s routine methods include dynamic tests, such as circular interpolation and free-form tests, thermal tests as described in ISO/DIS 230-3 and, for production machines, capability testing during acceptance or regular inspection.

The various influences of cutting processes, geometric and thermal accuracy, static and dynamic rigidity, and the positioning response of the feed axes on the attainable accuracy of the workpiece can now be analyzed more specifically. Machine errors are becoming increasingly obvious to the user.

Considering the increasing frequency of changing jobs and the concomitant reduction in batch sizes, reducing the thermal or systematic error of a machine tool through tedious optimization of individual production steps is seldom feasible. The accuracy of the first part is gaining in importance. In particular, the thermal error of machine tools is drawing ever more interest.

The following text shows that thermal error can be quite significant, especially for the feed axes. Unlike structural deformation, errors of the feed axes can be dramatically reduced through a choice of simple and readily available measuring techniques.

Feed-Drive System Design

An exact error analysis of position measurement via rotary encoder and feed screw begins with a consideration of prevalent mechanical feed-drive systems. Although machine tool designs vary immensely, the mechanical configuration of their feed drive is largely standardized (Figure 1).

In almost all cases, the recirculating ballscrew has established itself as the solution for converting the rotary motion of the servomotor into linear slide motion. Its bearing takes up all axial forces of the slide. The servomotor and ballscrew drive are usually directly coupled.

Toothed-belt drives are also widely used to achieve a compact design and better adapt the speed. For position measurement of feed axes on NC machine tools, it is possible to use either linear encoders or recirculating ballscrews in conjunction with rotary encoders.

A position control loop via rotary encoder and ballscrew includes only the servomotor (Figure 1; dotted line). In other words, there is no actual position control of the slide because only the position of the servomotor rotor is being controlled. To extrapolate the slide position, the mechanical system between the servomotor and the slide must have a known and, above all, reproducible mechanical transfer behavior.

A position control loop with a linear encoder, on the other hand, includes the entire mechanical feed-drive system. The linear encoder on the slide detects mechanical transmission errors which are compensated by the machine control unit.

Differing Terminology

Different terms distinguish between these two methods of position control. German-speaking and some English-speaking communities generally refer
to them (somewhat inaccurately) as “direct and indirect measurement.” However, these terms are rather poorly chosen because, strictly speaking, both methods are direct.

One method uses the line grating on the linear scale as the measuring standard, the other the pitch of the ballscrew. The rotary encoder simply serves as an interpolating aid. Here the Japanese concepts of “semiclosed-loop and closed-loop control” seem appropriate, since they more aptly describe the actual setup.

Trend Toward Digitally Driven Axes

As a result of the trend toward digital axes in drive technology, a large share of new servomotors feature rotary encoders which in principle can serve together with the feed screw for position control. With such a drive configuration, the decision must be made as to whether to add a linear encoder or simply to use a ballscrew working in combination with the already existing motor encoder.

Remember to consider the problems discussed in the following text regarding position measurement using a rotary encoder/ballscrew system. They can quickly increase the cost of an “economical” machine if the owner finds that its accuracy does not suffice in certain applications.

Kinematic Error

Kinematic error that can be directly attributed to position measurement using feed screw and rotary encoder result from ballscrew pitch error, from play
in the feed elements, and from the so-called pitch loss. Ballscrew pitch error directly influences the result of measurement because the pitch of the ballscrew is being used as a standard for linear measurement.

Play in the feed transfer elements causes backlash. The pitch loss [1] results from a shift of the balls during the positioning of ballscrew drives with two-point
preloading and can lead to reversal error on the order of 1 to 10 & 956 m.

Error compensation

Most controls are capable of compensating such pitch error and reversal error. However, to determine the compensation values it is necessary to make elaborate measurements with comparative measuring devices such as interferometers and grid encoders. In addition, the reversal error is often unstable over long periods of time and must be regularly recalibrated (Figure 2).

Strain in Drive Mechanisms

Forces leading to the deformation of feed drive mechanisms cause a shift in the actual axis slide position relative to the position measured with the ballscrew and rotary encoder. They are essentially inertia forces resulting from acceleration of the slide, cutting process forces, and friction in the guideways. The mean
axial rigidity of a feed drive mechanism as shown in Figure 1 lies in the range of 100 to 200 N/μm (with a distance between ball
nut and fixed bearing of 0.5 m and a ballscrew diameter of 40 mm).

Force of Acceleration

A typical slide mass of 500 kg and a moderate acceleration of 4 m/s2 result in deformations of 10 to 20 &#956m that cannot be recognized by the rotary encoder/ballscrew system. The present industry trend toward acceleration values in significantly higher ranges will result in increasingly large deformation values.

Cutting Force

The cutting force can quite possibly lie in the kN range, but its effect is distributed not only in the feed drive system, but also over the entire structure of the machine between the workpiece and the tool. The deformation of the feed drive system therefore normally has only a small share in the total deformation of the machine.

A linear encoder can recognize and correct only this small portion of the total deformation. Critical component dimensions, however, are normally finished at low feed rates with correspondingly low deformation of the feed drive system.

Force of Friction

The force of friction in the guideways lies between 1 percent and 2 percent of weight for roller guideways and 3 percent to 12 percent of weight for sliding guideways [2]. A weight exerting 5,000 N therefore results in feed drive deformation of only 0.25 to 6 μm.

Circular Test for Inspecting Machine Tools

A typical example for errors dependent on acceleration and velocity can be recorded in a circular interpolation test on a vertical machining center (Figure 3). Where position control is by rotary encoder and ballscrew, the circles traversed at higher velocities deviate significantly from the ideal path. The same machining center shows significantly better contour accuracy when equipped with linear encoders.

Positioning Error Due to Ballscrew Expansion

Positioning error resulting from thermal expansion of the ballscrew presents the greatest problem for position measurement via rotary encoder and ballscrew. This is because the ballscrew drive must serve a double function.

On the one hand, it must be as rigid as possible to convert the rotary motion of the servomotor to linear feed motion. On the other hand, it must serve as a precision measuring standard. This twofold function forces a compromise because both the rigidity and the thermal expansion depend on the preloading of the ball nut and the fixed bearing. Both the axial rigidity and the moment of friction are roughly proportional to the preloading.

Friction in the Ball Nut

The largest portion of the friction in a feed drive system is generated in the ball nut because of the complex kinematics of a recirculating ball nut. Although at first glance the balls may seem only to be rolling, they are in fact subjected to a great deal of friction. Besides the microslip resulting from relative motion
in the compressed contact areas, the greatest effect is from the macroslip due to kinematic exigencies.

The balls are not completely held in the races and wobble much like tennis balls rolling down a gutter. The result is a continual pressing and pushing, with occasional slipping of the balls. The friction among the balls is aggravated by high surface pressure due to the absence of a retaining device to separate them.

As in every angular-contact ball bearing, a spinning friction results from a contact diameter that is not orthogonal to the axis of ball rotation. Each ball therefore rotates about its contact diameter. Recent studies show that the balls can move in the thread only because of an additional slip component brought
on by the thread pitch [3].

The recirculation system is a special problem zone for ballscrews. With every entrance into the recirculation channel, just as with every exit, the movement
of the ball changes entirely. The rotational energy of the balls, which in rapid traverse typically rotate with 8,000 rpm, must be respectively started and stopped.

In contrast to the preloaded thread zone, in the recirculation zone the balls are not under stress. The play of energy causes the balls to collect in the recirculation channel.

Without elaborate measures to reintroduce the balls into the thread at the end of the channel it tends to congest, causing the familiar jamming of the ballscrew drive. The moment of friction of a ground precision recirculating ballscrew with 40 mm diameter and 10 mm pitch was measured by Golz [4] for various preload forces and rotational speeds (Figure 4).

The Stribeck characteristic of frictional moment is clearly recognizable. It confirms the high share of solid-body friction and mixed friction in ballscrew drives at low speeds. Viscous friction dominates at high speeds. It is interesting to note that for this typical ballscrew the normal machining feed rates lie
far below the speeds at which the moment of friction is at its minimum.

The rapid traverse feed rates, however, lie far above it. Therefore the feed rates at which this ballscrew is at optimum efficiency seldom occur. The moment of friction is only slightly dependent on the axis load of the ball nut [4].

Frictional Heat Generated in the Ball Nut

With a typical preload of 3 kN and allowing for the missing wiper, this results in a no-load or frictional moment of 0.5 to 1 Nm. This means that in rapid traverse at a ballscrew speed of 2,000 rpm, approximately 100 to 200 W of frictional heat is generated in the ball nut.

More Frictional Heat to Be Expected

To increase rapid traverse velocity, either the pitch or the rotational speed of the recirculating ballscrew must increase. The maximum permissible of speed recirculating ballscrews has doubled in the last five years. Due to the continually increasing requirement for acceleration, the preloading and therefore
the friction of the ball nut could not be reduced. Recirculating ballscrew drives therefore generate significantly more heat than before and will generate even more in the future.

Measurement of Positioning Accuracy According to ISO 230-3

The influence of this frictional heat on the positioning response of the feed axis becomes apparent when positioning tests are conducted in accordance with the new international standard ISO/ DIS 230-3. This standard contains proposals for making uniform measurements of thermal shifts of lathes and milling machines as a result of external and internal heat sources (Figure 5).

Deformation in the machine structure resulting from changes in ambient conditions or through heat generation in the spindle drive are recorded with the aid of five probes that measure against a cylinder mounted in the tool holder. This makes it possible to measure all five relevant degrees of freedom. To test the feed axes, it proposes a repeated positioning to two points that lie as near as possible to the traverse range at an agreed percentage of rapid traverse velocity.

The change of the positions with respect to the initial value is recorded. The test is to be conducted until a satiation effect is clearly observable. Dial
gauges or other test equipment simpler than the laser interferometer can also be used for the axis test. These tests enable any workshop to conduct such inspections at a reasonable cost.

Influence of the Ballscrew Bearing on Positioning Accuracy

Differing types of behavior can be expected depending on whether the ballscrew can expand freely. The various types of bearings for recirculating ballscrews are illustrated in Figure 6.

Fixed Bearing at One End

In the case of fixed/floating bearings, the ballscrew will expand freely away from the fixed bearing in accordance with its temperature profile. The thermal zero
point of such a feed axis lies at the location of the fixed bearing. This means that theoretically no thermal shift would be found if the ball nut is located at the fixed bearing. All other positions will be affected by the thermal expansion of the ballscrew.

Figure 7 shows the result of a positioning test per ISO/DIS 230-3 on a vertical machining center (built in 1998) equipped with linear encoders. The machine positioned to three positions on the 1 m long X-axis a total of 100 times with 10 m/min. Taking the standstill periods for measure value acquisition into account, the mean traversing speed during the test was approx. 4 m/min.

In addition to the two positions at the ends of traverse as recommended in the standard, a third position at the midpoint of traverse was measured.

Figure 7 shows the position values with respect to their initial values. At first the ballscrew/rotary encoder system was used for position control. In a second test under otherwise identical conditions, linear encoders were used. The comparator system was a VM 101 from Heidenhain.

In spite of the moderate feet rate of 10 m/min (rapid traverse 24 m/min), the position farthest from the fixed bearing of the ballscrew shifted by more than 110 μm within 40 minutes. It is interesting to note that the drift increases very quickly immediately after switch-on. Any change in the mean feed rate in a production process therefore immediately affects positioning accuracy. Similar results were published by Schmitt [5].

No Drift in Position Values Measured with Linear Encoders

The measured positioning accuracy therefore depends directly on the number of repetitions, particularly after the first few repetitions. The measurements taken by the retrofitted linear encoders show no drift.

Batch Production

To demonstrate the applicability of this experiment to actual production conditions, a small batch of aluminum workpieces were machined on the same machine. Eight 70 mm x 70 mm workpieces were fixed on a vertical machining center. Four pockets and two radii were machined using four tools with an infeed of 1 mm in the Z-axis (Figure 8).

After the 6-minute machining operation, the eight parts were not exchanged. Rather, the infeed in Z was increased by 1 mm and the operation repeated. As a result of the thermal expansion of the ballscrew, all workpieces show a step pattern on the left side. This pattern is particularly pronounced on the workpiece farthest away from the fixed bearing.

The right sides of the workpieces are smooth, because with each shift in the positive X direction the previous step was also removed. In principle, the same effect could be observed in the Y direction as in the X direction but, because of the lesser amount of movement in the Y axis, the step pattern is significantly less pronounced. In the X direction the comparative measurement of the step pattern shows a drift of approx. 90 μm with a time constant of thermal expansion slightly less than an hour (Figure 9).

If additional work is to be done on previously machined workpieces with critical dimensions, the machine datum must be continually inspected and corrected. The machine achieves thermal equilibrium after one hour, but after an interruption in machining it begins to drift in the reverse direction. If the part program and with it the mean feed rate are changed, it again takes approx. 1 hour for the ballscrew to regain thermal equilibrium.

Fixed Bearing at Both Ends

The situation is more complex in the case of fixed/fixed bearings. Ideally, rigid bearings would prevent expansion of the ballscrew at its end points. However, this would require considerable force. To prevent expansion of a ballscrew with a 40 mm diameter, 2.6 KN must be applied per degree Celsius of temperature

A typical angular-contact ball bearing would quickly fail under any large increase in temperature. Under real conditions, the rigidity of the purportedly fixed bearings with their seats lies in the area of 800 N/μm. This means that as the temperature of the ballscrew increases, the bearings deform significantly. The end points of the ballscrew do not remain at their original position.

The same experiment as in Figure 7 was conducted on a vertical machining center (built in 1998) with fixed bearings at both ends. The tested 1 m long feed axis was mechanically designed to be very rigid. At each end of the ballscrew the same bearing was built into seats that were machined directly into the machine’s cast frame.

The results of measurement in Figure 10 show curves as theoretically expected. The end points of the ballscrew cannot be kept in their original positions. They each move by 20 to 30 μm in the direction of the force generated by heat. The total expansion of the ballscrew is
about 50 percent less than that shown in Figure 7.

This means that by designing fixed bearings at both ends, the expansion could be halved. The thermal zero point of the feed axis seems to lie at the midpoint of the traverse range. This is also expected because the bearings have approximately equal rigidity and the ballscrew was heated evenly over its entire length.

Fixed/Preloaded Bearing

The fixed/fixed type of bearing causes problems for traversing programs with high mean velocities because the bearing load is detrimental to service life and the forces to be withstood result in deformation of the machine structure. A fixed/ preloaded bearing design (Figure 6) is therefore often used as a sort of pressure valve. With a typical preload of 50 μm/m, one would expect that such a bearing configuration would behave like a fixed/fixed combination up to a temperature increase of approx. 5 K and, beyond that, like a fixed/floating combination.

Figure 11 shows the results of a positioning test on a machining center with a ballscrew with fixed/preloaded bearings conducted along the pattern of the previously described experiments. Surprisingly, in spite of the fixed/preloaded bearing configuration, a position drift similar
to that in Figure 7 becomes apparent. This means that the feed axis behaves roughly like one with fixed/floating bearings.

The thermal zero point seems to lie near the fixed bearing. Unlike the axes in the two previous experiments, this axis had a travel of only 500 mm instead of one meter. The magnitude of the drift is therefore not comparable. This experiment shows that the simple model of the ballscrew with fixed/ preloaded bearings does not stand up to reality.

As a rule, the end with the movable bearing is much less rigid than the end with the fixed bearing. The cause lies in the difference in the bearing designs. While the first end with a genuine, inherently preloaded fixed bearing must continue to remain rigid when the second end has begun to move, with increasing temperature the second end loses its preload and therefore also its rigidity.

Next month we’ll finish our discussion by analyzing the influence of temperature distribution along the ball screw and comparing positioning error with other types of error.


1) Schröder, Wilhelm, “Fine Positioning with Kugelgewindetrieben,”

Progress Report VDI Row 1 NR. 277, Düsseldorf; VDI Verlag 1997.

2) VDW-Bericht 0153, “Investigation from Waelzfuehrungen to the Improvement of the
Static and Dynamic Behavior of Machine Tools.”

3) Weule, Hartmut, Rosum, Jens, “Optimization of the Friction Behavior of Ballscrew Drives
Through WC/C Coated Roller Bodies,” Production Engineering,
Vol. 1/1 (1993).

4) Golz, Hans Ulrich, “Analysis, Concept and Optimization of the Operational Behavior
of Kugelgewindetrieben,” University of Karlsruhe thesis, 1990.

5) Schmitt, Thomas, “Model of the Heat Transfer Procedures in the Mechanical Structure
of CNC Steered Feed Systems,” Shaker publishing house, 1996.

6) A. Frank, F. Ruech, “Position Measurement in CNC Machines . . .,” Lamdamap
Conference, Newcastle 1999.

Dr. Jan Braasch is the manager of linear encoder development at Dr. Johannes Heidenhain
Gmbh, Dr.-Johannes-Heidenhain-Straße 5, 83301 Traunreut, Germany, (0 86 69) 31-0, Fax:
(0 86 69) 3 86 09,


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