# Applying Neural Networks in Aerial Vehicle Guidance to Simplify Navigation Systems
## 2 Vehicle Modeling
### Equations of Flight Mechanics  
To construct the equations of flight, three reference frames are defined to project forces and moments: NED axes, working axes and body axes. NED axes, which are ground axes, are depicted by sub index \(NED\). They are defined by \(x_{NED}\) pointing north, \(z_{NED}\) orthogonal to \(x_{NED}\) and pointing nadir, and \(y_{NED}\) yielding a clockwise trihedron. Working axes are represented by sub index \(w\). They are given by \(x_{w}\) pointing to the destination point, \(y_{w}\) orthogonal to \(x_{w}\) and pointing apex, and \(z_{w}\) forming a clockwise trihedron. \(AZ_{0}\) is the initial azimuth, i.e., the azimuth between \(x_{e}\) and \(x_{w}\). Body axes are depicted by sub index \(b\). \(x_{b}\) pointing forward and contained in the plane of symmetry of the vehicle, \(z_{b}\) orthogonal to \(x_{b}\) pointing down and contained in the plane of symmetry of the vehicle, and \(y_{b}\) shaping a clockwise trihedron. The origin of body axes is located at the gravity center of the vehicle \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \multicolumn{2}{c}{**Maximum Thrust**} & \multicolumn{2}{c}{**Initial Mass**} \\ \hline Value & \multicolumn{2}{c}{29,160.00 N} & \multicolumn{2}{c}{62.40 kg} \\ \hline Parameter & Fuel mass & \(I_{x0}\) & \(I_{y0}\) \\ \hline Value & 21.00 kg & 0.19 kg m\({}^{2}\) & 18.85 kg m\({}^{2}\) \\ \hline M & \(C_{D_{0}}\) & \(C_{D_{x^{2}}}\) & \(C_{L_{a}}\) & \(C_{L_{y^{3}}}\) & \(C_{mf}\) & \(C_{N_{q}}\) \\ \hline
0.00 & 0.27 & 10.74 & 8.01 & 19.82 & \(-0.59\) & 50.81 \\
0.40 & 0.25 & 10.88 & 8.17 & 19.55 & \(-0.64\) & 53.25 \\
0.60 & 0.24 & 11.10 & 8.43 & 19.12 & \(-0.70\) & 57.43 \\
0.70 & 0.24 & 11.24 & 8.60 & 18.83 & \(-0.72\) & 60.31 \\
0.80 & 0.23 & 11.40 & 8.79 & 18.49 & \(-0.75\) & 63.80 \\
0.90 & 0.23 & 11.45 & 8.98 & 17.28 & \(-0.78\) & 67.93 \\
1.00 & 0.41 & 15.12 & 8.93 & 44.05 & \(-0.81\) & 71.38 \\ \hline M & \(C_{M_{a}}\) & \(C_{M_{a}3}\) & \(C_{M_{a}}\) & \(C_{mm}\) & \(C_{spin}\) & \(C_{Na_{w}}\) \\ \hline
0.00 & \(-35.58\) & \(-16.65\) & \(-225.73\) & 3.02 & \(-0.04\) & 0.00 \\
0.40 & \(-35.72\) & \(-18.09\) & \(-232.75\) & 3.29 & \(-0.04\) & 0.42 \\
0.60 & \(-36.00\) & \(-20.39\) & \(-245.32\) & 3.57 & \(-0.04\) & 0.43 \\
0.70 & \(-36.21\) & \(-21.82\) & \(-254.10\) & 3.71 & \(-0.03\) & 0.44 \\
0.80 & \(-36.51\) & \(-23.39\) & \(-264.85\) & 3.84 & \(-0.03\) & 0.44 \\
0.90 & \(-36.57\) & \(-18.48\) & \(-276.57\) & 3.98 & \(-0.03\) & 0.45 \\
1.00 & \(-35.99\) & 15.39 & \(-287.82\) & 4.12 & \(-0.03\) & 0.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Aerial vehicle parameters. and they are rigid coupled to the roll-decoupled control device. Figure 1 shows the previously defined axes systems. Next, flight dynamics equations are described. Note that these equations are compliant with the standard convention in [27]. Because the vehicle is assumed to be rigid, classical mechanics theory is employed. The Newton-Euler equations describe the combined translational and rotational dynamics of a rigid body. These laws, which are given by six equations, relate the motion of the center of gravity of a rigid body with the sum of forces and moments acting on the rigid body. \[\left[ \begin{array}{c}\overrightarrow{F_{ext}}\\ \overrightarrow{M_{ext}}\end{array} \right]=\left[ \begin{array}{c} \overrightarrow{L}+\overrightarrow{D}+\overrightarrow{P}+\overrightarrow{M}+ \overrightarrow{T}+\overrightarrow{W}+\overrightarrow{C}\\ \overrightarrow{P_{M}}+\overrightarrow{O}+\overrightarrow{M_{M}}+ \overrightarrow{S}\end{array}