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

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

The relationship between the wind angles, $${[\mu \gamma \chi ]}^{\text{T}}$$, can be determined by resolving the wind rates into the wind-fixed coordinate frame.

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

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

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

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

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

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

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

**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

`Fixed`

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

**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.

**Initial mass**The mass of the rigid body.

**Inertia**The 3-by-3 inertia tensor matrix

*I*, in body-fixed axes.

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. |

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 | Contains 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. |

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.

Stevens, B. L., and F. L. Lewis, *Aircraft Control
and Simulation*, John Wiley & Sons, New York, 1992.

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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