## 5 Simulink Results and Discussion
### The Results of CDR Movement on the Ground and the Water Surface
When there is no disturbance, the gain will be 0, and the gain will be adjusted in 20 s to suppress the influence of step disturbance on the system. However, when the ASMICB has adaptive compensation, a large sliding mode gain is not required, so the chattering of the state can be reduced. The adaptive output of ASMIBC with adaptive compensation is shown in Figure 13.  
The linear velocity of the robot moves on the ground is 2 m/s, and the linear velocity of the robot moves on the water surface is 1 m/s. To simulate the robot moving from the ground to the water surface, when the robot moves on the ground, the acceleration is subjected to a random disturbance with an average value \(-0.5\) m/s\({}^{2}\), which is used to simulate the lumped disturbance caused by air resistance and mechanical loss. At 15 s, the robot moves from the ground to the water surface, the acceleration is subjected to a random disturbance with an average value \(-3\) m/s\({}^{2}\), which is used to simulate the lumped disturbance caused by water resistance, water flow and the uncertain model parameters. The control result is shown in Figure 14.  
From 0 s to 15 s, the robot moves on the ground, all control methods except IBC can track the desired linear velocity of 2 m/s. This is because IBC is a model-based control method, and its robustness is weaker than other control methods. At 15 s, the robot reaches the water surface, the desired velocity speed of the robot is 1 m/s. Our proposed control method can track the desired linear speed, as shown by the red line. Without adaptive rate compensation, ASMIBC has a static error of about 0.05 m/s. The adaptive compensation and the adaptive sliding mode gain have been discussed in Figures 12 and 13, so they are not introduced here. The yaw angle of the robot on the water surface and the ground does not need to be constrained. Constraints on the yaw angular velocity are required to prevent the drive from saturating or the robot from being placed in a dangerous condition due to a large yaw angular velocity. The yaw angle control results on the ground is shown in Figure 15.  
Figure 10:0 Velocity control of CDR on the ground and water surface. From 0 s to 15 s, the robot moves on the ground, the desired velocity of the robot is 2 m/s. At 15 s, the robot reaches the water surface, the desired velocity of the robot on the water surface is 1 m/s. In Figure 15, the desired yaw angle is \(-1\) rad at 0 s and 1 rad at 15 s. A random disturbance with a mean value of 5 rad/s/s is applied to the acceleration of the yaw angle to simulate the impact of lumped uncertain disturbance on the system. Our proposed algorithm can control the actual yaw angle to follow the desired yaw angle. The angle control results of SMC, PID and IBC are shown in Figure 15b. Both PID and ASMIBC can follow the desired yaw angle. There is a static error of 0.3 rad in SMC and 0.4 rad in IBC. The yaw angular velocity control is shown in Figures 16 and 17. As shown in Figure 16a, the maximum yaw angular velocity is 2.5 rad/s. Our proposed control method can follow the desired yaw angular velocity without overshoot and without exceeding the set boundary of 2.6 rad/s. The results of the other four control methods are shown in Figure 16b. Without the adaptive compensation, ASMIBC has a static error of about 0.1 rad/s, but it does not exceed the set boundary of 2.6 rad/s. As mentioned before, our proposed ASMIBC can adaptively adjust the gain of the sliding surface to suppress the disturbance, but it cannot eliminate the influence of the disturbance in the system state. ASMIBC with adaptive compensation can completely eliminate the effect of the disturbance in the system. SMC and PID cannot constrain the yaw angular velocity,  
Figure 10:1 Yaw angular velocity control of robot on the ground. (**a**) The control result of proposed method. (**b**) The other four control method results. The maximum yaw angular velocity is 2.5 rad/s.