Objective  In intelligent ship motion control, accurate trajectory tracking of three-degree-of-freedom fully actuated ships poses a significant challenge. The complex and variable marine environment exposes ships to unpredictable and strongly coupled environmental forces, such as wind, waves, currents, and water resistance. Consequently, achieving precise trajectory tracking while ensuring system robustness is crucial. This study aims to tackle this challenge by proposing a cascade MPC control algorithm designed to enhance the efficiency of ship trajectory tracking control.

Methods The research methodology is both comprehensive and systematic. First, the model predictive control (MPC) controller is designed by integrating the ship's kinematics and dynamics equations separately. These two controllers are then combined to create a cascaded MPC controller, which addresses the characteristics of ship motion from both kinematic and dynamic perspectives. An optimization problem is then formulated, with state variables, input variables, and final value variables as objective functions, and system input and state constraints considered. To verify the simulation, a CyberShip II fully actuated ship is employed as a case study for circular trajectories, examining the impact of various cascaded MPC controller parameters in detail. Furthermore, the study investigates how controller parameter adjustments affect circular trajectory tracking under varying frequencies and amplitudes of environmental forces.

Results The simulation results highlight the controller's effectiveness in meeting trajectory tracking performance requirements. Fine-tuning the parameter matrix of the final value variable enables the controller to effectively withstand varying degrees of environmental interference, significantly enhancing the system's robustness and stability. For instance, increasing the state weight matrix parameter Q₁ of the kinematic MPC controller improves trajectory tracking accuracy, although it may introduce some initial oscillations. Additionally, the S₂ matrix parameter of the dynamic MPC controller greatly influences the system's robustness; a larger S₂ matrix parameter facilitates quicker stabilization of the system.

Conclusions The research provides valuable insights for improving trajectory tracking in fully dynamic ships. It has been established that in the cascaded MPC trajectory tracking controller, the optimal tracking results are achieved when the ratio of the state weight matrix to the input weight matrix in the kinematic MPC controller is between 5 and 15. Additionally, the final value parameter matrix in the dynamic MPC controller significantly influences system performance. Increasing the final value matrix parameter in the dynamic MPC controller can effectively mitigate oscillations caused by higher frequencies or amplitudes of external environmental forces, particularly those from increased amplitude. However, this study has its limitations, including using a linear equation for ship dynamics modeling applicable only under low-speed conditions and a linear MPC control algorithm. Future research should explore nonlinear MPC controllers for nonlinear ship dynamics, and investigate input increment control in the formulation of the objective function.