Screw Theory Cedric Fischer and Michael Mattmann Institute of - - PowerPoint PPT Presentation

Institute of Robotics and Intelligent Systems Department of Mechanical and Process Engineering (DMAVT) ETH Zurich

Screw Theory Cedric Fischer and Michael Mattmann

Screw Theory

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Every rigid body motion can be expressed by a rotation about an axis combined with a translation parallel to that axis. (= screw motion) Method is based on rigid body motion instead of location!  Time dependent now! Different ways to describe it:

• screws

(geometrical description: screw parameter)

• twists

(mathematical description: abstract)

• product of exponentials (mathematical description: homogeneous)

Screw Theory: Geometrical Description

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h l M

• Pitch h
• Ratio of translational motion to rotational

motion

• Axis l
• Axis of rotation, line through a point
• Direction of translation
• Magnitude M
• Amount of displacement
• Net rotation and/or translation

Screw parameters

Screw Theory: Mathematical Description

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Derivation: page 39 – A Mathematical Introduction to Robotic Manipulation 1994

• Twist Coordinates
• 6x1 vector
• Twist
• 4x4 matrix

with

General description

a´ b = ˆ ab = a2b3 - a3b2 a3b

1 - a1b3

a1b2 - a2b

1

é ë ê ê ê ê ù û ú ú ú ú =

• a3

a2 a3

• a1
• a2

a1 é ë ê ê ê ê ù û ú ú ú ú b

Skew-symmetric matrix

Screw Theory: Mathematical Description

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Derivation: page 39 – A Mathematical Introduction to Robotic Manipulation 1994

x =

• w ´ q

w é ë ê ê ù û ú ú x = v é ë ê ù û ú ˆ x = ˆ w

• w ´ q

é ë ê ê ù û ú ú ˆ x = v é ë ê ù û ú

• Twist Coordinates
• 6x1 vector
• Twist
• 4x4 matrix

Revolute joint Prismatic joint

a´ b = ˆ ab = a2b3 - a3b2 a3b

1 - a1b3

a1b2 - a2b

1

é ë ê ê ê ê ù û ú ú ú ú =

• a3

a2 a3

• a1
• a2

a1 é ë ê ê ê ê ù û ú ú ú ú b

Skew-symmetric matrix

Rodrigues’ Formula

• All rotation matrices can be written as a matrix exponential
• f a skew-symmetric matrix!
• Rodrigues’ Formula:

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• Homogeneous

Transformation

• 4x4 matrix

e

ˆ wq = I + ˆ

wsinq + ˆ w

2 1-cosq

( )

e

ˆ xq =

e

ˆ wq

I - e

ˆ wq

( ) w ´v

( )+ hqw

1 é ë ê ê ù û ú ú

Revolute joint Prismatic joint

Proof on extra slide!

e

ˆ xq =

e

ˆ wq

I - e

ˆ wq

( ) w ´v

( )

1 é ë ê ê ù û ú ú e

ˆ xq =

I qv 1 é ë ê ù û ú

Screw Theory: Mathematical Description

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If a coordinate frame B is attached to a rigid body undergoing a screw motion, the instantaneous configuration of the coordinate frame B, relative to a fixed frame A, is given by This transformation can be interpreted as follows:

• multiplication by gab(0) maps the coordinates of a point relative to the B frame into A’s

coordinates

• the exponential map transforms the point to its final location (still in A coordinates).

g(0):

• all joint angles defined as being zero
• Describes transformation from the base frame to tool frame

Assignment 3

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Two systems:

• a) g(0)
• Rotation and translation between CS
• b) Pitch, axis and magnitude
• c) Twist and twist coordinates
• d) Find total homogeneous transformation g12
• e) Matlab: use twist, twistexp
• plot trajectories of P1,P2,P3 ( g(θ)*Pi )

System 1: System 2:

• Figure, chart, video…

Kinematics Toolbox

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Magnitude M Axis l Pitch h Twist Coord. [6x1] x Twist [4x4] x^

• Hom. Transf.

[4x4] g Rotation matrix R Skew-symm. matrix w^ Rotation axis w Point q skew skewcoords skewexp skewlog createtwist twist twistcoords twistexp twistlog twistmagnitude twistaxis twistpitch