# 6DOF (Euler Angles)

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

• Library:
• Aerospace Blockset / Equations of Motion / 6DOF

## Description

The 6DOF (Euler Angles) block implements the Euler angle representation of six-degrees-of-freedom equations of motion, taking into consideration the rotation of a body-fixed coordinate frame (Xb, Yb, Zb) about a flat Earth reference frame (Xe, Ye, Ze). For more information about these reference points, see Algorithms.

## 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.

## Ports

### Input

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Applied forces, specified as three-element vector.

Data Types: `double`

Applied moments, specified as three-element vector.

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 the 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`

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 previously described equations of motion.

#### 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.

RepresentationDescriptionDefault For

`Euler Angles`

Use Euler angles within equations of motion.

`Quaternion`

Use quaternions within equations of motion.

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

#### Programmatic Use

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

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]`

The three-element vector for the initial velocity 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]`

The three-element vector for the initial Euler orientation angles [roll, pitch, yaw], in radians.

#### Programmatic Use

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

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

#### Programmatic Use

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

The initial mass of the rigid body.

#### Programmatic Use

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

A scalar value for the inertia of the body.

#### Programmatic Use

 Block Parameter: `inertia` Type: character vector Values: `eye(3)` | scalar Default: `eye(3)`

Select this check box to add an inertial acceleration port.

#### Programmatic Use

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

### 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.

Specify position state names.

#### Programmatic Use

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

Specify velocity state names.

#### Programmatic Use

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

Specify Euler rotation angle state names.

#### Programmatic Use

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

Specify body rotation rate state names.

#### Programmatic Use

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

## Algorithms

The 6DOF (Euler Angles) block uses these reference frame concepts.

• 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 motion relative to the "fixed stars" to be neglected.

• Translational motion of the body-fixed coordinate frame, where the applied forces [Fx Fy Fz]T are in the body-fixed frame, and the mass of the body m is assumed constant.

`$\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{b}+\overline{\omega }×{\overline{V}}_{b}\right)\\ {A}_{bb}=\left[\begin{array}{c}{\stackrel{˙}{u}}_{b}\\ {\stackrel{˙}{v}}_{b}\\ {\stackrel{˙}{w}}_{b}\end{array}\right]=\frac{1}{m}{\overline{F}}_{b}-\overline{\omega }×{\overline{V}}_{b}\\ {A}_{be}=\frac{1}{m}{F}_{b}\\ {\overline{V}}_{b}=\left[\begin{array}{c}{u}_{b}\\ {v}_{b}\\ {w}_{b}\end{array}\right],\overline{\omega }=\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\end{array}$`
• The rotational dynamics of the body-fixed frame, where the applied moments are [L M N]T, and the inertia tensor I is with respect to the origin O.

`$\begin{array}{l}{\overline{M}}_{B}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I\stackrel{˙}{\overline{\omega }}+\overline{\omega }×\left(I\overline{\omega }\right)\\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}$`
• The relationship between the body-fixed angular velocity vector, [p q r]T, and the rate of change of the Euler angles, $\left[\begin{array}{ccc}\stackrel{˙}{\varphi }\text{ }\text{\hspace{0.17em}}& \stackrel{˙}{\theta }\text{\hspace{0.17em}}\text{ }\text{ }& \stackrel{˙}{\psi }\end{array}{\right]}^{T}$, are determined by resolving the Euler rates into the body-fixed coordinate frame.

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

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

$\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]=J\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\text{\hspace{0.17em}}=\left[\begin{array}{ccc}1& \left(\mathrm{sin}\varphi \mathrm{tan}\theta \right)& \left(\mathrm{cos}\varphi \mathrm{tan}\theta \right)\\ 0& \mathrm{cos}\varphi & -\mathrm{sin}\varphi \\ 0& \frac{\mathrm{sin}\varphi }{\mathrm{cos}\theta }& \frac{\mathrm{cos}\varphi }{\mathrm{cos}\theta }\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]$

## References

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

[2] Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Reston, Va: Second Edition, AIAA Education Series, 2007.

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