# rod2dcm

Convert Euler-Rodrigues vector to direction cosine matrix

## 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.

## Version History

Introduced in R2017a