03_2013-10-14_ScrewTheory
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Transcript of 03_2013-10-14_ScrewTheory
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Institute of Robotics and Intelligent Systems Department of Mechanical and Process Engineering (DMAVT) ETH Zurich
Screw Theory Sandro Erni Ayoung Hong 14.10.2013
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Screw Theory
Geometrical description Screw Parameters
Mathematical (abstract) Twist
Mathematical (homogeneous) Matrix Exponentials
2
Every rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis.
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Screw Theory: Geometrical Description
Screw Parameters Pitch h
Ratio of translational motion to rotational motion
Axis l Axis of rotation, line through a point Direction of translation
Magnitude M Net rotation Net translation
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h l
M
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Skew-Symmetric Matrix
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a = a( ) ^
ab = ab =a2b3 a3b2a3b1 a1b3a1b2 a2b1
#
$
%%%%
&
'
((((
=
0 a3 a2a3 0 a1a2 a1 0
#
$
%%%%
&
'
((((
b
a = aT
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Screw Theory: Mathematical Description
Twist Coordinates 6x1 vector
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= q +h( )
#
$%%
&
'((
= q +h( )0 0
#
$%%
&
'((
= v0!
"#
$
%&
= 0 v0 0!
"#
$
%&
Twist 4x4 matrix
Homogeneous Transformation 4x4 matrix
e = e I e( ) v( )+ h0 1
#
$
%%
&
'
((e = I v0 1
!
"#
$
%&
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Rodrigues Formula
All rotation matrices can be written as a matrix exponential of a skew-symmetric matrix
Simple computation of the matrix exponential (where ||||=1)
Rodrigues Formula:
6
Homogeneous Transformation 4x4 matrix
e = I v0 1!
"#
$
%&
e = I + sin + 2 1 cos( )
e = e I e( ) v( )+ h0 1
#
$
%%
&
'
((
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Figure, chart, video
Kinematics Toolbox
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Magnitude M
Axis l
Pitch h
Twist Coord. [6x1]
Twist [4x4] ^
Hom. Transf. [4x4]
g
Rotation matrix R
Skew-symm. matrix ^
Rotation axis
Point q
skew
skewcoords
skewexp
skewlog
createtwist
twist
twistcoords
twistexp
twistlog
twistmagnitude
twistaxis
twistpitch
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Figure, chart, video
Assignment 3
a) (by inspection)
b) Screw parameters
c)
d)
e) plots
8
g12 0( )
= ?, = ?
g12 ( ) = e g12 0( )