Robotics: 3D Movements
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The document discusses different methods for representing 3D rotations and orientations, including rotation matrices, Euler angles, and quaternions. It explains that quaternions represent a rotation as a combination of a scalar and vector, and describe how to perform operations like rotation, composition, and normalization using quaternions. Quaternions use fewer parameters than rotation matrices but more easily represent arbitrary rotations and can be interpolated for smooth animation.
![Quaternions Proposed by Hamilton Euler parameters make up the components of the quaternion Quaternion Q has a scalar and a vector part Q=[w+v] If S =sin( /2) and C=cos( /2) then the rotation of around axis k where u is a unit vector.](https://image.slidesharecdn.com/lecture4-110104183354-phpapp02/85/Robotics-3D-Movements-12-320.jpg)
![Quaternions as Rotations Example : rotate point P = [ x, y, z, w ] about vector v by degrees. determine quaternion representation p of point P: create quaternion q for rotation: conjugate p by q : recover new rotated coordinate:](https://image.slidesharecdn.com/lecture4-110104183354-phpapp02/85/Robotics-3D-Movements-16-320.jpg)
Robotics: 3D Movements
- 2. 3-D Spaces and Points Axes xyz right hand rule X Y=Z X Z Y v v 1 v 2 v 3 X Y Z
- 3. 2 co-ordinate frames rotated about a common origin i R j is the rotation matrix from frame j to i X 0 Y 0 Z 0 X 1 Y 1 Z 1
- 4. Rotation about the axes X Y Z
- 5. Direction Cosine Form View rotation as the dot product between each of the equations for 1 x 0 , 1 y 0 , and 1 z 0 and the axes 0 x 0 , 0 y 0 , 0 z 0
- 6. Inverse of a rotation matrix
- 7. Composition of Rotations Consider three frames 0,1 and 2 rotated about a common origin with p a point relative to the common origin Assume all frames coincide Rotate 1 and 2 about y 0 by Rotate 2 about z 1 by Because rotations occurred in different frames
- 8. From origin P Left to right a From origin P Right to left a a a X Y Z X Y Z
- 9. Spatial Transformations Similar to 2-D case with and increase in dimensions Representation of Orientation Rotation matrix contains redundant information (9 terms) Only 3 terms independent 3 term representations Euler angles Rodrigues vectors 4 term representations Quaternions
- 10. Euler Angles Any non-redundant set of three successive rotations about principal axes 12 systems most common is ZYZ + Useful where they match the structure of the robot - Can have two solutions
- 11. Roll, Pitch Yaw angles R xyz ( ) = R z ( ) R y ( ) R x ( ) X Y Z Roll Yaw Pitch
- 12. Quaternions Proposed by Hamilton Euler parameters make up the components of the quaternion Quaternion Q has a scalar and a vector part Q=[w+v] If S =sin( /2) and C=cos( /2) then the rotation of around axis k where u is a unit vector.
- 13. Quaternion Operations Addition: Multiplication: Norm (length): Conjugate: Inverse:
- 14. Quaternions as Rotations We define the representation of a (homogeneous) Cartesian point in quaternion space as: Let q be a unit quaternion (i.e. N( q ) = 1 ). The conjugation of p by q is defined as:
- 15. Quaternions as Rotations then and if then the effect on p is to rotate it anti-clockwise about axis v by degrees. Therefore every unit quaternion represents a rotation about an axis. Note that the rotation q is the same as - q redundancy
- 16. Quaternions as Rotations Example : rotate point P = [ x, y, z, w ] about vector v by degrees. determine quaternion representation p of point P: create quaternion q for rotation: conjugate p by q : recover new rotated coordinate:
- 17. Rotation Matrix Quaternion Conversion
- 18. Deconstructing the Quaternion Given arbitrary unit quaternion we can recover the associated rotation axis and angle:
- 19. Normalisation With successive rotations, errors can accumulate making the quaternions of non-unit length Normalisation of Matrices is much harder
- 20. Quaternion Rotation Composition As with matrices, quaternion rotations may be composed of multiple basic rotations: note the order ( q 2 after q 1 ) property of conjugation
- 21. Advantages of Quaternions for Rotations Significantly less storage required (4 reals vs. 16 reals) Much easier to construct arbitrary rotations about vectors. Easily interpolated for smooth paths . Transformation via quaternions more efficient. Operation Matrices Quaternions Storage 9 4 Transformation 9M 6A 15M 15A Composition 27M 18A 16M 12A Normalisation complicated 8M 3A 1sqrt