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

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=

0 a3 a2a3 0 a1a2 a1 0

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b

a = aT

Screw Theory: Mathematical Description

Twist Coordinates 6x1 vector

5

= q +h( )

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= q +h( )0 0

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= v0!

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= 0 v0 0!

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Twist 4x4 matrix

Homogeneous Transformation 4x4 matrix

e = e I e( ) v( )+ h0 1

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((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!

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e = I + sin + 2 1 cos( )

e = e I e( ) v( )+ h0 1

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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( )