## 5 Further Discussion on the Tracking Control Algorithm
### The Proposed Controller versus Other Control Schemes
In simulation studies on the CyberShip II prototype, the authors focused on comparing the performance of different versions of the proposed control scheme for tracking three desired trajectories namely straight line, circle, and curve. The tracking accuracy strongly depended on the variant of the controller used. In paper [60], a decoupling and yaw control strategy was proposed for a USV to maintain a constant speed and a desired course for high performance of on-board sonar during seabed exploration. In addition to the direct Lyapunov method, a hierarchical stepwise control method was used to achieve the desired speed and yaw angle of the vehicle under the influence of external disturbances such as wind, waves, and currents. An disturbance observer was also constructed to compensate for oscillations occurring with external disturbances. The validity and robustness of the decoupling controller was verified by application to one model of USV and straight line tracking. [PERSON] et al. [40] proposed an adaptive online constructive fuzzy controller (AOCFC) for tracking USV trajectories while ensuring system stability. A new online constructive fuzzy approximator (OCFA) was integrated with the dynamic surface control (DSC) method to estimate the unknown time-variable uncertainty. Compared with the fixed-structure fuzzy regulator, the proposed online constructive fuzzy regulator strategy was able to adapt the online fuzzy rules to ensure the sufficiency and simplicity of the fuzzy system. In the simulation study (on one vehicle model), the fuzzy control algorithm (AOCFC) was compared with the adaptive structure-fixed fuzzy control (ASFFC) to show the effectiveness and advantages of the constructive online fuzzy strategy. [PERSON] and [PERSON] [46] presented three modified dynamic inversion methods for tracking control of underactuated ship. In the first called dynamic extension-based on dynamic expansion (DEDI) the input was treated as a state and dynamic expansion was used to obtain the relative degree. The second method, i.e., virtual input-based dynamic inversion (VIDI), treating the state as a virtual input to obtain the relative degree. The third method was output redefinition-based dynamic inversion (ORDI), selecting a specific variable as a new output to achieve relative degree. These three methods were generalizations of dynamic inversion control and eliminate some of its limitations, so they could be applied to a wide variety of underactuated systems. Numerical simulations performed on one selected vehicle demonstrated the effectiveness of the control approach. Furthermore, a comparison with a reference method was presented. In [56], a trajectory tracking strategy and and path control strategy for underactuated marine vehicles were presented. The strategy was developed as an extension of the results of the work on ground vehicles, which introduced the hand position point definition to marine vehicles (the definition of the hand position point was extended). An input-output feedback linearization method using hand position as output was applied and a trajectory tracking strategy and a path following strategy were developed for general paths, with straight line paths as a special case. The effects of environmental disturbances, such as a steady and non-rotating ocean current, on the vehicle were also considered. The effectiveness of the approach was confirmed using simulation studies on the model and experimental studies on the real vehicle. It is worthwhile to present the differences between the selected methods and the control scheme using quasi-velocities. The method proposed in this paper is intended not only to realize the trajectory tracking of the desired control but also to provide some insight into the dynamics of the vehicle in motion also under the influence of external disturbances. For this reason, only some of the control strategies would be suitable to realize the objective of the work. Methods based on the use of neural networks require additional mathematical knowledge.