## 5 Further Discussion on the Tracking Control Algorithm
### The Proposed Controller versus Other Control Schemes
Comparison of the proposed method with another taken from the literature (for one vehicle) showed slightly smaller tracking error results for the assumed complex trajectory. In [28], the problem of horizontal trajectory tracking control for underactuated marine vehicles subject to unknown internal and external disturbances was considered. The control design involved two stages. First, a kinematic control law of the desired surge speed and trajectory angle was derived. Then, a kinematic control law that satisfies the convergence requirement was designed. Specifically, the kinetic law was divided into two components: the control law of disturbance rejection control law and the stabilization control law. The former was obtained from the high gain ESO, which was designed to estimate unknown disturbances, and the latter was developed in a reduced model using backstepping method and composite-system method. The main advantage of the proposed approach is that the design and analysis process has been simplified to some extent. Simulation verification was performed for a model known from the literature and three trajectories: circular, sinusoidal and complex. Based on the results obtained, the authors concluded that the validity and effectiveness of the proposed method and the proposed control scheme have been sufficiently proven. [PERSON] et al. [57] demonstrated the design of an improved line-of-sight (LOS) based adaptive trajectory tracking controller for an underactuated marine vehicle subject to highly coupled nonlinearities, ocean currents-induced uncertainties, and input saturation. The strategy was composed of LOS, terminal sliding mode control (TSMC), and extended disturbance observers (EDO). The LOS guidance law was used to steer the vehicle towards the desired trajectory. The TSMC improved system robustness and asymptotic convergence, and was applied to weaken the influence of actuator saturation. The extended disturbance observers estimated the ocean current-induced disturbances in the kinematic model. Effectiveness and robustness of the proposed trajectory tracking was verified by numerical simulations on a well-known AUV REMUS vehicle. Four cases were considered for this control scheme, which confirmed the theoretical results. Paper [58] was devoted to solving a full-state control (FSRC) problem for an asymmetric underactuated surface vehicle (AUSV) subject to disturbances. The FSRC objective was divided into two subtasks, i.e., reaching trajectory (RT) guidance and synthesis of a tracking controller with underactuation and disturbances. After performing a series of coordinate transformations, the tracking error dynamics were shaped as a translational-rotational cascade with respect to the central frame of the circular orbit. Utilizing a finite-time method, the lumped disturbances were accurately estimated by accurate observers, facilitating the synthesis of the surge and yaw controllers. The effectiveness of the proposed approach was demonstrated using simulations on the CyberShip II prototype. However, no comparison with the results of another controller was shown. In [59], the problem of accurately controlling the trajectory tracking of an asymmetric underactuated surface vehicle (AUSV) was solved by guiding yaw dynamics, which were free of persistent excitation (PE). Next, a nested coordinate transformations was applied in order to formulate the AUSV model in a cascade structure consisting of translation and rotation subsystems with complex uncertainties. Finite time uncertainty observers (FUOs) were developed to accurately estimate the transformed uncertainties and enable the application of the separation principle in controller and observer syntheses. The global asymptotic stability of the whole translation-rotation tracking system was obtained and a FUO-based yaw-guided tracking control (FUO-YTC) scheme was developed for an AUSV with complex uncertainties.