ss 2007
DESCRIPTION
SS 2007. Geometrie in der Technik. H. Pottmann TU Wien. Kinematical Geometry. Overview. Kinematical Geometry Planar kinematics Quaternions Velocity field of a rigid body motion Helical motions Kinematic spaces. Planar Kinematical Geometry. Complex numbers and planar kinematics. - PowerPoint PPT PresentationTRANSCRIPT
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GEOMETRIE
Geometrie in der Technik
H. PottmannTU Wien
SS 2007
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Kinematical Geometry
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Overview
Kinematical Geometry Planar kinematics Quaternions Velocity field of a
rigid body motion Helical motions Kinematic spaces
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Planar Kinematical Geometry
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Complex numbers and planar kinematics
In the plane, a congruence transformation (x0,y0)(x,y), also called (discrete) motion, is given by
x=a1+x0 cos-y0 sin, y=a2+x0 sin+y0 cos
Collecting coordinates in complex numbers z=x+iy, we get with
a=a1+ia2, ei= cos +i sin
z=a+z0 ei, … rotational angle
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Planar kinematics
a sequence of congruence transformations, depending continuously on a real parameter t, form a one-parameter motion
z(t)=a(t)+z0 ei(t)
For a point z0 in the moving system 0, z(t) describes its path (trajectory) in the fixed system
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Example: trochoidal motion
Composition of two uniform rotations with angular velocities and , measured against the fixed system.
z(t) = aeit + z0eit
?
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trochoids
nephroid ellipse : = 1:-1
cardioid: = 1:2
cycloid (composed of rotation and translation)a=b
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velocity field
velocity vectors are found by first derivative,
z‘(t)=a‘(t)+z0i‘(t)ei(t)
at a fixed time instant t=t0 we set ‘(t0)=: (angular velocity) and obtain a linear relation between points z and their velocity z‘:
z‘=a‘+i(z-a)
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pole
For =0 we have an instantaneous translation
for ≠0 we get exactly one point p with vanishing velocity,
p=a+(i/)a‘ …. pole (expressed in the fixed system)
With p, the velocity field is z‘=i(z-p), i.e., the velocity field of a rotation about
p… instantaneous rotation
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polhodes
The locus of poles in the fixed (moving) system is called fixed (moving) polhode, respectively.
It can be shown that during the motion, the moving polhode rolls on the fixed polhode; the point of tangency being the instantaneous pole.
z‘
z
p
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Trochoidal motion
polhodes are circles Application of this motion in mechanical
engineering (e.g. construction of gears)
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Spatial Kinematics
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Quaternion representation ofrotations
Discrete rotation about the origin, in matrix notation
x=R.x0, R… orthogonal matrix Orthogonality constraint on R, R.RT=I, is
nonlinear. A simplified representation, which is an
extension of the use of complex numbers in planar kinematics, uses quaternions. This leads also to a parameterization of the set of orthogonal matrices.
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quaternions
A quaternion is a generalized complex number of the form
q=q0+iq1+jq2+kq3
The imaginary units i,j,k satisfy i2=j2=k2=-1 ij=-ji=k and cyclic permutations H=R4 with addition and
multiplication is a skew field
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quaternions
conjugate quaternion q*=q0-iq1-jq2-kq3
(ab)*=b*a* norm N(q)=q0
2+q12+q2
2+q32=qq*
inverse q-1=q*/N(q)
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Embedding R3 into H
We embed R3 into H as follows x=(x1,x2,x3) R3
x=ix1+jx2+kx3 Now take a fixed quaternion a of norm 1
and study the mapping x‘=a*xa We see: N(x‘) = x‘(x‘)*= a*xa a*x*a = = N(x)a*a = N(x)
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Mapping x‘=a*xa
This mapping is linear in x preserves the norm can be shown to have positive
determinant Therefore: x‘=a*xa represents a
rotation about the origin quaternion representation of
rotations
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Rotation with quaternions
From the quaternion a of norm 1, axis d (embedded in H) and angle of rotation follow by
a=cos(/2)-d sin(/2) The representation x‘=a*xa of a rotation
yields a parameterization of orthogonal matrices R with help of the parameters a0,a1,a2,a3 (see lecture notes)
They satisfy a02+a1
2+a22+a3
2=1, and thus we have a mapping between rotations and points a on the unit sphere S3R4
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applications
Examples for applications of the quaternion representation:
Design of motions (rotational part) via curve design in the 3-sphere S3 R4
Shoemake: Bezier-like curves in S3
Juettler and Wagner: rational curves in S3
Wallner (2004): nonlinear subdivision in S3
Explicit solution of the registration problem in R3 with known correspondences
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First order instantaneous kinematics
One-parameter motion in Euclidean 3-space (not just rotation about origin)
Velocity vector fieldis linear:
x0(t)
u(t)
x
v (x)
0
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derivation of velocity field
One-parameter motion x(t)=A(t).x0+a(t) velocity field x‘(t)=A‘(t).x0+a‘(t) express v(x)=x‘(t) in fixed frame
by using x0=AT.(x-a) v(x)=A‘.AT.x+a‘-A‘.AT.a=:C.x+c
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Derivation of velocity field
establish C as skew-symmetric by differentiation of the identity I=A.AT
0=A‘.AT+A.A‘T=C+CT
0 -c3 c2
C= c3 0 -c1
-c2 c1 0
with c=(c1,c2,c3): C.x=c x
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One-parameter motions with constant velocity vector field
1. translation
A
p
c
x
v(x)
2. uniform rotation
Rotation axis A has direction vector c and passes throughpoints p with .( … moment vector of the axis A)
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One-parameter motions with constant velocity vector field
General case: helical motion
Helical motion is the composition of • a rotation about an axis A and• a proportional translation parallel to A
A
tpt
p … pitch
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A spatial motion, composed of a einer uniform rotation abgout an axis a and a uniform translation parallel to a is called a uniform helical motion.
a ... Helical axis
Rotational angle and length of translation s proportional: a rotation with angle gedreht, so belongs to a translation of length s = p .
The constant quotient p = s / is called pitch.
h ...height
Continuous (uniform) helical motion
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Remark: discrete helical motion
Any two congruent positions of a rigid body can be mapped into each other by a discrete helical motion.
It is composed of a rotation about an axis and a translation parallel to this axis.
In special cases two positions are related by a pure rotation or a pure translation.
a
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Discrete and continuous case
In discrete case: Any two positions can be moved into each
other by a helical motion (or a special case of it)
In continuous case (consider two infinitesimally close positions) The velocity field of a one-parameter motion
at any time instant is that of a uniform helical motion (or a special case of it)
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Axis and pitchof a helical motion
From the vector the axis and the pitch p of the underlying helical motion are calculated by:
a … direction vector of axis A … moment vector of axis A
[independent of the choice of qsince qa = (q+a) a ]
6( , )C c c R( , )A a a
,ac
c
,
pa
c c
c
2
.pc c
c
a
A
aa
o q
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Euclidean motion group embedded in the affin group
A Euclidean displacement x= a0+ A.x0= a0+x1
0a1+x20a2+x3
0a3
is a special affine map; A has to be orthogonal
If A is an arbitrary matrix, we obtain an affine map.
Let us view an affine map as a point (a0,a1,a2,a3) in R12 (affine space)
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GEOMETRIEa kinematic space
Associate a point in R12 with an affine copy of the moving body (affine map)
Euclidean (rigid body) motions are mapped to points of six-dimensional manifold M6 in R12
Continuous motion is mapped to curve in M6
A12
R12
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Metric in R12
via feature points (1)
Moving body represented by feature points X: x1, x2, …
Squared distance d2(,) between two affine maps and := sum of squared distances of corresponding feature point positions
X
(xi)
(xi)
R3
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Metric in R12
via feature points (2)
Euclidean metric in R12 which only depends on barycenter covariance matrix
Replace X by 6 verticesf1, …, f6 of inertia ellipsoid
X
(fi)
(fi)
R3
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Properties and facts
Sufficient to choose some points on the moving body and define the metric with the sum of squared distances of their positions (don‘t need an integral); in fact, sufficient to take vertices of the inertia ellipsoid
In the defined metric, the orthogonal projection of a point onto M6 can be computed explicitly (4th degree problem; use quaternions; see lecture on registration)
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motion planning
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Curve approximation in robotics and animation
Interpolation or approximation of a set of positions by a smooth motion (Shoemake, Jüttler, Belta/Kumar,…) Equivalent to curve
interpolation/approximation in group of rigid body motion
Can use M6, S3,…
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Problem formulation
Given N positions (ti) of a moving body at time instances ti, compute a smooth rigid body motion (t) which interpolates or approximates the given positions
(t1)
(t2) (t3)
(t4)
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A simple solution (1)
the given positions correspond to points in M6
Interpolate them using a known curve design algorithm
results in affinely distorted copies of the moving body (called an affine motion)
R12 M6
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A simple solution (2)
R12 M6
c
c*
Perform orthogonal projection of c onto M6 (i.e., best approximate each affine position by a congruent copy of the moving body; see registration with known correspondences)
The resulting curve c* is the kinematic image of the designed motion
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Example: projection of a C2 spline
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Energy minimizing motions
The following examples have been computed with an algorithm for the computation of energy minimizing splines in manifolds
This algorithm has been applied to compute an energy minimizing curve on M6R12. Thus, we obtain an energy-minimizing motion in R3
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Cyclic motion minimizing cubic spline energy E2
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Cyclic motion minimizing tension spline energy Et
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Cyclic motion minimizing kinetic energy E1
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Motion smoothing
Curve smoothing on M6 yields motion smoothing