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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]
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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. Two formation scenarios (a single
formation consisting of a chief and a deputy and a double formation
consisting of a superchief, subchief, and deputy) are discussed in this
Note. It is also demonstrated that the performances of the doubleformation controller can be evaluated by using only the eccentricity
of the subchief no matter what the reference configuration is.
The HSP control theory has applications in first- and second-order
time-independent systems, as well as in time-periodic systems with
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