autonomous guidance and control of earth-orbiting formation flying spacecraft: closing the loop

Acta Astronautica 63 (2008) 1246–1258www.elsevier.com/locate/actaastro

Autonomous guidance and control of Earth-orbiting formation flyingspacecraft: Closing the loop

Jean-Francois Hamel∗, Jean de LafontaineDepartment of Electrical Engineering, Universite de Sherbrooke, Sherbrooke, Canada J1K 2R1

Received 5 December 2007; received in revised form 7 April 2008; accepted 6 May 2008Available online 11 July 2008

Abstract

Previous work on autonomous formation flying guidance and control identified three key challenges to overcome in orderto obtain a fully autonomous guidance and control loop: an accurate but simple model of relative motion about elliptical andperturbed orbits, an efficient way of performing conflicting requirements trade-off with power-limited on-board computers, andfinally an optimal or near-optimal control algorithm easy to implement on a flight computer. This paper first summarizes recentdevelopments on each of these subject that help to overcome these challenges, developments which are then used as buildingblocks for an autonomous formation flying guidance and control system. This system autonomously performs trade-offs betweenconflicting requirements, i.e. minimization of fuel cost, formation accuracy and equal repartition of the fuel expenditure withinthe formation. Simulation results show that a complete guidance and control loop can be established using mainly analyticalresults and with very few numerical optimization which facilitates on-board implementation.© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Formation flying will be a major trend in the up-coming years in space exploration. Indeed, replacing“conventional” large and expensive monolithic space-craft by several smaller and cheaper spacecraft presentsseveral advantages.

Firstly, formation flying presents operational advan-tages, such as mission reconfiguration capability and ro-bustness to system failures through failure recovery andgraceful degradation. In deep-space missions using mul-tiple spacecraft in formation, if a sub-system failure oc-curs in a spacecraft, another fully functional spacecraftcould support the disabled spacecraft. The capabilities

∗ Corresponding author.E-mail address: [email protected]

(J.-F. Hamel).

0094-5765/$ - see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.actaastro.2008.05.015

can be shared. For example, when a power, com-munication or navigation system failure occurs in aspacecraft, it may be possible to use another spacecraftsub-system either by physically linking the space-craft or by transmitting navigation information to thefailed spacecraft. In the case of a distributed spacecraftinterferometer or a distributed antenna mission, thefailure of one spacecraft would only cause a “gracefuldegradation” of the system, rather than compromisingthe whole mission. The second operational advantageis a mission restructuring capability. It is foreseeableto reconfigure the satellite formation on-orbit to follownew mission requirements. Moreover, if the missionhas multiple objectives, resources can be optimized bydispatching a certain group of spacecraft having specialattributes to achieve one objective, and then commandanother group of spacecraft to achieve another objectivein parallel.

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J.-F. Hamel, J. de Lafontaine / Acta Astronautica 63 (2008) 1246–1258 1247

Moreover, formation flying presents financial ad-vantages. Such a mission can potentially have a lowerproduction cost due to economics of scale, in thecase where a single large and complex satellite is re-placed by several “mass production” smaller spacecraft.Secondly, using a constellation of spacecraft could de-crease the cost of launch. Launching several smallerelements is potentially cheaper than launching a singlebig and heavy satellite, mainly because small satellitescan be launched piggy backed on a larger spacecraftflight support equipment.

However, using a formation of spacecraft involvesseveral challenges. The main one is an increase of therequired level of autonomy. In order to minimize the re-sources needed for ground support, it is required to limitthe command inputs to the system to high-level com-mands to the whole swarm of spacecraft. The swarmof spacecraft would then have to autonomously definelower-level commands for each of the spacecraft. How-ever, these low-level commands would have to remainfuel-optimal to maximize the lifetime of the whole for-mation. Moreover, if the consumption of fuel is not wellbalanced between spacecraft, some spacecraft could runout of fuel before other ones, and cause a prematuredegradation of the performance of the system.

This paper follows a paper presented in 2006 [1]where the main challenges in achieving a completelyautonomous formation flying guidance and control sys-tem were presented. In order to be autonomous andimplemented on-board, such guidance and control algo-rithms have to require as few computation as possible (tofacilitate implementation on limited-capacity on-boardcomputers) and require no (or few) inputs from theground segment. This paper summarizes recent devel-opments which solve some of the problems highlightedin the first paper and describes a new way of usingthese tools in a completely autonomous guidance andcontrol loop.

Section 2 summarizes recent developments in rela-tive motion theory that lead to an analytical model ofrelative motion for J2-perturbed elliptical orbits. Then,Section 3 presents the “Fuel-Equivalent Space”, a toolused to rapidly and analytically predict the fuel costof maneuvers and/or rapidly identify the most fuel-efficient way to attain a given formation. Section 4summarizes the results of the neighbouring optimumcontroller, a method for autonomous formation flyingcontrol. Section 5 describes how all these tools can bestitched together in a formation flying guidance andcontrol system that autonomously performs trade-offsbetween the formation flying conflicting requirementsusing a typical projected-circular formation (PCF) as

an example. Finally, Section 6 shows simulation resultsand demonstrates how a trade-off between conflictingrequirements can be done in a typical formation flyingreconfiguration problem.

2. Relative motion theories

Autonomous guidance and control systems requireaccurate but simple models of reality in their algorithms.Models have to be accurate enough to prevent unnec-essary fuel expenditure, but simple enough to allowon-board implementation. If perturbation models areincluded in the on-board model of reality, natural mo-tion induced by the perturbations can be used to sup-port maneuvers. If these perturbations are not included,the guidance and control system will most likely com-pensate for these perturbations, therefore leading to anunnecessary fuel expenditure.

The most widely used relative orbital motion modelis the Clohessy–Wiltshire–Hill model [2–4]. It providesa time-explicit closed-form analytical solution to rel-ative motion problem for circular unperturbed orbits.This model provides significant insights into the natu-ral relative motion about circular and unperturbed ref-erence orbits. However, assuming a circular referenceorbit yields considerable errors when the eccentricityof the reference orbit grows [5]. Several models havetherefore been proposed to model relative motion aboutunperturbed elliptical orbits [5–8]. In a recent publica-tion, Lane and Axelrad [9] developed a time-explicitclosed-form solution and studied the relative motionfor bounded relative elliptical orbits. Melton [10] alsoproposed an alternative solution for small-eccentricityorbits.

However, these models do not take into account or-bit perturbations. The most important perturbation en-countered for the relative motion problem, and also themost studied, is the perturbation caused by the oblate-ness of the Earth, referred to as the J2 perturbation.Schweighart and Sedwick [11,12] modified the clas-sic Clohessy–Wiltshire–Hill model to include the orbit-averaged impact of the J2 perturbation on a circularreference orbit. The most challenging problem is never-theless to consider an elliptical and perturbed referenceorbit. The most accurate way to model this problem isof course with numerical models [13,14]. In this case,solutions to the relative motion problem are obtainedthrough numerical integration of the dynamics equa-tions. However, numerical methods are not well suitedfor autonomous on-board applications because they typ-ically require large computing effort. Few publicationsactually provide an analytical solution to the relative

1248 J.-F. Hamel, J. de Lafontaine / Acta Astronautica 63 (2008) 1246–1258

motion around elliptical reference orbits taking into ac-count the J2 perturbation.

Gim and Alfriend [15] solve the problem by propos-ing a state transition matrix that provides a time-explicitsolution for the relative motion about a J2-perturbed el-liptical orbit. This model provides an accurate solutionto the problem. However, even though the model is fullyanalytical, the elements of the state transition matricesremain quite complex, the states of the reference tra-jectory still need to be numerically computed and ma-trix products and inversions remain. On the other hand,Schaub studied the relative motion about elliptical ref-erence orbits under J2 perturbation with very simpleexpressions using classical orbit elements [6]. However,this analysis is only performed in the mean orbit ele-ments space. This model cannot be written readily intoa state transition matrix form as the mapping betweeninstantaneous, or osculating, orbit elements remains tobe done.

Some recent work by the authors led to an analyticalstate transition matrix that accurately models relativemotion about elliptical reference orbits under J2 per-turbation, while using simpler expressions and withoutthe need to numerically propagate the states of the ref-erence trajectory [16,17]. It builds upon the approachof Schaub [6], but bridges the gap between osculatingrelative motion and relative mean orbit element drift.Desired formation relative dynamics are typically com-manded in terms of osculating, or “actual”, relativedynamics, which is why it is relevant to describe therelative motion in terms of osculating elements insteadof mean elements. This simplified model is oriented to-ward an on-board implementation for mission scenarioswhere computational power is limited, such as low-costand low-power scientific missions.

The proposed model uses a geometric approach, sim-ilar to the work of Gim–Alfriend [15] but with somesimplifying assumptions. The model neglects variationsin the short-periodic relative motion induced by the J2perturbation between the deputy and the chief, but in-cludes a osculating to mean orbit elements mapping. Inother words, it “adds” the relative mean orbit elementdrift to the natural osculating-element Keplerian dynam-ics, neglecting the impact of short-period variations onthe relative motion. This simplification is made at thecost of a prediction error as large as the short-periodicterms variations between the deputy and the chief. Fortwo spacecraft orbiting very close from one another, thiserror will remain small as the short-period oscillationscaused by the J2 perturbation will be nearly the samefor both spacecraft. However, in all cases, this error willremain bounded even for long-term prediction.

The main advantage of this approach is that the statesof the reference trajectory at the true anomaly where therelative dynamics need to be known are not required. Allthe elements of the state transition matrix are computedfrom the initial position of the reference trajectory andthe true anomaly for which the relative motion needs tobe predicted. Models that take into account short-periodvariations [15] need the states of the reference at thefinal time to do an accurate mapping between the meanelements and the osculating elements at this location.

The main use of such a model in the guidance andcontrol loop is to predict which relative state vector �e0is required to reach without any control effort the rela-tive Hill coordinates �X at a given true anomaly �, giventhe state vector of the reference trajectory (sometimesreferred to as “leader”):

�e0 = �(e0, �)−1�X(�) (1)

where the elements of the state transition matrix � as-sume a J2-perturbed elliptical reference orbit [16,17].For circular reference orbits, other models are bettersuited [11,12].

This state transition matrix is the first building blockof the autonomous guidance and control system pre-sented here, as it analytically provides which relativeorbit elements are required to “naturally” reach a de-sired set of Hill coordinates at any given true anomaly.

3. Fuel-equivalent space

Another recently developed tool, the fuel-equivalentspace [18,19], is the second building block of the guid-ance and control loop. The fuel-equivalent space is amathematical space where displacements on every axisare identical in terms of fuel cost. Thus, minimizingthe fuel cost of a maneuver is equivalent to minimiz-ing the distance in the fuel-equivalent space. Therefore,minimum-fuel problems are reduced to simple geomet-ric problems in the fuel-equivalent space.

This theory builds upon the results of the impul-sive feedback controller [20,21] which was proposedas a way to perform orbit element corrections whileminimizing the impact on the other orbit elements. Itmakes use of the Gauss variational equations and canperform any arbitrary small orbit correction with onlythree impulses. If only one or two elements are to becorrected, the controller provides essentially optimal re-sults in terms of fuel. If all six elements are to be cor-rected, the controller proposes maneuvers that are onlya few percents larger than the optimal multi-impulse so-lution. However, the most important advantage of thistechnique is that the impulses and their location can be


computed analytically with simple expressions, leadingquickly to a good approximation of the fuel cost of amaneuver, even if the spacecraft does not make use ofimpulsive thrusters.

The fuel-equivalent space maps the relative orbit ele-ment state vector �e into six fuel-equivalent coordinates

�V = [�Vtp �Vta �Vhi�Vh� �Vrp �Vra ]T (2)

where the coordinates represent the components ofthe three velocity impulses suggested by the impulsivefeedback controller. The fuel-equivalent coordinatesare computed through the linear mapping:

�V = S�e (3)

where the elements of S are defined by the state vectorof the reference trajectory. As presented in Ref. [18],the distance between �V1 and �V2 such that �V1 =S�e1and �V2 = S�e2 is a good approximation of the fuelcost, in terms of velocity impulse, required to go from�e1 to �e2. However, the distance in the fuel-equivalentspace is not the conventional Euclidean distance. In fact,because fuel-equivalent coordinates relate to the mag-nitude of the impulses predicted by the impulsive feed-back controller, and because these impulses occur atdifferent locations of the orbit, simultaneous displace-ments in some of the planes of the fuel-equivalent spaceare not possible. This so-called “fuel-equivalent” dis-tance dfe between �V1 and �V2 takes the form:

dfe =√

(�Vra2− �Vra1

)2 + (�Vta2− �Vta1

)2

+√

(�Vrp2− �Vrp1

)2 + (�Vtp2− �Vtp1

)2

+√

(�Vhi2− �Vhi1

)2 + (�Vh�2− �Vh�1

)2 (4)

This mapping is thus a very efficient way to estimatethe most fuel-efficient way to reach a formation becausethe problem is reduced to finding the minimum distancebetween a point (current spacecraft location) and a geo-metrical shape (the desired formation). For example, inthe fuel-equivalent space, all J2-invariant relative orbits[6,21] form a straight line, while all the relative statesforming a PCF about a leader form a six-dimension el-lipse. It will be shown later that this tool can be used inan autonomous formation flying guidance and controlsystem, as it is a straightforward way of predicting thefuel cost of a maneuver.

4. Relative motion control

The last missing element for a completely au-tonomous guidance and control loop was a simple but

fuel-efficient control algorithm. Such an algorithm willhave to efficiently and accurately bring the elementsof the formation from their initial set of relative orbitelements to the desired relative orbit elements withina reasonable time frame with as few computation aspossible.

Most of the formation flying control algorithms foundin the literature assume the “Leader/Follower” type ofarchitecture. Under this architecture, the relative motioncontrol problem is reduced to the tracking of a desiredtrajectory defined as a position relative to a referencetrajectory. The guidance system is typically responsiblefor defining a reference trajectory (that could be basedon the states of one member of the formation or any“virtual” point in space) and a position and velocityrelative to this reference trajectory.

The simplest controllers assume unperturbed rela-tive motion about a circular reference orbit. By usingthe CWH linear relative motion model, conventionallinear control can be applied to Earth-orbiting forma-tion flying spacecraft. The main advantage of linearcontrol theory is that it is a well-known method, withmeasurable performance and robustness assuming thelinearization conditions are valid. An example of suchan algorithm is the linear-quadratic regulator (LQR),that uses a constant feedback gain matrix that mini-mizes the infinite-horizon state error and the quadraticactuator command [22]. Rahmani [23] also developedan optimal reconfiguration maneuver of two spacecraftassuming CWH dynamics. The main conclusion of thework is that a balanced fuel-optimal maneuver of twospacecraft on unperturbed circular orbit is achievedthrough equal and opposite acceleration of both space-craft. However, those conclusions do not necessarilyapply to elliptical and perturbed orbits. In fact, Inalhan[5] demonstrated that assuming that the reference orbitis circular, even when the eccentricity is as small as0.005, leads to significant increase of fuel cost becausethe spacecraft “fights” the natural dynamics to keepthe same relative trajectory as it would in a circularorbit.

Therefore, other controllers have been developed forelliptical reference orbits. Such is the case for the Carte-sian coordinates feedback control law [21]. If the de-sired trajectory is described as an inertial position andan inertial velocity, a control feedback law based onCartesian coordinate errors can be used. Assuming therelative orbits are J2-invariant and that the distancebetween the spacecraft is small, this simple feedbackcontrol law can make use of the nonlinear dynamics(such as J2-perturbed dynamics) to compensate positionand velocity errors. On the other hand, the mean orbit


elements feedback control law, as developed by Schaubet al. [21,24], uses an error quantified in terms of relativeorbit element errors. It is thus possible to “cooperate”with the physics of orbital dynamics by acting directlyon the mean orbit elements to control specific orbitelements at specific instants on the orbitand increasethe fuel efficiency of the algorithm. For example, it ismuch more fuel efficient to correct an inclination er-ror at equator than at the pole, while an error in theascending node is easier to compensate near the poles.By carefully choosing the time-varying gain matrix ofthe controller, those effects can be accounted for. Simi-larly, the hybrid feedback law [21,25] uses desired statesdefined as a set of orbit element differences with a ref-erence orbit, while the tracking errors are Cartesian co-ordinate errors. The main advantage of this method isthat the controller uses inputs that are easily measured(relative position and velocity in orbital frame) whilethe reference is defined as orbit elements, which is moreconveniently expressed than rapidly evolving Cartesiancoordinates.

However, none of these feedback controllers canensure optimality for elliptical reference orbits. Thepresence of several constraints in the problem, the non-linearity of the dynamics and the need for optimalitymakes the optimal control theory a candidate of choicefor formation flying. This theory can fuel-optimizeor time-optimize any reconfiguration maneuver whileconsidering perturbed and nonlinear dynamics. For cir-cular reference orbits, an analytical solution can be ob-tained to get an analytical feedback law [26]. However,for reasonably complex dynamics, such as formationflying about an elliptical reference orbit, this methoddoes require highly demanding numerical optimizationwhich could not be implemented on-board. Howeversome near-optimal control methods, like the use ofneighbouring optimal paths, only require to solve theoptimal maneuver problem for one of the spacecraft.The other spacecraft of the formation can be consideredas “neighbours” of this optimal path, and the resultingcommand offsets can be easily computed. Therefore,the complexity of the problem does not necessarilygrow with the number of spacecraft in the formation,as is typically the case with optimal control solutions.Some more recent work [27] demonstrates how sucha neighbouring optimal feedback controller can be ap-plied to formation flying. This controller requires veryfew computation, which facilitates on-board imple-mentation. Furthermore, the formation accuracy/fuelconsumption trade-off can easily be implemented withthe selection of only one weight. Moreover, as opposedto the other formation flying feedback controllers, this

controller ensures near-optimality for all the elementsof the formation.

This controller seeks to minimize second-order vari-ations of the cost function J:

J = 1

2[�e(tf)]TK[�e(tf)] +

∫ tf

t0

1

2(u)TR(u) dt (5)

where �e(tf) is the error at final time tf , u(t) is thecommanded control effort and K and R are user-definedweighting matrices. The resulting feedback controller[27] takes the form:

u(t) = −H−1uu f T

u S(t)�e(t) (6)

where H−1uu is a constant diagonal matrix weighting the

control effort, fu is the derivative of the dynamics ofthe leader with respect to u (which is essentially thematrix form of the Gauss variational equations), S(t)

is a pre-computed time-varying 6 × 6 gain matrix and�e(t) is the instantaneous relative orbit elements error.Since both fu and S(t) are only based on the currentvalue of the uncontrolled leader orbit-element vector,the same gains can be applied to all the elements of theformation guaranteeing near-optimal results for all theelements of the formation. However, near-optimality isguaranteed only at the final time tf . The relative posi-tion error of the spacecraft between the initial time t0and the final time tf is not minimized, i.e. the controllerwill accept a large error at time t where t0 � t < tf ifthis is to minimize the cost function at the final time tf .Therefore, this controller is oriented toward formationreconfiguration as opposed to formation maintenance,the latter requiring a small position error for all the ele-ments of the formation at every moment. For formationmaintenance, other controllers such as the LQR or themean orbit element controller [21,24] are better suitedsince they only consider the current value of the errorin their feedback law.

This neighbouring optimum feedback control lawwill therefore be a part of the autonomous guidanceand control loop. It provides a fuel-efficient but nev-ertheless accurate way of taking the spacecraft fromtheir initial location to their desired configuration overa user-defined time frame.

5. Formation guidance

Now that all the building blocks are in place, the laststep to get a completely autonomous guidance and con-trol system is to build a guidance strategy, using thesetools, that is able to autonomously perform trade-offsbetween formation accuracy, fuel expenditure and equalsharing of the fuel expenditure within the formation.


The purpose of the guidance algorithm is to provide adesired set of relative orbit elements at every moment ofthe trajectory for all the elements of the formation. Thestrategy that is proposed here is to command the desiredrelative orbit elements to naturally bring the spacecraftfrom its current location to the desired location at thefinal true anomaly �. In other words, if the controller wasto be perfect and would instantaneously compensate anyerrors between the current and the desired relative orbitelements, no further control effort should be requiredto reach the targeted position at the final time. This canbe easily done with the state transition matrix presentedearlier:

�edes = �(e0, �)−1�Xdes(�) (7)

This way, the desired states �Xdes(�) of every spacecraftcan be mapped back to a current desired position �edes.The error to be compensated by the controller is there-fore the difference between the current relative orbit el-ements �e0 and the current desired location �edes. Sucha task can be performed by the neighbouring optimumcontroller, which will ensure that this error is reducedat the final time tf . Consequently, the main challenge inthe design of the guidance algorithm is to identify thisdesired position �Xdes(�) for every spacecraft.

The proposed guidance algorithm computes the set ofdesired positions for all spacecraft by minimizing a costfunction J that takes into account the total fuel expendi-ture, the accuracy of the formation and the inequalitiesbetween the fuel cost for each spacecraft:

J = Jfuel + Jfor + Jfdiff (8)

where Jfuel is a cost linked to the total fuel expenditureof the formation, Jfor is a cost linked to the accuracy ofthe formation and Jfdiff associates a cost to having dif-ferences between the planned fuel cost of the elementsof the formation.

For demonstration purposes, this guidance strategy isapplied here to the common PCF. The PCF is a forma-tion for which all members of the formation are at thesame distance from the centre of the formation in thenormal-tangential plane (Fig. 1). As seen from Earth,all members are distributed on a circle. This could haveseveral application for Earth observation.

In Hill coordinates (relative positions x, y and zand relative velocities Vx , Vy and Vz), the PCF isconstrained by

� =√

y2 + z2 (9)

where � is the radius of the PCF. The admissible setsof Hill coordinates for a given � can be obtained by

y

zx

β

Reference Orbit

Projected Circular

Formation

Toward Earth Center

Fig. 1. Projected circular formation in Hill coordinates.

sweeping the circular formation angular position �:

x(�) = −�

2cos � (10)

y(�) = � sin � (11)

z(�) = � cos � (12)

Vx(�) = �n

2sin � (13)

Vy(�) = �n cos � (14)

Vz(�) = �n sin � (15)

where n is the orbital mean motion. The required set ofrelative orbit elements �epcf(�) to reach the correspond-ing set of Hill coordinates at a desired orbit location canbe obtained:

�epcf(�) = �−1�X(�) (16)

where �X(�) = [x y z Vx Vy Vz]T. We shall assumehere a formation of three spacecraft (excluding the“leader”, that could be another spacecraft or a vir-tual point in space) and that the formation needs tobe reached exactly one orbit later. In this context, theguidance problem consists of identifying the angularposition vector � = [�1 �2 �3] containing the targetedangular position for each of the spacecraft that willglobally minimize the cost function J.

5.1. Absolute fuel cost minimization

Obviously, the first objective is to minimize the globalamount of propellant that is to be spent to reach theformation. That can be done by minimizing the sum ofthe predicted velocity impulses of all the spacecraft with

Jfuel = Kfuel

N∑i=1

1

mi

�Vi(�i ) (17)


where N is the total number of spacecraft of the for-mation (3 in the current example), mi is the remainingmass of fuel on-board the ith spacecraft and �Vi(�i )

is the predicted fuel cost, in the fuel-equivalent space,for the ith spacecraft to reach the angular position�i (by computing the distance between �epcf(�i ) andthe initial location of the ith spacecraft �ei0 ). Theremaining mass of propellant mi has been added asa way of increasing the cost for spacecraft with lessfuel remaining. This way, the guidance algorithmwill work toward an equal quantity of fuel remain-ing on-board all spacecraft in the case where not allspacecraft have the same amount of fuel remainingon-board before the maneuver. Finally Kfuel is a scalargain that the user can fit to particular needs depend-ing on how much importance is given to the total fuelexpenditure.

5.2. Formation accuracy optimization

Another important objective of the guidance algo-rithm is to make sure the spacecraft reach the formationdefined by the user. One of the strategies of doing so isby minimizing a potential function based on how wellthe targeted positions comply with the desired forma-tion. For the present example, the targeted formationconsists in placing all three elements of the formationuniformly distributed on the PCF, i.e. with a 120◦ an-gle between each of the spacecraft as seen from Earth.This desired formation will naturally minimize a po-tential function based on the sum of the squared angu-lar distances between all the elements of the formation,such as

Jfor = Kfor

N−1∑i=1

n=N∑n=i+1

1

(�i − �n)2

(18)

where Kfor, similarly to Kfuel is a user-defined gain.For other types of formation, Jfor could be any kind ofindicator of the formation accuracy, such as a qualityshape factor for a tetrahedral formation [28]. The onlyrequirement is for this value to be small if the proposedconfiguration is close to the desired formation and largeotherwise.

5.3. Minimization of the fuel cost difference

Finally, one would want the planned maneuver tominimize as much as possible the difference in theplanned fuel cost between all the elements of the for-mation. The simplest way of doing so is by comput-ing a cost Jfdiff based on the difference between the

�2

�1

�3

±d�±d�

±d�

S/C 1

S/C 2

S/C 3

PCF

Fig. 2. Fine formation optimization process.

Table 1Reference initial orbit elements

e0

a0 1.1Re

e0 0.05i0 �/4�0 00 0M0 0

Table 2Spacecraft initial orbit elements offset

S/C 1 S/C 2 S/C 3

�a0 0 0 0�e0 +0.0001 +0.0001 −0.0001�i0 +0.0001 +0.0001 +0.0001��0 +0.0001 −0.0001 +0.0001�0 −0.0001 −0.0001 0�M0 +0.0001 +0.0001 −0.0001

maximum and the planned �V :

Jfdiff = Kfdiff

[max

i(�Vi) − min

i(�Vi)

](19)

where Kfdiff is a third gain to be set by the user.The impact of this cost is not to be confused with

the use of the mi factor in the computation of Jfuel.The mi factor is added to compensate spacecraft havingdifferent amount of fuel remaining on-board before themaneuver, while the Jfdiff term ensures the planned ma-neuver will not induce a large difference between thefuel consumption of each spacecraft. The term Jfdiff canbe interpreted as a preventive measure, and the use ofmi as a corrective measure.


−1500 −1000 −500 0 500 1000 1500

−1000

−500

0

500

1000

y (m)

z (m

)

S/C 1

S/C 2

S/C 3

Fig. 3. Relative out-of-plane uncontrolled motion of the three spacecraft for one orbit. Circles mark initial locations of spacecraft. Diamondsmark final locations.

−1500 −1000 −500 0 500 1000 1500−1500

−1000

−500

0

500

1000

y (m)

z (m

)

Kfuel = 0

KPCF = 1

Kfdif f = 0

S/C 1S/C 2S/C 3

Fig. 4. Resulting trajectory if only formation cost is considered. Circles mark initial locations of spacecraft. Diamonds mark final locations.X’s mark targeted locations as computed by the guidance algorithm.

5.4. Optimization process

Now, with a clearly defined cost function, one hasto identify the set of angular positions � that optimizes

it. Obviously the optimization process is very mission-specific. Moreover, this optimization process is criticalas it is the only part of the guidance and control loopthat is not based on analytical solutions. An example of


how this could be done with a PCF with very few com-putation is shown next. However, this optimization pro-cess would have to be adapted to each particular missionscenario depending on the parameters that have to beoptimized and the number of elements of the formation.

The first step of the optimization is to identify aninitial guess. One could start from the set of angularpositions that will individually minimize the fuel con-sumption of each spacecraft. This can be done almost

0 1000 2000 3000 4000 5000 60000

1

2

V (

m/s

)

S/C1 S/C2 S/C3

2.5

Time (s)

1.5

0.5

Fig. 5. Cumulated fuel expenditure of each spacecraft if only for-mation accuracy is considered in the cost function.

−1500 −1000 −500 0 500 1000 1500

−1000

−500

0

500

1000

y (m)

z (m

)

Kfuel = 1

KPCF = 0

Kfdiff = 0

S/C 1 S/C 2 S/C 3

Fig. 6. Resulting trajectory if only total fuel expenditure is considered. Circles mark initial locations of spacecraft. Diamonds mark finallocations. X’s mark targeted locations as computed by the guidance algorithm.

analytically with very few numerical iterations in thefuel-equivalent space (see [18]). However, the dan-ger of using a fixed initial guess is that the searchprocess could remain trapped in a local minimum,therefore missing the global minimum. One efficientway of avoiding local minima in this case consistsin performing a global coarse search of all the possi-ble solutions. Sampling the complete solution spacewith 0.5 rad wide steps generates 2197 points wherethe cost function needs to be computed (13 positionsfor the three spacecraft), which is very reasonablesince the evaluation of the cost function is basedon simple analytical expressions. The initial loca-tion for the refinement process would be the vectorout of the 2197 with the lowest cost function value.

A refinement process is started from this initiallocation. Each of the three angular positions is movedby a smaller step d� forward then backward (Fig. 2).This leaves us with the evaluation of six new costfunction values (plus or minus 0.01 rad for each of thethree spacecraft in this particular case). If any of thesenew configurations lower the cost function value, thenthe configuration (out of the 6) with the lowest costfunction is used as the baseline. The process is thenrepeated until the cost function reaches a minimum. Onall cases simulated, this optimization process alwaystook less than 1 s on a common desktop computer.


6. Simulation examples

Finally, an example is given here of how this can beapplied to a LEO PCF. The problem consists in bring-ing three spacecraft initiated at arbitrary locations to anequally distributed 1 km size PCF, one orbit later, whileminimizing fuel expenditure and the difference in fuelconsumption between all spacecraft.

0 1000 2000 3000 4000 5000 60000

V (

m/s

)

S/C 1 S/C 2 S/C 3

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Time (s)

Fig. 7. Cumulated fuel expenditure of each spacecraft if only totalfuel expenditure is considered in the cost function.

−1500 −1000 −500 0 500 1000 1500

−1000

−500

0

500

1000

z (m

)

Kfuel = 0

KP C F = 0

Kfdiff = 1

S/C 1 S/C 2 S/C 3

y (m)

Fig. 8. Resulting trajectory if only differential fuel expenditure is considered. Circles mark initial locations of spacecraft. Diamonds mark finallocations. X’s mark targeted locations as computed by the guidance algorithm.

Table 1 presents the initial orbit elements of the chief,i.e. the semimajor axis a, the eccentricity e, the in-clination i, the right ascension of the ascending node�, the argument of perigee and finally the meananomaly M. Table 2 shows the initial relative orbit el-ements of the three spacecraft. As a reference, the re-sulting J2-perturbed uncontrolled relative motion of allthree spacecraft with respect to the leader are shown inFig. 3. On this figure, trajectories are also plotted withrespect to the targeted 1 km size PCF (dotted circle).

The first simulation scenario considers only forma-tion accuracy by setting Kfuel=0, Kfor=1 and Kfdiff =0.Fig. 4 shows the resulting out-of-plane relative trajec-tory of the three spacecraft (as seen from Earth), whileFig. 5 presents the time history of the total cumulatedfuel cost for each of the spacecraft. As expected, theguidance algorithm commands a uniform distributionof the spacecraft (all the spacecraft are located 120◦apart). However, fuel cost is not considered in the costfunction, which leads to very expensive maneuvers.

The second simulation scenario considers fuel costas the only criterion to optimize the function by settingKfuel = 1, Kfor = 0 and Kfdiff = 0. In this case, all thespacecraft are forced to end up on a 1 km size PCF, butthe selection of the angular location of every spacecraftis only based on the minimization of the fuel con-sumption. No importance is given to the distribution of


the spacecraft on the PCF. Fig. 6 shows the resulting out-of-plane relative trajectory of the three spacecraft andFig. 7 presents the time history of the total cumulatedvelocity impulse for each of the spacecraft. As can benoted in Fig. 6, considering only fuel cost does not leadto a well-balanced formation. In fact, spacecraft 1 and3 practically end up at the same location. However, fuelcost is much lower than if only the formation accuracyis considered.

0 1000 2000 3000 4000 5000 60000

1

2

3

V (

m/s

)

S/C 1S/C 2S/C 32.5

1.5

0.5

Time (s)

Fig. 9. Cumulated fuel expenditure of each spacecraft if only dif-ferential fuel expenditure is considered in the cost function.

−1500 −1000 −500 0 500 1000 1500

−1000

−500

0

500

1000

y (m)

z (m

)

Kfuel = 0.1

KPCF = 1

Kfdiff = 0.5

S/C 1S/C 2S/C 3

Fig. 10. Resulting trajectory if all terms of cost function are considered. Circles mark initial locations of spacecraft. Diamonds mark finallocations. X’s mark targeted locations as computed by the centralized guidance scheme.

The third simulation scenario consists in reachingthe PCF only by trying to minimize the fuel differencebetween all spacecraft (Figs. 8 and 9). In this case, thefuel expenditure is well balanced between spacecraft(Fig. 9), but the formation is not well balanced (Fig. 8).

Finally, Figs. 10 and 11 show the resulting trajectoryand fuel cost if all three terms of the cost function areused, i.e. Kfuel = 0.1, Kfor = 1, Kfdiff = 0.5. This showsa case where the obtained formation (Fig. 10) is close

0 1000 2000 3000 4000 5000 60000

1

2

V (

m/s

)

S/C 1S/C 2S/C 3

Time (s)

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

Fig. 11. Cumulated fuel expenditure of each spacecraft if all termsare considered in the cost function.


to the desired formation (Fig. 4) but with a much lowertotal fuel cost (Figs. 5 and 11).

Obviously any other trade-off can be achievedthrough the selection of the relative values of Kfuel,Kfor and Kfdiff . As the simulation results presentedhere have shown, these gains have a lot of influenceon the final configuration and the fuel spent by eachspacecraft to reach the formation.

7. Conclusion

This paper presented how an autonomous formationflying guidance and control loop can be established.This paper first summarized the design of three tools:a linearized relative motion model for J2-perturbedeccentric orbits, the fuel-equivalent space and theneighbouring optimum controller, which are the build-ing blocks of the system. Then, a guidance algorithmwhich autonomously performs trade-offs between fuelconsumption, formation accuracy and equality of thefuel expenditure was developed. Simulation resultshave shown that this guidance algorithm, linked withthe neighbouring optimum controller, is a very efficientway of autonomously performing maneuvers on-orbitover a 1 orbit time frame. Such a demonstration hasbeen completed with a typical 1 km size LEO PCF.

Even though this algorithm is applied here to the PCF,it can be adapted to other types of formation, such as thetetrahedral formation. The only requirement is to havean easy way of measuring the quality of the formationcommanded by the guidance system.

For other types of formation, the optimization pa-rameters and the optimization process would have tobe different. Even for a PCF, one could think of sev-eral other parameters that could be optimized, such aslocation of the leader (or the virtual centre) or the sizeof the formation. However, limiting the number of op-timization parameters drastically reduces the computa-tional power required (especially for a large formation)which largely facilitates implementation on a computa-tional power-limited on-board computer.

Further developments of such a system could includecollision avoidance and decentralization of the decisionprocess. Indeed, as of now, only the initial position andthe final location of the spacecraft are considered. Whathappens in-between is left to the controller, which isnot yet adapted to include collision avoidance in theplanning process. Furthermore, the guidance algorithmpresented here uses a centralized approach, i.e. the algo-rithm has all the information regarding all the spacecraftto make its decision. The decentralization of this algo-rithm could make it more robust to spacecraft failure,

since no single point would be responsible to make de-cisions for the whole fleet. This could be accomplishedby designing individual cost functions for each of thespacecraft that require a minimal information exchangebetween the spacecraft.

Acknowledgements

The authors would like to thank the National Scienceand Engineering Research Council of Canada and theCanadian Space Agency for financially supporting thisresearch.

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autonomous guidance and control of earth-orbiting formation flying spacecraft: closing the loop

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