Conversion between quaternions and Euler angles: Difference between revisions

Revision as of 17:25, 9 March 2023

Revision as of 09:00, 19 September 2023

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A direct formula for the conversion from a quaternion to Euler angles in any of the 12 possible sequences exists.<ref>{{cite journal |last1=Bernardes |first1=Evandro |last2=Viollet |first2=Stéphane |title=Quaternion to Euler angles conversion: A direct, general and computationally efficient method |journal=PLOS ONE |date=10 November 2022 |volume=17 |issue=11 |pages=e0276302 |doi=10.1371/journal.pone.0276302 |pmid=36355707 |pmc=9648712 |bibcode=2022PLoSO..1776302B |language=en |issn=1932-6203|doi-access=free }}</ref> For the rest of this section, the formula for the sequence '''Body 3-2-1''' will be shown.

If the quaternion is properly '''normalized''', the Euler angles can be obtained from the quaternions via the relations:

:<math>\begin{bmatrix}

\phi \\ \theta \\ \psi

\end{bmatrix} =

\begin{bmatrix}

\mbox{arctan} \left(\dfrac {2(q_w q_x + q_y q_z)} {1 - 2(q_x^2 + q_y^2)}\right)\\

- \pi/2 + 2 \, \mbox{arctan} \sqrt{\dfrac {1 + 2(q_w q_y - q_x q_z)} {1 - 2(q_w q_y - q_x q_z)}} \\

\mbox{arctan} \left(\dfrac {2(q_w q_z + q_x q_y)} {1 - 2(q_y^2 + q_z^2)}\right)

\end{bmatrix} </math>

However the [[arctan]] functions implemented in computer languages only produce results between −π/2 and [[right angle|π/2]], to generate all the orientations one needs to replace the arctan functions in computer code by [[atan2]]:

:<math>\begin{bmatrix}

Line 212 ⟶ 201:

\end{bmatrix} </math>

HoweverNote that the [[arctan]] functions implemented in computer languages only produce results between −π/2 and [[right angle|π/2]], which is why [[atan2]] is used to generate all the correct orientations. oneMoreover, needstypical toimplementations replace theof arctan functionsalso inmight computerhave codesome bynumerical disadvantages near zero and one. [[atan2]]:

Moreover, typical implementations of arctan also might have some numerical disadvantages near zero and one. Some implementations use the equivalent expression:<ref>{{cite journal|last1=Blanco|first1=Jose-Luis|title=A tutorial on se (3) transformation parameterizations and on-manifold optimization|journal=University of Malaga, Tech. Rep|date=2010|citeseerx=10.1.1.468.5407}}</ref>

Moreover, typical implementations of arctan also might have some numerical disadvantages near zero and one. Some implementations use the equivalent expression:<ref>{{cite journal|last1=Blanco|first1=Jose-Luis|title=A tutorial on se (3) transformation parameterizations and on-manifold optimization|journal=University of Malaga, Tech. Rep|date=2010|citeseerx=10.1.1.468.5407}}</ref>

:<math> \theta = \mbox{arcsin} (2(q_w q_y - q_x q_z)) </math>