# Research on Lazy Theta+
## 6 Simulation Experiment and Analysis
### Route Planning Algorithm Comparison Experiment
#### 6.2.1 Route Planning Algorithm Comparison Experiment  
1. Simulation  
This simulation compares and analyzes the route planning results of A*, Lazy Theta* and RRT* at three to five grid levels, and compares them with the algorithms under the grid points.  
The route planning simulation at the fifth grid level is shown in Figure 19, Figure 19a is the result of the grid Lazy Theta* and its top view, and Figure 19b is the result of grid point Lazy Theta* proposed in this paper.  
Figure 18: Variation trend of grid number and model fit.  
Other levels as well as the A* and RRT* simulation results are shown in Figures A2-A4 in Appendix B. Next, the experimental data of A*, Lazy Theta*, and RRT* route planning based on the grid and grid points at three to five levels are analyzed and compared.  
2 Analysis of results  
Table 2 displays the experimental results, and the data change trend is illustrated in Figure 20.  
Figure 19: Lazy Theta* route planning at grid level 5. (**a**) Route planning based on grid, and (**b**) Route planning based on grid point.  
The data change diagram in Figure 20 can be obtained through the simulation experiment. As the grid level increases, the path length and path cost obtained by A*, Lazy Theta*, and RRT* planning are reduced to varying degrees.  
Under the same grid level, the path length, path cost, and running time based on the grid points are significantly smaller than those based on the grid. A closer look at Figure 20 shows that the grid point-based Lazy Theta* algorithm has the lowest values in four aspects: path length, path cost, maximum pitch angle, and maximum turning angle, and the grid-based A* algorithm has the shortest running time. This indicates that the grid point-based Lazy Theta* algorithm can better control the aircraft's flight and turning angles than the competing algorithms, preserving the route to be as short as possible. In this case, the flight operation difficulty is reduced, and the flight safety and efficiency are improved. Similarly, we can find that as the grid level increases, the running time of the three algorithms increases in varying degrees.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithm** & **Grid Level** & **Path Length** & **Path Cost** & **Max Pitch Angle** & **Max Turn Angle** & **Running Time** \\ \hline \multirow{3}{*}{Grid A*} & 3 & 120.89 & 103.00 & 4.04 & 45.14 & 0.91 \\  & 4 & 113.48 & 93.93 & 4.04 & 45.14 & 1.53 \\  & 5 & 100.93 & 82.35 & 4.04 & 45.14 & 1.95 \\ \hline \multirow{3}{*}{Grid point A*} & 3 & 113.57 & 96.64 & 5.71 & 45.14 & 0.98 \\  & 4 & 95.71 & 82.60 & 3.02 & 45.14 & 1.02 \\  & 5 & 91.93 & 76.35 & 2.02 & 45.14 & 1.72 \\ \hline \multirow{3}{*}{Grid Lazy Theta*} & 3 & 116.40 & 97.35 & 4.04 & 63.46 & 1.34 \\  & 4 & 110.63 & 89.86 & 3.03 & 56.32 & 2.36 \\  & 5 & 98.15 & 79.37 & 1.59 & 22.83 & 2.54 \\ \hline \multirow{3}{*}{Grid point Lazy Theta*} & 3 & 105.02 & 86.39 & 2.86 & 63.64 & 1.25 \\  & 4 & 91.26 & 74.01 & 1.99 & 36.87 & 2.02 \\ \cline{1-1}  & 5 & 87.90 & 68.01 & 1.46 & 16.58 & 2.38 \\ \hline \multirow{3}{*}{RRT*} & step size = 25 & 164.97 & 132.02 & 14.33 & 62.41 & 1.91 \\ \cline{1-1}  & step size = 12.5 & 145.73 & 116.61 & 14.37 & 58.43 & 3.44 \\ \cline{1-1}  & step size = 6.25 & 105.10 & 84.14 & 10.58 & 58.59 & 5.23 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Multi-scale grid comparison data.  
Figure 20: _Cont._  
To summarize the above analysis: (1) Compared with the grid, the grid points are optimized in the path length, path cost, maximum turning angle, maximum pitch angle, and running time of the algorithm. (2) The running time increases as the grid level increases, but the path length, path cost, maximum turn angle, and maximum pitch angle are also optimized. (3) Among the five algorithms, the path planned by the grid point of Lazy Theta* has the highest flyability, and the grid point A* has the fastest planning time.