## 5 The Simulation Results and Discussion
### Simulation Results  
In order to evaluate the effectiveness of the \"Optimal_MFs_FRs_FLC\", in which both the MFs and the FRs are optimized by the optimization process, two motion FLCs are selected and implemented to compare them with the \"Optimal_MFs_FRs_FLC\". Two motion FLCs are the \"Thoa controller\" and the \"GA_FLC controller\" in [10]. In this study, the \"Thoa controller\" is called \"Expert_FLC\" and the \"GA_FLC controller\" is called \"Optimal_MFs_FLC\". The \"Expert_FLC\" is designed by an expert. The MFs and the FRs of the \"Expert_FLC\" are designed based on the heuristics and knowledge of an expert. For the \"Optimal_MFs_FLC\", only its MFs are optimized, while its FRs are designed by an expert. For more details refer to [10;13]. The main simulation results are shown in Figures 15-19 and listed in Tables 14-18. Figure 14: The suboptimal membership functions. Figure 15: The simulation results in the case of using the zigzag path. (**a**) The trajectory of the robot; (**b**) The robot’s travel distance over time; (**c**) The robot’s linear velocity over time; (**d**) The robot’s angular velocity over time; (**e**) The robot’s kinetic energy loss over time; and (**f**) The zoom of the robot’s kinetic energy loss over time. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Controllers**} & \multicolumn{2}{c}{**Energy Consumption**} & \multicolumn{2}{c}{**Travel Distance**} & \multicolumn{2}{c}{**Travel Time**} \\ \cline{2-5}  & **E (J)** & **E (\%)** & **s (m)** & **s (\%)** & **Time(s)** \\ \hline Expert\_FLC & 1.575194 & 798.56 & 8.096951 & 100.00 & 28.33 \\ Optimal\_MFs\_FLC & 0.720139 & 365.08 & 8.142723 & 100.57 & 22.53 \\ Optimal\_MFs\_FRs\_FLC & 0.197255 & 100.00 & 8.348827 & 103.11 & 34.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: The kinetic energy loss and the travel distance of the robot when the robot tracks the zigzag path by the different controllers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Controllers**} & \multicolumn{2}{c}{**Energy Consumption**} & \multicolumn{2}{c}{**Travel Distance**} \\ \cline{2-5}  & **E (J)** & **E (\%)** & **s (m)** & **s (\%)** \\ \hline Expert\_FLC & 1.350298 & 727.19 & 8.048057 & 100.74 \\ Optimal\_MFs\_FLC & 0.759424 & 408.98 & 7.989019 & 100.00 \\ Optimal\_MFs\_FRs\_FLC & 0.185687 & 100.00 & 8.202735 & 102.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: The kinetic energy loss and the travel distance of the robot when the robot tracks the square path by the different controllers. Figure 16: The simulation results in the case of using the square path. (**a**) The trajectory of the robot; (**b**) The robot’s travel distance over time; (**c**) The robot’s linear velocity over time; (**d**) The robot’s angular velocity over time; (**e**) The robot’s kinetic energy loss over time; and (**f**) The zoom of the robot’s kinetic energy loss over time. Figure 17: The simulation results in the case of using the sharp turn path. (**a**) The trajectory of the robot; (**b**) The robot’s travel distance over time; (**c**) The robot’s linear velocity over time; (**d**) The robot’s angular velocity over time; (**e**) The robot’s kinetic energy loss over time; and (**f**) The zoom of the robot’s kinetic energy loss over time. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Controllers**} & \multicolumn{2}{c}{**Energy Consumption**} & \multicolumn{2}{c}{**Travel Distance**} \\ \cline{2-5}  & **E (J)** & **E (\%)** & **s (m)** & **s (\%)** \\ \hline Expert\_FLC & 0.781463 & 568.37 & 4.279199 & 100.83 \\ Optimal\_MFs\_FLC & 0.393902 & 286.49 & 4.243791 & 100.00 \\ Optimal\_MFs\_FRs\_FLC & 0.137491 & 100.00 & 4.430193 & 104.39 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: The kinetic energy loss and the travel distance of the robot when the robot tracks the sharp turn path by the different controllers.