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Rodrigues to Direction Cosine Matrix

Convert Euler-Rodrigues vector to direction cosine matrix

  • Library:
  • Aerospace Blockset / Utilities / Axes Transformations

  • Rodrigues to Direction Cosine Matrix block


The Rodrigues to Direction Cosine Matrix 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 Euler-Rodrigues vectors, see Algorithms.



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

Data Types: double


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Direction cosine matrix determined from the Euler-Rodrigues vector.

Data Types: double


An Euler-Rodrigues vector b represents a rotation by integrating a direction cosine of a rotation axis with the tangent of half the rotation angle as follows:




are the Rodrigues parameters. Vector 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.


[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
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Introduced in R2017a