Rigid Body Velocity Cedric Fischer and Michael Mattmann Institute - - PowerPoint PPT Presentation

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

Rigid Body Velocity Cedric Fischer and Michael Mattmann

Angular velocity

Rotation only Rotation matrix has special properties!

Proof p.51

Instantaneous spatial angular velocity Instantaneous body angular velocity

We define:

From spatial to body frame

More detailed explanation p.51 Skew-symmetric

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Angular velocity - Example

• Calculate instantaneous angular velocity:

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• Derive rotation matrix:
• Calculate velocity of rigid body:
• r

Spatial and body velocity – General case

• Spatial velocity
• Body velocity

(in twist form) (in twist form) (in twist coordinates)

• Spatial velocity
• Body velocity

(in twist coordinates)

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As previously ̇ "#$(&) is not particularly useful But ̇ "#$"#$

() and "#$ () ̇

"#$have some special significance

• Rotation and translation

More detailed explanation p.54

Spatial and body velocity – General case

(in twist coordinates)

Spatial velocity Body velocity

(in twist coordinates)

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• Transformation from body to spatial velocity:

More detailed explanation p.54

) 6 6 ( )

ˆ ,

x g

R R p R Ad ! " # $ % & =

Adjoint transformation

• The adjoint transformation is invertible

Rigid Body Velocity

Spatial velocity § Point of the body § Expressed in spatial coordinates § Velocity of that point written in spatial coordinates

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Body velocity § Point of the body § Expressed in body coordinates § Velocity of that point with respect to the spatial frame written in body coordinates vabs is the velocity of a point (possibly

imaginary) attached to the body frame and passing through the origin of the spatial frame, written in spatial coordinates

wabs is the instantaneous angular velocity

• f the body as viewed in the spatial frame

vabb is the velocity of the origin of the

body frame (relative to the spatial frame) written in the body coordinates

wabb is the angular velocity of the body

frame, written in the body coordinates

Spatial and body velocity - Example

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Assignment 5

a) (by inspection) b) c) matrix exponentials d) e) Derivative of gst(q) f) Inverse of gst(q) g) Body velocity twist h) Body velocity twist coordinates i-k) Jacobians will be treated later in the lecture

§ Exercise part i) to k) not relevant for this week

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g0t(0)

ˆ ξi ξi

gbt(θ) = gb0g0t(θ) = gb0e

ˆ ξ1θ1e ˆ ξ2θ2g0t(0)