Calibration for multiple motion errors of X-Y table on micro-coordinate measuring machine(M-CMM) by utilizing multi-probe scanning method
Ping Yang1, Tomohiko Takamura1, Satoru Takahashi1, Kiyoshi Takamasu1,
Osamu Sato2; Sonko Osawa2, and Toshiyuki Takatsuji2
1

2

Department of Precision Engineering, The University of Tokyo, Tokyo, Japan
National Metrology Institute of Japan, National Institute of Advanced Industrial Science and Technology, Ibaraki, Japan

Abstract–A high-precision micro-CMM (M-CMM) is currently under development in the National Institute of Advanced Industrial Science andTechnology (AIST), in collaboration with the University of Tokyo. Our aim is to achieve 50-nm measurement uncertainty with a measuring volume of
30 × 30 × 10 mm (XYZ). In this paper, we have designed a multi-probe scanning system to calibrate the multiple motion errors of X-Y table stage on
the M-CMM and measure the profile of a reference bar mirror simultaneously. In the scanning system, three laser interferometers probe the surface ofthe flat bar mirror which is fixed on top of the X-Y table stage, while the autocollimator simultaneously measures the yaw error of the moving axis. We
have discussed the simulation results of the uncertainty value of the multi-probe scanning method and conducted pre-experiments based on a stepping
motor system to verify the simulation results. The results from the pre-experiment verify that the multi-probe measurement method performs the yaw
and straightness motion error measurement extremely well and also measure the reference bar mirror profile with a small standard deviation of 10 nm.
Finally, we are going to perform experiments on the M-CMM to calibrate the multiple motion errors of X-Y table stage in near future.
Key Words: Micro-CMM, multi-probe scanning method, uncertainty, motion errors, profile

1.

Institute of Advanced Industrial Science and Technology (AIST), in

Introduction

collaboration with the University of Tokyo. The moving volume of thefor

M-CMM is 160 × 160 × 100 mm (XYZ), and our aim is to achieve

three-dimensional (3D) metrologies for microsystem components have

50-nm measurement uncertainty with a measuring volume of 30 × 30 ×increased. However, conventional measuring methods cannot meet these

10 mm (XYZ). Since the motion errors of each stage of the M-CMM

requirements because the measurement scales of conventional

should be calibrated and compensated in order to develop a highcoordinate measuring machines (CMMs) are usually limited to several

precision M-CMM, we have proposed a multi-probe scanning method,tens of millimeters or more, which is not suitable for measuring small

simulated its measurement uncertainty results, and discussed its

parts of the order of submillimeters or even submicrometers. In addition,

applications for evaluating the multiple motion errors of each stage andconventional CMMs lack good 3D measurement uncertainty levels andmeasuring the profile of a reference bar mirror [7–9].

Inmordenmicrosystemtechnologies,

requirements

are often not supplied with the proper probing systems in many
applications [1]. Therefore, different kinds of micro-CMMs equippedwithspecialmicro-probe

systemsfor3D

metrology

2.

M-CMM configuration

having
The M-CMM configuration comprises three main parts: a crosshigh-aspect-ratio micro parts are currently being developed to satisfy the

XY-axis, a Z-axis, and a probe unit. Each axis has a linear motion stage

described requirements.
IBS Precision Engineering has developed the Isara CMM, anultra-precision 3D coordinate measuring machine, with a measuringsystem that comprises air-bearing sliders, a glass linear scale, a moving
table, a driving motor, and related parts.

volume of 100 × 100 × 40 mm and a 3D measuring uncertainty of 30

There are three special specifications in the M-CMM design. The

nm [2]. The F25 micro-CMM (Carl Zeiss) is another commerciallyframe table and main structure of each axis is made of alumina ceramic

available product with a measuring volume of one cubic decimeter andhaving high rigidity and low coefficient of thermal expansion (CTE; 7

measuring uncertainty of 250 nm at a resolution of 7.5 nm [3]. Further,

ppm/K). Further, the base plate of the M-CMM is made of granite

the National Physical Laboratory (NPL) is currently conducting

having CTE of 5 ppm/K, as shown in Fig. 1. In general, thermal effects

researches into developing a micro-scale probe that promises the fullare one of most significant reasons of non-repeatability in measuringpotentials

Physikalisch-Technischemachine accuracy. The M-CMM minimizes these effects by using theBundesanstalt (PTB) is working along with Carl Zeiss in the field of 3D

same alumina ceramic materials. Because of this optimized design,

measurements on micro parts with measurement uncertainties in thewhen the temperature changes, the main mechanical structures will

range of less than 0.1 µm [5]. M-NanoCoord designed by Mitutoyo candeform almost in the same deviation range. On the other hand, theequip with vision probe system and touch probe system with ameasurement area is covered with an enclosure to minimize heat inputs

measuring volume of 200 × 200 × 100 mm and accuray over 200 nm

from the machine environment (e.g., from operators). Therefore, the[6].

M-CMM exhibits good performance in response to temperature changes,ofmicro-CMMs

[4].

The

A high-precision micro-CMM (M-CMM) is currently underdevelopment at the National Metrology Institute of Japan in the National

and thermal deformations due to the driving heat and temperaturechanges are significantly reduced.

1

We subsequently divided the XYZ-axis into two mechanical parts: thereference and sensing points; the Abbe offsets are the distances between

XY-axis, which is stacked with two linear stages, and the Z-axis, which

the reference and sensing points. The Abbe errors are often the mostis separately designed and constructed at the center of the frame table.

important uncertainty sources in dimensional metrology applications

The primary reason for this division is that the measurementthat require measurement uncertainties of only a few nanometers

uncertainties of the Z-axis are larger than those of the XY-axis because

The motion errors of the M-CMM without any compensation are

the sensitivity of the 3D contacting micro-probes in the Z-direction isshown in Table 1. For instance, when the Abbe offset H is 50 mm, if

usually lower than that in the XY plane; this, in turn, is due to the effectthe tilt of the X axis, θ, to be 8 μrad, the Abbe error δ in the X directionof the length of the probe stylus on the horizontal probing direction.

will be 400 nm (Fig. 3). It cannot meet our expection of 50-nm

Finally, the probe unit has a changeable connector, and hence, themeasurement uncertainty. Therefore, the 6 DOFs of each stage of the

M-CMM can use different types of touch probe systems and conduct 3D

M-CMM are very important factors in the development of ameasurements with different levels of uncertainties.

high-precision M-CMM. Hence, the motion errors of the XY stage in
the M-CMM should be measured and calibrated.

Z-axisTable 1: Motion errors of M-CMM without compensation
Axis Degrees of freedom
Accuracy / range
X,YTranslation
Max: 0.5 µm / 160mm
X,Y
Rotation
Max: 8 µrad / 160 mm
Z
Translation
Max: 0.3 µm / 100 mm
Z
Rotation
Max: 5 µrad / 100 mm

Frame table
Counterbalancing weights

XY-axisZ

Work
100 mmProbe

X
H

Granite base plateθYFig. 1 Main structure of micro-coordinate measuring machine(M-CMM)

Moving direction

3.

Calibration for motion errors of X-Y table stage

Linear scaleFig. 3 Illustration of Abbe error without calibration

The XY-axis is a stacking-type mechanism having two linear stages

In the traditional calibration method, one displacement sensor is usedcomprising air-bearing sliders, ultrasonic motors (linear motors),

with a high-accuracy reference bar mirror. The accuracy of this method

moving tables, linear scales, and related parts, as shown in Fig. 2. The

is dependent on the accuracy of the reference mirror because the final

X-Y table stage is fixed on a granite base plate. Each stage of themeasured results include the profile of the reference bar mirror. In this

XY-axis is driven by an ultrasonic motor, and its movement is detected

paper, we have designed a multi-probe scanning system to calibrate theby a linear scale mounted on the side of the moving stage. A moving

multiple motion errors of X-Y table stage on the M-CMM and measuretable with a size of 280 mm × 280 mm has sufficient space to performthe profile of a reference bar mirror simultaneously.

calibrations.
4.
Z-axis with probe unit
50 mm

Measurement Principle

4.1 Principle of multi-probe scanning method

Y-axis moving table

The principle of the multi-probe scanning method is based on an error
Air-bearing sliders
Linear scale

separation technique originally proposed by Whitehouse in 1976 [10]and widely used in the precision measurement field [11–12].
Multi-probe methods for measuring straightness and roundness havebeen extensively developed. These methods began with research intosequential two-point and three-point methods [12–13] that not only

X-axis moving table

Ultrasonic motor

Fig. 2 Structure of XY stage and probe unit

measure the straightness motion error of the guideway, but also thestraightness of the objective surface. However, these methods arelimited in terms of horizontal resolution, and the results are affected bysystematic and random errors of the output sensors. In order to improve

Because of the mechanical design of X-Y table stage, we cannot

these, a multi-probe scanning method comprising three laser

satisfy the Abbe principle, and the Abbe error will exist. The Abbe errors

interferometers and one autocollimator has been proposed and anlysized.

are caused by the rotational errors of relative translations between the

In the multi-probe scanning system, the laser interferometers probe the2surface of a flat bar mirror that is fixed on top of an X-Y scanning stage,(2)

Y = AX

while the autocollimator simultaneously measures the yaw error of thescanning stage [7]. Unlike the case where the displacement sensors arewhere Y denotes the measured vectors

fixed on a moving scanner, the laser interferometers are mounted on
stationary housings, as shown in Fig. 4. During the measurement, thescanning stage moves in steps. And at each step, a computer

Y = m1 ( 1 ) , m,1 Ns() , m3 ,( 1 )m,3 sN,(m ) ,aT(m1 ) ,sNa(3)

automatically collects the data from the laser interferometers and the
autocollimator. Let the corresponding laser interferometers andautocollimator outputs be  () ,  () …  () and  () .
They can be expressed as follows:

and A is the Jacobian matrix constructed by the differential of Y and
X. The unknown vectors X are the flat bar mirror profile and the motion
errors.

m1 (n)  f ( xn )  e1 (n) m ( n)  f ( x  D )  e ( n )  D  e ( n )  c
 2
n
1
1
1
2
2
, n  1...N sm3 (n)  f ( xn  D2 )  e1 (n)  D2  e2 (n)  c3ma (n)  e2 (n)X = f ( x1 ),, f ( xN 2 )), e1 (1)(1),, e1 ( Ns )), e2 (1)(1),, e2 ( Ns )), c2 , c3

T

(4)

(1)
For a solution of the matrix X to exist according to Eq. (2), twoD2additional conditions must be satisfied. One is N s, whichs

Here f ( xn ) denotes the flat bar mirror profile; e1 (n) and e2 (n)denote horizontal straightness motion and yaw errors with systematicand random errors at the first step, respectively at each step n; D1 is themeans that the number of sampling points of the motion errors is greaterthan or equal to the normalized distance of the 3rd laser interferometer.
The least-squares solution exists when this condition is met. The other isthe greatest common divisor (GCD) of d1 and d2 should be 1. (GCD (d1,installation distance between the 1st and 2nd laser interferometers; D2 isd2) = 1) The parameters D1, D2, and s are chosen to satisfy these

the installation distance between the 1st and 3rd laser interferometers;

conditions.
From Eq. (2), the objective model Y comprises a linear combinationand c2 and c3 are supplementary parameters that are defined bycj  uju1

D j 1  ua , j  2,3 . The parameters u1 , u2 , u3 and uaof the parameters X. In the measurement system, Y is assumed to haverandom errors. The sources of error for each measured vector, which

are the offsets of each probe. The number of sampling points of the flat

follow the normal distribution, are dependent. Utilizing the linear

bar mirror profile is N and the number of sampling points of the motion

least-squares method to calculate the unknown matrix X, we obtainerrors is given by Ns  N d2 . The parameter d2 is defined as

d2  D2 sΧ   Α t S 1 A  A t S 1Y
1

that is the normalized distance of the 3rd laser(5)interferometer. The measuring step distance of the scanning stage is swhere S denotes the error variance between each measurement point.

and L  n s specifies the moving scale of the stage, as shown in Fig.
4. For the analysis here, the step distance s is determined by the numberof the laser interferometers and their relative distances.

From the reconstruction procedure, the measurement uncertainty
associated with the reconstructed vectors X can be derived. The
reliability of the reconstructed data and its associated measurement
uncertainty can be assessed via a criterion. The associated uncertainty of

Motion locus of X stage

the measurement process is calculated via the error propagation matrix
SX that is deformed in the least-squares method, as shown in Eq. (6).

Step nReference mirror
Top of X-Y linear stage

Starting step

The vectors on the diagonal of SX are the square values of themeasurement uncertainties of the flat bar mirror profile from 1 to N-2.

es (n)

ma
Autocollimator

 r11

S X  (A S A)  r
 Q1e y ( n)

T

Ln s

D1

Flat bar mirror Profile

D2

m1 m2 m3

Laser interferometers(Stationary)

11

r11QQ 
,
rQQQ (6)f ( xn )5.

Uncertainty Simulation

Fig. 4 Principle of multi-probe scanning method
In the simulation, we begin by setting up simultaneous linear
4.2 Data processing based on simultaneous linear equations andleast-squares method

equations that express the linear relationship between the measured
parameters and the unknowns. We then analyze the measurement
uncertainty for the flat bar mirror profile and calculate the unknown
parameters by applying the least-squares method [15]. We set up three

From Eq. (1), the whole measurement process is expressed simply asa set of simultaneous linear equations,laser interferometers and one autocollimator in the simulation model.
For the simulation,3f ( xn ) is predefined; e1 (n) and e2 (n) arerandom numbers from the initialization; L is set to 100 mm; and the

XY stage consisted of a reference bar mirror, a housing for the reference

standard deviation of the laser interferometers and autocollimator were

bar mirror, and a reference mirror for the autocollimator.

set to 1   2   3  3.5 nm and  a  0.23 μrad . These values were

The moving direction of the X-axis was from left to right. In thetaken from the model specifications (laser interferometers: 10705A,

equipment setup, the sampling length of the flat bar mirror is 100 mm

Agilent; autocollimator: Elcomat 3000, Moller-Wedel Optical). Eachbecause the valid size of the flat bar mirror is approximately 100 mm ×

sampling point of

30 mm with an accuracy of λ = 632.8 nm. Setting D1 = 10 mm, D2 = 21mm, and s = 1 mm, and for N = 100, gives Ns  N d2 as 79.

f ( xn ) is picked by the predefined function.

According to the GCD condition, we choose different sets of D1 and D2
values, with s set to 1 mm, as shown as Table 2, in order to analyze thefactors influencing the measurement uncertainty of the flat bar mirror
profile.

Flat bar mirror

Receivers

Beam splitters

Table 2 Different installation distances of laser interferometersGroup

D1 (mm)

D2 (mm)

No.1

2

5

No.2

4

9

No.3

5

11

No.4

10

21

No.5

15

31

Autocollimator

Laser head

X-Y stage

Mirrors
Laser interferometers

Fig. 6 Actual setup of pre-experiment
The simulation results are shown in Fig. 5.
6.2 Stability of each sensor
20

No.1

Uncertainty (2σ)
(nm)

No.2
No.3

The accuracies of the laser interferometers are sensitive to theNo.4

measurement environment and other external factors. Therefore, we

No.5

10

measured the stability of each sensor at the first sampling point before
performing the experiment. The standard deviations of the laserinterferometers and the autocollimator were calculated to be1   2   3  3.5 nm, and  a  0.23 μrad00

10 20 30 40 50 60 70 80 90 100
No. of sampling points of flat bar mirror (N)Fig. 5 Relationship between different installation distances of laserin 5 minutes that is

approximately the same duration as a whole measurement process.
6.3 Pre-experiment resultsinterferometers and measurement uncertainty of flat bar mirror profile
The yaw errors along the X-axis e2 (n) from ten repetitions of thefor 1   2   3  3.5 nm and  a  0.23 μrad .

experiments are presented in Fig. 7. The range ofe2 (n)

isFig. 5 shows the relationship between different sets of installation

approximately 40 µrad. The horizontal straightness motion errors along

distances of laser interferometers. From Fig. 5, the uncertainty curves

the X-axis e1 (n) , obtained by applying the simultaneous equation andincrease almost continuously with increasing installation distance (D1,least-squares methods, are shown in Fig. 8. The range of e1 (n) is

D2). Thus, the smaller the installation distances, the higher the accuracy.

approximately 2 µm. The reconstructed profiles of the flat bar mirror

However, it is difficult to arrange the installation distances to be lessf ( xn ) are shown in Fig. 9, and the profiles are reproduced well forthan 6 mm because the diameter of the laser beam is 6 mm. As a result,each of the ten measurements.

we chose group No. 4 (D1 = 10 mm, D2 = 21 mm) for our experiment.

The two standard deviation (95%) values of the flat bar mirror profile

The average measurement uncertainty is approximately 10 nm, which is

calculated over the ten experiments are shown in Fig. 10 indicated byacceptable for nanometer scale measurements.

the red curve; the average is approximately 10 nm. According to the Fig.
5, the simulated measurement uncertainty (2) is shown in Fig. 10 and

6.

Pre-experimentindicated by the black curve with parameters set as per group No. 4.
Comparing these two curves, we conclude that the two standard

6.1 Pre-experiment Setup

deviation of the flat bar mirror profile is mainly fitting the range of 2σ.
The multi-probe scanning method performs well with a small deviation

The pre-experiment of the multi-probe scanning method has beenof 10 nm in the measurement of the flat bar mirror profile, and measuresdesigned to measure the motion errors of an XY stage based on a

the horizontal straightness motion and yaw errors successfully with a

stepper motor system. Fig. 6 shows the actual setup of thehigh horizontal resolution. Moreover, in the comparison between the

pre-experiment that consisted of an XY stepper motor stage, laser

simulated measurement uncertainty (±2σ) and the difference curvesinterferometers, receivers, beam splitters, optical reflection mirrors, andbetween the average value of ten profiles and each profile (Fig.11), we

an autocollimator. The optical reflection parts that are fixed on top of thenote that the difference curves mainly lie with ±10 nm (±2σ) verifying

4

Fig. 11 Comparison of simulated measurement uncertainty (s2) of

nanometer accuracy.

flat bar mirror profile with difference between average value of ten
20

No.2

10
Yaw (μrad)profiles and each profile

No.1

No.3

0

No.4

-10

No.5

-20

No.6

-30

No.7

6.4 Comparison with ZYGO's interferometer system

To confirm the measurement accuracy of the flat bar mirror profile,

No.8

-40

No.9

-50

we compared our profile data with results measured by ZYGO's

No.10

0

10

20
30
40
50
60
No. of sampling points (Ns)70

80interferometer system. The average flat bar mirror profile for ten
experiments is indicated in Fig. 12 by the red curve. The profile

Fig. 7 Yaw error of X stage

measured by the ZYGO's interferometer system is indicated by the black
curve. The comparison shows that the two profiles are approximately
No.1
No.2

1000

the same within the deviation range of 10 nm, excluding some points at

No.3

500

No.4

the edge of the mirror. This is a limitation of the multi-probe scanning

No.5

0

No.6

method because these points can only be measured by one displacement

No.7

-500

No.8

-1000

sensor.

No.9

No.10

-1500
0

10

20
30
40
50
60
No. of sampling points (Ns)70

80

Flat bar mirror profiles (nm)Straightness motion error
(nm)

1500

Fig. 8 Straightness motion error of X stage
No.1

Flat bar mirror profiles
(nm)

400

No.2

300

No.3

200

No.4

100

No.5

100
0
-100

-100

No.8

-200

No.9

20
40
60
80
No. of sampling points of flat bar mirror (N)100

Fig. 12 Comparison of average flat bar mirror profile with profile

No.10

0

ZYGO
Measured result

-200

No.7

-300

measured by ZYGO's interferometer system

10 20 30 40 50 60 70 80 90 100
No. of sampling points of flat bar mirror (N)7.

Fig. 9 Profile of flat bar mirror

Conclusions and future works
We have devised a multi-probe scanning method to calibrate the40
Uncertainty (nm)200-300 䢢
0

No.6

0

䢢

300

2σ

multiple motion errors of X-Y table on M-CMM and measure a flat bar

2*Std.

30

mirror profile with nanometer accuracy simultaneously. From thesimulation and pre-experiment results, we can make the following

20

conclusions.
10(1)

The simulation results indicate that the average measurement
uncertainty (2σ) almost continuously with increasing installation

0
0

10

20

30

40

50

60

70

80

90

100

distances between the laser interferometers (D1, D2). When we set
D1 = 10 mm, D2 = 21 mm, s = 1 mm, 1   2   3  3.5 nm, andNo. of sampling points of flat bar mirror (N)Fig. 10 Comparison of two standard deviations of ten times flat bar a  0.23 μrad , the average measurement uncertainty is

mirror profiles (black line) with simulated measurement uncertainty (redline) (2)

approximately 10 nm.
(2)From the pre-experiment, the two standard deviation of the flat bar
mirror profile is mainly fitting the simulated measurement

No.1

Uncertainty (nm)40

uncertainty of 10 nm (2σ). Moreover, the difference curves

No.2
No.3

20

between the average value of ten profiles and each profile mainly

No.4
No.5

lie within the simulated measurement uncertainty of ±10 nm (±2σ).

No.6

0

In addition, the multi-probe scanning system can also measure the

No.7

No.8

-20

yaw and horizontal straightness motion errors successfully with a

No.9

No.10

high horizontal resolution.

±2σ

-40
0

10 20 30 40 50 60 70 80 90
No. of sampling points of flat bar mirror (N)

100

(3)

Comparing

with

the

results

measuredbythe

ZYGO'sinterferometer system, there are some variations in the flat bar

5http://www.zeiss.de/4125682000247242/Contents-Frame/5B1A8

mirror profile of our measurement results. They are due to some

EBC2A5E14B886257154006AE1A3

systematic error sources in the measurement process, such as theaccuracy of moving the scanning stage and the misalignment of[4]

Claverley, J. D, and Leach, R. K. A vibrating microscale CMM
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detail in M-CMM experiment.
The results from the pre-experiment verify the performance of[5]

M.Neugebauer, High-precision Calibration of Ball Plates for
Micro-Coordinate Measuring Machines,

multi-probe scanning method. Therefore, we have designed and built the

http://www.ptb.de/cms/en/fachabteilungen/abt5/nachrichten5/scie

optical parts on the M-CMM. Fig. 13 shows the actual optical setup on

ntific-news.html?tx_ttnews%5Btt_news%5D=86&tx_ttnews%5B
M-CMM. The laser head is fixed on top of the frame table because the

backPid%5D=752

space on granite base plate is limited. Laser light is devided into three[6]

Mitutoyo, UMAP Vision System, Bulletin No. 1631.

pathes though the 33% and 50% beam splitters. Other optical devices[7]

Yang P, Shibata S, Takahashi S, Takamasu K, Sato O, Osawa S,
and Takatsuji T. A Multi-Probe Measurement Method to Evaluateconsisted of several reflection mirrors, laser interferometers, receivers,

the Yaw and Straightness Errors of XY Stage on High Precisionreference bar mirror, and an autocollimator. Finally, we are going to

CMM. Key Engineering Materials 2010; 447–448; 590–594
perform experiments on the M-CMM to calibrate the multiple motion[8]

errors of X-Y table stage in near future.

Yang P, Shibata S, Takahashi S, Takamasu K, Sato O, Osawa S,
and Takatsuji T. Development of high precision Coordinate
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XY stage). 2009; JSPE autumn conference; 725–26[9]Yang P, Shibata S, Takahashi S, Takamasu K, Sato O, Osawa S,
and Takatsuji T. Development of high precision Coordinate
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Fig. 13 Actual setup on M-CMM

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Government (Monbukagakusho: MEXT) Scholarship and Japanese[15] Liu SJ, Watanabe K, Chen X, Takahashi S, and Takamasu K.

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6