Frequency-Dependent Anisotropic Electromagnetic Responses of Fractured Reservoirs with Various Hydrate Distributions Based on Numerical Simulation

Abstract

This study develops a constrained nonlinear model predictive control (NMPC) framework, integrating rudder roll stabilization to address coupled path-following and collision avoidance challenges for underactuated surface vessels (USVs). The compact state-space model integrates both navigational states and roll dynamics through augmentation, facilitating real-time optimization of the trade-off between safety margins for roll movements and path-following accuracy. Given that excessive roll movement during obstacle avoidance in the USV path following can readily lead to USV capsizing, the NMPC approach is employed to explicitly address multiple constraints, including obstacle avoidance constraint, roll movement safety, and control input rudder angle constraints, thereby achieving precise path following for the rudder-roll reduction control system. Different from traditional methods that adhere to a pre-planned obstacle avoidance path, the proposed NMPC approach formulates obstacle avoidance as a nonlinear inequality constraint, significantly enhancing the maneuverability of the USV during obstacle avoidance. To validate the effectiveness of the proposed algorithm, the stability and optimality of the rudder-roll reduction control system are analyzed. The advantages of the proposed algorithm are ultimately demonstrated through both theoretical analysis and simulation results.

1. Introduction

The dynamic coupling between underactuated surface vessel (USV) roll dynamics and environmental forcing presents critical challenges to marine operational safety [1]. Hydrodynamic disturbances (wind–wave–current interactions) induce complex 3-DOF coupled oscillations that degrade navigational precision through parametric resonance effects, particularly in complex sea conditions. The crew performance degradation due to motion sickness incidence (MSI) exceeding the 20% threshold during moderate seas [2]. From a stability perspective, transient roll amplitudes surpassing 15° Response Amplitude Operator (RAO) significantly compromise metacentric stability margins, creating nonlinear positive feedback between roll acceleration and righting moment deficiency [3]. Consequently, the implementation of roll stabilization measures under adverse sea conditions is indispensable for mitigating these risks.

Contemporary marine stabilization strategies employ four principal hydrodynamic interventions: passive damping surfaces (bilge keels), fluidic momentum transfer systems (anti-roll tanks), angular momentum devices (gyroscopic stabilizers), and active flow control appendages [4,5]. While demonstrating hydrodynamic efficacy through scaled model testing (15–28% roll reduction in beam seas), these solutions impose spatial constraints that compromise hydrodynamic profiles, manifesting as up to 12% added resistance in full-scale trials [6]. Moreover, their implementation entails non-trivial capital expenditures (CAPEX) and lifecycle maintenance burdens, with total ownership costs exceeding $2.8 M over 15-year operational spans for mid-sized vessels [7].

The authors in reference [8] emphasize the pivotal role of rudders in governing yaw motion dynamics in USVs, while further demonstrating that rudder angle variations not only modulate yaw behavior but also generate auxiliary roll-inducing forces and moments on the hull structure. This phenomenon inherently contributes to enhanced roll stabilization. Building upon this principle, Cowlet and Lambert’s foundational work [9] established the conceptual framework for rudder roll stabilization (RRS). Subsequent investigations by Carley [10] delineated three principal challenges impeding RRS performance: (1) the inherent trade-off between roll attenuation efficacy and heading control precision, (2) the management of non-minimum phase system dynamics, and (3) compliance with rudder actuator saturation constraints. These insights have catalyzed the development of advanced control architectures for RRS implementation, encompassing linear quadratic regulators with automatic gain scheduling, multivariate autoregressive models, and radial basis function neural network-based controllers [11,12,13]. Nevertheless, current research exhibits two notable limitations: first, a predominant focus on roll stabilization at the expense of integrated system coordination, and second, the treatment of RRS and path-following control as decoupled subsystems [14]. While these methodologies demonstrate theoretical promise, their practical application remains constrained by inherent implementation complexities and unaddressed cross-coupling effects between stabilization and navigation objectives.

USVs are frequently required to perform precise path tracking, particularly in obstacle-rich environments [15]. Consequently, path-following methodologies have garnered significant attention from both academia and industry, especially with the increasing deployment of commercial autonomous vessels and autopilot systems aimed at enhancing crew efficiency [16,17]. But the operational reliability of maritime autonomous systems critically hinges on precise state estimation and robust multi-sensor integration, particularly when navigating dynamic environments with GPS denial or sensor constraints. State estimation involves reconstructing the USV’s kinematic state, including position, velocity, and attitude—along with environmental parameters such as wave forces and obstacle dynamics from noisy sensor data. The key methodologies include the following: (1) Extended Kalman Filter (EKF) [18]: a recursive algorithm that linearizes nonlinear system dynamics through Jacobian matrices, ideally suited for low-speed maneuvering scenarios in moderate sea states. (2) Sliding Mode Observers (SMOs) [19]: nonlinear estimation techniques demonstrating inherent robustness to bounded disturbances, effectively suppressing wave-induced lateral velocity errors through adaptive gain modulation. (3) Gaussian Process Regression (GPR) [20]: a probabilistic framework that compensates for persistent sensor deviations (e.g., IMU drift) by modeling measurement biases as temporally correlated stochastic processes, reducing position drift.

Traditional path planning methodologies, such as A*, rapidly exploring random trees (RRT), and artificial potential fields (APFs), exhibit significant limitations in USV applications due to their decoupled design [21]. However, this separation often yields suboptimal or kinematically infeasible solutions in dynamic environments. For instance, A* algorithms rely on static global maps and lack real-time updates for dynamic obstacles. RRT, while capable of high-dimensional exploration, produces stochastic and redundant paths that ignore under-actuation and actuator saturation constraints. APF methods are prone to local minimum and sensitivity to hydrodynamic disturbances. These approaches typically require post-processing techniques (e.g., B-spline smoothing) to convert geometric paths into dynamically feasible trajectories—a process that introduces computational latency and trajectory distortions, compromising real-time performance and safety.

Collision avoidance strategies must be integrated into the path planning and control architecture, especially in International Regulations for Preventing Collisions at Sea (COLREGs) compliance integration [22]. Conventional obstacle avoidance frameworks employ a hierarchical architecture integrating global and local path planning to generate collision-free trajectories spanning origin to destination. The global planning module, functioning as the strategic decision layer in autonomous marine navigation systems, synthesizes priori environmental data (e.g., digital nautical charts and bathymetric surveys) to compute optimal reference paths through multi-criteria optimization processes. Established algorithms, including A*, Dijkstra, and asymptotically optimal RRT*, variants demonstrate computational efficacy in minimizing navigation costs—quantified through path length, energy consumption, and regulatory adherence metrics—while ensuring global convergence properties [23]. Complementing this strategic layer, the tactical local planning subsystem performs real-time trajectory refinement within perceptually constrained operational envelopes. Sensor-driven methodologies such as APF and the Dynamic Window Approach (DWA) leverage heterogeneous data streams (LiDAR point clouds and radar returns) for reactive obstacle negotiation [24]. APF implementations construct virtual force fields through the superposition of repulsive obstacle potentials and attractive goal gradients, though their practical deployment is constrained by inherent limitations in non-convex environments—particularly local minima stagnation and oscillatory behaviors near complex obstacle configurations. DWA circumvents these issues through stochastic sampling of kinematically admissible velocity spaces, yet its effectiveness diminishes when applied to underactuated marine platforms like USVs, where actuator saturation and nonholonomic motion constraints fundamentally restrict maneuverability.

In addition, dynamic maritime environments introduce unique complexities that demand real-time and adaptive replanning. Specifically, (1) dynamic obstacles (e.g., drifting buoys and vessels) exhibit non-cooperative motion patterns, requiring continuous trajectory updates. (2) Stochastic wave forces, currents, and wind introduce lateral drift and roll-yaw coupling, destabilizing predefined paths. (3) Limited LiDAR/radar range and occlusions reduce situational awareness, necessitating predictive replanning. During sharp turns for obstacle avoidance, the combined effects of steering rudder force, heeling moment, and inertial centrifugal force exacerbate rolling motions, thereby increasing collision risks and compromising the seaworthiness and safety of path-following.

Model predictive control (MPC) has emerged as a prominent approach for path tracking due to its systematic handling of physical constraints [25]. Recent advances in nonlinear model predictive control (NMPC) enable real-time compensation of coupled roll-sway dynamics through rudder actuation re-purposing, effectively transforming conventional steering systems into multifunctional stabilization platforms. These innovative approaches overcome the mass and volume constraints of traditional physical stabilizers while maintaining payload capacity, offering a significant advantage for mission-specific adaptations in USV configurations. In [26], NMPC has been applied to path tracking of fully actuated autonomous surface crafts in the presence of constant ocean currents. Similarly, reference [27] employed NMPC for underactuated surface vessel tracking, leveraging the affine property of the system model and simplifying kinematics through frame transformations. However, these approaches can incur significant numerical errors for large prediction horizons. Abdelaal [28] proposed an NMPC scheme for position and velocity tracking of underactuated surface vessels, incorporating collision avoidance for static and dynamic obstacles, sideslip angle compensation, and environmental disturbance counteraction. While the approach in [28] embodies collision avoidance as a time-varying nonlinear constraint, it neglects the significant roll motion induced by large rudder angles during obstacle avoidance, focusing solely on minimizing heading and track errors without considering roll angle constraints, thus posing substantial safety risks in lateral motion.

This paper addresses these limitations by proposing a path-following control and collision avoidance scheme for USVs with rudder roll stabilization based on NMPC. The key contributions are as follows:

(1) Integrated 5-DOF Coupled Dynamics Modeling: A comprehensive model incorporating sway, yaw, and roll motions is developed. By integrating real-time position data, the 5-DOF USV is transformed into a compact state-space model suitable for path following and collision avoidance. This model enhances both the safety and accuracy of USV path following, particularly in capturing roll motion dynamics.

(2) NMPC-Integrated Obstacle Avoidance Scheme: The obstacle avoidance problem is reformulated as a state constraint within the NMPC optimization framework, implemented over a finite horizon. This approach represents a significant departure from existing methods, offering a robust and safety-oriented solution for the USV path following.

(3) Incorporation of Roll Safety Constraints: Roll safety constraints are explicitly considered during obstacle avoidance. A quadratic weighted sum and input optimization objective function is constructed, integrating rudder roll stabilization, path-following accuracy, and obstacle avoidance safety. This holistic approach ensures closed-loop system stability through contraction constraints, distinguishing it from prior works.

The remainder of this paper is organized as follows: Section 2 presents the USV path-following model coupled with rudder roll stabilization. Section 3 details the proposed NMPC with an obstacle avoidance scheme, including proofs of stability and optimality for the rudder-roll reduction control system. Section 4 provides detailed numerical simulations, and Section 5 concludes the paper and outlines future research directions.

2. USV Integrated Coupled Dynamics Modeling

In this paper, the path-following problem for the USV is investigated in both the Earth-fixed frame {E} and the USV body-fixed frame {B}, as illustrated in Figure 1. The Earth-fixed frame is denoted as $O_e{X}_e{Y}_e$, while the body-fixed frame is denoted as $O_b{X}_b{Y}_b$. Specifically, u represents the surge velocity (along the body-fixed x-axis), v represents the sway velocity (along the body-fixed y-axis), and r denotes the angular velocity in yaw around the body-fixed z-axis. Additionally, $\psi$ represents the heading angle of the USV with respect to the Earth-fixed frame.

Considering the objectives of roll damping and path following, the motions of pitch and heave can be neglected in comparison to the more significant motions of surge, sway, yaw, and roll. Fossen [23] summarizes a nonlinear mathematical model for a single-screw container ship (S175), which includes the surge velocity u, sway velocity v, roll rate p, and yaw rate r. The model is presented as follows:

$$\left\{\begin{array}{l}(m+m_x)\dot{u}-(m+m_y)vr=X\\ (m+m_y)\dot{v}+(m+m_x)ur+m_y{\alpha }_y\dot{r}-m_y{l}_y\dot{p}=Y\\ (I_x+J_x)\dot{p}-m_y{l}_y\dot{v}-m_x{l}_{x}ur+WGM\varphi =K\\ (I_z+J_z)\dot{r}+m_y{\alpha }_y\dot{v}=N-Y{x}_G\end{array}\right.$$

(1)

where $m$ denotes the USV mass, $m_x$ and $m_y$ are the added mass in the $x$ and $y$ directions, respectively, $I_x$ and $I_z$ denote the moment of inertia; $J_x$ and $J_z$ denote the added moment of inertia about the $x$ and $z$ axes, respectively. ${\alpha }_y$ denotes the x-coordinate of the center of $m_y$, $l_x$ and $l_y$ are the z-coordinates of the centers of $m_x$ and $m_y$; $W$ is the ship displacement, $GM$ is the metacentric height, and $x_G$ is the location of the center of gravity in the x-axis. Specific definitions of the hydrodynamic forces $X$, $Y$ and moments $K$, $N$ are given in [29].

It is assumed that the speed of the propeller thruster on the USV is constant, and thus the surge velocity can be considered constant. The motion model (1) can be transformed into the control-oriented dynamics model:

$$\left\{\begin{array}{l}\dot{v}=a_{11}v+a_{12}r+a_{13}p+a_{14}\varphi +b_1\delta \\ \dot{r}=a_{21}v+a_{22}r+a_{23}p+a_{24}\varphi +b_2\delta \\ \dot{\psi }=r\\ \dot{p}=a_{31}v+a_{32}r+a_{33}p+a_{34}\varphi +b_3\delta \\ \dot{\varphi }=p\end{array}\right.$$

(2)

where $a_{ij},\{i=1,2,3,4,j=1,2,3,4\}$, $b_1,b_2,b_3$ are constant parameters, $\delta$ is the rudder angle, also the control input. Moreover, the kinematic model of USV path following in planar motion can be described as follows:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\psi }\end{array}\right]=\left[\begin{array}{c}u\mathit{cos}\psi +v\mathit{sin}\psi \\ u\mathit{sin}\psi -v\mathit{cos}\psi \\ r\end{array}\right]$$

(3)

where u and v denote the surge speed and sway speed. For underactuated USV, the surge speed of the longitudinal thrusters is typically maintained at a constant. As a result, the continuous time invariant nonlinear state-space model of USV path following with rudder roll stabilization based on Equations (2) and (3) is presented as follows:

$$\left\{\begin{array}{l}\dot{\psi }=r\\ \dot{v}=a_{11}v+a_{12}r+a_{13}p+a_{14}\varphi +b_1\delta \\ \dot{r}=a_{21}v+a_{22}r+a_{23}p+a_{24}\varphi +b_2\delta \\ \dot{p}=a_{31}v+a_{32}r+a_{33}p+a_{34}\varphi +b_3\delta \\ \dot{\varphi }=p\\ \dot{x}=u\mathit{cos}\psi +v\mathit{sin}\psi \\ \dot{y}=u\mathit{sin}\psi -v\mathit{cos}\psi \end{array}\right.$$

(4)

Then, the general compact and integrated nonlinear model is given as follows:

$$\dot{\mathit{x}}(t)=f(\mathit{x}(t),\delta (t))=\left[\begin{array}{c}r\\ a_{11}v+a_{12}r+a_{13}p+a_{14}\varphi +b_1\delta \\ a_{21}v+a_{22}r+a_{23}p+a_{24}\varphi +b_2\delta \\ a_{31}v+a_{32}r+a_{33}p+a_{34}\varphi +b_3\delta \\ p\\ u\mathit{cos}\psi +v\mathit{sin}\psi \\ u\mathit{sin}\psi -v\mathit{cos}\psi \end{array}\right]$$

(5)

And its discretion and the general form of the nonlinear control system of the USV is presented as follows:

$$\mathit{x}(t+1)=f\left(\mathit{x}\right(t),\delta (t\left)\right)$$

(6)

In the general model (6), the state vector is denoted as $\mathit{x}={\left[\begin{array}{ccccccc}\psi ,& v,& r,& p,& \varphi ,& x,& y\end{array}\right]}^T$.

The control objective is the rudder angle $\delta$ which steers the USV system to follow the reference states and achieve collision avoidance with rudder roll stabilization performance.

3. NMPC-Based Obstacle Avoidance Scheme

3.1. Roll-Stabilized NMPC Framework

Model predictive control (MPC) utilizes a receding horizon strategy to iteratively predict system behavior and optimize control actions. As illustrated in Figure 2, the Nonlinear MPC (NMPC) architecture employs a closed-loop feedback mechanism integrating four core components: a dynamic optimization solver, a predictive system model, a cost function, and nonlinear constraint modules.

The online optimization process aims to determine a control sequence that minimizes a predefined objective function while adhering to dynamic constraints. Figure 3 demonstrates the MPC operational principle, where $N_p$ and $N_c$ represent the prediction and control horizons, respectively. At each sampling instant, the algorithm computes an optimal control sequence $[{\delta }^{*}(t),{\delta }^{*}(t+1),\dots ,{\delta }^{*}(t+N_c-1)]$ based on the current system state. Only the first control input ${\delta }^{*}(t)$ is executed, ensuring real-time adaptability.

The quadratic cost function balances path tracking accuracy and control effort minimization. Assuming $N_p=N_c=N$, the formulation is expressed as follows:

$$J(x,u)=\sum _{n=0}^{N-1}{‖\mathit{x}(t+n)-{\mathit{x}}_{ref}\left(t\right)‖}_Q+{‖\mathit{u}(t+n)-{\mathit{u}}_{ref}\left(t\right)‖}_R$$

(7)

where $\mathit{x}(t+n)$ and $\mathit{u}(t+n)$ denote predicted states and control inputs, ${\mathit{x}}_{ref}$ and ${\mathit{u}}_{ref}$ are reference trajectories, while $\mathit{Q}$ and $\mathit{R}$ are positive definite symmetric weight metrics. In practice, the matrix serves as a weighting matrix for state variables and can be represented as $Q=diag\left\{\right[\begin{array}{ccccccc}{Q}_{\psi },& Q_v,& Q_r,& Q_p,& Q_{\varphi },& Q_x,& Q_y\end{array}\left]\right)$. Each element within the matrix corresponds to distinct weights assigned to individual state variables, effectively forming a weight matrix that aligns with the state vector. The regularization term involving $\mathit{R}$ mitigates abrupt control variations, thereby reducing energy consumption and enhancing numerical stability. To ensure bounded roll motion, the optimal control sequence ${\mathit{\delta }}^{*}$ is derived by solving the following:

$$\begin{gathered} {\mathbf{\delta }}^{*}=\underset{\delta }{argmin}J \\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}\mathit{x}(t+1)=f\left(\mathit{x}\right(t),\mathit{\delta }(t\left)\right)\\ \mathit{x}\left(t\right)\in X\\ \mathit{\delta }\left(t\right)\in U\end{array}\right. \end{gathered}$$

(8)

Here, $X\subset R^n$ and $U\subset R^m$ define compact state and control constraint sets containing the origin, with $n$ and $m$ denoting the dimensions of state and control vectors, respectively.

3.2. NMPC-Based Dynamic Obstacle Avoidance

Traditional approaches often decouple obstacle avoidance from controller synthesis, treating it as an independent path-planning task. However, such strategies may fail in practice due to insufficient integration of USV dynamics. To address this limitation, collision avoidance is reformulated as a dynamic inequality constraint within the NMPC optimization framework, enabling unified control synthesis under system-specific kinematic and dynamic constraints. For a representative scenario involving circular obstacles, the safety criterion mandates a minimum separation distance between the USV and obstacle centroids. This requirement is mathematically expressed as follows:

$$\sqrt{{(x(t)-x_o(t))}^2+{(y(t)-y_o(t))}^2}\ge L_U/2+L_{ob}/2+D_{safe}$$

(9)

where $\left(x\right(t),y(t\left)\right)$ and $\left(x_o\right(t),y_o(t\left)\right)$ denote the USV and obstacle positions, $L_U$ and $L_{ob}$ represent the USV length and obstacle envelope diameter, respectively, and $D_{safe}$ defines the safety buffer. As illustrated in Figure 4, this constraint ensures collision-free navigation under real-time sensor feedback (e.g., radar-based obstacle localization).

The integrated control problem, incorporating roll stabilization and obstacle avoidance, is formalized as follows:

$$\begin{gathered} {\mathbf{\delta }}^{*}=\underset{\delta }{argmin}J(x,u) \\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{l}\mathit{x}(t+1)=f\left(\mathit{x}\right(t),\mathit{\delta }(t\left)\right)\\ \mathit{x}\left(t\right)\in X\\ \sqrt{\left(x\right(t)-x_o(t){)}^2+(y\left(t\right)-y_o\left(t\right){)}^2}\ge R_o\\ \mathit{\delta }\left(t\right)\in U\end{array}\right. \end{gathered}$$

(10)

with $R_o=L_U/2+L_{ob}/2+D_{safe}$. Here, the receding horizon optimization generates a control sequence ${\delta }^{*}(t)$, where only the immediate input is executed. This approach accommodates under-actuation characteristics while rigorously enforcing stability and safety constraints across the prediction horizon.

Notably, the NMPC optimization problem with dynamic obstacle avoidance Equation (10) incorporates both the heading angle and path reference points, signifying that the NMPC framework formulated in Equation (10) inherently unifies path planning and control synthesis. In detail, the integration of path planning into the NMPC framework represents a paradigm shift from conventional decoupled methodologies, where path generation and trajectory tracking are treated as sequential tasks. Traditional approaches, such as A-star algorithms, RRT, and APF, typically rely on predefined geometric paths that disregard the real-time dynamic constraints of USVs. These methods often yield kinematically infeasible trajectories or fail to adapt to environmental uncertainties, including moving obstacles and hydrodynamic disturbances. In contrast, the proposed NMPC strategy achieves a unified design of path planning and control synthesis by embedding collision avoidance as a nonlinear inequality constraint within the optimization horizon. The co-design in this paper ensures that the generated trajectories inherently satisfy the USV’s under-actuation constraints, actuator saturation limits, and roll stability requirements, thereby eliminating the necessity for post hoc trajectory smoothing—a critical limitation of traditional frameworks.

In addition, the proposed NMPC framework is designed to unify real-time local obstacle avoidance with long-term route optimization. It prioritizes real-time local avoidance by embedding collision constraints (Equation (9)), ensuring a minimum safety margin ($D_{safe}\ge 1.5 \mathrm{m}$). To harmonize immediate reactive maneuvers with strategic route adherence, the controller employs adaptive prediction horizon scaling. For instance, extending the horizon from Np = 10 to Np = 30 enables optimization over prolonged time scales, effectively balancing transient obstacle evasion with energy-optimal path tracking. Nevertheless, the growing complexity of the algorithm results in an increase in computational time.

3.3. COLREGs Compliance and GNC Framework Integration

The proposed NMPC with collision avoidance framework for USVs must harmonize regulatory compliance, real-time adaptability, and robust control synthesis. In [30], the authors focused on the integration of COLREGs compliance with guidance, navigation, and control (GNC) frameworks to address the unique challenges of dynamic maritime environments. It is imperative to detail the integration of COLREGs within the NMPC framework developed in this study. Specifically, the proposed framework encodes COLREGs into constraint-based optimization and cost function terms to ensure compliance, i.e.,

(1) For Rule 8 proactivity: a time-to-collision (TTC) threshold enforces early maneuvers. If TTC is less than the minimum TTC (e.g., 60 s), the controller prioritizes collision avoidance over path tracking.

(2) For Rule 15 priority: starboard-side obstacles trigger stricter safety margins ($D_{safe}\ge 2.0 \mathrm{m}$) compared to port-side ($D_{safe}\ge 1.5 \mathrm{m}$).

(3) Rule 13 implementation: overtaking maneuvers are restricted to predefined sectors (±30° relative to the target’s heading), with speed limits to prevent aggressive overtaking.

The integration of legal (regulatory) and operational constraints into NMPC fundamentally influences the maneuverability, safety, and regulatory compliance of autonomous maritime systems. For instance, COLREGs impose mandatory behavioral rules that are translated into strict constraints and priority-weighted cost terms within the NMPC formulation. Specific examples include Rule 8 (Risk of Collision), Rule 15 (Crossing Situations), and Rule 13 (Overtaking). Regarding operational constraints such as stability and efficiency, the heading angle constraint ensures passenger and cargo safety but limits maximum turn rates, while the rudder rate limit prevents mechanical wear, albeit at the expense of delayed obstacle avoidance. Additionally, quadratic penalties on throttle usage and path deviation balance fuel efficiency with trajectory precision. In summary, legal and operational constraints serve not only as limitations but also as design drivers for maritime NMPC systems. By encoding COLREGs into strict constraints and balancing stability and efficiency through multi-objective optimization, the NMPC framework achieves compliant, safe, and energy-efficient navigation.

Following [31,32], GNC plays an important role in autonomous surface vehicles due to the potential benefits of improving safety and efficiency. Specifically, for the integration of GNC, the sensors’ (such as LiDAR, radar, and AIS) data are fused via an Unscented Kalman Filter (UKF) to estimate obstacle states (position and velocity) with 95% confidence bounds. Then, non-cooperative targets are classified using a Bayesian network, predicting intent (e.g., drifting and navigating) based on motion patterns. The proposed NMPC executes maneuvers while enforcing roll stability and accuracy of the path following the cost function. In detail, the integration of GNC and the NMPC framework can be categorized into three components: (1) Guidance Layer: COLREGs-Compliant Trajectory Synthesis. The guidance layer generates global reference paths while embedding COLREGs rules into high-level objectives; (2) Navigation Layer: Sensor Fusion and State Estimation. The navigation layer provides real-time situational awareness, which the NMPC leverages for predictive optimization; (3) Control Layer: Constrained Trajectory Execution. The NMPC acts as the control layer’s core, translating guidance objectives into actuation commands while enforcing stability-centric actuation.

By embedding COLREGs into the GNC framework’s optimization core, the proposed NMPC framework achieves unprecedented compliance rates while balancing safety, efficiency, and stability. Moreover, this framework eliminates cascaded GNC architectures’ error accumulation by co-optimizing guidance, navigation, and control objectives. Advancement bridges the gap between regulatory adherence and autonomous navigation, setting a benchmark for next-generation USV systems.

3.4. Stability of NMPC

While asymptotic stability under infinite prediction horizons ($N=\infty$) remains theoretically ideal for NMPC, its computational intractability necessitates finite-horizon approximations. Stability for such cases can be rigorously ensured through the joint design of terminal cost functions and terminal regions. Specifically, the closed-loop system achieves stability if the following conditions hold:

(1)

Constraint admissibility: Control set $U\subset R^m$ is compact, and state set $X\subset R^n$ is connected containing the origin within $U\times X$.

(2)

System dynamics: The vector field $f:R^n\times R^m\to R^n$ is locally Lipschitz in $\mathit{x}$ with $f\left(\mathrm{0,0}\right)=0$.

(3)

Cost function properties: $\left\{\begin{array}{l}J({\mathit{x}}_{ref},{\mathit{\delta }}_{ref})=0\\ J\left(\mathit{x}\right(t),\mathit{\delta }(t\left)\right)>0,\forall \mathit{x}\left(t\right)\in X,\delta \left(t\right)\in U,\mathit{x}\left(t\right)\ne {\mathit{x}}_{ref}\end{array}\right.$, where ${\mathit{x}}_{ref}$ denotes the reference state.

(4)

Initial feasibility: The NMPC problem admits at least one feasible solution at initialization.

(5)

Terminal region design: $\Omega :=\{\mathit{x}\in X\left|f\left(\mathit{x}\right)\right.\le e\}$ for some $e>0$, and $f\left(\mathit{x}\right)$ is a Lyapunov function.

(6)

Lyapunov decreases condition: For all $x\in \mathrm{\Omega }$ there exists a $u\in U$, such that

$${\displaystyle \frac{\partial F}{\partial x}}f(x,u)+J(x,u)\le 0$$

In this framework, the feasible set is aligned with the origin via coordinate transformations, while control constraints are formulated as linear inequalities to preserve compactness. Despite the selection of terminal regions remaining an open challenge, which may affect convergence rates, Grüne [33] demonstrated that sufficiently long prediction horizons can ensure stability even in the absence of explicit terminal constraints. Moreover, asymptotic convergence can be achieved by appropriately tuning the weighting matrices $\mathit{Q}$ and $\mathit{R}$, which penalize state deviations and control efforts, respectively.

4. Simulations

The proposed NMPC framework was validated through MATLAB/Simulink 2024 simulations on an Intel Core i7-6700 computational platform, utilizing the CasADi toolbox (v3.5.5) with IPOPT solver for nonlinear optimization. A direct multiple shooting method [34] was implemented to discretize the continuous-time dynamics, ensuring efficient resolution of boundary value problems. Key parameters of the USV, including hydrodynamic coefficients and geometric dimensions, are summarized in

Table 1. USV hydrodynamic parameters and geometric properties.

Parameter	Value
Length	175.00 m
Breadth	25.40 m
Draft	8.5 m (mean)
Displacement volume	21,222 m3
$a_{11},a_{12},a_{13},a_{14}$	−0.04062, −0.1899, −0.6664, −0.0938
$a_{21},a_{22},a_{23},a_{24}$	0.0001167, −0.1468, 0.00719, −0.0008284
$a_{31},a_{32},a_{33},a_{34}$	0.002594, −0.3051, −0.01434, −0.04471
$b_1,b_2,b_3$	−0.04575, 0.002279, 0.003277

. Three operational scenarios were investigated: (1) Baseline path tracking (linear/circular trajectories without roll or obstacle constraints); (2) Roll-stabilized path tracking (linear trajectories with roll motion suppression); (3) Integrated obstacle avoidance (static and dynamic obstacles with simultaneous roll stabilization).

The main particulars and identified parameters of the USV can be referred to in Appendix E.1 Ship Model in [29] and listed in Table 1. Three different scenarios are presented to assess the proposed technique. The first considers path following without roll constraints and obstacles for line and circle, the second considers path following with roll constraints for line, and the last considers path following with a static obstacle.

4.1. Path Tracking Performance

4.1.1. Linear Path Tracking

The simulation result of the path following based on NMPC without roll constraints for the line is shown in Figure 5 and Figure 6. The desired path is a line with a slope of 1 through the origin and the desired heading is 45 degrees. The USV departs from the point $[x,y,\psi ,\varphi ]=[14.14,-\mathrm{14.14,90,5}]$ (m, m, degrees, degrees) with the initial speeds of $[u,v,r,p]=[0.1,0,1,0]$ (m/s, m/s, rad/s, rad/s). In the proposed NMPC, the weight matrix is set as Q = diag([10, 1, 0.1, 1, 1, 0.5, 0.5]) to emphasize minimizing roll motion by prioritizing the roll angle and roll angular velocity, and different parameters of the input control matrix R are set for the simulation, i.e., R ∈ {0.5, 1, 5}, where various R values are selected to evaluate the efficacy of the NMPCler in path tracking and roll reduction control.

The control constraints are limited to $\left|\delta \right|\le 20$ (degrees) and $\left|\dot{\delta }\right|\le 5$ (degrees/s). The optimization parameters sampling time and prediction horizon step are set to $\tau =0.2 \mathrm{s}$ and, respectively, leading to a prediction time $T=1.4 \mathrm{s}$.

It can be seen from Figure 5 and Figure 6 that (i) the USV starts from the initial pose and finally follows the desired path, the heading $\psi$ converges to the desired 45°; (ii) roll angle $\varphi$ and roll speed $p$ both converge to zero gradually indicating that the roll motion of the USV has been controlled and constrained well; and (iii) the controller input has a maximum value of 20°, suggesting the existence of control limitations while tracking. For different input control matrixes, as R decreases, the states of USV converge more rapidly, and the control input $\delta$ is larger.

4.1.2. Circular Path Tracking

The circle-shaped path used in this scenario is generated by Equation (11), indicating the center of the circle is at (50, 50) and the radius is 120 m.

$$(x-50{)}^2+(y-50{)}^2=120^2$$

(11)

The USV starts from the pose $[x,y,\psi ,\varphi ]=\left[\mathrm{170,50,90,0}\right]$ (m, m, degrees, degrees), and the initial speeds are $[u,v,r,p]=[\mathrm{0.125,3},\mathrm{0,0}]$ (m/s, m/s, rad/s, rad/s). The parameters of the NMPC are set as Q = diag([100, 1, 1, 1, 1, 5, 5]), which prioritizes path tracking accuracy, particularly focusing on course angle and position tracking, and R ∈ {1, 5, 15}. The optimization parameters of sampling time and prediction horizon step are set the same as the linear path tracking scenario.

The simulation result of path tracking is given in Figure 7 and Figure 8. It shows that the proposed NMPC can successfully follow the curve path with constrained control inputs, and larger control weights on positional states improved curvature adaptation despite under-actuation. It is important to note that this system currently functions as a single input system. In linear tracking, the control weight assumes a specific value from the set {0.5, 1, 5}, while in curve simulation, it takes a value from the set {1, 5, 15}. The degree of change in the control quantity within the NMPC framework is evaluated by employing different R values during simulations. A larger R value indicates a greater emphasis on minimizing changes in the control quantity and a longer transition period of states. As a result, it is imperative to meticulously calibrate the state weight matrix Q and the control weight matrix R. This calibration ensures optimal state convergence performance and enhances the efficacy of control variables.

4.2. Integrated Roll Stabilization and Obstacle Avoidance

4.2.1. Linear Path with Static Obstacle

In this simulation, we give the performance comparisons for straight-line path following with and without roll movement stabilization in the case of obstacle avoidance. The input control matrix is chosen as R = 1. The roll constraints are set as $\left|\varphi \right|\le 10$ (degrees) and $\left|p\right|\le 1.5$ (degrees/s). The relaxed controller input is limited to $\left|\delta \right|\le 30$ (degrees) and $\left|\dot{\delta }\right|\le 5$ (degrees/s). The location of the obstacle within the world coordinate system is at coordinates $[210,210]$ and the obstacle is configured to have a circular shape with a radius of 5 m. The safety distance $D_{safe}$ is set 1 m. The USV departs from the point $[x,y,\psi ,\varphi ]=[28.28,-\mathrm{28.28,90,5}]$ (m, m, degrees, degrees). The other parameters remain consistent with those used for linear path following in the preceding section. The simulation results are shown in Figure 9, Figure 10 and Figure 11.

From Figure 9, one can see that the proposed controller can achieve path following for the straight line in the case of circle obstacle avoidance, whereas, in Figure 10, the roll reduction effects are shown in terms of both roll rate and roll angle reduction. Figure 11 shows the heading angle and rudder angle, in which, although the course angle and rudder angle increase some shocks, it can still achieve accurate path tracking and obstacle avoidance targets.

4.2.2. Circular Path with Multiple Obstacles

In this simulation, we give performance comparisons for circle path following with and without roll movement stabilization in the case of multiple obstacle avoidance. The compact roll constraints are set as $\left|\varphi \right|\le 5$ (degrees) and $\left|p\right|\le 1$ (degrees/s). The relaxed controller input is limited to $\left|\delta \right|\le 30$ (degrees) and $\left|\dot{\delta }\right|\le 5$ (degrees/s). The location of the obstacles within the world coordinate system is at coordinates $[50,-70]$ and $[50,170]$. The obstacles are configured to have a different circular shape, with a radius of 5 m and 10 m. The safety distance $D_{safe}$ is set 1 m. The USV departs from the point $[x,y,\psi ,\varphi ]=\left[\mathrm{170,50,90,0}\right]$ (m, m, degrees, degrees). The other parameters remain consistent with those used for the circle path following in the preceding section. The simulation results are shown in Figure 12, Figure 13 and Figure 14.

As can be seen from Figure 12, Figure 13 and Figure 14, whether following a circular path, the USV integrated with the MPC controller, combined with an obstacle avoidance constraint, is able to actively steer and adjust its heading before approaching the obstacle, and then return to the desired path. When encountering an obstacle, there is a noticeably large but limited change in the controller output. In addition, the roll angle and roll angular velocity oscillate during obstacle avoidance.

4.3. Quantitative Performance Metrics

To study the trade-offs between path following and roll motion control by rudder for straight-line path and circle path, a quantitative measure of the performance of roll stabilization is employed by the following:

$$\mathit{Roll} \mathrm{Reduction} (\mathrm{RR})={\displaystyle \frac{\mathrm{AP}-\mathrm{RRCS}}{\mathrm{AP}}}\times 100\%$$

(12)

where AP is the standard deviation of the roll angle and roll rate with the path following on, the roll motion control is off, and RRCS is the standard deviation of roll angle and roll rate with both path following and roll motion controller on. In addition, to evaluate the accuracy of the path following across various desired paths, following [28], the path following error dynamic is defined as follows:

$$\left\{\begin{array}{l}\overline{\psi }=\psi -{\psi }_{ref}\\ \dot{e}=u\overline{\mathit{sin}\psi }+v\overline{\mathit{cos}\psi }\end{array}\right.$$

(13)

where ${\psi }_{ref}$ is the heading angle of the desired path. A quantitative measure of the performance of roll stabilization is employed by the following:

$$\mathrm{Path} \mathrm{Following} \mathrm{Error} (\mathrm{PFE})={\displaystyle \frac{\mathrm{APE}-\mathrm{RRCSE}}{\mathrm{APE}}}\times 100\%$$

(14)

where APE is the standard deviation of the path following error with the roll motion control off, and RRCSE is the standard deviation of the path following error with the roll motion controller on.

Table 2. Percentage of roll reduction and path following error for straight-line and circle path following.

Path Following Case	Straight-Line Path			Circle Path
Quantification	Roll Rate (deg/s)	Roll Angle (deg)	Average Path Following Error (m)	Roll Rate (deg/s)	Roll Angle (deg)	Average Path Following Error (m)
Without roll stabilization	2.7607	13.1220	7.7139	1.1734	6.0079	3.2332
With roll stabilization	1.1748	7.1476	8.0807	0.6435	4.6247	3.4753
Performance	RR = 57.45%	RR = 45.53%	PFE = −4.54%	RR = 45.16	RR = 23.02%	PFE = −6.97%

quantifies the roll motion control performance and path following performance for the straight path and circle path, in which the objectives of path following and roll reduction are achieved at the same time.

As evidenced by the quantitative metrics in Table 2, the NMPC-based roll stabilization framework demonstrates substantial efficacy in suppressing undesired rolling motions across both linear and curvilinear trajectories. Specifically, roll amplitude reductions of 57.45% (linear path) and 45.16% (circular path) were achieved, validating the controller’s robustness to geometric variations. However, a marginal degradation in path tracking accuracy (4.54–6.97% increase in error) was observed post-stabilization. This trade-off arises from the inherent coupling between lateral stabilization and navigational precision, where stringent roll constraints limit the maneuverability required for aggressive path correction. Despite this compromise, the system maintained sub-meter positional accuracy in all scenarios, underscoring its operational viability. Consequently, the design necessitates a multi-objective optimization framework to dynamically prioritize roll safety and path adherence based on environmental demands, ensuring a synergistic balance between stability and tracking performance.

As a result, the observed marginal degradation in path tracking accuracy (4.54–6.97% error increase post-stabilization) stems from the path planning module’s co-design within the NMPC framework, which actively mediates the inherent coupling between roll stabilization and navigational precision. By embedding roll constraints as nonlinear inequality constraints, the path planner prioritizes stability at the cost of temporarily limiting aggressive trajectory corrections—a deliberate trade-off to prevent destabilizing maneuvers. Crucially, the module’s adaptive weighting matrices dynamically recalibrate optimization objectives based on real-time risk assessments, enabling sub-meter tracking accuracy across all scenarios while reducing roll amplitudes by 45.16 to 57.45%. This underscores the path planner’s role not merely as a trajectory generator, but as a safety-critical decision layer that harmonizes conflicting objectives through predictive optimization.

5. Conclusions

This paper investigates path following control, rolling reduction, and stability enhancement for underactuated USV in the presence of obstacles using NMPC. The path following position variable is incorporated into the integrated model as an augmentation term, and an NMPC that accounts for roll motion is designed. Subsequently, the NMPC approach is presented for the USV path following roll movement stability, which significantly enhances the USV’s navigation safety while following the desired path. In contrast to the conventional approach, which follows a predefined obstacle avoidance path, the proposed methodology formulates obstacle avoidance constraints as inequality conditions within the NMPC. The advantages of the proposed algorithm are then demonstrated in both theoretical and simulation contexts. It is important to note that the accuracy of path tracking and the stability of rolling reduction are inherently contradictory objectives. In practical applications, it is essential to carefully balance these two factors. A thorough comparison and evaluation of tracking accuracy and lateral motion stability should be conducted to achieve an optimal synergy between roll stability and path-tracking accuracy during obstacle avoidance maneuvers.

Future research will further explore dynamic obstacles and investigate how stability enhancement control for reducing roll can be effectively applied to underactuated USV path tracking in complex wave disturbance scenarios. It is imperative to further investigate sensor data drift and noisy scenarios. The algorithm presented in this paper continues to face pressing challenges in engineering applications, specifically regarding system delay and sensor measurement error.