Formation Flying for Electric Sails in Displaced Orbits. Part I: Geometrical Analysis

Abstract

We present a geometrical methodology for analyzing the formation flying of electric solar wind sail based spacecraft that operate in heliocentric, elliptic, displaced orbits. The spacecraft orbit is maintained by adjusting its propulsive acceleration modulus, whose value is estimated using a thrust model that takes into account a variation of the propulsive performance with the sail attitude. The properties of the relative motion of the spacecraft are studied in detail and a geometrical solution is obtained in terms of relative displaced orbital elements, assumed to be small quantities. In particular, for the small eccentricity case (i.e. for a near-circular displaced orbit), the bounds characterized by the extreme values of relative distances are analytically calculated, thus providing an useful mathematical tool for preliminary design of the spacecraft formation structure.

Full text

Formation flying for electric sails in displaced orbits.
Part I: Geometrical analysis
Wei Wang
a
, Giovanni Mengali
b,⇑
, Alessandro A. Quarta
b
, Jianping Yuan
a
a
National Key Laboratory of Aerospace of Flight Dynamics, Northwestern Polytechnical University, 710072 Xi’an, People’s Republic of China
b
Dipartimento di Ingegneria Civile e Industriale, University of Pisa, I-56122 Pisa, Italy
Received 14 February 2017; received in revised form 29 April 2017; accepted 9 May 2017
Available online 17 May 2017
Abstract
We present a geometrical methodology for analyzing the formation flying of electric solar wind sail based spacecraft that operate in
heliocentric, elliptic, displaced orbits. The spacecraft orbit is maintained by adjusting its propulsive acceleration modulus, whose value is
estimated using a thrust model that takes into account a variation of the propulsive performance with the sail attitude. The properties of
the relative motion of the spacecraft are studied in detail and a geometrical solution is obtained in terms of relative displaced orbital
elements, assumed to be small quantities. In particular, for the small eccentricity case (i.e. for a near-circular displaced orbit), the bounds
characterized by the extreme values of relative distances are analytically calculated, thus providing an useful mathematical tool for pre-
liminary design of the spacecraft formation structure.
Ó2017 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Electric solar wind sail; Heliocentric displaced orbits; Formation flying
1. Introduction
The Electric Solar Wind Sail (E-sail) is an innovative
propulsion concept that uses the natural solar wind
dynamic pressure to generate a continuous thrust, without
the need of any reaction mass nor any propellant. A space-
craft with an E-sail based propulsion system consists of a
number of thin, long and conducting tethers, which are
maintained at a high positive potential by means of an elec-
tron gun (Janhunen and Sandroos, 2007). Over the last
years, much effort has been devoted to study the physical
mechanisms of the E-sail concept by means of experimental
tests and through plasma dynamic simulations (Janhunen
et al., 2010; Sanchez-Torres, 2016). Theoretical analysis
results indicate that the E-sail thrust decays as the inverse
Sun-spacecraft distance r(i.e. as 1=r), with a decreasing
rate slower than that of a classical photonic solar sail (i.e.
as 1=r2). Due to this peculiarity, a number of mission sce-
narios have been analyzed using an E-sail as the primary
propulsion system (Quarta and Mengali, 2010; Mengali
et al., 2013; Yamaguchi and Yamakawa, 2016), especially
for what concerns the preliminary analysis of interplane-
tary transfers where the problem is formulated within a
time-optimal framework (Mengali et al., 2008; Quarta
et al., 2011).
Another promising application involving an E-sail based
spacecraft is to observe the polar region of a celestial body
by inserting the vehicle into a (closed) non-Keplerian dis-
placed orbit(McKay et al., 2011; Salazar et al., 2016;
Waters and McInnes, 2007), whose plane does not pass
through the Sun’s center-of-mass. Maintaining such a
desired displaced orbit requires a continuous propulsive
http://dx.doi.org/10.1016/j.asr.2017.05.015
0273-1177/Ó2017 COSPAR. Published by Elsevier Ltd. All rights reserved.
⇑
Corresponding author.
E-mail addresses: 418362467@qq.com (W. Wang), g.mengali@ing.
unipi.it (G. Mengali), a.quarta@ing.unipi.it (A.A. Quarta), jyuan@nwpu.
edu.cn (J. Yuan).
www.elsevier.com/locate/asr
Available online at www.sciencedirect.com
ScienceDirect
Advances in Space Research 60 (2017) 1115–1129

acceleration by suitably orienting the thrust vector direc-
tion in such a way to balance the centrifugal and gravita-
tional components of the acceleration(Mengali and
Quarta, 2009; McInnes and Simmons, 1992; Heiligers
et al., 2014), for a time-span on the order of some terres-
trial years. However, the propulsive requirements for this
kind of mission scenario, given in terms of maximum value
of propulsive acceleration modulus necessary to maintain
the displaced orbit, could be beyond the technological
capabilities of an E-sail propulsion system (Janhunen
et al., 2013).
A feasible solution to the problem is to (ideally) disag-
gregate the payload (Mazal and Gurfil, 2013) among mul-
tiple spacecraft, with each functional module conveying the
indispensable mass only. That way, with a reduction of the
payload mass, the reference propulsive acceleration of an
E-sail (the so called characteristic acceleration) may
become significantly high (Janhunen et al., 2013). In that
case, a number of different spacecraft are required to oper-
ate in close proximity, that is, in a formation flight(Mu
et al., 2015). In addition, the concept of formation flying
provides a means for increasing the observation capability,
by arranging multiple E-sail based spacecraft so as to con-
stitute a sort of large object with a predesigned geometry.
In some cases the formation geometry is loosely defined,
since the spacecraft are only required to operate within a
bounded region. This arrangement is usually termed cluster
flight or loosed formation (Mazal and Gurfil, 2013).
Finally, a formation flying system comprised of multiple
E-sails may contribute to a resolution improvement,
enabling promising scientific missions such as the Mer-
cury’s magnetotail measurement (Aliasi et al., 2015)or
the astronomical observation (Salazar et al., 2015).
Existing results on spacecraft formation flying around
displaced orbits are confined to cases in which the primary
propulsion system is a photonic solar sail. In this context,
Gong et al. (2007, 2008) have discussed the problem of a
chief spacecraft that covers a heliocentric circular displaced
orbit, while the deputy adjusts its thrust vector in order to
track a given trajectory by means of an active control
strategy.
An interesting contribution to the subject of spacecraft
formation flight is given by the study of the relative motion
of two spacecraft in displaced orbits using a geometrical
approach (Wang et al., 2016c,b,a). For example, Wang
et al. (2016c) analyzes the relative motion between two
solar sails that cover a circular displaced orbit, using a
set of (modified) displaced orbital elements. More recently,
the geometrical analysis has been also applied to elliptic
orbits using classical (Wang et al., 2016b) or non-singular
Nomenclature
asemimajor axis [au]
aspacecraft characteristic acceleration [mm=s2]
apropulsive acceleration [mm=s2]
eeccentricity
ftrue anomaly [rad]
hspecific angular momentum vector [au2=day]
Hdisplacement [au]
iinclination [rad]
Mmean anomaly [rad]
mspacecraft total mass [kg]
mpay payload mass [kg]
Nnumber of tethers
nmean motion [rad=day]
OSun’s center-of-mass
ofocus of displaced orbit
Rfocus-spacecraft distance [au]
rspacecraft position vector (with r¼r
kk
) [au]
Sspacecraft center-of-mass
ttime [days]
TIinertial reference frame
TPperifocal reference frame
TRrotating reference frame
TRI transformation matrix between TRand TI
TRDRCtransformation matrix between TRDand TRC
Tij ði;jÞentry of matrix TRDRC
^
x;^
y;^
zunit vectors of coordinate axes
acone angle [rad]
anpitch angle [rad]
celevation angle [rad]
hargument of latitude of displaced orbit [rad]
lSun’s gravitational parameter [au3=day2]
qx;qy;qzcomponents of relative position vector in the
chief’s rotating frame [au]
qrelative position vector [au]
Xright ascension of the ascending node of dis-
placed orbit [rad]
xangular velocity vector, (with x¼x
kk
) [rad=s]
-argument of periapsis of displaced orbit [rad]
Subscripts
Cchief
Ddeputy
Iinertial
max maximum
min minimum
Pcelestial body
Superscripts
T transpose
Hextreme value
time derivative
^unit vector
1116 W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129

orbital elements (Wang et al., 2016a). In both cases the
closed-form solution of the relative motion is parameter-
ized in the configuration space. Note, however, that in
Wang et al. (2016c,b,a) the problem of formation flight
has never been addressed.
The aim of this paper is to study, for the first time, the
formation flight of E-sail based spacecraft in elliptic dis-
placed orbits. The analysis uses the recent E-sail thrust
model (Yamaguchi and Yamakawa, 2013; Quarta and
Mengali, 2016), in which the propulsive acceleration mod-
ulus and the cone angle are both parameterized with
numerical fitting polynomial equations as functions of the
pitch angle. The relative motion of the spacecraft is
addressed in the configuration space using suitable coordi-
nate transformations that incorporate a set of displaced
orbital elements. Unlike the previous works (Wang et al.,
2016c,b,a), the formation geometry is studied assuming a
small distance between the spacecraft in the formation,
which allows the problem to be linearized. Accordingly,
the extreme values of the relative distances can be accu-
rately calculated in an analytical form. Maintaining a for-
mation flight with prescribed maximum relative distance
between the spacecraft requires, however, a cooperative
control strategy. This problem is discussed in detail in a
companion paper (Wang et al., 2017).
The paper is organized as follows. The next section
briefly summarizes the mathematical model used to
describe the propulsive thrust of an E-sail based spacecraft
and the relations necessary for maintaining an elliptic dis-
placed orbit. Section 3discusses the performance require-
ments, in terms of characteristic acceleration as a
function of the payload mass and the number of tethers.
Section 4illustrates the equation of relative motion and
introduces the approximations necessary to find an analyt-
ical form of the bounds of the relative motion. A case
study, discussed in Section 5, shows the effectiveness of
the proposed method, while Section 6contains the con-
cluding remarks.
2. E-sail requirements for displaced orbits
The formation flying problem consists of a chief space-
craft that is required to track a heliocentric displaced orbit
and a deputy spacecraft that orbits around the chief at a
small distance from it. To better describe the mathematical
model used in this paper, it is useful to start with some con-
cepts recently introduced in the literature.
2.1. E-sail thrust model
Each spacecraft is assumed to be equipped with an E-
sail, whose thrust model is taken from Yamaguchi and
Yamakawa (2013) and thoroughly discussed in Quarta
and Mengali (2016). In particular, Yamaguchi and
Yamakawa (2013) represent the more recent evolution of
the E-sail model used for mission analysis purposes, whose
first appearance is in the works of Janhunen and Sandroos
(2007).
To summarize this thrust model, we introduce the unit
vector ^
nnormal to the nominal plane containing the sail
tethers and pointing in the direction opposite to the Sun,
see Fig. 1. According to Yamaguchi and Yamakawa
(2013), the E-sail propulsive acceleration vector ais a func-
tion of both the Sun-spacecraft distance r, and the sail
pitch angle an2½0;90deg, the latter being the angle
between the unit vector ^
nand the Sun-spacecraft unit vec-
tor ^
r(where ^
r,r=r, with ris the Sun-spacecraft position
vector), viz.
a¼a
r
r

j^
að1Þ
where ais the spacecraft characteristic acceleration,
defined as the maximum modulus of aat a Sun-
spacecraft reference distance r,1 au, whereas
j¼jðanÞ2½0:5;1is the dimensionless propulsive acceler-
ation, defined as the ratio of the local value of akkat a cer-
tain pitch angle anto the local maximum propulsive
acceleration. Note that the value of acan be suitably var-
ied along the spacecraft trajectory, by adjusting the tethers
voltage (Toivanen et al., 2012; Toivanen and Janhunen,
2013), to maintain an elliptic displaced orbit (Niccolai
et al., in press-b).
Since abelongs to the plane ð^
n;^
rÞ, the propulsive accel-
eration can be conveniently written as a function of the
radial and normal unit vectors as (Quarta and Mengali,
2016)
a¼ar
r

jsin anaðÞ
sin an^
rþsin a
sin an^
n
hi
if an20;p=2ð
ar
r

^
rif an¼0
8
<
:ð2Þ
where a¼aðanÞ2½0;90deg is the sail cone angle, i.e. the
angle between the direction of ^
aand the Sun-spacecraft
line. Bearing in mind the data given by Yamaguchi and
Yamakawa (2013), the variations of jand awith the sail
pitch angle anare shown in Fig. 2. Note that the cone angle
reaches a maximum value of about 20 deg when
an’55 deg, whereas the dimensionless propulsive acceler-
ation is maximum (j¼1) at an¼0 (i.e., when ^
r^
nand
the sail nominal plane is normal to the Sun-spacecraft line),
and then decreases monotonically with anuntil j’0:5.
Further details about the thrust mathematical model for
mission analysis purposes can be found in Quarta and
Mengali (2016). Also, a more accurate performance model
could be obtained by taking into account the time delay
associated with turning the tether rig.
2.2. Chief’s elliptic displaced orbit
It is now useful to recall some mathematical relations
that give an estimate of the propulsive performance neces-
sary for maintaining a heliocentric, elliptic, displaced orbit
using an E-sail propulsion system. For an in-depth
W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129 1117

discussion about this problem the interested reader is
referred to Niccolai et al. (in press-b, in press-a).
Consider a Keplerian, elliptic, reference orbit with a
semimajor axis aPand an eccentricity eP. Without loss of
generality, assume the nominal orbit to belong to the refer-
ence plane and introduce a heliocentric-perifocal frame
TPðO;^
xP;^
yP;^
zPÞ, see Fig. 1. For example, the nominal
orbit could be the trajectory of a target celestial body (a
planet or an asteroid) in a scientific observation mission.
Note that the orbit of the celestial body is close to (but
not necessarily in) the ecliptic plane due to a (small) incli-
nation. A chief spacecraft (subscript C) is assumed to cover
an elliptic displaced orbit, with semimajor axis aC, eccen-
tricity eC¼eP, and whose orbital plane is parallel to that
of the celestial body. With reference again to Fig. 1, intro-
duce a rotating reference frame TRCðSC;^
xR;^
yR;^
zRÞ, where
^
zRaxis is positive in the direction of the spacecraft instan-
r
ˆP
z
ˆP
y
ˆP
x
Sun
celestial
body
O
ospacecraft
f
C
S
R
H
PFDO
γ
perihelion
perihelion
reference
plane
displaced orbit’s
plane
to Sun
α
ˆ
a
ˆ
n
n
α
ˆ
r
ˆR
x
ˆR
y
spacecraft
Fig. 1. Reference frames and E-sail characteristic angles.
Fig. 2. Sail cone angle aand dimensionless propulsive acceleration jas a
function of the sail pitch angle an.
1118 W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129

taneous angular velocity vector x. The focus-pericenter
direction of the displaced orbit is assumed to be parallel
to the focus-pericenter direction of the reference orbit,
whereas the chief’s instantaneous angular velocity xC
matches the instantaneous angular velocity of the celestial
body on the reference orbit. According to Niccolai et al.
(in press-b), such an orbit is referred to as Planet Following
Displaced Orbit (PFDO).
The spacecraft equation of motion in the rotating frame
TRCis
€
rCþ2xC_
rCþ_
xCrCþxCxCrC
ðÞ
¼l
r3
C
rCþaCð3Þ
where aCis the propulsive acceleration given by Eq. (2) as a
function of the E-sail characteristics, and lis the Sun’s
gravitational parameter. Note that xCcoincides, in this
case, with the angular velocity of TRCwith respect to
TP. The components of the chief’s position vector rCin
the frame TRCare
rC
½
TRC¼RC;0;HC
½
Tð4Þ
where HCis the (constant) displaced orbit’s plane distance
with respect to the Sun’s center-of-mass O,andRCis the o-
chief distance (i.e. the displaced orbit’s focus-chief dis-
tance), see Fig. 1. For an elliptic displaced orbit, the dis-
tance RCcan be written as
RC¼aC1e2
C

1þeCcos fC
ð5Þ
where fCis the chief’s true anomaly.
The components of Eq. (3) along the radial, circumfer-
ential and normal directions of the chief’s rotating frame
TRCare
€
RCþl
r2
C
cos cCRC_
f2
C¼ajC
r
rC

cos aCþcC
ðÞð6Þ
RC€
fCþ2_
RC_
fC¼0ð7Þ
l
r2
C
sin cC¼ajC
r
rC

sin aCþcC
ðÞ ð8Þ
where cCis the chief’s elevation angle defined as the angle
between the Sun-spacecraft line and the orbital plane,viz.
cC¼arctan HC
RC
 ð9Þ
Since the chief’s angular velocity _
fCequals by assump-
tion that of the reference celestial body xC
kk
, therefore
(Alfriend et al., 2010)
_
fC¼xC
kk
¼n1þeCcos fC
ðÞ
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1e2
C
ðÞ
3
qð10Þ
where n¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
l=a3
P
pis the mean motion along the reference
celestial orbit. The term €
RCin Eq. (6) can be calculated
from Eq. (5) as a function of faC;eC;fCg. In fact, taking
into account Eq. (10), the result is
€
RC¼aCeCn2
1e2
C
ðÞ
2cos fC1þeCcos fC
ðÞ
2ð11Þ
From Eq. (7), the component hCz,hC^
zRCof the chief’s
specific angular momentum hCalong the ^
zRCaxis is conser-
vative, which implies
hCz¼R2
C_
fC¼constant ð12Þ
The latter equation, taking into account Eqs. (5) and
(10), gives
hCz¼na
2
Cffiffiffiffiffiffiffiffiffiffiffiffiffi
1e2
C
q¼constant ð13Þ
Substituting Eqs. (10) and (11) into Eqs. (8) and (7) and
bearing in mind Eq. (13), the required cone angle aCand
characteristic acceleration afor maintaining a PFDO
are obtained as a function of cCas
tanaC¼n3
CtancCffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þtan2cC
p
1n3
Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þtan2cC
pð14Þ
a
l=r2

¼r
jCHCffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n6
Ctan2cC1þtan2cC
ðÞ2n3
C
tan2cC
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þtan2cC
pþtan2cC
1þtan2cC
s
ð15Þ
where
nC,aC
aP
ð16Þ
is the ratio between the semimajor axis of the (chief’s) dis-
placed and reference orbit. Eqs. (14)-(15) provide a com-
pact formulation of the conditions necessary for
maintaining a heliocentric, elliptic, displaced orbit using
an E-sail propulsion system. These equations are valid
for any elliptic displaced orbit and generalize the results
discussed in Niccolai et al. (in press-b), which are obtained
using a different approach. Note that in the circular case
(when eC¼0), Eqs. (14) and (15) reduce to the results of
Niccolai et al. (in press-a).
The procedure for calculating the propulsive accelera-
tion necessary for maintaining a given displaced orbit can
now be summarized. First, choose the parameters of the
reference Keplerian orbit faP;ePg, and those of the dis-
placed orbit to be reached fHC;aCg(recall that eC¼eP
by assumption). For a given point along the orbit charac-
terized by fC, calculate the value of RCfrom Eq. (5),cC
from Eq. (9), and the ratio nCfrom Eq. (16). The cone
angle aCis obtained from Eq. (14).IfaCturns out to be
greater than the maximum admissible value of about
20 deg, see Fig. 2, that means that the desired displaced
orbit cannot be maintained by an E-sail. Otherwise, the
dimensionless propulsive acceleration jCis derived from
Fig. 2. Finally, the dimensionless value of the spacecraft
characteristic acceleration is obtained from Eq. (15) (recall
that l=r2
’5:93 mm=s2).
W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129 1119

3. Typical performance requirements
According to Eqs. (5) and (9), in the elliptic case the ele-
vation angle cCvaries as a function of the chief’s true
anomaly fC. Therefore, the characteristic acceleration of
the chief spacecraft amust be slightly modified along
the trajectory to maintain the displaced orbit, and this
can be achieved by suitably adjusting (within certain limits)
the tether voltage (Toivanen et al., 2012; Toivanen and
Janhunen, 2013). For a given value of the displaced orbit’s
semimajor axis aCand displacement distance HC, the max-
imum value of the required characteristic acceleration
amax ¼maxðaÞis obtained when the chief is at its pericen-
ter (fC¼0), that is, when
tanðcCÞ¼tanðcCmax Þ,HC
aC1eC
ðÞ
HC
nCaP1eP
ðÞ
ð17Þ
Fig. 3. Feasible regions and contour lines of maximum value of characteristic accelerations amax as a function of HCand aCfor some reference celestial
bodies.
1120 W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129

Moreover, according to Eq. (14), the cone angle takes its
maximum value when cC¼cCmax and, therefore, fC¼0is
the most demanding condition in terms of propulsive
requirements to meet.
Fig. 3 illustrates the feasible regions and the contour
lines of the maximum value of the spacecraft characteristic
accelerations amax for some mission scenarios in which the
reference celestial body is a planet. The figure clearly shows
that, in all of the analyzed cases, the most favorable condi-
tion in terms of characteristic acceleration value it requires,
is obtained when HCis small and the semimajor axis aCis
near to that of the celestial body aP. Note that for an obser-
vation mission, a spacecraft close to the planet is a good
choice. However, in general, the spacecraft must avoid a
close approach to the planet since the E-sail is unable to
properly operate within the planetary magnetosphere that
shields and deflects the solar wind. This same problem is,
of course, much less important for other planets, such as
Venus and Mars.
Taking into account the latter remark, Fig. 3 shows that
the propulsive requirements for maintaining a displaced
orbit tend to increase quickly with the displacement HC,
especially in the case of the inner planets. For example, if
the reference celestial body is the Earth, a displaced orbit
with HC¼0:05 au and aC¼0:95 au can be obtained using
an E-sail with a maximum characteristic acceleration of
about 1:16 mm=s2, see Fig. 3. That value corresponds to
a medium-performance E-sail, whose technological level
could be reached in the near future (Janhunen et al.,
2010). However, with a characteristic acceleration of
1:16 mm=s2, a (relatively) small payload fraction can be
obtained with the current technology level.
A potential solution to this problem is to distribute the
payload among multiple E-sails in formation, with each
sail being comprised of indispensable modules (e.g. com-
munication and imaging module). In that way, the total
mass of a single E-sail based spacecraft can be reduced
and the propulsive performance, in terms of characteristic
acceleration, can be improved. This is exactly the concept
of disaggregated spacecraft, that is, to ideally split a mono-
lithic spacecraft into multiple closely free-flying compo-
nents that communicate with each other via inter-satellite
links (Mazal and Gurfil, 2013).
For example, using the mass breakdown model dis-
cussed by Janhunen et al. (2013) and assuming a tether
length of 15 km and a nominal tether voltage 25 kV, the
value of the characteristic acceleration and the total space-
craft mass mcan be calculated as a function of the payload
mass mpay and of the number of tethers N. The isocontour
lines of the two functions a¼aðmpay;NÞand
m¼mðmpay;NÞare drawn in Fig. 4. The figure shows that
a given value of spacecraft characteristic acceleration can
be obtained with different combinations of mpay and N, thus
obtaining different values of the total mass m. Under the
assumption that the characteristic acceleration is set equal
to the previous value of 1:16 mm=s2, it is possible to
numerically calculate (Janhunen et al., 2013) the relation
between the payload mass and the total spacecraft mass,
see Fig. 5. From the figure, there is a linear relation
between mand mpay for a given a. If, for example,
mpay ¼200 kg, the total spacecraft mass is about 750 kg.
Assuming a maximum launch mass of 500 kg per space-
craft (for example, the payload capability of the 7300-
series Delta II is about 600 kg for an interplanetary transfer
orbit insertion), the payload could be (ideally) split up into
two parts, each one with a mass of 1:15 200=2¼115 kg,
a value that includes a contingency factor of 15%. Accord-
Fig. 4. Characteristic acceleration aand total spacecraft mass mas a
function of the payload mass mpay and number of tethers N.
Fig. 5. Total spacecraft mass mas a function of the payload mass mpay
when a¼1:16 mm=s2.
W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129 1121

ingly, the mission could be accomplished using two space-
craft of 470 kg each (a value within the admissible range),
operating in a formation flight.
Note that the reduction of the payload mass (for a given
a) also implies a substantial simplification in the E-sail
propulsion system as the required number of tethers
decreases. In fact, Fig. 6 shows that if mpay ¼200 kg, a
characteristic acceleration of 1:16 mm=s2may be obtained
using about 100 tethers. If, instead, mpay ¼115 kg, the
number of tethers decreases to 60. Since the length of each
tether is by assumption equal to 15 km, a reduction of N
implies a reduction of the total tether length and a corre-
sponding simplification of the propulsion system.
4. Geometrical analysis of formation flying
Having analyzed the generation of an elliptic displaced
orbit for the chief spacecraft from the point of view of the
required propulsive performance, we are now in a position
to study the relative motion of two E-sail based spacecraft,
the chief and the deputy (subscript D), moving along two
different displaced orbits. In particular, the deputy tracks
a heliocentric displaced orbit whose orbital parameters
are slightly different from those of the chief spacecraft.
It has been pointed out by Wang et al. (2016c) that
when the semimajor axis and eccentricity of the two dis-
placed orbits are given, the relative motion between the
two spacecraft is univocally established, and evolves on
its invariant manifold. In the discussion to follow, the
geometry and bounds of the formation will be analyzed
in the configuration space using a purely geometric
approach, taking into account the recent results of
Wang et al. (2016b).
4.1. Approximate form of relative motion equations
Let q,rDrCdenote the relative position vector
between the two spacecraft, where rDis the Sun-deputy vec-
tor, and introduce a rotating reference frame
TRDðSD;^
xRD;^
yRD;^
zRDÞ, involving the deputy spacecraft,
defined in a perfectly similar way as that of the chief.
According to Wang et al. (2016b), the components of q
in the chief’s rotating reference frame TRCcan be written
as
½qTRC¼TT
RCIXC;iC;hC
ðÞTRDIXD;iD;hD
ðÞ½rDTRD½rCTRC
ð18Þ
where Xis the right ascension of the ascending node, iis
the inclination and h¼-þfis the argument of latitude.
The three angles fX;i;-g, which define the orientation of
the displaced orbit with respect to the inertial frame TI,
closely resemble the classical definition used for Keplerian
orbits, as is discussed by Wang et al. (2016b). The transfor-
mation matrix TRI from TRto TI, written as a function of
the three Euler’s angles fX;i;hg, is given by
For the sake of brevity, the following shorthand nota-
tion is now introduced
TRDRC,TT
RCIXC;iC;hC
ðÞTRDIXD;iD;hD
ðÞ ð20Þ
where TRDRC(with generic entry Tij ) represents the transfor-
mation matrix between the two rotating frames (from TRD
to TRC).
Assume that the chief and deputy fly close to each other.
Accordingly, the relative orbital elements of the two
spacecraft are first-order small quantities, that is,
Fig. 6. Number of tethers Nas a function of the payload mass mpay when
a¼1:16 mm=s2.
Table 1
Characteristics of the two PFDOs considered in the simulations.
a[au] H[au] ei[deg] X[deg] -[deg] f0[deg]
Chief 0.95 0.05 0.0167 0.001 224 237 0
Deputy 0.925 0.051 0.01 1 225 238 0
TRI X;i;hðÞ,
cos hcos Xsin hcos isin XðÞsin hcos Xcos hcos isin XðÞsin isin XðÞ
cos hsin Xþsin hcos icos XðÞsin hsin Xþcos hcos icos XðÞsin icos XðÞ
sin hsin icos hsin icos i
2
6
43
7
5ð19Þ
1122 W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129

Dh,hDhC1;DX ,XDXC1, and Di,iD
iC1. Therefore, the entries Tij of the transformation
matrix TRDRCcan be approximated as
T11 ¼T22 ¼T33 ’1;T12 ¼T21 ’Dhþcos iCDXðÞ;
T13 ¼T31 ’cos iCcos hCDX sin hCDi;
T23 ¼T32 ’cos iCsin hCDX þcos hCDiðÞ
ð21Þ
Substituting Eqs. (20) and (21) in Eq. (18), the radial,
along-track and normal components of the relative posi-
tion vector qcan be expressed in the chief’s rotating frame
TRCas
qx’DRþHCsin iCcos hCDX sin hCDiðÞð22Þ
qy’RCDhþcos iCDXðÞ
HCsin iCsin hCDX þcos hCDiðÞð23Þ
qz’RCsin hCDisin iCcos hCDXðÞþDHð24Þ
where DR,RDRCand DH,HDHC. Note that,
enforcing the condition HC¼HD¼0, Eqs. (22)–(24)
reduce to a form consistent with the Keplerian case
(Schaub, 2004; Jiang et al., 2008; Dang et al., 2014).
If the relative distance between the two spacecraft is suf-
ficiently small, the term DRmay be approximated using its
differential, that is
DR¼@R
@aC
Daþ@R
@eC
Deþ@R
@fC
Dfð25Þ
Fig. 7. Characteristic accelerations of the chief and deputy as a function
of the spacecraft true anomaly.
Fig. 8. Pitch angles of the chief and deputy as a function of the spacecraft
true anomaly.
Fig. 9. Spacecraft relative trajectory: approximate (dashed line) vs.
reference solution (solid line).
Fig. 10. Comparison between the reference (solid line) and approximate
(dashed line) solution as a function of the spacecraft true anomaly.
W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129 1123

with
@R
@aC
@RC
@aC
¼1e2
C
1þeCcos fC
ð26Þ
@R
@eC
@RC
@eC
¼aC2eCþ1þe2
C

cos fC

1þeCcos fC
ðÞ
2ð27Þ
@R
@fC
@RC
@fC
¼aCeC1e2
C

sin fC
1þeCcos fC
ðÞ
2ð28Þ
where the expression of RCgiven by Eq. (5) has been con-
sidered. Using the relation between the true anomaly fand
the mean anomaly M, it can be verified that (Schaub, 2004)
Df¼1þeCcos fC
ðÞ
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1e2
C
ðÞ
3
qDM
þ2þeCcos fC
ðÞsin fC
1e2
C
Deð29Þ
where DM,MDMCand De,eDeC.
For a formation flight mission, the two spacecraft
should satisfy the periodic (or 1:1 commensurability) con-
dition, otherwise the relative motion would be locally
unbounded. The mean motion of the two spacecraft must
take the same value, i.e. nC¼nD. Note that
M¼M0þntt0
ðÞ, therefore the periodic condition
implies DM¼DM0. The parametric solution to spacecraft
relative motion can be obtained in terms of displaced orbi-
tal elements differences by substituting Eqs. (25)–(29) into
Eqs. (22)–(24). Using the chief’s true anomaly fCas the
independent variable, the solution can be written as
qx’1e2
C
1þeCcos fC
DaaCcos fCDe
HCsin -CþfC
ðÞDi
þHCsin iCcos -CþfC
ðÞDX
þaCeCsin fC
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1e2
C
pDM0ð30Þ
Table 2
Extreme values of the relative distances bounds.
Component sHfH
Cmod 360 ½deg] Extreme values
qx3:1507 1023:1739 1013:6092 1:8361 1021:7443 1023:2529 102
qy1:0072 9:9282 1012:6959 1028:9587 1014:6712 1021:95728 102
qz3:4240 2:9206 1012:1256 1023:2562 1011:7802 1021:6282 102
q1:6213 1:0635 2:4333 1029:3525 1015:5850 1023:2627 102
Fig. 11. Approximate bounds (dashed lines) and simulated reference directional relative motion (solid lines).
1124 W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129

qy’aC2þeCcos fC
ðÞsin fC
1þeCcos fC
De
þaC1e2
C

cos iC
1þeCcos fC
HCsin iCsin -CþfC
ðÞ

DX
þaC1e2
C

1þeCcos fC
D-þaC1þeCcos fC
ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1e2
C
pDM0
ð31Þ
qz’aC1e2
C

1þeCcos fC
sin -CþfC
ðÞDi
aC1e2
C

1þeCcos fC
cos -CþfC
ðÞsin iCDX þDHð32Þ
Observe that the only assumption made so far to lin-
earize the relative motion equations is that the relative (dis-
placed) orbital elements of the two spacecraft are small
Fig. 12. Spacecraft relative trajectory (solid line: reference; dashed line: approximate) as a function of the chief’s eccentricity eC.
W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129 1125

quantities. In a general case with two highly elliptic dis-
placed orbits, the relative motion geometry represented
by Eqs. (30)–(32) is very complex, and the invariant mani-
fold along which it evolves resembles a quadric or a quartic
surface (Wang et al., 2016c,b). However, if the eccentricity
of the chief’s orbit is sufficiently small, i.e. if eC1, an
interesting (simplified) form of relative motion equations
can be obtained with the following approximations
1
1þeCcos fC
’1eCcos fCand ffiffiffiffiffiffiffiffiffiffiffiffiffi
1e2
C
q’1ð33Þ
In fact, substituting Eq. (33) into Eqs. (30)–(32), the
first-order approximation of the components of relative
position vector becomes
qx’DaþeCDaaCDeHCsin -CDið
þHCsin iCcos -CDXÞcos fCþHCcos -CDið
HCsin iCsin -CDX þaCeCDM0Þsin fCð34Þ
qy’aCcosiCDX þD-þDM0
ðÞ
þaCeCDM0cosiCDX D-
ðÞ½
HCcos-CDiþsiniCsin -CDXðÞcos fC
þ2aCDeþHCsin-CDisin iCcos -CDXðÞ½sinfC
aCeCDesin2fCð35Þ
qz’1
2aCeCsin iCcos -CDX sin -CDiþDHðÞ
þaCsin -CDisin iCcos -CDXðÞcos fC
þaCcos -CDiþsin iCsin -CDXðÞsin fC
þ1
2aCeCsin iCcos -CDX sin -CDiðÞcos 2fC
1
2aCeCcos -CDiþsin iCsin -CDXðÞsin 2fCð36Þ
In particular, if the chief spacecraft tracks a circular dis-
placed orbit, i.e. when eC¼0, Eqs. (34)–(36) reduce to the
compact form
qx’DaþAxsin fCþux
ðÞ ð37Þ
qy’aCcos iCDX þD-þDM0
ðÞþAysin fCþuy

ð38Þ
qz’DHþAzsin fCþuz
ðÞ ð39Þ
where
Az¼aCffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Di2þsin2iCDX2
qð42Þ
ux¼arctan HCsin -CDisin iCcos -CDXðÞþaCDe
HCcos -CDiþsin iCsin -CDXðÞ

ð43Þ
uy¼arctan HCcos -CDiþsin iCsin -CDXðÞ
HCsin iCcos -CDX sin -CDiðÞ2aCDe

ð44Þ
uz¼arctan sin -CDisin iCcos -CDX
cos -CDiþsin iCsin -CDX

ð45Þ
Eqs. (37)–(39) constitute a parametric representation of
an elliptic cylinder, which can be considered a degenerated
manifold of the full relative motion problem. Note that if
De¼0 and HC–0, the in-plane projection of the relative
motion is a circle with a radius Ax¼Ay¼HC
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Di2þsin2iCDX2
p. The other special case happens when
De–0 and HC¼0. In this case Ax=Ay¼1=2, which corre-
sponds to an in-plane projected ellipse with a semimajor
axis equal to 2 aCDeand a semiminor axis equal to aCDe.
4.2. Bounds of relative motion
An essential prerequisite for a formation flight is that
the relative distance of the two spacecraft be confined
within a well-defined region with some (assigned) lower
and upper bounds. These bounds play a fundamental role
in establishing the size of the formation. Unlike previous
studies (Wang et al., 2016b) in which these values were cal-
culated by solving some high-order (either quartic or sex-
tic) equations, in this paper they are obtained in an
Fig. 13. Maximum error (in modulus) achievable with the approximate
solution as a function of the orbital eccentricity.
Ax¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2
CDe2þ2aCHCDesin -CDisin iCcos -CDXðÞþH2
CDi2þsin2iCDX2

qð40Þ
Ay¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4a2
CDe2þ4aCHCDesin -CDisin iCcos -CDXðÞþH2
CDi2þsin2iCDX2

qð41Þ
1126 W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129

explicit form as a function of the displaced orbital
elements.
To this end, introduce the substitution s,tanðfC=2Þ
and assume a small eccentricity of the chief’s orbit. It can
be verified that the generic solution of the relative motion
described by Eqs. (30)–(32) can be approximated as
qx’k1þ2k2sþk3
s2þ1ð46Þ
qy’k4þ2k5sþk6
s2þ1ð47Þ
qz’k7þ2k8sþk9
s2þ1ð48Þ
where the coefficients k1;k2;...;k9are
k1¼1þeC
ðÞDaþaCDeþHCsin-CDiHCsiniCcos -CDX ð49Þ
k2¼HCcos-CDiHCsiniCsin -CDX þaCeCDM0ð50Þ
k3¼eCDaaCDeHCsin-CDiþHCsiniCcos -CDX ð51Þ
k4¼HCcos-CDiþaC1þeC
ðÞcosiCþHCsin iCsin-C
½DX
þaC1þeC
ðÞD-þaC1eC
ðÞDM0ð52Þ
k5¼aC2þeC
ðÞDeþHCsin-CDiHCsin iCcos -CDX ð53Þ
k6¼HCcos-CDiaCeC1þeC
ðÞcosiCþHCsin iCsin -C
½DX
aCeC1þeC
ðÞD-þaCeCDM0ð54Þ
k7¼aC1þeC
ðÞsin-CDiþaC1þeC
ðÞsiniCcos-CDX þDH
ð55Þ
k8¼aC1þeC
ðÞcos-CDiþaC1þeC
ðÞsin iCsin-CDX ð56Þ
k9¼aC1þ2eC
ðÞþHCsin-CDiHCsiniCcos -CDX ð57Þ
In general, the relative motion is three dimensional and
always evolves on its invariant manifold. However, under
the assumption eC1, as per Eqs. (46)–(48), the relative
motion lies on a plane parameterized by
CxqxþCyqyþCzqzþC0¼0ð58Þ
where
Cx¼k5k9k6k8ð59Þ
Cy¼k3k8k2k9ð60Þ
Cz¼k2k6k3k5ð61Þ
C0¼k1k6k8k1k5k9þk2k4k9k3k4k8þk3k5k7k2k6k7
ð62Þ
The bounds for each component of the relative position
vector can be found by enforcing the necessary conditions
@qi
@s¼0 with i¼x;y;z
fg ð63Þ
where the components qiare given by Eqs. (46)–(48). From
Eq. (63), the critical values of scorresponding to the radial
bounds (denoted with the superscript H) are obtained as
sH
x¼k3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2
2þk2
3
q
k2
ð64Þ
and the maximum and minimum radial distances are
qH
x¼k1þk3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2
2þk2
3
qð65Þ
Note that the mapping from fto sintroduces a number
of inherent singularities s¼1at f¼kp(where k2N),
which occur when k2¼0. In these cases, the extreme values
are identical, that is, qxmax ¼qxmin ¼k1.
Likewise, for the along-track and normal motion, the
critical values of sand the corresponding extreme values
of distances bounds are
sH
y¼k6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2
5þk2
6
q
k5
ð66Þ
qH
y¼k4þk6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2
5þk2
6
qð67Þ
sH
z¼k9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2
8þk2
9
q
k8
ð68Þ
qH
z¼k7þk9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2
8þk2
9
qð69Þ
The singularities corresponding to k5¼0ork8¼0 take
place when f¼kp. In these special cases, the extreme val-
ues for the along-track and normal motion reduce to
qymax ¼qymin ¼k4and qzmax ¼qzmin ¼k7, respectively.
The relative distance bounds of the two spacecraft can
also be found using Eqs. (46)–(48). Let q,qkk¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q2
xþq2
yþq2
z
qbe the two spacecraft relative distance, then
q2¼gþr3s3þr2s2þr1sþr0
s2þ1ðÞ
2ð70Þ
where
r0¼4k1k3þk4k6þk7k9þk2
3þk2
6þk2
9

ð71Þ
r1¼4k1k2þk4k5þk7k8þ2k2k3þ2k5k6þ2k8k9
ðÞð72Þ
r2¼4k1k3þk4k6þk7k9þk2
2þk2
5þk2
8

ð73Þ
r3¼4k1k2þk4k5þk7k8
ðÞ ð74Þ
g¼k2
1þk2
4þk2
7ð75Þ
The condition to be met for obtaining the extreme val-
ues of the relative distance is
@ðq2Þ
@s¼0ð76Þ
which amounts to solving a quartic equation in the variable
sgiven by.
r3s4þ2r2s3þ3r1r3
ðÞs2þ2r22r0
ðÞsþr1
¼0ð77Þ
whose real roots can be easily found numerically using a
standard approach.
5. Case study
To verify the proposed methodology for analyzing the
formation geometry and the distance bounds of relative
motion between two E-sail based spacecraft, Eq. (18) is
W. Wang et al. / Advances in Space Research 60 (2017) 1115–1129 1127

now assumed as the reference solution and is compared
with the (approximate) analytical results. The Earth is
taken as the reference celestial body with a semimajor axis
aP¼1 au and an eccentricity eP¼0:0167, with the chief
spacecraft covering a PFDO. The deputy’s orbital elements
are slightly different from those of the chief and, in order to
guarantee the periodicity of relative motion, the mean
motions of the two spacecraft are assumed
nC¼nD¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l=ð1auÞ3
q, a value that coincides with the
Earth’s mean motion. The relevant data of the two PFDOs
are summarized in Table 1.
The required characteristic accelerations necessary to
maintain the displaced orbits as a function of the true
anomaly for the two spacecraft are shown in Fig. 7, while
the variations of the chief and deputy’s pitch angles are
illustrated in Fig. 8.Fig. 9 depicts the formation geometry
obtained from the approximate solution of Eqs. (30)–(32),
which fits well the reference numerical results, as is clearly
shown in Fig. 10.
The extreme values of relative distances, as well as the
critical values of the chief’s true anomaly, are summarized
in Table 2. The performance of the parametric representa-
tion of Eqs. (46)–(48) is illustrated in Fig. 11, indicating
how the approximate extreme values are able to success-
fully predict the lower and upper bounds of the relative
distances.
In principle, the accuracy of the approximate solution
relies on the differences of the orbital elements and the
eccentricity of the chief (which is small by assumption).
To better emphasize the importance of eC,Fig. 12 shows
the accuracy level that can be obtained by varying the
chief’s orbital eccentricity in a range eC20;0:1½, while
maintaining unchanged the other parameters of Table 1.
Note that the accuracy decreases with eC, but the approx-
imate solution still performs well until eC60:04. The accu-
racy level is better illustrated in Fig. 13, which quantifies
the maximum error along a full orbit using the approxi-
mate solution. Note that the minimum value of the maxi-
mum error is approximately obtained when
eC¼eD¼0:0167, that is, when the relative eccentricity
(De) tends to zero.
6. Conclusions
The feasibility of maintaining an elliptic displaced orbit
using an E-sail has been investigated using a geometrical
approach. The problem of studying the formation flying
between two E-sail based spacecraft that cover displaced
orbits has been parameterized by a set of displaced orbital
elements using a semi-analytical procedure. Assuming a
small difference in the orbital elements, the closed-form
solution to the relative motion can be described in an alge-
braic form via relative orbital elements.
In particular, in the case of small eccentricity, the
extreme values for the relative distances can be calculated
analytically through the solution of an algebraic equation.
This result is useful for determining the formation bound-
aries with a given set of orbital elements or to estimate
the reachable domain when some uncertainties exist in
the orbital elements. However, maintaining the formation
requires an active control strategy where, in general, both
the characteristic acceleration and the sail attitudes vary
as a function of the spacecraft relative distance. The con-
trol strategy is closely related to the formation structure,
as is discussed in the companion paper (Wang et al.,
2017).
Acknowledgements
This work was funded by the National Natural Science
Foundation of China (No. 11472213) and Open Research
Foundation of Science and Technology in Aerospace
Flight Dynamics Laboratory of China (No. 2015afdl016).
This work was also supported by the Chinese Scholarship
Council.
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