Reentry Trajectory Optimization of Hypersonic Vehicle Based on Multi-Strategy Improved WOA Optimized Attention-LSTM Network

Abstract

Trajectory optimization of hypersonic vehicles face challenges from complex aerodynamic environments and multiple constraints, where traditional offline optimization methods struggle to meet real-time requirements. This study proposes a novel online trajectory optimization framework for hypersonic vehicles that integrates a multi-strategy improved whale optimization algorithm (MWOA) with an attention-mechanism Long Short-Term Memory (AM-LSTM) network. First, an offline trajectory dataset under aerodynamic uncertainties is generated using sequential second-order cone programming (SOCP). Subsequently, a multi-head attention mechanism is incorporated into the LSTM network to effectively capture sequential dependencies within the trajectory data. To automate the hyperparameter tuning of the AM-LSTM architecture, a multi-strategy improved whale optimization algorithm is developed, which incorporates circle chaotic mapping for population initialization, a nonlinear convergence factor to balance global and local search, and a dynamic golden-sine mutation strategy to enhance optimization robustness. The trained MWOA-AM-LSTM hybrid model is then employed for real-time trajectory generation. Numerical simulation results demonstrate that the proposed framework achieves superior terminal accuracy under aerodynamic perturbations, validating its effectiveness and robustness for hypersonic vehicle reentry trajectory optimization.

1. Introduction

Hypersonic vehicles represent a class of aircraft operating in the near-space region with flight velocity exceeding Mach 5, characterized by high maneuverability and extended operational range [1,2]. Due to the complex flight environment and numerous constraints, trajectory optimization is critical to ensuring the successful flight of hypersonic vehicles under multiple terminal and path constraints [3,4]. However, reference trajectories in engineering practice are typically generated through offline optimization methods, which struggle to accommodate diverse uncertainties during flight operations. Consequently, real-time reentry trajectory optimization remains a significant technical challenge.

Trajectory optimization is fundamentally a nonlinear optimal control problem, primarily addressed through indirect and direct methods. The indirect method, leveraging Pontryagin’s Minimum Principle, transforms the problem into a Hamiltonian boundary value problem, thereby offering theoretical guarantees of global optimality [5,6,7]. In contrast, the direct method discretizes the continuous problem into a nonlinear programming (NLP) formulation, which is then solved numerically. This approach has been successfully demonstrated in diverse applications, including atmospheric reentry [8,9], rocket landing [10,11], and planetary missions [12,13,14]. Despite their respective strengths, both methods face certain challenges in trajectory optimization. The indirect method grapples with problem size that grows exponentially under complex constraints, while the direct method often incurs prohibitive computational costs for large-scale NLPs.

In recent years, with the advancement of computational capabilities, deep learning has demonstrated tremendous application potential in the field of trajectory optimization [15,16,17]. Leveraging the powerful nonlinear mapping capabilities of neural networks, researchers have effectively reduced online computational burdens through offline training on historical trajectory datasets, thereby providing a promising alternative for real-time trajectory optimization in hypersonic vehicles [18,19,20,21]. Work [22] developed a dual-phase hypersonic reentry planner, incorporating offline fuzzy-optimized trajectory planning and online DNN execution. This strategy achieves real-time trajectory execution while maintaining solution feasibility under complex flight constraints, demonstrating superior computational efficiency compared to conventional optimization-based methods. Work [23] proposed an offline-trained Deep Neural Network (DNN) controller for hypersonic flight that learns state-to-optimal-action mappings from homotopy-generated trajectory data, enabling real-time near-optimal control with stable convergence. Work [24] proposed a DNN-constrained trajectory generator coupled with adaptive reinforcement learning control for air-breathing hypersonic vehicles, which achieves precise tracking control while satisfying path constraints. Although the aforementioned strategies achieve real-time computation through offline training, the static feedforward architecture of its DNN struggles to effectively capture the inherent temporal dependencies in trajectory data.

In this article, Long Short-Term Memory (LSTM) networks combined with multi-head attention mechanisms are proposed as the core architecture for trajectory planning. As a representative variant of recurrent neural networks, LSTM demonstrates exceptional capability in capturing complex temporal features within sequential data, exhibiting significant potential for predictive applications [25,26]. This specialized recurrent structure has proven effective in modeling data with strong nonlinearities and temporal dependencies. Furthermore, the multi-head attention mechanism improves prediction accuracy of LSTM by adaptively weighting critical information in a sequence [27,28]. The outstanding predictive capabilities of LSTM networks integrated with multi-head attention mechanisms have been documented across multiple disciplines, including anomaly detection [29,30], medical diagnosis [31,32], hydrology and water resources [33,34], and civil engineering applications [35,36].

Despite the proven efficacy of LSTM and multi-head attention mechanisms in temporal modeling, their hyperparameter configuration is still limited by a heavy reliance on empirical knowledge and inefficient tuning processes [37,38]. Such manual tuning paradigms fail to unlock the full potential of neural networks, proving inadequate for the rigorous demands of complex trajectory optimization tasks. While multiple approaches exist for hyperparameter optimization, the adoption of swarm intelligence algorithms represents a particularly promising solution [39,40]. Their principal advantage stems from replacing traditional gradient calculations with probabilistic search to find the global optimum solution in high-dimensional spaces.

This paper presents an online trajectory planning framework for hypersonic vehicles based on a multi-strategy improved whale optimization algorithm and an attention-enhanced Long Short-Term Memory (MWOA–AM-LSTM) model. The framework is designed to enable real-time onboard trajectory generation in complex reentry aerodynamic environments by learning an expert state–command mapping from offline solutions, while maintaining comparable solution quality with substantially reduced online computational cost. Specifically, the main contributions are:

(1)

Online learning-based trajectory generation guided by an offline expert database. We propose an integrated MWOA–AM-LSTM framework for hypersonic vehicle reentry trajectory planning, where sequential second-order cone programming (SOCP) is used offline to generate a reference trajectory–command dataset under bounded aerodynamic uncertainties. The AM-LSTM is trained in a supervised manner to approximate the expert state–command mapping—i.e., to infer the next-step bank-angle command from a short history of flight states—thereby enabling real-time online rollout with comparable performance to the SOCP-generated references. The resulting trajectory is propagated via numerical integration under admissible control bounds, allowing constraint-related quantities to be monitored during online execution and improving practical robustness in disturbed aerodynamic conditions.

(2)

Automated and robust hyperparameter tuning for AM-LSTM via an improved WOA. To avoid manual and empirical hyperparameter selection, we develop a multi-strategy improved whale optimization algorithm to automatically tune the AM-LSTM architecture. By incorporating circle chaotic mapping for diversified initialization, a nonlinear convergence factor for balancing exploration and exploitation, and a dynamic golden-sine mutation strategy to mitigate premature convergence, the proposed MWOA enhances the efficiency and robustness of hyperparameter search in high-dimensional spaces, thereby improving the reliability of the learned mapping for real-time deployment.

The article is structured as follows: Section 2 formulates the reentry trajectory optimization problem for hypersonic vehicles. Section 3 details a novel trajectory planning methodology based on the MWOA-AM-LSTM framework. Section 4 analyzes and evaluates simulation results. Section 5 provides concluding remarks.

2. Entry Problem Formulation

2.1. Trajectory Dynamics

This subsection presents the normalized dynamics for hypersonic vehicles through a dimensionless dynamics. By employing an energy-like variable e as the state variable and rigorously incorporating Earth’s rotational dynamics, the derived dimensionless trajectory dynamics are expressed as follows [19]:

$${\displaystyle \left\{\begin{array}{c}\frac{\mathrm{d}r}{\mathrm{d}e}=\frac{sin\gamma }{D}\hfill \\ \frac{\mathrm{d}\theta }{\mathrm{d}e}=\frac{cos\gamma sin\psi }{rDcos\varphi }\hfill \\ \frac{\mathrm{d}\varphi }{\mathrm{d}e}=\frac{cos\gamma cos\psi }{rD}\hfill \\ \frac{\mathrm{d}\gamma }{\mathrm{d}e}=\frac{1}{V^{2}D}\left(Lcos\sigma +\left(V^2-\frac{1}{r}\right)\frac{cos\gamma }{r}+V{\mathsf{\Omega }}_1\right)\hfill \\ \frac{\mathrm{d}\psi }{\mathrm{d}e}=\frac{1}{V^{2}D}\left(\frac{Lsin\sigma }{cos\gamma }+\frac{V^{2}cos\gamma sin\psi tan\varphi }{r}+V{\mathsf{\Omega }}_2\right)\hfill \end{array}\right.}$$

(1)

where

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_1=& [2V\mathsf{\Omega }sin\psi cos\varphi +{\mathsf{\Omega }}^{2}r{cos}^2\varphi cos\gamma \hfill \\ \hfill & +{\mathsf{\Omega }}^{2}rcos\varphi sin\varphi cos\psi sin\gamma ]/V\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_2=& [-2V\mathsf{\Omega }(cos\psi sin\gamma cos\varphi -cos\gamma sin\varphi )\hfill \\ \hfill & +{\mathsf{\Omega }}^{2}rcos\varphi sin\varphi sin\psi ]/Vcos\gamma \hfill \end{array}$$

(3)

where r represents the normalized radial distance from Earth’s center to the hypersonic vehicle, scaled by Earth’s reference radius $R_0$. $\theta$ and $\varphi$ correspond to geographic longitude and latitude; $\gamma$ signifies the flight path angle, while $\psi$ designates the heading angle. $\sigma$ is the bank angle. The dimensionless velocity V is scaled by $\sqrt{g_0{R}_0}$ and the dimensionless Earth rotation rate $\mathsf{\Omega }$ is scaled by $\sqrt{g_0/R_0}$. $g_0$ is the gravitational acceleration at $R_0$, where $R_0=6371$ km. Dimensionless Lift acceleration L and dimensionless drag acceleration D are characterized by expressions detailed as follows:

$$\left\{\begin{array}{c}L=0.5R_0\rho V^{2}S{C}_L/m\\ D=0.5R_0\rho V^{2}S{C}_D/m\end{array}\right.$$

(4)

where $C_L$ is the lift coefficient and $C_D$ is drag coefficient. S, m denote the reference area and mass, respectively. $\rho$ is atmospheric density, which is usually approximated as a local exponential function of altitude. Following [3], its expression is given by

$$\rho ={\rho }_{0}exp\left(-{\displaystyle \frac{r-1}{H}}\right)$$

(5)

where H is dimensionless density scale altitude, $H=H_s/R_0$. $H_s=7110$ m is the density scale height and treated as a constant.

The dimensionless energy in Equation (1) is formulated as:

$$e={\displaystyle \frac{1}{r}}-{\displaystyle \frac{V^2}{2}}\approx 1-{\displaystyle \frac{V^2}{2}}$$

(6)

Convert it to the derivative form with respect to dimensionless time $\tau$:

$${\displaystyle \frac{\mathrm{d}e}{\mathrm{d}\tau }}=DV>0,\tau ={\displaystyle \frac{t}{\sqrt{R_0/g_0}}}$$

(7)

For reentry trajectory optimization, the state and control vectors are formally defined as follows:

$$\left\{\begin{array}{c}\mathbf{x}={[r\left(e\right),\theta \left(e\right),\varphi \left(e\right),\gamma \left(e\right),\psi \left(e\right)]}^{\mathrm{T}}\hfill \\ \mathbf{u}={\left[\sigma \left(e\right)\right]}^{\mathrm{T}}\hfill \end{array}\right.$$

(8)

2.2. Reentry Constraints

Hypersonic vehicles are subjected to both terminal and path constraints throughout the reentry phase to ensure structural integrity and mission success.

Initial state constrain $x\left(e_0\right)$ and terminal state constrain $x\left(e_f\right)$ are expressed as:

$$\left\{\begin{array}{c}\mathbf{x}\left(e_0\right)={[r_0,{\theta }_0,{\varphi }_0,{\gamma }_0,{\psi }_0]}^{\mathrm{T}}\hfill \\ \mathbf{x}\left(e_f\right)={[r_f,{\theta }_f,{\varphi }_f,{\gamma }_f,{\psi }_f]}^{\mathrm{T}}\hfill \end{array}\right.$$

(9)

Heating rate $\dot{Q}$, dynamic pressure q, and normal load n represent common path constraints, formulated as follows:

$$\left\{\begin{array}{cc}\hfill \dot{Q}& =k_Q\sqrt{\rho }{\left(\sqrt{g_0{R}_0}V\right)}^{3.15}\le {\dot{Q}}_{\max }\hfill \\ \hfill q& ={\displaystyle \frac{1}{2}}\rho {\left(\sqrt{g_0{R}_0}V\right)}^2\le q_{\max }\hfill \\ \hfill n& =\sqrt{L^2+D^2}\le n_{\max }\hfill \end{array}\right.$$

(10)

where $k_Q=9.4369\times {10}^{-5}$.

The control variable is chosen as $\sigma$ which should be limited as follows:

$${\sigma }_{\min }\le \sigma \le {\sigma }_{\max }$$

(11)

The angle-of-attack profile is prescribed as a velocity-dependent piecewise linear function which is given as follows [21]:

$$\alpha \left(V\right)=\left\{\begin{array}{cc}{\alpha }_{\max },\hfill & V_1<V\le V_0,\hfill \\ {\displaystyle \frac{{\alpha }_g-{\alpha }_{\max }}{V_2-V_1}}(V-V_1)+{\alpha }_{\max },\hfill & V_2<V\le V_1,\hfill \\ {\alpha }_g,\hfill & V_f<V\le V_2,\hfill \end{array}\right.$$

(12)

where $V_0$ and $V_f$ denote the entry and terminal velocities, respectively; ${\alpha }_{\max }$ is the maximum angle of attack; ${\alpha }_g$ is the angle of attack corresponding to the maximum lift-to-drag ratio; and $V_1$ and $V_2$ are predefined velocity thresholds.

The dynamics Equation (1) can be expressed as an equality constraint:

$${\displaystyle \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}e}}=\mathbf{f}\left(\mathbf{x}\left(e\right),\mathbf{u}\left(e\right),e\right),e\in \left[e_0,e_f\right]$$

(13)

2.3. Trajectory Optimization Problem

The performance index for the optimization problem is selected as the minimum time:

$$J={\int }_{{\tau }_0}^{{\tau }_f}1d\tau ={\int }_{e_0}^{e_f}{\displaystyle \frac{1}{DV}}de$$

(14)

Combining the dimensionless dynamics Equation (1), reentry constraints Equations (9)–(13), and performance index Equation (14), the trajectory optimization problem is formally expressed as a nonlinear optimal control formulation:

$$\begin{array}{c}{P}_0:\min J={\int }_{e_0}^{e_f}{\displaystyle \frac{1}{DV}}de\hfill \\ \begin{array}{cc}\mathrm{s}.\mathrm{t}.& \left\{\begin{array}{c}\mathbf{x}\left(e_0\right)={\mathbf{x}}_0\hfill \\ \mathbf{x}\left(e_f\right)={\mathbf{x}}_f\hfill \\ \dot{Q}=k_Q\sqrt{\rho }{\left(\sqrt{g_0{R}_0}V\right)}^{3.15}\le {\dot{Q}}_{\max }\hfill \\ q=0.5\rho {\left(\sqrt{g_0{R}_0}V\right)}^2\le q_{\max }\hfill \\ n=\sqrt{L^2+D^2}\le n_{\max }\hfill \\ {\sigma }_{\min }\le \sigma \le {\sigma }_{\max }\hfill \\ {\displaystyle \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}e}}=f\left(\mathbf{x}\left(e\right),\mathbf{u}\left(e\right),e\right),e\in \left[e_0,e_f\right]\hfill \end{array}\right.\end{array}\hfill \end{array}$$

(15)

The optimal control problem predominantly features nonconvex constraints, precluding direct solution via sequential SOCP. Consequently, convexification and linearization treatments are applied to relevant constraint functions, thereby transforming the problem into a computationally tractable convex program.

Following the convexification and discretization procedures, the final convex programming problem $P_f$ is obtained. The specific process can be found in Appendix A.

$$\begin{array}{c}{P}_f:\min \begin{array}{c}\end{array}{R}^{\mathrm{T}}z\hfill \\ \begin{array}{cc}s.t.& \left\{\begin{array}{c}MZ=F\hfill \\ \left|x_i-x_i^k\right|⩽\delta \hfill \\ u_1^2+u_2^2⩽1\hfill \\ \cos {\sigma }_{\max }⩽{\left(u_1\right)}_i⩽cos{\sigma }_{\min }\hfill \\ R_i⩾\overline{L}\hfill \\ \left|\theta \left(e_f\right)-{\theta }_f\right|⩽\varsigma ,\left|\varphi \left(e_f\right)-{\varphi }_f\right|⩽\eta \hfill \\ r\left(e_f\right)=r_f,\gamma \left(e_f\right)={\gamma }_f,\psi \left(e_f\right)={\psi }_f\hfill \end{array}\right.\end{array}\hfill \end{array}$$

(16)

3. Online Trajectory Planning Framework Based on MWOA-AM-LSTM Network

3.1. Principles of Whale Optimization Algorithm

WOA constitutes a metaheuristic optimization methodology rooted in swarm intelligence principles, modeled after the distinctive bubble-net predation tactics of humpback whales. The algorithm is designed to balance global exploration and local exploitation in complex optimization through the emulation of three distinct behavioral patterns exhibited by whales: prey encircling, bubble-net foraging, and random search.

3.1.1. Encircling Prey

It has been demonstrated that humpback whales possess the cognitive ability to identify the whereabouts of their prey and subsequently encircle it. Since the optimal position in the search space remains unknown, the WOA algorithm thus presumes the current best candidate solution’s location as the target prey position. Following the determination of the target prey position, the remaining search agents coordinate encirclement maneuvers through positional updates governed by the following mathematical formulation:

$$\left\{\begin{array}{c}{X}_m\left(t+1\right)=X_m^{*}\left(t\right)-A_m\times D_m\hfill \\ A_m=2a\times rd-a\hfill \\ D_m=\left|C_m\times X_c^{*}\left(t\right)-X_c\left(t\right)\right|\hfill \\ C_m=2\times rd\hfill \end{array}\right.$$

(17)

where t is defined as the current number of iterations, $A_m$ and $C_m$ represents the coefficient vectors, $X_m$ is the current position of the solution, $X_m^{*}$ is the location of the current optimal solution. The control coefficient $\mathit{a}$ undergoes linear reduction from 2 to 0. rd is a random number from 0 to 1.

3.1.2. Bubble-Net Attacking Method

The bubble-net attacking strategy of humpback whales is computationally modeled through two synergistic mechanisms: prey encircling via shrinking encircling mechanism and spiral trajectory updating for position refinement. The Shrinking encircling mechanism is controlled through the coefficient $\mathit{a}$. As the coefficient A is bounded within $[-a,a]$, its fluctuation amplitude contracts proportionally with the reduction of a. The spiral update position mechanism employs a spiral function to connect the whale and its prey, replicating the helical bubble-net feeding behavior of a humpback whale. Its formulation is as follows:

$$\left\{\begin{array}{c}{X}_m\left(t+1\right)=X_m^{*}\left(t\right)+D_m^{*}\times e^{bl}\times cos\left(2\pi l\right)\hfill \\ D_m^{*}=\left|X_m^{*}\left(t\right)-X_m\left(t\right)\right|\hfill \end{array}\right.$$

(18)

where $D_m^{*}$ quantifies the distance between the i-th search agent and the incumbent optimal solution. Parameter b determines the curvature topology of the logarithmic spiral trajectory, while l represents a random variable uniformly distributed over $[-1, 1]$.

Assuming that the two mechanisms are executed with equal probability, the position update operator is written in the following piecewise form to account for both the shrink-encircling and spiral-feeding behaviors:

$${\mathbf{X}}_m(t+1)=\left\{\begin{array}{cc}{\mathbf{X}}_m^{*}\left(t\right)-{\mathbf{A}}_m\times {\mathbf{D}}_m,\hfill & \mathrm{if}p<0.5\hfill \\ {\mathbf{X}}_m^{*}\left(t\right)+cos\left(2\pi l\right)\times e^{bl}\times {\mathbf{D}}_m^{*},\hfill & \mathrm{if}p\ge 0.5\hfill \end{array}\right.$$

(19)

where p is a random number in the interval $[0, 1]$.

3.1.3. Search for Prey

Beyond the bubble-net attacking strategy, humpback whales also perform random prey exploration within the algorithm. Again, this is accomplished by changing the value of A. In instances where the absolute value of A exceeds 1, the whale will deviate from the trajectory of the target prey. Distinct from the Bubble-net attacking phase, this mode designates a random selected position vector as the update reference instead of the current global optimum. This exploration-oriented phase demonstrably augments the global search capacity, as formalized:

$$\left\{\begin{array}{c}{X}_m\left(t+1\right)=X_{rand}-A_m\times D_m\hfill \\ D_m=|C_m\times X_{rand}-X_m\left(t\right)|\hfill \end{array}\right.$$

(20)

where $x_{rand}$ is the location of a random individual in the whale population.

3.2. Multi-Strategy Improved WOA

The Whale Optimization Algorithm (WOA) is an efficient swarm-intelligence method that has demonstrated strong potential for complex optimization problems due to its concise mathematical formulation and robust global search capability. However, the canonical WOA still faces inherent limitations when tackling high-dimensional, nonlinear, multimodal, and dynamic optimization tasks. These limitations typically manifest as reduced population diversity, limited convergence accuracy, and a tendency to become trapped in local optima. Such issues are particularly pronounced in LSTM hyperparameter optimization within deep-learning frameworks, where WOA can be highly sensitive to parameter settings and may suffer from low search efficiency. To address these challenges, we propose a multi-strategy Improved WOA (IWOA) to enhance optimization performance. The pseudo-code is provided in Algorithm 1.

3.2.1. Circle Chaotic Mapping

The conventional WOA employs random initialization for the purpose of generating populations. However, this stochastic initialization strategy may result in an uneven distribution of search agents across the solution space, thereby compromising the algorithm’s global search space coverage capability.To address this limitation, we propose enhancing the population initialization strategy through the application of a Circle chaotic map. The mathematical formulation of the Circle chaotic mapping is defined as:

$$X^{i+1}=\mathrm{mod}\left(X^i+0.2-\left({\displaystyle \frac{1}{4\pi }}\right)sin\left(2\pi X^i\right),1\right)$$

(21)

where $X^i$, $X^{i+1}$ are denoted as the current individual as well as the subsequently generated individuals.

Algorithm 1 Multi-Strategy Improved Whale Optimization Algorithm (MWOA)

Require:  Objective function f, dimension D, population size N, maximum iterations $T_{\max }$

Require:  Lower bound $\mathbf{lb}$, upper bound $\mathbf{ub}$, damping factor b

Ensure:  Optimal solution $X_{\mathrm{best}}$

1:  Initialization Phase:

2:  Generate initial population using Circle chaotic map:

3:  ${\xi }_i^{\left(k\right)}\leftarrow \left({\xi }_i^{(k-1)}+0.2-{\displaystyle \frac{0.5}{2\pi }}sin\left(2\pi {\xi }_i^{(k-1)}\right)\right) \mathrm{mod} 1$

4:  $X_i\leftarrow \mathbf{lb}+{\xi }_i\odot (\mathbf{ub}-\mathbf{lb})$

5:  ▹ Map to parameter space

6:  $X_{\mathrm{best}}\leftarrow \arg \min f\left(X_i\right)$

7:  for  $t=1$ to  $T_{\max }$  do

8:  Update coefficient a

9:  for each individual $X_i$ do

10:   $r_1,r_2\sim U(0,1),l\sim U(-1,1),p\sim U(0,1)$

11:   if $p<0.5$ then

12:    $A\leftarrow 2a{r}_1-a$

13:    if $\left|A\right|<1$ then

14:     Update position using shrinking encircling by Equation (17)

15:    else

16:     Randomly select individual $X_{\mathrm{rand}}$

17:     Update position using searching for prey by Equation (20)

18:    end if

19:   else

20:    Update position using spiral update by Equation (18)

21:   end if

22:   if $\mathrm{rand}\left(\right)<0.3+0.5·(t/T_{\max })$ then

23:    Perform golden sine mutation by Equations (24)–(26)

24:   end if

25:   if $f\left(X_{\mathrm{new}}\right)<f\left(X_i\right)$ then

26:    $X_i\leftarrow X_{\mathrm{new}}$

27:    if $f\left(X_{\mathrm{new}}\right)<f\left(X_{\mathrm{best}}\right)$ then

28:     $X_{\mathrm{best}}\leftarrow X_{\mathrm{new}}$

29:    end if

30:   end if

31:  end for

32:  end for

3.2.2. Nonlinear Decay Factor

In the conventional WOA, the linear decay pattern of the convergence factor fails to effectively reconcile the algorithm’s distinct phase-dependent requirements: intensive global exploration during initial iterations and refined local exploitation in later stages. To address the limitations inherent in the linear decay mechanism of the convergence factor, this study proposes a dynamically adjusted non-linear decay factor based on an exponential function. The function is as follows:

$$a=2-{\displaystyle \frac{2}{1+\exp \left(-10b\left(\frac{2t}{T_{\max }}-1\right)\right)}}$$

(22)

where t represent the iteration index and $T_{max}$ defining the terminal iteration count. The image of this function is shown in Figure 1.

3.2.3. Dynamic Gold Sine Mutation Strategy

The WOA algorithm adopts a fixed spiral update mechanism during the local development stage. This makes it susceptible to falling into local optima and lacks an effective escape mechanism. Additionally, the general mutation strategy uses a fixed probability, making it challenging for the algorithm to balance global exploration and local development during later iterations. To overcome this defect, this paper introduces the dynamic golden sine mutation strategy. This strategy is based on a dynamic probabilistic perturbation triggering mechanism within an iterative process. It fuses the characteristics of the golden ratio and the sine function, thereby enhancing the algorithm’s searching ability at different stages. It also adaptively adjusts the searching step size and direction, balancing the algorithm’s global exploration and local exploitation abilities.

The steps are as follows. After the spiral update mechanism, a dynamic probabilistic perturbation mechanism is added to determine whether the golden sinusoidal mutation is performed. The expression for the mutation probability decaying over time is as follows:

$$p\left(t\right)=0.3+0.5t/T_{\max }$$

(23)

The golden sinusoidal variation mechanism first generates two sets of coefficients based on the golden ratio. These coefficients are then used to regulate the contraction and expansion of the sinusoidal function parameters, respectively. This process is expressed mathematically as follows:

$$\left\{\begin{array}{c}{\zeta }_1=a·\tau +(1-\tau )·b\\ {\zeta }_2=(1-\tau )·a+\tau ·b\end{array}\right.$$

(24)

where golden section ratio $\tau =(\sqrt{5}-1)/2$, a and b are the lower and upper bounds of the search space, respectively. Subsequently, the current solution is perturbed using a sinusoidal function, which is combined with the global optimal solution to adjust the current individual position:

$$\begin{array}{cc}\hfill X_{\mathrm{mut}}(t+1)=& -r_2·sin\left(r_1\right)·({\zeta }_1·X^{*}-{\zeta }_2·X\left(t\right))\hfill \\ \hfill & +X^{*}·|sin\left(r_1\right)|\hfill \end{array}$$

(25)

where, $r_1\in [0, 2\pi ]$ and $r_2\in [0, \pi ]$.

After the spiral update with the golden sine mutation mechanism, a reflection boundary treatment needs to be introduced to ensure that the newly generated solution $X_{mut}\left(t+1\right)$ satisfies the predefined feasible domain constraints. The expression is as follows:

$$X_{\mathrm{mut}}^i=\left\{\begin{array}{cc}2·{\mathrm{lb}}^i-X_{\mathrm{mut}}^i,\hfill & \mathrm{if}{X}_{\mathrm{mut}}^i<{\mathrm{lb}}^i\hfill \\ 2·{\mathrm{ub}}^i-X_{\mathrm{mut}}^i,\hfill & \mathrm{if}{X}_{\mathrm{mut}}^i>{\mathrm{ub}}^i\hfill \end{array}\right.$$

(26)

3.3. Principle of LSTM

LSTM represents a specialized deep learning architecture for processing sequential data, distinguished from conventional RNNs by three gating mechanisms: the input gate, forget gate, and output gate. These gates, coupled with a cell state that supersedes traditional hidden units, enable LSTM to model long-range temporal dependencies effectively.

As illustrated in Figure 2, the core components operate as follows: The cell state $c_t$ functions as the central memory conduit, preserving information across sequential timesteps. At each timestep, a candidate cell state ${\tilde{c}}_t$ is generated via tanh activation, representing potential updates to the memory. Concurrently, three gating units regulate information flow: The forget gate $f_t$ modulates retention of historical cell state $c_{t-1}$, the input gate $i_t$ controls assimilation of candidate state ${\tilde{c}}_t$, and the output gate $o_t$ gates the current cell state to yield hidden state $h_t$, which transmits temporal dependencies as the external output. The calculation process can be expressed as follows:

$$\left\{\begin{array}{c}{h}_t=o_t\odot \tanh \left(c_t\right)\hfill \\ c_t=f_t\odot c_{t-1}+i_t\odot {\tilde{c}}_t\hfill \\ {\tilde{c}}_t=\tanh (B_c+{\omega }_c[v_{t-1},x_t])\hfill \\ f_t=\sigma (B_f+{\omega }_f[v_{t-1},x_t])\hfill \\ i_t=\sigma (B_i+{\omega }_i[v_{t-1},x_t])\hfill \\ o_t=\sigma (B_o+{\omega }_o[v_{t-1},x_t])\hfill \end{array}\right.$$

(27)

where $\omega$ denotes trainable weight matrices, B represents corresponding bias terms, and $\sigma$ signifies the sigmoid activation function. These parameters collectively govern the gating mechanisms and state transformations.

3.4. Principle of Multi-Head Attention Mechanisms

Multi-head attention employs parallelized attention mechanisms to project input data into orthogonal feature subspaces, enabling selective feature weighting for critical information enhancement. The calculation process is shown as follows:

(1)

Input projection. The LSTM output sequence X through three linear layers:

$$Q=X{W}^Q,K=X{W}^K,V=X{W}^V$$

(28)

(2)

Subspace projection. Each matrix is partitioned into h heads and independently projected:

$$Q_i=Q{W}_i^Q,K_i=K{W}_i^K,V_i=V{W}_i^V$$

(29)

(3)

Parallel attention computation. Each head calculates the attention weights and weights the value vectors, using softmax to normalize and generate an attention weight matrix:

$${\mathrm{head}}_i=\mathrm{softmax}\left({\displaystyle \frac{Q_i{K}_i^T}{\sqrt{d_k}}}\right)V_i$$

(30)

(4)

Output fusion. Heads are concatenated and linearly transformed:

$$\begin{array}{cc}\hfill \mathrm{MutiHead}& =\mathrm{Concat}({\mathrm{head}}_1,{\mathrm{head}}_2,\dots ,{\mathrm{head}}_h)W^O\hfill \end{array}$$

(31)

Through the above process, the input sequence is mapped into h independent subspaces, where each head learns distinct feature patterns. The outputs of all heads are then concatenated and integrated via linear transformation, forming a global representation that captures complex dependencies between time steps, significantly enhancing the modeling capability for complex sequential relationships.

3.5. Reentry Trajectory Planning Method

While convex optimization provides theoretical solutions to trajectory optimization problems, its practical implementation is limited by computational burden. Convex optimization is employed to generate offline datasets supplying training samples for neural networks. Subsequently, the approximation capabilities of neural networks for complex nonlinear functions are leveraged to construct models mapping vehicle state variables to control commands. This study consequently proposes a hybrid framework enabling real-time trajectory planning and online command acquisition for hypersonic vehicles. The overall design scheme is shown in Figure 3.

3.5.1. Offline Dataset Generation

Deviations from expected aerodynamic parameter values occur during the entry of hypersonic vehicles into the atmosphere, which may lead to discrepancies between the actual flight path and the predefined ideal path. The deviation in aerodynamics can be modeled using a Gaussian distribution, and according to the $3\sigma$ rule of the Gaussian distribution, the deviation range of aerodynamic parameters (including drag and lift) is set to $\pm 20\%$. The drag deviation coefficient and lift deviation coefficient can be represented as ${\xi }_D$ and ${\xi }_L$, respectively. Based on this deviation model, 500 different sets of drag and lift coefficients are generated for further analysis. The drag coefficient and lift coefficient can be expressed respectively as $({\xi }_D+1)C_D$, $({\xi }_L+1)C_L$. Subsequently, the cvx solver was used to solve this problem and 500 trajectories with aerodynamic uncertainties were obtained. The convergence condition can be satisfied after seven iterations, and the entire SOCP solution process requires 18.779 s.

3.5.2. Offline Network Training

The proposed hybrid architecture integrates multi-head attention mechanisms with LSTM networks to enhance sequential data modeling. While LSTM’s gated structure and cell state design provide inherent advantages in gradient stability and long-term dependency capture, its performance remains highly sensitive to hyperparameter configurations. Critical parameters include: Hidden layer size (determining model capacity and feature extraction), Learning rate (controlling gradient descent convergence), and number of attention heads (controlling multi-scale feature weighting). The MWOA algorithm has been developed for the purpose of optimizing the hyperparameter. The resulting hybrid model MWOA-AM-LSTM, has the capacity to be utilized for the purpose of predicting time series data.

The overall training procedure of the proposed MWOA–AM-LSTM model is summarized in Algorithm 2.

Algorithm 2 MWOA–AM-LSTM model training process

Require:  Trajectory dataset $\mathcal{D}$ with aerodynamic uncertainties; MWOA configuration; AM-LSTM model structure.

Ensure:  Trained MWOA–AM-LSTM hybrid model; evaluation metrics (MSE, MAE).

1:  Step 1: Data preprocess.

2:  Split $\mathcal{D}$ into training set ${\mathcal{D}}_{\mathrm{train}}$ and test set ${\mathcal{D}}_{\mathrm{test}}$.

3:  Apply min–max normalization to all features for dimensional consistency.

4:  Step 2: Hyperparameter optimization.

5:  Use MWOA to optimize the AM-LSTM hyperparameters: learning rate, number of hidden neurons, and number of attention heads.

6:  Define the objective function as the model loss on a validation subset of ${\mathcal{D}}_{\mathrm{train}}$.

7:  Obtain the optimal hyperparameter set and configure the AM-LSTM network.

8:  Step 3: Model integration.

9:  Combine the optimized AM-LSTM and the MWOA-based tuning strategy to form the MWOA–AM-LSTM hybrid model.

10:  Train the optimized AM-LSTM on ${\mathcal{D}}_{\mathrm{train}}$ to obtain the final deployed model.

11:  Step 4: Control command prediction.

12:  Given historical flight states $(h,\theta ,\varphi ,\gamma ,\psi )$, predict the future bank-angle command $\sigma$.

13:  Step 5: Performance validation.

14:  Compare predicted and reference/measured quantities on ${\mathcal{D}}_{\mathrm{test}}$.

15:  Quantify model performance using MSE and MAE.

3.5.3. Online Trajectory Planning

The Runge–Kutta method is employed to simulate online trajectory planning. Commencing from the initial state, the trained neural network predicts control commands for the next time step. These commands are then numerically integrated to obtain subsequent flight states. Iteratively repeating this process enables rapid generation of the complete flight trajectory.

4. Numerical Simulation and Analysis

To comprehensively evaluate the performance of the proposed MWOA and its MWOA-driven AM-LSTM prediction framework, while validating the efficacy of the online trajectory planning methodology, the following simulation algorithms are performed: (1) Benchmark function performance comparison between MWOA and mainstream swarm intelligence algorithms; (2) Neural network prediction performance assessment; (3) Online trajectory planning verification under aerodynamic uncertainty. All numerical simulations are performed on a PC with Intel(R) Core(TM) i5-13490F CPU @ 2.50 GHz.

4.1. Testing and Analysis of MWOA Algorithm

This section evaluates the performance of the MWOA algorithm by comparing it with prominent algorithms that have gained popularity in recent years, which are listed below: GOOSE Algorithm (GO), Whale Optimization Algorithm (WOA), Harris Hawks Optimization Algorithm (HHO), Black-winged Kite Algorithm (BKA), Particle Swarm Optimization (PSO). The selected test functions include Sphere, Rastrigin, Ackley, Griewank, Zakharov, Levy, Rothyp, and Easom. Detailed mathematical definitions of each function can be found on this website: Optimization Test Functions (http://www.sfu.ca/~ssurjano/optimization.html (accessed on 12 March 2026)). The diagrams of these eight benchmark functions are presented in the Figure 4. These functions enable a comprehensive evaluation of the search and optimization capabilities of algorithms under diverse problem landscapes.

The comparative performance of MWOA against several swarm intelligence algorithms is summarized in

Table 1. Performance comparison of optimization algorithms on benchmark functions.

Function	Criteria	Algorithm
		MWOA	GO	WOA	HHO	BKA	PSO
Sphere	Mean	0.00	0.00	$7.41\times {10}^{-320}$	$1.90\times {10}^{-266}$	$1.37\times {10}^2$	$1.10\times {10}^0$
	Std	0.00	$3.35\times {10}^{-154}$	0.00	0.00	$1.61\times {10}^1$	$6.09\times {10}^{-1}$
Rastrigin	Mean	0.00	0.00	$8.50\times {10}^0$	0.00	$3.37\times {10}^2$	$9.50\times {10}^1$
	Std	0.00	0.00	$2.68\times {10}^1$	0.00	$1.82\times {10}^1$	$2.45\times {10}^1$
Ackley	Mean	$2.48\times {10}^{-16}$	$2.22\times {10}^{-15}$	$2.46\times {10}^{-15}$	$4.44\times {10}^{-16}$	$2.03\times {10}^1$	$7.73\times {10}^0$
	Std	0.00	$1.81\times {10}^{-15}$	$2.02\times {10}^{-15}$	0.00	$1.95\times {10}^{-1}$	$1.14\times {10}^0$
Griewank	Mean	0.00	0.00	$2.51\times {10}^{-4}$	0.00	$4.72\times {10}^2$	$4.78\times {10}^0$
	Std	0.00	0.00	$1.37\times {10}^{-3}$	0.00	$5.54\times {10}^1$	$2.09\times {10}^0$
Zakharov	Mean	0.00	0.00	$6.35\times {10}^2$	$7.03\times {10}^{-135}$	$9.39\times {10}^2$	$9.30\times {10}^1$
	Std	0.00	0.00	$1.60\times {10}^2$	$3.85\times {10}^{-134}$	$1.79\times {10}^2$	$6.72\times {10}^1$
Levy	Mean	$3.22\times {10}^{-3}$	$8.18\times {10}^{-1}$	$6.16\times {10}^0$	$1.03\times {10}^0$	$1.06\times {10}^2$	$7.01\times {10}^0$
	Std	$1.67\times {10}^{-2}$	$7.22\times {10}^{-2}$	$1.55\times {10}^1$	$2.69\times {10}^{-1}$	$1.34\times {10}^1$	$3.33\times {10}^0$
Rothyp	Mean	$2.42\times {10}^{-2}$	$4.25\times {10}^{-1}$	$1.02\times {10}^{-1}$	$2.55\times {10}^0$	$1.72\times {10}^4$	$5.98\times {10}^0$
	Std	$1.11\times {10}^{-2}$	$1.19\times {10}^{-1}$	$6.00\times {10}^{-2}$	$2.00\times {10}^0$	$4.64\times {10}^3$	$4.33\times {10}^0$
Easom	Mean	$-1.00\times {10}^0$	$-9.99\times {10}^{-1}$	$-1.00\times {10}^0$	$-1.00\times {10}^0$	$-1.00\times {10}^0$	$-1.00\times {10}^0$
	Std	$3.57\times {10}^{-17}$	$1.45\times {10}^{-3}$	$5.45\times {10}^{-17}$	$4.50\times {10}^{-5}$	0.00	0.00

, with all algorithms configured with a population size of 50 and a maximum of 1000 iterations, evaluated over 30 independent runs. The superior competitiveness and stability of MWOA are demonstrated by the results across all benchmark functions. The algorithm achieves perfect convergence on the Sphere, Zakharov, and Griewank functions, characterized by zero residual error. It also exhibits exceptional stability, registering a standard deviation of 0.00 across 30 runs on the highly multimodal Rastrigin and Griewank functions. On the Ackley function, MWOA attains a mean value of $2.48\times {10}^{-16}$ with zero standard deviation, outperforming the GO algorithm. The convergence behavior illustrated in Figure 5 further confirms these performance advantages. On unimodal functions such as Sphere, MWOA descends rapidly, approaching the theoretical optimum within 300 iterations. When applied to multimodal landscapes including Rastrigin and Ackley, the algorithm quickly escapes early oscillations and converges to extremely low error levels, as exemplified by its performance on Rastrigin where it reaches approximately ${10}^{-10}$ by 500 iterations. This behavior reflects a superior capacity to avoid local optima. The trend of high-precision convergence persists on complex functions such as Rothyp and Easom, where MWOA achieves significantly lower final fitness values and converges to the global optimum more rapidly and stably than all competing algorithms.

4.2. Neural Network Prediction Performance Experiment

Thissection verifies the effectiveness of the MWOA-AM-LSTM hybrid model by achieving adaptive optimization of AM-LSTM hyperparameters, aiming to enhance the prediction accuracy of bank angle $\sigma$.

The simulation object selected is the generic hypersonic aircraft CAV-H, with the initial conditions configured as specified in

Table 2. The initial conditions of the simulation.

State	Altitude	Longitude	Latitude	Velocity	Flight-Path Angle	Heading Angle
$x_0$	120.9 km	237°	$-25°$	7500 m/s	$-1.2°$	45°
$x_f$	30 km	279°	28.61°	900 m/s	$-6°$	90°

. To balance model capacity and computational burden, the hidden-layer neuron number is treated as an optimization variable and searched within [10, 200]. This interval is selected to span from under-parameterized to sufficiently expressive networks. Subsequently, the trajectory dataset is generated following the methodology detailed in Section 3.5.1. The dataset with the aerodynamic deviation consists of 500 trajectories, divided according to a fixed scale, with the training set containing 400 trajectories and the test set retaining 100 trajectories. Each trajectory consists of 201 discrete sampling points, each representing a six-dimensional feature vector containing the following flight states: altitude, longitude, latitude, flight path angle, heading angle and bank angle. The robustness of the neural network to biases caused by aerodynamic uncertainties is enhanced by offline network training.

Figure 5. Convergence curves of the algorithms.

Figure 5. Convergence curves of the algorithms.

Subsequently, a comparative analysis was conducted using hybrid models including MWOA-AM-LSTM, WOA-AM-LSTM, HHO-AM-LSTM, AM-LSTM, and the baseline LSTM model. The parameter boundaries for each model are detailed in

Table 3. Hyperparameter configurations for hybrid model and baseline LSTM.

Parameter	Value/Boundary	Definition
Population Size	20	Number of feasible solutions explored during optimization. Determines the search space coverage capability of the metaheuristic algorithm.
Iterations	100	Process of completing one full pass through the training dataset.
Batch Size	32	Number of samples selected from the training set to update model weights in a single parameter update step.
Hidden Layer Neurons	[10, 200]	Number of computational units in the hidden layer. For hybrid model: optimized parameter determining model capacity and feature extraction capability. For baseline model: fixed architecture with two layers (100 and 50 neurons).
Initial Learning Rate	[0.001, 0.02]	Base step size for parameter updates. For hybrid model: optimized parameter balancing convergence speed and stability. For baseline: fixed at 0.001.
Multi-head Attention	[4, 16]	Parallel attention heads capturing feature relationships in different subspaces. For hybrid model: optimization parameter. For baseline: fixed at 8 heads.

, while

Table 4. Model Performance Metrics.

Hybrid Model	Data Type	RMSE	MAE	${\mathit{R}}^2$
MWOA-AM-LSTM	Training dataset	0.0535	0.0241	0.9964
	Test dataset	0.0641	0.0261	0.9956
HHO-AM-LSTM	Training dataset	0.0672	0.0282	0.9942
	Test dataset	0.0784	0.0316	0.9933
WOA-AM-LSTM	Training dataset	0.0898	0.0341	0.9921
	Test dataset	0.0920	0.0357	0.9911
AM-LSTM	Training dataset	0.1123	0.0405	0.9852
	Test dataset	0.1174	0.0432	0.9846
LSTM	Training dataset	0.1211	0.0421	0.9838
	Test dataset	0.1224	0.0479	0.9833

presents performance comparisons across these architectures. The comparative evaluation reveals distinct performance characteristics across hybrid architectures. MWOA-AM-LSTM exhibits exceptional training accuracy, achieving the lowest RMSE of 0.0535 and highest R² of 0.9964 while maintaining competitive test performance with an RMSE of 0.0641 and R² of 0.9956. Its MAE values of 0.0241 for training and 0.0261 for testing represent the most balanced error distribution among hybrid models. The AM-LSTM baseline substantially outperforms conventional LSTM, reducing training RMSE by 7.3% and testing RMSE by 4.1%, thereby validating the efficacy of the attention mechanism. These superior metrics are clearly reflected in the bank angle predictions visualized in Figure 6, which demonstrates that the MWOA-AM-LSTM model achieves superior trajectory fitting.

4.3. Online Trajectory Planning Performance Verification

In this section, the the classical fourth-order Runge–Kutta method is employed to simulate the online trajectory optimization process. The trained neural network generates bank angle commands in real-time within 7 milliseconds. Subsequently, these commands are numerically integrated using the Runge–Kutta method to rapidly construct the complete flight trajectory. Figure 7 depicts the computational efficiency distribution from 1000 Monte Carlo simulations, validating the real-time processing efficacy of the MWOA-AM-LSTM framework.

Under nominal conditions, the comparative results of online trajectories generated by different neural networks and the reference trajectory are shown in Figure 8 and Figure 9. The MWOA-AM-LSTM, HHO-AM-LSTM, WOA-AM-LSTM, AM-LSTM, and baseline LSTM trajectories are denoted by distinct line styles with markers, while the reference trajectory is represented by the black solid line. Figure 8 demonstrates that all network-generated trajectories exhibit near-consistency with the offline trajectory. However, terminal state errors vary significantly across methods. Detailed terminal error metrics are listed in

Table 5. Terminal errors of different models under nominal conditions.

Model	Relative Error (%)
	Terminal Altitude	Terminal Longitude	Terminal Latitude
MWOA-AM-LSTM	0.1969	0.2043	1.47
HHO-AM-LSTM	0.3208	0.2545	2.19
WOA-AM-LSTM	0.4790	0.3441	3.06
AM-LSTM	0.7218	0.4229	3.27
LSTM	0.7874	0.5448	3.68

, confirming that the MWOA-AM-LSTM model achieves the smallest terminal errors.

Under no-nominal conditions with aerodynamic parameter perturbations, five aerodynamic deviation cases are considered to evaluate robustness, where each case corresponds to a specific combination of lift and drag deviation coefficients. The terminal position errors between trajectories generated online by the MWOA-AM-LSTM model and reference trajectories are quantified in

Table 6. The impact of lift and drag deviation coefficients on terminal position errors.

Case	Coefficient Deviation		Relative Error (%)
	Lift (${\mathsf{\xi }}_{\mathit{L}}$)	Drag (${\mathsf{\xi }}_{\mathit{D}}$)	Altitude	Longitude	Latitude
case1	0	0	0.1969	0.2043	1.58
case2	+20%	+20%	4.5243	1.4524	3.24
case3	−15%	+15%	2.2728	0.8521	2.03
case4	+10%	+20%	3.2486	1.1243	2.42
case5	+10%	−10%	0.4325	0.2452	1.78

and Figure 10, Figure 11 and Figure 12. To evaluate model robustness, multiple aerodynamic deviation combinations were tested. In the above scenarios, the relative errors of terminal altitude are lower than 4.52%, the relative errors of terminal longitude are lower than 1.45%, and the relative errors of terminal latitude are lower than 3.24%. This demonstrates that the proposed trajectory planning method can successfully complete online trajectory planning tasks under aerodynamic uncertainty while maintaining a certain level of accuracy and robustness.

5. Conclusions

This paper proposes an online trajectory planning framework for hypersonic vehicles based on a multi-strategy improved whale optimization algorithm and an attention-LSTM network. Considering aerodynamic uncertainties, trajectory samples are provided offline by the sequential second-order cone programming method. The hyperparameters of the AM-LSTM network, including the number of hidden layer neurons, the initial learning rate, and the number of multi-head attention heads, were first specified within predefined boundaries. Then, by minimizing the model loss function, the MWOA optimized these hyperparameters to their global optimum. Thus, the constructed MWOA-AM-LSTM model was able to generate optimal control commands online based on historical flight states and demonstrates outstanding generalization capabilities. Subsequently, it was used as a real-time trajectory generator for the hypersonic vehicle. Numerical simulations demonstrate remarkable performance of the proposed framework in computational efficiency and planning precision under both nominal and perturbated conditions.

In the future, we will investigate more complex reentry trajectory-planning scenarios and evaluate alternative deep learning backbones under the same training and onboard inference constraints.