ESREL 2003 European Safety and Reliability Conference

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ESREL 2003European Safety and Reliability ConferenceJune 15-18, 2003 - Maastricht, the Netherlands

Assessing Part Conformance

by Coordinate Measuring Machines

Daniele Romano

University of Cagliari (Italy) – Department of Mechanical Engineering

Grazia VicarioPolitecnico of Turin (Italy) – Department of Mathematics

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Problem Study of uncertainty of industrial measurement processes and its implications on process design

Objectives Analysis of uncertainty in position tolerance check of manufactured parts on Coordinate Measuring Machines

Optimal allocation of the measurement points on the part surfaces

Problem and objectives

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The research area (Metrology, Statistics, Engineering Design)

Analyze Uncertainty

Design better Product/Process

Regulations & Standards

Measurement Instrument

Measurement Process

Met

hods

& T

echn

ique

s

Simulation

Monte Carlo simulation

DOE

Computer Experiments

Robust Design

Statistical Inference

...

Product/Process

Objectives

Driv

ing

forc

e

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Errors in Measuremen

t

SYSTEMATIC

RANDOM

Orthogonality errors between slidesForm errors of slidesNon-linearity of amplifier responseErrors due to the approach angle of

the touch-ball….

What’s a CMM?

Inherent sampling error

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Planes A, B, C, definfìing the reference system, are ideal mating surfaces which real part surfaces are contacted with in the referencing order (A first, then B, then C)

Nominal hole axis is perpendicular to datum A and displaced by Xc and Yc from datum C and B respectively.

Actual hole axis isthe axis of the ideal largest size pin able to enter the hole perpendicular to plane A

The hole location problem

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Our measurement process

1. Estimation of datum A (envelope to the part surface)

2. Estimation of datum B (envelope)

3. Estimation of datum C (envelope)

5. Probing points on the hole surface

6. Projection of points on datum A

4. DRF origin is obtained by intersection of the three datums

7. Estimation of the largest size inscribed circle

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Calculating position error

X

Y

DRF origin

Plane of datum A

Xc

Yc

Measured points projected on datum A

Inscribed circle

Cnom

Cactep

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Acceptance rule

Deterministic Probabilistic

epeq = ep (dact dmin)/2 t/2 an uncertainty measure

Identifier of Maximum Material Condition (MMC)

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Outline of the study

The real measurement process is replaced by a stochastic simulation model (Romano and Vicario, 2000). In the model:

Measurement errors on the coordinates returned by the CMM are considered additive and described by i.i.d. normal random variables with zero mean and common variance, 2 = 0.0052 mm2.

The part has no error.

Experimentation is conducted on the simulation model investigating how uncertainty in the measure of the position error is affected by the number of points probed on the surfaces (control factors) and by part geometry (blocking factors).

A Monte Carlo simulation (N=104) is run at each experimental trial to have a reliable estimate of uncertainty.

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The experiment

Dimensions in mm

Labels Control factors Levels

nA, nB, nC, nH Number of points measured on 4 9 16

surfaces A, B, C and on the hole

Blocking factors

w Plate thickness 25 50 75Xc Horizontal boxed dimension 50 100 150Yc Vertical boxed dimension 50 100 150d Hole diameter 25 50 75

Simulation modelNumber of points probed

on each surface

Random error

Device variablesMeasurand geometry

uncertainty in the measure of position error

11 array

Patterns of measurement points

On planes On hole surface

Helix

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The uncertainty measure

I

)(|, drdt)t(f)t,r(fdrdt)t,r(f

2

0 0

A convenient representation for position error is the polar one, ep = ei and a suitable measure of uncertainty for ep is the area of a conjoint confidence region I of the two-dimensional random variable () at a (1-) level, defined as:

A useful way to solve the integral is by using conditional distribution f and marginal f:

1I

, drdt)t,r(f

A numerical solution is then provided by taking equally sized angular sectors and using the empirical distributions f and f (deriving from Monte Carlo simulations).

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0.005

0.01

0.015

0.02

30

210

60

240

90

270

120

300

150

330

180 0

[mm]

Uncertainty depends on the angle of the position error

Finding

Proposal of a different acceptance rule

Consequence

epeq (m) t/2 P( epeq /(m)

Empirical 95% confidence region of epeq

for two experimental settings

Solid boundary: all factors at high level

Dashed boundary: all factors at low level

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Empirical 95% confidence region of epeq for two experimental settings

Solid boundary: most polarized

Dashed boundary: least polarized

0.005

0.01

0.015

30

210

60

240

90

270

120

300

150

330

180 0

[mm]

Polarization depends on factors

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Role of Xc and Yc

0.005

0.01

0.015

0.02

30

210

60

240

90

270

120

300

150

330

180 0

[mm]

Solid boundary: Xc = Yc = 50 mm

Dashed boundary: Xc = Yc = 0 mm

(All other factors are at the low level)

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Factorial effects on uncertainty

Finding

Allotment of measurement points on the surfaces as adopted in industrial practice is not optimal. As an example, quota of points on the datums A, B, C are based on the 3:2:1 rule, disproved by results.

Best allotment also depends on the part geometry.

ConsequencesEffects on A95

No

rma

l sco

re

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Designing efficient measurement cycles

d)Yc,Xc,w,;n,n,n,(nA CBAH95 fˆ

integer

n1 1 1 1

:subject to

);(A

TOT

95

x

UBxLB

x

bx 0x

ˆmin

Given a prediction model for uncertainty

a simple optimization problem can be defined in order to find the allotment of probed points on the surfaces (x) that minimizes uncertainty for a given part (b0 ) and a given total number of probed points (nTOT):

x = (nH nA nB nC)T

b0 = (w0 Xc0 Yc0 d0)T

LB and UB are bounds on x

where

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Two design examples

b0 = (75mm 100mm 100mm 50mm)T

95A A quadratic response surface for is estimated from the experiment ( ) and used for optimization%.899R2

adj

The part geometry is defined by:

Solution is sought for in the experimental range: LB = (4 4 4 4)T

UB = (16 16 16 16)T

Solution

CasenTOTSolution typenH nA nB nC 95A 95ρ

#132OptimizedTypical

148

412

88

64

217355

8.310.6

#221Optimized 9 4 4 4 336 10.3

[m2] [m]

Results

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Conclusions

A statistical analysis of position error as measured by CMM has disproved a number of engineers beliefs:

Tolerance zone is a circle

Acceptance rule contains only the modulus of position error

The number of measurement points on planar datums A, B, C is best decided according to the 3:2:1 rule

The best allocation of measurement points on the surfaces does not depend on part geometry (plate thickness, boxed dimensions)

ALFSE

ALFSE

ALFSE

ALFSE

A comprehensive analysis of uncertainty is a prerequisite for an efficient design of the measurement process. Statistical methods and computer simulation seems a unique combination to cope with it.

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Scientific work on uncertainty in CMM measurements

Most of the work addresses the characterization of measurement errors due to the machine and the related calibration methods to compensate systematic errors.

The basic scenario for uncertainty analysis has been proposed by PTB and then adopted also by other metrology Institutes. In the approach the first measure is taken by the real machine, all other are obtained via a computer simulation model ( “virtual machine”).

We are not aware of applications of uncertainty analysis on the design of an efficient measurement process. Practitioners routinely select measurement cycles by applying simple rules of thumb where cost is the major concern.

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PCQK

O

O’

Plate thickness role in position error

Absolute reference

Datum Reference Frame

C: nominal position ofhole center on DRF

Case #1 Plate thickness = h4 points probed P = estimated center positionPC position error

Case #2 Plate thickness =2 h4 points probed Q = estimated center positionQC position error

Case #3 Plate thickness = 3h4 points probed K = estimated center positionKC position error

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z

x

y

3D plot of the origin of the Datum Reference Frame

270.000 points

Uncertainty depends on direction

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0.002 0.004 0.006 0.008 0.01 0.01200

5

10

15

20

25

30

35

40

45

Fre

quen

cy =2,5°

Case of the most polarized 95% confidence region

200

400

600

800

30

210

60

240

90

270

120

300

150

330

180 0

Frequency

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0.002 0.004 0.006 0.008 0.01 0.01200

5

10

15

20

25

30

35

40

45

Fre

quen

cy =2,5°

Case of maximum polarized 95% confidence region

0.005

0.01

0.015

30

210

60

240

90

270

120

300

150

330

180 0

[mm]

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Uncertainty Analysis

Basic

Product/Process Design

Take the same measurement N times

Estimate uncertainty of that measurement

Take M measurements according to an experimental design

Replicate the experiment N times

Estimate uncertainty in the whole sampling space

Knowledge of uncertainty and cost in the sampling space

Select hardware components

Select parameters of the measurement process

Design specifications (uncertainty, cost)

Comprehensive

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Planar datums in the referencing order with orthogonality constraint (Orthogonal Least Squares + shift) and estimation of the origin of the Datum Reference Frame (DFR)

Hole axis (Orthogonal Least Squares)Position error (distance between nominal and actual axis) in DRF

Monte Carlo simulation on the ideal parts (ideal form, perfect dimensions) with a measurement error N(0,2), 2= 0.0052

Study of the dependence of uncertainty of origin of DFR on the number of the inspected points on the surfaces through a 33 experimental design

Position Tolerance Check and its Uncertainty on CMM

Estimation

of features

Methodology

Evaluation

of uncertainty

of position error

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i x + i y + i z + i = 0 with i = 1,2,3

Mathematical models

Estimation of planar datums and origin of DRF

+=

+=

+=

i

i

i

Zii

Yii

Xii

zZ

yY

xX


( )I0 2 + ,N~

Probed points on surfaces

Ref. A

Ref. CRef. B

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Steps

•Maximum Likelihood estimators of parameters

•Orthogonal Least Squares

•Non-linear problem let use a constraint (Lagrange multiplier)

•Equivalent problem with

•Solution: unit norm eigenvector associated to the minimum eigenvalue

0131 nIAFir

st D

atu

m


1T

11 PPA

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•Maximum Likelihood estimators of parameters

•Orthogonal least Squares + orthogonality constraint with the first datum

•Same problem as the first datum unit norm eigenvector associated to the minimum eigenvalue

•...

Steps

Sec

ond

Dat

um

+T

hir

d D

atu

m


30

0

0

0

33333

22222

11111

TzˆyˆxˆˆTzˆyˆxˆˆ

Tzˆyˆxˆˆ

jmj

maxT 21

2 r

jmj

maxT 11

1 r

jmj

maxT 31

3 r

Step

Origin of DRF


Envelope rule

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Results: scatterplots of the origin of the DRF

Origins of estimated datumsas envelope surfaces

Origins of estimated datums withOrthogonal Least Squares

9 inspected pointson actual surfaces


Envelope rule, when form errors are comparable with measurements errors, produces a bias and increases uncertainty

Uncertainty depends on direction

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1122

2

2122

1

2

1

2

1

2

12

1

2

1

2

1

ytxxtyy

xtyytxx

O

O

Why does uncertainty depend on direction?


OLS lines with orthogonality constraint

OLS lines with no constraint

Orthogonality constraint makes a pattern!

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100

200

300

400

30

210

60

240

90

270

120

300

150

330

180 0

0 0.005 0.01 0.015 0.02

5101520253035404550

Fre

quen

cy

=135°

Frequency


d(Cnominal.,Cactual)=f(,)

Dependence on direction suggests to express position error by a polar (spherical) transformation in the two dimensional case

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Results: 95% Confidence Regions

0.01 mm

30°

210°

60°

240°

90°

270°

120°

300°

150°

330°

180° 0°

0.02 mm 0.03 mm


=0.005 mm

Measurement error is largely amplified

Reduction of uncertainty is heavily paid in terms of number of measurement point

n1=n2=n3=4; nc=4

n1=n2=n3=9; nc=9

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4 6 8 10 12 14 164

6

8

10

12

14

1655

60

65

70

75

80

5560

65

70

75

n1

n2

Results: effect of the number of measured points on the flat surfaces on uncertainty (of origin of DRF)


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22221

zyx OOOO )(Tr

with O = (XO, YO, ZO) DRF origins

A-optimality

with a 33 experimental design)( 3210 n,n,n

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Amount of uncertainty in the estimation of position error is not negligible and it may easily leads to incorrect decision about acceptance/rejection of the part, if not considered

Uncertainty depends on direction: a non trivial software module should be added to the machine

Results suggest some criticism of the envelope rule:

The tolerance zone (including uncertainty in the evaluation) looses the central symmetry

Envelope rule is unjustified and detrimental (biased estimates and increased uncertainty) when form errors of inspected surfaces are comparable with random error of CMM

Final Remarks


37Position Tolerance Check and its Uncertainty on CMM

CMM gives:

I. coordinates of a finite number of points pertaining to contact points between a touch probe and the planar datums according to a specific order

• coordinates of a finite number of points pertaining to contact points between a touch probe and the hole surface

CMM software computes coordinates and gives parameters “estimates” of probed surfaces, but the current practice does not include any uncertainty evaluation

Measurements process with CMM

Documents

ESREL 2003 European Safety and Reliability Conference