## 1 Introduction
[14] applied backstepping control to the trajectory tracking of an AUV to a docking station. Simulations of their controller showed that all tracking errors converge to a small neighborhood of the origin. [PERSON] et al. [15] proposed a solution for underactuated AUVs tracking using a combination of a Neural Network (NN) algorithm and a backstepping sliding mode controller. Simulations showed a smooth tracking of the desired three-dimensional trajectory. [PERSON] et al. [13] proposed a backstepping sliding mode controller with fuzzy switching gain. Simulation results showed that the controller designed can guarantee that the UUV accurately track the desired path in the presence of time-varying interference and effectively eliminate the chattering in the traditional Sliding Mode Control (SMC) method. SMC is another typical approach for trajectory tracking of UUVs. It has been extensively applied due to its simple form, robustness, and tolerance to model uncertainty. However, SMC has the problem of chattering. For UUV control, the chattering causes high-frequency changes in the speed of the thrusters, which leads to high energy consumption, increase friction, and thruster damage [16]. [PERSON] et al. [17] proposed an SMC with adaptative gain. In their controller, a continuous term is used instead of a discrete traditional sign function, a large gain for the sliding mode is applied to satisfy the reaching condition, as well as a small gain for the sliding mode to avoid the chattering phenomena. Simulation results showed the controller proposed can decrease the amplitude of chattering in the control signal and increase the convergence rate to the desired trajectory. [PERSON] et al. [18] proposed a fault-tolerant control method integrated with thrust allocation based on the sliding mode theory. Experimental results showed a reduction of up to 94% of the steady error compared with a conventional SMC. [PERSON] et al. [19] proposed a 2 nd order SMC combined with a backpropagation NN control scheme for UUVs to deal with external disturbances and parameter variations. Experimental results showed that the controller was able to reduce the position error to zero in a robust way while the online adjustments of the weights in the NN compensated for the variations in the vehicle hydrodynamics and perturbations. Despite the numerous approaches for trajectory tracking of UUVs encountered, just a few of them have considered a finite-time convergence. Finite-time control is an effective control strategy with strong robustness and fast convergence. It has many advantages such as faster convergence time, higher accuracy, and better anti-disturbance capability [20]. [PERSON] et al. [21] adopted the globally finite-time tracking control strategy, along with a PID-SMC model-based controller, to track a predefined trajectory for a UUV. Simulations showed that the control laws can achieve strongly robust and preferably control performance for the horizontal trajectory tracking control in the presence of parameter perturbation and unknown currents. Their work was extended to three-dimensional trajectory tracking by [PERSON] et al. [22]. Simulation results show that the vehicle can converge to the trajectory and remain in a small neighborhood even in the presence of ocean currents, wherein a small fluctuation in the tracking errors is observed. [PERSON] et al. [23] proposed a local recurrent NN for finite-time trajectory tracking of ROVs with an unknown dynamical model. The authors designed a state observer based on sliding-mode to estimate those variables that cannot be measured by the ROV and achieved finite-time convergence with the adaptative controller proposed. [PERSON] et al. [24] proposed two control schemes based on sliding mode to the tracking control of UUVs in presence of uncertainties and disturbances: an adaptive integral terminal sliding mode control and an adaptive fast integral terminal sliding mode control.