3D Kinematics. Consists of two parts

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1 D Kinematics Consists of two parts D rotation D translation The same as D D rotation is more complicated than D rotation (restricted to z-ais) Net, we will discuss the treatment for spatial (D) rotation

2 D Rotation Representations Euler angles Ais-angle X rotation matri Unit uaternion Learning Objectives Representation (uniueness) Perform rotation Composition Interpolation Conversion among representations

3 Euler Angles and GIMBAL LOCK Roll, pitch, yaw Gimbal lock: reduced DOF due to overlapping aes Ref:

4 Ais-Angle Representation

5 We create matri R for rotation Ais-Angle Representation Rot(n,) n: rotation ais (global) : rotation angle (rad. or deg.) follow right-handed rule Rot(n,)=Rot (-n,-) Problem with null rotation: rot(n,), any n Perform rotation Rodrigues formula Interpolation/Composition: poor Rot(n, )Rot(n, ) =?= Rot(n, )

6 Quaternions

7 Quaternion - Brief History Invented in 84 by Irish mathematician Sir William Rowan Hamilton Founded when attempting to etend comple numbers to the rd dimension Discovered on October 6 in the form of the euation: i j k ijk From: 7

8 Quaternion Brief History William Rowan Hamilton

9 Quaternion Definition i j i k ij ji k j k i jk kj ki ik i j k j

10 Applications of Quaternions Used to represent rotations and orientations of objects in three-dimensional space in: Computer graphics Control theory Signal processing Attitude controls Physics Orbital mechanics Quantum Computing, uantum circuit design From:

11 Advantages of Quaternions Avoids Gimbal Lock Faster multiplication algorithms to combine successive rotations than using rotation matrices Easier to normalize than rotation matrices Interpolation Mathematically stable suitable for statistics

12 Operators on Quaternions Operators Addition Multiplication Conjugate Length k p j p i p p p k j i k p j p p i p p p p p p p p * * * * p p *

13 Unit Quaternion Define unit uaternion as follows to represent rotation Rot ( n, ) cos sin n Eample Rot(z,9 ) and represent the same rotation Why unit? DOF point of view!

14 Quaternion scalar and vector parts = w + i + yj + zk w,, y, z are real numbers w is scalar part, y, z are vector parts Thus it can also be represented as: = (w, v(,y,z)) or = w + v

15 Quaternion Dimension and Transformation Scalar & Vector 4 dimensions of a uaternion: -dimensional space (vector) Angle of rotation (scalar) Quaternion can be transformed to other geometric algorithm: Rotation matri uaternion Rotation ais and angle uaternion Spherical rotation angles uaternion Euler rotation angles uaternion What are relations of uaternions to other topics in kinematics?

16 Details of Quaternion Operations

17 Quaternion Operations Addition/subtraction Multiplication Division Conjugate Magnitude Normalization Transformations Concatenation

18 Quaternion Operations Addition: Given two uaternions: = w + i + yj + zk = w + i + yj + zk The result uaternion is: = + = ( w + w) + ( + )i + ( y + y)j + ( z + z)k

19 Quaternion Operations Subtraction: Given two uaternions: = w + i + yj + zk = w + i + yj + zk The result uaternion is: = - = ( w w) + ( )i + ( y y)j + ( z z)k

20 Quaternion Operations Multiplication Distributive Associative Not commutative because of the i =j =k =- i i (-i) k j k (-k) (-j) j

21 Quaternion Operations Multiplication Given two uaternions: = w + i + yj + zk = w + i + yj + zk The result uaternion is: = * = w* w + w* i + w* yj + w* zk + i* w + i* * i+ etc

22 Quaternion Operations Multiplication Resulting uaternion is: = ( w w + + y y + z z) + ( w + w + y z z y)i + ( w y + y w + z z)j + ( w z + z w + y y )k

23 Quaternion Operations Multiplication Or, in scalar-vector format: = = ( w, v )( w, v ) = ( w w - v.v, wv + v + v v ) scalar vector or Dot product Cross product = w w - v.v + wv + v + v v

24 Quaternion Operations Magnitude Also called modulus Is the length of the uaternion from the origin Given a uaternion: = w + i+ yj + zk The magnitude of uaternion is, where: * w y z

25 Normalization Quaternion Operations Normalization results in a unit uaternion where: w + + y + z = Given a uaternion: = w + i+ yj + zk To normalize uaternion, divide it by its magnitude ( ): Also referred to as uaternion sign: sgn()

26 Quaternion Operations Conjugate: Given a uaternion: = w + i+ yj + zk The conjugate of uaternion is *, where: * = w i yj zk

27 Quaternion Operations Inverse Can be used for division Given a uaternion: = w + i+ yj + zk The inverse of uaternion is -, where: *

28 Quaternion Rotations

29 Matri Rotation Matri Rotation is based on rotations: On aes:, y, z Or yaw, pitch, roll (which one corresponds to which ais, depends on the orientation to the aes) Seuence matters (-y may not eual y-) z y

30 Matri Rotation cos sin sin cos cos sin sin cos z y cos sin sin cos rotation : ais cos sin sin cos rotation : ais y cos sin sin cos : rotation ais z y z cos cos cos sin sin sin cos sin sin cos sin cos cos sin cos cos sin sin sin sin cos cos sin sin sin sin cos cos cos : rotation matri Final

31 Quaternion Rotation Parts of uaternion w = cos(θ/) v = sin(θ/)û Where û is a unit/normalized vector u (i, j, k) Quaternion can be represented as: =cos(θ/)+sin(θ/)(i+yj+zk) or =cos(θ/)+sin(θ/) û w Matri rotation versus uaternion rotation

32 Let s do rotation! y z

33 Let s do rotation! y b c a z

34 Let s do another one! y z

35 Quaternion? v(,y,z) w Can create rotation by using arbitrary ais (v(,y,z)) and rotate the object by w amount.

36 Rotation of a Quaternion Calculation is still done in matri form Given a uaternion: = w + i+ yj + zk The matri form of uaternion is: w w wz y y z wy z w y wz y w yz z w wy yz w z y z Matri entries are taken all from uaternion

37 Quaternion Rotation When it is a unit uaternion: w y wz z y z wy y wz z w yz wy yz w z y Quaternion matri for unit uaternion: w=

38 Eample of unit uaternion Our notation: Rotation of vector v is v Where v is: v=ai+bj+ck By uaternion =(w,u) Where w = Vector (ais): u = i + j + k Rotation angle: = (π)/ radian (θ) Length of u = If we rotate a vector, the result should be a vector. uaternion cos sin u cos sin u cos sin u cos 6 sin 6u u ( i. i j k j k)

39 Eample of uaternion rotation (cont d) So to rotate v: v = v * Where * is conjugate of : * ( i j k) So now we can substitute v * to matri form of uaternion

40 Eample (cont d) v v v v v z y w yz w wy z w yz z y w y wz z wy wz y z y w v * * * Eample of uaternion rotation (cont d) Quaternion matri j ik ki i kj jk k ji ij k j i k j i

41 Gimbal Lock It happens when you turn this ais far enough until this ais aligns with this ais.

42 Gimbal Lock Final rotation matri : cos cos cos sin sin sin sin cos cos sin sin sin sin cos cos sin cos cos sin cos sin sin cos sin sin sin cos cos cos Gimbal lock occurs when rotated -9 or 9 on y-ais Remember that: = rotation on -ais = rotation on y-ais = rotation on z-ais Gimbal lock : sin cos cos sin cos cos sin sin sin sin cos cos cos sin sin cos

43 Eample of using uaternions in robotics: uaternion representing a rotation y z y z Rot (9,,,) OR Rot (-9,,,-) Rot(z,9 )

44 Eample of using uaternions in robotics y Rot(z,9 ) How to represent rotation? z y z Represented as uaternion Represented as matri Rot (9,,,) OR Rot (-9,,,-)

45 Operations on Unit Quaternions Perform Rotation Composition of rotations Interpolation Linear Spherical linear (more later) ) ( ) (... * * * * * * ) ( ) ( p p pp pp ) ( ) (, ) ( ) ( t p t p p tp p t t p

46 Eample of rotation using unit uaternions y, z,z y Rp p R Rot(z,9 ) p(,,) For comparison we first use matrices

47 Eample (cont) ) ( ) ( ) ( ) ( k j i p p p p p For comparison we use uaternions Net we convert to matrices We get the same result

48 New Eample: multiplication of uaternions y, z,z y ) ( ) ( sin cos ) ( ) ( sin cos , y z,y z ) ( ) ( ) ( ) ( ) ( ) )( ( n : Compositio k j i

49 Eample: Conversion of uaternion matri to rotation matri R ) ( ) ( ) ( ) ( ) ( ) ( R R Matri R represented with uaternions We substitute values of And we get R

50 Matri Conversion Formulas r r r r r r r r r r r r r r r r r r r r r r r r Relations between i and r ij

51 Rodrigues Formula r v v v =R v References: ml

52 Rotation Matri Meaning of three columns Perform rotation: linear algebra Composition: trivial orthogonalization might be reuired due to floating point errors Interpolation:? A Au Au Au u u u u u u a A ij R R R R R R

53 Gram-Schmidt Orthogonalization If rotation matri no longer orthonormal, metric properties might change! v v v u u u v v v v u v v v v u u v v v v v u u v u v Verify!

54 Spatial Displacement Any displacement can be decomposed into a rotation followed by a translation Matri Quaternion T d R T z y d R, d *

55 Spherical Linear Interpolation

56 Interpolation Interpolation produces an arc instead of a line

57 Spherical Linear Interpolation sin ( t) s( t) sin cos r r sin t r sin r r r uaternion unit sphere in R 4 The computed rotation uaternion rotates about a fied ais at constant speed References:

58 Spherical Linear Interpolation Slerp for unit-length uaternions sin(( t) ) sin( t) slerp( t;, ) sin - and are unit-length uaternions - θ = angle between and - t is interval [, ] - if and are the same uaternion, then θ =, but in this case, (t) = for all t.

59 Spherical Linear Interpolation - Slerp sin(( t) ) sin( t) slerp( t;, ) sin slerp( t;, ) sin( ( t)) sin( t) p sin( ( t)) p sin( ( t )) t [, t [ ],] - if = - then θ = π - use a third uaternion p perpendicular to (which could be infinite number of vectors) - interpolation is done from to p for and from p to for t [,] t [, ]

60 Spherical Linear Interpolation - Slerp. Used in joint animation by storing starting and ending joint position as uaternions.. Allows smooth rotations in keyframe animations.. Spherical Linear interpolation supports the animation of joints when starting and ending joint positions are stored as uaternions that represent the joint rotations from canonical positions Rob Saunders, Advanced Games Design, Theory and Practice, March 5, City University London

61 . The concept of canonical (or conjugate) variables is of major importance.. They always occur in complementary pairs, such as spatial location and linear momentum p, angle φ and angular momentum L, and energy E and time t.. They can be defined as any coordinates whose Poisson brackets give a Kronecker delta (or a Dirac delta incase of discrete variables).

62 Quaternion in Multi-Sensor Robot Navigation System (by S. Persa, P. Jonker, Technical University Delft, Netherlands) Used uaternions

63 Dimensional Synthesis of Spatial RR Robots (A. Perez, J.M. McCarthy, University of California, Irvine) Finish* Start* *assumption

64 Let s do rotation! y b c a z

65 Visualization of uaternions Difficult to visualize Not for the weak-on-math

66 Visualizing Quaternion Rotation Removing 7 degree twist without moving either end (from: J.C. Hart, G.K. Francis, L.H. Kauffman, Visualizing Quaternion Rotation, ACM Transactions on Graphics, Vol., No. July 994, p.67)

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69 Quaternion Eplained! By Mathias Sunardi for Quantum Research Group Seminar June 5, 6