## 1 Introduction
Usually, the importance of the AUV model and the low performance of the single-chip microcomputer equipped with ordinary AUVs are ignored. To deal with the complex calculation process of traditional neural networks, the real-time requirements of AUVs for control need to be guaranteed. The RBF neural network is used to compensate for plant nonlinearity adaptively, and it was certified that combining adaptive and robust control allows for acceptable approximation errors due to low-order networks [26]. RBF has a relatively simple recursive process, which can ensure that it can also be deployed on microcontrollers with low computing power. Model-based advanced controller design requires accurate models. However, the actual model may need to be more particular about adapting to different environments. Feedback controllers designed for such situations are not particularly effective, which makes adaptive algorithms more and more popular. The model reference adaptive control (MRAC) framework can artificially construct a reference model to represent the dynamic performance of the desired closed-loop system. Moreover, it can look for a dynamically adjusted feedback control law so that the closed-loop control performance of the system can be consistent with the performance of the reference model [27; 28]. It should be pointed out that this paper uses the RBF neural network as the feedback controller of the MRAC, and the actual error between the output and the reference output serves as the input to the RBF. The backpropagation algorithm is used to continuously adjust the weight coefficients of the hidden layer and the output layer to drive the dynamic trajectory of the AUV to be consistent with the reference trajectory. This design also guarantees the dynamic performance of the proposed controller since the weight coefficients are constantly adjusted. This paper aims to use a model reference adaptive controller on an AUV system and consider RBF neural networks as the adaptive mechanism. The main contributions of this paper include the following:  
* A new adaptive algorithm is proposed to solve the AUV trajectory tracking control problem. Using the proposed RBF adaptive algorithm, the performance and robustness of the tracking control can be significantly improved. Furthermore, its subproblems can be addressed simultaneously in the controller design. * A horizontal plane model for the AUV is proposed. And, for the underactuated underwater vehicle, which only provides surge force and sway moment, the boundedness of its velocity in the horizontal direction is proved. * For MRAC tracking control, it is neither feasible nor stable to only consider the control performance due to the influence of the optimal solution quality. Therefore, the proposed MRAC creates a trade-off between computational complexity and control performance, introducing an efficient mechanism for allocating computational resources to AUVs. This article is organized as follows. The second section introduces the kinematics and dynamics modeling of the robot, the third section tackles the controller design, the fourth section is on the underwater robot model and physical analysis, and the fifth section contains the summary and outlook.