Four different ways to represent rotation my head is spinning... - PowerPoint PPT Presentation

Four different ways to represent rotation my head is spinning...

The space of rotations { } ( ) × 3 3 T = ∈ = = + SO 3 R R | RR I , det( R ) 1 Special orthogonal group(3): = ± det( R ) 1 Why ? − = − Rp Rp p p Rotations preserve distance: 1 2 1 2 ( ) ( ) ( ) × = × Rp Rp R p p Rotations preserve orientation: 1 2 1 2

The space of rotations { } ( ) × 3 3 T = ∈ = = + SO 3 R R | RR I , det( R ) 1 Special orthogonal group(3): Why it’s a group: ( ) ( ) ∈ ∈ R , R SO 3 • R R SO 3 Closed under multiplication: if then 1 2 1 2 ( ) • Has an identity: ∃ ∈ = I SO 3 s.t. IR R 1 1 • Has a unique inverse… • Is associative… Why orthogonal: • vectors in matrix are orthogonal = ± = + det( R ) 1 det( R ) 1 Why it’s special: , NOT 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

ZYZ Euler Angles φ     = θ r   zyz   ψ   φ − φ cos sin 0     To get from A to B: φ = φ φ R ( ) sin cos 0   z   φ 0 0 1 1. Rotate about z axis   θ θ cos 0 sin   θ 2. Then rotate about y axis   θ = R ( ) 0 1 0   ψ y 3. Then rotate about z axis   − θ θ sin 0 cos   ψ − ψ  cos sin 0    ψ = ψ ψ R ( ) sin cos 0   z   0 0 1  

ZYZ Euler Angles ( φ ( θ ( ψ R ) R ) R ) Remember that encode the desired rotation in the pre- z y z rotation reference frame: − ( φ = pre rotation R ) R − z post rotation Therefore, the sequence of rotations is concatentated as follows: ( ) φ θ ψ = φ θ ψ R , , R ( ) R ( ) R ( ) zyz z y z φ − φ θ θ ψ − ψ  cos sin 0   cos 0 sin   cos sin 0        ( ) φ θ ψ = φ φ ψ ψ R , , sin cos 0 0 1 0 sin cos 0       zyz       − θ θ 0 0 1 sin 0 cos 0 0 1         − − − c c c s s c c s s c c s   φ θ ψ φ ψ φ θ ψ φ ψ φ θ ( ) φ θ ψ = + − + R zyz , , s c c c s s c s c c s s   φ θ ψ φ ψ φ θ ψ φ ψ φ θ   − s c s s c   θ ψ θ ψ θ

ZYX Euler Angles (roll, pitch, yaw) φ − φ  cos sin 0    To get from A to B: φ = φ φ R ( ) sin cos 0   z   φ 1. Rotate about z axis 0 0 1   θ θ  cos 0 sin  θ 2. Then rotate about y axis   θ = R ( ) 0 1 0   ψ y 3. Then rotate about x axis   − θ θ sin 0 cos   1 0 0     ψ = ψ − ψ R ( ) 0 cos sin   x   ψ ψ 0 sin cos   ( ) φ θ ψ = φ θ ψ R , , R ( ) R ( ) R ( ) zyx z y x φ − φ θ θ cos sin 0 cos 0 sin 1 0 0             ( ) φ θ ψ = φ φ ψ − ψ R , , sin cos 0 0 1 0 0 cos sin       zyz       − θ θ ψ ψ 0 0 1 sin 0 cos 0 sin cos      

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         = = r 90 r  89    1 2      0 90        − 90     − = r r  1  1 , but the actual distance is 1 2    − 90  

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 ( ) R ∈ SO 3 Theorem: (Euler). Any orientation, , is equivalent to a rotation 3 ω ∈ about a fixed axis, , through an angle R θ ∈ π [ 0 , 2 ) (also called exponential coordinates ) k   x   θ Angle: = k k Axis:   y   k   z Converting to a rotation matrix: ( ) ( ) ( ) ( ( ) ) ( ) 2 S k θ = = + θ + − θ R e I S k sin S k 1 cos k θ [ ] = that equation in the book... Rodrigues’ formula

Axis-angle representation Converting to axis angle: − trace ( R ) 1   cos 1 − θ = = k   Magnitude of rotation: 2   − r r   32 23   1 ˆ Axis of rotation: = − k r r   13 31 θ 2 sin   − r r   21 12  r r r  11 12 13   Where: = R r r r   12 22 23   r r r   13 23 33 = + + trace ( R ) r r r and: 11 22 33

Axis-angle problems Still suffers from the “edge” and distance preserving problems of Euler angles:  0    0    0  − = r r 0       1 2 =   r 0   = r 0    358 1   2      179  −   179    2 , but the actual distance is Distance metric changes as you get further from origin.

Axis-angle representation Axis angle is can be encoded by just three numbers instead of four: ≠ θ = k 0 k If then and If the three-number version of axis angle is used, then R = I 0 R For most orientations, , is unique. k  180 For rotations of , there are two equivalent representations:  = k 180 = R R If then k − k

Projection distortions

Example: differencing rotations 0     π 2     Calculate the difference between = k = π k 0     these two rotations: 2 2 1     0 0       π 2   − = − k k This is NOT the right answer: π   2 1 2   0   π According to that, this is the  − = = k k 127 . 27 1 2 magnitude of the difference: 2

Example: differencing rotations Convert to rotation matrices to solve this problem: 0     π 2     = k = π k 0     T 1 B B = R R R 2 2 1 2 1 2     0 0     1 0 0 1 0 0         ( ) ( ) ( ) b = = − = − R R 0 cos sin 0 0 1 π π π     2 2 2 1 x     ( ) ( ) 0 sin cos 0 1 0 π π     2 2 ( ) ( ) cos 0 sin 0 0 1     π π 2 2     ( ) b = = = R R 0 1 0 0 1 0 π     2 2 y     ( ) ( ) − − sin 0 cos 1 0 0 π π     2 2 1 0 0 0 0 1 0 0 1             T 1 B B = = = − R R R 0 0 1 0 1 0 1 0 0       2 1 2       − − − 0 1 0 1 0 0 0 1 0       − −  r r   1  − 1   32 23 2 π       1 1 −  trace ( R ) 1  − 1  ˆ = − = k r r 1 − − θ = 1 = 1 = π     = cos cos k 3 1     2   13 31 3 2 sin θ 2 2     3 3       − − r r 1 − 1       21 12

Quaternions 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? Answer: quaternions…