# quatrotate

Rotate vector by quaternion

## Description

example

n = quatrotate(q,r) calculates the resulting vector following the passive rotation of initial vector r by quaternion q and returns a final vector n. If quaternions are not yet normalized, the function normalizes them.

Aerospace Toolbox uses quaternions that are defined using the scalar-first convention. This function normalizes all quaternion inputs.

## Examples

collapse all

This example shows how to rotate a 1-by-3 vector by a 1-by-4 quaternion.

q = [1 0 1 0];
r = [1 1 1];
n = quatrotate(q, r)
n = 1×3

-1.0000    1.0000    1.0000

This example shows how to rotate two 1-by-3 vectors by a 1-by-4 quaternion.

q = [1 0 1 0];
r = [1 1 1; 2 3 4];
n = quatrotate(q, r)
n = 2×3

-1.0000    1.0000    1.0000
-4.0000    3.0000    2.0000

This example shows how to rotate a 1-by-3 vector by two 1-by-4 quaternions.

q = [1 0 1 0; 1 0.5 0.3 0.1];
r = [1 1 1];
n = quatrotate(q, r)
n = 2×3

-1.0000    1.0000    1.0000
0.8519    1.4741    0.3185

This example shows how to rotate multiple vectors by multiple quaternions.

q = [1 0 1 0; 1 0.5 0.3 0.1];
r = [1 1 1; 2 3 4];
n = quatrotate(q, r)
n = 2×3

-1.0000    1.0000    1.0000
1.3333    5.1333    0.9333

## Input Arguments

collapse all

Quaternion or set of quaternions, specified as an m-by-4 matrix containing m quaternions, or a single 1-by-4 quaternion. Each element must be real.

q must have its scalar number as the first column.

Data Types: double | single

Initial vector or set of vectors to be rotated, specified as an m-by-3 matrix, containing m vectors, or a single 1-by-3 array. Each element must be real.

Data Types: double | single

## Output Arguments

collapse all

Final vector, returned as an m-by-3 matrix.

collapse all

### q

The normalized quaternion q has the form:

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

### r

Vector r has the form:

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

The Aerospace Toolbox defines a passive quaternion rotation of the form:

${v}^{\prime }={q}^{-1}\otimes \left[\frac{0}{v}\right]\otimes q\text{,}$

where Ⓧ is the operator of a quaternion multiplication.

### n

Rotated vector n has the form:

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

The direction cosine matrix for this equation expects a normalized quaternion.

## References

[1] Stevens, Brian L., Frank L. Lewis. Aircraft Control and Simulation, 2nd Edition. Hoboken, NJ: John Wiley & Sons, 2003.

[2] Diebel, James. "Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors." Stanford University, Stanford, California, 2006.

## Version History

Introduced in R2006b