6DOF (Quaternion)

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 of six-degrees-of-freedom equations of motion with respect to body axes. For a description of the coordinate system and the translational dynamics, see the block description for the 6DOF (Euler Angles) block.

For more information on the integration of the rate of change of the quaternion vector, 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_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

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

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 the following table.

Representation	Description
Euler Angles	Use Euler angles within equations of motion.
Quaternion	Use quaternions within equations of motion.

The Quaternion selection conforms to the equations of motion in Algorithms.

Programmatic Use

Block Parameter: rep

Type: character vector

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 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-separated list surrounded by braces, for example, {'a', 'b', 'c'}. Each name must be unique.

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

• 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.

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. The gain K drives the norm of the quaternion state vector to 1.0 should $$\epsilon$$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 slows the simulation because fast dynamics are introduced. An error in the magnitude in one element of the quaternion vector is spread equally among all the elements, potentially increasing the error in the state vector.

$$\begin{array}{l}\left[\begin{array}{c}{\dot{q}}_0\\ {\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left[\begin{array}{cccc}0& -p& -q& -r\\ p& 0& r& -q\\ q& -r& 0& p\\ r& q& -p& 0\end{array}\right]\left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]+K\epsilon \left[\begin{array}{c}{q}_0\\ q_1\\ q_2\\ q_3\end{array}\right]\\ \epsilon =1-\left(q_0^2+q_1^2+q_2^2+q_3^2\right)\end{array}$$

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

Extended Capabilities

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

Version History

Introduced in R2006a