Adaptive on-Line Approximator-Based Finite-Time Trajectory Tracking Control for the Surface Vessel

An adaptive on-line approximator-based finite-time trajectory tracking scheme is proposed to solve the problem of trajectory tracking of underactuated surface vessel affected by dynamic uncertainties and external disturbances. In this scheme, underactuated transformation is performed by using kinematic virtual control law transformation and bounded constraint. By designing adaptive on-line approximator to approach the upper bound of uncertainty and unknown disturbance, the problem of parameter uncertainty and external disturbance are solved. It is proven by Lyapunov theory that the error signals in the system can converge quickly to the stable region in finite time. Finally, by comparing the simulation results, it is verified that the proposed control scheme can make the underactuated ship track the desired trajectory in finite time. Compared to the traditional control method, the convergence rate of the system error is faster, and it exhibits strong robustness in the face of unknown external disturbances. This has a certain reference value for practical engineering applications.


I. INTRODUCTION
The vessel's trajectory tracking can easily deviate from the expected trajectory under the disturbance of wind, waves and currents. Deviations from the desired trajectory can cause significant economic losses during shipping and transportation. Therefore, faster recovery to the desired trajectory under the influence of external environmental disturbances and unknown parameters is a problem worthy of study. In recent years, backstepping, sliding mode control, PID and other methods have been widely used in trajectory tracking control [1], [2], [3]. In [4], in order to realize vessel trajectory tracking, an event-triggered trajectory tracking control method for fully-actuated surface warship based on guidance The associate editor coordinating the review of this manuscript and approving it for publication was Jinquan Xu .
control structure was proposed. In [5], in order to realize the trajectory tracking of an surface vehicle subject to model uncertainties and time-varying disturbances, a robust adaptive controller is proposed.
The above schemes are all used for fully-actuated vessels. Compared to underactuated vessel, there are shortcomings such as poor overall integration, redundant control system, and complex vessel design. In [6] and [7], the authors proposed an adaptive neural tracking control method for the trajectory tracking control of underactuated vessel, which solves the problems of dynamic uncertainty and dynamic disturbance. In [8] and [9], the adaptive neural network technology was combined with sliding mode variable structure control to achieve global robustness. But neural networks have the problem of complex computation. In [10], in order to ensure the accuracy of underactuated vessel trajectory VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ tracking, an event-triggered robust fuzzy control scheme is proposed and the underactuated problem is solved. Reference [11] addresses the practical stabilization problem of arbitrary reference trajectories, a saturated robust adaptive controller was designed based on the inversion technique, the dynamic surface control method and the Lyapunov direct method.
In [12], aiming at the influence of parameter uncertainty and external time-varying disturbance on trajectory tracking of underactuated vessel, an improved sliding mode control algorithm is proposed. But in practical engineering, adaptive stabilization speed is considered, especially in the field of underactuated vessel trajectory tracking control. Finite-time control is an effective control strategy with strong robustness and fast convergence speed. It has the advantages of fast convergence, high precision and strong anti-interference ability. Therefore, we introduce finite time method into trajectory tracking control to improve its control performance. In [13], aiming at the influence of unknown external disturbances and parameter uncertainties in the trajectory tracking process of underactuated vessel, a new finite-time trajectory tracking control scheme is proposed by combining neural network technology, finitetime control theory and backstepping method. In [14], the time-varying obstacle Lyapunov function was introduced to solve the finite-time convergence problem. On this basis, the influence of dynamic uncertainty and uncertain disturbance is considered.
In [15], to ensure the accuracy of trajectory tracking of underactuated vessel, a new robust neural control method is proposed by combining neural network with robust neural damping technique. In [16], In order to solve the disturbance of parameter uncertainty and external unknown disturbance to trajectory tracking, a new sliding mode control based on finite time is proposed, which can make the system error tend to be stable in finite time. An adaptive robust finite-time tracking method was proposed in [17], and a new finite-time observer is introduced, which had a stronger convergence rate. In [18], a practical finite-time trajectory tracking control scheme is proposed for the problem of external disturbance and parameter uncertainty interference in the trajectory tracking process of underactuated surface vessel. A finite-time disturbance observer is designed to deal with the problem of compound disturbance, and the system error can stabilize in a finite time. In [19], aiming at the problem of external disturbance in the process of trajectory tracking of underactuated surface vessel, a finite time controller is designed by using integral sliding surface, which can make the system error converge to the stable region in finite time.
After analyzing the above research results, this paper designs an adaptive on-line approximator-based finite-time trajectory tracking control scheme for underactuated vessel. The main contributions are as follows: 1) Coordinate transformation and norm calculation are used to solve the problem of underactuated surface vessel trajectory tracking, and simplifies the design process.
2) In view of the problems of dynamic uncertainty and uncertain disturbance in the process of vessel navigation, an adaptive on-line approximator-based is proposed to approximate the upper bound of dynamic uncertainty and external disturbance.
3) The controller is designed based on the finite-time stability theory to solve the previous problem of the underactuated vessel's trajectory tracking error convergence time being too slow, so that the error can quickly converge to the stable region in finite time and improve the performance of the controller.
Compared with the traditional trajectory tracking control scheme for underactuated surface vessel, the control scheme designed in this paper avoids the problem of low work efficiency caused by slow trajectory tracking of underactuated svessel, reduces the economic cost, and is more suitable for practical engineering applications.

II. VESSEL MOTION MATHEMATICAL MODEL
The 3-DOF mathematical model of underactuated vessel tracking control can be expressed as equations (1) -(3) [20]: where x, y, ψ are the attitude variables of the vessel's position coordinates and heading angles.
where u, v and r are the speed variables of the vessel; τ u is the vessel surge control force, τ r is the yaw control moment; where f u (u, v, r), f v (u, v, r) and f r (u, v, r) are higherorder hydrodynamic terms. m i , i = u, v, r are the inertial mass parameters of the vessel; X u , Y v , Y r , N v , N r , and X u|u| , Y |v|v , Y |r|v , Y |v|r , Y |r|r , N |r|v , N |v|v , N |v|r , N |r|r are linear and nonlinear hydrodynamic damping parameters, respectively, which may lead to inaccurate modeling due to environmental effects. The relevant definitions and theorems are now introduced for the design of the control law: T and its derivative are bounded and there is an unknown positive number σ satisfying ||τ d ≤ σ .
Assumption 2: The surface vessel's reference trajectory T and its second-order derivative are available.
Definition 1: the nonlinear control system (4) is described as follows: where x ∈ R n is the state variable of the system, 0 is a spherical domain containing the origin, and f (x) is a continuous function. For an arbitrary initial condition x 0 , if there exists a constant I > 0 and a regulating practice function (4) can be said to be semi-globally practical finitetime stable [21]. Lemma 1 [22]: For nonlinear systems (4), suppose there exists a continuous Lyapunov function V : 0 → R which satisfies the following condition: (1) V is a positive definite function. (2) There is an open-loop neighborhood near the origin of real numbers λ 1 > 0, λ 2 > 0, l ∈ (0, 1) such that the following inequality satisfies.
Then the origin is finite time stable and the stable time is Lemma 2 [23]: For χ ∈ R, when a > 0 exists, the following inequality holds: The trajectory tracking control process of an underactuated surface vessel is shown in Fig. 1.

III. CONTROLLER DESIGN
In order to solve the problem of mismatch of lateral drive of underactuated vessel, the method of literature [24] is used for transformation.
First, the definition of underactuated surface vessel tracking error where x r , y r , ψ r are the desired position and heading angle.
Transforming equation (8), we can get Taking the derivation of equation (9) can geṫ For the stability error z e and ψ e , the following virtual control variables are designed: where p 11 , p 12 , p 31 , p 32 are design parameters greater than 0. The desired propulsion speed, yaw rate and heading angle are designed according to equation (11) as follows: Define equation (13) as the error variable Taking the derivative of equation (13) can get where, We can make Based on this design the control law as equation (17)  where,δ u ,δ r disturbances are estimates of upper bounds; p 21 , p 22 , p 41 , p 42 , ς are design parameters greater than 0. Design adaptive laws as shown in equations (18) and (19) where, γ d u , λ d u , γ d r , λ d r , u , r are design parameters greater than 0.

IV. STABILITY ANALYSIS
In order to verify the effectiveness and stability of the control scheme designed in this paper, the Lyapunov function is designed as equation (20).
T z e + ψ e ψ e + u e u e + r e r e +δ u γ d uδ u +δ r γ d rδ r (20) Then, derivation of equation (19) can be obtaineḋ where, According to literature [25], where, ρ 1 = min 2P 11 , 2 P 21 , P λ 2 P γ ,  (19) were designed, and the design parameters p 1i > 0, i = (1, 2), p 3i > 0, i = (1, 2) of the virtual control law equation (11), the designed control law and adaptive law can make the actual trajectory of the underactuated vessel track the desired trajectory, and the system error can converge to zero in a small field in finite time.

VOLUME 10, 2022
According to equation (34) if V > θ/ιρ 1 , we can geṫ According to Lemma 1, it can be known that the system will stabilize to the region V = {V : V ≤ θ/ιρ 1 }in a finite time, and the stabilization time is: is the initial value of V . If V < θ/ιρ 1 , then V will be in the range of V . Therefore, we prove that Theorem 1 holds.

V. SIMULATION
In order to verify the effectiveness of the control law designed in this paper, the Cybership II model of the Norwegian University of Science and Technology is used as the research object for computer simulation. The total length of the vessel model is L = 1.255m, and the mass is m = 23.8kg Other hydrodynamic parameters are detailed in [26]. In this paper, two different equations of circular trajectory and trapezoidal trajectory are used for simulation. Circular trajectory as equation (37).
Fold trajectory as equation (38) 47s < t ≤ 53s y r = 65−t 1.5 53s < t ≤ 62s y r = 10 − 100 − (t − 68) 2 62s < t ≤ 68s y r = 0 68s < t ≤ 112s y r = 10 − 100 − (t − 112) 2 112s < t ≤ 118s y r = t−115 The perturbation is given by equation (39) The proposed control strategy is compared with an adaptive control scheme that does not employ finite time. The virtual control law and the control law of comparison scheme is as equations (40) and (41):  The circular trajectory simulation results are shown in Figs 2-8. Fig. 2. shows the comparison results of the actual trajectory and the expected circular trajectory under the control law (17) and the control law (41). It can be seen that the two schemes can track the expected trajectory, but the control accuracy of the control law (17) is about 80% higher than that of the control law (41), and the tracking control performance is good.
The vessel position tracking is shown in Fig. 3. It can be seen that the two control schemes in two directions simultaneously track the desired trajectory. However, the control law (17) designed in this paper is closer to the desired trajectory,  and the tracking accuracy is 30% higher than that of the control law (41). Therefore, the control law designed in this paper has better control effect.
The vessel propulsion speed u and heading rate r are shown in Figs. 4-5. Obviously, the curve of control law (17) has no obvious buffeting phenomenon compared with the control law (41), curve stabilization time is shorter and closer to the desired vessel propulsion speed, the control effect of the control law (17) is better.
The pose error and velocity error of the vessel's navigation trajectory are shown in Figs. 6-7. In the control scheme of control law (17), x e and y e converge at about 30s, ψ e converges at about 9s, u e and r e converge at about 9s. Obviously, the control scheme using control law 17 can make the error converge in finite time, the convergence speed is faster than that of the control law (41), and the upper and lower bounds of the error are smaller and the curve is more stable. Fig. 8 is the time curve of the vessel's surge control force τ u and the yaw control moment τ r , which shows that both τ u and τ r tend to be bounded with time, and the control input of the control law (17) is slightly higher than that of the control law (41).
The temporal evolution of the adaptive parameters are shown in fig. 9. At the beginning of tracking the circular   desired trajectory, a large adaptive parameter is required, and then the adaptive parameter will become relatively flat when approaching the expected trajectory. To sum up, the control law (17) designed in this paper has better performance than the control law (41).
The simulation results of the trapezoidal trajectory are shown in Figs. 10-17. Fig. 10 shows the comparison between the actual trajectory and the expected trapezoidal trajectory under the control law (17) and control law (41). It can be seen that the control precision of the control law (17) tracking scheme designed in this paper is 70% higher than that of the control law (41), so the control scheme proposed in this paper has better tracking effect.     11 shows the tracking of the vessel in the plane. It can be seen that the two tracking schemes can track the desired trajectory in 13s. However, the control scheme designed in this paper is closer to the desired trajectory, and the tracking effect is better than that of the control law (41).
The vessel's propulsion speed u and heading rate r are shown in Figs. 12-13. It can be seen that both curves track   the desired curve in about 5s, but the control law (17) has no obvious chattering phenomenon compared with the control law (41), and the control effect is better.
Figs. 14-15 show the comparison effect of the trajectory error and velocity error of the system tracking. In the control scheme of control law ( 17 ), x e and y e converge at about 20s, ψ e converges at about 15s, u e converges at about 8s, r e converges at about 15s. It can be seen that the control scheme   in this paper can converge in finite time. Compared with the scheme of control law ( 41 ), the convergence speed is faster, the upper and lower bounds of error are smaller, and the curve is more stable. Fig. 16 is the time curve of the vessel's surge control force τ u and the yaw control moment τ r . The trapezoidal trajectory is composed of multiple trajectories. The overshoot occurs in both control schemes at the switching of the trajectory. However, compared with the control scheme of control law (41), the control input of the control scheme in this paper is significantly reduced, and the oscillation amplitude is smaller. Fig. 17 shows the time evolution curve of the adaptive parameter under the trapezoid trajectory. At the beginning of tracking the trapezoidal desired trajectory, a larger adaptive parameter is required, and then the adaptive parameter becomes relatively gentle when it is close to the desired trajectory. The adaptive parameters become larger during two turns to compensate for the system. In addition, under the same parameters, Obviously, the finite-time trajectory tracking control scheme proposed in this paper has stronger robustness by analyzing the tracking of different reference trajectories.

VI. CONCLUSION
In order to solve the problem of dynamic uncertainty and unknown time-varying disturbance in trajectory tracking control of underactuated surface vessel, an adaptive on-line approximator-based finite-time trajectory tracking control scheme is proposed. First of all, the method of coordinate transformation and norm calculation is used to solve the problem of trajectory tracking of underactuated vessel, thereby simplifying the design process. Secondly, an adaptive law is designed to estimate the upper bound of approximation error and external unknown disturbance, which enhances the robustness of the system, reduces the influence of external disturbance on trajectory tracking control, and makes the system have good anti-interference ability. Thirdly, by introducing the finite-time theory, the tracking error can quickly converge to the field near zero in a finite time compared with the traditional vessel trajectory tracking control. Finally, the effectiveness of the proposed control scheme for the trajectory tracking control of underactuated surface vessel is verified by theoretical analysis and simulation results. We will build rudder and propeller models in subsequent studies to make closer to engineering reality. And in the future, we will continue to pay attention to the application of adaptive finite time control scheme in practical projects such as automatic berthing and vessel collision avoidance.
YAN ZHANG received the Ph.D. degree in communication and information System from Dalian Maritime University. She is currently an Associate Professor with Shandong Jiaotong University. Her current research interests include nonlinear feedback control, ship control and safety, and wireless communication.
WENYI TAN is currently pursuing the Graduate degree in waterway transport with Shandong Jiaotong University. His current research interests include nonlinear feedback control, ship control, and safety.
XIANGFEI MENG is currently pursuing the Graduate degree in water-way transport and safety engineering with Shandong Jiaotong University. His current research interests include nonlinear feedback control, ship control, and safety.
QIANG ZHANG received the M.Sc. degree in navigation science and technology from Dalian Maritime University, Dalian, China, in 2008, where he is currently pursuing the Ph.D. degree in traffic information engineering and control. He is currently a Ship Captain and a Professor with Shandong Jiaotong University. His current research interests include nonlinear feedback control, ship automatic berthing control, and ship robust control.