## 1 Introduction
**Contributions**: The proposed work presents a way to improve the autonomous maneuvering capability of a four-degrees-of-freedom (4 DOF) AUV to perform trajectory tracking tasks in an underwater environment. The main contributions rely in the following three aspects:  
_Dynamic Model._ The present study considers that the vehicle has a nonlinear mathematical description, which is subject to modeling and parametric uncertainties as well as external disturbances. The dynamic model is developed specifically for the real AUV from BlueRobotics Inc called BlueROV2 [41] considering real operation conditions. The modelling process results in a decoupled translational dynamics (x,y,z) described by three decoupled input-affine second-order nonlinear subsystems. This is done by first solving for the heading angle dynamics stabilization. We integrated the unknown damping phenomena and external disturbances in a single expression, defined as lumped disturbances. In addition, we considered that the added mass and damping coefficients can be approximated by constant values. Additionally, we supposed that these constant values are unknown, taking into account the difficulty of finding them for complex-shape vehicles. Thus, the input-affine AUV model considered in this work resulted in unknown input gains dependent on constant added mass coefficients. Then, this fact does not represent a limitation for the neural network adaptive controller proposed in the present work, but rather, it contemplates a reasonable consideration of real conditions when these parameter uncertainties (added mass-dependent gains) are present in the system structure. _Control Strategy._ We proposed a novel control design for each dynamic based on a modification of the DNN-based parallel identifier structure in [42; 43] by replacing the DNN that is used to identify the function affecting the control input by a parametric estimate of the unknown constant input gain provided by an online parameter identification law. The proposed identifier structure merges a nonparametric identification of the lumped disturbances by using a DNN and a parametric identification of the constant input gain provided by a parameter adaptive law. It is worth noting that this type of combination,namely an identifier based on DNN plus parametric estimator, is not reported in the literature. Moreover, in contrast with the research related to such a parallel identifier, in our case, we add to this identification scheme a correction term based on the error between the AUV dynamics and the identifier to improve its approximation quality. The proposed indirect adaptive control structure makes use of such an identification error to update weights adapting itself to the dynamic changes of the vehicle and adjusts the parameter estimator simultaneously. Online neural network approximation of the unknown dynamic part (lumped disturbances) and estimation of the unknown gain of the control input are used to cancel the unknown and nonlinear effect from each vehicle dynamic via the designed control law. Then, for the resulting quasi-linear AUV's second-order subsystems, a feedback control law was calculated in order to force each subsystem to track the desired states provided by a reference system. This type of identification-control scheme is name the Dynamic Neural Control System (DNCS). _Stability Analysis._ The ultimate boundedness of the identification and tracking errors considering the proposed DNCS is guaranteed via the Lyapunov stability framework. Additionally, the stability proof allows us to derive the learning laws for the DNNs and the adaption law for the input (constant) gain parameter. The rest of the paper is structured as follows. Section 2 introduces some preliminaries and the mathematical model of the AUV for control purposes. In Section 3, the identification scheme based on nonparametric identification of the lumped disturbances and estimation of the parametric uncertainty of the input gains affecting the vehicle dynamics are addressed.