## 1 Introduction  
With the continuous progress and development of human society, the development and utilization of the ocean are critical. As the most resource-rich place in the world, the sea has always attracted people's exploration. Autonomous underwater vehicles (AUVs) have been the focus of many oceanographic studies in recent decades due to their wide-ranging applications in ocean exploration, ocean mapping, underwater pipeline inspection, and scientific and military missions [1; 2; 3]. The development of fluid mechanics also promotes the progress of AUV underwater dynamics analysis [4; 5]. Due to the model characteristics of AUV (nonlinearity, coupling, uncertainty), there are many external disturbances underwater in practical applications. Most AUVs are underdriven to reduce weight, which degrades the performance of AUVs [6]. The design of the control system becomes one of the most considerable difficulties in its development. Therefore, it is necessary to study the robust control of underactuated AUVs [7; 8; 9]. The autonomous underwater vehicle (AUV) is the latest achievement in the field of marine robot research. Because it does not require human intervention during its work, it dramatically reduces the risk and operating costs of various underwater projects, so it hasreceived attention from industry and academia. There is increasing attention in [10], which proves that well-designed control systems are necessary. Their limitations are known for existing control techniques such as proportional integral derivative (PID), adaptive control, and model predictive control. In early industrial process control, the system's parameters often changed because it was difficult to establish an accurate mathematical model of the controlled object. PID has many uses due to its simple control technology structure and convenient parameter adjustment, but it is only used in simple control systems [11; 12; 13]. The research object of adaptive control is a system whose parameters are not entirely determined, and the environment contains unknown and random factors. However, adaptive control is more complicated and costly than conventional control [14; 15; 16; 17]. Model predictive control (MPC) is an optimization-based time domain control method [18]. Compared with most traditional control methods, the distinguishing feature of MPC is its ability to deal systematically with system constraints in controller design [19]. Therefore, it is good at dealing with multi-input and multi-output systems. However, to solve the optimization problem, it needs to iterate at each time step, and the solution to the optimization problem is often time-consuming. The controller's action may have more vital requirements for real-time performance. This will produce a contradiction, a significant shortcoming of the current predictive control model [20; 21; 22; 23]. Some researchers did not consider the influence of propeller dynamics when designing AUV control schemes. The operating environment of an AUV is complex, and there are many degrees of freedom to be controlled. Existing control algorithms cannot adapt to environmental changes, weakening control effects. In recent years, the neural network represented by the BP algorithm has gradually become popular with the improvement of computing power. This further drives the control algorithm to become more intelligent. The most attractive property of neural networks is adaptability. Adaptability means the neural network can be appropriately organized and adjusted with the controlled object and environmental conditions through learning. [PERSON], [PERSON] used an AUV to study the feasibility of the neural network control of underwater vehicles in the presence of unpredictable dynamic changes [24]. Some researchers employ neural networks to approximate the complex AUV hydrodynamics and the difference in required tracking speeds [25]. A three-layer neural network can fit any function, but most neural networks require high computing power and cannot meet the real-time requirements of the controller.