International Journal of Aerospace Engineering

Volume 2018, Article ID 8793908, 17 pages

https://doi.org/10.1155/2018/8793908

## Reentry Trajectory Optimization for a Hypersonic Vehicle Based on an Improved Adaptive Fireworks Algorithm

^{1}National Key Laboratory of Science and Technology on Multispectral Information Processing, School of Automation, Huazhong University of Science and Technology (HUST), Wuhan 430074, China^{2}Beijing Aerospace Automatic Control Institute, Beijing 100854, China

Correspondence should be addressed to Lei Liu; nc.ude.tsuh@ieluil

Received 7 June 2017; Revised 4 February 2018; Accepted 21 February 2018; Published 26 April 2018

Academic Editor: Angel Velazquez

Copyright © 2018 Xing Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Generation of optimal reentry trajectory for a hypersonic vehicle (HV) satisfying both boundary conditions and path constraints is a challenging task. As a relatively new swarm intelligent algorithm, an adaptive fireworks algorithm (AFWA) has exhibited promising performance on some optimization problems. However, with respect to the optimal reentry trajectory generation under constraints, the AFWA may fall into local optimum, since the individuals including fireworks and sparks are not well informed by the whole swarm. In this paper, we propose an improved AFWA to generate the optimal reentry trajectory under constraints. First, via the Chebyshev polynomial interpolation, the trajectory optimization problem with infinite dimensions is transformed to a nonlinear programming problem (NLP) with finite dimension, and the scope of angle of attack (AOA) is obtained by path constraints to reduce the difficulty of the optimization. To solve the problem, an improved AFWA with a new mutation strategy is developed, where the fireworks can learn from more individuals by the new mutation operator. This strategy significantly enhances the interactions between the fireworks and sparks and thus increases the diversity of population and improves the global search capability. Besides, a constraint-handling technique based on an adaptive penalty function and distance measure is developed to deal with multiple constraints. The numerical simulations of two reentry scenarios for HV demonstrate the validity and effectiveness of the proposed improved AFWA optimization method, when compared with other optimization methods.

#### 1. Introduction

In recent decades, global strike and space transportation have spurred more and more interests in hypersonic vehicles (HVs) [1] for both military and civilian applications. The development of advanced guidance and control technologies for HV is promoted to meet the need for an effective and reliable access to the space. With aerodynamic control, the unpowered HV has the ability of reentering and gliding through the atmosphere. In order to steer an efficient and safety flight in the complex conditions, the reentry trajectory optimization problem of HV has been widely of concern. The aim of reentry trajectory optimization is to find an optimal solution under the reentry flight dynamics and physics, given the aerodynamics, structural strength, and the thermal protection system [2]. Besides, as the reentry dynamics is highly nonlinear, the reentry trajectory optimization is a nonconvex problem with multiple constraints, such as the control ability, heating rate, dynamic pressure, and aerodynamic load. Thus, it is difficult to solve these problems analytically, and numerical techniques are required to determine an approximation to the continuous solution.

There are two main kinds of numerical methods to solve trajectory optimization problems: indirect methods and direct methods [3]. In indirect methods [4], the original problem is transformed into a two-point boundary problem by applying the calculus of variations or the Pontryagin maximum principle. It has high accuracy, and it ensures that the solution satisfies the necessary optimality conditions. However, in indirect methods, it is quite complicated to derive the necessary optimality conditions. Besides, the radius of convergence is small, and it is difficult to guess the initial value of costate variable [5].

In order to overcome the disadvantages of indirect methods, direct methods have been developed, which are classified into two main types: direct shooting method and collocation methods. In the direct shooting method, only the control variables are parameterized and explicit numerical integration is used to satisfy the differential equation constraints. As an improvement, collocation methods [6–9] discretize both the states and control variables, and they use piecewise or global polynomials to approximate the differential equations at collocation points. Pseudospectral methods (PMs) are frequently used collocation methods, which include Gauss PM (GPM) [10, 11], Legendre PM (LPM), Radau PM (RPM) [12], and Chebyshev PM (CPM) [13]. By direct methods, the reentry trajectory optimization problem with infinite dimensions is transformed into the nonlinear programming problem (NLP) with finite dimensions, which is usually a high-dimensional multimodal nonsmooth problem. Some deterministic methods, such as sequential quadratic programming (SQP) [6, 14], second-order cone programming (SOCP) [15], and sequential convex programming method [16], are used to solve it. However, deterministic algorithms are sensitive, and they are easy to be stagnated at a local optimal point.

In the recent years, intelligent algorithms with the global searching ability, which are easy to implement, have been applied to solve the reentry trajectory optimization problems with the development of computational technology [14, 17–24]. In [17], a hybrid genetic algorithm (GA) collocation method was introduced for trajectory optimization, in which the initial guesses for the state and control variables are interpolated by the best solution of GA. Zhao and Zhou employed the particle swarm optimization (PSO) to obtain the end-to-end trajectory for hypersonic reentry vehicles in a quite simple formulation [19]. Zhang et al. applied the ant colony algorithm (ACO) to generate the optimal reentry trajectory for a reusable launch vehicle (RLV) [20]. Additionally, differential evolution (DE) [21, 22] that mimicked the process of nature evolution and simulated annealing (SA) inspired by annealing in metallurgy was also introduced to deal with reentry trajectory optimization problems.

Among these intelligent algorithms, the fireworks algorithm (FWA) is a relatively new swarm intelligence method firstly proposed by Tan and Zhu [25]. Inspired by the fireworks explosion, the algorithm selects some locations as the fireworks in space, each for exploding to generate a shower of sparks. Through the explosion procedure of the FWA, the diversity of population is enhanced. Besides, owing to the powerful local search capabilities and distributed parallel search mechanism, the FWA has a faster convergence speed compared with other intelligent algorithms.

The enhanced fireworks algorithm (EFWA) [26] is an improved version of the FWA in some operators. To improve the EFWA, the adaptive fireworks algorithm (AFWA) [27] was proposed with an adaptive explosion amplitude, which is calculated according to the fitness of individuals adaptively. Afterwards, some improved FWA were proposed and applied in solving various practical optimization problems. For example, Gao and Ming [28] combined the cultural algorithm (CA) with the FWA for the digital filter design. Zheng et al. [29] developed a hybrid FWA with DE operators to improve the diversity and avoid prematurity. Based on a self-adaption principle and bimodal Gaussian function, Xue et al. [30] proposed an advanced FWA to design the PID controllers with high-quality performances. From the previous research, it is recognized that the information interaction among all individuals of the FWA is relatively poor, whereas the interaction is vital in swarm intelligence algorithms. Thus, when solving complex multimodal problem of the optimal reentry trajectory generation, it may be easy to get trapped in a local optimum. As far as we are concerned, the application of the FWA for reentry trajectory optimization is scarce and has not been found in the published literature.

In this paper, we focus on the improvement of the AFWA and its application to the reentry trajectory optimization problems. Firstly, an improved version of the AFWA (I-AFWA) is developed by combining the standard AFWA (S-AFWA) with a new mutation strategy. In each iteration of the algorithm, the new mutation operator and auxiliary mutation individual selection strategy are applied to make Gaussian fireworks learn from more individuals (not only the best individual) and generate diverse sparks from all fireworks and explosion sparks. The interactions of the fireworks and sparks are enhanced to further improve the diversity of the population and avoid being trapped in local optima too early. Then, the I-AFWA is applied to the reentry trajectory optimization problems. The problem is transformed into NLP by using Chebyshev polynomial interpolation to discretize the angle of attack (AOA) and bank angle simultaneously, and the scope of the AOA is figured out by path constraints to simplify the problem. Next, a constraint-handling technique based on the adaptive penalty function and the distance measure is proposed to incorporate with the I-AFWA and used to deal with the multiconstrained parameter optimization problem. Finally, two reentry scenarios are given to illustrate the validity and effectiveness of the proposed I-AFWA method on the reentry trajectory optimization problem.

The remainder of this paper is organized as follows. The general reentry trajectory optimization problem is formulated in Section 2. Section 3 proposes the I-AFWA with a new mutation strategy, and a constraint-handling technique is incorporated with the I-AFWA for the reentry trajectory optimization. In Section 4, two different reentry tasks for HV are presented to evaluate the proposed algorithm. A few conclusions are made in Section 5.

#### 2. Problem Formulation

##### 2.1. Reentry Dynamics

Using a spherical rotating Earth model, the three-degree-of-freedom (3DOF) point mass dynamics equations [31] of a hypersonic reentry vehicle are described as follows: where is the radial distance from the center of the Earth to the vehicle. and are the longitude and latitude, respectively. is the Earth-relative velocity. is the velocity flight path angle and is the velocity heading angle measured from the North in a clockwise direction. is the bank angle, which is the angle between the vehicle longitudinal symmetry plane and the vertical plane [2]. is the Earth’s self-rotation angular velocity, and is the gravitational acceleration where is the gravitational constant. and are the aerodynamic lift force and drag force, respectively, which are given as follows: where is the density of atmosphere, with as the density of atmosphere at sea level, as the scalar height coefficient, and as the altitude from the sea level. , where denotes the Earth’s radius. and are the aerodynamic reference area and mass of the vehicle, respectively. and are the lift and drag coefficients determined by the AOA and Mach number . denotes the angle between the relative velocity and a reference line fixed with respect to the vehicle body [2], which does not occur explicitly in the equations and appear through the aerodynamic forces and . Given the altitude , velocity , and the AOA , the Mach number is calculated, and then the corresponding lift and drag coefficient and values are found by looking up the table functions with the value of and . The independent variable is the time , and control variables are and . The geometric sketch of reentry flight is shown in Figure 1.