## 2 Configuration Design and Problem Formulation
### Problem Formulation  
Due to the complexity of the actual water-entry process, with strong coupling and uncertainty, it is difficult to directly study the longitudinal and horizontal composite motion of the HAUV. Therefore, under certain assumptions, this paper only studies the longitudinal motion first. It should be noted that during the variable-swep process, the HAUV cannot be idealized as a rigid body anymore. Instead, it is divided into multiple independent parts as needed, and the force and moment equations of the morphing aircraft are analyzed with the help of multibody dynamics theory to obtain a simplified dynamic model in the longitudinal plane [22] as follows:  
Figure 2: Coordinate and depiction of the multibody representation. (**a**) Coordinate representation. (**b**) Level flight configuration. (**c**) Maneuver configuration. \[\left\{ \begin{array}{l}\dot{\theta}=q\\ \dot{\gamma}=L/(mv)-g\cos\gamma/v-F_{1z}/(mv)\\ \dot{\alpha}=\dot{\theta}-\dot{\gamma}\\ \dot{q}=(M_{1y}+M_{a}-S_{x}g\cos\theta-q\dot{I}_{y})/I_{y}\end{array} \right. \tag{1}\]  
The meaning and formulation represented by the symbols of all equations in this paper can be found in the Appendix A and Nomenclature. The core feature of the crossing process of the HAUV water-entry process is that the unknown and changing fluid force leads to the time-varying and uncertain parameters of the nonlinear system. There have been many discussions on the dynamics of the water-entry process of variant UAAVs [23], which are not repeated here. The attention of this paper is given to the control issues of uncertainty and external disturbances in the process of the HAUV during the completion of the variable-swept process and trans-media locomotion. The nonlinear control of an aircraft is a research hotspot in the field of control. Due to the nonlinearity and strong coupling in the mathematical description of the HAUV, designing control laws is even more difficult. The research direction of trans-media locomotion control problems is relatively rich in conventional multi-rotor HAUV configuration. [PERSON] et al. [24] simplified trans-media locomotion without considering the added mass and external disturbances in the air-water transition process, established a dynamic model of the trans-media process, transformed the nonlinear dynamic models in the air and underwater into a linear variable parameter model and designed a state feedback attitude controller to achieve attitude stability in the preset media, which was the first attempt at air-water transition control. Later, further consideration was given to the environmental impacts, and the situation of HAUV in water and air was piecewise linearized into two deterministic and continuous systems. This provides important reference significance for the continuous stability control research of HAUVs. Nowadays, more and more research institutions, such as Shanghai Jiao Tong University, have combined the advantages of multi-rotor and fixed-wing configurations to conduct specific control research on the multi-modal HAUV. The adaptive control (AC) method can identify systems with uncertainties and update controller parameters in comparison and decision making to ensure that the system keeps up with expected indicators [25]. AC exhibits reliable and effective characteristics in solving control problems of uncertain systems. Backstepping uses Lyapunov stability theory as a theoretical tool, adopts a step-by-step recursive strategy, backtracking from the output of the system to each subsystem to the input of the system, and in this process, constructs appropriate Lyapunov functions for each subsystem and designs virtual control variables to achieve system stabilization. The application characteristics of adaptive and backstepping control methods provide a feasible reference for the control of uncertain nonlinear systems in the next section of this paper. According to (1), there are two options for the selection of state variables: \((\gamma,\alpha,q)\) and \((\gamma,\theta,q)\).