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ESREL 2003 European Safety and Reliability
Conference June 15-18, 2003 - Maastricht, the
Netherlands
Assessing Part Conformance by Coordinate
Measuring Machines Daniele Romano University of
Cagliari (Italy) Department of Mechanical
Engineering Grazia Vicario Politecnico of Turin
(Italy) Department of Mathematics
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Problem and objectives

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

Problem

• Analysis of uncertainty in position tolerance
  check of manufactured parts on Coordinate
  Measuring Machines
• Optimal allocation of the measurement points on
  the part surfaces

Objectives
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The research area (Metrology, Statistics,
Engineering Design)
Product/Process
Measurement Instrument Measurement Process
Simulation Monte Carlo simulation DOE Computer
Experiments Robust Design Statistical
Inference ...
Regulations Standards
Methods Techniques
Driving force
Analyze Uncertainty Design better Product/Process
Objectives
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Whats a CMM?
Inherent sampling error

• Orthogonality errors between slides
• Form errors of slides
• Non-linearity of amplifier response
• Errors due to the approach angle of the
  touch-ball
• .

SYSTEMATIC
Errors in Measurement
RANDOM
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The hole location problem

• 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

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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)
4. DRF origin is obtained by intersection of the
three datums
5. Probing points on the hole surface
6. Projection of points on datum A
7. Estimation of the largest size inscribed
circle
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Calculating position error
Measured points projected on datum A
Plane of datum A
Y
ep
Cact
Yc
Cnom
Inscribed circle
DRF origin
X
Xc
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Acceptance rule
? epeq ? ? ep ? ? (dact ? dmin)/2 ? t/2
Deterministic
Probabilistic
? 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, s2 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 (N104) is run at each
  experimental trial to have a reliable estimate of
  uncertainty.

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The experiment
Random error
uncertainty in the measure of position error
Simulation model
Number of points probed on each surface
Measurand geometry
Device variables
Dimensions in mm
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(No Transcript)
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The uncertainty measure
A convenient representation for position error is
the polar one, ep reiq and a suitable measure
of uncertainty for ep is the area of a conjoint
confidence region I of the two-dimensional random
variable (r, q) at a (1-a) level, defined as
A useful way to solve the integral is by using
conditional distribution frq and marginal fq
A numerical solution is then provided by taking
equally sized angular sectors and using the
empirical distributions fq and frq (deriving
from Monte Carlo simulations).
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Uncertainty depends on the angle of the position
error
Empirical 95 confidence region of epeq for two
experimental settings Solid boundary all
factors at high level Dashed boundary all
factors at low level
Finding
Consequence
Proposal of a different acceptance rule
? epeq ?(m) ? t/2 ? P95(? epeq ? /q (m))
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Polarization depends on factors
Empirical 95 confidence region of epeq for two
experimental settings Solid boundary most
polarized Dashed boundary least polarized
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Role of Xc and Yc
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
Consequences

• 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 321 rule, disproved by
  results.
• Best allotment also depends on the part geometry.

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Designing efficient measurement cycles
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
where
LB and UB are bounds on x
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Two design examples
b0 (75mm 100mm 100mm 50mm)T

• The part geometry is defined by

LB (4 4 4 4)T UB (16 16 16 16)T

• Solution is sought for in the experimental range

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 321
  rule
• The best allocation of measurement points on the
  surfaces does not depend on part geometry (plate
  thickness, boxed dimensions)

• 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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Plate thickness role in position error
Case 1 Plate thickness h 4 points probed P
estimated center position PC position error
Absolute reference
Case 2 Plate thickness 2 h 4 points probed Q
estimated center position QC position error
Datum Reference Frame
Case 3 Plate thickness 3h 4 points probed K
estimated center position KC position error
C nominal position of hole center on DRF
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3D plot of the origin of the Datum Reference Frame
270.000 points

• Uncertainty depends on direction

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Case of the most polarized 95 confidence region
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Case of maximum polarized 95 confidence region
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Uncertainty Analysis
Comprehensive
Basic
Take M measurements according to an experimental
design
Take the same measurement N times
Replicate the experiment N times
Estimate uncertainty of that measurement
Estimate uncertainty in the whole sampling space
Product/Process Design
Knowledge of uncertainty and cost in the sampling
space
Design specifications (uncertainty, cost)

• Select hardware components
• Select parameters of the measurement process

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Methodology

• 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 e ? N(0,s2), s2 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

Estimation of features
Evaluation of uncertainty of position error
Position Tolerance Check and its Uncertainty on
CMM
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Estimation of planar datums and origin of DRF
Mathematical models
Probed points on surfaces
ai x bi y gi z di 0 with i 1,2,3
Ref. C
Ref. B
Ref. A
Position Tolerance Check and its Uncertainty on
CMM
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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

First Datum
Position Tolerance Check and its Uncertainty on
CMM
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Steps

• 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
• ...

Second Datum Third Datum
Position Tolerance Check and its Uncertainty on
CMM
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Step
Envelope rule
Origin of DRF
Position Tolerance Check and its Uncertainty on
CMM
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Results scatterplots of the origin of the DRF
9 inspected points on actual surfaces
Origins of estimated datums as envelope surfaces

• Envelope rule, when form errors are comparable
  with measurements errors, produces a bias and
  increases uncertainty
• Uncertainty depends on direction

Origins of estimated datums with Orthogonal Least
Squares
Position Tolerance Check and its Uncertainty on
CMM
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Why does uncertainty depend on direction?
OLS lines with orthogonality constraint
Orthogonality constraint makes a pattern!
OLS lines with no constraint
Position Tolerance Check and its Uncertainty on
CMM
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Dependence on direction suggests to express
position error by a polar (spherical)
transformation in the two dimensional case
d(Cnominal.,Cactual)f(?,?)
50
45
40
35
30
25
Frequency
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15
10
5
0
0.005
0.01
0.015
0.02
rq 135
Position Tolerance Check and its Uncertainty on
CMM
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Results 95 Confidence Regions
n1n2n34 nc4
n1n2n39 nc9

• Measurement error is largely amplified
• Reduction of uncertainty is heavily paid in terms
  of number of measurement point

s0.005 mm
Position Tolerance Check and its Uncertainty on
CMM
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Results effect of the number of measured points
on the flat surfaces on uncertainty (of origin of
DRF)
A-optimality
with O (XO, YO, ZO) DRF origins
with a 33 experimental design
Position Tolerance Check and its Uncertainty on
CMM
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Final Remarks

• 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

Position Tolerance Check and its Uncertainty on
CMM
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Measurements process with CMM

• CMM gives
• 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
Position Tolerance Check and its Uncertainty on
CMM