A Method for Evaluating Spindle Rotation Errors of Machine Tools Using a Laser Interferometer

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Measurement 41 (2008) 526–537

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A method for evaluating spindle rotation errors of machinetools using a laser interferometer

H.F.F. Castro *

Escola Politecnica, Universidade Federal da Bahia, CTAI, Rua Aristides Novis, 02–2� andar, Federacao,

Salvador, BA 40210-630, Brazil

Received 10 July 2006; received in revised form 10 June 2007; accepted 11 June 2007Available online 21 June 2007

Abstract

This paper presents a method for assessing radial and axial error motions of spindles. It uses the Hewlett Packard5529A laser interferometer. The measurement is made using reflection directly from a high-precision sphere. Such objectis used as the optical reflector. The sphere is affixed at the end of a wobble device, which is clamped in the spindle. Theprinciple of measurement is similar to that of a linear interferometer, except that the high-precision sphere is used in placeof the usual retroreflector. A convergent lens is utilized to focus the laser beam to a small spot on the sphere surface. Thisminimizes the dispersion of the beam due to the reflection on the spherical surface. A software package has been developedfor data acquisition and presentation of the error motion polar plots of the spindle. Application of this spindle error cal-ibrator on a CNC machining centre is undertaken. The results are presented and discussed.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Spindle rotation errors; Laser interferometer; Machine accuracy; Spindle metrology

1. Introduction

Nowadays, there is constant pressure on machinetool builders to improve the accuracy, manufactur-ing capabilities and productivity of their machines.Part tolerances in conventional machining havedecreased dramatically over the past two decades.Also, better surface finish has been required formost applications. The geometrical shape and sur-face roughness of workpieces depend substantially

0263-2241/$ - see front matter � 2007 Elsevier Ltd. All rights reserved

doi:10.1016/j.measurement.2007.06.002

* Tel.: +55 71 32359574; fax : +55 71 32039471.E-mail address: [email protected]

on the rotational accuracy of the machine toolspindle.

In view of the facts mentioned above, the evalu-ation of the spindle rotation errors has become veryimportant. Such errors cause degradation in surfacefinish, roundness, feature size and feature location.Furthermore, the analysis of spindle rotation errorscan predict the quality of the machined part. It canalso be used to evaluate the machine tool precisionfor purchasing and maintenance purposes.

The spindle errors comprise two parts as follows:(a) Error motions. These are small departures of theaxis of rotation relative to a stationary referencecoordinate axes (X, Y, Z). The five components ofthese error motions are the translations in the X,

.

H.F.F. Castro / Measurement 41 (2008) 526–537 527

Y and Z directions and the rotations (or tilting)about the X and Y-axes [1,2]. The error motionsin the X and Y directions at a specified axial loca-tion on the Z-axis are called radial error motions.Axial error motion is the error motion colinear withthe Z-axis. Tilt (angular) error motions are the errormotions in the X and Y angular directions relativeto the Z-axis [1,2]. (b) Spindle thermal drifts aremotions of the axis of rotation due to the thermalexpansion or contraction of the components belong-ing to the structural loop associated with the spin-dle. There are three types of spindle thermal drifts:axial, radial and tilt thermal drifts [1,3].

In the following, a brief review of the recentworks on the measurement of spindle rotationerrors is presented.

A simple technique is to use five LVDT (LinearVariable Displacement Transducer) sensors for mea-suring the radial, axial and angular error motions.This is achieved by means of a masterball and a mas-ter plate, which are employed as reference [4]. Theradial error motion can be determined by three-point [5] and four-point methods [6]. They are ableto separate the radial error motion of the spindlefrom the roundness error of a precision sphere. Fur-ther development of these works resulted in themulti-point method [7]. It permits the assessmentof the radial, axial and tilt error motions of thespindle.

Park and Kim [8] developed an optical Moiretechnique to evaluate the radial motion of a spindlewithout using a mechanical master (for example,masterball). Another approach to assess radial erroris offered by the Vector Indication Method [9]. Theradial error is evaluated by means of the vector ofradial error motion of the spindle. A measuring sys-tem based on a Fizeau interferometer [10] has beendeveloped for assessing the axial and angular errormotions of an ultra-precision air spindle.

In another technique, three capacitance-type sen-sors were employed to measure the three-dimen-sional positions of a masterball. This technique[11] traces the centre of the rotating masterball in3D space using polar plot. An angular three-pointmethod [12] that employs three 2D surface slopesensors is capable of measuring simultaneously theworkpiece out-of-roundness and the spindle radialand angular error motions. Liu et al. [13] developeda measuring system for evaluating the radial and tilterror motions of the spindle without using a mastersphere or cylinder. This system uses a rotational fix-ture with a built-in laser diode which is mounted on

the spindle. Two measuring devices with two posi-tion sensitive detectors (PSD) are fixed on themachine table in order to measure the laser pointposition from the laser diode.

Grejda et al. [14] have used a portable master axisof rotation (ultra high-precision air-bearing spindle)and a capacitance probe for assessing radial andaxial error motions of spindles at the nanometerlevel. Donaldson [15] and Estler [16] reversal tech-niques are employed to separate the master axiserror motion from that of the spindle under test.This reversal is achieved by using a rotary tableand a reversal chuck which eliminates the need forrelocating the capacitance probe.

Many methods described above use LVDT andcapacitance sensors that may interact with thedevice under test inducing noise in the measurementsignal, especially at spindles that run at very highspeed. Other ones are capable of assessing onlyradial error motion. The measuring system thatemploys a Fizeau interferometer is very complicatedand expensive. In the method that uses three 2Dslope sensors, it is difficult to align the optical com-ponents accurately around the cylindrical workpiecemounted on the spindle. These optical componentsare laser diodes, beam splitters, mirrors, autocolli-mator lenses, quadrant photodiodes, etc. The mea-suring system devised by Liu et al. [13] uses alsomany optics and devices that need to be mountedand aligned accurately on the machine. This systemis very time-consuming, thus resulting in largemachine downtime. Finally, the technique thatemploys the master axis requires a rotary table forsupporting and rotating the stator of the spindleunder test. However, for medium- and large-sizemachines, the mounting of the spindle headstockon a rotary table is not feasible because this head-stock is fixed on the machine structure. Therefore,this technique is only applicable for very restrictedcases, for example, for testing a small-size spindlethat can be fixed on a rotary table.

In order to overcome some problems associatedwith the methods mentioned above, a method basedon a laser interferometer is proposed in this paper.It is capable of evaluating the radial and axial errormotions of spindles at very high speed and at thenanometer level.

2. Method for evaluating spindle rotation errors

The principle of measurement is similar to that ofa linear interferometer, except that instead of having

528 H.F.F. Castro / Measurement 41 (2008) 526–537

the retroreflector, a masterball with high-surface fin-ish and accuracy is used to reflect the incident beamback to the interferometer. A convergent lens is uti-lized to focus the laser beam to a small spot on thesphere surface. This will minimize the dispersion ofthe beam due to the reflection on the spherical sur-face of the masterball.

A Hewlett Packard (HP) 5529A laser interferom-eter system (nowadays, ‘‘Agilent TechnologiesCompany’’) has been used in this research. It iscapable of performing dynamic calibration due toits high-sampling rate. In this application, the laserinterferometer makes the measurements by usingthe Time Base Generator (TBG). This is a circuitthat uses a 10 MHz crystal-controlled oscillator tocreate a timer circuit that causes a laser positionsample to occur repeatedly at a predetermined timeinterval. The TBG can produce an output frequencyof 10 MHz/(4JK), where J and K each may be setbetween 1 and 65535. When J and K assume value1, the frequency will be 2.5 MHz [17]. This is themaximum sampling rate.

The laser performance (accuracy, resolution, etc.)in this application is approximately the same as fora standard measurement with a linear interferome-ter. The resolution is about 1 nm. This method forevaluating the spindle errors offers some advantagesover other non-contact measurement techniques.They are the following:

(a) The laser interferometers provide higher reso-lution and precision. The spindle errors aremeasured in resolution of 1 nm.

(b) As mentioned above, by using the TBG, thesampling rate of 2.5 MHz can be achieved.Therefore, a very good picture of the spindlerotation errors can be obtained even at veryhigh-spindle speed.

(c) The spot size (measurement area) may bemade as small as necessary, whilst the mea-surement area (spatial resolution) of othermethods is often too large [18].

(d) Lasers do not interact with the devices undertest, so they do not produce measurementdefect like other techniques. For example,when measuring radial run-out of spindles atvery high speed, the air currents often set-upa vortex (air turbulence) trailing off mechani-cal transducers, inducing noise in the run-outsignal [18]. This is more probable to occurwhen the set-up is not stiff enough, especiallythe sensor holders.

(e) The method proposed here uses a masterballas a metrological reference. The masterballroundness errors are measured on a very highaccuracy roundness tester. Next, these errorsare input in the software by the user. The dataacquisition program subtracts the roundnesserrors from the radial run-out sampled bythe laser interferometer. Therefore, in thismeasuring system, the masterball roundnesserrors are compensated for via software.

With a laser interferometer system, the user/man-ufacturer can evaluate many other types of geomet-ric errors of the machine tools and coordinatemeasuring machines, such as, positional, straight-ness, yaw, pitch, flatness, parallelism, squareness(perpendicularity of the two axes of movement),etc. Therefore, in this respect, the laser interferome-ter is a cost-effective calibrator. It does not makesense to purchase this system only to calibrate spin-dle rotation errors.

2.1. Measuring set-up for assessing the radial motion

Fig. 1 depicts the set-up for measuring the spin-dle radial error on a universal lathe. The measuringsystem comprises an HP laser head, an HP linearinterferometer, a reference cube-corner, a screw dri-ven device, a convergent lens, a masterball, a mas-terblock, a plane mirror, a laser diode system, apulse generator, an HP 10887A card, a microcom-puter and a printer.

The laser interferometer measures the radial run-out of the spindle through the masterball. The lasermeasurements contain the eccentricity of the mas-terball (set-up error), the roundness errors of themasterball and the radial error motion of the spin-dle. The masterball is affixed at the end of the mas-terblock, which is mounted to the chuck of thespindle. The masterblock is utilized to reduce thecentring error of the masterball by means of a wob-ble device and adjusting screws.

The laser interferometer acquires the radial run-out during consecutive number of spindle revolu-tions, which was defined in the data acquisitionprogram by the user. The radial data are stored inmemory of the microcomputer. Next, the data areinterpolated and saved in a datafile in the hard diskof the computer. The number of samples per revolu-tion, which is desired, can be set in the software. Thedata acquisition program controls the HP 10887Acard, which is installed in the backplane slot of the

Fig. 1. Set-up for evaluating the spindle radial errors on a universal lathe.

H.F.F. Castro / Measurement 41 (2008) 526–537 529

microcomputer. The lens holder is attached to theback of the linear interferometer. The purpose ofthis procedure is to facilitate the alignment of theoptical system. The lens holder is designed for pro-viding high accuracy of mounting of the lens.

In order to start and end the sampling process atthe same point, it is necessary to detect precisely onerevolution of the spindle. For this purpose, a LaserDiode System (LDS) and a pulse generator areemployed. The former produces a pair of analoguesignal from a photo-quadrant sensor embedded inthe LDS. The latter converts the analogue inputsinto a narrow square pulse. This signal is sent tothe HP card in order to initiate the sampling processand count the number of revolutions by means ofthe software.

In order to achieve the beam alignment, the fol-lowing manual adjustments are provided: vertical(Vh) and transversal (Th) of the laser head; longitudi-nal (Lp) and transversal (Tp) of the cross-slide(machine toolpost); vertical (Vd) of the screw drivendevice. It is possible to obtain an adequate laser beamalignment (better than 90%) using this technique.

Owing to the eventual alteration of the spindlespeed during the test, the number of samples actu-ally acquired over each revolution may vary. Withthe purpose of plotting the error motion, the num-ber of samples should be the same over each revolu-tion. The number of data per revolution (rev) Nc,which is wanted by the user, is entered in the ‘‘Sys-tem Setup’’ program through the keyboard. It isstored in the configuration file of the software. In

order to convert the actual number of samples perrevolution in the number of data/rev Nc desiredby the user, an interpolation technique is employed.It consists of generating processing run-out data atthe angular resolution provided by Nc by meansof a linear interpolation between the two adjacentrun-out raw data.

Although the masterball (diameter = 12.000 mm)has a roundness accuracy within 0.1 lm, for high-precision spindles it may be necessary to separatethe masterball roundness error from the spindleradial motion. In this research, the roundness errorsof the masterball are input in the software by theuser. The program subtracts these roundness errorsfrom the interpolated radial data before saving thedata in the hard disk of the computer. The proce-dures for compensating for the masterball round-ness errors are the following:

(1) The masterball roundness errors are input inthe software in the anticlockwise directionas the viewer is facing the ball. To do so, itis necessary to establish a starting point byusing a very thin ink mark on the masterballsurface. From this point, going around in theanticlockwise direction, the roundness errors,previously measured on a very high-accuracyroundness tester, are picked at a uniformangular interval. Let us suppose 36 values ofthe roundness errors are considered over anangle of 360�. Thus, in this case, the angularresolution is 10�.

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(2) After the roundness errors are entered, thesoftware saves them in a file in two columns.In the first one, the errors are ordered in theanticlockwise direction. The second columncontains the roundness errors in the clockwisedirection. This is done so as to take intoaccount the direction of the spindle rotation.Therefore, if the spindle rotates in theclockwise direction (the viewer is facing it),the software uses the first column of the filefor compensating for the masterball roundnesserrors. If the spindle revolves in the anti-clockwise direction, the program uses thesecond column of roundness errors. Thedirection of spindle rotation is specified bythe user in the ‘‘System Setup’’ program.Such direction of rotation is evaluated as theviewer is facing the spindle. This informationis stored in the configuration file of thesoftware.

(3) As mentioned earlier, the sampled data, i.e.run-out data are interpolated so as to producethe number of samples per revolution Nc

desired by the user. Thereby, the numberof data/rev is made equal to Nc over eachrevolution. In order to compensate for themasterball roundness errors, the number ofroundness errors over 360� should be equalto Nc also. This requires that the masterballroundness errors be also interpolated so asto obtain the same angular resolution as previ-ously. For example, let us suppose Nc = 500data/rev and the number of roundness errors,which was entered previously in the softwarebe 36. To proceed with the compensation ofthe masterball errors, the computer programinterpolates the 36 values of roundness errorsso as to produce a new set of 500 roundnesserror data. Now, these masterball roundnesserrors are subtracted from the interpolatedradial data. Therefore, this new run-out datado not contain the roundness errors of themasterball anymore.

(4) Fig. 1 shows a plane mirror affixed on themasterblock. It is employed to reflect the laserdiode beam back to the Laser Diode System.Thereby, a pulse is sent to the HP card so asto initiate the data acquisition process. Inorder to compensate for the masterball errors,it is essential to begin the data acquisition atexactly the starting point (ink mark on themasterball), which was defined previously

(see Item (1) above). In this way, the measure-ments made by the laser interferometer will besynchronised with the right sequence of theroundness errors of the masterball. Therefore,to achieve this aim, the plane mirror has to bealigned with the ink mark on the masterballsurface. This alignment should be done withhigh accuracy.

As said earlier in this section, the radial run-outdata contain the eccentricity of the masterball.Therefore, before displaying the radial motion ofthe spindle, it is necessary to remove the masterballcentring error from the radial data. This is done byusing the least squares method [1]. The software dis-plays on the computer monitor the error motionpolar plots and their respective error motion valuesin respect to the least squares circle (LSC) centre [1].The following polar plots can be output to a printer:total error motion; average error motion; asynchro-nous error motion; outer error motion; and innererror motion. Also, the linear plot of the total errormotion can be printed.

2.2. Measuring set-up for assessing the axial motion

Basically, the set-up to assess the axial errormotion on a universal lathe is similar to thatdescribed in the previous section. Fig. 2 shows theset-up to undertake the test in the axial direction.The interferometer is mounted in the screw drivendevice so that the reference cube-corner is alignedwith the laser head. The optical axis of the lens ismade coincident with the spindle centre line.

The data acquisition program is the same as thecase of radial motion. The software displays thepolar plots of axial error motion and its componentswith respect to LSC centre. The polar graphs thatcan be printed are the following: total error motion;average error motion; asynchronous error motion;residual error motion; outer error motion; and innererror motion. The linear plot of the axial total errormotion can be also output as a function of the angu-lar position of the spindle.

As shown in Fig. 2, the laser beam is focused onthe surface of the masterball on a small area (laserspot size). The spindle centre line coincides withthe optical axis of the lens. Thus, theoreticallyspeaking, the laser measurements are made on thesame spot of the masterball. As consequence, forthis case, it is not necessary to compensate for themasterball roundness errors.

Fig. 2. Set-up for evaluating the spindle axial errors on a universal lathe.

H.F.F. Castro / Measurement 41 (2008) 526–537 531

3. Optical system

In order to return enough light energy to the laserhead, the laser beam must be focused to a small spotsize so that the loss of energy due to the reflectionon the spherical surface is minimized. Also, the sur-face must be smooth and reflective enough (overmeasurement area) to transmit an adequate signal.

As mentioned earlier, the measuring set-upemploys a single lens so as to focus the laser beamon the masterball surface. Two parameters areimportant in this technique: the spot size and thedepth of field of the single lens. The spot size is theradius of the beam waist at the focal point ofthe lens. The depth of field is the distance, centredabout the lens focal length, over which the object willbe in focus and a measurement can be made [18].

The selected lens is a positive achromatic dou-blet. It is composed of two elements which arecemented, a positive low-index (‘‘crown’’) elementand a negative high-index (‘‘flint’’) one. These ele-ments are chosen so as to minimize the sphericaland chromatic aberrations as well as coma. The lensdiameter of 25.4 mm was selected to fit the diameterof the interferometer (23 mm). Its focal length isf = 60 mm. This provides a laser spot radius of4.8 lm and a depth of field of ±50 lm.

It has been verified that a beam alignment of 96%reflection can be obtained using a tungsten carbide

masterball of 12 mm with surface finish Ra =0.02 lm (arithmetic average). This indicates thatthis spot size is quite satisfactory. The value of thedepth of field covers the range of the radial andaxial error motions which are intended to measure.

4. Application of the spindle error calibrator on a

CNC (computer numerically controlled) machining

centre

The tests were undertaken on an 11-year oldCNC machining centre which has the followingspecifications:

Type: Takisawa Mac-V3 machining centre ofthree-axis vertical spindle bed type.Working travel (mm): 510 · 400 · 360 (longitu-dinal · lateral · vertical spindle travel).Control system: FANUC System 6MB Series.Spindle: diameter = 55 mm, drive motor 7.5 hp,speed range 60–6000 rpm.

4.1. Radial and axial error motions

The origin of the polar graphs of Figs. 3–9 is theleast squares circle centre. The term ‘‘Error Band’’written in these graphs is the error motion value

[1]. According to ANSI/ASME B89.3.4M standard

Fig. 3. Radial total error motion polar plot of Takisawa spindle at speed of 1000 rpm.

Fig. 4. Radial average error motion polar plot of Takisawa spindle at speed of 1000 rpm.

532 H.F.F. Castro / Measurement 41 (2008) 526–537

[1], the error motion value is the difference in radiiof two concentric circles from a specified polar plotcentre that will enclose the corresponding errormotion plot.

For assessing the radial error motion, the laserhead was aligned with the X-axis of the machine.In order to assess the radial and axial motion, thespindle was run in anticlockwise direction (as theviewer faces it) at a speed of 1000 rpm. The radial

and axial motions were recorded over five revolu-tions at a sampling rate of 500 data/rev. The radialtotal and average error motion polar plots of themachining centre spindle are shown in Figs. 3 and4, respectively. In general, the radial average errormotion polar plot indicates the roundness errorsof the workpiece which is capable of being producedby the machine under ideal cutting conditions (toolwithout deflection, wear, built up edge, etc.) [1].

Fig. 5. Axial total error motion polar plot of Takisawa spindle at speed of 1000 rpm.

Fig. 6. Axial average error motion polar plot of Takisawa spindle at speed of 1000 rpm.

H.F.F. Castro / Measurement 41 (2008) 526–537 533

Figs. 5 and 6 depict, respectively, the axial total andaverage error motion polar plots of the machinespindle.

Experiments also were carried out at maximumspeed of 6000 rpm. Fig. 7 shows the radial totalerror motion polar plot of the spindle in that speed.The error motion value is 1.08 lm which is smallerthan the value 1.49 lm obtained with the spindlerunning at 1000 rpm (Fig. 3). Other tests have been

done with spindle speed of 2000, 3000, 4000 and5000 rpm. The test results indicated that as the spin-dle speed increases the radial error motion value isgradually reduced. Perhaps, this behaviour may beexplained on the basis of the increase of the preloador change in the dimensions and geometry of thespindle rolling bearings and their housings as therotational speed increases. As known, the preloadmay augment as a result of temperature rise in the

Fig. 7. Radial total error motion polar plot of Takisawa spindle at speed of 6000 rpm.

Fig. 8. Radial average error motion polar plot of Takisawa spindle at speed of 6000 rpm.

534 H.F.F. Castro / Measurement 41 (2008) 526–537

spindle. The preload not only removes play but alsoreduces the kinematic error caused by the error inthe reference part. This effect is known as ‘‘erroraveraging effect’’ or ‘‘principle of elastic averaging’’.

Fig. 8 shows the radial average error motionpolar plot at a speed of 6000 rpm. The average errormotion value is 0.21 lm which is less than half ofthe average error band of 0.51 lm at a speed of1000 rpm (Fig. 4). Theoretically, this means thatas the spindle speed increases, the machine produces

workpieces with better accuracy. Test result foraxial total error motion is presented in Fig. 9. Theerror band for total error motion is 0.87 lm whichis greater than the value 0.30 lm obtained withthe spindle running at 1000 rpm (Fig. 5). Tests car-ried out at various speeds have indicated that theaxial error motion value increases as the spindleruns faster. Also, it can be seen that the axial polargraphs at 6000 rpm (Fig. 9) are smoother than at1000 rpm (Fig. 5). All these features may have been

Fig. 9. Axial total error motion polar plot of Takisawa spindle at speed of 6000 rpm.

H.F.F. Castro / Measurement 41 (2008) 526–537 535

caused by the temperature rise and temperature gra-dient of the spindle, which in turn, produced theaxial growth and thermal instability of the spindle.

For evaluating the radial and axial errormotions, the deadpath error of the laser interferom-eter system was not compensated for in the soft-ware. This error is caused by an uncompensatedlength of the laser beam between the interferometerand retroreflector (masterball), with the machinestage at zero position. It occurs whenever environ-mental conditions change during a measurement.As the spindle tests were of short duration, theatmospheric conditions (air temperature, baromet-ric pressure and relative humidity) have not changedand, as consequence the deadpath error is null.

Similarly, the laser wavelength compensation wasnot implemented in the data acquisition program.The error in the laser reading due to the changeof the laser wavelength in the air is negligiblebecause the measured distance (spindle errormotions), in this case, is very small. This is demon-strated by considering the following example.

WCN is the Wavelength Compensation Number.It is the inverse of the index-of-refraction n of air,that is

WCN ¼ ka

kv

¼ 1

nð1Þ

where, kv and ka are, respectively, the laser wave-length in vacuum and in air. For the standard atmo-

spheric conditions, i.e. air temperature Tair = 20 �C,air barometric pressure Pair = 760 mm Hg and airrelative humidity = 50%, the WCNs = 0.9997288.Let us suppose that the spindle test is carried outin an environment which has the following atmo-spheric parameters : Tair = 29 �C, Pair = 750 mmHg and relative humidity = 50%. In this case, theWCNa is 0.9997407.

If the wavelength compensation is not consid-ered, the laser reading error will be

Laser reading error ¼ ðDWCNÞ� ðmeasured distanceÞ ð2Þ

For measured distance (spindle error motions) of10�5 m (10 lm), it obtains

Laser reading error ¼ ð0:9997407

� 0:9997288Þð10�5 mÞ¼ 0:12 nm ð3Þ

This error is very small, thus justifying not takinginto account the wavelength compensation in thisexperiment.

Air turbulence due to the rotation of the spindledoes not affect the laser measurement accuracy. Thelinear interferometer is a bit more than 60 mm farfrom the spindle and is mounted to the toolpostwith high stiffness. Therefore, the air turbulencedoes not produce strong pressure waves on it so asto cause vibration on this set-up. On the contrary,

536 H.F.F. Castro / Measurement 41 (2008) 526–537

in case of capacitive sensor, the distance betweenmasterball and sensor is of magnitude order ofmicrometer. For example, a C23-C capacitive sen-sor manufactured by Lion Precision Company withrange of operating of 50 lm, the values of near gapand far gap are 75 lm and 125 lm, respectively.Obviously, in this case, because of very small dis-tance between masterball and sensor, the waves ofpressure which hit the sensor are much strongerthan those which hit the laser system set-up. There-fore, the air pressure may cause noise in the sensorsignal at spindles that run at high speed.

The temperature gradient of the air due to thetemperature rise in the spindle does not produceinaccuracy in the laser measurement in this applica-tion. The reasons for that are described as follows:(1) The temperature of air affects the deadpath errorand the wavelength compensation error of the laserinterferometer system. However, the deadpath erroronly occurs if the atmospheric conditions vary dur-ing the measuring process. As the experiments toevaluate spindle error motion were of short dura-tion (less than 1 s), the temperature of air did notchange while the laser system made the measure-ments. (2) As shown in Eq. (3), wavelength compen-sation error is about 0.12 nm. This example showsthat air temperature, air barometric pressure andair relative humidity do not influence the values ofthe laser readings significantly. Particularly, this isdue to the very small measured distance (10 lm).Therefore, the temperature of air nearby the spindleis not relevant to the uncertainty in the laser inter-ferometer system in this application.

Finally, the method presented here is also capa-ble of measuring the radial and axial thermal driftsof spindles. This will be described in another paper.

5. Conclusions

(1) A method has been developed for evaluatingthe radial and axial error motions of machinetool spindles using a laser interferometer. Themeasurement is made using the reflection froma precision sphere, which is utilized as areflector.

(2) A convergent lens has been used to focus thelaser beam to a spot of radius 4.8 lm. Thedepth of field of the lens is about ±50 lm.This is sufficient to measure the spindle errorsdescribed in Item (1) above.

(3) The laser calibration system is able to evaluatethe accuracy performance of high-precision

spindles. The compensation of the masterballroundness errors from the spindle radial run-out has been carried out through software. Agood picture of the spindle errors at very highspeed can be obtained due to the high-sampling rate of the HP 5529A laserinterferometer.

(4) The use of low-cost components, such as a lin-ear interferometer, focus lens, etc. makes thislaser interferometer-based calibrator a cost-effective system.

Acknowledgement

I would like to express my thanks to Dr. CarlosC. Tu, from the ‘‘Universidade de Sao Paulo’’,Brazil, for his valuable suggestions.

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