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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.

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3-D Kinematics

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  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

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