Direction Cosine Matrix Body to Wind to Alpha and Beta

Convert direction cosine matrix to angle of attack and sideslip angle

Library

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) 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)

[ox2oy2oz2]=DCMwb[ox0oy0oz0][ox2oy2oz2]=[cosβsinβ0sinβcosβ0001][cosα0sinα010sinα0cosα][ox0oy0oz0]

Combining the two axis transformation matrices defines the following DCM.

DCMwb=[cosαcosβsinβsinαcosβcosαsinβcosβsinαsinβsinα0cosα]

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

α=asin(DCM(3,1))β=asin(DCM(1,2))

Parameters

Action for invalid DCM

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

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

  • Error — Displays error and indicates that the direction cosine matrix is invalid.

  • None — Does not display warning or error (default).

Tolerance for DCM validation

Tolerance of direction cosine matrix validity, specified as a scalar. Default is eps(2). 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).

Inputs and Outputs

InputDimension TypeDescription

First

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

First

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

Assumptions and Limitations

This implementation generates angles that lie between ±90 degrees.

Reference

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

Introduced before R2006a