Institute of Robotics and Intelligent Systems Department of Mechanical and Process Engineering (DMAVT) ETH Zurich
Screw Theory Sandro Erni Ayoung Hong 14.10.2013
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.
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
3
h l
M
Skew-Symmetric Matrix
4
a = a( ) ^
ab = ab =a2b3 a3b2a3b1 a1b3a1b2 a2b1
#
$
%%%%
&
'
((((
=
0 a3 a2a3 0 a1a2 a1 0
#
$
%%%%
&
'
((((
b
a = aT
Screw Theory: Mathematical Description
Twist Coordinates 6x1 vector
5
= 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
!
"#
$
%&
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
#
$
%%
&
'
((
Figure, chart, video
Kinematics Toolbox
7
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
Figure, chart, video
Assignment 3
a) (by inspection)
b) Screw parameters
c)
d)
e) plots
8
g12 0( )
= ?, = ?
g12 ( ) = e g12 0( )
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