Behaviour based, autonomous and distributed scatter manoeuvres for satellite swarms

Contents lists available at SciVerse ScienceDirect

Acta Astronautica

Acta Astronautica 82 (2013) 95–109

0094-57

http://d

n Corr

Massach

Tel.: þ1

E-m

journal homepage: www.elsevier.com/locate/actaastro

Behaviour based, autonomous and distributed scatter manoeuvresfor satellite swarms

Sreeja Nag a,b,n, Leopold Summerer b

a Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, USAb Advanced Concepts Team, European Space Agency (ESTEC), Noordwijk, The Netherlands

a r t i c l e i n f o

Article history:

Received 27 December 2011

Received in revised form

6 April 2012

Accepted 17 April 2012Available online 22 June 2012

Keywords:

Swarm intelligence

Satellite clusters

Equilibrium shaping

Formation flying

Collision avoidance

65/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.actaastro.2012.04.030

esponding author at: Department of Aeronauti

usetts Institute of Technology, Cambridge, USA

6177101845.

ail address: [email protected] (S. Nag).

a b s t r a c t

One of the key requirements of a satellite cluster is to maintain formation flight among

its physically distinct elements while at the same time being capable of collision

avoidance among each other and external threats. This paper addresses the capability of

clusters with tens and scores of satellites to perform the collision avoidance manoeuvre

in the event of an external, kinetic impact threat, via distributed autonomous control

and to return to its original configuration after the threat has passed. Various strategies

for response manoeuvres are proposed based on a path planning scheme called

‘‘equilibrium shaping’’. The satellites in the cluster, modelled as a swarm of agents,

follow biological rules of ‘‘avoidance’’ of each other and the threat, ‘‘gather’’ to maintain

the formation cluster and ‘‘attraction’’ towards target location according to pre-defined

artificial potential functions. The desired formation of this multi-agent system repre-

sents equilibrium points i.e., a minimum potential state, leading to predictable

emergent behaviour for the entire cluster. The dynamical system is defined by adding

a control feedback to the solution of the Hill–Clohessy–Wiltshire equations in order to

track the desired velocities (as returned by the kinematic swarm model for equilibrium

points). Various distributed path-planning, collision avoidance strategies are compared

to each other in terms of the following metrics: delta-V spent during the manoeuvre,

time required for the cluster to return to normal operations and distance of closest

approach with the threat. Actuation and technological feasibility of the above strategies

is benchmarked using available and potential CubeSAT system capabilities for propul-

sion, sensing and communication range. The significance of the results on designing

future responsive, distributed space systems is discussed.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

A satellite constellation is a group of artificialsatellites—a set of physically independent, ‘‘free-flying’’modules that collaborate on—orbit to collectively achievea certain level of system-wide functionality. They maycommunicate with each other, remain aware of each

ll rights reserved.

cs and Astronautics,

.

others’ states, operate with shared control and comple-ment each other in terms of overall functionality. Asatellite cluster is a constellation that needs to maintaina certain amount of proximity between the physicalelements and must fly in formation accordingly. Clustersmay be homogenous or heterogeneous in form and/orfunction.

While each satellite in a cluster has traditionally beenconsidered a self-sustaining entity in terms of the entirespacecraft bus (everything minus the payload), a newparadigm design in clusters called fractionated spacecraftallows for the distribution of almost all subsystems of a

www.elsevier.com/locate/actaastro

www.elsevier.com/locate/actaastro

dx.doi.org/10.1016/j.actaastro.2012.04.030



mailto:[email protected]


S. Nag, L. Summerer / Acta Astronautica 82 (2013) 95–10996

satellite among the different physical elements. Eachmodule in a fractionated spacecraft is composed ofvarious subsystems, and thus a fractionated spacecraftmight consist of separate modules responsible for powergeneration and storage, communications, payload, and soon. In 2008, DARPA in the USA began a programme calledthe System F6 Phase 1 (for Future, Fast, Flexible, Fractio-nated, Free-Flying Spacecraft) aiming to generate a newparadigm for space systems, especially in the responsivespace sector [1,2]. The large, monolithic spacecraft oftoday is not designed for responsiveness and has otherdrawbacks (e.g., delay cascading in manufacturing), whicha fractionated spacecraft approach could eliminate orreduce. This approach allows for quite a disruptive changein how satellites are built and how they will be used, sincethe establishing of space infrastructure lowers the entrybarrier for satellite building and allows for resourcesharing. The main idea is to modularize satellites up tothe point where the monolithic spacecraft can be decom-posed into a network of wirelessly linked modules, allseparate smaller spacecraft, flown in a cluster and provid-ing the same or more capabilities than a single spacecraft.The concept is assessed in [3] mainly regarding itsinfluences on the aerospace sector, resulting from stan-dardization and mass production. While DARPA’s stresswas on quantitative analysis of the fractionated space-craft, the European Space Agency conducted a morequalitative analysis using an internal GSP study [4]. Theoptions for fractionation for each of the subsystems wereevaluated, given the current technological capabilities andstrategies for networking and cluster dynamics proposed.Different fractionated architectures were benchmarkedbased on above analysis and 4 reference missions (LEO,GEO, Lagrange points and planetary missions). The studyconcluded that the system increases flexibility, reliabilityand is suitable for missions requiring continuity. On theother hand, it requires standardization of modules and ismore vulnerable to kinetic impact of space debris. Afractionated spacecraft may therefore be considered asatellite cluster, heterogeneous in both form and function.

Taking into account the findings in literature that oneof the chief drawbacks of satellite clusters and fractio-nated spacecraft is their vulnerability to impact, causingdegradation or loss of a (possibly important) module, thispaper addresses the problem of distributed evasive man-oeuvres by the cluster’s satellites, when approached by akinetic threat.

2. Research motivation

One of the key requirements of a satellite cluster withmultiple physical entities is the need for all the modulesto fly within a tight ellipsoid in orbit in order to befunctional. This requires solutions to multi-body pro-blems in Earth orbit, precise determination of position,attitude and time, advanced control algorithms, trajectoryplanning and a host of other issues. There have been manyinstances of cluster formation flight. In the late 1990s, theUS Air Force began the conceptual design of TechSAT-21,which was to demonstrate the ability of several satellitesto replace a large monolith in an interferometric mission.

Although the program was cancelled later, it provides arich resource to literature on formation flight technology.NASA demonstrated Enhanced Formation Flying (EFF) viatheir first formation flight mission in 2000 called theEarth Observation 1 (EO-1), which flew in formation withLandSat-7, an Earth environment satellite launched in1999. NASA’s New Millennium Program, of which EO-1was a part, was thus a great success and it paved the wayto many technological FF milestones, which has now ledto the plan of the Terrestrial Pathfinder Mission (TPF). InTPF, a virtual space interferometer system, with a 1 kmbaseline, will be implemented to detect and analyse thelight from stars. NASA has also planned MagnetosphericMulti-Space (MMS) and Solar Imaging Radio Array (SIRA)as future formation flying missions. In 2002, the GravityRecovery and Climate Experiment (GRACE) supported byNASA and DLR demonstrated formation flight by a pair ofsatellites to measure the Earth’s gravity field and itstemporal variations. The European Space Agency (ESA)has proposed the following formation flying projects:PROBA-3 with 3-axis stabilized pair of satellites [5],DARWIN to study the origins of life with one leader and4 follower satellites and SWARM to study the Earth’smagnetic field with 3 satellites.

In a space environment, that is getting increasinglycrowded, another key requirement of satellite clusters iscollision avoidance, from other satellites, orbital debris oreven anti-satellite missiles. The US Space SurveillanceNetwork is tracking more than 19,000 Earth-orbitingman-made objects more than 10 cm in diameter, of whichroughly 95% are debris [6]. There are also an estimated300,000 additional man-made objects in Earth orbitmeasuring 1–10 cm and more than a million smaller than1 cm. In February 2009, a defunct Russian Cosmos satellitecollided in space with a commercial Iridium satellite, notonly causing destruction but also adding to the debrisalready present in space. In June 2007, NASA reportedmanoeuvring its $1.3 billion Terra satellite to avoid apiece of Fengyun-1C debris. Antisatellite (ASAT) missileshave been technologically demonstrated since 1960,when a US U-2 spy plane was destroyed by a USSR ASAT[7]. The US tested its Air-launched miniature vehicle(ALMV) in the early 1980s to demonstrate ‘kinetic kill’.Thereafter, in spite of oscillating treaties such as the OuterSpace Treaty (1967), Anti-Ballistic Missile Treaty (1972),the ban on ASAT testing (1986), the third generation ASATsystems were developed. Most recently, in 2007, Chinatested the kinetic kill technology of its ASAT system byshooting down its own satellite.

Collision avoidance has been discussed in past literature,although very rarely for completely distributed systems. Themost popular approach has been linear programming wherethe obstacles and required formation is treated as a con-straint, modelled using Mixed Integer Linear Programming(MILP) [8]. NASA’s Jet Propulsion Laboratory published acollision avoidance approach based on the Bouncing Ballalgorithm (BB) and Stalemate which adopts a heuristicapproach to multiple satellite reconfigurations [9]. ESA’sPRISMA satellites, developed under the contract to theSwedish Space Corporation, have robust collision avoidancealgorithms for autonomous formation where separation and

S. Nag, L. Summerer / Acta Astronautica 82 (2013) 95–109 97

nominal guidance are solved for analytically. Analyticalsolutions are possible since the satellites have near-circularorbit [10,11]. Another approach of avoidance has been topropagate uncertainty covariances and calculate the prob-ability that the relative displacement between two objects isless than a ‘‘collision metric’’ [12]. Princeton Satellite Sys-tems has come up with distributed guidance laws for low,medium and high autonomy of constellations, solved usinglinear programming [13].

The above algorithms, are either centralized algorithmsor assume the presence of a ‘captain’ to assess the distrib-uted inputs from the less intelligent agents. Two uniquecollision avoidance techniques using distributed satellitesystems were implemented at MIT in the Space SystemsLaboratory. One technique used the artificial potentialconcept (APF) was used along with the Linear QuadraticRegulator (LQR) to demonstrate avoidance of both fixed andmoving obstacles [14]. Velocities were modelled accordingto mathematical Gaussian functions centred at the targetand obstacles. The other technique worked by predicting theclosest point of approach and then overriding regularsatellite controls to move the satellites in a directionperpendicular to the collision to avoid it [15]. Both theore-tical concepts have been tested on a nano-satellite testbedcalled the SPHERES facility [16] developed by MIT SSL.While both pieces of work have suggested methods bywhich the algorithms can be scaled up to multiple satellites,none have demonstrated it for hundreds of satellites. If thealgorithms were to be used for scores of satellites, imple-mentations promise to scale up computationally and arefurther complicated if the cluster is heterogeneous.

Principles of swarm intelligence are currently beingused to navigate decentralized groups of robots whereeach robot is treated like an intelligent agent that followsbiological behavioural rules to maintain functionality.NASA is planning to use swarm intelligence in its ANTSmission to explore Near Earth Asteroids with 1000 coop-erative autonomous spacecrafts [19]. ESA has plans to useswarm intelligence for its APIES mission [20]. Agent basedmodels for satellite constellations have been explored bythe Princeton Satellite Systems, which has made aMATLAB toolbox called TeamAgent for the purpose [17].Their agents represent software and remote terminalsconnect these with the hardware. The study demon-strated four MAS architectures (top-down coordination,central coordination, distributed coordination and fully-distributed coordination) and compares their perfor-mance (evaluated by CPU workload and communicationeffort) for simple functions such as de-orbiting andreconfiguration. Similar work has been done at CNES,where fully-distributed coordination for Earth observa-tion satellite constellations with limited communicationwere investigated in detail for Fault Detection Identifica-tion and Recovery (FDIR) based on an intricately definedtrust and collaboration model and subsequently definedprotocols [21].

This paper addresses the capability of satellite clusters,and the newly developing fractionated spacecraft, to per-form collision avoidance manoeuvres via distributed andautonomous path planning and control, in the event ofdirected or random kinetic-impact threats. It enumerates

several strategies of manoeuvres in response to differentmission constraints. For example, one of the missionconstraints could be to not lose communication linksbetween the entities at any time in the manoeuvre. Inthe path-planning approach proposed in this paper, thesatellites determine their states based on artificial poten-tial functions with coefficients determined through equi-librium shaping. The principles of swarm intelligence areused by each satellite in the cluster to react to an externalthreat.

The approach has several advantages over previouslypublished literature. The path planning and target assign-ment method is completely autonomous and distributed,no captain or leader in the cluster is required. It is scalableto tens and hundreds of satellites in a cluster with globalconvergence to target formation for any symmetricalgeometry. Asymmetrical geometries have also beendemonstrated (even within this paper), however, is sus-ceptible to local minima resulting in erroneous potentialsand target formations. Target acquisition is not dependenton initial conditions or states. The algorithm in eachsatellite’s software uses only relative states of the clusterelements to determine its target state.

3. Scatter manoeuvre implementation

In our approach, the satellite cluster has been treatedas a multi-agent system. While the proposed algorithmscan be scaled up for any number of satellites, symmetricgeometries and inter-satellite ranges, for the purpose ofthe demonstrations within this paper, the cluster ofsatellites has a limited number of heterogeneous satel-lites, a few candidate topologies, a predefined commu-nication and sensing range and distributed processing/computing abilities. Agent based autonomy has fourlevels of hierarchy [17] and only completely distributedsystems have been addressed in this paper. The rules ofpath-planning or determination of target states are thoseof swarm intelligence where scientists have attempted tocopy the simple rules used by individual agents such asbirds or bees within large flocks because these rules resultin evolved, coherent behaviour of the entire swarm as awhole. Some simple examples are: an agent must ‘‘avoid’’other agents that are too close, must ‘‘copy’’ the generaldirection of movement of the swarm, must move suchthat its ‘‘view’’ is not blocked by another agent, must tryto remain close to the ‘‘centre’’ of the swarm and so on[18]. These agents use self-organizing, decentralized con-trol mechanisms and form flexible, robust and scalablesystems that respond well to rapidly changing environ-ments. They do not require global communication andrely on individual simplicity to respond to local incon-tingencies either due to the environment or due to theactivities of other agents.

3.1. Artificial potential functions for kinematic path-

planning

The concepts described in this section are applicable toboth homogenous and heterogeneous (e.g. fractionatedspacecraft) satellite clusters. The ‘‘intelligence’’ of the

Fig. 1. Velocity fields using exponential functions. X-axis (dij)¼distance

between the satellite and the source, Y-axis¼velocity of the satellite

according to Eq. (2) where ls¼k in the figure and As¼1 (scaling factor).


satellites (agents in a multi-agent system) is a kinematicmodel which gives each satellite in the cluster the abilityto determine its target velocity vector as a response to itsenvironment, real-time targets at every control cycle. Thekinematic model comprises of artificial potential func-tions (APF) which can be programmed into each satellite’ssoftware and use only pre-defined constants and relativestates of the other entities in its environment as inputs.APFs can be derived from a general class of attraction/repulsion functions (depending on the polarity) used toachieve swarm aggregation [23,24]. These functions areodd and of the type g(y)¼�y[ga(|y|)�gr(|y|)], wherega(|y|) is the attracting function, gr(|y|) is the repellingfunction and ‘y’ is the distance between the satellites. Foreach gr(|y|) and ga(|y|), there exists a Jr(|y|) and Ja(|y|), suchthat rJr(|y|)¼ygr(|y|) and rJa(|y|)¼yga(|y|), where J(|y|) iscalled the artificial potential function.

A very common example of g(y), or a satellite’s velocityfield, is as follows:

gðyÞ ¼ �y½a�b exp�ð:y2:=cÞ� ð1Þ

There exists a unique distance d such that ga(d)¼gr(d),which means that the equilibrium condition of zerorelative velocity occurs at only one unique distancebetween two satellites. Thus, if such functions are definedfor all pairs of satellites in a formation, there can be only aunique geometry possible at the equilibrium condition.Theoretical studies on stability analysis have demon-strated convergence and an analytical value for time ofaggregation and maximum size of the swarm can also becalculated [24].

Just as pairs of satellites are modelled to be specificdistances apart in Eq. (1), a food source (or any entity tobe attracted to) can be modelled as an attractor and aphysical kinetic threat (or any entity to stay away from)can be modelled as a repellent. Some typical potentialfunctions for such sources are as follows [24,25]:

�

�

Linear: s(y)¼asTyþbs

�
Quadratic: s(y)¼ðAs=2Þ:y�cs:2þbs� �
Gaussian: sðyÞ ¼ As2 exp �

:y�cs:2

ls þbs

The target velocity vector of a satellite at a relativeposition ‘y’ with respect to a source cs is the differential ofthe selected potential function. For more than one source,the target velocity of a satellite will be the summation ofthe contribution of each potential function. For example,if all sources are modelled as Gaussians, then the totalpotential on any satellite due to the contribution of N suchsources is

sðyÞ ¼�XN

i ¼ 1

Ais

2exp

��ð:y�ci2

s Þ=lis

�þbs ð2Þ

An example of the Gaussian velocity fields, derivedfrom these potentials, is shown in Fig. 1.

According to the above theory, a set of equilibriumfunctions corresponds to a unique geometry for thesatellite cluster. Therefore, the vice versa must also betrue: Any geometry for a cluster can be achieved by using

appropriate potential functions and calculating the coeffi-cients (i.e., the free parameters) correctly, for each satel-lite. Thus, this method can be applied to swarmaggregation, social foraging and most importantly in ourcase, formation flying [25]. While the above theory wasoriginally developed for robots, the first instance of itsadaptation to suit satellite clusters was implemented atthe Advanced Concepts Team in the European SpaceAgency [26–28]. Homogenous satellites modelled aspoints in space were initiated at random states and weresuccessfully able to gather at particular geometries with-out colliding with each other by using the simple rules of‘‘dock’’ (modelled as an exponent), ‘‘gather’’ (modelledlinearly) and ‘‘avoid’’ (modelled as an exponent). Up tohundreds of homogenous satellites were able to achievethe required formation within 60,000 s, where each satel-lite was programmed to achieve target states usingspecific potential functions that followed the above rules.The coefficients or free parameters for the APFs werecalculated by the technique called ‘‘equilibrium shaping’’[27], where in a set of linear equations for the totalvelocity field of each satellite were equated to zero, tosolve for the free parameters at the equilibrium conditionof the system (minimum potential state). The equilibrium


condition in turn was pre-set as per the desired formationor relative geometry of the cluster.

The current study builds on past literature by introdu-cing the concept of an external threat, capable of destroy-ing or degrading the satellites by kinetic impact, andstudies the ability of the satellites to avoid the threatwhile also avoiding each other to prevent collisions. Thethreat is modelled using artificial potential functions, as amoving target that has to be avoided. The potentialfunction used for the threat is modelled as an exponentialpotential function and the target velocity of a satellite ‘i’at a given instant of time, as response to the threat, isgiven by

vðiÞ ¼ Ananexp �dij2

K

!ð3Þ

In Eq. (3), dij is the distance between the threat ‘j’ andthe satellite ‘i’ at the given instance of time, K is a functionof the sensing radius of the satellite, a is the unit vector inthe direction the satellite is programmed to move in orderto avoid the threat and A is a scaling factor. a can be eitherin the opposite direction of the approaching threat (whichis not a smart strategy because threats are usually fasterthan the escape speed of the satellite) or in the perpendi-cular direction to the approaching threat.

Fig. 2 shows a simple example of a simulation of fourheterogeneous satellites that were originally in a rhom-bus formation on the z¼0 plane. The rhombus has sides of6 m each, the angles are 601 and 1201 and the centroid liesat (0,0,0). The red circles show the original configurationof the cluster and the blue circles the final configuration.The green line shows the trajectory of the threat,approaching from the positive Z direction (black arrow),moving at a velocity of 1 m/s. The Gaussian distributionfor the APF is chosen as a function of the sensing radius ofthe satellites because the satellites will be able to sensethe threat only when it is within their sensing radii.Assuming a sensing radii of 10 m, the 3s point of the

Fig. 2. Trajectory of a satellite cluster (blue lines) in the event of an

approaching threat (green line with an arrow to indicate direction). The

red circles show the original configuration of the cluster and the blue

circles the final configuration. Note that one of the satellites would have

been hit, had the cluster not moved. (For interpretation of the references

to colour in this figure legend, the reader is referred to the web version

of this article.)

Gaussian distribution lies at 10 m. Since K¼2s2, K(ats¼20/3) in Eq. (3) is calculated to be 22.22 m. Thepotential functions (APF) used by each satellite to deter-mine its target velocity in Fig. 2 were that of the threat(exponential, K¼22.22, A¼40, a in the direction oppositeto the threat) and linear attraction and exponentialrepulsion as given in Eq. (1) (b¼20, c¼10) for theindividual satellites. ‘a’ in Eq. (1) is calculated from theequilibrium shaping formula i.e., by setting total velocityfor each satellite at the equilibrium condition to zero. Theequilibrium condition is determined by the final geometryto be acquired by the satellites in terms of their relativestates. For a rhombus with length 6 m, 60–1201 angles,the distance between any pair of satellites is 10.39 mbetween the apex/farthest satellites and 6 m for all otherpairs. The 6 m inter-satellite distances were chosen basedon typical 10 cm CubeSAT high bandwidth, inter-satellitecommunication ranges. Setting the velocity of Eq. (1) tozero for all satellites, a(i,j) for the ith–jth satellite pair isgiven by:

aði,jÞ ¼ bnexp�delði,jÞ2

cð4Þ

where del(i,j) is the distance required between the ith andthe jth satellite.

Once the parameters (i.e., the full matrices a, b and c) aredetermined by solving Eq. (4), each satellite can calculate itstarget velocity given by the summation of Eqs. (1) and (3)using the appropriate values from the determined para-meters. When the threat is within 10 m of the cluster, itmoves directly away from the threat, as is seen inFig. 2—the path planning algorithm is determined throughEqs. (1) and (3). When the threat is greater than 10 m awayfrom the cluster, only Eq. (1) governs the satellite’s targetvelocity and so its rhombus configuration is establishedagain within an error of 1mm. In a real world mission, theconstant parameters/matrices can be calculated beforehand,depending on the desired configuration of satellites andrequired aversion to the threat, and stored within thesoftware of every satellite. The satellites can use theseconstant parameters and the real-time relative states ofthe other satellites and threat (as per their sensing capabil-ities) and calculate their target velocities real-time. Alter-natively, if the mission designer wishes to keep the clusterconfiguration an open variable, appropriate APF equationsand a linear solver can made available to each satellite, sothat they may each determine the parameters as per therequired geometry and threat condition (e.g. multiplethreats vs. single threat). The system is therefore adaptable,dynamic and completely decentralized.

In the scheme described above, the cluster has about 10 sto react to the approaching threat, given that the threatmoves at a velocity of 1 m/s and the sensing radii is 10 mbecause it is relying completely on the individual sensors ofits satellites to detect the approaching threat. Real-worldthreats (esp. kinetic missiles) are much faster, so in order tosuccessfully avoid them, the satellites with CubeSAT capabil-ities must either have a threat sensing radii of 50 m or more,or must rely on a warning signal from a ground station. Fromthe next section onward, all modelling has taken into accounta warning from the ground station to all the satellites. At the


time of the warning, the ground is assumed to provide thefollowing information to the satellites: 3D position of thethreat at time of warning, its velocity vector and the timestamp of dispatching this information. Once warned, all thesatellites set the threat sensing radii values within their APFsoftware equal to their distance from the threat (at the timewhen the warning was broadcast) within their internalkinematic model and then begin to set target states accord-ingly (v(i)¼v(Eq. (1))þv(Eq. (3))). It is has been assumed inthis paper that the threat moves in a straight line, so thesatellites are able to forward propagate in time every sub-sequent position of the threat, hence ‘knows’ where it is. Themodel proposed can be easily adapted for real time missionsby either having the ground station send out the approximatetrajectory of the threat and/or send the threat state periodi-cally, such that the clusters may use the most recent state tocalculate their target behaviours. When the threat has passed,the ground station is expected to broadcast the ‘safe’ flag andthe satellites can reset the sensing radii values in the APFsoftware back to the true sensing radii (i.e., 10 m as demon-strated in Fig. 2). Thus, the ground station forecast allows thesatellites to have a virtual sensing radii for the purpose ofusage within the kinematic model, which gives the clustermore time to react to the threat and which would not havebeen possible if the satellites relied only on physical sensors.

For demonstration of the proposed algorithms, hetero-geneous satellites in the cluster are defined in terms ofCubeSat mass, power and subsystem requirements (�7 kg,3CU, 5 W per satellite), while the advancing threat is definedin terms of its approach velocity, head-time and circulararea probable (CEP). The satellite model chosen is similar tothe one used in NASA Ames GeneSAT. Algorithms have beendemonstrated using CubeSAT capabilities to keep with ESA’sincreased interest in small satellites (e.g. the NanoSat studyperformed at the CDF [22]).

3.2. Dynamics and controls for actuation

The example in the previous section demonstrated asimple kinematic model for determining target states of asatellite cluster treated as a swarm. The dynamical systemis defined by adding a control feedback, based on thekinematic model to track the desired velocities, to thesolution of the Hill–Clohessy–Wittshire equations [29] atthe mission altitude. The Hill’s equations for eccentricorbits (Eq. (5)) and their analytical solution (with aneccentric anomaly H(n)) are given in [30]. While it iscertainly possible to obtain the full solution as thedynamic system, it is beyond the scope of what this paperaims to demonstrate hence for the sake of simplicity,circular orbits are considered. The cluster is assumed tobe in Low Earth Orbit (LEO), 7000 km from the Earth’scentre with a period of 6.74e�4 days and an angularvelocity (n) of 1.078e�3 rad/s. Since the orbit is assumedto be circular, the three HCW equations simplify to Eq. (6)and an analytical solution is also available.

d

dt

_x

_y

_z

264

375¼�2

0 � _n 0

_n 0 0

0 0 0

264

375

_x

_y

_z

264

375�

0 � _n 0

0 � _n2 0

0 0 0

264

375

x

y

z

264375

�

0 � _n 0

_n 0 0

0 0 0

264

375

x

y

z

264375þn2 1þe cos n

1�e2

� �X

2x

�y

�z

264

375

þ

ax

ay

az

264

375 ð5Þ

€xi�2n _yi�3n2xi ¼uxi

mi

€yiþ2n _xi ¼uyi

mi

€ziþn2zi ¼uzi

mið6Þ

In Eq. (6), u(xi), is the control force for the x componentof the ith satellite and so on for u(yi) and u(zi). The controlforce depends on the control scheme selected for thedynamic system. For the purpose of swarm aggregation inspace, the following feedback control schemes have beendemonstrated in literature [26]:

�
Q-Guidance controls via Cross Product Steering Law(CPSL) � Q-Guidance controls via Velocity to be Gained Law
(VTBG)
� Sliding mode controls (SMC) � Artificial Potential approach controls (APC)
The results show that the delta-V spent via VTBG andSMC are the lowest for the average satellite in the swarm.For the purpose of demonstrations in this paper, a simpleproportional (to velocity) controller is used and thecontrol acceleration can be written as

ui ¼ kð _xi�viÞþ €xi�ain ð7Þ

where _xi is the kinematic velocity and €xi the kinematicacceleration obtained from the kinematic swarm modeldescribed in the previous section, vi is the actual dynamicvelocity of the satellite in Hill’s frame and ain are theinertial forces. The ð _xi�viÞ term is called velocity-to-be-gained, vg . The parameter k is analogous to the knownsteering laws [31]. For example, if k goes to infinity, thecontrol strategy is to thrust in the direction of the velocityto be gained vector regardless of the contribution of theuncontrollable term, €xi�ain. In our simulations using thedynamic framework, the control accelerations were deter-mined from the kinematic target velocities (Eq. (7)) andresulting dynamic positions and velocities were calcu-lated using a simple ODE 45 solver.

In real missions, as with the kinematic model, allparametric constants, a linear and differential equationsolver will be available within the software of eachsatellite. Each satellite will therefore be able to determineits kinematic target velocity in the same way as describedin Section 3.1 real-time, find the acceleration requiredby solving the control equation in Eq. (7) and thendetermine its dynamic target state by solving the differ-ential equation.

Technological feasibility of the proposed algorithmsand strategies have been benchmarked using technologiesavailable to small satellites, in terms of mass, thrust, etc.


Actuation feasibility of the control strategy is tested insimulation using numbers from available/potential Cube-Sat propulsion capabilities which puts a cap on themaximum thrusting ability and maximum fuel spent bythe spacecraft. Both electrical and chemical propulsionsystems are being developed for CubeSATs. The electricalsystems [32] offer very low thrust of ~70mN but a highspecific impulse of 43500 s within 0.5 kg of propulsionsystems. The chemical systems [33,34] offer a higherthrust of 2–10 mN per thruster (4 thrusters per cube)but a total delta-V of 35 m/s for the same system mass of0.5 kg. For the simulations enumerated in the forthcomingsections, when the control accelerations required weregreater than that possible to actuate by the thrusters, thethrust was capped to the maximum thrust available, i.e.,1.4 mm/s2 for chemical propulsion or 14mm/s2 for elec-trical propulsion. Initial simulations showed that electricpropulsion was not capable of thrusting enough to helpthe cluster manoeuvre out of threat’s way with more than1 m of miss distance using the path planning strategiesdemonstrated in the paper. All the results thus demon-strate the performance of a chemical propulsion system.

4. Demonstration of scatter manoeuvre strategies

The following section describes various strategies that canbe taken by a multi-satellite homogenous or heterogeneous(e.g. fractionated spacecraft) cluster in order to approach athreat approaching it using various types of artificial potentialfunctions (APF), calculated using equilibrium shaping (ES).The threat is modelled as a point object travelling in astraight line, but as explained in Section 3.1, it is possible tomodel and implement more complex trajectories ofapproaching threats. The strategies have been designed inresponse to specific mission requirements and flexibilityallowed of the geometric configuration of the cluster duringthe scatter manoeuvre. The various strategies are comparedto each other in terms of metrics: probability of successwithout loss of long-term functionality, minimum distance ofapproach between the satellites and the threat, delta-V spentand time required to return to normal operations. Since eachstrategy is unique to a mission requirement, the metricsshould be traded with mission-level metrics to understandand compare value of the strategies. In themselves, themetrics are not a measure of comparative value of thestrategies. Further, the mission requirements and the scatterstrategies are chosen as examples to demonstrate the applic-ability of the APF and ES to distributed scatter manoeuvres toseveral mission constraints. More requirements, strategiesand functions are certainly possible for larger clusters ofmore (tens or even hundreds) satellites in any symmetricalgeometrical configuration. The strategies described and com-pared in this section and the associated potential functionsused to model swarm behaviour and determine the kine-matic target velocities are listed below.

1.
Strategy 1: scatter and gatherMission requirement: Collision avoidance from the threatby direct exit from a predefined threat ellipse and asubsequent gather strategy that activates after the ground
station warning is switched off—‘safe’ flag signalled.APFs used for scatter: Lennard Jones potential function,Scatter potential function, viscosity functionAPFs used for gather: Gather potential, avoid potential,dock potential

2.
Strategy 2: adaptation to threat by moving clustercentroidMission requirement: Collision avoidance from thethreat with a shift in the formation centroid of thecluster allowed, however control over the relativegeometry of the cluster needs to be maintainedthroughout the scatter manoeuvres.APFs used: Gather potential, avoid potential, threatpotential
3.
Strategy 3: Adaptation to threat by fixed cluster centroidMission requirement: Collision avoidance from the threatwith the formation centroid of the cluster required to befixed at all times however no control over the relativegeometry for the cluster is required.APFs used: Gather potential, avoid potential, dock poten-tial, threat potential
4.
Strategy 4: Mix-and-match demonstrationMission requirement: Collision avoidance from threatby scattering such that a shift in the formation cen-troid of the cluster is allowed, i.e., Strategy 2 afterbeing warned by the ground station, followed by thegather strategy of Strategy 1 after the ground stationwarning is switched off—‘safe’ flag signalled. Strategy4 is not a novel strategy, instead demonstrates theability of the proposed methodology to improvise withthe usage of artificial potential functions to buildcluster behaviours as desired.APFs used for scatter: Gather potential, avoid potential,threat potentialAPFs used for gather: Gather potential, avoid potential,dock potential.
All the simulations were implemented in a dynamic Hill’sframe as described in Section 3.2 with a limit on themaximum control accelerations allowed to actuate theguidance and navigation, as determined by the dynamicmodel using APFs. The maximum control accelerations aredetermined by nominal CubeSAT chemical thruster forcesapplied to a typical CubeSAT mass. Since the space environ-ment has some other random forces such as solar radiation,air drag, lunar gravity and solar gravity, white noise of theamplitude of 1e�6 [35] was added to the Hill’s equations ofEq. (7). All the strategy demonstrations described have used4–6 satellites in the specific-geometry cluster as a proof ofconcept in simulation, but it is, in theory possible to scale thenumber of satellites up to hundreds in a symmetric swarmand the methodology would still hold good [36]. The onlyrequirement would be to recalculate the constant parameters(much larger matrices) such as those in Eq. (4) for thegeometry desired with the satellite swarm, and to makethem available to all the satellites in the cluster—the rest ofthe methodology, i.e., to calculate kinematic states and thendynamic states, remains the same. Since the constant para-meters and the dynamic accelerations can be determined forboth homogenous and heterogeneous clusters, it is easilypossible to adapt these strategies described for fractionated


spacecraft for increasing number of components and com-plexity. Heterogeneous cluster simulations have been suc-cessfully implemented in this regard and also describedbelow.

4.1. Strategy 1: scatter and gather

This is the simplest strategy using distributed decision-making and artificial potentials and has been demonstratedin the example below using a rhombus cluster formation. Themission scenario considered is: Four satellites in a rhombusformation on the plane z¼0 that has sides of 6 m each, theangles are 601 and 1201 and the centroid lies at (0,0,0). Att¼0, the ground station warns the satellites of an approach-ing threat and requires them to move out of a sphere ofradius 20 m centred at (0,0,0). In response to this scenario,the potential functions used to calculate the satellite targetstates for the scatter process after being warned by theground station are as follows:

�
Lennard Jones potential function determines theattraction–repulsion between the satellites. If V(r) isthe LJ potential between two bodies at a separation ‘r’,then its derivative, v1(r), will be the velocity of a singlesatellite, as explained in Section 3.1.
VðrÞ ¼ e sr

� �12�2

sr

� �6� �

ð8Þ

v1ðrÞ ¼12e

r

er

� �12

�er

� �6� �

ð9Þ

As per the LJ molecular theory, the attraction is verystrong when r is not much larger than s, but after acertain distance this force fades to zero. This meansthat two molecules, or in our case two satellites,interact strongly only when their mutual distance iswithin a certain value, thus explaining the reason whywe called this potential local. The stable arrangementof two molecules interacting with each other is suchthat they respect the mutual target distance s. In theabove equations, e is the depth of the potential well,which accounts for the attractiveness and stability ofthe minima located at distance s. It is set at 0.1 for thissimulation [28]. We set s to be 10.59 m for the furthesttwo satellites in the rhombus and 6 m for all otherpairs. The important feature of this potential is that thelattice is formed on the basis of positional informationonly: no communication is needed.
� Scatter potential function forces the satellites to move
radially outward from the centre of the danger spherei.e., the avoid region specified by the ground stationwhen forecasting the warning.

v2ðrÞ ¼ d½R2�r2� ð10Þ

For the purpose of the scatter potential, a Cartesiancoordinates are transformed to a new spherical

coordinate system whose centre is equal to the centreof the danger sphere (R¼20 m in our case). In Eq. (10),v2 represents the radial component of the velocity of asingle satellite in the new coordinate system, x theradial position of the satellite, and d the scaling factor.The angular components of the v2 are set to zero. Theavoid region can be extended from a sphere to anyother shape by modifying Eq. (10) accordingly.
� Viscosity function forces the satellites to stop motion
when they reach the danger sphere’s surface. Once theswarm converges to its final configuration, the grav-itational influence of the planet tends to disrupt theformation, therefore the satellites need to use theirthrusters to maintain their relative positions. Simula-tions show that the satellites actually oscillate aroundtheir equilibrium points, thus wasting propellant. Asolution to this problem can be found again withphysical considerations. The physical analogy is toimagine the satellites immersed in a viscous medium,such as for example air or some liquid. Therefore, themathematical expression of the velocity due to theviscosity potential is as follows:

vis¼K expð�x=2Þ if xoD

0 otherwise

(ð11Þ

This virtual viscosity is applied to the satellite’s velocity ifx i.e., the difference between the required radial positionand the actual radial position is less than a thresholdvalue, i.e., the satellite is almost at target. In our simula-tion, D¼0.5 m, which means that when the satellite iswithin 0.5 m of the surface of the danger sphere. Once theviscosity is known, velocity can be calculated as

v3¼�x _x ð12Þ

Here, _x is the actual velocity of the satellite. If thedanger sphere is a shape different from a sphere, Eq. (11)will also be modified accordingly.

The kinematic target velocity of each satellite in thisstrategy for the scatter is calculated at each control cycleby the summation of v1, v2 and v3. Since the satelliteshave the constants stored within their software before-hand, it is easy to determine all the velocities based on therelative distances to the other satellites and the threat.Results of this simulation, without and with a cap (due toactuation limits of hardware) on control acceleration inthe Hill’s frame, are shown in Fig. 3. The control profilesare shown in Fig. 4 that indicates that thrust at theirmaximum available thrust for 1500–2000 s after whichthe controls die down because the satellites have reachedtheir targeted sphere surface. The delta-V spent is�1.8 m/s which proves this strategy to be extremelydelta-V and thrust expensive.

Once the satellites have scattered and the threat haspassed over, the ground station is assumed to broadcastthe information (‘safe flag’) to the satellites. When thesatellites receive this information, a different strategy isused to model the return of the satellites to their originalpositions. The artificial potential functions used for this

Fig. 3. Strategy 1 results: (Top) Trajectories of the satellites when there

is no constraint on the control acceleration. (Bottom) Trajectories of

the satellites when control acceleration of each satellite is capped at

1.4 m/s2. The red circles show the original configuration of the cluster

and the blue circles the final configuration. (For interpretation of the

references to colour in this figure legend, the reader is referred to the

web version of this article.)

Fig. 4. Control acceleration (thrusting) profiles for Strategy 1 with an

imposed maximum control acceleration of 1.4 m/s2 per satellite

(RMS¼2.1 m/s2).


return phase, per control cycle, to determine their targetvelocities for each satellite are as follows:

�
Gather potential introduces 4 different and uniqueglobal attractors towards the sinks of the desiredformation, which in this demonstration are the nodesof the rhombus in Hill’s frame. Therefore each agenthas to know at each time where is the position of eachpoint of the final formation to be achieved. Theexpression for this kind of behaviour is defined as
v1ði,tÞ ¼�cði,tÞndxði,tÞ ð13Þ

Eq. (13) is the velocity of the ith satellite with respectto an attractor ‘‘t’’ (unique target, since the cluster isheterogeneous), with dx(i,t) as the distance betweenthe ith satellite and its target and c is the scaling factordetermined via equilibrium shaping.

�
Avoid potential establishes a keep-away relationshipbetween two different agents that are in proximitywith each other. In such a case a repulsive contributionwill assign to the desired velocity field a direction thatwill lead both the two agents away from each other.The expression that describes the assigned velocity ofa single satellite i with respect to another satellite j forthis kind of behaviour is given below:
v2ði,jÞ ¼ �dxði,jÞ bði,jÞnexp�:dxði,jÞ:

k2

� �� ð14Þ

In this equation, x(i,j) is the distance between the twoagents that are proximate and k2 describes the sphereof influence of this contribution (within 3o asdescribed in Section 3.1), i.e., the distance at whichthis behaviour would have a non-negligible effect. ‘‘b’’is calculated from the equilibrium shaping.
� Dock potential expresses the local attraction of each
agent towards each sink i.e., the final locations of thecluster’s satellites which in this demonstration are thenodes of the rhombus in Hill’s frame. The componentof the desired velocity field due to this behaviour has anon-negligible value only if the agent is in the vicinityof the sink. The parameter k3 determines the radius ofthe sphere of influence of the dock behaviour. Thevelocity of a satellite ‘‘i’’ with respect to a targetposition ‘‘t’’ for this potential is given as

v3ði,tÞ ¼�dxði,tÞ dði,tÞexp�:dxði,tÞ:

k3

� �� ð15Þ

in which again x(i,j) is the distance between the twoagents that are proximate and k3 describes the sphereof influence of this and ‘d’ is calculated from theequilibrium shaping.


velocity for each satellite (v1þv2þv3) is set to zero, forthe equilibrium geometrical configuration of a rhombus at

To calculate the constant parameters, the total satellite

z¼0. ‘c’ is set at 3.214e�3 for required target and ‘d’ at 40.‘k2’ and ‘k3’ to 6 for an assumed sensing radii of 6 m. Tocalculate the required ‘b’, Eq. (16) is used for the satellitesat the furthest ends of the rhombus and Eq. (17) is usedfor all other pairs of satellites.

Fig. 5. Trajectory of the rhombic cluster in Strategy 2 when it moves

opposite to the approaching threat (direction marked by an arrow). The

axes are the X-, Y-, Z-axis in the right-handed coordinate system in

metres. The red circles show the original configuration of the cluster and

the blue circles the final configuration. (For interpretation of the

references to colour in this figure legend, the reader is referred to the

web version of this article.)

bði,jÞ ¼3cþ3ndði,jÞnexpð�l2=k3Þþ

ffiffiffi3p

ndði,jÞnexpðð�3l2Þ=k3Þ�ffiffiffi3p

nbði,jÞnexpðð�3l2Þ=k2Þ

expð�l2=k2Þð16Þ

bði,jÞ ¼3cþ3ndði,jÞnexpð�l2=k3Þ

expð�l2=k2Þð17Þ

In both the above equations, it must be noted that thecalculation is an approximation to the original ones usedin simulation. For the original calculations, each x, y and z

components of the velocity were set to zero, in which casethere would be 3 independent equations to solve forinstead of 1 for each heterogeneous pair of satellites, afterassuming equal values for ‘c’ and ‘d’. For the full calcula-tion, the ‘l’ in Eqs. (16) and (17) would be replaced by thedifference between the x coordinates, y coordinates and z

coordinates for each set of satellites. As noted, the moreheterogeneous and fractionated the cluster, the more thenumber of constants to be determined, but easily possiblegiven a linear solver.

The desired kinematic velocity of any satellite in there-gather phase expressed as a sum of all three velocities,v1, v2 and v3 for all the targets and is calculated andactuated in the dynamic frame at every control cycle. There-gather strategy has a delta-V of �0.5 m/s per satelliteand takes 900–1200 s to complete.

4.2. Strategy 2: Adaptation to threat by moving cluster

centroid

This is a simple strategy in response to a missionscenario which requires the entire cluster to move out ofthe path of approach of the threat while maintainingrelative geometry between the component satellites. Thestrategy has been demonstrated using four satellites in arhombus formation on the plane z¼0 that has sides of 6 meach, the angles are 601 and 1201 and the centroid lies at(0,0,0). At t¼0, the ground station is assumed to warn thesatellites of an approaching threat and requires them tomove out of the direction of approach of the threat. Thereis with no restriction on the overall position of the clusterduring scatter, however, fixed relative distances isrequired to be maintained i.e., cluster geometry remainsintact (in this case a rhombus of side 6 m with a 60–1201angle between its sides). Like in Strategy 1, four satellitesare considered however unlike Strategy 1, an approachingthreat is modelled as per Eq. (3) where a is set to makethe cluster move in the direction of the approachingthreat i.e., away from the threat. The threat is modelledto move at a velocity of about 1 km/s. The original

position of the cluster had the rhombus on the z¼0 planewith the centroid at (0,0,0) but the final position is abouta 1 km away as seen in Fig. 5.

The potential functions used to calculate the kinematictarget velocities for each satellite are those of ‘‘Gather’’and ‘‘Avoid’’ (i.e., the attraction–repulsion function inEq. (1)) and ‘‘Threat’’ (Eq. (3)), as described in the previoussections. Fig. 5 shows the trajectories of the satellites for a

simulation when the ground station issued the warning2000 s before one of the satellites was to be hit by thethreat. In spite of such an early warning, the miss distancewas calculated as 7 m and an average delta-V of 1.6 m/s.This is because by escaping in the direction opposite tothe threat, the satellites thrust at their maximum buttheir velocity is naturally not enough to escape thevelocity of the threat. As a result, it is only when thethreat is very close that the satellites move out of its way.This is clearly not a good strategy because no matter howearly the warning is issued, the delta-V would be wastedonly to make the formation accelerate but not move outthe threat’s way. The warning is switched off at the 600thsecond as seen in the control profiles of Fig. 6 where thesecond leg of maximum thrust is to bring the satellitesback to their relative geometry within a simulation timeof 1000 s. The average error in geometry, calculated as theaverage RMS error per inter-satellite distance, is 8 cmafter the simulation is completed.

Strategy 2 can be modified such that in the threatpotential function in Eq. (3) is perpendicular to thedirection of approach of the threat (modify a). Since theground station informs the cluster of the velocity vector ofthe threat, the satellites can calculate an infinite number


of vectors perpendicular to that vector and randomly pickany one of the solutions, or additional mission constraintsmay exist to the direction of movement. The otherparameters of the threat potential function remain thesame and so do the other potential functions i.e., Gatherand Avoid. The simulation is run for several different timeperiods as shown in Table 1. In the table, ‘Headtime’ is thetime available between when the ground station issuedthe warning and when one of the satellites is calculated tobe hit by the threat i.e., the time that the cluster has toavoid being hit. ‘Warning off time’ is the time elapsedafter the warning was issued at which the warning isswitched off. The difference between ‘Headtime’ and‘Warning off time’ is the time when the cluster has tocontinue its scatter manoeuvre to avoid the threat.Average miss distance is the average distance of closestapproach between the threat and any satellite. Velocity ofattack is the RMS velocity of the threat in the Hill’s frame.Error is the formation error calculated as the average ofthe RMS errors between what the inter-satellite distancesare at the end of the simulation and what they should beas per mission geometry.

Fig. 7 (top) shows a test simulation where the greenline represents the approaching threat—direction markedwith an arrow. Fig. 7 (bottom) shows Sim # 2, Sim #3, Sim#4 in Table 1. It can be clearly seen from the table that themiss distance is a function of only headtime (given themaximum thrust of the satellites is limited), irrespectiveof the velocity of attack. Also, given more simulation time,the error of formation i.e., the average RMS error per

Fig. 6. Control acceleration (thrusting) profiles of each of the 4 satellites

in Strategy 2 when it moves opposite to the approaching threat

(direction marked by an arrow).

Table 1.Comparison between different metrics (described in the text) in Strategy 2.

Sim # Headtime (s) Velocity of attack (m/s) Warning off time (s)

1 50 1 100

2 200 1 400

3 200 1 400

4 200 1000 400

inter-satellite distance, reduces. For example, the errorreduces by 16 cm when the simulation is run for 200 smore—compare Sim #2 and Sim #3 in Table 1.

4.3. Strategy 3: Adaptation to threat by fixed cluster

centroid

This strategy is demonstrates the cluster scatter man-oeuvre using APFs for a mission that requires a fixedcluster centroid before and after the passage of the threatbut does not put any constraints on the relative config-uration of the satellites. The cluster geometry is allowed

Simulation time (s) Error (cm) Average miss distance (m)

200 4 4

500 45 45

700 29 45

700 22 45


perpendicular to the approaching threat (direction marked by an arrow)

and control acceleration limited to 1.4 m/s2. The red circles show the

original configuration of the cluster and the blue circles the final

configuration. (For interpretation of the references to colour in this

figure legend, the reader is referred to the web version of this article.)

Fig. 9. Control acceleration (thrusting) profiles of each of the 6 satellites

in Strategy 2 when the cluster moves perpendicular to the approaching

threat (direction marked by an arrow).

Fig. 10. Trajectory of the hexagonal cluster in Strategy 3 when it moves

perpendicular to the approaching threat (direction marked by an arrow).

The axes are the X-, Y-, Z-axis in the right-handed coordinate system.

The red circles show the original configuration of the cluster and the


to be broken during the scatter manoeuvre but the clusteris required to return to where it started in the Hill’s frameafter the threat has passed. The strategy is demonstratedusing six satellites in a hexagonal formation with a sidelength of 6 m. The potential functions used to calculatethe kinematic target velocities for each satellite are:‘‘Gather’’ (Eq. (13)), ‘‘Avoid’’ (Eq. (14)), ‘‘Dock’’ (Eq. (15))and ‘‘Threat’’ (Eq. (3)) potentials. The equilibrium shapingformula will now require a solution of 18 equationssimilar to the one below in Eq. (18) (obtained by summingEqs. (13)–(15) and setting to 0), for each satellite i¼1–6and for 3 x–y–z dimensions.

cði,tÞndxði,tÞþdxði,jÞ bði,jÞnexp�:dxði,jÞ:

k2

� ��

þdxði,tÞ dði,tÞexp�:dxði,tÞ:

k3

� �� ¼ 0 ð18Þ

Once all the constant matrices are calculated, they aremade available to all the satellites for scatter manoeuvr-ing in this specific mission scenario (Strategy 3). Asmentioned before, when the ground station warning isreceived, each satellite uses these constants and therelative states of all the other satellites and the approach-ing threat to calculate its target state at every controlcycle and actuates it in the dynamic environment. In thesimulation, the warning was forecast by the groundstation 500 s before the threat was to hit one of the 6satellites and turned off 130 s after the warning wasturned on. Fig. 8 shows the trajectory of the satellites inblue and the trajectory of the threat (with an arrow fordirection) in green. The control profiles in Fig. 9 show thesatellites thrusting at maximum for almost the entireperiod when the warning was on. After it is switched off,they began thrusting again after a time lag in order tocome back to their original positions. The entire simula-tion lasted 430 s, the average delta-V spent per satellitewas 0.86 m/s and the minimum distance of closestapproach with the threat was 62 m. Again, had the threatvelocity been modelled such that the satellites escapedopposite to the direction of approaching threat instead of

Fig. 8. Trajectory of the hexagonal cluster in Strategy 3 when it moves

perpendicular to the approaching threat. The red circles show the




blue circles the final configuration. The arrow shows the direction of

approaching threat. (For interpretation of the references to colour in this


perpendicular, the delta-V spent for the same warningand higher simulation time (1000 s) is higher (1.3 m/s) fora minimum miss distance of 9 m. The trajectory is shownin Fig. 10.

An important consideration in this strategy is thatwhen the satellites scatter, they go hundreds of metresaway from each other as seen in Fig. 10, which could beorders of magnitude greater than the range that the inter-satellite communications are designed to operate at. Toeffectively use this strategy, the subsystems must bedesigned to account for this drop in bandwidth duringthe scatter manoeuvre or switch to a different commu-nication system if such an opportunity arises. Maintaininga good communication link is especially important forfractionated spacecraft where there may be elements thatcritically need other elements to operate. That this strat-egy also causes the satellites to fly out of each others’


sensing radii is not a hindrance to navigation because allsatellites are programmed to swarm back toward thepredefined centroid and avoid potentials will automati-cally kick in when they sense each other in close vicinity.

4.4. Strategy 4: Mix-and-match demonstration

This strategy demonstrates that the artificial potentialfunctions or even full individual strategies described in allthe previous sections can be combined in convenientways to design different target behaviours of swarms ofsatellites as per different mission requirements. Strategy4’s demonstration does not use any new strategy or APFand is simply Strategy 2 followed by the re-gather part ofStrategy 1. A four satellite rhombic formation is used. Thepotential functions (APF) used to calculate kinematictarget velocities are the attraction-repulsion potential ofEq. (1) and the threat potential of Eq. (3), (satellites areforced to move in a direction perpendicular to threat),when the warning is switched on and the Gather, Avoidand Dock potentials (Eqs. (13)–(15)), when the warning isdeactivated by the ground station. The simulated trajec-tory of the satellites is shown in Fig. 11. The allowedhead-time for the simulation is 300 s and the warning isturned off 500 s after being turned on. Total simulationtime was 2000 s, which gave an average delta-V of 0.9 m/sper satellite and a minimum distance of closest approachas 33 m for a threat approach velocity of 1 m/s.


perpendicular to the approaching threat. The red circles show the




Table 2.Comparison of the strategies described on the basis of different metrics (descr

Strategy # Average time (s)

Strategy 1 (scatterþgather) 3000þ1000

Strategy 2 (parallel) 900

Strategy 2 (perpendicular) 700

Strategy 3 (parallel) 1000

Strategy 3 (perpendicular) 430

Strategy 4 (mix and match) 2000

4.5. Summary of APF and ES based demo-strategies

The strategy to use in a particular scenario depends onthe mission requirements and the scenario itself ratherthan how the strategies perform with respect to eachother. The four scatter manoeuvre strategies discussedfrom Sections 4.1 to 4.4 are compared in Table 2. Strate-gies 2 and 3 has been divided into two parts: one formovement parallel to direction of threat and two, forperpendicular movement i.e., different vectors a in Eq. (3).The metrics considered in Table 2 are average time(averaged over all satellites) required to reconfigure andreturn to normal operations, average delta-V per satelliterequired for the scatter manoeuvre and the minimumdistance of closest approach of the threat with anysatellite. Since we have used the ground station informa-tion to turn the threat warning on and off and thesatellites have responded to this information (ExceptStrategy 1 where the satellites are not made aware ofthe threat parameters), the average time required for thereconfiguration depends on the amount of time thesatellites have to react to the threat (headtime) and thetime after which the GS switches off the warning (warn-ing off time). Similarly, the miss distance depends only onthe headtime and the direction of escape. Delta-V spentdepends on the average time required for the reconfigura-tion and the maximum thrust available to the satellites.Probability of success of the scatter manoeuvre approachwithout loss of long-term functionality can be calculatedas a function of the miss distance and the error ellipse ofthe threat destruction trajectory. Since a point threatmoving in a straight line has been assumed for thepurpose of the paper, the calculations for error ellipsesand therefore probabilities of success, although possible,are beyond the scope of the paper’s goals. Initial simula-tions showed that the strategies (apart from the gatherstrategy which is very delta-V expensive) depend on theinput parameters and mission scenarios, but in them-selves, are not very different from each other in terms ofresource expenses. Hence, for the purpose of demonstrat-ing the applicability of AP functions to determine satellitekinematics for collision avoidance manoeuvres in adynamic on-orbit environment, varying mission scenarioshave been selected and consistent performance has beenshown for appropriately selected strategies and APFs.

Mission requirements apart from the ones described inthis section are also possible by designing strategies torespond to them and appropriate APFs for the strategies.It is also possible to program the satellites with manymodular strategies and APFs such that when the ground

ibed in the text).

Average delta-V (m/s) Minimum miss distance (m)

1.8þ0.5 N/A (equal to circle of threat)

1.6 7 (for headtime¼200 s)






station issues the threat warning (approach informationof the kinetic impact of the threat), it will also issue acommand to use one or a combination of the strategiesalong with information on the required final geometricalconfiguration. Each satellite can solve the linear equationsystem of equilibrium shaping using the commandedAPFs and inter-satellite distances for the required geome-try, find the constant parameters and then use the con-stant parameters at every subsequent control cycle withthe relative states of the other satellites to find its targetstate. The ground station may also update the threattrajectory at any point, requiring a simple recalculationof APF constants as described above and in Section 3.1.

5. Conclusion and future work

Satellite clusters aim to achieve, similar to autonomousmulti-robot/agent-systems, increased robustness (by takingadvantage of parallelism and redundancy), as well asprovide the heterogeneity of structures and functionsrequired to undertake different activities in hazardous anduncertain environmental conditions such as those present inspace. They are designed to require minimal communicationfrom the ground and within the agents, to reduce computa-tion loads and save on time spent to find optimal solutionsvia a centralized agent. The case of a homogenous orheterogeneous satellite cluster, even the new paradigm offractionated spacecraft, reacting to a kinetic impact threathas been taken up in this paper. The models suggested inthis paper have demonstrated only position control of thesatellites. Additional attitude control can be easily imple-mented by solving the 7 ES equations (for position – x, y, z –and quaternion – q1, q2, q3, q4) per satellite instead of 3 foronly position – x, y, z – and then actuating both attitude andposition control.

The applicability of artificial potential functions tonavigate large satellite clusters in the event of an externalkinetic impact threat by scatter manoeuvring their ele-ments in a dynamic on-orbit environment (approximatedby the Hill’s frame in this paper) has thus been clearlydemonstrated using several strategies. The strategy andAPFs chosen would depend on mission constraintsbecause that influences the absolute and/or relative pla-cement of satellites required by the cluster. All thestrategies are scalable and APFs can be easily calculatedand equilibrium shaping control implemented for sym-metric geometries of tens or hundreds of spacecrafts. Asmall number, 4 and then 6, was chosen to demonstratethe concept and larger clusters would only require calcu-lation of a larger number of parameters in the equilibriumshaping equation. This will not affect the reaction time orCPU load after a threat is declared because they need to besolved only once, depending on the cluster and missionrequirements, and uploaded onto the satellites (or ifrequired, calculated within satellites) before they evenbegin scatter operations. Thus, by tweaking local para-meters, it is possible to exhibit intelligent emergentcoordination at the global level. For example, if thefractionated spacecraft has different functionalities (suchas wireless power transfer) embedded or faces a change inmission needs, the strategies can be adapted accordingly.

Since the equilibrium shaping technique is based onminimizing the virtual potential of each satellite in orderto bring them to a halt at the required relative configura-tion, the biggest limitation of the technique is its ten-dency to get stuck in local virtual potential minima whichneed not necessarily correspond to the correct geometry.This limits its application to very large satellite swarms inan asymmetric geometry since these are most susceptibleto local minima errors. For such systems, global evolu-tionary path planning techniques using artificial neuralnetworks, although never demonstrated in space, is the-oretically a better approach.

The equilibrium shaping technique for position andattitude control is currently being tested on the SPHERESsatellites aboard the International Space Station [16].Algorithms for a triangular formation of three satellitesstarting at random initial locations have been successfullydemonstrated on the SPHERES simulator. Hardware testsare scheduled to be run on the SPHERES nanosatellites inmicrogravity this year to demonstrate feasibility forsatellite clusters in the real space environment as wellas robustness to excess disruption and noise, not mod-elled in the simulation. Results from the SPHERES testsessions will be made available in a later publication.

Acknowledgements

The authors are grateful to Dario Izzo and JurgenLeitner at the Advanced Concepts Team (ACT), ESTEC, forvery useful discussions on the topic of the paper and toProf. Olivier de Weck at MIT for suggesting the scattermanoeuvre problem as an important one in distributedspace systems. The primary author would like to thankthe International Relations Department of ESA for makingher internship at ESTEC possible in the summer of 2010and for funding her research stay in the ACT. Specialthanks to the Taiwan National Space Organization and theNational Central University for funding the author’s travelto Taiwan to present the first draft of this paper at theInternational Workshop on Satellite Constellations andFormation Flying in November 2010.

References

[1] O. Brown, P Eremenko, The Value Proposition for FractionatedSpace Architectures, AIAA-2006-7506, San Jose, CA, 2006.

[2] D.M. LoBosco, G.E. Cameron, R.A. Golding, T.M. Wong, The Pleiadesfractionated space system architecture and the future of nationalsecurity space, presented at the AIAA Space, Anaheim, CA, 2008.

[3] Charlotte Mathieu, Assessing the Fractionated Spacecraft Concept,Massachusetts Institute of Technology, Cambridge, Massachusetts,USA, 2006.

[4] ESA Internal Report on Fractionated Spacecraft, European SpaceAgency, Noordwijk, The Netherlands, ESA/ESTEC contract 22258/09/NL/AF, February 2010.

[5] Proba Mission Page on the European Space Agency website.[Online]. Available: /http://www.esa.int/esaMI/Proba/SEMXZ5ZVNUF_0.htmlS.

[6] L. David, Space Debris: A Growing Challenge, American Institute ofAeronautics and Astronautics, Aerospace America, October 2009,pp. 30–36.

[7] L. Grego, Short History of US and Soviet ASAT Programs, Union ofConcerned Scientists, April 2003.

http://www.esa.int/esaMI/Proba/SEMXZ5ZVNUF_0.html

http://www.esa.int/esaMI/Proba/SEMXZ5ZVNUF_0.html


[8] A. Richards, T. Schouwenaars, J.P. How, E. Feron, Spacecraft trajec-tory planning with avoidance constraints using Mixed-IntegerLinear Programming, J. Guidance Control Dyn. 25 (4) (2002).

[9] Y. Kim, M. Mesbahi, F.Y. Hadaegh, Multiple-spacecraft reconfigura-tion through collision avoidance, bouncing, and stalemate, J. Optim.Theory Appl. 122 (2004) 323–343.

[10] J. Mueller, R. Larsson, Collision avoidance maneuver planning withrobust optimization, presented at the ESA Guidance, Navigationand Control Conference, Tralee, Ireland, 2008.

[11] R. Larsson, J. Mueller, S. Thomas, B. Jakobsson, P. Bodin, Orbitconstellation safety on the PRISMA in-orbit formation flyingtestbed, presented at the ESA Formation Flying Symposium, TheNetherlands, 2008.

[12] G.L. Slater, S.M. Byram, T.W. Williams, Collision avoidance forsatellites in formation flight, J. Guidance Control Dyn. 29 (5) (2006).

[13] J.B. Mueller, A multiple-team organization for decentralized gui-dance and control of formation flying spacecraft, presented at theAIAA 1st Intelligent Systems Technical Conference, Chicago, IL,2004.

[14] Shawn B. McCarnish, Marcello Romano, Simon Nolet, ChritineEdwards, David W. Miller, Ground and flight testing of multiplespacecraft control on spheres during close proximity operations, J.Spacecr. Rockets 46(6) (2009).

[15] J.G. Katz, A. Saenz-Otero, D.W. Miller, Development and demon-stration of an autonomous collision avoidance algorithm aboardthe ISS, in: Aerospace Conference, IEEE, 2011, pp. 1–6.

[16] Alvar Saenz-Otero, et al., Distributed satellite systems algorithmmaturation with SPHERES aboard the ISS, presented at the Inter-national Astronautical Congress, Glasgow, Scotland, 2008.

[17] T. Schetter, M. Campbell, D. Surka, Multiple agent-based autonomyfor satellite constellations, Artif. Intell. 145 (2003) 147–180.

[18] E. Bonabeau, M. Dorigo, G. Theraulaz, Swarm Intelligence: FromNatural to Artificial Systems, Santa Fe Institute Studies on theSciences of Complexity, 1999.

[19] S.A. Curtis, J. Mica, J. Nuth, G. Marr, M. Rilee, M. Bhat, ANTS(autonomous nano-technology swarm): an artificial intelligenceapproach to asteroid belt resource exploration, in: Proceedings ofthe International Astronautical Federation, 51st Congress, 2000.

[20] P D’Arrigo, S Santandrea, The APIES mission to explore the asteroidbelt, Adv. Space Res. (2006).

[21] G. Bonnet, C. Tessier, Multi-agent collaboration: a satellite con-stellation case, in: Proceedings of the Fourth Starting AI Research-ers’ Symposium, 2008.

[22] NanoSat: Development of a Highly-Capable, Modular MiniaturisedPlatform, CDF Study Report CDF-84(A), European Space Agency,Noordwijk, The Netherlands, 2009.

[23] V. Gazi, K. M. Passino, Stability analysis of social foraging swarms:combined effects of attractant/repellent profiles, in: Proceedings ofthe 41st IEEE Conference on Decision and Control, Las Vegas, NV,USA, 2002.

[24] V. Gazi, K.M. Passino, Stability analysis of swarms, IEEE Trans.Automat. Control 48 (4) (2003).

[25] V. Gazi, B. Fidan, Y.S. Sanay, Aggregation, foraging, and formationcontrol of swarms with non-holonomic agents using potentialfunctions and sliding mode techniques, Turk. J. Electr. Eng. 15 (2)(2007).

[26] M. Ayre, D. Izzo, L. Pettazzi, Self assembly in space using behaviourbased intelligent components, in: Proceedings of the TAROS:Towards Autonomous Robotic Systems, London, United Kingdom,2005.

[27] D. Izzo, L. Pettazzi, Autonomous and distributed motion planningfor satellite swarm, J. Guidance Control Dyn. 30 (2) (2007).

[28] Carlo Pinciroli, Mauro Birattari, Elio Tuci, Marco Dorigo, Evolving aCollective Consciousness for a Swarm of Pico-Satellites, AdvancedConcepts Team Ariadna Report Ariadna Final report ID 07/8101under contract 20765/07/NL/CB, European Space Agency, Noord-wijk, The Netherlands, 2008.

[29] W.H. Clohessy, R.S. Wiltshire, Terminal guidance system for satel-lite rendezvous, J. Aerosp. Sci. 27 (1960) 653–658.

[30] M. Tillerson, G. Inalhan, J. How, Co-ordination and control ofdistributed spacecraft systems using convex optimization techni-ques, Int. J. Robust Nonlinear Control 12 (1) (2002) 207–242.

[31] R.H. Battin, An Introduction to the Mathematics and Methods ofAstrodynamics, New York, USA, 1987.

[32] P. Lozano, D. Courtney, On the development of high specificimpulse electric propulsion thrusters for small satellites, in: Pro-ceedings of the Symposium on Small Satellite Systems and Services,Funchal Madeira, Portugal, 2010.

[33] The CubeSAT Propulsion module from Delft University. [Online].Available: /http://www.lr.tudelft.nl/live/pagina.jsp?id=b01f8765-4b61-4c5a-8f20-96e5871934f5&lang=enS.

[34] K. Sarda, C.C. Grant, S. Eagleson, D.D. Kekez, R. Zee, Canadianadvanced nanospace experiment 2 orbit operations: two years ofpushing the nanosatellite performance envelope, in: Proceedings ofthe Symposium on Small Satellite Systems and Services, FunchalMadeira, Portugal, 2010.

[35] Jian Ma, Formation Flying of Spacecrafts for Monitoring andInspection, SpaceMaster Master Thesis, Lulea University of Tech-nology, Department of Space Science, Kiruna, 2007.

[36] D. Izzo, L. Pettazzi, Equilibrium shaping: distributed motion plan-ning for satellite swarm, in: Proceedings of the 8th InternationalSymposium on Artificial Intelligence, Robotics and Automation inSpace, 2005.

http://www.lr.tudelft.nl/live/pagina.jsp?id=b01f8765-4b61-4c5a-8f20-96e5871934f5&lang=en



Documents

Behaviour based, autonomous and distributed scatter manoeuvres for satellite swarms