Rigid Body Motion. Greg Hager Simon Leonard

Transcription

1 Rigid ody Motion Greg Hager Simon Leonard

2 Overview Different spaces used in robotics and why we need to get from one space to the other Focus on Cartesian space Transformation between two Cartesian coordinate systems Position and orientation Velocities

3 Transformation etween Spaces What is the state of a robot? vector of joint positions (angles and/or distances) Typically, we care about the position/orientation of the tool with respect to an inertial coordinate system? Cartesian space? Spherical space? Cylindrical space? Choose the space that fits your application: Frequency domain in signal processing Cylindrical, spherical spaces in electromagnetism

4 Task Spaces

5 Transformation etween Spaces FORWRD KINEMTICS Cartesian Space Tool (T) wrt ase () ( R T, t T ) R T :Orientation of T wrt t T : Position of Twrt Joint Space Joint 1 = q 1 Joint 2 = q 2... Joint N = q N INVERSE KINEMTICS FORWRD KINEMTICS Cylindrical Space Tool (T) wrt ase () ( z T, r T, θ T ) z T :Height of T wrt r T : radius of T wrt θ T :ngle of T wrt

6 Notes bout Symbols Each coordinate system is labeled by a letter,, etc. The coordinates of a point p are always expressed with respect to a coordinate system, i.e. p, p, etc. The coordinates of a point p are expressed in a coordinate frame by p = E p where E is a transformation that maps coordinates of coordinate system to coordinate system. P E The coordinates of a point P are expressed in p = E p p = E E p p = E p p = p

7 2D Cartesian Transformation Position Transformation within Cartesian Space Position component (2D translation) What are the (x,y) coordinates of p in the coordinate system? y y t P x p = p + t x

8 2D Cartesian Transformation Orientation Rotation component What are the (x,y) coordinates of p in the coordinate system? y y P p = R p x x where R is a Special Orthogonal Matrix SO(2) R ( R ) T = R ( R ) = I

9 2D Cartesian Transformation Position + Orientation ffine transformation What are the (x,y) coordinates of p in the coordinate system? p = R p + t y y Homogeneous coordinates x x

10 3D Cartesian Transformation Position Transformation within Cartesian Space Position component (3D translation) What are the coordinates of p in the coordinate system? t P z z y p = p + t x y x

11 3D Cartesian Transformation Orientation Rotation component What are the coordinates of P in the coordinate system P p = R p where R is a Special Orthogonal Matrix SO(3) R ( R ) T = R ( R ) = I

12 3D Cartesian Transformation Orientation There are several representations for 3D rotations Rotation about a 3D vector Quaternion Sequence of rotations about axes (XYZ, ZYX, ZYZ, etc.) Exponential of a skew-symmetric matrix

13 3D Cartesian Transformation Position and Orientation ffine transformation p = ( R p ) + t P Homogeneous coordinates p = E p

14 3D Cartesian Transformation Position and Orientation To invert E and obtain E we have

15 3D Cartesian Transformation Position and Orientation ffine transformation p = ( R p ) + t P Homogeneous coordinates p = E p

16 Cartesian Position/Orientation Summary 2D translation 3D translation 2D rotation 3D rotation Homogeneous coordinates Homogeneous coordinates

17 3D ngular Velocities The instantaneous angular velocity as seen from the coordinate frame is The instantaneous angular velocity as seen from the coordinate frame is

18 3D ngular Velocities How to transform between w, w? more concisely using

19 3D Cartesian Velocities We start again with p = E p nd the velocity of p is Substituting for p we obtain

20 3D Cartesian Velocities The instantaneous 3D velocity as seen from the coordinate frame is The instantaneous 3D velocity as seen from the coordinate frame is

21 Look further 3D Cartesian Velocities

22 3D Cartesian Velocities How to transform between V, V? after crunching we obtain