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

Rigid Body Velocity Cedric Fischer and Michael Mattmann Institute of Robotics and Intelligent Systems Department of Mechanical and Process Engineering (DMAVT) ETH Zurich

Angular velocity Rotation only Rotation matrix has special properties! Proof p.51 Skew-symmetric We define: Instantaneous spatial angular velocity Instantaneous body angular velocity From spatial to body frame More detailed explanation p.51 2

Angular velocity - Example • Derive rotation matrix: • Calculate instantaneous angular velocity: or • Calculate velocity of rigid body: 3

̇ ̇ Spatial and body velocity – General case More detailed explanation p.54 • Rotation and translation " #$ (&) is not particularly useful As previously () and " #$ () ̇ " #$ " #$ " #$ have some special significance But • Spatial velocity (in twist form) • Body velocity (in twist form) • Body velocity • Spatial velocity (in twist coordinates) (in twist coordinates) 4

Spatial and body velocity – General case Body velocity Spatial velocity (in twist coordinates) (in twist coordinates) • Transformation from body to spatial velocity: ) ( 6 x 6 ) & ˆ # R p R = , Ad Adjoint transformation $ ! g 0 R % " • The adjoint transformation is invertible More detailed explanation p.54 5

Rigid Body Velocity Spatial velocity Body velocity § Point of the body § Point of the body § Expressed in spatial coordinates § Expressed in body coordinates § Velocity of that point written in § Velocity of that point with respect to spatial coordinates the spatial frame written in body coordinates v abs is the velocity of a point (possibly v abb is the velocity of the origin of the imaginary) attached to the body frame and passing through the origin of the spatial body frame (relative to the spatial frame) frame, written in spatial coordinates written in the body coordinates w abs is the instantaneous angular velocity w abb is the angular velocity of the body of the body as viewed in the spatial frame frame, written in the body coordinates 6

Spatial and body velocity - Example 7

Assignment 5 g 0 t (0) a) (by inspection) ˆ b) ξ i ξ i c) matrix exponentials d) g bt ( θ ) = g b 0 g 0 t ( θ ) ˆ ˆ ξ 1 θ 1 e ξ 2 θ 2 g 0 t (0) = g b 0 e e) Derivative of g st ( q ) f) Inverse of g st ( 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 8