# Fixed-Wing Point Mass

Integrate fourth- or sixth-order point mass equations of motion in coordinated flight

Since R2021a

Libraries:
Aerospace Blockset / Equations of Motion / Point Mass
UAV Toolbox / Algorithms

## Description

The Fixed-Wing Point Mass block integrates fourth- or sixth-order point mass equations of motion in coordinated flight.

## Limitations

• The flat Earth reference frame is considered inertial, an approximation that allows the forces due to the Earth's motion relative to the "fixed stars" to be neglected.

• The block assumes that there is fully coordinated flight, that is, there is no side force (wind axes) and sideslip is always zero.

## Ports

### Input

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Lift, specified as a scalar in units of force.

Data Types: `double`

Drag, specified as a scalar in units of force.

Data Types: `double`

Weight, specified as a scalar in units of force.

Data Types: `double`

Thrust, specified as a scalar in units of force.

Data Types: `double`

Flight path angle relative to the air mass, specified as a scalar in radians.

Data Types: `double`

Bank angle, specified as a scalar in radians.

Data Types: `double`

Angle of attack, specified as a scalar in radians.

Data Types: `double`

Wind vector in the direction in which the air mass is moving, specified as a three-element vector.

Data Types: `double`

### Output

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Airspeed, returned as a scalar.

Data Types: `double`

Ground speed over the Earth (speed of motion over the ground), returned as a scalar.

Data Types: `double`

Velocity vector relative to the air mass, returned as a three-element vector.

Data Types: `double`

Velocity vector relative to Earth with ```[North East Down]``` orientation, returned as a three-element vector.

#### Dependencies

To enable this port, set Reference frame orientation to ```[North East Down]```.

Data Types: `double`

Velocity vector relative to Earth with ```[East North Up]``` orientation, returned as a three-element vector.

#### Dependencies

To enable this port, set Reference frame orientation to ```[East North Up]```.

Data Types: `double`

Position vector relative to Earth with ```[North East Down]``` orientation, returned as a three-element vector.

#### Dependencies

To enable this port, set Reference frame orientation to ```[North East Down]```.

Data Types: `double`

Position vector relative to Earth with ```[East North Up]``` orientation, returned as a three-element vector.

#### Dependencies

To enable this port, set Reference frame orientation to ```[East North Up]```.

Data Types: `double`

Flight path angle relative to the air mass, returned as a scalar.

Data Types: `double`

Flight path angle relative to Earth, returned as a scalar.

Data Types: `double`

Heading angle relative to air mass, returned as a scalar.

#### Dependencies

To enable this port, set Degrees of Freedom to `6th Order (Coordinated Flight)`.

Data Types: `double`

Heading angle relative to Earth, returned as a scalar.

#### Dependencies

To enable this port, set Degrees of Freedom to `6th Order (Coordinated Flight)`.

Data Types: `double`

## Parameters

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Input and output units, specified as follows:

Units

Forces

Velocity

Position

Mass

`Metric (MKS)`

newtons

meters per second

meters

kilograms

```English (velocity in ft/s)```

pounds

feet per second

feet

slugs

```English (velocity in kts)```

pounds

knots

feet

slugs

#### Programmatic Use

 Block Parameter: `units` Type: character vector Values: ```'Metric (MKS)'``` | ```'English (velocity in ft/s)'``` | ```'English (velocity in kts)'``` Default: ```'Metric (MKS)'```

Reference frames used for input ports and output ports, specified as `[East North Up]` or ```[North East Down]```.

#### Programmatic Use

 Block Parameter: `frame` Type: character vector Values: ```'[East North Up]'``` | `'[North East Down]'` Default: ```'[North East Down]'```

Degrees of freedom, specified as ```4th Order (Longitudinal)``` or ```6th Order (Coordinated Flight)```.

#### Programmatic Use

 Block Parameter: `order` Type: character vector Values: ```'4th Order (Longitudinal)'``` | ```'6th Order (Coordinated Flight)'``` Default: ```'6th Order (Coordinated Flight)'```

Initial East (Earth) location in the `[North East Down]` orientation, specified as a scalar.

#### Dependencies

The direction specification of this parameter depends on the Reference frame orientation and Degrees of Freedom setting:

Initial crossrangeReference frame orientationDegrees of freedom

East

[North East Down]

6th Order (Coordinated Flight)

North

[East North Up]

6th Order (Coordinated Flight)

#### Programmatic Use

 Block Parameter: `east` Type: character vector Values: scalar Default: `'0'`

Initial North (Earth) downrange of the point mass, specified as a scalar.

#### Dependencies

The direction specification of this parameter depends on the Reference frame orientation and Degrees of Freedom setting:

Initial downrangeReference frame orientationDegrees of freedom

North

[North East Down]

6th Order (Coordinated Flight)

North

[North East Down]

4th Order (Longitudinal)

East

[East North Up]

6th Order (Coordinated Flight)

East

[East North Up]

4th Order (Longitudinal)

#### Programmatic Use

 Block Parameter: `north` Type: character vector Values: scalar Default: `'0'`

Initial altitude of the point mass, specified as a scalar.

#### Programmatic Use

 Block Parameter: `altitude` Type: character vector Values: scalar Default: `'0'`

Initial airspeed of the point mass, specified as a scalar.

#### Programmatic Use

 Block Parameter: `'airspeed'` Type: character vector Values: scalar Default: `'50'`

Initial flight path angle of the point mass, specified as a scalar.

#### Programmatic Use

 Block Parameter: `gamma` Type: character vector Values: scalar Default: `'0'`

Initial heading angle of the point mass, specified as a scalar.

#### Dependencies

To enable this parameter, set Degrees of Freedom to `6th Order (Coordinated Flight)`.

#### Programmatic Use

 Block Parameter: `chi` Type: character vector Values: scalar Default: `'0'`

Mass of the point mass, specified as a scalar.

#### Programmatic Use

 Block Parameter: `mass` Type: character vector Values: scalar Default: `'10'`

## Algorithms

The integrated equations of motion for the point mass are:

`$\begin{array}{l}\stackrel{˙}{V}=\left(T\mathrm{cos}\alpha -D-W\mathrm{sin}{\gamma }_{ai}\right)/m\\ {\stackrel{˙}{\gamma }}_{a}=\left(\left(L+T\mathrm{sin}\alpha \right)\mathrm{cos}\mu -W\mathrm{cos}{\gamma }_{ai}\right)/\left(mV\right)\\ {\stackrel{˙}{X}}_{e}={V}_{a}+{V}_{w}\end{array}$`

6th order equations:

`$\begin{array}{l}{\stackrel{˙}{X}}_{a}=\left(\left(L+T\mathrm{sin}\alpha \right)\mathrm{sin}\mu \right)/\left(mV\mathrm{cos}{\gamma }_{a}\right)\\ {{\stackrel{˙}{X}}_{a}|}_{East}=V\mathrm{cos}{\chi }_{a}\mathrm{cos}{\gamma }_{a}\\ {{\stackrel{˙}{X}}_{a}|}_{North}=V\mathrm{sin}{\chi }_{a}\mathrm{cos}{\gamma }_{a}\\ {{\stackrel{˙}{X}}_{a}|}_{Up}=V\mathrm{sin}{\gamma }_{a}\end{array}$`

4th order equations:

`$\begin{array}{l}{\stackrel{˙}{\chi }}_{a}=0\\ {{\stackrel{˙}{X}}_{a}|}_{East}=V\mathrm{cos}{\gamma }_{a}\\ {{\stackrel{˙}{X}}_{a}|}_{North}=0\\ {{\stackrel{˙}{X}}_{a}|}_{Up}=V\mathrm{sin}{\gamma }_{a}\end{array}$`

where:

• m — Mass.

• g — Gravitational acceleration.

• W — Weight ( m*g).

• L — Lift force.

• D — Drag force.

• T — Thrust force.

• α — Angle of attack.

• μ — Angle of bank.

• γai — Input port value for the flight path angle.

• V — Airspeed, as measured on the aircraft, with respect to the air mass. It is also the magnitude of vector Va.

• Vw — Steady wind vector.

• Subscript a — For the variables, denotes that they are with respect to the steadily moving air mass:

• γa — Flight path angle.

• Xa — Position [East, North, Up].

• Subscript e — Flat Earth inertial frame such that so Xe is the position on the Earth after correcting Xa for the air mass movement.

`$\begin{array}{l}G=\sqrt{\left({{V}_{e}|}_{Eas{t}^{2}}+{{V}_{e}|}_{Nort{h}^{2}}\right)}\\ \gamma ={\mathrm{sin}}^{-1}\left(\frac{{{V}_{e}|}_{Up}}{‖\underset{¯}{{V}_{e}}‖}\right)\\ \chi ={\mathrm{tan}}^{-1}\left(\frac{{{V}_{e}|}_{North}}{{{V}_{e}|}_{East}}\right)\end{array}$`

where:

• The four-quadrant inverse tangent (`atan2`) calculates the heading angle.

• The groundspeed, G, is the speed over the flat Earth (a 2-D projection).

## Version History

Introduced in R2021a