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Implement Euler angle representation of six-degrees-of-freedom equations of motion of simple variable mass

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

The Simple Variable Mass 6DOF (Euler Angles) block considers
the rotation of a body-fixed coordinate frame
(*X _{b}*,

For a description of the coordinate system and the translational dynamics, see the description for the Simple Variable Mass 6DOF (Euler Angles) block. For more information on the body-fixed coordinate frame, see Algorithms.

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

The origin of the body-fixed 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 body-fixed coordinate frame is given below, where the
applied forces [F_{x} F_{y}_{}F_{z}]^{T
}are in the body-fixed frame. *Vre*_{b} is the relative velocity in the body axes at which the
mass flow ($$\dot{m}$$) is ejected or added to the body in body axes.

$$\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{l}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m({\dot{\overline{V}}}_{b}+\overline{\omega}\times {\overline{V}}_{b})+\dot{m}\overline{V}r{e}_{b}\\ {A}_{be}=\frac{{\overline{F}}_{b}-\dot{m}{\overline{V}}_{re}}{m}\\ {A}_{bb}=\left[\begin{array}{c}{\dot{u}}_{b}\\ {\dot{v}}_{b}\\ {\dot{\omega}}_{b}\end{array}\right]=\frac{{\overline{F}}_{b}-\dot{m}{\overline{V}}_{re}}{m}-\overline{\omega}\times {\overline{V}}_{b}\\ {\overline{V}}_{b}=\left[\begin{array}{l}{u}_{b}\\ {v}_{b}\\ {w}_{b}\end{array}\right],\overline{\omega}=\left[\begin{array}{l}p\\ q\\ r\end{array}\right]\end{array}$$

The rotational dynamics of the body-fixed 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.

$$\begin{array}{l}{\overline{M}}_{B}=\left[\begin{array}{l}L\\ M\\ N\end{array}\right]=I\dot{\overline{\omega}}+\overline{\omega}\times (I\overline{\omega})+\dot{I}\overline{\omega}\\ I=\left[\begin{array}{lll}{I}_{xx}\hfill & -{I}_{xy}\hfill & -{I}_{xz}\hfill \\ -{I}_{yx}\hfill & {I}_{yy}\hfill & -{I}_{yz}\hfill \\ -{I}_{zx}\hfill & -{I}_{zy}\hfill & {I}_{zz}\hfill \end{array}\right]\end{array}$$

The inertia tensor is determined using a table lookup which linearly interpolates
between *I _{full}* and

$$\dot{I}=\frac{{I}_{full}-{I}_{empty}}{{m}_{full}-{m}_{empty}}\dot{m}$$

The relationship between the body-fixed angular velocity vector, [p q
r]^{T}, and the rate of change of the Euler angles, [$$\dot{\varphi}\dot{\theta}\dot{\psi}$$]^{T}, can be determined by resolving the Euler
rates into the body-fixed coordinate frame.

$$\left[\begin{array}{l}p\\ q\\ r\end{array}\right]=\left[\begin{array}{l}\dot{\varphi}\\ 0\\ 0\end{array}\right]+\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & \mathrm{cos}\varphi \hfill & \mathrm{sin}\varphi \hfill \\ 0\hfill & -\mathrm{sin}\varphi \hfill & \mathrm{cos}\varphi \hfill \end{array}\right]\left[\begin{array}{l}0\\ \dot{\theta}\\ 0\end{array}\right]+\left[\begin{array}{lll}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & \mathrm{cos}\varphi \hfill & \mathrm{sin}\varphi \hfill \\ 0\hfill & -\mathrm{sin}\varphi \hfill & \mathrm{cos}\varphi \hfill \end{array}\right]\left[\begin{array}{lll}\mathrm{cos}\theta \hfill & 0\hfill & -\mathrm{sin}\theta \hfill \\ 0\hfill & 1\hfill & 0\hfill \\ \mathrm{sin}\theta \hfill & 0\hfill & \mathrm{cos}\theta \hfill \end{array}\right]\left[\begin{array}{l}0\\ 0\\ \dot{\psi}\end{array}\right]\equiv {J}^{-1}\left[\begin{array}{l}\dot{\varphi}\\ \dot{\theta}\\ \dot{\psi}\end{array}\right]$$

Inverting *J* then gives the required relationship to determine the
Euler rate vector.

$$\left[\begin{array}{l}\dot{\varphi}\\ \dot{\theta}\\ \dot{\psi}\end{array}\right]=J\left[\begin{array}{l}p\\ q\\ r\end{array}\right]=\left[\begin{array}{lll}1\hfill & (\mathrm{sin}\varphi \mathrm{tan}\theta )\hfill & (\mathrm{cos}\varphi \mathrm{tan}\theta )\hfill \\ 0\hfill & \mathrm{cos}\varphi \hfill & -\mathrm{sin}\varphi \hfill \\ 0\hfill & \frac{\mathrm{sin}\varphi}{\mathrm{cos}\theta}\hfill & \frac{\mathrm{cos}\varphi}{\mathrm{cos}\theta}\hfill \end{array}\right]\left[\begin{array}{l}p\\ q\\ r\end{array}\right]$$

[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.

6DOF (Euler Angles) | 6DOF (Quaternion) | 6DOF ECEF (Quaternion) | 6DOF Wind (Quaternion) | 6DOF Wind (Wind Angles) | 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 (Quaternion) | Simple Variable Mass 6DOF ECEF (Quaternion) | Simple Variable Mass 6DOF Wind (Quaternion) | Simple Variable Mass 6DOF Wind (Wind Angles)