## 1 Introduction  
A great amount of significant research has been made in the last decades with the development of advanced robotic systems, especially about autonomous underwater vehicles (AUVs) and unmanned aerial vehicles (UAVs). As AUVs and UAVs are good at completing their tasks in their respective fields, they can efficiently achieve marine and aerial observations and attack missions, respectively. However, when faced with a multi-domain task, it seems impossible to accomplish the mission whether AUVs or UAVs. The heterogeneous multi-robot systems have been used to accomplish multi-domain environment monitoring [1; 2; 3]. And the multi-robot cooperation enriches the observation and improvement in work efficiency. Nevertheless, some new challenges are brought by using such a multi-robot system for users. The difficulties of establishing and maintainingmay create multi-robot system operational difficulties. What is more, it is extremely difficult for the heterogeneous robots working in the different types of medium to communicate with each other because of the attenuation effect of water for the electromagnetic waves. Therefore, a single platform, that is, the hybrid aerial underwater vehicle (HAUV) which can be capable of moving in both air and water, is needed to be developed to complete multi-domain missions. Because of the huge differences in the physical properties between air and water, it is a great challenge to design an HAUV capable of both air flight and underwater navigation, especially the efficient control technique. According to the existing literature, many vehicles of various layouts such as fixed-wing [4; 5; 6], variable-swept wing [7], flapping wing [8], and multi-rotor systems [9; 10; 11; 12] have been adopted for aerial and underwater missions, which have shown the high practical value and fruitful results. And the controllability, payload, and practicality of HAUVs has been assessed [13]. However, the existing layouts of HAUVs are not compatible with the advantages of the aerial vehicles and the underwater vehicles. A transverse permeable slender body similar to the fuselage of a missile or a submarine is a suitable scheme for the fuselage of HAUVs. Then, a new HAUV layout is proposed which uses four rotors to realize the medium crossing motion of a transverse slender body similar to the fuselage of a missile or submarine, that is, the hybrid unmanned aerial underwater quadrotor (HAUQ) as shown in Figure 1. The HAUQ layout can help to use the morphing wing technology and the fixed-wing hybrid quadrotor technology to realize the underwater streamlined fuselage navigation and the fixed-wing flight. The key of the whole flight process is the trans-media motion control from water to air. The controller design of the hybrid aerial underwater quadrotor (HAUQ) is a typical gas-liquid coupling problem. The water-air interface crossing motion has strong non-linearities and uncertainties, such as the multivariable strong couplings, the nonlinear hydrodynamic characteristics of the water-air two-phase flow, the impact of water waves, the gust, and the passive drainage disturbance. For the HAUV, it includes the water-air transition section and the air-water transition section. To realize the stable and reliable conversion between the discontinuous media, the control scheme of the HAUV is one of the key issues of the cross-medium motion. The control models and the control arithmetics of the existing vehicles including the surface vessel and the underwater vehicle have some limitations and strong disturbances such as winds, waves, and currents which will challenge the control issues. For the marine environment interference, most of the research objects are underwater vehicles. The influence of the ocean current is studied on the motion control of the low-speed AUV [14]. A robust navigation algorithm is developed for the recovery of AUVs [15]. For the operability of an AUV, the least square method is introduced to estimate the ocean current [16].