Simulation and Optimization of Multi-Phase Terminal Trajectory for Three-Dimensional Anti-Ship Missiles Based on Hybrid MOPSO

Abstract

In high-dynamic battlefield environments, anti-ship missiles must perform intricate attitude adjustments and energy management within time constraints to hit a target accurately. Traditional optimization methods face challenges due to the high speed, flexibility, and varied constraints inherent to anti-ship missiles. To overcome these challenges, this research introduces a three-dimensional (3D) multi-stage trajectory optimization approach based on the hybrid multi-objective particle swarm optimization algorithm (MOPSO-h). A multi-stage optimization model is developed for terminal trajectory, dividing the flight process into three stages: cruising, altitude adjustment, and penetration dive. Dynamic equations are formulated for each stage, incorporating real-time observations and overload constraints and ensuring the trajectory remains smooth, continuous, and compliant with physical limitations. The proposed algorithm integrates an adaptive hybrid mutation strategy, effectively balancing global search with local exploitation, thus preventing premature convergence. The simulation results demonstrate that, in typical scenarios, the mean miss distance optimized by MOPSO-h remains no greater than 2.34 m, while the terminal landing angle is consistently no less than 85.68°. Furthermore, MOPSO-h enables the missile’s cruise altitude and speed, driven by multiple models, to maintain long-term stability, ensuring that the maneuver overload adheres to physical constraints. This research provides a rigorous and practical solution for anti-ship missile trajectory design and engagement with shipborne air defense systems in high-dynamic environments, achieved through a multi-stage collaborative optimization mechanism and error analysis.

1. Introduction

The central offensive–defensive dynamics in modern naval warfare revolve around the penetrating capabilities of anti-ship missiles and the continuous counter-enhancement of shipborne air defense systems. Advanced anti-ship missiles, such as Russia’s “Zircon”, are achieved terminal maximum speeds exceeding Mach 7 and possess multi-trajectory coordinated penetration capabilities, significantly reducing the interception success rate of traditional shipborne defense systems [1]. Designing terminal trajectories that are practical for real combat, highly concealed, and have a high hit rate can significantly enhance the penetration capability of anti-ship missiles [2]. Furthermore, these trajectories can be cleverly utilized in modern combat simulation systems, providing a competitive benchmark for the design and application of high-quality shipborne air defense systems [3,4,5,6]. Despite these advancements, trajectory optimization involves multidisciplinary challenges, such as aerodynamic parameter time-variant coupling and control order non-linear constraints, making it difficult to directly obtain an analytical solution [7,8].

The terminal trajectory optimization of anti-ship missiles presents unique challenges, as these systems must simultaneously complete attitude adjustment and energy management within an extremely constrained timeframe [9]. Compared to traditional ballistic missiles, the optimization of anti-ship missile trajectories involves significantly more complex parameters [10], necessitating the development of advanced optimization methodologies to effectively address the dynamic and unpredictable nature of modern naval combat scenarios. Current trajectory optimization methodologies are primarily categorized into indirect and direct approaches. The indirect method employs optimal control theory to formulate a Hamiltonian boundary value problem, which is subsequently solved using precise numerical integration techniques. This approach, grounded in Pontryagin’s Maximum Principle, provides theoretically optimal solutions but is heavily reliant on accurate initial value guesses and struggles to handle complex path constraints effectively [11,12,13,14]. In contrast, the direct optimization approach transforms the trajectory optimization problem into a non-linear programming (NLP) framework. By leveraging various intelligent optimization algorithms, this method seeks to identify the optimal trajectory parameters. While the direct approach has demonstrated a robust performance in complex engineering applications, it is not without limitations. Specifically, the “curse of dimensionality” becomes increasingly problematic as the number of time points and constraint variables increase, leading to exponential growth in computational demands [15,16].

In recent years, remarkable advancements in computer technology have significantly enhanced the performance of evolutionary computation techniques, including genetic algorithms (GA), particle swarm optimization (PSO), and ant colony optimization (ACO), in addressing complex nonlinear trajectory optimization challenges [17,18,19,20,21,22,23]. For instance, Kumar et al. proposed a GA-based approach to maximize the glide distance of a hypersonic aircraft [17]. Similarly, Dr. Özdil successfully applied GA and its variants to optimize tactical missile trajectories, ensuring the achievement of mission objectives [18]. Zheng et al. developed a unified PSO algorithm for the integrated design of long-range missile trajectories, propulsion systems, and aerodynamic parameters [19]. Furthermore, an enhanced ACO algorithm has demonstrated its effectiveness in solving six-degree-of-freedom missile trajectory modeling and optimization problems [20]. Xu et al. introduced a novel framework that combined GA with a twin delayed deep deterministic policy gradient to address trajectory planning for long-range air-to-ground strikes [21]. Compared to conventional optimization methods, these evolutionary algorithms exhibit superior capabilities in achieving global optimization within large-scale search spaces characterized by nonlinear constraints while maintaining strong adaptability and rapid convergence rates [22,23].

While significant advancements have been made in evolutionary computation methods for typical trajectory optimization problems, their application to the terminal trajectory optimization of anti-ship missiles remains limited. A primary challenge has arisen from the reliance of anti-ship missiles on single-pulse systems or seeker systems to continuously acquire target data during high-speed flight, necessitating trajectory optimization that satisfies the real-time updates required by the guidance loop [24,25,26]. Furthermore, during the terminal phase, anti-ship missiles execute multiple large-scale thrust maneuvers based on predefined trigger conditions [27]. Consequently, the optimization process must incorporate constraints related to flight time, motor overload, speed, and altitude [28], which substantially influence the performance of the optimization algorithm. This investigation specifically examines the terminal guidance loop of anti-ship missiles operating in mono-pulse tracking mode. To address these complexities, we have developed a multi-stage trajectory optimization model that integrates both the missile’s dynamic characteristics and real-time tracking data. The model accounts for various complex constraints inherent to the system. Additionally, through enhancements to the multi-objective particle swarm optimization algorithm (MOPSO) for handling complex objective functions, the missile can achieve terminal trajectories characterized by large impact angles and minimal miss distances across diverse simulation scenarios.

Current research is limited to the Data-Driven Modeling, Control, Optimization, and Decision-Making in Intelligent Systems, which is beneficial in providing a rigorous and practical solution for anti-ship missile trajectory design and interaction with shipborne air defense systems in high-dynamic environments, and does not pose a threat to public health or national security.

2. Theoretical Basis

2.1. The Terminal Trajectory of Anti-Ship Missiles

Terminal trajectories of anti-ship missiles generally refer to precise strikes achieved through motor-driven means within a limited approach distance to a target ship. If we define the launch coordinate system as $O{x}_m{y}_m{z}_m$ and the projectile coordinate system as $O{x}_b{y}_b{z}_b$, then the transformation matrix from $O{x}_m{y}_m{z}_m$ to $O{x}_b{y}_b{z}_b$ is as follows:

$$C_m^b=\left[\begin{array}{ccc}\cos {\beta }_m\cos {\alpha }_m& \cos {\beta }_m\sin {\alpha }_m& \sin {\beta }_m\\ -\sin {\alpha }_m& \cos {\alpha }_m& 0\\ -\sin {\beta }_m\cos {\alpha }_m& -\sin {\beta }_m\sin {\alpha }_m& \cos {\beta }_m\end{array}\right]\in {ℝ}^{3\times 3}.$$

(1)

The missile and ship position vectors are defined as $p_m={\left[x_m,y_m,z_m\right]}^{\mathrm{T}}$ and $p_s={\left[x_s,y_s,z_s\right]}^{\mathrm{T}}$, respectively, under the coordinate system $O{x}_m{y}_m{z}_m$. The line-of-sight vector $l=p_s-p_m$ is defined as the vector from the missile to the target ship at time $t$. The missile–target distance is denoted by $d=‖l‖$. The radar seeker’s observation value, $d^{\prime }$, is assumed to follow a normal distribution characterized by $d^{\prime }=d+{\epsilon }_d,{\epsilon }_d\sim \mathcal{N}\left(0,{\delta }_{{\epsilon }_d}^2\right)$. Within the projectile coordinate system, the target’s azimuth angles and elevation angles relative to the anti-ship missile are determined as follows:

$$\left\{\begin{array}{l}\theta ={\arctan }_2\left(l_{by},l_{bx}\right)\in (-\pi ,\pi ]\\ \varphi =\arcsin \left({\displaystyle \frac{l_{bz}}{d}}\right)\in (-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]\end{array}\right.,$$

(2)

where ${\left[l_{bx},l_{by},l_{bz}\right]}^{\mathrm{T}}=C_m^{b}l$ and ${\arctan }_2\left(\cdot \right)$ represent four-quadrant arctangent functions, ensuring consistency between the angle signs and their respective quadrants.

As illustrated in Figure 1, an anti-ship missile enhances its terminal penetration capability by employing ultra-low altitude, sea-skimming flight and a multi-stage propulsion system, thereby evading detection by shipborne radar during its terminal approach. Furthermore, the radar seeker is strategically activated only during Phase 3 and Phase 7 of the terminal trajectory to acquire the target ship’s kinematic data, mitigating the risk of jammer interference. In all other stages, the missile relies on pre-programmed intelligence and its onboard tracking system for target information.

The definition of the state variable $X={[p_m,{\dot{x}}_m,{\dot{y}}_m,{\dot{z}}_m,{\alpha }_m,{\beta }_m,{\dot{\alpha }}_m,{\dot{\beta }}_m,{\ddot{\alpha }}_m,{\ddot{\beta }}_m]}^{\mathrm{T}}$ has been provided, with subsequent sections elaborating the trajectory expressions for each phase.

(1) In Phase 1, when the missile–target distance is relatively large, the anti-ship missile maintains a fixed altitude and maximum cruising speed, and its desired course in the $x_{m}O{y}_m$-plane always aligns with the target. Based on the proportional-derivative (PD) feedback adjustment concept, the state equation for the motor-driven process of the anti-ship missile is designed as follows:

$$f_1\left(X\right)=\left\{\begin{array}{l}{\dot{x}}_m=V\cos \beta \cos \alpha \\ {\dot{y}}_m=V\cos \beta \sin \alpha \\ {\dot{z}}_m=V\sin \beta \\ \ddot{\alpha }=sat\left(-K_1\widehat{\theta }-K_2\dot{\alpha },{\dot{\alpha }}_{\max }-\dot{\alpha }\right)\\ \ddot{\beta }=sat\left(K_3\left(H_1-H\right),{\dot{\beta }}_{\max }-\dot{\beta }\right)\\ \widehat{\theta }=\theta +{\epsilon }_{\theta },{\epsilon }_{\theta }\sim \mathcal{N}\left(0,{\delta }_{{\epsilon }_{\theta }}^2\right)\end{array}\right.,$$

(3)

where $H=‖p_m‖-R$ represents the altitude variable, while ${\dot{\alpha }}_{\max }={\displaystyle \frac{n_{\max }g}{V}}\sqrt{1-{\left({\displaystyle \frac{\dot{\beta }V}{n_{\max }g}}\right)}^2}$ and ${\dot{\beta }}_{\max }={\displaystyle \frac{n_{\max }g}{V}}\sqrt{1-{\left({\displaystyle \frac{\dot{\alpha }V}{n_{\max }g}}\right)}^2}$ denote the boundary values for the azimuth angles’ speed and elevation angles’ speed, respectively. Additionally, $\mathrm{sat}\left(u,u_{\max }\right)=\left\{\begin{array}{l}{u}_{\max }, \mathrm{if} u>u_{\max }\hfill \\ -u_{\max }, \mathrm{if} u<-u_{\max }\hfill \\ u, \mathrm{else}\hfill \end{array}\right.$ signifies the saturation function, which is implemented to ensure compliance with overload constraints during missile operation.

(2) In Phase 2, when the missile–target distance diminishes to a specific threshold, the anti-ship missile executes a rapid ascent to a higher altitude, thereby enhancing its ability to acquire a superior vantage point for ship target detection. During the concluding phase, the missile adopts a stabilized flight attitude, with its speed vector aligned parallel to the sea surface. As Phase 2 follows seamlessly from Phase 1, adjustments are required for only a limited number of state variables, as illustrated in the following equations:

$$f_2\left(X\right){|}_{f_1}=\left\{\begin{array}{l}\dot{\alpha }=0\\ \ddot{\beta }=\left\{\begin{array}{l}sat\left(K_4\left(H_2-H\right),{\displaystyle \frac{n_{\max }g}{V}}-\dot{\beta }\right),\mathrm{if} H<H_1+c_1\left(H_2-H_1\right)\\ sat\left(-K_5\dot{\beta }+K_6\left(H_2-H\right),{\displaystyle \frac{n_{\max }g}{V}}-\dot{\beta }\right),\mathrm{else}\end{array}\right.\end{array}\right..$$

(4)

(3) In Phase 3, once the anti-ship missile’s attitude is stabilized, its radar seeker is activated. It is established that the missile employs the two-dimensional (2D) sum-difference amplitude ratio of the radar seeker head to acquire the target angle information ${\theta }^{\prime }$ and ${\varphi }^{\prime }$. Under the assumption of an ideal operational environment with a high signal-to-noise ratio (SNR) (≥10 dB), the angle estimation model can be formulated as follows:

$$\left\{\begin{array}{l}{\theta }^{\prime }=\theta +{{\epsilon }^{\prime }}_{\theta },{{\epsilon }^{\prime }}_{\theta }\sim \mathcal{N}\left(0,{\delta }_{{{\epsilon }^{\prime }}_{\theta }}^2\right)\\ {\varphi }^{\prime }=\varphi +{{\epsilon }^{\prime }}_{\varphi },{{\epsilon }^{\prime }}_{\varphi }\sim \mathcal{N}\left(0,{\delta }_{{{\epsilon }^{\prime }}_{\varphi }}^2\right)\end{array}\right..$$

(5)

Under the control of ${\theta }^{\prime }$ and ${\varphi }^{\prime }$, the anti-ship missile will adjust its flight path in accordance with the guidance law while maintaining altitude stabilization. The state equation for this phase follows from Phase 2 and is redefined as detailed below:

$$f_3\left(X\right){|}_{f_2}=\left\{\begin{array}{l}\ddot{\alpha }=sat\left(-K_1{\theta }^{\prime }-K_2\dot{\alpha },{\dot{\alpha }}_{\max }-\dot{\alpha }\right)\\ \ddot{\beta }=sat\left(K_3\left(H_2-H\right),{\dot{\beta }}_{\max }-\dot{\beta }\right)\end{array}\right..$$

(6)

(4) In Phase 4, the anti-ship missile has successfully obtained the necessary target tracking information for the terminal engagement. Owing to its elevated altitude, the missile must execute a rapid descent to an ultra-low altitude to evade detection by shipboard radar systems. By the conclusion of this phase, the missile maintains a stabilized flight attitude, with its speed vector aligned parallel to the sea surface, akin to Phase 2. Given the symmetrical nature of the trajectory, the state equations governing Phase 4 are as follows:

$$f_4\left(X\right){|}_{f_3}=\left\{\begin{array}{l}\dot{\alpha }=0\\ \ddot{\beta }=\left\{\begin{array}{l}sat\left(K_4\left(H_1-H\right),{\displaystyle \frac{n_{\max }g}{V}}-\dot{\beta }\right),\mathrm{if} H>H_1+\left(1-c_1\right)\left(H_2-H_1\right)\\ sat\left(-K_5\dot{\beta }+K_6\left(H_1-H\right),{\displaystyle \frac{n_{\max }g}{V}}-\dot{\beta }\right),\mathrm{else}\end{array}\right.\end{array}\right..$$

(7)

(5) In Phase 5, the anti-ship missile enters the final preparation stage for penetration. Similar to Phase 1, the missile maintains consistent altitude and cruising speed, with adjustments confined to speed adjustments within the $x_{m}O{y}_m$ plane. The sole distinction between the two phases lies in the following directional guidance mechanism: Phase 5 employs $\widehat{{\theta }^{\prime }}$ rather than $\widehat{\theta }$ as the directional reference:

$$f_5\left(X\right){|}_{f_4}=\left\{\begin{array}{l}\ddot{\alpha }=sat\left(-K_1\widehat{{\theta }^{\prime }}-K_2\dot{\alpha },{\dot{\alpha }}_{\max }-\dot{\alpha }\right)\\ \ddot{\beta }=sat\left(K_3\left(H_1-H\right),{\dot{\beta }}_{\max }-\dot{\beta }\right)\end{array}\right.,$$

(8)

where $\widehat{{\theta }^{\prime }}$ is derived from the predicted value of ${\theta }^{\prime }$.

(6) Phase 6 represents a critical operational phase. Upon achieving sufficient proximity to the target, the anti-ship missile executes a rapid ascent to designated elevation angles, subsequently redirecting its speed vector to a downward trajectory perpendicular to the sea surface in anticipation of a diving motor-driven maneuver. The corresponding state equation for this phase is expressed as follows:

$$f_6\left(X\right){|}_{f_5}=\left\{\begin{array}{l}\dot{\alpha }=0\\ \ddot{\beta }=\left\{\begin{array}{l}sat\left(K_7\left(H_3-H\right),{\displaystyle \frac{n_{\max }g}{V}}-\dot{\beta }\right), \mathrm{if} H<H_1+c_2\left(H_3-H_1\right)\\ sat\left(K_8(-{\displaystyle \frac{\pi }{2}}-\beta )-K_9\dot{\beta }-K_{10}\mathrm{sgn}(\beta +{\displaystyle \frac{\pi }{2}}),{\displaystyle \frac{n_{\max }g}{V}}-\dot{\beta }\right)\end{array}\right.\end{array}\right.,$$

(9)

where $\mathrm{sgn}\left(\cdot \right)$ represents the signum function.

(7) Phase 7 represents the most crucial phase of the terminal trajectory. Leveraging its superior aerodynamic configuration, the anti-ship missile executes a terminal dive with a trajectory perpendicular to the sea surface and is guided by the proportional navigation law. During this phase, the radar seeker is activated promptly to acquire enhanced target information. The corresponding state equation for this final stage is as follows:

$$f_7\left(X\right){|}_{f_6}=\left\{\begin{array}{l}\dot{\alpha }=\mathrm{sat}\left(-K_{11}{\displaystyle \frac{g}{V}}\mathrm{\Delta }{\theta }^{\prime }\mathrm{\Delta }{d}^{\prime }\cos \beta ,{\dot{\alpha }}_{\max }\right)\\ \dot{\beta }=\mathrm{sat}\left(-K_{11}{\displaystyle \frac{g}{V}}\mathrm{\Delta }{\varphi }^{\prime }\mathrm{\Delta }{d}^{\prime }\cos \beta ,{\dot{\beta }}_{\max }\right)\end{array}\right.,$$

(10)

where $\mathrm{\Delta }{\theta }^{\prime }$, $\mathrm{\Delta }{\varphi }^{\prime }$, and $\mathrm{\Delta }{d}^{\prime }$ denote the variations in target angle and distance information as perceived by the radar seeker over successive temporal intervals.

Through the comprehensive integration of all switching conditions associated with the final trajectory segments, the following result can be derived:

$$f\left(X,t\right)=\left\{\begin{array}{l}{f}_1\left(X\right), \mathrm{if} d^{\prime }>R_1\\ f_2\left(X\right),\mathrm{if} R_1\ge d^{\prime }>R_2\\ ⋮\\ f_k\left(X\right),\mathrm{if} R_{k-1}\ge d^{\prime }>R_k\\ ⋮\\ f_7\left(X\right),\mathrm{if} R_6\ge d^{\prime }\end{array}\right.,k=1,2,\dots ,6.$$

(11)

Based on $f\left(X\right)$, the termination criteria for the final phase trajectory include $t\ge N_{\max }/F$, $d_{\mathrm{end}}\le D$, and ${‖p_m‖}_{\mathrm{end}}\le R$, which correspond to missile fuel exhaustion, sea impact, and target engagement, respectively. Upon completing the seven-stage sequence, it is imperative that the anti-ship missile satisfies condition ${\beta }_{\mathrm{end}}\to -{\displaystyle \frac{\pi }{2}}$ simultaneously with the target impact.

Consequently, the objective function and constraints for the terminal trajectory optimization of anti-ship missiles are established as follows:

$$\begin{array}{ll}\mathrm{minmize} & J={\displaystyle \frac{F{d}_{\mathrm{end}}}{V_{\max }}}-{\displaystyle \frac{2{\beta }_{\mathrm{end}}}{\pi }}+t-N_{\max }/F\\ \mathrm{where} & d_{\mathrm{end}},{\beta }_{\mathrm{end}},{‖p_m‖}_{\mathrm{end}}\mathrm{are} \mathrm{generated} \mathrm{by} f\left(X,t\right)\\ \mathrm{s}.\mathrm{t}. & {\beta }_{\mathrm{end}}<0,V\le V_{\max }, {‖p_m‖}_{\mathrm{end}}\ge R\end{array}.$$

(12)

The objective function necessitates variable clamping in adherence to constraints, a process that intertwines the optimization gradients of missile miss distance and posture, thereby posing significant challenges to algorithmic optimization. In this study, we employ a methodological decomposition of Equation (12) and introduce a novel multi-objective framework for optimization purposes, thereby facilitating a more refined approach to the problem, as follows:

$$\begin{array}{ll}\mathrm{minmize} & J:J_1,J_2,J_3\\ \mathrm{where} & J_1={\displaystyle \frac{F{d}_{\mathrm{end}}}{V_{\max }}}\\ & J_2=-{\displaystyle \frac{2{\beta }_{\mathrm{end}}}{\pi }}\\ & J_3=t-N_{\max }/F\\ \mathrm{s}.\mathrm{t}. {\beta }_{\mathrm{end}}& <0,V\le V_{\max }, {‖p_m‖}_{\mathrm{end}}\ge R\end{array}.$$

(13)

2.2. MOPSO-h Algorithm and Applications

2.2.1. Algorithm Principles

Multi-objective optimization algorithms are designed to handle multiple conflicting objective functions simultaneously. Their primary aim is to identify the Pareto optimal solution set, thereby providing decision-makers with multi-dimensional trade-off solutions. In contrast to conventional single-objective optimization frameworks, the central challenge lies in balancing the convergence, distribution, and diversity of solutions. Prominent algorithms such as non-dominated sorting genetic algorithm-II (NSGA-II), non-dominated sorting genetic algorithm-III (NSGA-III), multi-objective evolutionary algorithm based on decomposition (MOEA/D), and MOPSO [29,30,31,32] employ advanced mechanisms like non-dominated sorting procedures, decomposition-based strategies, or indicator-based selection mechanisms to address intricate trade-offs. In the aerospace domain, multi-objective optimization finds extensive applications in aircraft aerodynamic optimization, trajectory optimization, and mission planning. This approach enables the coordination of conflicting requirements, such as fuel efficiency, penetration probability, and miss distance, ultimately delivering globally optimal solution clusters for complex aerospace systems.

The MOPSO algorithm integrates the efficient search mechanism of PSO with a multi-objective optimization framework, enabling it to rapidly approximate the Pareto front via iterative swarm intelligence. It is particularly effective for solving high-dimensional, nonlinear optimization problems and excels in handling physical constraints, such as overload limits and power models, making it highly suitable for aircraft trajectory optimization. In each iteration, the particle population is characterized by their position, $X_n^i\in {ℝ}^{\left|X\right|}$, and speed, $V_n^i\in {ℝ}^{\left|X\right|},$$i=1,2,\dots ,I,$$n=1,2,\dots ,N$. At the $i$-th iteration, the objective vector $J\left(X_n^i\right)=\left[J_1\left(X_n^i\right),J_2\left(X_n^i\right),J_3\left(X_n^i\right)\right]$ is computed for each particle. Subsequently, non-dominated sorting and diversity pruning are applied to generate a set of non-dominated solutions, $\left\{X_1^i,X_2^i,\dots ,X_{N_A}^i\right\}$ [33]. The global leader $X_A^i \sim {\displaystyle \sum _{a=1}^A{P}_a}\ge p \&{\displaystyle \sum _{a=1}^{A-1}{P}_a}<p,$$A\in \left\{1,2,\dots ,N_A\right\}$ is then selected from the current population based on a roulette wheel probability $P$, where the selection probability $P_a$ is determined by the crowding distance $CD\left(X_a^i\right)$, as follows:

$$P_a={\displaystyle \frac{CD\left(X_a^i\right)}{{\displaystyle \sum _{b=1}^{N_A}CD\left(X_b^i\right)}}}.$$

(14)

Furthermore, the MOPSO algorithm employs the mechanism $X_A^i$ for updating the speed and position of each particle, as follows:

$$\begin{array}{l}{V}_n^{i+1}=w^i{V}_n^i+a_1{r}_1\left(X_B^i-X_n^i\right)+a_2{r}_2\left(X_A^i-X_n^i\right)\\ X_n^{i+1}=X_n^i+V_n^{i+1}\end{array},$$

(15)

where $w^i=w_{\max }-\left(w_{\max }-w_{\min }\right)i/I$ is the dynamic inertia weight and $w_{\max }$ and $w_{\min }$ are the upper and lower bounds of the dynamic inertia weight, respectively.

At the same time, the individual historical best $X_B^i$ is updated based on the following calculations, and whether the iteration should be terminated is determined:

$$X_B^{i+1}=\left\{\begin{array}{l}{X}_n^{i+1},\mathrm{if} X_n^{i+1}<X_B^i \mathrm{or} \mathrm{incomparable}\\ X_B^i,\mathrm{else}\end{array}\right..$$

(16)

Given the inherent complexity of missile-trajectory optimization, characterized by its high-dimensional, multi-constraint, and multi-objective nature, the optimization process necessitates the implementation of probabilistic mutation operations during population evolution. These operations play a crucial role in maintaining population diversity, thereby mitigating the risk of premature convergence. This study integrates an adaptive Hybrid Mutation Strategy (HMS) into the standard MOPSO framework, serving as a replacement for conventional mutation operations. The strategy employs uniform mutation during initial iterations to enhance exploration capabilities, while subsequent stages utilize Gaussian mutation for precise optimization. The transition between these phases is dynamically determined based on the monitoring of population hypervolume (HV) trends, as follows:

$$X_n^i=\left\{\begin{array}{l}{X}_n^i+U\left(\left(1-{\displaystyle \frac{i}{I}}\right)\left(X_{\max }-X_{\min }\right)\right),\mathrm{if} {\displaystyle \frac{H{V}_i-H{V}_{i-\mathrm{\Delta }I}}{H{V}_{i-\mathrm{\Delta }I}}}>a_{HV}\hfill \\ X_n^i+\mathcal{N}\left(0,\exp \left(-{\displaystyle \frac{i}{I}}\right){\sigma }_0^2\right),\mathrm{else}\hfill \end{array}\right.,$$

(17)

where $U\left(\cdot \right)$ is the uniform distribution function, $n$ is the adjustment interval of $H{V}_i$, and $X_{\max }$ and $X_{\min }$ are the upper and lower boundaries of $X$, respectively. When the change in the hypervolume $H{V}_i$ surpasses the critical value $a_{HV}$, a uniform variation strategy is implemented. This involves random sampling within interval $\left(1-{\displaystyle \frac{i}{I}}\right)\left(X_{\max }-X_{\min }\right)$ to provide directional guidance for the mutation factor, thereby restricting its variation range and preventing premature algorithm convergence. The parameter $\left(1-{\displaystyle \frac{i}{I}}\right)$ exhibits linear decay as the iteration count increases, progressively diminishing its influence on $X_n^i$. In contrast, during the latter stages of iteration, $X_n^i$ approaches the optimal solution and only necessitates fine-tuning, while $\exp \left(-{\displaystyle \frac{i}{I}}\right){\sigma }_0^2$, an exponential decay factor, is introduced to induce disturbances around $X_n^i$ until its effect is negligible. The calculation of $H{V}_i$ refers to the method in reference [34].

Overall, MOPSO-h inherits several implementation techniques from MOPSO, including boundary clipping and constraint repair [35]. Nonetheless, these aspects are not central to the current investigation and, thus, are not elaborated upon in detail.

2.2.2. Design Variables

The primary objective of optimizing the terminal trajectory for multi-stage anti-ship missiles is to enhance the precision of the final strike within a constrained timeframe. By integrating Formulas (3) through (11), the control coefficients at each stage are formulated, and the control variables requiring optimization are combined to establish the ideal input for the MOPSO-h algorithm. Following this process, the theoretical variable dimensions identified for optimization amount to 20, designated as $X=[H_1,H_2,H_3,$$R_1,R_2,\dots ,R_6,$$K_1,K_2,\dots ,K_{11}]$. To mitigate the issue of prolonged algorithm optimization time resulting from high-dimensional variables, certain parameters are selected manually based on tuning experiments and expert insights. This selection is guided by the essential operational requirements for anti-ship missiles during low-altitude penetration, high-altitude reconnaissance, and vertical landing angle strikes, which collectively define the parameter set $H_1=10 \mathrm{m},H_2=1000 \mathrm{m}, H_3=800 \mathrm{m}.$ Furthermore, considering the diminishing influence of distance threshold during terminal penetration phases as the target proximity decreases and the inherent strong overload capacity of anti-ship missiles enabling greater maneuverability at closer ranges, the distance thresholds for Phase 1 to Phase 5 are sequentially established as $R_1=25 \mathrm{km},$$R_2=20 \mathrm{km}, R_3=17 \mathrm{km}, R_4=14 \mathrm{km}$. Additionally, given that stages 1, 3, and 5 involve constant-altitude cruising modes where rapid adjustments to the azimuth angles are critical, parameter $K_1=1$ is appropriately defined. Ultimately, the actual design variables involved in the optimization process are identified as $X=\left[R_5,R_6,K_2,\dots ,K_{11}\right]$, which constitutes a 12-dimensional variable.

3. Simulation Experiment

3.1. Scene Configuration

We have established a representative terminal penetration scenario involving anti-ship missiles to validate the developed algorithms and models, specifically focusing on missile engagement targeting the broadside of a ship. The missile’s geometric parameters are configured with a length of 4.6 m, a diameter of 0.3 m, and a wingspan of 0.9 m, based on the RGM-84 “Harpoon” missile. Similarly, the ship’s dimensions are defined as 172.8 m in length, 16.8 m in width, and approximately 40 m in height, following the specifications of the Navy’s Ticonderoga-class cruiser. In $O{x}_m{y}_m{z}_m$, the missile’s launch origin is set at the coordinate system’s origin, with its initial state denoted as $X=[0,0,10 \mathrm{m},0,-2 \mathrm{Ma},0,0,0,0,0,0,0]$, with a maximum velocity of 2 Ma and a maximum load factor of 15 g, where $\mathrm{g}=9.8{ \mathrm{m}/\mathrm{s}}^2$. The ship maintains a constant speed, with its initial position and speed designated as $p_s={[0,-\mathrm{30,000} \mathrm{m},0]}^{\mathrm{T}}$ and ${\dot{p}}_s={[20 \mathrm{m}/\mathrm{s},0,0]}^{\mathrm{T}}$, respectively. The missile maintains a collision proximity of 5 m, with a maximum mission duration of 60 s. The planning frequency is set to 100 Hz, and the total mission step count is 1000. Simulations are conducted on a computing workstation equipped with a Windows 11 operating system. The system features a 12th Gen Intel(R) Core(TM) i5-12500 CPU and 32.0 GB of RAM. The simulation environment is implemented using pyCharm v5.3.2, with Python v3.11 as the programming language. The essential software development kits include numpy v1.2.64 and matplot v0.1.9. The study currently does not incorporate external factors such as wind and waves, as the focus is on optimizing the missile’s flight performance and trajectory design in complex battlefield environments.

3.2. Algorithm Configuration

The process of generating the optimal terminal trajectory using MOPSO-h in this study is shown in Figure 2.

To align with reality, the entire simulation model and algorithm’s parameter configurations are shown in

Table 1. Parameter configurations of the simulation models and algorithms.

Model		MOPSO-h
Variable	Value	Variable	Value
${\delta }_{{\epsilon }_{\theta }}$	0.02 rad	I	300
${\delta }_{{{\epsilon }^{\prime }}_{\theta }}$	0.01 rad	N	50
${\delta }_{{{\epsilon }^{\prime }}_{\varphi }}$	0.01 rad	NA	10
R	6371 km	p	0.9
${\delta }_{{\epsilon }_d}$	1 m	a1	2
$n_{\max }$	15 g, g = 9.8 m/s2	a2	2
$V_{\max }$	2 Ma	a3	$U\left(-1,1\right)$
D	5 m	aHV	0.05
F	100 Hz	r1	$U\left(0,1\right)$
$N_{\max }$	60 s	r2	$U\left(0,1\right)$
$R_6$	[0, 1200 m]	${\delta }_0$	0.1
$K_2,K_3,\dots ,K_{11}$	[0, 10]	$\mathrm{\Delta }I$	5

.

4. Experimental Results Analysis

To evaluate the proposed algorithm’s performance in optimizing the terminal trajectory of anti-ship missiles, a comparative analysis has been conducted involving the following seven optimization algorithms: PSO [15], GA [17], ACO [34], NSGA-III [30], MOEA/D [31], MOPSO [32], and the enhanced MOPSO-h. All algorithms follow the parameter configurations of the original references without any modifications, and the experimental setup has the same initial conditions. The optimization criteria encompass the following six key performance metrics: mean miss distance (MMD), extra time percentage (ETP), terminal landing speed (TLS), terminal landing angle (TLA), peak surge altitude (PSA), and computational time cost (TC). Specifically, PSA denotes the maximum altitude required for a missile’s attitude adjustment during the terminal phase. To avoid interception, the smaller the PSA and the larger the TLA and TLS, the better. The calculation method for ETP is as follows:

$$ETP=\left({\displaystyle \frac{V_{\max }{t}_{\mathrm{end}}}{d_0}}-1\right)\times 100\%.$$

(18)

The particle populations of all algorithms are randomly initialized, and the average performance of each algorithm is statistically analyzed through 100 Monte Carlo simulations.

4.1. Algorithm Performance

The simulation results for two representative task scenarios, presented in

Table 2. Comparison of the algorithm performance indexes.

Algorithm	MOPSO-h	MOPSO	MOEA/D	NSGA-III	PSO	ACO	GA
MMD (m)	2.34 (1)	2.90 (5)	2.40 (2)	3.02 (6)	2.88 (4)	2.46 (3)	3.27 (7)
ETP (%)	8.91 (2)	9.88 (4)	9.02 (3)	10.38 (6)	8.42 (1)	10.70 (7)	10.22 (5)
TLS (m/s)	−542.1 (4)	−548.9 (3)	−521.2 (6)	−628.5 (1)	−493.7 (7)	−526.6 (5)	−619.3 (2)
TLA (°)	85.68 (2)	85.23 (4)	85.17 (5)	86.64 (1)	84.07 (7)	84.99 (6)	85.41 (3)
PSA (m)	999.6 (5)	952.3(4)	797.3 (2)	1197.1 (7)	703.2 (1)	839.0 (3)	1185.8 (6)
CT (s)	3184	3037	3058	3235	2856	2698	2763
Score	14	20	18	21	20	24	23

, reveal that MOPSO-h demonstrates superior performance compared to the other algorithms in terms of MMD and TLS, achieving a 25% improvement over traditional MOPSO. In comparison to the top-performing NSGA-III, MOPSO-h exhibits a 6% reduction in ETP. This can be attributed to MOPSO-h’s adaptive hybrid mutation strategy, which balances breakthrough velocity and strike precision by spending more time adjusting attitudes under physical constraints, thereby slightly relaxing the requirement for flight time. In consideration of task difficulty and convergence time requirements, the parameters of MOPSO-h are meticulously designed, with the specific configurations presented in Table 1. The experimental results demonstrate that the algorithm achieves complete convergence within 200 generations, as illustrated in Figure 3. MOPSO-h exhibits effective convergence behavior, avoiding premature entrapment in local optima while maintaining a balance between global exploration and local exploitation. However, this approach comes with a trade-off: the algorithm’s runtime is relatively lengthy, comparable to that of NSGA-III. Additionally, the performance gap between single-objective and multi-objective optimization algorithms is smaller than anticipated. The primary reasons may include: (1) the limited number of terminal trajectory optimization objectives for anti-ship missiles; and (2) the sufficient conditions adopted in single-objective optimization’s objective function design, where the coupling relationships under different conditions are weak, facilitating the identification of global optima.

4.2. Trajectory Analysis

Figure 4 illustrates the multi-stage terminal trajectories of anti-ship missiles generated by the various algorithms. While all algorithms successfully achieved target destruction within the specified iteration limits, the trajectory produced by the MOPSO-h demonstrates a superior performance. The optimal solution suggests that a larger entry angle and more stable attitude of the anti-ship missile impose higher requirements on H6, thereby allowing the missile to allocate sufficient time for attitude adjustment. The seven trajectories, governed by Equations (12) and (13), exhibit smoothness without any abnormal vibrations, and the altitude during the cruise phase remains stable, which corroborates the reasonableness of the design of $f\left(X,t\right)$.

Furthermore, Figure 5 illustrates the trend of distance decay between the missile and the ship, revealing that the decline of $d$ is nearly linear, which signifies that the missile’s motion state remains highly stable, consistently approaching the target at maximum speed. Figure 6, derived from the optimal solution of MOPSO-h, presents the speed and overload variation curves of the anti-ship missile. The results indicate that the missile tends to execute turning maneuvers at maximum overload and stabilizes upon reaching the expected altitude, effectively suppressing structural failure while adhering to fundamental physical constraints.

4.3. Error Impact Analysis

This study employs a mathematical model based on angle and distance observations to optimize the terminal trajectory of anti-ship missiles. Analyzing the impact of observational errors on optimization outcomes is essential. In scenario 1, the binding angle error ${\delta }_{{\epsilon }_{\theta }}$ varies within the interval $\left[0.01 \mathrm{rad},0.05 \mathrm{rad}\right]$, while the ranging error ${\delta }_{{\epsilon }_d}$ takes values of 1 m, 2 m, and 5 m. The variation curve of MMD is presented in Figure 7. When the binding angle error reaches 0.05 rad, ensuring that the optimal trajectory hits the target ($d\le D=5 \mathrm{m}$) becomes challenging. Moreover, when the ranging error exceeds 5 m, the missile’s hit accuracy significantly decreases.

With the parameters in Table 1 unchanged, the azimuth angles error ${\delta }_{{{\epsilon }^{\prime }}_{\theta }}$ and elevation angles error ${\delta }_{{{\epsilon }^{\prime }}_{\varphi }}$ are allowed to vary within the interval $\left[0,0.05 \mathrm{rad}\right]$. The resulting MMD surface is shown in Figure 8. Observations reveal that the influence of ${\delta }_{{{\epsilon }^{\prime }}_{\varphi }}$ is more pronounced than ${\delta }_{{{\epsilon }^{\prime }}_{\theta }}$. When ${\delta }_{{{\epsilon }^{\prime }}_{\varphi }}\ge 0.026 \mathrm{rad}$ occurs, the missile loses all reliability in achieving the target strike.

5. Conclusions

This study addresses the challenge of multi-phase terminal trajectory optimization for anti-ship missiles by proposing a 3D multi-phase trajectory optimization method based on MOPSO-h. The algorithm combines the efficient global search capability of PSO with a multi-objective optimization framework, incorporating an adaptive learning mechanism based on HMS to replace traditional mutation operations. This innovative approach enables missiles to achieve optimal trade-offs among various performance metrics in complex operational scenarios. The simulation results demonstrate that, in typical scenarios, the trajectory optimized by MOPSO-h achieves superior overall performance and significantly enhances terminal guidance accuracy compared to conventional algorithms. The state curve analysis indicates that MOPSO-h facilitates automatic speed control and overload adjustment for anti-ship missiles under constrained conditions. Additionally, the research examines the influence of angle-tracking errors on trajectory optimization performance, uncovering a nonlinear relationship between the two and offering valuable insights for improving missile guidance performance in complex battlefield environments.