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Cluster-Keeping Algorithms for the Satellite Swarm Sensor

Network Project

Eviatar Edlerman∗and Pini Gurfil†

Technion–Israel 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 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.

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, m2∕s

Isp = specific impulse, s

i= inclination, rad

ix,iy= inclination vector components

M= mean anomaly, rad

m= mass, kg

n= mean motion, rad∕s

Pcont = control acceleration vector created by the onboard

actuators, km∕s2

Pemp = empirical accelerations, km∕s2

Pg= acceleration vector due to the gravitational forces,

km∕s2

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, km∕s

yji = alongtrack coordinate of Sjrelative to Si,km

ΔV= velocity change, km∕s

ϑ= relative ascending node, rad

λ= argument of latitude, rad

μ= gravitational constant, km3∕s2

ρ= atmospheric density, kg∕m3

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

leader–follower 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) project‡has 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].

Article in Advance / 1

JOURNAL OF SPACECRAFT AND ROCKETS

Downloaded by TECHNION - ISRAEL INST OF TECH on January 6, 2019 | http://arc.aiaa.org | DOI: 10.2514/1.A34151

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 250–300 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 10–15 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

@

ad−a∕a

ud−uΩd−Ωcos i

exd−ex

eyd−ey

id−i

Ωd−Ωsin 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 3–52 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

Dawn–dusk LTAN 0600 hrs 30 min

Formation configuration Crosstrack cluster Alongtrack cluster Crosstrack cluster Alongtrack cluster

Relative position 17.5 km 10% 7km25% 200–2000 m 10% 100 m 10%

Satellite dry mass 100 kg 200 kg

Satellite power 250–300 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%

2Article in Advance / EDLERMAN AND GURFIL

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_

φ≈3

2π

TR2

e

a2J2cos2i−1(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

a0−Δex−Δey

Δλ−3Δa

2a−2Δey2Δex

00−ΔiyΔix

00−ΔeyΔex

3Δa

2a02Δex2Δey

00 ΔixΔiy

3

7

7

7

7

7

7

7

7

7

7

5

×

8

>

>

>

>

>

<

>

>

>

>

>

:

1

λ−λ0

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 sun’s 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

ha−ReμkTday

2πj21∕3−Re(8)

where μis the Earth gravitational parameter. Using the variational

equations

_

Ω−3

2J2

μ

pR2

ea−7∕21−e2cos i(9)

_

ω3

4J2

μ

pR2

ea−7∕21−e2−25cos2i−1(10)

_

M3

4J2

μ

pR2

ea−7∕21−e2−3∕23cos2i−1(11)

nj

kωe−_

Ω−_

M_

ω(12)

hμ

n21∕3−Re(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

icos−1_

Ωs

−3∕2J2

μ∕a3

pRe∕a1−e22(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:

Ωs0.98563N−80(15)

LTANhτ12 Ω−Ωs

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

Δiτr−τk(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.

Article in Advance / EDLERMAN AND GURFIL 3

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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 host’s 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 satellite’s 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 Brouwer–Lyddane 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

algorithms involved in the ADCS operation. The solar vector

algorithm calculates the topocentric sun vector, referred to the ECI

frame, by using the satellite’s geocentric state vector as estimated by

the orbital Kalman filter, as well as the data from the Jet Propulsion

Laboratory’s (JPL’s) planetary and lunar ephemeris as inputs. The

MGV algorithm calculates the geomagnetic field vector in the ECI

frame at a given point along the satellite’s orbit, using as inputs the

satellite’s 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 satellite’s PVTusing the previous step output

as the current step input. This is shown in Fig. 4, in which the host

satellite is denoted by “A”and the two other s are denoted by “B”and

“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 on–off 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

Article in Advance / EDLERMAN AND GURFIL 5

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.

6Article in Advance / EDLERMAN AND GURFIL

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 ≜rj−ri. 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 follower’s 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 5icm∕s.

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 satellite’s orientation

with respect to the velocity vector. The DD can be a means for passive

satellite cluster keeping [25,26,28–30].

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

satellite’s 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 satellite’s 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.

Article in Advance / EDLERMAN AND GURFIL 7

and Δ

λij. The differential mean semimajor axis is defined as follows:

Δ

aij ≜

ai−

aj(24)

The argument of latitude is defined as λ≜Mω, 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 controller’s 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,

A0.16–1m

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.

8Article in Advance / EDLERMAN AND GURFIL

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 ΩD−ΩC−1.5J2R2

e

μ

pcos i1

a7∕2

D

−1

a7∕2

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 deputy’s SMA. The required ΔVand the duration of the maneuver

can be calculated using the Hohmann transfer equations:

ΔV

μ

ai

r0

@

2

ζ1

s−1

ζ

p−

2ζ2

ζ1

s1

A;

Tman π

aiat3

8μ

s(28)

where aiis the initial orbit SMA, atis the target orbit SMA, and

ζai∕at. 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

a7∕2

D

−1

a7∕2

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

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 deputy’s 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.

10 Article in Advance / EDLERMAN AND GURFIL

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

aDaD−ϵT<y<ϵT

aDaC−Δay<−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

kgf7070;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

rsincos−1cos2isin2icos ΔΩ

2(33)

From Eqs. (2) and (6),

ΔrN

a−Δiycos uΔixsin u

−Ωd−Ωsin icos uid−isin u(34)

If Δi0, then

ΔrN

a−Ωd−Ωsin icos u→ΔΩ ΔrN

asin icos u(35)

We substitute the preceding values and find that

ΔvΩ18.58 m∕s. 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 m∕s. 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 deputy’s 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.

Article in Advance / EDLERMAN AND GURFIL 11

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 1cm∕sfor

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

0−3v∕aΔt0

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 cluster’s symmetry plane will be

the chief’s 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

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 ≥ΔrT≥Dmin

−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

4kDekDa

a;

ΔvT2−v

4kDek−Da

a(41)

along the orbit locations given by

λ1tan−1Dey

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

Δum−3ΔvT1Δt

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

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

ωπ∕2)

aΔe0

ΔrR→eDeCΔrR

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

ΩC−ΔrN

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 chief’s and deputies’initial 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 cm∕s. 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.

References

[1] Brown, O., and Eremenko, P., “Fractionated Space Architectures: a

Vision for Responsive Space,”4th Responsive Space Conference,

AIAA, Reston, VA, 2006.

[2] Moreira, A., Krieger, G., Hajnsek, I., Hounam, D., Werner, M., Riegger,

S., and Settelmeyer, E., “Tandem-X: A Terrasar-X Add-On Satellite for

Single-Pass SAR Interferometry,”2004 IEEE International Proceed-

ings.on Geoscience and Remote Sensing Symposium, 2004. IGARSS’04,

Vol. 2, IEEE Publ., Piscataway, NJ, 2004, pp. 1000–1003.

doi:10.1109/igarss.2004.1368578

[3] Mazal, L., and Gurfil, P., “Closed-Loop Distance-Keeping for Long-

Term Satellite Cluster Flight,”Acta Astronautica, Vol. 94, No. 1,

Jan. 2014, pp. 73–82.

doi:10.1016/j.actaastro.2013.08.002

[4] Mazal, L., Mingotti, G., and Gurfil, P., “Optimal On–Off Cooperative

Maneuvers for Long-Term Satellite Cluster Flight,”Journal of

Guidance, Control, and Dynamics, Vol. 37, No. 2, March 2014,

pp. 391–402.

doi:10.2514/1.61431

[5] Vaddi, S. S., Alfriend, K. T., Vadali, S. R., and Sengupta, P., “Formation

Establishment and Reconfiguration Using Impulsive Control,”Journal

of Guidance, Control, and Dynamics, Vol. 28, No. 2, March 2005,

pp. 262–268.

doi:10.2514/1.6687

[6] Schaub, H., Vadali, S.R., Junkins, J. L., and Alfriend, K. T., “Spacecraft

Formation Flying Control Using Mean Orbital Elements,”Journal of the

Astronautical Sciences, Vol. 48, No. 1, 2000, pp. 69–87.

[7] Mesbahi, M., and Hadaegh, F. Y., “Formation Flying Control of

Multiple Spacecraft via Graphs, Matrix Inequalities, and Switching,”

Journal of Guidance, Control, and Dynamics, Vol. 24, No. 2,

March 2001, pp. 369–377.

doi:10.2514/2.4721

[8] Dimarogonas, D. V., Tsiotras, P., and Kyriakopoulos, K. J., “Leader-

Follower Cooperative Attitude Control of Multiple Rigid Bodies,”

Systems and Control Letters, Vol. 58, No. 6, 2009, pp. 429–435.

doi:10.1016/j.sysconle.2009.02.002

[9] Rodriguez-Cassola, M., Prats, P., Schulze, D., Tous-Ramon, N.,

Steinbrecher, U., Marotti, L., Nannini, M., Younis, M., Lopez-Dekker,

P., Zink, M., et al., “First Bistatic Spaceborne SAR Experiments with

TanDEM-X,”IEEE Geoscience and Remote Sensing Letters, Vol. 9,

No. 1, Jan. 2012, pp. 33–37.

doi:10.1109/LGRS.2011.2158984

[10] Montenbruck, O., Kirschner, M., D’Amico, S., and Bettadpur, S.,

“E/I-Vector Separation for Safe Switching of the GRACE Formation,”

Aerospace Science and Technology, Vol. 10, No. 7, Oct. 2006,

pp. 628–635.

doi:10.1016/j.ast.2006.04.001

[11] Angelopoulos, V., The THEMIS Mission, Springer, New York, 2009,

pp. 5–34.

[12] D’Amico, S., “Autonomous Formation Flying in Low Earth Orbit,”

Ph.D. Thesis, Delft Univ.of Technology,Dept. of Earth Observation and

Space Systems at the Aerospace Engineering Faculty, Delft, The

Netherlands, 2010.

[13] D’Amico, S., and Montenbruck, O., “Proximity Operations of

Formation-Flying Spacecraft Using an Eccentricity/Inclination Vector

Separation,”Journal of Guidance, Control, and Dynamics, Vol. 29,

No. 3, May 2006, pp. 554–563.

doi:10.2514/1.15114

[14] Gurfil, P., and Seidelmann, P. K., Celestial Mechanics and

Astrodynamics: Theory and Practice, Springer, Berlin, 2016, Chap. 14.

doi:10.1007/978-3-662-50370-6

0.76 0.78 0.8 0.82 0.84 0.86 0.88

Time [day]

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Distance [km]

CTD1

CTD2

RADD1

RADD2

Fig. 18 Crosstrack and radial components of the deputies with respect

to the chief satellite (CT = cross-track, RAD = radial).

14 Article in Advance / EDLERMAN AND GURFIL

[15] Capderou, M., Handbook of Satellite Orbits, Springer International,

Cham, Switzerland, 2014, Chaps. 7,10.

doi:10.1007/978-3-319-03416-4

[16] Vallado, D. A., and McClain, W. D., Fundamentals of Astrodynamics

and Applications, 4th ed., Microcosm Press, Portland, OR, 2013,

pp. 349, 565–568.

[17] Battin, R. H., An Introduction to the Mathematics and Methods of

Astrodynamics, AIAA Education Series, AIAA, New York, 1987,

pp. 401–408, 495–508, Chaps. 8, 10.

[18] Ben-Yaacov, O., Edlerman, E., and Gurfil, P., “Analytical Technique for

Satellite Projected Cross-Sectional Area Calculation,”Advances in

Space Research, Vol. 56, No. 2, July 2015, pp. 205–217.

doi:10.1016/j.asr.2015.04.004

[19] Vallado, D., and Crawford, P., “SGP4 Orbit Determination,”AIAA/AAS

Astrodynamics Specialist Conference and Exhibit, AIAA Paper 2008-

6770, 2008.

[20] Petit, G., and Luzum, B. (eds.), “IERS Conventions (2010),”IERS TN

36, Verlagdes Bundesamts für Kartographie und Geodäsie, Frankfurt

am Main, Germany, 2010, http://tai.bipm.org/iers/conv2010/.

[21] Brouwer, D., “Solution of the Problem of Artificial Satellite Theory

Without Drag,”Astronomical Journal, Vol. 64, Nov. 1959, p. 378.

doi:10.1086/107958

[22] Lyddane, R. H., “Small Eccentricities or Inclinations in the Brouwer

Theory of the Artificial Satellite,”Astronomical Journal, Vol. 68,

Oct. 1963, p. 555.

doi:10.1086/109179

[23] Thébault, E., et al., “International Geomagnetic Reference Field: The

12th Generation,”Earth, Planets and Space, Vol. 67, No. 1, May 2015,

Paper 79.

doi:10.1186/s40623-015-0228-9

[24] Wen, C., Zhang, H., and Gurfil, P., “Orbit Injection Considerations for

Cluster Flight of Nanosatellites,”Journal of Spacecraft and Rockets,

Vol. 52, No. 1, Jan. 2015, pp. 196–208.

doi:10.2514/1.A32964

[25] Ben-Yaacov, O., and Gurfil, P., “Long-Term Cluster Flight of Multiple

Satellites Using Differential Drag,”Journal of Guidance, Control, and

Dynamics, Vol. 36, No. 6, Nov. 2013, pp. 1731–1740.

doi:10.2514/1.61496

[26] Ben-Yaacov, O., and Gurfil, P., “Stability and Performance of Orbital

Elements Feedback for Cluster Keeping Using Differential Drag,”

Journal of the Astronautical Sciences, Vol. 61, No. 2, June 2014,

pp. 198–226.

doi:10.1007/s40295-014-0022-0

[27] Alfriend, K., Vadali, S. R., and Gurfil, P., Spacecraft Formation Flying:

Dynamics, Control, and Navigation, Vol. 2, Butterworth Heinemann,

London, 2009, Chap. 4.

[28] Leonard, C. L., Hollister, W. M., and Bergmann, E. V., “Orbital

Formationkeeping with Differential Drag,”Journal of Guidance,

Control, and Dynamics, Vol. 12, No. 1, Jan. 1989, pp. 108–113.

doi:10.2514/3.20374

[29] Kumar, B. S., and Ng, A., “A Bang-Bang Control Approach to

Maneuver Spacecraft in a Formation with Differential Drag,”AIAA

Guidance, Navigation and Control Conference and Exhibit, AIAA

Paper 2008-6469, Aug. 2008.

doi:10.2514/6.2008-6469

[30] Kumar, B. S., Ng, A., Yoshihara, K., and Ruiter, A. D., “Differential

Drag as a Means of Spacecraft Formation Control,”IEEE Transactions

on Aerospace and Electronic Systems, Vol. 47, No. 2, April 2011,

pp. 1125–1135.

doi:10.1109/TAES.2011.5751247

[31] Ben-Yaacov, O., Ivantsov, A., and Gurfil, P., “Covariance Analysis of

Differential Drag-Based Satellite Cluster Flight,”Acta Astronautica,

Vol. 123, June 2016, pp. 387–396.

doi:10.1016/j.actaastro.2015.12.035

[32] Gurfil, P., Herscovitz, J., and Pariente, M., “The SAMSON Project-

Cluster Flight and Geolocation with Three Autonomous Nano-

Satellites,”Proceedings of the AIAA/USU Conference on Small

Satellites, Technical Session VII: Growing the Community, SSC12-

VII-2, Paper SSC12-VII-2, 2012, http://digitalcommons.usu.edu/

smallsat/2012/all2012/56/

[retrieved 2018].

R. Vingerhoeds

Associate Editor

Article in Advance / EDLERMAN AND GURFIL 15