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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)

$\begin{array}{l}\left[\begin{array}{c}o{x}_{3}\\ o{y}_{3}\\ o{z}_{3}\end{array}\right]=DC{M}_{we}\left[\begin{array}{c}o{x}_{0}\\ o{y}_{0}\\ o{z}_{0}\end{array}\right]\\ \\ \left[\begin{array}{c}o{x}_{3}\\ o{y}_{3}\\ o{z}_{3}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\mu & \mathrm{sin}\mu \\ 0& -\mathrm{sin}\mu & \mathrm{cos}\mu \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\gamma & 0& -\mathrm{sin}\gamma \\ 0& 1& 0\\ \mathrm{sin}\gamma & 0& \mathrm{cos}\gamma \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\chi & \mathrm{sin}\chi & 0\\ -\mathrm{sin}\chi & \mathrm{cos}\chi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}o{x}_{0}\\ o{y}_{0}\\ o{z}_{0}\end{array}\right]\end{array}$

Combining the three axis transformation matrices defines the following DCM.

$DC{M}_{we}=\left[\begin{array}{ccc}\mathrm{cos}\gamma \mathrm{cos}\chi & \mathrm{cos}\gamma \mathrm{sin}\chi & -\mathrm{sin}\gamma \\ \left(\mathrm{sin}\mu \mathrm{sin}\gamma \mathrm{cos}\chi -\mathrm{cos}\mu \mathrm{sin}\chi \right)& \left(\mathrm{sin}\mu \mathrm{sin}\gamma \mathrm{sin}\chi +\mathrm{cos}\mu \mathrm{cos}\chi \right)& \mathrm{sin}\mu \mathrm{cos}\gamma \\ \left(\mathrm{cos}\mu \mathrm{sin}\gamma \mathrm{cos}\chi +\mathrm{sin}\mu \mathrm{sin}\chi \right)& \left(\mathrm{cos}\mu \mathrm{sin}\gamma \mathrm{sin}\chi -\mathrm{sin}\mu \mathrm{cos}\chi \right)& \mathrm{cos}\mu \mathrm{cos}\gamma \end{array}\right]$

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

$\begin{array}{l}\mu =\text{atan}\left(\frac{DCM\left(2,3\right)}{DCM\left(3,3\right)}\right)\\ \\ \gamma =\text{asin}\left(-DCM\left(1,3\right)\right)\\ \\ \chi =\text{atan}\left(\frac{DCM\left(1,2\right)}{DCM\left(1,1\right)}\right)\end{array}$

## 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.