Implement quaternion representation of sixdegreesoffreedom equations of motion of simple variable mass with respect to wind axes
Equations of Motion/6DOF
The Simple Variable Mass 6DOF Wind (Quaternion) block considers the rotation of a windfixed coordinate frame (X_{w}, Y_{w}, Z_{w}) about an flat Earth reference frame (X_{e}, Y_{e}, Z_{e}). The origin of the windfixed coordinate frame is the center of gravity of the body, and the body is assumed to be rigid, an assumption that eliminates the need to consider the forces acting between individual elements of mass. The flat Earth reference frame is considered inertial, an excellent approximation that allows the forces due to the Earth's motion relative to the "fixed stars" to be neglected.
The translational motion of the windfixed coordinate frame is given below, where the applied forces [F_{x} F_{y} F_{z}]^{T} are in the windfixed frame. Vre_{w} is the relative velocity in the wind axes at which the mass flow ($$\dot{m}$$) is ejected or added to the body.
$$\begin{array}{l}{\overline{F}}_{w}=\left[\begin{array}{l}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m({\dot{\overline{V}}}_{w}+{\overline{\omega}}_{w}\times {\overline{V}}_{w})+\dot{m}\overline{V}r{e}_{w}\\ {\overline{V}}_{w}=\left[\begin{array}{l}V\\ 0\\ 0\end{array}\right],{\overline{\omega}}_{w}=\left[\begin{array}{l}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}\dot{\beta}\mathrm{sin}\alpha \\ {q}_{b}\dot{\alpha}\\ {r}_{b}+\dot{\beta}\mathrm{cos}\alpha \end{array}\right],{\overline{w}}_{b}\left[\begin{array}{l}{p}_{b}\\ {q}_{b}\\ {r}_{b}\end{array}\right]\end{array}$$
The rotational dynamics of the bodyfixed frame are given below, where the applied moments are [L M_{ }N]^{T}, and the inertia tensor I is with respect to the origin O. Inertia tensor I is much easier to define in bodyfixed frame.
$$\begin{array}{l}{\overline{M}}_{b}=\left[\begin{array}{l}L\\ M\\ N\end{array}\right]=I{\dot{\overline{\omega}}}_{b}+{\overline{\omega}}_{b}\times (I{\overline{\omega}}_{b})+\dot{I}{\overline{\omega}}_{b}\\ I=\left[\begin{array}{lll}{I}_{xx}\hfill & {I}_{xy}\hfill & {I}_{xz}\hfill \\ {I}_{yx}\hfill & {I}_{yy}\hfill & {I}_{yz}\hfill \\ {I}_{zx}\hfill & {I}_{zy}\hfill & {I}_{zz}\hfill \end{array}\right]\end{array}$$
The inertia tensor is determined using a table lookup which linearly interpolates between I_{full} and I_{empty} based on mass (m). While the rate of change of the inertia tensor is estimated by the following equation.
$$\dot{I}=\frac{{I}_{full}{I}_{empty}}{{m}_{full}{m}_{empty}}\dot{m}$$
The integration of the rate of change of the quaternion vector is given below.
$$\left[\begin{array}{l}{\dot{q}}_{0}\\ {\dot{q}}_{1}\\ {\dot{q}}_{2}\\ {\dot{q}}_{3}\end{array}\right]=1/2\left[\begin{array}{llll}0\hfill & p\hfill & q\hfill & r\hfill \\ p\hfill & 0\hfill & r\hfill & q\hfill \\ q\hfill & r\hfill & 0\hfill & p\hfill \\ r\hfill & q\hfill & p\hfill & 0\hfill \end{array}\right]\left[\begin{array}{l}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]$$
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:
Wind Angles  Use wind 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 containing the initial airspeed, initial sideslip angle and initial angle of attack.
The threeelement vector containing the initial wind angles [bank, flight path, and heading], 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 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, in bodyfixed axes.
A 3by3 inertia tensor matrix for the full inertia of the body, in bodyfixed axes.
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 in windfixed axes. 
Second  Vector  Contains the three applied moments in bodyfixed axes. 
Third  Scalar or vector  Contains one or more rates of change of mass. 
Fourth (Optional)  Vector  Contains one or more relative velocities at which the mass is accreted to or ablated from the body in wind 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 wind rotation angles [bank, flight path, heading], in radians. 
Fourth  3by3 matrix  Applies to the coordinate transformation from flat Earth axes to windfixed axes. 
Fifth  Threeelement vector  Contains the velocity in the windfixed frame. 
Sixth  Twoelement vector  Contains the angle of attack and sideslip angle, in radians. 
Seventh  Twoelement vector  Contains the rate of change of angle of attack and rate of change of sideslip angle, in radians per second. 
Eighth  Threeelement vector  Contains the angular rates in bodyfixed axes, in radians per second. 
Ninth  Threeelement vector  Contains the angular accelerations in bodyfixed axes, in radians per second squared. 
Tenth  Threeelement vector  Contains the accelerations in bodyfixed axes. 
Eleventh  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 (Quaternion)