Direction Cosine Matrix Body to Wind to Alpha and Beta

Convert direction cosine matrix to angle of attack and sideslip angle


Utilities/Axes Transformations


The Direction Cosine Matrix Body to Wind to Alpha and Beta block converts a 3-by-3 direction cosine matrix (DCM) into angle of attack and sideslip angle. The DCM matrix performs the coordinate transformation of a vector in body axes (ox0, oy0, oz0) into a vector in wind axes (ox2, oy2, oz2). The order of the axis rotations required to bring this about is:

  1. A rotation about oy0 through the angle of attack (α) to axes (ox1, oy1, oz1)

  2. A rotation about oz1 through the sideslip angle (β) to axes (ox2, oy2, oz2)


Combining the two axis transformation matrices defines the following DCM.


To determine angles from the DCM, the following equations are used:


Inputs and Outputs

InputDimension TypeDescription


3-by-3 direction cosine matrixTransforms body-fixed vectors to wind-fixed vectors.

OutputDimension TypeDescription


2-by-1 vectorContains angle of attack and sideslip angle, in radians.

Assumptions and Limitations

This implementation generates angles that lie between ±90 degrees.


Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.

Introduced before R2006a

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