Force-Free Lateral-Directional Motion

Defining the State

Let's start by defining the state vector that we'll be tracking when analyzing the lateral-directional motion.

\mathbf{x} = \begin{bmatrix} x_I \\ y_I \\ z_I \\ \phi \\ \theta \\ \psi \\ u \\ v \\ w \\ p \\ r \end{bmatrix}

We can define each of these variables:

Variables	Physical meaning
x_I, y_I, z_I	x, y, z position in inertial frame
\phi, \theta, \psi	Euler angles
u, v, w	x, y, z velocity in body frame
p, r	Body rates about the x and z axes

Note that the pitch rate q is not included in the state.

Force-free equations of motion

The equations of motion for the force-free case are as follows:

\begin{aligned} \dot{x}_I &= u (\cos \theta \cos \psi) + v (\sin \phi \sin \theta \cos \psi - \cos \phi \sin \psi) + w (\cos \phi \sin \theta \cos \psi + \sin \phi \sin \psi) \\ \dot{y}_I &= u (\cos \theta \sin \psi) + v (\sin \phi \sin \theta \cos \psi + \cos \phi \cos \psi) + w (\cos \phi \sin \theta \sin \psi - \sin \phi \cos \psi) \\ \dot{z}_I &= - u \sin \theta + v \sin \phi \cos \theta + w \cos \phi \cos \theta \\ \dot{\phi} &= p + r \cos \phi \tan \theta \\ \dot{\theta} &= - r \sin \phi \\ \dot{\psi} &= r \cos \phi \sec \theta \\ \dot{u} &= rv \\ \dot{v} &= pw - ru \\ \dot{w} &= -pv \\ \dot{p} &= 0 \\ \dot{r} &= 0 \end{aligned}

Reminder: Fixed Wing Cheat Sheet

You can find all of the equations for this module in the Fixed Wing Cheat Sheet.