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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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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