1. 3D Kinematics Consists of two parts 3D rotation 3D translation  The same as 2D 3D rotation is more complicated than 2D rotation (restricted to z- axis) Next, we will focus on the spatial (3D) rotation 2D (rigid body) kinematics C.M. translation and rotation Fall 20121

2. 3D Rotation Representations Euler angles Axis-angle 3X3 rotation matrix Unit quaternion Learning Objectives Representation (uniqueness) Perform rotation Composition Interpolation Conversion among representations … Fall 20122

3. Euler Angles Specify orientation in rotation along 3 axes Variation: which axes? global or local? Fall 20123

4.    Write a program … From Mason’s book Fall 20124

5. Euler Angles Roll, pitch, yaw Ref: http://www.fho-emden.de/~hoffmann/gimbal09082002.pdf Gimbal lock: reduced DOF due to overlapping axes Why gimbal lock a problem? Fall 20125

6. Axis-Angle Representation Rot(n,  ) n: rotation axis (global)  : rotation angle (rad. or deg.) follow right-handed rule Rot(n,  )=Rot (-n,-  ) Problem with null rotation: rot(n,0), any n Perform rotation Rodrigues formula Interpolation/Composition: poor Rot(n 2,  2 )Rot(n 1,  1 ) =?= Rot(n 3,  3 ) Fall 20126

7. Rodrigues Formula v ’ =R v  r v v’v’ References: http://mesh.caltech.edu/ee148/notes/rotations.pdf http://www.cs.berkeley.edu/~ug/slide/pipeline/assignments/as5/rotation.ht ml Fall 20127

8. Rotation Matrix Meaning of three columns Perform rotation: linear algebra Composition: trivial orthogonalization might be required due to FP errors Interpolation: ? Fall 20128

9. Gram-Schmidt Orthogonalization If 3x3 rotation matrix no longer orthonormal, metric properties might change! Verify! Fall 20129

10. Quaternion A mathematical entity invented by Hamilton Definition i j k Fall 201210

11. Quaternion (cont) Operators Addition Multiplication Conjugate Length Fall 201211

12. Unit Quaternion Define unit quaternion as follows to represent rotation Example Rot(z,90°)  q and – q represent the same rotation Why “ unit ” ? DOF point of view! Fall 201212

13. quaternion → axis-angle Fall 201213 Use both values of sine and cosine to determine the angle!!

14. Ex: q and –q are the same! Fall 201214

15. Example x y z x y z Rot (90, 0,0,1) OR Rot (-90,0,0,-1) Fall 201215

16. Unit Quaternion (cont) Perform Rotation Composition Interpolation Linear Spherical linear Fall 201216

17. Example x y,x ’ z,z ’ y’y’ Rot(z,90°) p(2,1,1) Fall 201217

18. Example (cont) Fall 201218

19. Example x y,x ’ z,z ’ y’y’ x,x ’ y z,y ’ z’z’ Fall 201219

20. Matrix Conversion Fall 201220

21. Matrix Conversion (cont) Find largest qi 2 ; solve the rest Fall 201221

22. Slerp (Spherical Linear Interpolation) The computed rotation quaternion rotates about a fixed axis at constant speed References: http://www.gamedev.net/reference/articles/article1095.asp http://www.diku.dk/research-groups/image/teaching/Studentprojects/Quaternion/ http://www.sjbrown.co.uk/quaternions.html http://www.theory.org/software/qfa/writeup/node12.html q r unit sphere in R 4 Fall 201222

23. Spatial Displacement Any displacement can be decomposed into a rotation followed by a translation Matrix Quaternion Fall 201223

24.    Write a program … From Mason’s book Fall 201224

25.    y z x x’’’ y’’’ z’’’ Fall 201225

26. Fall 201226

27. From previous page Fall 201227

28. From Lee and Koh (1995) Euler angles in ASF In v’=Mv convention Fall 201228