Axis angle

From Wikipedia, the free encyclopedia

The axis angle (or rotation vector) representation of a rotation, also known as the exponential coordinates of a rotation, parameterizes a rotation by two values: a unit vector indicating the direction of a directed axis (straight line), and an angle describing the magnitude of the rotation about the axis. The rotation occurs in the sense prescribed by the right hand grip rule.

This representation evolves from Euler's rotation theorem, which implies that any rotation or sequence of rotations of a rigid body in a three-dimensional space is equivalent to a pure rotation about a single fixed axis.

The axis angle representation is equivalent to the more concise rotation vector representation. In this case, both the axis and the angle are represented by a non-normalized vector codirectional with the axis whose magnitude is the rotation angle.

[edit] Uses

The axis angle representation is convenient when dealing with rigid body dynamics. It is useful to both characterize rotations, and also for converting between different representations of rigid body motion, such as homogeneous transformations and twists.

[edit] Example

Say you are standing on the ground and you pick the direction of gravity to be the negative z direction. Then if you turn to your left, you will travel $\tfrac{\pi}{2}$ radians (or 90 degrees) about the z axis. In axis angle representation, this would be

$\langle \mathrm{axis}, \mathrm{angle} \rangle = \left( \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix},\theta \right) = \left( \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix},\frac{\pi}{2}\right)$

This can be represented as a rotation vector with a magnitude of $\tfrac{\pi}{2}$ pointing in the z direction.

$\begin{bmatrix} 0 \\ 0 \\ \frac{\pi}{2} \end{bmatrix}$

[edit] Relationship to other representations

There are many ways to represent a rotation. It is useful to understand how different representation relate to one another, and how to convert between them.

[edit] Exponential map from so(3) to SO(3)

The exponential map is used as a transformation from axis angle representation of rotations to rotation matrices.

$\exp\colon so(3) \to SO(3)$

Essentially, by using a Taylor expansion you can derive a closed form relationship between these two representations. Given an axis, $\omega \in \Bbb{R}^{3}$ having length 1, and an angle, $\theta \in \Bbb{R}$ , an equivalent rotation matrix is given by the following:

$R = \exp(\hat{\omega} \theta) = \sum_{k=0}^\infty\frac{(\hat{\omega}\theta)^k}{k!} = I + \hat{\omega} \theta + \frac{1}{2}(\hat{\omega}\theta)^2 + \frac{1}{6}(\hat{\omega}\theta)^3 + \cdots$

$R = I + \hat{\omega}\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right) + \hat{\omega}^2 \left(\frac{\theta^2}{2!} - \frac{\theta^4}{4!} + \frac{\theta^6}{6!} - \cdots\right)$

$R = I + \hat{\omega} \sin(\theta) + \hat{\omega}^2 (1-\cos(\theta))$

where R is a 3x3 rotation matrix and the hat operator gives the antisymmetric matrix equivalent of the cross product. This is the Rodrigues' rotation formula.

[edit] Log map from SO(3) to so(3)

To retrieve the axis angle representation of a rotation matrix calculate the angle of rotation:

$\theta = \arccos\left( \frac{\mathrm{trace}(R) - 1}{2} \right)$

and then use it to find the normalized axis:

$\omega = \frac{1}{2 \sin(\theta)} \begin{bmatrix} R(3,2)-R(2,3) \\ R(1,3)-R(3,1) \\ R(2,1)-R(1,2) \end{bmatrix}$

Note, also that the Matrix logarithm of the rotation matrix R is:

$\log R = \left\{ \begin{matrix} 0 & \mathrm{if} \; \theta = 0 \\ \frac{\theta}{2 \sin(\theta)} (R - R^T) & \mathrm{if} \; \theta \ne 0 \; \mathrm{and} \; \theta \in (-\pi, \pi) \end{matrix}\right.$

Except when R has eigenvalues equal to -1 where the log is not unique. However, even in the case where $θ = π$ the Frobenius norm of the log is:

$\| \log(R) \|_F = \sqrt{2} | \theta |$

Note that given rotation matrices A and B:

$d_g(A,B) := \| \log(A^TB)\|_F$

is the geodesic distance on the 3D manifold of rotation matrices.

[edit] Unit Quaternions

Main article: Quaternions and spatial rotation

To transform from axis angle coordinates to unit quaternions use the following expression:

$Q = \left(\cos\left(\frac{\theta}{2}\right), \omega \sin\left(\frac{\theta}{2}\right)\right)$

Given a unit quaternion, the axis angle coordinates can be extracted using the following:

$\theta = 2\,\arccos(q_0)\,$

$\omega = \left\{ \begin{matrix} \frac{q}{ \sin( \theta/2 ) } , & \mathrm{if} \; \theta \neq 0 \\ 0, & \mathrm{otherwise} \end{matrix} \right.$

It may also be useful to know:

$\frac{q}{ \sin( \theta/2 ) } = \frac{[q_1 q_2 q_3]^{\mathrm{T}}}{ \|{[q_1 q_2 q_3]^{\mathrm{T}}}\|}$

[edit] See also

SO(3) - the group of all rotations in three dimensional space
Charts on SO(3) for other representations
rotation group - a mathematical look at rotations
homogeneous coordinate transformations - a mathematical representation of rigid body motions, including both translation and rotation.
screw theory - a representation of rigid body motions and velocities using the concepts of twists, screws and wrenches
Rotation around a fixed axis
Rotation representation (mathematics)
Rodrigues' rotation formula

Axis angle

From Wikipedia, the free encyclopedia

Contents

[edit] Uses

[edit] Example

[edit] Relationship to other representations

[edit] Exponential map from so(3) to SO(3)

[edit] Log map from SO(3) to so(3)

[edit] Unit Quaternions

[edit] See also

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