Trajectory tracking control for uncertain underactuated surface vessels with guaranteed prescribed performance under stochastic disturbances

The environmental disturbances acting on marine surface vessels are naturally stochastic. The control performance may be seriously deteriorated if the stochastic components are ignored in control design. In this paper, we aim to address the trajectory tracking control problem for uncertain underactuated surface vessels exposed to stochastic disturbances under prescribed performance constraints. The environmental disturbances are divided into a deterministic part and a stochastic component modeled as standard Wiener process. Subsequently, a novel tan-type quartic barrier Lyapunov function (BLF) is incorporated into the control design to handle the Hessian terms induced by stochastic di ﬀ erentiation rule and simultaneously guarantee prescribed performance of tracking errors in probability. Meanwhile, a transverse function control method is developed to introduce an additional control input so as to deal with the di ﬃ culties caused by underactuation and nonzero o ﬀ -diagonal terms. Based on transverse function control, backstepping procedure, neural networks (NNs), and quartic barrier Lyapunov function, an adaptive NN-based trajectory tracking control protocol is ﬁnally developed, whose advantage is that prescribed performance of tracking errors is guaranteed in probability despite the presence of stochastic environmental disturbances. Simulations are carried out to validate the e ﬀ ectiveness of the proposed control protocol.


Introduction
In the past decades, trajectory tracking control of marine surface vessel has received much attention from researchers in control and marine engineering communities due to its important applications, such as marine transportation, environmental surveying, and ocean sampling [1,2,3,4].In general, most of marine surface vessels are underactuated, which means that they have fewer independent control inputs than the degrees of freedom to be controlled [5,6,7].It is challenging to design control laws for underactuated surface vessels to achieve trajectory tracking with prescribed performance and simultaneously stabilize the sway dynamics in the absence of a direct actuation in sway direction [8].The challenge would be even harder when the environmental disturbances induced by wind, waves, and ocean currents are taken into consideration [9].
To address the underactuation issue, many elegant control design techniques have been proposed for underactuated surface vessels, e.g., [10,11,12,13,14,15].A synchronization motion tracking control was developed in [10], where a dynamical variable is introduced to overcome the underactuation in the sway dynamics.Based on the constructed time-varying dynamic os-cillator, a tracking controller for an underactuated surface vessel was designed in [11].Inspired by this dynamic-oscillatorbased method, a transverse function control approach was firstly proposed in [16] for controllable driftless nonlinear systems, where this control design approach was further applied to underactuated surface vehicle [12].Motivated by the transverse function control approach [16], the authors in [13] constructed an elegant coordinate transformation in kinematic control design to introduce an additional control input, such that the additional control input instead of sway velocity was applied to force the sway displacement to track the corresponding desired displacement.To provide prescribed specifications on the transient and steady-state performances for tracking errors, two transverse functions were introduced in the kinetic control design [14], where the advantages include: 1) the tracking errors would not be directly affected by the transverse functions, and 2) no separate stability proof of the sway dynamics would be required.
It should be pointed out that the environmental disturbances in the above-mentioned works are neglected or assumed to be deterministic, i.e., either constant or time-varying but continuously differentiable.However, these assumptions might be unrealistic since the environmental disturbances caused by waves, wind, and ocean currents generally contain stochastic components [17,18].Consequently, the deterministic control design methods proposed in [11,12,13] cannot directly apply for the vessel exposed to stochastic environmental disturbances, since direct application of these deterministic methods to stochastic vessel system would seriously deteriorate the system's performance and even cause instability of the closed-loop system in stochastic sense [18].Thus, the environmental loads were treated as stochastic in [18], and then the Itô's differentiation rule was introduced to facilitate the control design and stability analysis, where the quartic Lyapunov functions were employed to overcome the difficulties caused by Hessian terms resulting from Itô's formula.However, the vehicle system was assumed to be fully actuated in [18].The authors in [18] further developed path-tracking control strategy for underactuated ship subject to stochastic disturbances in [19], where the path parameter was used as an additional control input, and together with surge velocity and yaw angle was utilized to stabilize the path-tracking error dynamic system.
Despite the great progress made in tracking control of ocean vehicle under stochastic disturbances, an important concern related to output constraints has not been resolved in the literature, e.g., [18,19].In many practical applications, the actual trajectory of the vessel should be restrained to certain range around the desired trajectory.For example, the vehicle is required to keep a feasible distance from the desired trajectory when passing through a narrow waterway [20], such that the collision with obstacle could be avoided.Although the constraint imposed on the system output could be transformed into the tracking error constraints, guaranteeing the satisfaction of prescribed performance for tracking errors under stochastic disturbances is challenging.Recently, a barrier Lyapunov function (BLF) was originally constructed in [21] and was employed to ensure the constraint requirement on the tracking errors in the sense of probability for a class of nonlinear systems with stochastic components.However, it is not directly applicable to an underactuated system including the underactuated vessel.
Based on the above discussion, in this work, we aim to address the trajectory tracking control problem for uncertain underactuated surface vessels subject to stochastic disturbances under tracking error constraints.To overcome the difficulties caused by underactuation and nonzero off-diagonal terms, we develop the transverse function control approach to introduce an additional control input, such that the sway dynamics could be stabilized by the additional control input.To tackle the Hessian terms induced by stochastic differentiation rule, the quartic BLF is incorporated into the control design, which could simultaneously guarantee the prescribed performance constraint in probability.To improve the robustness against the modeling uncertainty, we construct the radial basis function (RBF) neural network (NN) approximators to approximate the uncertain dynamics, e.g., hydrodynamic damping terms.Compared with the existing tracking control approaches of underactuated surface vessels [11,12,13], we develop a unified tracking control approach in the sense that the proposed control method can handle both deterministic and stochastic environmental disturbances.Different from the works on tracking control for marine vehicle under stochastic environmental loads [18,19], we propose a performance-guaranteed control strategy in the sense that the prescribed performance of the tracking errors is ensured in probability during the whole motion.The major contribution of this work is summarized as follows.
(1) In the kinetic design, an additional control input is introduced by developing novel transverse functions, which ensures that the stability of the sway dynamics does not need to be proved separately.
(2) A novel tan-type quartic barrier Lyapunov function (BLF) is presented for underactuated surface vessels with stochastic disturbances.
(3) By incorporating the tan-type quartic BLF into the control design, convergence of tracking errors into a small neighborhood of zero is guaranteed, and the prescribed transient performance of the tracking errors are not violated during operation in probability.

Underactuated vessel model
Assume that the heave, pitch and roll of vessel system are neglected.Consider an underactuated surface vessel moving in a horizontal plane, whose kinematics is given by ẋ where (x, y) and ψ denote the vessel position and orientation in the earth-fixed frame, respectively; u, υ, and r represent the surge velocity, sway velocity, and yaw angular velocity in the body-fixed frame, respectively.Denote η = [x, y, ψ] T and ν = [u, υ, r].Following the results in [22], the kinetics of an underactuated surface vessel is written as with denote the hydrodynamic damping terms, including potential damping, skin friction, wave drift damping, and the damping due to vortex shedding; X (•) , Y (•) , and N (•) denote the linear and quadratic hydrodynamic damping coefficients; τ u and τ r represent the surge force and yaw moment, respectively; τ wu , τ wv , and τ wr denote the disturbances caused by waves, wind, and ocean currents on the vessel along the surge, sway, and yaw axes, respectively, and they consist of constant components and time-varying parts, which are stochastic [19].Since the two independent control inputs (surge force and yaw moment) are less than the three freedom degrees of vehicle system (2) to be controlled, and thus system (2) is underactuated.

Stochastic equations of motion
For vessel system (2), the marine environment disturbances τ wu , τ wυ , and τ wr can be rewritten as τ wu = τ wu + τwu , τ wυ = τ wυ + τwυ , τ wr = τ wr + τwr (3) where • and • represent the constant and stochastic terms of •, respectively.For the unknown constant terms, we denote Meanwhile, we employ the Wiener process to model the stochastic terms where Λ i (t) and ω i denote the time-varying covariance and Wiener process with i = 1, 2, 3, respectively.Substituting equations (3)-(5) into system (2) and considering system (1), the stochastic differential equations of the underactuated vessel system can be obtained by Assumption 1 ( [19]).The stochastic disturbances τ wu , τ wυ , and τ wr in (2)  Remark 1.In this paper, the environmental disturbances are assumed to be divided into a deterministic part and a stochastic component modeled as a standard Wiener process [19].The stochastic component may be more complicated in practical maritime environments, which may contain the random jump term.Thus, the Lévy process consisting of drift (deterministic), diffusion (standard Brownian motion), and random jump (Poisson) terms may be more suitable to model the stochastic sea load [23,24].Control design for the stochastic systems driven by Lévy processes is much more challenging.On the other hand, the growth conditions on the unknown drift ϑ i and the time-varying covariance Λ i could be reduced to polynomials of arbitrary order of system state, as discussed in [25], where the Young's inequality could be utilized to handle these unknown terms.

Stochastic stability
Consider the following nonlinear stochastic system where x ∈ R n and ω denote the system state and r-dimensional independent Wiener process defined on a complete probability space (Ω, G, P) [19], respectively; Λ(t) : R + → R r×r is Borel measurable and bounded, and nonnegative definite for each t ∈ R + ; and f : R n × R + → R n and G : R n × R + → R n×r are locally Lipschitz continuous in x, uniformly in t ∈ R + , and locally bounded.Let y(x) : R n → R be a C 2 function with respect to x, the stochastic differential equation (Itô's formula) of y(x(t)) along system (7) is where Tr(X) denotes the trace of X.
For a given function V(x) ∈ C 2 , the infinitesimal generator LV(x) along the stochastic system ( 7) is defined as ) .

Definition 2 ([19]
).The equilibrium x = 0 of the system ( 7) is 1) globally stable in probability if for all ε > 0 there exists a class K function γ ′ such that for all t ≥ t 0 ≥ 0 and x(t 0 ) ∈ R n \{0}; 2) globally asymptotically stable in probability if for all ε > 0 there exists a class KL function γ ′′ (•, •) such that Lemma 1 ( [26]).If there exist a C 2 function V(x), and class K ∞ functions γ 1 and γ 2 , and a class K function γ 3 such that where ε 0 is a positive constant.Then there exists a unique strong solution of (7) for each x(t 0 ) ∈ R n .

Lemma 2 ([19]
).If there exist a C 2 function V(x), and class K ∞ functions γ 1 and γ 2 such that where W : R n → R is continuous and nonnegative, then there exists a unique strong solution of (7), and the equilibrium x ≡ 0 is globally stable in probability and P(lim

Prescribed performance constraint
Let η d = [x d , y d , ψ d ] T be the desired trajectory, the trajectory tracking error is defined as To improve the transient and steady-state performances, the tracking errors are required to satisfy the following performance constraints in probability where k b1i (t) are taken as the exponentially decaying functions of time with k b1i,0 > k b1i,∞ > 0 and κ bi > 0 being designer-specified parameters, and their maximum value occurs at the initial instant t = 0.
Remark 2. According to the properties of performance constraint (11), the prescribed transient and steady-state performances of the tracking errors are described as follows: 1) the convergence rates of the tracking errors are faster than the exponential decay rates −κ bi ; 2) the lower and the upper bounds of overshoots are −k b1i,0 and k b1i,0 , respectively; and 3) the steady-state errors evolve within the predefined region . Thus, the transient and steady-state performance on the tracking errors can be improved by adjusting the preselected parameters k b1i,0 , k b1i,∞ , and κ bi > 0.
Assumption 3. The reference trajectory η d and its derivatives ηd and ηd are smooth and bounded.
Control Objective: Under Assumptions 1-3, our objective is to design the control laws τ u and τ r for underactuated vessel with system dynamics (1) and (2), such that (i) the tracking errors converge to small neighborhoods around zero in probability, and (ii) the prescribed performance constraint (11) with ( 12) is guaranteed during the whole operation.

Control design for known vessel model
In this section, we assume that the exact model of underactuated surface vessels is available a priori, i.e., the hydrodynamic damping dynamics f u , f υ , and f r are known.For system (6), we employ the backstepping procedure with two steps to design the model-based tracking controllers τ u and τ r for achieving the control objective.In the first step, the (x, y)-subsystem and ψ-subsystem are stabilized by viewing (u, υ), and r as virtual control inputs, respectively.In the second step, the actual control input τ u is designed to stabilize the u-subsystem, and the actual control input τ r with additional control input β are designed to stabilize the (υ, r)-subsystem.
Step 1: The derivative of (10 Following the transverse function control approach [14], we define the following error coordinate transformations ) where α i (i = 1, 2, 3) denote the virtual control inputs, h 1 (β) and h 2 (β) are referred to as transverse functions that are bounded and differentiable with respect to β [14].
Remark 3. It can be seen in ( 6) that there are only two independent control inputs, i.e., surge force τ u and yaw moment τ r .The yaw moment τ r acts directly on the sway dynamics since the hull is not fore/aft symmetry.Under this configuration, it is difficult to simultaneously stabilize the sway dynamics and yaw dynamics.To overcome this difficulty, we employ the transverse function control method [14] to introduce an additional control input β in (26), such that the yaw moment τ r and the additional control β can be utilized to stabilize the sway dynamics and yaw dynamics.
To guarantee the nonviolation of the performance constraint (11), we consider the following tan-type barrier Lyapunov function (BLF) candidate whose derivative along system ( 13) is given by where ν z1i = p i z 3 1i with p i = 1/ cos 2 (
Remark 4. Compared with the quadratic Lyapunov function, the tan-type BLF V 1 in (15) has the property of finite escape, i.e., V 1 approaches infinity whenever the tracking errors z 1i , i = 1, 2, 3, approach the prescribed bound of the set When the performance function k b1i tends to infinity, using L' Hospital rules, the following condition holds: which implies that the tan-type BLF can be replaced with the quartic Lyapunov function 1 4 z 4 1i when there is no constraint requirement on the tracking errors z 1i .Thus, the developed tantype BLF analysis for tracking control with constraint on the system outputs is a general approach that is applied for systems without prescribed performance constraint.
Step 2: Using (6), the stochastic differential equations of ( 14) are described by where , and the derivatives of the virtual controls ( 20)-( 22) are available for controller implementation.
3) We have shown in 2) that P(lim given in ( 15) is a part of W( Xe ).Thus, there exists a nonnegative constant c * such that P(lim [z 11 , z 12 , z 13 ] T , which shows convergence of V 1 (z 1 ) to a bounded value.Hence, the prescribed performance constraint (11) with ( 12) is not violated.

Control design under modeling uncertainties
Assumption 4. The hydrodynamic terms f u , f υ , and f r given in (2) are unknown.
Under Assumption 4, the control laws τ * u in (33) and τ * r in (34), and the additional control β * in (35) can not be implemented because they include the unavailable hydrodynamic terms f u , f υ , and f r .Hence, this section presents an adaptive NN-based control strategy for uncertain underactuated vessels subject to stochastic disturbances under performance constraint (11).
The uncertain dynamics f u , f υ , and f r can be approximated by RBF NNs [27,28,29] as T ∈ Ω Z 2 are the NN input vectors with Ω Z 1 and Ω Z 2 being compact sets; W * i (i = 1, 2, 3) are the ideal constant weight vectors; ϵ i (•) is the approximation error satisfying |ϵ i (•)| < ϵ * i with ϵ * i > 0 being a constant; and S i (•) denotes the activation function vector satisfying ∥S i (•)∥ ≤ s * i with s * i > 0 being a constant.According to (43), systems ( 25) and ( 26) can be rewritten as Based on the NN ŴT i S i (i = 1, 2, 3), the actual control laws τ u , τ r , and additional control β are redesigned as follows where Γ i > 0 denotes the adaptation gain matrix and σ i > 0 represents the σ-modification parameter.Using [ The infinitesimal generator LV 2 along (23), and ( 44)-( 49) is derived as Based on the above inequalities, (51) could be rewritten as where γ = min{k 11 , k 12 , k 13 , 4(k 41 − 2), 4(λ min (K 5 ) − 4), k θ1 ι 1 , ) .The following theorem describes the NN-based tracking results for underactuated surface vessels subject to stochastic disturbances, modeling uncertainties, and performance constraint.The schematic diagram of the proposed control strategy is given in Fig. 1. x(m) 1) the closed-loop system consisting of ( 24), ( 53)-( 54), and the update laws (36) and ( 49) has a unique strong solution; 2) the tracking errors converge to a ball centered at the origin; and 3) the prescribed performance constraint (11) with ( 12) is not violated.
Proof: The proof is similar to that of Theorem 1 and is therefore omitted.

Simulation studies
To demonstrate the effectiveness and superiority of the proposed control protocol, we conduct a numerical simulation on the Cybership-II model in [22], a benchmark surface vessel with length L = 1.255 m and mass m = 23.8kg.The vessel system parameters in International System of Units are given in Table I.The desired trajectories are selected as: taken as 004, µ 2 = 0.005, and µ 3 = 0.006.

Adaptive NN Control Protocol
In what follows, we perform the simulation for two cases: 1) the exact model of underactuated surface vessel systems can be obtained a priori; and 2) the hydrodynamic terms f u , f υ , and f r given in (2) are unknown.
Model-Based Control (MBC): Assume that the accurate vessel model can be obtained a priori, we develop the modelbased tracking controllers given in (33) and (34) with additional control β in (35) and adaptive laws (36) to achieve the control objective.The stochastic disturbances are chosen as follows: τ wu = m 11 sin(0.01t),τ wυ = m 22 sin(0.01t),τ wr = m 33 sin(0.01t),τwu = 0.5m 11 sin(0.03t)ω1 , τwυ = 0.5m 22 cos(0.03t)ω2 , and τwr = 0.5m 33 cos(0.03t)ω3 .According to Assumption 1, we can obtain ϑ max Adaptive NN Control (ANNC): Suppose that the hydrodynamic damping terms f u , f υ , and f r are uncertain.The adaptive NN controllers τ u and τ r given in ( 46) and (47) with additional control β in (48), adaptive laws (36), and NN weight update laws in (49) are designed to achieve the control objective.The Simulation results for both MBC and ANNC are presented in Figs.2-8.Fig. 2 shows the phase-plane trajectories of the vessel system, it can be seen from Fig. 2 that the vessel position (x, y) can successfully track the desired position (x d , y d ) with small tracking error under both two controllers.The evolutions of the tracking errors z 1i (i = 1, 2, 3) are shown in Figs.3-5, which indicate that the tacking errors converge to a small neighborhood of zero, while the error constraint will not be violated despite the presence of stochastic disturbances.Figs.6-7 show the control input signals τ u and τ r , respectively, which are all bounded.The two norm of NN weight estimates are illustrated in Fig. 8 to show the boundedness of the NN weight estimates.Comparing the simulation results in Figs.3-7, we can observe that 1) the overshoots/undershoots of the tracking errors obtained by ANNC are larger than the ones obtained by  MBC, and 2) the amplitudes of the control inputs obtained by ANNC are also larger in transient state.This is because the adaptive NN controller with less system knowledge of the vessel model need to adapt to the modeling uncertainties f u , f υ , and f r by the online adjustment of NN weight estimates.

Conclusion
This paper presented a trajectory tracking control design technique for an uncertain underactuated surface vessel subject to prescribed performance constraint in the presence of stochastic disturbances.To overcome the difficulties raised by underactuation and nonzero off-diagonal system matrix, the transverse function control approach was introduced to provide an additional control input.A tan-type quartic barrier Lyapunov function instead of the quadratic form was incorporated into the control design to deal with the Hessian terms induced by stochastic differentiation rule and simultaneously guarantee the satisfaction of prescribed performance constraint in probability.For the case that the hydrodynamic damping terms are unknown, the NN approximators were constructed and embedded into the controllers.The merit of the proposed control protocol is that it is capable of handling the stochastic disturbances induced by wind, wave, and ocean current.Future efforts will be devoted to addressing the formation control problem for a group of uncertain underactuated surface vessel with limited communication sources under stochastic disturbances.under Grant SML2022008, in part by the National Natural Science Foundation of China under Grant 42227901.

Figure 1 :
Figure 1: Schematic diagram of the proposed control strategy.

Figure 2 :Figure 3 :
Figure 2: Vessel position (x, y) tracks the reference position (x d , y d ) in the phase plane.

Figure 9 :
Figure 9: Vessel position (x, y) tracks the reference position (x d , y d ) in the phase plane.