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

The 6DOF (Euler Angles) block considers the rotation of a body-fixed
coordinate frame (*X _{b} , Y_{b} ,
Z_{b}* ) about
a flat Earth reference frame (

The translational motion of the body-fixed coordinate frame
is given below, where the applied forces [*F _{x} F_{y} F_{z} *]

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

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.

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

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

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

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

**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*.

Input | Dimension Type | Description |
---|---|---|

First | Vector | Contains the three applied forces in body-fixed coordinate frame. |

Second | Vector | Contains the three applied moments in body-fixed coordinate frame. |

Output | Dimension Type | Description |
---|---|---|

First | Three-element vector | Contains the velocity in the flat Earth reference frame. |

Second | Three-element vector | Contains the position in the flat Earth reference frame. |

Third | Three-element vector | Contains the Euler rotation angles [roll, pitch, yaw], in radians. |

Fourth | 3-by-3 matrix | Contains the coordinate transformation from flat Earth axes to body-fixed axes. |

Fifth | Three-element vector | Contains the velocity in the body-fixed frame. |

Sixth | Three-element vector | Contains the angular rates in body-fixed axes, in radians per second. |

Seventh | Three-element vector | Contains the angular accelerations in body-fixed axes, in radians per second squared. |

Eighth | Three-element vector | Contains the accelerations in body-fixed axes. |

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.

See the `aeroblk_six_dof`

airframe
in `aeroblk_HL20`

and `asbhl20`

for examples
of this block.

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.

6th Order Point Mass (Coordinated Flight)

Custom Variable Mass 6DOF (Euler Angles)

Custom Variable Mass 6DOF (Quaternion)

Custom Variable Mass 6DOF ECEF (Quaternion)

Custom Variable Mass 6DOF Wind (Quaternion)

Custom Variable Mass 6DOF Wind (Wind Angles)

Simple Variable Mass 6DOF (Euler Angles)

Simple Variable Mass 6DOF (Quaternion)

Simple Variable Mass 6DOF ECEF (Quaternion)

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