1 / 36

3-D Kinematics

3-D Kinematics. Position and Orientation of a Rigid Body. Position and Orientation of a Rigid Body. The position of origin O’ with respect to O-xyz is expressed by the relation. The component of each unit vector are the direction cosines of the axes of frame O’-x’y’z’. Rotation Matrix.

kiana
Télécharger la présentation

3-D Kinematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 3-D Kinematics

  2. Position and Orientation of a Rigid Body

  3. Position and Orientation of a Rigid Body • The position of origin O’ with respect to O-xyz is expressed by the relation • The component of each unit vector are the direction cosines of the axes of frame O’-x’y’z’

  4. Rotation Matrix • Orientation can be described by rotation matrix • R is orthogonal matrix

  5. Elementary Rotations Rotation by an angle about axis z

  6. Elementary Rotations • Rotation by an angle about axis y • Rotation by an angle about axis x

  7. Representation of a Vector

  8. Representation of a Vector • Representation of p w.r.t O-xyz • Representation of p w.r.t O-x’y’z’

  9. Rotation of a Vector

  10. Equivalent Geometrical Meaningsof Rotation Matrix

  11. Composition of Rotation Matrices • Let Rijdenote the rotation matrix of Frame i with respect to Frame j • Post-multiplication interpretation • Refer to current frame • Pre-multiplication interpretation • Refer to fixed frame

  12. Euler Angles • Minimal representation of orientation • Three parameters are sufficient • Euler Angles • Two successive rotations are not made about parallel axes • How many kinds of Euler angles are there?

  13. ZYZ Angles • The rotation described by ZYZ angles is

  14. ZYZ Angles

  15. ZYZ Angles • The rotation matrix is

  16. ZYZ Angles • Inverse problem: determine the Euler angles corresponding to a given rotation matrix • Solution 1: theta is in the range (0, pi)

  17. ZYZ Angles y=1 x=1; y=-1 x=1; y=1 x=-1; y=-1 x=-1;

  18. ZYZ Angles • Solution 1: theta is in the range (0, pi)

  19. ZYZ Angles • Solution 1: theta is in the range (0, pi)

  20. ZYZ Angles • Solution 2: theta is in the range (-pi, 0)

  21. ZYZ Angles • Solution 2: theta is in the range (-pi, 0)

  22. ZYZ Angles • Solution 2: theta is in the range (-pi, 0)

  23. ZYZ Angles • What will happen if sin(theta) = 0? • Matlab: eul2tr, tr2eul

  24. Roll-Pitch-Yaw Angles • Originate from (aero)nautical field

  25. Roll-Pitch-Yaw Angles MATLAB: QUATDEMO

  26. Roll-Pitch-Yaw Angles • The rotation matrix is

  27. Roll-Pitch-Yaw Angles • Inverse problem: determine the Euler angles corresponding to a given rotation matrix • Solution 1: theta is in the range (-pi/2, pi/2)

  28. Roll-Pitch-Yaw Angles • Solution 2: theta is in the range (pi/2, 3pi/2)

  29. Roll-Pitch-Yaw Angles • What will happen if cos(theta) = 0? • Matlab: rpy2tr, tr2rpy

  30. Angle and Axis • Non-minimal representation: four parameters • The unit vector of a rotation axis w.r.t O-xyz • The angle theta about the axis • Matlab: quatdemo

  31. Angle and Axis • Align r with z • Rotate by theta about z • Realign with the initial direction of r Attention: always refer to the fixed frame

  32. Angle and Axis • The resulting rotation matrix is

  33. Angle and Axis • The inverse problem • Remember: the three component of r is not independent

  34. Angle and Axis • Problems: • solution is not unique • r is arbitrary when theta = 0

  35. Unit Quaternion • Unit quaternion is defined as

  36. Unit Quaternion • Inverse problem: • Matlab:quaternion, plot, quaternion.t

More Related