6DOF (Euler Angles)

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

• 

Libraries:
Aerospace Blockset / Equations of Motion / 6DOF

Alternative Configurations of 6DOF (Euler Angles) Block:
Simple Variable Mass 6DOF (Euler Angles) | Custom Variable Mass 6DOF (Euler Angles)

Description

The 6DOF (Euler Angles) block implements the Euler angle representation of six-degrees-of-freedom equations of motion, taking into consideration the rotation of a body-fixed coordinate frame (X_b, Y_b, Z_b) about a flat Earth reference frame (X_e, Y_e, Z_e). For more information about these reference points, see Algorithms.

The 6DOF (Euler Angles), Simple Variable Mass 6DOF (Euler Angles), and Custom Variable Mass 6DOF (Euler Angles) blocks are alternative configurations of the same block.

• 6DOF (Euler Angles) — Implement Euler angle representation of six-degrees-of-freedom equations of motion

• Simple Variable Mass 6DOF (Euler Angles) — Implement Euler angle representation of six-degrees-of-freedom equations of motion of simple variable mass

• Custom Variable Mass 6DOF (Euler Angles) — Implement Euler angle representation of six-degrees-of-freedom equations of motion of custom variable mass

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_xyz(N) — Applied forces

three-element vector

Applied forces, specified as a three-element vector in body-fixed axes. For more information on the frame, see Body Coordinates.

Data Types: double

M_xyz(N-m) — Applied moments

three-element vector

Applied moments, specified as a three-element vector in body-fixed axes. For more information on the frame, see Body Coordinates.

Data Types: double

Output

V_e — Velocity in flat Earth reference frame

three-element vector

Velocity in the flat Earth reference frame, returned as a three-element vector.

Data Types: double

X_e — Position in flat Earth reference frame

three-element vector

Position in the flat Earth reference frame, returned as a three-element vector.

Data Types: double

φ θ ψ (rad) — Euler rotation angles

three-element vector

Euler rotation angles [roll, pitch, yaw] defining an intrinsic x-y-z rotation, as a three-element vector, in radians. Yaw, pitch, and roll angles are applied using the z-y-x rotation sequence, such as angle2dcm(yaw,pitch,roll,"ZYX").

Data Types: double

DCM_be — Coordinate transformation

3-by-3 matrix

Coordinate transformation from flat Earth axes to body-fixed axes, returned as a 3-by-3 matrix.

Data Types: double

V_b — Velocity in the body-fixed frame

three-element vector

Velocity in the body-fixed frame, returned as a three-element vector.

Data Types: double

ω_b (rad/s) — Angular rates in body-fixed axes

three-element vector

Angular rates in body-fixed axes, returned as a three-element vector, in radians per second.

Data Types: double

m — Mass

scalar

Mass, returned as a scalar.

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)

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.	6DOF (Euler Angles) 6DOF (Quaternion) 6DOF Wind (Wind Angles) 6DOF Wind (Quaternion) 6DOF ECEF (Quaternion)
Simple Variable	Mass and inertia vary linearly as a function of mass rate.	Simple Variable Mass 6DOF (Euler Angles) Simple Variable Mass 6DOF (Quaternion) Simple Variable Mass 6DOF Wind (Wind Angles) Simple Variable Mass 6DOF Wind (Quaternion) Simple Variable Mass 6DOF ECEF (Quaternion)
Custom Variable	Mass and inertia variations are customizable.	Custom Variable Mass 6DOF (Euler Angles) Custom Variable Mass 6DOF (Quaternion) Custom Variable Mass 6DOF Wind (Wind Angles) Custom Variable Mass 6DOF Wind (Quaternion) Custom Variable Mass 6DOF ECEF (Quaternion)

The Simple Variable 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 position in inertial axes [Xe,Ye,Ze] — Position in inertial axes

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

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

Programmatic Use

Block Parameter: xme_0

Type: character vector

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

Default: '[0 0 0]'

Initial velocity in body axes [U,v,w] — Velocity in body axes

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

Initial velocity in body axes, specified as a three-element vector, in the body-fixed coordinate frame.

Programmatic Use

Block Parameter: Vm_0

Type: character vector

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

Default: '[0 0 0]'

Initial Euler orientation [roll, pitch, yaw] — Initial Euler orientation

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

Initial Euler orientation angles [roll, pitch, yaw], specified as a three-element vector, in radians. Euler rotation angles are those between the body and north-east-down (NED) coordinate systems.

Programmatic Use

Block Parameter: eul_0

Type: character vector

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

Default: '[0 0 0]'

Initial body rotation rates [p,q,r] — Initial body rotation

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

Initial body-fixed angular rates with respect to the NED frame, specified as a three-element vector, in radians per second.

Programmatic Use

Block Parameter: pm_0

Type: character vector

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

Default: '[0 0 0]'

Initial mass — Initial mass

1.0 (default) | scalar

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

Programmatic Use

Block Parameter: mass_0

Type: character vector

Values: '1.0' | double scalar

Default: '1.0'

Empty mass — Empty mass

0.5 (default) | scalar

Empty mass of the body, specified as a double scalar.

Programmatic Use

Block Parameter: mass_e

Type: character vector

Values: double scalar

Default: '0.5'

Full mass — Full mass of body

2.0 (default) | scalar

Full mass of the body, specified as a double scalar.

Programmatic Use

Block Parameter: mass_f

Type: character vector

Values: double scalar

Default: '2.0'

Inertia — Inertia

eye(3) (default) | scalar

Inertia of the body, specified as a double scalar.

Dependencies

To enable this parameter, set Mass type to Fixed.

Programmatic Use

Block Parameter: inertia

Type: character vector

Values: eye(3) | double scalar

Default: eye(3)

Empty inertia — Empty inertia matrix

eye(3) (default) | 3-by-3 matrix

Inertia tensor matrix for the empty inertia of the body, specified as 3-by-3 matrix.

Programmatic Use

Block Parameter: inertia_e

Type: character vector

Values: 'eye(3)' | 3-by-3 matrix

Default: 'eye(3)'

Full inertia — Full inertia of body

2*eye(3) (default) | 3-by-3 matrix

Inertia tensor matrix for the full inertia of the body, specified as 3-by-3 matrix.

Programmatic Use

Block Parameter: inertia_f

Type: character vector

Values: '2*eye(3)' | 3-by-3 matrix

Default: '2*eye(3)'

Include mass flow relative velocity — Mass flow relative velocity port

off (default) | on

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.

Programmatic Use

Block Parameter: vre_flag

Type: character vector

Values: off | on

Default: off

Output mass properties for acceleration computation — 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:

• 6DOF Acceleration — m output port

• 6DOF 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 unique name to each state. You can use state names instead of block paths during linearization.

• 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.

Velocity: e.g., {'U','v','w'} — Velocity state name

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

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

Programmatic Use

Block Parameter: Vm_statename

Type: character vector

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

Default: ''

Position: e.g., {'Xe','Ye','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: xme_statename

Type: character vector

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

Default: ''

Body rotation rates: e.g., {'p','q','r'} — Body rotation state names

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

Body rotation rate state names, specified comma-separated list surrounded by braces.

Programmatic Use

Block Parameter: pm_statename

Type: character vector

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

Default: ''

Euler rotation angles: e.g., {'phi','theta','psi'} — Euler rotation state name

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

Euler rotation angle state names, specified as a comma-separated list surrounded by braces.

Programmatic Use

Block Parameter: eul_statename

Type: character vector

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

Default: ''

Mass: e.g., 'mass' — Mass state name

'' (default) | character vector

Mass state name, specified as a character vector.

Programmatic Use

Block Parameter: mass_statename

Type: character vector

Values: '' | character vector

Default: ''

Alternative Configurations

Simple Variable Mass 6DOF (Euler Angles) — Implement Euler angle representation of six-degrees-of-freedom equations of motion of simple variable mass

The Simple Variable Mass 6DOF (Euler Angles) block implements the Euler angle representation of six-degrees-of-freedom equations of motion of simple variable mass. To enable this block, set Mass type to Simple Variable.

Libraries:
Aerospace Blockset / Equations of Motion / 6DOF

Custom Variable Mass 6DOF (Euler Angles) — Implement Euler angle representation of six-degrees-of-freedom equations of motion of custom variable mass

The Custom Variable Mass 6DOF (Euler Angles) block implements the Euler angle representation of six-degrees-of-freedom equations of motion of custom variable mass. To enable this block, set Mass type to Custom Variable.

Libraries:
Aerospace Blockset / Equations of Motion / 6DOF

Algorithms

The 6DOF (Euler Angles) block uses these reference frame concepts.

• The origin of the body-fixed coordinate frame is the center of gravity of the body, and the body is assumed to be rigid, an assumption that eliminates the need to consider the forces acting between individual elements of mass.

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

  

• Translational motion of the body-fixed coordinate frame, where the applied forces [F_x F_y F_z]^T are in the body-fixed frame, and the mass of the body m is assumed constant.

  $$\begin{array}{l}{\overline{F}}_b=\left[\begin{array}{c}{F}_x\\ F_y\\ F_z\end{array}\right]=m\left({\dot{\overline{V}}}_b+\overline{\omega }\times {\overline{V}}_b\right)\\ A_{bb}=\left[\begin{array}{c}{\dot{u}}_b\\ {\dot{v}}_b\\ {\dot{w}}_b\end{array}\right]=\frac{1}{m}{\overline{F}}_b-\overline{\omega }\times {\overline{V}}_b\\ A_{be}=\frac{1}{m}{F}_b\\ {\overline{V}}_b=\left[\begin{array}{c}{u}_b\\ v_b\\ w_b\end{array}\right],\overline{\omega }=\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\end{array}$$

• The rotational dynamics of the body-fixed frame, where the applied moments are [L M N]^T, and the inertia tensor I is with respect to the origin O.

  $$\begin{array}{l}{\overline{M}}_B=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I\dot{\overline{\omega }}+\overline{\omega }\times (I\overline{\omega })\\ I=\left[\begin{array}{ccc}{I}_{xx}& -I_{xy}& -I_{xz}\\ -I_{yx}& I_{yy}& -I_{yz}\\ -I_{zx}& -I_{zy}& I_{zz}\end{array}\right]\end{array}$$

• The relationship between the body-fixed angular velocity vector, [p q r]^T, and the rate of change of the Euler angles, $$[\begin{array}{ccc}\dot{\varphi }& \dot{\theta }& \dot{\psi }\end{array}{]}^T$$, are determined by resolving the Euler rates into the body-fixed coordinate frame.

  $$\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{c}\dot{\varphi }\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & \sin \varphi \\ 0& -\sin \varphi & \cos \varphi \end{array}\right]\left[\begin{array}{c}0\\ \dot{\theta }\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & \sin \varphi \\ 0& -\sin \varphi & \cos \varphi \end{array}\right]\left[\begin{array}{ccc}\cos \theta & 0& -\sin \theta \\ 0& 1& 0\\ \sin \theta & 0& \cos \theta \end{array}\right]\left[\begin{array}{c}0\\ 0\\ \dot{\psi }\end{array}\right]\equiv J^{-1}\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]$$

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

  $$\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]=J\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{ccc}1& (\sin \varphi \tan \theta )& (\cos \varphi \tan \theta )\\ 0& \cos \varphi & -\sin \varphi \\ 0& \frac{\sin \varphi }{\cos \theta }& \frac{\cos \varphi }{\cos \theta }\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]$$

References

[1] Stevens, Brian, and Frank Lewis, Aircraft Control and Simulation. Hoboken, NJ: Second Edition, John Wiley & Sons, 2003.

[2] Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Reston, Va: Second Edition, AIAA Education Series, 2007.

Extended Capabilities

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

Version History

Introduced in R2006a

R2025a: 6DOF 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 6DOF (Euler Angles).

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

Block	Ports	Parameters
6DOF (Euler Angles)	dωb/dt removed Abb removed Abe removed	Representation removed

As a result of the changes, the block output dimension has changed from a 2D row vector, such as a 2-by-N vector, to a 1D vector, N. N is the number of elements.