Rotation and Orientation: Fundamentals Jehee Lee Seoul National University

What is Rotation ? Not intuitive Many different ways to describe 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 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) point : the 3d location of the bunny vector : translational movement Reference coordinate system

Analogy (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

2D Orientation Polar Coordinates

2D Orientation Although the motion is continuous, its representation could be discontinuous

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

Extra Parameter

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

Complex Number Imaginary Real

Complex Exponentiation Imaginary Real

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 Imaginary Real

2D Rotation and Orientation 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 Imaginary Real

2D Rotation and Displacement Imaginary Real

2D Orientation Composition Imaginary Real

2D Rotation Composition Imaginary Real

Analogy

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

Euler’s Rotation Theorem The general displacement of a rigid body with one point fixed is a rotation about some axis Leonhard Euler (1707-1783) In other words, Arbitrary 3D rotation equals to one rotation around an axis Any 3D rotation leaves one vector unchanged

Euler angles Gimble Hardware implementation of Euler angles Aircraft, Camera

Euler Angles Rotation about three orthogonal axes Gimble lock 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.

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

Dependence Compare Rotations about x-, y-, and z-axes are dependent 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)

3D Orientation Unhappy with three parameters Euler angles Discontinuity (or many-to-one correspondences) Gimble lock Rotation vector (a.k.a Axis/Angle)

Using an Extra Parameter Euler parameters

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

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

Quaternions William Thomson Arthur Cayley “… 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

Rotation about an Arbitrary Axis Rotation about axis by angle where Purely Imaginary Quaternion

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

Unit Quaternion Algebra 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

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)

Tangent Vector (Infinitesimal Rotation)

Tangent Vector (Infinitesimal Rotation) Angular Velocity

Exp and Log log exp

Exp and Log Euler parameters log exp

Rotation Vector

Rotation Vector

Rotation Vector

Rotation Vector Finite rotation Infinitesimal 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

Spherical Linear Interpolation SLERP [Shoemake 1985] Linear interpolation of two orientations

Spherical Linear Interpolation

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