## 1 Introduction  
In recent years, with the development of science and technology and the traction of marine resource exploitation demand, marine robots, such as ocean unmanned aerial vehicles (UAVs) [1], unmanned surface vehicles (USVs) [2], ocean buoys (OBs) [3], and unmanned underwater vehicles (UUVs) [4], have been rapidly developed and achieved great success. This technology is increasingly playing an important role in marine surveillance, offshore rescue, resource exploration, and even military operations. As research results on diverse marine robots continue to emerge, researchers are beginning to think of and design new robots for seemingly impractical amphibious tasks. The hybrid aerial-underwater vehicle (HAUV), which is capable of performing air and water amphibious missions, belongs to a new concept of amphibious vehicles. The concept of amphibious vehicles can even be traced back to the 1930s, when the United States Navy and the Soviet Union, respectively, conceived of flying submarines and underwater aircraft [5] and carried out a series of studies [6; 7], but early research was not smooth. After decades of silence, with the progress of marine robotics technology and the increasingly urgent demand for amphibious missions, the development and testing of prototype HAUVs and the pre-research of related technologies have attracted more and more scholars' research interest. There are two main ways to implement HAUVs: flying submarines and diving aircraft. At this stage, the most mature scheme mainly refers to existing aviation UAV design schemes, such as fixed-wing structure forms, multi-rotor forms, and bionic design forms [8]. Of course, there are also composite wing HUAVs designed by combining the advantages of various configurations [9; 10]. In addition, schemes to realize cross-media motion are mainly divided into fast and slow water surface crossing [10]. The main means of rapid water exiting are compressed gas injection [11] and combustible gas explosion [12; 13], while slow water exit processes are mainly achieved with propeller propulsion. In particular, HAUVs with a multi-rotor configuration can achieve vertical take-off and landing (VTOL), which has won the attention of many researchers and achieved many research results. In 2014, the authors of [14] first proposed the double-layer quadrotor configuration HAUV, and the researchers designed a simple proportion derivation (PD) controller for it. Researchers at Rutgers University designed a variable parameter based on a sensor readings proportion integration derivation (PID) controller for their HAUV (Navigator) [15]. At the same time, the team established the attitude model for Navigator by using the quaternion method and successfully carried out high-maneuverability operation testing [16]. In 2019, researchers designed a classical PD flight controller for an HAUV (Loon Copter) equipped with a buoyancy regulation system and successfully achieved take-off and landing on the water surface [17]. For the purpose of further improving the working characteristics of classical linear controllers, the improved PID control strategy is gradually derived. In [18], the researchers proposed an active disturbance rejection control (ADRC) strategy to achieve attitude stability control for a double-layer HAUV and conducted experimental tests near the water. Fuzzy P + ID [19] has also been applied to HAUV systems. In addition, the authors of [20] proposed an improved PID controller, which combines an intelligent genetic algorithm (GA) and radial basis function (RBFNN) to deal with the height control problem of cross-domain processes. The limitation of poor robustness of linear controllers can not be completely overcome by the above improvements. With the progress of control technology, some nonlinear control strategies that have been widely used in quadrotor control have been applied in the HAUV field.