Www.geometrie.tuwien.ac.at IMA Tutorial, 20. 4. 2001 Instantaneous Motions - Applications to...
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Transcript of Www.geometrie.tuwien.ac.at IMA Tutorial, 20. 4. 2001 Instantaneous Motions - Applications to...
![Page 1: Www.geometrie.tuwien.ac.at IMA Tutorial, 20. 4. 2001 Instantaneous Motions - Applications to Problems in Reverse Engineering and 3D Inspection H. Pottmann.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d435503460f94a1f364/html5/thumbnails/1.jpg)
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Instantaneous Motions -
Applications to Problems in
Reverse Engineering
and 3D Inspection
H. PottmannInstitut für Geometrie, TU Wien
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
One parameter motions
moving system
fixed system 0
( )t0x
( )tu
x
( ) = ( ) + ( )t t t0 Ax u x
with .TA A E
Trajectory of a point x:
Velocity vector field of x:
( ) = ( ) + ( )t t t0 Ax u x
Representation in the same system, e.g. moving system:
+( ) T T T0
Sc
x u Av A Ax A x
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
One parameter motions
1
2
3
with : there is
c
c
c
S xc c x
With we have T T TA A E A A A A 0
3 2
3 1
2 1
0
0
0
c c
c c
c c
S TS S 0 skew symmetricS
( ) cv cx x
velocity vect ( )or of :v x x
Linear vector field!
+ ( ) T T
Sc
Av uA Ax x
![Page 4: Www.geometrie.tuwien.ac.at IMA Tutorial, 20. 4. 2001 Instantaneous Motions - Applications to Problems in Reverse Engineering and 3D Inspection H. Pottmann.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d435503460f94a1f364/html5/thumbnails/4.jpg)
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Discussion of the geometry of the velocity vector field
Special case 1: c o
( )v x c instantaneous translation
c oSpecial case 2:
( )v x c x
: there is ( ) .A x v xc o
For all points of the line
instantaneous rotation about axis A
A
o
c
x
v(x)
![Page 5: Www.geometrie.tuwien.ac.at IMA Tutorial, 20. 4. 2001 Instantaneous Motions - Applications to Problems in Reverse Engineering and 3D Inspection H. Pottmann.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d435503460f94a1f364/html5/thumbnails/5.jpg)
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Discussion of the geometryof the velocity vector field
General case:Composition of an instantaneous rotationand an instantaneous translation… instantaneous screw motion
A
p
c
x
v(x)
c c p
c
Rotation axis A has direction vector c and passes throughpoints p with .
( … moment vector of the axis A)
Special case 3 (geometrically equivalent to case 2):
.c c o instantaneous rotation ( ) mitc cv x x
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Discussion of the geometryof the velocity vector field
• translation with constant velocity
• uniform rotation about an axis
• uniform screw motion
One parameter motions with a constant
(time independent) velocity vector field:
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Uniform screw motion
Composition of a rotation aboutan axis A and a proportionaltranslation parallel to A.
0 0
0 0
0
( ) cos( ) sin( )
( ) sin( ) cos( )
( )
x t x t y t
y t x t y t
z t z ptp … pitch
A curve generates a helical surface.
A
tpt
0 :p pure rotation, generates rotational surface.
' ' :p pure translation, generates cylindrical surface.
![Page 8: Www.geometrie.tuwien.ac.at IMA Tutorial, 20. 4. 2001 Instantaneous Motions - Applications to Problems in Reverse Engineering and 3D Inspection H. Pottmann.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d435503460f94a1f364/html5/thumbnails/8.jpg)
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Reverse Engineering
Problem description:
Usually, in CAD/CAM systems are used todesign and produce technical objects.
Conversely, reverse engineering deals with thereconstruction of a computer model from anexisting object.
Different 3D scanning tools givea large number of data points(with measurement errors) on the surface of the technical object.
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Reverse Engineering
Goal:
Automatic detection of simple surfaces in thereconstruction process the CAD model. This is important for a precise CAD representation!
Simple classes of surfaces include:
• planes,• cylinders & cones of revolution,• spheres,• tori,• general rotational surfaces,• helical surfaces.
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Reconstruction ofhelical surfaces
Given: data points di
1. step: estimation of surface normals ni
di
ni
2. step: calculate an appropriate motion with the velocity vector field ( .) c cv x x
with ) ( .i i i i ii in ncv dnc n nd
Minimization of 2cos i is nonlinear in ( , ) : .c c C
v(di)i
instead: minimize
2
1
( )N
ii i
Tnc Cn Nc C
21 ,T DC C Dcwith side condition
1
1
1
0
0
0
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Generalized eigenvalues
Lagrange: ( ) 0, 1.TD C CN DC
det( ) 0, cubic in .DN
For a gen. eigenvalue and gen. eigenvector C we have
( ) .F T TC DC C CNC
Therefore, the solution of F(C) = min is the generalized eigenvector C to the smallest generalized eigenvalue.
From the solution vector 6( , )C c c the axis ( , )A a aand the pitch p of the ‘best fitting’ screw motion is calculated:
,ac
c
,
pa
c c
c
2
.pc c
c
![Page 12: Www.geometrie.tuwien.ac.at IMA Tutorial, 20. 4. 2001 Instantaneous Motions - Applications to Problems in Reverse Engineering and 3D Inspection H. Pottmann.](https://reader035.fdocuments.net/reader035/viewer/2022062407/56649d435503460f94a1f364/html5/thumbnails/12.jpg)
IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Reconstruction process
• Computation of the one parameter motion whose velocity vector field fits the estimated normals best.
• Estimation of surface normals in the data points.
• Reconstruction of the profile curve by moving the data into an appropriate plane.
• Reconstruction of the surface by applying the one parameter motion to the profile curve.
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Curve reconstruction fromunorganized data points
• Spline curve through selected ordered data points.
• Compute minimal spanning tree of the complete graph by Delauny triangulation.
• Apply ‘moving least squares’ algorithm (see next slide).
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Moving least squares
• Choose a point p and its neighborhood (distance <= H) in the ‘minimum spanning tree’.
• Fit parabola (weighted point data).
• Project p onto parabola.
p
x
y
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Example: rotational surface
• Data points and estimated surface normal vectors
• Data points and computed axis of rotation
• Data points, rotated into a plane, and approximating curve
• Reconstructed surface of revolution
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Example: tubular surface
• Data points and estimated surface normal vectors
• Reconstruction of spine curve with locally approximating tori
• Reconstructed tubular surface
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Example: developable surface
• Developable surface
• Disturbed data points of the original surface
• Computation of locally best- fitting cones of revolution by region-growing
• Reconstruction of a G1-surface composed of segments of cones of revolution (right circular cones)
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Example: developable surface
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
3D Inspection
Problem description:
Determine a one parameter motion that matches a point cloudto a surface (or another pointcloud) as good as possible.
Motivation:
• Inspection of technical objects / quality checks,
• 3D matching, e.g. joining different 3D laser scanner images to a 3D object.
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
3D inspection
xi
ni
di
v(xi)
( )d i in vi
xi ... data points
velocity vector fieldv(xi) of a one parameter motion:
( .) c cv x x
minimize 2( ( ))d iin c xci
determine distance di to surfaceand surface normal ni in thecorresponding surface point.
quadratic in , !c c
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
CAD-modelof a technicalobject
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Image of theexisting object
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Cloud of data pointsfrom the surface ofthe existing object
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Point cloud withtransparentsurface
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Input to thematchingalgorithm
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Output of thematchingalgorithm
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Industrial inspection
Color codingof deviations
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IMA Tutorial, 20. 4. 2001 www.geometrie.tuwien.ac.at
Instantaneous Motions -
Applications to Problems in
Reverse Engineering
and 3D Inspection
H. PottmannInstitut für Geometrie, TU Wien