## 1 Introduction
However, the research on the modeling and control algorithm design of the trans-media motion in a complex environment is extremely rare. Figure 1: Take-off for a semi-submersible hybrid aerial underwater quadrotor on the water. For the control design problem of the water-to-air transition motion process of HAUQs, a water-air crossing motion control design scheme is proposed based on the lift provided by four rotors. The basic idea of the control algorithm of the HAUQ is to use four rotors in the air to pull up the streamlined body which is submerged in the water and drain the water inside the body in a short time. The traditional four-rotor UAVs is the most common and representative UAV in a multi-rotor system, and it is a multiple-input and multiple-output nonlinear system, including the nonlinearities, the multiple variables, the underactuated characteristics, the weak anti-jamming ability, and the complex couplings [17; 18]. The HAUQ inherits these characteristics. The HAUQ is facing a flight environment different from the traditional quadrotor UAV. When the permeable slender body of the HAUQ crosses from water to air, the medium changes, and there are many complex changes whose mechanism is not clear enough, such as the additional mass caused by the fast drainage, the multi-media dynamics, and the fast-changing buoyancy. And the body is filled with water and needs to be discharged in a very short time such that the mass of the whole HAUQ changes dramatically when it takes off from the water to the air. The nonuniform drainage will also cause the change in the center of mass and the arm of force. The main contributions of the manuscript can be summarized as follows:  
1. A new water-air trans-medium pattern is proposed for the HAUQ with a permeable slender body. Compared with the existing layouts of the HAUVs, the HAUQs with a permeable slender body can help to keep the streamlined fuselage needed for underwater and air navigation. 2. A general mathematical model is established for HAUQs by employing the Newton-Euler formulation as the factors exist, including the strong uncertainties caused by the fluid dynamics in the complex water-air mixed environment, the fast time-varying added mass caused by the fluid dynamics and the residual water inside the slender body, the influence of the passive drainage, and the external disturbance. 3. A disturbance observer-based robust control scheme is proposed for HAUQs. The robust control is adopted to compensate for the fast time-varying mass uncertainty. For the uncertainties of the multi-media complex dynamics modeling on the position and attitude dynamic equations, it is estimated by considering it as a combination of the specific basis functions, and an adaptive method is used to estimate the unknown weight parameters. The rapid and uncontrollable drainage will cause the mass and the center of mass to change during take-off on the water surface. Meanwhile, the length of the arm of force and the moment of inertia matrix will change unpredictably, and they are considered as the bounded uncertainty of the moment of inertia matrix and the force arm variation. Then, a comprehensive dynamic disturbance term is formed together with the bounded additional disturbances, and a disturbance observer structure in [19] is introduced to estimate it under the assumption that the total disturbance is measurable. The idea of using a disturbance observer to estimate the system disturbances is to introduce feedforward compensation in the process of controller design to improve the control performance of the system, and it is widely used in aircraft control [20; 21; 22; 23; 24; 25; 26; 27; 28; 29]. The input-to-state stability theorem is an effective method to study nonlinear systems with noises and disturbances [30]. This method can obtain the bounded states by suppressing the bounded disturbances; therefore, the stability of the position and attitude control of the HAUQ is analyzed by the input-state stability theorem.