Conversion between quaternions and Euler angles

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

A unit quaternion can be described as:

${\displaystyle \mathbf {q} ={\begin{bmatrix}q_{0}&q_{1}&q_{2}&q_{3}^{2}\end{bmatrix}}^{T}}$

where

${\displaystyle |\mathbf {q} |^{2}=q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}=1}$

Simmilarly for Euler angles, we use:

• Roll - ${\displaystyle \phi }$: rotation about the X-axis
• Pitch - ${\displaystyle \theta }$: rotation about the Y-axis
• Yaw - ${\displaystyle \psi }$: 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

${\displaystyle {\begin{bmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2bc-2ad&2ac+2bd\\2ad+2bc&a^{2}-b^{2}+c^{2}-d^{2}&2cd-2ab\\2bd-2ac&2ab+2cd&a^{2}-b^{2}-c^{2}+d^{2}\\\end{bmatrix}}}$ 

The orthogonal matrix corresponding to a rotation with Euler angles ${\displaystyle \phi (roll),\theta (pitch)and\psi (yaw)}$ , is given by

${\displaystyle {\begin{bmatrix}cos\theta cos\psi &-cos\phi sin\psi +sin\phi sin\theta cos\psi &sin\phi sin\psi +cos\phi sin\theta cos\psi \\cos\theta sin\psi &cos\phi cos\psi +cos\phi sin\theta sin\psi &-sin\phi cos\psi +cos\phi sin\theta sin\psi \\-sin\theta &sin\phi cos\theta &cos\phi cos\theta \\\end{bmatrix}}}$ 

Conversion

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