Screw Theory Cedric Fischer and Michael Mattmann

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

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)

2 Screw Theory: Geometrical Description

l M

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 . Amount of displacement . Net rotation and/or translation

3 Screw Theory: Mathematical Description

é a b - a b ù é 0 -a a ù ê 2 3 3 2 ú ê 3 2 ú Skew-symmetric matrix a ´ b = abˆ = ê a3b1 - a1b3 ú = ê a3 0 -a1 úb ê ú ê ú a b - a b -a a 0 ëê 1 2 2 1 ûú ëê 2 1 ûú

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

General description

with

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

4 Screw Theory: Mathematical Description

é a b - a b ù é 0 -a a ù ê 2 3 3 2 ú ê 3 2 ú Skew-symmetric matrix a ´ b = abˆ = ê a3b1 - a1b3 ú = ê a3 0 -a1 úb ê ú ê ú a b - a b -a a 0 ëê 1 2 2 1 ûú ëê 2 1 ûú

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

é -w ´ q ù é wˆ -w ´ q ù x = ê ú xˆ = ê ú ê w ú ê 0 0 ú Revolute joint ë û ë û

é v ù é 0 v ù x = ê ú xˆ = ê ú ë 0 û ë 0 0 û Prismatic joint

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

5 Rodrigues’ Formula

. All rotation matrices can be written as a matrix exponential of a skew-symmetric matrix!

wqˆ 2 . Rodrigues’ Formula: e = I +wˆ sinq +wˆ (1- cosq) Proof on extra slide!

. Homogeneous é wqˆ wqˆ ù ˆ e I - e w ´ v + hqw Transformation exq = ê ( )( ) ú ê ú . 4x4 matrix ë 0 1 û

Revolute joint Prismatic joint

é wqˆ wqˆ ù é ù xqˆ e I - e (w ´ v ) xqˆ I qv e = ê ( ) ú e = ê ú ê ú 0 1 ë 0 1 û ë û

6 Screw Theory: Mathematical Description

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

g(0): - all joint angles defined as being zero - Describes transformation from the base frame to tool frame

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).

7 Assignment 3

Two systems: System 1:

. a) g(0) . Rotation and translation between CS

. b) Pitch, axis and magnitude System 2: . 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 )

9 Kinematics Toolbox

• Figure, chart, video… Point Rotation axis skew Skew-symm. matrix skewlog q w w^ skewcoords Rotation matrix skewexp R createtwist

Magnitude twistmagnitude M Twist Coord. twist Twist Axis twistaxis [6x1] [4x4] twistlog l x twistcoords x^ Pitch h twistpitch

Hom. Transf. [4x4] twistexp g