Rodrigues to Direction Cosine Matrix

Convert Euler-Rodrigues vector to direction cosine matrix

Libraries:
Aerospace Blockset / Utilities / Axes Transformations

Description

The Rodrigues to Direction Cosine Matrix block determines the 3-by-3 direction cosine matrix from a three-element Euler-Rodrigues vector. The Euler-Rodrigues vector input and resulting direction cosine matrix represent a right-hand passive transformation from frame A to frame B. The resulting direction cosine matrix represent a series of right-hand intrinsic passive rotations from frame A to frame B. For more information on Euler-Rodrigues vectors, see Algorithms.

Ports

Input

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rod — Euler-Rodrigues vector
three-element vector

Euler-Rodrigues vector from which to determine the direction cosine matrix.

Data Types: double

Output

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DCM — Direction cosine matrix
3-by-3 matrix

Direction cosine matrix determined from the Euler-Rodrigues vector.

Data Types: double

Algorithms

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

$\vec{b} = [\begin{matrix} b_{x} & b_{y} & b_{z} \end{matrix}]$

where:

$\begin{array}{l} b_{x} = \tan (\frac{1}{2} θ) s_{x}, \\ b_{y} = \tan (\frac{1}{2} θ) s_{y}, \\ b_{z} = \tan (\frac{1}{2} θ) s_{z} \end{array}$

are the Rodrigues parameters. Vector $\overset{⇀}{s}$ 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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2017a