IAC-12,C1,3,3,x13767

CLUSTER-KEEPING ALGORITHMS FOR THESAMSON PROJECT

Leonel Mazal,and Pini GurlyTechnionIsrael Institute of Technology, Haifa 32000, Israel

Abstract

Space Autonomous Mission for Swarming and Geolocation with Nanosatellites (SAMSON)is a new satellite mission, led by the Distributed Space Systems Lab at the Technion Is-rael Institute of Technology. SAMSON will include three nanosatellites, built based on theCubeSat standard. The mission is planned for at least one year, and has two main goals: (i)Demonstrate long-term autonomous cluster ight of multiple satellites, and (ii) Determine theposition of a radiating electromagnetic terrestrial source based on time dierence of arrival(TDOA) and/or frequency dierence of arrival (FDOA). In this paper, the cluster ight orbitcontrol law for SAMSON is discussed. The control law is aimed at regulating the mean semi-major axis, eccentricity and inclination so as to provide long-term bounded relative motionwhile utilizing small amount of fuel. The considered actuators are constant-thrust-magnitudethrusters. Simulations for 1 year are shown, validating the potential implementability of theproposed algorithm on-board the SAMSON satellites.

I. Introduction

Disaggregated satellites constitute an emergingconcept in the realm of distributed space sys-tems. The main idea is to replace a monolithicsatellite by multiple free-ying, physically-separated modules interacting through wirelesscross-links. The disaggregated satellites con-cept enables novel space architectures that canpotentially outperform the conventional mono-lithic architectures.

Unlike traditional satellite formation yingmissions, in disaggregated systems, the mod-ules do not necessarily have to operate in atightly-controlled formation; instead, they arerequired to maintain the inter-module distances

bounded (typically between 100m and 100 km)for the entire mission lifetime. This concept istermed cluster ight [1].

One of the rst cluster ight demonstra-tion missions is the Space Autonomous Missionfor Swarming and Geolocation with Nanosatel-lites (SAMSON). The SAMSON project is ledby the Distributed Space Systems Lab (DSSL)at the Technion[2]. It includes the rst-evernanosatellite trio, and has two main goals:(1) demonstrate long-term autonomous clusteright and (2) geolocate a radiating electromag-netic terrestrial source based on time dier-ence of arrival (TDOA) and/or frequency dif-ference of arrival (FDOA). The 3 nanosatel-lites will be launched into a low Earth orbit

Doctoral Student, Distributed Space Systems Lab, Faculty of Aerospace Engineering. email: leo-mazal@techunix.technion.ac.il

yAssociate Professor, Distributed Space Systems Lab, Faculty of Aerospace Engineering. email: pgur-l@technion.ac.il

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(LEO) together, to form an autonomous clus-ter, which is required to hold inter-satellite dis-tances ranging from 100 m to 250 km. Thenanosatellites will perform relative orbital ele-ment corrections to satisfy the relative distanceconstraints based on GPS measurements. Theplanned lifetime for the mission is 1 year. Eachof the three nanosatellites will be equipped witha single cold-gas thruster, providing a bang-o-bang thrust on the order of tens of millinewtons.

Without control forces, initially-close satel-lites will drift apart due to dierential acceler-ations. It is thus imperative to derive eectivecontrol strategies for keeping the cluster opera-tional with limited fuel budgets. In the absenceof perturbations, equal semimajor axes guar-antee bounded motion[3]. But in more realis-tic scenarios, the problem is more complicated.The most signicant perturbations aecting lowEarth-orbit satellites are the Earth oblatenessand drag. Many works presented strategies tomitigate or avoid relative drifts among satel-lites subject to the main perturbations see e.g.[4, 5, 6, 7] and references therein. Mazal andGurl[8] recently derived constraints on the rel-ative states, which ensure bounded motion un-der zonal harmonics and drag, in the context oflong-term cluster ight. Ref. [8] also introducedan algorithm to steer a cluster to the desiredrelative states assuming impulsive cooperativemaneuvers. Mazal, Mingotti and Gurl intro-duced the rst cooperative optimal low-thrustconstant-magnitude guidance law for cluster-establishment, considering bang-o-bang thrustproles[9].

Since the SAMSON satellites are equippedwith constant-magnitude thrusters, the cur-rent work presents a cluster-keeping algorithmdesigned considering these harsh technologicalconstraint. This closed-loop control algorithmis described and assessed. Although the actualon-board control algorithm for SAMSON hasnot been nalized yet, it will likely be based onthe principles described throughout the paper.

II. Global DecentralizedCluster-Keeping Strategy and

Algorithm

Consider a cluster composed of N satellites. Itis required to hold the distances between any

two satellites i and j within prescribed upperand lower thresholds, i.e Dmin dij(t) Dmax.dij(t) , krj(t) ri(t)k denotes the distance be-tween the satellites i and j at any time t, andri and rj represent the position vectors of satel-lites i and j respectively. Hence, the controllogic of the entire cluster should simultaneouslymonitor N (N 1) =2 inter-satellite distances.Most approaches on formation and cluster ightdeal with only two satellites, and thus a singledistance to be considered. However, if N = 3 asin SAMSON, the problem becomes more com-plex as the number of distances to simultane-ously control is three, d12(t); d23(t); d31(t).

Consider two satellites, i and j, orbiting theEarth, under natural perturbations. If they areinitially moving in proximity, the inter-satellitedistance dij(t) will exhibit a pattern similar tothat illustrated in Fig. 1.

0 5 10 150

500

1000

1500

2000

2500

3000

Time [days]

d [k

m]

Figure 1: Secular behavior of uncontrolled inter-satellite distance.

In Fig. 1, one can distinguish a secular be-havior superimposed with oscillations. The sec-ular behavior reaches a minimum of s 0 km.The distance reaches a maximum at s 14000km (in this example the semimajor axes of thesatellites are s 7000 km). The maxima reachedwhen when the angle between the position vec-tors of the satellites, denoted by , satises s 180. Conversely, the minima are reachedat times in which s 0.

The distance between the satellites can bewritten as

dij(t) =qr2i (t) + r

2j (t) 2 ri(t) rj(t) cos (t)

(1)

2

where ri = krik and rj = krjk. For operationalclusters in LEO, ri and rj are both boundedfrom above and below as ai(t) (1 ei(t)) ri(t) ai(t) (1 + ei(t)) and aj(t) (1 ej(t)) rj(t) aj(t) (1 + ej(t)). Yet, dij(t) can attainvery large or very small values if the angle (t) isnot controlled. Furthermore, for practical sce-narios of cluster ight, it holds that

r(t)

ri(t)=

rj(t) ri(t)ri(t)

1 (2)

Expanding Eq. (1) into a rst-order Taylor se-ries about rri = 0 yields

d(t) ' ri(t)p2p1 cos ((t))

1 +

1

2

r(t)

ri(t)

(3)

which shows the critical inuence of on therelative distance.

The main cause for the motion to seemlocally-unbounded is a dierence in the semi-major axes of the satellites, aij , ai aj 6=0. Indeed, even in the unperturbed two-bodyproblem, dierent semimajor axes lead to sec-ular drifts of the distance between the satel-lites. For the perturbed two-body problem,the scenario is more complex, but still holdingaij , aiaj > 0 yields locally-unbounded sec-ular behavior of dij(t). In the presence of drag,dierent semimajor axes evolutions would af-fect the rate of the secular behavior. Generallyspeaking, for cluster ight purposes, these sec-ular drifts must be avoided. However, they canalso be exploited to hold the distances dij(t)between the satellites of a cluster within giventhresholds, as will be seen in the sequel.

To proceed, let (oe) denote the mean value(secular component) of the orbital element (oe).It is well known that for mitigating the secu-lar growth of the dierential orbital elementsij , i j , !ij , !i !j , and M ij ,M i M j due to the J2 eect, one could setaij = ai aj = 0, eij , ei ej = 0, andiij , ii ij = 0[3]. However, in the presenceof other eects such as dierential drag, aij isaected, leading to aij ? 0 and thus a sec-ular drift in the distance takes place. Assumethat aij > 0, which implies the time that takessatellite i to complete one revolution is longerthan that of satellite j. Thus, if satellite i is be-hind satellite j, they will get farther from each

other, i.e. the angle will grow. However, ifdue to a maneuver, sgn(aij) is inverted, satel-lite i will complete one revolution faster thansatellite j. Consequently, they will start get-ting closer.

2.1 Global Strategy

The main strategy of the cluster control logicconsists of performing corrective maneuvers,when at any time t = ti the inter-satellite dis-tance between any two satellites of the cluster,dij(ti), reaches the upper or lower threshold.Recalling the secular behavior seen in Fig. 1,unless sgn(aij) is switched, it is expected thatdij(t) 8t > ti will continue increasing. Thus,a control action aimed at switching sgn(aij)must be performed. In case that ai(ti) > aj(ti),the control logic performs a maneuver so thata+i < a

+j , where the superscript (oe)

+ denotesthe value of the orbital element (oe) immedi-ately after the maneuver. Fig. 2 illustrates theidea, for an upper bound of 70 km.

0 2.31 4.62 6.93 9.260.1

0.05

0

0.05

0.1

a [K

m]

0 2.31 4.62 6.93 9.260

20

40

60

80

Dist

ance

[Km]

Time [Days]

Figure 2: Maneuvering switching the secular be-havior of dij(t).

The dierences in the semimajor axes afterthe maneuver, a+ij = a

+i a+j , are intended

to be slightly dierent from zero, to properlyaect the secular behavior of dij . However,they should not be too large, because the largerja+ij j, the steeper the rate of the secular-likebehavior. The goal of each collective maneuverwill be to generate proper dierences betweenthe semimajor axes of the three satellites, suchthat the distances between any pair of satellites

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are held within the given range.When the cluster is composed of more than

two satellites, the issue is more involved, be-cause changes in one of the semimajor axes,say ai, aects the behavior of the distances be-tween every pair of satellites involving satellitei, dij(t) 8 j 6= i. For SAMSON, before per-forming a maneuver, a control logic properlysets a+i 8i = 1 : : : 3, to obtain the desired be-havior of dij(t) 8 i; j after the maneuver.

2.2 Dening Terminal StatesThis section shows how the terminal states ofthe satellites are determined, according to thestates of the three satellites at t = ti. The ter-minal states are the orbital elements that thesatellites are steered to during the maneuver.Each time a maneuver is required this proce-dure takes place, setting the terminal states e+,i+, and a+k k = 1; 2; 3.

Assume that at ti one of the three dis-tances reached either the upper or the lowerbound, and that ak(ti) are known. Withoutloss of generality, assume that jdij(ti)DM j >jdjk(ti) DM j > jdki(ti) DM j, where DM ,(Dmax +Dmin) =2. Hence, it must hold thatdij(ti) reached either the upper or the lowerbound. Therefore, the rst priority is to cor-rect the secular behavior of dij , for which a con-straint is imposed to switch the semimajor axesai and aj as follows.(

if ai(ti) > aj(ti) =) a+i < a+jif ai(ti) < aj(ti) =) a+i > a+j

(4)Modifying ai and aj will also aect the behav-ior of dik and djk. Since jdjk(ti) DM j >jdki(ti) DM j, djk(ti) is closer to one of thethresholds than dki(ti). Therefore, the secondpriority is to generate the proper secular behav-ior of djk(t). To that end, the logic given in (6)yields a constraint between a+j and a

+k , where

_djk denotes the secular rate of change of thedistance djk.

With the constraints imposed to the pairs(a+i ; a

+j ) and (a

+j ; a

+k ), it might suce or not

to completely sort the three terminal semima-jor axes. For instance, if it was obtained thata+i > a

+j and a

+j > a

+k , then one concludes

that a+i > a+j > a

+k . However, if a

+i > a

+j and

a+j < a+k is obtained, either a

+i > a

+k > a

+j or

a+k > a+i > a

+j are admissible. Thus, only in

case that with the two constraints, (4) and (6),the terminal semimajor axes cannot be com-pletely sorted because there are still two op-tions, the logic (7) establishes a constraint be-tween a+k and a

+i .

Applying the logic described in (4), (6), and(7), the terminal semimajor axes are properlysorted, such that it prevents the most criticaldistance dij from exceeding the allowed lowerand upper bounds, as well as moving djk anddki away from the bounds.

To obtain slight dierences in the semima-jor axes, a+ij 8 i; j , the control logic proceedsin the following manner. First (oe)p is denedand computed as

(oe)p ,max

(oe)1(ti); (oe)2(ti); (oe)3(ti)

2

+

min(oe)1(ti); (oe)2(ti); (oe)3(ti)

2

(5)where (oe) stands for a, e, and i. Then, accord-ing to how the terminal semimajor axes weresorted, the satellite required to have the highestsemimajor axis is steered to a+ = ap +a; thesatellite required to have the lowest semimajoraxis is steered to a+ = ap a; and the satel-lite required to have a semimajor axis in be-tween the former is steered to a+ = ap. a is apositive user-dened parameter that aects thelong-term performance of the algorithm. Thelarger a, the steeper are the secular drifts.However, if a is small, due to the tolerances ofimplementation, it might occur that in the endof the maneuver the constraints (4), (6), and(7) are not satised.

On the other hand, the eccentricities ek aresteered to e+ = ep and the inclinations ik aresteered to i+ = ip, k = 1; 2; 3.The mean incli-nations of the satellites are steered to an equalterminal state to avoid large out-of-plane sepa-rations.

4

8>>>>>>>:if djk(ti)DM > ( ( ak(ti) =) a+j < a+kif djk(ti)DM > ()0 and aj(ti) > ak(ti) =) a+j > a+kif djk(ti)DM > ( ( a+kif djk(ti)DM > ()0 and aj(ti) < ak(ti) =) a+j < a+k

(6)

8>>>>>>>:if dki(ti)DM > ( ( ai(ti) =) a+k < a+iif dki(ti)DM > ()0 and ak(ti) > ai(ti) =) a+k > a+iif dki(ti)DM > ( ( a+iif dki(ti)DM > ()0 and ak(ti) < ai(ti) =) a+k < a+i

(7)The rationale behind the computation of

(oe)p is to minimize the largest required ele-ment variation, i.e.

(oe)p = min(oe)

max

24j(oe)1(ti) (oe) jj(oe)2(ti) (oe) jj(oe)3(ti) (oe) j

35T (8)2.3 Control Law for Reaching the Terminal StatesAssume that at time t = ti the satellite khave mean elements ak(ti), ek(ti) and ik(ti),k = 1; 2; 3. It is desired to steer the satellitesuch that at the end of maneuver it reaches a+k ,e+ and i+.

Dene the satellite-xed frame Vk as fol-lows: The X^k direction lies along the instan-taneous velocity vector vk, the Z^k directionlies along the instantaneous angular momentumvector hk = rkvk, and Y^k completes the right-handed triad.

Assume that each satellite is equipped witha single engine that exerts constant-magnitudethrust Tk. Dening k and k as the respectiveright ascension and declination angles, with re-spect to the frame Vk, the thrust vector is re-solved in Vk as

Tk = Tk

24cos(k) cos(k)sin(k) cos(k)sin(k)

35 (9)

To design a control law for k and k that steersthe system to the desired state, the followingLyapunov function is dened:

Vk =1

2

24ak a+kek e+ik i+

35T 24ka 0 00 ke 00 0 ki

3524ak a+kek e+ik i+

35(10)

Dierentiating Vk yields

_Vk = ka(ak a+k ) _ak + ke(ek e+)_ek + ki(ik i+)_ik

(11)For any orbital element (oe), the time

derivative of its secular behavior _(oe) can bewritten as

_(oe) = _(oe)p +

_(oe)c (12)

where the sub-index ()p denotes variations dueto natural perturbations and ()c denotes vari-ations due to control actions.

It is important to stress that the maneu-vers usually last for short time intervals, shorterthan an orbital period, and thus the eect ofperturbations during the maneuver is small.Yet, the eects due to Earth oblateness (mainlythe J2 term) are taken into account during themaneuvers. Under J2 perturbations, _akp =_ekp =

_ikp = 0. Thus, the terms remaining to

be computed are _(oe)c. According to Ref. [10]the eect of the control thrust on the mean el-ements can be approximated by the eect ofthe same thrust on the corresponding osculat-ing elements. The Gauss Variational Equations

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(GVE) resolved in the frame V are given by[11]

_ak =2 a2k vkmk

Tk cos(k) cos(k) (13)

_ek =Tk cos(k)

vkmk2 (ek + cos fk) cos(k)

Tk cos(k)vkmk

rkak

sin fk sin(k)

(14)

_ik =rk

hkmkcos (fk + !k)Tk sin(k) (15)

where vk is the norm of the inertial velocity vec-tor,mk is the mass, fk is the true anomaly, !k isthe argument of perigee, and hk is the norm ofthe angular momentum vector, all correspond-ing to satellite k.

Introducing Eqs. (13)-(15) into Eq. (11)yields

_Vk =Tkm

"1k cosk cos k + 2k sink cos k

+ 3k sin k

#(16)

where

1k = ka(ak a+k )2 a2k vk

+ ke(ek e+)2 (ek + cos fk)vk

(17)

2k = ke(ek e+)rk sin fkak vk

(18)

3k = ki(ik i+)rk cos (fk + !k)hk

(19)

In order to guarantee that _V will be alwaysnegative at any point dierent from the desiredone, and are selected as

(k; k) = arg min

(k; k)

241k2k3k

35T 24cosk cos ksink cos ksin k

35(20)

It can be proved that for any value of 1k, 2k,and 3k, the minimum of _V will be negative,unless 1k = 2k = 3k = 0 in which case_Vk = 0. However, 1k, 2k, and 3k will si-multaneously vanish and remain zero only for(ak a+k ) = (ek e+) = (ik i

+) = 0, which is

the desired terminal condition.

The global minimum of _Vk is found eitherat8>>>:

1k =arctan2k1k

1k =arctan

3k

1k cos1k + 2k sin1k

(21)

or at (2k =

1k +

2k = 1k(22)

For given 1k, 2k, 3k, if the mini-mum is found at

1k ;

1k

and it attains a

value _Vk(1k ; 1k ) < 0, then the maximum is

found at2k ;

2k

attaining _Vk(2k ;

2k ) =

_Vk(1k ; 1k ) > 0.

2.3.1 Implementation Issues

Actual implementation of this control law re-quires to dene tolerances to terminate the ma-neuver. Generally speaking, these tolerancesgenerate a trade-o between the accuracy in theresults of the maneuver and required fuel. Theuser should select them in such a way that theygenerate good performances without demand-ing too large fuel amounts for each maneuver.

In the control strategy of the whole cluster,the terminal desired elements a+k , e

+, i+ arecomputed at time t = ti, once the necessity ofthe maneuver was detected. Since then, eachsatellite is required to maneuver until certainterminal conditions are achieved. Two dier-ent approaches can be considered to dene thetermination of the maneuver of each satellite.

The rst approach is that each satellite k ex-ecutes its maneuver until it achieves jaka+k j a

+3 > a

+2 . The maneuver achieves this

requirement and the cluster is held within thebounds. At ti = 212:4 days a new maneuveris required as d12(ti) reaches the lower thresh-old. Before the maneuver, the semimajor axeswere a1(ti) = 6969:454 km, a2(ti) = 6969:443km, and a3(ti) = 6969:447 km. According to(4)-(7), the terminal semimajor axes after themaneuver should satisfy a+2 > a

+1 > a

+3 . The

new maneuver achieves this requirement andthe cluster is held within the bounds. Afterthe two corrective maneuvers, the total massconsumption was 10:5 grams. For this exam-ple the propagation of the satellite orbits be-tween maneuvers was performed by using anastrodynamical model including gravitationalgeopotential up to degree and order 21, dragaccording to the Jacchia-Roberts model, so-lar radiation pressure, and lunisolar attraction.The cross-sectional area of the satellites was as-sumed to be 0:1 m2.

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III. Conclusions

The cluster ight control method presentedherein is simple, which makes it feasible for on-board autonomous implementation. The orbitcontrol law for each satellite is robust in thesense that even if the thrust level were lowerthan the nominal value, it would take longerto reach the required terminal conditions, buteventually they would be attained. Even dier-ent ballistic coecients would not represent ahard obstacle for the algorithm, as these pos-sible dierences would simply increase the fre-quency of the corrective maneuvers. Since thecontrol strategy does not rely on very accurate

targeting of the terminal conditions, unmodeledperturbations can still be rejected. Finally, theproposed algorithm could be easily extended toa larger number of satellites.

IV. Acknowledgements

This work was supported by the EuropeanResearch Council Starting Independent Re-searcher Grant - 278231: Flight Algorithms forDisaggregated Space Architectures (FADER),the Ministry of Science of the State of Israel,and the Technion Graduate Fellowship Pro-gram.

Table 1: Initial orbital elements for satellite 1 (row 1), 2 (row 2), and 3 (row 3).

Satellite a [km] e [-] i [deg] [deg] ! [deg] f [deg] m [kg]

#1 @ (t0) 6978:966 0:001704 51:422 293:492 62:185 273:643 8

#2 @ (t0) 6978:952 0:001665 51:422 293:489 63:813 272:008 8

#3 @ (t0) 6979:016 0:001253 51:427 293:475 75:330 260:951 8

Table 2: Initial orbital elements for satellite 1 (row 1), 2 (row 2), and 3 (row 3).

Satellite a [km] e [-] i [deg] [deg] ! [deg] f [deg] m [kg]

#1 @ (t0) 6979:984 0:001717 51:427 293:489 57:302 286:349 7:98

#2 @ (t0) 6979:997 0:001688 51:427 293:486 58:785 284:857 7:98

#3 @ (t0) 6980:023 0:001550 51:428 293:474 69:065 275:033 7:98

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IV.References

[1] Brown, O. and Eremenko, P., Fraction-ated Space Architectures: A Vision forResponsive Space, 4th Responsive SpaceConference, Los Angeles, CA, 2006.

[2] Gurl, P., Herscovitz, J., and Pariente, M.,The SAMSON Project - Cluster Flightand Geolocation with Three AutonomousNano-satellites, 26th AIAA/USU Confer-ence on Small Satellites, Salt Lake City,UT, 2012.

[3] Alfriend, K., Vadali, S., Gurl, P., How, J.,and Breger, L., Spacecraft Formation Fly-ing: Dynamics,Control and Navigation,Elseiver, Oxford, UK, 2010, Chapter 2, pp.24, 35-36.

[4] Vadali, S. R., Schaub, H., and Al-friend, K. T., Initial Conditions and Fuel-Optimal Control for Formation Flying ofSatellites, AIAA Guidance, Navigation,and Control Conference and Exhibit , Port-land, OR, 1999.

[5] Schaub, H. and Alfriend, K., J2 Invari-ant Relative Orbits for Spacecraft Forma-tions, Celestial Mechanics and DynamicalAstronomy , Vol. 79, No. 2, February 2001.

[6] Mishne, D., Formation Control of Satel-lites Subject to Drag Variations and J2

Perturbations, Journal of Guidance, Con-trol and Dynamics, Vol. 27, No. 4, July-August 2004, pp. 685692.

[7] Beigelman, I. and Gurl, P., OptimalFuel-Balanced Impulsive Formationkeep-ing for Perturbed Spacecraft Orbits, Jour-nal of Guidance, Control and Dynamics,Vol. 31, No. 5, September-October 2008,pp. 12661283.

[8] Mazal, L. and Gurl, P., Cluster FlightAlgorithms for Disaggregated Satellites,Journal of Guidance, Control and Dynam-ics, Vol. to appear, 2012.

[9] Mazal, L., Mingotti, G., and Gur-l, P., Continuous-Thrust CooperativeGuidance Law for Disaggregated Satel-lites, AIAA/AAS Astrodynamics Special-ist Conference, Minneapolis, MN, 2012,paper AIAA-2012-4742.

[10] Schaub, H. and Junkins, J., Analytical me-chanics of space systems, Vol. 1, Aiaa,2003, Chapter 14, pp. 731.

[11] Battin, R., An Introduction to the Math-ematics and Methods of Astrodynamics,Revised Edition, AIAA Education Series,1999, Chapter 10, pp. 489.

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