## 5 Simulation Analysis
### Simulation of the Hybrid Algorithm  
In order to enhance the simulation performance of the path planning for an amphibious robot, it is very important to take into consideration the impact of different motion modes on path planning. The dynamic window method is combined with the improved potential field method for simulation, because its algorithm requires the dynamics and kinematics equations of the robot as its own constraints, which is more in line with the actual situation. Figures 12-15 are the simulation results as follows, the right side figures show the curve which means the distance between the robot and the obstacle, and the left side is the whole process of the robot running.  
Here, the numbers from 1 to 10 represent the obstacles' serial numbers. The horizontal axis represents the number of iterations, and the vertical axis represents the distance between the robotic fish and the obstacle. Every curve has this pattern: with an increase in time, the robot approaches the obstacle and then moves away from it, and finally, the fluctuation is caused by the constant correction of distance (essentially local minima problem). In the end, the vibration can approach and flatten quickly, indicating that at this time, it keeps a certain distance from the obstacle, and does not take a repeated road section. In other words, it can continue to move towards the target point. Meanwhile, it can also prove the correctness of the establishment of the artificial potential field equation. The bottom horizontal blue line indicates the minimum boundary distance, and our simulation is always within the safe range. In addition, the intersection of 10 lines with different distances in the figure is precisely due to the change of the environment, which changes the motion mode.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Figure** & **Execution Time** & **Path Length** \\ \hline traditional artificial & 10a & 0.7830 s & 2.6717 m \\ potential field method & 11a & 0.7770 s & 1.3853 m \\ improved artificial & 10b & 0.8030 s & 4.1231 m \\ potential field method & 11b & 0.7940 s & 4.4283 m \\ \hline \hline \end{tabular}
\end{table}
Table 4: The comparison results between the improved and traditional algorithm.  
Figure 12: Case without time step in group 1. (**a**) Path planning of 3D map. (**b**) The distance from obstacles.  
We divide these into two groups. Figures 12 and 13 are group 1, and Figures 14 and 15 are group 2. Among all the environmental parameters and the parameters of robot fish itself, the only difference lies in the transformation time of the dynamic equation. This is because, according to the improved equation, only this parameter is easy to control. Figures 12 and 14 show that there is no time step for the transformation progress; however, Figures 13 and 15 show the results of the transformation motion mode with 20 times steps. It can be seen that the robot can reach the goal in all cases, which proves that the hybrid algorithm is effective. As shown in group 1, the transformation time has little impact on the performance of the system.  
Figure 14: Case without time step in group 2. (**a**) Path planning of 3D map. (**b**) The distance from obstacles.  
Figure 13: Case with time step in group 1. (**a**) Path planning of 3D map. (**b**) The distance from obstacles.  
Figure 15: Case with time step in group 2. (**a**) Path planning of 3D map. (**b**) The distance from obstacles.  
on path planning; however, in the second group, which has a local minimum problem, the transformation time has a significant impact on the efficiency and stability of the whole path planning. We set the duration of the conversion so that the wavelength is shorter, faster and more stable. Therefore, the hybrid algorithm increases the transformation time as much as possible to ensure that it can be more efficient and effective.