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# 6DOF Wind (Wind Angles)

Implement wind angle representation of six-degrees-of-freedom equations of motion

## Library

Equations of Motion/6DOF

## Description

For a description of the coordinate system employed and the translational dynamics, see the block description for the 6DOF Wind (Quaternion) block.

The relationship between the wind angles, ${\left[\mu \gamma \chi \right]}^{\text{T}}$, can be determined by resolving the wind rates into the wind-fixed coordinate frame.

 `$\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{c}\stackrel{˙}{\mu }\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\mu & \mathrm{sin}\mu \\ 0& -\mathrm{sin}\mu & \mathrm{cos}\mu \end{array}\right]\left[\begin{array}{c}0\\ \stackrel{˙}{\gamma }\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\mu & \mathrm{sin}\mu \\ 0& -\mathrm{sin}\mu & \mathrm{cos}\mu \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\gamma & 0& -\mathrm{sin}\gamma \\ 0& 1& 0\\ \mathrm{sin}\gamma & 0& \mathrm{cos}\gamma \end{array}\right]\left[\begin{array}{c}0\\ 0\\ \stackrel{˙}{\chi }\end{array}\right]\equiv {J}^{-1}\left[\begin{array}{c}\stackrel{˙}{\mu }\\ \stackrel{˙}{\gamma }\\ \stackrel{˙}{\chi }\end{array}\right]$`

Inverting J then gives the required relationship to determine the wind rate vector.

`$\left[\begin{array}{c}\stackrel{˙}{\mu }\\ \stackrel{˙}{\gamma }\\ \stackrel{˙}{\chi }\end{array}\right]=J\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{ccc}1& \left(\mathrm{sin}\mu \mathrm{tan}\gamma \right)& \left(\mathrm{cos}\mu \mathrm{tan}\gamma \right)\\ 0& \mathrm{cos}\mu & -\mathrm{sin}\mu \\ 0& \frac{\mathrm{sin}\mu }{\mathrm{cos}\gamma }& \frac{\mathrm{cos}\mu }{\mathrm{cos}\gamma }\end{array}\right]\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]$`

The body-fixed angular rates are related to the wind-fixed angular rate by the following equation.

`$\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\stackrel{˙}{\beta }\mathrm{sin}\alpha \\ {q}_{b}-\stackrel{˙}{\alpha }\\ {r}_{b}+\stackrel{˙}{\beta }\mathrm{cos}\alpha \end{array}\right]$`

Using this relationship in the wind rate vector equations, gives the relationship between the wind rate vector and the body-fixed angular rates.

`$\left[\begin{array}{c}\stackrel{˙}{\mu }\\ \stackrel{˙}{\gamma }\\ \stackrel{˙}{\chi }\end{array}\right]=J\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{ccc}1& \left(\mathrm{sin}\mu \mathrm{tan}\gamma \right)& \left(\mathrm{cos}\mu \mathrm{tan}\gamma \right)\\ 0& \mathrm{cos}\mu & -\mathrm{sin}\mu \\ 0& \frac{\mathrm{sin}\mu }{\mathrm{cos}\gamma }& \frac{\mathrm{cos}\mu }{\mathrm{cos}\gamma }\end{array}\right]DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\stackrel{˙}{\beta }\mathrm{sin}\alpha \\ {q}_{b}-\stackrel{˙}{\alpha }\\ {r}_{b}+\stackrel{˙}{\beta }\mathrm{cos}\alpha \end{array}\right]$`

## Parameters

### Main

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 `Wind Angles` 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

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

Include inertial acceleration

Select this check box to enable an additional output port for the accelerations in body-fixed axes with respect to the inertial frame. You typically connect this signal to the accelerometer.

### State Attributes

Assign unique name to each state. You can use state names instead of block paths during linearization.

• To assign a name to a single state, enter a unique name between quotes, for example, `'velocity'`.

• To assign names to multiple states, enter a comma-delimited list surrounded by braces, for example, `{'a', 'b', 'c'}`. Each name must be unique.

• If a parameter is empty (```' '```), no name assignment occurs.

• The state names apply only to the selected block with the name parameter.

• The number of states must divide evenly among the number of state names.

• You can specify fewer names than states, but you cannot specify more names than states.

For example, you can specify two names in a system with four states. The first name applies to the first two states and the second name to the last two states.

• To assign state names with a variable in the MATLAB® workspace, enter the variable without quotes. A variable can be a character vector, cell array, or structure.

Position: e.g., {'Xe', 'Ye', 'Ze'}

Specify position state names.

Default value is `''`.

Velocity: e.g., 'V'

Specify velocity state name.

Default value is `''`.

Incidence angle: e.g., 'alpha'

Specify incidence angle state name.

Default value is `''`.

Sideslip angle: e.g., 'beta'

Specify sideslip angle state name.

Default value is `''`.

Wind orientation: e.g., {'mu', 'gamma', 'chi'}

Specify wind orientation state names. This parameter appears if the Representation parameter is set to ```Wind Angles```.

Default value is `''`.

Body rotation rates: e.g., {'p', 'q', 'r'}

Specify body rotation rate state names.

Default value is `''`.

## 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], within ±pi, 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.

Eighth

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 with respect to body frame.

Eleventh (Optional)

Three-element vectorContains the accelerations in body-fixed axes with respect to inertial frame (flat Earth). You typically connect this signal to the accelerometer.

## 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.

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