1 esrel 2003 european safety and reliability conference june 15-18, 2003 - maastricht, the...
TRANSCRIPT
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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
Position Tolerance Check and its Uncertainty on CMM
( )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
Position Tolerance Check and its Uncertainty on CMM
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
Position Tolerance Check and its Uncertainty on CMM
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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
Position Tolerance Check and its Uncertainty on CMM
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
Position Tolerance Check and its Uncertainty on CMM
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?
Position Tolerance Check and its Uncertainty on CMM
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
Position Tolerance Check and its Uncertainty on CMM
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
Position Tolerance Check and its Uncertainty on CMM
=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)
Position Tolerance Check and its Uncertainty on CMM
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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
Position Tolerance Check and its Uncertainty on CMM
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