Minimizing the Effects of Navigation Uncertainties on the Spacecraft Rendezvous Precision

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Minimizing the Effects of Navigation Uncertainties on the Spacecraft Rendezvous Precision

Georgia Deaconu, Christophe Louembet, Alain Théron

To cite this version:

Georgia Deaconu, Christophe Louembet, Alain Théron. Minimizing the Effects of Navigation Un- certainties on the Spacecraft Rendezvous Precision. Journal of Guidance, Control, and Dynamics, American Institute of Aeronautics and Astronautics, 2014, 37 (2), pp.695-700. �10.2514/1.62219�.

�hal-01078527�

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the spacecraft rendezvous precision

Georgia Deaconu 1

and Christophe Louembet 2

and Alain Théron 3

CNRS; LAAS; 7 avenue du colonel Roche, F-31400 Toulouse; France

Univ de Toulouse; UPS, LAAS; F-31400 Toulouse; France

I. Introduction

Theabilitytorobustlyandpreciselycontrolthespacecraftrelativemotionwillplayanimportant

role in future on-orbit inspection and on-orbit servicing missions [1]. Model Predictive Control

(MPC) is considered to be an eective control strategy for these types of spacecraft operations,

thatcaneasilyhandlemissionspecicconstraintswhileexplicitlyminimizingthefuelconsumption

[2]. Themaneuversplan isobtainedbysolvinganitehorizonopen-loopoptimalcontrol problem

starting from the spacecraft relative state and the optimal solution consists of aseries of control

actions {u1, u2, ..., uN}^,ôutôf^whichônly^the^rstôneîsêxecuted^[3].

TheMPCstrategiesareinherentlyrobustto arbitrarilysmall perturbations [4]andnewmea-

surementinformation isincluded everytime themaneuversplanis re-computed. However,in the

caseofspacecraftrelativemotion,navigationuncertainties andorbitalperturbationscancausethe

realrelativetrajectorytodiersignicantlyfromthepredictionusedforobtainingthecontrolplan.

Reference[5]showsthat smallerrorsintheestimationofthespacecraftrelativevelocitycanresult

in verylargepredictionerrorsfortherelativestate. Sincetrajectoryplanningreliesheavilyonthe

knowledge of the relative state, not accounting for navigationerrors may have some undesirable

eectssuchaspoorperformances,constraintsviolationsand/orinfeasibilityofthecontrolproblem.

For circular reference orbits, a method for constraints tightening in the MPC problem was

1

MethodsandAlgorithmsinControl,LAAS-CNRS,gdeaconu@laas.fr

2

MethodsandAlgorithmsinControl,LAAS-CNRS,christophe.louembet@laas.fr

3

MethodsandAlgorithmsinControl,LAAS-CNRS,atheron@laas.fr

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nilpotentin at mostN ^steps. ^It^guaranteesconstraintssatisfactionand recursivefeasibilityofthe optimisationproblem, evenwhensensornoiseisconsidered. Themethodhoweverisnotapplicable

foreccentricreferenceorbits,wherethespacecraftrelativedynamicsareLinearTimeVarying(LTV).

Thepresenceofunknownbutboundednavigationerrorsinthecaseofeccentricreferenceorbits

is dealtwithin [8]bypropagatingtheuncertainties setoverthepredictionhorizonand tightening

the constraints to account for their eects. A similar idea is used in [9], combined with an on-

lineestimation oftheuncertaintiesbounds. Reference[10]identiesthedisturbancesequencethat

can cause the maximum variation of the spacecraftrelative state and then uses this sequence to

tightentheconstraintsofadeterministicMPCspacecrafttrajectorycontrolproblem. Thesetypes

ofmethodsdoprovidemoreaccurateinformationabouttheevolutionofthesystem,buttheyhave

nocontroloverthespreadofthepredictedtrajectories. Moreover,tighteningconstraintsusingthe

open-looppropagationoftheuncertaintiesimposesthechoice ofshort predictionhorizonsin order

toensurethefeasibilityoftheproblem.

A methodthat directlyoptimizesthenal spacecraftrendezvousprecisionwithoutrestricting

thedurationofthemaneuversisproposedinthisarticle. Themethodisbasedontheso-calledfeed-

back MPC [11],whichusesasequenceoffeedbackpolicies {µ1(·), ..., µN−1(·)}^as^decision^variables,

insteadofasequenceofcontrolactions {u1, ..., uN−1}^. ^Thecomputationofsuchfeedbacklawscan beextremely dicultin the generalcasesincethe decisionvariables areinnite dimensional[12].

However, restricting the admissiblefeedback policies to the class of ane state feedback control

lawscanreduce thecomplexityoftheproblem.

Forcircularreferenceorbits,astaticfeedbacktermcombinedwiththenominaloptimalsolution

totheclassicalMPCproblemcouldensurethat,inpresenceofsensingnoise,allpossiblespacecraft

relativetrajectoriesremaininsideatubecenteredaroundthenominaltrajectory[12]. Wheneccen-

tricreferenceorbitsareconsidered,time-varyingfeedbackpoliciesneedtobecomputedinthesame

time asthe nominalcontrol andtheproblem maybecome non-convex. However,re-parametrising

thecontrolas anedisturbancefeedback policies canremovethisissue[13].

Someideasfromtube-basedMPCareusedinthispapertosolvetherobustxed-timespacecraft

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policies that steers the spacecraftfrom an initial relative state towards an ellipsoidalset centred

around a desired nal state, in presence of navigation uncertainties. This must be done while

respectingtheactuatorssaturationconstraintsandwhilepursuingadoubleobjective: minimisethe

fuel cost of the mission and minimise the size of the arrival set to guarantee agood rendezvous

precision. The control policies are restricted to ane disturbance feedback policies to ensure a

convexformulation of the control synthesis problem. The obtained sequence of feedback policies

drivesthesystemtotheguaranteedarrivalsetwithoutanyneedforrecurrentoptimization.

II. Spacecraftrelativedynamics

Thexed-timerendezvousbetweentwospacecraftonarbitraryellipticalorbitsconsistsofbring-

ingthesystemfromaninitialstateX1^toâ^desired^nal^stateXf âtâ^specied^time. ^This^needs^to

beachievedbyringthethrustersaxednumberoftimesN ât^some^predenedînstants. Ôur^goal

istodetermineasequenceoffeedbackpoliciesthatguaranteesthebestnalrendezvousprecisionin

presenceofnavigationuncertainties,whileminimisingthemission'sfuelcostandrobustlysatisfying

theactuatorssaturationconstraints.

PSfragreplacements

O

ML

MF

P~ Q~

~xL

~zL

~yL

x

z ν

Fig.1: Thelocal Cartesianframeattachedtotheleaderspacecraft

Therelativestatebetweenthetwospacecraftisdenedastherelativepositionandtherelative

velocity, expressedinalocalCartesianframe (Figure1)attachedto theleaderspacecraftX ∈R⁶^, X = [x y z vx vy vz]^T^. ^The ^following ^operation^can^be ^used^to ^transform^the^time ^domain ^state

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variablesinto scaledstatevariablesdependingonthetrueanomalyoftheleaderν^: X˜(ν) =







(1 +ecosν)I3 03

−esinνI3

qa³(1−e²)³ µ

1 1+ecosνI3







X(t) ⁽¹⁾

where µ îs ^the Êarth's gravitational parameter, e îs ^the eccentricity of the orbit of the leader spacecraft and a îts ^semi-major âxis. ^The ^variable ^change ⁽¹⁾ ^has ^been ûsed ^by ^T^schauner ând

Hempel in [14] to obtain linearised spacecraft relativedynamics valid for arbitrary values of the

eccentricity. Yamanakaand Ankersen presentedin [15] aclosed form solution for theTschauner-

Hempelequations undertheform ofatransition matrixΦ∈R^6×6^. ^This ^transition^matrix^is ^used

heretopropagate thespacecraftrelativetrajectorywithoutnumericalintegration. The spacecraft

relativedynamicsunder impulsivecontrol∆v∈R³ ^can^be^written ^as:

X˜(νk+1) = Φ(νk+1, νk)( ˜X(νk) +B∆vk) ⁽²⁾

where the matrix B = [03 I3]^T ^models ^the instantaneous relative velocity change caused by the impulsivecontrol∆vk^. ^The^spacecraft^relative^dynamicsⁱⁿ ⁽²⁾^canâlso^be^seenâsân^LTV^system

dened as:

xk+1=Akxk+Bkuk ⁽³⁾

where xk = ˜X(νk)^, Ak = Φ(νk+1, νk)^, Bk = Φ(νk+1, νk)B ^and uk = ∆vk^. ^Thisdescriptionof the relativedynamicswill beusednexttowritetherobustxed-timespacecraftrendezvousproblem.

III. Disturbancefeedback control

Considerthe spacecraftrelative dynamics in (3) where the knowledge of the state is aected

at each instant k ^by ûnknown ^navigation êrrors δxk^. ^The ^navigation êrrors δxk ^belong ^to ^some

ellipsoidal sets E(0, Qk) = {x ∈ R⁶ | x^TQkx ≤ 1}^. ^Note ^that ^the ^dening ^matrices Qk ∈ R^6×6

are notnecessarilyidentical. Thereal spacecraftrelative statexk îsûnknown ^but ât êach^step îs

connectedtothemeasuredstatex^m_k ^through:

xk =x^m_k +δxk, δxk ∈E(0, Qk) ⁽⁴⁾

Thepredictionofthestatethatwillbemeasuredat thenextstepx^m_k|k+1 ^is^given^by:

x^m_k|k+1=xk+1−δxk+1 =Akxk+Bkuk−δxk+1, δxk+1∈E(0, Qk+1) ⁽⁵⁾

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x^m_k|k+1=Akx^m_k +Bkuk+Akδxk−δxk+1 ⁽⁶⁾

Letwk ^be^the^totalcontribution ofthe measurementnoiseoveroneprediction step. From(6) wk

canbedenedas:

wk=Akδxk−δxk+1, δxk ∈E(0, Qk), δxk+1∈E(0, Qk+1) ⁽⁷⁾

Sincethenavigationerrorsbelongtoellipsoidalsetsthataresymmetricwithrespectto theorigin,

thedomainforwk ^is^dened^by:

wk∈E(0, A^−T_k QkA⁻¹_k )⊕E(0, Qk+1)⊆E(0, Q^w_k) ⁽⁸⁾

where ⊕^denotes ^the^Minkowski ^sum ^of ^the ^two^sets. ^The ellipsoidal set E(0, Q^w_k)^represents ^an

outerapproximationoftherealdomainanditcanbecomputedanalytically,usingforinstancethe

proceduredescribedin[16].

Denotingx= [x^m_1|2 x^m_1|3... x^m_1|N]^T^,u= [u1u2 u3 ... uN−1]^T ^andw= [w1w2 w3... wN−1]^T^,^the

evolutionofthemeasuredstateoverthepredictionhorizonisgivenby:

x=Ax^m₁ +Bu+Cw (9)

wherethematricesA∈R^6(N−1)×6^,B∈R^6(N^{−1)×3(N−1)}^,C∈R^6(N^−1)×6(N⁻¹⁾^are^dened ^as:

A=





 A1

A2A1

.

AN−1...A1





 B=







B1 0 0 0 ...

A2B1 B2 0 0 ...

.

AN−1...A2B1 ... ... BN−1





 C=







I 0 0 0 ...

A2 I 0 0 ...

.

. .

.

AN−1...A2 AN−1 I







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The structure chosen for the control policies uk îs ^based ôn ^the ^results ⁱⁿ ^[13] ând ît ^consists ôf

an impulsivecomponent plussome disturbance feedback termsused to compensate theeects of

navigationerrors:

uk = ∆vk+

k−1

X

i=1

Lk,iwi ⁽¹¹⁾

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Pastdisturbancetermsuntilk−1âre^consideredⁱⁿôrder^toûse^the^maximumâmountôfâvailable

informationwhilemakingsurethatthecontrollawiscausal. Thisalsoensuresthatthedisturbances

willnotbepropagatedin openloopforlongerthantheintervalbetweentwoconsecutivecontrols.

Denoting∆v= [∆v1 ∆v2 ...∆vN−1]^T^,â^more^compactêxpression^can^beôbtained:

u=∆v+Lw (12)

wherethematrixL∈R^{3(N−1)×6(N}⁻¹⁾ ^is^dened^by:

L=







0 0 0 ... 0

L2,1 0 0 ... 0

L3,1 L3,2 0 ... 0

.

LN−1,1 ... ... LN−1,N−2 0







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Thespacecraftclosed-looprelativedynamicscanbewrittenas:

x=Ax^m₁ +B∆v+ (BL+C)w (14)

Let¯x= [¯x_1|2 x¯_1|3 ...x¯_1|N]^T ^be^the^nominal ^trajectoryôbtained^whenâssuming^perfect^stateînfor-

mationandusingonlytheimpulsivepartofthecontrol:

¯

x=Ax^m₁ +B∆v (15)

The impulsive plan ∆v must be such that the nominal trajectory satises the nal rendezvous

objective:

¯

x_1|N =Xf ⁽¹⁶⁾

Inthiscase,theerrorbetweentheperturbedtrajectoryandthenominaltrajectorycanbewritten

asalinearfunctionofthedisturbancevectorw:

e=x−x¯= (BL+C)w (17)

The purpose is to ndthe impulsiveplan ∆v andthe correctiongains matrixL that guarantees

thesmallesterrorattheend ofthepredictionhorizon eN ^and^the ^lowest ^fuel^cost ^for^the^nominal

trajectory,allwhilerobustlysatisfyingthesaturationconstraintsonthethrusters.

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thecostfunctioncorrespondingtothefuelconsumptionforthenominaltrajectoryisdenedbythe

sumofthe`1^-norm^of^the^thrust^vector^[17]:

J∆v=k∆vk1 ⁽¹⁸⁾

Thevariableszk ∈R³areintroducedtolinearisetheobjectivefunctionJ∆v ^[18]^and^are^such^that:

|∆vk| ≤zk, k= 1..N−1 ⁽¹⁹⁾

Theactuatorssaturationsare denedby|uk| ≤umax^where uk ^is^dened ^by^(11). ^Theconstraints must hold for all admissible values of the disturbances w ∈ E(0, Q^w)^. ^This ^can ^be ^written ^as

row-wiseconic constraintsonzk ^that^account^for^the^presence^of^the^correction^terms:

zk≤umax−

k−1

X

i=1

kLk,iP_i^wk2, k = 1..N−1^whereP_i^w= (Q^w_i)^−1/2 ⁽²⁰⁾

TheobjectiveforthenalerroreN ^translates^into^computing^the^smallestellipsoidalsetE(0, Q⁻¹_f )

that boundseN ^forâllâdmissible^valuesôf^thedisturbancesw:

min^tr(Qf)^s.t. e^T_NQ⁻¹_f eN ≤1, ∀wi∈E(0, Q^w_i), i= 1..N −1 ⁽²¹⁾

Minimizing the traceof Qf correspondsto minimizing the sumof squares ofthe semi-axisof the ellipsoidalsetE(0, Q⁻¹_f )^. ^F^rom^(17),^the^quadratic^constraint^oneN ^can^be^written ^as:

w^T(B_NL+C_N)^TQ⁻¹_f (B_NL+C_N)w≤1, ∀wi∈E(0, Q^w_i ), i= 1..N−1 ⁽²²⁾

whereB_N andC_N areobtainedbyselectingtheappropriatelinesintheBandCmatrices. Using

the S-procedure [19] andthe Schur complement, constraint(22) canbetransformed into alinear

matrixinequality:

∃τ1, τ2, ... τN−1≥0, Qf ≥0







1−PN−1

k=1 τk 0 0

0 Qw (B_NL+C_N)^T 0 (B_NL+C_N) Qf







≥0, Qw=





 τ1Q^w₁

.

τN−1Q^w_N₋₁







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Theconicoptimizationproblemthat mustnallybesolvedisgivenby:

Qf,zmink,∆vk,L

tr(Qf) +PN−1 k=1 zk

s.t. (16),⁽¹⁹⁾,⁽²⁰⁾,⁽²³⁾

(24)

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Thestructurechosenforthecontroluk^,^other^than^leading^to^a^convexoptimization problem,also presentstheadvantagethatthecorrectionsfortheeectsofpreviousmeasurementerrorsaremade

in thesame time asthe burnsfor the nominal trajectory. This ensures that the totalnumber of

maneuversremainsconstantandthat noextrastressisputonthethrusters.

IV. Simulation results

The xed-time rendezvous scenarios summarized in Table 1 and 2 are used to illustrate the

eectivenessofdisturbancefeedbackcontrolinsolvingthistypeofproblem. Themaneuversplanis

computedbasedonthelinearisedspacecraftdynamicsand thentestedonanindustrial non-linear

simulator. Theinitialspacecraftrelativestateusedinthesimulationsisobtainedbyaddingrandom

noiseto theinitialstateusedfor controlcomputation. Random noiseis alsoaddedto everyother

state measure and the magnitudeof the noiseis bounded by a0.02 m ellipsoidal set for position

and0.002m/sforvelocity.

The control uk îs ôbtained ^from ^(11), ^where ∆v and L have been obtained by solving (24) before thesimulation. Thedisturbance termsw âreêstimated ^during ^the ^closed ^loopsimulations usingtheperturbedstatemeasures and(6)and(7),: wk−1=x^m_k −Ak−1x^m_k−1−Bk−1uk−1^.

Table1: Prismamissionsimulationdata

e a^[km] ⁱ^[^◦^] Ω^[^◦^] ω^[^◦^] ν1[

◦

] X1 [m,m/s] Xf ^[m,m/s] ^duration^[s] umax[m/s]

0.004 7011 98 190 0 0 [10000,0,0,0,0,0] [330,0,30,0,0,-0.0158 ] 18000 0.26

Table2: SimbolXmissionsimulationdata

e a^[km] ⁱ^[^◦^] Ω^[^◦^] ω^[^◦^] ν1^[◦

] X1 ^[m,m/s] Xf ^[m,m/s] ^duration ^[s] umax^[m/s]

0.7988 106246.975 5.2 180 90 135 [-305,0,396,0,0,0] [-60.2,0,79.85,0,0,0] 8000 0.8

First,thedimensionoftheguaranteed ellipsoidalarrivalsetdened bythematrixQf ^is^com-

paredagainsttheestimationprovidedbyaclassicalMPCapproachbasedontheopen-looppropa-

gationofthestateuncertainty[8]. Theresultsobtainedforthetwomissions usingN = 10^control

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Prismamission,thedisturbancefeedbackcontrolmethodguaranteesineverycaseasmallerarrival

set if thesum ofthe semi-axes isconsidered. FortheSimbol X mission,the durationof themis-

sion isalot smallerthanthe orbitalperiod ofthe leader. In thiscase, theestimates provided by

thedisturbanceMPCmethod arelargerforshorter horizonsbut thischanges asthelengthofthe

durationofthemissionincreases.

Table3: Semi-axesofthearrivalsetinthexz ^plane^for^the^Prisma^mission^andN = 10

missionduration[s] 6000 9000 12000 18000

disturbancefeedbackMPC[m] 3.66 2.83 5.77 3.53 8.76 4.31 17.49 5.41

open-loopMPC[m] 35.03 0.31 56.82 1.4 70.08 0.61 105.2 0.9

Table4: Semi-axesofthearrivalset inthexz ^plane^for^the^Simbol^X^mission ^andN= 10

missionduration[s] 8000 12000 16000 20000

disturbancefeedbackMPC[m] 22.61 22.61 26.51 26.51 29.02 29.02 31.34 31.33

open-loopMPC[m] 16.15 15.92 24.48 23.75 33.04 31.5 41.86 39.08

It shouldbenotedthat whilethesize ofthearrivalset fortheopen-loopMPCdependssolely

on thedurationof themission,the performances ofthe disturbancefeedback schemealso depend

on the number of control instants. The trajectories obtained for the Prisma rendezvousmission

whenusingthedisturbancefeedbackcontroltechniquewithN = 6ândN = 10^controlînstantsâre

presentedin Figure2. Fifty perturbedinitial conditionsareconsidered and thecontrol is applied

withoutanyre-computationoftheimpulsivepartorofthecorrectiongains. Theerrorswithrespect

tothedesirednalpositionXf ^belong^each^time^to^theellipsoidalarrivalsetdenedbythematrix

Qf^, âs^guaranteed ^by^theâlgorithm. ^The^dimensionsôf ^theêstimatedârrival^set âre^bigger ^when

N = 6^,^reecting^the^fact ^that ^the^interval^between^twoconsecutivecontrolsislarger(3600swith

respectto 2000sforN = 10^).

Figure 3 presents the closed loop trajectories obtainedfor the Simbol X rendezvousmission.

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−2000 0 2000 4000 6000 8000 10000 12000

−1200

−1000

−800

−600

−400

−200 0 200 400

x [m]

z [m]

N=6 N=10

−40 −30 −20 −10 0 10 20 30 40

−15

−10

−5 0 5 10 15

x [m]

z [m]

E(0,Q_f) for (N=6) final errors for N=6 E(0,Q_f) for N=10 final errors for N=10

Fig.2: Closed-looptrajectoriesandnal errorsforthePrismamission

The nalerrorsare allcontainedinside theguaranteedarrivalset, but for thismission where the

referenceorbitishighlyeccentrictheestimationgivenforthenalsetseemstobemoreconservative.

−35050 −300 −250 −200 −150 −100 −50

100 150 200 250 300 350 400

x [m]

z [m]

N=6 N=10

−40 −30 −20 −10 0 10 20 30 40

−40

−30

−20

−10 0 10 20 30 40

x [m]

z [m]

E(0,Q_f) for N=6 final errors for N=6 E(0,Q_f) for N=10 final errors for N=10

Fig.3: Closed-looptrajectoriesandnalerrorsfortheSimbolXmission

V. Conclusion

ThearticleshowsthatthedisturbancefeedbackModelPredictiveControl (MPC)canbeused

foreectivelysolvingthexed timerendezvousprobleminpresenceofnavigationuncertainties. It

providesseveral advantages overtheclassical MPCapproachessuch asbetter aprioriguarantees

for the closed-loop system behaviour for any admissible value of the uncertainty. It also avoids

problems linked to infeasibility when the satellites are in close range since it does not rely on

repeated re-computations to achieve robustness. If a feasible solution is found, the rendezvous

maneuvers plan can be applied without modications, at the cost of only some simple algebraic

computationsfortheon-lineestimationofthedisturbanceterms. Theplancanbecomputedbythe