On-machine roundness measurement of cylindrical workpieces by the combined three-point method

ELSEVIER

PII: S0263-2241 (97)00060-2

MeasurementVol. 21, No. 4, pp. 147-156, 1997 © 1998 Elsevier Science Limited. All rights reserved Printed in The Netherlands 0263-2241/97 $17.00 +0.00

On-machine roundness measurement of cylindrical workpieces by the combined three-point method Wei Gao *, Satoshi Kiyono Department of Mechatronics, Faculty of Engineering, Tohoku University, Aramaki Aza Aoba. Aoba-ku, Sendai, 980. Japan

Abstract

In this paper, we describe a new differential method for on-machine roundness measurement of cylindrical workpieces, which is called the combined three-point method. This method combines the advantages of the generalized three-point method and the sequential three-point method and can accurately measure roundness profiles including stepwise variations. In the combined threepoint method, some data points in the roundness profile evaluated by the generalized three-point method are chosen as the reference points of the standard area and used to determine the relationships among the data groups of the sequential three-point method. An interpolation technique is employed in the data processing of the generalized three-point method to improve the accuracy of the standard area. Theoretical analyses and computer simulations confirming the feasibility of the combined three-point method are shown in this paper. A roundness measurement system using three capacitance-type displacement probes is constructed. The measurement system and the experimental results are also presented. © 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Combined method; Error separation; Fourier analysis; Measurement; Roundness; Spindle error; Stepwise profile; Threepoint method

1. Introduction

errors. However, it cannot express stepwise variations of the profile accurately because the number of data points in this method is too few. On the other hand, the generalized three-point method yields high accuracy in the long wavelength range and provides more information of the profile by sampling the surface with a sampling period far shorter than the probe interval. Even in this method, however, profiles including stepwise variations cannot be expressed correctly. In the G3P method, the roundness profile is evaluated from the differential output of the probes through integration calculation or discrete Fourier analysis. In the integration-type G3P method, stepwise variations are smoothed by the integration. In the DFTtype G3P method, stepwise variations cannot be treated correctly by the Fourier analysis. To measure profiles of straightness surfaces including stepwise variations, we have proposed a method named as the combined two-point method [8 10], which combines the advantages of the

The three-point method has been proposed to measure roundness of machined workpieces accurately under on-machine conditions [1-7]. In this method, three displacement probes are fixed around the workpiece to detect roundness profile and two-dimensional spindle error components simultaneously. The effect of the spindle error is canceled in the differential output of the probes and the correct roundness profile can be evaluated from the differential data. We refer to the threepoint method as the sequential three-point (S3P) method [6,7] when the sampling period is equal to the probe interval and as the generalized threepoint (G3P) method [1-5] when they are not equal. The S3P method is capable of measuring a discrete roundness profile without data processing * Corresponding author. Tel: +81 22 217 6953; Fax: +81 22 217 7027; e-mail: [email protected] 147

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W. Gao, S. Kiyono

sequential two-point method and the generalized two-point method of integration-type. This combined two-point method can express stepwise variations accurately. However, it cannot be applied to the roundness measurement directly because this method is based on the two-point method and the two-dimensional spindle error components cannot be canceled completely. To solve this problem, in this paper we propose a new method named the combined three-point (C3P) method by combining the S3P method and the G3P method. The proposed C3P method cannot only express accurately roundness profiles including stepwise variations, but can also completely cancel the two-dimensional spindle error components. Considering that G3P method generally employs discrete Fourier analysis in its data processing, we choose the DFT-type G3P method in the G3P method. An interpolation technique is used in the DFT-type G3P method to avoid the influence of stepwise variations. Computer simulations and experiments of roundness measurement by using the proposed C3P method have been performed.

Y e~[(0n)

~

~ C

Probe C pProbe

\" o I "~.]._.~/Workpiece /t~f[-

r(0n)

Step-wise variation Fig. 1. Principle o f the three-point m e t h o d measurement.

for r o u n d n e s s

the probe outputs can be expressed as:

mA(O,)=r(O,)+ex(O,),

2.1. The sequential three-point (S3P) method Figure 1 gives the principle of the three-point method schematically. Three probes are fixed around a cylindrical workpiece with an equal angular probe interval 4) and scan the workpiece while it is rotating. Assume that point O is the intersection of the three probes and is near the rotational center of the workpiece. The fixed coordinate axes X Y are also shown in the same figure. Let P be a representative point of the workpiece and the roundness profile be described by the function r(O,), where 0, is the angle between point P and the Y-axis. Let 0s be the sampling period and N be the number of sampled data. If the displacement outputs of the probes are denoted by mA(O,), ms(O,) and mc(O,), respectively,

(1)

mB( O,) = r( O, -- 4)) + er( O,) sin 4)+ ex( O,) cos 0,

(2) mc(O,) = r(O, - 24))+ er(0,) sin 24) +ex(O,)cos2(~, n=0, 1..... N - l ,

2. The three-point method for roundness measurement

B

(3)

where ex(O,) and ey(0,) are the X-directional component and Y-directional component of the spindle error, respectively. The differential output m3(O,) of the three-point method can be denoted as: m 3 (On) = m a ( O n ) - -

2 cos 4)mB(O.) + mc(O.)

= r(0,)--2 cos 4)r(O, -4))+r(O, -24)).

(4)

Consequently, the spindle error is canceled. We refer to the three-point method as the sequential three-point method (S3P method) when 0s =4). In this case, the roundness profile can be evaluated as:

P(O,) = -p(O,

_ 2 ) + 2 c o s ~ P ( O n _ 1) q- m 3 (On)

= - P ( 0 , _ 2 ) + 2 cos 4)p(O,-1)+mA(O,) --2 COS4)mB(O,) + mc(O,) (n=2, 3..... N - - l ) .

(5)

It can be seen that there is no errors in the data processing with Eq. (5) in the S3P method. This

149

w. Gao, S. Kiyono

indicates that the S3P method is capable of expressing a discrete roundness profile correctly.

2.2. The generalized three-point (G3P) m e t h o d

The three-point method (S3P method) is defined as the generalized three-point method (G3P method) when 0~¢¢. In the G3P method, the roundness profile is evaluated from the differential output of the probes through integration calculation or discrete Fourier analysis. We refer to the former as the I-G3P method and the latter as the D F T - G 3 P method.

2.2.1. The I - G 3 P m e t h o d In the I-G3P method, an approximate double derivative M~'(O,) of r(O,) is defined by m3(O,)/(Rr¢) 2 as follows: m;(O,) = m3(O,)/(Rrq}) 2 = {[r(0,) - cos c~r(O, - ¢)1

- [cos Cr(O, - ¢) - r(O, - 2~b)]I/(R~b)2 {Jr(0,) - r(O, - ¢)] - [r(O, - O) -

r(O,

-

2¢)]}/(R~¢) 2 = {Jr(0,)

--

r(O,

--

¢)]/(Rr ¢ ) -

[r(O

n -

(6)

2.2.2. The D F T - G 3 P m e t h o d In the D F T - G 3 P method, discrete Fourier analysis is employed. Consider Eq. (4) as a system. In this system, r(O,) is treated as the input and m~(O,) as the output. The relationship between the input r(O,) and the output M3(o),) can be defined by the following transfer function [11,12]:

H3(co,) = m 3 ( ~ o , ) / R ( e ) , ) = 1 - 2 cos q~-e -j~'-~ + d 2'°-~ ( n = 2 ..... N - 1),

(8)

where co, is the spatial frequency and M3(%) and R ( % ) are the Fourier transforms of m3(0,) and r(O,), respectively. R(~o,) can be obtained from M3(%) and H3(~o,) and r(O,) can be evaluated by reverse D F T of R(%). Figure 2 shows the amplitude of H3(~o,) when ¢ = 18 °. As can be seen in this figure, the amplitude at some frequencies in the transfer function approaches zero. This prevents the D F T - G 3 P method from measuring the corresponding frequency components correctly. However, most of the frequency components can be obtained accurately. Considering that the G3P method generally employs discrete Fourier analysis in its data processing, we chose the D F T - G 3 P method in the combined three-point method as shown in the next section.

O)

-- r( O. - 2¢ )]/(R~¢ ) } /(R~¢ ) 3. The combined three-point method for roundness measurement

[r'( On) -- r'( On -- ~b)]/(Rr¢ ) ~ r"( On) ,

where R r is the radius of the workpiece. The approximated roundness profile z(O,) can be described as follows:

Although profile heights at discrete points can be measured without the errors inherent in data i

Z(0,) = k ~

~

= [

j = l

m~(Oj)RrOs

x R r O s ( n = l , 2 ..... N - l ) .

J

J

i

i

i

i

~

i

~

r

i

i

i

i

J

i

i

i

~

i

L

i

i

t

r

t

i

i

i

i

k

i

i

i

i

i

g (7)

In the G3P method, the approximation error of z(O,) to r(O,) is introduced from two kinds of operations. One is shown in Eq. (6), where r"(O,) is approximated by m,,r• 3t~,). The other is shown in Eq. (7), where numerical integration is performed. Just as the generalized two-point method of integration type, the I-G3P method yields high accuracy in the long wavelength range.

=

3

O

~

2

~. E <

1

o -1

10

i

i

i

i

i

2o 30 Frequency m/2~

i

4o

h

i

,

i

5o

tl

Fig. 2. Transfer function of the generalizedthree-pointmethod.

150

W. Gao, S. K i y o n o

processing in the S3P method, the number of data points is too few to express stepwise variations accurately. In order to increase the number of points in the S3P method, many sets of the discrete curves are obtained by shifting the starting points as shown in Fig. 3b. If the amount of the shift is equal to the sampling period s, M sets of data groups can be obtained. The position of data points along the circumference can be expressed as:

Oij = i s + j M (i=0, 1..... M - 1,

(9)

j = 0 , 1..... N / M - l ) , where i is the group number and j is the data number in each group. For a fixed i, a discrete roundness profile Pij can be expressed by using one data group of the S3P method as follows: Pij = --

eli- 2 "Ol-2p U- 1 + [mc( Oij) -- 2mB( Oij)

+2ma(Oij)] ( j = 2 , 3 ..... N / M - - 1 ) ,

(10)

where pio=0, p a = 0 and i=0, 1..... M - 1 . The relative heights of the data points in each data group of the S3P method can be determined exactly. However, since the relative heights among different groups and the inclination in each group is indeterminate, we set the heights of starting data points and second data points to zero (Fig. 3c) as

in the usual S3P method. The resulting curve can be seen to have an error in Fig. 3c. Therefore, the profile still cannot be calculated properly, even with an abundant amount of data, without first determining the relationships among the data groups. Here, we choose some data points in the roundness profile evaluated by the DFT-G3P method as the reference points of the standard are to determine the relationships among the data groups of the sequential three-point method. However, because stepwise variations cannot be treated properly by the Fourier analysis, errors will be caused over the entire circumference if the data of the stepwise part are involved in the data processing of the DTF-G3P method. To solve this problem, we employ an interpolation technique as shown in Fig. 4. The stepwise part in the differential output is interpolated by using the least-squares approximation method to get an improved differential output without including stepwise variations (Fig. 4b and c). As can be seen in Fig. 4d, the profile z(O,) of the DFT-G3P method can be evaluated accurately in parts except that of the stepwise variation by using the improved differential output. Here, we call such a DFT-G3P method as the improved DFT-G3P method. Those

(a) Real roundness profile (a) Real roundness profile 0 0

2n

(b) Sampled data

..___...,_

2~

(b) Differential?utput ~

..... 1 - ~

Data Group of S3P

°

1 0

2n

I

II

i

I Interpolation I

(c) Improved differential output

i o

(c) Evaluated results by the S3P method ~,--I~---i

•

•

2~

(d) Profile Z(On) evaluated by G3P method , L

~

I

I

~,__.~-_~_ ~ -"J 0~ Starting points Secondpoints -" 0

*--o---~ . . . .

Fig. 3. Data groups of the S3P method.

0 2=

0

. . . . .

, I I

,-, ,~' Part of interpolation i

I

Fig. 4. The improved DFT-G3P method.

2ax

W. Gao, S. Kiyono

parts that are determined to have high accuracy by the improved DFT-G3P method are used as a standard area to determine the relationships among the data groups obtained by the S3P method. Relating all data groups with the standard area, the entire roundness profile r~j can be expressed precisely (Fig. 5). The roundness profile r~j has three degrees of freedom: C~o, c, and c~2 and can be expressed as:

151

I

E t~

2

e-

'LJ i

( 11 )

rij = Cio + Cil Oij + ci202j + Pij.

60

120

i

240

300

360

Rotating angle deg.

Using the results of the improved DFT-G3P method and the S3P method, C~o, cia and c~2 can be obtained from the following equations: =

180

6(a) Rectilinearplot

(12)

(z j - rij) J

8Ei/~Cio = O,

(13)

63Ei/Ocil = O,

(14)

6~Ei/8ci2

(15)

= O.

E.

4. Computer simulation

The feasibility of the proposed C3P method for roundness measurements was confirmed by computer simulations. The parameters of the simulation were as follows: probe interval ~b sampling period 0s number of sampled data

18 ° 1.8 °

200

Figure 6 shows the input roundness profile. Figure 6a shows the rectilinear plot and Fig. 6b

6(b) Polarph)t Fig. 6. Input profile: (a) rectilinear plot; (b) polar plot.

shows the corresponding polar plot. A stepwise variation was included in the profile. The differential output m3(0,) of the three-point method defined in Eq. (4) is shown in Fig. 7. The improved

(a) Real roundness profile :zk

Differential output

o .-~

O

,~ .~ 0 (b) Evaluated roundness profile by the C3P method ~ ]

2~ Improved differential output .

O

2~

Given relations by using result of Fig. 4(d) Fig. 5. Profile evaluated by the C3P method.

~ ~rq

..-~

~

Part of interpolation

0

I

I

I

I

I

60

120

180

240

300

Rotating angle deg Fig. 7. Differential outputs.

k

360

152

W. Gao, S. Kiyono

differential output is also plotted in this figure. The part of stepwise variations in the differential output was interpolated by using the least-squares approximation method. It can be seen that the improved differential output becomes a continuous curve and can be treated accurately by Fourier analysis. Figure 8 shows the errors of the profiles evaluated by the DFT-G3P method from the differential output and the improved differential output, respectively. As can be seen in this figure, large profile evaluation errors were caused in the entire circumference when the data in the part of the stepwise variations were used in the data processing. In contrast, the improved DFT-G3P method provided an accurate profile in the parts except that of the stepwise variation. Figure 9 shows the roundness profiles evaluated by the improved DFT-G3P method, the S3P method and the C3P method, respectively. The

Generalized 3-point~

~

(D

whole profile including the stepwise variation can be seen to be well expressed by the C3P method.

5. M e a s u r e m e n t

system

5. I. Experiment

Figure 10 shows the system for roundness measurement. Three capacitance-type displacement probes (Microsense) were employed. Each probe had a measurement range of 100 gm. The footprint size of the probe was 1.7 mm. The probes were fixed and the cylindrical workpiece was driven by a servomotor with an optical encoder. A ball bearing was used in the spindle. The angular probe interval q~was set to 18°. The output signals of the probes were taken into a personnel computer via a 12-bit A/D converter. The positional signal of the optical encoder of the servomotor was sent to the A/D converter as a trigger signal. The output signals were sampled simultaneously so that the errors attributable the sampling time delay can be avoided. 5.2. Basic performance of the measurement system

Improved generalized 3.point method [ J

i

.o D.,

Partof

interpolation

I

I

I

I

I

60

120

180

240

300

360

Rotating angle deg Fig. 8. Profileevaluation errors of the G3P method. i

'~

i

|

1

ImprOvedgeneralized3-point method : \ i Part of interpolation

~.o

==

i

Probes [ Workpiece x f ~ C I ~ , ~ B i

-

0

YI _ I

Combined3-pointmethod I -

60

-

,

120 180 240 300 Rotating angle deg.

Fig. 9. Evaluated results by the three methods.

---

A/D Converter

motor r

Sequential 3,pointm e~ - -t h o d / ~

O

Long- and short-term stability tests of the differential output of the three-point method were carried out without any temperature control or vibration isolation. In these tests, the cylindrical workpieces were not rotated. Figures 11 and 12 show the results of long- and short-term tests, respectively. The probe outputs and the differential

360

Trigger sign

Positional signal PC-Computer Fig. 10. Measurement system.

153

W. Gao, S. Kiyono

PrOA

E

.

.

.

.

.

Probe B

~5

.= ~

Probe C

,.-

._

-

Different!a! output

I

10

0

,;

2'0

~.

o

•

2'5

I

I

I

30

I

I

I

I

I

I

L

j

L

20

10

I

J

J

I

L

i

I

i

I

L

i

I

30 40 Number

Time sec. Fig. 11. Short-term stability test.

I

I

I

~

I

50

l

60

Fig. 13. Spindle error reduction.

E~

=k

linearity of the probes. Figure 14 shows the result when vibration errors were introduced. It can be seen that the vibration error with an amplitude of 10 ~m was almost reduced to the same level as the probe resolution. These results confirmed that the measurement system has a good performance. 5.3. Roundness measurement

.o 0

l0

15

2(1 25 Time min.

30

35

40

Fig. 12. Long-term stability test. output of the three-point method are plotted in these figures. The test terms were 30 s and 40 min, respectively. In Fig. 11, because the test term was short and the influence of the thermal drift was small, the output of each probe was almost the same level as the probe resolution. The stability of the differential output was also the same level. In the long-term test because of the influence of the thermal drift, the maximum variation of the outputs of individual probes was 0.4 gm. However, the differential output was almost the same level as that in the short-term test. The characteristics of the differential output to cancel spindle errors were investigated. Figure 13 shows the result when ex(O) was introduced by moving the cylindrical workpiece shown in Fig. 10 in the Xdirection. It can be seen that the movement of the workpiece, which was detected by the probes correctly, was removed in the differential output. The residual error was mainly caused by the non-

Two samples were measured. Sample 1 was a cylindrical workpiece with a diameter of 80 mm. Sample 2 was made by sticking one piece of thin aluminum film on sample 1. The parameters of the roundness measurement were the same as those in the computer simulation. The data for sample 1 are shown from Figs. 15-18. Figure 15 shows the probe outputs and the differential output. Figures 16 and 17 show the profiles and the spindle errors evaluated by the D F T G 3 P method and the combined i

i

,

,

i

r

i

L

i

i

,

,

,

,

,

i

,

r

i

,

,

,

Probe A •Probe B "5

•Probe C

e-, ¢D

Diff nt i

i

r

I

0.2

i

i

i

,

i

0 ,

,

• ,

0.4 0.6 Time sec.

,

,

t

i

0.8

Fig. 14. Vibration error reduction•

154

W. Gao, S. Kiyono '

./

'

'

I

!

.....

~I

i

0

I

60

i

i

i

120 180 240 300 Rotating angle deg.

0

360

~ ' I Generalized3-point method

0

6'0

120 180 ' 240 ' 300 Rotating angle deg.

360

Fig. 16. Evaluated roundness profiles of sample 1. /

~. |

'

'

'

i

Generalized3-pointmethod

r~=~~ 0

i

60

120 180 240 300 Rotating angle deg.

120 180 240 300 Rotating angle deg

360

mately 5 gin and the spindle error was approximately 10 gm. Figure 18 shows the repeatability errors of the roundness profile and the spindle error measured by the C3P method, respectively. It can be seen that the repeatability error of the roundness profile is much smaller than that of the spindle error. The spindle error has no repeatability because a ball bearing was used in the spindle. The results shown in this figure confirmed that the roundness profile and the spindle error were separated with each other correctly. The data for sample 2 are shown from Figs. 19-23. The output data of the probes are shown in Fig. 19. One stepwise variation caused by the thin aluminum film can be seen. Figure 20 shows the profile evaluated by the G3P method without the data processing of interpolation. It can be seen that large evaluation errors were caused in the entire circumference and the result cannot be used in the C3P method. Figure 21

Cimbined i-p0int 7eth°d > 60

Sipndlee.~ -'--- ............. .

Fig. 18. Repeatabilityerrors of sample 1.

Fig. 15. Output data of the probes. ..~

a

!

!

!

360 t"q

Fig. 17. Evaluated spindle errors of sample 1. method, respectively. Because there is no stepwise variations in the profile of sample 1, the DFTG3P method can also evaluate the profile correctly. It can be seen that the results of the DFT-G3P method and the C3P method show good agreement with each other. The roundness error was approxi-

2

Differenfialloutput i ~ / ~

,

k_J t_.j I

6

I

I

120 180 240 300 Rotating angle deg.

Fig. 19. Output data of the probes.

360

155

W.. Gao, S. K i y o n o

Generalized3-pointmeth0~

:zL 0

cO

._= 0 I

0

60

I

120

I

I

180

240

I

300

360

I

I

I

I

I

60

120

180

240

300

Rotating angle deg. Fig. 20. Evaluated profile of sample 2 without the data processing of interpolation. i

Fig. 22. Evaluated spindle errors of sample 2.

E

Improvedgeneralized3-pointmethod i

¢..q

360

Rotating angle deg.

\

Roundnessprofile

:

Part of interpolation Sequential3-pointmethod

@

Combined3~pointmethod

© I

0

60

I

I

i

I

120 180 240 300 Rotating angle deg.

360

0

60

120

180

240

300

360

Rotating angle deg. Fig. 23. Repeatability errors of sample 2.

21 (a) Rectilinear plot

©

@ o c~ 21(b) Polar plot

method is plotted in this figure. The C3P method combined the advantages of the improved G3P method and the S3P method and the whole profile can be seen to be expressed most properly by the C3P method. Figure 22 shows the spindle error evaluated by the C3P method. The spindle error was approximately 10 gm. The repeatability errors of the roundness profile and the spindle error evaluated by the C3P method arc shown in Fig. 23. The repeatability error of the roundness profile is slightly larger than that shown in Fig. 18. This is mainly caused by the positioning error of sampling [10].

Fig. 21. Evaluated roundness profiles of sample 2: (a) rectilinear plot; (b) polar plot.

6. Conclusions

shows the results of evaluation by the improved G3P method, the S3P method and the C3P method, respectively. One group of the S3P

The results of this paper are summarized as follows.

156

I41. Gao, S. Kiyono

(1) A new method named the combined threepoint method for on-machine measurement of roundness profiles has been proposed. Some accurate parts of the profile evaluated by the generalized three-point method are used as the standard area to determine the relationships among the data groups of the sequential three-point method. (2) The proposed combined three-point method, which combines the advantages of the generalized three-point method and the sequential three-point method, cannot only express precisely roundness profiles including stepwise variations, but also completely cancel the two-dimensional spindle error components. (3) To assure the accuracy of the combined three-point method, an interpolation technique has been employed in the data processing of the generalized three-point method. The part of stepwise variations in the differential output is interpolated with the least-squares approximation method to let the differential output to be a continuous curve. As a result, the differential output can he treated accurately by the Fourier analysis in the data processing of the generalized three-point method and the profile data in the standard area can be evaluated accurately. (4) The feasibility of the proposed combined three-point method for roundness measurement has been confirmed by computer simulations. (5) An experimental system employing three capacitance-type displacement probes has been constructed. Roundness measurement of cylindrical workpieces including stepwise variations has also been measured by the combined three-point method.

Acknowledgements The authors would like to thank Mr Yoichi Shirahata of Tohoku Richo Co., Ltd and Mr

Taizou Touyama of Toyoda Machine Works, Ltd for providing the cylindrical workpiece and the spindle, respectively. A part of this research was funded by a grant-in-aid from the Osawa Scientific Studies Foundation.

References 1. S. Ozono, On a new method of roundness measurement based on the three points method, in: Proceedings of the International Conference on Production Engineering, Tokyo, 1974, pp. 457-462. 2. Shinno, H., Mitui, K., Tatsue, Y., Tanaka, N. and Tabata, T., A new method for evaluating error motion of ultra precision spindle. Ann. CIRP, 1987, 36(1), 381 384. 3. Moore, D., Design considerations in multiprobe roundness measurement. J. Phys. E. Sci. Instrum., 1989, 9, 38b-384. 4. Kato, H., Song, R. and Nomura, Y., In-situ measuring system of circularity using an industrial robot and a piezoactuator. Int. J. JSPE, 1991, 25(2), 130 135. 5. E. Gleason, H. Schwenke, Spindleless instrument for the roundness measurement of precision balls, in: Proceedings of the ASPE l l t h Annual Meeting, 1996, pp. 167-171. 6. Obi, M., Kobayashi, T. and Furukawa, S., On a new method of roundness measurement based on the sequentialthree-points method. J. JSME, 1988, 54(506), 2475-2480. (in Japanese). 7. Obi, M. and Kobayashi, T., A new method of the displacement along a circle based on the sequential-three-point method. J. JSME, 1992, 58(548), 1278 1283. (in Japanese). 8. Kiyono, S. and Ogaki, H., Measurement of stepwise profile of machined surface with software datum. Precision Engng, 1993, 59(8), 1319--1324. (in Japanese). 9. Kiyono, S. and Gao, W., Profile measurement of machined surface with a new differential method, precision engineering. Precision Engng, 1994, 16(3), 212 218. 10. Gao, W. and Kiyono, S., High accuracy profile measurement of a machined surface by the combined method. Measurement, 1996, 19(1), 55 64. 11. R.W. Hanmming, Digital Filters, Prentice-Hall, New York, 1977, Chapters 2 and 3 (translated into Japanese by H. Miyagawa and H. Jideki, Kagaku-Gijyutu Publishing, 1980). 12. Gao, W., Kiyono, S. and Nomura, T., New multi-probe method of roundness measurements. Precision Engng, 1996, 19(1), 37 45.