3DOF (Wind Axes)

R2026b

Implement three-degrees-of-freedom equations of motion with respect to wind axes

• 

3DOF (Wind Axes) block

To add a block to a model, double-click the canvas and start typing the block name. Then, select the block from the list.

Libraries:
Aerospace Blockset / Equations of Motion / 3DOF

Description

The 3DOF (Wind Axes) block implements three-degrees-of-freedom equations of motion with respect to wind axes. It considers the rotation in the vertical plane of a wind-fixed coordinate frame about a flat Earth reference frame. For more information about the rotation and equations of motion, see Algorithms.

Limitations

The block assumes that the applied forces act at the center of gravity of the body, and that the mass and inertia are constant.

Ports

Input

F_x — Applied force along wind x-axis

scalar

Applied force along the wind x-axis, specified as a scalar, in the units selected in Units.

Data Types: double

F_z — Applied force along wind z-axis

scalar

Applied force along the wind z-axis, specified as a scalar.

Data Types: double

M — Applied pitching moment

scalar

Applied pitching moment, specified as a scalar, in the units selected in Units.

Data Types: double

g — Gravity

scalar

Gravity, specified as a scalar.

Dependencies

To enable this port, set Gravity source to External.

Data Types: double

Output

γ — Flight path angle

scalar

Flight path angle, within ±pi, returned as a scalar, in radians.

Data Types: double

q — Pitch angular rate

scalar

Pitch angular rate, returned as a scalar, in radians per second.

Data Types: double

X_eZ_e — Location of body

two-element vector

Location of the body in the flat Earth reference frame, (Xe, Ze), returned as a two-element vector.

Data Types: double

V_w — Velocity in wind-fixed frame

two-element vector

Velocity of the body resolved into the wind-fixed coordinate frame, (V, 0), returned as a two-element vector.

Data Types: double

α — Angle of attack

scalar

Angle of attack, returned as a scalar, in radians.

Data Types: double

g — Gravity

scalar

Gravity, returned as a scalar.

Dependencies

To enable this port, select the Output mass properties for acceleration computation parameter.

Data Types: double

m — Mass

scalar

Mass, returned as a scalar, in the units selected in Units.

Dependencies

To enable this port, select the Output mass properties for acceleration computation parameter.

Data Types: double

I — Inertia tensor matrix

3-by-3 matrix

Inertia tensor matrix, returned as a 3-by-3 matrix.

Dependencies

To enable this port, select the Output mass properties for acceleration computation parameter.

Data Types: double

Parameters

Main

Units — Input and output units

Metric (MKS) (default) | English (Velocity in ft/s) | English (Velocity in kts)

Input and output units, specified as Metric (MKS), English (Velocity in ft/s), or English (Velocity in kts).

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

Programmatic Use

Block Parameter: units

Type: character vector

Values: Metric (MKS) | English (Velocity in ft/s) | English (Velocity in kts)

Default: Metric (MKS)

Axes — Body or wind axes

Wind (default) | Body

Body or wind axes, specified as Wind or Body.

Programmatic Use

Block Parameter: axes

Type: character vector

Values: Wind | Body

Default: Wind

Mass type — Mass type

Fixed (default) | Simple Variable | Custom Variable

Mass type, specified according to the following table.

Mass Type	Description	Default for
Fixed	Mass is constant throughout the simulation.	3DOF (Body Axes) 3DOF (Wind Axes)
Simple Variable	Mass and inertia vary linearly as a function of mass rate.	Simple Variable Mass 3DOF (Body Axes) Simple Variable Mass 3DOF (Wind Axes)
Custom Variable	Mass and inertia variations are customizable.	Custom Variable Mass 3DOF (Body Axes) Custom Variable Mass 3DOF (Wind Axes)

The Fixed selection conforms to the previously described equations of motion.

Programmatic Use

Block Parameter: mtype

Type: character vector

Values: Fixed | Simple Variable | Custom Variable

Default: 'Fixed'

Initial airspeed — Initial speed

100 (default) | scalar greater than 0

Initial speed of the body, (V₀), specified as a scalar greater than 0.

Programmatic Use

Block Parameter: V_ini

Type: character vector

Values: '100' | scalar

Default: '100'

Initial flight path angle — Initial flight path angle

0 (default) | scalar

Initial flight path angle of the body, (γ₀), specified as a scalar.

Programmatic Use

Block Parameter: gamma_ini

Type: character vector

Values: '0' | scalar

Default: '0'

Initial body rotation rate — Initial pitch rotation rate

0 (default) | scalar

Initial pitch rotation rate, (q₀), specified as a scalar.

Programmatic Use

Block Parameter: q_ini

Type: character vector

Values: '0' | scalar

Default: '0'

Initial incidence — Initial angle

0 (default) | scalar

Initial angle between the velocity vector and the body, (α₀), specified as a scalar.

Programmatic Use

Block Parameter: alpha_ini

Type: character vector

Values: '0' | scalar

Default: '0'

Initial position (x,z) — Initial location

[0 0] (default) | two-element vector

Initial location of the body in the flat Earth reference frame, specified as a two-element vector.

Programmatic Use

Block Parameter: pos_ini

Type: character vector

Values: '[0 0]' | two-element vector

Default: '[0 0]'

Initial mass — Initial mass

1.0 (default) | scalar

Initial mass of the rigid body, specified as a scalar.

Programmatic Use

Block Parameter: mass

Type: character vector

Values: '1.0' | scalar

Default: '1.0'

Inertia body axes — Inertia of body

1.0 (default) | scalar

Inertia of the body, specified as a scalar.

Dependencies

To enable this parameter, set Mass type to Fixed.

Programmatic Use

Block Parameter: Iyy

Type: character vector

Values: '1.0' | scalar

Default: '1.0'

Gravity Source — Gravity source

Internal (default) | External

Gravity source, specified as:

External	Variable gravity input to block
Internal	Constant gravity specified in mask

Programmatic Use

Block Parameter: g_in

Type: character vector

Values: 'Internal' | 'External'

Default: 'Internal'

Acceleration due to gravity — Gravity source

9.81 (default) | scalar

Acceleration due to gravity, specified as a double scalar and used if internal gravity source is selected. If gravity is to be neglected in the simulation, this value can be set to 0.

Dependencies

• To enable this parameter, set Gravity Source to Internal.

Programmatic Use

Block Parameter: g

Type: character vector

Values: '9.81' | scalar

Default: '9.81'

Output mass properties for acceleration computation — Option to include mass properties for acceleration port

off (default) | on

Select this check box to enable ports for mass properties for acceleration. You can then use these ports as inputs for these blocks:

• 3DOF Acceleration — m output port

• 3DOF Angular Acceleration — I and dI/dt coeff ports.

Programmatic Use

Block Parameter: mass_flag

Type: character vector

Values: 'off' | 'on'

Default: off

State Attributes

Assign a unique name to each state. You can use state names instead of block paths during linearization.

• The number of names must match the number of states, as shown for each item, or be empty. Set all or none of the block states.

• To assign names to single-variable states, enter unique names between quotes, for example, 'q' or "q".

• To assign names to two-variable states, enter a comma-separated list surrounded by braces, for example, {'Xe','Ze'}.

• If a state parameter is empty (' '), no name is assigned.

• 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 of character vectors, or string.

Velocity: e.g., 'V' — Velocity state name

'' (default) | character vector

Velocity state name, specified as a character vector or string.

Programmatic Use

Block Parameter: V_statename

Type: character vector | string

Values: '' | scalar

Default: ''

Position: e.g., {'Xe', 'Ze'} — Position state name

'' (default) | comma-separated list surrounded by braces

Position state names, specified as a comma-separated list surrounded by braces.

Programmatic Use

Block Parameter: pos_statename

Type: character vector

Values: '' | comma-separated list surrounded by braces

Default: ''

Body rotation rate: e.g., 'q' — Body rotation state name

'' (default) | scalar

Body rotation rate state names, specified as a character vector or string.

Programmatic Use

Block Parameter: q_statename

Type: character vector | string

Values: '' | scalar

Default: ''

Flight path angle: e.g., 'gamma' — Flight path angle state name

'' (default)

Flight path angle state name, specified as a character vector or string.

Programmatic Use

Block Parameter: gamma_statename

Type: character vector | string

Values: '' | scalar

Default: ''

Incidence angle e.g., 'alpha' — Incidence angle state name

'' (default) | scalar

Incidence angle state name, specified as a character vector or string.

Programmatic Use

Block Parameter: alpha_statename

Type: character vector | string

Values: '' | scalar

Default: ''

Programmatic Use: Add Block

To add the block programmatically, use the add_block function with one of these library block paths as the source.

"aerolib3dof2/3dof (Wind Axes)"

Example: add_block("aerolib3dof2/3dof (Wind Axes)","myModel/myBlock",MakeNameUnique="on")

Algorithms

The block considers the rotation in the vertical plane of a wind-fixed coordinate frame about a flat Earth reference frame.

The equations of motion are

$$\begin{array}{l}{A}_{xb}=A_{xe}-qV\sin \alpha \\ A_{zb}=A_{ze}+qV\cos \alpha \\ A_{xe}=\left(\frac{F_x}{m}-g\sin \gamma \right)\cos \alpha -\left(\frac{F_z}{m}+g\cos \gamma \right)\sin \alpha \\ A_{ze}=\left(\frac{F_x}{m}-g\sin \gamma \right)\sin \alpha +\left(\frac{F_z}{m}+g\cos \gamma \right)\cos \alpha \\ \dot{V}=\frac{F_x}{m}-g\sin \gamma \\ {\dot{X}}_e=V\cos \gamma \\ {\dot{Z}}_e=-V\sin \gamma \\ \dot{q}=\frac{M_y}{I_{yy}}\\ \dot{\gamma }=q-\dot{\alpha }\\ \dot{\alpha }=\frac{F_z}{mV}+\frac{g}{V}\cos \gamma +q\end{array}$$

where the applied forces are assumed to act at the center of gravity of the body. Input variables are wind-axes forces F_x and F_z and body moment M_y. g is an optional input variable.

References

[1] Stevens, Brian, and Frank Lewis. Aircraft Control and Simulation. New York: John Wiley & Sons, 1992.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2006a

R2026a: 3DOF equations of motion block acceleration ports removed

As a result of the removal of the acceleration ports, the Include inertial acceleration parameter has been replaced with Output mass properties for acceleration computation for 3DOF (Wind Axes).

Related to this change, the block output ports have changed:

Block	Output Ports Removed	Output Ports Added
3DOF (Wind Axes)	dq/dt AxbAzb AxeAze	g m I

R2021b: 3DOF (Wind Axes) Block Changes

The 3DOF equations of motion have been updated. Existing models created prior to R2021b that contain 3DOF equations of motion blocks continue to run. If you replace a pre-R2021b version of a 3DOF equation of motion block with an R2021b or later version, your updated model might have a higher tendency for algebraic loops. For an example of how to remove algebraic loops using unit delays, see Remove Algebraic Loops. For further information about algebraic loops, see Identify Algebraic Loops in Your Model.