3D Kinematics

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 discuss the treatment for spatial
  (3D) rotation

2
3D Rotation Representations

• Euler angles
• Axis-angle
• 3X3 rotation matrix
• Unit quaternion

• Learning Objectives
• Representation (uniqueness)
• Perform rotation
• Composition
• Interpolation
• Conversion among representations

3
Euler Angles

• Roll, pitch, yaw

• Gimbal lock reduced DOF due to overlapping axes

Ref http//www.fho-emden.de/hoffmann/gimbal09082
002.pdf
4
Axis-Angle Representation

• Rot(n,q)
• n rotation axis (global)
• q rotation angle (rad. or deg.)
• follow right-handed rule
• Rot(n,q)Rot (-n,-q)
• Problem with null rotation rot(n,0), any n
• Perform rotation
• Rodrigues formula
• Interpolation/Composition poor
• Rot(n2,q2)Rot(n1,q1) ? Rot(n3,q3)

5
Rodrigues Formula
r
v
v
vR v
References http//mesh.caltech.edu/ee148/notes/ro
tations.pdf http//www.cs.berkeley.edu/ug/slide/p
ipeline/assignments/as5/rotation.html
6
Rotation Matrix

• Meaning of three columns
• Perform rotation linear algebra
• Composition trivial
• orthogonalization might be required due to FP
  errors
• Interpolation ?

7
Gram-Schmidt Orthogonalization

• If 3x3 rotation matrix no longer orthonormal,
  metric properties might change!

Verify!
8
Quaternion

• A mathematical entity invented by Hamilton
• Definition

9
Quaternion (cont)

• Operators
• Addition
• Multiplication
• Conjugate
• Length

10
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!
11
Example
x
y
z
Rot (90, 0,0,1) OR Rot (-90,0,0,-1)
12
Unit Quaternion (cont)

• Perform Rotation
• Composition
• Interpolation
• Linear
• Spherical linear (more later)

13
Example
p(2,1,1)
Rot(z,90)
14
Example (cont)
15
Example
y
x,x
z,y
z
16
Matrix Conversion
17
Matrix Conversion (cont)
Find largest qi2 solve the rest
18
Spherical Linear Interpolation
unit sphere in R4
The computed rotation quaternion rotates about a
fixed axis at constant speed
References http//www.gamedev.net/reference/artic
les/article1095.asp http//www.diku.dk/research-gr
oups/image/teaching/Studentprojects/Quaternion/ ht
tp//www.sjbrown.co.uk/quaternions.html http//www
.theory.org/software/qfa/writeup/node12.html
19
Spatial Displacement

• Any displacement can be decomposed into a
  rotation followed by a translation
• Matrix
• Quaternion