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

Authors: Mingying Huo; He Liao; Yanfang Liu; Naiming Qi
Journal: International Journal of Aerospace Engineering
 (2017)

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## 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.


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## Body

## 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 Figure1, 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.Figure 1
Conceptual sketch of an electric sail.Since the propulsive thrust is induced by the sail attitude, the orbital and attitude dynamics of electric-sail-based spacecraft are coupled. However, the coupled orbit-attitude control of electric sails has not been discussed in most published literatures. In our previous research [16], the propulsive thrust of electric sail is described as a function of the orbital radius and the sail angle. Consequently, the coupled attitude-orbit dynamics of an electric-sail-based spacecraft are obtained. For the heliocentric transfer mission, the flight control is investigated, wherein the orbital control is studied in an optimal framework via a hybrid optimization method and the attitude controller is designed based on feedback linearization control. In fact, the above attitude and orbital control are carried out separately. Differently, in this paper, the attitude and orbital control of electric sail in displaced solar orbits will be investigated together, instead of being implanted separately.This paper is organized as follows. First of all, the coupled orbit-attitude dynamics of electric sail are discussed in the heliocentric-ecliptic inertial frame and the body frame. Secondly, the equilibrium point of the coupled dynamical system in displaced orbit is obtained, and its stability is analyzed through a linearization. Finally, the linear quadratic regulator is employed to control the coupled orbit-attitude system for unstable displaced orbits.


## 2. Coupled Orbit-Attitude Dynamics of Electric Sail

In our previous research [16], the coupled orbit-attitude dynamics of electric sail are obtained. For displaced orbits design and coupled control of electric sail, the coupled dynamics are discussed briefly in this paper.


### 2.1. Reference Frame

Before the description of coupled dynamics, three reference frames are introduced, which are the body frameobxbybzb, the orbital frame ooxoyozo, and the heliocentric-ecliptic inertial frame oixiyizi. Considering an electric sail consists of N tethers as seen in Figure 2, these tethers can be numbered counterclockwise. The origin of the body frame obxbybzb is at the center of mass of the sail, and the xb-axis is in the direction of a given reference tether. The zb-axis is along the normal of the sail, and the yb-axis forms a right-handed system.Figure 2
Body reference frame.The orbital frameooxoyozo and the inertial frame oixiyizi are shown in Figure 3. The origin of the orbital frame ooxoyozo is at the center of mass of the sail and the zo-axis is along the sun-spacecraft direction. The yo-axis is perpendicular to the normal of the ecliptic plane and the zo-axis, and the xo-axis forms a right-handed triad. The origin of the inertial frame oixiyizi is at the center of mass of the sun, and the xi-axis is in the direction of sun-equinox. The zi-axis is along the normal of the ecliptic plane, and the yi-axis forms a right-handed system. The attitude of the sail in the orbital frame can be described by three angles, ϕ, θ, and ψ, and the rotation sequence from the orbital frame to the body frame is x(ϕ)→y(θ)→z(ψ).Figure 3
Heliocentric-ecliptic inertial frame and orbital reference frame.


### 2.2. Coupled Orbit-Attitude Dynamical Equations

In order to get greater propulsive acceleration and reduce the launch cost, the electric sail must employ large-scale light-weight charged tethers. With these tethers, the electric sail has some special properties of extreme flexibility, low damping, and low frequency. The flexibility of tethers will influence the attitude operation. When the flexible tethers are in motion, the vibration of the structure influences the dynamic characteristics of the entire structure and the coupled control, especially when the structure undergoes large-angle maneuvering. Attitude manoeuvres combined with gyroscopic effects might induce motions that cannot be captured by the rigid body model. However, the coupling mechanism of orbit-attitude-deformation of flexible electric sails is quite complicated. For simplicity, the flexibility of charged tethers is ignored, because the angle of maneuver is small in this paper. As discussed in our previous research [16], the coupled orbit-attitude dynamical scalar equations of electric-sail-based spacecraft, under the assumption of rigid body model, can be written as(1)r˙=vrΘ˙=ωΘΨ˙=ωΨv˙r=rωΨ2sin2⁡Θ+rωΘ2-μ⊙r2+κa⊕r⊕2rcos2⁡ϕcos2θ+1ω˙Θ=ωΨ2sin⁡Θcos⁡Θ-2vrωΘr+κa⊕r⊕2r2cos⁡ϕsin⁡θcos⁡θω˙Ψ=-2vrωΨr-2ωΘωΨcot⁡Θ-κa⊕r⊕2r2sin⁡ϕcos⁡ϕcos2θϕ˙=ωbxcos⁡ψcos⁡θ-ωbysin⁡ψcos⁡θ+ωΨsin⁡Θ-ωΘsin⁡ϕtan⁡θ+ωΨcos⁡Θcos⁡ϕtan⁡θθ˙=ωbxsin⁡ψ+ωbycos⁡ψ-ωΘcos⁡ϕ-ωΨcos⁡Θsin⁡ϕψ˙=-ωbxtan⁡θcos⁡ψ+ωbytan⁡θsin⁡ψ+ωbz+ωΘsin⁡ϕcos⁡θ-ωΨcos⁡Θcos⁡ϕcos⁡θω˙bx=ωbyωbzIy-IzIx+TxIxω˙by=ωbxωbzIz-IxIy+TyIyω˙bz=ωbxωbyIx-IyIz+TzIz,where r is the distance between sun and sailcraft, Ψ is the ecliptic longitude and Θ is the ecliptic latitude as shown in Figure 3, μ⊙ is sun’s gravitational parameter, thrust control coefficient κ∈[0,1], as the thrust of the electric sail can be adjusted by the electron gun, a⊕ is the characteristic acceleration of the electric sail, when the sun-spacecraft distance is r⊕=1au (the characteristic acceleration of the electric sail can be adjusted by the electron gun within certain range, ωbxωbyωbz is the vector of angular velocity described in the body frame obxbybzb, TxTyTz is the vector of control torque, which can be generated by adjusting the voltage distribution of charged tethers [16], and the inertia matrix of sailcraft I=diag(Ix,Iy,Iz), wherein Iz=2Ix=2Iy=14.666×108kgm2 in this paper.


## 2.1. Reference Frame

Before the description of coupled dynamics, three reference frames are introduced, which are the body frameobxbybzb, the orbital frame ooxoyozo, and the heliocentric-ecliptic inertial frame oixiyizi. Considering an electric sail consists of N tethers as seen in Figure 2, these tethers can be numbered counterclockwise. The origin of the body frame obxbybzb is at the center of mass of the sail, and the xb-axis is in the direction of a given reference tether. The zb-axis is along the normal of the sail, and the yb-axis forms a right-handed system.Figure 2
Body reference frame.The orbital frameooxoyozo and the inertial frame oixiyizi are shown in Figure 3. The origin of the orbital frame ooxoyozo is at the center of mass of the sail and the zo-axis is along the sun-spacecraft direction. The yo-axis is perpendicular to the normal of the ecliptic plane and the zo-axis, and the xo-axis forms a right-handed triad. The origin of the inertial frame oixiyizi is at the center of mass of the sun, and the xi-axis is in the direction of sun-equinox. The zi-axis is along the normal of the ecliptic plane, and the yi-axis forms a right-handed system. The attitude of the sail in the orbital frame can be described by three angles, ϕ, θ, and ψ, and the rotation sequence from the orbital frame to the body frame is x(ϕ)→y(θ)→z(ψ).Figure 3
Heliocentric-ecliptic inertial frame and orbital reference frame.


## 2.2. Coupled Orbit-Attitude Dynamical Equations

In order to get greater propulsive acceleration and reduce the launch cost, the electric sail must employ large-scale light-weight charged tethers. With these tethers, the electric sail has some special properties of extreme flexibility, low damping, and low frequency. The flexibility of tethers will influence the attitude operation. When the flexible tethers are in motion, the vibration of the structure influences the dynamic characteristics of the entire structure and the coupled control, especially when the structure undergoes large-angle maneuvering. Attitude manoeuvres combined with gyroscopic effects might induce motions that cannot be captured by the rigid body model. However, the coupling mechanism of orbit-attitude-deformation of flexible electric sails is quite complicated. For simplicity, the flexibility of charged tethers is ignored, because the angle of maneuver is small in this paper. As discussed in our previous research [16], the coupled orbit-attitude dynamical scalar equations of electric-sail-based spacecraft, under the assumption of rigid body model, can be written as(1)r˙=vrΘ˙=ωΘΨ˙=ωΨv˙r=rωΨ2sin2⁡Θ+rωΘ2-μ⊙r2+κa⊕r⊕2rcos2⁡ϕcos2θ+1ω˙Θ=ωΨ2sin⁡Θcos⁡Θ-2vrωΘr+κa⊕r⊕2r2cos⁡ϕsin⁡θcos⁡θω˙Ψ=-2vrωΨr-2ωΘωΨcot⁡Θ-κa⊕r⊕2r2sin⁡ϕcos⁡ϕcos2θϕ˙=ωbxcos⁡ψcos⁡θ-ωbysin⁡ψcos⁡θ+ωΨsin⁡Θ-ωΘsin⁡ϕtan⁡θ+ωΨcos⁡Θcos⁡ϕtan⁡θθ˙=ωbxsin⁡ψ+ωbycos⁡ψ-ωΘcos⁡ϕ-ωΨcos⁡Θsin⁡ϕψ˙=-ωbxtan⁡θcos⁡ψ+ωbytan⁡θsin⁡ψ+ωbz+ωΘsin⁡ϕcos⁡θ-ωΨcos⁡Θcos⁡ϕcos⁡θω˙bx=ωbyωbzIy-IzIx+TxIxω˙by=ωbxωbzIz-IxIy+TyIyω˙bz=ωbxωbyIx-IyIz+TzIz,where r is the distance between sun and sailcraft, Ψ is the ecliptic longitude and Θ is the ecliptic latitude as shown in Figure 3, μ⊙ is sun’s gravitational parameter, thrust control coefficient κ∈[0,1], as the thrust of the electric sail can be adjusted by the electron gun, a⊕ is the characteristic acceleration of the electric sail, when the sun-spacecraft distance is r⊕=1au (the characteristic acceleration of the electric sail can be adjusted by the electron gun within certain range, ωbxωbyωbz is the vector of angular velocity described in the body frame obxbybzb, TxTyTz is the vector of control torque, which can be generated by adjusting the voltage distribution of charged tethers [16], and the inertia matrix of sailcraft I=diag(Ix,Iy,Iz), wherein Iz=2Ix=2Iy=14.666×108kgm2 in this paper.


## 3. Displaced Electric Sail Orbits

### 3.1. Displaced Orbits in General

Solutions of displaced electric sail orbits can be found by seeking equilibrium solutions to the two-body problem. Considering the definition of displaced orbits in [2], the orbital radius r, the ecliptic latitude Θ, and the ecliptic longitude angle velocity ωΨ should be constant and can be chosen to be some particular fixed value. Therefore, the displaced orbits in general should have the following characteristics:(2)r=rdΘ=ΘdωΨ=ωdr˙d=r¨d=Θ˙d=Θ¨d=ω˙d=0,where rd, Θd, and ωd are the selected orbital radius, ecliptic latitude, and ecliptic longitude velocity of displaced orbit, respectively.Substituting (2) in the orbital dynamical scalar equations (1), the required characteristic acceleration a⊕d and required attitude angles can be obtained as(3)a⊕d=3μ⊙-3rd3ωd2sin2⁡Θd-χ2rdr⊕ϕd=0sin⁡θd=χ+μ⊙2+rd6ωd4sin2⁡Θd3sin2Θd-2-rd3ωd2sin2⁡Θdχ+2μ⊙2rd6ωd4sin2⁡Θd-4rd3ωd2sin2⁡Θd+2μ⊙2μ⊙+χ-rd3ωd2sin2⁡Θd2rd3ωd2sin⁡Θdcos⁡Θd,where χ = rd6ωd4sin2⁡Θd(9sin2⁡Θd-8)-2μ⊙rd3ωd2sin2⁡Θd+μ⊙2.As shown in (3), the attitude angle ϕd should be equal to zero for keeping the required displaced orbit. It means that the normal of the electric sail should be in the same plane as the position vector r and the orbital angular velocity vector ω. This requirement is consistent with the requirement for keeping displaced orbit using a solar sail [5]. In addition, through a series of analyses, it is known that the chosen parameters of displaced orbit (rd,Θd,ωd) should meet the following criteria:(4)rd6ωd4sin2⁡Θd9sin2⁡Θd-8-2μ⊙rd3ωd2sin2⁡Θd+μ⊙2≥0.If the selected parameters cannot meet the requirements in (4), electric sails would not be used in keeping the selected displaced orbit, due to the limitation of the maximum thrust cone angle of the electric sail. As discussed in our previous research [16], the thrust cone angle of the electric sail reaches the maximum value 19.47∘, when the light incident angle β is 54.75∘.


### 3.2. Displaced Geostationary Orbits

If the orbital period of displaced orbit is chosen to be consistent with the orbital period of Earth, this kind of displaced orbits is named displaced geostationary orbits. Displaced geostationary orbits have the following characteristics:(5)ωd=μ⊙r⊕3.Substituting (5) in (3), the requirements of displaced geostationary orbits are obtained as follows:(6)a⊕=3μ⊙r⊕3-3μ⊙rd3sin2⁡Θd-χr⊕32rdr⊕4ϕd=0sin⁡θ=χ+μ⊙2r⊕6+μ⊙2rd6sin2⁡Θd3sin2⁡Θd-2-μ⊙rd3r⊕3sin2⁡Θdχ+2μ⊙2μ⊙2rd6sin2⁡Θd-4μ⊙rd3r⊕3sin2⁡Θd+2μ⊙2r⊕6μ⊙+χr⊕3-μ⊙rd3sin2⁡Θd2μ⊙rd3sin⁡Θdcos⁡Θd,where χ = μ⊙2rd6sin2⁡Θd(9sin2⁡Θd-8)/r⊕6-2μ⊙2rd3sin2⁡Θd/r⊕3+μ⊙2.According to the previous discussion, the chosen parameters of displaced geostationary orbit (rd,Θd) should meet the following criteria: (7)μ⊙2rd6sin2⁡Θd9sin2⁡Θd-8r⊕6-2μ⊙2rd3sin2⁡Θdr⊕3+μ⊙2≥0.Based on (7), we can obtain the feasible region for the displaced geostationary orbit, which is shown in Figure 4.Figure 4
Feasible region for the displaced geostationary orbit.Based on (6), the required characteristic acceleration a⊕d and attitude angle θd, for keeping displaced geostationary orbits with different parameters (rd,Θd) in feasible region, are calculated and shown in Figures 5 and 6, respectively. As shown in Figure 5, the required characteristic acceleration a⊕d increases with the decrease of the sun-sailcraft distance and increases with the increase of Θd-90∘.Figure 5
Required characteristic acceleration for the displaced geostationary orbit.Figure 6
Required attitude angleθ for the displaced geostationary orbit.


## 3.1. Displaced Orbits in General

Solutions of displaced electric sail orbits can be found by seeking equilibrium solutions to the two-body problem. Considering the definition of displaced orbits in [2], the orbital radius r, the ecliptic latitude Θ, and the ecliptic longitude angle velocity ωΨ should be constant and can be chosen to be some particular fixed value. Therefore, the displaced orbits in general should have the following characteristics:(2)r=rdΘ=ΘdωΨ=ωdr˙d=r¨d=Θ˙d=Θ¨d=ω˙d=0,where rd, Θd, and ωd are the selected orbital radius, ecliptic latitude, and ecliptic longitude velocity of displaced orbit, respectively.Substituting (2) in the orbital dynamical scalar equations (1), the required characteristic acceleration a⊕d and required attitude angles can be obtained as(3)a⊕d=3μ⊙-3rd3ωd2sin2⁡Θd-χ2rdr⊕ϕd=0sin⁡θd=χ+μ⊙2+rd6ωd4sin2⁡Θd3sin2Θd-2-rd3ωd2sin2⁡Θdχ+2μ⊙2rd6ωd4sin2⁡Θd-4rd3ωd2sin2⁡Θd+2μ⊙2μ⊙+χ-rd3ωd2sin2⁡Θd2rd3ωd2sin⁡Θdcos⁡Θd,where χ = rd6ωd4sin2⁡Θd(9sin2⁡Θd-8)-2μ⊙rd3ωd2sin2⁡Θd+μ⊙2.As shown in (3), the attitude angle ϕd should be equal to zero for keeping the required displaced orbit. It means that the normal of the electric sail should be in the same plane as the position vector r and the orbital angular velocity vector ω. This requirement is consistent with the requirement for keeping displaced orbit using a solar sail [5]. In addition, through a series of analyses, it is known that the chosen parameters of displaced orbit (rd,Θd,ωd) should meet the following criteria:(4)rd6ωd4sin2⁡Θd9sin2⁡Θd-8-2μ⊙rd3ωd2sin2⁡Θd+μ⊙2≥0.If the selected parameters cannot meet the requirements in (4), electric sails would not be used in keeping the selected displaced orbit, due to the limitation of the maximum thrust cone angle of the electric sail. As discussed in our previous research [16], the thrust cone angle of the electric sail reaches the maximum value 19.47∘, when the light incident angle β is 54.75∘.


## 3.2. Displaced Geostationary Orbits

If the orbital period of displaced orbit is chosen to be consistent with the orbital period of Earth, this kind of displaced orbits is named displaced geostationary orbits. Displaced geostationary orbits have the following characteristics:(5)ωd=μ⊙r⊕3.Substituting (5) in (3), the requirements of displaced geostationary orbits are obtained as follows:(6)a⊕=3μ⊙r⊕3-3μ⊙rd3sin2⁡Θd-χr⊕32rdr⊕4ϕd=0sin⁡θ=χ+μ⊙2r⊕6+μ⊙2rd6sin2⁡Θd3sin2⁡Θd-2-μ⊙rd3r⊕3sin2⁡Θdχ+2μ⊙2μ⊙2rd6sin2⁡Θd-4μ⊙rd3r⊕3sin2⁡Θd+2μ⊙2r⊕6μ⊙+χr⊕3-μ⊙rd3sin2⁡Θd2μ⊙rd3sin⁡Θdcos⁡Θd,where χ = μ⊙2rd6sin2⁡Θd(9sin2⁡Θd-8)/r⊕6-2μ⊙2rd3sin2⁡Θd/r⊕3+μ⊙2.According to the previous discussion, the chosen parameters of displaced geostationary orbit (rd,Θd) should meet the following criteria: (7)μ⊙2rd6sin2⁡Θd9sin2⁡Θd-8r⊕6-2μ⊙2rd3sin2⁡Θdr⊕3+μ⊙2≥0.Based on (7), we can obtain the feasible region for the displaced geostationary orbit, which is shown in Figure 4.Figure 4
Feasible region for the displaced geostationary orbit.Based on (6), the required characteristic acceleration a⊕d and attitude angle θd, for keeping displaced geostationary orbits with different parameters (rd,Θd) in feasible region, are calculated and shown in Figures 5 and 6, respectively. As shown in Figure 5, the required characteristic acceleration a⊕d increases with the decrease of the sun-sailcraft distance and increases with the increase of Θd-90∘.Figure 5
Required characteristic acceleration for the displaced geostationary orbit.Figure 6
Required attitude angleθ for the displaced geostationary orbit.


## 4. Stability Analysis

### 4.1. Analysis of the Stability

In order to facilitate the analysis of the stability of displaced orbits, letX1=[r,Θ,Ψ]T, X2=[vr,ωΘ,ωΨ]T, X3=[ϕ,θ,ψ]T, and X4=[ωbx,ωby,ωbz]T. Then, the coupled orbit-attitude dynamical scalar equations of electric-sail-based spacecraft can be written as(8)X˙1=X2X˙2=ΓX1,X2,X3X˙3=ΛX1,X2,X3,X4X˙4=ΠX4,where (9)Γ=rωΨ2sin2⁡Θ+rωΘ2-μ⊙r2+a⊕r⊕2rcos2⁡ϕcos2⁡θ+1ωΨ2sin⁡Θcos⁡Θ-2vrωΘr+a⊕r⊕2r2cos⁡ϕsin⁡θcos⁡θ-2vrωΨr-2ωΘωΨcot⁡Θ-a⊕r⊕2r2sin⁡ϕcos⁡ϕcos2⁡θΛ=ωbxcos⁡ψcos⁡θ-ωbysin⁡ψcos⁡θ+ωΨsin⁡Θ-ωΘsin⁡ϕtan⁡θ+ωΨcos⁡Θcos⁡ϕtan⁡θωbxsin⁡ψ+ωbycos⁡ψ-ωΘcos⁡ϕ-ωΨcos⁡Θsin⁡ϕ-ωbxtan⁡θcos⁡ψ+ωbytan⁡θsin⁡ψ+ωbz+ωΘsin⁡ϕcos⁡θ-ωΨcos⁡Θcos⁡ϕcos⁡θΠ=ωbyωbzIy-IzIxωbxωbzIz-IxIyωbxωbyIx-IyIz.According to the previous discussion, there are equilibrium points of coupled orbit-attitude system in feasible region. By the linearization of the coupled system at the equilibrium pointXref=[Xref1T,Xref2T,Xref3T,Xref4T]T, the variational equations are obtained as(10)δX˙1=δX2δX˙2=∂Γ∂X1X1=Xref1X2=Xref2X3=Xref3δX1+∂Γ∂X2X1=Xref1X2=Xref2X3=Xref3δX2+∂Γ∂X3X1=Xref1X2=Xref2X3=Xref3δX3=A1δX1+A2δX2+A3δX3δX˙3=∂Λ∂X1X1=Xref1X2=Xref2X3=Xref3X4=Xref4δX1+∂Λ∂X2X1=Xref1X2=Xref2X3=Xref3X4=Xref4δX2+∂Λ∂X3X1=Xref1X2=Xref2X3=Xref3X4=Xref4δX3+∂Λ∂ωX1=Xref1X2=Xref2X3=Xref3X4=Xref4δX4=A4δX1+A5δX2+A6δX3+A7δX4δX˙4=∂Π∂X4X4=Xref4δX4=A8δX4,where Ai(i=1,2,…,8) are 3×3 matrixes.Letδ = X1T-Xref1TX2T-Xref2TX3T-Xref3TX4T-Xref4TT=δX1TδX2TδX3TδX4TT; the variational equations can be written as (11)δ˙=0E3×300A1A2A30A4A5A6A7000A8δ=Hcδ,where E3×3 is a 3×3 identity matrix.The stability of the equilibrium point can be checked by calculating the eigenvalues of the variational coefficient matrix of coupled systemHc. To quantify the stability of the coupled system, a parameter labeled as characteristic index is defined as the maximal real part of all eigenvalues, given by(12)Ξc=max1≤i≤12Re⁡λi,where λi(i=1,2,…,12) are eigenvalues of matrix Hc.According to Lyapunov’s stability theory, ifΞc<0, the coupled system is asymptotically stable. If Ξc=0, the coupled system is marginally stable. If Ξc>0, the coupled system is unstable. Referring to (11), the orbital variational coefficient matrix Ho and the attitude variational coefficient matrix Ha can be written as(13)Ho=0EA1A2Ha=A6A70A8.We can also define the characteristic index of orbit and attitude as(14)Ξo=max1≤i≤6Re⁡λoiΞa=max1≤i≤6Re⁡λai,where λoi(i=1,2,…,6) are eigenvalues of the orbital variational coefficient matrix Ho and λai(i=1,2,…,6) are eigenvalues of the attitude variational coefficient matrix Ha.


### 4.2. Numerical Analysis of the Stability

In this section, the linearized coupled dynamical equations are employed to investigate the stability of the coupled equilibrium numerically. An arbitrary displaced orbit is selected to investigate the relation between the stability of the coupled system and the parameterωd. The selected displaced orbit for simulation is determined by rd=0.9au and Θd=86∘. The range of selected orbital angular velocity for simulation is ωd∈0.5ω⊕,5ω⊕. Other displaced orbits can be analyzed in the same way. As shown in Figure 7, it is found that the orbital system of selected displaced orbit is unstable when the selected orbital angular velocity ωd is less than a critical value (the critical value is 0.84ω⊕ for rd=0.9au and Θd=86∘). It is marginally stable when the selected orbital angular velocity ωd is more than the critical value. Figure 8 indicates that the attitude system is always marginally stable and ωd does not influence the stability.Figure 7
Orbital characteristic indexΞo as a function of the angular velocity ωd.Figure 8
Attitude characteristic indexΞa as a function of the angular velocity ωd.Figure9 indicates that the coupled system of selected displaced orbit is unstable when the selected orbital angular velocity ωd is less than a critical value (the critical value is 1.32ω⊕ in this simulation). The coupled system is marginally stable when the selected orbital angular velocity ωd is more than the critical value. Therefore, it is easy to be known that the displaced geostationary orbit (ωd=ω⊕) with rd=0.9au and Θd=86∘ is unstable for a fixed attitude.Figure 9
Coupled-system characteristic indexesΞc as a function of the angular velocity ωd.


## 4.1. Analysis of the Stability

In order to facilitate the analysis of the stability of displaced orbits, letX1=[r,Θ,Ψ]T, X2=[vr,ωΘ,ωΨ]T, X3=[ϕ,θ,ψ]T, and X4=[ωbx,ωby,ωbz]T. Then, the coupled orbit-attitude dynamical scalar equations of electric-sail-based spacecraft can be written as(8)X˙1=X2X˙2=ΓX1,X2,X3X˙3=ΛX1,X2,X3,X4X˙4=ΠX4,where (9)Γ=rωΨ2sin2⁡Θ+rωΘ2-μ⊙r2+a⊕r⊕2rcos2⁡ϕcos2⁡θ+1ωΨ2sin⁡Θcos⁡Θ-2vrωΘr+a⊕r⊕2r2cos⁡ϕsin⁡θcos⁡θ-2vrωΨr-2ωΘωΨcot⁡Θ-a⊕r⊕2r2sin⁡ϕcos⁡ϕcos2⁡θΛ=ωbxcos⁡ψcos⁡θ-ωbysin⁡ψcos⁡θ+ωΨsin⁡Θ-ωΘsin⁡ϕtan⁡θ+ωΨcos⁡Θcos⁡ϕtan⁡θωbxsin⁡ψ+ωbycos⁡ψ-ωΘcos⁡ϕ-ωΨcos⁡Θsin⁡ϕ-ωbxtan⁡θcos⁡ψ+ωbytan⁡θsin⁡ψ+ωbz+ωΘsin⁡ϕcos⁡θ-ωΨcos⁡Θcos⁡ϕcos⁡θΠ=ωbyωbzIy-IzIxωbxωbzIz-IxIyωbxωbyIx-IyIz.According to the previous discussion, there are equilibrium points of coupled orbit-attitude system in feasible region. By the linearization of the coupled system at the equilibrium pointXref=[Xref1T,Xref2T,Xref3T,Xref4T]T, the variational equations are obtained as(10)δX˙1=δX2δX˙2=∂Γ∂X1X1=Xref1X2=Xref2X3=Xref3δX1+∂Γ∂X2X1=Xref1X2=Xref2X3=Xref3δX2+∂Γ∂X3X1=Xref1X2=Xref2X3=Xref3δX3=A1δX1+A2δX2+A3δX3δX˙3=∂Λ∂X1X1=Xref1X2=Xref2X3=Xref3X4=Xref4δX1+∂Λ∂X2X1=Xref1X2=Xref2X3=Xref3X4=Xref4δX2+∂Λ∂X3X1=Xref1X2=Xref2X3=Xref3X4=Xref4δX3+∂Λ∂ωX1=Xref1X2=Xref2X3=Xref3X4=Xref4δX4=A4δX1+A5δX2+A6δX3+A7δX4δX˙4=∂Π∂X4X4=Xref4δX4=A8δX4,where Ai(i=1,2,…,8) are 3×3 matrixes.Letδ = X1T-Xref1TX2T-Xref2TX3T-Xref3TX4T-Xref4TT=δX1TδX2TδX3TδX4TT; the variational equations can be written as (11)δ˙=0E3×300A1A2A30A4A5A6A7000A8δ=Hcδ,where E3×3 is a 3×3 identity matrix.The stability of the equilibrium point can be checked by calculating the eigenvalues of the variational coefficient matrix of coupled systemHc. To quantify the stability of the coupled system, a parameter labeled as characteristic index is defined as the maximal real part of all eigenvalues, given by(12)Ξc=max1≤i≤12Re⁡λi,where λi(i=1,2,…,12) are eigenvalues of matrix Hc.According to Lyapunov’s stability theory, ifΞc<0, the coupled system is asymptotically stable. If Ξc=0, the coupled system is marginally stable. If Ξc>0, the coupled system is unstable. Referring to (11), the orbital variational coefficient matrix Ho and the attitude variational coefficient matrix Ha can be written as(13)Ho=0EA1A2Ha=A6A70A8.We can also define the characteristic index of orbit and attitude as(14)Ξo=max1≤i≤6Re⁡λoiΞa=max1≤i≤6Re⁡λai,where λoi(i=1,2,…,6) are eigenvalues of the orbital variational coefficient matrix Ho and λai(i=1,2,…,6) are eigenvalues of the attitude variational coefficient matrix Ha.


## 4.2. Numerical Analysis of the Stability

In this section, the linearized coupled dynamical equations are employed to investigate the stability of the coupled equilibrium numerically. An arbitrary displaced orbit is selected to investigate the relation between the stability of the coupled system and the parameterωd. The selected displaced orbit for simulation is determined by rd=0.9au and Θd=86∘. The range of selected orbital angular velocity for simulation is ωd∈0.5ω⊕,5ω⊕. Other displaced orbits can be analyzed in the same way. As shown in Figure 7, it is found that the orbital system of selected displaced orbit is unstable when the selected orbital angular velocity ωd is less than a critical value (the critical value is 0.84ω⊕ for rd=0.9au and Θd=86∘). It is marginally stable when the selected orbital angular velocity ωd is more than the critical value. Figure 8 indicates that the attitude system is always marginally stable and ωd does not influence the stability.Figure 7
Orbital characteristic indexΞo as a function of the angular velocity ωd.Figure 8
Attitude characteristic indexΞa as a function of the angular velocity ωd.Figure9 indicates that the coupled system of selected displaced orbit is unstable when the selected orbital angular velocity ωd is less than a critical value (the critical value is 1.32ω⊕ in this simulation). The coupled system is marginally stable when the selected orbital angular velocity ωd is more than the critical value. Therefore, it is easy to be known that the displaced geostationary orbit (ωd=ω⊕) with rd=0.9au and Θd=86∘ is unstable for a fixed attitude.Figure 9
Coupled-system characteristic indexesΞc as a function of the angular velocity ωd.


## 5. Coupled Orbit-Attitude Control of Electric Sail

### 5.1. Control Algorithm

The stability of the displaced orbit has been discussed above. The numerical results show that only some of displaced orbits are marginally stable. Therefore, active control is necessary to stabilize the unstable displaced orbits. In this paper, only control torque is employed to stabilize both the attitude and orbit of sailcraft. Then, the coupled system can be linearized and the corresponding linear system is employed for control design. The coupled equations linearized at the equilibrium point with control torque can be written as(15)δ˙=Hcδ+Bu,where δ = [δr,δΘ,δΨ,δvr,δωΘ,δωΨ,δϕ,δθ,δψ,δωx,δωy,δωz]T is state perturbation, Hc is the variational coefficient matrix of coupled system, B=[03×3,03×3,03×3,1/I]T (I is the inertia matrix of sailcraft), and u=T=[Tx,Ty,Tz]T is the vector of control torque.In order to determine the controllability of this system, the rank of the controllability matrix of the linearized system has been calculated. The results show that the system is controllable, because the controllability matrix has full rank. In this paper, linear quadratic regulator (LQR) is employed to design the control torque to track the desired attitude and orbit. The performance function is selected as(16)J=∫0∞δTQδ+uTRudt,where the matrix Q∈R12×12 is symmetric positive definite and represents the weight of the state errors. The matrix R∈R3×3 is also a symmetric positive definite matrix and represents the weight of the control input.The linear state feedback can be obtained by solving the following algebraic Riccati matrix:(17)PHc+HcP-PBR-1BTP+Q=0.Then, the control torque can be written in the state-feedback form as(18)u=-R-1BTPδ.


### 5.2. Numerical Simulation

In order to investigate the stability of the displaced orbit with linear quadratic regulator control, a numerical example is adopted in this section, using the full nonlinear dynamical equations. The selected displaced orbit for simulation is a displaced geostationary orbit (ωd=ω⊕) and is determined by rd=0.9au and Θd=86∘. For keeping the selected displaced geostationary orbit, the required characteristic acceleration a⊕d should be equal to 1.942mm/s2, and attitude angle θd should be equal to -21.749∘. As discussed above, the chosen displaced orbit is unstable without control. Assume that the orbit and the attitude have a small perturbation to the required orbit and attitude at the initial time. The initial values of the perturbation are given by δX1=[4×104m,0,0]T, δX2=[0,0,0]T, δX3=[0,0.2∘,0]T, and δX4=[0,0,0]T. Figures 10 and 11 give the responses of attitude and orbit, respectively. Numerical simulation shows that the LQR control law can control the coupled system. As shown in Figure 12, a small control torque can stabilize both the attitude and orbit.Figure 10
The responses of the orbit errors.Figure 11
The responses of the attitude errors.Figure 12
The control torques.


### 5.3. Preliminary Study of Voltage Distribution

The control torque of electric sail can be realized by adjusting the voltage distribution of charged tethers of electric sail [16, 17]. In this paper, the generation of control torques is preliminarily discussed in an optimal framework. Referring to Eq. (10) and Eq. (13) in [16], the vector of propulsive force F and the vector of control torque T can be written as functions of the vertical thrust magnitude of charged tethers σ⊕1,…,σ⊕N.(19)Fϕ,θ,ψ,r,l,σ⊕1,…,σ⊕N=∑k=1NFkϕ,θ,ψ,r,l,σ⊕kTϕ,θ,ψ,r,l,σ⊕1,…,σ⊕N=∑k=1NTkϕ,θ,ψ,r,l,σ⊕k,where σ⊕k,k∈[1,N], is the kth tether’s magnitude of the force per unit length, when the sun-spacecraft distance is r⊕=1au and the solar wind is perpendicular to tether. As discussed in [17], σ⊕k can be controlled by altering the voltage ofkth charged tether Vk and is given by (20)σ⊕k=0.18max⁡0,Vk-V0ε0pdyn,where V0 is the electric potential corresponding to the kinetic energy of the solar wind ions, ε0 is the vacuum permittivity, and pdyn is the solar wind dynamic pressure.In order to generate the required control forceFc and control torque Tc, the following equality constraints should be satisfied:(21)∑k=1NFkϕ,θ,ψ,r,l,σ⊕k=Fc∑k=1NTkϕ,θ,ψ,r,l,σ⊕k=Tc.As shown in (21), the quantity of equality constraints is 6 in voltage distribution problem. Comparatively, the number of design variables σ⊕1,…,σ⊕N is N∈[20,100] in general, which is decided by the quantity of charged tethers of electric sail. Therefore, the definite solution cannot be obtained by solving (21). In this paper, the voltage distribution problem is converted into a nonlinear programming problem (NLP). Equation (21) is processed into equality constraints in optimal framework. In order to reduce the required power, the optimization performance index is selected as follows:(22)J=∑k=1Nσ⊕k.The above NLP can be solved within the MATLAB environment usingfmincon, which is a constrained optimization routine available with the Optimization Toolbox. The selected optimization algorithm is sequential quadratic programming (SQP). Consider an electric sail, which is composed of N = 100 20 km-long tether. At the beginning of the mission, the required control force can be calculated according to (1), with the required characteristic acceleration a⊕d=1.942mm/s2 and attitude angle θd=-21.949∘. As seen in Figure 12, the required control torque is T=[9.3941×10-5,-6.1×10-3,1.8116×10-6]Nm at the beginning of the mission.This NLP was solved in 2.453 seconds using a personal computer with 8 GB RAM running at 2.20 GHz processor speed. The obtained thrust magnitudes per unit lengthσ⊕k,k∈[1,N], are shown in Figure 13. Considering (20), the required voltage distribution Vk,k∈[1,N], can be obtained and is shown in Figure 14, where Vaver=∑k=1NVk is the average value of Vk,k∈[1,N]. Through the preliminary study of voltage distribution, it can be found that the control force and torque of electric sail can be realized by adjusting the voltage distribution of charged tethers. This feature is quite different between traditional torque-generated methods, in which the actuator is at the bus center in general.Figure 13
The thrust magnitudes per unit length of tethers at the beginning of the mission.Figure 14
The voltage distribution of tethers at the beginning of the mission.Even though a solar sail and an electric sail are both capable of producing a propulsive thrust without the need of any propellant, these two propulsion systems are substantially different in terms of performance, shape, and dimensions [18]. Compared to the dynamics and control of solar sail discussed in [6], the difference is mainly reflected in the following aspects. In terms of propulsion mechanism, electric sails are propelled by the solar wind dynamic pressure using charged tethers, instead of the solar photon momentum. In the thrust model of solar sail, the thrust cone angle of ideal solar sail is equal to the pitch angle. Differently, the thrust cone angle of electric sail reaches a maximum value of 19.47°, when pitch angle is equal to 54.75°. This feature causes the fact that there are infeasible regions for the displaced geostationary orbit using electric sail, as shown in Figure 4. About the difference on control, the control force and control torque of electric sail can be realized by adjusting the voltage distribution of charged tethers, as discussed in this section. This feature is quite different from solar sail.


## 5.1. Control Algorithm

The stability of the displaced orbit has been discussed above. The numerical results show that only some of displaced orbits are marginally stable. Therefore, active control is necessary to stabilize the unstable displaced orbits. In this paper, only control torque is employed to stabilize both the attitude and orbit of sailcraft. Then, the coupled system can be linearized and the corresponding linear system is employed for control design. The coupled equations linearized at the equilibrium point with control torque can be written as(15)δ˙=Hcδ+Bu,where δ = [δr,δΘ,δΨ,δvr,δωΘ,δωΨ,δϕ,δθ,δψ,δωx,δωy,δωz]T is state perturbation, Hc is the variational coefficient matrix of coupled system, B=[03×3,03×3,03×3,1/I]T (I is the inertia matrix of sailcraft), and u=T=[Tx,Ty,Tz]T is the vector of control torque.In order to determine the controllability of this system, the rank of the controllability matrix of the linearized system has been calculated. The results show that the system is controllable, because the controllability matrix has full rank. In this paper, linear quadratic regulator (LQR) is employed to design the control torque to track the desired attitude and orbit. The performance function is selected as(16)J=∫0∞δTQδ+uTRudt,where the matrix Q∈R12×12 is symmetric positive definite and represents the weight of the state errors. The matrix R∈R3×3 is also a symmetric positive definite matrix and represents the weight of the control input.The linear state feedback can be obtained by solving the following algebraic Riccati matrix:(17)PHc+HcP-PBR-1BTP+Q=0.Then, the control torque can be written in the state-feedback form as(18)u=-R-1BTPδ.


## 5.2. Numerical Simulation

In order to investigate the stability of the displaced orbit with linear quadratic regulator control, a numerical example is adopted in this section, using the full nonlinear dynamical equations. The selected displaced orbit for simulation is a displaced geostationary orbit (ωd=ω⊕) and is determined by rd=0.9au and Θd=86∘. For keeping the selected displaced geostationary orbit, the required characteristic acceleration a⊕d should be equal to 1.942mm/s2, and attitude angle θd should be equal to -21.749∘. As discussed above, the chosen displaced orbit is unstable without control. Assume that the orbit and the attitude have a small perturbation to the required orbit and attitude at the initial time. The initial values of the perturbation are given by δX1=[4×104m,0,0]T, δX2=[0,0,0]T, δX3=[0,0.2∘,0]T, and δX4=[0,0,0]T. Figures 10 and 11 give the responses of attitude and orbit, respectively. Numerical simulation shows that the LQR control law can control the coupled system. As shown in Figure 12, a small control torque can stabilize both the attitude and orbit.Figure 10
The responses of the orbit errors.Figure 11
The responses of the attitude errors.Figure 12
The control torques.


## 5.3. Preliminary Study of Voltage Distribution

The control torque of electric sail can be realized by adjusting the voltage distribution of charged tethers of electric sail [16, 17]. In this paper, the generation of control torques is preliminarily discussed in an optimal framework. Referring to Eq. (10) and Eq. (13) in [16], the vector of propulsive force F and the vector of control torque T can be written as functions of the vertical thrust magnitude of charged tethers σ⊕1,…,σ⊕N.(19)Fϕ,θ,ψ,r,l,σ⊕1,…,σ⊕N=∑k=1NFkϕ,θ,ψ,r,l,σ⊕kTϕ,θ,ψ,r,l,σ⊕1,…,σ⊕N=∑k=1NTkϕ,θ,ψ,r,l,σ⊕k,where σ⊕k,k∈[1,N], is the kth tether’s magnitude of the force per unit length, when the sun-spacecraft distance is r⊕=1au and the solar wind is perpendicular to tether. As discussed in [17], σ⊕k can be controlled by altering the voltage ofkth charged tether Vk and is given by (20)σ⊕k=0.18max⁡0,Vk-V0ε0pdyn,where V0 is the electric potential corresponding to the kinetic energy of the solar wind ions, ε0 is the vacuum permittivity, and pdyn is the solar wind dynamic pressure.In order to generate the required control forceFc and control torque Tc, the following equality constraints should be satisfied:(21)∑k=1NFkϕ,θ,ψ,r,l,σ⊕k=Fc∑k=1NTkϕ,θ,ψ,r,l,σ⊕k=Tc.As shown in (21), the quantity of equality constraints is 6 in voltage distribution problem. Comparatively, the number of design variables σ⊕1,…,σ⊕N is N∈[20,100] in general, which is decided by the quantity of charged tethers of electric sail. Therefore, the definite solution cannot be obtained by solving (21). In this paper, the voltage distribution problem is converted into a nonlinear programming problem (NLP). Equation (21) is processed into equality constraints in optimal framework. In order to reduce the required power, the optimization performance index is selected as follows:(22)J=∑k=1Nσ⊕k.The above NLP can be solved within the MATLAB environment usingfmincon, which is a constrained optimization routine available with the Optimization Toolbox. The selected optimization algorithm is sequential quadratic programming (SQP). Consider an electric sail, which is composed of N = 100 20 km-long tether. At the beginning of the mission, the required control force can be calculated according to (1), with the required characteristic acceleration a⊕d=1.942mm/s2 and attitude angle θd=-21.949∘. As seen in Figure 12, the required control torque is T=[9.3941×10-5,-6.1×10-3,1.8116×10-6]Nm at the beginning of the mission.This NLP was solved in 2.453 seconds using a personal computer with 8 GB RAM running at 2.20 GHz processor speed. The obtained thrust magnitudes per unit lengthσ⊕k,k∈[1,N], are shown in Figure 13. Considering (20), the required voltage distribution Vk,k∈[1,N], can be obtained and is shown in Figure 14, where Vaver=∑k=1NVk is the average value of Vk,k∈[1,N]. Through the preliminary study of voltage distribution, it can be found that the control force and torque of electric sail can be realized by adjusting the voltage distribution of charged tethers. This feature is quite different between traditional torque-generated methods, in which the actuator is at the bus center in general.Figure 13
The thrust magnitudes per unit length of tethers at the beginning of the mission.Figure 14
The voltage distribution of tethers at the beginning of the mission.Even though a solar sail and an electric sail are both capable of producing a propulsive thrust without the need of any propellant, these two propulsion systems are substantially different in terms of performance, shape, and dimensions [18]. Compared to the dynamics and control of solar sail discussed in [6], the difference is mainly reflected in the following aspects. In terms of propulsion mechanism, electric sails are propelled by the solar wind dynamic pressure using charged tethers, instead of the solar photon momentum. In the thrust model of solar sail, the thrust cone angle of ideal solar sail is equal to the pitch angle. Differently, the thrust cone angle of electric sail reaches a maximum value of 19.47°, when pitch angle is equal to 54.75°. This feature causes the fact that there are infeasible regions for the displaced geostationary orbit using electric sail, as shown in Figure 4. About the difference on control, the control force and control torque of electric sail can be realized by adjusting the voltage distribution of charged tethers, as discussed in this section. This feature is quite different from solar sail.


## 6. Conclusions

In this paper, displaced solar orbits for spacecraft propelled by electric sails are investigated as an alternative to solar sails. Since the propulsive thrust is induced by the sail attitude, the orbital and attitude dynamics of electric-sail-based spacecraft are coupled and investigated together. 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 orbit-attitude system. Numerical simulation shows that the linear quadratic regulator can control the coupled system and a small torque can stabilize both the attitude and orbit. To generate the control force and torque, the voltage distribution problem is converted into a nonlinear programming problem and solved using sequential quadratic programming 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.



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*Source: 3812397-2017-07-31.xml*