Rotation and Orientation: Fundamentals

1. Rotation and Orientation:Fundamentals Jehee Lee Seoul National University

2. What is Rotation ? • Not intuitive • Formal definitions are also confusing • Many different ways to describe • Rotation (direction cosine) matrix • Euler angles • Axis-angle • Rotation vector • Helical angles • Unit quaternions

3. Orientation vs. Rotation • Rotation • Circular movement • Orientation • The state of being oriented • Given a coordinate system, the orientation of an object can be represented as a rotation from a reference pose

4. Analogy (point : vector) is similar to (orientation : rotation) Both represent a sort of (state : movement) Reference coordinate system

5. Analogy (point : vector) is similar to (orientation : rotation) Both represent a sort of (state : movement) point: the 3d location of the bunny vector: translational movement Reference coordinate system

6. Analogy (point : vector) is similar to (orientation : rotation) Both represent a sort of (state : movement) point: the 3d location of the bunny vector: translational movement orientation: the 3d orientation of the bunny rotation : circular movement Reference coordinate system

7. 2D Orientation Polar Coordinates

8. 2D Orientation Although the motion is continuous, its representation could be discontinuous

9. 2D Orientation Many-to-one correspondences between 2D orientations and their representations

10. Extra Parameter

11. Extra Parameter 2x2 Rotation matrix is yet another method of using extra parameters

12. Complex Number Imaginary Real

13. Complex Exponentiation Imaginary Real

14. 2D Rotation • Complex numbers are good for representing 2D orientations, but inadequate for 2D rotations • A complex number cannot distinguish different rotational movements that result in the same final orientation • Turn 120 degree counter-clockwise • Turn -240 degree clockwise • Turn 480 degree counter-clockwise Imaginary Real

15. 2D Rotation and Orientation • 2D Rotation • The consequence of any 2D rotational movement can be uniquely represented by a turning angle • 2D Orientation • The non-singular parameterization of 2D orientations requires extra parameters • Eg) Complex numbers, 2x2 rotation matrices

16. Operations in 2D • (orientation) : complex number • (rotation) : scalar value • exp(rotation) : complex number

17. 2D Rotation and Displacement Imaginary Real

18. 2D Rotation and Displacement Imaginary Real

19. 2D Orientation Composition Imaginary Real

20. 2D Rotation Composition Imaginary Real

21. Analogy

22. 3D Rotation • Given two arbitrary orientations of a rigid object,

23. 3D Rotation • We can always find a fixed axis of rotation and an angle about the axis

24. Euler’s Rotation Theorem In other words, • Arbitrary 3D rotation equals to one rotation around an axis • Any 3D rotation leaves one vector unchanged The general displacement of a rigid body with one point fixed is a rotation about some axis Leonhard Euler (1707-1783)

25. Euler angles • Gimble • Hardware implementation of Euler angles • Aircraft, Camera

26. Euler Angles • Rotation about three orthogonal axes • 12 combinations • XYZ, XYX, XZY, XZX • YZX, YZY, YXZ, YXY • ZXY, ZXZ, ZYX, ZYZ • Gimble lock • Coincidence of inner most and outmost gimbles’ rotation axes • Loss of degree of freedom

27. Euler Angles • Euler angles are ambiguous • Two different Euler angles can represent the same orientation • This ambiguity brings unexpected results of animation where frames are generated by interpolation.

28. Ranges • Euler angles in ZXZ • Z-axis by a  ( -p , p ] • X’-axis by b  ( -p/2 , p/2 ] • Z’’-axis by g  ( -p , p ]

29. Dependence • Compare • Rotation about z-axis by 180 degree • Rotation about y-axis by 180 degree, followed by another rotation about x-axis by 180 degree • Rotations about x-, y-, and z-axes are dependent

30. Rotation Vector • Rotation vector(3 parameters) • Axis-Angle(2+1 parameters)

31. 3D Orientation • Unhappy with three parameters • Euler angles • Discontinuity (or many-to-one correspondences) • Gimble lock • Rotation vector (a.k.a Axis/Angle) • Discontinuity (or many-to-one correspondences)

32. Using an Extra Parameter • Euler parameters

33. Quaternions • William Rowan Hamilton (1805-1865) • Algebraic couples (complex number) 1833 where

34. Quaternions • William Rowan Hamilton (1805-1865) • Algebraic couples (complex number) 1833 • Quaternions 1843 where where

35. Quaternions William Thomson “… though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.” Arthur Cayley “… which contained everything but had to be unfolded into another form before it could be understood.”

36. Unit Quaternions • Unit quaternions represent 3D rotations

37. Rotation about an Arbitrary Axis • Rotation about axis by angle where Purely Imaginary Quaternion

38. Unit Quaternion Algebra • Identity • Multiplication • Inverse • Opposite axis or negative angle • Unit quaternion space is • closed under multiplication and inverse, • but not closed under addition and subtraction

39. Unit Quaternion Algebra • Antipodal equivalence • q and –q represent the same rotation • ex) rotation by p-q about opposite direction • 2-to-1 mapping between S and SO(3) • Twice as fast as in SO(3) 3

40. 3D Orientations and Rotations Orientations and rotations are different in coordinate-invariant geometric programming Use unit quaternions for orientation representation • 3x3 orthogonal matrix is theoretically identical Use 3-vectors for rotation representation

41. Tangent Vector(Infinitesimal Rotation)

42. Tangent Vector(Infinitesimal Rotation)

43. Tangent Vector(Infinitesimal Rotation) Angular Velocity

44. Exp and Log log exp

45. Exp and Log Euler parameters log exp

46. Rotation Vector

47. Rotation Vector

48. Rotation Vector

49. Rotation Vector • Finite rotation • Eg) Angular displacement • Be careful when you add two rotation vectors • Infinitesimal rotation • Eg) Instantaneous angular velocity • Addition of angular velocity vectors are meaningful

50. Spherical Linear Interpolation • SLERP [Shoemake 1985] • Linear interpolation of two orientations