# Direction Cosine Matrix to Rodrigues

Convert direction cosine matrix to Euler-Rodrigues vector

• Library:
• Aerospace Blockset / Utilities / Axes Transformations

## Description

The Direction Cosine Matrix to Rodrigues block determines the 3-by-3 direction cosine matrix from a three-element Euler-Rodrigues vector. The rotation used in this block is a passive transformation between two coordinate systems. For more information on the direction cosine matrix, see Algorithms.

## Ports

### Input

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Direction cosine matrix, specified as a 3-by-3 matrix, from which to determine the Euler-Rodrigues vector.

Data Types: double

### Output

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Euler-Rodrigues vector, returned as a three-element vector.

Data Types: double

## Parameters

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Block behavior when direction cosine matrix is invalid (not orthogonal).

• Warning — Displays warning and indicates that the direction cosine matrix is invalid.

• Error — Displays error and indicates that the direction cosine matrix is invalid.

• None — Does not display warning or error (default).

#### Programmatic Use

 Block Parameter: action Type: character vector Values: 'None' | 'Warning' | 'Error' Default: 'None'

Data Types: char | string

Tolerance of direction cosine matrix validity, specified as a scalar. The block considers the direction cosine matrix valid if these conditions are true:

• The transpose of the direction cosine matrix times itself equals 1 within the specified tolerance (transpose(n)*n == 1±tolerance)

• The determinant of the direction cosine matrix equals 1 within the specified tolerance (det(n) == 1±tolerance).

#### Programmatic Use

 Block Parameter: tolerance Type: character vector Values: 'eps(2)' | scalar Default: 'eps(2)'

Data Types: double

## 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