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Moon system
Accepted Manuscript Formation Flying on Quasi-halo Orbits in Restricted Sun-Earth/Moon System
Ming Xu, Yuying Liang, Xiaoyu Fu
PII: DOI: Reference:
S1270-9638(16)30383-2 http://dx.doi.org/10.1016/j.ast.2017.03.038 AESCTE 3977
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
3 August 2016 8 December 2016 30 March 2017
Please cite this article in press as: M. Xu et al., Formation Flying on Quasi-halo Orbits in Restricted Sun-Earth/Moon System, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.03.038
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Formation Flying on Quasi-halo Orbits in Restricted SunEarth/Moon System Ming Xu1, Yuying Liang2, and Xiaoyu Fu3 Beihang University, Beijing, 100191, People’s Republic of China
Due to the strong nonlinear perturbations near the libration points, the continuous low-thrust technique has many potential applications in stationkeeping relative motions. The Hamiltonian structure-preserving (HSP) control is employed in this paper to stabilize formation flying on quasi-periodic orbits near LL1 of the restricted Sun-Earth-Moon-spacecraft system. In the bi-circular model (BCM), a multiple shooting corrector is developed to refine quasi-periodic orbits as the chief spacecraft’s reference trajectories. The linearized variation equation in BCM is used to design the stationkeeping control. A HSP controller is constructed to change the topology of the equilibrium from hyperbolic to elliptic using only relative position feedbacks consisting of stable, unstable and center manifolds. The critical control gains for transient and long-term stabilities are presented to guide the selection of control gains.
Hamiltonian structure-preserving control; formation flying; bi-circular model; quasi-halo orbits; libration point
I. Introduction Spacecraft formation flying in the context of two-body and multiple-body problems has received much attention recently. Adopted in NASA’s future missions, e.g., Terrestrial Planet Finder (TPF) and Micro-Arcsecond X-ray Imaginary Mission (MAXIM), formation flying in the vicinity of libration points is considered as one of important technologies in future deep-space exploration. Recent literature on relative dynamics and control becomes extensive. Perea et al. [1] developed a spacecraft control strategy, which was applied to the transfer orbit of a set of loose-formation spacecraft following a natural trajectory to a libration point. Guibout and Scheeres [2] studied a reconfiguration method of spacecraft formation about a 1
Associate Professor, School of Astronautics, [email protected]. PhD Candidate, School of Astronautics, [email protected]. 3 Master Candidate, School of Astronautics, [email protected]. 2
1
Sun-Earth libration point and demonstrated that optimal transfer time of the reconfiguration depends largely on initial and final positions. Howell et al. [3] investigated a non-natural formation-keeping problem near libration points of SunEarth or Earth-Moon systems based on continuous control. Gurfil et al. [4] put forward a nonlinear adaptive neural control method for deep-space spacecraft formation flying, which is applicable to formation flying near libration points of three body problem. Marchand and Howell [5] set fundamental propulsive requirements near libration points of circular restricted three body problem (abbr. CR3BP) and introduced related constrictions on formation control strategy. Qi et al. [6] studied an impulsive control technique to station-keep formation flying in CR3BP and proposed a formation-keeping strategy based on tangent targeting method to deal with uncertainty issue. A probabilistic evaluation method was introduced by Miller and Campbell [7] to validate a simplified model of formation dynamical system in the vicinity of libration point L2 of Sun-Earth or Earth-Moon systems. Heritier and Howell [8] investigated the dynamics of natural formation and eigenstructure of a reference trajectory in a multiple-body regime for formation flying. Catlin and McLaughlin [9] demonstrated the existence and nature of trajectories near Earth-Moon triangular libration points in CR3BP and designed a chief/deputy formation for long-period cases. An optimal periodic controller based on continuous low-thrust technique was developed by Peng et al. [10] for spacecraft station-maintaining and formationkeeping near periodic libration-point orbits of Sun-Earth system. Hughes and Steven [11] introduced a general approach for formation flying, which is adaptive to orbital perturbations and applicable to multiple fight regimes including libration point orbits of CR3BP. Finite elements for formation flying computation method was introduced by Garcia et al. [12] in a libration point regime of Sun-Earth system. A tangent targeting method was exploited by Qi et al. [13] to control the leader/follower architecture for maintaining a prescribed orientation near libration points of CR3BP. Pernicka et al. [14] investigated a formation-maintaining problem during both passive and active modes of operations near a libration point of CR3BP. Shahid and Kumar [15] looked into the same problem and designed a stable controller based on Lyapunov theory. To achieve high-precision formation maneuver and station keeping in CR3BP, Li et al. [16] employed a nonlinear adaptive control scheme based on nonzero set-point linear quadratic regular method and neural network. Darvish et al. [17] utilized a nonlinear integral sliding mode to investigate precision formation control of spacecraft on halo orbit around libration point L2 of Sun-Earth system. By linearized motion equations relative to periodic orbits of L4 or L5 of CR3BP, Salazar et al. [18] studied regions of zero, minimum and maximum relative radial acceleration. Inampudi et al. [19] studied a two-spacecraft tether formation associated by line-of sight elastic forces moving in CR3BP and demonstrated that equilibria deflections in the formation are trivial. Huang et al. [20] constructed
2
a reconfiguration model for a two-satellite Coulomb tether formation near Earth-Moon libration points and proved the model’s practicability. Most of the aforementioned works focused on the formation flying in the vicinity of libration points in Sun-Earth or Earth-Moon systems, and periodic orbits of CR3BP (e.g., halo orbits) were used as reference orbits. However, considering the strict condition of halo orbits, there exist no periodic orbits in the restricted four-body problem. Theoretically, quasi-halo orbits can be computed in Sun-Earth-Moon system, while algorithms correlated with the computation of quasi-halo orbits is very complicated in formation flying in the context of restricted four-body problem. Different from the aforementioned literature, this paper studies a station-keeping control method for formation flying on quasi-halo orbits in the restricted Sun-Earth-Moon-spacecraft system. Firstly, a simplified model of restricted four-body problem, i.e., bi-circular model (BCM), is adopted in the Earth-Moon rotating coordinate frame and a useful multiple shooting corrector is developed to improve the accuracy of quasi-halo orbits which acts as the reference orbits of the chief spacecraft. Linearized relative dynamics in BCM is derived to stabilize relative motions near quasi-halo orbits around libration point LL1. Subsequently, a Hamiltonian structure-preserving (HSP) controller is constructed to change topologies of equilibria from hyperbolic to elliptic using the stable, unstable and center manifolds. Furthermore, the critical control gains of the transient stability and the long-term stability are presented on G2-G3 plane with the tint of colour indicating the corresponding value of G1. The stability condition of the controlled system is then numerically demonstrated. Due to the effects of the strong nonlinear perturbations near libration points, continuous low-thrust technique is used to achieve the designed HSP control in formation-maintaining and station-keeping missions.
II. Relative Dynamics on Quasi-halo Orbits in Restricted Sun-Earth-Moon-Spacecraft System A. Bi-Circular Model of Restricted Sun-Earth-Moon-Spacecraft System In a bi-circular model (BCM), the Earth and the Moon are supposed to move in a circular orbit about their common barycenter with a mutual separation d1. Considering that the mass of Earth-Moon System is concentrated at its barycenter, we assume that the Sun and Earth-Moon system are also in circular motion about their barycenter, with a mutual separation d2>d1. Based on the aforementioned assumptions, a BCM of the restricted Sun-Earth-Moon-spacecraft system has the following characteristics: i) the Earth and Moon are regarded as independent gravitational points, moving in their Keplerian circular orbits about their own barycenter; ii) the barycenter of the Earth-Moon system moves in a circumsolar
3
orbit, the eccentricity of which is also ignored; iii) the inclination of the lunar-orbit plane to ecliptic plane is also ignored. Thus, the orbits of the four bodies (Sun, Earth, Moon and spacecraft) are restricted in a same plane. To avoid heavy computation, the normalization of this system is defined as follows: the unit of length is chosen to be one Earth-Moon distance; the unit of mass is chosen to be the total mass of Earth-Moon System; the unit of time is chosen such that the orbital period of the Earth and the Moon around their barycenter is 2ʌ. Subsequently, the universal constant of gravitation is equal to one. A rotating coordinate frame [21] of the Earth-Moon system is defined as follows: the origin is located at barycenter of the Earth and the Moon; the x-axis is along the line which points from the Earth to the Moon; the y-axis is chosen to make the system orthogonal and positive oriented; the z-axis is given by the right-hand rule. In this frame, positions of the Earth and the Moon are fixed on the x-axis respectively. Position and velocity vectors of spacecraft are defined as r=[x y z]T and r = [x y z ]T respectively. As shown in Fig.1, the Sun is rotating clockwise in a circular orbit of a normalized constant radius aS = 388.81114 at an angular velocity of ωs = 0.925195985520347. The Sun’s phase θS is time-varying.
Fig. 1 BCM in the Earth-Moon rotating coordinate frame. After normalization, equations of motion in BCM of the restricted Sun-Earth-Moon-spacecraft system is rሷ +2Jrሶ =
U r
(1)
where J is symplectic operator and U = U(r) can be written as: 1
ȝ
ȝ
ȝ
ȝ
ȝS
S
as
U= ሺx2 +y2 ሻ+ ԡr Eԡ + ԡr Mԡ + ԡr Sԡ - ԡa Sԡ3 aS T ·r+ 2
E
M
4
S
where the Sun’s position vector is aS = aS ⋅[cos(θS), sin(θS), 0]T. Since that the dynamical equations are established in the Earth-Moon rotating frame, the Sun’s convected velocity must be take into consideration, which is indicated by the term ȝS
a ԡaS ԡ3 S
T
·r. The term
ȝS is a constant, which guarantees that the value of U at LL1 is zero. as
B. A Quasi-halo Orbit by Multiple Shooting Method Due to the eccentricity of Moon’s orbit and the perturbations caused by other celestial bodies, such as the Sun and Jupiter, the CR3BP is not accurate enough in many cases and even libration points cannot stay stationary. In order to construct natural orbits around libration points, a multiple shooting method is applied in full ephemeris model, which is widely used in the computation of periodic orbits with long periods, such as solar sail’s displaced orbits [22] and J2 invariant relative orbits [23]. The multiple shooting method employed in this paper is efficient in refining long-period orbits. A full time span of iteration procedure [t0, tf ] is divided into N sub-intervals by N-1 equal-spaced time t1, t2, ..., tN-1. Each equal-length subinterval is defined as Δt=ti+1−ti (i =0,1,2,..., N-1) and a point on quasi-halo trajectory can be expressed as Qi = [ti, xi, yi, zi, xi , y i , zi ]T (i =0,1,2,..., N-1). (Qi) is the image of Qi under the flow related to ephemeris dynamics after time step Δt, i.e., (Qi)=Qi+1. Without perturbation terms, e.g., solar perturbation in BCM, this equation holds for each Qi. Hence, our goal is to minimize the difference between (Qi) and Qi+1, i.e., to solve the following nonlinear problems,
§ Q1 · § φ (Q1 ) · § Q 2 · ¨ ¸ ¨ ¸ ¨ ¸ ¨ Q 2 ¸ ¨ φ (Q 2 ) ¸ ¨ Q 3 ¸ = F¨ ¸−¨ ¸ = 0 ¸ ¨ ¨ ¸ ¨ ¸ ¨ ¸ ¨ Q ¸ ¨ φ (Q ) ¸ ¨ Q ¸ N −1 ¹ © N ¹ © N¹ ©
(2)
T
Defining Q(j) =(Qሺjሻ , Qሺjሻ ,֥,Qሺjሻ ) , Newton’s iteration method is adopted to solve Eq. (2). The jth iteration of this 1 2 N procedure has the following expression:
∂F ∂Q
[
]
⋅ Q ( j +1) − Q ( j ) = − F Q( j )
and
5
Q( j )
(3)
§ ∂ij ¨ ¨ ∂Q Q1( j ) ¨ ∂F ¨ =¨ ∂Q ¨ ¨ ¨ ¨¨ © where
ij
ቚ
Q Q(j) i
−I ∂ij ∂Q Q ( j )
−I
2
∂ij ∂Q Q ( j )
N −1
· ¸ ¸ ¸ ¸. ¸ ¸ ¸ ¸ −I¸ ¸ ¹
(4)
(i=1,2,…,N-1) is a 6×6 state transition matrix. After each iteration, there are 6×(N-1) equations and 6×N
unknowns, indicating that other conditions are required to solve these unknowns. Thus, we take the state at t0 or tN into consideration. Define ǻQ(j)= Q(j+1)- Q(j). In order to minimize ||ǻQ(j)||2, a Lagrange function is set as Lሺ¨Q,IJሻ=¨QT ·¨Q+IJ(FሺQሻ+DF(Q)·¨Q)
(5)
Hence, in the jth iteration, T -1
T
¨Qሺjሻ =-DF(Qሺjሻ ) ·[DF(Qሺjሻ )·DF(Qሺjሻ ) ] ·F(Qሺjሻ )
(6)
Defining M=DF(Q(j))·DF(Q(j))T and Z(j)=M-1·F(Q(j)) yields M·Z(j)= F(Q(j)), ǻQ(j)=-DF(Q(j))T·Z(j)
(7)
Rewrite M in matrix form as D1
I
L ൦ 2
I ֧
D2
൪·൦
֧ LN-1
T
I
I
֧
L ൪·൦ 2 DN
I ֧
֧ LN-1
൪
(8)
I
where D1 =I+
ij
ቚ
· (j)
Q Q 1
ij
ቚ
Q Q(j) 1
T
, Li =-
ij
ቚ
·D-1 i-1 , Di =I+ (j)
Q Q i
ij
ቚ
· (j)
Q Q 1
ij
ቚ
Q Q(j) 1
T
ǦLi ·Di-1 ·LTi , X1=F1(Q(j)), Xk=Fk (Q(j))-Lk·Xk-1, YN-1 =D-1 N-1 XN-1 ,
T and Yk =D-1 k Xk -Lk+1 Yk+1 .
An initial iteration of the dynamics’ natural solution is essential in order to increase the convergence speed. For a quasi-halo orbit near LL1, a 3-order Richardson expansion derived from Lindstedt-Poincaré method [24][25] can perform as an effective initial iteration:
(
)
( )
)
x = a 21 Ax2 + a 22 Az2 − Ax cos ω t + a 23 Ax2 − a 24 Az2 cos 2ωt + a31 Az3 − a32 Ax Az2 cos 3ω t ° 2 2 3 2 (9) ® y = κAx sin ωt + b21 Ax − b22 Az sin 2ωt + b31 Ax − b32 Ax Az sin 3ωt ° 2 3 ¯ z = ± Az cos ωt ± d 21 Az [cos 2ωt − 3] ± d 32 Az Ax − d 31 Az cos 3ω t
(
)
(
(
)
where the values of coefficients a, b, c, ω, and κ can be found in Ref. [24]. The amplitudes Ax and Az, which are dependent on each other, need to be larger than some critical values so that frequency along the z-direction can be equal
6
to those along x-direction and y-direction. Therefore, a halo orbit with a minimum amplitude Ax=11500km is constructed by the iteration algorithm where amplitude Ax is selected as the only parameter to identify different halo orbits in this paper. As demonstrated in Fig.2, using the initial iteration proposed for LL1, a bounded trajectory in BCM is refined by
0.1
0.1
0.05
0.05 y [LE-M]
y [LE-M]
the multiple shooting method.
0
0
-0.05
-0.05
-0.1
-0.1 0.8
0.9 x [LE-M]
1
-0.1
0.1
0.05 z [LE-M]
0.05 z [LE-M]
0 z [LE-M]
0
0 -0.05 -0.1 0.1
-0.05
0
-0.1 0.75
0.8
0.85 0.9 x [LE-M]
y [LE-M] -0.1
0.95
0.88 0.86 0.84 0.82 x [LE-M]
Fig.2 A quasi-halo orbit near LL1 computed by the multiple shooting corrector: the Sun’s angle θs0 at epoch time is set as 0, and initial amplitude Ax of this quasi-halo orbit before iterations is set as 1×105km. C. Relative Dynamics of Bi-Circular Model Formation flying on a quasi-halo orbit of LL1 is studied in the restricted Sun-Earth-Moon-spacecraft system in this paper. In the Earth-Moon rotating frame, position vectors of the chief and deputy spacecraft are denoted by rc=[xc, yc, zc]T and rd =[xd, yd, zd]T, respectively. Thus, the relative position vector Δr=[Δx, Δy, Δz]T from the chief to the deputy satisfies Δr=rd-rc . Because |Δr| << |r0|, the relative dynamics in BCM is derived as
¨rሷ +2J¨rሶ =Urr ȁr=r0 ¨r+o(¨r)
(10)
where r0 denotes the phase space location of equilibrium point and Urr denotes the second derivative matrix of U with respect to r at r0. Thus, it is yielded that
7
įrሶ =Ȟሺrc +įrሻ-Ȟ(rc )
(11)
where Ȟ is the dynamics function corresponding to the relative dynamical equations above, which can be expanded as įrሶ =Ȟሺrc ሻ+
Ȟ rc
įr+֥-Ȟ(rc ) or įrሶ ̱
Ȟ (t)įr rc
Thus the solution for relative motion can then be expressed as įr=ĭ(t,t0 )įr0 where the state transition matrix defined as ĭ(t,t0 )=
(12) rc (0,rc ȁt=0 ) can be solved from the following matrix differential rc
equation as Ȟ ĭሶ (t,t0 )ൌ (t)ĭ(t,t0 )
(13)
rc
where the initial condition at t0=0 as ĭ(t0 ,t0 )=I6×6. Performing eigenvalue decomposition upon the state transition matrix of the uncontrolled system yields three pairs of eigenvalues, which are located on either real or imaginary axis with stable/unstable/center manifolds denoted as u+1 u-1 , u+2 u-2 and u+3 u-3 . As demonstrated in Fig.3, the quasi-halo orbits of LL1 has one eigenvalue located on real axis, denoted by ±ı1, and two eigenvalues located on imaginary axis, denoted by ±γ2 and ±γ3. 6
σ1 γ2 γ3
5
eigenvalue
4
3
2
1
0
0
20
40
60
80 100 time [day]
120
140
160
Fig.3 Time history of one real and two imaginary eigenvalues of uncontrolled relative dynamic system.
8
III. Application of Hamiltonian Structure-Preserving Control to Stabilize Relative Motion A.
Hamiltonian Structure-Preserving Stabilization for Quasi-Periodic Formation Flying Using hyperbolic Hamiltonian dynamics with stable/unstable and center manifolds is efficient and effective in
many classical astrodynamical problems, including true-anomaly-based formation flying on elliptic orbit, CR3BP, and the restricted Hill three-body problem. Bounded orbits near hyperbolic equilibrium points have been successfully applied in various astronautical missions, e.g., maximizing the coverage time of specific ground targets for an on-orbit satellite and minimizing the loss of telecommunication signals in sunlight. Acquirement of the bounded orbits can be efficiently associated with Hamiltonian structure-preserving (HSP) controller, which is able to generate anticipatory bounded trajectories for both time-independent [26][27] and time-periodic Hamiltonian systems [28][23]. The concept of HSP controller was firstly introduced by Scheeres [28], whose work focused on stabilizing of unstable orbits in Hill restricted three-body problem. Subsequently, the HSP control have been successfully applied in stabilizing relative motions on a J2-perturbed mean circular orbit [23], solar sails [26], frozen high-eccentricity orbits [27], elliptic orbit [29], and halo orbits [30]. However, the relative dynamics on a quasi-halo orbit in the restricted fourbody problem are time-varying, not time-periodic. Thus, some concepts defined in classical dynamic systems, such as Floquet stability, are required to be extended to a time-varying Hamiltonian system. Dynamics of the most astrodynamical problems has general form as Eq. (1). Thus, the design of a HSP controller in this section is based on the linearized relative dynamics, i.e., Eq. (10). As the indication of r0 varies in different problems (e.g., r0 is defined as the fixed artificial libration point in solar sail’s dynamics [26] and a reference periodic orbit of chief spacecraft in formation flying around the Earth [23][29]), in this case, r0 is the osculating state of a quasi-halo orbit refined by the multiple-shooting method presented in Section II. Thus, its eigenvalues and the corresponding manifolds (i.e., {σ1u+1 u-1 }, {γ2 u+2 u-2 }, and {γ3 u+3 u-3 }) of the uncontrolled dynamic system are time-varying, demonstrated in Fig.3. In Ref.[28], a preliminary HSP controller is built up using stable/unstable manifolds with the same control gains. As an important extension, Ref.[23] shows a HSP controller using center manifolds as well as stable/unstable manifolds and proved that the stability of such controller. Inherited from their works but using position feedbacks only, a HSP controller is constructed by stable/unstable/center manifolds as follows: Tc =Ȇ·¨rҧ
(14)
where ¨rҧ denotes the relative position feedback and the coefficient matrix Π is expressed as ାு ି ିு ଶ ା ାு ି ିு Ȇ=-G1 ıଵଶ (uଵା uଵାு +uଵି uଵିு ሻ െ G2 ߛଶଶ (uା ଶ uଶ +uଶ uଶ ሻǦG͵ ߛଷ (uଷ uଷ +uଷ uଷ ሻ
9
(15)
where Gi (i=1,2,3) are control gains correlated with manifolds. Hence, the linearized dynamical equations of the controlled system are ¨rҧሷ+2J¨rҧሶ=Urr ¨rҧ+Ȇ·¨rҧ
(16)
The fact that Urr and Ȇ are symmetry matrices guarantees the linear feedback controller to preserve the Hamiltonian structure [31]. It can be verified that only the feedback of relative position does not change the location of equilibrium point but its topological type. Thus, for a two-dimensional Hamiltonian system, assumptions and theorems have been proposed in [26] for poles assignment and optimization of control gains. Based on the propositions, topological types of equilibrium points can be further changed by the optimized control gains. B. Stability of HSP Controller Eq. (15) contains six coefficients, i.e., G1, G2, G3, ı1, Ȗ2 and Ȗ3. The latter three are set as 1 in this paper. The former three are control gains. The HSP controller used in this paper is inherited from results of Xu et al. [26] and Scheeres et al. [28]. It has been analytically proved that the controller’s capability of stabilizing the system can be guaranteed, as the control gains are large enough. For simplicity, in this paper, this conclusion is applied without proving and a group of large enough control gains (G1, G2, G3), i.e., G1=G2=G3=2, is used to stabilize the relative motion on a quasi-halo orbit around LL1 point. As demonstrated in Fig.3, the original uncontrolled system has three eigenvalues, real number σ1 and pure imaginary numbers γ2 and γ3, which are changed to three pure imaginary eigenvalues ω1, ω2 and ω3 of the controlled dynamic system, presented in Fig.4, indicating that the topological type of transient equilibrium is changed from hyperbolic to elliptic.
10
10
ω1 ω2 ω3
9 8
eigenvalue
7 6 5 4 3 2 1 0
0
20
40
60
80 100 120 140 160 time [day] Fig. 4 Time history of three pure imaginary eigenvalues of the controlled system.
To accurately estimate the relationship between transient stability and the control gains, groups of critical gains (G1, G2, G3) achieving transient stability is shown in Fig.5, where the critical gains are presented on G2-G3 plane and the tint of colour at each point indicates the corresponding value of G1. In (G1, G2, G3) space, such critical groups form a twodimensional surface, dividing the whole space into stable and unstable sub-spaces. The larger control gains, which are located on the upper right side above the critical groups, are capable of stabilizing the system while the smaller control gains, located on the lower left side of the critical groups, fail in stabilizing the system.
11
Critial G1 5
4.5
4
4
3.5 3
G3
3 2 2.5 1 2 0
1.5
-1 -1
0
1
2 G2
3
4
5
1
Fig.5 Critical control gains of transient stability presented on G2-G3 space: the tint of colour indicates the corresponding value of G1. In terms of a time-periodic Hamiltonian system [23][29], even if its equilibrium points are always elliptic during its period, its correlated controlled system may be still unstable. As a result, the concept of Floquet stability is introduced to examine if Floquet multipliers lie on a unit circle in complex plane and whether moduli of all the multipliers are equal to one. However, in time-quasi-periodic Hamiltonian system studied in this section, Floquet multiplier is unavailable. Long-term eigenvalues of the linearized controlled system over the whole time span, instead of one-period eigenvalues used to check regular Floquet multipliers, are solved from the following matrix equation: 0 ĭሶ = ቈU | halo rr r=r (t) 0
I 2J ĭ
(17)
where r0halo(t) is denoted as position vector of the quasi-halo orbit, and the initial value of Φ at epoch time is an identity matrix, i.e., I6×6. In order to investigate the long-term stability of the aforementioned controlled system, three eigenvalues with positive real or positive imaginary part are selected respectively from all three pairs of conjugate eigenvalues of the controlled system. As shown in Fig.6, long-term eigenvalues grow linearly with respect to time no
12
matter whether they are stable or not. Two eigenvalues are located steadily on imaginary axis while the other one’s position switches from real axis to imaginary axis with the change of chosen control gains.
a)
b)
Fig. 6 Time history of three long-term eigenvalues of controlled system with different control gains: a) G1=0.5, G2=0 and G3=0; b) G1=2, G2=0 and G3=0 in Fig. 6b; the unstable eigenvalue Σ1 with real part and other two eigenvalues in Fig. 6a is replaced by the three stable ones with pure imaginary part in Fig. 6b. The long-term eigenvalues of above linearized matrix equation can qualitatively demonstrate how relative motion can be stabilized expectantly by the HSP controller: for the unstable eigenvalue Σ1 with real part, the term e+Ȉ1 t will gradually enlarge instability; for the stable eigenvalues Ωj (j=1,2,3) with imaginary parts, periodic terms eiΩj⋅t =cos(Ωj⋅t)+i⋅sin(Ωj⋅t) are the expected ones to stabilize relative motion. Therefore, to stabilize the system, a HSP controller must eliminate unstable real eigenvalue Σ1. To demonstrate the relationship between long-term stability and the control gains, groups of critical gains (G1, G2, G3) of long-term stability is presented in Fig.7, where the critical gains are presented on G2-G3 plane and the tint of colour at each point indicates the corresponding value of G1. In (G1, G2, G3) space, such critical groups form a two-dimensional surface, dividing the whole space into stable and unstable sub-spaces. The larger control gains, which are located on the upper right side above the critical groups, are capable of stabilizing the system while the smaller control gains, located on the lower left side of the critical groups, fail in stabilizing the system. In conclusion, the sufficient and necessary condition for the stability of the controlled system by a HSP controller is to choose control gain group guaranteeing both transient stability and long-term stability. Additionally, the smaller control gains are selected, the less fuel consumption is required.
13
Critial G1 4
5
3.5 4 3
G3
3
2.5 2
2
1.5 1
1 0.5
0
0 -1 -1
0
1
2 G2
3
4
5
Fig. 7 Critical gain pairs of long-term stability presented on G2-G3 plane: the tint of colour at each point indicates the corresponding value of G1.
IV.
Numerical Simulations
A natural quasi-halo orbit acting as the reference orbit of the chief spacecraft is integrated from an initial state and refined by the multiple shooting method, as introduced in Section II. Only using the feedback of relative position in Section III, a non-natural relative trajectory of the deputy spacecraft in BCM is obtained from an initial relative position and velocity by HSP control. The Sun’s angle θs at epoch time is set as 0, and the initial amplitude Ax of the quasi-halo orbit is set as 1×105 km. Arbitrary initial relative position and velocity can be used to generate a non-natural relative trajectory of the deputy spacecraft. In this section, x0=y0=z0=1×10-5 LE-M (i.e., 3.844km) and x0 = y 0 = z0 = 1× 10−5 LE −M / TE −M . Besides, the control gains are selected large enough, as G1=2, G2=1, and G3=1, to stabilize the relative motion near the chief spacecraft, as illustrated in Fig.8.
14
5
0 -5 -10
0.05
0 -5
0.05 y [LE-M]
10
5
y [LE-M]
15
10 y [km]
y [km]
15
0 -0.05
0 -0.05
-10
-15 -10
-5
0 x [km]
5
-15
10
-20
0 z [km]
0.75
20
0.8
0.85 0.9 x [LE-M]
0.95
-0.1 -0.05 0 0.05 z [LE-M]
30
z [LE-M]
z [km]
z [km]
0 -10
z [LE-M]
20
10
0 -20
-20 -30 -10
-5
0 x [km]
a)
5
10
0
0
y [km]
0.05
10 -10
-0.1
0 x [km]
-10
0 -0.05 -0.1
-0.05
10
Chief Deputy
0.05
0.05
20
0.1
0.75
0.8
b)
0.85 0.9 x [LE-M]
0.95
0 -0.05 y [LE-M]
0.88 0.86 0.84 x [LE-M]
Fig. 8 A relative trajectory generated by HSP control about a quasi-halo orbit of LL1: a) the relative trajectory of the deputy with respect to the chief over six months; b) absolute trajectories of the deputy and the chief over one month. The cluster flight based on continuous low-thrust technique greatly advanced the utilization of electromagnetic and electrostatic forces on innovative electric propellers. According to the continuous control acceleration by HSP control shown in Fig.9a, the maximum acceleration is supposed to be no more than 1×10-6m/s2 in order to obtain the expected bounded trajectories. Fuel consumption can be calculated by the following cost function: t
ΔV = ³ Tc dt .
(18)
0
As shown in Fig.9b, the fuel consumption for station-keeping over 6 months is 2.8m/s. Taking PPS-1350 Hall engine equipped by SMART-1 with a high specific impulse 1643.4s as an example, fuel consumption in one year is no more than 1.0% of the entire mass. -7
14
x 10
12
2.5
2
8
ΔV [m/s]
control acceleration [m/s 2]
10
6 4
1.5
1
2 0
0.5 -2 -4
a)
0
20
40
60
80 100 time [day]
120
140
0
160
b)
15
20
40
60
80 100 time [day]
120
140
160
Fig.9 Time history of the continuous acceleration and fuel consumption function over six months by HSP control of G1=2, G2=1, and G3=1.
V.
Conclusion
In this paper, the Hamiltonian structure-preserving control (HSP) is applied to stabilize the formation flying in the restricted Sun-Earth-Moon-spacecraft system. A simplified model of Sun-Earth-Moon-spacecraft system, i.e., bi-circular model (BCM), is employed in Earth-Moon rotating frame, where a quasi-halo orbit of LL1 is generated and corrected by a multiple shooting method acting as reference orbit of the chief spacecraft. The equilibrium points of the linearized relative dynamic in bi-circular model are studied and their topologies are changed from hyperbolic to elliptic by HSP control with feedbacks of stable, unstable and center manifolds. The stability of the controlled system by HSP control is analyzed. Such conclusion can be drawn that whether all three eigenvalues of the controlled system are located on imaginary axis is critical for transient stability. The long-term eigenvalues solved from state-transition matrix of the controlled system over the whole time span determine long-term stability of the controlled system. The sufficient and necessary condition for stability of the controlled system is that control gains locate inside the region where both transient stability and long-term stability is guaranteed. The station-keeping controller developed in this paper has much potential in formation flying around other libration points of restricted four-body problem, such as LL2 and ELi (i=1,2,3). It can also be used to deal with formation flying in weak stability boundary theory, the flight duration of which is long enough (about half a year, such as Hiten and NASA’s GRAIL) to accomplish explorations in deep space.
Acknowledgments The research is supported by the National Natural Science Foundation of China (11172020 and 11432001), Beijing Natural Science Foundation (4153060), and the Fundamental Research Funds for the Central Universities.
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[2] Guibout, V. M., and Scheeres, D. J., “Solving Relative Two-point Boundary Value Problems: Spacecraft Formation Flight Transfers Application,” Journal of Guidance, Control, and Dynamics, Vol. 27, No. 4, 2004, pp. 693-704. doi: 10.2514/1.11164 [3] Howell, K. C., and Marchand, B. G., “Natural and Non-natural Spacecraft Formations near the L-1 and L-2 Libration Points in the Sun-Earth/Moon Ephemeris System,” Dynamical Systems: An International Journal, Vol. 20, No. 1, 2005, pp. 149-173. doi:10.1080/1468936042000298224 [4] Gurfil, P., Idan, M., and Kasdin, N. J., “Adaptive Neural Control of Deep-space Formation Flying,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 3, 2003, pp. 491-501. doi: 10.2514/2.5072 [5] Marchand, B. G., and Howell, K. C., “Control Strategies for Formation Flight in the Vicinity of the Libration Points,” Journal of Guidance, Control, and Dynamics, Vol. 28, No. 6, 2005, pp. 1210-1219. doi: 10.2514/1.11016 [6] Qi, R., Xu, S. J., and Chen, T., “Control of Orientation for Spacecraft Formations in the Vicinity of the Sun-Earth L2 Libration Point,” Science China: Physics, Mechanics, and Astronomy, Vol. 57, No. 9, 2014, pp. 1778-1787. doi: 10.1007/s11433-014-5398-0 [7] Miller, I., and Campbell, M., “Probabilistic Validation of Simplified Models of Spacecraft Formations near Libration Points,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 47, Issue. 4, 2011, pp. 2779-2791. doi: 10.1109/taes.2011.6034664 [8] Heritier, A., and Howell, K. C., “Dynamical Evolution of Natural Formations in Libration Point Orbits in a Multi-body Regime,” Acta Astronautica, Vol. 102, 2014, pp. 332-340. doi: 10.1016/j.actaastro.2013.10.017 [9] Catlin, K. A., and McLaughlin, C. A., “Earth-Moon Triangular Libration Point Spacecraft Formations,” Journal of Spacecraft and Rockets, Vol. 44, No. 3, 2007, pp. 660-670. doi: 10.2514/1.20152 [10] Peng, H. J., Zhao, J., Wu, Z. G., and Zhong, W. X., “Optimal Periodic Controller for Formation Flying on Libration Point Orbits,” Acta Astronautica,Vol. 69, No. 7-8, 2011, pp. 537-550. doi: 10.1016/j.actaastro.2011.04.020 [11] Hughes, S. P., “General Method for Optimal Guidance of Spacecraft Formations,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 2, 2008, pp. 414-423. doi: 10.2514/1.23731
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[12] Garcia-Taberner, L., and Masdemont, J. J., “FEFF Methodology for Spacecraft Formations Reconfiguration in the Vicinity of Libration Points,” Acta Astronautica, Vol. 67, No. 7-8, 2010, pp. 810-817. doi: 10.1016/j.actaastro.2010.05.007 [13] Qi, R., Xu, S. J., and Xu, M., “Impulsive Control for Formation Flight about Libration Points,” Journal of Guidance, Control, and Dynamics, Vol. 35, No. 2, 2012, pp. 484-496. doi: 10.2514/1.54383 [14] Pernicka, H. J., Carlson, B. A., and Balakrishnan, S. N., “Spacecraft Formation Flight about Libration Points Using Impulsive Maneuvering,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 5, 2006, pp. 1122-1130. doi: 10.2514/1.18813 [15] Shahid, K., and Kumar, K. D., “Formation Control at the Sun-Earth L2 Libration Point Using Solar Radiation Pressure,” Journal of Spacecraft and Rockets, Vol. 47, No. 4, 2010, pp. 614-626. doi: 10.2514/1.47342 [16] Li, P., Cui, P. Y., and Cui, H. T., “An Improved Nonlinear Control Strategy for Deep Space Formation Flying Spacecraft,” Acta Mechanica Sinica, Vol. 25, No. 6, 2009, pp. 847-856. doi: 10.1007/s10409-009-0274-2 [17] Darvish, K., Pourtakdoust, S. H., and Assadian, N., “Linear and Nonlinear Control Strategies for Formation and Station Keeping of Spacecrafts within the Context of the Three Body Problem,” Aerospace Science and Technology, Vol. 42, 2015, pp. 12-24. doi: 10.1016/j.ast.2014.12.030 [18] Salazar, F. J. T., Masdemont, J. J., Gomez, G., Macau, E. E., and Winter, O. C., “Zero, Minimum and Maximum Relative Radial Acceleration for Planar Formation Flight Dynamics near Triangular Libration Points in the Earth-Moon System,” Advances in Space Research, Vol. 54, No. 9, 2014, pp. 1838-1857. doi: 10.1016/j.asr.2014.07.018 [19] Inampudi, R., and Schaub, H., “Two-craft Tether Formation Relative Equilibria about Circular Orbits and Libration Points,” Acta Astronautica, Vol. 68, No. 11-12, 2011, pp. 1761-1773. doi: 10.1016/j.actaastro.2011.02.003 [20] Huang, J., Ma, G. F., and Liu, G., “Nonlinear Dynamics and Reconfiguration Control of Two-satellite Coulomb Tether Formation at Libration Points,” Aerospace Science and Technology, Vol. 39, 2014, pp. 501-512. doi: 10.1016/j.ast.2014.06.001 [21] Koon, W.S., Lo, M.W., Marsden, J.E., and Ross, S.D., Dynamical Systems, the Three-Body Problem and Space Mission Design, Springer-Verlag, New York, 2011, Chap. 5.
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[22] Xu, M., and Xu, S. J., “Nonlinear Dynamical Analysis for Displaced Orbits Above a Planet,” Celestial Mechanics and Dynamical Astronomy, Vol. 102, No. 4, 2008, pp. 327–353. doi: 10.1007/s10569-008-9171-4 [23] Xu, M., Wang, Y., and Xu, S., “On the Existence of J2 Invariant Relative Orbits from the Dynamical System Point of View,” Celestial Mechanics and Dynamical Astronomy, Vol. 112, No. 4, 2012, pp. 427–444. doi: 10.1007/s10569-012-9401-7 [24] Richardson, D. L., “Analytic Construction of Periodic Orbits about the Collinear Points,” Celestial Mechanics, Vol. 22, Issue. 3, 1980, pp. 241–253. doi: 10.1007/BF01229511 [25] Popescu, M., and Cardos, M., “The Domain of Initial Conditions for the Class of Three-dimensional Halo Periodical Orbits,” Acta Astronautica, Vol. 36, No. 4, 1995, pp. 193–196. doi: 10.1016/0094-5765(95)00107-B [26] Xu, M., and Xu, S., “Structure-Preserving Stabilization for Hamiltonian System and its Applications in Solar Sail,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 3, 2009, pp. 997–1004. doi: 10.2514/1.34757 [27] Colombo, C., Xu, M., and McInnes, C., “Stabilisation of the Hyperbolic Equilibrium of High Area-to-Mass Spacecraft,” 63rd International Astronautical Congress, IAC-12-C1.1.13, Naples, Italy, 2012. [28] Scheeres, D. J., Hsiao, F. Y., and Vinh, N. X., “Stabilizing Motion Relative to an Unstable Orbit: Applications to Spacecraft Formation Flight,” Journal of Guidance, Control, and Dynamics, Vol. 26, No.1, 2003, pp. 62-73. doi: 10.2514/2.5015 [29] Xu, M., Liang, Y., Tan, T., and Wei, L., “Cluster Fight Control for Fractionated Spacecraft on an Elliptic Orbit,” Celestial Mechanics and Dynamical Astronomy, Vol. 125, No. 4, 2016, pp. 383-412. doi: 10.1007/s10569-016-9685-0 [30] Soldini, S., Colombo, C., and Walker, S., “Comparison of Hamiltonian Structure-Preserving and Floquét Mode Station-keeping for Libration-point Orbits,” AIAA/AAS Astrodynamics Specialist Conference, AIAA-2014-4118, San Diego, California, 2014. doi: 10.2514/6.2014-4118 [31] Abeber, H., and Katzschmann, K., “Structure-Preserving Stabilization of Hamiltonian Control Systems,” Systems and Control Letters, Vol. 22, No. 4, 1994, pp. 281-285.
19
Ming Xu, Yuying Liang, Xiaoyu Fu
PII: DOI: Reference:
S1270-9638(16)30383-2 http://dx.doi.org/10.1016/j.ast.2017.03.038 AESCTE 3977
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
3 August 2016 8 December 2016 30 March 2017
Please cite this article in press as: M. Xu et al., Formation Flying on Quasi-halo Orbits in Restricted Sun-Earth/Moon System, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.03.038
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Formation Flying on Quasi-halo Orbits in Restricted SunEarth/Moon System Ming Xu1, Yuying Liang2, and Xiaoyu Fu3 Beihang University, Beijing, 100191, People’s Republic of China
Due to the strong nonlinear perturbations near the libration points, the continuous low-thrust technique has many potential applications in stationkeeping relative motions. The Hamiltonian structure-preserving (HSP) control is employed in this paper to stabilize formation flying on quasi-periodic orbits near LL1 of the restricted Sun-Earth-Moon-spacecraft system. In the bi-circular model (BCM), a multiple shooting corrector is developed to refine quasi-periodic orbits as the chief spacecraft’s reference trajectories. The linearized variation equation in BCM is used to design the stationkeeping control. A HSP controller is constructed to change the topology of the equilibrium from hyperbolic to elliptic using only relative position feedbacks consisting of stable, unstable and center manifolds. The critical control gains for transient and long-term stabilities are presented to guide the selection of control gains.
Hamiltonian structure-preserving control; formation flying; bi-circular model; quasi-halo orbits; libration point
I. Introduction Spacecraft formation flying in the context of two-body and multiple-body problems has received much attention recently. Adopted in NASA’s future missions, e.g., Terrestrial Planet Finder (TPF) and Micro-Arcsecond X-ray Imaginary Mission (MAXIM), formation flying in the vicinity of libration points is considered as one of important technologies in future deep-space exploration. Recent literature on relative dynamics and control becomes extensive. Perea et al. [1] developed a spacecraft control strategy, which was applied to the transfer orbit of a set of loose-formation spacecraft following a natural trajectory to a libration point. Guibout and Scheeres [2] studied a reconfiguration method of spacecraft formation about a 1
Associate Professor, School of Astronautics, [email protected]. PhD Candidate, School of Astronautics, [email protected]. 3 Master Candidate, School of Astronautics, [email protected]. 2
1
Sun-Earth libration point and demonstrated that optimal transfer time of the reconfiguration depends largely on initial and final positions. Howell et al. [3] investigated a non-natural formation-keeping problem near libration points of SunEarth or Earth-Moon systems based on continuous control. Gurfil et al. [4] put forward a nonlinear adaptive neural control method for deep-space spacecraft formation flying, which is applicable to formation flying near libration points of three body problem. Marchand and Howell [5] set fundamental propulsive requirements near libration points of circular restricted three body problem (abbr. CR3BP) and introduced related constrictions on formation control strategy. Qi et al. [6] studied an impulsive control technique to station-keep formation flying in CR3BP and proposed a formation-keeping strategy based on tangent targeting method to deal with uncertainty issue. A probabilistic evaluation method was introduced by Miller and Campbell [7] to validate a simplified model of formation dynamical system in the vicinity of libration point L2 of Sun-Earth or Earth-Moon systems. Heritier and Howell [8] investigated the dynamics of natural formation and eigenstructure of a reference trajectory in a multiple-body regime for formation flying. Catlin and McLaughlin [9] demonstrated the existence and nature of trajectories near Earth-Moon triangular libration points in CR3BP and designed a chief/deputy formation for long-period cases. An optimal periodic controller based on continuous low-thrust technique was developed by Peng et al. [10] for spacecraft station-maintaining and formationkeeping near periodic libration-point orbits of Sun-Earth system. Hughes and Steven [11] introduced a general approach for formation flying, which is adaptive to orbital perturbations and applicable to multiple fight regimes including libration point orbits of CR3BP. Finite elements for formation flying computation method was introduced by Garcia et al. [12] in a libration point regime of Sun-Earth system. A tangent targeting method was exploited by Qi et al. [13] to control the leader/follower architecture for maintaining a prescribed orientation near libration points of CR3BP. Pernicka et al. [14] investigated a formation-maintaining problem during both passive and active modes of operations near a libration point of CR3BP. Shahid and Kumar [15] looked into the same problem and designed a stable controller based on Lyapunov theory. To achieve high-precision formation maneuver and station keeping in CR3BP, Li et al. [16] employed a nonlinear adaptive control scheme based on nonzero set-point linear quadratic regular method and neural network. Darvish et al. [17] utilized a nonlinear integral sliding mode to investigate precision formation control of spacecraft on halo orbit around libration point L2 of Sun-Earth system. By linearized motion equations relative to periodic orbits of L4 or L5 of CR3BP, Salazar et al. [18] studied regions of zero, minimum and maximum relative radial acceleration. Inampudi et al. [19] studied a two-spacecraft tether formation associated by line-of sight elastic forces moving in CR3BP and demonstrated that equilibria deflections in the formation are trivial. Huang et al. [20] constructed
2
a reconfiguration model for a two-satellite Coulomb tether formation near Earth-Moon libration points and proved the model’s practicability. Most of the aforementioned works focused on the formation flying in the vicinity of libration points in Sun-Earth or Earth-Moon systems, and periodic orbits of CR3BP (e.g., halo orbits) were used as reference orbits. However, considering the strict condition of halo orbits, there exist no periodic orbits in the restricted four-body problem. Theoretically, quasi-halo orbits can be computed in Sun-Earth-Moon system, while algorithms correlated with the computation of quasi-halo orbits is very complicated in formation flying in the context of restricted four-body problem. Different from the aforementioned literature, this paper studies a station-keeping control method for formation flying on quasi-halo orbits in the restricted Sun-Earth-Moon-spacecraft system. Firstly, a simplified model of restricted four-body problem, i.e., bi-circular model (BCM), is adopted in the Earth-Moon rotating coordinate frame and a useful multiple shooting corrector is developed to improve the accuracy of quasi-halo orbits which acts as the reference orbits of the chief spacecraft. Linearized relative dynamics in BCM is derived to stabilize relative motions near quasi-halo orbits around libration point LL1. Subsequently, a Hamiltonian structure-preserving (HSP) controller is constructed to change topologies of equilibria from hyperbolic to elliptic using the stable, unstable and center manifolds. Furthermore, the critical control gains of the transient stability and the long-term stability are presented on G2-G3 plane with the tint of colour indicating the corresponding value of G1. The stability condition of the controlled system is then numerically demonstrated. Due to the effects of the strong nonlinear perturbations near libration points, continuous low-thrust technique is used to achieve the designed HSP control in formation-maintaining and station-keeping missions.
II. Relative Dynamics on Quasi-halo Orbits in Restricted Sun-Earth-Moon-Spacecraft System A. Bi-Circular Model of Restricted Sun-Earth-Moon-Spacecraft System In a bi-circular model (BCM), the Earth and the Moon are supposed to move in a circular orbit about their common barycenter with a mutual separation d1. Considering that the mass of Earth-Moon System is concentrated at its barycenter, we assume that the Sun and Earth-Moon system are also in circular motion about their barycenter, with a mutual separation d2>d1. Based on the aforementioned assumptions, a BCM of the restricted Sun-Earth-Moon-spacecraft system has the following characteristics: i) the Earth and Moon are regarded as independent gravitational points, moving in their Keplerian circular orbits about their own barycenter; ii) the barycenter of the Earth-Moon system moves in a circumsolar
3
orbit, the eccentricity of which is also ignored; iii) the inclination of the lunar-orbit plane to ecliptic plane is also ignored. Thus, the orbits of the four bodies (Sun, Earth, Moon and spacecraft) are restricted in a same plane. To avoid heavy computation, the normalization of this system is defined as follows: the unit of length is chosen to be one Earth-Moon distance; the unit of mass is chosen to be the total mass of Earth-Moon System; the unit of time is chosen such that the orbital period of the Earth and the Moon around their barycenter is 2ʌ. Subsequently, the universal constant of gravitation is equal to one. A rotating coordinate frame [21] of the Earth-Moon system is defined as follows: the origin is located at barycenter of the Earth and the Moon; the x-axis is along the line which points from the Earth to the Moon; the y-axis is chosen to make the system orthogonal and positive oriented; the z-axis is given by the right-hand rule. In this frame, positions of the Earth and the Moon are fixed on the x-axis respectively. Position and velocity vectors of spacecraft are defined as r=[x y z]T and r = [x y z ]T respectively. As shown in Fig.1, the Sun is rotating clockwise in a circular orbit of a normalized constant radius aS = 388.81114 at an angular velocity of ωs = 0.925195985520347. The Sun’s phase θS is time-varying.
Fig. 1 BCM in the Earth-Moon rotating coordinate frame. After normalization, equations of motion in BCM of the restricted Sun-Earth-Moon-spacecraft system is rሷ +2Jrሶ =
U r
(1)
where J is symplectic operator and U = U(r) can be written as: 1
ȝ
ȝ
ȝ
ȝ
ȝS
S
as
U= ሺx2 +y2 ሻ+ ԡr Eԡ + ԡr Mԡ + ԡr Sԡ - ԡa Sԡ3 aS T ·r+ 2
E
M
4
S
where the Sun’s position vector is aS = aS ⋅[cos(θS), sin(θS), 0]T. Since that the dynamical equations are established in the Earth-Moon rotating frame, the Sun’s convected velocity must be take into consideration, which is indicated by the term ȝS
a ԡaS ԡ3 S
T
·r. The term
ȝS is a constant, which guarantees that the value of U at LL1 is zero. as
B. A Quasi-halo Orbit by Multiple Shooting Method Due to the eccentricity of Moon’s orbit and the perturbations caused by other celestial bodies, such as the Sun and Jupiter, the CR3BP is not accurate enough in many cases and even libration points cannot stay stationary. In order to construct natural orbits around libration points, a multiple shooting method is applied in full ephemeris model, which is widely used in the computation of periodic orbits with long periods, such as solar sail’s displaced orbits [22] and J2 invariant relative orbits [23]. The multiple shooting method employed in this paper is efficient in refining long-period orbits. A full time span of iteration procedure [t0, tf ] is divided into N sub-intervals by N-1 equal-spaced time t1, t2, ..., tN-1. Each equal-length subinterval is defined as Δt=ti+1−ti (i =0,1,2,..., N-1) and a point on quasi-halo trajectory can be expressed as Qi = [ti, xi, yi, zi, xi , y i , zi ]T (i =0,1,2,..., N-1). (Qi) is the image of Qi under the flow related to ephemeris dynamics after time step Δt, i.e., (Qi)=Qi+1. Without perturbation terms, e.g., solar perturbation in BCM, this equation holds for each Qi. Hence, our goal is to minimize the difference between (Qi) and Qi+1, i.e., to solve the following nonlinear problems,
§ Q1 · § φ (Q1 ) · § Q 2 · ¨ ¸ ¨ ¸ ¨ ¸ ¨ Q 2 ¸ ¨ φ (Q 2 ) ¸ ¨ Q 3 ¸ = F¨ ¸−¨ ¸ = 0 ¸ ¨ ¨ ¸ ¨ ¸ ¨ ¸ ¨ Q ¸ ¨ φ (Q ) ¸ ¨ Q ¸ N −1 ¹ © N ¹ © N¹ ©
(2)
T
Defining Q(j) =(Qሺjሻ , Qሺjሻ ,֥,Qሺjሻ ) , Newton’s iteration method is adopted to solve Eq. (2). The jth iteration of this 1 2 N procedure has the following expression:
∂F ∂Q
[
]
⋅ Q ( j +1) − Q ( j ) = − F Q( j )
and
5
Q( j )
(3)
§ ∂ij ¨ ¨ ∂Q Q1( j ) ¨ ∂F ¨ =¨ ∂Q ¨ ¨ ¨ ¨¨ © where
ij
ቚ
Q Q(j) i
−I ∂ij ∂Q Q ( j )
−I
2
∂ij ∂Q Q ( j )
N −1
· ¸ ¸ ¸ ¸. ¸ ¸ ¸ ¸ −I¸ ¸ ¹
(4)
(i=1,2,…,N-1) is a 6×6 state transition matrix. After each iteration, there are 6×(N-1) equations and 6×N
unknowns, indicating that other conditions are required to solve these unknowns. Thus, we take the state at t0 or tN into consideration. Define ǻQ(j)= Q(j+1)- Q(j). In order to minimize ||ǻQ(j)||2, a Lagrange function is set as Lሺ¨Q,IJሻ=¨QT ·¨Q+IJ(FሺQሻ+DF(Q)·¨Q)
(5)
Hence, in the jth iteration, T -1
T
¨Qሺjሻ =-DF(Qሺjሻ ) ·[DF(Qሺjሻ )·DF(Qሺjሻ ) ] ·F(Qሺjሻ )
(6)
Defining M=DF(Q(j))·DF(Q(j))T and Z(j)=M-1·F(Q(j)) yields M·Z(j)= F(Q(j)), ǻQ(j)=-DF(Q(j))T·Z(j)
(7)
Rewrite M in matrix form as D1
I
L ൦ 2
I ֧
D2
൪·൦
֧ LN-1
T
I
I
֧
L ൪·൦ 2 DN
I ֧
֧ LN-1
൪
(8)
I
where D1 =I+
ij
ቚ
· (j)
Q Q 1
ij
ቚ
Q Q(j) 1
T
, Li =-
ij
ቚ
·D-1 i-1 , Di =I+ (j)
Q Q i
ij
ቚ
· (j)
Q Q 1
ij
ቚ
Q Q(j) 1
T
ǦLi ·Di-1 ·LTi , X1=F1(Q(j)), Xk=Fk (Q(j))-Lk·Xk-1, YN-1 =D-1 N-1 XN-1 ,
T and Yk =D-1 k Xk -Lk+1 Yk+1 .
An initial iteration of the dynamics’ natural solution is essential in order to increase the convergence speed. For a quasi-halo orbit near LL1, a 3-order Richardson expansion derived from Lindstedt-Poincaré method [24][25] can perform as an effective initial iteration:
(
)
( )
)
x = a 21 Ax2 + a 22 Az2 − Ax cos ω t + a 23 Ax2 − a 24 Az2 cos 2ωt + a31 Az3 − a32 Ax Az2 cos 3ω t ° 2 2 3 2 (9) ® y = κAx sin ωt + b21 Ax − b22 Az sin 2ωt + b31 Ax − b32 Ax Az sin 3ωt ° 2 3 ¯ z = ± Az cos ωt ± d 21 Az [cos 2ωt − 3] ± d 32 Az Ax − d 31 Az cos 3ω t
(
)
(
(
)
where the values of coefficients a, b, c, ω, and κ can be found in Ref. [24]. The amplitudes Ax and Az, which are dependent on each other, need to be larger than some critical values so that frequency along the z-direction can be equal
6
to those along x-direction and y-direction. Therefore, a halo orbit with a minimum amplitude Ax=11500km is constructed by the iteration algorithm where amplitude Ax is selected as the only parameter to identify different halo orbits in this paper. As demonstrated in Fig.2, using the initial iteration proposed for LL1, a bounded trajectory in BCM is refined by
0.1
0.1
0.05
0.05 y [LE-M]
y [LE-M]
the multiple shooting method.
0
0
-0.05
-0.05
-0.1
-0.1 0.8
0.9 x [LE-M]
1
-0.1
0.1
0.05 z [LE-M]
0.05 z [LE-M]
0 z [LE-M]
0
0 -0.05 -0.1 0.1
-0.05
0
-0.1 0.75
0.8
0.85 0.9 x [LE-M]
y [LE-M] -0.1
0.95
0.88 0.86 0.84 0.82 x [LE-M]
Fig.2 A quasi-halo orbit near LL1 computed by the multiple shooting corrector: the Sun’s angle θs0 at epoch time is set as 0, and initial amplitude Ax of this quasi-halo orbit before iterations is set as 1×105km. C. Relative Dynamics of Bi-Circular Model Formation flying on a quasi-halo orbit of LL1 is studied in the restricted Sun-Earth-Moon-spacecraft system in this paper. In the Earth-Moon rotating frame, position vectors of the chief and deputy spacecraft are denoted by rc=[xc, yc, zc]T and rd =[xd, yd, zd]T, respectively. Thus, the relative position vector Δr=[Δx, Δy, Δz]T from the chief to the deputy satisfies Δr=rd-rc . Because |Δr| << |r0|, the relative dynamics in BCM is derived as
¨rሷ +2J¨rሶ =Urr ȁr=r0 ¨r+o(¨r)
(10)
where r0 denotes the phase space location of equilibrium point and Urr denotes the second derivative matrix of U with respect to r at r0. Thus, it is yielded that
7
įrሶ =Ȟሺrc +įrሻ-Ȟ(rc )
(11)
where Ȟ is the dynamics function corresponding to the relative dynamical equations above, which can be expanded as įrሶ =Ȟሺrc ሻ+
Ȟ rc
įr+֥-Ȟ(rc ) or įrሶ ̱
Ȟ (t)įr rc
Thus the solution for relative motion can then be expressed as įr=ĭ(t,t0 )įr0 where the state transition matrix defined as ĭ(t,t0 )=
(12) rc (0,rc ȁt=0 ) can be solved from the following matrix differential rc
equation as Ȟ ĭሶ (t,t0 )ൌ (t)ĭ(t,t0 )
(13)
rc
where the initial condition at t0=0 as ĭ(t0 ,t0 )=I6×6. Performing eigenvalue decomposition upon the state transition matrix of the uncontrolled system yields three pairs of eigenvalues, which are located on either real or imaginary axis with stable/unstable/center manifolds denoted as u+1 u-1 , u+2 u-2 and u+3 u-3 . As demonstrated in Fig.3, the quasi-halo orbits of LL1 has one eigenvalue located on real axis, denoted by ±ı1, and two eigenvalues located on imaginary axis, denoted by ±γ2 and ±γ3. 6
σ1 γ2 γ3
5
eigenvalue
4
3
2
1
0
0
20
40
60
80 100 time [day]
120
140
160
Fig.3 Time history of one real and two imaginary eigenvalues of uncontrolled relative dynamic system.
8
III. Application of Hamiltonian Structure-Preserving Control to Stabilize Relative Motion A.
Hamiltonian Structure-Preserving Stabilization for Quasi-Periodic Formation Flying Using hyperbolic Hamiltonian dynamics with stable/unstable and center manifolds is efficient and effective in
many classical astrodynamical problems, including true-anomaly-based formation flying on elliptic orbit, CR3BP, and the restricted Hill three-body problem. Bounded orbits near hyperbolic equilibrium points have been successfully applied in various astronautical missions, e.g., maximizing the coverage time of specific ground targets for an on-orbit satellite and minimizing the loss of telecommunication signals in sunlight. Acquirement of the bounded orbits can be efficiently associated with Hamiltonian structure-preserving (HSP) controller, which is able to generate anticipatory bounded trajectories for both time-independent [26][27] and time-periodic Hamiltonian systems [28][23]. The concept of HSP controller was firstly introduced by Scheeres [28], whose work focused on stabilizing of unstable orbits in Hill restricted three-body problem. Subsequently, the HSP control have been successfully applied in stabilizing relative motions on a J2-perturbed mean circular orbit [23], solar sails [26], frozen high-eccentricity orbits [27], elliptic orbit [29], and halo orbits [30]. However, the relative dynamics on a quasi-halo orbit in the restricted fourbody problem are time-varying, not time-periodic. Thus, some concepts defined in classical dynamic systems, such as Floquet stability, are required to be extended to a time-varying Hamiltonian system. Dynamics of the most astrodynamical problems has general form as Eq. (1). Thus, the design of a HSP controller in this section is based on the linearized relative dynamics, i.e., Eq. (10). As the indication of r0 varies in different problems (e.g., r0 is defined as the fixed artificial libration point in solar sail’s dynamics [26] and a reference periodic orbit of chief spacecraft in formation flying around the Earth [23][29]), in this case, r0 is the osculating state of a quasi-halo orbit refined by the multiple-shooting method presented in Section II. Thus, its eigenvalues and the corresponding manifolds (i.e., {σ1u+1 u-1 }, {γ2 u+2 u-2 }, and {γ3 u+3 u-3 }) of the uncontrolled dynamic system are time-varying, demonstrated in Fig.3. In Ref.[28], a preliminary HSP controller is built up using stable/unstable manifolds with the same control gains. As an important extension, Ref.[23] shows a HSP controller using center manifolds as well as stable/unstable manifolds and proved that the stability of such controller. Inherited from their works but using position feedbacks only, a HSP controller is constructed by stable/unstable/center manifolds as follows: Tc =Ȇ·¨rҧ
(14)
where ¨rҧ denotes the relative position feedback and the coefficient matrix Π is expressed as ାு ି ିு ଶ ା ାு ି ିு Ȇ=-G1 ıଵଶ (uଵା uଵାு +uଵି uଵିு ሻ െ G2 ߛଶଶ (uା ଶ uଶ +uଶ uଶ ሻǦG͵ ߛଷ (uଷ uଷ +uଷ uଷ ሻ
9
(15)
where Gi (i=1,2,3) are control gains correlated with manifolds. Hence, the linearized dynamical equations of the controlled system are ¨rҧሷ+2J¨rҧሶ=Urr ¨rҧ+Ȇ·¨rҧ
(16)
The fact that Urr and Ȇ are symmetry matrices guarantees the linear feedback controller to preserve the Hamiltonian structure [31]. It can be verified that only the feedback of relative position does not change the location of equilibrium point but its topological type. Thus, for a two-dimensional Hamiltonian system, assumptions and theorems have been proposed in [26] for poles assignment and optimization of control gains. Based on the propositions, topological types of equilibrium points can be further changed by the optimized control gains. B. Stability of HSP Controller Eq. (15) contains six coefficients, i.e., G1, G2, G3, ı1, Ȗ2 and Ȗ3. The latter three are set as 1 in this paper. The former three are control gains. The HSP controller used in this paper is inherited from results of Xu et al. [26] and Scheeres et al. [28]. It has been analytically proved that the controller’s capability of stabilizing the system can be guaranteed, as the control gains are large enough. For simplicity, in this paper, this conclusion is applied without proving and a group of large enough control gains (G1, G2, G3), i.e., G1=G2=G3=2, is used to stabilize the relative motion on a quasi-halo orbit around LL1 point. As demonstrated in Fig.3, the original uncontrolled system has three eigenvalues, real number σ1 and pure imaginary numbers γ2 and γ3, which are changed to three pure imaginary eigenvalues ω1, ω2 and ω3 of the controlled dynamic system, presented in Fig.4, indicating that the topological type of transient equilibrium is changed from hyperbolic to elliptic.
10
10
ω1 ω2 ω3
9 8
eigenvalue
7 6 5 4 3 2 1 0
0
20
40
60
80 100 120 140 160 time [day] Fig. 4 Time history of three pure imaginary eigenvalues of the controlled system.
To accurately estimate the relationship between transient stability and the control gains, groups of critical gains (G1, G2, G3) achieving transient stability is shown in Fig.5, where the critical gains are presented on G2-G3 plane and the tint of colour at each point indicates the corresponding value of G1. In (G1, G2, G3) space, such critical groups form a twodimensional surface, dividing the whole space into stable and unstable sub-spaces. The larger control gains, which are located on the upper right side above the critical groups, are capable of stabilizing the system while the smaller control gains, located on the lower left side of the critical groups, fail in stabilizing the system.
11
Critial G1 5
4.5
4
4
3.5 3
G3
3 2 2.5 1 2 0
1.5
-1 -1
0
1
2 G2
3
4
5
1
Fig.5 Critical control gains of transient stability presented on G2-G3 space: the tint of colour indicates the corresponding value of G1. In terms of a time-periodic Hamiltonian system [23][29], even if its equilibrium points are always elliptic during its period, its correlated controlled system may be still unstable. As a result, the concept of Floquet stability is introduced to examine if Floquet multipliers lie on a unit circle in complex plane and whether moduli of all the multipliers are equal to one. However, in time-quasi-periodic Hamiltonian system studied in this section, Floquet multiplier is unavailable. Long-term eigenvalues of the linearized controlled system over the whole time span, instead of one-period eigenvalues used to check regular Floquet multipliers, are solved from the following matrix equation: 0 ĭሶ = ቈU | halo rr r=r (t) 0
I 2J ĭ
(17)
where r0halo(t) is denoted as position vector of the quasi-halo orbit, and the initial value of Φ at epoch time is an identity matrix, i.e., I6×6. In order to investigate the long-term stability of the aforementioned controlled system, three eigenvalues with positive real or positive imaginary part are selected respectively from all three pairs of conjugate eigenvalues of the controlled system. As shown in Fig.6, long-term eigenvalues grow linearly with respect to time no
12
matter whether they are stable or not. Two eigenvalues are located steadily on imaginary axis while the other one’s position switches from real axis to imaginary axis with the change of chosen control gains.
a)
b)
Fig. 6 Time history of three long-term eigenvalues of controlled system with different control gains: a) G1=0.5, G2=0 and G3=0; b) G1=2, G2=0 and G3=0 in Fig. 6b; the unstable eigenvalue Σ1 with real part and other two eigenvalues in Fig. 6a is replaced by the three stable ones with pure imaginary part in Fig. 6b. The long-term eigenvalues of above linearized matrix equation can qualitatively demonstrate how relative motion can be stabilized expectantly by the HSP controller: for the unstable eigenvalue Σ1 with real part, the term e+Ȉ1 t will gradually enlarge instability; for the stable eigenvalues Ωj (j=1,2,3) with imaginary parts, periodic terms eiΩj⋅t =cos(Ωj⋅t)+i⋅sin(Ωj⋅t) are the expected ones to stabilize relative motion. Therefore, to stabilize the system, a HSP controller must eliminate unstable real eigenvalue Σ1. To demonstrate the relationship between long-term stability and the control gains, groups of critical gains (G1, G2, G3) of long-term stability is presented in Fig.7, where the critical gains are presented on G2-G3 plane and the tint of colour at each point indicates the corresponding value of G1. In (G1, G2, G3) space, such critical groups form a two-dimensional surface, dividing the whole space into stable and unstable sub-spaces. The larger control gains, which are located on the upper right side above the critical groups, are capable of stabilizing the system while the smaller control gains, located on the lower left side of the critical groups, fail in stabilizing the system. In conclusion, the sufficient and necessary condition for the stability of the controlled system by a HSP controller is to choose control gain group guaranteeing both transient stability and long-term stability. Additionally, the smaller control gains are selected, the less fuel consumption is required.
13
Critial G1 4
5
3.5 4 3
G3
3
2.5 2
2
1.5 1
1 0.5
0
0 -1 -1
0
1
2 G2
3
4
5
Fig. 7 Critical gain pairs of long-term stability presented on G2-G3 plane: the tint of colour at each point indicates the corresponding value of G1.
IV.
Numerical Simulations
A natural quasi-halo orbit acting as the reference orbit of the chief spacecraft is integrated from an initial state and refined by the multiple shooting method, as introduced in Section II. Only using the feedback of relative position in Section III, a non-natural relative trajectory of the deputy spacecraft in BCM is obtained from an initial relative position and velocity by HSP control. The Sun’s angle θs at epoch time is set as 0, and the initial amplitude Ax of the quasi-halo orbit is set as 1×105 km. Arbitrary initial relative position and velocity can be used to generate a non-natural relative trajectory of the deputy spacecraft. In this section, x0=y0=z0=1×10-5 LE-M (i.e., 3.844km) and x0 = y 0 = z0 = 1× 10−5 LE −M / TE −M . Besides, the control gains are selected large enough, as G1=2, G2=1, and G3=1, to stabilize the relative motion near the chief spacecraft, as illustrated in Fig.8.
14
5
0 -5 -10
0.05
0 -5
0.05 y [LE-M]
10
5
y [LE-M]
15
10 y [km]
y [km]
15
0 -0.05
0 -0.05
-10
-15 -10
-5
0 x [km]
5
-15
10
-20
0 z [km]
0.75
20
0.8
0.85 0.9 x [LE-M]
0.95
-0.1 -0.05 0 0.05 z [LE-M]
30
z [LE-M]
z [km]
z [km]
0 -10
z [LE-M]
20
10
0 -20
-20 -30 -10
-5
0 x [km]
a)
5
10
0
0
y [km]
0.05
10 -10
-0.1
0 x [km]
-10
0 -0.05 -0.1
-0.05
10
Chief Deputy
0.05
0.05
20
0.1
0.75
0.8
b)
0.85 0.9 x [LE-M]
0.95
0 -0.05 y [LE-M]
0.88 0.86 0.84 x [LE-M]
Fig. 8 A relative trajectory generated by HSP control about a quasi-halo orbit of LL1: a) the relative trajectory of the deputy with respect to the chief over six months; b) absolute trajectories of the deputy and the chief over one month. The cluster flight based on continuous low-thrust technique greatly advanced the utilization of electromagnetic and electrostatic forces on innovative electric propellers. According to the continuous control acceleration by HSP control shown in Fig.9a, the maximum acceleration is supposed to be no more than 1×10-6m/s2 in order to obtain the expected bounded trajectories. Fuel consumption can be calculated by the following cost function: t
ΔV = ³ Tc dt .
(18)
0
As shown in Fig.9b, the fuel consumption for station-keeping over 6 months is 2.8m/s. Taking PPS-1350 Hall engine equipped by SMART-1 with a high specific impulse 1643.4s as an example, fuel consumption in one year is no more than 1.0% of the entire mass. -7
14
x 10
12
2.5
2
8
ΔV [m/s]
control acceleration [m/s 2]
10
6 4
1.5
1
2 0
0.5 -2 -4
a)
0
20
40
60
80 100 time [day]
120
140
0
160
b)
15
20
40
60
80 100 time [day]
120
140
160
Fig.9 Time history of the continuous acceleration and fuel consumption function over six months by HSP control of G1=2, G2=1, and G3=1.
V.
Conclusion
In this paper, the Hamiltonian structure-preserving control (HSP) is applied to stabilize the formation flying in the restricted Sun-Earth-Moon-spacecraft system. A simplified model of Sun-Earth-Moon-spacecraft system, i.e., bi-circular model (BCM), is employed in Earth-Moon rotating frame, where a quasi-halo orbit of LL1 is generated and corrected by a multiple shooting method acting as reference orbit of the chief spacecraft. The equilibrium points of the linearized relative dynamic in bi-circular model are studied and their topologies are changed from hyperbolic to elliptic by HSP control with feedbacks of stable, unstable and center manifolds. The stability of the controlled system by HSP control is analyzed. Such conclusion can be drawn that whether all three eigenvalues of the controlled system are located on imaginary axis is critical for transient stability. The long-term eigenvalues solved from state-transition matrix of the controlled system over the whole time span determine long-term stability of the controlled system. The sufficient and necessary condition for stability of the controlled system is that control gains locate inside the region where both transient stability and long-term stability is guaranteed. The station-keeping controller developed in this paper has much potential in formation flying around other libration points of restricted four-body problem, such as LL2 and ELi (i=1,2,3). It can also be used to deal with formation flying in weak stability boundary theory, the flight duration of which is long enough (about half a year, such as Hiten and NASA’s GRAIL) to accomplish explorations in deep space.
Acknowledgments The research is supported by the National Natural Science Foundation of China (11172020 and 11432001), Beijing Natural Science Foundation (4153060), and the Fundamental Research Funds for the Central Universities.
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