Documentation

# rod2dcm

Convert Euler-Rodrigues vector to direction cosine matrix

## Syntax

``dcm=rod2dcm(R)``

## Description

example

````dcm=rod2dcm(R)` function calculates the direction cosine matrix, for a given Euler-Rodrigues (also known as Rodrigues) vector, `R`.```

## Examples

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Determine the direction cosine matrix from the Euler-Rodrigues vector.

```r = [.1 .2 -.1]; DCM = rod2dcm(r)```
```DCM = 0.9057 -0.1509 -0.3962 0.2264 0.9623 0.1509 0.3585 -0.2264 0.9057```

## Input Arguments

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M-by-3 matrix containing M Rodrigues vectors.

Data Types: `double`

## Output Arguments

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3-by-3-by-M containing M direction cosine matrices.

## 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.

## References

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