## 1 Introduction  
An air-cushion vehicle (ACV), which is a high-performance ship, has a flexible skirt system at the bottom and can be lifted by an air cushion force. Compared with traditional marine surface vessels, its unique design makes it perform better at high speeds with a high wave resistance, strong flexibility and amphibious performance. Therefore, the ACV is used in marine transportation [1], military, rescue mission [2], scientific and marine development projects [3]. The ACV can easily exhibit stalling, sideslip, tail swing, roll and drift phenomena during turning at high speeds; thus, high-precision control and partial state safety constraints are necessary to avoid these phenomena. An accurate mathematical model is the basis for realizing safety and precision control. From a review of the available literature about the ACV, only mathematical models with three DOFs were adopted [4; 5; 6; 7; 8; 9; 10]. However, the roll angle has a substantial influence on the turning radius of the ACV. In addition, the roll angle is also an important safety indicator. In actual sailing, the roll angle must be determined within a certain range; otherwise, the ship deck will enter the water and cause the ship to overturn. Four DOFs models, including surge, sway, roll and yaw motion, are established in these papers [11; 12]. It is also shown that the four DOFs model is more suitable for engineering applications than the three DOFs model. In these articles, the idea of separation modelling is used. The forces and moments including hydrodynamics, aerodynamics, air momentum, air propeller forces, rudder forces and skirt drag are represented in the four DOFs. However, there is far less research literature on hovercrafts than on ordinary surface vessels [13]. If the model of the ACV is transformed into a vector form similar to the ordinary surface ship, not only can the differences between them be compared, but this can also facilitate the promotion and application of the latter in the former. The safe control of the ACV can also be understood as the constraint problem of some state variables during high-speed sailing. The preset performance function method, model predictive control approach, barrier Lyapunov functions, reference setting method and control skill based on invariant settings or permissible settings are typical techniques for solving constraint problems [14; 15; 16; 17; 18; 19]. A preset performance path tracking controller for under-actuated surface ships with input saturation and external disturbances was proposed to assure that the positional errors were limited to a predetermined range [20]. For a class of strict feedback nonlinear systems [21], a backstepping controller was proposed based on the preset performance, taking into account both the steady-state and temporary state performance. In [22], surface ships with full-state constraints and dynamic uncertainties were controlled based on the barrier Lyapunov function, which was used to prevent state violation constraints. For the longitudinal model of the hypersonic vehicle, the adaptive neural control method was designed by using a neural network to constrain the angle of attack within the specified safety range [23]. In [24], the author proposed Lyapunov functions to solve the problem of trajectory tracking control of fully actuated unmanned vessels with input and output asymmetric constraints. An adaptive neural network tracking controller that combined the backstepping method, adaptive neural network and preset performance was designed to approximate unmoulded dynamics and external disturbances [25]. For the ACV, when it turns at high speeds, the yaw rate changes too much, which will cause side-slip. At the same time, the comfort of the driver will also be affected. Therefore, this paper, is the first to combine the BLF, Nussbaum function and backstepping methods to address the rudder nonlinearity and constraints of the yaw rate error.