## 5 Numerical Examples for Reachable Zone Prediction
### Example 2: DBM Case  
In this subsection, five examples are set up to illustrate the advantages of the DBM method in terms of computational accuracy and time efficiency. The simulation configuration parameters set for these cases are shown in Tables 1 and 4.  
**Case 0: Reachable zone history. The reachable zone's history is shown in Figure 6.  
As shown in Figure 6, the area of the reachable zone of the vehicle decreases as the predicted time of the vehicle progresses. A polynomial curve was fitted to each boundary point versus the predicted time to obtain the trajectories of the changes in position of the six boundary points, as shown in Figure 7, and the positions of the six boundary points of the reachable zone are related to the predicted time.  
What impresses researchers deeply about the proposed method is its accuracy and applicability in the online calculation of the reachable zone. The performance of the DBM in these two aspects will be verified next.  
It was already known in advance that the gliding vehicle would fly under the standard condition that it would track the pre-planned D-V profile when \(t<t_{p}\). Additionally, assume that an unusual event occurs in the vehicle at the moment \(t=t_{p}\); here, we choose a predicted time \(t_{p}\) randomly to replace this incident. Then, six boundary points would be interpolated at the moment \(t=t_{p}\) based on the database that has already been created offline, and the close reachable zone would be fitted to the current state of the vehicle  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\lambda\)/\((^{\circ})\) & \(\phi\)/\((^{\circ})\) & \(h\)/\((\mathbf{km})\) & \(v\)/\((\mathbf{m}\cdot\mathbf{s}^{-1})\) \\ \hline Case 0 & 85 & 30 & 35 & 3000 \\ Case 1 & 70 & 10 & 35 & 3000 \\ Case 2 & 70 & 15 & 35 & 3000 \\ Case 3 & 70 & 20 & 35 & 3000 \\ Case 4 & 70 & 15 & 37 & 3200 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Terminal condition configuration for five cases.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(\lambda\)/\((^{\circ})\) & 48.12 & 65.39 & 48.12 & 92.74 & 97.93 & 92.74 \\ \(\phi\)/\((^{\circ})\) & 29.33 & 0.32 & \(-\)29.33 & 32.78 & 0.94 & \(-\)32.78 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The six boundary points of the reachable zone.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & \(\lambda\)/\((^{\circ})\) & \(\phi\)/\((^{\circ})\) & **Reachability** \\ \hline
**1** & 78.86 & 14.32 & Yes \\
**2** & 78.15 & 35.61 & No \\
**3** & 52.54 & 16.74 & No \\
**4** & 54.17 & \(-\)30.45 & No \\
**5** & 76.44 & \(-\)3.99 & Yes \\
**6** & 67.52 & 12.96 & Yes \\
**7** & 86.22 & \(-\)11.98 & Yes \\
**8** & 79.72 & \(-\)6.71 & Yes \\
**9** & 90.52 & 26.63 & Yes \\
**10** & 55.39 & 9.08 & No \\ \hline \hline \end{tabular}
\end{table}
Table 3: Reachability tests for the random points.  
online. Furthermore, a new target point would be quickly discriminated against whether it is within the reachable zone or not. As a comparison, the reachable zone determined by the trajectory integration method (TIM) will be employed. In addition, the error of distance is determined by the ratio of the absolute distance value of (DBM-TIM) to the range to go.  
The following properties of the DBM are inferred compared with the simulation results mentioned above:  
* **Adaptability:** The reachable zone database can be created offline in advance using the typical trajectories of various initial states and terminal states. As shown in Figure 8, the reachable zone at the present time can be immediately predicted by the DBM when the aircraft is in an emergency. As a result, this method has good adaptability and robustness.