## 4 Simulation Results and Analysis
Based on the results presented in Table 5, the path determined using the modified A* algorithm was longer than that determined using the basic A* algorithm. However, the differences were not significant (except for case 3), and the largest difference of 82.2 m in case 5 was 2.8% of the path length determined by the basic A*. However, the difference in the sum of the path length for both algorithms for cases 1-2 and 4-5 was 198.2 m, which was 1.25% of the value obtained as a result of the basic A* calculation. \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{4}{c}{**Calculation Time**} \\ \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**A* with Map Processing [s]**} & \multirow{2}{*}{**A* [s]**} & **Ratio (A*/A* with** **Map Processing)** \\ \hline
**1** & 89 (1 m 29 s) & 59,542 (2 days 10 h 22 m) & 669 \\ \hline
**2** & 356 (5 m 56 s) & 210,122 (2 days 10 h 22 m) & 590 \\ \hline
**3** & 33 & 24,083 (6 h 41 m) & 730 \\ \hline
**4** & 56 & 40,250 (11 h 10 m) & 719 \\ \hline
**5** & 207 (3 m 27 s) & 120,262 (1 day 9 h 24 m) & 584 \\ \hline
**6** & 1399 (23 m 19 s) & 728,163 (8 days 10 h 16 m) & 521 \\ \hline
**Total**\({}^{1}\) & 2098 (34 m) & 1,156,797 (13 days 9 h 19 m) & 551 \\ \hline \end{tabular} \({}^{1}\) Without taking into account case 3.  
\end{table} 
Table 6: Results of path calculations in terms of computation time. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Minimal Distance to the Obstacle**} & \multicolumn{2}{c}{**Number of Waypoints**} \\ \cline{2-5}  & \multicolumn{2}{c}{**A* with Map Processing [m]**} & \multirow{2}{*}{**A* [m]**} & **A* with Map Processing [s]** & **A* [s]** \\ \hline
**1** & 12.37 & 6.08 & 370 & 1846 \\ \hline
**2** & 4.47 & 1.00 & 492 & 2388 \\ \hline
**3** & 3.16 & 1.00 & 174 & 590 \\ \hline
**4** & 4.00 & 1.00 & 459 & 2288 \\ \hline
**5** & 4.00 & 1.00 & 543 & 26 Figure 10: Visualisation of determining the path between subsequent waypoints: (**a**) case 1, (**b**) case 2, (**c**) case 3, (**d**) case 4, (**e**) case 5, (**f**) case 6. The path calculated using the basic version of the A* algorithm is marked in yellow, while the route calculated using A* with map image processing is marked in green. The blue frames highlight zoomed-in areas of the map. Table 6 shows the calculation time for path determination using both methods. It can be observed that the longest calculation time for A* with map image processing was 23 m 19 s, while for the basic A*, it was 8 days, 10 h, and 16 s, indicating that the calculation time using the modified method was reduced by a factor of 521. Based on the results, it can also be concluded that usually, the shorter the time for calculating the path, the greater the ratio of the calculation time of both methods. This is due to the long-term creation of the list of obstacles by the A* method, which essentially determines the total path calculation time. In case 4, A* with map processing achieved a result 719 times faster than the basic algorithm with a difference in path length of 7.1 m (0.29%). In the worst case (not counting case 3) in terms of the calculation time ratio, in case 6, the ratio was 521, and the difference in path length was 37.8 m (0.67%). The total calculation time (excluding case 3) was 34 m and 13 days 9 h 19 m for the modified and basic methods, respectively. The main reason for such a large difference in calculation time between the basic and proposed approaches is reducing the map's resolution, which means that 1 pixel after processing represents an area of 5 \(\times\) 5 m. In order to compensate for losses in the area occupied by obstacles resulting from averaging the values of neighbouring pixels, mathematical morphology was used to enlarge areas occupied by obstacles artificially. Applying a greater dimension reduction results in a greater gain in computation time. However, excessive reduction in dimensions may lead to limited access to some areas of the map due to merging obstacles into one large obstacle.