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Direction Cosine Matrix to Wind Angles

Convert direction cosine matrix to wind angles

Library

Utilities/Axes Transformations

Description

The Direction Cosine Matrix to Wind Angles block converts a 3-by-3 direction cosine matrix (DCM) into three wind rotation angles. The DCM matrix performs the coordinate transformation of a vector in earth axes (ox0, oy0, oz0) into a vector in wind axes (ox3, oy3, oz3). The order of the axis rotations required to bring this about is:

  1. A rotation about oz0 through the heading angle (χ) to axes (ox1, oy1, oz1)

  2. A rotation about oy1 through the flight path angle (γ) to axes (ox2, oy2, oz2)

  3. A rotation about ox2 through the bank angle (μ) to axes (ox3, oy3, oz3)

[ox3oy3oz3]=DCMwe[ox0oy0oz0][ox3oy3oz3]=[1000cosμsinμ0sinμcosμ][cosγ0sinγ010sinγ0cosγ][cosχsinχ0sinχcosχ0001][ox0oy0oz0]

Combining the three axis transformation matrices defines the following DCM.

DCMwe=[cosγcosχcosγsinχsinγ(sinμsinγcosχcosμsinχ)(sinμsinγsinχ+cosμcosχ)sinμcosγ(cosμsinγcosχ+sinμsinχ)(cosμsinγsinχsinμcosχ)cosμcosγ]

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

μ=atan(DCM(2,3)DCM(3,3))γ=asin(DCM(1,3))χ=atan(DCM(1,2)DCM(1,1))

Inputs and Outputs

InputDimension TypeDescription

First

3-by-3 direction cosine matrixTransforms earth vectors to wind vectors.

OutputDimension TypeDescription

First

3-by-1 vectorContains the wind angles, in radians.

Assumptions and Limitations

This implementation generates a flight path angle that lies between ±90 degrees, and bank and heading angles that lie between ±180 degrees.

Introduced before R2006a

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