# Applying Neural Networks in Aerial Vehicle Guidance to Simplify Navigation Systems
## 4 Numerical Simulations
### Discussion  
Results for noncontrolled trajectories are shown in Figure 3, which depicts destination point dispersion patterns. This figure shows the ballistic trajectory that the vehicle follows for three different launch angles without using the control system at all, that is, it shows the flight of the system before implementing the improvements provided by the GNC system. The circular error probable (CEP), which is defined as the radius of a circle, centered on the mean, whose boundary is expected to include the landing points of 50% of the flights. The CEP is employed as a quality check parameter at the final step of the simulation as it is a valid reference for any utilized method. Indeed, the lower the CEP is the better the global GNC device is. Table 7 displays the CEP for noncontrolled flights, Kalman hybridization based controlled flights and ideally controlled flights for each of the initial pitch angles. Analyzing Table 7, it may be concluded that the CEP increases with target distance for noncontrolled flights, as expected. However, it almost remains constant for either Kalman based or ideally controlled flights, which means the use of an appropriate GNC device eliminates the correlation between the CEP and the distance to the objective. Recall, the main purpose is to show that these results are reproducible when employing machine learning and when reducing the availability of sensors. Table 8 shows again the CEP for different trajectories, now as obtained by the novel presented approach. Each row, excluding the first one, which shows the headings of the columns, displays the resulting CEPs for every combination of trajectory and NN training algorithm. The first column in the table identifies the trajectory, the second one the training algorithm, the third one the CEP for the NN architecture in method 1, and the last one the CEP for the NN architecture in method 2. One of the main conclusions drawn from Table 8 is that the results for the SCG training algorithm present an unacceptable big CEP, which means low accuracy when reaching destination. Consequently, this training algorithm should be discarded in this case. Indeed, we recover results equivalent to a defective GNC system, even showing worse performance than ballistic flights. Several tests and a hyperparametric analysis were conducted, and it is observed that these kinds of errors were systematic, concluding that this training algorithm does not match well with the fed data to the NN. However, the results from BR and LM training algorithms show a good behavior throughout the trajectory, which means level of accuracy at destination is high. Diving into the numerical results in Table 8, it can be stated that the presented novel approach, which relaxes sensor requirements, is even able of outperforming the Kalman hybridization based approach. It can also be observed that the results are coherent with the training results depicted in Table 4. For example, the poor MSE and R values for the SCG training algorithm are reflected in the unacceptable GNC device results. Comparing the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter (deg)** & **Initial \(\boldsymbol{\phi}\)** & **Initial Pitch** & **Wind Speed** & **Wind Direction** & **Thrust** & **Initial Azimuth** \\ \hline Mean & 0\({}^{\circ}\) & Nom. & 10 m/s & 0\({}^{\circ}\) & T(t) & Nom. \\ \hline Standard Deviation & 20\({}^{\circ}\) & 0.01\({}^{\circ}\) & 5 m/s & 20\({}^{\circ}\) & 10 N & 0.01\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Monte Carlo simulation parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Initial Pitch Angle (deg)** & **Noncontrolled CEP (m)** & **Kalman Based CEP (m)** & **Ideal CEP (m)** \\ \hline
20 & 169.34 & 1.28 & 1.18 \\ \hline
30 & 239.37 & 1.18 & 1.06 \\ \hline
45 & 281.59 & 0.98 & 0.83 \\ \hline \hline \end{tabular}
\end{table}