Discrete Dynamics in Nature and Society

Volume 2018, Article ID 6916848, 14 pages

https://doi.org/10.1155/2018/6916848

## The Earth-Mars Transfer Trajectory Optimization of Solar Sail Based on* hp*-Adaptive Pseudospectral Method

^{1}School of Aerospace Science and Technology, Xidian University, Xi’an, Shaanxi 710071, China^{2}School of Astronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Correspondence should be addressed to Dongzhu Feng; moc.361@tengnefuhzgnod

Received 7 March 2018; Revised 29 June 2018; Accepted 26 July 2018; Published 2 September 2018

Academic Editor: Miguel Ángel López

Copyright © 2018 Yufei Guo 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

Solar sails have many advantages over traditional chemical propulsion spacecraft, such as needlessness of fuel, high payload ratio, long service life, and great application potential in deep space exploration, interstellar voyage, and other aerospace application fields. However, the period of solar sail transfer from the initial orbit to the desired orbit is relatively long. Thus, it is necessary to optimize the transfer trajectory of solar sail according to specific tasks. The* hp*-adaptive pseudospectral method combines the global pseudospectral method with the finite element method, which adopts double layer optimization strategy to solve the optimal control problem and has higher computational efficiency and accuracy than the traditional global pseudospectral method. In this paper, the pressure of solar light acting on the solar sail is analyzed, and the kinematic equation of the solar sail is established in polar coordinate system first; then the basic principle of the* hp*-adaptive pseudospectral method is introduced, and the steps of solving the transfer trajectory optimization problem by* hp*-adaptive pseudospectral method are proposed; finally, the trajectory optimization of the solar sail from Earth orbit to Sun-centered Mars orbit is simulated as an example to demonstrate the effectiveness of* hp*-adaptive pseudospectral method in the orbit transfer optimization problem of solar sail. The simulation results show that the adopted method is not sensitive to the initial values and has more reasonable distribution and less computational cost than the Gauss pseudospectral method.

#### 1. Introduction

With the increasing complexity and space distance of deep space exploration missions, the conventional propulsion methods hardly meet the demands. Solar sail is a new type of spacecraft that produces propulsion by a large area of light film that reflects solar photons, where the power source is natural solar energy, without consuming the chemical fuel. Therefore, the lifespan of the solar sail in space is not limited by fuel. The impact force of solar photons is very weak, but they continuously act on the solar sail, which will generate continuous small propulsion and persistent acceleration. The speed of the solar sail can reach up to 93km/s, which is 5 to 7 times the speed of the current fastest spacecraft [1]. In addition, the characteristics of light structure, low cost, and small launching risk make the solar sail have great advantages in long period space missions such as deep space exploration and interplanetary navigation.

The key technologies of solar sail include structure design, material selection, folding and deployment of solar sail, attitude control, and trajectory optimization. After nearly a century of development, solar sail technologies have advanced greatly. On the one hand, many scholars have made in-depth theoretical analysis of solar sail; on the other hand, a number of space agencies have carried out some ground or space tests on the key technologies of solar sail. Trajectory optimization for spacecraft can prolong the orbital life and increase the ability of performing tasks [2]. Thus, it is necessary to optimize the transfer trajectory of solar sail to shorten the orbital transfer time during deep space exploration missions.

Trajectory optimization of solar sail is essentially a nonlinear dynamic optimal control problem with state constraints and control constraints and is very difficult to solve. At present, the research on the trajectory optimization of solar sail is still in the early stage, which needs further theoretical research and experimental verification. Carl G. Sauer [3] (in 1976) used the Calculus of Variations of the classical indirect method to optimize the solar sail rendezvous trajectories for each of the terrestrial planets (Venus, Mercury, and Mars) and an asteroid roundtrip mission of a solar sail. The results proved the versatility of the solar sail spacecraft. However, the indirect method has some disadvantages including the difficulty of deducing the optimal condition, the small radius of convergence, and the difficulty of predicting the initial value of the covariate variable. Michiel Otten [4] (in 2001) obtained the nearly minimum transfer time for Earth-Mars coplanar trajectory of solar sail by using the direct discrete method. The author divided the whole orbital transfer process into equal periods of , and the control variables in each segment (i.e., the solar sail steering angle ) were constants, so that the optimized parameters were the values of and the terminal time of the time periods. Although the steering law obtained by this method is easier to implement for a realistic solar sail, the orbit transfer time of the optimal solution is longer than that of the indirect method. Nassiri [5] (in 2005) introduced the force model and orbital dynamics model for an ideal solar sail and used the direct collocation method to cast the solar sail trajectory optimization problem as a nonlinear programming problem. It is concluded that the direct collocation method is capable of finding accurate solutions to the optimal control problem of solar sail interplanetary trajectories. Gau [6] (in 2015) proposed a method using a solar sail to change the orbit of a near Earth asteroid. He derived the dynamic model for a tethered system formed by an asteroid and a solar sail and utilized the Legendre pseudospectral method to solve the optimal control problem of its trajectory. As an improvement of the direct collocation method, the pseudospectral method can obtain higher solution accuracy with less computational cost. Moreover, some scholars have applied intelligent optimization algorithms to solve the interplanetary optimal transfer trajectory problem for a solar sail, such as Evolutionary Neurocontrol [7] and Generalized Extremal Optimization [8, 9]. The intelligent optimization algorithms have the advantages of strong robustness and global convergence, but they are prone to produce premature phenomenon, and their local optimization ability is poor.

Since the collocation points of the pseudospectral method are fixed (dense near the boundaries and sparse in the middle) [10], the interpolation polynomial with higher dimensions is often needed to obtain an ideal approximate solution [11]. The* hp*-adaptive pseudospectral method [12–14] proposed by C. L. Darby and A. V. Rao et al. has the characteristics of flexibility of collocation points distribution, sparsity of calculation, and fast convergence and can adjust adaptively the number of the units or the degree of the polynomial in the corresponding segments according to the requirement of calculation accuracy. This method combines the advantages of the global pseudospectral method and the finite element method and can greatly reduce the computational cost. Here, the* hp*-adaptive pseudospectral method is adopted to solve the trajectory optimization problem for a solar sail during the transfer from Earth orbit to Mars orbit; furthermore, the collocation points distribution and time-consuming of* hp*-adaptive pseudospectral method and Gauss pseudospectral method are compared and analyzed in this paper. The results show that the* hp*-adaptive pseudospectral method is an effective trajectory optimization method.

This paper is organized as follows. In Section 2 we establish the kinematic equation of the solar sail. In Section 3 we present the steps for solving the transfer trajectory optimization problem based on the* hp*-adaptive pseudospectral method. In Section 4 we take the solar sail Earth-Mars transfer trajectory optimization as an example for the simulation and provide a discussion and analysis of the results. In Section 5 we provide concluding remarks.

#### 2. Kinematics Model of the Ideal Solar Sail

We assume that the solar sail is an ideal plane and perfectly reflecting, and the orbital inclination of the desired planetary orbit is ignored. Therefore, the three-dimensional trajectory optimization problem can be simplified into a two-dimensional trajectory optimization problem.

##### 2.1. Solar-Radiation-Pressure Model

The driving force of the solar sail is produced by momentum exchange between the solar photons and the highly reflective sail, so the solar-radiation-pressure (SRP) plays a dominant role on solar sail [15]. As shown in Figure 1, considering an ideal solar sail, the solar pressure energy is generated by the momentum transfer of incident light and reflected light with the sail. Since the solar photons are ideally reflected on the ideal plane sails, the magnitudes of the forces generated by the incident and reflected photons are equal, and they are given bywhere is the SRP, and is the effective area of solar sail, , and is the sail area. The pitch angle is the angle between the unit vector of incident rays and the unit normal vector of the solar sail , and is perpendicular to the sail and opposite to the Sun. The unit tangent vector of the solar sail is perpendicular to the normal and its positive direction is counterclockwise. Then, the SRP resultant force acting on a solar sail can be described as [16]