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

Equations of Motion/6DOF

For a description of the coordinate system employed and the translational dynamics, see the block description for the Custom Variable Mass 6DOF Wind (Quaternion) block.

The relationship between the wind angles, [*μ
γ χ*]^{T}, can be determined
by resolving the wind rates into the wind-fixed coordinate frame.

$$\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{l}\dot{\mu}\\ 0\\ 0\end{array}\right]+\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & \mathrm{cos}\mu \hfill & \mathrm{sin}\mu \hfill \\ 0\hfill & -\mathrm{sin}\mu \hfill & \mathrm{cos}\mu \hfill \end{array}\right]\left[\begin{array}{l}0\\ \dot{\gamma}\\ 0\end{array}\right]+\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & \mathrm{cos}\mu \hfill & \mathrm{sin}\mu \hfill \\ 0\hfill & -\mathrm{sin}\mu \hfill & \mathrm{cos}\mu \hfill \end{array}\right]\left[\begin{array}{lll}\mathrm{cos}\gamma \hfill & 0\hfill & -\mathrm{sin}\gamma \hfill \\ 0\hfill & 1\hfill & 0\hfill \\ \mathrm{sin}\gamma \hfill & 0\hfill & \mathrm{cos}\gamma \hfill \end{array}\right]\left[\begin{array}{l}0\\ 0\\ \dot{\chi}\end{array}\right]\equiv {J}^{-1}\left[\begin{array}{l}\dot{\mu}\\ \dot{\gamma}\\ \dot{\chi}\end{array}\right]$$

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

$$\left[\begin{array}{l}\dot{\mu}\\ \dot{\gamma}\\ \dot{\chi}\end{array}\right]=J\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{lll}1\hfill & (\mathrm{sin}\mu \mathrm{tan}\gamma )\hfill & (\mathrm{cos}\mu \mathrm{tan}\gamma )\hfill \\ 0\hfill & \mathrm{cos}\mu \hfill & -\mathrm{sin}\mu \hfill \\ 0\hfill & \frac{\mathrm{sin}\mu}{\mathrm{cos}\gamma}\hfill & \frac{\mathrm{cos}\mu}{\mathrm{cos}\gamma}\hfill \end{array}\right]\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]$$

The body-fixed angular rates are related to the wind-fixed angular rate by the following equation.

$$\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\dot{\beta}\mathrm{sin}\alpha \\ {q}_{b}-\dot{\alpha}\\ {r}_{b}+\dot{\beta}\mathrm{cos}\alpha \end{array}\right]$$

Using this relationship in the wind rate vector equations, gives the relationship between the wind rate vector and the body-fixed angular rates.

$$\left[\begin{array}{l}\dot{\mu}\\ \dot{\gamma}\\ \dot{\chi}\end{array}\right]=J\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=\left[\begin{array}{lll}1\hfill & (\mathrm{sin}\mu \mathrm{tan}\gamma )\hfill & (\mathrm{cos}\mu \mathrm{tan}\gamma )\hfill \\ 0\hfill & \mathrm{cos}\mu \hfill & -\mathrm{sin}\mu \hfill \\ 0\hfill & \frac{\mathrm{sin}\mu}{\mathrm{cos}\gamma}\hfill & \frac{\mathrm{cos}\mu}{\mathrm{cos}\gamma}\hfill \end{array}\right]DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\dot{\beta}\mathrm{sin}\alpha \\ {q}_{b}-\dot{\alpha}\\ {r}_{b}+\dot{\beta}\mathrm{cos}\alpha \end{array}\right]$$

**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

`Wind 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 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.

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. |

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) | 1-by-1-by-m array | Contains one or more relative velocities at which the mass
is accreted to or ablated from the body in wind axes. m is
three times the size of the third input vector. |

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 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 (Quaternion)

Simple Variable Mass 6DOF (Euler Angles)

Simple Variable Mass 6DOF (Quaternion)

Simple Variable Mass 6DOF ECEF (Quaternion)

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