Quaternions and rotations

Introduction

Quaternion is a generalization of complex numbers with three imaginary numbers ($i$, $j$ and $k$ ). It is a four-dimensional complex number that can be used to represent the orientation of a rigid body or coordinate frame in three-dimensional space. The general definition of a quaternion is given by:

$$ Q=a+b.i+c.j+d.k = \left[ \begin{matrix} a && b && c && d \end{matrix} \right] $$

Representation

Quaternions represents a rotation tranformation in 3D. It can be expressed from Euler angles as on this online visualization. Therefore, the easiest way to represent a quaternion is to imagine the rotation of a given angle around a given vector. The following figure illustrates the rotation of angle $ \theta $ around vector $ \vec{V} $ defined by 3 scalars ( $ V_x $, $ V_y $ and $ V_z $) :

The quaternion associated to this transformation is given by:

$$ Q = \left[ \begin{matrix} q_w && q_x && q_y && q_z \end{matrix} \right] $$

$$ Q = \left[ \begin{matrix} cos \frac{\theta}{2} && V_x.sin \frac{\theta}{2} && V_y.sin \frac{\theta}{2} && V_z.sin \frac{\theta}{2} \end{matrix} \right] $$

Rotation around axes

Based on the previous formula, we can now calculate the quaternion defining a rotation around each axis:

Rotation around X

$$ Q_X=\left[ \begin{matrix} cos \frac{\theta}{2} && sin \frac{\theta}{2} && 0 && 0 \end{matrix} \right] $$

Rotation around Y

$$ Q_Y=\left[ \begin{matrix} cos \frac{\theta}{2} && 0 && sin \frac{\theta}{2} && 0 \end{matrix} \right] $$

Rotation around Z

$$ Q_Z=\left[ \begin{matrix} cos \frac{\theta}{2} && 0 && 0 && sin \frac{\theta}{2} \end{matrix} \right] $$

General rotation

Let's now assume we want to calculate the coordinates of a given vector $ \vec{v}_A $ (or point) rotated according to the quaternion $ {}^BQ_A $. The resulting vector $ \vec{v}_B $ ) can be calculated by the following formula based on the quaternion product and quaternion conjugate.

$$ \vec{V}_B = {}^BQ_A \otimes \vec{V}_A \otimes \overline {{}^BQ_A} $$

Note that $ \vec{V}_A $ and $ \vec{V}_B $ live in $ \mathbb{R}^4 $ while $ \vec{v}_A $ and $ \vec{v}_B $ live in $ \mathbb{R}^3 $. $ \vec{V}_A $ and $ \vec{V}_B $ are pure quaternions whose real parts are zero:

$$ \vec{V}_A = \begin{bmatrix} 0 \\ \vec{v}_A \end{bmatrix} = \begin{bmatrix} 0 \\ x_A \\ y_A \\ z_A \end{bmatrix} $$

$$ \vec{V}_B = \begin{bmatrix} 0 \\ \vec{v}_B \end{bmatrix} = \begin{bmatrix} 0 \\ x_B \\ y_B \\ z_B \end{bmatrix} $$

Another way to visualize the previous result is to consider that $ \vec{Q_{V_A}} $ and $ \vec{Q_{V_B}} $ are the same vector described in frame $ A $ and frame $ B $ respectively. The quaternion $ {}^BQ_A $ represents the transformation from frame $ A $ and frame $ B $.

Example

This simple example shows the rotation of an ellipse defined in the plane { $ \vec{X} $ , $ \vec{Y} $ }. The green and red points are respectively the initial and final position of the ellipses. The black dash-dotted line is the rotation axis. The rotation angle is $ \frac {\pi}{4} $ and the rotation vector is given by:

$$ V_R = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} $$

Download

Files for the previous example can be downloaded here:

ellipse_rotation.zip

Credit: based on the Madgwick's quaternion library for Matlab.