www.geometrie.tuwien.ac.at ima tutorial, 20. 4. 2001 instantaneous motions - applications to...

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


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


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


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)


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


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:


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.


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.


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.


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


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


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.


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).


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


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


Example: tubular surface

• Data points and estimated surface normal vectors

• Reconstruction of spine curve with locally approximating tori

• Reconstructed tubular surface


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)


Example: developable surface


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.


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


Industrial inspection

CAD-modelof a technicalobject



Image of theexisting object



Cloud of data pointsfrom the surface ofthe existing object



Point cloud withtransparentsurface



Input to thematchingalgorithm



Output of thematchingalgorithm



Color codingof deviations


Instantaneous Motions -

Applications to Problems in

Reverse Engineering

and 3D Inspection

H. PottmannInstitut für Geometrie, TU Wien

www.geometrie.tuwien.ac.at ima tutorial, 20. 4. 2001 instantaneous motions - applications to...

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