## 1 Introduction  
Unmanned Underwater Vehicles (UUVs) are used in multiple tasks such as ship's hull inspection [1], ecological surveys [2, 3], underwater inspection and mapping [4], among others. The use of UUVs avoid risks for human divers and reduce operational costs. In addition, they can perform some tasks that cannot be done by humans in an underwater environment. The motion of these vehicles can be remotely operated (ROVs) or autonomous (AUVs). For ROVs, there is a human pilot that commands the vehicle's movements and actions, usually through a tether. Human presence makes complex exploration and intervention tasks possible since humans can react to changes in the mission plan caused by the unpredictable nature of the underwater environments. However, certain operations such as high precision navigation require some form of autonomy from the vehicle. That is a complex task to achieve and one of the main problems for AUVs [5] since their six Degree of Freedom (DOF) dynamics is highly non-linear and, therefore, difficult to control [6]. Motion control of a UUV has different goals such as waypoint tracking, path following, and trajectory tracking [7]. For waypoint tracking, the controller should guide the vehicle from a starting position to the desired destination throughout a set of pre-defined waypoints. For path following, the aim is that the vehicle accurately follows the desired path in terms of a geometric description while the trajectory tracking problem requires the vehicle to converge to a desired time-parametrized trajectory. For trajectory tracking, convergence can be either asymptotic, exponential, or in finite-time. Several control strategies have been applied in the past to achieve high precision trajectory tracking. The Proportional Integral Derivative (PID) control is one of the most employed control strategies for UUVs due to their simple structure. They work well for systems where it is possible to control all outputs through the inputs--meaning they are not underactuated--and are widely used for heading, depth, and surge control [8]. However, one limitation for PID controllers is that they are tuned to deal with specific conditions and, if those conditions change, the performance will be affected. In addition, PID controllers do not consider nonlinearities, which will eventually deteriorate the controller performance. Some other strategies are used with the PID controller to compensate some of the uncertainties faced in the underwater environment. [PERSON] et al. [9] designed a fuzzy rules-based strategy to tune a PID depth controller. Experiments were carried out to test the controller, showing a depth control accuracy of up to 3 cm. [PERSON] et al. [10] adopted a cascade PID structure to deal with horizontal displacements in the heading control. Experimental results showed better performance when compared with a traditional PID controller since oscillations were reduced and even eliminated when controlling the heading of the ROV. [PERSON] et al. [11] used Particle Swarm Optimization (PSO) to optimize the parameters of a PID controller for horizontal displacements. Simulation results showed better performance than a PID controller, but weaker compared with a fuzzy controller. [PERSON] et al. [12] proposed a Neural Network (NN) based self-tuning PID controller where the NN automatically estimates a suitable set of gains for the PID controller to achieve system stability. The controller was validated with numerical simulations and experimentation, showing better performance when subject to unknown disturbances and over 3% of energy-saving against a traditional PID controller. Backstepping Control (BC) is a model-based nonlinear controller that is commonly used for trajectory tracking of UUVs. It is based on the Lyapunov theory [13] and works by sectioning the dynamic system of the vehicle into subsystems, so the controller begins with a known stable system and then iterates to obtain proper controllers for the rest of the subsystems. [PERSON] et al.