## 1 Introduction  
A typical motion control problem for an underactuated vessel is path following, which is concerned with the design of control laws that force a vessel to reach and follow a time-independent geometric path. Pioneering works [1; 2; 3; 4; 5; 6] in this area proposed many solutions for a vessel sailing in open waters. When a vessel sails in restricted waters (e.g., coastal areas), vessel-bank and vessel-bottom interactions have significant influences on the maneuverability and make it difficult to maintain a steady course [7; 8]. Many studies have pointed out that a vessel sailing in restricted waters is unstable in the absence of reasonable control [9]. Therefore, the motion controller design for an underactuated vessel sailing in restricted waters is required. Some researchers [7; 10; 11; 12; 13; 14; 15] made extensive studies on vessel behaviors in restricted waters, including the detailed hydrodynamic coefficients in the mathematical models of vessel motion and analysis of directional stability. Their research laid the foundation for the study and design of the controller for an underactuated vessel sailing in restricted waters. Several control methods were introduced and satisfactory results were obtained assuming the water depth being constant [7; 14; 15; 16]. While in real life, the actual seabed usually has a varying bottom, resulting in varying water depths. The straight-line path following control problem with varying water depths is needed, considering the poor maneuverability in restricted waters. The straight-line path following control for an underactuated surface vessel without sway actuator is studied. The absence of sway actuator brings about significant challenges on the path following control design. To overcome this difficulty, an effective way is to implement the path following control scheme, which consists of a guidance system and a control system. In the guidance system, the reference signals for the successive control system are produced according to the path information and environmental conditions. A commonly used guidance system is designed based on the traditional line-of-sight (LOS) guidance law. Despite its simplicity and effectiveness, it has limitations when the vessel is subject to model uncertainties and external disturbances acting in the underactuated lateral direction. Variations in the vessel's lateral position generate a nonzero sideslip angle for the vessel. Using the traditional LOS, the desired path cannot be followed by an underactuated vessel subjecting to the effect of sideslip angle. To solve this problem, scholars have devoted many efforts to the field in the past. [PERSON] and [PERSON] measured the sideslip angle directly using sensors [17; 18], while the data from sensors is affected by noise and sensor errors, which is not precise enough. Another method to solve this problem is the integral line-of-sight (ILOS) guidance. It was first put forward by [PERSON] et al. [19], and the integral term was added to the traditional LOS to compensate for the constant current to achieve precise path following. Furthermore, [PERSON] et al. [20] adopted the adaptive control theory to design the guidance law, they assumed the sideslip angle to be an unknown constant. They also proposed both direct and indirect adaptive LOS (ALOS) guidance laws in [21] to deal with the time-varying current. Nevertheless, choosing adaptive gains is a tough problem because of the intrinsic tradeoff between the convergence speed and the chattering. A reduced order state observer was proposed to design the guidance law to conquer the intrinsic weakness of the ALOS guidance [22]. However, the sideslip angle is assumed small, and only one design parameter is available in the observer design, it is hard to guarantee better estimation performance. Furthermore, none of the above work considered designing a guidance system for a vessel sailing in restricted waters. In the guidance system, a guidance angle is calculated to serve as a reference input for the yaw tracking control system.