形状誤差における測定点の直接測定結果の分布

(1)

愛知工業大学研究報告第40号 B平成 17年 111

D

i

s

i

r

i

b

u

t

i

o

n

o

f

D

i

r

e

c

t

Measurement R

e

s

u

l

t

o

f

Measurement P

o

i

n

t

o

f

Form E

r

o

r

形状誤差における測定点の直接測定結果の分布

Xiaohua NI

，

t

Yoshihisa UCHIDAtt

侃騒騨

T

内田敬久↑十

Absiract The uncertainty of measurement result of form 回 oris influenced by the uncertainties of

measurement data that is used during the processing of the calculation. According to th巴

prope此iesof the measぽementof form error， by means of Maximum en仕opymethod and the

theory of probability and statistics， the probability di蹴ibutionsof reading valu巴anddirect m巴asurementresult of measurement points are deduced. It is concluded that it is reasonable to regard the uncertainty distributions of reading value as uniform di柑ibution，and the uncertainty distributions of direct measurement result of measurement points as normal dis仕ibution. 1.

I

ntroduct i on The uncertainザ ofmeasurement result of form error is influenced by the data at measurement point that is used during the processing of the calculation of form error. During the measurement of form error

，

the data is called reading value that is read by operator or collected by computer at the measurement point.Generally it could not be used to estimate the form error， especially when the measurement is un-dir巴ction. Usually the reading value should b巴dealwith to get rid of those e町orsthat are caused by the movement of instrument， the estimation reading of operator

，

m巴asurement circumstance， the impaction and destination of parts and ↑ Department of M巴chanical Engineering， Yancheng Insti印te of Technology (Yancheng， China) and Visiting Scholar of Aichi Institute of Technology (Toyota， Japan) 什 Department of Mechanical Engineering， Aichi Institute of T巴chnology (Toyota

，

Japan) so on. The data is called direct measurement result that could be directly used to estimate the form e町orand any transform is not n巴cessary. Sometimes the direct m巴asurement result of each point is calculated by cumulation or coordinates位ansformwith reading values. Certainly the direct measurement result is equal to the reading value for some measurement method. In the simplified calculation， sometimes the reading value of instrument is regarded as direct measurement result， and do not elirr由mteanye立ors Since the direct measurement result is directly used to estimate the forrn e汀orto calculate the final measurement result， so the uncertainty of it has great con仕ibutionto the uncertainty of final measurement result.During the measurement of form error， the total uncertainty of reading value of measurement point is also influenc巴dby many factors. Itis gained by the combination with the uncertainties of these factors. The decision of the coverage factor k of B type uncert出nty evaluation is influenced by the probability distribution of these factors1l

_，

_a_l_s_o_t_h_e_{combined u}_n_c_e_r_t_a_i_n_t_y_o_f reading value must be decided

，

because it will influence the decision of direct measurement result of point and

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112 _{愛知工業大学研究報告，第 40号 B，平成 1}_7年_{， Vo}_l_.₄₀ 開B，Mar.， 2005 the estimation of the uncertainty of final measurement result. Although the probability distribution of measurement result may be gained by statistic of rep巴at experiments and assumption test， such as Histogram method and Probability paper method， it is use白lonly to the limited dis仕ibutionsand do not have explicit

judgment boundary.

During the measurement of form e立or，at the

same measurement point， measurement times is usually no more than one time， th巳otherinfluence factors is only known as a value between an interval. So the probability distribution of direct measurement result of form e汀or could not be estimated with popular probability estimation method. Inthis paper， Maximum 即位opymethod is used to estimate the probability distr・ibutionof direct measurement result of form eηor.

2.Probab i I i ty d i str i but i on est i mat i on by means of Maximum entropy method

A discrete information source may be expresses as follows:

x

:

[

;

:

)

噌 ] E i r -- L That's to say， if discrete variable x is given the valueXi， the probability p円 i= 1，2，・..，n.Here，

p(x=xiI x=x)=O

，

i

*J

_[₂_]

LPi=l

[3] Then the en仕opyis defined

H(x)

=

H(PPP2

，

A

，

Pn)

=

-kLPi

l

o

g

p

i

_[₄_] The constant

k

is decided by the unit， usually， it is taken ask= 1. The different base of logarithm is given， the entropy will have di町erentunit. From the convenience of calculation， it is given the naturallogarithm. Thus the unit ofH(x) is“N at" and the quantum H is called information en仕opy. It is used to describe the uncertainty of information source. To continuous information source， the distribution of x is d巳scribed with probability densityp(x). So in the case of continuous distribution， the entropy is expressed as 、、 ‘ ，，ノ χ r J a a ‘ 、、 ρ A

n

耳

一一

X 7 0

ω w

p

n

助

川

W

? 十

一

、、 . 附，ノ χ 〆 ' ' a_・ ‘ 、

H

[5] ーベヌD In another words， the means of the logarithm of dis仕ibutiondensityp(x)is entropy.

The maximum of en仕opymay be used to estimate

the probability distribution， the actual calculation method is introduced in references (2)開 (4).With this method， no other subjective assumption is needed but measurement data e汀orthat contains the all information and constrained condition of samples. That is to say， in the most of uncertainty， the actual probability dis仕ibution and its paramet巴rs are estimated with maximum entropy rule4_l.

3. The distribution of reading value of measurement point

During the m巴asurementラtheuncertainザinterval

of reading value may be talcen from the handbook of instrument， suppose the estimation of reading valu巴xis

μラsothe true value of it is among an interval， suppos巳

the interval is [α， b]. Generally the interval is symm巴trical where μis center point. Now let the

interval to be changed to make the center point value to zero， thus the shape of the dis仕ibution do not be influenced. Also letL = (b-a) / 2， so the interval becomes [-L， L]. Approximately the probability densityp(x)和国1s

レ

ω

=

1

_[₆_]

From formu1a [5]， there wiU be the en仕opyfunction

H(

功

= H

(

P

(

x

)

二一

j

内

)

l

n

p

(

χ

)

d

x

(3)

Distribution ofDirect Measurement Result ofMeasurement Point ofForm Error 113 To get extremum， Lagrang巴methodof multipliers is used， suppose

D=-f

p

(

x

)

l

n

p

(

吟

-L

+(ゐ+

1 )

[

レ付命

-

1 ]

[8] Let，

8D

-

=

-

=

-

=

0

8

J7 [9] Then ハ U

一一

' χ L F l r J 、、 . . _{，，} 2E J 4 E E A

+

A

，， ZE 官、、

x

，d -E -E ﹂噌 E E A

+

、、 . ，ノ χ / ， E_‘ _、、

p

n

L P H_ド u _[₁₀_] 一L -L So

-

l

n

p

(

x

)

+

λ

。

+1

ニ

O

1

1 1 1 i [ Thus

p

(

x

)

=

e

x

p

(

ー

1 -

λ

。

)

[12] From equation [6]，

e

x

p

(

-1λ

。)=土

V /

2L

[l3] So we get ー一江

一一

1 1 '

x

/ ， a_圃 _置 _、 _、

p

ヲxE[-L，L] [14] This is uniform distribution. That is to say， in the interva1 [-L， L]， it is uniform distribution that have the maximum entropy， so the distribution白nction of reading va1ue may be deduced that

f(x)

=

了

1 。

-a

，x

ε

[a， b] [15]

4. The d i str i but i on of d i rect measurement resu I t of measurement point During the calcu1ation of uncertainty of direct measurement result of measurement point， the uncertainty of reading va1ue is on1y a factor of白e uncertainty of measurement point data.Itis a1so influenced by movement of instrument， measurement circumstance and the impaction and destination of parts and so on. Although the numerica1 va1ue of measurement point data is equa1 to reading value， the uncertainty is 1arger. Itis known that the uncertainty of measurement point data is combined with the uncertainties of many factors， the distribution of these factors may obeys uniform dis仕ibution，norma1 distribution or other non-norma1 dis仕ibution，since the m巴asurementtimes is few. So the distribution of combined resu1t cou1d not b巴 estimated with popular statistic m巴thod.Therefore， the Maximumen仕opymethod is a1so used. Suppose the uncertainty of measurement point data is knownラsoit is considered that the variance is

known， 1et it be

c

i

， the estimation ofmeasur巴mentpoint

data is仏 not1et it be zero，μbecome variab1e x， then the variance c/ is second origin moment， suppose the distributing bound of x is xε[ー∞，+∞J.Then

f

p

(

榊

=1

[16] ←-00

f

x

p

(

x

)

d

x

=

0

[17] -00

f

x

2

p

(

x

)

d

x

=

_σ2 [18] -00 From reference (4)ラ it is known that normal

distribution make the en仕opyto maximum. So

内 ) ニで」叫

(-4)

σ

、

ん

2

Jr ~'2σ“

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114 愛知工業大学研究報告，第 40号 B，平成 17年， Vol.40-B， Mar.， 2005 Then the dis仕ibution伽lctionof total uncertainty of measurement point data that makes the en仕opyto get a maximum shou1d be

(x-

μ)

f(x)=~芹=expト ---τ一]

σ

、

白

JL

π

ム

σ

， xE [・∞，+∞] Itis a1so norma1 distribution. [20] During the measurement of form error， whether the measurement is direct or not， the direct measurement result at each measurem巴ntpoints may be expressed as 1inear function of meas町ementpoint data， because the measurement point data is normal probability variab1e and independent企omeach other; from the theory of probabi1ity， the 1inear function of norma1 random variab1e is also normal random variable. Therefore， it is deduced that the direct measurement result of measurement point obeys to normal dis仕ibution 5. Summary The uncertainty of measurement result of form error is influenced by the data at measurement point that is us巴dduring the processing of the calculation of form e汀or.According to the properties of the measurement of form error， the modem probability dis仕ibution

estimation method ----Maximum entropy method is used to deduce the probability distribution of reading value and direct measurement result.Itis deduced that the distribution of reading value obeys uniform distribution and the distribution of direct measurement result obeys normal dis出bution.The conclusion wi1l give convenience for estimation of total uncertainty of the measurement result of form error. References 1) BIPM-IEC-IFCC-ISO-IUP AP欄OIML.Guide to也e Expression of Uncertainty in Measurement.ISO， pp.ト6，1995 2) Ling Honghua，“Instruction of the methods of non-normal error statistics processing and estimation" Special Papers for Measurement Uncertainty， Statute Depa抗ment of National Metrology Bureaus， pp.67田74，1985

3) Wang Zhong戸1，Xia Xintao， Zhu Jianmin，“The

Un嗣StatisticTheory of Measurement Uncertainty

ヘ

National Defense Publishing Company， pp.100-119， 2000

4) Zhou Zhaojin，“A princip1巴ofMEM to estimate the

measurement uncertainty"， Measurement Technique， No. p，4 p.1-3， 1989