rotation and orientation: fundamentals jehee lee seoul national university
TRANSCRIPT
Rotation and Orientation:Fundamentals
Jehee Lee
Seoul 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)
Reference coordinate system
Analogy
(point : vector) is similar to (orientation : rotation)
Both represent a sort of (state : movement)
Reference coordinate system
point : the 3d location of the bunnyvector : translational movement
Analogy
(point : vector) is similar to (orientation : rotation)
Both represent a sort of (state : movement)
Reference coordinate system
point : the 3d location of the bunnyvector : translational movement
orientation : the 3d orientation of the bunnyrotation : circular movement
2D Orientation
0
2
2
or
Polar Coordinates
2D Orientation
Although the motion is continuous, its representation could be discontinuous
0
2
2
or
)(t
Time
2D Orientation
Many-to-one correspondences between 2D orientations and their representations
0
2
2
or
)(t
Time
Extra Parameter
),( yx
X
Y122 yx
Extra Parameter
),( yx
X
Y122 yx
cossin
sincos
2x2 Rotation matrix is yet another method of using extra parameters
Complex Number
iyx
Real
Imaginary
Complex Exponentiation
iyx
Real
Imaginary
ie
iiyx
sincos
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
2
2D Rotation and Orientation
• 2D Rotation– The consequence of any 2D rotational movement can be
uniquely represented by a turning angle
• 2D Orientation– The non-singular parameterization of 2D orientations requires
extra parameters• Eg) Complex numbers, 2x2 rotation matrices
Operations in 2D
• (orientation) : complex number• (rotation) : scalar value• exp(rotation) : complex number
2D Rotation and Displacement
iyxc 1
Real
Imaginary
2D Rotation and Displacement
iyxc 1
Real
Imaginary
i
i
ecc
or
ecc
2
11
12
2c
2D Orientation Composition
Real
Imaginary
1c2c
)(21 undefinedcc
2D Rotation Composition
iii eee )(
Real
Imaginary
Analogy
X
Y
Z
X
Y
Z
3D Rotation
• Given two arbitrary orientations of a rigid object,
3D Rotation
• We can always find a fixed axis of rotation and an angle about the axis
v̂
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
• Gimble– 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
• Gimble lock– Coincidence of inner most
and outmost gimbles’ rotation axes
– Loss of degree of freedom
Euler Angles
• Euler angles are ambiguous– Two different Euler angles can represent the same
orientation
– This ambiguity brings unexpected results of animation where frames are generated by interpolation.
),2
,0 and )0,2
, 21 (R(θ),r,r(rR zyx
Ranges
• Euler angles in ZXZ– Z-axis by – X’-axis by – Z’’-axis by
Dependence
• Compare– Rotation about z-axis by 180 degree– Rotation about y-axis by 180 degree, followed by
another rotation about x-axis by 180 degree
• Rotations about x-, y-, and z-axes are dependent
Rotation Vector
• Rotation vector (3 parameters)
• Axis-Angle (2+1 parameters)
v̂
),,(ˆ zyx vv )ˆ,( v
anglescalar :
runit vecto : ˆ
v
3D Orientation
• Unhappy with three parameters – Euler angles
• Discontinuity (or many-to-one correspondences)• Gimble lock
– Rotation vector (a.k.a Axis/Angle)• Discontinuity (or many-to-one correspondences)
Using an Extra Parameter
• Euler parameters
2sinˆ
2cos
3
2
1
0
v
e
e
e
e
axisrotation :ˆ
anglerotation :
v
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
jikikjkji
jkiijkkij
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
w
zyxw
kzjyixw
12222 zyxw
3S
v̂
Rotation about an Arbitrary Axis
• Rotation about axis by angle
1 qpqp ),,,0( zyxpwhere
),,( zyx
),,( zyx
Purely Imaginary Quaternion
v̂
2sinˆ,
2cos
vq
Unit Quaternion Algebra
• Identity
• Multiplication
• Inverse
– Opposite axis or negative angle
• Unit quaternion space is– closed under multiplication and inverse,
– but not closed under addition and subtraction
),(
),)(,(
2112212121
221121
vvvvvv
vvqq
wwww
ww
)/(),,,(
)/(),,,(2222
22221
zyxwzyxw
zyxwzyxw
q
)0,0,0,1(q
Unit Quaternion Algebra
• Antipodal equivalence– q and –q represent the same rotation
– ex) rotation by about opposite direction– 2-to-1 mapping between S and SO(3)– Twice as fast as in SO(3)
)()( pp qq RR
3
3D Orientations and Rotations
Orientations and rotations are different in coordinate-invariant geometric programming
Use unit quaternions for orientation representation– 3x3 orthogonal matrix is theoretically identical
Use 3-vectors for rotation representation
Tangent Vector(Infinitesimal Rotation)
3SqT
q
Tangent Vector(Infinitesimal Rotation)
3SqT
q
1q
Tangent Vector(Infinitesimal Rotation)
3SIT
),,,( 0001I
),,,( zyx0
qq 1
2
1
Angular Velocity
Exp and Log
log exp
I
)sinˆ,(cos)ˆexp()exp( vvv
Exp and Log
log exp
I
)sinˆ,(cos)ˆexp()exp( vvv
2sinˆ
2cos
3
2
1
0
v
e
e
e
e
Eulerparameters
Rotation Vector
1p
12 ppu
2p
2q
)( 121
12
ppp
upp
1q3S3R
21
11
12
logexp
)exp(
qqq
vqq
Rotation Vector
3S
21
1 qq
11
1 qqI 1p
12 ppu
2p
)( 121
12
ppp
upp
3R
21
11
12
logexp
)exp(
qqq
vqq
Rotation Vector
21
1 qq
11
1 qqI
21
11
12
logexp
)exp(
qqq
vqq
21
1log qqv
1p
12 ppu
2p
)( 121
12
ppp
upp
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
Spherical Linear Interpolation
• SLERP [Shoemake 1985]– Linear interpolation of two orientations
t
t1
))(logexp(t
)(),(slerp
21
11
21
1121
qqq
qqqqq
tt
Spherical Linear Interpolation
))(logexp(t
)(),(slerp
21
11
21
1121
qqq
qqqqq
tt
Coordinate-Invariant Operations
Adding Rotation Vectors
Affine Combination of Orientations
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
• Cf) matrix orthogonalization (Gram-shmidt process, QR, SVD decomposition)
• 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 always possible
• In practice, conversion is not straightward because of difference in convention
• Quaternion to Matrix
1000
02222
02222
02222
2222000
02222
00
002222
0
zyxxzyyzx
xzyzyxzyx
yzxzyxzyx
qqqqqqqqqqqq
qqqqqqqqqqqq
qqqqqqqqqqqq
R