3D Kinematics
Eric Whitman
1/24/2010
Rigid Body State: 2D

p
ŷ
x̂
Rigid Body State: 3D
p
ŷ
ẑ
x̂
Add a Reference Frame
ŷ '
p
ŷ
ẑ
x̂
ẑ '
x̂'
Rotation Matrix
• Linear Algebra definition
– Orthogonal matrix: R-1 = RT
• square
– det(R) = 1
• 2D: 4 numbers
• 3D: 9 numbers
Unit Vectors
xˆ '  x' x xˆ  x' y yˆ  x' z zˆ
ŷ '
p
ŷ
ẑ
x̂
ẑ '
yˆ '  y ' x xˆ  y ' y yˆ  y ' z zˆ
zˆ '  z ' x xˆ  z ' y yˆ  z ' z zˆ
x̂'
 x' x

R   x' y
 x' z
y'x
y' y
y'z
z'x 

z' y 
z ' z 
Using the Rotation Matrix
ŷ '
a
A
p
ŷ
ẑ
x̂
ẑ '
x̂'
 

A  p  Ra
Pros
and
• Rotates Vectors Directly
• Easy composition
Cons
• 9 numbers
• Difficult to enforce
constraints
Simple Rotation Matrices
2D
cos( )  sin(  )
R( )  

sin(

)
cos(

)


3D
cos( )
Rz ( )   sin(  )
 0
 cos( )
R y ( )   0
 sin(  )
 sin(  ) 0
cos( ) 0
0
1
0 sin(  ) 
1
0 
0 cos( )
0
0 
1
Rx ( )  0 cos( )  sin(  )
0 sin(  ) cos( ) 
Degrees of Freedom
• 2D
– 2x2 matrix has 4
numbers
– Only one DoF
• 3D
– 3x3 matrix has 9
numbers
– 6 constraints
– 3 DoF
Euler Angle Combinations
• Can use body or world coordinates
• 2 consecutive angles must be different
– Can alternate (3-1-3) or be all different (3-1-2)
• 24 possibilities (12 pairs of equivalent)
• For aircraft, 3-2-1 body is common
– Yaw, pitch, roll
• For spacecraft, 3-1-3 body is common
Construct a Rotation Matrix
3-1-3 Body Convention – Common for spacecraft
 cc  scs
R( , , )   sc  ccs

ss
cs  scc
ss 
 ss  ccc cs 
 sc
c 
Recover Euler Angles
 cc  scs
R( , , )   sc  ccs

ss
cs  scc
ss 
 ss  ccc cs 
 sc
c 
  arccos( R33 )
  arctan( R31,  R32 )
  arctan( R13, R23 )
Gimbal Lock
• Physically: two gimbal axes line up, making
movement in one direction impossible
• Mathematically describes a singularity in Euler
angle systems
• For the 3-1-3 body convention, this occurs
when angle 2 equals 0 or pi
• For the 3-1-2 body convention, this occurs
when angle 2 +/- pi/2
• Switching helps
Pros
and
• Minimal Representation
• Human readable
Cons
• Gimbal Lock
• Must convert to RM to
rotate a vector
• No easy composition
Axis Angle (4 numbers)
• A special case of Euler’s Rotation Theorem:
any combination of rotations can be
represented as a single rotation
• 3 numbers to represent the axis of rotation
• 1 number to represent the angle of rotation
• Has singularity for small rotations

a, 
Rotation Vector (3 numbers)
• The axis can be a unit vector (only 2 DoF)
• Multiply axis by angle of rotation
• Can easily extract axis angle
– Axis = rotation vector
• Normalize if desired
– Angle = ||rotation vector||
• Same singularity – small rotations
â
Pros
and
• Minimal Representation
• Human readable (sort
of)
Cons
• Singularity for small
rotations
• Must convert to RM to
rotate a vector
• No easy composition
(Unit) Quaternions
• All schemes with 3 numbers will have a
singularity
q 
– So says math (topology)
1
q 
q


2


q
 
 q3   q 4 
 
 q4 
 q1 
q  q2   sin(  / 2) aˆ
 q3 
q4  cos( / 2)
Constraint
q q  q  q  q  q 1
T
• Easy to enforce
2
1
2
2
2
3
2
4
Conversion with RM
q12  q22  q32  q42

R   2(q2 q1  q4 q3 )
 2(q3 q1  q4 q2 )

2(q1q2  q4 q3 )
 q12  q22  q32  q42
2(q3 q2  q4 q1 )
1
q4 
1  R11  R22  R33
2
1
q1 
( R23  R32 )
4q4
q2 
1
( R31  R13 )
4q4
q3 
1
( R12  R21 )
4q4
2(q1q3  q4 q2 ) 

2(q2 q3  q4 q1 ) 
 q12  q22  q32  q42 
Composition
q'4 q  q4q'q'q
q'q  

 q'4 q4  q'q 
Pros
and
• No Singularity
• Almost minimal
representation
• Easy to enforce
constraint
• Easy composition
• Interpolation possible
Cons
• Not quite minimal
• Somewhat confusing
Summary of Rotation Representations
• Need rotation matrix to rotate vectors
• Often more convenient to use something else
and convert to rotation matrix
• Euler angles good for small angular deviations
• Quaternions good for free rotation
Homogeneous Transformations
p o R p
0
0
1
0
1
1
Define:
 
 p
~
p 
 
 
1


0
A1  



R10
0T

0
o1 


1
Composition
0
0 1
n 1 n
~
p  A1 A2 ... An p