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Transcript of 1 Profile Coordinate Metrology Based on Maximum Conformance to Tolerances Faculty of Engineering and...
1
Profile Coordinate Metrology Based on Maximum Conformance
to Tolerances
Faculty of Engineering and Applied Science University of Ontario Institute of Technology
Oshawa, Ontario, Canada
Dr. Ahmad [email protected]
CMSC - Coordinate Metrology Systems Conference (CMSC 2011), Phoenix, Arizona, 25 – 28 July 2011
II. Substitute Geometry Estimation (SGE)
The Best Substituted Geometry
εCi
eiεTi
Desired Geometry in the Reference coordinate System
pi
I. Point Measurement Planning (PMP)
Substituted Geometry in the Reference coordinate System
pi : i th Measured point
ri
Tolerance envelope
III. Deviation Zone Evaluation (DZE)
Probability Density Function of Geometric Deviations
f(e)
e
Three Basic Computation Tasks in Coordinate Metrology
3
Integration Inspection System: Current Research and Final Goal
SGEPMP
DZE
c) Integrated computational tasks
SGEPMP
DZE
b) Recent research on integration of tasks
PMP SGE DZE
a) Sequential tasks in traditional coordinate metrology
SGEPMP
DZE
d) Integrated Inspection System
Design & Manufact. Data
G Nominal geometryG′ Actual geometryG″ Substitute geometryG* Optimum substitute geometry
pi'
G*
Fitting Process
G″
pi ″
Δ(pi ′,G
″)
ΔG′ (G″)
G
G'
pi
εi
XZ
Y
1000
))cos(cos())cos(sin()sin(
))cos(sin(
))cos(cos(
621212
52,32,223
41,31,223
ζζζζζζ
ζζTζTζζ
ζζTζTζζ
ζT
))sin()sin(sin())cos(cos(
))sin()sin(cos())cos(sin(
123132,2
123131,2
ζζζζζζT
ζζζζζζT
))cos()sin(sin())sin(cos(
))cos()sin(cos())sin(sin(
123132,3
123131,3
ζζζζζζT
ζζζζζζT
Six-DOF rigid body transformation
Geometric Deviations
Substitute Geometry
Objective Function
Geometric Deviation
The Best Substituted Geometry
ei
Desired Geometry in the Reference coordinate System
pi
pi*
pi*: Corresponding point
pi: i th measured point
ni: normal vector at the corresponding point
DG: Desired geometry
T: Transformation matrix
Deviation Zone Evaluation of a Single Geometric Feature
GζTG
GζTpiζ
i
maxminObj
( ) GζTG ** ×=
*
**
i
ii
n
nppe ii *
i Gp ⊂*
6
Tolerance Zone & Residual Deviations
GpnnpG
GpnnpG
l
u
l
u
t
t
Tolerance Zone Definition(ASME Y14.5)
Upper tolerance limit
Lower tolerance limit
Normal vector
Gu ”
Gl
”
G'
pi'
XZ
Y
G ”pi″(u* ,v*)
Δ(p i′, G
” )
Res
idua
l D
evia
tion
Pi(u* ,v*)
7
A Drawback of Common Fitting Methods
A Measured Point that Complies to the Tolerance Zone
0
0
l
u
Gp
GpZone Tolerance p ,Gp
,Δ
,Δif
p1'
p 1 ″ Gu″
Gl″G″
Δ(p5′,G″)
p2'
p3' p4'
p5'p2 ″
XZ
Y
Measured pointCorresponding point
p1'
p1 ″
Gu″
Gl″
G″
Δ(p4′,G″)
p2'
p3' p4'
p5'p2 ″
XZ
Y
Measured pointCorresponding pointCorresponding point in the previous tolerance zone
Accept
Reject
8
Application: Over-Cut & Under-Cut in Closed Loop Machining
G'
pi'
XZ
Y
G *pi″
Δ(p i′, G
* )
Over-Cut
Und
er-C
ut
Under-Cut
G
Closed-Loop Machining Strategy:
Correction of Under-Cut Regions Elimination of Over-Cut Regions
Common fitting methods are not suitable for closed-loop machining & inspection.
9
Required Properties for the Fitting Function
Fitting to the tolerance zone and not to the nominal geometryFitting to eliminate the over-cut situation Fitting to minimize the under-cut by minimizing the residual deviations
p/1n
i
p
ip en
1C
0,Δif
0,Δif,Δ
0,Δ0,Δif0
e
i
ii
ii
i
l
uu
lu
Gp
GpGp
GpGp
Residual Deviation Function
Minimization Objective
Fitting Function
G
G'pi'
pi ″
Δ(p i′,
G l″)
Δ G′(
G″)
G u
G l
G″Gu″
Gl″
Δ(pi′,Gu″)
XZ
Y
G
G'
pi'
XZ
Y
G u
G l
Gu *
Gl
*
G *
Λ G′(Gu
* )
pi″
Maximum Conformance to Tolerance (MCT) function (p→∞)
ii
emaxC
10
Objectives in Closed-Loop Inspection and Machining
Fitting criteria for Closed-Loop Machining & Inspection:
Inspection Based on Machining:
Machining Based on Inspection:
To develop a fitting methodology to construct the substitute geometry that minimizes the required compensation operations but maximizes the compensation capability of the geometric deviations.
To develop a method to determine number and location of the measured points based on the characteristics and properties of the actual machined surface, to reduce uncertainty of the process.
To develop a compensation procedure based on the inspection results. The procedure should be capable to interpolate the compensation requirements between the measured points for the entire machined surface.
11
Barrier ensures that a feasible solution never becomes infeasible. However, this objective function can by highly non-linear with discontinuities.
In practice, the optimization may have an infeasible initial condition and stuck there.It is likely to be stuck in a feasible pocket with a local minimum.
Modification of MCT Function
Two drawbacks of this objective function are:
Adding a penalty condition instead of the barrier condition to avoid straying too far from the feasible solutions. Utilizing a method for iterative data capturing to escape from the local
minima by increasing the energy level.
Solutions:
12
General Form of Penalty Function (Juliff, 1993; Patton et al., 1995; Back et al., 1997)
Penalty Function
xx MCP P
Distance metric
function
Penalty factor
satisfiedisintconstraif0
violatedisintconstraif1P
0
0
000
ll
uu
lu
GpGp
GpGp
GpGp
,Δif,ΔC
,Δif,Δ
,Δ,Δif
e
ii
ii
ii
i
For any point with the over cut condition the Δ(pi',Gl″) is a negative value that monotonically decreases when the point moves further from the tolerance zone. Therefore it can be a good choice for the distance metric function.
Modified Residual Deviation Function
Gradi
ent
Transformation of Substitute Geometry
Pena
lized
zone
Res
idu
al D
evia
tion
Fu
nct
ion
MCT Function
i
iemaxminCminObj
ζζ
13
Adaptive Penalty Function
Error Model-Local two directional sinusoid waves
14
Distribution of Geometric Deviations
Error Model-Local two directional sinusoid waves
MDZ MCTHalf- Normal Distribution
xx2)x(f
x2
x
dttx,e2
1x
2
Shape factor
α=0 : standard normal densityα→∞: half-normal density
1,01 2
15
Penalty factor, C, controls the velocity of transition from the state of the standard normal to the target state of extremely skewed
Violation of the Feasible Solution Area
Ct
Penalty Factor
Very Sever Penalty Function Convergence problem
C
,Δe
ii
max *uGp
Violation of the Feasible Solution
Large
Small
selecting a relatively small penalty factor (C=10).compensate the violation of the feasible solution by adoption of lower
tolerance limit
Solutions:
ett ll
16
Distribution of the Geometric Deviations (DGD)
Problem 2: is the region with maximum deviation sampled?
p'
Problem 1: how much is the deviation of an unmeasured point?
Approach: Search-Guided Sampling (Adoptive)Assumption: Distribution of deviations on the manufactured surface has a continues Probability Density Function .
Approach: Using Surface’s Geometric CharacteristicsAssumption: Gradient of the deviations is a direct function of the proximity of the Surface points with a high confidence level.
17
Example: Effect of Systematic Machining Errors
Kinematic Modeling of Generic Orthogonal Machine Tools Using Homogeneous Transformation Matrices
Homogeneous Transformation of X-AxisIMxAxHx
misalignment homogeneous
matrix
Motion homogeneous
matrix
Identity matrix
Homogeneous Transformation of Workpiece HzHy,Hx,Hw 1f
Homogeneous Transformation of Tool HzHy,Hx,Ht 2f Actual Machined Point
hwHwtHtp offsetoffset
Zero offset Homogenous vector
18
NURBS Presentation of Machined Surface
Quasistatic Linear Operator
Jacobian Matrix of the Actual Machining Point
1000
3GY
3
GY
3
GX
3
2GY
2
GY
2
GX
2
1GY
1
GY
1
GX
1
pppp
pppp
pppp
p
pJ
ThphIJΩ
Explicit Form of Machined Geometry
GΩG
NURBS Presentation of the Machined Surface:
1vu,0vu,vu,m
0i
n
0jji,ji,
QΩRG
NURBS piecewise rational basis
functions
Matrix of the surface control points
19
Behavior of Systematic ErrorsA Typical Vertical Machining Center
(Calibration using laser interferometer, electronic levels, optical squares)
00000.100000.000000.000000.0
0.02549-1.000000.0000200000.0
0.11600-0.000001.0000000000.0
0.05100-0.000000.00005-00000.1
XYWZTΩ
A Typical Horizontal Machining Center (Calibration using laser interferometer, electronic
levels, optical squares)
00000.100000.000000.000000.0
0.025001.000000.0000000000.0
0.024000.00006-1.0000000000.0
0.010000.000040.0000000000.1
XWYZTΩ
Nominal Geometry
20
Geometric Deviations Resulting from Systematic ErrorsGeometric Deviations
Hor
izon
tal M
achi
ning
Cen
ter
Systematic Error Vectors
Vert
ical
Mac
hini
ng C
ente
r
21
Search-Guided Sampling
Monitoring Continuity in Probability Density Function (PDF) of Geometric Deviations
eee d)(fRgPrRg
i
On-Line Estimation of PDFIn contrast, the only assumption in this works is the continuity of the true probability density function.
Probability Pr that a given deviation ei
will fall in a region Rg
Window Function 2
x2
e2
1x
Range of Windowing
hh
h
floori
i
)2min
(e
e1
h
h
h
ceili
i
)2max
(e
em
Parzen Windows Method (Parzen, 1962 )
n
1i
in hh
1
n
1f
eee
Window width
22
Iterative Search Hessian Function
Positive Maximum Absolute Hessian
mjh
)(f)(f2)(f
1m,...,2jh
)(f)(f)(f2)(f
1jh
)(f)(f2)(f
2
1mmm
2
1j1jjj
2
211
eee
eeee
eee
Negative Maximum Absolute Hessian
23
Fitting Uncertainty Using the Search Method Error Models (magnification: 100×):
1-Quasistatic errors of a vertical machine tool
2-One directional sinusoid wave
3-Two directional sinusoid waves 4-Local two directional sinusoid waves
24
Stratified Sampling
Error Model-Local two directional sinusoid waves
144 Stratified Points64 Stratified Points64 Random Points
25
Result of Search-Guided Sampling
Error Model-Local two directional sinusoid waves
26
Estimation of Uncertainty-Results
100 Times MiniMax Inspection Using Five Different Data Capturing Method (2000 Experiments)
27
Plug-In Uncertainty – Bootstrap Estimation
Plug-in uncertainty comes from the fact that it is always unknown how much of the captured dataset is a good representation of the real distribution function
The plug-in uncertainty is very much related to the probability of capturing critical points. A Bootstrap method is used to evaluate this probability.
Estimation of Maximum Geometric Deviation
P=(P1, P2, P3,…,Pn) and Θ =θ(P)Empirical probability density function of f: =1/n on each of the observed values
*n
*3
*2
*1
* P,...,P,P,PPf~
B,...,3,2,1kP~ k**
k
Bootstrap estimate of the standard error
21
B
1k
2*av
*k
*~
~~1B
1sd
B
~~
B
1k
*k
*av
28
Estimation of Plug-In Uncertainty-Bootstrap Results
100 Bootstrap Replications of Inspection of Five Different Data Capturing Method (2000 Experiments)
29
Distribution of the Geometric Deviations (DGD)
Pragmatic Space )v,u(GG
X
Z
Y
Cartesian Workspace Space
p'
Parametric Spaceu
e
v
Mapping
u*
v*
es
30
Interpolation of Geometric Deviations
Recall:
the variations of non-rigid transformation vectors of the machined point has a direct relationship with the distance of the nominal points.
A Proximity Problem
SxxsxsxsV jii ,ij:
Voronoi Diagram
Delaunay Triangle (O’Rourke, 1998; Okabe at el., 2000)
Variation of Non-Rigid Body Transformations
pζT-Ω
pζT-ΩppζT-Ωε*
**n
31
Interpolation Procedure
Position in the Parametric Space:
A Location between Sites any location on the uv parametric plane belongs to an individual Delaunay triangle
u
e
vek
sjsi
sk
oi oj
ok
ejei
er
r
0
-eevvuu
-eevvuu
-eevvuu
ik
ij
i
ikik
ijij
ii
ssss
ssss
ss
ir eC
BAe
uuvvvvuuC
vv-ee-eevvuuB
uu-ee-eeuuvvA
ijik
ijik
ikijikij
ikiji
ikiji
ssssssss
sssssr
sssssr
32
Case Study: DGD of a NURBS surface
Stamping Die of front door of a vehicle with the general dimensions of 1150mm×1080mm×35mm
(Forth order uniform, non-periodic NURBS surface with 16 control points)
Tolerance Specification
33
Simulation of Machining (Vertical Machine Tool)
PDF of Residual DeviationsInspection (Search procedure captures in 163 data points)
Step 1
Step 2
Mean of geometric deviation (mm)
Standard deviation of
geometricdeviation
Maximum geometric deviation Minimum geometric deviation
Deviation (mm)Parameters in the substitute surface
[u v]
Deviation (mm)
Parameters in the substitute surface
[u v]
0.031126 0.031716 0.083133 [0.0201 0.0211] -0.040000 [0.0229 0.9629]
34
Interpolation of Deviations
Development of DRD
Step 3
35
Inspection
Machining
Application: Closed-Loop of Machining and Inspection
Brown & Sharpe CMM Renishaw PH9 Probe HeadHorizontal Spindle Rotary TableReported displacement accuracies: ±0.004mmRoom Temperature: 20°C
Whali CNC Machine Tool Five AxesHorizontal Spindle Rotary TableReported Positional Accuracy: ±0.015mmRoom Temperature: 20°CMaterial:Aluminum 6061 Coolant: OilTool: 12mm Ball-noseDepth of Cut: 0.25mmft= 0.015(mm/min) and V=110 (m/min)
)rpm(3000)mm(12
min)/m(1101000)rpm(N s
min)/mm(90
)rpm(30002)tooth/mm(015.0min)/mm(fm
36
Setup & Machining Phase
Alignment Setup Roughing
Reference Patch Finishing CC-Lines Finished Part
37
Inspection Phase
Physical Measurement Cylindrical Fit DGD
Virtual Data Capturing Final Inspection
38
Experiment #1:Flexible Knot LocationsBefore
After
%8.73)mm(0332.0
)mm(0087.0)mm(0332.0
RangeErrorTotalofductionRe
39
Experiment #2- First Degree NURB Surface
Control Net Second Degree NURBS Surface
First Degree NURBS SurfaceCC-Lines
Upper Tolerance=0.006 mmLower Tolerance=-0.007 mm
40
Setup & Machining Phase
Alignment Setup Roughing
Reference Patch Finishing CC-Lines Finished Part
41
Inspection Phase
Physical Measurement Cylindrical Fit DGD
Virtual Data Capturing Final Inspection
42
Experiment #ResultsBefore
After%2.64
)mm(0358.0
)mm(0128.0)mm(0358.0
RangeErrorTotalofductionRe
43
Conclusions
A new fitting methodology for coordinate methodology is developed that maximizes conformance of the measured points to a given tolerance zone.
Generating detailed information of the deviation zone on the measured surfaces should be based on the needs of the upstream processes such as compensating machining, finishing or reverse engineering.
A methodology is developed to estimate distribution of the geometric deviations on a surface that is measured using discrete point sampling.
Developed search method is an alternative approach in coordinate data capturing which significantly reduces plug-in uncertainty.
Integration of computational tasks in coordinate metrology significantly reduces measurement uncertainties.