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Implement quaternion representation of six-degrees-of-freedom equations of motion of custom variable mass with respect to wind axes

Equations of Motion/6DOF

The Custom Variable Mass 6DOF Wind (Quaternion)
block considers the rotation of a wind-fixed coordinate frame (*X _{w}*,

The translational motion of the wind-fixed coordinate frame
is given below, where the applied forces [*F _{x}*,

$$\begin{array}{l}{\overline{F}}_{w}=\left[\begin{array}{l}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m({\dot{\overline{V}}}_{w}+{\overline{\omega}}_{w}\times {\overline{V}}_{w})+\dot{m}\overline{V}r{e}_{w}\\ {A}_{be}=DC{M}_{wb}\frac{\left[{\overline{F}}_{w}-\dot{m}{V}_{re}\right]}{m}\\ {\overline{V}}_{w}=\left[\begin{array}{l}V\\ 0\\ 0\end{array}\right],{\overline{\omega}}_{w}=\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\dot{\beta}\mathrm{sin}\alpha \\ {q}_{b}-\dot{\alpha}\\ {r}_{b}+\dot{\beta}\mathrm{cos}\alpha \end{array}\right],{\overline{w}}_{b}=\left[\begin{array}{l}{p}_{b}\\ {q}_{b}\\ {r}_{b}\end{array}\right]\\ {A}_{bb}=DC{M}_{wb}\left[\frac{\overline{F}w-\dot{m}{V}_{re}}{m}-{\overline{\omega}}_{w}\times {\overline{V}}_{w}\right]\end{array}$$

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. Inertia tensor *I* is much easier to define
in body-fixed frame.

$$\begin{array}{l}{\overline{M}}_{b}=\left[\begin{array}{l}L\\ M\\ N\end{array}\right]=I{\dot{\overline{\omega}}}_{b}+{\overline{\omega}}_{b}\times (I{\overline{\omega}}_{b})+\dot{I}{\overline{\omega}}_{b}\\ {A}_{bb}=\left[\begin{array}{l}{\dot{U}}_{b}\\ {\dot{V}}_{b}\\ {\dot{W}}_{b}\end{array}\right]=DC{M}_{wb}\left[\frac{\overline{F}w-\dot{m}{V}_{re}}{m}-{\overline{\omega}}_{w}\times {\overline{V}}_{w}\right]\\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}$$

The integration of the rate of change of the quaternion vector is given below.

$$\left[\begin{array}{l}{\dot{q}}_{0}\\ {\dot{q}}_{1}\\ {\dot{q}}_{2}\\ {\dot{q}}_{3}\end{array}\right]=-\frac{1}{2}\left[\begin{array}{cccc}0& p& q& r\\ -p& 0& -r& q\\ -q& r& 0& -p\\ -r& -q& p& 0\end{array}\right]\left[\begin{array}{l}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]$$

**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:

`Wind Angles`

Use wind angles within equations of motion.

`Quaternion`

Use quaternions within equations of motion.

The

`Quaternion`

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 airspeed, sideslip angle, and angle of attack**The three-element vector containing the initial airspeed, initial sideslip angle and initial angle of attack.

**Initial wind orientation**The three-element vector containing the initial wind angles [bank, flight path, and heading], 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. Use state names instead of block paths throughout the linearization process.

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 name for the state.

Default value is

`' '`

.**Velocity: e.g., 'V'**Specify velocity name for the state.

Default value is

`' '`

.**Incidence angle: e.g., 'alpha'**Specify incidence angle name for the state.

Default value is

`' '`

.**Sideslip angle: e.g., 'beta'**Specify sideslip angle name for the state.

Default value is

`' '`

.**Quaternion vector: e.g., {'qr', 'qi', 'qj', 'qk'}**Specify quaternion vector name for the state.

Default value is

`' '`

.**Body rotation rates: e.g., {'p', 'q', 'r'}**Specify body rotation rates name for the state.

Default value is

`' '`

.

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

First | Vector | Contains the three applied forces in wind-fixed axes. |

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

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

Fourth | Scalar | Contains the mass of the body |

Fifth | 3-by-3 matrix | Applies to the rate of change of inertia tensor matrix in body-fixed axes. |

Sixth | 3-by-3 matrix | Applies to the inertia tensor matrix in body-fixed axes. |

Seventh (Optional) | Three-element vector | Contains one or more relative velocities at which the mass is accreted to or ablated from the body in wind 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 wind rotation angles [bank, flight path, heading], in radians. |

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

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

Sixth | Two-element vector | Contains the angle of attack and sideslip angle, in radians. |

Seventh | Two-element vector | Contains the rate of change of angle of attack and rate of change of sideslip angle, in radians per second. |

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

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

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

Eleventh (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 (Euler Angles)

Custom Variable Mass 6DOF (Quaternion)

Custom Variable Mass 6DOF ECEF (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)