## 1 Introduction
For a class of underactuated systems dependent on model parameters, a robust interconnection and damping assignment-passivity-based control (IDA-PBC) strategy was designed [21]. The superiority of the proposed control scheme is verified by the simulation of the quadrotor system with an unknown load. Furthermore, an improved IDA-PBC strategy based on filter observer is proposed to solve the interference and model uncertainty in quadrotor trajectory tracking control. The proposed algorithm can deal with measurement noise and model uncertainty [22]. In particular, sliding mode control (SMC) is widely used in the controller designs of various robot systems due to its insensitivity to model perturbation and strong robustness. The authors of [23] designed a classic SMC controller for a tilting HAUV to verify the effectiveness of the SMC method in altitude and attitude control. [PERSON] et al. [24] designed a second-order SMC controller for a Nezha series multi-rotor. In order to improve the response speed of the control system, the author also designed an adaptive nonsingular fast terminal sliding mode controller for a coaxial HAUV in previous work [25]. Although the above research has fruitfully explored HAUVs to achieve cross-media motion, in practice, the key difficulty is that the additional variables, such as added mass, damping, and the restoring force, change dramatically with the operating state of HAUVs in the trans-medium process. In addition, the combined disturbance of wind and wave load at the water-air interface brings further challenges to the controller design [26]. In [27], it is mentioned that the support arm structure of multi-rotor HAUVs will cause great interference to the dynamics of the vehicle when it breaks through the water surface. Compared to multi-rotor HAUV systems, coaxial HAUV systems have attractive advantages such as a compact structure, excellent hover performance, and so on [28]. Coaxial HAUVs do not require arm structures, which reduces the forward projection area. Therefore, coaxial HAUVs have attracted the interest of researchers in recent years [29; 30]. It turns out that this new type of coaxial HAUV has great potential for future development. Inspired by the above research and referring to existing coaxial aircraft, the author developed a coaxial HAUV system in [25]. Figure 1 shows the operation flow chart of the designed coaxial HAUV. As can be seen in Figure 1, coaxial HAUV can use two operating modes according to different tasks. In the initial phase of controller design, the mathematical model of an HAUV should be established first. However, most existing HAUVs mainly adopt the modeling idea of quadrotor UAV, that is, modeling based on the small-angle assumption. This limits the model description of HAUV in the large-angle state. Mainstream methods for describing attitude parameters include the rotation matrix, Euler angle, Rodriguez parameters (RPs), modified Rodriguez parameters (MRPs), quaternion, etc. [4]. Because it is relatively easy to understand in terms of physical interpretation, most UUV and UAV control schemes are based on the Euler angle representation for attitude control. However, the use of the Euler angle to represent the attitude may cause a singularity when the vehicle approaches a singular direction. In most UAV and AUV scenarios, this is usually not a serious problem, as the vehicle only needs to operate at angles close to the equilibrium point and away from the singularity [31]. For the sake of effectively overcoming the large drag, the propulsion force needs to be more distributed in the forward direction, which requires a large inclination angle. To study the stability of HAUVs under a large angle of inclination under harsh sea conditions, it is a valuable option to choose the attitude resolution method based on quaternion representation [32].