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:

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

where

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

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 about 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 q is given by

${\displaystyle {\begin{bmatrix}1-2(q_{2}^{2}+q_{3}^{2})&2(q_{1}q_{2}-q_{0}q_{3})&2(q_{0}q_{2}+q_{1}q_{3})\\2(q_{1}q_{2}+q_{0}q_{3})&1-2(q_{1}^{2}+q_{3}^{2})&2(q_{2}q_{3}-q_{0}q_{1})\\2(q_{1}q_{3}-q_{0}q_{2})&2(q_{0}q_{1}+q_{2}q_{3})&1-2(q_{1}^{2}+q_{2}^{2})\end{bmatrix}}}$ 

The orthogonal matrix corresponding to a rotation with Euler angles ${\displaystyle \phi ,\theta \,and\,\psi }$ , 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....

${\displaystyle \mathbf {q} ={\begin{bmatrix}cos_{2}\phi cos_{2}\theta cos_{2}\psi +sin_{2}\phi sin_{2}\theta sin_{2}\psi \\sin_{2}\phi cos_{2}\theta cos_{2}\psi +cos_{2}\phi sin_{2}\theta sin_{2}\psi \\cos_{2}\phi sin_{2}\theta cos_{2}\psi +sin_{2}\phi cos_{2}\theta sin_{2}\psi \\cos_{2}\phi cos_{2}\theta sin_{2}\psi +sin_{2}\phi sin_{2}\theta cos_{2}\psi \\\end{bmatrix}}}$ 

where ${\displaystyle cos_{2}\alpha }$  is a notational shortcut for ${\displaystyle cos{\frac {\alpha }{2}}}$ , and ${\displaystyle sin_{2}\alpha }$  for ${\displaystyle sin{\frac {\alpha }{2}}}$ . And for Euler angles we get:

${\displaystyle {\begin{bmatrix}\phi \\\theta \\\psi \end{bmatrix}}={\begin{bmatrix}atan{\frac {2(e_{0}e_{1}+e_{2}e_{3})}{1-2(e_{1}^{2}+e_{2}^{2})}}\\asin(2(e_{0}e_{2}-e_{3}e_{1}))\\atan{\frac {2(e_{0}e_{3}+e_{1}e_{2})}{1-2(e_{2}^{2}+e_{3}^{2})}}\end{bmatrix}}}$ 

Singularities

One must be aware of singularities in the Euler angle parametrizartion when the pitch approaches ${\displaystyle \pm 90^{o}}$  (north/south pole). These cases must be handled specially