Reentry trajectory optimization of hypersonic glide vehicle based on improved particle swarm algorithm

The reentry process of hypersonic glide vehicle presents a host of intricate constraint issues. A method for optimizing the reentry trajectory of hypersonic glide vehicle is presented to achieve the shortest possible reentry time. The approach utilizes the improved particle swarm optimization (IPSO) algorithm. Firstly, population initialization is performed using the quasi-oppositional differential evolution (QODE) strategy to promote diversity within the population. Then an adaptive adjustment algorithm is designed for the inertia weight coefficient and learning factor to achieve a balance between global and local search capabilities. Finally, utilizing the IPSO algorithm for optimizing the flight parameters of the reentry trajectory, the reentry time is reduced from 338 seconds to 335 seconds. The simulations results indicate that the IPSO algorithm outperforms both the standard PSO algorithm and genetic algorithm (GA) in terms of optimization performance index.


Introduction
The hypersonic glide vehicle, owing to its unique advantages such as high-precision targeting and rapid response capabilities, can achieve long-range glide flight through aerodynamic forces, making it an vehicle with significant military application potential [1,2] .In recent years, the optimization of reentry trajectories for hypersonic glide vehicle has received a lot of attentions [3] .
In recent years, intelligent optimization algorithms have found widespread application in the field of vehicle trajectory optimization [4] .Intelligent optimization algorithms, known for their strong global optimization capabilities and robustness, offer advantages in solving complex problems and are well-suited for addressing the non-linear, multi-objective, and multi-constraint complexities often encountered in vehicle trajectory optimization problems.Some of the key intelligent optimization algorithms encompass genetic algorithms, particle swarm optimization (PSO), sparrow search algorithm (SSA) [5] , whale optimization algorithm (WOA) [6] , and more.In each guidance cycle, computational efficiency is significantly enhanced by optimizing the proportional guidance weights using an improved PSO algorithm in [7].In [8], an enhanced WOA is presented, which translated the optimization of reentry trajectories into a parameterized design problem involving angle-of-attack and bank angle profiles.In [9], the global search capabilities of the WOA are improved by employing Tent chaotic mapping and a control factor cosine variation strategy.
Although swarm intelligence algorithms have made significant progress in the field of hypersonic trajectory optimization, they still face challenges related to efficiency, premature convergence, and 2 susceptibility to local optima when dealing with complex constraint problems.

Equations of motion for reentry gliding trajectories
Assuming the Earth is a uniform sphere and neglecting rotation of the Earth, the equations of motion for the vehicle can be represented as: Where r denotes the distance of the vehicle from the Earth's center, v stands for velocity, γ indicates the flight path angle, ψ indicates the heading angle, θ signifies longitude, ϕ corresponds to latitude, m is the mass, g denotes gravitational acceleration, σ stands for the bank angle, t represents time, L refers to lift, and D refers to drag.
The expressions for L and D can be formulated as follows: Where CL and CD are the coefficients of L and D, dependent on the Mach number (Ma) and angle of attack (α); Sref represents the aerodynamic reference area; ρ signifies atmospheric density, which varies with altitude (h).For this study, the U.S. 1976 Standard Atmosphere Model is employed for interpolating atmospheric parameters.

Constraints
The constraints that the vehicle needs to consider during the reentry process include the following: Where • Q, q, and n represent heat flux, dynamic pressure, and load factor, respectively, and • Qmax, qmax, and nmax correspond to their respective maximum values.
The objective of the vehicle is to precisely guide the vehicle to a specified location and meet the corresponding flight mission criteria.Therefore, the terminal constraints are: IOP Publishing doi:10.1088/1742-6596/2764/1/0120693 Where tf denotes the terminal time, θf denotes the longitude of the target point, ϕf represents the latitude of the target point, vf stands for the minimum velocity at which the vehicle reaches the target point, and γmin and γmax signify the smallest and largest values of the flight path angle when the vehicle reaches the target point.

Objective function
The reentry process is segmented into five phases.In the final phase, the proportional navigation method is employed for precise target interception, the optimization parameters for this phase include the longitudinal and lateral guidance coefficients, i.e. k1 and k2.For the other phases, the optimization parameters consist of α, σ, and the time duration of each phase.
The optimization objective examined in this study is the minimization of the time duration of the reentry process.The objective function is given as follows: minimize: where tf represents the total time required for the entire reentry process.
3 Improved particle swarm optimization algorithm PSO is a heuristic optimization method that is inspired by simulating collaboration and information exchange among a flock of birds.The PSO algorithm is particularly well-suited for optimizing tasks in dynamic and multi-objective environments.When compared to conventional optimization algorithms, it boasts faster computational speed and superior global search capabilities.
In the population, the best position is considered in each particle it has encountered during its flight as the best solution it has found, which is referred to as individual best (pi).The best position encountered by the entire swarm of particles is considered the best solution found by the entire group and is referred to as the global best (g).Assuming a search space of D dimensions and a population consisting of N particles, the following can be obtained: ( , , , ), In the standard PSO algorithm, the procedures for updating particle velocity and position are as follows: (0,1) ( ) In the equation, during the t-th iteration, xi,j represents position, vi,j denotes velocity, pi,j is individual best, gj stands for the global best, w stands for the inertia weight coefficient, signifying the extent to which particles retain their current velocity, c1 and c2 are learning coefficients, and rand(0,1) is random numbers uniformly distributed in the range [0,1], introducing randomness to the particle's movement.

Algorithm improvement methods
PSO is a frequently employed for solving optimization problems.To overcome the issues of slow convergence and vulnerability to local optima in the later stages of the PSO algorithm, this paper presents an adaptive adjustment algorithm designed to fine-tune the inertia weight and learning factors used in PSO.Furthermore, initializing the population through a learning strategy.
Tizhoosh first introduced the idea of Opposition-Based Learning (OBL) [10] .Let xi be the solution of an individual, and xi O be its reverse solution, the specific calculation formula is obtained as follows: One variant of OBL known as quasi-oppositional differential evolution (QODE) was introduced by Rahnamayan [11] .The calculation formula for the quasi-reverse solution xi QO is as follows: Where Mi is the midpoint of the upper and lower limits of xi, and Mi = (xl + xu)/2.The inertia weight coefficient w assumes a pivotal role in harmonizing the ability to search.To enhance the algorithm's global search ability, a larger value of w is set, but this may not be conducive to local optimization.On the other hand, for improving local fine-grained solution searches and optimizing the local region, a smaller value of w is used, which increases the likelihood of getting stuck in local optima [12] .Therefore, an adaptive weight coefficient is designed as follows: Where wmax and wmin denote the upper and lower bounds of the w, respectively, t signifies the present iteration count, and Tmax represents the maximum count of iterations.The variation of the inertia weight coefficient w is depicted in Figure 1.Appropriately tuning the values of these two learning factors can improve the searching ability and accelerate the convergence process.When c1 is small, the convergence speed is faster, but it's more prone to local optima.When c2 is small, it can help avoid premature convergence to some extent.
To improve global search capability in the initial stages and strengthen local search capability in the later iterations, a cosine function is used to control c1 and c2 in a nonlinear manner.The expressions are as follows: Where cmin and cmax represent the smallest and largest values of c1 and c2.The change curves of c1 and c2 are illustrated in Figure 2 and Figure 3.

Simulation results and analysis
Taking reference from CAV data [13] , the vehicle's mass is m at 907.2 kg, and the aerodynamic reference area is Sref at 0.4839 m².The maximum heat flux is • Q max = 500 kW/(m²), the maximum dynamic pressure is qmax = 500 kPa, and the maximum load factor is nmax = 5.The reentry point data for the hypersonic vehicle and landing constraints are shown in Table 1.The IPSO is used to optimize the reentry trajectory's time, with a maximum of 100 iterations and a population size of 50 individuals.Referring to [14], in order to achieve a stronger early-stage global convergence capability and better later-stage local search ability, wmax = 0.9, wmin = 0.4, cmax = 2, and cmin = 0.8 are set in this paper.2. Figure 4 shows the three-dimensional representation of the optimized trajectory.Figure 5 to Figure 7 depict the changes in load factor, dynamic pressure, and heat flux throughout the reentry process, all of which remain within the specified constraints.To assess the optimization performance of the IPSO, a comparative examination was conducted with three intelligent algorithms: PSO, IPSO, and GA.
For the reentry trajectory, each algorithm underwent 100 optimizations, with the largest count of iterations and the size of the population set to match the IPSO.The average values of the 100 optimization iterations for each algorithm were computed and depicted in Figure 8. Table 3 provides the mean of the reentry time and standard deviation of the 100 optimizations for different algorithms.The iterative process in Figure 8 indicates that in the early stages of iteration, the IPSO algorithm's convergence speed is slower than that of the PSO algorithm.This is attributed to the initial setting of relatively large values for w and c1, and a relatively small value for c2, which grants IPSO stronger local search capabilities in the early stages.After around 20 iterations, the IPSO algorithm's convergence speed surpasses that of the PSO algorithm.This shift is due to the gradual reduction of w and c1 values in the middle to later stages and the gradual increase of the c2 value.Consequently, the IPSO algorithm exhibits stronger local search capabilities during the middle to later stages compared to the PSO algorithm.5067 For the optimization of the shortest reentry trajectory time, the PSO algorithm optimized a reentry time of 338 seconds, while the GA algorithm resulted in an optimized time of 339.36 seconds.The final optimization results of the PSO and GA algorithms are quite close, but the PSO algorithm exhibits a lower standard deviation, indicating better stability than the GA algorithm.On the other hand, the IPSO algorithm shortened the reentry time by 3 seconds compared to the PSO algorithm, achieving significantly better optimization results than both PSO and GA algorithms.Furthermore, the standard deviation of the results from multiple optimizations is the lowest for the IPSO algorithm, indicating good stability.

Conclusion
This paper transforms the optimization of hypersonic glide trajectories into a parameter optimization problem.The IPSO approach is used for parameter optimization.Simulation results indicate that the IPSO algorithm performs well in optimizing reentry trajectories and can handle complex constraint conditions.Compared to PSO and GA algorithms, the IPSO algorithm provides a higher quality of optimization for the objective function and demonstrates superior stability.

Figure 1 .
Figure 1.Evolution of w with iteration.

Figure 3 .
Figure 3. Evolution of c2with Iterations.The learning coefficients c1 and c2 are responsible for controlling the maximum step size when approaching the individual best and global best, respectively.They determine the influence of the individual experience and the collective information of the group on the particle's movement.Appropriately tuning the values of these two learning factors can improve the searching ability and accelerate the convergence process.When c1 is small, the convergence speed is faster, but it's more prone to local optima.When c2 is small, it can help avoid premature convergence to some extent.To improve global search capability in the initial stages and strengthen local search capability in the later iterations, a cosine function is used to control c1 and c2 in a nonlinear manner.The expressions are as follows:

Figure 4 .
Figure 4. Three-dimensional representation of the optimized trajectory.Figure 5. Load factor change over time curve.

Figure 5 .
Figure 4. Three-dimensional representation of the optimized trajectory.Figure 5. Load factor change over time curve.

Figure 6 .
Figure 6.Dynamic pressure change over time curve.Figure 7. Heat flux change over time curve.

Figure 7 .
Figure 6.Dynamic pressure change over time curve.Figure 7. Heat flux change over time curve.

Figure 8 .
Figure 8. Fitness change curves for different optimization algorithms.

Table 1 .
Data of reentry trajectory starting and ending point.

Table 2 .
Optimization results for parameters.

Table 3 .
Results from different optimization algorithms.