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# Quaternion Rotation

Rotate vector by quaternion

## Library

Utilities/Math Operations

## Description

The Quaternion Rotation block rotates a vector by a quaternion.

The quaternion has the form of

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

The vector has the form of

$v=i{v}_{1}+j{v}_{2}+k{v}_{3}.$

The rotated vector has the form of

${v}^{\prime }=\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\end{array}\right]=\left[\begin{array}{ccc}\left(1-2{q}_{2}^{2}-2{q}_{3}^{2}\right)& 2\left({q}_{1}{q}_{2}+{q}_{0}{q}_{3}\right)& 2\left({q}_{1}{q}_{3}-{q}_{0}{q}_{2}\right)\\ 2\left({q}_{1}{q}_{2}-{q}_{0}{q}_{3}\right)& \left(1-2{q}_{1}^{2}-2{q}_{3}^{2}\right)& 2\left({q}_{2}{q}_{3}+{q}_{0}{q}_{1}\right)\\ 2\left({q}_{1}{q}_{3}+{q}_{0}{q}_{2}\right)& 2\left({q}_{2}{q}_{3}-{q}_{0}{q}_{1}\right)& \left(1-2{q}_{1}^{2}-2{q}_{2}^{2}\right)\end{array}\right]\left[\begin{array}{c}{v}_{1}\\ {v}_{2}\\ {v}_{3}\end{array}\right]$

## Inputs and Outputs

InputDimension TypeDescription

First

Quaternion or vectorContains quaternions in the form of [q0, r0, ..., q1, r1, ... , q2, r2, ... , q3, r3, ...].

Second

VectorContains vector or vector of vectors in the form of [v1, u1, ... , v2, u2, ... , v3, u3, ...].

OutputDimension TypeDescription

First

Rotated quaternion or vectorContains rotated vector or vector of rotated vectors.

## References

Stevens, Brian L., Frank L. Lewis, Aircraft Control and Simulation, Wiley–Interscience, 2nd Edition.