# 6DOF (Euler Angles)

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

## Library

Equations of Motion/6DOF

## Description

The 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 ). 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 Fy Fz ]T are in the body-fixed frame, and the mass of the body m is assumed constant.

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

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

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

The relationship between the body-fixed angular velocity vector, [p q r]T, and the rate of change of the Euler angles, ${\left[\stackrel{˙}{\varphi }\stackrel{˙}{\theta }\stackrel{˙}{\psi }\right]}^{\text{T}}$, can be 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.

## 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:

 `Euler Angles` Use Euler angles within equations of motion. `Quaternion` Use quaternions within equations of motion.

The `Euler 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 velocity in body axes

The three-element vector for the initial velocity in the body-fixed coordinate frame.

Initial Euler rotation

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

## Inputs and Outputs

InputDimension TypeDescription

First

VectorContains the three applied forces in body-fixed coordinate frame.

Second

VectorContains the three applied moments in body-fixed coordinate frame.

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 Euler rotation angles [roll, pitch, yaw], in radians.

Fourth

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

Fifth

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

Sixth

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

Seventh

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

Eighth

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.

## Examples

See the `aeroblk_six_dof` airframe in `aeroblk_HL20` and `asbhl20` for examples of this block.

## Reference

Stevens, Brian, and Frank Lewis, Aircraft Control and Simulation, Second Edition, John Wiley & Sons, 2003.

Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Second Edition, AIAA Education Series, 2007.