# Applying Neural Networks in Aerial Vehicle Guidance to Simplify Navigation Systems
## 2 Vehicle Modeling
### Equations of Flight Mechanics
\right], \tag{1}\]  
Equation (1) shows total external forces and moments acting on the vehicle: \(\overrightarrow{L}\) is the lift force, \(\overrightarrow{D}\) is the drag force, \(\overrightarrow{P}\) is the pitch damping force, \(\overrightarrow{M}\) is the Magnus force, \(\overrightarrow{T}\) is the thrust force, \(\overrightarrow{W}\) is the weight force, \(\overrightarrow{C}\) is the Coriolis force, \(\overrightarrow{P_{M}}\) the is pitch damping moment, \(\overrightarrow{O}\) is the overturn moment, \(\overrightarrow{M_{M}}\) is the Magnus moment, and \(\overrightarrow{S}\) is the spin damping moment. \[\left[ \begin{array}{c}\overrightarrow{L}\\ \overrightarrow{D}\\ \overrightarrow{D}\\ \overrightarrow{M}\\ \overrightarrow{T}\\ \overrightarrow{W}\\ \overrightarrow{C}\end{array} \right]=\left[ \begin{array}{c}\left(C_{L_ {a}}(M)\cdot\alpha+C_{L_{a^{3}}}(M)\alpha^{2}\right)\left(\|\overrightarrow {v_{w}}\|^{2}\overrightarrow{x_{w}}-\left(\overrightarrow{x_{w}}\cdot \overrightarrow{v_{w}}\right)\overrightarrow{v_{w}}\right)\\ \left(C_{D_{0}}(M)+C_{D_{a^{2}}}(M)\alpha^{2}\right)\|\overrightarrow{v_{w}}\| \overrightarrow{v_{w}}\\ -d\frac{C_{N_{t}}(M)}{l_{y}}\|\overrightarrow{v_{w}}\|^{2}\ \left(\overrightarrow{L_{w}}\times \overrightarrow{x_{w}}\right)\\ d\frac{C_{m\overrightarrow{f}}(M)}{l_{x}}\left(\overrightarrow{L_{w}} \cdot\overrightarrow{x_{w}}\right)\ \left(\overrightarrow{x_{w}}\times \overrightarrow{v_{w}}\right)\\ T\left(t\right)\overrightarrow{x_{w}}\\ m\overrightarrow{g_{w}^{\ast}}\\ -2m\overrightarrow{\Omega}\times\overrightarrow{v_{w}}\end{array} \right] \tag{2}\]  
For the computational experiments in this paper, the external forces in working axes are shown in expression (2), where \(C_{L_{a}}\) is the lift force linear coefficient, \(C_{L_{a^{3}}}\) is the lift force cubic coefficient, \(\alpha\) is the total angle of attack, \(C_{D_{0}}\) is the drag force linear coefficient, \(C_{D_{a^{2}}}\) is the drag force square coefficient, \(\overrightarrow{L_{w}}\) is the vehicle angular momentum in working axes, \(I_{x}\)_and \(I_{y}\)_ are the vehicle inertia  
Figure 1: The three employed reference frames. moments in body axes, \(C_{Nq}\) is the pitch damping force coefficient, \(C_{mf}\) is the Magnus force coefficient, \(\overrightarrow{x_{w}}\) is vehicle pointing vector in working axes, \(\overrightarrow{g_{w}}\) is the gravity vector in working axes, \(\overrightarrow{\Omega}\) is earth's angular speed vector, \(\overrightarrow{v_{w}}\) is vehicle velocity in working axes, \(d\) is the reference surface of the vehicle, \(\rho\) is the air density, and \(m\) is the mass of the vehicle. Note that, \(\overrightarrow{x_{w}}\) is the unitary vector of the \(x_{w}\) axis, and \(\overrightarrow{g_{w}}\) is the gravity vector in working axes. Be aware they are nonlinear functions of the variables describing the movement of the vehicle, such as aerodynamic speed, total angle of attack, Mach number, and aerodynamic parameters. \[\left[ \begin{array}{c}\overrightarrow{P_{M}}\\ \overrightarrow{O}\\ \overrightarrow{S}\end{array} \right]=\frac{\pi}{8}d^{3}\rho\left[ \begin{array} []{c}\frac{1}{l_{y}}C_{M_{q}}(M)\|\overrightarrow{v_{w}}\|\ \left(\overrightarrow{L_{w}}-\left(\overrightarrow{L_{w}}\cdot \overrightarrow{x_{w}}\right)\overrightarrow{x_{w}}\right)\\ \left(C_{M_{u}}(M)+C_{M_{u^{3}}}(M)a^{2}\right)\|\overrightarrow{v_{w}}\|^{ 2}\ \left(\overrightarrow{v_{w}}\times\overrightarrow{x_{w}}\right)\\ -\frac{d}{l_{x}}C_{mm}(M)\left(\left(\overrightarrow{L_{w}}^{\prime}\cdot \overrightarrow{x_{w}}\right)\left(\left(\overrightarrow{v_{w}}^{\prime}\cdot \overrightarrow{x_{w}}\right)\overrightarrow{x_{w}}\right)-\overrightarrow{v _{w}}\right)\\ \frac{d}{l_{x}}C_{spin}(M)\|\overrightarrow{v_{w}}\|^{2}\left(\overrightarrow {L_{w}}\cdot\overrightarrow{x_{w}}\right)\overrightarrow{x_{w}}\end{array}  \right], \tag{3}\]  
Similarly, the equations in (3) show the mathematical expressions for the moments, including overturning, pitch damping, Magnus, spin damping, and variables and parameters.