02. Force-Free Motion
Nd787 C5 L3 A02 L Force Free Dynamics V3
Force-Free Motion
When tracking the longitudinal dynamics, we use the following state vector:
\mathbf{x} = \begin{bmatrix}
x_I\\
z_I\\
\theta \\
u \\
w \\
q
\end{bmatrix}
x_I and z_I are the position in the inertial frame (note that z is positive UPWARDS for this lesson).
\theta is the pitch angle.
u and w are the body frame velocities.
q is this pitch rate.
When there are no forces acting on the vehicle, the equations of motion are as follows:
\begin{aligned}
\dot{x}_I &= u \cos \theta + w \sin \theta \\
\dot{z}_I &= -u \sin \theta + w \cos \theta \\
\dot{\theta} &= q \\
\dot{u} &=- qw \\
\dot{w} &= qu \\
\dot{q} &= 0 \\
\end{aligned}
Reminder: Fixed Wing Cheat Sheet
You can find all of the equations for this module in the Fixed Wing Cheat Sheet.