## 8 Errors and Approach Analysis
The curves were traced, using the calculated aircraft displacement as a function of the steps used for analysis; all these curves tended to adopt a final value of the displacement to the corresponding axis. Invariability is a criterion used to consider a good approach during simulation. Accordingly, with this, it is possible to confirm for simulations under 300 steps whether there are significant fluctuations along the curves; then, the approach is not satisfactory. But over 300 steps, the variation is not significant; thus, a good approach can be obtained. Otherwise, the calculation time is very short computationally, and only a few seconds are required; consequently, a pilot or a user can use 2000 steps or more for simulation and results will be ready in just a few seconds. All curves shown in Figures 15-17 tend towards a final true value. Here, the calculation of the aircraft displacement along horizontal and vertical direction is performed as a function of the steps used, and the invariability of these values is taken as a criterion to affirm that more steps are required to obtain a better approach. The values of the final displacements, calculated at 10,000 steps for the 3 flights simulated, are shown in Table 2. This number of steps is enough for a very good approach. Total displacement describes the total aircraft path measured since the flight starts at the original position on (0,0,0) until the end of the latest maneuver. Here, the largest displacement along the axis (x) was obtained on flight (3). The largest total displacement was also for flight (3). However, but the aircraft in flight (2) had the largest altitude and showed a negative value upon its displacement along axis (y); according to the positive reference, this just meant that the aircraft flew in the opposite direction. Moreover, total displacements are the true aircraft path; thus, if the aircraft changes its direction during any maneuvers, this value will always be larger than that of the final aircraft position as measured from the origin. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Flight**} & \multicolumn{3}{c}{**Axis**} \\ \cline{2-5}  & **“X” Displacement** & **“Y” Displacement** & **“Z” Displacement** & **Total Displacement** \\  & **North–South** & **East–West** & **Altitude** & **(m)** \\  & **(m)** & **(m)** & **(m)** & **(m)** \\ \hline
1 & 20,721 & 10,432 & 3136 & 39,123 \\ \hline
2 & 21,061 & \(-\)24,921 & 5720 & 78,227 \\ \hline
3 & 54,450 & 14,855 & 4085 & 72,902 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculated true displacements along horizontal and vertical axes for every simulated flight. Figure 14: Simulation of aircraft displacement—calculated using algorithms developed during maneuvers (1) and (2). Shows reduction in step time (\(\Delta t\)) as function of steps used for calculation. Figure 15a shows the displacement of the aircraft for the proposed flight (1) along the axis (x), which corresponds to the north-south direction; the calculation of distance at the beginning is a very high value when less than 100 steps are used for calculation; then, an increment of the steps is required to achieve a better approach. The displacement along this direction is always larger than the final displacement value due to the aircraft being preferentially in one single direction. In Figure 15b, the calculation for the aircraft flight along the axis (y) along the east-west direction is shown. Here, high fluctuations can be appreciated when less than 200 steps are used for calculation. In addition, fluctuations go up and down; these fluctuations are attributed to the aircraft's positive and negative displacement along this direction. In other words, they occur due to the changes along the heading angle; moreover, it is possible to appreciate the aircraft's speed influence due to the size of the form in the movement of every maneuver and the influence of the time of endurance, especially on the latest maneuver.