Documentation |
Implement quaternion representation of six-degrees-of-freedom equations of motion with respect to body axes
For a description of the coordinate system and the translational dynamics, see the block description for the 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 $$\epsilon $$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.
$$\begin{array}{l}\left[\begin{array}{c}{\dot{q}}_{0}\\ {\dot{q}}_{1}\\ {\dot{q}}_{2}\\ {\dot{q}}_{3}\end{array}\right]=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left[\begin{array}{cccc}0& -p& -q& -r\\ p& 0& r& -q\\ q& -r& 0& p\\ r& q& -p& 0\end{array}\right]\left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]+K\epsilon \left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]\\ \epsilon =1-\left({q}_{0}^{2}+{q}_{1}^{2}+{q}_{2}^{2}+{q}_{3}^{2}\right)\end{array}$$
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 Fixed 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 three-element vector for the initial location of the body in the flat Earth reference frame.
The three-element vector for the initial velocity in the body-fixed coordinate frame.
The three-element vector for the initial Euler rotation angles [roll, pitch, yaw], 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.
The gain to maintain the norm of the quaternion vector equal to 1.0.
Input | Dimension Type | Description |
---|---|---|
First | Vector | Contains the three applied forces. |
Second | Vector | Contains the three applied moments. |
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 Euler rotation angles [roll, pitch, yaw], in radians. |
Fourth | 3-by-3 matrix | Contains the coordinate transformation from flat Earth axes to body-fixed axes. |
Fifth | Three-element vector | Contains the velocity in the body-fixed frame. |
Sixth | Three-element vector | Contains the angular rates in body-fixed axes, in radians per second. |
Seventh | Three-element vector | Contains the angular accelerations in body-fixed axes, in radians per second squared. |
Eight | 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.
Mangiacasale, L., Flight Mechanics of a μ-Airplane with a MATLAB Simulink Helper, Edizioni Libreria CLUP, Milan, 1998.
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)