Complexity

Volume 2018, Article ID 5036791, 13 pages

https://doi.org/10.1155/2018/5036791

## Legendre Cooperative PSO Strategies for Trajectory Optimization

^{1}National Key Laboratory of Science and Technology on Multispectral Information Processing, School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China^{2}School of Electric Engineering, University of South China, Hengyang, Hunan, China

Correspondence should be addressed to Yongji Wang; nc.ude.tsuh@hcjygnaw

Received 29 December 2017; Revised 14 February 2018; Accepted 19 February 2018; Published 12 April 2018

Academic Editor: Jesus Medina

Copyright © 2018 Lei Liu 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

Particle swarm optimization (PSO) is a population-based stochastic optimization technique in a smooth search space. However, in a category of trajectory optimization problem with arbitrary final time and multiple control variables, the smoothness of variables cannot be satisfied since the linear interpolation is widely used. In the paper, a novel Legendre cooperative PSO (LCPSO) is proposed by introducing Legendre orthogonal polynomials instead of the linear interpolation. An additional control variable is introduced to transcribe the original optimal problem with arbitrary final time to the fixed one. Then, a practical fast one-dimensional interval search algorithm is designed to optimize the additional control variable. Furthermore, to improve the convergence and prevent explosion of the LCPSO, a theorem on how to determine the boundaries of the coefficient of polynomials is given and proven. Finally, in the numeral simulations, compared with the ordinary PSO and other typical intelligent optimization algorithms GA and DE, the proposed LCPSO has traits of lower dimension, faster speed of convergence, and higher accuracy, while providing smoother control variables.

#### 1. Introduction

Swarm intelligence is a collective dynamic behavior of distributed, self-organized systems, natural or artificial, employed in work on artificial intelligence. It introduces many simple agents with very general rules to achieve an “intelligent” global optimal behavior. Swarm intelligence-based techniques can be used in a number of applications on optimization. The US military is investigating the swarming techniques to control unmanned vehicles. The European Space Agency is thinking about an orbital swarm for self-assembly and interferometry. NASA is investigating the use of swarm technology for planetary mapping.

In particular, trajectory optimization problem is one of the most important tasks in the preliminary design of the next generation of high speed vehicles, such as NASA’s X-43 unmanned hypersonic vehicle (HV), and has a great effect on the choice of conceptual design [1]. It is a daunting work to get the solution of nonlinear optimal control problem with the arbitrary final time and multiconstraints.

Trajectory optimization with multiconstraints has been linked with some stochastic search algorithms. The typical approaches include genetic algorithms (GA) [2], differential evolution (DE) [3], and particle swarm optimization (PSO) [4–6]. Besides, several novel bionic optimization algorithms spring up in these years and show remarkable efficiency in the industrial domain, such as the honeybee mating optimization [7], harmony search algorithm [8], and ants swarm optimization [9]; however, they are rarely used in the aerospace field because of their excessive novelty. The PSO algorithm, as an optimization algorithm based on swarm intelligence, similar to GA and DE for their origination from population-based heuristic search mechanisms, has recently become more popular due to its simplicity and effectiveness. It is widely used because of its simple principle, small number of parameters to be adjusted, and easy realization. Considering that the dynamic parameter optimization needs to be transformed into the static format, it is validated to evaluate the trajectory parameters optimization using a spline function [10].

Under the guidance of thought of the hybrid algorithm [11], a novel method of trajectory optimization is proposed to improve the global convergence ability in this paper, the Legendre cooperative PSO (LCPSO) method, which is a kind of cooperative PSO based on Legendre orthogonal polynomials.

In LCPSO, the following improvements make it promising in solving complex problems. Firstly, the search space is divided into certain subspaces and different swarms are arranged to optimize the different parts of the search space. In this way, the optimized scale for each swarm can be reduced directly. The algorithm is suitable for the problems with larger scale and higher dimension.

Secondly, for the optimal control problem by PSO, the discretization is necessary before solving the parameter optimization problem. Herein, the Legendre orthogonal polynomial approximation can achieve higher smoothness of control variables with lower dimensions. Additionally, a theorem on how to find the precise range of optimal parameters is given and proven, as well as how to find the boundaries of the coefficient of polynomials. Therefore, the proposed LCPSO is expected to realize the optimization with higher accuracy and efficiency.

Thirdly, in order to solve the optimal control problems with arbitrary final time rather than the fixed one, the proposed Legendre orthogonal polynomial approximation method introduces an additional control variable to transcribe the original optimal problem to the one with fixed final time. Then, a traditional one-dimensional search method based on the interval analysis is proposed to optimize the additional control variable. This way, the specific optimal problem with single boundary can be solved.

Finally, we use two typical trajectory optimization problems to illustrate efficiency of the proposed LCPSO algorithm. One is the ascent trajectory optimization of X-43 hypersonic vehicle, and the other one is the classic optimal orbit transfer problem. The simulation results demonstrate the advantages of the LCPSO in terms of solution accuracy and convergence rate by comparing with some traditional intelligent optimization algorithms.

The organization of the remainder of this paper is as follows. Section 2 formulates the optimal control problem by 3-DOF mass point dynamics. Section 3 describes the algorithm of novel Legendre cooperative PSO and some properties. Section 4 presents the numerical simulations on the performance of the proposed optimal algorithms. Section 5 draws conclusions to the paper.

#### 2. Problem Description

HV is one of the main workhorses for most of the nations of the world for scientific studies and military and commercial applications [1]. Efforts are ongoing in the 21st century to enhance flexibility and reliability and reduce the overall cost of such systems. Here, without loss of generality, we provide the proposed algorithm to solve a trajectory optimal problem of HV. Its 3-DOF dynamics over a spherical earth are described by the following motion equations [12, 13]:

The 3-DOF dynamics described in (1) are dimensionless equations of motion with six state variables and two control variables. The real variables are normalized as follows:where is the radial distance from the earth’s center to the vehicle, is the longitude, is the latitude, is the velocity of the vehicle, is the flight path angle, is the velocity azimuth angle measured clockwise from the north, the control variables are the angle of attack and the fuel throttle opening , is the mass of the vehicle, and the terms and are the aerodynamic drag and lift acceleration, which are defined byHere, is the reference area of the vehicle. The terms of and are the coefficients of drag and lift, which are also functions of and .

The control variables and are approximated by Legendre orthogonal polynomial functions, respectively. Assuming is the Legendre polynomial of degree , the continuous function of variable can be approximated as

This way, the dynamic parameter optimization problem is transformed into a static one before the PSO algorithm performed.

The evaluation of the performance index starts when the scramjet is launched. The vehicle releases from the boost phase and then enters into the later ascent and cruise phase. Here, we define the switching time as . By analyzing the dynamic characteristics of HV, the fuel throttle opening shall be minimized during the later ascent stage.

In this problem, the optimal time of ascent stage is free but satisfying some constraints. Thus, an additional control variable is introduced to transcribe the proposed problem to a fixed final time problem. The switching step is available and then the ascent trajectory can be divided into two phases. In the first phase, only the fuel throttle opening needs to be optimized to guarantee that it is a minimized constant value at the beginning of the second phase. In order to simplify the equations and problem, we give a reasonable assumption that the switching step satisfies , , with a fixed time interval , and the system’s continuous state does not jump at each switching point. Therefore, the arbitrary terminal time optimal control problem is turned into a two-point boundary value problem with the fixed terminal time. The optimal time would be obtained as long as all the terminal conditions are fulfilled.

The design space of HV ascent trajectory optimization using LCPSO can be defined by the following vector:

Here, the left two parts in the vector represent the two groups of coefficients of the orthogonal polynomials which approximate the control variables and , while the last parameter represents the ratio between the switching step and the supposed final time. Hence, the problem space can be divided into three subspaces , , and . Then, in LCPSO, we provide two groups of particles to optimize the subspaces and , respectively, and use a one-dimensional search method based on interval analysis to optimize the variable .

To seek the solutions of both the continuous input and the switching step according to the given initial state , we define a performance function as follows:In this case, the above nonlinear optimal problem can be described as a basic optimal control problem of nonlinear Bolza type.

Considering that the performance function here is selected to minimize the fuel expendable ratio (FER) through the control variables and , normally, we use the following relationship to describe it directly:

Furthermore, the ascent trajectory terminate conditions can be represented by the following inequalities:where , , and are three state variables of the terminal point, while the tolerance is given by the upper bound and the lower bounds , , and .

The terminal conditions are added to the performance function to ensure that all the constraints are satisfied. Then, the performance function of LCPSO is defined as follows:Here, in (9), the first part represents the optimal index of minimum FER described in (7). The three terms , , and represent the terminal constraints index. is the radial distance from the center of the earth to the vehicle and represents the velocity at the final time, while the flight path angle index is .

Considering that the proposed LCPSO provides two groups of particles based on Legendre orthogonal polynomial approximation, the assumption of the fixed time interval can be transformed into a closed interval of Legendre orthogonal polynomials to facilitate subsequent processing. Meanwhile, the additional switching step can make the optimal interval of more accurate. After that, the PSO algorithm is employed to solve the optimal control problem about in the fixed interval and the next interval , . Furthermore, the 4th-order Runge-Kutta numeric integration is employed to evaluate the performance function, while the optimal time can be obtained as long as all the terminal conditions are satisfied.

#### 3. Algorithm Formulation

##### 3.1. LCPSO

The proposed LCPSO is designed as a double iteration scheme: the inner iteration of dual cooperative PSO (CPSO) [14] and the outer iteration of interval analysis. The mathematical formulation of the CPSO algorithm is redesigned in (10) and (11).with , , , ,where is the coordinate of particle in dimension at time as subinterval at interval iteration step .

The pseudocode of CPSO algorithm is reported in Pseudocode 1.