Rotation and Orientation:Fundamentals
Jehee LeeSeoul National University
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
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
• Analogy– (point : vector) is similar to (orientation : rotation)– Both represent a sort of (state : movement)
2D Orientation
0
2p
2p
-
p
p
-or
q
Polar Coordinates
2D Orientation
Although the motion is continuous, its representation could be discontinuous
0
2p
2p
-
p
p
-or
)(tq p
p-
q
Time
2D Orientation
Many-to-one correspondences between 2D orientations and their representations
0
2p
2p
-
p
p
-or
)(tq p
p-
q
Time
Extra Parameter
q
),( yx
X
Y 122 =+ yx
Extra Parameter
q
),( yx
X
Y 122 =+ yx
÷÷ø
öççè
æ -qqqq
cossinsincos
2x2 Rotation matrix is yet another method of using extra parameters
Complex Number
q
iyx +
Real
Imaginary
Complex Exponentiation
q
iyx +
Real
Imaginary
q
qqie
iiyx=
+=+ sincos
Rotation Composition
fqfq iii eee =+ )(
Real
Imaginary
fq +
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
q
pq 2-
2D Rotation and Orientation• 2D Rotation
– The consequence of any 2D rotational movement can be uniquely represented by a turning angle
– A turning angle is independent of the choice of the reference orientation
• 2D Orientation– The non-singular parameterization of 2D orientations requires
extra parameters• Eg) Complex numbers, 2x2 rotation matrices
– The parameterization is dependent on the choice of the reference orientation
X
Y
Z
X ¢
Y ¢
Z ¢
3D Rotation
• Given two arbitrary orientations of a rigid object,
How many rotations do we need to transform one
orientation to the other ?
v̂
q
3D Rotation
• Given two arbitrary orientations of a rigid object,
we can always find a fixed axis of rotation and a rotation angle about the axis
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 withone point fixed is a rotation about some axis
Leonhard Euler (1707-1783)
Euler angles• Gimbal
– Hardware implementation of Euler angles
– Aircraft, Camera
Euler Angles
• Rotation about three orthogonal axes– 12 combinations
• XYZ, XYX, XZY, XZX• YZX, YZY, YXZ, YXY• ZXY, ZXZ, ZYX, ZYZ
• Gimbal lock– Coincidence of inner most
and outmost gimbals’ rotation axes
– Loss of degree of freedom
Rotation Vector
• Rotation vector (3 parameters)
• Axis-Angle (2+1 parameters)
v̂
q
),,(ˆ zyx== vv q)ˆ,( vq
anglescalar : runit vecto : ˆ
qv
3D Orientation• Unhappy with three parameters
– Euler angles• Discontinuity (or many-to-one correspondences)• Gimbal lock
– Rotation vector (a.k.a Axis/Angle)• Discontinuity (or many-to-one correspondences)
pp-
Using an Extra Parameter
• Euler parameters
÷øö
çèæ=
úúú
û
ù
êêê
ë
é
÷øö
çèæ=
2sinˆ
2cos
3
2
1
0
q
q
veee
e
axisrotation :ˆ
anglerotation :
v
q
Quaternions
• William Rowan Hamilton (1805-1865)– Algebraic couples (complex number) 1833
iyx + 12 -=iwhere
Quaternions
• William Rowan Hamilton (1805-1865)– Algebraic couples (complex number) 1833
– Quaternions 1843
iyx + 12 -=iwhere
kzjyixw +++
jikikjkjijkiijkkij
ijkkji
-=-=-====-====
,,,,
1222where
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.”
Unit Quaternions
• Unit quaternions represent 3D rotations
),(),,,(
v
q
wzyxw
kzjyixw
==
+++=
12222 =+++ zyxw
3S
v̂
q
Rotation about an Arbitrary Axis
• Rotation about axis by angle
1-=¢ qpqp ),,,0( zyx=pwhere
),,( zyx),,( zyx ¢¢¢
Purely Imaginary Quaternion
v̂ q
÷øö
çèæ=
2sinˆ,
2cos qq vq
Unit Quaternion Algebra
• Identity
• Multiplication
• Inverse
• Unit quaternion space is– closed under multiplication and inverse,– but not closed under addition and subtraction
),(),)(,(
2112212121
221121
vvvvvvvvqq
´++×-==
wwwwww
)/(),,,()/(),,,(
2222
22221
zyxwzyxwzyxwzyxw
+++-=
+++---=-q
)0,0,0,1(=q
Unit Quaternion Algebra
• Antipodal equivalence– q and –q represent the same rotation
– 2-to-1 mapping between S and SO(3)– Twice as fast as in SO(3)
)()( pp qq -= RR
3
Coordinate Free Concepts
• Orientation– Unit quaternions parameterize 3D orientations without
singularity• (Finite) Rotation
– Unit quaternions can represent 3D rotation, but– More convenient to use 3D vectors
• Infinitesimal Rotation– Can be represented by 3D vectors
Tangent Vector(Infinitesimal Rotation)
3SqT
q
Tangent Vector(Infinitesimal Rotation)
3SqT
q
1-q
Tangent Vector(Infinitesimal Rotation)
3SIT),,,( 0001=I
),,,( zyx0
qq !12 -=wAngular Velocity
Exp and Log
log exp
I
)sinˆ,(cos)ˆexp()exp( qqq vvv ==
Exp and Log
log exp
I
)sinˆ,(cos)ˆexp()exp( qqq vvv == ÷øö
çèæ=
úúú
û
ù
êêê
ë
é
÷øö
çèæ=
2sinˆ
2cos
3
2
1
0
q
q
veee
e
Eulerparameters
Rotation Vector
1p
12 ppu -=
2p2q
)( 121
12
pppupp
-+=+=
1q3S3R
Rotation Vector
3S
211 qq-
111 qqI -=
1p
12 ppu -=
2p
)( 121
12
pppupp
-+=+=
3R
Rotation Vector
211 qq-
111 qqI -=
( )( )2111
12
logexp
)exp(
qqqvqq
-=
=
( )211log qqv -=
1p
12 ppu -=
2p
)( 121
12
pppupp
-+=+=
3R
Rotation Vector
• 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
vuvu eee +¹
Spherical Linear Interpolation
• SLERP [Shoemake 1985]– Linear interpolation of two orientations
tt-1
))(logexp(t
)(),(slerp
2111
211121
qqqqqqqq
-
-
×=
= tt
Spherical Linear Interpolation
))(logexp(t
)(),(slerp
2111
211121
qqqqqqqq
-
-
×=
= tt
Coordinate-Invariant Operations
Adding Rotation Vectors
Analogy
• (point : vector) is similar to (orientation : rotation)
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
• Why rotation matrices ?– One-to-one correspondence– Handle rotation and translation in a uniform way
• Eg) 4x4 homogeneous matrices
Rotation Conversions
• In theory, conversion between any representations is possible
• In practice, conversion is not simple because of different conventions
• Quaternion to Matrix
÷÷÷÷÷
ø
ö
ççççç
è
æ
+--+---+-++---+
=
1000022220222202222
2222000
02222
00
002222
0
zyxxzyyzx
xzyzyxzyx
yzxzyxzyx
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
R