International Journal of Aerospace Engineering

Volume 2017, Article ID 3812397, 12 pages

https://doi.org/10.1155/2017/3812397

## The Coupled Orbit-Attitude Dynamics and Control of Electric Sail in Displaced Solar Orbits

^{1}Department of Aerospace Engineering, Harbin Institute of Technology, Harbin 150001, China^{2}Shanghai Satellite Engineering Research Institute, Shanghai 200240, China

Correspondence should be addressed to Mingying Huo; moc.liamg@321gniygnimouh

Received 22 March 2017; Revised 24 May 2017; Accepted 2 July 2017; Published 31 July 2017

Academic Editor: Christian Circi

Copyright © 2017 Mingying Huo 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

Displaced solar orbits for spacecraft propelled by electric sails are investigated. Since the propulsive thrust is induced by the sail attitude, the orbital and attitude dynamics of electric-sail-based spacecraft are coupled and required to be investigated together. However, the coupled dynamics and control of electric sails have not been discussed in most published literatures. In this paper, the equilibrium point of the coupled dynamical system in displaced orbit is obtained, and its stability is analyzed through a linearization. The results of stability analysis show that only some of the orbits are marginally stable. For unstable displaced orbits, linear quadratic regulator is employed to control the coupled attitude-orbit system. Numerical simulations show that the proposed strategy can control the coupled system and a small torque can stabilize both the attitude and orbit. In order to generate the control force and torque, the voltage distribution problem is studied in an optimal framework. The numerical results show that the control force and torque of electric sail can be realized by adjusting the voltage distribution of charged tethers.

#### 1. Introduction

Displaced solar orbit is a kind of non-Keplerian orbits, which is lifted above the ecliptic plane by applying a continuous thrust to counterbalance the gravity. From displaced orbit, one would have a continuous view to the polar region of the sun, or a long time scale uninterrupted helioseismological coverage [1]. As a result of that the maintenance of displaced orbit requires continuous propulsive thrust; this mission is impossible for most of conventional (either chemical or electrical) propulsion systems. The solar sail is firstly proposed to maintain the displaced orbits as it utilizes solar radiation pressure to generate continuous and propellant-less thrust. As early as 1929, Oberth mentioned that the solar radiation pressure can be used to generate a displacement between the orbital plane and the ecliptic plane. More recently, large families of displaced solar sail orbits are proposed by McInnes and Simmons [2–4]. The dynamics, stability, and control of displaced solar sail orbits were investigated by considering a solar sail in a rotating frame. Gong et al. did a lot of work on the coupled attitude-orbit dynamics and control of a solar sail in displaced orbits [5, 6] and proposed the solar sail formation flying around displaced orbits [7–10].

However, the thrust acceleration level of solar sails cannot meet the requirements of maintaining high displacement orbits, because its reflection film is not light enough [11]. In addition, as the thrust acceleration of solar sail cannot be adjusted at will between zero and some maximum, the displaced orbit maintained by solar sail is not flexible enough. In light of these problems, the electric solar wind sail (electric sail for short) is used, as an alternative to the use of solar sail, to maintain displaced orbits in this paper. The electric sail, which is first proposed by Janhunen [12] in 2004, is an innovative concept for spacecraft propulsion. Similar to solar sails, electric sails can produce continuous thrust without the need of propellant. Unlike solar sails, electric sails are propelled by the solar wind dynamic pressure, instead of the solar photon momentum.

As shown in Figure 1, the electric sail consists of many tethers, which are held at a high positive potential by a solar-powered electron gun. The electrostatic field generated by the charged tethers can reflect the solar wind protons to generate a continuous thrust without any propellant. The deployment and maintenance of electric sail are implemented by spinning the sailcraft about the symmetry axis. It is noticed that an electrostatic field potential structure with a spatial scale larger than 100 m can be created around a thin tether with thickness of a few tens of micrometers. Therefore, the characteristic acceleration of electric sails can be higher than that of solar sails. Recent results show that electric sails can generate 1 N thrust with only 100 kg propulsion system mass [13]. In addition, the thrust of electric sail can be adjusted at will between zero and some maximum by controlling the power of the electron gun [1]. Consequently, the displaced orbit maintained by electric sail is more flexible than that maintained by solar sail. The displaced non-Keplerian orbits for electric sails have been studied by Mengali and Quarta [14]. In their paper, the electric sail capabilities of generating a class of displaced non-Keplerian orbits are analyzed, and a comparison with a solar sail is made. Qi et al. [15] investigated displaced electric sail orbits and the transition trajectory optimization. However, in the above literatures, the displaced orbits of electric sail are researched based on a classical thrust model, and the coupled orbit-attitude dynamics and control of electric sail are not considered. In the classical thrust model, the effects of electric sail attitude on the propulsive thrust modulus and direction were neglected. The thrust modulus was assumed to be invariable with the change of pitch angle, and the thrust cone angle was assumed to be approximately equal to one-half of the pitch angle. Obviously, the above models are not accurate enough to describe the thrust vector of an electric sail for mission analysis. In this paper, the coupled orbit-attitude dynamics and control of electric sail will be considered together based on an advanced thrust model.