Direction Cosine Matrix Body to Wind

Convert angle of attack and sideslip angle to direction cosine matrix

Libraries:
Aerospace Blockset / Utilities / Axes Transformations

Description

The Direction Cosine Matrix Body to Wind block converts angle of attack and sideslip angles into a 3-by-3 direction cosine matrix (DCM). This direction cosine matrix is helpful for vector body axes to wind axes coordinate transformations. To transform the coordinates of a vector in body axes (ox₀, oy₀, oz₀) to a vector in wind axes (ox₂, oy₂, oz₂), multiply the block output direction cosine matrix with a vector in body axes. For information on the axis rotations for this transformation, see Algorithms.

Examples

Lightweight Airplane Design

Address the technical and process challenges of aircraft design using the design of a lightweight aircraft.

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Ports

Input

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ɑ β — Angle of attack and sideslip angle
2-by-1 vector

Angle of attack and sideslip angle, specified as a 2-by-1 vector, in radians.

Data Types: double

Output

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DCM_wb — Direction cosine matrix
3-by-3 direction cosine matrix

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

Data Types: double

Algorithms

The order of the axis rotations required to bring this transformation about is:

A rotation about oy₀ through the angle of attack (α) to axes (ox₁, oy₁, oz₁)
A rotation about oz₁ through the sideslip angle (β) to axes (ox₂, oy₂, oz₂)

$\begin{array}{l} [\begin{matrix} o x_{2} \\ o y_{2} \\ o z_{2} \end{matrix}] = D C M_{w b} [\begin{matrix} o x_{0} \\ o y_{0} \\ o z_{0} \end{matrix}] \\ [\begin{matrix} o x_{2} \\ o y_{2} \\ o z_{2} \end{matrix}] = [\begin{matrix} \cos β & \sin β & 0 \\ - \sin β & \cos β & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} \cos α & 0 & \sin α \\ 0 & 1 & 0 \\ - \sin α & 0 & \cos α \end{matrix}] [\begin{matrix} o x_{0} \\ o y_{0} \\ o z_{0} \end{matrix}] \end{array}$

Combining the two axis transformation matrices defines the following DCM.

$D C M_{w b} = [\begin{matrix} \cos α \cos β & \sin β & \sin α \cos β \\ - \cos α \sin β & \cos β & - \sin α \sin β \\ - \sin α & 0 & \cos α \end{matrix}]$

References

[1] Stevens, B. L., and F. L. Lewis. Aircraft Control and Simulation. Hoboken, NJ: John Wiley & Sons, 1992.

Extended Capabilities

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

Version History

Introduced before R2006a