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. Dong 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. Yang 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. Zhang 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. Hernández-Alvarado 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. Yan et al. [
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. Zhou 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. Yan 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]. Ramezani-al 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. Lv 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. García-Valdovinos et al. [
19] proposed a 2nd 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]. Yan 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 Yu 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. Chu 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. Qiao 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. Each controller is composed of a kinematic and a dynamic controller, where the kinematic can achieve local finite-time convergence of the tracking error to zero. Guerrero et al. [
25] proposed an adaptative high order SMC with finite-time convergence for an AUV. The authors validated the proposed controller with real experiments for depth and yaw trajectories, considering parametric uncertainties and external disturbances.
Finite-time tracking controllers have been widely and successfully used to control fully actuated robot arms [
26,
27], but little research has been done for underactuated unmanned underwater vehicles (UUVs), whose particular nonlinear dynamics have other challenges, for instance, the hydrodynamics effects inherent to the vehicle’s geometry and water density. Considering the advantages of a controller with finite-time convergence, and that most of the controllers designed to achieve that for trajectory tracking in UUVs navigation are model-based 1st order SMC, this paper presents the formulation and validation of a model-free 2nd order SMC based on a Time Base Generator (TBG) that guarantees finite-time three-dimensional trajectory tracking of underactuated UUVs subject to ocean currents. The resulting control law does neither depend on the hydrodynamics nor other parameters of the vehicle; therefore, it is considered as a model-free controller. In addition, formulation of the control law of this 2nd order SMC eliminates the chattering effect which is common in conventional 1st order SMC [
28,
29]. To achieve finite-time convergence of the tracking errors of three-dimensional trajectories, the 2nd order SMC is parametrized by a TBG which allows the user to set an arbitrary base-time in which the vehicle will meet the desired trajectory. The performance of the proposed controller is evaluated by numerical simulations including the effects of external disturbances. All this is structured in this paper as:
Section 2 describes the BlueROV2 UUV and its dynamic model, as well as the proposed control scheme including the TBG.
Section 3 describes the method and parameters used to validate the proposed controller.
Section 4 shows the results of the testing and validation, and
Section 5 contains the concluding remarks.
4. Results and Discussion
A simulation was performed so that a computational model of the BlueROV2 follow the spiral trajectory described before, the parameters and gains for the proposed model-free 2nd order SMC with finite-time convergence are shown in
Table 4. The initial position for the vehicle was set to
,
, and
so it started slightly away from the desired trajectory. The effects of ocean currents were neglected for this simulation.
Results for trajectory tracking and errors of the proposed controller are presented in
Figure 5. The controlled trajectory smoothly converged with the desired trajectory at the time-base proposed and without overshoots, after that, the vehicle followed the desired trajectory without deviations. The errors decreased over time to become zero at the designated
except for the pitch, which, as established in
Section 2.1, was not actuated and the resulting increase in its error—smaller than 0.5°—was caused by the vehicle’s dynamics while moving forward.
A three-dimensional trajectory of the BlueROV2 simulation along with the control trajectory can be seen in
Figure 6.
The control coefficients for each of the BlueROV2 thrusters are shown in
Figure 7. These control coefficients were in a range of
, where
represents that the controller is demanding 100% of the thruster force. The maximum force was demanded at the start of the simulation and, after reaching the time-base, the control signals remained constant and no chattering was observed.
Since
can be tuned arbitrarily for the user, further simulations were performed for the proposed controller under the same gains and parameters but for different time-bases. Simulation results for trajectory tracking and errors are shown in
Figure 8 and
Figure 9 for a time-base of 3 and 7 s, respectively. In both cases, the vehicle met the desired trajectory at the designated time-base
.
The controllers presented and described in
Section 3.3 were simulated for comparing their error convergence to zero and the control coefficient computed for the BlueROV2 thrusters. As can be seen in
Figure 10, the surge and sway error of the proposed controller converges to zero at the desired time. Meanwhile, the rest of the controllers converged at different times and, for the PID and FBL, there was not a stationary error but an oscillation in its value was observed through the simulated time. For the heave error, despite the oscillations resulting in the PID and the FBL, all the controllers converged to zero. Finally, for the yaw orientation, the FBL exhibited some troubles converging to zero during the first 20 s to finally converge to it exponentially. In general, the performance of the proposed model-free 2nd order SMC with finite-time convergence surpassed the rest of the controllers, even when FBL and Lyapunov-based were model-based.
To evaluate the performance of the proposed controller in terms of energy consumption from the thrusters, the Root Mean Square (RMS) value of the control coefficients computed for the thrusters was obtained and results are shown in
Table 5. The mean column of the table represents the mean value of the coefficients considering all six thrusters. As it can be seen in the table, the average thruster force demanded by the proposed model-free 2nd order SMC was in the range of 0.1004 and 0.1296—1.0 being the maximum—this was up to 50% less than the forces demanded by a traditional PID or FBL controller even when it converged faster. The Lyapunov-based controller converged to the trajectory approximately at 7 s, and its mean value for control coefficient of the thrusters was 23% bigger than the 2nd Order SMC with
.
To test the controller response to external disturbances, the effects of ocean currents were included as described in
Section 3.2. Results for trajectory tracking and errors of a simulation with the proposed model-free 2nd order SMC with
are shown in
Figure 11. Despite the high ocean currents introduced, the BlueROV2 still smoothly converged to the desired trajectory at the designated
, there were not overshoots at all, and the vehicle followed the designed trajectory with an error of zero after the time-base was reached.
A three-dimensional trajectory of the BlueROV2 simulation along with the control trajectory when subject to ocean currents can be seen in
Figure 12.
Control signals for each of the BlueROV2 thrusters are shown in
Figure 13. Contrary to the simulation results shown in
Figure 7, the control signal did not remain constant after the time-base was reached. Thruster control signals changed over time to overcome the perturbations introduced by the ocean currents, which allowed the BlueROV2 to follow the desired trajectory as designed.
Ocean currents were also included in simulations for the rest of the controllers considered in the past section. Results for tracking errors are presented in
Figure 14. None of the classic controllers tested were able to overcome the perturbations introduced, while the proposed model-free 2nd order SMC was able to compensate for them and keeping the finite-time convergence as designated by the user.
To evaluate the performance of the proposed controller in terms of energy consumption from the thrusters, the Root Mean Square (RMS) value of the control coefficients computed for the thrusters was obtained when the vehicle was subject to high ocean currents. Results are shown in
Table 6.
The average thruster force demanded by the model-free 2nd order SMC with finite-time convergence was in a range of 0.1903 and 0.2105, which resulted in up to 30% less than the average force demanded by the PID controller and up to 64% less than the demanded by the FBL controller, even when they were not able to meet the trajectory and maintain the vehicle there.