Implement Euler angle representation of sixdegreesoffreedom equations of motion of custom variable mass
Aerospace Blockset / Equations of Motion / 6DOF
The Custom Variable Mass 6DOF (Euler Angles) block implements the Euler angle representation of sixdegreesoffreedom equations of motion of custom variable mass. It considers the rotation of a bodyfixed coordinate frame (X_{b}, Y_{b}, Z_{b}) about a flat Earth reference frame (X_{e}, Y_{e}, Z_{e}). For more information on Euler angles, see Algorithms.
The block assumes that the applied forces act at the center of gravity of the body.
The origin of the bodyfixed coordinate frame is the center of gravity of the body. The body is assumed to be rigid, which 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 bodyfixed coordinate frame is given below, where the applied forces [F_{x} F_{y} F_{z}]^{T} are in the bodyfixed frame. Vre_{b} is the relative velocity in the body axes at which the mass flow ($$\dot{m}$$) is ejected or added to the bodyfixed axes.
$$\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m({\dot{\overline{V}}}_{b}+\overline{\omega}\times {\overline{V}}_{b})+\dot{m}\overline{V}r{e}_{b}\\ {A}_{be}=\frac{{\overline{F}}_{b}\dot{m}{\overline{V}}_{r{e}_{b}}}{m}\\ {A}_{bb}=\left[\begin{array}{c}{\dot{u}}_{b}\\ {\dot{v}}_{b}\\ {\dot{w}}_{b}\end{array}\right]=\frac{{\overline{F}}_{b}\dot{m}{\overline{V}}_{r{e}_{b}}}{m}\overline{\omega}\times {\overline{V}}_{b}\\ {\overline{V}}_{b}=\left[\begin{array}{c}{u}_{b}\\ {v}_{b}\\ {w}_{b}\end{array}\right],\overline{\omega}=\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\end{array}$$
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.
$$\begin{array}{l}{\overline{M}}_{B}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I\dot{\overline{\omega}}+\overline{\omega}\times (I\overline{\omega})+\dot{I}\overline{\omega}\\ \\ I=\left[\begin{array}{ccc}{I}_{xx}& {I}_{xy}& {I}_{xz}\\ {I}_{yx}& {I}_{yy}& {I}_{yz}\\ {I}_{zx}& {I}_{zy}& {I}_{zz}\end{array}\right]\\ \\ \dot{I}=\left[\begin{array}{ccc}{\dot{I}}_{xx}& {\dot{I}}_{xy}& {\dot{I}}_{xz}\\ {\dot{I}}_{yx}& {\dot{I}}_{yy}& {\dot{I}}_{yz}\\ {\dot{I}}_{zx}& {\dot{I}}_{zy}& {\dot{I}}_{zz}\end{array}\right]\end{array}$$
The relationship between the bodyfixed angular velocity vector, [p q r]^{T}, and the rate of change of the Euler angles, $${[\dot{\varphi}\dot{\theta}\dot{\psi}]}^{\text{T}}$$, can be determined by resolving the Euler rates into the bodyfixed coordinate frame.
$$\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{c}\dot{\varphi}\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& \mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{c}0\\ \dot{\theta}\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& \mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& \mathrm{sin}\theta \\ 0& 1& 0\\ \mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}0\\ 0\\ \dot{\psi}\end{array}\right]={J}^{1}\left[\begin{array}{c}\dot{\varphi}\\ \dot{\theta}\\ \dot{\psi}\end{array}\right]$$

Inverting J then gives the required relationship to determine the Euler rate vector.
$$\left[\begin{array}{c}\dot{\varphi}\\ \dot{\theta}\\ \dot{\psi}\end{array}\right]=J\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{ccc}1& (\mathrm{sin}\varphi \mathrm{tan}\theta )& (\mathrm{cos}\varphi \mathrm{tan}\theta )\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& \frac{\mathrm{sin}\varphi}{\mathrm{cos}\theta}& \frac{\mathrm{cos}\varphi}{\mathrm{cos}\theta}\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]$$
For more information on aerospace coordinate systems, see About Aerospace Coordinate Systems.
[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 (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)  Simple Variable Mass 6DOF ECEF (Quaternion)  Simple Variable Mass 6DOF Wind (Quaternion)  Simple Variable Mass 6DOF Wind (Wind Angles)