# NESO-Based Path Following Control for
## 1 Introduction  
Nonlinear motion control of marine surface vehicles, which includes underactuated hovercrafts, unmanned surface vehicles, and unmanned underwater vehicles, has attracted the attention of many researchers [1]. Compared with motion control of fully actuated marine surface vehicles, it presents challenging control problems. Hovercrafts, different from the traditional surface marine vehicles, are characterized by underactuated, high velocity and a large size [2]. As shown in Figure 0(a), an underactuated hovercraft is supported by a pressurized air cushion, which makes the hard structure just far from the water surface, reducing the water drag, surface interference, and wave-making [3]. This makes it able to withstand motion on different surfaces. Therefore, underactuated hovercrafts are widely used in military and civil fields, such as delivery of landing soldiers, mine clearance, and rescue [4]. Although the underactuated hovercraft has the above advantages, its maneuverabilityand safety are poor due to its special cushion physical structure. In addition, hovercrafts are usually underactuated, which brings a challenging nonlinear motion automatic control problem [5]. The kinetics of underactuated hovercrafts is a complex nonlinear system, and it is very difficult to obtain an accurate model of underactuated hovercrafts for the control system design [6]. In addition, underactuated hovercrafts are suffering from the complex external ocean disturbances during navigation, thus there are assorted uncertainties, such as external disturbances, unmodeled hydrodynamics, and parametric perturbations present in an underactuated hovercraft motion model [7]. Therefore, control of underactuated hovercrafts faces challenges due to the external ocean disturbances and internal unmodeled dynamics, which has drawn considerable attention from researchers. In recent years, a lot of research, such as fuzzy logic [8] and neural network [9], has been done. However, these methods rely too much on prior knowledge, which makes their implementation more complicated in practice. The extended state observer (ESO), initially proposed by [PERSON] in 1988, provides the possibility to overcome the difficulty of control of uncertain systems [10]. ESOs are designed to approximate the unknown \"total uncertainties\", which introduce excellent challenges to design a motion controller for marine vehicles that usually has nonlinear uncertainties. The ESO does not rely on any model information so that the fully model-free control for the nonlinear systems with unknown uncertainties are achieved by [PERSON] in [11]. In the past three decades, many studies on ESOs have been carried out [12; 13]. The linear extended state observers (LESOs) were designed to estimate the states and the general external disturbances for the autonomous underwater vehicles and unmanned surface vehicles, respectively, by [PERSON] in [14] and [PERSON] in [15]. Although both LESOs and nonlinear extended state observers (NESOs) can estimate the uncertainties, the accuracy of the NESO is better than that of the LESO [14; 16]. To estimate the fast-varying disturbances, a harmonic ESO was proposed to achieve a high-accuracy estimation of disturbances for a fully-actuated autonomous underwater vehicle by [PERSON] in [17], where the designed harmonic ESO can only estimate the periodic disturbances. However, for the underactuated hovercraft suffering from unmodeled hydrodynamics and complex oceanic disturbances, the total uncertainties are usually time-varying, which cannot be estimated above ESOs. To approximate the unknown time-varying uncertainties of underactuated hovercrafts, a novel homogenous nonlinear extended state observer (NESO) is first proposed for the underactuated hovercraft to estimate and compensate the unknown time-varying uncertainties in this paper.