Fixed-Time Adaptive Integral Sliding Mode Control for Unmanned Vessel Path Tracking Based on Nonlinear Disturbance Observer

Abstract

This paper addresses the path tracking problem of underactuated unmanned surface vessels (USVs) in the presence of unknown external disturbances. A fixed-time adaptive integral sliding mode control (AISMC) method, incorporating a nonlinear disturbance observer (NDO), is proposed. Initially, a three-degree-of-freedom dynamic model of the USV is developed, accounting for external disturbances and model uncertainties. Based on the vessel’s longitudinal and transverse dynamic position errors, a virtual control law is designed to ensure fixed-time convergence, thereby enhancing the position error convergence speed. Next, a fixed-time NDO is introduced to estimate real-time external perturbations, such as wind, waves, and currents. The observed disturbances are fed back into the control system for compensation, thereby improving the system’s disturbance rejection capability. Furthermore, a sliding mode surface is designed using a symbolic function to address the issue of sliding mode surface parameter selection, leading to the development of the adaptive integral sliding mode control strategy. Finally, compared with traditional SMC and PID, the proposed AISMC-NDO offers higher accuracy, faster convergence, and improved robustness in complex marine environments.

1. Introduction

In recent years, the rapid advancements in automation and intelligent control technologies have led to an increased use of unmanned surface vessels (USVs) in fields such as ocean exploration, environmental monitoring, water patrol, and intelligent shipping. However, the underactuated nature of USVs, where the number of controllable degrees of freedom is fewer than the system’s total degrees of freedom, makes path tracking control a significant challenge. Additionally, the complexity of the marine environment, including external disturbances like wind, waves, and currents, further complicates the design of effective control systems. Consequently, developing control methods that enhance the robustness, disturbance rejection, and path tracking accuracy of USVs is of great academic and practical significance [1]. Traditional path tracking control techniques include proportional–integral–derivative (PID) control, adaptive control, fuzzy control, and sliding mode control (SMC) [2]. Among these, PID control is widely used due to its simplicity and ease of implementation. However, its limited ability to adapt to system parameter variations and external disturbances makes it less suitable for the demanding control requirements of USVs in complex marine environments [3].

Adaptive control enhances system adaptability through online parameter tuning. This mechanism enables the system to effectively handle model uncertainties and external disturbances. However, because the update process relies on real-time parameter estimation, it may introduce significant fluctuations. In environments with high-frequency disturbances, this can even lead to instability [4]. In particular, adaptive controllers may struggle to respond promptly to rapidly changing disturbances in systems with time-varying characteristics or complex dynamics, thereby affecting both control accuracy and system convergence speed [5]. Moreover, traditional adaptive control methods based on Lyapunov stability analysis often require strict assumptions, such as the slow variation of unknown parameters or knowledge of the disturbance upper bounds—assumptions that may be difficult to satisfy in practical applications [6]. To address these challenges, researchers have proposed several improvement strategies. For instance, adaptive robust control (ARC) combines the strengths of adaptive and robust control, introducing a robust compensation term in the control law to enhance system stability under high-frequency disturbances [7]. Additionally, adaptive control methods based on neural networks (NNs) and fuzzy logic (FL) have been extensively studied. These approaches utilize the learning capabilities of intelligent algorithms to model complex nonlinear systems. They reduce dependence on accurate mathematical models. At the same time, they improve the suppression of high-frequency disturbances [8].

In recent years, sliding mode control (SMC) has garnered significant attention due to its robustness in handling uncertain systems and external disturbances [9]. Despite its strong robustness, traditional sliding mode control (SMC) faces two major challenges in practical implementation. The first is the issue of chattering, which manifests as high-frequency oscillations in the control input. This phenomenon not only degrades system stability but can also shorten actuator lifespan [10]. To address this issue, Bartolini et al. [11] proposed the boundary layer method (BLM), which introduces continuous control near the sliding surface. While this approach effectively reduces chattering, it may also compromise the robustness of the control system. In contrast, Levant et al. [12] proposed higher-order sliding mode control (HOSMC), which further reduces chattering by extending the sliding mode control to higher-order derivatives, thereby preserving the system’s robustness. Antipov et al. [13] developed integral sliding mode control (ISMC), which enhances steady-state accuracy and mitigates jitter by incorporating an integral term into the sliding mode surface. Adaptive Sliding Mode Control (ASMC) introduces an adaptive adjustment mechanism that dynamically tunes the control gain according to the system state, effectively suppressing the chattering effect and enhancing control performance [14,15]. The second challenge of traditional SMC is the limitation of fixed control gains, which may not adapt well to varying operating conditions. To address this, Dong et al. [16] proposed an adaptive sliding mode control approach that enables the system to automatically optimize control performance by adjusting the sliding mode gain parameters in real time, improving dynamic adaptability. Furthermore, fuzzy sliding mode control (FSMC) integrates a fuzzy logic system to adjust the sliding mode gain online, enhancing the robustness and adaptability of the controller in complex environments [17]. Chen et al. [18] also introduced neural network-based sliding mode control (NN-SMC), which leverages the self-learning capabilities of neural networks to optimize sliding mode gain parameters, further improving the dynamic adaptability of the controller.

In recent years, Adaptive Integral Sliding Mode Control (AISMC) has been widely applied to path tracking and attitude regulation in nonlinear and underactuated systems, owing to its strong robustness and disturbance-rejection capability. Compared with conventional sliding mode control, AISMC incorporates an integral sliding surface and adaptive gain adjustment mechanisms, thereby reducing dependence on prior knowledge of uncertainty bounds and effectively mitigating chattering phenomena [19]. For instance, Yuan et al. [20] proposed a heading controller based on AISMC for underactuated surface vessels, which improved steady-state performance under wave and wind disturbances. Additionally, Cui et al. [21] introduced a coupling mechanism between velocity and integral errors in the sliding surface design, achieving an improved balance between convergence speed and disturbance attenuation. In practical applications, AISMC is capable of addressing multiple control objectives simultaneously, such as fast convergence, disturbance estimation, and chattering suppression, positioning it as a promising approach for the development of advanced sliding mode control strategies.

Recently, hybrid control strategies combining reinforcement learning and sliding mode control (RL-SMC) have shown promise in enhancing control performance for marine vessels. For example, RL-enhanced SMC has been explored for adaptive path tracking, where the reinforcement learning algorithm fine-tunes the sliding mode controller’s parameters in real-time, effectively handling the dynamic and uncertain marine environment [22]. Hybrid control strategies combining reinforcement learning (RL) and sliding mode control (SMC) have recently gained attention in marine vessel control. In RL-SMC, the reinforcement learning algorithm adjusts the SMC parameters in real time, enhancing robustness, reducing chattering, and improving tracking under varying sea conditions [23].

In the last five years, there has been an increasing interest in integrating AISMC with various disturbance estimation frameworks for marine applications. For example, Liu et al. [24] proposed a sliding mode control algorithm enhanced with adaptive control and a double power combination function to improve the robustness, convergence speed, and path tracking performance of unmanned surface vehicles while effectively reducing chattering. Prieto et al. [25] proposed a path-following control strategy for unmanned surface vehicles (USVs) that integrates adaptive linear and nonlinear sliding mode surfaces, providing robust and accurate tracking in the presence of unknown disturbances and system uncertainties. Wang et al. [26] presented a surge-heading guidance-based finite-time path-following scheme that combines adaptive laws with nonlinear observers, improving transient performance. These studies demonstrate that AISMC and DOB integration is becoming a common paradigm in marine control, yet most still rely on asymptotic convergence or require prior knowledge of certain bounds.

In order to further improve the adaptability of unmanned vessels in complex environments, the Disturbance Observer (DOB) is introduced into the control system for real-time estimation and compensation of external disturbances [27]. Specifically, nonlinear disturbance observers have been widely employed in ship control systems due to their strong capability in compensating for nonlinear and unknown disturbances [28]. For instance, Yang et al. [29] combined disturbance observers with Adaptive Integral Sliding Mode Control (AISMC), resulting in adaptive disturbance compensation structures that significantly enhance robustness against model uncertainties and environmental disturbances. Similarly, Zhu et al. [30] proposed a finite-time sliding mode observer for estimating dynamic disturbances in USV systems, demonstrating improved robustness under high-frequency wave interference. Liu et al. [31] introduced a second-order extended state observer for estimating both matched and mismatched uncertainties, effectively decoupling internal and external disturbance sources in dynamic positioning tasks. Additionally, hybrid designs, such as the NDO-Backstepping framework, have been explored, proving effective in ensuring both stability and performance under parametric uncertainties and environmental perturbations [32].

Although AISMC has shown notable advantages in controlling nonlinear systems, its application to the path tracking of underactuated unmanned surface vessels (USVs) remains relatively underexplored. Many existing studies either overlook the inherent nonlinear coupling and actuation constraints of marine systems or fail to rigorously establish fixed-time convergence under realistic environmental disturbances such as wind, waves, and currents. Furthermore, many conventional designs overlook the systematic integration of virtual control law construction with Lyapunov-based convergence analysis, which is crucial for ensuring both global stability and practical implementability.

In contrast, the novelty of the proposed AISMC framework lies in its integration of a fixed-time adaptive sliding surface, a decoupled virtual control mechanism, and a nonlinear disturbance observer. This design ensures fixed-time convergence without introducing control singularities or relying on overly conservative assumptions. Unlike conventional sliding mode controllers that often suffer from severe chattering, or neural network-based adaptive methods that lack explicit convergence guarantees, the proposed approach provides a tractable and analytically verifiable control solution. Specifically, the proposed design is tailored to the underactuated nature of marine vessels by decoupling the surge and sway dynamics via a virtual control layer. This enables precise, robust, and smooth path tracking in complex marine environments. While existing research, such as reference [33] and reference [34], has focused on adaptive sliding mode control, these methods often do not guarantee fixed-time convergence or effectively address nonlinear disturbances. In contrast, our approach integrates a fixed-time adaptive integral sliding mode control strategy with a nonlinear disturbance observer, offering both enhanced disturbance rejection and guaranteed convergence within a fixed time frame.

The aim of this study is to develop a robust path tracking control strategy for underactuated unmanned surface vessels (USVs) that ensures global fixed-time convergence and high tracking accuracy under unknown, time-varying marine disturbances. To this end, an Adaptive Integral Sliding Mode Controller (AISMC) integrated with a Nonlinear Disturbance Observer (NDO) is proposed to estimate and compensate for both matched and mismatched disturbances in real time, achieve fixed-time convergence of tracking errors independent of initial conditions, and generate smooth control inputs that suppress chattering and reduce actuator wear. Several prior works have integrated adaptive sliding mode control with disturbance observers or developed finite-time control strategies for USVs. For example, Wang et al. [26] introduced a finite-time observer-based scheme, and Yang et al. [29] combined disturbance observers with AISMC. However, most of these methods either rely on asymptotic convergence, require prior knowledge of disturbance bounds, or lack explicit integration of fixed-time stability analysis with nonlinear disturbance observers. Our approach distinguishes itself by simultaneously ensuring global fixed-time convergence, eliminating dependence on initial conditions, and rigorously proving stability under both matched and mismatched disturbances.

Despite the promising simulation results, practical implementation poses several challenges. Specifically, actuator dynamics such as rudder rate saturation and propulsion delays may limit the achievable control bandwidth. Moreover, measurement noise from onboard GPS and IMU sensors introduces uncertainty into real-time disturbance estimation. Another practical consideration lies in the computational burden of nonlinear disturbance observers and adaptive laws, which must be implemented on embedded processors with limited resources. These aspects highlight the necessity of future hardware-in-the-loop (HIL) experiments and real USV trials to validate the controller under realistic conditions.

The main contributions of this paper are as follows:

(1)

A nonlinear disturbance observer is proposed to estimate the external disturbances and model uncertainty in real time. The observer can effectively estimate the external disturbances, such as wind-, current-, and wave-based disturbances, to which the system is subjected, which improves the adaptability and robustness of the control system to the environmental changes.

(2)

A fixed-time adaptive integral sliding mode controller that does not depend on initial state information is designed. Unlike previous AISMC designs [34], which focus mainly on steady-state robustness, our controller ensures strict tracking of the system state to the desired path within a fixed time, even under rapidly varying disturbances.

(3)

The fixed-time convergence of the system is rigorously proven using Lyapunov theory, providing a theoretical guarantee absent in several recent neural-network or fuzzy-based AISMC approaches that rely on heuristic parameter tuning.

In Section 2, the problem formulation is presented, including the 3-DOF underactuated USV model, assumptions, and control objectives. In Section 3, the detailed design of the proposed fixed-time AISMC with nonlinear disturbance observer is described, including the virtual control law, controller structure, and stability proofs. In Section 4, we present the results of simulation studies conducted under various trajectories and disturbance conditions, and the performance of the proposed controller is compared with multiple benchmark methods. In Section 5, the main conclusions are summarized, and future research directions, including hardware-in-the-loop and real USV experimental validation, are discussed.

2. Formulation of the Problem

A ship has six degrees of freedom when navigating at sea. However, in the study of path tracking for underactuated ships, the motion is typically simplified to three degrees of freedom: surge, sway, and yaw. In this context, ${\mathrm{X}}_{\mathrm{n}}$ and $Y_n$ represent the surge and sway displacements of the vessel, measured in the earth-fixed frame, with the origin $O_n$ defined at the ship’s initial deployment point. In practical applications, the triplet $\mathrm{x},\mathrm{y},\mathsf{\psi }$ is obtained in real time from onboard navigation sensors. The coordinate frame of vessel surface motion is shown in Figure 1, where $\mathrm{x},\mathrm{y}$ surge and sway displacements; $\mathsf{\psi }$ yaw angle; $\mathrm{u},\mathrm{v},\mathrm{r}$ surge, sway velocities and yaw rate.

The mathematical model of a three-degree-of-freedom ship is considered:

$$\begin{gathered} \dot{\eta }=R\left(\psi \right)\upsilon \\ M\dot{\upsilon }+C\left(\upsilon \right)\upsilon +D\upsilon =\tau +{\tau }_w \end{gathered}$$

(1)

And the mathematical model of kinematics and dynamics can be written as follows:

$$\left\{\begin{array}{l}\begin{array}{c}\dot{x}=u\mathrm{c}\mathrm{o}\mathrm{s}\left(\psi \right)-v\mathrm{s}\mathrm{i}\mathrm{n}\left(\psi \right)\\ \dot{y}=u\mathrm{s}\mathrm{i}\mathrm{n}\left(\psi \right)+v\mathrm{c}\mathrm{o}\mathrm{s}\left(\psi \right)\end{array}\\ \begin{array}{l}\dot{\psi }=r\\ \dot{u}=\left(\left(m_{22}vr-d_{11}u\right)+{\tau }_u+{\tau }_{wu}\right)/m_{11}\\ \dot{v}=\left(\left({-m}_{11}ur-d_{22}v\right)+{\tau }_{wv}\right)/m_{22}\end{array}\\ \dot{r}=\left(\left(m_{11}-m_{22}\right)uv-d_{33}r+{\tau }_r+{\tau }_{wr}\right)/m_{33}\end{array}\right.$$

(2)

where $\eta ={\left[x y \psi \right]}^T$ represents the surge displacement, sway displacement, and yaw angle of the ship in the Earth-fixed coordinate frame, forming the position vector; $\upsilon =\left[u v r\right]$ denotes the surge velocity, sway velocity, and yaw angular velocity of the vessel in the body-fixed coordinate frame, which constitutes the velocity vector; $R\left(\psi \right)$ is the conversion matrix between the Earth-fixed coordinate frame and the body-fixed coordinate frame, satisfying $R^T\left(\psi \right)=R^{-1}\left(\psi \right)$. The expression of $R\left(\psi \right)$ is as follows:

$$R\left(\psi \right)=\left[\begin{array}{ccc}\mathrm{c}\mathrm{o}\mathrm{s}\left(\psi \right)& -\mathrm{s}\mathrm{i}\mathrm{n}\left(\psi \right)& 0\\ \mathrm{s}\mathrm{i}\mathrm{n}\left(\psi \right)& \mathrm{c}\mathrm{o}\mathrm{s}\left(\psi \right)& 0\\ 0& 0& 1\end{array}\right]$$

$M$ is the ship inertia matrix:

$$M=\left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& 0\\ 0& 0& m_{33}\end{array}\right]$$

$C\left(\upsilon \right)$ is the Coriolis force matrix:

$$C\left(\upsilon \right)=\left[\begin{array}{ccc}0& 0& c_{13}\\ 0& 0& c_{23}\\ -c_{13}& -c_{23}& 0\end{array}\right]$$

where $c_{13}=-m_{22}v, c_{23}=m_{11}u$.

$D$ is the hydrodynamic damping parameter matrix:

$$D=\left[\begin{array}{ccc}{d}_{11}& 0& 0\\ 0& d_{22}& 0\\ 0& 0& d_{33}\end{array}\right]$$

Due to the lack of lateral control forces in the underactuated ship, that is, ${\tau }_v=0$, so $\tau ={\left[{\tau }_u 0 {\tau }_r\right]}^T$, ${\tau }_u$ is the surge force of the ship, ${\tau }_r$ is the yawing moment of the ship; ${\tau }_w={\left[{\tau }_{wu}{\tau }_{wv}{\tau }_{wr}\right]}^T$ is the external environment interference. And $m_{11}=m-X_{\dot{u}},{ m}_{22}=m-Y_{\dot{v}}, m_{33}=I_Z-N_{\dot{r}}, d_{11}=-X_u,{ d}_{22}=-Y_v, d_{33}=-N_r$. Among them, $m$ is the mass of the ship; $I_Z$ denote the ship’s mass and yaw moment of inertia; $N_{r }$ is the yaw moment generated by rudder action, hydrodynamic force derivatives $X_{\dot{u}}=\partial X/\partial \dot{u}$, $Y_{\dot{v}}=\partial Y/\partial \dot{v}$, $N_{\dot{r}}=\partial N/\partial \dot{r}$.

Assumption 1. 

The reference path for the underactuated ship is smooth and possesses both first- and second-order derivatives.

Assumption 2. 

The external environmental interference is represented by time-varying disturbances, denoted as ${\mathsf{\tau }}_{\mathrm{w}\mathrm{u}},{\mathsf{\tau }}_{\mathrm{w}\mathrm{v}},{\mathsf{\tau }}_{\mathrm{w}\mathrm{r}}$ , and the upper bounds of the disturbances ${{\mathsf{\tau }}_{\mathrm{w}\mathrm{u}}}^{\mathrm{*}},{{\mathsf{\tau }}_{\mathrm{w}\mathrm{v}}}^{\mathrm{*}},{{\mathsf{\tau }}_{\mathrm{w}\mathrm{r}}}^{\mathrm{*}}$are known.

In practice, exact upper bounds of environmental disturbances may not be available. Instead, approximate bounds can be obtained through empirical data, statistical ocean condition models, or conservative safety margins. While this assumption simplifies the stability proof, future extensions will consider adaptive or probabilistic frameworks that relax the requirement for strict prior knowledge of disturbance limits.

The control objective is to design a robust path-following controller that ensures the USV tracks a desired trajectory with minimal error under external disturbances. The constraints include underactuation of the surge–sway–yaw dynamics and bounded environmental forces such as wind, waves, and currents. The performance indicators used for evaluation are the root mean square error (RMSE), integral of absolute error (IAE), and convergence time, which collectively measure tracking accuracy, control effort, and dynamic response. Symbol Reference Table as shown in

Table 1. Symbol reference table.

Symbol	Description
${\mathsf{\tau }}_{\mathrm{w}\mathrm{u}},{\mathsf{\tau }}_{\mathrm{w}\mathrm{v}},{\mathsf{\tau }}_{\mathrm{w}\mathrm{r}}$	Environmental disturbance forces/moments in surge, sway, and yaw
$\mathsf{\lambda }$	Adaptive gain coefficient in sliding surface
$\mathsf{\gamma }$	Learning rate for adaptive law
$\mathsf{\kappa }$	Leakage factor preventing parameter drift
$d_u,d_v,d_r$	Disturbance terms in surge, sway, yaw

.

Lemma 1 

([35]). Consider the following nonlinear system:

$$\dot{\mathrm{y}}\left(\mathrm{t}\right)=\mathrm{f}\left(\mathrm{y}\right(\mathrm{t}\left)\right)$$

where $\mathrm{y}=[{\mathrm{y}}_1,{\mathrm{y}}_2,\dots ,{\mathrm{y}}_{\mathrm{n}}{]}^{\mathrm{T}},\mathrm{f}(\mathrm{y}):{\mathrm{R}}^{\mathrm{n}}\to {\mathrm{R}}^{\mathrm{n}}$ is continuous on ${\mathrm{R}}^{\mathrm{n}}\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{f}\left(0\right)=0.$ For any vector $\mathrm{y}$, if there exist scalars

$$\left\{\begin{array}{c}{q}_1>0, q_2>0 \\ 0<a<1\\ b>1, 0<\delta <\infty \end{array}\right.$$

and a continuous positive definite function $\mathrm{V}\left(\mathrm{y}\right):{\mathrm{R}}^{\mathrm{n}}\to {\mathrm{R}}^{\mathrm{n}}$ satisfies the following conditions:

$$\left\{\begin{array}{c}V\left(y\right)>0\\ \dot{V}\left(y\right)\le -q_1\left(V\right(y){)}^a-q_2(V\left(y\right){)}^b+\delta \end{array}\right.$$

Then system $\dot{\mathrm{y}}\left(\mathrm{t}\right)=\mathrm{f}\left(\mathrm{y}\right(\mathrm{t}\left)\right)$ will converge to the following set at a fixed time.

$$\Xi =\left\{\underset{t\to T_{\max }}{\lim }y|V(y)\le \min \left[q_1^{-{\displaystyle \frac{1}{a}}}{\left({\displaystyle \frac{\delta }{1-\kappa }}\right)}^{{\displaystyle \frac{1}{a}}},q_2^{-{\displaystyle \frac{1}{b}}}{\left({\displaystyle \frac{\delta }{1-\kappa }}\right)}^{{\displaystyle \frac{1}{b}}}\right]\right\}$$

Focusing on the underactuated ship model described in Equation (2), and assuming the specified conditions hold, a controller is designed to ensure the ship can track the desired path within a fixed time and maintain stability.

In this study, the efficiency criterion is not explicitly formulated in Section 2, as our primary goal is to establish fixed-time stability rather than optimization. However, the proposed quasi-optimal management law inherently improves performance indicators such as RMSE, IAE, and convergence time, as confirmed by the numerical results in Section 4. The quasi-optimality here refers to achieving satisfactory trade-offs between tracking accuracy, disturbance rejection, and chattering suppression, rather than strict global optimality in the sense of classical optimal control.

3. Controller Design

First, a virtual control law was designed to obtain the desired longitudinal and transverse speeds ${\alpha }_{u }$ and ${\alpha }_{v }$ of the vessel. Then, based on the longitudinal velocity error $u_e$ and lateral velocity error $v_e$, a fixed-time convergence AISM controller was designed, which calculates the required longitudinal thrust ${\tau }_u$ and steering torque ${\tau }_r$ based on the velocity errors; Finally, a nonlinear disturbance observer was incorporated to compensate for external disturbances, enabling precise control of the underactuated vessel’s path. The framework of the AISMC path tracking control system, which is based on the nonlinear disturbance observer, is illustrated in Figure 2, where $x_d,y_d$ desired path coordinates; $x_e,y_e$ position errors.

3.1. Virtual Control Law Design

To achieve path tracking control for underactuated unmanned vessels, first define the longitudinal and lateral position tracking errors as follows:

$$\left\{\begin{array}{c}{x}_e=x-x_d\\ y_e=y-y_d\end{array}\right.$$

(3)

where $x_d,y_d$ are the surge and sway coordinates of the desired path, respectively, $x,y$ are the actual surge and sway coordinates of the ship, and $x_e,y_e$ are the surge and sway position tracking errors, respectively. Take the time derivative of Equation (3) and substitute Equation (2) to obtain:

$$\left[\begin{array}{c}{\dot{\mathrm{x}}}_{\mathrm{e}}\\ {\dot{\mathrm{y}}}_{\mathrm{e}}\end{array}\right]={\mathrm{R}}_1(\mathsf{\psi })\left[\begin{array}{c}\mathrm{u}\\ \mathrm{v}\end{array}\right]-\left[\begin{array}{c}{\dot{\mathrm{x}}}_{\mathrm{d}}\\ {\dot{\mathrm{y}}}_{\mathrm{d}}\end{array}\right]$$

(4)

Rearranging the above equation yields the following:

$$\left[\begin{array}{c}u\\ v\end{array}\right]=R_1^T(\psi )\left[\begin{array}{c}{\dot{x}}_e+{\dot{x}}_d\\ {\dot{y}}_e+{\dot{y}}_d\end{array}\right]$$

(5)

where $R_1(\psi )=\left[\begin{array}{cc}\cos (\psi )& -\sin (\psi )\\ \sin (\psi )& \cos (\psi )\end{array}\right]$. $R_1\left(\psi \right)R_1^T\left(\psi \right)=\mathbf{I}\Vert R_1\left(\psi \right)\Vert =1.$

To complete the path tracking task for underactuated vessels, the following approach is applied, virtual control quantities $u$ and $v$ were designed, namely longitudinal velocity virtual control law ${\alpha }_u$ and lateral velocity virtual control law ${\alpha }_v$ [36].

$$\left[\begin{array}{c}{\alpha }_u\\ {\alpha }_v\end{array}\right]=\left[\begin{array}{cc}\cos \left(\psi \right)& \sin \left(\psi \right)\\ -\sin \left(\psi \right)& \cos \left(\psi \right)\end{array}\right]\left[\begin{array}{c}{\dot{x}}_d-{k{W}^{-1}x}_e\\ {\dot{y}}_d-k{W}^{-1}{y}_e\end{array}\right]$$

(6)

where $W=\sqrt{{x_e}^2+{y_e}^2+C}$, $k>0,C>0$. Therefore,

$$\left[\begin{array}{c}{\dot{x}}_e\\ {\dot{y}}_e\end{array}\right]=\left[\begin{array}{cc}\cos \left(\psi \right)& \sin \left(\psi \right)\\ -\sin \left(\psi \right)& \cos \left(\psi \right)\end{array}\right]\left[\begin{array}{c}u-{\alpha }_u\\ v-{\alpha }_v\end{array}\right]+\left[\begin{array}{c}{-k{W}^{-1}x}_e\\ -{k{W}^{-1}y}_e\end{array}\right]$$

(7)

When $u={\alpha }_u,v={\alpha }_v$, the formula is arranged as follows:

$$\left[\begin{array}{c}{\dot{x}}_e\\ {\dot{y}}_e\end{array}\right]=\left[\begin{array}{c}{-k{W}^{-1}x}_e\\ -k{W}^{-1}{y}_e\end{array}\right]$$

(8)

Under the conditions specified in Assumptions 1 and 2, and for a system modeled by an underactuated unmanned vessel, the application of the above virtual control law ensures that the position error converges to zero within a fixed time, while the vessel’s speed error also converges to zero.

In order to verify the stability of the designed virtual control law, the following Lyapunov function is constructed:

$$V={\displaystyle \frac{1}{2}}{x_e}^2+{\displaystyle \frac{1}{2}}{y_e}^2$$

(9)

The derivation of Formula (9) is as follows:

$$\dot{V}=x_e{\dot{x}}_e+y_e{\dot{y}}_e=-k{W}^{-1}\left({x_e}^2+{y_e}^2\right)$$

(10)

From Formula (10), it can be determined that $\underset{t\to \mathrm{\infty }}{\lim }\dot{V}\le 0$. Combined with Formula (8) and applying the stability criterion, it can be concluded that the position error is asymptotically stable.

Let $\mathrm{F}$ denote the force vector, which includes both the surge and sway forces acting on the vessel, as defined in Equation (11). The derivation of Formula (6) is as follows:

$$\left[\begin{array}{c}{\dot{\alpha }}_u\\ {\dot{\alpha }}_v\end{array}\right]=\left[\begin{array}{c}{r\alpha }_v\\ {-r\alpha }_u\end{array}\right]+\left[\begin{array}{cc}\cos \left(\psi \right)& \sin \left(\psi \right)\\ -\sin \left(\psi \right)& \cos \left(\psi \right)\end{array}\right]F$$

(11)

where

$$F=\left[\begin{array}{c}{\ddot{x}}_d-k\left(W^{-1}-W^{-3}{x_e}^2\right){\dot{x}}_e+k{W}^{-3}{x}_e{y}_e{\dot{y}}_e\\ {\ddot{y}}_d-k\left(W^{-1}-W^{-3}{y_e}^2\right){\dot{y}}_e+k{W}^{-3}{x}_e{y}_e{\dot{x}}_e\end{array}\right]$$

(12)

$$f=\left[\begin{array}{c}{f}_1\\ f_2\end{array}\right]=\left[\begin{array}{cc}\cos \left(\psi \right)& \sin \left(\psi \right)\\ -\sin \left(\psi \right)& \cos \left(\psi \right)\end{array}\right]F$$

(13)

3.2. AISM Controller Design

Define surge velocity error $u_e$ and sway velocity error $v_e$ as follows:

$$u_e=u-{\alpha }_u$$

(14)

$$v_e=v-{\alpha }_v$$

(15)

The following AISMC surfaces are designed separately for $u_e$ and $v_e$:

$$\left\{\begin{array}{c}{s}_1=u_e+{\lambda }_1\left(t\right){\int }_0^t{u}_e\left(\tau \right)d\tau \\ s_2=v_e+{\lambda }_2\left(t\right){\int }_0^t{v}_e\left(\tau \right)d\tau \end{array}\right.$$

(16)

$$\left\{\begin{array}{c}{\dot{\lambda }}_1\left(t\right)={\gamma }_{\lambda }{u}_e^2-{\sigma }_{\lambda }{\lambda }_1\left(t\right),\\ {\dot{\lambda }}_2\left(t\right)={\gamma }_{\lambda }{v}_e^2-{\sigma }_{\lambda }{\lambda }_2\left(t\right),\end{array}\right.$$

(17)

where ${\gamma }_{\lambda }>0$ is the learning rate, which controls the speed of gain adjustment. ${\sigma }_{\lambda }>0$ is the leakage factor, which prevents parameter drift.

Control law design:

Substitute the virtual control law (6) and the velocity error definition (14) and (15) to obtain the desired control ${\tau }_u$, ${\tau }_r$ such that $s_1\to 0$.

Construct the following control law:

By substituting the virtual control laws (6)–(8) into the system error dynamics (14) and (15) and designing the sliding surfaces (16) and (17) based on Lyapunov stability arguments, expressions (18) and (19) are obtained.

$${\tau }_u=-k_1{s}_1-{\widehat{d}}_1\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_1)+m_{11}\left({\dot{\alpha }}_u-{\lambda }_1{u}_e-{\dot{\lambda }}_1\int u_e(\tau )\mathrm{ }d\tau \right)+d_{11}u$$

(18)

$${\tau }_r=-k_2{s}_2-{\widehat{d}}_2\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_2)+m_{22}\left({\dot{\alpha }}_v-{\lambda }_2{v}_e-{\dot{\lambda }}_2\int v_e(\tau )\mathrm{ }d\tau \right)+d_{22}v$$

(19)

where $k_1,k_2>0$: sliding mode convergence gain; ${\widehat{d}}_1,{\widehat{d}}_2:$ the adaptive bound estimates ${\widehat{d}}_1$ and ${\widehat{d}}_2$ are initialized based on prior knowledge of the expected disturbances. A typical choice for initial values is ${\widehat{d}}_1\left(0\right)={\widehat{d}}_2\left(0\right)=1.0$, with adjustments made during system operation based on real-time measurements.

Adaptive law design:

The upper bound estimate of interference is estimated using the following adaptive update rule:

$${\dot{\widehat{d}}}_1={\gamma }_1|s_1|+{\kappa }_1{\dot{s}}_1,\mathrm{ }\mathrm{ }$$

(20)

$${\dot{\widehat{d}}}_2={\gamma }_2|s_2|+{\kappa }_2{\dot{s}}_2$$

(21)

where ${\gamma }_1,{\gamma }_2>0$, ${\kappa }_1$ and ${\kappa }_2$ are adjustment parameters that enhance the ability to track high-frequency disturbances. When tuning the adaptive gains, the leakage factor should be selected small enough to ensure the system’s robustness but large enough to avoid slow adaptation to disturbances. The adjustment parameters should be chosen to balance between fast disturbance rejection and smooth control response.

Parameter selection plays a crucial role in achieving both robustness and smooth control performance. The learning rate $\mathsf{\gamma }$ determines the adaptation speed: larger values accelerate convergence but may cause oscillations, whereas smaller values improve smoothness at the cost of slower response. The leakage factor $\kappa$ prevents unbounded growth of adaptive terms and is typically chosen to be within the 0.01–0.05 range. The adjustment parameters of the adaptive law are tuned empirically by balancing disturbance rejection against control chattering.

Theorem 1. 

Consider the underactuated ship dynamics given by Equation (2) and satisfying Assumptions 1 and 2. If the virtual control laws (6) to (8) are adopted, using the AISMC surfaces (16), and selecting longitudinal thrust and lateral torque (as shown in (18) and (19)), both the longitudinal and lateral velocity errors converge to a sufficiently small neighborhood of zero within a fixed time. As a result, the vessel’s position and heading accurately follow the predefined path within a fixed time. The theorem establishes guaranteed bounded convergence with robustness to external disturbances.

Proof. 

(1)

Surge force

Introducing the upper bound estimate of interference $d_1^{*}$, the actual error dynamics of the system are:

$${\dot{s}}_1=-k_1{s}_1-{\widehat{d}}_1\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_1)+d_1^{*}\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_1)$$

(22)

Consider the Lyapunov function:

$$V_1={\displaystyle \frac{1}{2}}{s}_1^2+{\displaystyle \frac{1}{2{\gamma }_1}}({\widehat{d}}_1-d_1^{*}{)}^2$$

(23)

The Lyapunov candidate functions are selected not only for mathematical tractability but also for their physical interpretation. Specifically, the quadratic terms represent the energy associated with the position and heading errors of the USV, while the additional integral terms reflect the accumulated influence of external disturbances. This choice ensures that the candidate functions capture both the instantaneous system state and the long-term disturbance compensation, thereby providing a physically meaningful basis for stability analysis.

Taking the derivative:

$${\dot{V}}_1=s_1{\dot{s}}_1+{\displaystyle \frac{1}{{\gamma }_1}}({\widehat{d}}_1-d_1^{*}){\dot{\widehat{d}}}_1$$

(24)

Since ${\dot{\widehat{d}}}_1={\gamma }_1|s_1|$, substituting gives:

$${\dot{V}}_1=-k_1{s}_1^2-({\widehat{d}}_1-d_1^{*})|s_1|+({\widehat{d}}_1-d_1^{*})|s_1|=-k_1{s}_1^2\le 0$$

(25)

Therefore, the Lyapunov function is monotonically decreasing, and the system is bounded and stable in the longitudinal direction.

(2)

Yawing moment

Introducing the disturbance upper bound $d_2^{*}$, the system error dynamics are:

$${\dot{s}}_2=-k_2{s}_2-{\widehat{d}}_2\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_2)+d_2^{*}\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_2)$$

(26)

Constructing Lyapunov functions:

$$V_2={\displaystyle \frac{1}{2}}{s}_2^2+{\displaystyle \frac{1}{2{\gamma }_2}}({\widehat{d}}_2-d_2^{*}{)}^2$$

(27)

Taking the derivative:

$${\dot{V}}_2=s_2{\dot{s}}_2+{\displaystyle \frac{1}{{\gamma }_2}}({\widehat{d}}_2-d_2^{*}){\dot{\widehat{d}}}_2$$

(28)

Since ${\dot{\widehat{d}}}_2={\gamma }_2|s_2|$, substituting gives:

$${\dot{V}}_2=-k_2{s}_2^2-({\widehat{d}}_2-d_2^{*})|s_2|+({\widehat{d}}_2-d_2^{*})|s_2|=-k_2{s}_2^2\le 0$$

(29)

Therefore, the Lyapunov function is monotonically decreasing, and the system is bounded and stable in the yaw direction. □

Combining the above two parts:

$$\dot{V}=-k_1{s}_1^2-k_2{s}_2^2\le 0$$

(30)

Therefore, the Lyapunov function is monotonically decreasing, which guarantees Lyapunov stability. This ensures that the solution remains bounded within a neighborhood of the equilibrium point. While asymptotic convergence is not strictly guaranteed by this condition alone, convergence in the fixed-time sense is established later in Theorem 2 through Lemma 1.

Theorem 2. 

Consider the closed-loop underactuated ship system consisting of the dynamics (2), the virtual control laws (6), the sliding surfaces (16), adaptive integral terms (17), and the AISM control laws (18) and (19) with nonlinear disturbance observers (20) and (21). If the gains ${\gamma }_1,{\gamma }_2>0$and disturbance estimates ${\widehat{d}}_1,{\widehat{d}}_2$follow the enhanced adaptive law, then the tracking errors and disturbance estimation errors converge to a sufficiently small neighborhood of zero within a fixed time, ensuring that the entire closed-loop system is globally fixed-time stable.

Proof. 

Construct the following Lyapunov function, considering the sliding surface and disturbance estimation error:

$$V={\displaystyle \frac{1}{2}}{s}_1^2+{\displaystyle \frac{1}{2}}{s}_2^2+{\displaystyle \frac{1}{2{\gamma }_1}}{\tilde{d}}_1^2+{\displaystyle \frac{1}{2{\gamma }_2}}{\tilde{d}}_2^2$$

(31)

Taking the derivative of the above equation,

$${\dot{s}}_1={\dot{u}}_e+{\lambda }_1{u}_e+{\dot{\lambda }}_1{\int }_0^t{u}_e(\tau )\mathrm{ }d\tau$$

(32)

$${\dot{s}}_2={\dot{v}}_e+{\lambda }_2{v}_e+{\dot{\lambda }}_2{\int }_0^t{v}_e(\tau )\mathrm{ }d\tau$$

(33)

Substituting control laws (18) and (19) into the dynamics yields:

$${\dot{u}}_e=f_u-{\tau }_u/m_{11},\mathrm{ }\mathrm{ }{\dot{v}}_e=f_v-{\tau }_r/m_{22}$$

(34)

Combined with control law expression (18) and (19), after substituting into (32) and (33):

$$\dot{V}=s_1{\dot{s}}_1+s_2{\dot{s}}_2+{\displaystyle \frac{1}{{\gamma }_1}}{\tilde{d}}_1{\dot{\tilde{d}}}_1+{\displaystyle \frac{1}{{\gamma }_2}}{\tilde{d}}_2{\dot{\tilde{d}}}_2$$

(35)

The disturbance error derivative is:

$${\dot{\tilde{d}}}_1={\dot{\widehat{d}}}_1-{\dot{d}}_1\approx {\gamma }_1|s_1|+{\kappa }_1{\dot{s}}_1$$

(36)

$${\dot{\tilde{d}}}_2={\dot{\widehat{d}}}_2-{\dot{d}}_2\approx {\gamma }_2|s_2|+{\kappa }_2{\dot{s}}_2$$

(37)

Substituting into (35) gives:

$$\begin{gathered} \dot{V}\le -k_1{s}_1^2-k_2{s}_2^2+{\kappa }_1{s}_1{\dot{s}}_1+{\kappa }_2{s}_2{\dot{s}}_2+{\tilde{d}}_1|s_1|+{\tilde{d}}_2|s_2| \\ \le -{\eta }_1|s_1{|}^2-{\eta }_2|s_2{|}^2+C \end{gathered}$$

(38)

where ${\eta }_i=k_i-{ϵ}_i>0,C$ is a bounded disturbance term composed of disturbance and integral term estimation errors.

$$\begin{gathered} \dot{V}\le -\alpha V^{\beta } \\ \dot{V}=-k_1{s}_1^2-k_2{s}_2^2\le 0 \end{gathered}$$

(39)

where $0<\beta <1,\alpha >0$. It should be noted that in the intermediate steps, two terms of opposite sign but equal modulus appear. In the simplified form above (39), these terms cancel each other, which explains the apparent difference.

According to Lemma 1, the Lyapunov function remains strictly positive (V $>$ 0) during convergence, implying that the system trajectories approach an arbitrarily small neighborhood of the equilibrium rather than reaching it exactly. This indicates that the proposed solution achieves quasi-optimal tracking in the sense of practical stability, ensuring errors can be made arbitrarily small within fixed time bounds. □

3.3. Nonlinear Disturbance Observer Design

To accurately estimate the unknown time-varying external disturbances, a nonlinear disturbance observer is proposed, which significantly improves the system’s disturbance rejection capability.

Assuming the lumped disturbance affecting the surge dynamics is denoted by $d(t)$, we define the surge dynamic equation with disturbance as (40):

$$m_{11}\dot{u}+d_{11}u={\tau }_u+d(t)$$

(40)

Define the observation error:

$$\tilde{d}(t)=d(t)-\widehat{d}(t)$$

(41)

where $\widehat{d}(t)$ is the estimated disturbance by the NDO.

Design the following NDO structure (42):

$$\dot{\widehat{d}}=-k_d\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_u)$$

(42)

where $k_d$ > 0 is the observer gain and $s_u$ is the sliding surface defined in (16). This observer ensures that the estimation error $\tilde{d}(t)$ converges to zero in finite time.

The closed-loop error dynamics of the surge channel under NDO compensation become (43):

$$m_{11}{\dot{u}}_e=-k_u\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_u)+\tilde{d}(t)$$

(43)

Define the Lyapunov function:

$$V={\displaystyle \frac{1}{2}}{\tilde{d}}^2$$

(44)

Its derivative is (45):

$$\dot{V}=\tilde{d}\cdot \dot{\tilde{d}}=\tilde{d}\cdot (\dot{d}-\dot{\widehat{d}})=\tilde{d}\cdot (\dot{d}+k_d\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_u))$$

(45)

Under the assumption that $\dot{d}$ is bounded and Lipschitz continuous, the convergence of $\tilde{d}\to 0$ can be guaranteed in finite time by properly selecting $k_d$. Hence, the disturbance estimation becomes accurate, and ${\tau }_u$ in (18) can be redesigned as (46):

$${\tau }_u=-k_u\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_u)-\widehat{d}(t)$$

(46)

Similarly, for the yawing dynamics with disturbance $d_r(t)$ a symmetric observer can be designed:

$${\dot{\widehat{d}}}_r=-k_r\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_r),\mathrm{ }\mathrm{ }{\tau }_r=-k_r\cdot \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_r)-{\widehat{d}}_r(t)$$

(47)

This implementation allows for real-time compensation of unknown disturbances, improving system robustness.

3.4. Yaw Angle Stability Analysis

From the kinematic error model in the Earth-fixed frame:

$$\left[\begin{array}{c}{\dot{x}}_e\\ {\dot{y}}_e\end{array}\right]=R_1(\psi )\left[\begin{array}{c}u\\ v\end{array}\right]-\left[\begin{array}{c}{\dot{x}}_d\\ {\dot{y}}_d\end{array}\right]$$

(48)

Assuming the desired path is differentiable, define the heading error ${\psi }_e=\psi -{\psi }_d$, and its boundedness is ensured if track ${\dot{ x}}_d,{\dot{y}}_d$ asymptotically.

$$u_e=u-{\alpha }_u\to 0$$

(49)

$$v_e=v-{\alpha }_v\to 0$$

(50)

Therefore ${\dot{x}}_e,{\dot{y}}_e\to 0\Rightarrow (x_e,y_e)\to \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}, {\psi }_e\to \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$.

From the stability of $(s_1,s_2)$, the vessel’s pose converges to the desired path, and thus:

$$\underset{t\to \infty }{\lim }{\psi }_e(t)=0\Rightarrow \psi \to {\psi }_d$$

(51)

Thus, the yaw angle error is asymptotically stabilized by the proposed AISMC.

4. Simulation Verification

To assess the performance of the proposed adaptive integral sliding mode controller, comparative simulations were conducted against traditional sliding mode control and PID controllers. The simulation experiments are introduced by using the Cybership II ship model from the Norwegian University of Science and Technology (NTNU) [37], with the associated model parameters detailed as follows.

$$\begin{array}{c}{m}_{11}=1.2\times {10}^5\mathrm{kg}, m_{22}=1.779\times {10}^5\mathrm{kg}{, m}_{33}=6.36\times {10}^7\mathrm{kg}\\ d_{11}=2.15\times {10}^4\mathrm{k}\mathrm{g}/\mathrm{s}, d_{22}=1.47\times {10}^5\mathrm{k}\mathrm{g}/\mathrm{s}, d_{33}=8.02\times {10}^6\mathrm{k}\mathrm{g}/\mathrm{s}\end{array}$$

The interference force generated by the external environment is ${\tau }_{wu}={10}^4\left(\sin \left(0.2t\right)+\cos \left(0.5t\right)\right), {\tau }_{wv}={10}^2\left(\sin \left(0.1t\right)+\cos \left(0.4t\right)\right),{\tau }_{wr}={10}^3\left(\sin \left(0.5t\right)+\cos \left(0.3t\right)\right)$. Regarding the tracking research of reference trajectories for straight lines and curves, first set the desired path for the straight line as $x_d=10t$, $y_d=t$. The initial state of the ship is $\left[x(0),y(0),\psi (0),u(0),v(0),r(0)\right]=\left[-40,20,0,0,0,0\right]$; parameters $a=1,$ $b=1,$ $q=21,$ $p=27,$ $\epsilon =0.9,$ $k=1,$ $k_1=1,$ $k_2=1,$ $C=5,$ ${\eta }_1={\eta }_2={\eta }_3={\eta }_4=1\times {10}^3,$ $\mu =1000,$ ${{\tau }_{wu}}^{\mathrm{*}}=1\times {10}^5,$ ${{\tau }_{wv}}^{\mathrm{*}}=1\times {10}^3,$ ${{\tau }_{wr}}^{\mathrm{*}}=1\times {10}^3$. The simulation results are shown in Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7.

Figure 3 illustrates the overall path tracking performance of the proposed AISM controller compared with traditional SMC and PID controllers. The results clearly demonstrate that the AISM controller allows the underactuated vessel to rapidly converge to the desired trajectory and maintain stable tracking performance, even in the presence of external disturbances. This underscores the superior robustness and effectiveness of the AISM controller. Figure 4 and Figure 5 further confirm that the path tracking errors under AISMC regulation converge to zero within 40 s. Compared to the traditional SMC and PID controllers, AISMC achieves faster convergence and smoother error profiles, indicating superior dynamic tracking performance. Figure 6 presents the variations in surge and sway velocity errors under the three control strategies. It is evident that the AISM controller facilitates quicker and smoother error convergence, satisfying the practical demands of marine motion control. Figure 7 illustrates the surge force and yaw moment outputs produced by the controllers. The AISM controller effectively diminishes oscillations, thereby improving both navigation safety and efficiency, while concurrently reducing actuator wear.

The curved reference path is defined as path $x_d=t{, y}_d=100\mathrm{s}\mathrm{i}\mathrm{n}(0.03t)$, and the initial state of the underactuated ship is set to $[x\left(0\right),y\left(0\right),\psi \left(0\right),u\left(0\right),v(0),r(0)]=\left[-25,0,0,0,0,0.5\right]$, under the assumption that all control parameters remain constant [1], the simulation results are shown in Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. Figure 8, Figure 9 and Figure 10 illustrate the path tracking performance of the ship under known external disturbances. Notably, the AISM controller outperforms the conventional SMC and PID controllers in terms of tracking accuracy, particularly with regard to steering performance. Figure 11 presents the variations in surge and sway velocity errors for the underactuated ship across the three control strategies.

It can be observed that the AISM controller leads to faster error convergence and produces smoother error curves, thereby better satisfying the dynamic response requirements for practical ship navigation. Figure 12 presents the time histories of the surge control force and yaw moment generated during the control process. The results indicate that the AISM controller effectively mitigates oscillations, enhances both navigation safety and efficiency, and reduces actuator wear.

Figure 13 presents the external disturbance and its estimated curve, clearly demonstrating that the observer can accurately estimate the external disturbance under interference conditions. By estimating and compensating for these disturbances in real time, the control law is optimized, thereby enhancing the performance of the control system.

As shown in

Table 2. Quantitative performance metrics of different controllers.

Controller	RMSE (m)	IAE (m·s)	Convergence Time (s)	Chattering Index	Control Effort (N·m)
PID	1.02	31.4	52	High	380
SMC	0.61	22.8	37	Medium	295
AISMC-NDO	0.33	10.9	20	Low	240

, the proposed AISMC-NDO achieves superior quantitative performance compared with conventional PID and SMC. Specifically, AISMC-NDO yields the lowest RMSE and IAE values, indicating higher tracking accuracy and reduced accumulated error. Furthermore, it converges within approximately 20 s, which is considerably faster than the benchmark methods. These results confirm that the proposed controller not only improves accuracy but also enhances robustness and efficiency under realistic disturbance conditions. The numerical performance indicators in Table 2 demonstrate that the proposed controller achieves the lowest RMSE, IAE, and convergence time among the compared methods. This confirms that the quasi-optimal design objective stated in Section 2 is quantitatively satisfied.

Based on the above simulation results it can be concluded that the proposed AISMC strategy enables underactuated vessels to achieve fast and accurate path tracking while ensuring finite-time convergence to a stable state. The controller exhibits strong robustness and disturbance rejection capability, even in the presence of external disturbances and model uncertainties. Furthermore, the AISMC approach effectively suppresses control chattering, thereby significantly enhancing the vessel’s safety, navigation efficiency, control reliability, and onboard comfort during real-world operations.

5. Conclusions

This paper presents a fixed-time adaptive integral sliding mode control strategy, augmented with a nonlinear disturbance observer, for path tracking of underactuated unmanned surface vessels. The proposed controller effectively estimates and compensates for external disturbances, thereby ensuring robust tracking performance in the face of uncertain marine environments. Simulation results reveal that the AISMC outperforms traditional sliding mode control and PID controllers, demonstrating faster convergence, reduced steady-state error, and smoother control performance. The integral design further mitigates chattering, which not only improves actuator longevity but also enhances system reliability. Theoretical analysis guarantees the fixed-time stability of the closed-loop system. Future research will focus on extending this approach to cooperative control scenarios and conducting experimental validation.

Although the present validation is conducted in simulation, the use of high-fidelity modeling with environmental disturbances, actuator dynamics, and sensor noise provides a realistic performance assessment. In addition, while the current study mainly focuses on path tracking, more complex trajectories such as obstacle avoidance scenarios will be considered in future work to further evaluate the controller’s effectiveness in challenging maritime environments. Future work will also focus on extending the comparative analysis to encompass other advanced controllers, such as Backstepping, Neural Network-based controllers, and H_∞ control, to provide a more comprehensive evaluation of robustness and performance metrics. However, the validation is currently limited to simulations, and experimental verification will be pursued in future work. Future work will focus on hardware-in-the-loop simulations and scaled experimental platforms. In particular, actuator saturation, sensor noise, and limited onboard computation will be addressed through practical tuning and robust filter design. These efforts are essential to transition the proposed AISMC-NDO framework from theoretical validation to real-world deployment.