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This paper develops cluster control algorithms for the Satellite Swarm Sensor Network project, for which the main aim is to enable disaggregation of space-based remote sensing, imaging, and observation satellites. A methodological development of orbit control algorithms is provided, supporting the various use cases of the mission. Emphasis is given on outlining the algorithm’s structure, information flow, and implementation. The methodology presented herein enables operation of multiple satellites in coordination to facilitate disaggregation of space sensors and augmentation of data provided therefrom.
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Cluster-Keeping Algorithms for the Satellite Swarm Sensor
Network Project
Eviatar Edlermanand Pini Gurfil
TechnionIsrael Institute of Technology, 32000 Haifa, Israel
DOI: 10.2514/1.A34151
This paper develops cluster control algorithms for the Satellite Swarm Sensor Network project, for which the main
aim is to enable disaggregation of space-based remote sensing, imaging, and observation satellites. A methodological
development of orbit control algorithms is provided, supporting the various use cases of the mission. Emphasis is given
on outlining the algorithms structure, information flow, and implementation. The methodology presented herein
enables operation of multiple satellites in coordination to facilitate disaggregation of space sensors and augmentation
of data provided therefrom.
Nomenclature
A= effective cross section, m2
a= semimajor axis, km
CD= drag coefficient
Dmax = distance upper bound, km
Dmin = distance lower bound, km
e= eccentricity
ex,ey= eccentricity vector components
h= orbital specific angular momentum vector, m2s
Isp = specific impulse, s
i= inclination, rad
ix,iy= inclination vector components
M= mean anomaly, rad
m= mass, kg
n= mean motion, rads
Pcont = control acceleration vector created by the onboard
actuators, kms2
Pemp = empirical accelerations, kms2
Pg= acceleration vector due to the gravitational forces,
kms2
Re= radius of the Earth, km
r= inertial position vector, km
rR,rT,rN= radial, tangential, and crosstrack components, km
T= orbit period, s
v= velocity, kms
yji = alongtrack coordinate of Sjrelative to Si,km
ΔV= velocity change, kms
ϑ= relative ascending node, rad
λ= argument of latitude, rad
μ= gravitational constant, km3s2
ρ= atmospheric density, kgm3
φ= relative perigee, rad
Ω= right ascension of the ascending node, rad
Ωs= right ascension of the sun, rad
ω= argument of perigee, rad
I. Introduction
MODERN approaches in Earth observation show a trend of
moving away from single-satellite missions, in which one
satellite includes a complete set of sensors and instruments, toward
disaggregated and distributed sensor missions [1]. Such missions
promise an improved imaging quality: for example, elimination of
radar or optical shadow effects by observing the same region with
multiple sensors from different angles. Moreover, incremental
deployment of different sensors can lead to improved quality of
service; for example, single pass interferometry with multiple
satellites can result in reduced time for image delivery from days
down to hours [2].
However, there are major technical challenges associated with
creating clusters of space sensors, such as interfacing, communication
within the cluster and to ground stations, synchronization, cluster
keeping, precision thrusting, and pointing, as well as the realization
of sensor fusion. Further research is required to determine the
effectiveness of disaggregated and distributed space sensors: in
particular, the feasibility of cluster-keeping approaches needed for
various distributed sensor mission scenarios [3]. These approaches are
strongly dependent on communication and processing capabilities,
time synchronization, and the harsh limitations on propellant
consumption, which are required for precision thrusting and pointing.
To fully use the potential of disaggregated sensors, different
satellites must be capable of flying in geometrically defined clusters
according to the sensing requirements (e.g., pointing to the same spot
or scanning the same area, potentially from different angles). In terms
of cluster flight, the precision thrusting and pointing strategies will
determine the duration of satellite employment through the usage of
limited amounts of fuel, thus constituting a critical factor for future
satellite swarms [4].
Formation flight and cluster flight algorithms have been conceived
previously, either with open-loop or closed-loop designs [5,6]. The
majority of these methods were implemented in a centralized manner
by correcting the orbit of each satellite independently to known offset
values with respect to a real or virtual satellite. A centralized
methodology with a prechosen reference orbit also implies that the
control cannot be implemented in an autonomous manner because all
the satellites should have some common knowledge a priori to make
decisions. Another disadvantage is that the control performance
strongly depends on the reference, especially under the situation of
limited thrust magnitude; a poorly selected reference orbit may result
in divergence of control. A variation of the centralized scheme is the
leaderfollower method [7,8]. Some satellites are assigned to be
leaders and are allowed to make decisions; the other satellites are
followers. Because the information flows only from leaders to
followers, this method is prone to failure of the leaders.
The Satellite Swarm Sensor Network (S3NET) projecthas been
established to work in parallel on key enablers required to develop the
efficient and autonomous use of disaggregated satellite swarms. One
of the objectives of S3NET is to achieve breakthrough progress in
distributed cluster flight algorithms by proposing cluster-keeping
Received 27 November 2017; revision received 21 March 2018; accepted
for publication 12 October 2018; published online 27 December 2018.
Copyright © 2018 by the authors. Published by the American Institute of
Aeronautics and Astronautics, Inc., with permission. All requests for copying
and permission to reprint should be submitted to CCC at www.copyright.com;
employ the ISSN 0022-4650 (print) or 1533-6794 (online) to initiate your
request. See also AIAA Rights and Permissions www.aiaa.org/randp.
*Research Staff Member, Asher Space Research Institute; eviatar@
technion.ac.il.
Professor, Faculty of Aerospace Engineering; pgurfil@technion.ac.il.
Associate Fellow AIAA. Data available online at http://s3net-h2020.eu/ [retrieved 2018].
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methods that will permit sharing resources across the cluster network
with real-time guarantees, including distributed data processing,
optimal resource allocation, and distributed control under time
delays; by semiautonomous long-duration maintenance of a cluster
network, including addition/incremental deployment and removal of
spacecraft modules to/from the cluster network under collision
avoidance constraints; and by autonomously reconfiguring the
cluster to retain critical functionality in the face of network
degradation, component failures, or space debris damage.
In this paper, we present the cluster-keeping methodology of
S3NET, which was designed for both optical and synthetic aperture
radar (SAR) use cases. In each case, the unique constraints on cluster
geometry and maintenance, as well as the related computational and
implementation considerations, are analyzed; and a corresponding
cluster control algorithm is designed and tested in a high-fidelity
simulation. The main advances compared to past formation-flying
missions such as TanDEM-X [9], GRACE [10], and Time History of
Events and Macroscale Interactions during Substorms (THEMIS)
[11] are the versatility of the algorithms, which are capable of
adjusting to multiple use cases, and the operation time of the cluster-
keeping algorithms, which can maintain the satellite cluster
configuration autonomously for prolonged mission lifetimes.
II. Cluster-Keeping Requirements
Table 1 summarizes the cluster-keeping requirements of the
different use cases in S3NET. In all cases, the semimajor axis (SMA)
is about 7070 km, and the orbit is near circular and sun synchronous.
The local time of the ascending node (LTAN) is different; this is
related to the launcher, and it has a limited effect on the cluster-
keeping algorithms. The different use cases are based on satellites
with the same mass range and similar electrical power. From the
cluster geometry perspective, the main differences stems from the
need to have alongtrack and crosstrack clusters. The alongtrack
clusters (such as multispectral and ocean currents monitoring) can
use the same cluster-keeping algorithms. The alongtrack baseline and
allowed tolerances can be used as inputs to the orbit control
algorithm. The two crosstrack clusters cannot be grouped due to the
large difference in the required crosstrack baseline. In this case, two
different cluster-keeping algorithms are needed. This paper presents
cluster-keeping algorithms that match the requirements in Table 1.
The power estimates are based on previous experience with larger
satellites. Whereas an optical payload is a passive sensor, the SAR
payload needs a powerful transmitter, hence the large difference in
power consumption: 110 W of power are required for the payload
alone in the SAR case, whereas 250300 W are required in the optical
case for the entire satellite. Although, in some use cases, we assume
that the SAR satellites will work in receive-only mode, 110 W are
required for the transmit mode. A typical continuous on time of the
radar is about 1015 min per orbit. The radar operation time is
constrained primarily by the spacecraft power system and the thermal
response. In addition, the SAR payload creates a massive amount of
data, and so downlink communication is another bottleneck. The
optical payload, on the other hand, needs a smaller amount of power;
and the main limitations on the active time are communication and
disoperation while in umbra.
III. Preliminaries
We start the discussion with definitions of the state variables used
for cluster control, and we proceed with the definitions of the required
reference orbits and astrodynamical models.
A. Relative Orbital Elements
To avoid the singularities associated with the classical elements
(ais the semimajor axis; eis eccentricity; iis inclination; Ωis the
right ascension of the ascending node; ωis the argument of perigee;
and Mis the mean anomaly) for near-circular orbits, the following set
of orbital elements can be used [12]:
α0
B
B
B
B
B
@
a
λ
ex
ey
i
Ω
1
C
C
C
C
C
A
0
B
B
B
B
B
@
a
ωM
ecos ω
esin ω
i
Ω
1
C
C
C
C
C
A
(1)
where exand eyare the components of the eccentricity vector.
Equation (1) is used to define a new set of relative orbital elements:
δα
0
B
B
B
B
B
B
@
δa
δλ
δex
δey
δix
δiy
1
C
C
C
C
C
C
A
0
B
B
B
B
B
B
@
adaa
uduΩdΩcos i
exdex
eydey
idi
ΩdΩsin i
1
C
C
C
C
C
C
A
(2)
where ixand iyare the components of the inclination vector. The
subscript ddenotes the orbital elements related to the deputy satellite.
We can also calculate the relative perigee φand relative ascending
node ϑby using the relations
δeδex
δeyδecos φ
sin φ(3)
δiδix
δiyδicos ϑ
sin ϑ(4)
Under the effect of the J2zonal harmonic, we have
Table 1 Formation-flying requirements summary
Optical Radar
Panchromatic Multispectral
Arctic, Antarctica, solid Earth
digital elevation model (DEM) Ocean currents/ship detection
Use case no. 1 2 3 4
Number of satellites 3 352 4
Orbit type Near-circular sun-synchronous low Earth orbit
Orbit altitude 693 km
Orbit inclination 98.16 deg
Light conditions Observation in light LTAN 1030 hrs
30 min
Dawndusk LTAN 0600 hrs 30 min
Formation configuration Crosstrack cluster Alongtrack cluster Crosstrack cluster Alongtrack cluster
Relative position 17.5 km 10% 7km25% 2002000 m 10% 100 m 10%
Satellite dry mass 100 kg 200 kg
Satellite power 250300 W 110 W (payload power consumption)
Satellite size 40 ×40 ×40 cm 85 cm ×120 cm ×57 (height) cm (stowed, estimated envelope)
Active time, % 50% Maximum 10%
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_
φ3
2π
TR2
e
a2J2cos2i1(5)
where Tis the orbital period, and Reis the equatorial radius.
Equation (5) is an approximation of _
φ, which is valid for close
formations with a short baseline. Using the definition of relative
elements and denoting by rR,rT, and rNthe radial, tangential, and
crosstrack separations, respectively, Ref. [13] presented a set of
linearized equations for spacecraft relative motion:
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
ΔrR
a
ΔrT
a
ΔrN
a
Δ_
rR
v
Δ_
rT
v
Δ_
rN
v
9
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
;
2
6
6
6
6
6
6
6
6
6
6
4
Δa
a0ΔexΔey
Δλ3Δa
2a2Δey2Δex
00ΔiyΔix
00ΔeyΔex
3Δa
2a02Δex2Δey
00 ΔixΔiy
3
7
7
7
7
7
7
7
7
7
7
5
×
8
>
>
>
>
>
<
>
>
>
>
>
:
1
λλ0
cos λ
sin λ
9
>
>
>
>
>
=
>
>
>
>
>
;
(6)
B. Near-Polar Sun-Synchronous Orbit with Repeating Ground Track
Based on Ref. [14], we define the following procedure to design a
sun-synchronous orbit (SSO) repeating ground track (RGT) orbit,
which is required for both SAR and optical satellites. In an RGTorbit,
the satellite tracks the same trace on the ground with a given
periodicity pattern. In an SSO, the satellite nodal rate matches the
average rate of the suns motion projected on the equator ( _
Ωs). This
means that the satellite passes over the same part of the Earth at
roughly the same local time each day. Combining these two features
into the same orbit is possible by finding the right values of a,e, and i.
In most cases, the SSO is almost circular, and the effect of eis minor.
By controlling aand i, we can control the SSO drift rate. The RGT
parameters jand k, defining the RGT period, satisfy
TkTday
j(7)
and the corresponding altitude is
haReμkTday
2πj213Re(8)
where μis the Earth gravitational parameter. Using the variational
equations
_
Ω3
2J2
μ
pR2
ea721e2cos i(9)
_
ω3
4J2
μ
pR2
ea721e225cos2i1(10)
_
M3
4J2
μ
pR2
ea721e2323cos2i1(11)
nj
kωe_
Ω_
M_
ω(12)
hμ
n213Re(13)
we can improve the height calculation by taking into account the
effect of the Earth oblateness. Notice that the designer should provide
values for eand iin the initial step by using the relation
icos1_
Ωs
32J2
μa3
pRea1e22(14)
The new height is substituted into Eq. (14) to get the inclination
value that provides the required nodal rate. This process is repeated
until aand iconverge. To find Ω, the following equations are used:
Ωs0.98563N80(15)
LTANhτ12 ΩΩs
15 (16)
where Ωsis the right ascension of the sun, and Nis the number of the
day in the year. The remaining orbital elements will be defined later to
construct the required geometry of the cluster.
C. Sun-Synchronous Orbit Control
SSO maintenance means keeping the LTAN fixed. Due to the
nature of the problem, the LTAN value is constantly changing. To
have a feasible control scheme, it is better to define an allowed
tolerance around the nominal LTAN value. For example, the SAR
Earth Observation Satellite (TerraSAR-X) satellite uses an LTAN of
1800 hrs with tolerances of 15 min. For the optical application, we
use the Advanced Earth Observing Satellite 1 (ADEOS I) satellite of
the Japan Aerospace Exploration Agency with LTAN of 1030 hrs
with tolerances of 15 min [15].
The secular rate of Ωdepends on a,e, and i. It is particularly
sensitive to changes in the inclination [16]. A non-coplanar burn at
the ascending node can be used to reset Ωto the required value; but, in
this case, the inclination will have to be corrected to match the SSO
constraint. Another approach is to take advantage of the fact that Δi
causes nodal drift. We can add Δito the nominal inclination, and we
use the excessive drift to correct the LTAN. We can define a control
box around the nominal LTAN (τΔτ). When the LTAN reaches
the bound, a burn is performed at the ascending node to change the
inclination, and we create a nodal drift in the opposite direction. The
required Δican be calculated by
Δiτrτk(17)
where τris the required LTAN, and kis a positive constant. For near-
circular orbits, the connection between the required inclination
change and ΔVis
ΔV2VsinΔi
2(18)
This maneuver takes place at the ascending node.
D. Atmospheric Drag Modeling
Atmospheric drag is one of the main perturbations for low-Earth-
orbit (LEO) spacecraft. The specific force due to atmospheric drag is
modeled in the Earth-centered inertial (ECI) coordinate system frame
as [17]
Fdrag 1
2ρkvkvACD
m(19)
where ρis the atmospheric density, Ais the projected cross-sectional
area (PCSA) (normal to the satellite velocity vector), CDis the drag
coefficient, and mis the mass. The vector vis the velocity in ECI
coordinates. Many atmospheric models have been developed over the
past few decades, but there are numerous uncertainties. In addition,
drag models contain many parameters that are difficult to estimate,
including CD,ρ, and A[16].
Reference [18] developed an analytical technique for calculating
the PCSA of satellites in any attitude. This technique takes into
account the geometrical shape of the satellite and shading. The input
to the algorithm is the velocity vector in body axes, and the output of
the algorithm is the PCSA.
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IV. Orbit and Attitude Determination Algorithm
Structure
Soon after the separation from the launcher (typically within less
than an orbit), the satellites will initiate their cluster-keeping
algorithm and maintain it throughout the mission. On each satellite,
the cluster control will be carried out by the orbit control module
(OCM) with both hardware and software components. The orbital
control module is in charge of a number of tasks, such as orbit
determination and mean orbital elements estimation of the host and
other satellites from Global Positioning System (GPS) measure-
ments, determination of the relative positions and velocities of all the
satellites in the cluster, management of the orbit and cluster control,
and delivery of the relevant inputs and commands to the attitude
determination and control system (ADCS). The structure and the
interfaces of the orbital control module are shown in the block
diagram of Fig. 1. The OCM is composed of three main submodules:
the orbit submodule, the intersat submodule, and the control
submodule.
A. Orbit Submodule
The orbit submodule is responsible for determination of the
position, velocity, and osculating orbital elements from GPS
measurements, determination of mean orbital elements, and
performance of various auxiliary calculations (e.g., the sun vector).
The orbital Kalman filter (OKF) algorithm estimates the absolute
orbit by the online processing of the respective GPS data. The GPS
fixes, obtained by the hosts onboard GPS receiver, include the
velocity and time (PVT) in the conventional earth-centered, earth-
fixed (ECEF) format, as well as the GPS time tags. The extended
Kalman filter (EKF) algorithm consists of two parts, as shown in
Fig. 2: time the update, and the measurement update. At the time
update stage, the satellite state vector and state covariance matrix are
propagated to the time of the latest PVT set to calculate their a priori
estimates. The orbit propagation is carried out by the numerical
integration of the differential equations:
rPgPng Pemp Pcont (20)
where ris the position vector of the satellite center of mass referred to
the J2000 ECI frame; Pgis the acceleration vector due to the
gravitational forces acting on the satellite; Png is the acceleration
vector due to the nongravitational (surface) forces; Pemp denotes
empirical accelerations, accounting for the unmodeled perturbations;
and Pcont is the control acceleration vector created by the onboard
actuators.
The gravitational accelerations include the effects of the
geopotential; the gravity of the sun, moon, and major planets;
the solid Earth tides; ocean tides; and general relativity. The
nongravitational accelerations taken into account are atmospheric
drag and solar radiation pressure. Details on the used models can be
found in Table 2. To calculate the nongravitational accelerations
precisely, the satellite cross sections with respect to the relevant
directions are to be determined using the information from the
ADCS, as well as an analytic mapping algorithm [18] that calculates
the satellites projected cross-sectional area in real time. To start the
orbit propagation, the first available PVT dataset should be taken as
the initial state vector. The backup option to calculate the initial
conditions and propagate the orbit is represented by the Simplified
perturbations models 4 (SGP4) orbit predictor [19], assuming the
availability of the relevant two-line element (TLE) set files. At the
measurement update stage, both the satellite state vector and state
covariance are corrected by the latest GPS fix to obtain their a
posteriori estimates. The orbital Kalman filter output includes time as
well as estimated position and velocity in the ECI frame. It also
includes osculating orbital elements. The basic models employed in
the orbit propagation are mostly those recommended in Ref. [20].
For the mean element estimator (MEE), which is a formulation that
transforms raw GPS measurements into mean orbital elements, a
first-order mapping based on the BrouwerLyddane theory [21,22] is
used. The inputs are the estimated orbital elements from the OKF.
The auxiliary algorithms include the solar vector (SV), the Earth
magnetic vector (MGV) and the ground station vector (GSV)
Fig. 1 Block diagram of the orbit control module (GS = ground station, PVT = position velocity and time, GSV = ground station vector, i, j, k are the
indexes of the satellite).
Fig. 2 Block diagram of the orbital Kalman filter algorithm (EOP =
Earth Orientation Parameter, ISD = inter-satellite distance).
4Article in Advance / EDLERMAN AND GURFIL
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algorithms involved in the ADCS operation. The solar vector
algorithm calculates the topocentric sun vector, referred to the ECI
frame, by using the satellites geocentric state vector as estimated by
the orbital Kalman filter, as well as the data from the Jet Propulsion
Laboratorys (JPLs) planetary and lunar ephemeris as inputs. The
MGV algorithm calculates the geomagnetic field vector in the ECI
frame at a given point along the satellites orbit, using as inputs the
satellites position and the international geomagnetic reference field
(IGRF) model [23], represented by the Gaussian coefficients and
their time derivatives. The GSValgorithm calculates the line of sight
vector from the satellite to the ground station. The block diagrams of
both algorithms are shown in Figs. 3a and 3b, respectively.
B. Intersat Submodule
The intersat submodule processes data obtained from the
intersatellite communication. The intersat submodule uses the same
orbital Kalman filter and mean element estimator, but the inputs to
the orbital Kalman filter are estimated PVT, which are obtained
from the intersatellite link. The intersat submodule orbital Kalman
filter uses the drag cross-section area and the sun cross-section area
of the other satellites, as received via intersatellite communication.
In the case of missing data, the intersat submodule will keep
propagating the other satellites PVTusing the previous step output
as the current step input. This is shown in Fig. 4, in which the host
satellite is denoted by Aand the two other s are denoted by Band
C. If the time since the last intersatellite update crosses a
predetermined threshold, then these data will be replaced by the
most updated TLE data.
C. Control Submodule
The control submodule determines thruster commands to control
the distance among the satellites and manages the active cluster
control law, i.e., switching between the cluster control algorithm
(CCA), the differential drag algorithm (DDA), and contingencies.
Because even initially close satellites will gradually drift apart
without active control, the cluster control algorithm [3,4,24] has two
tasks to perform: cluster establishment at the initial phase of the
mission; and cluster keeping throughout the whole mission, which
determines the satellite allowed relative drift within prespecified
upper and lower limits.
The cooperative cluster-keeping controller steers the mean orbital
elements, as defined by the cluster orbital dynamics, to their desired
values. The closed-loop control law operates in an onoff mode,
turning on the relevant thruster once a certain intersatellite distance is
about to exceed the limits, and turning it off otherwise. The block
diagram of the cluster control algorithm is shown in Fig. 5a. The
inputs to cluster control algorithm are the position and mean orbital
elements (from the orbital Kalman filter and mean element estimator
algorithms) and the relative position and velocity vectors with respect
to the other satellites of the cluster (from the intersat submodule). The
outputs from the cluster control algorithm include commands to the
ADCS and commands to the propulsion module.
Being a backup for the cluster control algorithm, the differential
drag algorithm [25,26] is aimed at keeping the relative distances
between the satellites inside the prescribed limits by a proper
variation of their effective cross-sectional areas perpendicular to their
respective velocity vectors in order to change their atmospheric drag
accelerations, and thus cause the required changes in their semimajor
axes and alongtrack positions. The block diagram of the differential
drag algorithm is shown in Fig. 5b. The inputs to the differential drag
algorithm are the mean orbital elements of all the satellites from the
mean elements estimator, and the output is the command to the
ADCS actuators.
Fig. 3 Inter-satellite distance (TDB = Barycentric Dynamical Time).
Fig. 4 Block diagram of the intersat submodule (EPVT = estimated
position velocity and time).
Table 2 Basic models employed in the orbit propagation
Model Source
Coordinate transformations SOFA
Earth gravity field EGM2008
Ephemerides of the solar system major bodies DE421/LE421
Solid Earth tides Ref. [20]
Ocean tides FES2004
Atmospheric density NRLMSISE-00
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V. Alongtrack Cluster Control in Low Earth Orbit
The goal of keeping multiple satellites in an alongtrack cluster can
be achieved in many ways. The satellites can be equipped with a wide
range of propulsion systems that provide different thrust and Isp
values. The satellites can also perform nonpropulsive maneuvers
using natural forces, such as atmospheric drag or solar radiation
pressure. These maneuvers do not require any fuel, but the maneuver
duration is long. In this study, we divide the alongtrack cluster control
into two cases. The first case is the case in which the allowed error in
the alongtrack distance is small (tight cluster flight). In this case,
maneuvers should be carried out frequently; therefore, the maneuver
duration should be short. In the second case, the allowed error is large
(loose cluster flight). In this case, the maneuver duration can be
increased and, with the right design, these maneuvers can be carried
out using limited amounts of fuel.
A. Tight Cluster Flight
From Table 1, scenario 4 has a 100 m alongtrack baseline with only
10 m of allowed error. To fulfill these strict requirements, one of the
satellites in the cluster will be the leader and the rest will be followers.
The leader satellite will not perform any orbital maneuvers. Thus, the
leader can be replaced during the mission without affecting the cluster
performance. Figure 6 shows the proposed control concept for this use
case. The different parts will be described in the following sections.
1. Control Law for Cluster Establishment
For simplicity, the discussion assumes two satellite: leader L, and
follower F. It is assumed that the satellites are injected into orbit from
a single launch vehicle (LV). Once inorbit testing is complete (usually
within a day), the follower will apply a set of four maneuvers in the
following order:
1) A dual-impulse maneuver will be applied to reduce the
alongtrack distance by introducing a small mean SMA difference so
that aLϵaF. The magnitude of ϵcan be used as a design
parameter with a tradeoff between maneuver duration and ΔV.
2) Once the relative distance decreases below 1 km, the follower
applies another dual-impulse maneuver to eliminate any SMA
difference so that
aL
aF.
2. Control Law for Cluster Keeping
After these four maneuvers, the alongtrack distance is small, with a
small residual alongtrack drift. The alongtrack separation yji of
Fig. 5 Cluster control algorithms.
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satellite Sjrelative to satellite Siis given by [27]
yji dT
ji
hi×ri
khi×rik(21)
where his the orbital specific angular momentum vector, ris the
inertial position vector, and dji rjri. The term yji from Eq. (21)
is osculating, and therefore cannot be a used as controlled variable,
because even small eccentricity differences created by the orbit
injection will cause an oscillation that is larger by an order of
magnitude as compared to the allowed baseline error. The mean
alongtrack distance can be obtained using the relation
yji
a
ω
M
aΔu(22)
Figure 7 shows the cluster-keeping logic where
λFL is the mean
argument of latitude difference between the follower and the leader,
Δ
aFL is the mean SMA difference,
aFis the followers mean SMA,
yFLRis the required mean alongtrack distance, and ϵis the allowed
intersatellite distance error.
3. Simulation
In this section, we use the following parameters (assuming all
satellites are identical): m200 kg,CD2.2, and A1m
2. The
launch scenario assumes the following:
1) All satellites are launched by the same LV.
2) The satellite are released into the negative velocity direction
with a 10 s gap between each satellite.
3) The LV release mechanism provides each satellite the release
velocity of ΔVi10 5icms.
4) The satellite drifts for one day after the injection before any
maneuver is activated.
The simulation duration is 50 days.
Figure 8a shows the alongtrack separation after orbit injection over
50 days. Figure 8b shows an enlarged view of the cluster-keeping
phase, in which we can notice about a 100 m difference between each
pair. Figure 8c shows an enlarged view of one pair during the cluster-
keeping phase. In this figure, we see the difference between the
alongtrack distance and the mean alongtrack distance. The controller
manages to keep the mean alongtrack distance within the desired
tolerance but not the instantaneous alongtrack distance. Figure 8d
shows the ΔVbudget for the simulation duration. In the initial time,
we notice a large change in ΔVover a short duration due to the cluster
establishment maneuver. The slope represents accumulated cluster-
keeping maneuvers that are performed on a daily basis.
B. Loose Cluster Flight
Scenario 2 in Table 1 has an alongtrack baseline of a few
kilometers, and the allowed error is about 2 km. These requirements
can be achieved by using nonpropulsive maneuvers or low-thrust
maneuvers. The proposed method in this section is designed to
minimize the required ΔVby using differential drag (DD). DD is the
difference in drag per unit mass acting on each satellite. If the
satellites are passing through a similar density with similar velocities,
then any DD is due to different ballistic coefficients of the vehicles
[28]. Changing the ballistic coefficients creates a DD force on each
satellite, which can be used as a control force. Changing the ballistic
coefficient can be obtained by changing the satellites orientation
with respect to the velocity vector. The DD can be a means for passive
satellite cluster keeping [25,26,2830].
In this study, we will consider the control laws presented in
previous papers [25,26,31], wherein the controlled variables were
differential mean orbital elements. The controller can be
automatically activated, and it does not require a predetermined
activation time.
Using the DD adds constraints on the ADCS. For example, setting
a strict constraint on nadir pointing will limit the ability to change the
satellites attitude that is required for DD. A requirement for pointing
one of the satellite axes to a specific direction allows only one
rotational degree of freedom. By fixing two of the satellite axes, we
fully determine the satellites attitude. Therefore, in most cases, the
satellite can only achieve one hard constraint and one soft constraint.
Another possible option is to fulfill two soft constraints in parallel.
For the optical use case, we can consider operating the differential
drag algorithm during umbra only, and we can leave the sunlit portion
of the orbit for camera operation and sun acquisition. This will make
the differential drag algorithm work slowly, but the primary mission
will not be affected by the cluster control requirements.
The differential drag algorithm can apply forces mostly inside the
orbital plane, and therefore out-of-plane maneuvers cannot be done in
practice using this technique. If the lifetime is longer than five years,
the constellation might have to perform orbit correction maneuvers to
maintain the nominal sun-synchronous orbit. Hence, each satellite
will also be equipped with an impulsive propulsion system.
1. Control Law
The DD controller can be written using the following equation:
Si8
>
>
<
>
>
:
Sref Δ
aijΔ
λij 0
Smax Δ
aijΔ
λij <0;Δ
aij >0
Smin Δ
aijΔ
λij <0;Δ
aij <0
j(i1i1;2;3; :::;N 1
1;iN(23)
This is a cyclic controller, meaning that each satellite must receive
data from one satellite only. The data include two parameters: Δ
aij
Fig. 7 Tight cluster flight cluster-keeping logic.
Fig. 6 Tight cluster flight control concept.
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and Δ
λij. The differential mean semimajor axis is defined as follows:
Δ
aij
ai
aj(24)
The argument of latitude is defined as λMω, and the
differential mean argument of latitude is
Δ
λij
λi
λj(25)
By monitoring Δ
aand Δ
λ, the controller in Eq. (23) determines the
required PCSA.
The basic DDA logic is presented in Fig. 9a. Figure 9b shows that
same logic but, this time, the DDA operates differently during the day
and night. From Eq. (23), it is evident that the controllers goal is to
minimize Δ
aijΔ
λij. In the present case, it is desired to create a few-
kilometer gap between each pair of satellites. To do this, the term
Δ
aijΔ
λij can be replaced by the term Δ
aijΔ
λij offset.
Over time, the SMA will change, and the LTAN will drift beyond
the allowed tolerance. To fix this, the control scheme described in
Sec. V.C should be employed.
2. Simulation
In this section, we assume the parameters given in Sec. III but, here,
A0.161m
2. Figure 10 shows the alongtrack separation after
orbit injection over one day. The relative alongtrack separation is
nearly constant. The crosstrack and radial components are minor in
this case.
The initial conditions for this scenario are the satellite state one day
after orbit injection. The following figures show the result of a one-
year simulation with five satellites. In this simulation, we assume that
the first 50 days are dedicated to orbit establishment and that the DDA
has a higher priority than other mission goals. Figure 11a shows the
intersatellite distances. We see that, during the orbit establishment
phase, some of the distances crossed the required value. Once this
phase is finished, the intersatellite distances remain nearly constant,
and the controller is only activated during umbra. Figures` 11b and
11c show Δ
aij and Δ
λij. Although Δ
aij converges to zero, Δ
λij
converges to a positive value, and it ensures the required alongtrack
distance. Figure 11d shows the LTAN of the satellites. There is a
minor drift in the LTAN value over the year but, during a lifetime of
10 years, it is possible that a correction maneuver will be required.
VI. Crosstrack Cluster Control in Low Earth Orbit
Distributed sensing applications such as synthetic aperture radar
[13], geolocation [32], and optical imaging require crosstrack
separation among satellites. One option would be using multiple
launch vehicles that can provide the required orbit for each satellite.
This solution may not be realistic due to high cost. Another option is
to apply an out-of-plane impulsive maneuver. This option may be
feasible if the required crosstrack is small but, in general, these
maneuvers demand a high ΔV. We propose another option, wherein
the Earth geopotential is used to achieve the required crosstrack
separation with a reasonable amount of ΔV.
A. Loose Cluster Flight
1. Control Law for Cluster Establishment
We assume a two-satellite cluster, including a chief Cand a deputy
D. Both satellites have similar orbits, and the chief orbit remains
unchanged. The goal is to create a crosstrack difference between the
0102030405060
0102030405060
Time [day]
0
10
20
30
40
50
60
Distance [km]
d15
d25
d35
d45
a) Intersatellite distance
40 42 44 46 48 50
40 42 44 46 48 50
Time [day]
0
0.1
0.2
0.3
0.4
0.5
Distance [km]
d15 d25 d35 d45
b) Intersatellite distance during cluster keeping
Time [day]
380
385
390
395
400
405
410
415
420
Distance [m]
c) Intersatellite distance during cluster keeping:
zoom-in to one pair
Time [day]
0
1
2
3
4
5
6
7
[cm/s]
d) V budget for 50 days
Fig. 8 Tight cluster flight simulation results.
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chief and the deputy with a small amount of ΔV. The idea is to use
Eq. (9) to create the required crosstrack. For a near-circular orbit,
assuming the same initial orbital plane, Eq. (9) yields
ΔΩ ΩDΩC1.5J2R2
e
μ
pcos i1
a72
D
1
a72
Ct(26)
Thus, small differences in the SMAwill induce the right ascension
of the ascending node difference. This effect becomes stronger when
the orbit inclination is smaller. The SMA difference also creates an
alongtrack relative drift, which is determined by Eqs. (10) and (11).
The alongtrack separation angle can be represented by the
difference in the mean argument of latitude:
Δ
λ
M
wD
M
wC(27)
The proposed crosstrack generation scheme is composed of three
parts. The first part includes two in-plane orbit maneuvers to change
the deputys SMA. The required ΔVand the duration of the maneuver
can be calculated using the Hohmann transfer equations:
ΔV
μ
ai
r0
@
2
ζ1
s1
ζ
p
2ζ2
ζ1
s1
A;
Tman π
aiat3
8μ
s(28)
where aiis the initial orbit SMA, atis the target orbit SMA, and
ζaiat. The second part is a coasting phase, wherein both the
alongtrack and crosstrack distances increase. Equations (26) and (27)
provide expressions for the crosstrack and alongtrack separations.
The crosstrack drift is slow as compared to the alongtrack drift. By the
time the deputy reaches the required crosstrack separation, the in-
plane distance might be a few thousand kilometers. Therefore, a
constraint on the final in-plane separation is added. Let
t2πk
Δn(29)
where kis the number of orbits, and Δnis the difference in mean
motions. We use Eq. (27) to find the required time for which
Δ
λ2πk, and we substitute into Eq. (26)
ΔΩ 1.5J2R2
e
μ
pcos i1
a72
D
1
a72
C2πk
Δn(30)
Fig. 9 DD logic.
0 0.2 0.4 0.6 0.8 1
Time [days]
-1.5
-1
-0.5
0
0.5
Alongtrack separation [km]
AT separation1
AT separation2
AT separation3
AT separation4
Fig. 10 Alongtrack separation after orbit injection (AT = along-track).
Article in Advance / EDLERMAN AND GURFIL 9
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Figure 12 shows the required ΔVto obtain a specific crosstrack
difference. Figure 12a shows a minimal crosstrack of about 18 km
with an inclination of 97.8 deg. Figure 12b shows a minimal
crosstrack of about 86 km with an inclination of 51.5 deg.
The third part includes another two in-plane orbit maneuvers to
raise the deputys orbit back to nominal and restrain the drift.
At the end of the maneuver, the crosstrack distance should be as
required, whereas the alongtrack distance is close to minimal. To
verify that the alongtrack distance is close to minimal at the end of
the maneuver, an appropriate timing should be determined.
Equation (28) is used to calculate the maneuver duration and the
required ΔV.
Figure 13 summarizes the preceding method.
2. Control Law for Cluster Keeping
Whereas the crosstrack distance will hardly change during the
mission, the alongtrack distance will drift mostly due to differential
drag effects. To avoid unwanted alongtrack distance, we propose to
10 20 30 40 50 60 70 80
Crosstrack [km]
0
50
100
150
200
Maneuver duration [day]
4 m/s
8 m/s
12 m/s
16 m/s
20 m/s
a) i = 97.8 b) i = 51.5
50 100 150 200 250 300 350
Crosstrack [km]
0
50
100
150
200
Maneuver duration [day]
4 m/s
8 m/s
12 m/s
16 m/s
20 m/s
Fig. 12 Maneuver duration as a function of the required crosstrack and ΔV.
a) Intersatellite distance
c) Argument of latitude difference d) Mean local time at the ascending node
b) Semimajor axis difference
Fig. 11 DDA simulation results (MLT = mean local time).
Fig. 13 Two Hohmann transfers with a coasting phase to create a
required crosstrack separation.
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create a small SMA difference between the deputy and the chief to
induce controlled alongtrack drift. We should define the allowed
alongtrack distance between the chief and the deputy ϵT. If this
distance is violated, we apply a Hohmann transfer maneuver to
change the SMA according to the following logic:
8
<
:
aDaCΔay>ϵT
aDaDϵT<y<ϵT
aDaCΔay<ϵT
(31)
3. Simulation and Results
The numerical simulation is based on use case 1, described in
Table 1. The cluster includes three satellites with a mass of 100 kg.
The nominal orbit height is 693 km, the required crosstrack
separation is 17.5 10%, and the alongtrack separation should be
minimized. We assume that the three satellites are injected into the
same orbit using a single launch vehicle, with the initial orbital
elements
fa; e; i; Ω;ω;M
kgf7070;0.001;98.16°;257.71°;0°;0.01k°g
(32)
In Eq. (32), k0, 1, 2 denotes the satellite number. For reference,
the required ΔVfor a direct out-of-plane maneuver is [16]
ΔvΩ2
μ
a
rsincos1cos2isin2icos ΔΩ
2(33)
From Eqs. (2) and (6),
ΔrN
aΔiycos uΔixsin u
ΩdΩsin icos uidisin u(34)
If Δi0, then
ΔrN
aΩdΩsin icos uΔΩ ΔrN
asin icos u(35)
We substitute the preceding values and find that
ΔvΩ18.58 ms. The simulation is performed using the
FreeFlyer® software, using a WGS84 21x21 Earth gravity potential
model, solar radiation pressure, drag, and lunisolar gravity. In this
example, ΔV6.5 ms. Using Eq. (28), the change in SMA is
about 6 km. Based on Eq. (28), each of the two double impulsive
maneuvers takes about 2950 s. Based on Eq. (29), the coasting phase
duration is about 53 days. The alongtrack distance change during the
second Hohmann transfer is about 14 km. This value is used for
timing of the second set of maneuvers.
The simulation results can be found in Fig. 14. Figure 14a shows
the mean SMA evolution during the maneuver. The chief SMA
is slightly decreasing due to drag. The SMA of the first deputy
decreases due to the maneuver, whereas the SMA of the second
deputy increases due to an opposite maneuver. The SMA of deputy 1
returns to the nominal value faster than the SMA of deputy 2. Due to
the SMA differences and the resulting differential drag, the argument
of latitude period is slightly shorter. The growth in the crosstrack
distance through the maneuver can be seen in Figs. 14b and 14c, and
the final value satisfies the requirements. Figure 14d shows the
relative position between the chief and the two deputies.
Figure 15 shows the results of the cluster keeping, which starts
immediately after the cluster establishment. To be more realistic, we
modified the second deputys drag cross-sectional area and increased
Time [day]
7054
7056
7058
7060
7062
7064
7066
7068
Mean SMA [km]
Mean SMA chief
Mean SMA deputy 1
Mean SMA deputy 2
a) Mean SMA evolution during the maneuver
Time [day]
-20
-15
-10
-5
0
5
10
15
20
Crosstrack from chief to deputy 1 [km]
Crosstrack target
Crosstrack upper bound
Crosstrack lower bound
b) The crosstrack distance between sat 1 and sat 2
0 10203040506070
0 10203040506070
0 10203040506070
Time [day]
-20
-15
-10
-5
0
5
10
15
20
Crosstrack from chief to deputy 2 [km]
Crosstrack target
Crosstrack upper bound
Crosstrack lower bound
c) The crosstrack distance between sat 1 and sat 3
53.9 53.95 54 54.05 54.1 54.15 54.2 54.25
Time [day]
-30
-20
-10
0
10
20
30
Distance [km]
Crosstrack C2D1
Alongtrack C2D1
Crosstrack C2D2
Alongtrack C2D2
d) Relative position at the end of the maneuver
Fig. 14 Numerical simulation results of the cluster establishment scenario.
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it by 5%. We set the alongtrack allowed error to be ϵT1kmand
Δa0.001 km. We notice two different behaviors: Deputy 1
changes its SMA only due to the maneuver, whereas deputy 2 has an
SMA drift due to the differential drag effect. The values of ϵTand Δa
can be changed according to the specific mission requirement and
satellite design. The required ΔVfor this phase is less than 1cmsfor
the simulation duration.
B. Tight Cluster Flight
1. Cluster Establishment Maneuver
This section will provide an estimate of the required ΔVfor the
orbit establishment. This is a simplified calculation, and it should be
revaluated to match a specific launcher. We assume that the chief
satellite is already in orbit for several years. We wish to add two
deputies to the same orbit to increase the mission return. The deputies
are injected into an orbit with the same orbital elements as the chief,
but with a 1 deg offset in true anomaly. The offset creates an
alongtrack distance of 120 km between the chief and the deputies. We
assume a 0.001 deg offset in true anomaly between the deputies,
creating a 120 m alongtrack distance between them.
Wewish to create a 300 m radial distance and a 1000 m crosstrack
distance between the deputies with respect to the master orbital plane.
To find the connection between the maneuver and the required
changes in the orbital elements, a set of simplified variational
equations is used [13]:
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
Da
Dex
Dey
Dix
Diy
Du
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
1
v
2
6
6
6
6
6
6
4
02a0
sin λ2 cos λ0
cos λ2 sin λ0
0 0 cos λ
0 0 sin λ
03vaΔt0
3
7
7
7
7
7
7
5
8
<
:
ΔvR
ΔvT
ΔvN
9
=
;
(36)
where {ΔvR,ΔvT,ΔvN} are the radial, tangential, and crosstrack
velocity increments, respectively. Equation (36) implies that there is
more than one way to create the required radial distance. We can
apply thrust in the radial direction, the tangential direction, or a
combination thereof. The tangential direction needs less ΔV, but it
has a side effect of creating an unwanted SMA difference, which will
later create an alongtrack drift. The radial maneuver requires twice as
much ΔV, but it has no effect on the SMA. In this study, we assume a
radial maneuver to avoid the alongtrack drift. The maneuver can be
split between the ascending and descending nodes, as well as
between the two deputies, and so the clusters symmetry plane will be
the chiefs orbital plane. Thus,
ΔvR8
>
>
>
<
>
>
>
:
ΔrRv
2au0
ΔrRv
2auπ
(37)
Equation (37) shows the required ΔVin the radial direction of one
of the deputies. The other one should apply the same thrust, but in the
opposite direction, to obtain the mentioned symmetry:
ΔvN8
>
>
>
<
>
>
>
:
ΔrNv
2aλπ
2
ΔrNv
2aλ3π
2
(38)
For the crosstrack distance, we apply thrust in the normal direction
according to Eq. (38):
ΔvΔrRv
aΔrNv
a(39)
The combined ΔV(per satellite) required for orbit establishment is
calculated using Eq. (39).
2. Control Law for Cluster Keeping
The cluster control for this use case can be separated into two parts:
alongtrack separation, and eccentricity/inclination vector separation.
The alongtrack distance is created mostly by the differential drag
perturbation. The chief and deputies have different configurations,
and therefore different ballistic coefficients. Controlling the
alongtrack distance of satellites on similar near-circular orbits can
be done by introducing small SMA changes. The trigger of this logic
will be a violation of the upper/lower bound of the alongtrack
distance between the chief and one of the deputies. Once the trigger is
60 80 100 120 140 160 180 60 80 100 120 140 160 180
Time [day]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Mean SMA difference [m]
Mean SMA chief - Mean SMA deputy 1
Mean SMA chief - Mean SMA deputy 2
a) Mean SMA evolution during the maneuver
Time [day]
-1.5
-1
-0.5
0
0.5
1
1.5
Distance [km]
Alongtrack C2D1
Alongtrack C2D2
b) The alongtrack distance between the chief and the deputies
Fig. 15 Numerical simulation results of the cluster-keeping scenario.
Fig. 16 Alongtrack controller.
12 Article in Advance / EDLERMAN AND GURFIL
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activated, both deputies will apply similar thrust in the alongtrack
direction.
Figure 16 describes the following logic: if the upper bound Dmax is
reached, thrust will be applied in the negative alongtrack direction,
and a lower bound Dmin crossing will activate the opposite maneuver
[Eq. (40)]:
Δv8
<
:
ΔvΔrT<Dmin
0Dmax ΔrTDmin
ΔvΔrT>D
max
(40)
The maneuver magnitude should be set to minimum to create a
slow alongtrack drift and save fuel. To reduce the control effort and
avoid chattering, these maneuvers can be limited to one maneuver per
day. The synchronization between the two maneuvers can tolerate a
few hundred seconds of difference without causing damage to the
cluster.
Based on Eq. (36), the alongtrack velocity correction required to
keep the eccentricity vector parallel to the inclination vector and to
deal with differential drag can be calculated. To keep the eccentricity
vector parallel to the inclination vector (to avoid collision risk), we
use the impulses [13]
ΔvT1v
4kDekDa
a;
ΔvT2v
4kDekDa
a(41)
along the orbit locations given by
λ1tan1Dey
Dex;λ2λ1π(42)
Based on Eq. (5), we can calculate the required time between the
maneuver pairs as a function of the allowed φerror. The control also
changes the relative argument of latitude, which is expressed as
Δum3ΔvT1Δt
a(43)
where Δtis the time between pairs of maneuvers. We can adjust the
required ΔVso that aΔuwill be bounded. The controller in Eq. (41)
Table 3 Spacecraft parameters
Chief Deputy 1 Deputy 2
m, kg 2000 200 200
A,m25 1.5 1.5
CD2.3 2.2 2.2
-800 -600 -400 -200 200 400 600 8000
Radial separation [m]
-1500
-1000
-500
0
500
1000
1500
Crosstrack [m]
a) Alongtrack and cross-track plane
Time [day]
0
10
20
30
40
50
60
70
80
90
100
V [cm/s]
Delta V of Deputy 1
Delta V of Deputy 2
020 40 60 80 100 120
02040
60 80 100 120
Time [day]
8
10
12
14
16
18
20
Alongtrack distance [km]
Chief to Deputy 1
Chief to Deputy 2
c) Alongtrack distance between the chief and deputies
Fig. 17 Cluster-keeping simulation results.
Table 4 Initial conditions
Chief Deputy 1 Deputy 2
a, km 7070.96 7070.96 7070.96
i, deg 98.158 98.158 98.158
ω, deg 90 90 90
Ω, deg 338.9941 338.99 338.997
e0.001 0.00104 0.0009
f, deg 0 0.15 0.15
Article in Advance / EDLERMAN AND GURFIL 13
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applies thrust on only one of the deputies. To reduce the mass
difference between the satellites, the thrusting deputy can be
alternated. This alternation can be done after a long period or after
each maneuver pair.
The relative radial component is controlled by using (assuming
ωπ2)
aΔe0
ΔrReDeCΔrR
aC
(44)
Based on previous work [13], it is known that any inclination
differences between the satellites should be avoided. Thus, we can
achieve crosstrack separation using the relative right ascension of the
ascending node:
aΔi0
ΔrN
ΩD
ΩCΔrN
acsin
iC
(45)
3. Simulation and Results
In this section, we apply the control logic mentioned previously
and test it in a high-fidelity simulation. The spacecraft parameters are
given in Table 3.
Table 3 summarizes the chiefs and deputiesinitial conditions
based on the RGT and SSO requirements. The true anomaly
difference of 1.5 deg creates the required alongtrack separation, and
ωis set to be 90 deg.
Figure 17a shows the cluster-keeping simulation results.
To improve our estimate of the required ΔV, we ran a simulation
including a simplified cluster establishment maneuver. We
performed the analysis using the launch scenario described in
Sec. III and the maneuver described in Sec. I. Using Eq. (37), we
calculated the required ΔVto create ΔrR300 m and, using
Eq. (38), we calculated the required ΔVto create ΔrN1000 m.
The required ΔVfor the cluster establishment maneuver amounted to
about 70 cms. Figure 18 shows the crosstrack and radial
components of the deputies with respect to the chief satellite.
VII. Conclusions
A methodological design enabling cluster flight and cluster
keeping of a swarm of cooperative satellites has been presented. This
design methodology supported all use cases relevant for fractionated
sensors. Implementation included a new simulator, which was
capable of supporting the development of software modules and the
synchronization of time among each satellite. It is concluded that
both the SAR and the optical disaggregated sensors could be
controlled using a reasonable amount of fuel. This was true for
predefined reference orbits, such as sun-synchronous orbits, and for
relative orbit maintenance. An important conclusion was that using
the natural effects such as the Earth geopotential and atmospheric
drag was an enabler for fractionated space sensors. Without a
judicious utilization of these effects, long-term cluster flight missions
could not be implemented in practice.
Acknowledgments
This work was supported by the European Commission Horizon
2020 Program in the framework of the Satellite Swarm Sensor
Network project under grant agreement 687351. We would also like
to acknowledge David Mishna and Alexander Shirayev for their
work on the orbit control module.
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Article in Advance / EDLERMAN AND GURFIL 15
Downloaded by TECHNION - ISRAEL INST OF TECH on January 6, 2019 | http://arc.aiaa.org | DOI: 10.2514/1.A34151
... The satellites in the cluster operate loosely in close proximity within a bounded space area and do not need to maintain strict spatial geometry configuration. Most satellite clusters are heterogeneous, relying on local information interaction to maintain the relative motion bounded [1][2][3], which are loose clusters that accomplish space tasks through autonomous cooperation. At present, the National Aeronautics and Space Administration (NASA), European Space Agency (ESA), and other space agencies have developed or planned several satellite cluster systems for detection, remote sensing, communication, and surveillance. ...
... relative to the Earth. The CW equation is a linearized equation proposed by Clohessy et al. [28] to describe the close relative motion of two satellites, which is expressed in the target orbital frame (defined in Figure 5) as: 3 2 , ...
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The satellite cluster formation reconfiguration has received considerable attention in recent years. However, the traditional centralized control methods are challenging to apply to satellite clusters because of the enormous fuel consumption, and few studies have addressed the mathematical characterization of satellite clusters. This research aims to propose a mathematical characterization method for satellite clusters and investigate the formation reconfiguration control of satellite clusters. This study provided the five-element characterization method to represent the cluster characteristics and internal correlation characteristics of orbiting satellite clusters. In addition, a control method based on bifurcating potential fields was proposed to realize satellite cluster formation’s dynamic migration and rapid reconfiguration. A cluster with 50 satellites was simulated to verify the feasibility and effectiveness of the proposed formation control algorithm. The results show that various formation topologies were achieved by simply changing the bifurcation parameter and configuration adjustment parameters. The five descriptive elements of the satellite cluster can intuitively and effectively reflect the running state of the satellite cluster.
... As a result, the design and control of space swarms with 63 large quantities (hundreds to thousands) of spacecraft is of 64 interest. 13 In a sense, spacecraft swarms can be considered as 65 loose formations. Compared with the spacecraft formations, 66 spacecraft swarms pay more attention to bounded relative 67 motion instead of precise geometry configuration. ...
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Space swarms, enabled by the miniaturization of spacecraft, have the potential capability to lower costs, increase efficiencies, and broaden the horizons of space missions. The formation control problem of large-scale spacecraft swarms flying around an elliptic orbit is considered. The objective is to drive the entire formation to produce a specified spatial pattern. The relative motion between agents becomes complicated as the number of agents increases. Hence, a density-based method is adopted, which concerns the density evolution of the entire swarm instead of the trajectories of individuals. The density-based method manipulates the density evolution with Partial Differential Equations (PDEs). This density-based control in this work has two aspects, global pattern control of the whole swarm and local collision-avoidance between nearby agents. The global behavior of the swarm is driven via designing velocity fields. For each spacecraft, the Q-guidance steering law is adopted to track the desired velocity with accelerations in a distributed manner. However, the final stable velocity field is required to be zero in the classical density-based approach, which appears as an obstacle from the viewpoint of astrodynamics since the periodic relative motion is always time-varying. To solve this issue, a novel transformation is constructed based on the periodic solutions of Tschauner-Hempel (TH) equations. The relative motion in Cartesian coordinates is then transformed into a new coordinate system, which permits zero-velocity in a stable configuration. The local behavior of the swarm, such as achieving collision avoidance, is achieved via a carefully-designed local density estimation algorithm. Numerical simulations are provided to demonstrate the performance of this approach.
... Clusters, formations or swarms of low-cost miniaturized spacecraft with distributed sensing and measurement capabilities have been explored for the past several years and they are envisaged to offer promising outcomes in applications involving the imaging of the Earth and the heavenly bodies, the collection of atmospheric and meteorological data and other scientific observations [1]. Several missions involving fleets of small satellites have been considered for various topographical, meteorological and astronomical applications that require space-based sensors with large aperture sizes (e.g., [2][3][4][5]). The motivation for the use of a fleet of small satellites instead of a single large satellite in such applications includes simple and low-cost design, increased redundancy, reconfigurability and mission flexibility [6]. ...
... As a result, the spacecraft swarm control is of concern [4]. One type of control method stems from multi-agent systems theory and is based on algebraic graph theory [5,6]. ...
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This volume is designed as an introductory text and reference book for graduate students, researchers and practitioners in the fields of astronomy, astrodynamics, satellite systems, space sciences and astrophysics. The purpose of the book is to emphasize the similarities between celestial mechanics and astrodynamics, and to present recent advances in these two fields so that the reader can understand the inter-relations and mutual influences. The juxtaposition of celestial mechanics and astrodynamics is a unique approach that is expected to be a refreshing attempt to discuss both the mechanics of space flight and the dynamics of celestial objects. “Celestial Mechanics and Astrodynamics: Theory and Practice” also presents the main challenges and future prospects for the two fields in an elaborate, comprehensive and rigorous manner. The book presents homogenous and fluent discussions of the key problems, rendering a portrayal of recent advances in the field together with some basic concepts and essential infrastructure in orbital mechanics. The text contains introductory material followed by a gradual development of ideas interweaved to yield a coherent presentation of advanced topics.
Chapter
The Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission is the fifth NASA Medium-class Explorer (MIDEX), launched on February 17, 2007 to determine the trigger and large-scale evolution of substorms. The mission employs five identical micro-satellites (hereafter termed “probes”) which line up along the Earth’s magnetotail to track the motion of particles, plasma and waves from one point to another and for the first time resolve space–time ambiguities in key regions of the magnetosphere on a global scale. The probes are equipped with comprehensive in-situ particles and fields instruments that measure the thermal and super-thermal ions and electrons, and electromagnetic fields from DC to beyond the electron cyclotron frequency in the regions of interest. The primary goal of THEMIS, which drove the mission design, is to elucidate which magnetotail process is responsible for substorm onset at the region where substorm auroras map (∼10 RE): (i) a local disruption of the plasma sheet current (current disruption) or (ii) the interaction of the current sheet with the rapid influx of plasma emanating from reconnection at ∼25 RE. However, the probes also traverse the radiation belts and the dayside magnetosphere, allowing THEMIS to address additional baseline objectives, namely: how the radiation belts are energized on time scales of 2–4 hours during the recovery phase of storms, and how the pristine solar wind’s interaction with upstream beams, waves and the bow shock affects Sun–Earth coupling. THEMIS’s open data policy, platform-independent dataset, open-source analysis software, automated plotting and dissemination of data within hours of receipt, dedicated ground-based observatory network and strong links to ancillary space-based and ground-based programs. promote a grass-roots integration of relevant NASA, NSF and international assets in the context of an international Heliophysics Observatory over the next decade. The mission has demonstrated spacecraft and mission design strategies ideal for Constellation-class missions and its science is complementary to Cluster and MMS. THEMIS, the first NASA micro-satellite constellation, is a technological pathfinder for future Sun-Earth Connections missions and a stepping stone towards understanding Space Weather.