1. 3D Kinematics Consists of two parts
2D (rigid body) kinematics
C.M. translation and rotation
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)
Spring 2016

2. Homogeneous Coordinates (齊次座標)
Translate, rotate with matrices
Homogeneous Coordinates (齊次座標)
Fall 2016

3. Homogeneous Coordinate
Point in 3D:
Translate(a,b,c)
with homo. coord.:
Fall 2016

4. Rotate a Point about Origin
Rotate (z, q) r z
Polar Coordinate:
P (x, y) = (r cosf, r sinf)
Fall 2016

5. Rotation Matrices
v = Hu
Fall 2016

6. Order Matters?!
Translate(t1) Translate (t2) = Translate (t2) Translate (t1)
Translate(t) Rotate(R) ≠ Rotate (R) Translate (t)
Rotate (R1) Rotate (R2) ≠ Rotate (R2) Rotate (R1)
Fall 2016

7. Order Matters! V V Rotate (0,0,1, 30); Translate (10, 0, 0);
drawPlant( );
Translate (10, 0, 0);
Rotate (0,0,1, 30);
drawPlant( );
Fall 2016

8. 兩種詮釋法 Translate (10,5) Rotate (z, 30) draw [transform點座標] V
Fall 2016
Also give ways to compute the world pos, given local coord

9. Case 1: Translate (10, 0, 0); Translate (10, 0, 0);
Rotate (0,0,1, 30);
drawPlant( );
Translate (10, 0, 0);
Rotate (0,0,1, 30);
drawPlant( ); y x z
Fall 2016

10. Case 2: Rotate (0,0,1, 30); Rotate (0,0,1, 30); Translate (10, 0, 0);
drawPlant( );
Rotate (0,0,1, 30);
Translate (10, 0, 0);
drawPlant( ); y x z
Fall 2016

11. 3D Rotation Representations
Euler angles
Axis-angle
3X3 rotation matrix
Unit quaternion
Learning Objectives
Representation (uniqueness)
Perform rotation
Composition
Interpolation
Conversion among representations …
Spring 2016

12. Euler Angles Specify orientation in rotation angles along three axes
Usually, refer to local axes
Common used:
RPY (roll, pitch, yaw)
ZYZ (next page)
Fall 2014

13. Euler Angles (ZYZ) a b g
Fall 2014

14. [Euler Angles (ZYZ)]
a=0 y’
z’’’
b=90
z’’’
z’’
g=90
y’’
Fall 2014
y’’’

15. [Performing Rotation with (a, b, g) ]
Rot(z”,g) Rot(y’,b) Rot(z,a)
= Rot(z,a) Rot(y,b) Rot(z,g)
A three-page CAD article (1995) by Lee and Koh talked about this
Fall 2014

16. Euler Angles Roll, pitch, yaw
Why gimbal lock a problem?
Roll, pitch, yaw
Gimbal lock: reduced DOF due to overlapping axes
在目前這個configuration,飛機無法yaw
Spring 2016

17. Loss of DOF in gimbal lock (url)
Rotation (a, b, g, ‘XYZ’)
When
Rotation (0.4, 1.57, 0, ‘XYZ’) = Rotation (0, 1.57, 0.4, ‘XYZ’)
Spring 2016

18. Axis-Angle Representation
Rot(r,a)
r: rotation axis (unit vector)
a: rotation angle (rad. or deg.)
follow right-handed rule
DOF? 3
Rot(r,a)=Rot (-r,-a)
Problem with null rotation:
rot(r, 0), any r (many-to-one mapping)
單位向量:長度為1之向量 a r v’ v
Fall 2014

19. Axis Angle (cont) Perform rotation
Rodrigues rotation formula (next pages)
Fall 2014

20. Rodrigues Rotation Formula
Compute the transformed point v’, after the axis-angle rotation, Rot(r, a)
r: rotation axis, unit vector
𝑣 ′ =𝑣 cos 𝛼+ 𝑟×𝑣 sin 𝛼+𝑟 𝑟∙𝑣 1− cos 𝛼 a r v’ v
Alternative formula
v’=R v

21. Example (Rodrigues Formula)
𝑣 ′ =𝑣 cos 𝛼+ 𝑟×𝑣 sin 𝛼+𝑟 𝑟∙𝑣 1− cos 𝛼 r v’ v y
𝑅𝑜𝑡 1,0,0 , 𝜋
= cos 𝜋 × sin 𝜋 ∙ − cos 𝜋 2 x
=0+ 𝑖 𝑗 𝑘 = = z
Fall 2014

22. Axis-Angle Interpolation: poor Composition: poor
Rot(n1,q1) & Rot(n2,q2)  Rot(n1.5, q1.5) = ?
Composition: poor
Rot(n2,q2) Rot(n1,q1) =?= Rot(n3,q3)
Fall 2014

23. [Euler’s Rotation Theorem]
in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point.
Spring 2016

24. 2D Rotation orthogonal basis (after rotation) 𝑥=𝑟 cos ∅ 𝑦=𝑟 sin ∅ q fp
𝑥=𝑟 cos ∅
𝑦=𝑟 sin ∅ r p’ q p f
orthogonal basis
(after rotation)
Fall 2014

25. Extend to 3D Rotation Matricesy z x y z x y z
𝑅𝑜𝑡 𝑥,𝜃 = cos 𝜃 − sin 𝜃 0 sin 𝜃 cos 𝜃
𝑅𝑜𝑡 𝑦,𝜃 = cos 𝜃 0 sin 𝜃 − sin 𝜃 0 cos 𝜃
𝑅𝑜𝑡 𝑧,𝜃 = cos 𝜃 −sin 𝜃 0 sin 𝜃 cos 𝜃
orthogonal basis
Fall 2014

26. Rotation Matrix Three columns represent the basis
Perform rotation: linear algebra
Composition: trivial
orthogonalization might be required due to FP errors
Fall 2014

27. Example x
y,x’
z,z’ y’
p(2,1,1)
Rot(z,90°)
Fall 2014

28. Rotation Matrix DOF? Interpolation:
9 numbers, but with these constraints, leaves 3
Interpolation:
R1 & R2  R1.5 = ?
Fall 2014

29. Gram-Schmidt Orthogonalization
Given three independent vectors: u1, u2, u3
If 3x3 rotation matrix is no longer orthonormal, metric properties(volume, length, angle) might change!
Fall 2014

30. [Proof (matrix version)]
Rotation matrix is an orthogonal matrix
[Proof (matrix version)]
1 is an eigenvalue of R
x is the corresponding eigenvector (the rotation axis)
Spring 2016

31. Quaternion A mathematical entity invented by Hamilton Definition i k j
Fall 2014

32. Quaternion (cont) Operators Addition Multiplication Conjugate Length
Fall 2014

33. Unit Quaternion as Rotation
Define unit quaternion as follows to represent rotation
Example
Rot(z,90°)
q and –q represent the same rotation
DOF? 3
Fall 2014

34. In-class examples Rot(z,90°) Rot(z,-90°) Rot(x,180°) Rot(-z,-90°)
Same as rot(z,90)
Spring 2016

35. Important values & equalities
Sin(q)
Cos(q) 1
p/6
1/2
Sqrt(3)/2
p/4
Sqrt(2)/2
p/3
p/2
Spring 2016

36. Unit Quaternion: rotating
Fall 2014

37. Unit Quaternion: composition
Fall 2014

38. Slerp (Spherical Linear Interpolation)q r
unit sphere in R4
The computed rotation quaternion rotates about a fixed axis at constant speed
Fall 2014

39. Unit Quaternion: slerp
Fall 2014

40. Null Rotation (q = 0)
Fall 2014

41. Ex: q and –q are the same!
Fall 2014

42. q & -q in slerp May want to consider
slerp (t, -q, r) instead of slerp (t, q, r), for smoother interpolation
Fall 2014

43. Unit Quaternion Conversion
Axis-angle
3x3 matrix
Fall 2014

44. Unit Quaternion & Axis-Angle
Axis-angle  quaternion:
Quaternion axis-angle
First, find axis (unit vector)
Then, use both values of sine and cosine to find angle
Example
Fall 2014

45. Unit Quaternion3x3 Matrix
Example
Fall 2014

46. 3x3 MatrixUnit Quaternion
Find largest qi2; solve the rest
Fall 2014

47. Exercise x y z
Fall 2014

48. In-class Exercise y x z As we have shown,
This corresponds to rot(z,90)
(take the positive one)
Spring 2016

49. quaternion→axis-angle
Use both values of sine and cosine to determine the angle!!
Spring 2016

50. Example x y z x y z
Rot (90, 0,0,1) OR Rot (-90,0,0,-1)
Spring 2016

51. Example x
y,x’
z,z’ y’
p(2,1,1)
Rot(z,90°)
Spring 2016

52. Example x
y,x’
z,z’ y’ y
x,x’
z,y’ z’
Spring 2016

53. Matrix Conversion
Spring 2016

54. Spatial Displacement
Any displacement can be decomposed into a rotation followed by a translation
Matrix
Quaternion
Spring 2016

55. Extra’s
Spring 2016

56. From Mason’s book a g b
Write a program …
Spring 2016

57. y z x a
x’’’
y’’’
z’’’ g b
Spring 2016

58. Spring 2016

59. From previous page
Spring 2016

60. From Lee and Koh (1995) Euler angles in ASF In v’=Mv convention
Spring 2016