## 1 Introduction
The quaternion unit has been widely used in spacecraft attitude control or satellite attitude control [33; 34], but spacecraft attitude control does  
Figure 1: Schematic diagram of coaxial HAUV deployment and operation. not consider translational motion. Both translational and rotational motion should be considered for the trans-medium motion control problem of HAUVs. Therefore, these control schemes cannot be directly applied to the motion control of HAUVs. In [35], the quaternion-based PD attitude control method was applied to the motion control of a six-degrees of freedom (DOF) AUV, but it lacks measures to further enhance its robustness. In [16], the PD algorithm based on quaternion is also adopted in the attitude control of HAUV. This solves the problem of large-angle maneuvering but also lacks sufficient consideration of the robustness of the controller. Motivated by the above discussion, an appropriate strategy must be proposed to solve this singular constraint to improve HAUV mobility. According to the underactuated characteristics of a coaxial HAUV system, it was divided into inner (attitude) and outer (position) loops [36], and controllers were designed, respectively. In the outer loop trajectory tracking control loop, the adaptive sliding mode control (ASMC) technology enhanced by disturbance observer (DO) [37] was used to calculate the thrust and the expected attitude angle. The disturbance observer was combined with an adaptive technique to compensate for the lumped uncertainty, which effectively weakens the inherent chattering phenomenon of sliding mode control. The calculated attitude angle was converted into a desired quaternion, and a DO-enhanced robust feedforward PD control algorithm was used to track the target quaternion. Finally, a variable parameter design scheme based on the learning rate was introduced into the cascaded SMC-PD controller, and the error convergence speed of the system was further improved by adaptively changing the controller parameters [38]. In order to realize fast and stable cross-domain tracking control of coaxial HAUV systems, a DO-based SMC-PD control strategy is proposed in this paper. The contributions of this paper are as follows:  
1. Removing the limitation of the small-angle assumption, a six-DOF cross-domain dynamic model that can perform large-angle maneuvering is established. The quaternion attitude modeling scheme is adopted to avoid the singularity problem of HAUV. 2. In the hierarchical cascade SMC-PD controller, the control gain parameters are adaptively changed by using the learning rate method, which improves the error convergence speed of the system. 3. A DO with a simple structure is adopted to supplement the uncertainty in the dynamic model and enhance the robustness of the system. The arrangement of the rest of this manuscript is outlined as follows: in Section 2, the dynamics model of the cross-domain process of a coaxial HAUV is analyzed and established. The attitude dynamics model based on quaternion representation is introduced. In Section 3, the design process of the control strategy is given, and the stability of the proposed control scheme is proved by using the Lyapunov theory. The control performance of the designed controller is analyzed by using the numerical simulation study in Section 4. Finally, Section 5 presents some conclusions and perspectives as well as an outlook for future work.