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) Reference coordinate system point : the 3d location of the bunny vector : translational movement

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

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

12. Complex Number Real Imaginary

13. Complex Exponentiation Real Imaginary

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 Real Imaginary

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 Real Imaginary

18. 2D Rotation and Displacement Real Imaginary

19. 2D Orientation Composition Real Imaginary

20. 2D Rotation Composition Real Imaginary

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  –X’-axis by  –Z’’-axis by 

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

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  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)

43. Angular Velocity

44. Exp and Log logexp

45. Exp and Log logexp Euler parameters

46. Rotation Vector

49. 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

51. Spherical Linear Interpolation

52. Coordinate-Invariant Operations

53. Adding Rotation Vectors

54. Affine Combination of Orientations

55. Analogy (point : vector) is similar to (orientation : rotation)

56. Rotation Matrix vs. Unit Quaternion Equivalent in many aspects –Redundant, No singularity –Exp & Log, Special tangent space Why quaternions ? –Fewer parameters, Simpler algebra –Easy to fix numerical error Cf) matrix orthogonalization (Gram-shmidt process, QR, SVD decomposition) Why rotation matrices ? –One-to-one correspondence –Handle rotation and translation in a uniform way Eg) 4x4 homogeneous matrices

57. Rotation Conversions In theory, conversion between any representations is always possible In practice, conversion is not straightward because of difference in convention Quaternion to Matrix