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Implement quaternion representation of six-degrees-of-freedom equations of motion with respect to wind axes
The 6DOF Wind (Quaternion) block considers the rotation of a wind-fixed coordinate frame (X_{w} , Y_{w} , Z_{w} ) about an flat Earth reference frame (X_{e} , Y_{e} , Z_{e} ). The origin of the wind-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 wind-fixed coordinate frame is given below, where the applied forces [F_{x} F_{y} F_{z}]^{T }are in the wind-fixed frame, and the mass of the body m is assumed constant.
$$\begin{array}{l}{\overline{F}}_{w}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m({\dot{\overline{V}}}_{w}+{\overline{\omega}}_{w}\times {\overline{V}}_{w})\\ \\ {\overline{V}}_{w}=\left[\begin{array}{c}V\\ 0\\ 0\end{array}\right],{\overline{\omega}}_{w}=\left[\begin{array}{c}{p}_{w}\\ {q}_{w}\\ {r}_{w}\end{array}\right]=DM{C}_{wb}\left[\begin{array}{c}{p}_{b}-\dot{\beta}\mathrm{sin}\alpha \\ {q}_{b}-\dot{\alpha}\\ {r}_{b}+\dot{\beta}\mathrm{cos}\alpha \end{array}\right],{\overline{\omega}}_{b}=\left[\begin{array}{c}{p}_{b}\\ {q}_{b}\\ {r}_{b}\end{array}\right]\\ \end{array}$$
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. Inertia tensor I is much easier to define in body-fixed frame.
$$\begin{array}{l}{\overline{M}}_{b}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I{\dot{\overline{\omega}}}_{b}+{\overline{\omega}}_{b}\times \left(I{\overline{\omega}}_{b}\right)\\ \\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\end{array}$$
The integration of the rate of change of the quaternion vector is given below.
$$\left[\begin{array}{c}{\dot{q}}_{0}\\ {\dot{q}}_{1}\\ {\dot{q}}_{2}\\ {\dot{q}}_{3}\end{array}\right]=-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left[\begin{array}{cccc}0& p& q& r\\ -p& 0& -r& q\\ -q& r& 0& -p\\ -r& -q& p& 0\end{array}\right]\left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]$$
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 Fixed 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 three-element vector for the initial location of the body in the flat Earth reference frame.
The three-element vector containing the initial airspeed, initial angle of attack and initial sideslip angle.
The three-element vector containing the initial wind angles [bank, flight path, and heading], in radians.
The three-element vector for the initial body-fixed angular rates, in radians per second.
The mass of the rigid body.
The 3-by-3 inertia tensor matrix I, in body-fixed axes.
Input | Dimension Type | Description |
---|---|---|
First | Vector | Contains the three applied forces in wind-fixed axes. |
Second | Vector | Contains the three applied moments in body-fixed axes. |
Output | Dimension Type | Description |
---|---|---|
First | Three-element vector | Contains the velocity in the flat Earth reference frame. |
Second | Three-element vector | Contains the position in the flat Earth reference frame. |
Third | Three-element vector | Contains the wind rotation angles [bank, flight path, heading], in radians. |
Fourth | 3-by-3 matrix | Contains the coordinate transformation from flat Earth axes to wind-fixed axes. |
Fifth | Three-element vector | Contains the velocity in the wind-fixed frame. |
Sixth | Two-element vector | Contains the angle of attack and sideslip angle, in radians. |
Seventh | Two-element vector | Contains the rate of change of angle of attack and rate of change of sideslip angle, in radians per second. |
Eight | Three-element vector | Contains the angular rates in body-fixed axes, in radians per second. |
Ninth | Three-element vector | Contains the angular accelerations in body-fixed axes, in radians per second squared. |
Tenth | Three-element vector | Contains the accelerations in body-fixed axes. |
The block assumes that the applied forces are acting at the center of gravity of the body, and that the mass and inertia are constant.
Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.
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)
Simple Variable Mass 6DOF ECEF (Quaternion)