Mathematical Problems in Engineering

Volume 2018, Article ID 8135274, 13 pages

https://doi.org/10.1155/2018/8135274

## Improved Chicken Swarm Optimization Method for Reentry Trajectory Optimization

^{1}College of Aerospace Engineering, Chongqing University, Chongqing 400044, China^{2}School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

Correspondence should be addressed to Yu Wu; nc.ude.uqc@uyuwuqc

Received 18 September 2017; Revised 17 December 2017; Accepted 2 January 2018; Published 31 January 2018

Academic Editor: Antonino Laudani

Copyright © 2018 Yu Wu 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

Reentry trajectory optimization has been researched as a popular topic because of its wide applications in both military and civilian use. It is a challenging problem owing to its strong nonlinearity in motion equations and constraints. Besides, it is a high-dimensional optimization problem. In this paper, an improved chicken swarm optimization (ICSO) method is proposed considering that the chicken swarm optimization (CSO) method is easy to fall into local optimum when solving high-dimensional optimization problem. Firstly, the model used in this study is described, including its characteristic, the nonlinear constraints, and cost function. Then, by introducing the crossover operator, the principles and the advantages of the ICSO algorithm are explained. Finally, the ICSO method solving the reentry trajectory optimization problem is proposed. The control variables are discretized at a set of Chebyshev collocation points, and the angle of attack is set to fit with the flight velocity to make the optimization efficient. Based on those operations, the process of ICSO method is depicted. Experiments are conducted using five algorithms under different indexes, and the superiority of the proposed ICSO algorithm is demonstrated. Another group of experiments are carried out to investigate the influence of hen percentage on the algorithm’s performance.

#### 1. Introduction

In recent years, the study on hypersonic vehicle has a rapid development with human’s exploration in space [1]. The hypersonic vehicle has a wide range of applications in both military and civilian use, such as global strike and space transportation. To realize an effective and reliable flight, generation of an optimal trajectory is one of the key technologies to enhance the global reach capability of hypersonic vehicle. Reentry trajectory optimization has always been considered as a difficult problem due to its strong nonlinearity in motion equations and constraints [2]. Reentry trajectory optimization is an optimal control problem, and there are two categories of methods to solve such problems, that is, indirect methods and direct methods [3]. In indirect methods, the solution model is transformed into a Hamiltonian boundary-value problem through variation method or Pontryagin maximum principle, and then the problem can be solved with the Newton method, conjugate gradient method, and so on [4]. A high accuracy of solution is the main advantage of the indirect method, while a relatively accurate guess of initial solution limits its application. The main idea of the direct methods is to discretize the dynamic process of system and convert the optimal control problem to a nonlinear programming problem [5]. Compared to the indirect methods, the constraints and the objective function are not necessarily continuous and differentiable in the direct methods. However, the direct methods have a slower convergence rate and cannot guarantee finding the global optimal solution. Among the direct methods, the metaheuristic algorithms are widely applied in solving the combinatorial optimization problem. Those algorithms are a combination of stochastic algorithm and local search algorithm, and the deviation of the optimal solution from the feasible solution usually cannot be estimated.

Swarm intelligence algorithms are a class of metaheuristic algorithms which have been paid more and more attention over the past decade. These algorithms mimic the mechanism of information exchange, information sharing, and learning among swarm animals in natural environment [6]. What these algorithms have in common is that they get the solution with good quality after iteration [7]. Swarm intelligence algorithms have attracted great research interest because they can make a balance between the convergence rate and the solution quality [8]. For example, the artificial bee colony (ABC) algorithm was originally developed by Karaboga and Basturk in 2007 and was inspired by the collective behavior of honey bees [9]. The ABC algorithm has shown a better performance in the function optimization problem compared to the genetic algorithm (GA), the differential evolution (DE) algorithm, and the particle swarm optimization (PSO) algorithm [10]. Inspired by the behavior of homing pigeons, pigeon-inspired optimization (PIO) algorithm was first proposed by Duan and Qiao in 2014 [11]. Based on the elements for pigeons to find their homes, the algorithm consists of two operators, that is, map and compass operator and landmark operator. The feasibility and rapidness of the algorithm have been proven in controlling parameters optimization problem [12].

New algorithms are still being studied. The chicken swarm optimization (CSO) algorithm was originally proposed by Meng et al. in 2014 [13]. It is a stochastic optimization algorithm and mimics the hierarchical order and the behaviors of the chicken swarm. The whole chicken swarm is divided into several subgroups, and there are one rooster, some hens, and several chicks in each subgroup. The three kinds of chicken search for food, respectively, according to their different movement principles. Besides, hierarchy exists in each subgroup, and there is competition between different subgroups. The above characteristics of chicken swarm make the algorithm search for the global optimum. In [13], CSO algorithm was used to solve the benchmark function optimization problem, and the experimental results demonstrate that CSO algorithm performs better in terms of convergence rate and optimization accuracy compared to PSO, DE algorithm, and bat algorithm (BA). However, the benchmark functions were tested only in a low-dimensional case. To verify the validity of CSO algorithm in high-dimensional case, standard test functions are used in the experiment of [14]. The results indicate that the CSO algorithm is easy to fall into local optimum, and premature convergence occurs [14].

As far as we know, the CSO algorithm has not been applied in solving the reentry trajectory optimization problem in the reported studies. Other swarm intelligence algorithms, such as PSO algorithm [15], ant colony optimization (ACO) algorithm [16], and artificial bee colony (ABC) algorithm [17] have been used to obtain the reentry trajectories for hypersonic vehicle. In these studies, the problem is formulated into a high-dimensional optimization problem with strong nonlinearity and multiple constraints, and the control variables of vehicle in different moments are optimized by various swarm intelligence algorithms, respectively. Compared to the low-dimensional cases, the solution space is expanded rapidly in the reentry trajectory optimization problem. As the scale of the chicken swarm is fixed in the original CSO, it is more likely to fail to obtain the global optimum. Therefore, the CSO algorithm is required to be modified to improve its performance in solving the high-dimensional optimization problem. Also, the improved version of CSO is expected to show its superiority in solution quality and convergence rate over other swarm intelligence algorithms.

To this end, an improved chicken swarm optimization (ICSO) method solving the reentry trajectory optimization problem is proposed for the first time in this paper. To be specific, the movement principle of hen is modified based on that in original CSO algorithm. The reasons are as follows. Firstly, the number of hens accounts for the majority of the chicken swarm, so the movement of hen has a great influence on the algorithm’s performance. Secondly, there are more parameters in the equation of updating the hen’s movement, which also affects the algorithm’s performance. A crossover operator is introduced in the ICSO algorithm after the hens finish updating their positions. With the crossover operator, the hens with the worst fitness values are substituted for the new offspring, and the new offspring have certain probability to be better solutions compared to those in the original CSO algorithm. Thus, the local optimal solution and the premature convergence are avoided. Besides, the proposed ICSO algorithm can speed up the convergence rate.

The remainder of the paper is organized as follows. Section 2 presents the dynamic equations and the aerodynamic model of the hypersonic vehicle studied in this paper. The constraints and the cost function are also formulated in this section. Section 3 introduces the basic principle of CSO algorithm, and how the CSO algorithm is improved is also highlighted. In Section 4, the details of ICSO method solving the reentry trajectory optimization problem are elaborated. Comparison experiments among several swarm intelligence algorithms are conducted in Section 5, and the influence of the hen percentage on the algorithm’s performance is further explained. The concluding remarks are contained in the last section.

#### 2. Description of Reentry Trajectory Optimization Problem

In this section, the reentry trajectory optimization problem is formulated into the optimal control problem. Then the model used in this study, namely, the common aero vehicle (CAV), is described, including the dynamic model and the aerodynamic model. Besides, the constraints and the cost function are proposed to establish the optimization model of the problem.

##### 2.1. Characteristics of CAV

There are two kinds of CAV, that is, CAV-L and CAV-H. CAV-L means low-performance CAV and has a hypersonic lift to drag ratio (L/D) in the 2.0–2.5 range. CAV-H is a lifting body design applicable to high performance with hypersonic L/Ds in the 3.5–5.0 range [18]. In this study, the model of CAV-H is used. Compared to CAV-L, CAV-H has greater down-range and cross-range gliding ability.

In the reentry trajectory optimization problem discussed in this paper, the CAV-H is assumed to fly over aspherical rotating Earth. The 3-DOF motion equations of CAV-H are as follows [19]:where is the radial distance from the center of the Earth to the vehicle, is the vehicle’s longitude, and is the latitude. In addition, is the flight velocity, is the flight path angle, and is the heading angle. and represent the gravity constant and rotational angular velocity of the Earth. According to (1), the state vector is defined as . The terms and are aerodynamic lift and drag and have forms as follows:where is the dynamic pressure and can be further expressed as ; is the reference area, and is the density of the atmosphere. and are both nonlinear functions of angle of attack and Mach number (Ma). Aerodynamic data can be obtained by consulting [20].

From (1) to (2), when the values of and the bank angle at every moment are known, the state of vehicle can be calculated with numerical method under a given initial state vector . In the reentry trajectory optimization problem, and are the control variables required to be optimized for making the vehicle from a given initial state to a specified terminal state.

##### 2.2. Constraints of Reentry

During the reentry phase, the constraints of trajectory, terminal state, and control variables must be obeyed to ensure a safe and accurate flight. These constraints are described as follows.

*(1) Trajectory Constraints.* This category of constraints includes those on dynamic pressure , overload , and aerodynamic heating rate . They must be satisfied every moment to ensure the flight safety and structural safety of vehicle. The mathematical expressions of these constraints are listed as follows:

The dynamic pressure* q* is in Newton per square meter, and the overload is in . The aerodynamic heating rate , in watt per square meter, is defined at the stagnation point on the surface of vehicle because the stagnation point is the most seriously heated region. In (11),* K* is a constant and it is common to take [21].

*(**2) Terminal Constraints.* To guarantee precise guide in terminal stage of landing, the terminal constraints must be met after reentry phase. The error of radius, longitude, and latitude must satisfy the following constraints:

*(**3) Constraints of Control Variables.* The control variables and must be kept within a certain range to meet the physical characteristics of the actuator. The above constraints are expressed in the following equation:

##### 2.3. Cost Function in Reentry Trajectory Optimization Problem

The goal of reentry tasks is not always the same. It should be determined based on the demand and preference of decision-makers. Usually, to make a better performance of vehicle and an efficient flight, the following three cost functions are proposed:

In (6), the integral of the aerodynamic heating rate of the stagnation point is the optimization goal to reduce the burden and the weight of the thermal protection system. ensures that the reentry phase is finished within the shortest time in (18). To develop the down-range gliding ability of vehicle, maximum down-range in longitude profile is regarded as the cost function in (19). In experimental studies, any of the above three cost functions can be selected to further analyze the results.

#### 3. Principles of the ICSO Algorithm

Chickens are one kind of social animals and have their unique way to search for food. Although one chicken has a low efficiency of searching for food, the chicken swarm can always find food quickly by cooperation [22]. Motivated by such self-organized social behavior, Meng proposed the CSO algorithm in 2014. In this section, the basics of CSO algorithm are introduced, and then a detailed description of how the CSO algorithm is improved will be presented.

##### 3.1. Basics of CSO Algorithm

To make a better understanding of CSO algorithm, the basics of CSO algorithm will be introduced from three aspects, that is, the structure of chicken swarm, the rules of updating positions, and the change of hierarchical order.

*(**1) The Structure of Chicken Swarm.* There are several subgroups in the chicken swarm, and, in each subgroup, one rooster, some hens, and several chicks are included. The chickens with the best several fitness values would be selected as roosters; that is to say, the number of roosters equals the number of subgroups. The chickens with the worst several fitness values would be assumed as chicks, and other chickens are the hens. The number of hens is usually the largest. In a subgroup, hens and chicks are randomly assigned, and a certain number of hens are chosen to establish mother-child relationship with the chicks randomly.

*(**2) The Rules of Updating Positions.* Chickens have different behaviors when they are searching for food according to their roles in the subgroup. Roosters are the strongest in each subgroup, so they search for food for their subgroups selflessly. Besides, they have duty on the safety of chickens and territory in the subgroup. Hens follow the rooster in the same subgroup to search for food, and they can also randomly steal the food from other chickens and prevent others from eating their own food. Chicks are the weakest and are raised by their mother, so they imitate their mother’s behavior to search for food. Since the chickens’ behaviors vary in the subgroup, the rules of updating their positions are also different according to their roles, as seen in the following equations:

In the above equations, denotes the th element of the th chicken in the th iteration, where is the number of chickens in the swarm, is the dimension of the solution space, and Iter is the maximum iteration times. Equations (7) and (8) are the updating rules of rooster’s position. is a random number obeying normal distribution with mean 0 and standard deviation . In (8), is a randomly selected rooster’s index; is the fitness value of the corresponding chicken ; is the smallest constant in the computer and makes the equation meaningful. Such a position update rule enables the roosters to search for food in a wider range of places. Equations (9)–(11) are the updating rules of hen’s position. Rand2 and Rand3 are two random numbers in the interval . 1 is the index of a rooster, which is in the same subgroup as the th hen. 2 is the index of a randomly selected chicken (rooster or hen). This rule shows two behaviors of hens when searching for food, that is, following the rooster in the same subgroup to search for food and stealing the food from other chickens. As for the chicks, they move around their mother to search for food, and this behavior can be expressed as (12), where is the index of chick ’s mother and Rand4 is a parameter that usually takes the value in the interval .

*(**3) The Change of Hierarchical Order.* In the chicken swarm, hierarchical order will not be kept unchanged all the time. To reflect the change of each chicken’s fitness value timely, parameter is introduced. The roosters, hens, chicks, and mothers will be redefined according to their fitness values every iterations. This mechanism can avoid the solution trapping into the local optimum to some extent and make the search move towards the global optimum.

##### 3.2. Description of ICSO Algorithm

Reentry trajectory optimization is a high-dimensional optimization problem, and the results in [14] indicate that the CSO algorithm is easy to fall into local optimum. Therefore, the CSO algorithm needs to be modified to qualify for solving the reentry trajectory optimization problem.

In CSO algorithm, the number of hens is the largest in the chicken swarm, and there are more variable parameters in hen’s position updating rule. The hens’ fitness values have great influence on the algorithm performance. In hen’s position updating rule, the rooster’s position is referred. Compared to other intelligent optimization algorithms, for example, PSO algorithm, this operation can make the hens keep away from the local optimum, and ultimately the algorithm has less possibility to be trapped into local optimum. However, when more roosters are trapped into the local optimum, the hens will also face the same situation, which results in missing the global optimum.

To overcome the weakness of CSO algorithm, a crossover operator is introduced in ICSO algorithm. After the hens update their position, a random number is generated. If is smaller than the default crossover probability, the two hens with the best fitness values are selected to do the crossover operation. The information of new offspring can be expressed as the following equations:

In (13), parameter represents the information percentage that the offspring can inherit the corresponding hen. After the crossover operation, the two new offspring will substitute the two hens with the worst fitness values and continue the iterative process with other chickens. When there are hens trapped into the local optimum, the ICSO algorithm can change the position of hens with the crossover operator. Therefore, the fitness values of the new offspring are better than those of roosters at a certain probability. With the mechanism of hierarchical order change in CSO algorithm, the roosters will be replaced by the hens with better fitness values. This mechanism can not only substitute the roosters which may be trapped into the local optimum but also improve the search speed. Ultimately, the roosters have faster speed towards the global optimum, which accelerate the convergence rate.

#### 4. The ICSO Method Solving the Reentry Trajectory Optimization Problem

When using the ICSO algorithm to solve the reentry trajectory optimization problem, the control variable is discretized at a set of Chebyshev collocation points based on the flight time . Only the control variable at these moments is optimized, and the control variable at other moments can be obtained by interpolation. Thus, the reentry trajectory optimization problem is transformed into a high-dimensional control variable optimization problem and can be solved with the ICSO algorithm.

##### 4.1. Determination of Collocation Points

In the Chebyshev pseudospectral method [23], the collocation points are distributed according to the values of , as shown in the following equation:

The number of collocation points is , and is the value of collocation point . The Chebyshev points are dense at both ends and sparse in the middle. This feature exactly meets the demand of precise control near the destination. It is obvious that , and should be converted to the time interval using the following equation:

Ultimately, the control variable at the moments needs to be optimized.

##### 4.2. Operation of Control Variables

There are two control variables in the reentry trajectory optimization problem, that is, angle of attack and bank angle . To have a lower peak heating rate, is usually set to the value with the maximum ratio. A nominal angle of attack profile corresponding to the maximum ratio of CAV-H [24] is shown in Figure 1.