# quat2rod

Convert quaternion to Euler-Rodrigues vector

## Syntax

``rod=quat2rod(quat)``

## Description

example

````rod=quat2rod(quat)` function calculates the Euler-Rodrigues vector, `rod`, for a given quaternion `quat`. Aerospace Toolbox uses quaternions that are defined using the scalar-first convention. This function normalizes all quaternion inputs.```

## Examples

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Determine the Euler-Rodrigues vector from the quaternion.

```q = [-0.7071 0 0.7071 0] r = quat2rod( q )```
```q = -0.7071 0 0.7071 0 r = 0 -1.0000 0```

## Input Arguments

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M-by-4 array of quaternions. `quat` has its scalar number as the first column.

Data Types: `double`

## Output Arguments

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M-by-3 matrix containing M Euler-Rodrigues vectors.

## Algorithms

An Euler-Rodrigues vector $\stackrel{⇀}{b}$ represents a rotation by integrating a direction cosine of a rotation axis with the tangent of half the rotation angle as follows:

`$\stackrel{\to }{b}=\left[\begin{array}{ccc}{b}_{x}& {b}_{y}& {b}_{z}\end{array}\right]$`

where:

`$\begin{array}{l}{b}_{x}=\mathrm{tan}\left(\frac{1}{2}\theta \right){s}_{x},\\ {b}_{y}=\mathrm{tan}\left(\frac{1}{2}\theta \right){s}_{y},\\ {b}_{z}=\mathrm{tan}\left(\frac{1}{2}\theta \right){s}_{z}\end{array}$`

are the Rodrigues parameters. Vector $\stackrel{⇀}{s}$ represents a unit vector around which the rotation is performed. Due to the tangent, the rotation vector is indeterminate when the rotation angle equals ±pi radians or ±180 deg. Values can be negative or positive.

 Dai, J.S. "Euler-Rodrigues formula variations, quaternion conjugation and intrinsic connections." Mechanism and Machine Theory, 92, 144-152. Elsevier, 2015.