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# 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

Equations of Motion/6DOF

## Description

For a description of the coordinate system and the translational dynamics, see the block description for the Custom Variable Mass 6DOF (Euler Angles) block.

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.

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

## Dialog Box

Units

Specifies the input and output units:

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

Mass Type

Select the type of mass to use:

 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.

Representation

Select the representation to use:

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

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

Initial position in inertial axes

The three-element vector for the initial location of the body in the flat Earth reference frame.

Initial velocity in body axes

The three-element vector for the initial velocity in the body-fixed coordinate frame.

Initial Euler rotation

The three-element vector for the initial Euler rotation angles [roll, pitch, yaw], in radians.

Initial body rotation rates

The three-element vector for the initial body-fixed angular rates, in radians per second.

Gain for quaternion normalization

The gain to maintain the norm of the quaternion vector equal to 1.0.

Include mass flow relative velocity

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.

## Inputs and Outputs

InputDimension TypeDescription
FirstVectorContains the three applied forces.
SecondVectorContains the three applied moments.
Third (Optional)VectorContains one or more rates of change of mass (positive if accreted, negative if ablated).
FourthScalarContains the mass.
Fifth3-by-3 matrixContains rate of change of inertia tensor matrix.
Sixth3-by-3 matrixContains the inertia tensor matrix.

Seventh (Optional)

Three-element vectorContains one or more relative velocities at which the mass is accreted to or ablated from the body in body-fixed axes.

OutputDimension TypeDescription
FirstThree-element vectorContains the velocity in the flat Earth reference frame.
SecondThree-element vectorContains the position in the flat Earth reference frame.
ThirdThree-element vectorContains the Euler rotation angles [roll, pitch, yaw], in radians.
Fourth3-by-3 matrixContains the coordinate transformation from flat Earth axes to body-fixed axes.
FifthThree-element vectorContains the velocity in the body-fixed frame.
SixthThree-element vectorContains the angular rates in body-fixed axes, in radians per second.
SeventhThree-element vectorContains the angular accelerations in body-fixed axes, in radians per second squared.
EighthThree-element vectorContains the accelerations in body-fixed axes.

## Assumptions and Limitations

The block assumes that the applied forces are acting at the center of gravity of the body.

## Reference

Mangiacasale, L., Flight Mechanics of a u-Airplane with a MATLAB Simulink Helper, Edizioni Libreria CLUP, Milan, 1998.