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# Simple Variable Mass 6DOF (Quaternion)

Implement quaternion representation of six-degrees-of-freedom equations of motion of simple 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 Simple 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.

## Dialog Box

Units

Specifies the input and output units:

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

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 Simple 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.

Initial mass

The initial mass of the rigid body.

Empty mass

A scalar value for the empty mass of the body.

Full mass

A scalar value for the full mass of the body.

Empty inertia matrix

A 3-by-3 inertia tensor matrix for the empty inertia of the body.

Full inertia matrix

A 3-by-3 inertia tensor matrix for the full inertia of the body.

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

First

VectorContains the three applied forces.

Second

VectorContains the three applied moments.

Third

ScalarContains one or more rates of change of mass (positive if accreted, negative if ablated).

Fourth (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

First

Three-element vectorContains the velocity in the flat Earth reference frame.

Second

Three-element vectorContains the position in the flat Earth reference frame.

Third

Three-element vectorContains the Euler rotation angles [roll, pitch, yaw], in radians.

Fourth

3-by-3 matrixApplies to the coordinate transformation from flat Earth axes to body-fixed axes.

Fifth

Three-element vectorContains the velocity in the body-fixed frame.

Sixth

Three-element vectorContains the angular rates in body-fixed axes, in radians per second.

Seventh

Three-element vectorContains the angular accelerations in body-fixed axes, in radians per second squared.

Eight

Three-element vectorContains the accelerations in body-fixed axes.

Ninth

Scalar elementContains a flag for fuel tank status:
• 1 indicates that the tank is full.

• 0 indicates that the integral is neither full nor empty.

• -1 indicates that the tank is empty.

## 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 μ-Airplane with a MATLAB Simulink Helper, Edizioni Libreria CLUP, Milan, 1998.