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Precision Engineering 43 (2016) 493–504

Contents lists available at ScienceDirect

Precision Engineering

jo ur nal ho me p age: www.elsev ier .com/ locate /prec is ion

inematic errors identification of three-axis machine tools based onachined work pieces

ostafa Pezeshki1, Behrooz Arezoo ∗

AD/CAPP/CAM Research Center, Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave,ehran 15875-4413, Iran

r t i c l e i n f o

rticle history:eceived 28 March 2015eceived in revised form 1 August 2015ccepted 23 September 2015vailable online 30 September 2015

eywords:olumetric error model

a b s t r a c t

Evaluating machine tool performance under machining conditions is generally used as the final test inmachine tool industry. The seventh part of ISO-10791 describes a machining test using the accuracy of afinished work piece to determine the accuracy of three-axis machine tools. However the kinematic errorscannot be distinguished from each other by means of these test pieces. In this paper a new method toidentify the kinematic errors of three-axis machine tool is proposed. A set of test pieces are designedwhere the kinematic errors of a machine tool can be measured separately along X, Y and Z directions. Avolumetric error model is also presented based on the measured errors. This method is initially evaluated

inematic errorstraightness errorsngular errorsinished work pieceshree-axis machine tool

in virtual environment and then with some test pieces designed for this purpose. The results are comparedwith the laser interferometry measurements. It is shown that the measured positioning and straightnesserrors are consistent with the laser interferometry results. Angular errors measured by the test piecesare also complied with the laser interferometry results as long as the angular error magnitudes are largeenough.

© 2015 Elsevier Inc. All rights reserved.

. Introduction

CNC machine tools are widely used to reach required dimen-ional and geometrical tolerances. Geometric and kinematicccuracy of a machine tool moving axes is a significant criterionhich is considered by machine tool manufacturers [1–4]. To eval-ate and improve the kinematic accuracy of a machine tool, theinematic errors should be identified accurately. Volumetric errors then modeled to compute the relative error between tool and

ork piece. Kinematic errors identification and modeling are twoajor factors to compute the volumetric error accurately. It is

mportant to identify the relationship between kinematic errorsuring error identification and modeling [5,6]. The measurementethod and procedure are described in ISO 230-1 and ISO 230-

, respectively [7,8]. All of the machine tests are under no loadondition and do not indicate the machine performance under

achining conditions. The seventh part of ISO 10791 describes aachining test using the accuracy of a finished work piece to deter-ine the accuracy of the machine tool. This standard considers the

∗ Corresponding author. Tel.: +98 912197997.E-mail addresses: Mostafa.pezeshki@yahoo.com (M. Pezeshki),

arezoo@yahoo.com (B. Arezoo).1 Tel: +98 9139301912.

ttp://dx.doi.org/10.1016/j.precisioneng.2015.09.018141-6359/© 2015 Elsevier Inc. All rights reserved.

machining of two components to assess the machine tool accuracyunder finishing conditions [9]. NCG/VDI 5211-1 also proposes a testpiece to assess the performance of high speed three-axis machines,especially when the axes are used simultaneously [10,11]. The testpieces used in these standards are applied to indicate the machineperformance as the final machine test and not used to calibrate themachines. It is difficult to identify the sources of kinematic errorby using the geometry of machined work pieces. The importantissue is that the 21 kinematic error component of the machine toolcannot be identified separately by studying the test pieces. Indeed,some kinematic errors deliver the same effects on the work pieceand it is not possible to distinguish between them. Consequently,the interpretation of the influence of kinematic errors on the geom-etry of machined profiles is difficult. Hong et al. [12] described theeffects of position-dependent errors of rotary axes on cone frustum.Ibaraki et al. [13] proposed a test piece to indicate the influence ofkinematic errors of five-axis machining center. Using the machinedtest piece, the kinematic errors of rotary axes are estimated.

According to the knowledge of the authors there has been nowork to distinguish and identify the kinematic errors of linear axesof a machine tool, using machining test pieces.

In this paper a new method to identify the kinematic errorsof three-axis machine tools is presented. A set of test pieces aredesigned from which the influence of kinematic errors can be dis-tinguished separately from the machined profiles. To compute the

dx.doi.org/10.1016/j.precisioneng.2015.09.018http://www.sciencedirect.com/science/journal/01416359http://www.elsevier.com/locate/precisionhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.precisioneng.2015.09.018&domain=pdfmailto:Mostafa.pezeshki@yahoo.commailto:barezoo@yahoo.comdx.doi.org/10.1016/j.precisioneng.2015.09.018

494 M. Pezeshki, B. Arezoo / Precision Engineering 43 (2016) 493–504

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work piece. To estimate the errors some perpendicular slots areconsidered to machine. As it is shown in Fig. 1b two side cutting

ig. 1. Proposed test pieces to identify kinematic errors of three-axis milling machrrors.

olumetric error of a machine tool, a kinematic error model is putorward based on the measured errors of the test pieces. Usinghis method the volumetric error caused by kinematic errors areetermined. In this approach the kinematic errors are identifiednder finishing conditions and the machine tool performance cane assessed under real loading conditions. To evaluate the approachsed in the present work, the test pieces are machined in thexperiments and the results are compared with the results of lasernterferometry of the machine tool.

. Test piece specification to identify kinematic errors

In this section a set of test pieces are proposed to identifyinematic errors of three-axis milling machines. One test piece isesigned to determine X axis kinematic errors (Fig. 1a); another isesigned to determine Y axis kinematics errors as well as square-ess between X & Y and two others are designed to determine Zxis kinematic errors as well as squareness between Z & X axis andlso Z & Y axis (Fig. 1b).

As it is shown in Fig. 1 a number of patterns (tool path) are indi-ated to identify the kinematic errors of each axis. The presentedatterns should be as simple as possible so that the axes movement

s not engaged in simultaneous movements (It should be noted thaturing the travelling of an axis along its direction, the machine toolrror compensation system should be switched off; Otherwise thether axes may perform compensation movements to counteracthe straightness errors of X-axis).

In the present research the slots which are aligned with movingxis (the axis which is intended to measure the kinematic errors)re called main slots and the slots which are perpendicular toain slots are called side slots (Fig. 1a). All of these slots should

e machined under finishing conditions. A set of points is theneasured by a coordinate measuring machine (CMM) and kine-atic errors are derived. In this section the machining procedures

nd measurement methods are explained and the kinematic errorodel is then derived in Section 3.

.1. Machining procedure

Slots are first pre-machined using roughing and semi finish-ng conditions. An amount of 0.1 mm material is left on the wallsnd bottom of the slots. The finishing step is then carried out to

) Related to X axis (aligned with X axis), (b) related to Y and Z axes and squareness

project the machine tool kinematic errors on the work piece. Thisis explained as follows:

2.1.1. Machining tool path to project straightness and roll errorsof X and Y axes

To project the straightness error of X axis in Z direction (EZX),the bottom of main slots are machined by face milling operation. Tominimize the effect of vibration on the work piece the tool shouldnot be engaged with the walls. Pattern 2 indicates the machiningtool path in forward and backward direction (Fig. 1a). The straight-ness error of X axis in Y direction (EYX) and roll error (EAX) canbe projected using the same tool path. This time the tool shouldsweep the walls of the main slots without engaging with the bot-tom. Here, the walls are machined with a side cutting operationin forward and backward directions. In these operations the toolis moved along the X axis and other axes are not involved (If themachine tool error compensation system is switched off). There-fore, Y and Z kinematic errors do not interfere with X axis kinematicerrors.

2.1.2. Machining tool path to project positioning, yaw and pitcherror of X and Y axes

In order to identify the positioning error (EXX), pitch error (EBX)and yaw error (ECX) side slots should be machined. In Fig. 1a, pat-tern 1 indicates the machining tool path to machine side slots.Side slots are consistent with target positions according to laserinterferometry procedures and the distance between them can bedetermined based on ISO230-2. The kinematic errors of Y axis canbe projected with the same manner (Fig. 1b).

2.1.3. Machining tool path to project squareness errorsIn a three-axis milling machine there are three squareness

errors: the squareness errors Y relative to X, Z relative to X and Z rel-ative to Y which are called COY, BOZ and AOZ, respectively. Theseerrors can change the geometrical characteristics of a machined

operations are considered to estimate COY, and a combination of aside cutting operation and a boring operation are considered to esti-mate AOZ as well as BOZ. The measurement procedure and model-ing of the volumetric errors are described in Sections 2.2 and 3.

M. Pezeshki, B. Arezoo / Precision Engineering 43 (2016) 493–504 495

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Fig. 2. Set of points to identify (a) EZX

.1.4. Machining tool path to project kinematic errors of Z axisThe proposed test pieces and tool path for projecting the kine-

atic errors of Z axis are shown in Fig. 1b. The test piece 1hich is considered for Y axis kinematic errors is placed as the

ase of Z axis related test pieces 2 and 3. Two slots in Z direc-ion are machined by a boring operation at first on test piece 2.est piece 3 is then placed and fixed on test piece 2 to machinehe other two slots in Z direction. Pattern 3 in Fig. 1b shows the

achining tool paths for Z direction. As it is shown only Z axis isnvolved.

.2. Measurement procedure

In order to analyze the kinematic errors using machined testieces, the coordinate of some points on the machined slots shoulde measured by a CMM. A set of points should be considered toetermine each error. In this section several sets of points are pro-osed to measure and characterize kinematic errors.

.2.1. Straightness and roll errors IdentificationTo measure EZX a set of points in the bottom of main slots are

ecorded (Fig. 2a). This area is machined by face milling opera-ion in forward and backward directions. The target positions areetermined based on ISO230-2. The selected points should be on

surface which is created by conventional milling. The deviationsetween real and nominal values in Z direction are used to calculateZX.

The walls of main slots contribute to identify EYX and EAX byecording target positions from upper and lower side of the walls

Fig. 2b). A set of points are recorded from upper side of the wallso identify EYX. The deviations between real and nominal valuesn Y direction are used to calculate EYX. Another set of points areecorded from the lower side of the walls to identify EAX (Fig. 2c).

Fig. 3. Set of points to identify

YX, EAX and (c) the effect of roll error.

Therefore, in every target position EAX can be obtained as the fol-lowing equation

EAX = dy�z

= yb′ − ya′zb′ − za′

(1)

where dy, the deviation between upper and lower target positionsin Y direction; and �z, indicates their vertical distance.

2.2.2. Positioning, yaw and pitch errors identificationEXX, EBX and ECX can be derived by recording data from side

slots walls (Fig. 3). The data should be recorded from upper andlower side of the slots from both sides of the workpiece (Fig. 3a andb). The deviation between real and nominal values on upper set ofpoints in X direction is used to identify EXX.

As shown in Fig. 4a pitch error is calculated as the followingequation:

EBX = dx�z

= xb′ − xa′zb′ − za′

(2)

where dx, the deviation between upper and lower target positionsin X direction; and �z, indicates their vertical distance.

Yaw error of X axis is obtained as the following equation:

ECX = dx�y

= xb′ − xa′yb′ − ya′

(3)

where dx, the deviation between upper target positions in bothsides in X direction; and �y, indicates their horizontal distance(Fig. 4b).

The measurement procedure and error identification of Y axis iscarried out in the same manner.

2.2.3. Squareness errors identificationTo identify COY, the walls of two perpendicular slots in X and

Y directions are measured by CMM and the angle between least

(a) EXX, EBX and (b) ECX.

496 M. Pezeshki, B. Arezoo / Precision En

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Fig. 4. The effect of (a) pitch and (b) yaw error.

quare lines which are passed through recorded points along X and directions is computed (Fig. 5a). To calculate BOZ, the wall of thelot along X direction and the wall of half-cylinder along Z directionhould also be measured. The normal vector of the points on thealf-cylinder is along the X direction (Fig. 5b). The angle between

east square lines which are passed through recorded points along and Z directions indicates BOZ. AOZ can also be calculated fromhe angle of least square lines along Y and Z directions.

.2.4. Kinematic errors of Z axis identificationIn order to identify the effects of angular and straightness errors

f Z axis, walls of half-cylinders are recorded by CMM. As it is shownn Fig. 1b three half-cylinders exist on two sides of the test pieces, 2 and 3. A number of points on the half-cylinders are recordedy CMM as Z target positions. The normal vectors of the points

n one side is along X direction and on the other side is along Yirection. Therefore the deviations of Z target positions in X and Yirections can be measured. To identify positioning errors a pointn the bottom of half-cylinders are recorded by CMM (Fig. 6a and). Fig. 6a and b indicates the recorded points to identify positioningrrors and kinematic errors in X and Y directions, respectively, andig. 6c indicates measurement method of the machined test pieceso record data.

. Derivation of kinematic error model

Kinematic error model illustrates the effects of kine-atic/geometric errors of a machine component on the position

Fig. 5. Set of points to identify (a

gineering 43 (2016) 493–504

and orientation of tool with respect to work piece. The errormodel computes an error vector in all positions of the machinework space. The error vectors are interpreted by homogeneoustransformation matrix [6,14–16] and also by vector representation[1,17]. In modeling of kinematic error of a three-axis machinetool it is important to consider the effect of 21 kinematic errors,correctly. In this section the development of a kinematic errormodel for a three-axis milling machine is described, where theX and Y axes carry the work piece and Z axis carries the tool.The developed kinematic error model is then generalized for thekinematic errors obtained by machined test pieces.

3.1. Modeling kinematic errors of three-axis machine tool

Positioning error of X axis creates a vector along X directionand also straightness errors make two vectors normal to move-ment in Y and Z directions. In order to measure straightness errorsa functional point is determined on the moving component and itslateral displacements are measured. Eq. (4) indicates the linear andpositioning error magnitudes of X axis:

⎧⎪⎨⎪⎩

ex = EXXey = −EYXez = −EZX

(4)

Every angular error vector is resolved in two components in twodifferent directions (Fig. 7). The magnitudes of angular errors canbe obtained as the following equation:

⎧⎪⎨⎪⎩

ex = ECX × yo − EBX × zoey = −ECX × xo + EAX × zoez = EBX × xo − EAX × yo

(5)

where xo, yo and zo are Abbe offsets. Therefore the kinematic errorsof X axis can be obtained as the following equation:

⎧⎪⎨⎪⎩

ex = EXX + ECX × yo − EBX × zoey = −EYX − ECX × xo + EAX × zoez = −EZX + EBX × xo − EAX × yo

(6)

The same procedure is also carried out in order to compute theerror magnitudes of Y axis.

When Z axis positioning and straightness errors are being mea-sured, the measurement system is always aligned with functionalpoints where the displacement and straightness are to be mea-sured. So, according to Abbe principle [18,19] it is not necessary

) COY, (b) BOZ and (c) AOZ.

M. Pezeshki, B. Arezoo / Precision Engineering 43 (2016) 493–504 497

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Fig. 6. Set of points to measure kinematic errors of Z axis along (a) X d

o measure the angular errors. Therefore the kinematic errors of Zxis are obtained as the following equation:

ex = EXZey = EYZez = EZZ

(7)

Three squareness errors can also be calculated as the followingquation:

ex = COY × yw − BOZ × zwey = AOZ × zw − COY × xwez = BOZ × xw − AOZ × yw

(8)

here xw, yw, zw are Abbe offsets. Therefore by adding X, Y and Zinematic errors, the kinematic error model of a three-axis machineool is obtained.

Fig. 7. The effects of X-axis (a) pitch, (b) roll a

on, (b) Y direction and (c) CMM set up to measure half-cylinder walls.

3.2. Modeling kinematic errors of three-axis machine tool basedon machined work pieces

To compute the EZX and EYX the bottom part and the wall of themain slots are measured by CMM, respectively. However angularerrors also have an effect on these areas. For example, the errorcomponent of ECX which is along the Y direction (Fig. 7c) affectsthe wall of main slots in addition to EYX. Also the error componentof EBX which is along the Z direction affects the bottom part of mainslots in addition to EZX (Fig. 7a). It is very difficult to distinguish theeffects of angular errors and straightness errors from machined testpieces. So, in X axis the measured data EYX and EZX are combinedwith angular errors.

As shown in Fig. 7, some error components do not change in con-stant X position and different Y or Z position. However a numberof angular error component values do change in a constant X posi-

tion (target position) and different Y or Z positions. On the otherhand, the magnitudes of constant components only depend on Xposition and are independent of Z and Y positions. For example inFig. 7c two points with same X positions and different Y positions

nd (c) yaw error in a certain x position.

4 ion Engineering 43 (2016) 493–504

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re considered (a,b). Affecting the yaw error the points are trans-erred to new positions (a′,b′). After resolving the

−→aa′ and

−→bb′ in to

heir components, it is indicated that the components which arelong Y direction are constant whereas the components which arelong X directions change depending on their Y positions.

The error components which affect the linear errors are thoserror components which do not change in constant X position.herefore in kinematic error model the straightness errors andhe angular error components which are in the same direction areonsidered together as the following equations:

z (x) = EZX + EBX × xo (9)y (x) = EYX + ECX × xo (10)here subscript letter represents the direction of deviation, and let-

er in bracket represents the intended direction of motion. Dz(x) andy(x) are two error components which are measured by CMM fromottoms and walls of main slots. These components are constant inach X position and different Y and Z positions. Consequently, theinematic errors of X axis are obtained as the following equation:

ex = EXX + ECX × yo − EBX × zoey = −Dy (x) + EAX × zoez = −Dz (x) − EAX × yo

(11)

The kinematic errors of Y axis can be calculated in the same man-er. To measure Z axis kinematic errors, the wall of half-cylinderswhere the normal vector is along X or on the other side along

direction) and the bottom of half-cylinder are also measured.ince the considered set of points are aligned with functional pointswhere the displacement and straightness are to be measured), the

easuring system is consistent with Abbe principle [18,19] and sohe angular errors should not be measured. Therefore the kinematicrrors of Z axis are obtained as the following equation:

ex = EXZey = EYZez = EZZ

(12)

The squareness errors are also calculated as Eq. (8). As a resulthe kinematic error model of a three-axis machine tool is given ashe following equation:

ex = EXX + ECX × yo − EBX × zo − Dx (y) − EBY × zo + EXZ+ COY × yw − BOZ × zw

ey = −Dy (x) + EAX × z0 + EYY − ECY × xo + EAY × zo + EYZ+ AOZ × zw − COY × xw

ez = −Dz (x) − EAX × yo − Dz (y) + EBY × xo + EZZ + BOZ × xw− AOZ × yw

(13)

It should be noted that all of the terms in Eq. (13) can be mea-ured by machined test pieces and all effects of the machine toolinematic errors are considered in the presented kinematic model.

. Results

Kinematic errors measurement based on machining test piecesan be affected by many factors such as machining conditions,ool and environmental aspects. Therefore the repeatability condi-ions should be observed in test piece machining and the dynamic

ffects of machining operations should be reduced as much asossible. However, the dynamic effects of machining operationannot be eliminated totally and so it can influence the valida-ion of the developed model in the present work. Therefore to

Fig. 8. Test piece related to X axis.

assess the proposed method of kinematic errors identification, anenhanced virtual machining is applied at first. This is defined asthe integration of machine tool kinematic errors model into NCmachining simulation [20]. The workpieces are machined by meansof a NC machining simulator, using the nominal machine coordi-nates altered by the kinematic errors model (applying a developedsoftware package). The machined virtual parts have the effects ofthe kinematic errors on their geometry when compared with nomi-nal work pieces. The dynamic properties of the machine are ignoredin a NC-simulator, so the virtual machined parts are only influencedby kinematic errors. The kinematic errors are estimated by mea-suring the machined test pieces in the NC simulator according tothe proposed method and are compared with the kinematic errorswhich are input to software package.

Finally to complete the verification process the test pieces aremachined in real environment and kinematic errors of the machinetool are estimated. The estimated errors are then compared withthe results of the laser interferometer.

4.1. Errors identification method verification

In this section the verification of the error identification methodin virtual environment is explained. Nominal machine coordinatescan be altered with kinematic errors using the developed softwarepackage. NC-codes of the test piece which is related to the X axisare fed to the software. Twenty-one hypothetical kinematic errorsof a machine tool are then fed to the software package and nominalmachine coordinates are altered accordingly by the developed soft-ware package. The X axis related test piece is machined in VericutNC-simulator [21] using the generated NC-codes by the softwarepackage (Fig. 8). The virtual machined test piece is saved in STLformat.

The STL format of the virtual machined test piece is input to aCAD software and its dimensions are compared with the nominaltest piece. The discrepancy of the virtual machined and nominal testpieces are measured based on the developed method in the CADsoftware and the kinematic errors of X-axis is then calculated. InFig. 9 the six X-axis related errors (out of 21 hypothetical kinematicerrors) are compared with the six X-axis kinematic errors estimatedby measuring the test piece.

As shown in Fig. 9 the blue lines indicate the hypothetical kine-

matic errors of X-axis and the red lines indicate the errors estimatedby comparing dimensions of the simulated and the nominal testpieces which are called identified errors. The hypothetical angu-lar errors which are loaded to the software package are very small

M. Pezeshki, B. Arezoo / Precision Engineering 43 (2016) 493–504 499

Fig. 9. Kinematic errors of X axis (a) EXX, (b) Dy(x), (c) Dz(x), (d) EBX, (e) ECX, (f) EAX. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

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1 �rad to 20 �rad or 0.2′′ to 4′′). This may have caused some of theismatching between the results. Another cause may be due to the

TL format estimation errors by the CAD/CAM software used in theresent work. However the matching between results seems goodnough for verification purposes. The results indicate that the kine-atic errors identified by virtual machined test piece are consistentith simulator software input errors.

.2. Machine test method validation

In order to validate the presented machine test method a DROOP REIN machine tool is considered. The test pieces are machinedccording to the procedure described in Section 2.1 which arehown in Fig. 10.

Since a rough surface can easily affect the results, it is essen-tial to minimize the vibration and tool deflection by improvingthe machining conditions. The surfaces should be finish machinedby conventional milling and repeatability conditions should alsobe considered. Table 1 indicates the machining conditions andrepeatability conditions.The sets of points which are described inSection 2.2 are measured by a DEA coordinate measuring machine.The positioning, linear and angular errors of the X axis are calcu-lated and compared with the laser interferometry results which areshown in Fig. 11.

The distribution of measuring points is added to EXX graph inFig. 11a as an example. In this graph the first 6 points are the distri-bution of measurements taken from the test pieces. The last 4 pointsare distribution of measurements by the laser interferometer. In

500 M. Pezeshki, B. Arezoo / Precision Engineering 43 (2016) 493–504

Fig. 10. Machining of (a) X, (b) Y, ((c) a

Table 1Machining conditions of machined test pieces.

Parameter Magnitude

Spindle speed in side and face cutting operation (rpm) 650Spindle speed in boring operation (rpm) 400Feed rate in side and face cutting operation (mm/min) 90Feed rate in boring operation (mm/min) 35Radial depth of cut (mm) 0.1

oaT

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Axial depth of cut (mm) 0.1Air temperature (◦C) 19–22

rder to keep up with the standards for each point 5 measurementsre carried out and their mean values are plotted in these graphs.he rest of the graphs are formed in exactly the same way.

The discrepancy between kinematic errors measured by lasernterferometry test and test piece along Y direction are shown inig. 12.

The test pieces dimensions can be designed in a way that theyover all the machine workspace. However, this will result in timeonsuming and costly machining operations. Therefore the testieces used in the present wok have the common workspace forhe machine tool used in the experiments. The dimensions of theest pieces used in the present study are shown in Appendix A. Theistance between each target position is 50 mm.

As shown in Figs. 11 and 12 the measured positioning errors of Xnd Y axes are consistent with laser interferometry test results andtraightness errors are also complied with the laser interferometry

esults. The accuracy of straightness errors measured by the testieces along Z direction is less than X and Y direction because theurfaces machined by the face milling operation are rougher thanhose which are machined by side cutting operation. The angular

able 2est results of roll error of Y axis (EBY).

I 1 2

Mean unidirectional deviation x̄i (�m/m) 84.28 21.15 Estimator of standard uncertainty si (�m/m) 4.84 5.43 x̄i − 2si (�m/m) 74.58 10.29 x̄i + 2si (�m/m) 93.97 32.02 Unidirectional repeatability Ri = 4si (�m/m) 19.39 21.72

nd (d)) Z axes related test pieces.

errors measured by the test pieces are also complied with the laserinterferometry results as long as the angular error magnitudesare large enough. If the angular errors are small, their effectsmay become insignificant in comparison to other volumetricerror sources such as cutting force induced errors, thermal errorsand fixture dependent errors. Although the effects of these errorsources are reduced as much as possible, these can be largerthan the machine tool angular errors magnitudes (Figs. 11e and fand 12e and f). In other words, the test pieces dimensions causerestrictions for measuring very small angular errors. �z and �y inEqs. (1)–(3) have restricted values (50–75 mm). These values aresmall and cannot project very little angular errors. �y (or �x intest piece related to Y-axis) is 75 mm for obtaining the yaw errorsand �z is 50 mm and 55 mm for obtaining the pitch and roll errors,respectively. For example when the EBX is 10 �m/m and when�z is 50 mm, dx in Eq. (2) is just 0.5 �m. Since dx is influenced bydynamic effects of machining operation, it is difficult to projectthis value on the test pieces.

However, according to ISO 10791-2 [22], the permissible angularerrors tolerance of the X and Y axes of the considered machine tool is60 �m/m or 12′′ (the difference between maximum and minimummeasured errors should not exceed the tolerance). If the differenceis less than 60 �rad, the values of the angular errors are within thepermissible range. Therefore, the error magnitude of 10 �m/m (or2′′) is very small and ignored in most industries. As it is shown inFigs. 11d, e, and f and 12e and f, when the angular errors mag-nitudes of a machine tool measured by laser interferometer (or

electronic level) are within permissible tolerances, they carry verysmall values and cannot be treated with the method in the presentresearch. However if the errors are large and exceed the permissi-ble tolerances and become large enough compared with the effects

3 4 5 6

17.09 0 −0.11 −48.653.39 5.37 3.87 3.39

10.31 −10.75 −7.86 −55.4423.88 10.75 7.63 −41.8713.56 21.49 15.49 13.56

M. Pezeshki, B. Arezoo / Precision Engineering 43 (2016) 493–504 501

(b) D

ou

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Fig. 11. Kinematic errors of X axis (a) EXX,

f other error sources, the present method can be used with a lowncertainty factor (Fig. 12d).

The test results for roll errors for Y axis, estimated by measuringhe test pieces are shown in Table 2. All other test results for angular

rrors for X and Y axes are estimated in the same way. Tables 3 and 4re the final results of uncertainty analysis of angular errors of X and

axes according to ISO 230-2 [8].

able 3ncertainty analysis of angular errors of X axis.

Unidirectional deviation ↑ (�m/m) EAX EBX ECXSystematic deviation (E) 59.73 41.03 4.53Repeatability of error (R) 27.86 23.83 12.41Accuracy (A) 79.14 58.78 15.71

y(x), (c) Dz(x), (d) EBX, (e) ECX and (f) EAX.

As shown in Tables 3 and 4 among the angular errors measuredby the test pieces, only the systematic deviation of EBY exceedsfrom the permissible tolerance range. This error exceeded from per-missible tolerance range when is measured by electronic level.A

boring operation along Z direction is carried out to project thekinematic errors of Z axis. As mentioned the considered points formeasuring the errors are always aligned with the functional points

Table 4Uncertainty analysis of angular errors of Y axis.

Unidirectional deviation ↑ (�m/m) EAY EBY ECYSystematic deviation (E) 54.54 132.93 27.7Repeatability of error (R) 28 21.72 24.16Accuracy (A) 81.14 149.141 51.87

502 M. Pezeshki, B. Arezoo / Precision Engineering 43 (2016) 493–504

Fig. 12. Kinematic errors of Y axis (a) EYY, (b) Dx(y), (c) Dz(y), (d) EBY, (e) ECY and (f) EAY.

Fig. 13. Kinematic errors of Z axis (a) EYZ and (b) EXZ.

M. Pezeshki, B. Arezoo / Precision En

Table 5The squareness errors of the machine tool measured by test piece.

wSmla

tpdmar

5

kap

AOZ (�m/m) BOZ (�m/m) COY (�m/m)

29.3 28.9 42.7

here the straightness and positioning errors are to be measured.o, all effects of the angular errors are measured during linear erroreasuring. The straightness errors of Z axis are compared with

aser interferometry results in Fig. 13. The squareness errors arelso shown in Table 5.

Considering the results which are shown in Figs. 11–13, posi-ioning errors measured by the laser interferometer resemble theositioning error determined by the test pieces within 73%. Theata also indicates that the straightness errors and angular errorseasured by laser interferometer resemble the straightness errors

nd angular errors measured by test pieces within 66% and 38%,espectively.

. Conclusion

In this paper a new method was developed to identify theinematic errors of three-axis machine tools. Some test piecesre designed and machining procedures to project errors on testieces are explained. Using this method the kinematic errors can be

Fig. A1. The dimensions of the test piece related to X-axis

the best consistency with using the laser interferometry results(about 73%). In addition straightness and angular errors are alsoconsistent and resemble within 66% and 38%, respectively. Sincethe machine test is under loading conditions, other sources of vol-umetric errors such as thermal errors, cutting force induced errorsand fixture dependent errors can affect results. These sources cancause the deviation between error measured by the test pieces andlaser interferometry test. The new method is proposed to assessthe machine tool kinematic errors under loading conditions as thefinal test. Indeed the new method can evaluate machine tool per-formance under real condition. The test pieces can be used severaltimes as long as the test pieces walls are thick enough.

Acknowledgements

The authors would like to thank Iran Khodro Industrial Dies(IKID) and LAKSAR Companies for providing the machine tooland CMM and the related instruments for the experiments in thepresent research.

Appendix A. The test pieces dimensions

Figs. A1–A3.

Fig. A2. The dimensions of the t

gineering 43 (2016) 493–504 503

distinguished from each other. The errors are measured and com-pared with laser interferometry results. Positioning errors have

est piece related to Y-axis

504 M. Pezeshki, B. Arezoo / Precision En

R

[

[

[

[

[

[

[

[

[

[

[

Fig. A3. The dimensions of the test pieces related to Z-axis

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Kinematic errors identification of three-axis machine tools based on machined work pieces1 Introduction2 Test piece specification to identify kinematic errors2.1 Machining procedure2.1.1 Machining tool path to project straightness and roll errors of X and Y axes2.1.2 Machining tool path to project positioning, yaw and pitch error of X and Y axes2.1.3 Machining tool path to project squareness errors2.1.4 Machining tool path to project kinematic errors of Z axis

2.2 Measurement procedure2.2.1 Straightness and roll errors Identification2.2.2 Positioning, yaw and pitch errors identification2.2.3 Squareness errors identification2.2.4 Kinematic errors of Z axis identification

3 Derivation of kinematic error model3.1 Modeling kinematic errors of three-axis machine tool3.2 Modeling kinematic errors of three-axis machine tool based on machined work pieces

4 Results4.1 Errors identification method verification4.2 Machine test method validation

5 ConclusionAcknowledgementsAppendix A The test pieces dimensionsReferences

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