JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Engineering Notes Formation Flying on Elliptic Orbits by Hamiltonian Structure-Preserving Control Ming Xu∗ and Yuying Liang† Beihang University, 100191 Beijing, People’s Republic of China Downloaded by TUFTS UNIVERSITY on October 28, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G002141 DOI: 10.2514/1.G002141 I. Introduction F ORMATION flying on elliptic orbits is drawing increasingly more attention from the astronautics society. Several missions have already been launched or are in progress, such as GRACE [1], TerraSAR-X, and TanDEM-X [2]. The advantages of bounded formation configurations include the ability to observe the Earth’s surface by interferometric synthetic aperture radar, monitoring other spacecraft, and constituting a fractionated spacecraft system. Using formation flying on an elliptic orbit for on-orbit surveillance for inspection or repair requires rapid changes in formation configuration to achieve full three-dimensional imaging or to create a trajectory that is unpredictable to noncooperative spacecraft for military tasks [3]. Therefore, a large body of literature on station-keeping control of formation flying on an elliptic orbit has been published. Several control design models have been introduced to cope with orbital eccentricity including time-explicit solutions such as Melton’s equation [4] and trueanomaly-based solutions, such as the Lawden equation [5], TschaunerHempel (T-H) equations [6], and the relative orbital element method [7,8]. Sinclair et al. [9] interpreted the fundamental solutions of the T-H equations as generalizations of the drifting two-by-one ellipse, which describe the motion of a deputy satellite relative to a chief satellite with arbitrary eccentricity. The T-H equations have a Hamiltonian structure, whereas Melton’s equation does not. Hence, more fuel is saved using a station-keeping controller based on the T-H equations than the Melton formation controller due to the elimination of its non-Hamiltonian term. To generate control accelerations, impulsive thrust and low thrust are widely used to control a formation. Impulsive thrust has been further studied to achieve formation flying on elliptic orbits. For example, Yin and Han [10] derived a new elliptic formation-flying model using a new set of relative orbital elements and developed impulsive feedback control for both in-plane and out-of-plane relative motions. Gurfil [11] presented the necessary conditions for bounded formation flying on elliptic Keplerian orbits and derived an optimal single-impulse formation-keeping maneuver based on relative state variables. With the development of astronautic techniques, low thrust has become a popular research topic from engineering and theoretical viewpoints. Zanon and Campbell [12] developed a fast solution to an individual spacecraft minimum time or fuel maneuver using the Hamilton–Jacobi–Bellman formulation. They also formulated an Received 4 April 2016; revision received 11 September 2017; accepted for publication 18 September 2017; published online 19 October 2017. Copyright © 2017 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0731-5090 (print) or 1533-3884 (online) to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. *Associate Professor, School of Astronautics; xuming@buaa.edu.cn. † Ph.D. Candidate, School of Astronautics; buaa_liangyuying@163.com. optimal planner for spacecraft formations on elliptic reference orbits. Liu et al. [13] considered the control of entire-formation maneuvering in low-thrust Earth-orbiting spacecraft formation flying and developed a coordinated control scheme based on the leader–follower approach. Cho et al. [14] presented and examined a general analytical solution to the optimal reconfiguration problem of satellite formation flying on an arbitrary elliptic orbit by variable low thrust. Particularly, ion thrusters provide excellent thruster performance and have successfully been implemented as the primary propulsion system on several science missions (e.g., Deep Space and Dawn by NASA [15,16], GOCE and SMART-1 from the ESA [17,18], and Hayabusa2 by the Japan Aerospace Exploration Agency [19]. According to the structure of a controlled system, all controllers can be classified into dissipative and Hamiltonian structure-preserving (HSP) controllers. The dissipative controllers can change the unstable positive real part of eigenvalues into stable negatives (i.e., change the topological type of equilibrium points from unstable points to focal points or nodes). From the viewpoint of stability, a dissipative controller can achieve asymptotic stability. Thus, the deputy approaches the chief gradually. The dissipative controllers are built to track a reference relative configuration. According to the variable structure model reference adaptive control theory, Lee and Singh [20] designed a new output feedback control system for the formation flying of satellites on elliptic orbits. Wei et al. [21] constructed a two-degree-of-freedom signal-based optimal H ∞ robust output feedback controller for satellite formations in an arbitrary elliptical reference orbit. To improve the system robustness in the presence of external disturbances and J2 perturbation, Chen et al. [22] proposed a reconfiguration and formationkeeping control law based on the nonlinear sliding mode function, where nonlinear gains of exponential growth are used to replace the constant gain of the traditional sliding mode function. Guerman et al. [23] proposed a Newton-type method to reduce the modeling error of a dissipative control law to create periodic relative trajectories in the case of both passive magnetic and spin stabilization. In contrast, the Hamiltonian system does not allow a focal or node point due to its conservative energy [24]. The equilibrium points change into stable elliptic points that are surrounded by bounded trajectories, which do not converge onto them. Bounded motions have various advantages and applications. These include the quasi periodicity of subsatellite points for nonrepeating ground-track orbits to achieve full coverage over the Earth, rather than covering only specific regions by repeat orbits [25]. In addition, bounded motions enabled the placement of the ISEE-3 in a halo orbit with a nonzero z amplitude to subtend 4.5 deg from the L1 point, avoiding the loss of communication signals, due to the sunrays, at the L1 point [26]. The HSP controller is a powerful tool for producing bounded trajectories near an equilibrium point in both time-independent [27,28] and timedependent Hamiltonian systems [4,29]. The original application of the HSP controller was to stabilize the motions of a spacecraft in the Hill restricted problem [27]. Subsequently, the controller has been applied to stabilize the motions of solar sails [28], frozen high-eccentricity orbits [30], halo orbits [31], and formation flying in a J 2 -perturbed mean circular orbit [32] or on an elliptic orbit. Xu et al. [33] developed an HSP control to stabilize the relative motions on an elliptic orbit based on the time-based Melton’s equation. However, the authors declare that extra fuel was consumed to stabilize the relative motion of formation flying by HSP control due to the non-Hamiltonian terms in Melton’s equation [33]. To overcome this problem, we studied the stabilization of formation flying using T-H equations. As an extension of [33], this Note addresses two different types of formation flying on a reference elliptic orbit that are stabilized by an HSP controller, in which no perturbation is considered except for the control acceleration provided by continuous low-thrust propulsion. The first type of formation (single formation) considers the deputy Article in Advance / 1 Downloaded by TUFTS UNIVERSITY on October 28, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G002141 Two different formation scenarios with different reference configurations are discussed in this Note. In the first scenario, the deputy is not required to track any specific reference configuration but is expected to fly around the chief. This scenario has potential applications in on-orbit surveillance, inspection, or repair missions for only the chief. In the second scenario with double-formation flying, the deputy is controlled to fly on a controlled bounded trajectory around the subchief without ultimately approaching it, whereas the subchief flies around the superchief in a natural two-spacecraft formation. This scenario can be used to on-orbit monitor or observe the subchief. Figure 1 illustrates the single- and double-formations. Compared with the first scenario using only a single chief, the second scenario involves both the superchief and subchief. The relative dynamics of the deputy are modeled in either the subchief’s double-formation or the superchief’s single-formation localvertical/local-horizontal (LVLH) frame, and the relative dynamics of the subchief is modeled in the superchief’s LVLH frame. These frames are defined as follows. In the subchief’s LVLH frame, the origin of the frame is set at the origin of the subchief, the x axis points from the Earth to the subchief, the z axis points normal to the orbital plane, and the y axis is determined by the right-hand rule, shown in Fig. 2. In the superchief’s LVLH frame, the origin of the frame is set at the origin of the superchief, the x axis points from the Earth to the superchief, the z axis points normal to the orbital plane, and the y axis is determined by the right-hand rule. The subchief (or superchief) is located at 0; 0; 0T in the corresponding LVLH frame. y z ECI frame e Periodic Relative Dynamics for Formation Flying on an Elliptic Orbit x LVLH frame lan II. zi lp spacecraft in controlled flight around the chief spacecraft. The second type of formation considers three spacecraft, where the deputy is controlled to fly around the subchief spacecraft in a specific reference configuration, and the subchief spacecraft flies on natural bounded trajectories around the superchief spacecraft. This formation is referred to as double-formation flying because of the coexisting configurations. It will be shown that the double-formation controller can be transformed into a single-formation controller through the relative orbital elements between the superchief and the subchief spacecraft. Furthermore, in contrast to the feedback of full-dimensional manifolds [33], only the center manifolds are used in this study, without the help of stable or unstable manifolds. The performance of the HSP controller is systematically evaluated by the T-H formation, including the critical gain, controlled frequencies, foundational motions, and the maximumlikelihood optimizations, which are only affected by the eccentricity and control gain. The semimajor axis has no effect on the performance because it is eliminated from the controller by length normalization. Furthermore, it is demonstrated that the performance of the doubleformation controller can be evaluated by the eccentricity of the subchief regardless of the reference configuration. / ENGINEERING NOTES ita Article in Advance orb 2 equator yi Xi orbital plane Fig. 2 LVLH frame and ECI frame Ixi ;yi ;zi . For a given reference elliptic orbit, the true anomaly, semimajor axis, eccentricity, and orbital radius are denoted as f, a, e, and r a1 − e2 ∕1 e cos f, respectively. The full nonlinear relative true-anomaly-based dynamics derived by Szebehely and Giacaglia [34] are used in this study, differing from the time-based analysis in [31]. The true-anomaly-based dynamics are represented in the chief’s LVLH frame as ρ 0 0 Aρ 0 W ρ Tc (1) ~ y; ~ z where the physical relative position vector ρ~ x; ~ T points from the subchief to the deputy in the subchief’s LVLH frame or from the superchief to the subchief in the superchief’s LVLH frame and is then normalized by ρ x; y; zT ρ~ r (2) Tc is the HSP control acceleration imposed only on the deputy, and 2 3 0 −2 0 A 42 0 05 0 0 0 In this Note, the continuous control acceleration is provided by an ion thruster. All the first and second derivatives in Eq. (1) are with respect to f except for W ρ, which is the partial derivative of the pseudopotential function W with respect to ρ, where " 1 1 2 x y2 −z2 ⋅ e cos f 1 e cos f 2 # 1 p − 1 x 1 x2 y2 z2 W a) b) Fig. 1 Schematic for a) single- and b) double-formation flying. The red, blue, and green marks indicate the superchief, subchief, and deputy, respectively; the dashed and solid lines indicate the quasi-periodic and periodic trajectories, respectively. (3) Because of the length normalization in Eq. (2), the semimajor axis is removed from the relative dynamics and the HSP controller. Thus, the semimajor axis has no effect on the controller’s performance except on the physical three-dimensional (3-D) trajectory and the fuel cost. In the double-formation case, the periodic reference configuration that the subchief uses to fly around the superchief is denoted as ρ0 f with ρ0 f 2p ρ0 f. This periodic configuration can be achieved by an appropriate initial condition, such as setting the relative semimajor axis between the subchief and superchief’s Δa to zero and setting Δe, Δω, Δi, ΔΩ, and Δf arbitrarily using the relative orbital element method [35], or setting the relative position and velocity according to the Lawden or Carter condition [36]. Thus, the chief’s location at 0; 0; 0T and the reference configuration ρ0 f are the equilibrium solutions to Eq. (1). Moreover, if the value of the configuration is equal Article in Advance to zero (i.e., the superchief shares its location with the subchief), the double-formation degenerates into the single formation. Defining the general form of relative motion around a reference configuration as δρ, where ρ ρ0 δρ, and substituting it into Eq. (1) and ignoring highorder terms yields δρ 0 0 Aδρ 0 Bfδρ Tc it is preferable to decompose the singular matrix D into the Jordan canonical form rather than the diagonal matrix using eigenvalue decomposition. In the Jordan-Chevalley decomposition [38], there exists a nonsingular matrix P to decompose D such that 2 0 60 6 60 P−1 DP 6 60 6 40 0 (4) where B is a periodic matrix given as 2 3 W xz W yz 5 W zz W xy W yy W yz (5) 2 3 0 0 0 0 5 0 −1 3∕1 e cos f ~ 0 Bf −4 0 (6) d δρT ; δ_ρT T D ⋅ δρT ; δ_ρT T dt L1 e f 350 L2 t 1 t and 0 1 2 cos nt − sin nt 6 6 sin nt cos nt 6 6 6 0 0 4 0 0 0 cos nt sin nt 0 3 7 7 7 7 − sin nt 7 5 cos nt 0 0.5 0 -1 300 3 0 0 0 0 7 7 0 −n 5 n 0 Because of the transient instability caused by the double zero eigenvalues shown in Fig. 3, it is necessary to impose an HSP controller on the full relative dynamics ρ 0 0 Aρ 0 W ρ Tc , changing the topology of the equilibrium point. The classical HSP theory is based on the feedback of the stable and unstable manifolds of the equilibrium point as the topology transforms from an unstable -1 250 0 −n 6n 0 and L2 6 40 0 0 0 A. HSP Station-Keeping Control -0.5 200 (7) III. Hamiltonian Structure-Preserving Control to Stabilize Relative Motions on an Elliptic Orbit -0.5 [0 ] The second term composed of cos nt and sin nt is related to the periodic modes, whereas the first term eL1 t containing t is related to the linearly growing modes. imaginary part 0 2 0 1 150 0 1 0 0 eL1 t 1 100 0 L2 where 1.5 50 3 0 0 7 7 0 7 7 L1 0 1 7 7 −n 5 0 0 0 0 0 −n 1 0 0 0 0 0 n Thus, the solution to the C-W equation can be obtained as Lt δρt e 1 0 eDt P ⋅ ⋅ P−1 δ_ρt 0 eL2 t 1.5 0.5 a) Whereas Eqs. (4) and (6) are referred to as the T-H equation [37], Eqs. (4) and (5) will be referred to as the general T-H equation. Although B and B~ have different forms, they are both periodic with a period of 2π. The topology of the relative dynamics of the subchief’s reference configuration ρ0 f or superchief’s zero position can be indicated by the six eigenvalues of the T-H equation, all of which have zero real parts and two of which are always zero. Thus, the nonzero imaginary eigenvalues are denoted as fσ 1 i; σ 2 ig with their − − eigenvectors, called center manifolds, denoted as u 1 , u1 and u2 , u2 , respectively. Figure 3 depicts the uncontrolled eigenvalues of the superchief on a reference elliptic orbit with e 0.2. Although there is no real eigenvalue for the general T-H equation, the linearized dynamics are still unstable due to their linearly growing mode induced by the double zero eigenvalues. This conclusion can be derived from the special case of e 0 [i.e., the Clohessy–Wiltshire (C-W) equation], which has two pairs of double eigenvalues: 0, 0, ni, and −ni, where n is the orbital rate of the circular orbit. Differing from the eigenvalues of the time-based Melton’s equation [28], the double ni and double zeros induce the linearly growing modes in the C-W equation. In the general form -1.5 0 1 0 0 0 0 0 0 n 0 0 0 0 where where the elements of Bf are the second partial derivatives of W with respect to x, y, or z. Any reference to xf, yf, and zf in B comes from the position vector of the reference configuration ρ0 f. Hence, Bf 2π Bf can be derived from ρ0 f 2π ρ0 f. Compared with the time-based Melton’s equation containing the skewsymmetric B, the true-anomaly-based B listed in Eq. (5) is symmetric. In the degenerate case, where the superchief and subchief are both at ρ0 0; 0; 0T , B becomes real part Downloaded by TUFTS UNIVERSITY on October 28, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G002141 Bf −W ρρ W xx −4 W xy W xz 3 / ENGINEERING NOTES -1.5 0 50 100 150 200 f [0 ] 250 b) Fig. 3 Topology of the chief’s position at 0;0;0T using uncontrolled T-H equation. 300 350 4 Article in Advance hyperbolic type into a stable elliptic type. In the Melton formation, the skew-symmetric matrix B results in nonzero eigenvalues except at perigee and apogee, which indicates that the stable and unstable manifolds can be used by the classical HSP controller to stabilize this formation [33]. However, due to the existence of zero eigenvalues, there are no stable or unstable manifolds in the T-H formation. Therefore, unlike the feedback of the full-dimensional manifolds [33], only the center manifolds are employed in this study to stabilize the T-H formation. No research has been published on the use of the center manifolds to stabilize a system without the help of stable or unstable manifolds, although it has been proven to be effective in time-independent systems [28]. In this time-periodic T-H formation system, the feedback of the two-dimensional center manifolds is designed as Tc N ⋅ δρ (8) Downloaded by TUFTS UNIVERSITY on October 28, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G002141 where the coefficient matrix of the controller N is expressed as − − − − − − N −G1 σ 1 σ 1 u1 u1 u1 u1 −G2 σ 2 σ 2 u2 u2 u2 u2 (9) where the overbar indicates the complex conjugate of the vectors, and − − the two branches of the center manifolds u 1 u1 and u2 u2 have their own control gains G1 and G2 . For simplicity, G1 is set equal to G2 , thus only one gain G is used to denote the two identical gains in the following sections. Correspondingly, Eq. (9) can be rewritten as − − − − − − N −Gσ 1 σ 1 u1 u1 u1 u1 σ 2 σ 2 u2 u2 u2 u2 (10) Some propositions have been postulated for the two-dimensional Hamiltonian system [28] regarding the poles assignment and the optimization of the control gain to change the topological type of the equilibrium points. Comparing the equilibrium points with the nonzero eigenvalues in other applications, such as the Melton formation [33] or the formation on a J 2 -perturbed circular orbit [32], the T-H formation can be considered as an application that uses the center manifolds without the use of stable or unstable manifolds. In addition, from a practical engineering perspective, although the design of the controller is time dependent, its application is simpler than other controllers because only the feedback of the relative positions is employed. The true-anomaly-based coefficient matrix Nf of the controller relies only on the topological structure of the subchief’s reference configuration or the superchief’s position at 0; 0; 0T ; thus, it can be calculated offline and stored in the onboard computer. The controller then reads the matrix Nf according to the space-borne clock and creates the control output from the feedback of δρ. B. HSP Station-Keeping Controller Performance Unlike dissipative controllers that require a certain time to drive the deputy to its expected position or state, such as a proportionalintegral-derivative controller, an HSP controller has an immediate Fig. 4 / ENGINEERING NOTES effect on stabilizing the relative motion once it is activated. Thus, problems such as steady-state error, overshoot, or slow response speed do not occur with an HSP controller. Instead, its advantages include critical gain, controlled frequencies, foundational motions, and maximum-likelihood optimizations. Compared with the time-independent Hamiltonian system, the controlled periodic system may be unstable, even though the equilibrium is always elliptic during its periods [39]. Therefore, it is necessary to evaluate the eigenvalues of the monodromy matrix over one period, which provides the information about Floquet multipliers and the controlled frequencies. The Floquet stability is guaranteed by a control gain larger than the critical value Gc . For the time-periodic T-H formation discussed in this Note, the formation is stabilized only if the Floquet multipliers (i.e., the eigenvalues of the monodromy matrix) are located on a unit circle in the complex plane. The Floquet multipliers are denoted as eθk i (k 1; 2; 3) and the normalized eigenvector pairs as η1 η1 , η2 η2 , and η3 η3 , where θk characterizes the controlled relative trajectories, and the eigenvectors characterize the six periodic foundational motions with the periods of 2π∕θk . The ergodic relationship among θk , the control gain G, and the eccentricity e is shown in Fig. 4, where the boundary indicating the critical gain Gc is marked by blue lines. The larger the eccentricity is, the larger the critical gain is. The minimum critical gain is 7.9 at e 0. All three frequencies are considerably affected by G and e and have a similar polyline-like distribution for the Melton formation, as shown in [33]. The six eigenvectors of the Floquet multipliers span the entire space of the initial conditions δρT ; δ_ρT T , that is, any initial relative position and velocity vector can be decomposed as δρT ; δ_ρT T 6 X αi νi i1 where αi is the coordinate of the initial state vector under the basis set of (ν1 , ν2 , ν3 , ν4 , ν5 , ν6 ), and νi (i 1; 2; 3) is defined as the real part of ηj (j 1; 2; 3) and νi (i 4; 5; 6) is defined as the imaginary part of ηj (j 1; 2; 3). Thus, all bounded relative trajectories near the equilibrium can be considered as a combination of six foundational motions. Only three of these motions are presented because the trajectory generated by the initial conditions vj , j 1; 2; 3 has the same shape as vj3 but with opposite directions. The general trajectories are quasi periodic due to the different frequencies of the foundational motions. Figure 5 illustrates the deputy’s foundational motions and quasi-periodic relative trajectory created by a control gain G 8 on a reference elliptic orbit with a 8178.137 km, e 0.2 in the chief’s single-formation LVLH frame, and Fig. 6 shows the controlled eigenvalues for the chief’s position at 0; 0; 0T . These values are selected to allow a comparison with the results in [33]. In Fig. 5, the initial relative position and velocity of the deputy in the chief’s LVLH frame are set as x0 y0 z0 1 km and y_ 0 2nx0 ⋅ η, x_ 0 ny0 ∕2 ⋅ η, z_0 0, where n is the mean orbital rate of the elliptic orbit, and the constant η is set to 1.5 to avoid having the initial relative conditions become the generalized energy-matching Ergodic representation of controlled frequencies θk (k 1;2;3) in the G–e space. Article in Advance 5 / ENGINEERING NOTES 1 1 y [km] y [km] 0.5 0 0 -0.5 -1 -1 -2 -1 0 x [km] 1 2 v -1 v 1 1 3 1 z [km] Downloaded by TUFTS UNIVERSITY on October 28, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G002141 z [km] 1 v 2 0 z [km] 0 0 -1 -1 2 1 0 0 -2 Fig. 5 -1 0 x [km] 1 2 -1 y [km] x [km] -2 Foundational motions and quasi-periodic relative trajectory created by the HSP controller. 1.5 8 6 1 4 imaginary part real part 0.5 0 -0.5 2 0 -2 -4 -1 -1.5 -6 0 50 100 150 a) Fig. 6 200 f [0 ] 250 300 1 J~ 2f1 f1 0 TTc Tc df 1 2f1 Z f1 -8 0 50 100 150 200 f [0 ] 250 300 350 b) Topology of the chief at 0;0;0T for the controlled T-H equation. condition by Gurfil [40]. Compared with the uncontrolled eigenvalues shown in Fig. 3, the HSP controller can change the topology of the equilibrium point into an elliptic type. Because the controller can stabilize any initial relative position and velocity, the foundational motions are used to evaluate its performances in terms of the fuel consumption. An averaging quadratic cost function is defined to measure the fuel consumption on the true anomaly interval 0; f1 [41,42]: Z 350 δρT GT CT fCfGδρ df 0 Zf 6 X 6 1 1 X αi αj ζ i fT GT CT fCfGζ j f df 2f1 i1 j1 0 (11) where αi is defined in the first half of Sec. III.B, and ζ i are the position components of the periodic orbit generated by the eigenvector vi. The coefficient matrix of the controller N can be decomposed as Nf GċCf given by 2 − − − σ 1 σ 1 u11 u11 u11 u11 − − − σ 1 σ 1 u11 u12 u11 u12 − − − σ 2 σ 2 u21 u21 u21 u21 − − − σ 2 σ 2 u21 u22 u21 u22 # 6 6 σ σ − u u u− u− 6 2 2 21 21 21 21 Cf 6 6 σ σ − u u u− u− 4 1 1 11 11 11 11 " G −G1 −G2 0 0 0 0 −G1 −G2 3 7 − − − 7 σ 2 σ 2 u21 u22 u21 u22 7 7; − − − 7 σ 1 σ 1 u11 u12 u11 u22 5 (12) − − and u ij (or uij ) is the jth component of ui (or ui ). In Eq. (10), ζ i f is the position component of any foundational motions generated by the eigenvector vi for i 1; 2; 3, and f1 is chosen as the maximum of 2π∕θ1 , 2π∕θ2 , and 2π∕θ3 . The cost matrix MG; e is defined by its component in the ith row and jth column as Mij 1 f1 Z f1 0 ζ i fT GT CT fCfGζ j f df (13) 6 Article in Advance / ENGINEERING NOTES 1 0.98 0.96 trM TH / trM M 0.94 0.92 0.9 0.88 0.86 0.84 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Downloaded by TUFTS UNIVERSITY on October 28, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G002141 e Fig. 8 Relationship between eccentricity and ratio of the traces of the T-H formation and the Melton formation at their respective critical gains. Fig. 7 Ergodic representation of maximum-likelihood optimization index in the G–e space. The cost function J~ can be rewritten as 1 ~ JG; e; α αT MG; eα 2 (14) where α α1 ; α2 ; : : : ; α6 T is an arbitrary constant vector. In the general case, the control gain minimizing the trace of MG; e, that is, trM 6 X λi i1 will minimize J~ in terms of the maximum likelihood. The relationship between the trace of MG; e, the control gain and the eccentricity is shown in Fig. 7, which represents the maximum-likelihood optimization index defined as the ratio of the trace for any pair G; e to the trace for zero eccentricity and its critical gain. It is shown that an increase in both gain and eccentricity will increase the fuel requirement. However, the gain has a greater effect on J~ than does the eccentricity e. The T-H equation has a Hamiltonian structure, whereas Melton’s equation does not. Therefore, it is interesting to investigate whether the Melton formation controller requires more fuel to eliminate its nonHamiltonian term. Although the two formations have different critical control gains for a reference elliptic orbit, the ratios of the two traces at the respective critical gains are illustrated for a range of eccentricities in Fig. 8. However, when e is equal to zero, the T-H equation has the same form as Melton’s equation, which indicates that they have the same trace of M. For any reference elliptic orbit, the T-H formation controller shows a smaller fuel consumption than the Melton formation from the viewpoint of a maximum-likelihood optimization. C. Double-Formation by HSP Station-Keeping Control For some traditional formation-flying missions using low-impulse chemical propulsion, such as the TerraSAR-X and TanDEM-X [43] formation, and time-difference-of-arrival navigation found in the Naval Ocean Surveillance System [44], the spacecraft are guided to a specific initial condition by impulsive control and then move in a natural periodic configuration under no control (i.e., without fuel consumption). However, to on-orbit monitor a noncooperative spacecraft, the quasi-periodic orbits are preferable to periodic ones because they can be maintained more easily by ion-based propulsion and achieve full three-dimensional imaging. Thus, the monitoring deputy flies on a quasi-periodic trajectory about the subchief under HSP control and the noncooperative superchief and subchief serving traditional formation missions fly in a periodic configuration without any control. In the following, e, i, Ω, and ω represent the eccentricity, inclination, longitude of the ascending node, and argument of periapsis of the superchief’s elliptic orbit, respectively. The relative elements, Δa, Δe, Δi, ΔΩ, Δω, and Δf are defined as the subchief’s elements subtracted from the superchief’s elements. Generally, Δa is selected as zero to maintain the periodic configuration, whereas the other relative orbital elements are used to design different types of configuration. Because the natural subchief’s configuration is without any control, the double formation can be considered as a single-formation scenario with respect to the subchief’s elliptic orbit with e Δe. However, the double formation requires a different controller for the deputy than the single formation because the former uses the general T-H equations (2) and (5), whereas the latter uses Eqs. (2) and (6). To determine whether the two controllers result in the same performance, the relative position of the deputy with respect to the subchief can be expressed in two ways: in the subchief’s LVLH frame as δρ and in the superchief’s LVLH frame as Δρ. These frames have the following relationship: δρ εΔe; Δf ⋅ RΔi; ΔΩ; Δω Δf ⋅ Δρ (15) where εΔe; Δf 1 e cos f1 − e Δe2 1 e Δe cosf Δf1 − e2 due to the different length normalizations in Eq. (2) and can be approximated as one for small Δe. R is a coordinate matrix that transforms the superchief’s LVLH frame into the inertial frame and then into the subchief’s LVLH frame and can be approximated as R Rz Δω ΔfRx ΔiRz ΔΩ, where Rx and Rz are the unit transformation matrix along the x axis and z axis, respectively. Substituting Eq. (15) into the linearized uncontrolled equation δρ 0 0 Aδρ 0 W ρρ δρ 0 yields RΔρ 0 0 R−1 ARΔρ 0 R−1 W ρρ RΔρ 0 0 (16) When comparing Eq. (16) with the linearized uncontrolled equation with respect to the superchief (i.e., Δρ 0 0 AΔρ 0 BΔρ 0), it follows that R−1 AR A; R−1 W rr R B (17) Therefore, it can be proven that the linearized equation δρ 0 0 Aδρ 0 W ρρ δρ 0 has eigenvalues σ i for i 1; 2; 3 and eigenvectors − 00 0 R ⋅ u i and R ⋅ ui if Δρ AΔρ BΔρ 0 has eigenvalues σ i for i 1; 2; 3 and eigenvectors u and u−i . Thus, substituting Eq. (15) into i Eq. (8) yields Article in Advance T − − − T Tc −G ⋅ σ 1 σ 1 Ru1 u1 R Ru1 u1 R T − − − T σ 2 σ 2 Ru2 u2 R Ru2 u2 R ⋅ εRΔρ − − − − − − εR ⋅ −Gσ 1 σ 1 u1 u1 u1 u1 σ 2 σ 2 u2 u2 u2 u2 Δρ (18) deputy’s controlled trajectory in the Earth-centered inertial (ECI) frame and then by transforming the relative motions from the ECI frame to the subchief or superchief’s LVLH rotating frame to present the 3-D relative trajectories. Before imposing the true-anomalybased controller formulized in Eqs. (8) and (9) onto the deputy’s twobody dynamics, the dimensionless Tc is transformed into the physical control acceleration Substituting Eqs. (15) and (18) into the control equation δρ 0 0 Aδρ 0 W ρρ δρ Tc yields the other control equation Δρ 0 0 BΔρ − σ 2 σ 2 u2 u2 − −Gσ 1 σ 1 u1 u1 IV. μ1 e cos f T~ c Tc a2 1 − e2 2 3 u−1 u−1 u−2 u−2 Δρ where the term on the right side has the same form as that of the HSP controller, which addresses the single-formation scenario that stabilizes Δρ. Therefore, both controllers are equivalent and have the same performances. In other words, the double-formation HSP controller can be evaluated by the eccentricity (e Δe) and gain G according to the G − e-space ergodic representations of the critical gain and controlled frequencies in Fig. 4, or by the maximum-likelihood optimization in Fig. 7, and is independent of Δi, ΔΩ, Δω, and Δf. Although the performance of the double-formation controller depends only on the eccentricity of the subchief regardless of the reference configuration, the deputy is stabilized by the HSP controller with G 9 following the reference configuration of the subchief with Δe 4.6 × 10−4 , Δi 1 × 10−3 , ΔΩ 0, Δω 0, and Δf 0.6 × 10−4 with respect to the superchief, and both the deputy and the subchief fly around the superchief on a reference orbit of a 12;000.137 km and e 0.5, as shown in the superchief LVLH frame in Fig. 9. The quasi-periodic trajectory of the deputy around the subchief is propagated from the same initial values of position and velocity as those (but expressed in the subchief’s LVLH frame) in Fig. 5. Z is used as an indication for the fuel consumption and is demonstrated in Fig. 11, where tf represents the final time of 30 days. In conclusion, the T-H formation HSP controller can save approximately 6–7% fuel mass compared with the Melton formation for this particular test case. Taking the PPS-1350 Hall engine, which was used in ESA’s SMART-1 mission and has a specific impulse I sp of 1643.4 s, as an example [45], the ratio of the fuel consumption mass to the initial mass for every month is Δm 1 ΔV ≈ 1.4‰ mtotal I sp g where g is the standard gravity. 5 5 y [km] y [km] deputy's controlled trajectory 10 0 -5 -5 -10 -10 0 x [km] 10 kT~ c k dt 0 10 -10 tf ΔV subchief's reference configuration 0 -20 -10 z [km] 0 10 5 z [km] 0 -5 -10 5 0 -5 -10 -15 10 -15 5 0 0 0 10 -5 y [km] -10 x [km] x [km] Fig. 9 Application of the HSP controller for a double-formation scenario with the subchief on the elliptic reference orbit. -10 (19) and then transformed from the LVLH frame into the ECI frame. In Eq. (19), μ is the gravitational constant. To create the quasi-periodic trajectories in Fig. 5, the HSP controller with G 8 requires control accelerations less than 6 × 10−6 m∕s2 , as shown in Fig. 10. As a comparison with the Melton formation [33] with the same control gain, the velocity increment Numerical Simulations The numerical simulations in this Note are achieved by propagating both the chief’s uncontrolled reference trajectory and the z [km] Downloaded by TUFTS UNIVERSITY on October 28, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G002141 AΔρ 0 7 / ENGINEERING NOTES 8 Article in Advance / ENGINEERING NOTES single frequency. Future research could focus on time-periodic systems with double frequencies, such as the stabilization of formation flying in a J2 -perturbed elliptic orbit, which have a fast frequency associated with the orbital period and a slow frequency associated with the evolution period of the mean argument of perigee. Acknowledgments The research is supported by the National Natural Science Foundation of China (11772024 and 11432001) and the Fundamental Research Funds for the Central Universities. The authors thank George Knox, Ramily Santos, and the reviewers for their suggestions to improve this Note. Fig. 10 Time history of the continuous acceleration implemented by the controller over one day. 35 T-H Formation Melton Formation 30 25 V [m/s] Downloaded by TUFTS UNIVERSITY on October 28, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G002141 References 20 15 10 5 0 0 5 10 15 time [day] 20 25 30 Fig. 11 Time history of the fuel consumption using the T-H and Melton formation controllers over 30 days. V. Conclusions The stabilization of relative trajectories about an elliptic reference orbit using only feedback from relative positions is investigated in this Note. Compared with previous controllers based on Melton’s equation, the current controller derived from the Tschauner–Hempel equation consumes less fuel due to its Hamiltonian structure, which uses the center manifolds as the only feedback without the help of stable or unstable manifolds. In contrast to dissipative controllers, the proposed Hamiltonian structure-preserving (HSP) controller has an immediate effect of stabilizing the relative motion once activated. Its performances, such as the critical gain, controlled frequencies, foundational motions, and the maximum-likelihood optimizations are manifested in the G–e space. The semimajor axis has no effects on these performances except on the physical 3-D trajectories and fuel costs because it is eliminated from the formulations of the controller by length normalization. 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