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 between the two representations.

A unit quaternion can be described as:

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

where

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

Similarly 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 terms 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:

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

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.