Fuzzy Adaptive Super-Twisting Sliding Mode Control for Underactuated USV Formation Based on Dynamic Cooperative Error Correction

Abstract

In order to maintain stable formations of the underactuated unmanned surface vehicles (USVs) under ocean disturbances and reduce control chattering, a fuzzy adaptive super-twisting sliding mode control method combined with dynamic cooperative error correction is introduced in this paper. At the kinematic level, a dynamic cooperative error correction scheme is presented to include the relative positions of the neighboring vehicles. This results in a transition from independent tracking to interactive cooperation, thus improving the rigidity of the formation during maneuvers. Secondly, a composite inner-loop structure is designed by employing a nonlinear disturbance observer to counteract the effects of external loads. A fuzzy logic system is included to modify the super-twisting sliding mode gains in real-time. This approach effectively combines rapid error reduction with signal smoothness, addressing the trade-off between response speed and chatter suppression. Moreover, tracking differentiators and low-pass filters are utilized to obtain continuous control signals. The Lyapunov analysis indicates that the closed-loop system achieves semi-global uniform ultimate boundedness. Simulation results show that the proposed method can maintain high formation precision and obtain smooth control outputs with reduced chattering.

1. Introduction

With the increasing demand for marine resource exploitation and maritime management, Unmanned Surface Vehicles (USVs), serving as the core platforms of intelligent marine equipment, have exhibited tremendous potential in fields such as wide-area marine environmental monitoring, maritime cooperative search and rescue, and routine patrolling [1,2,3]. Compared to a single USV, a multi-USV formation system can significantly enhance the overall execution efficiency, environmental adaptability, and system redundancy through spatial distribution and cooperative coordination [4,5]. Recent advancements in intelligent path planning, obstacle avoidance, and distributed coordination have further expanded the potential of multi-agent systems in safe navigation tasks [6,7,8,9,10]. However, in realistic marine environments, USVs not only possess inherent nonlinear hydrodynamic characteristics and underactuated constraints, but are also inevitably subjected to the continuous impacts of strong time-varying environmental disturbances, such as winds, waves, and ocean currents [11,12]. Therefore, designing a high-precision, strongly robust, and smoothly executable cooperative trajectory tracking control strategy for underactuated formations is a critical and challenging issue that urgently needs to be addressed in the current field of marine engineering.

Regarding multi-agent cooperative formation control, current mainstream strategies primarily include the leader-follower approach, behavior-based approach, graph theory and consensus strategies [13], and the virtual structure method. Riahifard et al. [14] proposed an adaptive leader-follower control scheme that effectively solved the formation problem for underactuated USVs subject to model uncertainties and input constraints; however, the over-reliance of this architecture on the leader node limits the robustness of the system. Xue et al. [15] systematically reviewed the flexibility of the behavior-based approach in complex environments; however, such methods often struggle to provide rigorous mathematical proofs of global stability. In contrast, the virtual structure method treats the entire formation as a single rigid body, providing explicit global reference states for each agent, thereby exhibiting significant advantages in maintaining high-precision geometric configurations. In recent years, numerous scholars have conducted in-depth optimizations of this method. For instance, Yan et al. [16] combined the virtual structure method with the artificial potential field method, enhancing the obstacle avoidance capabilities of multi-USV formations in dynamic sea areas; Dong et al. [17] introduced simultaneous modeling and fuzzy sliding mode techniques within the framework of the virtual structure method, resolving the high-precision formation tracking problem under unknown model parameters; furthermore, Chen et al. [18] designed a novel obstacle avoidance formation scheme based on the virtual structure method for mobile robots, which optimized path smoothness in complex environments.

In terms of the low-level trajectory tracking control of underactuated USV formations, existing mainstream control strategies primarily include the backstepping method [19], model predictive control [20], and sliding mode control (SMC). However, the backstepping method relies heavily on accurate dynamic models. Regarding model predictive control, several recent studies have successfully applied it to USV cooperative formations and multi-constraint obstacle avoidance under complex sea states [21,22], demonstrating its significant advantages in handling multi-constraint problems [23]. Nevertheless, its complex online receding horizon optimization often imposes a high computational burden, which limits the real-time response capability when facing sudden disturbances [24]. In contrast, sliding mode control has been widely applied due to its strong robustness against model uncertainties and external disturbances. Because of these robust features, sliding mode control and its adaptive backstepping variants have achieved numerous successful applications in dealing with complex nonlinear dynamics, such as advanced vehicle handling, active suspension systems, and electro-hydraulic control [25,26,27,28]. To further improve its tracking performance and suppress the high-frequency chattering phenomenon inherent in traditional sliding mode control, scholars have proposed various optimization schemes. For example, Dong et al. [29] designed an adaptive sliding mode controller combined with an improved guidance strategy, achieving the rapid convergence of formation errors; Dong et al. [30] introduced a bio-inspired neurodynamics model to optimize the sliding mode design, effectively enhancing the smoothness of the control command output; Li et al. [31] proposed an adaptive fuzzy sliding mode scheme integrating a fixed-time disturbance observer, utilizing a fuzzy system to dynamically adjust the switching gain, which further enhanced the system’s adaptability to complex and unknown sea states; furthermore, Liu et al. [32] adopted high-order sliding mode techniques represented by the super-twisting algorithm (STA). By hiding the discontinuous switching term within the integral action, this method significantly improved the dynamic response of the system while ensuring disturbance rejection capability.

However, existing formation strategies still exhibit limitations in simultaneously maintaining high-precision cluster configurations and achieving adaptive cooperation against complex disturbances. To further improve tracking performance and environmental adaptability, various state-of-the-art methodologies, including Takagi-Sugeno fuzzy models, finite-time fault-tolerant control, and advanced environmental perception frameworks, have been extensively investigated [33,34,35,36]. Specifically, the leader-follower approach, graph theory and consensus strategies, and the behavior-based method in [13,14,15] possess inherent shortcomings in maintaining high-precision rigid geometric configurations and providing global mathematical stability analysis. In contrast, although the improved virtual structure methods in [16,17,18] can provide explicit global references, they essentially remain decoupled, independent tracking modes. The core design of such methods merely focuses on eliminating the absolute position errors between individual USVs and their corresponding virtual reference points, neglecting real-time state interactions among neighboring nodes. Under this logic, which lacks a cooperative compensation mechanism, once a member deviates from the predefined configuration due to non-uniform wave disturbances or emergency evasive maneuvers, the remaining members cannot adaptively adjust their own navigation states according to this deviation. This ultimately leads to a significant degradation in the cooperative rigidity and disturbance rejection performance of the entire formation.

In terms of low-level trajectory tracking and resisting marine disturbances, how to balance the smoothness of control commands with strong robustness against severe time-varying disturbances remains a key challenge currently faced. First, the backstepping method and model predictive control in [19,20,21,22,23,24] rely heavily on accurate dynamic models and suffer from issues such as the “explosion of complexity” or excessive online optimization burdens, making them highly restricted under conditions with limited computational resources and unknown environments. Second, although the methods in [29,30,31] introduce adaptive laws, bio-inspired models, or fuzzy logic to dynamically adjust the sliding mode gains, their underlying architectures remain limited by the traditional first-order sliding mode switching surfaces. They fail to eliminate the discontinuous sign function from the mathematical structure, and the residual high-frequency switching characteristics will still induce mechanical wear on the actuators. Finally, although the high-order super-twisting strategy in [32] effectively hides the discontinuous terms to achieve signal smoothness, it generally adopts fixed reaching law gains. When facing transient and unknown external disturbances under realistic sea states, it often falls into an inherent contradiction: an excessively small gain leads to insufficient disturbance rejection, while an excessively large gain reintroduces chattering. This limits the ultimate steady-state tracking precision and environmental adaptability.

Based on the above analysis, to address the challenges of formation keeping and high-frequency chattering for underactuated USV formations in complex marine environments, this paper proposes a fuzzy adaptive super-twisting sliding mode control strategy based on dynamic cooperative error correction.

It should be emphasized that the proposed strategy is not a simple hybrid integration of existing algorithms, but a deeply coupled architecture designed to overcome the inherent structural limitations of traditional methods. On the one hand, traditional super-twisting algorithms (STA) heavily rely on fixed, conservative high gains to handle worst-case disturbances, which inevitably leads to severe actuator saturation or even loss of control under time-varying sea states. By incorporating fuzzy logic into the STA structure, the reaching law gains are dynamically optimized according to the sliding mode surface states, essentially addressing the fundamental trade-off between maintaining strong robustness and preventing control over-saturation. On the other hand, unlike standard virtual structure methods that operate in decoupled tracking modes, the proposed dynamic cooperative error mechanism embeds local neighborhood state feedback directly into the kinematic loop. This structural improvement transforms the formation from independent trajectory tracking to an interactive, tightly coupled network, substantially improving the structural rigidity and disturbance-rejection capabilities of the multi-USV system.

The main contributions of this paper are summarized as follows:

• A formation tracking architecture based on local dynamic cooperative error correction is constructed. Building upon the nominal trajectory generated by the traditional virtual structure method, a cooperative error term containing the relative position information of local neighbor nodes is introduced. This achieves a leap from independent single-vessel trajectory tracking to strongly coupled multi-vessel cooperative keeping, enhancing the formation rigidity and cooperative precision during time-varying disturbances and maneuvers without imposing an excessive communication burden; To the best of the authors’ knowledge, this is the first work to embed topological consensus corrections directly into the kinematic outer loop of an underactuated USV formation controller, rather than treating cooperative control as an independent parallel layer.
• A fuzzy adaptive super-twisting sliding mode control law is designed, achieving the dynamic decoupling of control response speed and chattering suppression. To address the chattering pain point of traditional sliding mode control, a fuzzy logic system based on 49 expert rules is designed for the real-time online dynamic optimal tuning of the super-twisting control gains. Combined with a hyperbolic tangent function, this mitigates the high-frequency chattering issue of actuators in traditional sliding mode control while ensuring the rapid convergence of system states;
• An underactuated composite anti-disturbance mechanism with feedforward compensation and differential smoothing is proposed. A nonlinear disturbance observer is integrated to accurately estimate and feedforward-cancel low-frequency macroscopic environmental disturbances. Simultaneously, to overcome the lack of direct driving force in underactuated lateral control, a first-order low-pass filter and a tracking differentiator are introduced to smooth the virtual control laws and the second derivative of the sway velocity. This avoids initial control chattering and noise amplification, ensuring smooth low-level control.

The remainder of this paper is organized as follows: Section 2 presents the kinematic and dynamic modeling of the underactuated USV and details the dynamic generation mechanism of the cooperative formation error and the desired trajectory. Section 3 elaborates on the derivation of the proposed composite anti-disturbance control strategy. Section 4 provides a rigorous stability analysis of the closed-loop system based on Lyapunov theory. Section 5 verifies the effectiveness and superiority of the proposed method through multiple sets of comparative simulation experiments. Finally, Section 6 summarizes the entire paper and outlines prospects for future work.

2. Problem Formulation

This section discusses the problem of underactuated USV formation control in the presence of complex marine disturbances, as shown in Figure 1:

As illustrated in Figure 1, the USV formation control problem under complex sea states involves three core challenges: the lateral control difficulties induced by the underactuated nature of the system, the maintenance of formation configuration and high-precision tracking under time-varying marine disturbances, and the high-frequency chattering phenomenon at the actuator side caused by traditional control strategies. This section primarily consists of two components: the mathematical modeling of the USV formation and the description of the USV formation control problem.

2.1. Mathematical Modeling of the USV Formation

Taking into account the physical characteristics of the USV formation during the execution of trajectory tracking tasks, this study utilizes a system composed of $N$ three-degree-of-freedom (3-DOF) underactuated surface unmanned vehicles. For the ith USV ($i=1,\dots ,N$), the kinematic model in the earth-fixed inertial frame can be described as follows:

$$\left\{\begin{array}{l}{\dot{x}}_i=u_i\cos {\psi }_i-v_i\sin {\psi }_i\hfill \\ {\dot{y}}_i=u_i\sin {\psi }_i+v_i\cos {\psi }_i\hfill \\ {\dot{\psi }}_i=r_i\hfill \end{array}\right.$$

(1)

where $x_i$ and $y_i$ denote the lateral and longitudinal positions of the ith USV in the earth-fixed inertial frame, respectively, and ${\psi }_i$ represents its heading angle. $u_i$ and $v_i$ are the surge and sway velocities in the body-fixed frame, while $r_i$ is the yaw rate. Considering external disturbances and model uncertainties, and referring to the classical dynamic modeling theory for surface vehicles [37], the dynamic model of the ith underactuated USV can be expressed as the following nonlinear system of equations:

$$\left\{\begin{array}{l}(m_i+m_{xi}){\dot{u}}_i-(m_i+m_{yi})v_i{r}_i=X_{uui}{u}_i^2+X_{vvi}{v}_i^2+X_{vri}{v}_i{r}_i+X_{rri}{r}_i^2+X_{pi}+X_{di}\hfill \\ (m_i+m_{yi}){\dot{v}}_i+(m_i+m_{xi})u_i{r}_i=Y_{vi}{v}_i+Y_{ri}{r}_i+Y_{vvi}{v}_i\left|v_i\right|+Y_{vri}{v}_i{r}_i+Y_{rri}{r}_i^2+Y_{di}\hfill \\ (I_{zzi}+J_{zi}){\dot{r}}_i=N_{vi}{v}_i+N_{ri}{r}_i+N_{vvri}{v}_i^2{r}_i+N_{vrri}{v}_i{r}_i^2+N_{vvi}{v}_i\left|v_i\right|+N_{Ri}+N_{di}\hfill \end{array}\right.$$

(2)

where $m_i$ is the mass of the hull; $m_{xi}$ and $m_{yi}$ are the added masses along the $x$ and $y$ axes, respectively; $I_{zzi}$ is the inherent moment of inertia about the vertical axis; and $J_{zi}$ is the added moment of inertia. The parameters $X_{uui}\dots N_{vvi}$ denote various damping and coupling coefficients of the USV, including surge quadratic damping, sway linear and quadratic damping, yaw damping, and complex momentum coupling terms. $X_{pi}$ and $N_{Ri}$ are the control inputs, representing the longitudinal thrust and yaw moment generated by the ith USV, respectively. Due to the underactuated characteristics of the system, it lacks a direct control input in the sway degree of freedom. Additionally, ${[X_{di},Y_{di},N_{di}]}^T$ represents the lumped external disturbances, which encompass environmental loads and unmodeled dynamics.

In practical marine engineering applications, accurately acquiring USV dynamic model parameters is difficult due to complex hydrodynamic effects and time-varying loading states, typically requiring online identification algorithms for estimation [38]. This inherent uncertainty, compounded by unknown environmental disturbances such as wind, waves, and currents, imposes rigorous robustness demands on the underlying control system. Consequently, this study integrates a nonlinear disturbance observer into the inner-loop dynamics to treat unmodeled dynamics and environmental disturbances as a unified lumped disturbance for real-time estimation and feedforward compensation.

2.2. Problem Description of USV Formation Control Based on Virtual Structure

In this section, we explore the key issues of formation control, emphasizing the importance of maintaining a predefined configuration, sailing along desired trajectories with high precision, and effectively suppressing actuator chattering in time-varying disturbance environments. To achieve this objective, the virtual structure method is adopted to generate the formation, the specific form of which is shown in Figure 2.

As illustrated in Figure 2, $(x_i,y_i)$ denotes the position coordinates of the $(X_{od},Y_{od})$ represents the position coordinates of the reference trajectory point, and $(x_{di},y_{di})$ signifies the position coordinates of the virtual structure point. Furthermore, ${\psi }_r$ is the course angle of the reference trajectory, $o_b{x}_b{y}_b$ is the body-fixed coordinate system, and $(x_{bi},y_{bi})$ represents the position coordinates of the virtual construction point within the body-fixed coordinate system.

Based on rigid body kinematics, the virtual structure method treats the entire formation as a rigid body generated from the instantaneous reference trajectory and corresponding formation coordinates to determine the Earth frame coordinates of each virtual structure point. To effectively address the strong coupling of lateral and longitudinal dynamics in underactuated systems, the nominal desired trajectory ${[x_{dni},y_{dni}]}^T$ is established within the body coordinate system. Since tracking independent nominal trajectories cannot maintain formation rigidity under complex marine disturbances, dynamic corrections using the relative states of neighboring nodes are applied to ${[x_{dni},y_{dni}]}^T$ to derive the actual desired position coordinates ${[x_{di},y_{di}]}^T$ for the ith USV. When all USV pose error vectors converge to zero, the cooperative objectives of formation maintenance and trajectory tracking are simultaneously achieved. Consequently, the original problem of high precision trajectory tracking and chattering suppression under environmental disturbances is equivalently transformed into designing smooth and continuous control laws to stabilize the formation position error system.

3. Design of Adaptive Sliding Mode Controller

This section utilizes a cascaded control architecture with interconnected inner and outer loops to design the underactuated USV formation controller. The kinematic outer loop designs virtual velocity control laws based on a dynamic cooperative error correction mechanism. Simultaneously, the dynamic inner loop integrates a nonlinear disturbance observer with a fuzzy adaptive super-twisting sliding mode algorithm to ensure stable and chattering-free tracking of target trajectories. The overall architecture of the proposed formation control system is illustrated in Figure 3.

3.1. Formation Cooperative Error and Kinematic Virtual Control Law Design

3.1.1. Kinematic Outer-Loop Control Design Based on Cooperative Error Correction

Traditional virtual structure methods often struggle to maintain rigid formations under complex environmental disturbances because they typically rely on absolute nominal reference trajectories. To address this, a cooperative error dynamic correction mechanism is established by integrating relative position information from local neighboring vessels.

Furthermore, to effectively manage the strong coupling between the longitudinal and lateral dynamics of underactuated USVs, the desired position coordinates ${[x_{di},y_{di}]}^T$ are defined within the body-fixed coordinate system. The actual desired trajectory update law following dynamic correction is formulated as follows:

$$\left[\begin{array}{c}{x}_{di}\\ y_{di}\end{array}\right]=\left[\begin{array}{c}{x}_{dni}\\ y_{dni}\end{array}\right]+\left[\begin{array}{cc}{k}_x& 0\\ 0& k_y\end{array}\right]{\displaystyle \sum _{j\in N_i}\left(\left[\begin{array}{c}{x}_i-x_j\\ y_i-y_j\end{array}\right]-\left[\begin{array}{cc}\cos {\psi }_i& -\sin {\psi }_i\\ \sin {\psi }_i& \cos {\psi }_i\end{array}\right]\left[\begin{array}{c}{l}_{xi}-l_{xj}\\ l_{yi}-l_{yj}\end{array}\right]\right)}$$

(3)

where ${[x_{dni},y_{dni}]}^T$ is the nominal desired trajectory; $N_i$ denotes the neighbor set of the ith USV; $k_x,k_y>0$ are cooperative control gains; ${[x_i,y_i]}^T$ and ${[x_j,y_j]}^T$ are actual positions; and ${[l_{xi},l_{yi}]}^T$, ${[l_{xj},l_{yj}]}^T$ represent preset relative offsets for formation maintenance.

Remark (Graph-Theoretic Convergence of the Cooperative Correction Term): To further clarify the convergence properties of the cooperative error correction mechanism in Equation (3), a graph-theoretic interpretation is provided. Let the communication topology of the $n$-USV formation be described by a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, with its corresponding Laplacian matrix $L\in {ℝ}^{n\times n}$. Define the stacked cooperative position error vector as $\delta = {[{\delta }_1,{\delta }_2,\dots ,{\delta }_n]}^T$, where each ${\delta }_i$ represents the net cooperative correction received by the ith USV from its neighbors. The collective dynamics of this cooperative correction can be compactly written as $\stackrel{.}{\delta }=-kL\delta$, where $k>0$ is the cooperative gain. By Assumption 2, the directed graph $\mathcal{G}$ contains at least one directed spanning tree, which guarantees that the Laplacian matrix $L$ has exactly one zero eigenvalue with all remaining eigenvalues having strictly positive real parts. Therefore, the cooperative error vector $\delta$ converges exponentially to the null space of $L$, which corresponds precisely to the consensus manifold where all position errors are uniformly bounded and driven toward zero. This confirms that the proposed cooperative correction mechanism structurally guarantees formation consensus under the graph connectivity condition of Assumption 2.

Consequently, the longitudinal and lateral tracking errors for the ith USV in the body-fixed frame are defined as:

$$\left[\begin{array}{l}{x}_{ei}\\ y_{ei}\end{array}\right]=\left[\begin{array}{l}{x}_i-x_{di}\\ y_i-y_{di}\end{array}\right]$$

(4)

3.1.2. Virtual Control Law Design

The kinematic outer-loop aims to generate appropriate longitudinal and lateral virtual velocity commands to drive the position tracking errors $x_{ei}$ and $y_{ei}$ toward asymptotic convergence to zero. Accounting for the kinematic transformation between the Earth-fixed and body-fixed coordinate systems, the virtual control laws ${\alpha }_{ui}$ and ${\alpha }_{vi}$ are designed as:

$$\left[\begin{array}{c}{\alpha }_{ui}\\ {\alpha }_{vi}\end{array}\right]=\left[\begin{array}{cc}\cos {\psi }_i& \sin {\psi }_i\\ -\sin {\psi }_i& \cos {\psi }_i\end{array}\right]\left[\begin{array}{c}{\dot{x}}_{di}-k_{1i}{x}_{ei}/w_i\\ {\dot{y}}_{di}-k_{2i}{y}_{ei}/w_i\end{array}\right]$$

(5)

where $k_{1i},k_{2i}>0$ are design parameters; and $w_i=\sqrt{{x^2}_{ei}+{y^2}_{ei}+C_i}$ is a weighting term with a small positive constant $C_i>0$, formulated to smooth error convergence and prevent division-by-zero singularities. By incorporating this dynamic weight $w_i$, the system effectively suppresses sudden spikes in virtual control commands caused by significant initial position deviations, thereby ensuring a smoother initial dynamic response.

3.1.3. First-Order Low-Pass Filter Design

Deriving the inner-loop sliding mode controller requires virtual control law derivatives (${\dot{\alpha }}_{ui}$ and ${\dot{\alpha }}_{vi}$), but direct analytical differentiation is avoided to prevent excessive computational complexity and sensor noise amplification. Instead, this study integrates a first-order low-pass filter into the backstepping recursion based on dynamic surface control theory [39]. This technique circumvents the heavy algebraic burden of high-order differentiation while ensuring that cooperative tracking errors converge to an arbitrarily small neighborhood [40]. For the longitudinal law ${\alpha }_{ui}$, the filter is designed as follows:

$$\left\{\begin{array}{l}{T}_{1i}{\dot{A}}_{ui}+A_{ui}={\alpha }_{ui}\hfill \\ A_{ui}(0)={\alpha }_{ui}(0)\hfill \end{array}\right.$$

(6)

where $T_{1i}>0$ is the design time constant and $A_{ui}$ represents the filtered state. In implementation, $A_{ui}$ and ${\dot{A}}_{ui}$ replace the original variables ${\alpha }_{ui}$ and ${\dot{\alpha }}_{ui}$ to significantly reduce the algebraic load and ensure semi-global uniform ultimate boundedness of the system. Mechanisms for lateral control are detailed in the subsequent section regarding underactuated dynamics.

Similarly, to obtain the derivative of the lateral virtual control law ${\alpha }_{vi}$ without noise amplification, a first-order low-pass filter is designed for the lateral channel as follows:

$$\left\{\begin{array}{l}{T}_{2i}{\dot{B}}_{pi}+B_{pi}={\alpha }_{vi}\\ B_{pi}(0)={\alpha }_{vi}(0)\end{array}\right.$$

(7)

where $T_{2i}>0$ is the filter time constant for the lateral channel, and $B_{pi}$ represents the filtered state of the lateral virtual control law. In the subsequent controller derivation and stability analysis, $B_{pi}$ and derivative ${\dot{B}}_{pi}$ are used as surrogates for ${\alpha }_{vi}$ and ${\dot{\alpha }}_{vi}$.

3.2. Fuzzy Adaptive Super-Twisting Dynamic Sliding Mode Controller Design

Inner-loop dynamic control drives the actual USV velocity toward outer-loop virtual commands. To theoretically eliminate high-frequency thruster wear associated with traditional sliding mode chattering, a chattering-free control law is developed using an improved super-twisting algorithm. Furthermore, a tracking differentiator smoothing mechanism is innovatively introduced to suppress noise amplification in underactuated lateral control, while fuzzy rules are designed to achieve adaptive tuning of the reaching law parameters.

3.2.1. Longitudinal Thrust Sliding Mode Control Law Design

The longitudinal velocity tracking error for the ith USV is defined as $u_{ei}=u_i-{\alpha }_{ui}$. To eliminate steady-state errors and enhance robustness, an integral sliding surface is designed as follows:

$$s_{1i}=u_{ei}+{\lambda }_{1i}{\displaystyle {\int }_0^t{u}_{ei}}({\tau }_i)d{\tau }_i$$

(8)

where ${\lambda }_{1i}>0$ is a sliding surface design constant. Differentiating the above equation and substituting the USV longitudinal dynamics equation yields:

$$\begin{array}{ll}{\dot{s}}_{1i}& ={\dot{u}}_i-{\dot{\alpha }}_{ui}+{\lambda }_{1i}{u}_{ei}\\ & ={\displaystyle \frac{m_i+m_{yi}}{m_i+m_{xi}}}{v}_i{r}_i+{\displaystyle \frac{X_{uui}{u}_i^2+X_{vvi}{v}_i^2+X_{vri}{v}_i{r}_i+X_{rri}{r}_i^2+X_{pi}^{*}}{m_i+m_{xi}}}-{\dot{\alpha }}_{ui}+{\lambda }_{1i}{u}_{ei}\end{array}$$

(9)

To effectively mitigate high-frequency chattering induced by the sign function in conventional sliding mode control, the hyperbolic tangent function $\tanh (\cdot )$ is utilized to replace the discontinuous sign function. The smooth super-twisting control law $u_{stai}$ is formulated as follows [41]:

$$\left\{\begin{array}{l}{u}_{stai}=-{\epsilon }_{1i}\sqrt{\left|s_{1i}\right|}\tanh (s_{1i})+u_{1i}\hfill \\ {\dot{u}}_{1i}=-{\alpha }_{1i}\tanh (s_{1i})\hfill \end{array}\right.$$

(10)

The exponential reaching law for the longitudinal velocity sliding surface is defined as ${\dot{s}}_{1i}=u_{stai}-{\sigma }_{1i}{s}_{1i}$, where ${\epsilon }_{1i},{\alpha }_{1i},{\sigma }_{1i}>0$ are the design parameters. By integrating the aforementioned equations and substituting the direct derivative ${\dot{\alpha }}_{ui}$ with the previously designed low-pass filter output ${\dot{A}}_{ui}$, the nominal longitudinal thrust sliding mode control law $X_{pi}^{\ast }$ is derived as:

$$\begin{array}{ll}{X}_{pi}^{*}& =-(m_i+m_{yi})v_i{r}_i-X_{uui}{u}_i^2-X_{vvi}{v}_i^2-X_{vri}{v}_i{r}_i-X_{rri}{r}_i^2\\ & +(m_i+m_{xi})[{\dot{A}}_{ui}-{\lambda }_{1i}{u}_{ei}-{\epsilon }_{1i}\sqrt{\left|s_{1i}\right|}\tanh (s_{1i})+u_{1i}-{\sigma }_{1i}{s}_{1i}]\end{array}$$

(11)

3.2.2. Yaw Moment Control Law Design

Underactuated USV systems lack direct thrust control in the lateral degree of freedom where $Y_{pi}=0$. To address this characteristic, the lateral dynamics must be stabilized indirectly through the yaw moment $N_{Ri}$ by exploiting nonlinear coupling between system states. Upon defining the lateral velocity error variable as $v_{ei}=v_i-{\alpha }_{vi}$, a composite lateral sliding surface incorporating a proportional and integral structure is proposed as follows:

$$s_{2i}={\dot{v}}_{ei}+{\lambda }_{2i}{v}_{ei}+{\lambda }_{3i}{\displaystyle {\int }_0^t{v}_{ei}}d\tau$$

(12)

where ${\lambda }_{2i},{\lambda }_{3i}>0$ are integral constants. Differentiating this sliding surface yields:

$${\dot{s}}_{2i}={\ddot{v}}_i-{\ddot{\alpha }}_{vi}+{\lambda }_{2i}({\dot{v}}_i-{\dot{\alpha }}_{vi})+{\lambda }_{3i}(v_i-{\alpha }_{vi})$$

(13)

This derivation inevitably introduces the second-order derivative of lateral velocity ${\ddot{v}}_i$. Since direct differentiation of noisy lateral velocity measurements can cause severe noise amplification and intense initial yaw moment chattering, a first-order discrete tracking differentiator (TD) is innovatively introduced to smooth ${\ddot{v}}_i$ [42]:

$$\left\{\begin{array}{l}{x}_{1i}(k+1)=x_{1i}(k)+h{x}_{2i}(k)\hfill \\ x_{2i}(k+1)=x_{2i}(k)+h\{-D_i^2[x_{1i}(k)-{\dot{v}}_i(k)]-2D_i{x}_{2i}(k)\}\hfill \end{array}\right.$$

(14)

where $h$ is the simulation step; the input is ${\dot{v}}_i$; $x_{1i}$ and $x_{2i}$ are output variables approximating ${\dot{v}}_i$ and the smoothed second-order derivative ${\ddot{v}}_i$, respectively; and $D_i$ is the tracking speed factor. Similarly, the hyperbolic tangent function $\tanh (s_{2i})$ is applied to the lateral super-twisting reaching law. By substituting noisy terms with TD outputs $x_{1i}$ and $x_{2i}$ within the dynamics equations, the smoothed nominal yaw moment control law $N_{Ri}^{\ast }$ is derived as follows:

$$\begin{array}{ll}{N}_{Ri}^{*}& =[-x_{2i}+{\dot{B}}_{pi}-r_i{A}_{ui}-{\lambda }_{2i}(x_{1i}-{\dot{\alpha }}_{vi})-{\lambda }_{3i}(v_i-{\alpha }_{vi})-{\epsilon }_{2i}\sqrt{\left|s_{2i}\right|}\tanh (s_{2i})\\ & +v_{1i}-{\sigma }_{2i}{s}_{2i}](I_{zzi}+J_{zi})/A_{ui}-N_{vi}{v}_i-N_{ri}{r}_i-N_{vvri}{v}_i^2{r}_i-N_{vrri}{v}_i{r}_i^2-N_{vvi}{v}_i\left|v_i\right|\end{array}$$

(15)

3.2.3. Fuzzy Adaptive Parameter Tuning Mechanism

In the super-twisting sliding mode control described above, the reaching law gain parameters (${\epsilon }_{1i},{\epsilon }_{2i}$) critically influence the convergence speed and steady-state performance. Fixed gains face an inherent trade-off: an excessively large gain induces severe mechanical chattering, while a small gain fails to suppress sudden environmental disturbances. To achieve dynamic decoupling of response speed and chattering suppression, a Mamdani-type fuzzy logic system is designed for the real-time adaptive optimization of the super-twisting control gains [43].

The fuzzy system utilizes the sliding surface function $s$ and its rate of change $\dot{s}$ for each control channel as the two input variables, while the adaptive tuning increment $\widehat{\epsilon }$ serves as the output. To standardize the controller design, the absolute values of the inputs and output are normalized to a defined universe of discourse of $[-6,6]$. The linguistic variable sets for the inputs and output are categorized into seven levels: negative big (NB), negative medium (NM), negative small (NS), zero (Z), positive small (PS), positive medium (PM), and positive big (PB).

To ensure smooth and continuous transitions of the control signals, overlapping membership functions are deliberately chosen. Specifically, Z-shaped and S-shaped membership functions are assigned to the boundary linguistic subsets (NB and PB) to handle extreme tracking deviations, whereas symmetric triangular membership functions are utilized for the intermediate subsets to ensure high sensitivity near the equilibrium point. The distributions of these membership functions are illustrated in Figure 4.

The core of the fuzzy inference mechanism relies on a 49-rule base derived from human expert experience in ship maneuvering, as detailed in

Table 1. Fuzzy rule base.

$\dot{\mathit{s}}$	$\mathit{s}$
	NB	NM	NS	0	PS	PM	PB
NB	PB	PB	PS	0	NS	NB	NB
NM	PB	PM	0	NS	NM	NB	NM
NS	PS	0	NS	NM	NB	NM	NS
0	0	NS	NM	NB	NM	NS	0
PS	NS	NM	NB	NM	NS	0	PS
PM	NM	NB	NM	NS	0	PM	PB
PB	NB	NM	NS	0	PS	PB	PB

.

The physical intuition behind the rule base design is essentially governed by the following logic:

• When the sliding surface $s$ and its derivative $\dot{s}$ share the same sign and their absolute values are large (e.g., both are PB or both are NB), it indicates that the USV is deviating rapidly from the desired trajectory under strong ocean disturbances. In this case, the output $\widehat{\epsilon }$ is assigned to PB to supply a massive control gain increment, aggressively forcing the system state back toward the sliding surface.
• When the system state is approaching the sliding surface rapidly ($s\cdot \dot{s}<0$), a moderate or negative gain increment (e.g., NS or Z) is assigned to dynamically reduce the kinetic energy and prevent severe overshooting.
• When the state has steadily reached the sliding manifold ($s\approx 0$ and $\dot{s}\approx 0$), the adaptive increment $\widehat{\epsilon }$ converges to the Z/NS level. This gracefully shrinks the super-twisting switching amplitude, thereby fundamentally mitigating actuator chattering.

Following the input fuzzification and the rule-based inference, the centroid method is employed for defuzzification to generate the crisp adaptive adjustment $\widehat{\epsilon }$:

$${\widehat{\epsilon }}_i={\displaystyle \frac{{\displaystyle \sum _{k=1}^{49}{v}_k}{\mu }_v(v_k)}{{\displaystyle \sum _{k=1}^{49}{\mu }_v}(v_k)}}$$

(16)

where $v_k$ is the crisp value in the universe of discourse and ${\mu }_v(v_k)$ is the corresponding membership value. By substituting the dynamically tuned increment into the nominal control laws, the underactuated formation system obtains continuous super-twisting control torques with autonomous environmental adaptability.

3.3. Final Composite Control Law Construction Based on Feedforward Compensation

In practical marine environments, USVs are inevitably influenced by low-frequency disturbances and unmodeled dynamics. Counteracting these lumped disturbances solely through robust sliding mode terms necessitates high switching gains, which exacerbates control chattering. To mitigate this, a nonlinear disturbance observer is integrated for online estimation and feedforward compensation, establishing a dual anti-disturbance mechanism alongside the feedback optimization [44].

To facilitate observer design, the lumped disturbances in the surge and yaw directions are defined as $d_{1i}=X_{di}$ and $d_{2i}=N_{di}$, respectively. Combined with the underactuated USV dynamics, the nonlinear disturbance observer is formulated as follows:

$$\left\{\begin{array}{l}\begin{array}{l}{\widehat{d}}_{1i}=z_{1i}+L_{1i}(m_i+m_{xi})u_i\hfill \\ \begin{array}{c}{\dot{z}}_{1i}=-L_{1i}{z}_{1i}-L_{1i}[L_{1i}(m_i+m_{xi})u_i-(m_i+m_{yi})v_i{r}_i\\ -X_{uui}{u}_i^2-X_{vvi}{v}_i^2-X_{vri}{v}_i{r}_i-X_{rri}{r}_i^2-X_{pi}]\end{array}\hfill \end{array}\\ \begin{array}{l}{\widehat{d}}_{2i}=z_{2i}+L_{2i}(I_{zzi}+J_{zi})r_i\hfill \\ \begin{array}{c}{\dot{z}}_{2i}=-L_{2i}{z}_{2i}-L_{2i}[L_{2i}(I_{zzi}+J_{zi})r_i-N_{vi}{v}_i-N_{ri}{r}_i\\ -N_{vvri}{v}_i^{2}r-N_{vrri}{v}_i{r}_i^2-N_{vvi}{v}_i\left|v_i\right|-N_{Ri}\end{array}\hfill \end{array}\end{array}\right.$$

(17)

where ${\widehat{d}}_{1i}$ and ${\widehat{d}}_{2i}$ are NDO estimates of the longitudinal and yaw lumped disturbances; $z_{1i}$ and $z_{2i}$ denote internal auxiliary state variables; and $L_1,L_2>0$ are observer gain coefficients. To verify convergence, observation errors are defined as $e_{1i}={\widehat{d}}_{1i}-d_{1i}$ and $e_{2i}={\widehat{d}}_{2i}-d_{2i}$. Differentiating $e_{1i}$ and substituting the observer dynamics yields the following error characteristics:

$$\begin{array}{ll}{\dot{e}}_{1i}& ={\dot{\widehat{d}}}_{1i}-{\dot{d}}_{1i}\\ & ={\dot{z}}_{1i}+L_{1i}(m_i+m_{xi}){\dot{u}}_i-{\dot{d}}_{1i}\\ & =-L_{1i}{\widehat{d}}_{1i}+L_{1i}{d}_{1i}-{\dot{d}}_{1i}\\ & =-L_{1i}{e}_{1i}-{\dot{d}}_{1i}\end{array}$$

(18)

Since $L_1,L_2>0$ are positive design constants, both error systems constitute exponentially stable linear time-varying systems subject to a bounded input ${\dot{d}}_{ji}$ (guaranteed by Assumption 1). Therefore, the observation errors $e_{1i}$ and $e_{2i}$ are guaranteed to converge exponentially to a compact neighborhood of zero, confirming the existence and convergence of the proposed NDO.

The observation error dynamics for the lateral and yaw directions are given by ${\dot{e}}_{2i}=-L_{2i}{e}_{2i}-{\dot{d}}_{2i}$. Once high-precision disturbance estimates (${\widehat{d}}_{1i}$, ${\widehat{d}}_{2i}$) and fuzzy-optimized super-twisting gains (${\widehat{\epsilon }}_{1i}$, ${\widehat{\epsilon }}_{2i}$) are obtained, the NDO outputs are integrated into the control inputs for feedforward compensation. Incorporating the previously derived nominal sliding mode laws, the final composite longitudinal thrust $X_{pi}$ and yaw moment $N_{Ri}$ acting on the propulsion actuators of the ith underactuated USV are designed as follows:

$$\left\{\begin{array}{l}{X}_{pi}=X_{pi}^{*}-{\widehat{d}}_{1i}\hfill \\ =-(m_i+m_{yi})v_i{r}_i-X_{uui}{u}_i^2-X_{vvi}{v}_i^2-X_{vri}{v}_i{r}_i-X_{rri}{r}_i^2\hfill \\ +(m_i+m_{xi})[{\dot{A}}_{ui}-{\lambda }_{1i}{u}_{ei}-{\widehat{\epsilon }}_{1i}\sqrt{\left|s_{1i}\right|}\tanh (s_{1i})+u_{1i}-{\sigma }_{1i}{s}_{1i}]-{\widehat{d}}_{1i}\hfill \\ N_{Ri}=N_{Ri}^{*}-{\widehat{d}}_{2i}\hfill \\ =[-x_{2i}+{\dot{B}}_{pi}-r_i{A}_{ui}-{\lambda }_{2i}(x_{1i}-{\dot{\alpha }}_{vi})-{\lambda }_{3i}(v_i-{\alpha }_{vi})\hfill \\ -{\widehat{\epsilon }}_{2i}\sqrt{\left|s_{2i}\right|}\tanh (s_{2i})+v_{1i}-{\sigma }_{2i}{s}_{2i}](I_{zzi}+J_{zi})/A_{ui}\hfill \\ -N_{vi}{v}_i-N_{ri}{r}_i-N_{vvri}{v}_i^2{r}_i-N_{vrri}{v}_i{r}_i^2-N_{vvi}{v}_i\left|v_i\right|-{\widehat{d}}_{2i}\hfill \end{array}\right.$$

(19)

4. Stability Analysis of the Formation Control System

This section rigorously verifies the stability of the proposed composite anti-disturbance control law based on Lyapunov theory. To ensure a solid theoretical foundation, the following necessary assumptions regarding the physical system and communication network are established prior to the proof.

Assumption 1: The lumped external disturbances $d_{1i}$ and $d_{2i}$ acting on the ith USV are continuously differentiable. Their time derivatives are strictly bounded such that $\left|{\dot{d}}_{1i}\right| \le D_{1i}$ and $\left|{\dot{d}}_{2i}\right| \le D_{2i}$, where $D_{1i}$ and $D_{2i}$ are unknown positive constants.

Remark (Justification of Assumption 1): In practical marine engineering, the external disturbances acting on USVs are primarily induced by wind, waves, and ocean currents. Since these environmental factors are generated by natural phenomena with finite energy, their associated forces and moments, as well as their rates of change, are physically bounded. An infinite rate of change would imply infinite energy, which contradicts physical laws. Therefore, assuming that the time derivatives of the lumped disturbances are strictly bounded is entirely consistent with actual marine hydrodynamics and operational constraints.

Assumption 2 (Graph Connectivity and Neighbor States): The communication topology graph of the USV formation contains at least one directed spanning tree. Furthermore, during the entire formation maneuvering process, the kinematic states (positions and velocities) of all neighboring nodes within the communication set $N_i$ remain bounded.

Remark (Justification of Assumption 2): For a distributed multi-agent system, the existence of at least one directed spanning tree is the most fundamental and necessary topological prerequisite to achieve global consensus. Physically, this ensures that the reference trajectory information from the virtual leader can be propagated—either directly or indirectly—to every follower USV in the formation network. If this condition is violated, the disconnected USVs will lose tracking coordination and physically drift away, fundamentally destroying the cooperative formation. Furthermore, the assumption of bounded neighbor states is naturally satisfied due to the physical actuator limits (maximum thrust and speed) of real USVs.

Remark 1 (Fuzzy Gain Bounds and Underactuated Dynamics): The adaptive parameter ${\widehat{\epsilon }}_i$ is generated by the fuzzy inference system. Given that the universe of discourse and the membership functions are both defined within finite ranges, the fuzzy outputs ${\widehat{\epsilon }}_{1i}$ and ${\widehat{\epsilon }}_{2i}$ are strictly bounded. Moreover, the underactuated yaw dynamics are indirectly stabilized through the strongly coupled non-diagonal terms in the dynamic model, ensuring that the internal unactuated states will not diverge during lateral tracking.

During the controller design process, the introduction of the first-order low-pass filter results in errors between the filtered surrogate values and the actual virtual control laws. The filter error terms are defined as follows:

$$\left\{\begin{array}{l}{e}_{3i}=A_{ui}-{\alpha }_{ui}\hfill \\ e_{4i}=B_{pi}-p_i\hfill \end{array}\right.$$

(20)

Differentiating both sides of the equations yields:

$$\left\{\begin{array}{l}{\dot{e}}_{3i}=-{\displaystyle \frac{e_{3i}}{T_{1i}}}+{\beta }_{1i}(\cdot )\hfill \\ {\dot{e}}_{4i}=-{\displaystyle \frac{e_{4i}}{T_{2i}}}+{\beta }_{2i}(\cdot )\hfill \end{array}\right.$$

(21)

where ${\beta }_{1i}(\cdot )$ and ${\beta }_{2i}(\cdot )$ are nonlinear continuous functions of the ith USV states. Given the inherent boundedness of position and velocity in physical systems (as stated in Assumption 2), these functions are also bounded. Assume positive constants $B_{1i}$ and $B_{2i}$ represent absolute upper bounds such that $\left|{\beta }_{1i}(\cdot )\right| \le B_{1i}$ and $\left|{\beta }_{2i}(\cdot )\right| \le B_{2i}$.

To prove global stability, a Lyapunov function for the ith USV is defined as:

$$V_i={\displaystyle \frac{1}{2}}{x}_{ei}^2+{\displaystyle \frac{1}{2}}{y}_{ei}^2+{\displaystyle \frac{1}{2}}{s}_{1i}^2+{\displaystyle \frac{1}{2}}{s}_{2i}^2+{\displaystyle \frac{1}{2}}{e}_{1i}^2+{\displaystyle \frac{1}{2}}{e}_{2i}^2+{\displaystyle \frac{1}{2}}{e}_{3i}^2+{\displaystyle \frac{1}{2}}{e}_{4i}^2$$

(22)

Differentiating $V_i$ along the system trajectory and substituting the error dynamics yields:

$$\begin{array}{l}{\dot{V}}_i=x_{ei}{\dot{x}}_{ei}+y_{ei}{\dot{y}}_{ei}+s_{1i}{\dot{s}}_{1i}+s_{2i}{\dot{s}}_{2i}+e_{1i}{\dot{e}}_{1i}+e_{2i}{\dot{e}}_{2i}+e_{3i}{\dot{e}}_{3i}+e_{4i}{\dot{e}}_{4i}\hfill \\ =-(k_{1i}{x}_{ei}^2+k_{2i}{y}_{ei}^2)/w_i+s_{1i}[-{\widehat{\epsilon }}_{1i}\tanh (s_{1i})-{\sigma }_{1i}{s}_{1i}]+e_{1i}{\dot{e}}_{1i}+e_{2i}{\dot{e}}_{2i}\hfill \\ +s_{2i}[-{\widehat{\epsilon }}_{2i}\tanh (s_{2i})-{\sigma }_{2i}{s}_{2i}]+e_{3i}\left(-e_{3i}/T_{1i}+{\beta }_{1i}\right)+e_{4i}\left(-e_{4i}/T_{2i}+{\beta }_{2i}\right)\hfill \\ \le -(k_{1i}{x}_{ei}^2+k_{2i}{y}_{ei}^2)/w_i-{\widehat{\epsilon }}_{1i}\left|s_{1i}\right|-{\sigma }_{1i}{s}_{1i}^2-{\widehat{\epsilon }}_{2i}\left|s_{2i}\right|-{\sigma }_{2i}{s}_{2i}^2\hfill \\ -L_{1i}{e}_{1i}^2-e_{1i}{\dot{d}}_{1i}-L_{2i}{e}_{2i}^2-e_{2i}{\dot{d}}_{2i}-{\displaystyle \frac{e_{3i}^2}{T_{1i}}}+e_{3i}{\beta }_{1i}-{\displaystyle \frac{e_{4i}^2}{T_{2i}}}+e_{4i}{\beta }_{2i}\hfill \end{array}$$

(23)

To properly handle the cross-coupling terms and establish a strict negative definite boundary, Young’s inequality is applied. Based on Assumption 1 and the bounds of $\beta$, the intermediate variable transformations are executed as follows:

$$\left\{\begin{array}{l}-e_{1i}{\dot{d}}_{1i}\le {\displaystyle \frac{1}{2}}{e_{1i}}^2+{\displaystyle \frac{1}{2}}{{\dot{d}}_{1i}}^2\le {\displaystyle \frac{1}{2}}{e_{1i}}^2+{\displaystyle \frac{1}{2}}{D}_{1i}^2\\ e_{3i}{\beta }_{1i}\le {\alpha }_{1i}{e}_{3i}^2+{\displaystyle \frac{{\beta }_{1i}^2}{4{\alpha }_{1i}}}\le {\alpha }_{1i}{e}_{3i}^2+{\displaystyle \frac{B_{1i}^2}{4{\alpha }_{1i}}}\end{array}\right.$$

By substituting these intermediate bounds back, the derivative of the Lyapunov function can be further simplified as:

$$\begin{array}{l}{\dot{V}}_i\le -{\displaystyle \frac{k_{1i}}{\sqrt{x_{ei}^2+y_{ei}^2+C_i}}}{x}_{ei}^2-{\displaystyle \frac{k_{2i}}{\sqrt{x_{ei}^2+y_{ei}^2+C_i}}}{y}_{ei}^2-{\sigma }_{1i}{s}_{1i}^2-{\sigma }_{2i}{s}_{2i}^2-\left(L_{1i}-{\displaystyle \frac{1}{2}}\right)e_{1i}^2\hfill \\ -\left(L_{2i}-{\displaystyle \frac{1}{2}}\right)e_{2i}^2-\left({\displaystyle \frac{1}{T_{1i}}}-{\alpha }_{1i}\right)e_{3i}^2-\left({\displaystyle \frac{1}{T_{2i}}}-{\alpha }_{2i}\right)e_{4i}^2+{\displaystyle \frac{1}{2}}{\dot{d}}_{1i}^2+{\displaystyle \frac{1}{2}}{\dot{d}}_{2i}^2+{\displaystyle \frac{B_{1i}^2}{4{\alpha }_{1i}}}+{\displaystyle \frac{B_{2i}^2}{4{\alpha }_{2i}}}\hfill \\ \le -{\mu }_i{V}_i+N_i\hfill \end{array}$$

(24)

where ${\alpha }_{1i},{\alpha }_{2i}>0$ are positive constants; and the system convergence rate ${\mu }_i$ and the bounded constant $N_i$ are defined as:

$$\begin{array}{l}{\mu }_i=2\min \left\{\begin{array}{l}{\displaystyle \frac{k_{1i}}{\sqrt{x_{ei}^2+y_{ei}^2+C_i}}},{\displaystyle \frac{k_{2i}}{\sqrt{x_{ei}^2+y_{ei}^2+C_i}}},{\sigma }_{1i},{\sigma }_{2i},\\ L_{1i}-0.5,L_{2i}-0.5,{\displaystyle \frac{1}{T_{1i}}}-{\alpha }_{1i},{\displaystyle \frac{1}{T_{2i}}}-{\alpha }_{2i}\end{array}\right\}\\ N_i={\displaystyle \frac{1}{2}}{D}_{1i}^2+{\displaystyle \frac{1}{2}}{D}_{2i}^2+{\displaystyle \frac{B_{1i}^2}{4{\alpha }_{1i}}}+{\displaystyle \frac{B_{2i}^2}{4{\alpha }_{2i}}}\end{array}$$

(25)

Solving this differential inequality ${\dot{V}}_i\le -{\mu }_i{V}_i+N_i$ yields:

$$0\le V_i\le {\displaystyle \frac{N_i}{{\mu }_i}}+\left[V_i(0)-{\displaystyle \frac{N_i}{{\mu }_i}}\right]e^{-{\mu }_{i}t}$$

(26)

It is worth noting that the Lyapunov function defined in Equation (22) simultaneously incorporates both the position/sliding surface errors ($x_{ei}$, $y_{ei}$, $s_{1i}$, $s_{2i}$) and the observer errors ($e_{3i}$, $e_{4i}$) as well as the disturbance estimation errors ($e_{1i}$, $e_{2i}$). This unified Lyapunov framework explicitly accounts for the observer-controller interaction: the disturbance estimation errors enter the sliding surface dynamics as bounded perturbations (treated via Assumption 1), and their attenuation by the NDO directly reduces the required switching gain magnitude in the STA, thereby jointly guaranteeing closed-loop SGUUB.

According to Lyapunov stability theory, the system exhibits exponential convergence within a domain centered at the origin with a radius of $N_i/{\mu }_i$. Analysis confirms that $B_{1i}$ and $B_{2i}$ are positive definite bounded parameters, while ${\dot{d}}_{1i}$ and ${\dot{d}}_{2i}$ represent the time-varying rates of disturbances and unmodeled dynamics. Given bounded energy assumptions (Assumption 1), strict upper bounds $D_{1i}$ and $D_{2i}$ exist for the lumped disturbance rates. By increasing ${\alpha }_{1i}$ and ${\alpha }_{2i}$, the magnitude of $N_i$ is effectively compressed. Furthermore, systematic adjustment of filter time constants, observer bandwidths, and sliding mode reaching coefficients yields a larger convergence rate ${\mu }_i$. Consequently, the convergence radius $N_i/{\mu }_i$ is maintained at a minimal level to ensure that all error terms asymptotically converge to a compact hypersphere. Specifically, from Equation (25), the radius of this ultimate boundedness set can be explicitly quantified as $r_i=\sqrt{N_i/{\mu }_i}$, where $N_i$ and ${\mu }_i$ are defined in terms of the controller gains, observer bandwidth, and disturbance bounds. This quantitative characterization confirms that the convergence region can be made arbitrarily small by appropriately increasing the cooperative gains $k_x$, $k_y$, the observer gains $L_{1i}$, $L_{2i}$, and the sliding mode reaching coefficients. Theoretical derivation strictly verifies that the proposed USV formation error system achieves semi-globally uniform ultimate boundedness (SGUUB) [45]. It explicitly demonstrates how underactuated USVs can accurately track dynamic desired trajectories while maintaining cooperative error bounds, thereby mathematically guaranteeing high-precision robust formation control.

5. Simulation Verification and Results Analysis

To validate the effectiveness, robustness, and superior anti-chattering performance of the proposed fuzzy adaptive super-twisting sliding mode control based on dynamic coordinated error correction, a multi-USV formation cooperative control simulation system is developed. To rigorously respond to practical marine engineering challenges, this section employs a comprehensive ablation study framework under realistic physical constraints.

5.1. Simulation Parameter Settings and Realistic Marine Environment Modeling

The simulation system comprises three underactuated USVs with identical dynamic characteristics. The simulation time step is set to dt = 0.1 s. To ensure the authenticity and reproducibility of the numerical simulations, the specific hydrodynamic and hull parameters of the underactuated USVs are directly adopted from the established benchmark model in [17].

In contrast to conventional ideal simulations, this study constructs a highly realistic marine environment by introducing multiple stringent physical constraints. To evaluate the robustness of the proposed composite anti-disturbance mechanism against complex sea states, a hybrid disturbance model is constructed. This model simultaneously incorporates low-frequency sinusoidal variations (ocean currents), high-frequency stochastic Gaussian white noise $d_{wave}(t)$ (random wave loads), and sudden step signals $d_{gust}(t)$ (transient wind gusts). The specific expressions for the lumped external disturbances are defined as:

$$\left\{\begin{array}{l}\begin{array}{l}{X}_d(t)=8\sin (0.1t)+1+d_{X_{w}ave}(t)+d_{X_{g}ust}(t) \\ Y_d(t)=0.1\cos (0.1t+\pi /3)+d_{Y_{w}ave}(t)+d_{Y_{g}ust}(t) \\ N_d(t)=4\sin (0.1t+\pi /4)+1+d_{N_{w}ave}(t)+d_{N_{g}ust}(t)\end{array}\hfill \end{array}\right.$$

(27)

To evaluate the real-time feasibility of the proposed controller, its computational complexity was assessed. The proposed algorithm consists of a Nonlinear Disturbance Observer, a first-order low-pass filter, a fuzzy inference system, and a Super-Twisting sliding mode law. All operations involve only scalar multiplications, additions, and lookup-table evaluations, without any matrix inversions or high-dimensional integrals. On a standard Intel Core i7 CPU, the average execution time per control step was measured to be approximately $2.3159$ ms, which is significantly lower than the simulation time step of $dt=0.1$ s (100 ms). This confirms a $43.2$$\times$ computational margin, demonstrating that the proposed controller is fully compatible with real-time implementation on embedded USV hardware, where typical control loop frequencies range from $10$ Hz.

Furthermore, to reflect the hardware limitations in practical implementations, Gaussian white noise is injected into the kinematic feedback loops to simulate sensor measurement inaccuracies. An actuator physical saturation constraint is also imposed, strictly limiting the maximum thrust and yaw moment to $\pm 500 \mathrm{N}$ and $\pm 500 \mathrm{N}\cdot \mathrm{m}$, respectively. Finally, a transport delay of τ_d = 0.1 s is applied to the state information transmitted between neighboring USVs to simulate the latency inherent in maritime wireless ad hoc networks. By integrating these severe non-ideal factors, the simulation environment closely mimics actual sea trials, providing a rigorous testbed for evaluating the proposed algorithms.

5.2. Ablation Study Scenario Design

To systematically validate the performance enhancements brought by each proposed module (the dynamic cooperative mechanism, the super-twisting algorithm, and the fuzzy adaptive tuning), an ablation study is designed.

5.2.1. Trajectory and Formation Geometry

The USV formation is commanded to track a persistent sinusoidal trajectory defined by $x_d(t)=0.5t$ and $y_d(t)=40\sin (0.012t)$, which simulates complex maneuvering in restricted waters. The formation is required to maintain a rigid equilateral triangular configuration with a side length of L = 30 m. Relative to the formation center in the body-fixed frame, the position offsets for the leader (USV1) and followers (USV2, USV3) are assigned as $[17.32, 0]$, $[-8.66,15]$, and $[-8.66,-15]$, respectively.

To evaluate the transient convergence capability of the controllers, the initial states of the three USVs are set with predefined positional deviations from the ideal formation structure, specified as follows:

$$\begin{array}{l}\left[x_1,y_1,{\psi }_1,u_1,v_1,r_1\right]=\left[12.5,12.0,\pi /4,0.01,0,0\right]\\ \left[x_2,y_2,{\psi }_2,u_2,v_2,r_2\right]=\left[-16.6,4.8,\pi /4,0.01,0,0\right]\\ \left[x_3,y_3,{\psi }_3,u_3,v_3,r_3\right]=\left[4.1,-16.8,0,0.01,0,0\right]\end{array}$$

5.2.2. Definition of Ablation Scenarios

Four comparative scenarios are established to isolate and evaluate the contribution of specific control modules. To ensure the fairness and rigor of the ablation study, all shared kinematic and dynamic control parameters across the four groups are kept strictly consistent, as detailed in Table 1. The specific differences for the control variables are defined as follows:

Proposed Method (Complete Algorithm): The fuzzy adaptive super-twisting sliding mode control with dynamic cooperative error correction is fully activated. The cooperative error gains are set to $k_x=100$ and $k_y=20$ to provide strong formation rigidity against disturbances.

Baseline 1 (Fixed-gain STA): The fuzzy logic tuning mechanism is disabled. To ensure comparable initial reaching speeds for a fair comparison, the fixed super-twisting gains are empirically selected as ${\epsilon }_{1i}=3.0$ and ${\epsilon }_{2i}=5.0$. This baseline primarily highlights the anti-chattering advantage of the fuzzy adaptive system.

Baseline 2 (Fuzzy First-order SMC): The hidden integral action of the super-twisting algorithm is replaced by a conventional $K\mathrm{sign}(s)$ reaching law. To withstand the same intensity of composite disturbances, the robust switching gains are configured as $K_{1i}=50$ and $K_{2i}=50$. This highlights the superiority of STA in generating smooth, continuous control signals.

Baseline 3 (Non-cooperative): The dynamic cooperative error terms in the kinematic outer loop are deactivated by setting $k_x=0$ and $k_y=0$. The USVs track their respective virtual reference points independently. This demonstrates the necessity of state coupling for maintaining structural rigidity during maneuvers and sudden disturbances.

The specific numerical values of the controller and observer parameters used in the simulation are summarized in

Table 2. Control parameter settings for the ablation study.

Term	USV1	USV2	USV3	Term	USV1	USV2	USV3
$k_1$	2.7	4.0	2.7	${\lambda }_1$	0.1	0.1	0.1
$k_2$	0.18	1.5	0.1	${\lambda }_2$	50	50	50
$T_1$	0.1	0.1	0.1	${\lambda }_3$	0.01	0.01	0.01
$T_2$	0.1	0.1	0.1	${\sigma }_1$	0.1	0.1	0.1
$L_1$	10	10	10	${\sigma }_2$	0.1	0.1	0.1
$L_2$	5	10	10	$D$	8	8	8
${\alpha }_1$	0.1	0.1	0.1	$C$	10,000	10,000	10,000
${\alpha }_2$	0.1	0.1	0.1

.

The control gains are selected following a hierarchical tuning procedure. The kinematic gains $k_1$, $k_2$ are first tuned to ensure adequate position error convergence. The sliding mode reaching gains ($\epsilon$, $\sigma$) and observer bandwidths ($L_1$, $L_2$) are then set to ensure the stability conditions derived in Section 4 are satisfied with sufficient margin. The fuzzy membership functions are defined empirically to cover the operating range of the sliding surface. Due to the structural robustness provided by the NDO and the fuzzy adaptive mechanism, the closed-loop performance is observed to be insensitive to moderate gain perturbations.

5.3. Simulation Results and Discussion

To comprehensively evaluate the control performance, the simulation results are analyzed from qualitative graphical perspectives (Figure 5, Figure 6, Figure 7 and Figure 8) and quantitative metrics. A comprehensive statistical comparison of the four scenarios is summarized in

Table 3. Quantitative performance comparison of the ablation study.

Algorithms	Position RMSE (m)	Formation RMSE (m)	Chattering (RMS dU)
Proposed	0.7416	0.2545	0.6996
Baseline 1 (Fixed-gain STA)	3.5581	1.9011	1.7806
Baseline 2 (Fuzzy SMC)	1.2673	0.7887	2.2378
Baseline 3 (No Cooperative)	2.0889	2.0861	0.7251

.

Position RMSE evaluates the absolute trajectory tracking accuracy. Formation RMSE evaluates the formation rigidity and topology maintenance. Chattering (RMS dU) represents the root mean square of the control increments, punishing high-frequency jumps.

5.3.1. Trajectory Tracking and Formation Rigidity Analysis

The two-dimensional trajectory tracking performances of the four control schemes are illustrated in Figure 5.

Figure 5. Trajectory tracking and formation-keeping performances: (a) Proposed method; (b) Baseline 1 (Fixed-gain STA); (c) Baseline 2 (Fuzzy SMC); (d) Baseline 3 (No Cooperative).

Figure 5. Trajectory tracking and formation-keeping performances: (a) Proposed method; (b) Baseline 1 (Fixed-gain STA); (c) Baseline 2 (Fuzzy SMC); (d) Baseline 3 (No Cooperative).

As shown in Figure 5a, the proposed method achieves accurate trajectory tracking. The equilateral triangular formation is rigidly maintained despite the presence of composite environmental disturbances. Figure 5b,c show the trajectories for Baseline 1 (fixed-gain STA) and Baseline 2 (Fuzzy SMC), respectively. While they can follow the general path, their trajectories exhibit noticeable oscillations compared to the proposed method. More importantly, Figure 5d demonstrates the result of Baseline 3, where the cooperative error mechanism is deactivated. In Figure 5d, when USV2 encounters a sudden simulated wind gust at approximately t = 250 s, it is blown significantly off course. Since USV1 and USV3 lack the cooperative feedback to adjust their movements accordingly, the formation topology is completely destroyed.

This observation is supported by the quantitative spatial performance in Figure 6.

Figure 6. Quantitative spatial performance analysis of the multi-USV formation: (a) Absolute position tracking error; (b) Relative distances.

Figure 6. Quantitative spatial performance analysis of the multi-USV formation: (a) Absolute position tracking error; (b) Relative distances.

Figure 6a presents the absolute position tracking error ($e_{xy}$) for all three USVs. The proposed algorithm (blue solid line) converges quickly and maintains an error near zero. Figure 6b shows the relative distances ($d_{ij}$) between neighboring USVs. The proposed method strictly confines the relative distances near the desired 30 m. As detailed in Table 3, the proposed method achieves a Formation RMSE of 0.2545 m. Without the cooperative mechanism (Baseline 3), the Formation RMSE increases to 2.0861 m. This confirms that incorporating neighborhood state synchronization into the kinematic loop is essential for maintaining the structural rigidity of the multi-USV formation under asymmetric disturbances.

5.3.2. Velocity Tracking and Control Smoothness Analysis

As seen in Figure 7a (surge velocity error) and Figure 7b (sway velocity error), the proposed method ensures rapid convergence with minimal fluctuation. Conversely, Baseline 1 suffers from severe velocity oscillations throughout the simulation. This indicates that a fixed high-gain switching term struggles to adapt to the time-varying nature of marine disturbances. Baseline 2 also shows noticeable steady-state errors and oscillations.

Figure 7. Velocity tracking errors under different control schemes: (a) Surge velocity error; (b) Sway velocity error.

Figure 7. Velocity tracking errors under different control schemes: (a) Surge velocity error; (b) Sway velocity error.

The fundamental cause of these velocity oscillations is evident in the control inputs. Figure 8 depicts the surge thrust ($X_{P1}$) and yaw moment ($N_{R1}$) applied to the actuators of USV1.

Figure 8. Control inputs and chattering analysis for USV1: (a) Surge thrust; (b) Yaw moment.

Figure 8. Control inputs and chattering analysis for USV1: (a) Surge thrust; (b) Yaw moment.

In Figure 8a,b, Baseline 2 (Fuzzy SMC) induces persistent high-frequency chattering due to the use of a conventional signum function. Baseline 1 (Fixed-gain STA) utilizes conservatively large fixed gains to handle worst-case disturbances, which leads to violent actuator oscillations. As seen in Figure 8b, the grey line frequently hits the physical saturation limits (±30 N·m).

By integrating fuzzy logic with the super-twisting algorithm, the proposed method dynamically adjusts the switching gains based on the sliding mode reaching state. As observed in Figure 8, the proposed controller produces smooth and continuous control commands, avoiding high-frequency chattering and respecting actuator saturation limits. According to Table 3, the Chattering (RMS dU) metric for the proposed method is 0.6996, which is significantly lower than Baseline 1 (1.7806) and Baseline 2 (2.2378). In summary, the ablation study demonstrates that the proposed fuzzy adaptive super-twisting sliding mode control with cooperative error correction successfully balances robustness and control smoothness.

5.4. Robustness Test Against Communication Uncertainty

To verify the robustness of the proposed framework against realistic communication uncertainties, a stress test featuring a 20% random packet dropout rate is conducted under extreme gust disturbances. The proposed algorithm is evaluated under these degraded communication conditions and compared with the ideal transmission scenario.

As depicted in Figure 9, the formation successfully follows the desired path even with 20% packet loss. Furthermore, Figure 10 reveals that despite the intermittent loss of neighbor states, the relative distances remain strictly bounded around the desired offsets. This robust performance demonstrates that the cross-layer cooperative term can effectively maintain formation rigidity under communication uncertainties, proving its parameter insensitivity and high practical value.

5.5. Comparative Analysis with State-of-the-Art Algorithm

To further demonstrate the superiority of the proposed method, an additional simulation is conducted against a recent State-of-the-Art methodology: the Fixed-Time Sliding Mode Control (FxTSMC). The reaching law of the implemented FxTSMC baseline is constructed as $\dot{s}=-k_1|s{|}^{\alpha }\mathrm{sgn}(s)-k_2|s{|}^{\beta }\mathrm{sgn}(s)$, with fractional parameters $\alpha =0.5$ and $\beta =1.2$.

As shown in Figure 11, both methods achieve excellent path tracking and formation keeping. However, a critical divergence is revealed in their control inputs (Figure 12). Relying on the discontinuous signum function and a super-linear term ($\beta >1$), the FxTSMC induces severe, high-frequency control chattering under discrete digital implementation, which is unacceptable for physical actuators. In contrast, by incorporating the Super-Twisting integral action, the proposed method completely eliminates this destructive chattering while maintaining the same high-precision SOTA tracking performance, highlighting its distinct engineering practicality.

6. Conclusions

To solve the problems of formation dispersion and high-frequency chattering in the control of underactuated USVs under time-varying marine disturbances, this paper introduces a fuzzy adaptive super-twisting sliding mode control scheme combined with dynamic cooperative error correction. In the kinematic outer loop, a coordinated error correction mechanism using nearby information is proposed to improve the dynamic interaction, thus effectively strengthening the formation rigidity and accuracy during complicated maneuvers. In the dynamic inner loop, a composite anti-disturbance structure that includes a nonlinear disturbance observer and differential smoothing is used to carry out precise feedforward compensation for low-frequency disturbances while effectively preventing the explosion of complexity and noise amplification that usually occurs in backstepping control. Moreover, a fuzzy logic system is designed to optimize the gains in real-time in order to avoid the occurrence of actuator chattering due to discontinuous switching rules. The simulation results show that the proposed method can make the system states converge quickly and maintain stable formation configurations. It performs well in producing smooth control signals and reducing the mechanical wear on the propulsion system. Rather than a simple combination of existing approaches, this framework achieves a deep structural integration by directly coupling topological consensus errors into the kinematic loop and eliminating the dependency on predefined disturbance bounds. Furthermore, extensive robustness verifications, including a severe 20% packet loss scenario and a comparative analysis with a state-of-the-art fixed-time controller, explicitly confirm the engineering practicality of the proposed method in complex communication and physical environments. However, the current research has not yet considered the internal collision constraints and external obstacle avoidance in complex restricted areas. In the future, efforts will be made to incorporate multi-constraint intelligent obstacle avoidance mechanisms into the coordination of USV swarms, along with verification and practical application on real ships.