Implement quaternion representation of six-degrees-of-freedom equations of motion with respect to body axes (2024)

Table of Contents

Description Limitations Ports Input Fxyz(N) — Applied forcesthree-element vector Mxyz(N-m) — Applied moments three-element vector Output Ve — Velocity in flat Earth reference framethree-element vector Xe — Position in flat Earth reference frame three-element vector φ θ ψ (rad) — Euler rotation angles three-element vector DCMbe — Coordinate transformation 3-by-3 matrix Vb — Velocity in the body-fixed frame three-element vector ωb (rad/s) — Angular rates in body-fixed axes three-element vector dωb/dt — Angular accelerations three-element vector Abb — Accelerations in body-fixed axesthree-element vector Abe — Accelerations with respect to inertial frame three-element vector Parameters Main Units — Input and output unitsMetric (MKS) (default) | English (Velocity in ft/s) | English (Velocity in kts) Mass Type — Mass type Fixed (default) | Simple Variable | Custom Variable Representation — Equations of motion representation Quaternion (default) | Euler Angles Initial position in inertial axes [Xe,Ye,Ze] — Position in inertial axes [0 0 0] (default) | three-element vector Initial velocity in body axes [U,v,w] — Velocity in body axes [0 0 0] (default) | three-element vector Initial Euler orientation [roll, pitch, yaw] — Initial Euler orientation [0 0 0] (default) | three-element vector Initial body rotation rates [p,q,r] — Initial body rotation [0 0 0] (default) | three-element vector Initial mass — Initial mass 1.0 (default) | scalar Inertia — Inertia eye(3) (default) | scalar Gain for quaternion normalization — Gain 1.0 (default) | scalar Include inertial acceleration — Include inertial acceleration port off (default) | on State Attributes Position: e.g., {'Xe', 'Ye', 'Ze'} — Position state name '' (default) | comma-separated list surrounded by braces Velocity: e.g., {'U', 'v', 'w'} — Velocity state name '' (default) | comma-separated list surrounded by braces Quaternion vector: e.g., {'qr', 'qi', 'qj', 'qk'} — Quaternion vector state name '' (default) | comma-separated list surrounded by braces Body rotation rates: e.g., {'p', 'q', 'r'} — Body rotation state names '' (default) | comma-separated list surrounded by braces Algorithms Extended Capabilities C/C++ Code Generation Generate C and C++ code using Simulink® Coder™. Version History See Also MATLAB Command Americas Europe Asia Pacific FAQs

Implement quaternion representation of six-degrees-of-freedom equations of motion with respect to body axes

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Libraries:
Aerospace Blockset / Equations of Motion / 6DOF

Description

The 6DOF (Quaternion) block implements quaternion representation ofsix-degrees-of-freedom equations of motion with respect to body axes. For a descriptionof the coordinate system and the translational dynamics, see the block description forthe 6DOF (Euler Angles) block.

For more information on the integration of the rate of change of the quaternionvector, 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

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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

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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").

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

dω_b/dt — Angular accelerations
three-element vector

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

Data Types: double

A_bb — Accelerations in body-fixed axes
three-element vector

Accelerations in body-fixed axes with respect to body frame, returned as a three-element vector.

Data Types: double

A_be — Accelerations with respect to inertial frame
three-element vector

Accelerations in body-fixed axes with respect to inertial frame (flat Earth), returned as a three-element vector. You typically connect this signal to the accelerometer.

Dependencies

This port appears only when the Include inertial acceleration check box is selected.

Data Types: double

Parameters

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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: Simple Variable

Representation — Equations of motion representation
`Quaternion` (default) | `Euler Angles`

Equations of motion representation, specified according to thefollowing table.

Representation	Description
`Euler Angles`	Use Euler angles within equations ofmotion.
`Quaternion`	Use quaternions within equations ofmotion.

The Quaternion selection conforms to the equations of motion inAlgorithms.

Programmatic Use

Block Parameter:rep

Type: charactervector

Values:Euler Angles |Quaternion

Default:'Quaternion'

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'

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)

Gain for quaternion normalization — Gain
`1.0` (default) | scalar

Gain to maintain the norm of the quaternion vector equal to 1.0, specified as a double scalar.

Programmatic Use

Block Parameter: k_quat

Type: character vector

Values: 1.0 | double scalar

Default: 1.0

Include inertial acceleration — Include inertial acceleration port
`off` (default) | `on`

Select this check box to add an inertial acceleration port.

Dependencies

To enable the A_{b ff} port, select this parameter.

Programmatic Use

Block Parameter: abi_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 duringlinearization.

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-separated listsurrounded by braces, for example, {'a', 'b', 'c'}. Eachname must be unique.
If a parameter is empty (' '), no name isassigned.
The state names apply only to the selected block with the nameparameter.
The number of states must divide evenly among the number of statenames.
You can specify fewer names than states, but you cannot specify more namesthan states.
For example, you can specify two names in a system with four states. Thefirst name applies to the first two states and the second name to the lasttwo states.
To assign state names with a variable in the MATLAB^® workspace, enter the variable without quotes. A variable canbe a character vector, cell array, or structure.

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: ''

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: ''

Quaternion vector: e.g., {'qr', 'qi', 'qj', 'qk'} — Quaternion vector state name
`''` (default) | comma-separated list surrounded by braces

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

Programmatic Use

Block Parameter: quat_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: ''

Algorithms

The integration of the rate of change of the quaternion vector is given below. Thegain K drives the norm of the quaternion state vector to 1.0 should $ε$ become nonzero. You must choose the value of this gain with care,because a large value improves the decay rate of the error in the norm, but also slowsthe simulation because fast dynamics are introduced. An error in the magnitude in oneelement of the quaternion vector is spread equally among all the elements, potentiallyincreasing the error in the state vector.

$\begin{array}{l} [\begin{matrix} {\dot{q}}_{0} \\ {\dot{q}}_{1} \\ {\dot{q}}_{2} \\ {\dot{q}}_{3} \end{matrix}] = \frac{1}{2} [\begin{matrix} 0 & - p & - q & - r \\ p & 0 & r & - q \\ q & - r & 0 & p \\ r & q & - p & 0 \end{matrix}] [\begin{matrix} q_{0} \\ q_{1} \\ q_{2} \\ q_{3} \end{matrix}] + K ε [\begin{matrix} q_{0} \\ q_{1} \\ q_{2} \\ q_{3} \end{matrix}] \\ ε = 1 - (q_{0}^{2} + q_{1}^{2} + q_{2}^{2} + q_{3}^{2}) \end{array}$

Aerospace Blockset™ uses quaternions that are defined using the scalar-firstconvention.

Extended Capabilities

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

Version History

Introduced in R2006a

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Implement quaternion representation of six-degrees-of-freedom equations of motion with respect to body axes (2024)

FAQs

What is six degrees of freedom 6 dof motion platform? ›

A 6DOF (Six Degrees of Freedom) platform is a mechanical system that provides movement in all six spatial directions: forward/backward, left/right, up/down, pitch, roll, and yaw.

View Details ›

What is the 6DOF algorithm? ›

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.