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Rotation and Orientation: Fundamentals

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Rotation and Orientation:Fundamentals

Jehee Lee

Seoul National University

- 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

- 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

(point : vector) is similar to (orientation : rotation)

Both represent a sort of (state : movement)

Reference coordinate system

(point : vector) is similar to (orientation : rotation)

Both represent a sort of (state : movement)

point: the 3d location of the bunny

vector: translational movement

Reference coordinate system

(point : vector) is similar to (orientation : rotation)

Both represent a sort of (state : movement)

point: the 3d location of the bunny

vector: translational movement

orientation: the 3d orientation of the bunny

rotation : circular movement

Reference coordinate system

Polar Coordinates

Although the motion is continuous,

its representation could be discontinuous

Many-to-one correspondences between 2D orientations and their representations

2x2 Rotation matrix is yet another method of using extra parameters

Imaginary

Real

Imaginary

Real

- 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

Imaginary

Real

- 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

- The non-singular parameterization of 2D orientations requires extra parameters

- (orientation) : complex number
- (rotation) : scalar value
- exp(rotation) : complex number

Imaginary

Real

Imaginary

Real

Imaginary

Real

Imaginary

Real

- Given two arbitrary orientations of a rigid object,

- We can always find a fixed axis of rotation and an angle about the axis

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 with

one point fixed is a rotation about some axis

Leonhard Euler (1707-1783)

- Gimble
- Hardware implementation of Euler angles
- Aircraft, Camera

- Rotation about three orthogonal axes
- 12 combinations
- XYZ, XYX, XZY, XZX
- YZX, YZY, YXZ, YXY
- ZXY, ZXZ, ZYX, ZYZ

- 12 combinations
- Gimble lock
- Coincidence of inner most and outmost gimbles’ rotation axes
- Loss of degree of freedom

- 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.

- Euler angles in ZXZ
- Z-axis by a ( -p , p ]
- X’-axis by b ( -p/2 , p/2 ]
- Z’’-axis by g ( -p , p ]

- 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(3 parameters)
- Axis-Angle(2+1 parameters)

- 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)

- Euler angles

- Euler parameters

- William Rowan Hamilton (1805-1865)
- Algebraic couples (complex number) 1833

where

- William Rowan Hamilton (1805-1865)
- Algebraic couples (complex number) 1833
- Quaternions 1843

where

where

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 represent 3D rotations

- Rotation about axis by angle

where

Purely Imaginary Quaternion

- Identity
- Multiplication
- Inverse
- Opposite axis or negative angle

- Unit quaternion space is
- closed under multiplication and inverse,
- but not closed under addition and subtraction

- Antipodal equivalence
- q and –q represent the same rotation
- ex) rotation by p-q about opposite direction
- 2-to-1 mapping between S and SO(3)
- Twice as fast as in SO(3)

3

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

Angular Velocity

log

exp

Euler

parameters

log

exp

- 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

- SLERP [Shoemake 1985]
- Linear interpolation of two orientations

- (point : vector) is similar to (orientation : rotation)

- 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

- In theory, conversion between any representations is always possible
- In practice, conversion is not straightward because of difference in convention
- Quaternion to Matrix