, asand

ise is

theand

is a

outort,

A istion

inalces

CIRP Annals - Manufacturing Technology 62 (2013) 499–502

ped

the

een

tial

een

IRP.

A study on machine calibration techniques

Xinghua Li, Guoxiong Zhang (1)*, Shugui Liu, Zurong Qiu, Haitao Zhang, Zhikun Su, Honglei Ran

State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin, China

1. Introduction

To improve the quality of machined parts and reduce wasteparts a coordinate measuring machine (CMM) for on machine insitu measurement of just machined workpieces has been devel-oped [1]. For CMMs with articulated arms the dimensions andgeometrical features of the measured parts cannot be determinedwithout machine parameter calibration [2,3]. The diagram of thedeveloped CMM is shown in Fig. 1. The x carriage can be movedalong its guideway on the frame. The z carriage movesperpendicular to the x motion. The articulated arm rotates about

the rotary axis C1. A motorized probe head is mounted on the frontend of the arm and can be rotated about both vertical axis C2 andhorizontal axis A [4]. For determining the dimensions andgeometrical features of the measured parts the length L of thearm and the length t of the stylus must be calibrated. It is alsoessential to know the zero pose angles for u of the arm and w and cof the probe head. For enhancing the accuracy of the CMM all theparallelism and squareness errors between motion axes and their

determination of zero positions of the rotating arm and styluswell as parallelism and squareness errors between rotational

linear axes will be discussed.For the convenience of discussion a coordinate system

established, in which the average line traveled by the x carriagdefined as the x-axis. The plane passing through the x-axis andaverage line traveled by the z carriage is defined as the xz plane

the y-axis is perpendicular to the xz plane. The z-axisperpendicular to the x and y axes and these three axes formright hand coordinate system.

2. Stylus calibration

The perpendicular line from ball tip center P to axis A abwhich the line rotates is defined as the stylus axis or, for shstylus. The perpendicular distance from ball tip center P to axiscalled stylus length t. The zero position of the stylus is the posiat which the stylus is parallel to the xy plane.

As shown in Fig. 2 when the stylus rotates p/2 from its origposition Dw the ball tip center moves from P1 to P2. The distan

rot ary axis

C1 C2

styl us

armframe x carriage z carri age

prob e headxz

θ

P

ψ

φL tAx

z

Fig. 1. Diagram of the developed CMM.

A R T I C L E I N F O

Keywords:

Coordinate measuring machine

Calibration

Arm

A B S T R A C T

A coordinate measuring machine (CMM) with two linear axes and an articulated arm has been develo

for on machine in situ measurement of just machined workpieces. Some techniques for calibrating

CMM parameters, such as the arm and stylus lengths, offsets, parallelism and squareness errors betw

axes, as well as zero positions of the articulated arm and stylus, have been worked out, which are essen

for assuring high accuracy of the CMM. The effectiveness of the proposed calibration techniques has b

proved by experiments and practice.

� 2013 C

z P1styl uszero

posi tionxπ/2

t2

Δφ

t

A

Contents lists available at SciVerse ScienceDirect

CIRP Annals - Manufacturing Technology

journal homepage: http: / /ees.elsevier.com/cirp/default .asp

P2

error motions also should be calibrated for introducing errorcompensation. Since the methods for calibrating error motions oflinear and rotational axes can be found in literatures [5–9] onlyarm and stylus length calibrations, their offset calibration, and

t1

Fig. 2. Movement of the ball center P during rotation about axis A.* Corresponding author.

E-mail address: gxzhang@tju.edu.cn (G. Zhang).

0007-8506/$ – see front matter � 2013 CIRP.

http://dx.doi.org/10.1016/j.cirp.2013.03.014

mov

t1 ¼

t2 ¼

The

Dw

t ¼ t

D’

Fig.

meapositriggblacand

direits x

t

the

CMMby bin thCMMshow

Azeroaccucom

3. Z

Tthe pwhiposiaxis

Tin F

X. Li et al. / CIRP Annals - Manufacturing Technology 62 (2013) 499–502500

ed in x and z directions are as follows.

t cos D’ � t sin D’ (1)

t cos D’ þ t sin D’ (2)

zero position of the stylus is the position at which t1 = t2. Whenis small

1 þ t2

2(3)

¼ t2 � t1

t1 þ t2(4)

3a shows the diagram for measuring t1. It consists of twosurements. In first measurement the stylus is at horizontaltion. The CMM moves in the +x direction until the probeers (or indicates zero in case of analog probe) as shown by

k lines and its x reading is x1. After that the CMM moves up by t

then the stylus rotates p/2. Finally the CMM moves in the +x

ction until the probe triggers again as shown by red lines and reading is x2. t1 = x2 � x1.

2 measurement also consists of two steps. As shown in Fig. 3bstylus is at its vertical position in the first measurement. The

moves in the �z direction until the probe triggers as shownlack lines and its z reading is z1. After that the CMM moves backe �x direction by t and then the stylus rotates p/2. Finally the

moves in the �z direction until the probe triggers again asn by red lines and its z reading is z2. t2 = z1 � z2.

fter t1 and t2 have been measured both stylus length t and its position can be determined. For enhancing their calibrationracy errors of x and z motions should be previously

aligned horizontal and perpendicular to the x-axis of the CMM byusing a mechanical square. The arm rotates about axis C1 until theprobe triggers. Its angle reading is u1. Then the CMM moves in the+z direction and the arm rotates above the straightedge, then backdown and rotates to trigger on the straightedge from the oppositeside as shown by red lines in Fig. 4a. Its angle reading is u2. The zeroposition of the arm is where u = (u1 + u2)/2.

In the ideal case the projection of ball tip center P on the xy

plane should be on the axis C2 when w = p/2. However there is anoffset e1 between axes C2 and A in the direction perpendicular toboth these axes as shown in Fig. 4b. Besides, the stylus does notintersect with axis C2, but has an offset e2 in the direction parallelto axis A. The ball tip center Pp/2 has a vector offset (e1, e2) withreference to axis C2 and this vector changes its direction during twomeasurements shown in Fig. 4a. To avoid the error caused by thisoffset both u1 and u2 should be measured twice. One at the originalposition, another one after rotating the probe head p about axis C2

[10,11]. Four readings u11, u12, u21, u22 are obtained. The zeroposition of the arm is where u = (u11 + u12 + u21 + u22)/4.

The zero position of axis C2 is defined as the position at which thestylus is parallel to the arm when w = 0. The calibration is carried outat u = 0. A straightedge is aligned parallel to the x-axis while theprobe moves in x direction as shown in Fig. 5a. The angle reading isc1. Then the stylus rotates about axis C2 clockwise until the probetriggers again as shown in Fig. 5b and the angle reading is c2. Zeroposition of axis C2 locates at position c = (c1 + c2)/2 + p/2.

(b) t2 measurement.

(a) t1 measurement.

x

z

triggerrotate

x1

x2

block

x

z

trigger

rotate

z2

z1

block

back

trigger

probe

head

trigger

up

probe

head

t

styl us

styl us

t

Fig. 3. t1 and t2 measurements.

C1

arm

x

y

θ1

θ2

straig htedge

x

yP0Pπ/2

e1e2

C2

A

styl us

square

Fig. 4. Zero position determination of the articulated arm.

straig htedge

C2ψ2

x

y

styl us

straig htedge

C2

(b) First step.

x

y

styl us

ψ1

probe

probe

Fig. 5. Zero position determination of axis C2.

pensated.

ero position determination

he line perpendicular to both the rotary axis C1 and axis C2 ofrobe head is defined as the articulated arm (Fig. 1). The pose at

ch the arm is parallel to the xz plane is defined as the zerotion of the arm. The distance between the rotary axis C1 and

C2 of the probe head is the arm length L.he diagram for determining zero position of the arm is shownig. 4a. A straightedge is mounted on the machine table and

4. Arm length calibration

Since the arm cannot rotate to u = p the arm length is calibratedat positions u = p/2 and u = �p/2. First at position u = w = p/2 thecube corner moves along a guide straightedge in the �y directionuntil the probe head triggers as shown by black lines in Fig. 6. The

ess For

bewill

d z

ands as

bes.nesnes,are

theare byonstionead

le w the

thedericalheythelateandand

bederical

arenal

aree z

and

ad

X. Li et al. / CIRP Annals - Manufacturing Technology 62 (2013) 499–502 501

position of ball tip center P is measured by a laser interferometer.The distance between P and rotary axis C1 equals

L1 ¼ L sin u þ ðe1 þ t cos ’Þ sinðu þ cÞ (5)

For eliminating the effect of (e1 + t cos w) sin(u + c) the probe headis rotated p about axis C2 and the measured distance will beL1 = L sin u � (e1 + t cos w) sin(u + c). The average obtained fromthese two measurements is taken as L1.

Then the arm is rotated to position u = �p/2 and the cube cornermoves along the guide straightedge until the probe triggers againas shown by red lines. The probe head should be rotated p aboutaxis C2 as well to eliminate the error caused by (e1 + t cos w)sin(u + c) and the average obtained from these two measurementsis taken as L2.

The displacement of the cube corner measured by laserinterferometer equals L1 � L2 = 2L sin u � 2L. Both departures ofu from p/2 and �p/2, and e2 cause only a second order error.

5. Offset calibration

As shown in Fig. 4b there are two types of offsets e1 and e2 in theprobe head. e1 might be obtained during the arm lengthcalibration. In the first run the arm length L is calibrated atw = p/2 and c = 0. In the second run it is calibrated at w = p/2 andc = p. The difference between the interferometer readingsobtained in these two runs equals 4e1.

Offsets e1 can be also obtained by rotating the stylus around axisC2 at w = 0. The position of P changes 2(t + e1).

Fig. 7 shows a diagram to calibrate e1 separately. A gage block isfixed on the machine table. At position u = c = 0, w = p/2 the probehead moves in the x direction until it triggers on the gage block. The x

reading is x1. Then probe head rotates p about axis C2 and moves totrigger on the gage block again. The x reading is x2. e1 = (x2 � x1)/2.

However there is another offset e3 = tg caused by squarenerror g between axes A and C2 when w = p/2 as shown in Fig. 7b.obtaining e2 squareness error g between axes A and C2 shouldcalibrated to separate e3 from e2. The method for calibrating g

be discussed in Section 6.

6. Parallelism and squareness calibration

To calibrate the parallelism between the rotary axis C1 anmotion a flat plate is mounted underneath of the probe head

aligned with reference to either axis by adjusting three jackshown in Fig. 8. The important thing is that two axes shouldmeasured with a common reference without readjusting jack

A mechanical square is mounted parallel to the xz and yz plasuccessively as shown in Fig. 8 by black solid and red dash lirespectively. The plate should be thick enough and the squshould be mounted close to the middle of two jacks to makeangle deformation of the plate caused by the weight of the squsmall. The angle deformation in two directions can be checkedan electronic level put on the plate in corresponding directisuccessively. The directional cosines (l0, m0, n0) of the z mowith reference to the plate normal are measured by the probe hwhile z carriage moves.

After that the square is taken off. The stylus rotates to an angclose to 0 and the z carriage goes down until the stylus touchesplate as shown in Fig. 9. For calibrating the parallelism betweenrotary axis C1 and z motion the arm rotates about its axis C1 unc = 0 as shown by . A set of data (L3, ui, z) in cylindrcoordinate system are gathered, where L3 = L + (t + e1) cos w. Tare fitted to a plane by using least square algorithm and

directional cosines (l1, m1, n1) of C1 with reference to the pnormal are calculated. The parallelism errors between axis C1

the z motion in x and y directions can be expressed as l1 � l0m1 � m0, respectively.

The parallelism between axis C2 and the z motion cancalibrated in a similar way. The stylus rotates about axis C2 unu = 0 as shown by . A set of data (L4, ci, z) in the cylindrcoordinate system are gathered, where L4 = (t + e1) cos w. Theyfitted to a plane by using least square algorithm and the directiocosines (l2, m2, n2) of C2 with reference to the plate normalcalculated. The parallelism errors between the axis C2 and thmotion in x and y directions can be expressed as l2 � l0m2 � m0, respectively.

straig htedgerot ary axis

C1 C2

probe headarm

interf ero meter

θ

P

ψ

φL

t

A

y

z

e1

cube corner

Fig. 6. Arm length calibration.

y

rot ary axis

C1 C2

probe head

gage block

θ=0

P

ψ

φ=π/2A

e1

x

z ψ=π/2

y

z

e3

γ

γ

A

C2

t

rot ary axis

C1 C2

level

arm

jack

square

z carri age prob e head

zAx

z

Fig. 8. Plate alignment.

ψz carri age prob e heC1 C2

arm A

θ Lx

z

the in

MM

(a) e1 calibr ation. (b) Off set e3.

Fig. 7. Offset calibration.jack

rot ary axisz

plate

φ styl ust

Fig. 9. Parallelism calibration.

Offset e1 is perpendicular to both axes A and C2. Axis A is in the y

direction while c = 0, so e1 is in the x direction and e2 is in the y

direction at c = 0. e2 has little effect on the measured result whenthe calibration is carried out in the x direction at c = 0 and p.

The diagram shown in Fig. 7 can be also used for calibrating e2.

The calibration should be taken at position u = 0, w = p/2, c = p/2and �p/2.

For determining the directional cosines (l3, m3, n3) of axis Aplate should be mounted parallel to the yz plane as shownFig. 10. The directional cosines of the plate normal in the C

systundcoorleasnormsqual02l03 þtionresp

7. C

Awell0.00posierromeaof th

TableArm

Bef

Aft

Ave

TableDirec

Lar

Sm

Ave

Var

TableDirec

Lar

Sm

Ave

Var

X. Li et al. / CIRP Annals - Manufacturing Technology 62 (2013) 499–502502

em should be remeasured. The stylus rotates about axis Aer u = 0 and c = p/2. A set of data (t, wi, z) in the cylindricaldinate system are gathered. They are fitted to a plane by usingt square algorithm and the directional cosines (l3, m3, n3) of its

al in the Cartesian coordinate system are calculated. Thereness error between the axes A and C2 is determined as

m02m03 þ n02n03, where ðl02; m02; n02Þ and ðl03; m03; n03Þ are direc-al cosines of axes A and C2 in CMM coordinate system,ectively.

alibration results

ll the linear parameters, including arm and stylus lengths, as as offsets, are calibrated with dispersions no larger than3 mm. Dispersions of angular parameters, including zerotions of three rotational axes, parallelism and squarenessrs between rotational and linear axes during repeatedsurements are no larger than 1.5 arcseconds. Below are somee calibrated results (Tables 1–4).

8. Summary

(1) Parameter calibration is essential for obtaining dimensions andgeometrical features of the measured workpieces for non-orthogonal CMMs. It is also critical for enhancing themeasuring accuracy of both Cartesian and nonorthogonalCMMs.

(2) The main parameters for a rotational axis include arm length,zero position, offsets, orientation of its axis and error motions.

(3) The kinematics and geometry of the CMM should be carefullyanalyzed before calibration. Ignoring some parameters withzero nominal values might introduce significant errors in thecalibration of other parameters.

(4) The calibration methodology should be designed in a way tomake the variations of other parameters cause only secondorder errors. Reversal is often used to eliminate the effects ofcertain parameters on others.

(5) The proposed calibration techniques are convenient and canmeet the accuracy requirement for the developed CMM aimedat a measuring uncertainty no larger than 0.01 mm.These techniques can be applied for CMMs with differentconfigurations.

Acknowledgment

The author would like to express their sincere thanks to Prof.Jimmie Miller from UNC Charlotte, USA for his kind help inimproving the English writing.

References

[1] Kunzmann H, Pfeifer T, Schmitt R, Schwenke H, Weckenmann A (2005)Productive Metrology: Adding Value to Manufacture. Annals of the CIRP54(4):691–704.

[2] ASME B89.4.22-2004 (2004) Methods for Performance Evaluation of ArticulatedArm Coordinate Measuring Machines, America Society of Mechanical Engineers,New York.

[3] Werner L (1997) ScanMax—A Novel 3D Coordinate Measuring Machine for theShop-floor Environment. Measurement 18(1):17–25.

[4] http://www.renishaw.com, REVO five-axis measurement system.[5] Hocken RJ, Pereira PH (2011) Coordinate Measuring Machines and Systems,

Second ed. CRC Press, Boca Raton, USA; New York, USA; Abingdon, UK.[6] Zhang G, Veale R, Charlton T, Borchardt B, Hocken R (1985) Error Compensa-

tion of Coordinate Measuring Machines. Annals of the CIRP 34(1):445–448.[7] Sartori S, Zhang G (1995) Geometric Error Measurements and Compensation of

Machines. Annals of the CIRP 44(2):599–609.[8] Schwenke H, Knapp W, Haitjema H, Weckenmann A, Schmitt R, Delbressine F

(2008) Geometric Error Measurement and Compensation of Machines—AnUpdate. Annals of the CIRP 57(2):660–675.

[9] ASME B89.3.4-2010 (2010) Axes of Rotation, Methods for Specifying and Testing,America Society of Mechanical Engineers, New York.

[10] Evans C, Hocken R, Estler W (1996) Self-Calibration: Reversal, Redundancy,Error Separation, and ‘Absolute Testing’. Annals of the CIRP 45(2):617–634.

[11] Donaldson RR (1972) A Simple Method for Separating Spindle Error from Test

C2

probe headψ=π/2

φA

P

y

z

styl us

Fig. 10. Squareness calibration.

1length calibration (mm).

Runs

1 2 3

ore rotation of c 600.9191 600.9181 600.9201

er rotation of c 600.9742 600.9745 600.9711

rages 600.9467 600.9463 600.9456

3tional cosines of the rotational axis of the arm C1.

Directional cosines

l1 m1 n1

gest inclination 0.001068 �0.000152 0.999999

allest inclination 0.001064 �0.000158 0.999999

rages 0.001066 �0.000155 0.999999

iations 0.000004 0.000006 0

2tional cosines of z motion.

Directional cosines

l0 m0 n0

gest inclination 0.001157 0.007176 0.999974

allest inclination 0.001156 0.007172 0.999974

rages 0.0011565 0.007174 0.999974

iations 0.000001 0.000004 0

Table 4Directional cosines of axis C2.

Directional cosines

l2 m2 n2

Largest inclination 0.004167 �0.002449 0.999988

Smallest inclination 0.004166 �0.002448 0.999988

Averages 0.0041665 �0.0024485 0.999988

Variations 0.000001 0.000001 0

Ball Roundness Error. Annals of the CIRP 21(1):125–128.