my head is spinning...

Four different ways to represent rotation

The space of rotations

Special orthogonal group(3):

( ) 1)det(,|3 33 +==∈= × RIRRRRSO T

Rotations preserve distance: 2121 ppRpRp −=−

Rotations preserve orientation: ( ) ( ) ( )2121 ppRRpRp ×=×

Why ? 1)det( ±=R

The space of rotations

Special orthogonal group(3):

( ) 1)det(,|3 33 +==∈= × RIRRRRSO T

Why it’s a group:

• Closed under multiplication: if then

• Has an identity:

• Has a unique inverse…

• Is associative…

( )321 SORR ∈( )3, 21 SORR ∈( ) 11 s.t. 3 RIRSOI =∈∃

Why orthogonal:

• vectors in matrix are orthogonal

Why it’s special: , NOT1)det( +=R 1)det( ±=R

Right hand coordinate system

Possible rotation representations

You need at least three numbers to represent an arbitrary rotation in SO(3) (Euler theorem). Some three-number representations:

• ZYZ Euler angles

• ZYX Euler angles (roll, pitch, yaw)

• Axis angle

One four-number representation:

• quaternions

To get from A to B:

1. Rotate about z axis

2. Then rotate about y axis

3. Then rotate about z axis

ZYZ Euler Angles

=

ψθφ

zyzr

−=

100

0cossin

0sincos

)( φφφφ

φzR

−=

θθ

θθθ

cos0sin

010

sin0cos

)(yR

φθψ

−=

100

0cossin

0sincos

)( ψψψψ

ψzR

( ) )()()(,, ψθφψθφ zyzzyz RRRR =

Therefore, the sequence of rotations is concatentated as follows:

ZYZ Euler Angles

( )

−

−

−=

100

0cossin

0sincos

cos0sin

010

sin0cos

100

0cossin

0sincos

,, ψψψψ

θθ

θθφφφφ

ψθφzyzR

Remember that encode the desired rotation in the pre-rotation reference frame:

)(φzR )(θyR )(ψzR

rotationpostrotationpre

z RR −−=)(φ

( )

−+−+−−−

=

θψθψθ

θφψφψθφψφψθφ

θφψφψθφψφψθφ

ψθφcsscs

ssccscsscccs

sccssccssccc

Rzyz ,,

−=

100

0cossin

0sincos

)( φφφφ

φzR

−=

θθ

θθθ

cos0sin

010

sin0cos

)(yR

−=

ψψψψψ

cossin0

sincos0

001

)(xR

φTo get from A to B:

1. Rotate about z axis

2. Then rotate about y axis

3. Then rotate about x axis

θψ

( ) )()()(,, ψθφψθφ xyzzyx RRRR =

ZYX Euler Angles (roll, pitch, yaw)

( )

−

−

−=

ψψψψ

θθ

θθφφφφ

ψθφcossin0

sincos0

001

cos0sin

010

sin0cos

100

0cossin

0sincos

,,zyzR

In Euler angles, the each rotation is imagined to be represented in the post-rotation coordinate frame of the last rotation

( ) )()()(,, ψθφψθφ xyzzyx RRRR =

ZYX Euler Angles (roll, pitch, yaw)

In Fixed angles, all rotations are imagined to be represented in the original (fixed) coordinate frame.

ZYX Euler angles can be thought of as:

1. ZYX Euler

2. XYZ Fixed

Problems w/ Euler Angles

If two axes are aligned, then there is a “don’t care” manifold of Euler angles that represent the same orientation

• The system loses one DOF

=

0

90

0

1r

=

90

89

90

2r

−

−=−

90

1

90

21 rr , but the actual distance is 1

Problem w/ Euler Angles: gimbal lock

1. When a small change in orientation is associated with a large change in rotation representation

2. Happens in “singular configurations” of the rotational representation (similar to singular configurations of a manipulator)

3. This is a problem w/ any Euler angle representation

Problem w/ Euler Angles: gimbal lock

Problem w/ Euler Angles: gimbal lock

Axis-angle representation

Theorem: (Euler). Any orientation, , is equivalent to a rotation about a fixed axis, , through an angle

=

z

y

x

k

k

k

k

( )3SOR∈3R∈ω )2,0[ πθ ∈

θAxis: Angle:

( ) ( ) ( ) ( ) ( )( )θθθθ cos1sin 2 −++== kkkk SSIeR S

(also called exponential coordinates)

[ ]book... in theequation that =

Rodrigues’ formula

Converting to a rotation matrix:

Axis-angle representation

−== −

2

1)(cos 1 Rtrace

kθ

=

332313

232212

131211

rrr

rrr

rrr

R

Magnitude of rotation:

332211)( rrrRtrace ++=

Where:

−−−

=

1221

3113

2332

sin2

1ˆ

rr

rr

rr

kθ

Axis of rotation:

and:

Converting to axis angle:

Axis-angle representation

Axis angle is can be encoded by just three numbers instead of four:

k

kk

=ˆ k=θIf then0≠k and

For most orientations, , is unique.kR

If the three-number version of axis angle is used, then

IR =0

For rotations of , there are two equivalent representations:180

kk RR −=If then180=k

Axis-angle problems

Still suffers from the “edge” and distance preserving problems of Euler angles:

=

179

0

0

1r

−=

179

0

0

2r

=−

358

0

0

21 rr

, but the actual distance is 2

Distance metric changes as you get further from origin.

Projection distortions

Example: differencing rotations

=

0

02

1

π

kCalculate the difference between

these two rotations:

=

0

0

22πk

−=−

02

2

21π

π

kkThis is NOT the right answer:

27.1272

21 ==− πkkAccording to that, this is the

magnitude of the difference:

=

0

02

1

π

k

Convert to rotation matrices to solve this problem:

=

0

0

22πk

−−=

−

−==

010

001

100

001

010

100

010

100

001

2121 RRR BTB

( ) ( ) ( )( ) ( )

−=

−==

010

100

001

cossin0

sincos0

001

22

2221

ππ

πππx

b RR

( )( ) ( )

( ) ( )

−=

−==

001

010

100

cos0sin

010

sin0cos

22

22

22

ππ

ππ

πy

b RR

πθ 3211

2

1cos

2

1)(cos =

−=

−= −− Rtrace

Example: differencing rotations

−

−=

−−−

=1

1

1

3

1

sin2

1ˆ

1221

3113

2332

rr

rr

rr

kθ

−

−=

1

1

1

332 π

k

2121 RRR BTB=

So far, rotation matrices seem to be the most reliable method of manipulating rotations. But there are problems:

• Over a long series of computations, numerical errors can cause these 3x3 matrices to no longer be orthogonal (you need to “orthogonalize” them from time to time).

• Although you can accurately calculate rotation differences, you can’t interpolate over a difference.’

• Suppose you wanted to smoothly rotate from one orientation to another – how would you do it?

Quaternions

Answer: quaternions…

Quaternions

3210 kqjqiqqQ +++=Generalization of complex numbers:

( )qqQ ,0=

( ) ( )qqqqQ −== ,, 0*

0*

Essentially a 4-dimensional quantity

( )( )32103210 kpjpippkqjqiqqQP ++++++=Multiplication:

( )qppqqpqpqpQP ×++⋅−= 0000 ,

Complex conjugate:

1−==== ijkkkjjii

kjiij =−=

ikjjk =−=

jikki =−=

Properties of complex dimensions:

Quaternions

Invented by Hamilton in 1843:

Along the royal canal in Dublin…

Quaternions

123

22

21

20

2 =+++= qqqqQLet’s consider the set of unit

quaternions:

This is a four-dimensional hypersphere, i.e. the 3-sphere 3S

Therefore, the inverse of a unit quaternion is: 1* −=QQ

( ) ( ) ( ) )0,1(,,, 002

0000* =×+−−=−= qqqqqqqqqqqqqQQ

The identity quaternion is: ( )0,1=Q

Since:

You can rotate from frame a to b:

Quaternions

Pa *ba

aba

b PQQP =

Given a unit axis, , and an angle, :

Associate a rotation with a unit quaternion as follows:

k θ (just like axis angle)

=

2sinˆ,

2cos

,ˆ

θθθ kQk

The associated quaternion is:

Therefore, represents the same rotation asQ Q−

Let be the quaternion associated with the vector ( )pP ii ,0= pi

bacbca QQQ =Composition:

1−= bacacb QQQInversion:

Rotate by

Example: Quaternions

=

0

0

,2

12

1Q

=

0

0

1

,0Pa

−

==

0

0

,

0

0

1

,0

0

0

,2

12

12

12

1*PQQP ab

−

=

21

21

21

21 0,0

0

0

,

−=

−

−+

−=

1

0

0

,000,0

21

21

21

21

=

0

02

1

π

kFind the difference between these two axis angle

rotations:

=

0

0

22πk

Example: Quaternions

=

0

0

,2

12

1cbQ

=

0

0,2

1

21

baQ( ) ( )2

144 cossin == ππ

−

== −

0

0,

0

0

,2

1

21

21

211

bacacb QQQ

( )qppqqpqpqpQP ×++⋅−= 0000 ,

−

−=

−

−=

21

21

21

21

21

21

21

21

21 ,,

( ) πθ 32

211cos == −

cb

−

−=

31

31

31

cbk

Quaternions: Interpolation

Suppose you’re given two rotations, and

How do you calculate intermediate rotations?1R 2R

( ) 21 1 RRRi αα −+= This does not even result in a rotation matrix

Do quaternions help?

( )( ) 21

21

1

1

QQ

QQQi αα

αα−+−+=

Suprisingly, this actually works

• Finds a geodesic

This method normalizes automatically (SLERP):

( )Ω

Ω+Ω−=sin

sin1sin 21 αα QQQi