More than Just Lines on a Map: Best Practices for U.S Bike Routes
Ph D Thesis Defense Presentation
1. Computer-Aided
Inspection
D. ElKott
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension
case study
CAD-Based Inspection of Sculptured Surfaces
ASSESSMENT
approach
probe path
localization
sub. geometry Diaa F. ElKott
demo
CONTRIBUTIONS
Department of Mechanical Engineering
BIBLIOGRAPHY McMaster University
November 3, 2006
2. Computer-Aided
Inspection
D. ElKott
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach
probe path
localization I. Background
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
2
3. Computer-Aided
Inspection
D. ElKott
Background
Sculptured vs Primitive Surfaces
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT Part with primitive surfaces Part with sculptured surfaces
approach
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
3
4. Computer-Aided
Inspection
D. ElKott
Background
The Manufacture of Sculptured Surfaces
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs. CAD/CAM/CAE
extension
case study
ASSESSMENT
6. Computer-Aided
Inspection
D. ElKott
Background
The Manufacture of Sculptured Surfaces
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs. CAD/CAM/CAE
extension
case study clamp CMM
probe
7. ASSESSMENT
approach
probe
probe path
(3) scanning path
localization
(1)
sub. geometry clamp
demo
(2)
CONTRIBUTIONS
physical
BIBLIOGRAPHY
!
model
Research Issues
Limitations in the reported literature
Sampling of sculptured surfaces.
Assessment of the physical object geometry.
Sampling strategy implementation.
5
8. Computer-Aided
Inspection
D. ElKott
Background
Objectives, and Scope of Research
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs. Objectives
extension
case study 1 Develop a CAD-based system to generate the sampling strategy of
ASSESSMENT
sculptured surfaces.
approach
probe path
localization
2 Address the issues associated with the implementation of the
sub. geometry sampling plan.
demo
CONTRIBUTIONS
3 Develop methods for the physical object assessment using the CMM
BIBLIOGRAPHY measurement data.
Scope of research Continuous probe scanning
clamp CMM
Answer the following questions: probe
1 Where to measure a model (3)
probe
scanning path
with sculptured features? (1)
clamp
2 How to assess the quality of (2)
the physical object? physical
model
6
9. Computer-Aided
Inspection
D. ElKott
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach
probe path
localization II. Sampling of Sculptured Surfaces
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
7
10. Computer-Aided
Inspection
D. ElKott
The Methodology
BACKGROUND
SAMPLING Criterion
methodology Isoparametric line sampling
quadrilateral upper sampling plan
surfs. tolerance bound
extension x u , v
case study
isoparametric
ASSESSMENT u=const. curves substitute
approach geometry
probe path
localization CAD
v =const.
sub. geometry model
demo
sampling
CONTRIBUTIONS X3
plan error
BIBLIOGRAPHY
CAD model
X2 lower sampling plan
X1
tolerance bound
Challenges
Shape complexity.
Substitute geometry construction.
Tools
8
C++, SMLibTM geometric modelling kernel.
11. Computer-Aided
Inspection
D. ElKott
The Problem, and Approach to Solution
BACKGROUND
SAMPLING
methodology
quadrilateral
The problem
surfs.
extension Find the sample line distribution that results in an acceptable error
case study
between the substitute geometry and the CAD model.
ASSESSMENT
approach
probe path
localization Approach to sampling
sub. geometry
demo NURBS parameters sample curves
curvature analyses sample size
CONTRIBUTIONS Geometry point-based rep. Preliminary substitute geom. Sampling Plan
CAD model
BIBLIOGRAPHY Processing Sampling Assessment
max. no. of
scanning curves
sample curves
sampling plan deviations
sample size
error tolerance matrix
substitute geom.
Modify NO Terminate
Sampling Plan
?
sample curves
sample size
Output YES
substitute geometry
deviations matrices Sampling Plan
continuity checks
9
12. Computer-Aided
Inspection
D. ElKott
Sampling Quadrilateral Surfaces
Sampling Algorithms
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs. Automatic sampling
extension
case study
1 Preliminary sampling plan.
ASSESSMENT
approach 2 Assess plan → ∆.
probe path
localization 3 IF (!TERM) GOTO 4 ELSE END
sub. geometry
demo 4 Add sample at ∆max , GOTO 2.
CONTRIBUTIONS
BIBLIOGRAPHY
Example: 15 lines, ∆max = 5µm
10
13. Computer-Aided
Inspection
D. ElKott
Sampling Quadrilateral Surfaces
Sampling Algorithms
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs. Curvature-based sampling
extension
case study
1 Assess surface → κ.
ASSESSMENT
approach 2 Preliminary sampling plan.
probe path
localization 3 WHILE (!TERM) GOTO ??
sub. geometry
demo 4 Add sample at κmax . DO WHILE
CONTRIBUTIONS
BIBLIOGRAPHY
Example: 23 lines, ∆max = 50µm
11
15. Computer-Aided
Inspection
D. ElKott
Sampling Quadrilateral Surfaces
Limitations
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension Algorithm limitations
case study
ASSESSMENT
Quadrilateral surface sampling algorithms cannot generate sampling plans
approach for the following features:
probe path
localization Trimmed surfaces.
sub. geometry
demo n-sided surface patches.
CONTRIBUTIONS
BIBLIOGRAPHY
Models composed of multiple surfaces.
Research issues
CAD model representation.
Sampling algorithm.
Substitute geometry construction algorithm.
Sampling plan assessment.
13
16. Computer-Aided
Inspection
D. ElKott
Extending the Work
Trimmed Surfaces – The Building Block
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
Point-based representation
extension
case study
ASSESSMENT
approach
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
14
17. Computer-Aided
Inspection
D. ElKott
Extending the Work
Trimmed Surfaces – The Building Block
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
Point-based representation
extension
case study
ASSESSMENT
approach
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
¡T
Pi,j = ui vj Oij ∆κu
ij ∆κv
ij , i = 1, . . . , N u , j = 1, . . . , N v
@
1 for Si,j ∈ ST
Oij =
15 0 for Si,j ∈ ST
/
18. Computer-Aided
Inspection
D. ElKott
Extending the Work
Trimmed Surfaces – The Building Block
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
Hybrid sampling
extension
case study 1 Criteria for ranking of isoparametric lines (dynamic!):
ASSESSMENT The point occurrence, R1.
approach The surface mean curvature, R2.
probe path
localization The deviation between the substitute geometry and CAD, R3.
sub. geometry
demo
2 A combined weight is calculated for each isoparametric curve:
CONTRIBUTIONS @
BIBLIOGRAPHY (αR1 + βR2) / (α + β) → preliminary.
R=
(αR1 + βR2 + γR3) / (α + β + γ) → update.
Demonstration
16
19. Computer-Aided
Inspection
D. ElKott
Extending the Work
Trimmed Surfaces – The Building Block
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
Substitute geometry algorithm
extension
case study Requirement: Fit adjacent surface patches such that cross-boundary
ASSESSMENT derivative continuity is maintained.
approach
probe path Algorithm: skinning of cross sectional curves with boundary
localization
sub. geometry
constraints [1].
demo
CONTRIBUTIONS
BIBLIOGRAPHY
Sampling plan assessment
The sampling plan is assessed according to two criteria:
1 Deviation between the substitute geometry and the CAD model
(automatic).
2 Visual continuity between adjacent surface patches (semi-automatic).
17
20. Computer-Aided
Inspection
D. ElKott
Extending the Work
Example – The Human Ear Model
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
The model
extension
case study Model composed of two surfaces, sewed along their partition line.
ASSESSMENT
approach
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
18
21. Computer-Aided
Inspection
D. ElKott
Extending the Work
Example – The Human Ear Model
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
u-Sampling plan and deviations
extension
case study Output of sampling algorithm when sampling both surfaces in the
ASSESSMENT u-direction.
approach
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
19
22. Computer-Aided
Inspection
D. ElKott
Extending the Work
Example – The Human Ear Model
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
Visual continuity check
extension
case study VC 1 check to compare u−, and v − sampling plans.
ASSESSMENT
1
approach
probe path
localization 0.9
sub. geometry
VC1 continuity measure for substitute geometry
demo 0.8
CONTRIBUTIONS
0.7
BIBLIOGRAPHY
0.6
0.5
0.4
0.3
0.2
0.1
20 u-sampling plan
v-sampling plan
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
VC1 continuity measure for CAD model
23. Computer-Aided
Inspection
D. ElKott
Extending the Work
Example – The Human Ear Model
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
Visual continuity check
extension
case study VC 2 check to compare u−, and v − sampling plans.
ASSESSMENT
1
approach
probe path
localization 0.9
sub. geometry
demo
VC2 continuity measure for substitute geometry
0.8
CONTRIBUTIONS
0.7
BIBLIOGRAPHY
0.6
0.5
0.4
0.3
0.2
0.1
21 u-sampling plan
v-sampling plan
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
VC2 continuity measure for CAD model
24. Computer-Aided
Inspection
D. ElKott
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach
probe path
localization III. Physical Model Assessment
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
22
25. Computer-Aided
Inspection
D. ElKott
The Problem and Approach to Solution
BACKGROUND
SAMPLING
methodology
quadrilateral The problem
surfs.
extension
case study
Develop a framework for the implementation of the sampling strategy, and
ASSESSMENT assessment of the physical object using the CMM measurement.
approach
probe path
localization
sub. geometry
Approach to physical object assessment
demo
sampling hardware errors optimization algorithm
CONTRIBUTIONS plan software errors objective function
BIBLIOGRAPHY
Sampling modified sampling plan
CAD CMM measurement Data
model Plan
System data (filtered) Localization
Refinement
measurement
physical
data (filtered,
model
aligned)
tolerances
Deviations of Physical substitute
Substitute
fitting algorithm
physical model Model Geometry fitting tolerances
from CAD geometry
Assessment Algorithm
23
26. Computer-Aided
Inspection
D. ElKott
Sampling Strategy Implementation
Probe Scanning Path
BACKGROUND
SAMPLING
methodology
quadrilateral The problem
surfs.
extension
case study
Calculate a nominal path for the continuous probe-scanning of the physical
ASSESSMENT
object.
approach
probe path
localization
sub. geometry
Nominal probe scanning paths
demo
v
CONTRIBUTIONS
UV trimming valid UV
BIBLIOGRAPHY curve probe path
BRep model
trimming
curve
edge
avoidance
distance
modified
probe
intersection
invalid UV scanning
point Z
probe path paths
UV sample
curve
24 u
Y X
27. Computer-Aided
Inspection
D. ElKott
Localization of CMM Measurement
Problem and Solution Approach
BACKGROUND
SAMPLING
methodology The problem
quadrilateral
surfs.
extension
Find the rigid body transformation that optimizes the alignment of the
case study measurement data with the CAD model so as to minimize the influence of
ASSESSMENT
the CMM system, and machining errors on the accuracy of the substitute
approach
probe path geometry.
localization
sub. geometry
demo
CONTRIBUTIONS
Why localization?!
BIBLIOGRAPHY physical model nominal
coordinate frame scanning
Z point
M
Y
M
X
M
physical
model
actual
ZCAD
measurement
CAD point
CAD model
25 YCAD
XCAD coordinate
frame
28. Computer-Aided
Inspection
D. ElKott
Localization of CMM Measurement
Problem Formulation
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension Problem formulation
case study
ASSESSMENT
Align a cloud of scanned data points P with the BRep CAD model of the
approach physical object such that:
probe path
localization
sub. geometry P = {P0 , . . . , Pn−1 }
demo
Pi = {pi0 , . . . , pimi −1 }
CONTRIBUTIONS
BIBLIOGRAPHY using the objective function:
H I
€
MinimizeFunction d n−1
1
Pm × dl2 e
i ∀pl ∈P
i=0
dl = GetShortestDistance(S, R × pl + T)
H I H I H I
xlnew xlold δx
pnew
l = d ylnew e = R(O, θx )×R(O, θy )×R(O, θz )× d ylold e + d δy e
26 zlnew zlold δz
29. Computer-Aided
Inspection
D. ElKott
Localization of CMM Measurement
Problem Formulation
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension Problem and solution
case study
ASSESSMENT
approach
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
27
30. Computer-Aided
Inspection
D. ElKott
Localization of CMM Measurement
Problem Formulation
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension Challenge
case study
ASSESSMENT
1 Shape complexity.
approach
probe path
2 Data localization without prior knowledge of correspondence.
localization
sub. geometry
3 Calculation of initial conditions for optimization.
demo
CONTRIBUTIONS
BIBLIOGRAPHY
Tools
1 C++.
2 Gnu Scientific Library (GSL).
3 SMLibTM .
28
31. Computer-Aided
Inspection
D. ElKott
Constructing the Substitute Geometry
Problem
BACKGROUND
SAMPLING
methodology The problem
quadrilateral
surfs.
extension
Use the aligned CMM measurement to contruct a substitute geometry for
case study the physical object using the algorithms for surface skinning of
ASSESSMENT
cross-section curves.
approach
probe path
localization
sub. geometry Challenge
demo
CONTRIBUTIONS Construct substite geometry for trimmed cross-section curves.
BIBLIOGRAPHY Approach:
localized measurement points valid model
points extracted from CAD model space
substitute
geometry
points at
model edges
points at
model edges
ZCAD
29 YCAD
trimmed
region
XCAD
32. Computer-Aided
Inspection
D. ElKott
Sampling Plan Implementation–An
BACKGROUND Example
SAMPLING Trimmed Surface Sampling
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach Surface Model
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
30
33. Computer-Aided
Inspection
D. ElKott
Sampling Plan Implementation–An
BACKGROUND Example
SAMPLING Trimmed Surface Sampling
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach Solid Model, and Physical Object
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
31
34. Computer-Aided
Inspection
D. ElKott
Sampling Plan Implementation–An
BACKGROUND Example
SAMPLING Trimmed Surface Sampling
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach Sampling Plan–virtual
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
32
35. Computer-Aided
Inspection
D. ElKott
Sampling Plan Implementation–An
BACKGROUND Example
SAMPLING Trimmed Surface Sampling
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach Measurement Data, Object Assessment Results
probe path
localization
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY 0.080 mm
0.030 mm
0.005 mm
0.000 mm
33
36. Computer-Aided
Inspection
D. ElKott
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach
probe path
localization IV. Conclusions
sub. geometry
demo
CONTRIBUTIONS
BIBLIOGRAPHY
34
37. Computer-Aided
Inspection
D. ElKott
Conclusions
Contributions, Limitations, Recommendations
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach Contributions
probe path
localization Research
sub. geometry
demo 1 New and generalized methods for sampling sculptured surfaces.
CONTRIBUTIONS
2 A framework for physical object assessment.
BIBLIOGRAPHY Development
1 BRep-based approach.
2 Addressing several shape complexity issues
3 Linking the inspection planning and CMM measurement plan
implementation.
4 Automation of several functions.
35
38. Computer-Aided
Inspection
D. ElKott
Conclusions
Contributions, Limitations, Recommendations
BACKGROUND
SAMPLING
methodology
quadrilateral
surfs.
extension
case study
ASSESSMENT
approach Limitations
probe path
localization 1 Sampling is limited to isoparametric curves.
sub. geometry
demo 2 Accessibility assumption.
CONTRIBUTIONS
3 Data localization is limited to cloud WRT CAD.
BIBLIOGRAPHY
Recommendations
1 Use part-specific shape features to guide sampling.
2 Consider CAD models which are based on data clouds.
3 Extend the work on the construction of substitute geometry.
36
39. Computer-Aided
Inspection
D. ElKott
Bibliography
BACKGROUND
SAMPLING L. Piegl and W. Tiller.
methodology Cross-sectional design with boundary constraints.
quadrilateral Engineering With Computers, 15:171–180, 1999.
surfs.
extension D. ElKott and S. Veldhuis.
case study Isoparametric line sampling for the inspection planning of sculptured surfaces.
Computer Aided Design, 37:189–200, 2005.
ASSESSMENT
approach D. ElKott and S. Veldhuis.
probe path
A CAD-based approach to CMM inspection of models with sculptured features.
localization
Accepted for Publication Engineering With Computers.
sub. geometry
demo
J. Nelder and R. Mead.
CONTRIBUTIONS A simplex method for function minimization.
Computer Journal, 7:308–315, 1965.
BIBLIOGRAPHY
M. Galassi and J. Davies and J. Theiler and B. Gough and G. Jungman and M. Booth and F. Rossi.
Gnu Scientific Library Reference Manual.
December 2004.
L. Piegl and W. Tiller.
The N U RBSbook.
Springer-Verlaag, 1997.
37
41. Computer-Aided
Inspection
D. ElKott
Curvature effect on manufacturing
CURVATURE
manufacture
measures
UNCERTAINTY
ERRORS
GEOMETRY Effect of surface curvature
diff.geometry
BRep Effect of Step Length Effect of Path Interval
n-sided surfs.
Overcut (τi )
continuity
vis.continuity
sub.geometry
SCANNING
LOCALIZATION λi
inititial corr.
CCi CCi+1
optimisation
λi+1 ωi
Cusp (η )
Undercut (τo )
CCi+2
39
42. Computer-Aided
Inspection
D. ElKott
Curvature Measures
CURVATURE
manufacture
measures
UNCERTAINTY
Surface vs. curve curvature
ERRORS
GEOMETRY
Iso-curve
diff.geometry Osculating Surface
BRep Circle Normal
n-sided surfs.
continuity Iso-curve
vis.continuity Principal
sub.geometry Normal
Isoparametric
SCANNING Surface Curve
LOCALIZATION
inititial corr.
optimisation
Surface Minimum
Principal Curvature
Arc and Direction
Surface Maximum
Principal Curvature
Arc and Direction
Surface
40
44. Computer-Aided
Inspection
D. ElKott
Measurement Uncertainty
CURVATURE
manufacture
measures
UNCERTAINTY
CMM measurement uncertainty
ERRORS
GEOMETRY
A representation of all possible measurement errors in the CMM
diff.geometry measurement.
BRep
n-sided surfs.
continuity
vis.continuity true point
sub.geometry Z (unknown)
SCANNING
LOCALIZATION measured
error
inititial corr. point
optimisation
X
Y
42
45. Computer-Aided
Inspection
D. ElKott
Sources of Measurement Errors
CURVATURE
manufacture
measures
UNCERTAINTY
ERRORS
GEOMETRY Effect of probing direction
diff.geometry
BRep
n-sided surfs. probing in surface probe probing in vertical probe
normal direction direction
continuity
vis.continuity
sub.geometry r r
cp cp
SCANNING
n
n'
LOCALIZATION
p p
inititial corr. p'
optimisation work work
piece piece
(a) Correct contact point, (b) Erroneous contact point,
p = cp + r × n p = cp + r × n
43
46. Computer-Aided
Inspection
D. ElKott
Sources of Measurement Errors
CURVATURE
manufacture
measures
UNCERTAINTY
ERRORS
GEOMETRY Effect of software
diff.geometry
BRep
n-sided surfs. work work
sample points piece
continuity piece
vis.continuity with measurement
sub.geometry errors
SCANNING
C1
LOCALIZATION
C2
inititial corr.
optimisation
work piece sampling C2
strategy #1
sampling
strategy #2
C1
(g) Effect of fitting algorithm se- (h) Sampling strategy effect
lection
44
47. Computer-Aided
Inspection
D. ElKott
Geometry of Curves and Surfaces
CURVATURE
manufacture
measures
UNCERTAINTY
ERRORS
GEOMETRY
osculating plane
diff.geometry
normal plane
n
BRep xu
n-sided surfs.
continuity
vis.continuity b p xv
sub.geometry x u , v
SCANNING rectifying plane x ui , v j tangent
LOCALIZATION
x s t plane
inititial corr.
optimisation X3 X3
X1 X2 X1 X2
(i) Moving trihedron of a space curve (j) Local frame, and tangent plane of a
surface
45
48. Computer-Aided
Inspection
D. ElKott
Nonmanifold Boundary Representation
CURVATURE
manufacture
measures
UNCERTAINTY
BRep
ERRORS
GEOMETRY
diff.geometry Region
BRep
n-sided surfs.
continuity
vis.continuity Shell
sub.geometry
SCANNING
FaceUse Face Surface
LOCALIZATION
inititial corr.
optimisation
LoopUse Loop
EdgeUse Edge Curve
VertexUse Vertex
46 Adapted from the SMLibTM online documentation.
54. Computer-Aided
Inspection
D. ElKott
Initial Correspondence Points
CURVATURE
manufacture
measures
UNCERTAINTY
ERRORS
GEOMETRY curvature CMM
2, j
circle for probe
v
diff.geometry
BRep
n-sided surfs. Rp
continuity nj
vis.continuity
sub.geometry
SCANNING
S
LOCALIZATION
inititial corr.
optimisation pj
curvature
1, j
circle for
52
55. Computer-Aided
Inspection
D. ElKott
Localization of CMM Measurement
Data Post Procesing
CURVATURE
manufacture
measures
UNCERTAINTY
ERRORS
GEOMETRY Post processing of localized data
diff.geometry
BRep localized
n-sided surfs. measurement
continuity data points
vis.continuity model
sub.geometry boundary
excluded
SCANNING
points
LOCALIZATION
inititial corr.
optimisation CAD
coordinate model
frame boundary
ZCAD
CAD
YCAD model
XCAD
53
56. Computer-Aided
Inspection
D. ElKott
Nelder-Mead Simplex Algorithm
Minimization of function of n-variables
CURVATURE
manufacture START
measures Initial Pi
UNCERTAINTY y i = f (Pi ), i = 0 , , n → Get l , h, P
P * = (1 + α )P − αPh
ERRORS
y * = f (P * )
GEOMETRY
diff.geometry
No Yes No
BRep y * yl y * yi ,i ≠ h y * yh Ph = P *
n-sided surfs.
continuity Yes Yes
vis.continuity P ** = (1 + γ )P * − γP P ** = β Ph + (1 − β )P
sub.geometry y ** = f (P ** ) No y ** = f (P ** )
SCANNING
LOCALIZATION No Yes
y ** y l y ** y h
inititial corr.
optimisation Yes No
Pi + Pl
Pi = , i = 0, ,n
Ph = P ** Ph = P * Ph = P ** 2
No Minimum
Reached
Yes
EXIT
Reproduced from Nelder and Mead, 1965 [4]
54
57. Computer-Aided
Inspection
D. ElKott
Nelder-Mead Simplex Algorithm
Minimization of function of n-variables
CURVATURE
START
manufacture
Initial Pi
measures
y i = f (Pi ), i = 0 , , n → Get l , h, P
P * = (1 + α )P − αPh
UNCERTAINTY
y * = f (P * )
ERRORS
No Yes No
y * yl y * yi ,i ≠ h y * yh Ph = P *
GEOMETRY
Yes Yes
diff.geometry
P ** = (1 + γ )P * − γP P ** = β Ph + (1 − β )P
BRep y ** = f (P ** ) No y ** = f (P ** )
n-sided surfs.
continuity y ** y l
No
y ** y h
Yes
vis.continuity
Yes No
sub.geometry Pi =
Pi + Pl
, i = 0, ,n
Ph = P **
Ph = P *
Ph = P **
2
SCANNING
LOCALIZATION No Minimum
Reached
inititial corr.
Yes
optimisation
EXIT
The simplex [5]
P0 = (x0 , x1 , ..., xn )
P1 = (x0 + δ0 , x1 , ..., xn )
P2 = (x0 , x1 + δ1 , ..., xn )
... ...
Pn = (x0 , x1 , ..., xn + δn )
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