Research Journal of Applied Sciences, Engineering and Technology 4(16): 2809-2812, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 26, 2012 Accepted: April 17, 2012 Published: August 15, 2012
1
1, 2
1
2
Abstract: In the next generation of GPS (Geometrical Product Specification and Verification) standard system, measurement uncertainty and specification uncertainty collectively known as conformance uncertainty. A method to estimate the conformance uncertainty according to the next generation of GPS standard system is proposed. A case study of straightness is given. Based on the model given in the next generation of GPS, the calculation equations of the conformance uncertainty for different methods are deduced. The experimental result indicates that the method can not only assure the integrity of the verification result, but also improve the reliability of verification.
Keywords: Conformance uncertainty, Geometrical Product Specification (GPS), measurement
INTRODUCTION
The next generation of Geometrical Product
Specification (GPS) is an internationally accepted and rapidly developing standard system. It includes macroand micro-geometry specifications covering dimensional and geometrical tolerance, surface properties and the related verification principles, measuring equipment and calibration requirements (Dovmark, 2001). It is the foundation of the technology standards and metrology specifications of mechanical and electric products.
ISO/TR 14 638 give the overall structure of the next generation of GPS system, known as the GPS Masterplan, it includes four types of GPS standard (ISO/TR, 1995): global, basic, general and additional GPS standards. The relationship between them is shown in Fig. 1. Where, general GPS standards are the main body of the standard system of the next generation of GPS.
Uncertainty is one of the most important concepts in general GPS standards. In the next generation of GPS standard system, uncertainty is used as an economic tool to enable optimum allocation of resources amongst specification, production and verification. So uncertainty theory and its application is one of the key technologies in the next generation of GPS. The concept of uncertainty is expanded broadly in the next generation of GPS (Wang et al.
, 2005). It is not only the measurement uncertainty, but also includes: correlation uncertainty, specification uncertainty and so on. In the next generation of GPS standard system, measurement uncertainty is divided into method uncertainty and implementation uncertainty and measurement uncertainty and specification uncertainty collectively known as conformance uncertainty.
Uncertainties are existent in the whole process of design, production and verification/measurement of a product.
METHODOLOGY
The principle of uncertainty GPS estimation: The uncertainty for the same measurement process can in many cases be evaluated on several levels of details or models. In ISO/TS 14253-2 (ISO/TS, 1999), black box and transparent box models of uncertainty estimation are given as two extreme cases of all the models. In the black box method the total measurement process is modeled as a black box with unknown content. The uncertainty budget and the uncertainty components are only describing the total effect on the measurement process. In this choice of model it may be very difficult to determine the functional relationship between uncertainty components and individual error contributors.
To have the full benefit of uncertainty budgeting it may be necessary to open the black box and make a more detailed uncertainty budget. This could either be based on several smaller black boxes or the behavior of all the details in the measurement process, the transparent box model of uncertainty estimation. The black box may also be characterized as a low resolution method and the transparent box model as a high resolution method.
In conclusion, black box model is a method for uncertainty estimation in which the output value of a measurement is obtained in the same unit as the input, rather than by the measurement of other quantities functionally related to the measurand; transparent box model is a method for uncertainty estimation in which the value of a measurand is obtained by measurement of other quantities functionally related to the measurand.
Corresponding Author: Feng Gang, Zhejiang Industry Polytechnic College, 312000, ShaoXing, China
2809

Res. J. Appl. Sci. Eng. Technol., 4(16): 2809-2812, 2012
General GPS standards
+
Basic GPS
standard
Globle GPS
standard
Fig. 1: Relationship between GPS standards
Black box method of uncertainty estimation: In the black box method of uncertainty estimation the result of the measurement is the reading corrected by an eventually known correction:
Y = X+C (1) where, X is the reading of the measuring instrument and
C =
E
Ci is the sum of the corresponding additive corrections known from e.g. calibration, temperature correction, deformation correction, etc.
The combined standard uncertainty of measurement is given by the equation: u c
 u r
2 
 p
1 u i
2
(2)
Where, p is the number of uncorrected uncertainty contributors; u r
is the “sum” of the strongly corrected uncertainty contributors. According to ISO/TS 14253-2, a conservative estimate is to take as strongly correlated all components which are not known to be fully uncorrelated and the correlation coefficient
D
may be equal to 1 or -1 according to the correlation is positive or negative. u r
is calculated by the equation: u r

 p
1 u i
(3)
The uncorrelated (
D
= 0) uncertainty contributors are to added geometrically (the square root of the sum of squares). The strongly correlated uncertainty contributors are to be added arithmetically.
Transparent box model of uncertainty estimation: To have the full benefit of uncertainty budgeting, the transparent box model of uncertainty estimation based on the behavior of all the details in the measurement process should be selected.
In the transparent box method of uncertainty estimation the value of the measurand is modeled as a function of several measured values X i contributors):
(uncertainty
Y = G(X
1
, X
2
..., X i
..., X p+r
) (4)
The combined standard uncertainty of measurement is given by the equation: u c
 u r
2  i
 p

1




Y

X i
 u
Xi



2
(5) where, p is the number of uncorrected uncertainty contributors; u r
is the “sum” of the strongly correlated components of measuring uncertainty: u r
 i
 r

1

Y

X i
 u
Xi
(6)
As can be seen from the analysis, we can know that the uncertainty estimation based on transparent box method is more accurate than the estimation based on black box method. Consequently, the uncertainty should be estimated based on transparent box model if we can.
RESULTS AND DISCUSSION
GPS estimation of uncertainty in spatial straightness measurement: In spatial straightness verification, there are several fitting algorithms. Least-square fitting and mini-zone fitting are usually selected. The estimation method is studied according to the GPS standards 14253
(ISO, 1998).
Verification of straightness by least-square fitting: In spatial straightness least-square verification, every measured value in the verification process can be estimated separately and the partial derivatives of the measurand to every measured value which are required by the transparent box method can be calculated, so the transparent box model can be adopted.
An axis was measured and verified as an experimental research. The spatial straightness specification of the axis is 0.030 .
2810

where, x y

 b b
1
2

 k z k z
Res. J. Appl. Sci. Eng. Technol., 4(16): 2809-2812, 2012
We measured the axis by a Coordinate Measuring
Machine (CMM).
The fitted axis by least-square method can be represented as:
(7) u
 



 z
1 u z 1


2
  



 x
1 u
X 1 
 2
  



 y
1 u y 1 
 2
 




 z
1 u b 1


2
  



 b
2 u b 2


2
  



 k
1 u k 1


2
 
2
2


  k

2 u k

 b b

1

2


 b
2
 b
2 
 2
 
2 u b 1 u k
2
2 u u k 2
 
 b
1
 b
2


2
2

 b
2 u u

 b
2
 b
1
 
 b
2 b 2

2 u u k 1 u u k 2

 
 b
1
 b
1 u u k 1
 b
1


( x z i
)
 z

 
2 z i

 n

 z z i
2 i
2
 x i
; b
2


( y z i
)

 
2 z i

 n

 z i
2 z i
2
 y i k
1
 k
2

 x i
 z i
 n
 x z i i
  i
2
 n
 z i
2
;
 y i

  i z i
2

 n

n
 z y z i
2 i i
(11)
The axis was multiply sampled by the same method of sampling, so the uncertainty of the coordinate value of peak points can be obtained. Then, the professional software Table Curve 3D was used to obtain the equation of fitted axis and the uncertainties of the coefficients. And then, the uncertainty generated by the correlation between the uncertainty contributors was also calculated according to ISO/TS 14253-2. Finally, all the uncertainty contributors were combined and the compliance uncertainty of the result of the spatial straightness measurement was obtained.
Through calculation, we can obtain that the result of the spatial straightness is 0.0224 and the uncertainty of the spatial straightness is 0.0037 . This uncertainty value is just the compliance uncertainty in the spatial straightness verification by least-square fitting.
The distance from a sampled point to the fitted axis is: d i

( x i
 b
1
 k z
1 i
) 2

( y i
 b
2
 k z
2 i
) 2
The spatial straightness estimated by least-square method can be represented as:
*
= max d i
!
min d i
(9)
 
( x
1
 b
1
 k z )
2 
( y
1
 b
2
 k z )
2 
( x
2
 b
1
 k z ) 2 
( y
2
 b
2
 k z
2
) 2
(8)
So the result of the spatial straightness measurement is
(10) where, z
1
, z
2
, b
1
, b
2
, k
1
, k
2
are the uncertainty contributors of the measurand
*
. According to the transparent box model, the measurand
*
can be represented as a function of all the uncertainty contributors:
*
= G ( z
1
, x
1
, y
1
, b
1
, b
2
, k
1
, k
2
), In all the uncertainty contributors, generally speaking, we can consider that only b
1
, b
2
, k
1
, k
2
are correlated. According to formula (5), (6) and (10), the uncertainty of the spatial straightness can be deduced according to the transparent box model:
Verification of straightness by mini-zone fitting: In straightness mini-zone fitting, according to ASME standards, in the evaluation of straightness error, the position of the fitting elements should meet the minimum conditions. This can be achieved by minimum zone method. We should find two parallel lines to include the measured profile and the distance between the parallel lines is the least distance. Although there are innumerable pairs of parallel lines which can include the measured profile in theory, only one pair of parallel lines has the smallest distance.
In straightness mini-zone verification, every measured value in the verification process can be estimated separately and the partial derivatives of the measurand to every measured value which are required by the transparent box method can be calculated, so the transparent box model can be adopted.
An axis was measured and verified as an experimental research. The straightness specification of the axis is 0.020 .
We measured the axis by a Coordinate Measuring
Machine (CMM).
The fitted axis by mini-zone method can be represented as: y = ax+b (12)
2811

Res. J. Appl. Sci. Eng. Technol., 4(16): 2809-2812, 2012
CONCLUSION that:
According to the mini-zone condition, we can obtain a
 n

 x y n x i i
2




 x i
 y i x i

2 b

 y i
 a
 x i
The distance from a sampled point to the fitted axis is: d i
 y i
 ax i
 b / 1
 a 2 (13)
The calculation method for the conformance uncertainty of the spatial straightness verification is proposed in this study. The method is accordant with the requirements of the next generation of GPS system. By different fitting method, the conformance uncertainty is different. For the same measurement processing, the conformance uncertainty of the result fitted by mini-zone method is less than the conformance uncertainty of the result by least-square method. Consequently, the veracity of the verification can be improved and the verification result by different method can be likely to compared based on the GPS standards in the internationally trade of geometrical products.
The straightness estimated by mini-zone method can be represented as:
*
= max d i
!
min d i
(14) where, a , b , y are the uncertainty contributors of the measurand
*
. According to the transparent box model, the measurand
*
can be represented as a function of all the uncertainty contributors:
*
= G ( a , b , y ). In all the uncertainty contributors, generally speaking, we can consider that only a , b are correlated. According to formula (5), (6) and (9), the uncertainty of the straightness can be deduced according to the transparent box model: u

2  

 
 t y u y


2
  



 t y u a


2
  



 t y u b


2
 
2 u u a b where, t = y !
ax !
b
(15)
Through calculation, we can obtain that the result of the straightness is 0.0223 and the uncertainty of the straightness is 0.0026. This uncertainty value is just the compliance uncertainty in the straightness mini-zone verification.
REFERENCES
Dovmark, J., 2001. The New Interesting Concepts from
ISO/TC 213. In: ACMC Annual General Meeting in
Ottawa, Canada.
ISO, 14253-1, 1998. Decision Rules for Proving conformance or Non-Conformance with
Specifications. International Organization for
Standardization, Geneva.
ISO/TR, 14638, 1995. Geometrical Product
Specifications(GPS)-Geo me trical Product
Specification (GPS)-Masterplan.
ISO/TS, 14253-2, 1999. Guide to the Estimation of
Uncertainty in GPS Measurement, in Calibration of measuring Equipment and in Product Verification.
International Organization for Standardization,
Geneva.
Wang, J.X., X. Jiang, L.M. Ma, Z.G. Xu and Z. Li, 2005.
Decision rules for workpieces based on compliance uncertainty. Int. J. Adv. Manuf. Technol., DOI:
10.1007/s00170-004-2477-9.
2812