# Custom Variable Mass 6DOF (Quaternion)

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

• Library:
• Aerospace Blockset / Equations of Motion / 6DOF

## Description

The Custom Variable Mass 6DOF (Quaternion) block implements a quaternion representation of six-degrees-of-freedom equations of motion of custom variable mass with respect to body axes. For a description of the coordinate system and the translational dynamics, see the block description for the Custom Variable Mass 6DOF (Euler Angles) block.

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

## Ports

### Input

expand all

Applied forces, specified as a three-element vector.

Data Types: double

Applied moments, specified as a three-element vector.

Data Types: double

One or more rates of change of mass (positive if accreted, negative if ablated), specified as a three-element vector.

#### Dependencies

To enable this port, select Include mass flow relative velocity.

Data Types: double

Mass, specified as a scalar.

Data Types: double

Rate of change of inertia tensor matrix, specified as a 3-by-3 matrix.

Data Types: double

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

Data Types: double

One or more relative velocities at which the mass is accreted to or ablated from the body in body-fixed axes, specified as a three-element vector.

#### Dependencies

To enable this port, select Include mass flow relative velocity.

Data Types: double

### Output

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

Data Types: double

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

Data Types: double

Euler rotation angles [roll, pitch, yaw], returned as a three-element vector, in radians.

Data Types: double

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

Data Types: double

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

Data Types: double

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

Data Types: double

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

Data Types: double

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

Data Types: double

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

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

UnitsForcesMomentAccelerationVelocityPositionMassInertia
Metric (MKS) NewtonNewton-meterMeters per second squaredMeters per secondMetersKilogramKilogram meter squared
English (Velocity in ft/s) PoundFoot-poundFeet per second squaredFeet per secondFeetSlugSlug foot squared
English (Velocity in kts) PoundFoot-poundFeet per second squaredKnotsFeetSlugSlug 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, specified according to the following table.

Mass TypeDescriptionDefault for
Fixed

Mass is constant throughout the simulation.

Simple Variable

Mass and inertia vary linearly as a function of mass rate.

Custom Variable

Mass and inertia variations are customizable.

The Custom 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: 'Custom Variable'

Equations of motion representation, specified according to the following table.

RepresentationDescription

Quaternion

Use quaternions within equations of motion.

Euler Angles

Use Euler angles 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: 'Euler Angles'

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, 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 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-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]'

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

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

Select this check box to add an inertial acceleration port.

#### Dependencies

To enable the Abe 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 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 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 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 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 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 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.

$\begin{array}{l}\left[\begin{array}{c}{\stackrel{˙}{q}}_{0}\\ {\stackrel{˙}{q}}_{1}\\ {\stackrel{˙}{q}}_{2}\\ {\stackrel{˙}{q}}_{3}\end{array}\right]=1}{2}\left[\begin{array}{cccc}0& -p& -q& -r\\ p& 0& r& -q\\ q& -r& 0& p\\ r& q& -p& 0\end{array}\right]\left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]+K\epsilon \left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]\\ \epsilon =1-\left({q}_{0}^{2}+{q}_{1}^{2}+{q}_{2}^{2}+{q}_{3}^{2}\right).\end{array}$

## References

[1] Stevens, Brian, and Frank Lewis. Aircraft Control and Simulation, 2nd ed. Hoboken, NJ: John Wiley & Sons, 2003.

[2] Zipfel, Peter H. Modeling and Simulation of Aerospace Vehicle Dynamics. 2nd ed. Reston, VA: AIAA Education Series, 2007.

## Extended Capabilities

### C/C++ Code GenerationGenerate C and C++ code using Simulink® Coder™.

Introduced in R2006a

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