The Path Tracking Control of Unmanned Surface Vehicles Based on an Improved Non-Dominated Sorting Genetic Algorithm II-Based Multi-Objective Nonlinear Model Predictive Control Method

Abstract

This paper proposes a multi-objective nonlinear model predictive control (MOMPC) method based on an improved non-dominated sorting genetic algorithm II (NSGAII) for the path tracking problem of unmanned surface vehicles (USVs). To enhance performance in cross-track error, a varying look-ahead distance is utilized in the line of sight (LOS) algorithm, which allows the MPC control algorithm to compute the look-ahead distance and desired speed rather than directly calculating the control input. Since the cost function of the MPC algorithm includes multiple objective terms, a multi-objective model predictive control algorithm is employed to improve overall control performance. Additionally, an adaptive rotation-based simulated binary crossover (ARSBX) is integrated into the NSGAII algorithm, and the non-dominated sorting method is optimized to reduce computation time. These enhancements increase diversity and exploration in the solution space, enabling the algorithm to find the optimal solution more efficiently. Simulations conducted in two different scenarios demonstrate that the nonlinear MPC method based on the improved NSGAII successfully tracks the desired path; it achieved an improvement of approximately 41% in time performance and about 5% in path-tracking error performance, exhibiting strong control performance and robustness.

1. Introduction

With the development of global intelligent technology, unmanned systems have rapidly advanced and gained significant attention from both the technical field and commercial market. In recent years, the application and research of unmanned ships have received increasing attention [1,2]. However, the development of unmanned ship technology, as an important application of unmanned systems, remains relatively lagging, though it holds immense potential. Unmanned surface vehicles (USVs) are small surface platforms equipped with autonomous navigation capabilities, enabling them to independently carry out tasks such as environmental sensing and target detection [3,4].

Autonomous path tracking for unmanned surface vehicles (USVs) is essential for completing their missions. Depending on the ship model, different control methods are used, and the model can be simplified based on the characteristics of the tracked path [5,6,7]. Intelligent ship motion control typically addresses three main problems: target tracking, trajectory tracking, and path tracking [8]. This paper focuses on path tracking, where the goal is to follow a time-independent geometric path. Unlike trajectory tracking, which has strict timing requirements, path tracking allows for smoother convergence to the desired path and is less prone to controller saturation, making it ideal for tasks where timing is less critical [9].

In the field of trajectory tracking control for unmanned vessels, various control algorithms have been implemented, such as PID control [10], active disturbance rejection control (ADRC), and sliding mode control (SMC). PID control achieves target tracking through its proportional, integral, and derivative components. Liu, Y. et al. proposed an improved nonlinear control algorithm rooted in sliding mode control theory to address trajectory tracking challenges for underactuated vessels with static constraints. Through the integration of a straightforward incremental feedback control law, they developed a dynamic control strategy aimed at achieving both tracking and stabilization for underactuated systems. Simulations and theoretical analysis conducted on a training ship validated the proposed controller’s stability and robustness [11]. Yang Z. et al. presented a trajectory control method based on the active disturbance rejection control algorithm, with simulation results demonstrating that this approach effectively meets track control requirements and allows for rapid stabilization of the heading. However, these methods lack the capability for dynamic optimization of future behavior, making it difficult to respond effectively to dynamic changes in the system, and they are not flexible enough in handling constraints, often leading to violations of the limits [12].

Artificial neural networks (ANNs) have garnered significant attention as a robust control method. When compared to traditional control techniques, artificial neural networks demonstrate superior adaptability and control efficacy [13,14]. Sun, W. et al. proposed a dual neural network approach for USV trajectory tracking, where one deep neural network (DNN) evaluates the navigation performance of the unmanned surface vehicle, and the other DNN estimates the cross-track error for the guidance law. Experimental results indicate that this approach achieves improvements in trajectory tracking error performance to varying degrees [15]. Sun, W. et al. developed two deep neural network (DNN) models, one for real-time evaluation of navigation performance and the other for estimating guidance law parameters. The two DNNs are integrated in parallel with the unmanned surface vehicle’s control loop to provide predictive supervision and assist decision-making for traditional control methods [16]. Zhang, G. et al. proposed a practical adaptive neural control algorithm using dynamic surface control techniques, neural network approximation, and polar coordinates. The developed neural controller can effectively account for vehicle uncertainties without requiring precise information about the hydrodynamic damping structure and external sea disturbances. Although using neural networks as control algorithms performs well, training these models requires significant resources, and obtaining data for practical applications can be challenging [17].

Model predictive control (MPC) effectively predicts system behavior by optimizing the control inputs over a future time horizon, thereby providing the optimal input at the current moment, which enhances trajectory tracking accuracy and better accommodates system constraints [18,19]. Compared to control methods utilizing artificial neural networks, MPC requires fewer resources and offers higher computational efficiency [20,21]. Dong, Z. et al. designed a controller for ship path tracking based on MPC, investigating the boundary of environmental disturbances within which the MPC controller can maintain acceptable accuracy [22]. Similarly, Zhou, X. et al. developed a course tracking controller based on MPC, demonstrating that the designed heading tracking control algorithm accurately follows waypoints and can withstand certain levels of environmental disturbances [23].

In this paper, while considering trajectory tracking error, the control of USV velocity is also taken into account. Employing a single-objective optimization algorithm may not effectively balance these two objectives simultaneously. Based on the analysis presented, this paper proposes a multi-objective model predictive control method based on an improved NSGAII algorithm for path tracking. The main contributions are as follows:

• To enhance the trajectory tracking performance of unmanned surface vessels (USVs), this paper employs a time-varying look-ahead distance, treating it alongside the desired velocity and acceleration as parameters to be calculated within the MPC algorithm;
• Given that the cost function of the MPC algorithm comprises multiple objective terms that may conflict with one another, a single-objective optimization approach could lead to a decline in the performance of other metrics. Therefore, this paper utilizes a multi-objective MPC algorithm, enabling comprehensive consideration of multiple performance indicators through multi-objective optimization, thereby identifying the optimal balance among the various objectives;
• This paper adopts an improved NSGAII algorithm within the multi-objective MPC framework, incorporating an adaptive rotation-based simulated binary crossover operation to enhance diversity and convergence. Additionally, the method of non-dominated sorting has been refined. These improvements help to prevent premature convergence while also reducing computational time.

2. Mathematical Modeling

2.1. Models Used to Simulate the Vessel

The simulation model used in this paper can be found in [24]. The ship dynamics model mainly describes the ship’s motion behavior under the action of external forces, including the balance of force and moment [25], as well as the resulting changes in acceleration, velocity, and position. By considering the effects of external forces (such as thrust, drag, and wave forces) on the ship’s hull, the dynamic model can predict the ship’s motion response. This model is typically based on the Newton–Euler equations, deriving the ship’s equations of motion by relating external forces and moments to the ship’s mass and inertia.

In this formula, translational motion corresponds to surge, sway, and heave, which includes the position of the ship $x, y$ and the linear velocity $u, v$ along the direction of the position. The rotational motion corresponds to the yaw angle $\psi$, which is the attitude angle of the ship and the angular velocity $r$ in the direction of the attitude angle.

$$\begin{array}{c}\mathit{M}={\mathit{M}}_{RB}+{\mathit{M}}_A\\ {\mathit{M}}_{RB}=\left[\begin{array}{ccc}m& 0& 0\\ 0& m& m{{\rm X}}_g\\ 0& m{{\rm X}}_g& I_z\end{array}\right]\\ {\mathit{M}}_A=\left[\begin{array}{ccc}-X_{\dot{u}}& 0& 0\\ 0& -Y_{\dot{v}}& -Y_{\dot{r}}\\ 0& -N_{\dot{v}}& -N_{\dot{r}}\end{array}\right]\end{array}$$

(1)

$$\begin{array}{l}\mathit{C}\left(\mathit{\upsilon }\right)={\mathit{C}}_{RB}\left(\mathit{\upsilon }\right)+{\mathit{C}}_A\left(\mathit{\upsilon }\right)\\ {\mathbf{C}}_A(\mathbf{v})=-\left[\begin{array}{ccc}0& 0& -Y_{\dot{v}}v-{\displaystyle \frac{1}{2}}(Y_{\dot{r}}+N_{\dot{v}})r\\ 0& 0& X_{\dot{u}}u\\ Y_{\dot{v}}v+{\displaystyle \frac{1}{2}}(Y_{\dot{r}}+N_{\dot{v}})r& -X_{\dot{u}}u& 0\end{array}\right]\\ {\mathbf{C}}_{RB}(\mathbf{v})=\left[\begin{array}{ccc}0& 0& -m({{\rm X}}_{g}r+v)\\ 0& 0& mu\\ m({{\rm X}}_{g}r+v)& -mu& 0\end{array}\right]\end{array}$$

(2)

$$\mathit{M}\dot{\mathit{\upsilon }}+\mathit{C}\left(\mathit{\upsilon }\right)\mathit{\upsilon }+\mathit{D}\left(\mathit{\upsilon }\right)\mathit{\upsilon }=\mathit{\tau }$$

(3)

$$\begin{array}{l}\mathit{M}=\left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& m_{23}\\ 0& m_{32}& m_{33}\end{array}\right]=\left[\begin{array}{ccc}m-X_{\dot{u}}& 0& 0\\ 0& m-Y_{\dot{u}}& -Y_{\dot{r}}\\ 0& -N_{\dot{v}}& I_z-N_{\dot{r}}\end{array}\right]\\ \mathit{C}\left(\mathit{\upsilon }\right)=\left[\begin{array}{ccc}0& 0& -m_{22}v-m_{23}r\\ 0& 0& m_{11}u\\ m_{22}v+m_{23}r& -m_{11}u& 0\end{array}\right]\\ \mathit{D}\left(\mathit{\upsilon }\right)=\left[\begin{array}{ccc}{d}_{11}& 0& 0\\ 0& d_{22}& d_{23}\\ 0& d_{32}& d_{33}\end{array}\right]=\left[\begin{array}{ccc}-X_u& 0& 0\\ 0& -Y_v& -Y_r\\ 0& -N_v& -N_r\end{array}\right]\end{array}$$

(4)

$\mathit{M}, \mathit{C}\left(\mathit{\upsilon }\right), \mathit{D}\left(\mathit{\upsilon }\right), \mathit{\upsilon }, {\mathit{\tau }}_w, \mathit{\tau }$ all represent six degrees of freedom matrices or vectors, where $\mathit{M}$ is a positive definite matrix with $\mathit{M}={\mathit{M}}^{\mathrm{T}}$. $\mathit{C}\left(\mathit{\upsilon }\right)=-{\mathit{C}}^{\mathrm{T}}\left(\mathit{\upsilon }\right)$, $\mathit{\upsilon }={\left[\begin{array}{ccc}u& v& r\end{array}\right]}^{\mathrm{T}}$. For underactuated ships, due to the lack of lateral propulsion devices, the control force $\tau$ is given by:

$$\tau ={[{\tau }_u N_{\delta }\delta Y_{\delta }\delta ]}^T$$

(5)

where $\delta$ is the control input for yaw. The specific expressions are provided in Section 3.2.1. The parameters $N_{\delta }$ and $Y_{\delta }$ are key factors that determine the maneuverability and hydrodynamic response of the ship. $N_{\delta }$ primarily affects the ship’s steering capability and yaw dynamics, while $Y_{\delta }$ influences the lateral force, playing a role in drift and contributing to trajectory correction. The higher the ratio ${\displaystyle \frac{N_{\delta }}{Y_{\delta }}}$, the stronger the steering ability of the USV, but it may weaken the lateral stability.

2.2. Models Used for Predictions

The unactuated sway dynamics are influenced by the rudder through $F(\delta )$, which introduces complexity to the model. To eliminate this effect on the sway dynamics, a coordinate transformation proposed by [26] can be applied. To simplify the cascaded control design, we introduce several coordinate transformations. Initially, we translate the desired equilibrium point to the origin. Specifically, we convert the surge velocity into the surge deviation, which is expressed as follows:

$$\overline{u}=u-u_d$$

(6)

$$\overline{x}=x+\epsilon \cos \psi$$

(7)

$$\overline{y}=y+\epsilon \sin \psi$$

(8)

$$\overline{v}=v+\epsilon r$$

(9)

where,

$$\epsilon =-\left({\displaystyle \frac{m_{33}{Y}_{\delta }-m_{23}{N}_{\delta }}{m_{22}{N}_{\delta }-m_{23}{Y}_{\delta }}}\right)$$

(10)

This involves shifting the origin along the x-axis of the body-fixed coordinate system to the point where the rudder generates only a rotational moment without producing any sway force. The equations of the transformed system are then given by:

$$\mathrm{\Gamma }=m_{22}{m}_{33}-m_{23}^2>0$$

(11)

$$\dot{\overline{y}}=\sin (\psi )(\overline{u}+u_d)+\cos (\psi )\overline{v}$$

(12)

$$\dot{\psi }=r$$

(13)

$$\dot{\overline{v}}=\dot{v}+\epsilon \dot{r}$$

(14)

$$\dot{r}={\displaystyle \frac{\delta }{\mathrm{\Gamma }}}(m_{22}{N}_{\delta }-m_{23}{Y}_{\delta })+\mathrm{\Omega }r+F\overline{v}$$

(15)

$$\dot{\overline{u}}={\displaystyle \frac{1}{m_{11}}}({\tau }_u+(m_{22}v+m_{23}r)r-d_{11}u)$$

(16)

$$\begin{array}{c}\mathrm{\Omega }=(m_{23}{m}_{11}(\overline{u}+u_d)+m_{22}^2(\overline{u}+u_d)\epsilon -m_{23}{d}_{22}\epsilon +m_{23}{d}_{23}-m_{22}{m}_{11}(\overline{u}+u_d)\epsilon -\\ m_{22}{m}_{23}(\overline{u}+u_d)-m_{22}{d}_{33}+m_{22}{d}_{32}\epsilon )\end{array}$$

(17)

$$F={\displaystyle \frac{1}{\mathrm{\Gamma }}}(m_{23}{d}_{22}-m_{22}^2(\overline{u}+u_d)-m_{22}{d}_{32}+m_{22}{m}_{11}(\overline{u}+u_d))$$

(18)

3. Path Tracking

3.1. LOS

The line of sight (LOS) algorithm is a simple and effective guidance method widely used in fields such as missiles, unmanned ships, and unmanned vehicles [27]. Its goal is to calculate the desired heading angle of the ship by combining the desired path points and current position information, as shown in Figure 1. In the application within the maritime field, by keeping the ship’s heading aligned with the LOS angle and maintaining control for a period of time, the ship can accurately track the desired path.

The cross-track error $e$, desired heading ${\psi }_d$, and look-ahead distance $\mathrm{\Delta }$ can be found in Figure 2a, where $\left[x, y\right]$ represents the Cartesian coordinates in the inertial reference frame, and $\left[x_b, y_b\right]$ represents the Cartesian coordinates in the body-fixed reference frame. If the coordinate system is rotated by an angle $\theta$ around the z-axis to align the x-axis with the path, the expression for the cross-track error $e$ will be simplified. Figure 2b shows the rotated coordinate system and the new definitions of $e$ and ${\psi }_d$. The desired heading ${\psi }_d$ and cross-track error $e$ are now defined as follows:

$${\psi }_d=-arctan({\displaystyle \frac{e}{\mathrm{\Delta }}})$$

(19)

$$e=y-D_y$$

(20)

$$\dot{e}=\dot{y}=u\sin \psi +v\cos \psi$$

(21)

$$\begin{array}{l}where:\\ {\dot{D}}_y=0,since D_y is constant.\end{array}$$

In this paper, a time-varying look-ahead distance $\mathrm{\Delta }$ is utilized to achieve better control performance during trajectory changes.

3.2. Controller

3.2.1. MPC

The design of the controller is based on the MPC concept, establishing a nonlinear predictive model and introducing an improved NSGAII algorithm. The controller solves the constrained optimization problem online at each time step to obtain the optimal control sequence, applying the first control component to the unmanned surface vehicle (USV) system [28,29], meanwhile rolling the prediction horizon forward to predict future possible states [30]. The MPC path tracking control block diagram is shown in Figure 3.

First, the waypoints and their corresponding reference trajectories are defined:

$$Y_{\mathrm{ref}}=[x_{\mathrm{ref}}, y_{\mathrm{ref}}·{\psi }_{\mathrm{ref}}]$$

(22)

The control algorithm used in this paper does not compute control inputs but rather calculates parameters related to the look-ahead distance, as well as the reference velocity and acceleration. A larger look-ahead distance value will result in a slower reduction of cross-track error, and if used in a constant look-ahead distance LOS algorithm, it will lead to little or no overshoot. On the other hand, a smaller look-ahead distance value will cause the cross-track error to decrease more quickly but will result in a larger overshoot [31]. Therefore, this paper utilizes a varying look-ahead distance to achieve more accurate tracking of the desired path.

The MPC algorithm is allowed to adjust the desired velocity of the USV. When the vessel is far from the desired path, the desired velocity can be increased to help the USV quickly converge to the expected trajectory. Conversely, when the vessel is closer to the path, the desired velocity remains fixed, allowing the USV to maintain a relatively stable surge velocity during trajectory tracking. The influence of the look-ahead distance on the USV’s velocity can be found in n the following equation, which is used to form the system equation for prediction.

$${\psi }_d=-arctan({\displaystyle \frac{\overline{y}}{\mathrm{\Delta }}})$$

(23)

$$r_d={\dot{\psi }}_d=-{\displaystyle \frac{\mathrm{\Delta }\dot{\overline{y}}}{{\mathrm{\Delta }}^2+{\overline{y}}^2}}$$

(24)

$$z_1=\psi -{\psi }_d$$

(25)

$$z_2={\dot{z}}_1=r-r_d$$

(26)

$${\dot{r}}_d={\displaystyle \frac{2\mathrm{\Delta }\overline{y}{(\dot{\overline{y}})}^2}{{({\mathrm{\Delta }}^2+{\overline{y}}^2)}^2}}-{\displaystyle \frac{\mathrm{\Delta }\ddot{\overline{y}}}{{\mathrm{\Delta }}^2+{\overline{y}}^2}}$$

(27)

$${\tau }_u=-(m_{22}v+m_{23}r)r+d_{11}u-m_{11}{k}_u\overline{u}$$

(28)

$$\delta ={\displaystyle \frac{\mathrm{\Gamma }}{(m_{22}{N}_{\delta }-m_{23}{Y}_{\delta })}}(-\mathrm{\Omega }r-F\overline{v}+{\displaystyle \frac{2\mathsf{\Delta }\overline{y}{(\dot{\overline{y}})}^2}{{({\mathsf{\Delta }}^2+{\overline{y}}^2)}^2}}-{\displaystyle \frac{\mathsf{\Delta }\ddot{\overline{y}}}{{\mathsf{\Delta }}^2+{\overline{y}}^2}}-k_1{z}_2-k_0{z}_1)$$

(29)

$$\dot{y}=\sin (\psi )(\overline{u}+u_d)+\cos (\psi )v$$

(30)

$k_0$, $k_1$, and $k_u$ are the control parameters which are provided in the experimental results section. In the control process, a well-defined loss function is essential for calculating the current state error. This allows the system to follow the predefined trajectory, minimize the error, and optimize the control strategy. The state error is defined as follows:

$$X_e({\iota }_{\kappa })=\left[\begin{array}{c}{X}_e({\iota }_{\kappa } | {\iota }_{\kappa })\\ X_e({\iota }_{\kappa }+1 | {\iota }_{\kappa })\\ ⋮\\ X_e({\iota }_{\kappa }+H_u | {\iota }_{\kappa })\\ ⋮\\ X_e({\iota }_{\kappa }+H_{\mathrm{p}}-1 | {\iota }_{\kappa })\end{array}\right]\hspace{1em}{U}_e({\iota }_{\kappa })=\left[\begin{array}{c}{U}_e({\iota }_{\kappa } | {\iota }_{\kappa })\\ U_e({\iota }_{\kappa }+1 | {\iota }_{\kappa })\\ ⋮\\ U_e({\iota }_{\kappa }+H_u | {\iota }_{\kappa })\end{array}\right]$$

(31)

where $H_u$ and $H_{\mathrm{p}}$ control the control and prediction horizons, respectively; $X_e({\iota }_{\kappa }+1 | {\iota }_{\kappa })$ represents the trajectory error at the next time step, and $U_e({\iota }_{\kappa }+1 | {\iota }_{\kappa })$ denotes the predicted ship control variables at the next time step. The cost function includes a term aimed at reducing rapid changes in the expected heave velocity.

To enhance the performance of the USV in cross-track error, the following cost function is constructed:

$${\displaystyle \sum _{i=k}^{k+H_p}{k}_y}{y}_i^2+k_{{\overline{u}}_d}{\overline{u}}_{d,i}^2+k_{{\dot{u}}_d}{\dot{u}}_{d,i}^2$$

(32)

When the linear time-varying model is used, the cost function takes the form:

$${\mathit{J}}^{\ast }({\iota }_{\kappa })={\displaystyle \sum _{i=0}^{H_p}{X}_e}{({\iota }_{\kappa }+i | {\iota }_{\kappa })}^{T}Q{X}_e({\iota }_{\kappa }+i | {\iota }_{\kappa })+{\displaystyle \sum _{i=0}^{H_u}\mathrm{\Delta }}{U}^{T}R\mathrm{\Delta }U+{\displaystyle \sum _{i=0}^{H_u}{U_e}^T}S{U}_e$$

(33)

Due to the potential suboptimality of various factors in single-objective optimization, this paper employs the NSGAII algorithm and splits the cost function into a bi-objective problem. The two objective functions are split as follows:

$${{\mathit{J}}_1}^{\ast }({\iota }_{\kappa })={\displaystyle \sum _{i=0}^{H_p}{X}_e}{({\iota }_{\kappa }+i | {\iota }_{\kappa })}^{T}Q{X}_e({\iota }_{\kappa }+i | {\iota }_{\kappa })$$

(34)

$${{\mathit{J}}_2}^{\ast }({\iota }_{\kappa })={\displaystyle \sum _{i=0}^{H_u}\mathrm{\Delta }}{U}^{T}R\mathrm{\Delta }U+{\displaystyle \sum _{i=0}^{H_u}{U_e}^T}S{U}_e$$

(35)

In Formula (30), $Q$, $R$, and $S$ are weight matrices, corresponding to the path error, velocity error, and acceleration error, respectively. To stabilize the navigation speed of the USV, this paper increases the weight of speed control in the objective function without compromising control accuracy. In the objective function, the first objective represents the tracking error, while the second objective corresponds to speed control, which is the control effort. In terms of weight design, since a multi-objective optimization algorithm is employed, the Pareto sorting method in the algorithm directly obtains trade-off solutions for multiple objectives by comparing the dominance relationships of solutions, thereby relatively reducing the overall impact of the weights.

3.2.2. Constraints

The constraints of the control algorithm should be expressed in terms of changes in input variables. Additionally, a constraint is introduced to prevent the look-ahead distance from becoming too large when the vessel attempts to avoid overshooting. This constraint is added because an excessively large look-ahead distance will reduce reaction time when it becomes necessary to reduce the distance again. It also prevents excessive velocity from causing large cross-track errors during trajectory changes. The controller input is expressed as:

$$G=\left[\begin{array}{c}{g}_{1,i}\\ g_{2,i}\\ g_{3,i}\\ g_{4,i}\\ g_{5,i}\end{array}\right]$$

(36)

$${\mathrm{\Delta }}_{min}{<\mathrm{g}}_1<{\mathrm{\Delta }}_{max}$$

(37)

$$u_{d,min}<g_4<u_{d,max}$$

(38)

$$g_2(t)=\dot{\mathrm{\Delta }}(t)$$

(39)

$$g_3(t)=\ddot{\mathrm{\Delta }}(t)$$

(40)

$$g_5(t)={\dot{u}}_d(t)$$

(41)

${\mathrm{\Delta }}_{min}, {\mathrm{\Delta }}_{max}$ represent the lower and upper bounds of the look-ahead distance, while $u_{d,min}, u_{d,max}$ denote the lower and upper limits of the surge velocity.

3.2.3. NSGAII

NSGAII is a genetic algorithm designed to solve multi-objective optimization problems [32]. NSGAII is an improvement of the original NSGA, primarily addressing the shortcomings of NSGA in terms of computational complexity and diversity [33]. These enhancements make NSGAII more suitable for solving complex multi-objective optimization problems [34]. Incorporating the ARSBX algorithm into NSGAII can enhance the exploration capability and diversity of the algorithm. ARSBX combines adaptive rotation and simulated binary crossover (SBX) techniques to improve the performance of multi-objective evolutionary algorithms, particularly on rotated problems where the Pareto sets are not aligned with the decision variables [35]. The specific process is illustrated in Figure 4.

The main steps of the improved non-dominated sorting method in the improved NSGAII algorithm are as follows: In a minimization problem, this method first arranges the solutions in the population $P$ in ascending order based on the first objective value, where $N$ represents the population size. If two solutions have the same value for the first objective, they are then sorted by the second objective value. This process continues until all individuals in the population are fully sorted. For solutions with identical values across all objectives, their order can be assigned arbitrarily. Once the sorting of individuals in population $P$ is complete, the algorithm begins assigning solutions to the different fronts of the sorted population sequentially. Since a solution in the sorted population cannot be dominated by any solution that follows it, determining the front of each solution only requires comparison with those solutions already assigned to a front. This reduces the computational effort needed to place each solution in the correct front.

• Initialize population $P$, mean vector $m$, and rotation matrix $V$. Generate an initial population with population size $N$. For each individual $x_i$ in the population, calculate the objective function values $f\left(x_i\right)$. For multi-objective optimization, there will be multiple objectives.
• Non-dominated sorting and parent selection. Perform non-dominated sorting to divide the population into fronts and compute crowding distance for each individual to ensure diversity in the population. Select individuals from the current population to serve as parents for crossover.
• Integrating ARSBX crossover and mutation. Select $N/2$ pairs of parents for crossover and use ARSBX to generate offspring. Decide whether to use the rotation matrix based on probability $p_s$; if $rand<p_s$, use the identity matrix and zero mean vector:

  $$V=I, m=0$$

  (42)

Otherwise, compute the rotation matrix and mean vector based on the current population:

$$m={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{x}_i}$$

(43)

$$C{v}_i={\lambda }_i{v}_i$$

(44)

where,

$$C_{i,j}={\displaystyle \frac{{\displaystyle \sum _{k=1}^D(x_{i,k}-m_k)(x_{j,k}-m_k)}}{N-1}}$$

(45)

Apply the rotation matrix and mean vector to transform parent solutions into a rotated coordinate system.

$$P_{\mathrm{rotated}}=V\cdot (P^{(g)}-m)$$

(46)

Perform simulated binary crossover (SBX) on the transformed individuals to generate offspring $q$.

$$q_i=0.5\times \left[(1+{\beta }_q)p_{\mathrm{rotated},i}^{(1)}+(1-{\beta }_q)p_{\mathrm{rotated},i}^{(2)}\right]$$

(47)

where ${\beta }_q$ is a parameter that controls the crossover distribution. Then, transform the offspring back to the original coordinate space.

$$q_{\mathrm{original}}=V^{-1}\cdot q+m$$

(48)

Add the newly generated offspring to the offspring population $Q$.

$$Q=Q\cup \left\{q_{\mathrm{original}}\right\}$$

(49)

• NSGA selection and sorting remain unchanged. Merge the parent population and offspring population:

  $$P^{(g+1)}=P^{(g)}\cup Q$$

  (50)

Perform non-dominated sorting on the merged population and use crowding distance to select the top $N$ individuals for the next generation. According to the new population, the rotation matrix and mean vector are updated according to Formulas (35) and (36). Update the probability $p_s$ based on the change in the objective function values.

$$p_s=p_s\cdot \exp (\gamma \cdot \mathrm{\Delta }f)$$

(51)

where $\gamma$ is a learning rate and $\mathrm{\Delta }f$ is the difference in the objective function values.

• Repeat the above steps until the maximum generation is reached or another stopping criterion is satisfied.

3.2.4. Stability

This paper proposes an intelligent predictive control method that incorporates terminal constraints within the finite horizon rolling optimization process [36]. The improved NSGAII results are used as the optimal performance index and are treated as a Lyapunov function. By leveraging the monotonicity of the Lyapunov function, the stability of the system is proven.

To simplify the proof process, it is assumed that the control horizon and prediction horizon are equal, both denoted as $N$, and the optimal performance index obtained through the improved NSGAII at time t is used for the proof.

$${\mathit{J}}^{\ast }({\iota }_{\kappa })=\underset{U_{\mathrm{e}}}{\min }({\displaystyle \sum _{i=1}^N}\parallel X_{\mathrm{e}}({\iota }_{\kappa }+i | {\iota }_{\kappa }){\parallel }_Q+\parallel {\mathit{U}}_e({\iota }_{\kappa }+i | {\iota }_{\kappa }){\parallel }_R+\parallel ∆\mathit{U}({\iota }_{\kappa }+i | {\iota }_{\kappa }){\parallel }_S)$$

(52)

where ${\mathit{J}}^{\ast }({\iota }_{\kappa })⩾0,{\mathit{J}}^{\ast }({\iota }_{\kappa })=0$. It holds if and only if $X_{\mathrm{e}}=0$, $U_{\mathrm{e}}=0$. Assuming the origin represents the system’s equilibrium state, and the system meets the iterative feasibility conditions. Terminal equality constraints are introduced, which set the state error to zero at the end of the rolling optimization. The formula is as follows:

$$X_{\mathrm{e}}({\iota }_{\kappa }+i | {\iota }_{\kappa })\approx 0,i⩾N$$

(53)

$$\begin{array}{cc}\hfill {\mathit{J}}^{\ast }({\iota }_{\kappa })& =\underset{U_{\mathrm{e}}}{\min }({\displaystyle \sum _{i=1}^N}\parallel X_{\mathrm{e}}({\iota }_{\kappa }+i | {\iota }_{\kappa }){\parallel }_Q+\parallel {\mathit{U}}_{\mathrm{e}}({\iota }_{\kappa }+i | {\iota }_{\kappa }){\parallel }_R+\parallel ∆\mathit{U}({\iota }_{\kappa }+i | {\iota }_{\kappa }){\parallel }_S)+\hfill \\ & \parallel X_{\mathrm{e}}({\iota }_{\kappa }+N | {\iota }_{\kappa }){\parallel }_Q+\parallel {\mathit{U}}_{\mathrm{e}}({\iota }_{\kappa }+N | {\iota }_{\kappa }){\parallel }_R+\parallel ∆\mathit{U}({\iota }_{\kappa }+N | {\iota }_{\kappa }){\parallel }_S\hfill \end{array}$$

(54)

The stability proof is as follows: at time ${\iota }_{\kappa }+1$, the optimal performance index value after particle swarm optimization is,

$$\begin{array}{cc}\hfill {\mathit{J}}^{\ast }({\iota }_{\kappa }+1)& =\underset{U_{\mathrm{e}}}{\min }({\displaystyle \sum _{i=1}^N}\parallel X_{\mathrm{e}}({\iota }_{\kappa }+i+1 | {\iota }_{\kappa }+1){\parallel }_Q+\parallel {\mathit{U}}_{\mathrm{e}}({\iota }_{\kappa }+i+1 | {\iota }_{\kappa }+1){\parallel }_R+\parallel ∆\mathit{U}({\iota }_{\kappa }+i+1 | {\iota }_{\kappa }+1){\parallel }_S)\hfill \\ & +\parallel X_{\mathrm{e}}({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_Q+\parallel {\mathit{U}}_{\mathrm{e}}({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_R+\parallel ∆\mathit{U}({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_S\hfill \end{array}$$

(55)

Each term is rewritten as follows:

$${\displaystyle \sum _{i=1}^N}(\parallel X_{\mathrm{e}}({\iota }_{\kappa }+i+1 | {\iota }_{\kappa }){\parallel }_Q={\displaystyle \sum _{i=1}^N}(\parallel X_{\mathrm{e}}({\iota }_{\kappa }+i | {\iota }_{\kappa }){\parallel }_Q-\parallel X_{\mathrm{e}}({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_Q$$

(56)

Considering that the independent variable optimization starts from time ${\iota }_{\kappa }+1$ and does not affect the performance index at time ${\iota }_{\kappa }$ and adding the terminal constraint, Equation (48) is converted to:

$$\begin{array}{c}{J}^{\ast }({\iota }_{\kappa }+1)=\underset{U_{\mathrm{e}}}{\min }({\displaystyle \sum _{i=1}^N}\parallel X_{\mathrm{e}}({\iota }_{\kappa }+i | {\iota }_{\kappa }){\parallel }_Q-\parallel X_{\mathrm{e}}({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_Q+\parallel U_{\mathrm{e}}({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_R-\parallel U_{\mathrm{e}}({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_R+\\ \parallel ∆U({\iota }_{\kappa }+i | {\iota }_{\kappa }){\parallel }_S-∆U({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_S)+X_{\mathrm{e}}({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_Q+U_{\mathrm{e}}({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_R+\\ \parallel ∆U({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_S\le \underset{U_{\mathrm{e}}}{\min }({\displaystyle \sum _{i=1}^N}-\parallel X_{\mathrm{e}}({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_Q-\parallel U_{\mathrm{e}}({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_R-\parallel ∆U({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_S)+\\ J^{\ast }({\iota }_{\kappa })+\parallel X_{\mathrm{e}}({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_Q+\parallel U_{\mathrm{e}}({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_R+∆U({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_S\end{array}$$

(57)

Because of the terminal constraint added in Equation (52), the following holds:

$$\parallel X_e({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_Q+\parallel U_e({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_R+\parallel ∆U({\iota }_{\kappa }+N+1 | {\iota }_{\kappa }){\parallel }_S\approx 0$$

(58)

Simplifying Equation (49) yields:

$$\begin{array}{c}{J}^{\ast }({\iota }_{\kappa }+1)-J^{\ast }({\iota }_{\kappa })⩽-\parallel X_{\mathrm{e}}({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_Q-\parallel U_{\mathrm{e}}({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_R\\ -\parallel \mathrm{\Delta }U({\iota }_{\kappa }+1 | {\iota }_{\kappa }){\parallel }_S\end{array}$$

(59)

Furthermore, this implies:

$$J^{\ast }({\iota }_{\kappa }+1)<J^{\ast }({\iota }_{\kappa })$$

(60)

From this, it is concluded that the control algorithm ensures the global performance index is monotonically decreasing, and the proposed method guarantees the asymptotic stability of the closed-loop system. If inevitable violations occur in practical applications, terminal constraints can be incorporated into the cost function, with weighted penalties applied to the errors.

4. Results

To validate the performance of the proposed trajectory tracking control method in following the reference trajectory, this paper conducted path-tracking simulations under two scenarios and analyzed the results. The design of the experiments includes evaluating trajectory tracking performance with different look-ahead distances and comparing these results with those obtained using the NSGAII algorithm, as well as the case with a fixed look-ahead distance and single-objective. The parameters of CyberShip II are shown in

Table 1. Parameters of CyberShip II.

Parameters	Value	Unit	Parameters	Value	Unit
$m$	23.8	kg	$Y_r$	0.1079	N · s/m
${{\rm X}}_g$	0.046	m	$Y_{\dot{r}}$	−0	kg
$I_z$	1.76	kg · m	$N_v$	0.1052	N · s/m
$X_u$	−0.7225	N · s/m	$N_{\dot{v}}$	−0	kg
$X_{\dot{u}}$	−2.0	kg	$N_r$	−0.5	N · s/m
$Y_v$	−0.8612	N · s/m	$N_{\dot{r}}$	−1.0	kg
$Y_{\dot{v}}$	−10	kg

. The cross-track error for USV path tracking is recorded every 0.5 s. The comparison of run times for the original NSGAII algorithm and the improved algorithm for each execution is shown in

Table 2. Comparison of algorithm computation times.

Approach	Average Single Run Time (s)
NSGAII	2.557
Improved NSGAII	0.89
QPSO	2.210

. It achieved an improvement of approximately 41% in time performance. The results of the Quantum Particle Swarm Optimization (QPSO) algorithm have also been included in the comparison. The population size of NSGAII is set to 200, and the number of iterations is set to 5000. All the simulations are performed on an Intel i9-13900HX processor (14-core, 32 GB RAM) with an NVIDIA RTX 4050 GPU.

In practical engineering, the calculation frequency of ship control algorithms is typically around 1 Hz. The improved algorithm is capable of meeting the practical requirements.

4.1. Scenario 1

First, we selected a desired path with relatively small changes in path angle and minimal wave disturbances to conduct the USV’s path-tracking simulation. The control parameters are provided in Table 2. The sea current affecting the ship is modeled using the following parameters, with the impact of sea waves on the ship’s speed considered as well:

$$\begin{array}{l}{u}_{cb}=V_c\cos ({\beta }_c-\psi )\\ v_{cb}=V_c\sin ({\beta }_c-\psi )\end{array}$$

(61)

$$\begin{array}{l}{V}_c=0.01 \mathrm{m}/\mathrm{s}\\ {\beta }_c=\pi /3\end{array}$$

(62)

The system matrix of the ship model is expressed as follows:

$$\begin{array}{l}\mathrm{M}=\left[\begin{array}{ccc}25.8& 0& 0\\ 0& 33.8& 1.0115\\ 0& 1.0115& 2.76\end{array}\right]\\ \mathrm{D}=\left[\begin{array}{ccc}0.9257& 0& 0\\ 0& 2.8909& -0.2601\\ 0& -0.2602& 0.5\end{array}\right]\end{array}$$

(63)

Using 11 reference points to construct the desired tracking path, the generated reference path 1 is shown in Figure 5.

In the simulation experiments, the controller parameters are set as shown in

Table 3. Controller parameters.

Controller Parameter	Value	Unit
$\mathrm{Predictive} \mathrm{horizon} H_{\mathrm{p}}$	45
$\mathrm{Control} \mathrm{horizon} H_u$	15
$\mathrm{Maximum} \mathrm{control} \mathrm{input} [Ta{u}_{u,max},Ta{u}_{\delta ,max}]$	[2, 1.5]	[N, N · m]
$\mathrm{Minimum} \mathrm{control} \mathrm{input} [Ta{u}_{u,\min },Ta{u}_{\delta ,\min }]$	[−2, −1.5]	[N, N · m]
$\mathrm{Rudder} \mathrm{force} \mathrm{coefficient} \left[N_{\delta },Y_{\delta }\right]$	[1, −0.2]
$\mathrm{Path} \mathrm{error} \mathrm{weight} \mathrm{coefficient} k_y$	4
$\mathrm{Speed} \mathrm{error} \mathrm{weight} \mathrm{coefficient} k_{{\overline{u}}_d}$	40
$\mathrm{Acceleration} \mathrm{error} \mathrm{weight} \mathrm{coefficient} k_{{\dot{u}}_d}$	5
$\mathrm{The} \mathrm{control} \mathrm{parameters} [k_0$$,k_1$$,k_u$]	[10, 6.32,5]
The range of surge velocity of USV u	[0, 0.5]	m/s

. The simulation time is 660 s, and the step size is 0.5 s. For both algorithms, the population size is set to 200, the number of iterations to 200, and the initial position of the USV is set to (−1,4).

Figure 6 shows the trajectory tracking results for three different look-ahead distances, with performance metrics illustrated in Figure 7, Figure 8 and Figure 9.

Figure 7 shows the cross-track error curves for four scenarios: using the NSGAII algorithm, improved NSGAII algorithm, QPSO algorithm, single-objective algorithm, and two cases where the latter algorithm is used with fixed look-ahead distances. The overshoot generated during trajectory changes is minimal, effectively responding to path variations while maintaining a relatively stable navigation state. From the figure, it can be seen that the curve of the improved NSGAII algorithm with a varying look-ahead distance quickly converges the cross-track error to near zero in the initial phase and performs well throughout the entire path. The specific numerical results will be analyzed later.

Figure 8 and Figure 9 illustrate the changes in heading angle and USV speed for the two algorithms. It can be observed that as the cross-track error increases, the USV’s surge speed also increases, allowing the trajectory to quickly converge to the desired path, with the speed remaining stable at around 0.2 m/s for most of the time. Since the trajectory tracking performance with a fixed look-ahead distance differs significantly from the case with a varying look-ahead distance, the analysis of the heading angle and speed will not cover the fixed distance cases.

The total cross-track error for the entire path for each algorithm is shown in

Table 4. Simulation results of scenario 1.

Method	Sum of the Absolute Values of the Cross-Track Errors (m)
NSGAII	220.52
Improved NSGAII	204.31
Improved NSGAII and look-ahead distance = 1	298.57
Improved NSGAII and constant look-ahead distance = 0.5	246.81
Single-objective	294.70
QPSO	225.26

. The cross-track error for the NSGAII algorithm is 220.52, while for the improved NSGAII algorithm, it is 204.31. For the QPSO algorithm, the error is 225.26. Under the same algorithm, the cross-track error is 246.81 when the look-ahead distance is fixed at 1, and 298.57 when the look-ahead distance is fixed at 0.5. The cross-track error using the single-objective algorithm is 294.70.

From the above data, it can be seen that the performance of tracking the desired path with a fixed look-ahead distance is significantly worse compared to using a varying look-ahead distance. Additionally, the proposed improved NSGAII algorithm achieves a smaller error than the other algorithm.

4.2. Scenario 2

In this section, we select a desired path with faster changes in angle and use higher current speeds in the wave parameter settings.

$$\begin{array}{l}{V}_c=0.02 \mathrm{m}/\mathrm{s}\\ {\beta }_c=\pi /3\end{array}$$

(64)

Using four reference points to construct the desired tracking path, the generated reference path 2 is shown in Figure 10.

Figure 11, Figure 12, Figure 13 and Figure 14 show the tracking results under scenario 2. From the figures, it can be observed that despite significant wave disturbances and large changes in the desired path angles, the tracking of the desired path remains relatively accurate. Additionally, the improved NSGAII algorithm still performs slightly better than other algorithms in tracking the path.

The path-tracking results for scenario 2 are shown in Table 4. The tracking error for the NSGAII algorithm is 78.67, while for the improved NSGAII algorithm, it is 72.17. The cross-track error for the NSGAII algorithm is 220.52, while for the improved NSGAII algorithm, it is 204.31. For the QPSO algorithm, the error is 79.61. With a fixed look-ahead distance of 1, the error is 104.68, and with a fixed look-ahead distance of 0.5, the error is 79.54. The cross-track error using the single-objective algorithm is 112.29. Compared to the NSGAII algorithm, it achieved an improvement of about 5% in path-tracking error performance.

From the data in

Table 5. Simulation results of scenario 2.

Method	Sum of the Absolute Values of the Cross-Track Errors (m)
NSGAII	78.67
Improved NSGAII	72.17
Improved NSGAII and look-ahead distance = 1	104.68
Improved NSGAII and constant look-ahead distance = 0.5	79.54
Single-objective	112.29
QPSO	79.61

, it is evident that the improved NSGAII algorithm outperforms other algorithms in terms of cross-track error performance in scenario 2. Even under conditions of high environmental disturbance, the cross-track error remains relatively stable without significant increase.

Based on the results above, we can conclude that:

• The MPC algorithm with variable look-ahead distance demonstrates superior cross-track error performance compared to the fixed look-ahead distance algorithm, validating the effectiveness of this approach.
• A comparison between multi-objective and single-objective algorithms in terms of trajectory tracking performance shows that the multi-objective approach based on the improved NSGAII algorithm achieves a balanced consideration of both cross-track error and USV tracking speed, resulting in reduced cross-track error.
• By integrating an adaptive rotation-based simulated binary crossover and enhancing the non-dominated sorting method in the NSGAII algorithm, the improved NSGAII demonstrates superior tracking performance with reduced computation time, as observed from the trajectory tracking results.

5. Conclusions

This paper addresses the path-tracking problem of USV systems by designing an improved NSGAII-based MOMPC method for path tracking. Initially, a variable look-ahead distance is introduced in the LOS algorithm, and both look-ahead distance and desired velocity are computed through the MPC control algorithm, enhancing control accuracy in terms of cross-track error for USVs.

To enhance the global search capability of the algorithm, this study incorporates an ARSBX approach into the NSGAII algorithm, along with optimizations to the non-dominated sorting mechanism. These improvements effectively reduce computation time, increase solution space diversity and exploration, and strengthen the algorithm’s ability to solve multi-objective optimization problems.

Simulation results conducted in two distinct scenarios demonstrate that the improved NSGAII-based nonlinear MPC method successfully achieves precise path tracking, showing advantages in cross-track error performance compared to other algorithms. Although the proposed method performs well in simulations, further experiments are necessary to verify its stability under varied environmental conditions. Although the optimization process involved in NSGAII is relatively complex, considering the real-time requirements for practical applications, the computation time can be appropriately reduced by methods such as decreasing the number of iterations to meet time constraints. Through effective resource management, the algorithm can ensure system stability while maintaining control accuracy. Future research should focus on applying the proposed path tracking control method to practical control systems. Additionally, exploring the use of reinforcement learning to dynamically adjust the weights of the MPC algorithm could further enhance its applicability to USV path tracking. This approach could also be extended to areas such as cooperative control of multiple USVs or autonomous berthing operations.