# Rotation Angles to Quaternions

Calculate quaternion from rotation angles

## Library

Utilities/Axes Transformations

## Description

The Rotation Angles to Quaternions block converts the rotation described by the three rotation angles (R1, R2, R3) into the four-element quaternion vector (q0, q1, q2, q3). A quaternion vector represents a rotation about a unit vector $\left({\mu }_{x},{\mu }_{y},{\mu }_{z}\right)$ through an angle θ. A unit quaternion itself has unit magnitude, and can be written in the following vector format.

$q=\left[\begin{array}{l}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]=\left[\begin{array}{c}\mathrm{cos}\left(\theta /2\right)\\ \mathrm{sin}\left(\theta /2\right){\mu }_{x}\\ \mathrm{sin}\left(\theta /2\right){\mu }_{y}\\ \mathrm{sin}\left(\theta /2\right){\mu }_{z}\end{array}\right]$

An alternative representation of a quaternion is as a complex number,

$q={q}_{0}+i{q}_{1}+j{q}_{2}+k{q}_{3}$

where, for the purposes of multiplication,

$\begin{array}{l}{i}^{2}={j}^{2}={k}^{2}=-1\\ ij=-ji=k\\ jk=-kj=i\\ ki=-ik=j\end{array}$

The benefit of representing the quaternion in this way is the ease with which the quaternion product can represent the resulting transformation after two or more rotations.

## Parameters

Rotation Order

Specifies the output rotation order for three wind rotation angles. From the list, select ZYX, ZYZ, ZXY, ZXZ, YXZ, YXY, YZX, YZY, XYZ, XYX, XZY, or XZX. The default is ZYX.

## Inputs and Outputs

InputDimension TypeDescription

First

3-by-1 vectorContains the rotation angles, in radians.

OutputDimension TypeDescription

First

4-by-1 quaternion vectorContains the quaternion vector.

## Assumptions and Limitations

The limitations for the 'ZYX', 'ZXY', 'YXZ', 'YZX', 'XYZ', and 'XZY' implementations generate an R2 angle that is between ±90 degrees, and R1 and R3 angles that are between ±180 degrees.

The limitations for the 'ZYZ', 'ZXZ', 'YXY', 'YZY', 'XYX', and 'XZX' implementations generate an R2 angle that is between 0 and 180 degrees, and R1 and R3 angles that are between ±180 degrees.

Direction Cosine Matrix to Quaternions

Quaternions to Direction Cosine Matrix

Quaternions to Rotation Angles

Rotation Angles to Direction Cosine Matrix