Optimal Fuel-Balanced Impulsive Formationkeeping for Perturbed Spacecraft Orbits

Abstract

This paper develops an impulsive spacecraft formation-flying control algorithm using relative-orbital-element corrections. This formalism introduces an inherent freedom that is used for deriving an optimal formation keeping law, balancing the fuel consumption among the spacecraft based on the impulsive Gauss variational equations. The main idea is that formulating the problem of formationkeeping in terms of relative-orbital-element corrections leaves the final values of the orbital elements unconstrained, thus allowing the spacecraft to create natural energy-balanced formation. The freedom rendered by this modeling is used to find optimal impulsive maneuvers, minimizing the squared l(2)-norm of the velocity-correction vector, which can be used for formation initialization and control. The optimization is solved using the least-squares method. The optimal formationkeeping method is designed to accommodate the effects of oblateness and drag. Based on graph theory, it is shown that the spacecraft will naturally form a stable energy-balanced formation and that the optimal formationkeeping strategy is invariant to the spanning tree. The algorithm is illustrated by simulating the motion of a formation of spacecraft possessing different ballistic coefficients subject to oblateness and drag.

Full text

Optimal Fuel-Balanced Impulsive Formationkeeping
for Perturbed Spacecraft Orbits
Igor Beigelman∗and Pini Gurfil†
Technion–Israel Institute of Technology, 32000 Haifa, Israel
DOI: 10.2514/1.34266
This paper develops an impulsive spacecraft formation-flying control algorithm using relative-orbital-element
corrections. This formalism introduces an inherent freedom that is used for deriving an optimal formationkeeping
law, balancing the fuel consumption among the spacecraft based on the impulsive Gauss variational equations. The
main idea is that formulating the problem of formationkeeping in terms of relative-orbital-element corrections leaves
the final values of the orbital elements unconstrained, thus allowing the spacecraft to create a natural energy-
balanced formation. The freedom rendered by this modeling is used to find optimal impulsive maneuvers,
minimizing the squared l2-norm of the velocity-correction vector, which can be used for formation initialization and
control. The optimization is solved using the least-squares method. The optimal formationkeeping method is
designed to accommodate the effects of oblateness and drag. Based on graph theory, it is shown that the spacecraft
will naturally form a stable energy-balanced formation and that the optimal formationkeeping strategy is invariant
to the spanning tree. The algorithm is illustrated by simulating the motion of a formation of spacecraft possessing
different ballistic coefficients subject to oblateness and drag.
Nomenclature
A= system matrix
a= semimajor axis
CD= drag coefficient
e= eccentricity
f= true anomaly
G= graph
g0= gravitational acceleration at sea level
H= cost functional
h= orbital angular momentum
I= inclination
In= modified Bessel function of the first kind
Isp = specific impulse
KD= ballistic coefficient
M= mean anomaly
m= mass
n= mean motion
p= semilatus rectum
r= orbit radius
Si= spacecraft i
s= reference area
= classical orbital-element vector
= inverse of the atmospheric scale factor
v= velocity-correction vector
:= variation of :
= Lagrange multiplier vector
= gravitational constant
= right ascension of the ascending node
!= argument of perigee
kk
m=m-norm of a vector
I. Introduction
CONTROL of relative spacecraft motion, known as
formationkeeping, is an enabling technology for spacecraft
formation-flying missions. Modeling relative-motion dynamics, an
essential infrastructure for formationkeeping, has seen significant
progress in recent years, since Clohessy and Wiltshire [1] published
their linearized relative-motion approximation in the early 1960s.
For example, Karlgaard and Lutze [2] derived an approximate
solution to second-order relative-motion equations for spacecraft in
near-circular Keplerian orbits. Gurfil and Kasdin [3] presented a
methodology for obtaining high-order approximations of the relative
motion between spacecraft by using the Cartesian configuration
space in conjunction with the classical orbital elements. Chichka [4]
used the linear Clohessy–Wiltshire equations to find natural
formations of satellites with a constant apparent distribution, an
important application of the relative-motion equations for remote
sensing.
However, natural formations may not survive, due to orbital
perturbations. The need for formationkeeping, therefore, stems from
the fact that space is a perturbed environment, thus inducing
separation of the formation-flying spacecraft orbital planes. In low
Earth orbits, the dominant perturbations are the Earth oblateness and
drag, and at higher altitudes, the spacecraft are perturbed by the
attraction forces of the sun and the moon.
The growing number of autonomous spacecraft missions that are
to perform complicated missions in close proximity requires
accurate, robust, and fuel-efficient formationkeeping algorithms.
Thus, attention should be given to developing a new rigorous control
approach, taking into account the effect of all dominant orbital
perturbations.
One such perturbation is Earth’s oblateness. Because J2
perturbations affect the orbital elements of each spacecraft in the
formation differently, the formation will tend to separate. An
analytical method developed by Alfriend et al. [5] can be used to
evaluate the differential forces on the formation-flying spacecraft.
Moreover, Schaub and Alfriend [6] also found that J2-invariant
relative orbits can be designed analytically; spacecraft placed on the
Schaub–Alfriend orbits will be influenced by the same forces and
will therefore have the same drift, thus yielding bounded relative
motion in the configuration space.
Formationkeeping can be implemented by using either impulsive
control, relying on chemical thrusters, or continuous control, using
low-thrust electric propulsion. Impulsive formationkeeping usually
assumes that the orbital elements of some reference orbit are known
Presented as Paper 6544 at the AIAA Guidance, Navigation and Control
Conference and Exhibit, Hilton Head, SC, 20–23 August 2007; received 26
August 2007; revision received 8 January 2008; accepted for publication 8
January 2008. Copyright © 2008 by The Authors. Published by the American
Institute of Aeronautics and Astronautics, Inc., with permission. Copies of
this paper may be made for personal or internal use, on condition that the
copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc.,
222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/08
$10.00 in correspondence with the CCC.
∗Graduate Student, Faculty of Aerospace Engineering; beigel@tx.
technion.ac.il.
†Senior Lecturer, Faculty of Aerospace Engineering; pgurfil@technion.
ac.il. Associate Fellow AIAA.
JOURNAL OF GUIDANCE,CONTROL,AND DYNAMICS
Vol. 31, No. 5, September–October 2008
1266
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and attempts to generate control commands that will match the
instantaneous orbital elements to some desired values [7], even under
orbital perturbations [8]. However, in many cases, such as in situ field
mapping and astronomical observations, setting an a priori reference
orbit is not a mission requirement (although orbit knowledge is
required). More important, on some occasions, specifying the
reference orbit a priori can result in a greater fuel consumption.
At this point, one may ask what can be done to the improve the
existing [7–10] impulsive-formationkeeping schemes. The answer is
that the overall amount of propellent used for formationkeeping can
be optimally distributed among the spacecraft if the reference orbit
for the formation-flying spacecraft is not specified. In other words, if
one used the relative-orbital-element corrections to define the
formationkeeping strategy, the formation could be designed so that
the relative geometry is determined without imposing an absolute
configuration on the formation. This approach is conceptually
different from existing distributed spacecraft system optimization
methods [11].
Relative-orbital-element corrections are the differences between
the successive corrections of orbital elements of any two formation-
flying spacecraft. In essence, these elements differ from the orbital-
element differences proposed by Schaub [12], because they do not
depend on a particular reference orbit. The proposed modeling
approach remains valid in the presence of orbital perturbations.
In this work, we develop a generic method for the impulsive
control of formation-flying spacecraft using the Gauss variational
equations (GVEs) [13] as the dynamic model and the classical
relative-orbital-element corrections as the variables representing the
formation structure. As opposed to other works [14–16], we shall not
attempt to minimize the overall amount of fuel consumed by the
formation-flying spacecraft, but rather to optimally balance a given
amount of fuel among the spacecraft. This prevents an undesirable
situation in which some of the spacecraft maneuver much more than
others, thus considerably increasing the a priori propellent margin of
the entire formation.
This approach relies on the key observation that formulating the
problem of formationkeeping in terms of relative-orbital-element
corrections leaves the final values of the corrected elements
unconstrained. This freedom can be used to design optimal impulsive
maneuvers for formation initialization, reconfiguration and
maintenance. We show that our impulsive control scheme is a
natural scheme in the sense that it creates a stable formation that
corresponds to an optimum balancing of the overall amount of
propellent. This approach is substantially different from other fuel-
balancing methods [17], wherein the control design exploits the
approximate dynamics given by the Clohessy–Wiltshire equations
and an approximate perturbation model.
In our analysis, we use elementary graph theory [18] to classify the
formation topologies and to guide the quest for optimal maneuvers.
Graph theory is a useful tool for analyzing multiple spacecraft
formations; thus far, however, it has been mostly used in the
continuous-formationkeeping setting [19]. We offer to apply a
graph-theory-inspired analysis for impulsive-formationkeeping
design.
II. Dynamic Model
In this section, we briefly outline the underlying dynamic model
used for the design of optimal formationkeeping, including the
effects of J2and drag.
A. Coordinate Systems
We first define the following coordinate systems [20]:
1) Ris a rotating polar coordinate system, centered at the
spacecraft. The fundamental plane is the orbital plane. The unit
vector ^
ris directed from the spacecraft radially outward, ^
his normal
to the fundamental plane, and ^
completes the right-hand setup.
2) Tis a rotating tangential-normal frame, centered at the
spacecraft. The fundamental plane is the orbital plane. The unit
vector ^
tlies along the spacecraft velocity vector, ^
hcoincides with the
instantaneous angular momentum vector, and ^
ncompletes the right-
hand setup.
3) Pis a perifocal coordinate system, centered at the primary. The
fundamental plane is the orbital plane. The unit vector ^
xpis directed
from the primary’s center to the periapsis, ^
zpis normal to the
fundamental plane, and ^
ypcompletes the right-hand setup.
4) Iis an inertial coordinate system, centered at the Earth. The
fundamental plane is the equator, ^
Xis directed from the Earth’s
center to the vernal equinox, ^
Zis normal to the fundamental plane,
and ^
Ycompletes the right-hand setup.
B. Gauss Variational Equations
The GVEs model the effect of a control and/or a disturbance
acceleration vector
uut;u
n;u
hT
on the osculating orbital element’s time derivatives. This vector is
represented in the reference frame T, and so utand unare the input
components in the plane of the osculating orbit along the velocity
vector and perpendicular to it, respectively, and uhlies along the
instantaneous angular momentum vector, normal to the orbital plane.
In the case of impulsive maneuvers, we can write
_
t; uttVt;u
ntVn;
uhtVh
(1)
where is an orbital element; is an orbital-element correction;
Vt,Vn, and Vhare the components of the velocity-correction
vector in frame T; and the impulsive time interval t!0. The
impulsive form of the GVEs defines algebraic relationships between
the orbital-element corrections and the components of the velocity-
correction vector [20]:
a2a2v
Vt(2a)
e1
v2ecos fVtr
asin fVn(2b)
Ircosf!
hVh(2c)
 rsinf!
hsin IVh(2d)
!1
ev 2 sin fVt2er
acos fVn
rsinf!cos I
hsin IVh(2e)
M b
eav 21e2r
psin fVtr
acos fVn(2f)
where ais the semimajor axis, eis the eccentricity, Iis the
inclination, is the right ascension of the ascending node, !is the
argument of perigee, Mis the mean anomaly, the parameter fis the
true anomaly, ris the scalar orbit radius, and pis the semilatus
rectum. The mean motion nsatisfies n
=a3
p, where is the
gravitational parameter. The semiminor axis bsatisfies
ba
1e2
p.
Consequently, in matrix form, GVEs (2) for some spacecraft Si
can be written as
BEIGELMAN AND GURFIL 1267
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iGivi(3)
where
i
a
e
I

!
M
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5i
;Gi
cT
a
cT
e
cT
I
cT

cT
!
cT
M
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5i
;vi
Vt
Vn
Vh
2
43
5i
(4)
where cTdenotes a coefficients vector of the controlled orbital
elements emanating from the GVEs, as seen in Eqs. (2), and vis the
concomitant impulsive velocity-correction vector.
C. Mean Orbital Elements Subject to a J2Perturbation
In the absence of perturbations, the six orbital elements remain
constant. Because of the influence of J2, some orbital elements will
oscillate about a nominal value and others will exhibit secular drifts.
The averaged values of the orbital elements are called the mean
elements. If all the spacecraft in the formation are of equal type and
build (i.e., have the same ballistic coefficient), then the differential J2
acceleration is the dominant perturbation experienced by the
spacecraft. In this case, the differential drag effect on the relative
motion is negligible over a time period of several orbits.
The mean orbital elements under the effect of J2are determined by
the differential equations [20]:
da
dt0(5)
de
dt0(6)
di
dt0(7)
d
dt3
2J2Re
p2
ncos I(8)
d!
dt3
4J2Re
p2
n5cos2I1(9)
dM
dtn3
4J2Re
p2
n
1e2
p3cos2I1(10)
where Reis the equatorial radius.
D. Mean Change of the Orbital Elements Because of Drag
If atmospheric rotation is neglected, then the inclination, the right
ascension of the ascending node, and the argument of perigee are not
affected by drag. The relationships for the averaged rates of the
semimajor axis and the eccentricity, developed for a small
eccentricity (expanding up to second-order in e), are given by [8]
da
dt2pa2nKDI02eI13e2
4I0I2expae
(11)
de
dtpKDan2I1eI0I2e2
45II1I3expae
(12)
where Ikare modified Bessel functions of the first kind of order kand
argument ae,defined in the integral form,
Ikz 1
2Z2
0
ezcos coskd(13)
pis the atmospheric density at perigee, is the inverse of the
atmospheric scale height, and the ballistic coefficient is
KDsCD
2m
where CDis the drag coefficient, mis the spacecraft mass, and sis a
reference area.
E. Relative Position and Velocity in Inertial Coordinates
To model the relative motion of the spacecraft under the preceding
orbital perturbations, we will first write the inertial position and
velocity of each spacecraft. This can be done by solving the
Keplerian two-body problem in inertial coordinates and then
extending the solution to include orbital perturbations using a
standard variation-of-parameters procedure. The inertial position r
and inertial velocity v_
rare given by [20]
ra1e2
1ecos f
cosf!cos  cos Isinf!sin 
cos Icos  sinf!cosf!sin 
sin Isinf!
2
43
5
(14)
v_
r

a1e2
rVx;V
y;V
zT(15)
where
Vxcos  sinf!sin  cos Icosf!
ecos  sin !sin  cos !cos I
Vycos  cos Icosf!sin  sinf!esin  sin !
cos  cos !cos I
Vzsin Icosf!ecos !
In the presence of orbital perturbations and velocity corrections,
the classical orbital elements vary with time. In this case, the inertial
position and velocity are determined as follows:
1) Integrate the orbital elements with respect to time under orbital
perturbations.
2) Calculate the inertial velocity until the moment of velocity
correction.
3) Compute the inertial position by quadrature: rR_
rdt.
4) Perform an impulsive correction v(the required velocity-
correction components are transformed from frame Tto I).
5) Determine the inertial velocity after the impulsive correction:
vvv.
6) Integrate the orbital elements (step 1).
The relative velocity and relative position vectors in frame I
between any two spacecraft Sjand Siare defined as follows:
_
ri;j Vxi;j;Vy
i;j;Vz
i;jT_
rj_
ri(16)
ri;j Xi;j;Y
i;j;Z
i;jTrjri(17)
III. Formation Topologies and Optimal
Formationkeeping
In this section, we consider the relative motion of a group of N
spacecraft Si(i1;...;N). We will develop an impulsive-
formationkeeping maneuver strategy for consuming minimum fuel.
1268 BEIGELMAN AND GURFIL
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The impulsive maneuver will be represented by a velocity-correction
vector performed by each spacecraft, and the formation topology will
be represented using elementary graph theory, briefly described in
the following subsection.
A. Elementary Graph Theory
A graph is a pair [GV; E] of sets such that the elements of E
are two-element subsets of V. The elements of Vare the vertices or
nodes, and the elements of Eare the edges or arcs. The degree of a
vertex v,dGv, is the number of edges at v. A labeled graph is a
graph with each node labeled differently (but arbitrarily), so that all
nodes are considered to be distinct for the purpose of enumeration. A
path PV; Eon a labeled graph is a sequence
Vfv1;v
2;...;v
Ng
such that
Efv1v2;v2v3;...;vN1vNg
are the graph edges of the graph GV; E. A closed path
Pv1;v
2;...;v
N;v
1
with N3is called a graph cycle. The length of a cycle is its number
of edges; a cycle of length mis denoted by Cm. A directed graph, or
digraph, Gis an ordered pair [GV; A], where Vis a set of
vertices or nodes and Ais a set of ordered pairs of vertices (called
directed edges, arcs, or arrows). An edge ex; yis considered to
be directed from xto y(yis called the head and xis called the tail of
the edge); yis said to be a direct successor of x, and xis said to be a
direct predecessor of y. If a path leads from xto y, then yis said to be a
successor of x, and xis said to be a predecessor of y. A tree Tis a set of
straight-line segments connected at their ends, containing no closed
loops (cycles). In other words, it is a simple, undirected, connected,
acyclic graph. A tree with Nnodes has N1graph edges.
Conversely, a connected graph with Nnodes and N1edges is a
tree. A labeled tree is a tree in which each vertex is given a unique
label. The vertices of a labeled tree with nvertices are typically given
the labels 1;2;...;n. The leader graph is denoted by Lk; the leader
graph is a tree, which is a directed graph. A spanning tree of a
connected graph Gis either the maximal set of edges of Gthat
contains no cycles or the minimal set of edges that connect all
vertices.
B. Relative-Orbital-Element Corrections
To develop an impulsive control for a group of Nspacecraft, we
shall treat this group as a digraph GNand will term it the formation
graph. Each spacecraft Si(i1;...;N) constitutes a vertex in GN.A
relative-orbital-element correction of Siand Sj, denoted by i;j ,
constitutes an edge in GNand is defined by
i;j ji(18)
where the orbital-element corrections of Siand Sjin the presence of
an impulsive velocity correction are, respectively,
i
i
i(19)
j
j
j(20)
where is a value of the orbital element before an impulsive
correction, and is the value of an orbital element after the
impulsive correction. Therefore, a relative-orbital-element correc-
tion can be written as
i;j ji
j
j
i
i(21)
The relative-orbital-element corrections are indifferent to a particular
predefined reference value chosen for an orbital element, ref (i.e.,
they are reference-orbit-independent); this can be seen by writing
i
iref ;j
jref (22)
and
i;j ji
j
i(23)
If the corrected orbital elements of Siare to be equal to the orbital
elements of Sj, then in the absence of perturbations, a formation will
be created; this case is referred to as perturbation-free orbital-element
matching. Semimajor axis (energy) matching in the case of Keplerian
motion is a typical example [10]:
a
ia
j)ai;j a
ia
j(24)
If perturbations are present, their effect will generally induce a
relative drift. The basic requirement from the formation control law is
to cancel the mean relative drift. When the relative deviation exceeds
a maximum allowed value, a velocity correction should be applied.
Each velocity correction sets the current value of the orbital-element
correction to a new value that cancels future drifts. In this case, the
corrected orbital elements of Siand Sjsatisfy the following
condition:

j
i
i;j (25)
and the relative element correction is
i;j 
i;j 
i;j (26)
where 
i;j is the required relative orbital element of Siand Sj,defined
as
i;j ji
after the impulsive correction (this correction is determined based on
the effect of drag and J2, as will be shown in the sequel). Similarly,

i;j is the relative orbital element before the impulsive correction.
C. Tree Topology
Let us assume for the moment that there are no perturbations
present and that the formation is created by orbital-element
matching. The spacecraft form a leader graph (i.e., a tree), as defined
in Sec. III.A. Let us label the leader of this graph by S1, with SNbeing
the root. Per this definition, the spacecraft form a leader formation
LN, and the resulting formation topology assumes the following
form:
1;2
2;3
.
.
.
N1;N
2
6
6
6
4
3
7
7
7
5
110 0
011 0
   
00 1 1
2
6
6
43
7
7
5

1

2
.
.
.

N
2
6
6
6
4
3
7
7
7
5
(27)
Using GVEs (2), we note that the orbital-element corrections can be
expressed as a function of the velocity correction in the following
manner:
icT
ivi;jcT
jvj(28)
where ciand cjare GVE coefficient vectors of Siand Sj,
respectively, as seen in Eqs. (2), and vi,vjare the concomitant
impulsive velocity-correction vectors. The relative-orbital-element
correction can be thus written as follows:
i;j jicT
jvjcT
ivi(29)
The state-space representation of this tree can be written as
Av b(30)
where
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A
cT
1cT
20 0
0cT
2cT
3 0
   
00 cT
N1cT
N
2
6
6
43
7
7
5
(31)
is an N13Nmatrix, and
b
1;2
2;3
.
.
.
N1;N
2
6
6
6
4
3
7
7
7
5
(32)
is a N11vector.
D. Balancing the Fuel Consumption
The use of relative-orbital-element corrections as edges in the tree
topology has introduced an excess freedom into the formation graph;
this can be plainly seen by the fact that the number of unknowns in
Eq. (30) exceeds the number of constraints. This implies that a static
parameter optimization problem with equality constrains can be
solved; the only remaining issue is to choose the proper cost
functional.
Choosing a cost functional that minimizes the total vof the
formation seems to be the most straightforward selection [14–16].
For a formation of spacecraft equipped with separate impulsive
thrusters for all three axes, this cost functional is [21]
v



v


1
≜X
N
i1



vi


1
(33)
For spacecraft equipped with gimbaled thrusters, the minimum-fuel
cost functional for the formation is
v



v


2
≜X
N
i1



vi


2
(34)
In both cases, the cost functional constitutes the overall vof the
formation. The rationale for choosing the overall vas the cost stems
from the fact that the vis proportional to the amount of propellant
m. This can be readily seen from the rocket equation:
mm01expv
Ispg0 (35)
where m0is the initial fuel mass, Isp is the specific impulse, and g0is
the gravitational acceleration at sea level. For a first-order small m,
Eq. (35) can be written as
vmIspg0
m0Om2(36)
However, minimizing either Eq. (33) or Eq. (34) may lead to a
situation wherein some of the spacecraft consume much more
propellant than the others. This unbalance can cause some of the
spacecraft to run out of fuel faster than the others; if the formation is
to be designed with identical or nearly identical spacecraft, each
spacecraft must be designed for the worst-case fuel consumption,
thus resulting in a conservative fuel budget for the entire formation.
To alleviate this situation, we propose using a penalty not on m,
but rather on m2. Although not necessarily minimizing the overall
fuel consumption, this penalizes large individual fuel usage and
balances the fuel among all the formation members. In this case, the
spacecraft will tend to form a natural fuel-balanced formation: that is,
to converge onto an orbit that does not require excessive fuel
consumption from some of the formation members while preventing
other spacecraft from maneuvering.
Penalizing m2implies penalizing v2; again, this can be seen
from Eq. (35):
v2m2I2
spg2
0
m2
0Om3(37)
Thus, the problem of fuel-balanced optimal formationkeeping can be
cast as follows. Find an optimal impulsive maneuver v, satisfying
varg min
v



v



2
2arg min
vX
N
i1



vi



2
2
(38)
such that
Av b(39)
Augmenting the cost functional with the equality constrains using the
Lagrange multiplier vector
1;
2;...;
N1T
yields
Hkvk2
2TAvb(40)
The necessary and sufficient conditions for the existence of minima
are
@H
@v0(41)
@2H
@v2>0(42)
The Hessian appearing in Eq. (42) is
@2H
@v22IN(43)
where INis an NNidentity matrix. Thus, the solution of Eq. (41) is
always a minimum, meaning that the solution of the optimization
problem (38) and (39) yields an optimal impulsive velocity-
correction vector v. This observation holds regardless of the
particular spanning tree used to define the formation, as shown in
Appendix A.
A straightforward approach for solving the optimization problem
formulated in Eqs. (38) and (39) is to use the least-squares method.
To that end, we rewrite expression (39) into
AvbAvbTAvAvTbAvTAvbT
bbTAvvTAT2bvTATbbT(44)
Taking the derivative with respect to vyields
2AATv2ATb0(45)
Therefore, the minimizing vector vis a solution of the equation
AATvATb(46)
If the rows of Aare linearly independent, then AATis invertible. In
that case, the optimal solution of the system of linear equations is
unique and is given by
vATAAT1bAb(47)
where Ais the pseudoinverse of A.
IV. Control Strategy
The first goal of the control strategy is to initialize the spacecraft
formation and then to keep the spacecraft in the leader graph LNeven
in the presence of perturbations. In the absence of perturbations, the
necessary and sufficient condition guaranteeing a stable formation is
that the mean motions of Siand Sjform a 1:1 resonant motion.
Because the periods of Keplerian elliptic orbits are determined by the
1270 BEIGELMAN AND GURFIL
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orbital energy, this requirement can be transformed into an energy
matching condition [10] or, equivalently, semimajor-axis matching.
However, orbital perturbations induce temporal changes in the
orbital elements of each spacecraft. In general, the effects of
perturbations on each spacecraft are not the same. The Earth
oblateness effect is a function of the semimajor axis, eccentricity, and
inclination. Because of initialization errors, small initial differences
of the semimajor axis, eccentricity, and inclination lead to secular
changes in both the nodal rate _
and the mean latitude rate _
M_
!.
These differences lead to a buildup of out-of-plane and in-plane
angular separations. In addition, not all the spacecraft involved in the
formation are of equal type, mass, shape, and attitude profile. These
physical variations lead to drag differences that cause secular
changes in the semimajor axis and the eccentricity, which in turn lead
to secular changes in the nodal rate and the mean latitude rate due to
the Earth oblateness.
The formationkeeping control law is required to mitigate the
relative deviations of the mean drift of _
and _
M_
!. That could be
done by persistently matching semimajor axis, eccentricity, and
inclination.
A. Optimal Corrections of the Semimajor Axis, Eccentricity, and
Inclination
Two different scenarios should be considered. The first is when
all the spacecraft in the formation experience the same drag; the
relative drift of the orbital elements is then zero. The second scenario
is when the effect of natural perturbations on each spacecraft is
different.
To preserve the formation, a periodic velocity correction should
be applied. As already discussed, the corrected elements a
i,e
i, and
I
i(i1;...;N) would theoretically remain unchanged in the
absence of drag perturbations, and no further correction of these
elements would be needed. However, when a drag difference exists,
the elements a
i,e
i, and I
i(i1;...;N) slowly deviate from the
initial values until an additional velocity correction must be applied.
The time interval between subsequent corrections is chosen
according to the allowed deviation. When the deviation exceeds
the allowed maximum, a new velocity correction is performed.
Each velocity correction sets the element to cancel future drifts due
to drag.
Choosing to control semimajor axis, eccentricity, and inclination
leads to a relationship in the form of Eq. (30). Let us define the
matrices Aa,Ae, and AIrepresenting the mapping of the velocity
corrections to the relative-orbital-element corrections ai;j ,ei;j,
and Ii;j (i1;...;N) based on GVEs (2):
Aa
ca1ca20 01N01N
0ca2ca3 01N01N
.
.
...
..
.
...
..
.
.
0 caN1caN01N01N
2
6
6
6
4
3
7
7
7
5
(48)
where caiis given by [compare with Eqs. (2)]
cai2via2
i
(49)
Similarly,
Ae
ce1ce20 se1se20 01N
0ce2ce3 0se1se3 01N
.
.
...
...
...
...
..
.
..
.
.
2
6
43
7
5
(50)
where
cei2eicos fi
vi
;seirisin fi
viai
(51)
and
Ai
01N01NkI1kI20
01N01N0kI2kI3
.
.
...
...
...
..
.
.
2
6
43
7
5(52)
where
kIiricosfi!i
hi
(53)
Finally, in the absence of perturbations, Eq. (30) assumes the
following form:
Aa
Ae
AI
2
43
5
Vt1
Vt2
.
.
.
VtN
Vn1
Vn2
.
.
.
VnN
Vh1
Vh2
.
.
.
VhN
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5

a1;2
a2;3
.
.
.
aN;N1
e1;2
e2;3
.
.
.
eN;N1
I1;2
I2;3
.
.
.
IN;N1
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(54)
which complies with the general form of Eq. (39).
When perturbations are present, the vector bbecomes a
superposition of the precorrected relative orbital elements and the
final relative orbital elements, as shown in Eq. (26). Thus, the
formationkeeping topology developed here is generic; the only
difference between the perturbed and unperturbed cases is the entries
of b.
B. Closed-Form Expressions for Optimal Impulsive
Formationkeeping
Consider the following scenario: A formation graph G3is located
in nearby orbits with different initial semimajor-axis values. In the
absence of perturbations, this graph will form a stable formation if the
energies of S1,S2, and S3are equal or, equivalently, if the semimajor
axes of their orbits are identical. We will show that the preceding
optimal formationkeeping scheme can yield analytic closed-form
expressions for the optimal impulsive maneuvers required to match
the semimajor axes using the least-squares method.
The initial values of semimajor axes and velocities of G3are a1,
a2, and a3and v1,v2, and v3, respectively, and the relative semimajor
axes corrections are
a2;1a2a1(55)
a3;2a3a2(56)
The matrix Aa, which is given by Eqs. (48) and (49), assumes the
following form:
Aa2v1a2
1

2v2a2
2
0
02v2a2
2

2v3a2
3

2
43
5(57)
The optimal impulsive maneuver is calculated using the least-
squares method:
vAT
aAaAT
a1b;ba2;1;a3;2T(58)
The optimal velocity corrections required to match the semimajor
axes of the spacecraft are given by
BEIGELMAN AND GURFIL 1271
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V?
t11
2
v1a2
1v2
2a4
2a3;2v2
2a4
2a2;1v2
3a4
3a2;1
v2
1a4
1v2
3a4
3v2
1a4
1v2
2a4
2v2
3a4
3v2
2a4
2
(59)
V?
t21
2
v2a2
2v2
3a4
3a2;1v2
1a4
1a3;2
v2
1a4
1v2
3a4
3v2
1a4
1v2
2a4
2v2
3a4
3v2
2a4
2
(60)
V?
t31
2
v3a2
3v2
1a4
1a3;2v2
2a4
2a3;2v2
2a4
2a2;1
v2
1a4
1v2
3a4
3v2
1a4
1v2
2a4
2v2
3a4
3v2
2a4
2
(61)
and the resulting semimajor-axis corrections are
a?
i2a2
ivi
V?
ti(62)
yielding
a?
1
v2
1a4
1v2
2a4
2a3;2v2
2a4
2a2;1v2
3a4
3a2;1
v2
1a4
1v2
3a4
3v2
1a4
1v2
2a4
2v2
3a4
3v2
2a4
2
(63)
a?
2
a4
2v2
2v2
3a4
3a2;1v2
1a4
1a3;2
v2
1a4
1v2
3a4
3v2
1a4
1v2
2a4
2v2
3a4
3v2
2a4
2
(64)
a?
3
a4
3v2
3v2
1a4
1a3;2v2
2a4
2a3;2v2
2a4
2a2;1
v2
1a4
1v2
3a4
3v2
1a4
1v2
2a4
2v2
3a4
3v2
2a4
2
(65)
All the spacecraft in G3will match their semimajor axes to some
value a?. The value of a?corresponds to the balanced velocity
correction required to form a formation in the absence of
perturbations and is given by
a?aia?
i(66)
It can be shown that a?is the centroid of the initial values of the
semimajor axis:
a?1
NX
N
i0
ai(67)
This value of a?yields the globally optimal total velocity correction
in the sense of minimizing kvk2
2for all possible values of a
reference semimajor axis (this result is valid only for controlling each
orbital element separately).
V. Optimal Formationkeeping Under Perturbations:
Identical Ballistic Coefficients
In this section, we will implement the new control strategy on a
formation of J2-perturbed spacecraft having identical ballistic
coefficients. We will extend the method to the general case in the next
section. To illustrate the main idea, consider a formation of four
spacecraft flying in a low Earth orbit. All spacecraft have the same
ballistic coefficient and a small initial difference in the values of the
semimajor axis and eccentricity. In this case, as was described in
Sec. IV, to form a stable formation, one can match semimajor axis
and eccentricity, then the relative deviations of the mean drifts _
and
_
M_
!would be zero. The initial orbital elements and ballistic
coefficients are summarized in Table 1.
The initial differences in semimajor axis and eccentricity lead to a
buildup of angular separation between the orbital planes of the
spacecraft, i;j, and of the in-plane angular separation M!i;j ,
which increases the distance between spacecraft, di;j. We shall now
demonstrate the implementation of the proposed formationkeeping
method (matching semimajor axis, eccentricity, and inclination).
Figures 1 and 2 depict the time history of the mean relative orbital
elements. The impulsive maneuver is applied at f90 deg after
1.25 orbits. It can be seen that relative drift of the mean relative
orbital elements is stopped. Although the values of semimajor axis
and eccentricity decrease due to drag, the formation remains intact,
and the distances between spacecraft are bounded. Figure 3 shows
the time history of the relative velocity components Vxi;j,Vyi;j, and
Vzi;j in inertial coordinates [Eq. (16)]. The discontinuity in the
relative velocity components is a result of the impulsive velocity
correction. Following the impulsive maneuver, the position
components exhibit a bounded periodic motion. This can be seen
in Fig. 4, showing the three-dimensional relative orbit in inertial
coordinates and the projections thereof on the X–Y,X–Z, and Y–Z
planes.
VI. Optimal Formationkeeping of Spacecraft with
Different Ballistic Coefficients
In this section, we study the motion of a formation of spacecraft
having different ballistic coefficients, and we develop an optimal
formationkeeping algorithm based on the least-squares formalism
developed in Sec. III. Different ballistic coefficients cause different
secular changes of semimajor axis and eccentricity for each
spacecraft, which in turn lead to different secular changes of the
nodal rate and the mean latitude rate, due to the Earth oblateness.
Periodic velocity corrections must then be applied, because the
corrected orbital elements will continue to deviate from the reference
values.
To that end, we will implement the optimal impulsive scheme
developed herein while relying on an auxiliary dynamic equations
developed by Mishne [8]. The auxiliary relations will be used to
calculate the required corrections of the semimajor axis and the
eccentricity that will cancel relative drifts in the nodal angle and the
mean latitude angle until the next velocity correction. We shall also
suggest a new method for mitigating the cross-coupling of the
orbital-element corrections (semimajor axis and eccentricity are
corrected by applying tangential and normal maneuvers; this
correction affects the mean latitude rate as well).
Before commencing with our development of the formationkeep-
ing strategy, we will highlight some important differences between
the current work and the method of Mishne [8]:
1) We propose controlling nodal rate and mean latitude rate
indirectly, by selecting suitable values of semimajor axis and
eccentricity. These values are chosen to cancel future drifts until the
next velocity correction. It is possible to choose which two of the
three elements to control because there are only two constraints:
namely, equalizing the out-of-plane and in-plane drifts.
2) Fuel optimization is done according to the method described in
Sec. III. Controlling semimajor axis and eccentricity leads to in-plane
velocity corrections only. By finding a suitable timing for these
corrections, it is possible to minimize the effect of changes in !and
Mdue to the impulsive corrections in the tangential and normal
directions.
3) We offer an optimal solution by not imposing a global reference
orbit a priori.
Table 1 Initial orbital elements and ballistic coefficients of a formation
in a low Earth orbit.
Sia,km eI, rad , rad !, rad f0, rad KD
S16928.2 0.0012 0.01 0 0 0 0.0355
S26928.3 0.0013 0.01 0 0 0 0.0355
S36928.5 0.0014 0.01 0 0 0 0.0355
S46928.8 0.0015 0.01 0 0 0 0.0355
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0123456
−0.4
−0.3
−0.2
−0.1
0
0.1
ai,j [km]
Time [orbits]
0123456
−15
−10
−5
0
5x 10−5
ei,j
Time [orbits]
0123456
−0.03
−0.02
−0.01
0
0.01
fi,j [rad]
Time [orbits]
f2,1
f3,2
f4,3
e2,1
e3,2
e4,3
a2,1
a3,2
a4,3
Fig. 1 Mean values of the relative semimajor axis and eccentricity and the true anomaly, with an optimal velocity correction applied.
0123456
−2
0
2
4
6x 10−4
(ω+M)i,j [rad]
Time [orbits]
0123456
−2
−1.5
−1
−0.5
0x 10−6
Ωi,j [rad]
Time [orbits]
Ω2,1
Ω3,2
Ω4,3
(ω+M)2,1
(ω+M)3,2
(ω+M)4,3
Fig. 2 Relative drift in !Mand is arrested after a velocity correction.
BEIGELMAN AND GURFIL 1273
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0123456
−4
−2
0
2
4x 10−3
Vxi,j [km/sec]
Time [orbits]
Vx2,1
Vx3,2
Vx4,3
0123456
−4
−2
0
2
4x 10−3
Vyi,j [km/sec]
Time [orbits]
Vy2,1
Vy3,2
Vy4,3
0123456
−2
−1
0
1
2x 10−3
Vzi,j [km/sec]
Time [orbits]
Vz2,1
Vz3,2
Vz4,3
Fig. 3 Relative velocity components in the inertial frame, with an impulsive correction, become bounded.
−3 −2 −1 0 1 2 3 4
−3
−2
−1
0
1
2
3
Yi,j [km]
Xi,j [km]
−3 −2 −1 0 1 2 3 4
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
Zi,j [km]
Xi,j [km]
−3 −2 −1 0 1 2 3
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
Zi,j [km]
Yi,j [km]
−5
0
5
−5
0
5
−0.04
−0.02
0
0.02
0.04
Xi,j [km]
Yi,j [km]
Zi,j [km]
S2,1
S3,2
S4,3
Fig. 4 Relative motion in inertial coordinates exhibits bounded orbits following the formationkeeping maneuver, performed after 1.25 orbits.
1274 BEIGELMAN AND GURFIL
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A. Calculation of the Required Correction of Semimajor Axis and
Eccentricity
In the presence of J2and drag, the relative orbital elements of any
Siare determined by the GVEs given in Sec. II. The perturbed
dynamics of these elements can be written in the general form:
_
F(68)
where FF1;...;F
6Tis a vector-valued function F:R6!R6.If
the initial orbital elements of Siare different from those of Sj, and in
addition there exists a difference KDin the ballistic coefficients,
then linearizing Eq. (68) about ientails [8]
_
i;j Qi;j qKD(69)
where
QfQlmg@F=@i2R66
and
qq1;...;q
6T@F=@KDi2R6
The entries of Qand qpertinent for the subsequent calculations are
given by [8]
Q41 @F4
@a 21
4J2Re
p2n
acos I(70)
Q42 @F4
@e 6J2R2
e
nae
p3cos I(71)
Q51 @F5
@a Q41
5cos2I1
2 cos I(72)
Q52 @F5
@e Q42
5cos2I1
2 cos I(73)
Q61 @F6
@a 3n
2a17
4J2
1e2
pRe
p2
3cos2I1(74)
Q62 @F6
@e 9
4J2nRe
p2
3cos2I1e
1e25
2
(75)
and
q4@F4
@KD0;q
5@F5
@KD0;q
6@F6
@KD0(76)
To control the orbital-element deviations, impulsive velocity
corrections must be applied. An impulsive velocity correction
causes an instantaneous change of the orbital elements d. The
relationship between the velocity change and the element change
is given by GVEs (3). The velocity correction is chosen to satisfy
[8]:
_
i;j _

i;j;_
M_
!i;j _
M_
!
i;j (77)
When a drag difference exists, 
i;j constitutes an initial condition,
and the deviation changes according to Eq. (69). Thus, the desired
drift rates are set to values that will compensate for the effect of
drag differences until next correction.
By combining Eqs. (3) and (69) the following two conditions for
the desired correction values of semimajor axis and eccentricity are
obtained [8]:
L1
i;j q4KD_

i;j (78)
L2
i;j q5q6KD_
M_
!
i;j (79)
where the vector L1is the fourth row of Q,L2is the sum of the fifth
and sixth rows of Q, and qiare the elements of q. This procedure
results in
Q41a
i;j Q42e
i;j Q43I
i;j _

i;j (80)
Q51 S61a
i;j Q52 Q62 e
i;j Q53 Q63 I
i;j _
M_
!
i;j
(81)
Equations (81) are two linear equations for the three desired relative
orbital elements a
i;j;e

i;j;I

i;j. This means that there is a single
degree-of-freedom. Because a
i;j and e
i;j depend upon the ballistic
coefficient and I
i;j is unaffected by either J2or by KD, we will
assume that I
i;j 0. This yields two linear equations with two
unknowns, a
i;j and e
i;j, for which the solution is
e
i;j _
M_
!
i;j Q51 Q61_

i;j=Q41
Q52 Q62Q42 =Q41Q51 Q61(82a)
a
i;j 
i;j Q42e
i;j
Q41
(82b)
Finally, the optimal velocity correction vis calculated using
Eq. (47) and is given by [compare with Eq. (54)]:
vAa
Ae
a
1;2a
1;2
a
2;3a
2;3
.
.
.
a
N;N1a
2;3
e
1;2e
1;2
e
2;3e
1;2
.
.
.
e
N;N1e
N;N1
2
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
5
Aa
Ae
a1;2
a2;3
.
.
.
aN;N1
e1;2
e2;3
.
.
.
eN;N1
2
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
5
Aa
Aeb(83)
where Aaand Aeare matrices defined in Eqs. (48–51), and the vector
bis a superposition of the precorrected relative orbital elements and
the desired corrections in semimajor axis and eccentricity given by
Eq. (82).
B. Calculation of the Desired Drifts
The desired drifts immediately following the velocity correction,
_

i;j and _
M_
!
i;j, are chosen to be in the opposite direction of the
expected drift due to drag that will be accumulated until the next
correction. These values will be used in Eq. (82). The velocity
correction will yield initial drift rates that are opposite to the drift
due to drag. These drifts can be found using a linear approximation,
valid for a short time interval Tbetween two subsequent corrections
[8]:
_

i;j 1
2Q41F1Q42 F2KD
KD
T(84)
_
M_
!
i;j 1
2Q51 Q61F1Q52 Q62 F2KD
KD
T
(85)
BEIGELMAN AND GURFIL 1275
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C. Minimization of the Cross-Coupling Effect
A well-known difficulty in controlling spacecraft orbits using
impulsive velocity corrections is the cross-coupling between orbital
elements. In formation-flying problems, this effect renders finding
conditions for bounded relative motion quite difficult.
Semimajor axis and eccentricity are corrected by applying in-
plane velocity corrections Vtand Vn. These velocity corrections
in turn cause undesirable changes in Mand !, creating an in-plane
separation M!i;j each time the velocity correction is performed.
However, the orbital elements depend upon true anomaly. This fact
can be used to mitigate the coupling of orbital-element corrections.
For example, a tangential velocity correction applied at periapsis or
apoapsis will not cause any change in !, and a normal velocity
correction applied at f90 or 270 deg will not influence M.
We will now derive a method for mitigating the in-plane cross-
coupling. This can be done if the semimajor axis and eccentricity
corrections are performed twice per orbit. The first correction is
performed at f90 deg and the second is performed at
f270 deg. Let us denote the correction of an orbital element 
at these points by Iand II, respectively.
Because the change in semimajor axis and eccentricity due to drag
is slow, we can adopt the following approximation:
aIaII (86)
eIeII (87)
This gives
aI2aI2vI
VI
t2aII2vII
VII
taII (88)
From which it immediately follows that VI
tVII
t(under the
assumption that vIvII vand aIaII a). Similarly,
eI2eI
vIVI
trI
aIVI
n2eII
vII VII
trII
aII VII
neII (89)
yielding VI
nVII
n(assuming that eIeII eand
rIrII r).
The parasitic change in !and in Mdue to the first velocity
correction is given by
!I2
ev VtI2eVI
n(90a)
MI b
eav 21e2r
pVI
t(90b)
and the unwanted change in !and in Mdue to second velocity
correction is given by
!II 2
ev VII
t2eVII
n(91a)
MII b
eav 21e2r
pVII
t(91b)
Table 2 Initial orbital elements of four spacecraft with different
ballistic coefficients
Sia,km eI, rad , rad !, rad f0, rad KD
S16803 0.0012 0.01 0 0 0 0.0355
S26803 0.0012 0.01 0 0 0 0.03905
S36803 0.0012 0.01 0 0 0 0.036
S46803 0.0012 0.01 0 0 0 0.0396
0246810121416
−0.04
−0.02
0
0.02
0.04
ai,j [km]
Time [orbits]
0246810121416
−4
−2
0
2
4x 10−7
ei,j
Time [orbits]
0246810121416
−4
−2
0
2
4x 10−4
fi,j[rad]
Time [orbits]
f2,1
f3,2
f4,3
e2,1
e3,2
e4,3
a2,1
a3,2
a4,3
Fig. 5 Growth of the mean relative semimajor axis, eccentricity and true anomaly due to different ballistic coefficients, without applying
formationkeeping maneuvers.
1276 BEIGELMAN AND GURFIL
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0246810121416
−4
−2
0
2
4x 10−4
(ω+M)i,j [rad]
Time [orbits]
(ω+M)2,1
(ω+M)3,2
(ω+M)4,3
0246810121416
−1.5
−1
−0.5
0
0.5
1
1.5 x 10−6
Ωi,j [rad]
Time [orbits]
Ω2,1
Ω3,2
Ω4,3
Fig. 6 In-plane and out-of-plane angular separation due to different ballistic coefficients, without applying a formationkeeping maneuver.
0246810121416
−4
−2
0
2
4
Xi,j [km]
Time [orbits]
X2,1
X3,2
X4,3
0246810121416
−4
−2
0
2
4
Yi,j [km]
Time [orbits]
Y2,1
Y3,2
Y4,3
0246810121416
−0.04
−0.02
0
0.02
0.04
Zi,j [km]
Time [orbits]
Z2,1
Z3,2
Z4,3
Fig. 7 Relative drift of the relative position components due to different ballistic coefficients, without applying a velocity correction.
BEIGELMAN AND GURFIL 1277
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0246810121416
−1
−0.5
0
0.5
1x 10−3
ai,j [km]
Time [orbits]
a2,1
a3,2
a4,3
0246810121416
−1
−0.5
0
0.5
1x 10−8
ei,j
Time [orbits]
e2,1
e3,2
e4,3
0246810121416
−2
0
2x 10−4
fi,j [rad]
Time [orbits]
f2,1
f3,2
f4,3
Fig. 8 Time history of the mean relative values of the semimajor axis and eccentricity, and the true anomaly with different ballistic coefficients, using
optimal velocity corrections.
0 2 4 6 8 10121416
−1.5
−1
−0.5
0
0.5
1x 10−7
(ω+M)i,j [rad]
Time [orbits]
(ω+M)2,1
(ω+M)3,2
(ω+M)4,3
0 2 4 6 8 10121416
−3
−2
−1
0
1
2
3
4x 10−10
Ωi,j [rad]
Time [orbits]
Ω2,1
Ω3,2
Ω4,3
Fig. 9 Time history of the in-plane and out-of-plane angular separation with different ballistic coefficients using periodic velocity corrections. The
angular drifts are minuscule.
1278 BEIGELMAN AND GURFIL
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0246810121416
−2
−1
0
1
2x 10−3
Vxi,j [m/sec]
Time [orbits]
Vx2,1
Vx3,2
Vx4,3
0246810121416
−2
−1
0
1
2x 10−3
Vyi,j [m/sec]
Time [orbits]
Vy2,1
Vy3,2
Vy4,3
0246810121416
−1
−0.5
0
0.5
1x 10−4
Vzi,j [m/sec]
Time [orbits]
Vz2,1
Vz3,2
Vz4,3
Fig. 10 Relative velocity components with different ballistic coefficients and periodic velocity corrections.
0246810121416
−2
0
2
4
Xi,j [m]
Time [orbits]
X2,1
X3,2
X4,3
0246810121416
−2
−1
0
1
2
Yi,j [m]
Time [orbits]
Y2,1
Y3,2
Y4,3
0246810121416
−0.02
−0.01
0
0.01
0.02
Zi,j [m]
Time [orbits]
Z2,1
Z3,2
Z4,3
Fig. 11 The position components in inertial coordinates with different ballistic coefficients under periodic velocity corrections, optimally calculated to
mitigate the effect of differential perturbations.
BEIGELMAN AND GURFIL 1279
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By combining Eqs. (90) with Eqs. (91), we get the total in-plane
deviation during an orbital period:
!I!IIMIMII 0(92)
which means that if we perform in-plane corrections at these points, a
second in-plane change will cancel out the first parasitic in-plane
change.
D. Simulation
First, we will simulate the uncontrolled motion of the spacecraft in
a 425-km low Earth orbit. The differences in ballistic coefficients
between the spacecraft are about 10%. The initial orbital elements
and ballistic coefficients are summarized in Table 2. As can be seen in
Figs. 5 and 6, different ballistic coefficients cause different drifts in
semimajor axis and eccentricity, which in turn induces secular differ-
ences in the out-of-plane and in-plane angular motions due to the J2
effect. Figure 7 depicts the relative position components in inertial
coordinates. The difference in ballistic coefficients increases the rela-
tive position components so that the average distance between the
formation-flying spacecraft amount to about 2.5 km after 16 orbits.
Next, we demonstrate that the proposed formationkeeping method
[Eq. (83)] considerably reduces the relative drift. The orbital elements
are corrected twice per orbit, at f90 and 270 deg, thus minimizing
the effect of cross-coupling between the orbital elements. As shown in
Figs. 8 and 9, the periodic impulsive corrections almost nullify the
relative drift in the relative orbital elements. Figures 10 and 11 show
the time history of the relative velocity and position components,
respectively, in inertial coordinates. The discontinuities in the relative
velocity components are a result of the impulsive velocity corrections
that are performed twice per orbit. Figure 12 shows the three-
dimensional relative orbits and the projections of the relative orbits
onto the X–Y,X–Z, and Y–Zplanes in inertial coordinates. Figure 13
depicts the distance between spacecraft. The drift is not zero, but it is
much smaller than the uncorrected case (about three orders of
magnitude, relative to the uncorrected case). A small in-plane drift
remains because the cross-coupling effect causes a small residual drift
in relative position. The actual in-plane drift is not identically zero
because of the assumption we made that the change in semimajor axis
and eccentricity due to drag is slow. The maximum separation is
between S4and S3and is about 2.5 m after 16 orbits (one day).
E. Fuel Consumption
The fuel required for formationkeeping is related to the total veloc-
itychangeby the rocket equation(35).The total velocity changecanbe
calculated by either Eq. (33) or Eq. (34), depending on the type of
propulsion system. For a mission lifetime of five years, the total veloc-
ity corrections of each spacecraft, calculated according to Eq. (34), are
vtot119:89 m=s;vtot214:78 m=s
vtot314:97 m=s;vtot420:26 m=s(93)
−2 −1 0 1 2 3
−2
−1
0
1
2
Yi,j [m]
X i,j [m]
−2 −1 0 1 2 3
− 0.02
− 0.01
0
0.01
0.02
Zi,j [m]
Xi,j [m]
−2 −1 0 1 2
− 0.02
− 0.01
0
0.01
0.02
Zi,j [m]
Yi,j [m]
−5
0
5
−2
0
2
− 0.02
0
0.02
Xi,j [m]
Yi,j [m]
Zi,j [m]
S2,1
S3,2
S4,3
Fig. 12 Relative motion between spacecraft with different ballistic coefficients in inertial coordinates, using periodic velocity corrections. The drift is
reduced by three orders of magnitude compared with the uncontrolled case.
1280 BEIGELMAN AND GURFIL
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and the total vis about 70 m=s. The actual propellent consumption
can be calculated according to the rocket equation (35). Taking
Isp 220 s and m0500 kg, the required fuel mass for each
spacecraft is as follows:
m14:59 kg;m23:41 kg
m33:46 kg;m44:67 kg (94)
We thus see that the propellent mass is properly balanced among the
formation spacecraft. For comparison, the total five-year velocity
corrections calculated according to Eq. (33) are
vtot123:12 m=s;vtot217:16 m=s
vtot317:45 m=s;vtot423:41 m=s(95)
and the total velocity correction is about 81 m=s.
VII. Conclusions
We developed a generic fuel-balanced impulsive maneuver
scheme for multiple spacecraft formationkeeping using relative-
orbital-element corrections. The spacecraft formation was modeled
as a directed graph, an approach that accommodates any number of
spacecraft. More important, formulating the problem of
formationkeeping in terms of relative-orbital-element corrections
left the final values of the controlled elements unconstrained. This
freedom was used to find optimal impulsive maneuvers, in the sense
of minimizing the l2-norm, for formation initialization and control in
the presence of perturbations.
The main conclusions are as follows:
1) A controlled group of spacecraft will form a pattern in the
position space that corresponds to the optimal velocity correction.
2) Leaving the reference orbit free results in a globally optimal
velocity correction.
3) This velocity correction can be analytically computed using a
simple least-squares procedure.
Because the formationkeeping formalism is formulated in terms of
relative-orbital-element corrections, one can choose which orbital
elements to control. The final differences in the controlled elements
can be constrained to minimize the effect of orbital perturbations and
cross-coupling.
The newly developed algorithm was illustrated by a simulation of
a formation of four spacecraft flying in low Earth orbits subject to J2
and drag perturbations. Two different scenarios were examined.
The first is when all spacecraft possess identical ballistic
coefficients, and the second is when the spacecraft have different
ballistic coefficients.
In the first scenario, the most dominant perturbation is J2. Because
of initialization errors, small initial differences of the semimajor axis,
eccentricity, and inclination lead to secular changes in both the nodal
rate and the mean latitude rate. The formationkeeping algorithm is
then required to cancel theses relative mean drifts. This was achieved
by matching semimajor axis, eccentricity, and inclination.
In the second scenario, a periodic velocity correction should be
applied to preserve the formation, because the corrected orbital
elements deviate from the reference values due to different ballistic
coefficients. We proposed controlling the nodal rate and the mean
latitude rate indirectly, by selecting suitable values of semimajor axis
and eccentricity. These values were selected to cancel future drifts
until the next velocity correction. The simulation results showed that
applying the newly developed formationkeeping reduced the relative
drift by three orders of magnitude.
Appendix A: Effect of Different Spanning Trees on the
Velocity-Correction Vector
According to Cayley’s formula, the number of different spanning
trees of the connected graph GNis NN2, where each spanning tree
forms a different labeled graph [22]. In the previous sections, we
treated the spacecraft formation as a leader formation LN, which is
also a labeled graph. The leader of this formation was labeled S1.
This means that for a formation of Nspacecraft, we can form NN2
different Amatrices and bvectors. The analysis in this paper was
performed for a formation represented by the particular spanning
tree
SN!SN1!!S1
We will now examine how different spanning trees affect the
velocity correction required from each spacecraft and the total
velocity correction. For simplicity, we will study only semimajor-
axis corrections. To that end, consider all 16 possible spanning trees
for a formation of four spacecraft, as shown in Fig. A1. We
arbitrarily choose four different spanning trees: L4
1,L4
7,L4
13, and
L4
16. These trees are shown in Fig. A1. The initial values of
semimajor axes for this formation are as follows:
a142;160 km;a
242;167 km;
a342;170 km;a
442;185 km (A1)
The matrices Aa1,Aa7,Aa13, and Aa16 and the vectors b1,b7,b13 ,
and b16, corresponding to the trees L4
1,L4
7,L4
13, and L4
16 are as
follows:
Aa1
2v1a2
1

2v2a2
2
00
02v2a2
2

2v3a2
3
0
00
2v3a2
3

2v4a2
4

2
6
6
43
7
7
5
;b1
a2;1
a3;2
a4;3
2
43
5
(A2)
Aa7
2v1a2
1

2v2a2
2
00
2v1a2
1
02v3a2
3
0
00
2v3a2
3

2v4a2
4

2
6
6
43
7
7
5
;b7
a2;1
a3;1
a4;3
2
43
5
(A3)
Aa13 
2v1a2
1

2v2a2
2
00
02v2a2
2

2v3a2
3
0
02v2a2
2
02v4a2
4

2
6
6
43
7
7
5
;b13 
a2;1
a3;2
a4;2
2
43
5
(A4)
0246810121416
0
0.5
1
1.5
2
2.5
Relative distance [m]
Time [orbits]
d2,1
d3,2
d4,3
Fig. 13 Relative distance between spacecraft with different ballistic
coefficients shows that a maximum separation of only 2.5 m after 16
orbits.
BEIGELMAN AND GURFIL 1281
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Aa16 
2v1a2
1

2v2a2
2
00
2v1a2
1
02v3a2
3
0
2v1a2
1
00
2v4a2
4

2
6
6
43
7
7
5
;b16 
a2;1
a3;1
a4;1
2
43
5
(A5)
Substitution into Eq. (47) yields the optimal velocity-correction
vector, which is equal for the four different trees already described, as
is the final value of the semimajor axis of this formation (without
perturbations):
v?
0:3826
0:1273
0:0180
0:5285
2
6
6
43
7
7
5
m=s;a
a?42;170:49 km (A6)
This example shows that different spanning trees do not affect the
control vector and the required velocity corrections of each
spacecraft. This is because different spanning trees are merely a
different parametrization of the same physical values of orbital
elements. The final value of the orbital element is an optimal value
for given initial conditions. Choosing different spanning trees does
not change these initial values and the concomitant required
corrections .
Acknowledgments
This research was partially supported by the Israel Ministry of
Science and Technology Infrastructure Program. The authors owe a
debt of gratitude to Moshe Guelman and David Mishne for providing
important insights.
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1282 BEIGELMAN AND GURFIL
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