## 4 Simulation Results
### Comparative Simulations  
In order to verify the advantages of the proposed MPC, a comparison with the traditional MPC algorithm that considers the control input directly and no constraint to the increase of the control input was conducted. Figure 12 depicts the comparison results. The two algorithms were able to drive the robot to track the desired trajectory quickly, as shown in Figure 12. In order to clearly observe the tracking results, the end of each line stage is amplified. The proposed method converges to the desired trajectory faster, as depicted in the enlarged part of the figure. Besides, the traditional MPC-based tracking trajectory exhibits a small error with the desired trajectory when the BSR converges to the desired trajectory. Figure 13 plots a comparison of the state errors. As the limit of the increase in control, the state errors are slightly larger when the yaw angle suddenly changes. As the tracking result converges to the desired trajectory, the errors that are based on the improved MPC tend toward zero. While, as compared with the improved MPC, the traditional MPC-based tracking results exhibits few static errors at the end of each line stage. Figure 14 depicts a comparison of the control increments. When compared with the improved MPC, the traditional MPC generates a sharp increase in control when the yaw angel of the robot suddenly changes, which is likely to damage the thrusters. It was verified that the improved MPC respects the constraints of the control input and its increase while achieving the trajectory tracking task.  
Figure 11: Control input signals-Case II.  
Figure 10: The state errors of the biomimetic spherical robot-Case II.  
Figure 14: The comparison of control increment in square tracking.  
Figure 12: The comparison result in square tracking.  
Figure 13: The comparison of the state errors in square tracking.  
Furthermore, a comparative simulation using a typical backstepping algorithm [19] that performs well in trajectory tracking was conducted. The simulation results are depicted in Figures 15-17. The simulation trajectories based on the improved MPC and the backstepping methods both catch up and land on the reference trajectory at the end, as indicated in Figure 15. However, the improved MPC is able to drive the robot in order to track the desired trajectory more smoothly. The state errors during the tracking process are plotted in Figure 16. As the constraints of the input increase, the position errors at the corner of the square are slightly larger than that based on the backstepping method. The error of the yaw angle based on the proposed MPC is clearly smaller during the entire tracking process. Figure 17 shows the comparison results of the control inputs. When compared with the backstepping method, owing to the ability to deal with the constraints of the proposed method, the control inputs are smooth and they strictly respect the saturation of the thruster. While, the control inputs obtained by the backstepping method jump sharply and reach \(-6\)\(N\) and \(-0.18\)\(N\times m\) when the tracking errors occur at the initial time, which overshoots the constraints of the thruster. A comparison of the control inputs reveals the advantage of the proposed algorithm in coping with the constraints.  
Figure 16: The comparison of the state errors with the backstepping method in square tracking.  
Figure 15: The simulation results compared with the backstepping method in square tracking.