# 6DOF Wind (Quaternion)

Implement quaternion representation of six-degrees-of-freedom equations of motion with respect to wind axes

## Library

Equations of Motion/6DOF

## Description

The 6DOF Wind (Quaternion) block considers the rotation of a wind-fixed coordinate frame (Xw , Yw , Zw ) about an flat Earth reference frame (Xe , Ye , Ze ). The origin of the wind-fixed coordinate frame is the center of gravity of the body, and the body is assumed to be rigid, an assumption that eliminates the need to consider the forces acting between individual elements of mass. The flat Earth reference frame is considered inertial, an excellent approximation that allows the forces due to the Earth's motion relative to the "fixed stars" to be neglected.

The translational motion of the wind-fixed coordinate frame is given below, where the applied forces [Fx Fy Fz]T are in the wind-fixed frame, and the mass of the body m is assumed constant.

`$\begin{array}{l}{\overline{F}}_{w}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{w}+{\overline{\omega }}_{w}×{\overline{V}}_{w}\right)\\ \\ {\overline{V}}_{w}=\left[\begin{array}{c}V\\ 0\\ 0\end{array}\right],{\overline{\omega }}_{w}=\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\stackrel{˙}{\beta }\mathrm{sin}\alpha \\ {q}_{b}-\stackrel{˙}{\alpha }\\ {r}_{b}+\stackrel{˙}{\beta }\mathrm{cos}\alpha \end{array}\right],{\overline{\omega }}_{b}=\left[\begin{array}{c}{p}_{b}\\ {q}_{b}\\ {r}_{b}\end{array}\right]\\ \end{array}$`

The rotational dynamics of the body-fixed frame are given below, where the applied moments are [L M N]T, and the inertia tensor I is with respect to the origin O. Inertia tensor I is much easier to define in body-fixed frame.

`$\begin{array}{l}{\overline{M}}_{b}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I{\stackrel{˙}{\overline{\omega }}}_{b}+{\overline{\omega }}_{b}×\left(I{\overline{\omega }}_{b}\right)\\ \\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}$`

The integration of the rate of change of the quaternion vector is given below.

`$\left[\begin{array}{c}{\stackrel{˙}{q}}_{0}\\ {\stackrel{˙}{q}}_{1}\\ {\stackrel{˙}{q}}_{2}\\ {\stackrel{˙}{q}}_{3}\end{array}\right]=-1}{2}\left[\begin{array}{cccc}0& p& q& r\\ -p& 0& -r& q\\ -q& r& 0& -p\\ -r& -q& p& 0\end{array}\right]\left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]$`

## Parameters

Units

Specifies the input and output units:

Units

Forces

Moment

Acceleration

Velocity

Position

Mass

Inertia

`Metric (MKS)`

Newton

Newton meter

Meters per second squared

Meters per second

Meters

Kilogram

Kilogram meter squared

`English (Velocity in ft/s)`

Pound

Foot pound

Feet per second squared

Feet per second

Feet

Slug

Slug foot squared

`English (Velocity in kts)`

Pound

Foot pound

Feet per second squared

Knots

Feet

Slug

Slug foot squared

Mass Type

Select the type of mass to use:

 `Fixed` Mass is constant throughout the simulation. `Simple Variable` Mass and inertia vary linearly as a function of mass rate. `Custom Variable` Mass and inertia variations are customizable.

The `Fixed` selection conforms to the previously described equations of motion.

Representation

Select the representation to use:

 `Wind Angles` Use wind angles within equations of motion. `Quaternion` Use quaternions within equations of motion.

The `Quaternion` selection conforms to the previously described equations of motion.

Initial position in inertial axes

The three-element vector for the initial location of the body in the flat Earth reference frame.

Initial airspeed, angle of attack, and sideslip angle

The three-element vector containing the initial airspeed, initial angle of attack and initial sideslip angle.

Initial wind orientation

The three-element vector containing the initial wind angles [bank, flight path, and heading], in radians.

Initial body rotation rates

The three-element vector for the initial body-fixed angular rates, in radians per second.

Initial mass

The mass of the rigid body.

Inertia matrix

The 3-by-3 inertia tensor matrix I, in body-fixed axes.

## Inputs and Outputs

InputDimension TypeDescription

First

VectorContains the three applied forces in wind-fixed axes.

Second

VectorContains the three applied moments in body-fixed axes.

OutputDimension TypeDescription

First

Three-element vectorContains the velocity in the flat Earth reference frame.

Second

Three-element vectorContains the position in the flat Earth reference frame.

Third

Three-element vectorContains the wind rotation angles [bank, flight path, heading], in radians.

Fourth

3-by-3 matrixContains the coordinate transformation from flat Earth axes to wind-fixed axes.

Fifth

Three-element vectorContains the velocity in the wind-fixed frame.

Sixth

Two-element vectorContains the angle of attack and sideslip angle, in radians.

Seventh

Two-element vectorContains the rate of change of angle of attack and rate of change of sideslip angle, in radians per second.

Eight

Three-element vectorContains the angular rates in body-fixed axes, in radians per second.

Ninth

Three-element vectorContains the angular accelerations in body-fixed axes, in radians per second squared.

Tenth

Three-element vectorContains the accelerations in body-fixed axes.

## Assumptions and Limitations

The block assumes that the applied forces are acting at the center of gravity of the body, and that the mass and inertia are constant.

## Reference

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