Implement quaternion representation of sixdegreesoffreedom equations of motion of simple variable mass with respect to body axes
Equations of Motion/6DOF
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.
$$\begin{array}{l}\left[\begin{array}{c}{\dot{q}}_{0}\\ {\dot{q}}_{1}\\ {\dot{q}}_{2}\\ {\dot{q}}_{3}\end{array}\right]=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{1ex}{$2$}\right.\left[\begin{array}{cccc}0& p& q& r\\ p& 0& r& q\\ q& r& 0& p\\ r& q& p& 0\end{array}\right]\left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]+K\epsilon \left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]\\ \\ \epsilon =1\left({q}_{0}{}^{2}+{q}_{1}{}^{2}+{q}_{3}{}^{2}+{q}_{4}{}^{2}\right)\end{array}$$
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 
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.
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.
The threeelement vector for the initial location of the body in the flat Earth reference frame.
The threeelement vector for the initial velocity in the bodyfixed coordinate frame.
The threeelement vector for the initial Euler rotation angles [roll, pitch, yaw], in radians.
The threeelement vector for the initial bodyfixed angular rates, in radians per second.
The initial mass of the rigid body.
A scalar value for the inertia of the body.
A scalar value for the empty mass of the body.
A scalar value for the full mass of the body.
A 3by3 inertia tensor matrix for the empty inertia of the body.
A 3by3 inertia tensor matrix for the full inertia of the body.
The gain to maintain the norm of the quaternion vector equal to 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.
Input  Dimension Type  Description 

First  Vector  Contains the three applied forces. 
Second  Vector  Contains the three applied moments. 
Third  Scalar  Contains one or more rates of change of mass (positive if accreted, negative if ablated). 
Fourth (Optional)  Threeelement vector  Contains one or more relative velocities at which the mass is accreted to or ablated from the body in bodyfixed axes. 
Output  Dimension Type  Description 

First  Threeelement vector  Contains the velocity in the flat Earth reference frame. 
Second  Threeelement vector  Contains the position in the flat Earth reference frame. 
Third  Threeelement vector  Contains the Euler rotation angles [roll, pitch, yaw], in radians. 
Fourth  3by3 matrix  Applies to the coordinate transformation from flat Earth axes to bodyfixed axes. 
Fifth  Threeelement vector  Contains the velocity in the bodyfixed frame. 
Sixth  Threeelement vector  Contains the angular rates in bodyfixed axes, in radians per second. 
Seventh  Threeelement vector  Contains the angular accelerations in bodyfixed axes, in radians per second squared. 
Eight  Threeelement vector  Contains the accelerations in bodyfixed axes. 
Ninth  Scalar element  Contains a flag for fuel tank status:

The block assumes that the applied forces are acting at the center of gravity of the body.
Stevens, Brian, and Frank Lewis, Aircraft Control and Simulation, Second Edition, John Wiley & Sons, 2003.
Zipfel, Peter H., Modeling and Simulation of Aerospace Vehicle Dynamics. Second Edition, AIAA Education Series, 2007.
6th Order Point Mass (Coordinated Flight)
Custom Variable Mass 6DOF (Euler Angles)
Custom Variable Mass 6DOF (Quaternion)
Custom Variable Mass 6DOF ECEF (Quaternion)
Custom Variable Mass 6DOF Wind (Quaternion)
Custom Variable Mass 6DOF Wind (Wind Angles)
Simple Variable Mass 6DOF (Euler Angles)
Simple Variable Mass 6DOF ECEF (Quaternion)