## 5 Conclusions  
Finite-time trajectory tracking of underactuated UUVs is a considerable challenge. Their nonlinear dynamics and the challenging conditions of the underwater environment make it more complex for the controller to achieve a stable and robust performance as they do in other systems and environments. In this work, for the first time, a model-free 2 nd order SMC and a TBG were synthesized and implemented to achieve finite-time convergence of an underactuated nonlinear UUV to a desired trajectory. Such a controller has some advantages compared with other finite-time model-based 1 st order SMC reported in literature since it neither requires the dynamic model structure nor the knowledge of any hydrodynamical parameter from the vehicle. In addition, this controller does not present the chattering effect common in other conventional SMC. The TBG was included in the controller by the parametrization of the \(\alpha\) gain at the sliding surface, making it time-varying instead of constant. The hydrodynamical model of the BlueROV2 UUV was implemented in MATLAB/Simulink(r). There, numerical simulations were performed to validate the performance of the proposed controller. Results have shown that the controller was able to manage the BlueROV2 to smoothly converge to the desired trajectory at a specific time without overshoots, and then follow it with a tracking error of zero. Other classic controllers such as PID and FBL could not achieve a similar performance. Since the time-base can be tuned arbitrarily for the user, simulations were repeated with different time-bases and the results remained consistent and satisfactory. The performance of the solution proposed was also evaluated in terms of the energy consumption regarding the thruster forces demanded by the different controllers. It was shown that the proposed model-free 2 nd order SMC demands up to 50% less power from the thrusters than other of the controllers simulated and it does not present the chattering effect common in some SMC controllers. Finally, the vehicle was subject to high ocean currents. Simulation results showed that the controller stills able to meet the desired trajectory at the designated time, smoothly and without overshoots, and then maintain the vehicle in the desired trajectory when other classic controllers could not. In future work, authors expect to verify the finite-time convergence of the model-free 2 nd order SMC by performing experimentation under controlled conditions with the BlueROV2 vehicle whose hydrodynamical model was considered for the simulations presented in this work. Further research will also include the obstacle avoidance problem during the trajectory tracking.  
Conceptualization, [PERSON], [PERSON], and [PERSON]; methodology, [PERSON], and [PERSON]; investigation, [PERSON], N.A.N.[PERSON], and [PERSON]; software, [PERSON], and N.A.N.-N.; writing--original draft preparation, [PERSON]; visualization, [PERSON], and N.A.N.-N.; writing--review and editing, [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON]; supervision, [PERSON], [PERSON], and [PERSON]; All authors have read and agreed to the published version of the manuscript.  
The authors would like to acknowledge the financial support of Tecnologico de Monterrey, in the production of this work.  
Not applicable.  
Data Availability StatementData sharing is not applicable to this article.  
The authors would like to acknowledge support from CONACyT for PhD studies of first author and MSc studies of second author.  
The authors declare no conflict of interest.