# 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 act 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 a three-element vector in body-fixed axes. For more information on the frame, see Body Coordinates.

Data Types: double

Applied moments, specified as a three-element vector in body-fixed axes. For more information on the frame, see Body Coordinates.

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] defining an intrinsic x-y-z rotation, as a three-element vector, in radians. Yaw, pitch, and roll angles are applied using the z-y-x rotation sequence, such as angle2dcm(yaw,pitch,roll,"ZYX").

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.

RepresentationDescription

Euler Angles

Use Euler angles within equations of motion.

Quaternion

Use quaternions within equations of motion.

The Quaternion selection conforms 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'

Inertia of the body, specified as a double scalar.

#### Dependencies

To enable this parameter, set Mass type to Fixed.

#### Programmatic Use

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

Select this check box to add an inertial acceleration port.

#### Dependencies

To enable the Abe port, select this parameter.

#### 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.

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: ''

## 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.

## Version History

Introduced in R2006a