## 1 Introduction
However, this study only focused on the flight control of the HAUV under a complex flow field near the water surface and did not carry out a complete traverse-water maneuver. In 2018, to achieve vertical takeoff and landing of a double-layer HAUV in disturbed environments, the team further developed an adaptive sliding-mode control (ASMC) algorithm to achieve real-time estimation and compensation of system parameters and determine the impact of time-varying environments [15]. In 2019, [PERSON] et al. from Dalian Maritime University developed a tilting quadrotor HAUV with positive buoyancy and designed a sliding-mode controller (SMC) for the vehicle to realize stable control [16]. In 2021, the first HAUV with a coaxial rotor system and foldable propeller was proposed, and an MPC-SMC cascade controller was designed for it, which further improved the controller design through feedforward compensation for the influence of buoyancy [17]. In the same year, researchers from the Shanghai Maritime University designed a \"fuzzy P+ID\" attitude controller for their single-layer quadrotor HAUV prototype. Since the controller mechanism is simple and easy to adapt in traditional flight control systems, the authors experimentally demonstrated the effectiveness of the method [18]. [PERSON] et al. modeled and analyzed the cross-medium process of a multi-rotor HAUV in detail, introduced a performance function into the HAUV control system for the first time, and adopted the adaptive dynamic surface control (ADSC) strategy to realize prescribed performance control in the trans-medium maneuver [19]. Subsequently, the team designed a dynamic surface controller based on a nonlinear disturbance observer (NDO) [20], which could accurately estimate the total uncertainty in real time and improve the robustness of the system. Recently, this team studied the instantaneous impact of periodic wave force acting on propellers and designed a second-order sliding-mode controller (SMC) to achieve a faster convergence rate and better robustness than traditional PID controllers [21]. A team from the Harbin Institute of Technology developed an amphibious multi-rotor UAV prototype [22], and the position and attitude controllers were established using an active disturbance rejection algorithm. The HAUVs based on the multi-rotor configuration mentioned above have achieved fruitful research results. It is mentioned in references [23; 24; 25] that the arm of a multi-rotor vehicle causes a large disturbance to the body dynamics in the process of breaking through the water surface, which is the main shortcoming of the multi-rotor configuration. Coaxial systems offer a more compact structure and higher efficiency, as well as a greater payload and rotor area, than multi-rotor HAUV systems. A coaxial HAUV does not need to use the arm used by a multi-rotor HAUV, which reduces the forward projection area. Due to the application of the vector platform, a coaxial HAUV propeller does not need to accelerate and decelerate frequently during operation, thus reducing energy consumption and counteracting the gyroscopic effect of the propeller. Therefore, coaxial HAUVs have attracted the interest of researchers in recent years. In the literature, a missile-type trans-domain amphibious vehicle (TDAV) was proposed and successfully verified using a trans-medium motion flight test [26]. In another study, a coaxial HAUV with a folding propeller was proposed, and a cascaded predictive and sliding-mode controller was designed for the HAUV [17]. The new coaxial HAUV proved to have great potential for future development. Based on the above research, we find that (1) it is very important to study an applicable HAUV modeling scheme and establish a reliable and accurate mathematical model for subsequent controller design work; (2) PD/PID and other linear controllers have a simple structure and are convenient for engineering applications.