Conversion between quaternions and Euler angles: Difference between revisions

Revision as of 23:23, 27 May 2005

Revision as of 23:33, 27 May 2005

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Spatial rotations in three dimensions can be parametrized using both [[Euler angles]] and [[Quaternions and spatial rotation|unit quaternions]]. This article explains how to convert betweeen the two representatios.

A unit quaternion can be described as:

:<math>\mathbf{q} = \begin{bmatrix} q_0 & q_1 & q_2 & q_3 \end{bmatrix}^T</math>

where

:<math>|\mathbf{q}|^2 = q_0^2 + q_1^2 + q_2^2 + q_3 = 1</math>

Simmilarly for Euler angles, we use:

* Roll - <math>\phi</math>: rotation about the X-axis

* Pitch - <math>\theta</math>: rotation about the Y-axis

* Yaw - <math>\psi</math>: rotation the Z-axis

where the X-axis points forward, Y-axis to the right and Z-axis downward

== Rotation matrices ==

The [[orthogonal matrix]] corresponding to a rotation by the unit quaternion ''z'' = ''a'' + ''bi'' + ''cj'' + ''dk'' (with |z| = 1) is given by

:<math>\begin{bmatrix}

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-sin\theta & sin\phi cos\theta & cos\phi cos\theta \\

\end{bmatrix}</math>

== Conversion ==

By comparing the tems in the two matrices, we get....