Attitude Dynamics and Control for a Solar Sail with Individually Controllable Elements

Abstract

Open AccessEngineering NotesAttitude Dynamics and Control for a Solar Sail with Individually Controllable ElementsTong Luo, Chuang Yao, Ming Xu and Qingyu QuTong LuoBeihang University, 100191 Beijing, People’s Republic of China*Ph.D. Candidate, School of Astronautics; .Search for more papers by this author, Chuang YaoShanghai Institute of Satellite Engineering, 201109 Shanghai, People’s Republic of China†Engineer; .Search for more papers by this author, Ming XuBeihang University, 100191 Beijing, People’s Republic of China‡Associate Professor, School of Astronautics; (Corresponding Author).Search for more papers by this author and Qingyu QuBeihang University, 100191 Beijing, People’s Republic of China§Master’s Student, School of Astronautics; .Search for more papers by this authorPublished Online:11 Feb 2019https://doi.org/10.2514/1.G003957SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutI. IntroductionSince 2010, several solar-sail spacecraft missions have been completed, such as the Interplanetary Kite-Craft Accelerated by Radiation of the Sun (IKAROS) [1] and NanoSail-D2 [2]. Also, many future missions for solar-sail spacecraft [3–7] have been proposed. The trajectory of a solar-sail spacecraft is generally controlled by adjusting the attitude of the solar sail, and this can be accomplished in a variety of ways. The most common method is using control vanes [8–10]. In addition, a sliding mass mechanism is used for attitude control of flexible solar sails [11,12]. The solar-sail spacecraft, IKAROS, achieved attitude control by changing the optical reflectivity of the sail membrane [13], leading to further study of this control method [14,15].In the aforementioned work, the position of the solar sails relative to the spacecraft is fixed. However, the solar sail in a heliogyro [16] can be adjusted by pitching the blades. Heiligers et al. [16] exploited this characteristic and proposed a linear quadratic regulator (LQR) feedback controller for orbital control. A new design for a solar sail with individually controllable elements (SSICE), using rotatable blades on a square solar sail, was proposed by Luo et al. [17]. The initial folded structure of the SSICE before deployment is shown in Fig. 1a. The expanded square configuration of the SSICE after deployment is shown in Fig. 1b. The SSICE consists of four different components: component A is the blade used for reflecting the solar radiation pressure (SRP); component B is the basic cell, including a motor that controls the attitude of the connected blade; component C is the inflatable frame used for the expansion process; and component D is the cube central body where the main effective loads (such as space-based equipment and gas cylinder) are placed. Compared to conventional solar sail, the SSICE has three advantages. First, two basic cells and the connected blade between them compose an individually controllable element. This decentralized layout allows for adjustment of the SSICE size. Second, motors control the linear torsional deformation of the blades (Fig. 1c) to provide a resultant torque, which then determines the attitude of the SSICE. Thus, attitude control of the SSICE is achieved by rotating the motors, which is much simpler than a conventional reaction control system and the complex reflectance control device on IKAROS. Third, a pneumatic expansion mechanism is introduced to deploy the solar sail for the spacecraft. Specifically, when the compressed gas stored in a cylinder is released, the inflatable frame is filled with the gas, expands within the structure of four rigid constraints, and unfolds into a square shape.Fig. 1 a) Initial structure before deploying, b) expanded structure after deploying, and c) torsional deformation of a blade.This Note will develop the orbit and attitude dynamic models for the SSICE, and propose a feasible orbit/attitude-tracking controller. The Note is organized as follows: Sec. II reviews the configuration design and establishes the models of SRP and SRP torque exerted on the SSICE. Section III provides a nominal trajectory around an artificial equilibrium point in the sun–Earth system and designs an orbit-tracking controller using the LQR method. Section IV determines a group of feasible nominal Euler angles and designs an attitude-tracking controller. Finally, Sec. V gives simulation results for the constructed controller.II. SSICE ModelA. Configuration ParametersA body coordinate system Sb is defined with the origin located at the center of the SSICE; xb axis along the normal direction of the frame plane, pointing to the front; yb axis in the frame plane, perpendicular to the blades; and zb axis determined by the right-hand rule. The elements of the inertia matrix of SSICE in Sb [i.e., J=diag(Jx,Jy,Jz)] are given by Jx=m1s26+∑k=1n/2(m2+m32)[s+(2k−1)w]2+nm2l22+nm2a23+nm312(l2+w2)+11m460l2Jy=m1s26+nm2l22+nm2a26+nm3l212+m410l2Jz=m1s26+∑k=1n/2(m2+m32)[s+(2k−1)w]2+nm2a26+nm3w212+m412l2(1)in which m1 is the mass of the cube central body, m2 is the mass of each cell, m3 is the mass of each blade, and m4 is the mass of the frame. Here, l and w are the length and width of each blade, respectively; a is the side length of each cell; s is the side length of the cube central body; and n is the total number of blades. The specific values of the SSICE parameters are given in Table 1.B. SRP and SRP TorqueThe anticlockwise rotation around the zb axis is defined as the positive rotation, and the rotation angles of two motors are defined as zero when the blade is parallel with the frame plane. In addition, the rotation angles are assumed to linearly vary along a blade. Thus, the rotation angle of the ith blade at zb is expressed as δi(zb)=δ2i−1+δ2i2+δ2i−1−δ2ilzb,(−l2≤zb≤l2)(2)in which δ2i−1 and δ2i are the rotation angles of two motors. The normal vector of the ith blade at zb is {n}b=[cosδi(zb),sinδi(zb),0]T.An orbit coordinate system So is defined with the origin at the center of the SSICE; xo axis pointing from the sun to the SSICE; yo axis in the ecliptic plane, perpendicular to the xo axis; and zo axis determined by the right-hand rule. The Sb is obtained with three basic coordinate rotations from the So, namely, So→Lx(φ)∘→Ly(θ)∘→Lz(ψ)Sb, in which the three rotation angles are defined as Euler angles. Thus, the SRP vector r1 in the Sb is calculated as {r1}b=Lbo[100]T=[cosθcosψ−sinγsinψsinθ]T(3)in which Lbo is the coordinate transformation matrix from the So to the Sb. The angle between {n}b and {r1}b is calculated as ηi=arccos[cosθ⋅cos(δi(zb)+ψ)]. According to Ref. [18], the SRP dFi=[dFix,dFiy,dFiz]T on the ith blade at zb is calculated as dFi=2Pwcos2θcos2(δi(zb)+ψ)[cosδi(zb)sinδi(zb)0]Tdzb(4)in which P is the SRP. Note that, because the normal vectors of the blades must be on the plane vertical to the zb axis, dFzi is always zero. The SRP torques generated by dFi are dMi=[dMxidMyidMzi]T=[−dFyi⋅zbdFxi⋅zb−dFxi⋅yi]T(5)in which yi is the coordinate value of the blade midline at the y axis.By integrating Eqs. (4) and (5), the SRP on the entire blade and the resulting SRP torques are calculated. Then, the SRP and SRP torque on the SSICE are obtained by summation: {F}b=[∑i=14∫−(l/2)l/2 dFxi∑i=14∫−(l/2)l/2dFyi0]T,{M}b=[∑i=14∫−(l/2)l/2 dMxi∑i=14∫−(l/2)l/2 dMyi∑i=14∫−(l/2)l/2 dMzi]T(6)and can be taken as functions of the rotation angles of eight motors. At a given moment, if the required SRP and SRP torque are known and the rotation angles of eight motors are regarded as eight unknown variables, then Eq. (6) is an indeterminate equation group, with infinite solutions. The required values for SRP and SRP torque will be provided by controllers described in Secs. III and IV. In this section, a group of feasible solutions for rotation angles are generated on the condition that the SRP and SRP torque are known.If the rotation angles of eight motors at initial moment ta are δa1, δa2, δa3, δa4, δa5, δa6, δa7, and δa8, we determine the values of rotation angles δb1, δb2, δb3, δb4, δb5, δb6, δb7, and δb8 at the next moment tb with the finite difference method: [δb1−δa1⋯δb8−δa8]T=A+[Fbx−FaxFby−FayMbx−MaxMby−MayMbz−Maz]T(7)in which Fax, Fay, Max, May, and Maz denote the required SRP and SRP torque at moment ta; Fbx, Fby, Mbx, Mby, and Mbz denote the required SRP and SRP torque at moment tb; and A+ denotes the Moore–Penrose generalized inverse of a Jacobian matrix A=[∂Fx/∂δ1⋯∂Fx/∂δ8⋮⋮∂Mz/∂δ1⋯∂Mz/∂δ8]. Note that the continuity of the rotation angles is guaranteed by the finite difference methods, and the uniqueness of the rotation angles at moment tb is guaranteed by the unique form of A+. Through numerical calculation, we determine the solutions of δb1…δb8, which are related to the rotation angles at moment ta, and the SRP and SRP torque at moments ta and tb. Next, we calculate the values of the rotation angles in the entire time sequence through recursion, using the aforementioned method. To minimize the change of rotation angles, an optimization index is chosen as J=−(∑j=1n∑k=18‖σk(AHA)‖), in which σk(AHA) denotes the kth eigenvalue of AHA, and n denotes adjustment times for the rotation angles of the motors. The optimal initial rotation angles are obtained by solving the nonlinear constrained optimization problem: min Jsubject to−90 deg≤δ1,δ2,δ3,δ4,δ5,δ6,δ7,δ8≤90 deg}(8)III. Orbital Dynamics and Control Around Artificial Equilibrium PointsA. Nominal Trajectory DesignSuppose that Earth and the sun are two point masses moving in circular orbits around their common center of mass. An inertial coordinate system Si (O-Xi-Yi-Zi) and a rotating coordinate system Sr (O-X-Y-Z) are shown in Fig. 2a. The units of mass, length, and time are normalized, so that the total mass of the system, the Earth–sun distance, and the period of the motion of the primaries are 1, 1 AU, and 2π, respectively. After normalization, the gravitational constant G is 1, and the mass ratio is defined as μ=msun/(mEarth+msun), in which msun and mEarth are the actual masses of the sun and Earth, respectively. The motion of the SSICE in the Sr is described by R¨−2JR˙−∂V∂R=0,J=[010−100000](9)in which R=[X,Y,Z]T is the position vector, and V is the potential function. Including the SRP, V is expressed as V(X,Y,Z)=U(X,Y,Z)+a⋅R,U(X,Y,Z)=12(X2+Y2)+(1−μ)r1+μr2(10)in which r1 and r2 are the distances from the SSICE to the sun and Earth, respectively, and a is the radiation pressure acceleration. Because of the SRP, infinitely, many artificial equilibrium points can be generated [18].Fig. 2 a) Diagrams of Si and Sr, and b) diagrams of ξ1, ξ2, α, and γ.For saddle×center×center type artificial equilibrium points, their unstable manifolds will lead to instability in the sense of Lyapunov. Consequently, additional maneuvers are required to obtain stable bounded trajectories around these artificial equilibrium points. Scheeres et al. [19] proposed a Hamiltonian structure-preserving (HSP) controller, which can use the local invariant manifolds of a hyperbolic system, to transform a hyperbolic equilibrium point into an elliptic equilibrium point. The stabilization of the controlled system has been theoretically proven in Refs. [20,21]. Here, we adopt a three-dimensional HSP controller from Ref. [21]: Tc=T⋅δR,T=−σ2G1(u+u+T+u−u−T)−γ2G2(uuH+u¯u¯H)−δ2G3(vvH+v¯v¯H)(11)in which δR=[δX,δY,δZ]T denotes the position error vector; G1, G2, and G3 are control gains; u+ and u− represent the local stable and unstable manifolds; and u, u¯, v, and v¯ represent the local center manifolds. After adding the constructed HSP controller, a nominal trajectory Rn is solved by integrating the following equation: R¨n−2JR˙n−dV|R=Rn=Tc(12)B. Design of an Orbit-Tracking ControllerThe controlled motion of the SSICE near the calculated nominal trajectory Rn is described by (R¨n+δr¨)−2J(R˙n+δr˙)−dV|R=Rn+δr=F(δr,δr˙,t)(13)in which δr=[δx,δy,δz]T denotes the position error vector between the actual and nominal trajectories, which is different from δR in Eq. (11), and F is the orbit-tracking controller. By substituting Eq. (12) into Eq. (13) and ignoring the higher-order terms, we obtain the variation equation near the calculated nominal trajectory Rn as δr˙−2Jδr˙−VRR|R=Rnδr=F(δr,δr˙,t)−Tc(14)A quadratic cost function is defined as J=12∫0∞[δrT(t)Q1(t)δr(t)+δr˙T(t)Q2(t)δr˙(t)+(F−Tc)TR1(t)(F−Tc)] dt(15)in which Q1(t) and Q2(t) are nonnegative weight matrices, and R1(t) is a positive weight matrix. The orbit-tracking controller F can be solved using the LQR method.IV. Attitude Dynamics and ControlA. Attitude Kinematics and DynamicsThe angular velocity of the Sr relative to the Si is expressed as {ωri}r=[0,0,Ω]T, in which Ω denotes Earth’s angular velocity around the sun. The So is obtained by two basic coordinate rotations from the Sr, namely, Sr→Lz(ξ1)∘→Ly(−ξ2)So. As shown in Fig. 2b, two rotation angles, ξ1 and ξ2, are determined by the position of the SSICE in the Sr. Then, the angular velocity of the So relative to the Sr is expressed as {ωor}o=Ly(−ξ2)[00ξ˙]T+[0−ξ˙20]T=[ξ˙1sinξ2−ξ˙2ξ˙1cosξ2]T(16)Based on the relationship between the So and the Sb provided in Sec. II, the angular velocity of the Sb relative to the So is expressed as {ωbo}b=Lz(ψ)Ly(θ)Lx(φ)[φ˙00]+Lz(ψ)Ly(θ)[0θ˙0]+Lz(ψ)[00ψ˙]=[θ˙sinψ+φ˙cosθcosψθ˙cosψ−φ˙cosθsinψψ˙+φ˙sinθ](17)in which φ˙, θ˙, and ψ˙ are the change rates of Euler angles. Thus, the absolute angular velocity of the SSICE is {ω}b=LboLor{ωri}r+Lbo{ωor}o+{ωbo}b(18)According to the theorem of angular momentum, the attitude dynamics of the SSICE described in the Sb is {J}b{ω˙}b+[{ω}b]×{J}b{ω}b={M}b(19)in which {J}b is the inertia matrix of the SSICE, {ω}b is the absolute angular velocity of the SSICE, and {M}b is the external torque. Note that {J}b, {ω}b, and {M}b were described in Eqs. (1), (18), and (6), respectively.B. Nominal Attitude DesignAssuming that the rotation angles of all motors are always zero, the SSICE is considered to be a general solar sail, with only one sail. As shown in Fig. 2b, the normal vector n of the simplified SSICE is determined by the sail’s two attitude angles, α and γ, and its expression in the So is {n}o=[cosαsinαcosγsinαsinγ]T(20)Based on the geometric relationship, the angle between the vectors r1 and n is exactly α. According to Ref. [18], the expression for the acceleration {a}r is written as a function of α, γ, and lightness number β: {a}r(α,γ,β)=β(1−μ)cos2αr12Lro{n}o(21)Then, the orbit-tracking controller F is approximated with a Taylor expansion near [α0,γ0,β0]T as F=a(α,γ,β)+∇U|R=R0=a(α,γ,β)−a(α0,γ0,β0)=∂a(α,γ,β)∂Θ|Θ0δΘ+o(δΘ2)(22)in which Θ=[α,γ,β]T is the control parameter vector, and Θ0=[α0,γ0,β0]T is the reference value that can stabilize the SSICE at artificial equilibrium point R0. Thus, the control parameter vector Θ is approximated as Θ=Θ0+∂a(α,γ,β)∂Θ|Θ0−1F(23)Note that the normal vector n of the simplified SSICE can also be expressed by three Euler angles as {n}o=Lob[100]=[cosθcosψsinφsinθcosψ+cosφsinψsinφsinψ−cosφsinθcosψ](24)in which Lob is the coordinate transformation matrix from the Sb to the So, and [1,0,0]T is the expression of n in the Sb. By comparing Eqs. (20) and (24), we obtain a group of feasible nominal Euler angles as φn=γ, θn=0, and ψn=α. In addition, the actual SSICE can adjust its lightness number by changing the attitude of the sail. Specifically, the attitude of the decentralized blades can be individually adjusted by rotating the connected motors, which can provide the required control force without changing the force direction. In terms of effect, we consider that the lightness number of the actual SSICE is changed.C. Design of an Attitude-Tracking ControllerThe errors δζ between the actual Euler angles ζ=[φ,θ,ψ]T and the nominal Euler angles ζn=[φn,θn,ψn]T are considered to be small variables. By linearizing the attitude dynamic equations near the nominal Euler angles ζn, we describe the motion of δζ with the following variation equation: [JxcosψnJxsinψn0−JysinψnJycosψn000Jz]δζ¨+[b11b12b13b21b22b23b31b32b33]δζ˙+[c11c12c13c21c22c23c31c32c33]δζ+D=M(25)in which M is the attitude-tracking controller, and other parameters are given as b11=Ω[Jxcosξ2cosφnsinψn+(Jz−Jy)(cosξ2sinφncos2ψn−sinξ2sin2ψn)],b12=Ω[−Jxcosξ2cosφncosψn+(Jz−Jy)(cosξ2sinφnsin2ψn+sinξ2cos2ψn)],b13=Ω⋅Jx[cosξ2(sinφncosψn+cosφncosψn)−sinξ2sinψn],b21=Ω[Jycosξ2cosφncosψn−(Jx−Jz)cosξ2cosφnsinψn],b22=Ω[Jycosξ2cosφnsinψn+(Jx−Jz)cosξ2cosφncosψn],b23=Ω[Jy(cosξ2cosφncosψn−cosξ2sinφnsinψn−sinξ2cosψn)−(Jx−Jz)(cosξ2sinφncosψn−sinξ2sinψn)],b31=−Ω(Jy−Jx)cosξ2cosφncosψn,b32=Ω[Jzsinξ2+(Jy−Jx)cosξ2cosφnsinψn],b33=Ω(Jy−Jx)(cosξ2sinφnsinψn−sinξ2cosψn)(26)c11=(Jz−Jy)Ω2(cos2ξ2sinφncosφnsin2ψn−cos2ξ2cos2φncos2ψn+sinξ2cosξ2cosφncos2ψn),c12=(Jz−Jy)Ω2(cos2ξ2sinφncosφnsin2ψn−sin2ξ2cosφnsin2ψn),c13=(Jz−Jy)Ω2(cos2ξ2sin2φncos2ψn−cos2ξ2sinφncosφnsin2ψn−sin2ξ2sinφnsin2ψn−sin2ξ2cos2ψn),c21=(Jx−Jz)Ω2cos2ξ2cos2φncosψn,c22=(Jx−Jz)Ω2(sinξ2cosξ2sinφncosψn+cos2ξ2cos2φnsinψn−sin2ξ2sinψn),c23=(Jx−Jz)Ω2(−cos2ξ2sinφncosφnsinψn−sinξ2cosξ2cos2ψn),c31=(Jy−Jx)Ω2(cos2ξ2cos2φnsinψn+sinξ2cosξ2sinφncosψn),c32=(Jy−Jx)Ω2(sin2ξ2cosψn−cos2ξ2cos2φncosψn+sinξ2cosξ2sinφnsinψn),c33=(Jy−Jx)Ω2(cos2ξ2sinφncosφncosψn+sinξ2cosξ2cosφnsinψn)(27)D=[d1d2d3]T,d1=(Jz−Jy)[Ω(cosξ2sinφncos2ψn−sinξ2sin2ψn)φ˙n+Ω22(cos2ξ2sin2φnsin2ψn−cos2ξ2sinφncos2ψn+sin2ξ2sinφncos2ψn−sin2ξ2sin2ψn)]+Jx(φ¨ncosφn+Ωφ˙ncosξ2cosφnsinψn),d2=(Jx−Jz)[Ω(−cosξ2cosφnsinψnφ˙n+cosξ2sinφncosψnψ˙n−sinξ2sinψnψ˙n)+Ω2(cos2ξ2sinφncosφncosψn−sinξ2cosξ2sinψncosψn)]+Jy[Ωψ˙n(−cosξ2sinφnsinψn+cosξ2cosφncosψn−sinξ2cosψn)−φ¨nsinφn+Ωφ˙ncosξ2cosφncosψn],d3=(Jy−Jx)[Ω(cosξ2cosφncosψnφ˙n+cosξ2sinφnsinψnψ˙n−sinξ2cosψnψ˙n)+Ω2(cos2ξ2sinφncosφnsinψn−sinξ2cosξ2cosψncosψn)]+Jzψ¨n(28)Similarly, a quadratic cost function is defined as J=12∫0∞[δξT(t)Q3(t)δξ(t)+δξ˙T(t)Q4(t)δξ˙(t)+(M−D)TR2(t)(M−D)] dt(29)in which Q3(t) and Q4(t) are nonnegative weight matrices, and R2(t) is a positive weight matrix. The attitude-tracking controller M can also be solved using the LQR method.Now that the orbit- and attitude-tracking controllers are both obtained, a group of feasible solutions for the rotation angles of eight motors can be generated using the method introduced in Sec. II.V. Simulation ResultsThe position of the equilibrium point is chosen as R0=[0.985,0.01,0.05]T AU, and the initial states of the SSICE are set as δR0=[10−3,10−3,10−3]T AU and δR˙0=[10−4,1.25×10−4,2.5×10−4]T. When an HSP controller with control gains G1=20, G2=10, and G3=10 is applied to the SSICE, a bounded nominal trajectory Rn is obtained, as shown in Fig. 3. Assuming that the initial errors between the actual states and the nominal states are δr0=[2×10−7,−1.5×10−7,−1×10−7]T AU and δr˙0=[−1×10−8,2×10−8,5×10−8]T, we set three weight matrices in Eq. (15) as Q1(t)=diag(1,1,1), Q2(t)=diag(1,1,1), and R1(t)=diag(1,1,1), and adopt the LQR method to design an orbit-tracking controller F. The changing value of F is shown in Fig. 4. As illustrated in Fig. 5, the distance error between the tracking trajectory and the nominal trajectory decreases to zero after 2 years.Fig. 3 Bounded nominal trajectory.Fig. 4 Change rule of orbit-tracking controller.Fig. 5 Distance error between the actual and nominal trajectories.With the orbit-tracking controller F obtained, we generate a group of feasible nominal Euler angles and the required lightness number, based on the method introduced in Sec. IV. Nominal Euler angle θn is always zero, and the change rules of nominal Euler angles φn and ψn are shown in Fig. 6a. The time history of the lightness number is shown in Fig. 6b. Note that, because the attitude of the SSICE’s frame changes with the nominal Euler angles, the change rule of the lightness number will not coincide with that of the orbit-tracking controller. Assuming that the initial errors between the actual Euler angles and the nominal Euler angles are δζ0=[3,2,4]T deg and δζ˙0=[0.01,0.02,−0.01]T deg/s, we set three weight matrices in Eq. (29) as Q3(t)=diag(0.001,0.001,0.001), Q4(t)=diag(0,0,0), and R2(t)=diag(100,100,100), and adopt the LQR method to design the attitude control torques M. As shown in Fig. 7, three components of M all approximately converge to zero after 2000 s. Figure 8a presents the change rules of three Euler angular velocities, and Fig. 8b presents the change rules of the errors between the actual and nominal Euler angles. Results demonstrate that the actual Euler angles can converge to the nominal Euler angles after 2000 s. Figure 9 gives a group of solutions for the rotation angles of the motors, which verifies the feasibility of using motors to achieve orbit and attitude control of the SSICE.Fig. 6 Change rules of three parameters: a) nominal Euler angle φn and ψn, and b) lightness number β.Fig. 7 Time history of the required attitude control torques.Fig. 8 a) Euler angular velocity, and b) errors between the actual and nominal Euler angles.Fig. 9 Group of feasible solutions of the rotation angles of the motors.Finally, we examine the effect of the gravitational gradient moments of the sun and Earth as disturbance torques on the attitude control. According to Ref. [22], the gravitational gradient moment of the sun is approximated as Msun=3μsunr13[(Jz−Jy)cosα2cosα3(Jx−Jz)cosα3cosα1(Jy−Jx)cosα1cosα2]T(30)in which μsun is the gravitational constant of the sun; r1 is the distance between the sun and the SSICE; α1, α2, and α3 are three angles between the sun–SSICE connected line and xb axis, yb axis, and zb axis; and Jx, Jy, Jz are three moments of inertia for the SSICE in the Sb. Then, the elements in Msun are estimated as Msun≤3μsunr13[Jz−JyJx−JzJy−Jx]T=[3.184×10−121.113×10−11−7.940×10−12]T N⋅m(31)In the same way, the gravitational gradient moment of Earth is estimated as MEarth≤3μEarthr23[Jz−JyJx−JzJy−Jx]T=[2.708×10−129.467×10−12−6.753×10−12]T N⋅m(32)Compared to 10−4 N⋅m SRP torque, these gravitational gradient moments can be ignored.VI. ConclusionsA novel type of solar sail, the solar sail with individually controllable elements (SSICE), allows for convenient adjustment of the magnitude and direction of solar radiation pressure through rotatable blades. This study demonstrated that rotating the motors connected to the blades is an effective method of orbit and attitude control for the SSICE. To track a given nominal trajectory and the nominal Euler angles, it is possible to design orbit- and attitude-tracking controllers with traditional control methods, such as the linear quadratic regulator method used in this work. Once the SRP and torque required by the orbit- and attitude-tracking controllers are determined, a group of feasible solutions can be generated for the rotation angles of the eight motors, using an optimization method. According to the simulation results, it is possible for an SSICE to orbit on a calculated trajectory around the chosen artificial equilibrium point, in the sun–Earth system. Thus, the SSICE has the potential to be applied to halo orbit missions around the sun–Earth Lagrangian points, as well as other interplanetary missions.AcknowledgmentsThis work was supported by the National Natural Science Foundation of China (11432001 and 11772024), the Academic Excellence Foundation of Beihang University for Ph.D. Students, and the Innovation Practice Foundation of Beihang University for Postgraduate Students (YCSJ-01-2018-08).