## 5 Simulink Results and Discussion
### The Results of CDR Movement on the Ground and the Water Surface  
The FLOS algorithm is used to plan the yaw angle and the linear velocity for the robot. It is necessary to design a controller for the robot to track the desired yaw angle and linear velocity. To verify the effectiveness of the proposed control algorithm, the proposed ASMIBC is applied to an ideal second-order system, such as Equation (27). There is no unknown and uncertain disturbance in the system from 0 to 20 s. At 20 s, the second-order state of the system is subjected to a step disturbance with a value of 3. The control results are compared with PID, SMC and IBC, and the results are shown in Figures 11-13. \begin{table}
\begin{tabular}{c c c c c} \hline
**Performance Parameter** & **FLOS** & **LOS** & **ALOS** & **ELOS** \\ \hline Standard deviation \(y_{e}\) (m) & 0.2780 & 0.3193 & 0.3329 & 0.2942 \\
0–15 s max \(y_{e}\) (m) & 0 & 0 & 0.7443 & 0.3009 \\
0–15 s \(y_{e}\) convergence time (s), & 4.2190 & 5.722 & 11.311 & 11.286 \\ (\(y_{e}\)! 0.01 m) & 3.2380 & 4.4880 & 7.7780 & 6.8250 \\
0–15 s heading angle planning time (s), (abs(\(y_{e}\))! 0.05 rad) & 3.2380 & 4.4880 & 7.7780 & 6.8250 \\
15–70 s \(y_{e}\) average convergence time (s) & 2.3950 & 2.4200 & 3.2040 & 5.1020 \\
15–70 s mean time (s) for heading angle planning, abs(\(y_{e}\))! 0.05 rad & 4.1810 & 4.2340 & 6.9530 & 5.7670 \\ \hline \end{tabular}  
* In Table 3, the \(y_{e}\) standard deviation of FLOS (0.278 m) is the smallest compared with the other three algorithms. Compared to other algorithms, the FLOS algorithm speeds up the convergence speed because it introduces the exponential term of the position error for planning heading angle.  
\end{table} 
Table 3: Quantitative comparison of performances for the proposed FLOS with LOS, ALOS, and ELOS in Case 2. Figure 10: Heading angle and position error \(y_{e}\) of the CDR. (**a**) Heading angle changes when robot tracks the desired trajectory. (**b**) The CDR hexagonal trajectory tracking of \(y_{e}\). As shown in Figure 10. The desired state of 1 is a constant, IBC loses ability to constrain the system state as shown by the pink line. The static error of SMC is smaller than that of IBC, which is about 1.5. However, the gain of the sliding surface is a fixed constant and the robustness of the controller is limited. The PID control has an integral term, the error can be calculated as  
Figure 11: ASMIBC for an ideal second-order control model. Only the proposed algorithm can track the desired value 1. As the adaptive rate is designed in the ASMIBC, there is no static error. Figure 12: ASMIBC adaptive sliding mode gain. (**a**) ASMIBC without adaptive compensation. Our proposed method can adaptively adjust the gain of sliding mode. When there is no disturbance, the gain will be 0, and the gain will increase when the system is influenced by disturbance. (**b**) ASMIBC with adaptive compensation. With the ASMICB has adaptive compensation, a large sliding mode gain is not required. Figure 13: ASMIBC adaptive compensation output. The second-order state of the system is subjected to a step disturbance with a value of 3. The disturbance of the system can be obtained through the adaptive rate. converge slowly, while it also has the same maximum tracking error as IBC. Our proposed ASMIBC has a static error of about 0.05 when there is no adaptive compensation (the blue line). As we design an adaptive sliding surface gain, the closer the system state is to the set boundary, the larger the sliding surface gain will be. Therefore, the robustness of the controller is improved. When adaptive compensation exists, our proposed control method (the red line) can ensure the error convergence is achieved within 6 s. The gain of sliding mode surface is shown in Figure 12.  
The robustness of SMC can be improved by increasing the gain of the switching function, but if the gain is too large, the system state will tremble. Our proposed method can adaptively adjust the gain of sliding mode switching function.