Wind Angles to Direction Cosine Matrix

Convert wind angles to direction cosine matrix

Libraries:
Aerospace Blockset / Utilities / Axes Transformations

Description

The Wind Angles to Direction Cosine Matrix block converts three wind rotation angles into a 3-by-3 direction cosine matrix (DCM). 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). For more information on the direction cosine matrix, see Algorithms.

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

Ports

Input

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μ γ χ — Wind rotation angles
3-by-1 vector

Wind rotation angles, specified as a 3-by-1 vector, in radians.

Data Types: double

Output

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DCM_we — Direction cosine matrix
3-by-3 matrix

Direction cosine matrix, returned as a 3-by-3 matrix.

Data Types: double

Algorithms

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:

A rotation about oz₀ through the heading angle (χ) to axes (ox₁, oy₁, oz₁)
A rotation about oy₁ through the flight path angle (γ) to axes (ox₂, oy₂, oz₂)
A rotation about ox₂ through the bank angle (μ) to axes (ox₃, oy₃, oz₃)

$\begin{array}{l} [\begin{matrix} o x_{3} \\ o y_{3} \\ o z_{3} \end{matrix}] = D C M_{w e} [\begin{matrix} o x_{0} \\ o y_{0} \\ o z_{0} \end{matrix}] \\ [\begin{matrix} o x_{3} \\ o y_{3} \\ o z_{3} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos μ & \sin μ \\ 0 & - \sin μ & \cos μ \end{matrix}] [\begin{matrix} \cos γ & 0 & - \sin γ \\ 0 & 1 & 0 \\ \sin γ & 0 & \cos γ \end{matrix}] [\begin{matrix} \cos χ & \sin χ & 0 \\ - \sin χ & \cos χ & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} o x_{0} \\ o y_{0} \\ o z_{0} \end{matrix}] \end{array}$

Combining the three axis transformation matrices defines the following DCM:

$D C M_{w e} = [\begin{matrix} \cos γ \cos χ & \cos γ \sin χ & - \sin γ \\ (\sin μ \sin γ \cos χ - \cos μ \sin χ) & (\sin μ \sin γ \sin χ + \cos μ \cos χ) & \sin μ \cos γ \\ (\cos μ \sin γ \cos χ + \sin μ \sin χ) & (\cos μ \sin γ \sin χ - \sin μ \cos χ & \cos μ \cos γ \end{matrix}]$

Extended Capabilities

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

Version History

Introduced before R2006a