03. Incorporating Forces

\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} &= -g \sin \theta + \frac{\rho V^2 S}{2m}\left [ C_{X_0} + C_{X_\alpha}\alpha \right ] + T/m +rv \\ \dot{v} &= g \cos \theta \sin \phi + \frac{\rho V^2 S}{2m}\left [ C_{Y_0} + C_{Y_\beta}\beta + C_{Y_p}\frac{bp}{2V} + C_{Y_r}\frac{br}{2V} + C_{Y_{\delta_a}\delta_a} + C_{Y_{\delta_r}\delta_r} \right ] +pw - ru \\ \dot{w} &= g \cos \theta \cos \phi + \frac{\rho V^2 S}{2m}\left [ C_{Z_0} + C_{Z_\alpha}\alpha \right ] -pv \\ \dot{p} &= \left (I_{zz}L + I_{xz}N \right ) / (I_{xx} I_{zz} - I^2_{xz}) \\ \dot{r} &= \left (I_{xz}L + I_{xx}N \right ) / (I_{xx} I_{zz} - I^2_{xz}) \end{aligned}

\begin{aligned} L&=\frac{1}{2}\rho V^2 S b \left [ C_{l_0} + C_{l_\beta}\beta + C_{l_p}\frac{bp}{2V} + C_{l_r}\frac{br}{2V} + C_{l_{\delta a}} \delta a + C_{l_{\delta r}} \delta r \right ] \\ N&=\frac{1}{2}\rho V^2 S b \left [ C_{r_0} + C_{r_\beta}\beta + C_{r_p}\frac{bp}{2V} + C_{r_r}\frac{br}{2V} + C_{r_{\delta a}} \delta a + C_{r_{\delta r}} \delta r \right ] \end{aligned}