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

Equations of Motion/6DOF

The Custom Variable Mass 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}*]

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

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

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

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.

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

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

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

For more information on aerospace coordinate systems, see About Aerospace Coordinate Systems.

**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

`Custom Variable`

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.

**Include mass flow relative velocity**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.

**Include inertial acceleration**Select this check box to enable an additional output port for the accelerations in body-fixed axes with respect to the inertial frame. You typically connect this signal to the accelerometer.

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: e.g., {'Xe', 'Ye', 'Ze'}**Specify position state names.

Default value is

`''`

.**Velocity: e.g., {'U', 'v', 'w'}**Specify velocity state names.

Default value is

`''`

.**Euler rotation angles: e.g., {'phi', 'theta', 'psi'}**Specify Euler rotation angles state names. This parameter appears if the

**Representation**parameter is set to`Euler Angles`

.Default value is

`''`

.**Body rotation rates: e.g., {'p', 'q', 'r'}**Specify body rotation rate state names.

Default value is

`''`

.

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

First | Vector | Contains the three applied forces. |

Second | Vector | Contains the three applied moments. |

Third (Optional) | Vector | Contains one or more rates of change of mass (positive if accreted, negative if ablated). |

Fourth | Scalar | Contains the mass. |

Fifth | 3-by-3 matrix | Contains the rate of change of inertia tensor matrix. |

Sixth | 3-by-3 matrix | Contains the inertia tensor matrix. |

Seventh (Optional) | Three-element vector | Contains one or more relative velocities at which the mass is accreted to or ablated from the body in body-fixed axes. |

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], within ±pi, 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. |

Eight | Three-element vector | Contains the accelerations in body-fixed axes with respect to body frame. |

Ninth (Optional) | Three-element vector | Contains the accelerations in body-fixed axes with respect to inertial frame (flat Earth). You typically connect this signal to the accelerometer. |

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

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 (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)

Simple Variable Mass 6DOF Wind (Quaternion)