# Simple Variable Mass 6DOF (Euler Angles)

Implement Euler angle representation of six-degrees-of-freedom equations of motion of simple variable mass

Libraries:
Aerospace Blockset / Equations of Motion / 6DOF

## Description

The Simple Variable Mass 6DOF (Euler Angles) block considers the rotation of a body-fixed coordinate frame (Xb, Yb, Zb) about a flat Earth reference frame (Xe, Ye, Ze).

For a description of the coordinate system and the translational dynamics, see the description for the Simple Variable Mass 6DOF (Euler Angles) block. For more information on the body-fixed coordinate frame, see Algorithms.

## Limitations

The block assumes that the applied forces are acting at the center of gravity of the body.

## Ports

### Input

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Applied forces, specified as a three-element vector.

Data Types: `double`

Applied moments, specified as a three-element vector.

Data Types: `double`

One or more rates of change of mass (positive if accreted, negative if ablated), specified as a scalar.

Data Types: `double`

One or more relative velocities, specified as a three-element vector, at which the mass is accreted to or ablated from the body in body-fixed axes.

#### Dependencies

To enable this port, select Include mass flow relative velocity.

Data Types: `double`

### Output

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Velocity in the flat Earth reference frame, returned as a three-element vector.

Data Types: `double`

Position in the flat Earth reference frame, returned as a three-element vector.

Data Types: `double`

Euler rotation angles [roll, pitch, yaw], returned as three-element vector, in radians.

Data Types: `double`

Coordinate transformation from flat Earth axes to body-fixed axes, returned as a 3-by-3 matrix.

Data Types: `double`

Velocity in body-fixed frame, returned as a three-element vector.

Data Types: `double`

Angular rates in body-fixed axes, returned as a three-element vector, in radians per second.

Data Types: `double`

Angular accelerations in body-fixed axes, returned as a three-element vector, in radians per second squared.

Data Types: `double`

Accelerations in body-fixed axes with respect to body frame, returned as a three-element vector.

Data Types: `double`

Fuel tank status, returned as:

• `1` — Tank is full.

• `0` — Tank is neither full nor empty.

• `-1` — Tank is empty.

Data Types: `double`

Accelerations in body-fixed axes with respect to inertial frame (flat Earth), returned as a three-element vector. You typically connect this signal to the accelerometer.

#### Dependencies

This port appears only when the Include inertial acceleration check box is selected.

Data Types: `double`

## Parameters

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### Main

Input and output units, specified as `Metric (MKS)`, `English (Velocity in ft/s)`, or `English (Velocity in kts)`.

UnitsForcesMomentAccelerationVelocityPositionMassInertia
`Metric (MKS)` NewtonNewton-meterMeters per second squaredMeters per secondMetersKilogramKilogram meter squared
`English (Velocity in ft/s)` PoundFoot-poundFeet per second squaredFeet per secondFeetSlugSlug foot squared
`English (Velocity in kts)` PoundFoot-poundFeet per second squaredKnotsFeetSlugSlug foot squared

#### Programmatic Use

 Block Parameter: `units` Type: character vector Values: `Metric (MKS)` | `English (Velocity in ft/s)` | `English (Velocity in kts)` Default: `Metric (MKS)`

Mass type, specified according to the following table.

Mass TypeDescriptionDefault For
`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 `Simple Variable` selection conforms to the equations of motion in Algorithms.

#### Programmatic Use

 Block Parameter: `mtype` Type: character vector Values: `Fixed` | `Simple Variable` | `Custom Variable` Default: `Simple Variable`

Equations of motion representation, specified according to the following table.

RepresentationDescription

`Euler Angles`

Use Euler angles within equations of motion.

`Quaternion`

Use quaternions within equations of motion.

The `Euler Angles` selection conforms to the equations of motion in Algorithms.

#### Programmatic Use

 Block Parameter: `rep` Type: character vector Values: `Euler Angles` | `Quaternion` Default: `'Euler Angles'`

Initial location of the body in the flat Earth reference frame, specified as a three-element vector.

#### Programmatic Use

 Block Parameter: `xme_0` Type: character vector Values: `'[0 0 0]'` | three-element vector Default: `'[0 0 0]'`

Initial velocity in body axes, specified as a three-element vector, in the body-fixed coordinate frame.

#### Programmatic Use

 Block Parameter: `Vm_0` Type: character vector Values: `'[0 0 0]'` | three-element vector Default: `'[0 0 0]'`

Initial Euler orientation angles [roll, pitch, yaw], specified as a three-element vector, in radians. Euler rotation angles are those between the body and north-east-down (NED) coordinate systems.

#### Programmatic Use

 Block Parameter: `eul_0` Type: character vector Values: `'[0 0 0]'` | three-element vector Default: `'[0 0 0]'`

Initial body-fixed angular rates with respect to the NED frame, specified as a three-element vector, in radians per second.

#### Programmatic Use

 Block Parameter: `pm_0` Type: character vector Values: `'[0 0 0]'` | three-element vector Default: `'[0 0 0]'`

Initial mass of the rigid body, specified as a double scalar.

#### Programmatic Use

 Block Parameter: `mass_0` Type: character vector Values: `'1.0'` | double scalar Default: `'1.0'`

Empty mass of the body, specified as a double scalar.

#### Programmatic Use

 Block Parameter: `mass_e` Type: character vector Values: double scalar Default: `'0.5'`

Full mass of the body, specified as a double scalar.

#### Programmatic Use

 Block Parameter: `mass_f` Type: character vector Values: double scalar Default: `'2.0'`

Inertia tensor matrix for the empty inertia of the body, specified as 3-by-3 matrix.

#### Programmatic Use

 Block Parameter: `inertia_e` Type: character vector Values: `'eye(3)'` | 3-by-3 matrix Default: `'eye(3)'`

Inertia tensor matrix for the full inertia of the body, specified as 3-by-3 matrix.

#### Programmatic Use

 Block Parameter: `inertia_f` Type: character vector Values: `'2*eye(3)'` | 3-by-3 matrix Default: `'2*eye(3)'`

Select this check box to add a mass flow relative velocity port. This is the relative velocity at which the mass is accreted or ablated.

#### Programmatic Use

 Block Parameter: `vre_flag` Type: character vector Values: `off` | `on` Default: `off`

Select this check box to add an inertial acceleration port.

#### Dependencies

To enable the Ab ff port, select this parameter.

#### Programmatic Use

 Block Parameter: `abi_flag` Type: character vector Values: `'off'` | `'on'` Default: `off`

### State Attributes

Assign a 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-separated list surrounded by braces, for example, `{'a', 'b', 'c'}`. Each name must be unique.

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

• 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 state names, specified as a comma-separated list surrounded by braces.

#### Programmatic Use

 Block Parameter: `xme_statename` Type: character vector Values: `''` | comma-separated list surrounded by braces Default: `''`

Velocity state names, specified as comma-separated list surrounded by braces.

#### Programmatic Use

 Block Parameter: `Vm_statename` Type: character vector Values: `''` | comma-separated list surrounded by braces Default: `''`

Euler rotation angle state names, specified as a comma-separated list surrounded by braces.

#### Programmatic Use

 Block Parameter: `eul_statename` Type: character vector Values: `''` | comma-separated list surrounded by braces Default: `''`

Body rotation rate state names, specified comma-separated list surrounded by braces.

#### Programmatic Use

 Block Parameter: `pm_statename` Type: character vector Values: `''` | comma-separated list surrounded by braces Default: `''`

Mass state name, specified as a character vector.

#### Programmatic Use

 Block Parameter: `mass_statename` Type: character vector Values: `''` | character vector Default: `''`

## Algorithms

The origin of the body-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 body-fixed coordinate frame is given below, where the applied forces [Fx FyFz]T are in the body-fixed frame. Vreb is the relative velocity in the body axes at which the mass flow ($\stackrel{˙}{m}$) is ejected or added to the body in body axes.

`$\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{l}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{b}+\overline{\omega }×{\overline{V}}_{b}\right)+\stackrel{˙}{m}\overline{V}r{e}_{b}\\ {A}_{be}=\frac{{\overline{F}}_{b}-\stackrel{˙}{m}{\overline{V}}_{re}}{m}\\ {A}_{bb}=\left[\begin{array}{c}{\stackrel{˙}{u}}_{b}\\ {\stackrel{˙}{v}}_{b}\\ {\stackrel{˙}{\omega }}_{b}\end{array}\right]=\frac{{\overline{F}}_{b}-\stackrel{˙}{m}{\overline{V}}_{re}}{m}-\overline{\omega }×{\overline{V}}_{b}\\ {\overline{V}}_{b}=\left[\begin{array}{l}{u}_{b}\\ {v}_{b}\\ {w}_{b}\end{array}\right],\overline{\omega }=\left[\begin{array}{l}p\\ q\\ r\end{array}\right]\end{array}$`

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.

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

The inertia tensor is determined using a table lookup which linearly interpolates between Ifull and Iempty based on mass (m). While the rate of change of the inertia tensor is estimated by the following equation.

`$\stackrel{˙}{I}=\frac{{I}_{full}-{I}_{empty}}{{m}_{full}-{m}_{empty}}\stackrel{˙}{m}$`

The relationship between the body-fixed angular velocity vector, [p q r]T, and the rate of change of the Euler angles, [$\stackrel{˙}{\varphi }\stackrel{˙}{\theta }\stackrel{˙}{\psi }$]T, can be determined by resolving the Euler rates into the body-fixed coordinate frame.

`$\left[\begin{array}{l}p\\ q\\ r\end{array}\right]=\left[\begin{array}{l}\stackrel{˙}{\varphi }\\ 0\\ 0\end{array}\right]+\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & \mathrm{cos}\varphi \hfill & \mathrm{sin}\varphi \hfill \\ 0\hfill & -\mathrm{sin}\varphi \hfill & \mathrm{cos}\varphi \hfill \end{array}\right]\left[\begin{array}{l}0\\ \stackrel{˙}{\theta }\\ 0\end{array}\right]+\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & \mathrm{cos}\varphi \hfill & \mathrm{sin}\varphi \hfill \\ 0\hfill & -\mathrm{sin}\varphi \hfill & \mathrm{cos}\varphi \hfill \end{array}\right]\left[\begin{array}{lll}\mathrm{cos}\theta \hfill & 0\hfill & -\mathrm{sin}\theta \hfill \\ 0\hfill & 1\hfill & 0\hfill \\ \mathrm{sin}\theta \hfill & 0\hfill & \mathrm{cos}\theta \hfill \end{array}\right]\left[\begin{array}{l}0\\ 0\\ \stackrel{˙}{\psi }\end{array}\right]\equiv {J}^{-1}\left[\begin{array}{l}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]$`

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

`$\left[\begin{array}{l}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]=J\left[\begin{array}{l}p\\ q\\ r\end{array}\right]=\left[\begin{array}{lll}1\hfill & \left(\mathrm{sin}\varphi \mathrm{tan}\theta \right)\hfill & \left(\mathrm{cos}\varphi \mathrm{tan}\theta \right)\hfill \\ 0\hfill & \mathrm{cos}\varphi \hfill & -\mathrm{sin}\varphi \hfill \\ 0\hfill & \frac{\mathrm{sin}\varphi }{\mathrm{cos}\theta }\hfill & \frac{\mathrm{cos}\varphi }{\mathrm{cos}\theta }\hfill \end{array}\right]\left[\begin{array}{l}p\\ q\\ r\end{array}\right]$`

## References

[1] Stevens, Brian, and Frank Lewis. Aircraft Control and Simulation. 2nd ed. Hoboken, NJ: John Wiley & Sons, 2003.

[2] Zipfel, Peter H. Modeling and Simulation of Aerospace Vehicle Dynamics. 2nd ed. Reston, VA: AIAA Education Series, 2007.

## Version History

Introduced in R2006a