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Direction Cosine Matrix Body to Wind to Alpha and Beta

Convert direction cosine matrix to angle of attack and sideslip angle

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  • Direction Cosine Matrix Body to Wind to Alpha and Beta block

Libraries:
Aerospace Blockset / Utilities / Axes Transformations

Description

The Direction Cosine Matrix Body to Wind to Alpha and Beta block converts a 3-by-3 direction cosine matrix (DCM) to angle of attack and sideslip angle. The DCM performs the coordinate transformation of a vector in body axes (ox0, oy0, oz0) into a vector in wind axes (ox2, oy2, oz2). For more information on the direction cosine matrix, see Algorithms.

Limitations

  • This implementation generates angles that lie between ±90 degrees.

Ports

Input

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Direction cosine matrix to transform body-fixed vectors to wind-fixed vectors, specified as a 3-by-3 direct cosine matrix.

Data Types: double

Output

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Angle of attack and sideslip angle, returned as a vector, in radians.

Data Types: double

Parameters

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

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

  • Error — Displays error indicating 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 the 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

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)

[ox2oy2oz2]=DCMwb[ox0oy0oz0][ox2oy2oz2]=[cosβsinβ0−sinβcosβ0001][cosα0sinα010−sinα0cosα][ox0oy0oz0]

Combining the two axis transformation matrices defines the following DCM.

DCMwb=[cosαcosβsinβsinαcosβ−cosαsinβcosβ−sinαsinβ−sinα0cosα]

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

α=asin(−DCM(3,1))β=asin(DCM(1,2))

References

[1] Stevens, Brian L., Frank L. Lewis. Aircraft Control and Simulation, Second Edition. Hoboken, NJ: Wiley–Interscience.

Extended Capabilities

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

Version History

Introduced before R2006a

See Also

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