Designing Continuously Constrained Spacecraft Relative Trajectories for Proximity Operations

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Designing Continuously Constrained Spacecraft Relative

Trajectories for Proximity Operations

Georgia Deaconu, Christophe Louembet, Alain Théron

To cite this version:

Georgia Deaconu, Christophe Louembet, Alain Théron. Designing Continuously Constrained

Space-craft Relative Trajectories for Proximity Operations. Journal of Guidance, Control, and Dynamics,

American Institute of Aeronautics and Astronautics, 2015, 38 (7), pp.1208-1217. �10.2514/1.G000283�.

�hal-01078528�

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using non negative polynomials

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

Thearticlepresentsanewmethodfordesigninganoptimalplanofimpulsive maneu-versforspacecraft rendezvousthataccountsforthepresence ofcontinuousconstraints on the relative trajectory. Impulsive control and continuous constraints are brought togetherthroughtheparameterizationofthespacecraftrelativepositionsbetweentwo consecutive maneuvers. By using a variable change and a polynomial approximation of the integral term in the expressions of the relative positions, the continuous con-straintson the trajectory can betransformed into nonnegativity constraintsof some polynomialsona given interval. Theresultingoptimal controlproblemissolved using semi-deniteprogramming.

I. Introduction

Inthe recentyears, theneedsand requirementsofspacecrafton-orbitservicing missionshave beenthoroughlyanalysed [1]. Thiskindof operationsdemandexible controlalgorithms, capable of handling the type of hard constraints associated to spacecraft proximity maneuvers. During close vicinity rendezvousoperations, in additionto the inputconstraintsusually considered,hard constraintson the trajectory must also beaddressed[25]. If, for instance, the LIDAR sensor is used in the estimation of the spacecraftrelativestate, the limited eld of view of the equipment

1

MethodsandAlgorithmsinControl,LAAS-CNRS,gdeaconu@laas.fr 2

MethodsandAlgorithmsinControl,LAAS-CNRS,christophe.louembet@laas.fr 3

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within thevisibilityconeofthesensor[68].

The ability of the spacecraftto hover inside a specied volume in a fuel optimal manner is also a major feature in the on-orbitservicing missions. Target monitoring introduces some hard constraints on the relative trajectory since it requires for the chaser spacecraft to remain in a specied zone dened in the target-centered frame. This problem has been recently studied by [9, 10] with thepurpose of maximizing the timespentby thespacecraftin aspeciedcylindrical region. ThehoveringcapabilitiesarealsousedduringtheATV(AutomatedTransferVehicle)ights totheInternationalSpaceStation. Forthesemissions,therendezvoustrajectoryconsistsofseveral way-pointsthat thespacecraftmustreachandwhereitmustwaitfortheauthorisationtoproceed [11]. Thespacecraftisin fact placedonaperiodic parkingorbitaroundthe desiredwaiting point sinceit is moreecientfrom afuelconsumption point ofviewthan maintainingaxed position. Inthis paper,thehoveringoperation refersto periodicrelativeorbitsexclusivelysincetheyallow thechaserspacecraftto remaininside thespeciedregiononan innitetimehorizon withnofuel consumption.

Thepreviousexampleshaveincommonthenecessityofimposingconstraintsonthespacecraft relativetrajectory. Theyexpresstheneedforagenericalgorithmthatcanproviderigoroussolutions totheconstrainedoptimalcontrolproblemsthatfollowfromthesedierentguidancescenarios. An algorithm that can be used for the design of such constrained spacecraft relative trajectories is presentedin thispaper.

The spacecraft relativemotion with respect to arbitrary elliptical orbits was investigated by Tschauner and Hempel in [12]. A transition matrix that enables the usage of the closed form solutionsfor the spacecraftrelativemotionwaspresented in [13]. Conventional methods use this tool andthe discretization of the constraintsin order to obtaina desiredrelative trajectory: the relative motionis propagated at specied instants where the trajectory constraints are explicitly checked [5]. The main advantage of these methods is that they transform the optimal control problemintoatractableprogram. However,theydonotaccountforthebehaviouroftheobtained trajectoryinbetweenthediscretizationpointsandviolationsoftheconstraintsmightoccuronthese

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Preciseparametriccharacterisationofthegeometryofthespacecraftrelativetrajectorieswould enable the designer to choose only those that continuously satisfythe given set of guidance con-straints. Works in this area focused mainly on the case of periodic relative motion. Periodic trajectoriescanbeobtainedbyimposingtheequalitybetweenthesemi-majoraxesofthespacecraft orbits[14]. AperiodicityconditionvalidfortheCartesiannon-linearmodelofrelativemotionwas proposedby Gurllin [15]. However,theperiodicityconditionalone doesnotprovideanyinsight with respect to the geometry of the obtainedtrajectory and further investigations are necessary. Analyticalexpressionsforthetheminimumandthemaximumdistancebetweentwospacecrafton elliptical orbitsare given in [16] asa function of the orbital elements ofthe satellites. An eighth degreetrigonometricpolynomialmustbesolvedinorderto obtainthetrueanomalies correspond-ing to theworst caseextremal distances, renderingthemethod toocomplexand tooconservative to be used within a guidance algorithm. A step forward into the study of the geometry of the spacecraft relative motion was achieved in [17] where the eects of the eccentricity on the shape of theperiodictrajectories areanalyzed bymeans ofaparametrization of therelativemotion. A similarparametrizationispresentedin[18]in ordertoshowthatwhensatellitesonellipticalorbits are considered the relative periodic trajectories are usually three-dimensional and then compute the numberof self-intersections. In [19], the periodic trajectory is expressed asa function of the dierencebetweenthespacecraftorbitalelementsbuttheextremalseparationdistancesare analyti-callycalculatedonlyforsomeparticularcases. Aconstraints-discretizationbasedmethodisusedin [20]inordertoobtainperiodicrelativetrajectoriesthatrespecttheimposeddimensionconstraints. The resultspresent thesamebenetsand drawbacksinherentto thistypes ofmethods that were previouslymentioned.

It becomesclearthat thegeometry of spacecraftrelativeperiodicorbitscanbe studied using dierentkindsofparametricrepresentations. Whilesomeinterestingconclusionscanbedrawn,these parametrizationsstilllackaclearlinkbetweentheparametersand thedimensionsoftheresulting trajectoriesinthegeneralcase. Weproposeaparametricrepresentationfor thespacecraftrelative positionthatisbasedontheYamanaka-Ankersentransitionmatrix[13]andweshowhowitcanbe

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initial conditionsof therelativemotionofthesatellitesthroughalinearfunction andtheperiodic motion canbe treatedasa particularcase byrequiring oneof theparameters to be zero. Inour approach,thelinearconstraintsimposedonthespacecraftrelativetrajectoryaretranslatedintoa linearrelationbetweentheproposedvectorofparametersandtheconeofthepositivesemi-denite matrices. The obtained solution is guaranteed to satisfy the trajectory constraints continuously in time. This representsamajorimprovementovertheconventionaldiscretization basedmethods which, for a similar algorithmic complexity, requirea specic a posteriori checkout procedure in order to validate thesolution. Furthermore,in thecaseof periodic relativemotion, theproposed approach provides an analyticaldescription of the set of states belonging to periodic trajectories that respectagivenset oflinearconstraints.

The articleis organizedasfollows. Section II presentsthe parametrizationfor the spacecraft relativetrajectory, with thehighlighton thelink betweenthe vectorof parametersand themain designparameter,theinitialspacecraftrelativestate. Rationalexpressionsaregivenfortherelative motionbyusinganappropriatevariablechangeandapolynomialapproximationofthedriftingterm

J

oftheYamanaka-Ankersen transitionmatrix. Insection III the spacecraftrendezvousguidance

problemisstatedas anoptimalcontrolproblemwithimpulsiveinputundersaturationconstraints and undercontinuous stateconstraints. Byusingthe rationalexpressionsfortherelativemotion, the constraints on the state are translated into non negativity constraintson some polynomials. Results onnon negative polynomialsare then used to formulate theguidance problem as a semi-deniteprogram. InsectionIV,severalexamplesdemonstratethelargerangeofrendezvousguidance problemsthat canbeaddressedusingtheproposedmethod.

II. Parameterization ofthespacecraft relative motion

Considertwospacecraftonarbitraryelliptic Keplerianorbits,onespacecraftcalled theleader andtheothercalled thefollowerorthechaser. Thespacecraftaredepictedas

M

L

and

M

F

respec-tivelyin gure1. Thetrueanomaly

ν

expressesat eachmomentthe position of theleader onits orbitanditismeasuredintheperifocalbasis

~

P , ~

Q, ~

W

(6)

rotatingCartesianlocal-vertical/local-horizontal(LVLH) basisattached tothe leader,

(~x

L

, ~y

L

, ~z

L

)

in gure1. PSfragreplacements

O

M

L

M

F

~

P

~

Q

~x

L

~z

L

~

y

L

x

z

ν

Fig.1: Thelocal LVLHframeattachedtotheleaderandthespacecraftrelativeposition

We assume that the distance betweenthe leader and the follower spacecraftis much smaller than the distance between the leader and the center of the Earth. In this case, the linearized spacecraftrelativemotion expressed in the leader's LVLH frame canbe described using the well-knownTschauner-Hempelequations:

¨

x = 2 ˙ν ˙z + ¨

ν z + ˙ν

2 _{x −}

µ

R

3 x + u

x

¨

y = −

_R

µ

3 y + u

y

¨

z = −2 ˙ν ˙x − ¨ν x + ˙ν

2 _{z + 2}

µ

R

3 z + u

z

(1)

where

µ

isthegravitationalconstantoftheEarth,

˙ν =

r

_µ

a

3 _{(1 − e}

2 ₎

3 (1 + e cos ν)

2 _,

_{R =}

a (1 − e

2 )

1 + e cos ν

(2)

and

a

and

e

are the semi-major axis and the eccentricity of the leader spacecraft. To obtain a simplied form of equations (1), the independent variable can be changed from the time to the leader'strueanomaly

ν

through:

(7)







˜

x

˜

y

˜

z







= (1 + e cos ν)







x

y

z







,







˜

x

0 ˜

y

0 ˜

z

0 





= −e sin ν







x

y

z







+

1 + e cos ν

˙ν







˙x

˙y

˙z







(4)

Thisleadstothefollowingequationsfortherelativemotion:

˜

x

00 _{= 2 ˜}

_z

0 _{+ ˜}

_u

x

˜

y

00 = −˜y + ˜u

y

˜

z

00 =

3 1 + e cos ν

z − 2 ˜x

˜

0 + ˜

u

z

(5)

Startingfromequations(5) ,YamanakaandAnkersen[13]presentedatransitionmatrixthatcanbe usedforthepropagationoftherelativemotionstartingfromaninitialstate

X(ν

˜

0 )

,underimpulsive control

∆ ˜

V

:

˜

X(ν) = Φ

ν

0 X(ν

˜

0 ) +

X

i

Φ

ν

i

B ∆ ˜

˜

V

i

.

(6) where

X(ν) = [˜

˜

x(ν) ˜

y(ν) ˜

z(ν) ˜

x

0 _{(ν) ˜}

_y

0 _{(ν) ˜}

_z

0 _(ν)]

T

. Thematrix

B

˜

isgivenby

B = [0

˜

3 I

3 ]

T

,sincethe eect oftheimpulsivecontrolismodelledasaninstantaneouschangein therelativespeed.

Basedonthedenitionofthetransitionmatrixgivenin[13],aparametrizationforthespacecraft relative positions is presented next. This parametrization is then used for designing impulsive maneuvers leading to spacecraft relative trajectories that respect continuously in time dierent typesoflinearconstraintsthat areusuallyassociatedtospacecraftproximityoperations.

A. Nonperiodic spacecraft relative motion

Theevolutionofthespacecraftrelativetrajectorybetweentwoconsecutiveimpulsivecontrols, applied at

ν

0

and

ν

f

respectively,can beseenastheopenlooppropagationof therelativemotion startingfromthestaterightafterthattherstimpulseisred:

˜

X(ν) = Φ

ν

0 ( ˜

X(ν

0 ) + B ∆ ˜

V

0 ) = Φ

ν

0 ˜

X

+

_(ν

0 ),

ν ∈ [ν

0 , ν

f

]

(7)

(8)

Starting fromthedenition givenin [13]forthetransition matrix

Φ

, theparametricequationsfor theautonomousrelativetrajectoryaregivenby:

˜

x(ν) = (2 + e cos ν)(d

1 sin ν − d

2 cos ν) + d

3 + 3 d

4 J(ν) (1 + e cos ν)

2 ˜

y(ν) = d

5 cos ν + d

6 sin ν

˜

z(ν) = (1 + e cos ν)(d

2 sin ν + d

1 cos ν) − 3 e d

4 J(ν) sin ν (1 + e cos ν) + 2 d

4

(8)

where the independent variable

ν

belongs to

[ν

0 , ν

f

]

and the vector of parameters

D =

[d

1 d

2 d

3 d

4 d

5 d

6 ]

T

depends linearly on

X

˜

0 = ˜

X

+

_(ν

0 )

, the relative state from which the

satel-litesmotionispropagated:

D = C(ν

0 ) ˜

X

0

(9) where:

C(ν) =







0

0 3(e+cos ν)

_e

2 ₋

₁

−(2 cos ν+e cos

2 ν+e)

e

2 ₋

₁

0 sin ν(1+e cos ν)

e

2 ₋

₁

0

0 3 sin ν(1+e cos ν+e

_(e

2 _{−1)(1+e cos ν)}

2 )

−

sin ν(2+e cos ν)

e

2 ₋₁

0 −

(cos ν+e cos

2 ν−2e)

e

2 ₋₁

1

0 −3e sin ν(2+e cos ν)

_(e

2 _{−1)(1+e cos ν)}

e sin ν(2+e cos ν)

e

2 ₋₁

0 e

2 cos

2 ν+e cos ν−2

e

2 ₋₁

0

0 −(3e cos ν+e

_e

2 ₋₁

2 +2)

(1+e cos ν)

2 e

2 ₋₁

0 −e sin ν(1+e cos ν)

e

2 ₋₁

0 cos ν

0

0 − sin ν

0 0 sin ν

0

0 cos ν

0 





(10)

The relative motion on the

y

axis is naturallyperiodic, while the motion in the orbital plane is dened byacombination betweenperiodictrigonometric termsandadriftingtermdenoted

J(ν)

. Thislattergrowslinearlyintimeandexpressedasafunctionoftheindependentvariable

ν

isgiven by:

J(ν) =

Z

ν

0 dτ

(1 + e cos τ )

2 =

n(t − t

0 )

(1 − e

2 ₎

3/2

(11)

Thefollowingvariablechangecanbeusedin ordertoremovethetrigonometrictermsin(8):

w = tan

ν

2 cos ν =

1 − w

2 1 + w

2 sin ν =

2w

1 + w

2

(12)

(9)

˜

x(w) =

1 (1 + w

2 ₎

2 [P

x

(w) + 3 d

4 P

Jx

(w) J(w)]

˜

y(w) =

1 1 + w

2 P

y

(w)

˜

z(w) =

1 (1 + w

2 ₎

2 [P

z

(w) + 2 d

4 P

Jz

(w) J(w)]

(13) where

w ∈ [w

0 , w

f

]

and:

J(w) =

Z

w

0 2 τ

2 _{+ 2}

((1 − e) τ

2 _{+ e + 1)}

2 dτ

(14)

P

x

(w) =

4 X

i=0

p

xi

w

i

, P

y

(w) =

2 X

i=0

p

yi

w

i

, P

z

(w) =

4 X

i=0

p

zi

w

i

,

(15)

P

Jx

(w) = (1 + e) + (1 − e)w

2

2 ,

P

Jz

(w) = −3e (1 + e) + (1 − e)w

2 w

(16) Thecoecients

p

x

= [p

x0

p

x1

p

x2

p

x3

p

x4

]

T

,

p

y

= [p

y0

p

y1

p

y2

]

T

and

p

z

= [p

z0

p

z1

p

z2

p

z3

p

z4

]

T

of thepolynomials

P

x

(w)

,

P

y

(w)

and

P

z

(w)

respectivelydepend linearlyonthevectorofparameters

D

:

p

x

= C

x

D p

y

= C

y

D p

z

= C

z

D

(17) with:

C

x

=







0 −2 − e 1 0 0 0

4 + 2e

0 0 0 0 0

0 2e

2 0 0 0

4 − 2e

0 0 0 0 0

0 2 − e 1 0 0 0







C

y

=







0 0 0 0 1 0

0 0 0 0 0 2

0 0 0 0 −1 0







C

z

=







e + 1

0 0 2 0 0

0 2e + 2 0 0 0 0

−2e

0 0 4 0 0

0 2 − 2e 0 0 0 0

e − 1

0 0 2 0 0







(18)

Hence thecoecientsof thesepolynomialsdepend linearlyon

X

˜

0

, therelativestatestartingfrom whichthesatellitesmotionispropagated.

Thepresenceofthedriftingterm

J

causesthespacecraftrelativetrajectorytohaveanirrational form,regardlessofwhetherthevariable

ν

orthevariable

w

isused. Analyticalrationalexpressions fortherelativemotionareinsteadneededinordertoimposecontinuousconstraintsontherelative

(10)

periodicrelativemotion. Onewayofobtainingthenecessaryrationalexpressionsispresentednext anditisbasedonusingapolynomialapproximationoftheterm

J

.

B. Polynomial approximations for the driftingterm

J(w)

Theclosedformexpressionoftheintegral(14)isgivenby:

J(w) =







2 e w

(e

2 _{− 1) ((1 − e) w}

2 _{+ e + 1)}

−

2 arctanh

√e − 1

√

e + 1

w

(e

2 _{− 1)}

3

2 





w

0 Θ

u

(w)

respectively, such that: Itmustbenotedthatthe intervalonwhich

J(w)

maybe

approxi-matedbyapolynomialmust beanite subsetof

R

. Infact,

J(w)

isdiscontinuousonthebounds of itsdenition set i.e.

lim

w→−∞

J(w) 6= lim

w→+∞

J(w)

since

lim

t→−1

arctanh(t) 6= lim

t→+1

arctanh(t)

. Con-sequently, no polynomial or rational function can approximate

J(w)

on

R

. If the term

J(w)

is replaced in (13) bya polynomialapproximation

Θ

r

(w)

, valid on anite interval

W = [w

0 , w

f

]

, thentheexpressionsdescribingthespacecraftrelativetrajectorybecomerational:

˜

x(w) =

1 (1 + w

2 ₎

2 [P

x

(w) + 3 d

4 P

Jx

(w) Θ

r

(w)]

˜

y(w) =

1 1 + w

2 P

y

(w)

˜

z(w) =

1 (1 + w

2 ₎

2 [P

z

(w) + 2 d

4 P

Jz

(w) Θ

r

(w)]

, ∀w ∈ [w

0 , w

f

]

(19)

Recentresultsfrom [22]showthataxed-degreepolynomialapproximation

Θ

r

(w)

withacertied maximumerror canbeobtainedfortheterm

J(w)

:

J(w) = Θ

r

(w) + ε(w)

(20)

where

Θ

r

isofdegree

r

andfor

w

0 = tan

ν

0

2

wehave:

Θ

r

(w

0 ) = J(w

0 ) = 0

Thecertied maximumerror

ε

¯

providedby thealgorithm in [22]

ε = max

¯

w∈W

ε(w)

isusedin order to obtainupperandlowerpolynomialbounds ontheterm

J(ν)

:

(11)

Θ

l

(w) = Θ

r

(w) − ¯ε

Θ

u

(w) = Θ

r

(w) + ¯

ε

(22)

C. Periodic spacecraft relative motion

Fromexpressions(8)itcanbenoticed thattheinuenceofthenonperiodicterm

J(w)

onthe relativetrajectorydependsonthevalueoftheparameter

d

4

. Aperiodicrelativetrajectorymaybe obtainedbyrequiringtheparameter

d

4

tobezeroandthusremovingthedriftingterm. Imposing

d

4 = 0

in (9)leads toalinearperiodicity constraintontherelativestatestarting from which the

spacecraftmotionispropagated:

M (ν

0 ) X

0 = 0

(23)

wherethematrix

M (ν) ∈ R

1×6

isdenedby:

M (ν) =

0 0 3e cos ν + e

2 _{+ 2 −(1 + e cos ν)}

2 _{0 e sin ν(1 + e cos ν)}

(24)

Assuming that the initial state

X

˜

0

satises the constraint(23), the trigonometric expressions for thepropagationoftheperiodictrajectoryaregivenby:

˜

x(ν) = (2 + e cos ν)(d

p

₁

sin ν − d

p

2 cos ν) + d

p

3 ˜

y(ν) = d

p

5 cos ν + d

p

6 sin ν

˜

z(ν) = (1 + e cos ν)(d

p

₂

sin ν + d

p

₁

cos ν)

,

∀ν ≥ ν

0

(25)

where thevectorofparameters

D

p

= [d

p

1 d

p

2 d

p

3 d

p

5 d

p

6 ]

T

isobtainedfrom thesubstitution in(9) of

theperiodicitycondition(23):

D

p

= C

p

(ν

0 ) ˜

X

0

(26) and:

C

p

(ν) =







0

0 2e cos

_{(1+e cos ν)}

2 ν−e+cos ν

2

0

0 −

sin ν

1+e cos ν

0

0 sin ν(1+2e cos ν)

_{(1+e cos ν)}

2

0

0 cos ν

1+e cos ν

1

0 e sin ν(2+e cos ν)

_{(1+e cos ν)}

2

0

0 2+e cos ν

1+e cos ν

0 cos ν

0 0 − sin ν

0 0 sin ν

0

0 cos ν

0 





(27)

(12)

tothenonperiodiccase. Asexpected, theseexpressionsnolongercontaintheterm

J(w)

:

˜

x(w) =

1 (1 + w

2 ₎

2 P

xp

(w), ˜

y(w) =

1 1 + w

2 P

y

(w), ˜

z(w) =

1 (1 + w

2 ₎

2 P

zp

(w).

(28)

The coecientsof the polynomials

P

xp

(w)

and

P

zp

(w)

depend linearlyon the initial stateof the propagationoftheperiodicmotionthroughthevectorofparameters

D

p

:

p

xp

= C

x

D

p

zp

= C

z

D

p

(29)

wherematrices

C

x

and

C

z

aredened in(18).

Itisimportanttonoticethatintheperiodiccasetheexpressionsfortherelativepositionsarepurely rational. Approximationsare no longer required sincethe drifting term

J(w)

has been removed. Expressions(28) arethesameasthose presentedin [21] andweshowedherethat they arejust a particularcaseofamoregeneralparametrizationofthespacecraftrelativetrajectory.

III. Constrained spacecraft relative trajectory design

Theprevioussectionshowedthatthespacecraftrelativemotionbetweentwoimpulsivecontrols canbeparametrizedwithrespecttothestateimmediatelyaftertherstthrust:

X

˜

+

_(ν

0 ) = ˜

X(ν

0 ) +

˜

B ∆ ˜

V

0

. Thus the choice of

∆ ˜

V

0

plays a crucial role in obtaining a trajectory that continuously

satises aspeciedset ofconstraintsontheintervalbetweenthe twoconsecutivecontrols. Anew method for calculating the impulsive control

∆ ˜

V

0

leading to admissible trajectoriesis presented. The methodis basedonthe previouslydeveloped parametrizationforthe relativemotion. Let us assume that theconstraintson thespacecraftrelativepath canbewritten in thegeneral form of linearinequalities:

V







˜

x(ν)

˜

y(ν)

˜

z(ν)







≤ ˜

K, ∀ν ∈ [ν

0 , ν

f

]

(30) where

V ∈ R

s×3

,

K ∈ R

˜

s

,

s

isthenumberofconstraintsand

K = (1 + e cos ν)K

˜

sincetheusageof thevariablechange(4) mustbetakenintoaccountwhenwritingtheconstraints.

(13)

v

i,1

x(w) + v

˜

i,2

y(w) + v

˜

i,3

z(w) ≤

˜

1 + e + (1 − e) w

2 1 + w

2 k

i

, ∀ w ∈ [w

0 , w

f

], i = 1..s

(31)

Letusdenetherationalexpressions

Ξ

i

(w)

as:

Ξ

i

(w) = −v

i,1

x(w) − v

˜

i,2

y(w) − v

˜

i,3

˜

z(w) +

1 + e + (1 − e) w

2 1 + w

2 k

i

=

1 (1 + w

2 ₎

2 Γ

i

(w), i = 1..s

(32)

where:

Γ

i

(w) = −v

i,1

[P

x

(w) + 3 d

4 P

Jx

(w) J(w)] − v

i,2

P

¯

y

(w) − v

i,3

[P

z

(w) + 2 d

4 P

Jz

(w) J(w)] + k

i

T (w)

(33)

¯

P

y

(w) = (1 + w

2 ), P

y

(w)

andthecoecientsofthepolynomial

T (w)

are

t = [1+e 0 2 0 1−e]

T

. As previouslyshown, thecoecients ofthe polynomials

P

x

(w)

,

P

¯

y

(w)

and

P

z

(w)

depend linearlyon

˜

X

+

_(ν

0 )

, therelativestatefromwhichthesatellitesmotionispropagated:

p

x

= C

x

C(ν

0 )

˜

X(ν

0 ) + ˜

B ∆ ˜

V

0 ¯

p

y

= ¯

C

y

C(ν

0 )

˜

X(ν

0 ) + ˜

B ∆ ˜

V

0 p

z

= C

z

C(ν

0 )

˜

X(ν

0 ) + ˜

B ∆ ˜

V

0

(34) Hence, expressions

Γ

i

(w)

depend linearly on the decision variable

∆ ˜

V

0

. Constraints (30) on the relativetrajectoryaresatisedifthereexists animpulsivecontrol

∆ ˜

V

0

suchthat:

∃ ∆ ˜

V

0

s.t.

Ξ

i

(w) ≥ 0, ∀w ∈ [w

0 , w

f

], ∀i = 1..s

(35)

Sincethecommondenominatorfor

Ξ

i

(w)

is

(1 + w

2 ₎

2

which isnonnegativeforall

w ∈ R

,nding

∆ ˜

V

0

such that the expressions

Γ

i

(w)

are non negativeon the given intervalguarantees that the

constraints(30)arerespected:

∃ ∆ ˜

V

0

s.t.

Γ

i

(w) ≥ 0, ∀ w ∈ [w

0 , w

f

], ∀ i = 1..s

(36)

It has been evidenced in the previous section the fact that the expressions

Γ

i

(w)

are irrational functions. Theusageofthepolynomialapproximationsfortheterm

J(ν)

allowsconstraints(36)to becomepolynomialnonnegativityconstraintswhichcanbereformulatedaslinearmatrixinequality constraints [23]. These latter canbe solved eciently using convex programmingmethods. The uncertaintiesresulting from theapproximationprocess canbedirectly accountedfor. expressions

Γ

i

(w)

contain the integral term

J(w)

, they can be transformed into polynomial expressions by

(14)

A. Thenon-periodic case: using the polynomial approximation ofthedrifting term

J(w)

Expressions(33)canbetransformed into polynomialsby using theapproximation(20). This introducesanunknownbut bounded error

ε(w)

suchthat:

Γ

i

≡ Γ

i

(w, Θ

r

(w), ε(w))

Thusthesatisfactionoftheuncertainconstraints:

∃ ∆ ˜

V

0

s.t.

Γ

i

(w, Θ

r

(w), ε(w))) ≥ 0, ∀ ε(w) ∈ [−¯ε ¯ε], w ∈ [w

0 , w

f

], i = 1..s

(37)

isasucientconditionforthesatisfactionofconstraints(36). Resultsfromconvexrobustanalysis [24]providearobustcounterpartto(37):

∃ ∆ ˜

V

0

s.t.







Γ

l

i

(w, Θ

l

(w)) ≥ 0

Γ

u

i

(w, Θ

u

(w)) ≥ 0







, w ∈ [w

0 , w

f

], i = 1..s

(38) where

Γ

u

i

(w, Θ

u

(w))

and

Γ

l

i

(w, Θ

l

(w))

arethepolynomialsobtainedafter replacingtheterm

J(w)

in (33) with its certied upper and lower polynomial bounds,

Θ

u

(w)

and

Θ

l

(w)

respectively (22).Therefore ndingan impulsive control

∆ ˜

V

0

such that polynomials

Γ

u

i

(w)

and

Γ

l

i

(w)

are non

negativeguarantees thatinitial constraints(36)aresatised, although solvingthis problem intro-ducessomeconservatism.

Reference [23] provides necessary and sucient LMI conditionsto check whether the coecients ofanunivariatepolynomialbelong totheconeofcoecientsofpolynomialsthat arenonnegative on anite interval. Since the coecients

γ

l

i

and

γ

u

i

ofthe polynomials

Γ

l

i

(w)

and

Γ

u

i

(w)

depend

linearlyon

∆ ˜

V

0

,these conditionscanbeusedtondasuitable

∆ ˜

V

0

suchthat theconstraints(38) onthepropagatedtrajectoryaresatised.

Theorem1 (Nonnegativepolynomialonniteinterval).

Let

K

a,b

be the convex, closed and pointed cone of polynomials that are non negative on a nite interval

[a, b] ∈ R

:

K

a,b

= {p ∈ R

n+1

, P (w) =

n

X

i=0

p

i

w

i

≥ 0, ∀ w ∈ [a, b]}

A polynomial

P (w)

,representedthrough its vectorof coecients

p = [p

0 , . . . , p

n

]

T

, belongsto

K

a,b

if andonlyif thereexisttwosymmetricpositive semi-denitematrices

Y

1

and

Y

2

suchthat:

(15)

wherethe operator

Λ

∗

isdenedbelow.

Using thisresult,designingarelativetrajectoryforwhichtheinnitelymanyconstraints(30)are satisedbecomesequivalenttondinganimpulsivecontrol

∆ ˜

V

0

suchthat:

∃ Y

i1

l

0, Y

i2

l

0

s.t.

γ

l

i

= Λ

∗

(Y

i1

l

, Y

i2

l

)

∃ (Y

i1

u

0, Y

i2

u

0)

s.t.

γ

u

i

= Λ

∗

(Y

i1

u

, Y

i2

u

)

, i = 1..s

(40)

Thedenitionoftheoperator

Λ

∗

dependsonwhetherthepolynomial

P (w)

hasanoddoraneven degree. For

n

odd,take

m = (n − 1)/2

and

Y

1 , Y

2 ∈ R

(m+1)×(m+1)

₀

. Let

H

k,i

∈ R

(k+1)×(k+1)

be someHenkelmatrices thatcontainones onthei-th anti-diagonalandzeroseverywhereelse. Then theoperator

Λ

∗

is denedby:

Λ

∗

(Y

1 , Y

2 ) =







tr

(Y

1 (−aH

m,1

)) +

tr

(Y

2 (bH

m,1

))

tr

(Y

1 (H

m,1

− aH

m,2

)) +

tr

(Y

2 (bH

m,2

− H

m,1

))

. . .

tr

(Y

1 (H

m,i−1

− aH

m,i

)) +

tr

(Y

2 (bH

m,i

− H

m,i−1

))

. . . tr

(Y

1 H

m,2m+1

) +

tr

(Y

2 (−H

m,2m+1

))







(41)

For

n

even,take

m = n/2

and

Y

1 ∈ R

(m+1)×(m+1)

₀

,

Y

2 ∈ R

m×m

₀

. Then theoperator

Λ

∗

is dened by:

Λ

∗

(Y

1 , Y

2 ) =







tr

(Y

1 H

m,1

) +

tr

(Y

2 (−abH

m−1,1

))

tr

(Y

1 H

m,2

) +

tr

(Y

2 ((b + a)H

m−1,1

− abH

m−1,2

))

tr

(Y

1 H

m,3

) +

tr

(Y

2 ((b + a)H

m−1,2

− H

m−1,1

− abH

m−1,3

))

. . .

tr

(Y

1 H

m,i

) +

tr

(Y

2 ((b + a)H

m−1,i−1

− H

m−1,i−2

− abH

m−1,i

))

. . . tr

(Y

1 H

m,2m

) +

tr

(Y

2 ((b + a)H

m−1,2m−1

− H

m−1,2m−2

))

tr

(Y

1 , H

m,2m+1

) +

tr

(Y

2 (−H

m−1,2m−1

))







(42)

(16)

Whenconstraints(30)mustbeimposedonaperiodicrelativetrajectory,expressions

Γ

i

(w)

are directly polynomialandnoapproximationisneeded:

Γ

i

(w) = −v

i,1

P

xp

(w) − v

i,2

P

¯

y

(w) − v

i,3

P

zp

(w) + k

i

T (w) ≥ 0, i = 1..s

(43)

Theircoecientscanbewritten directlyasalinearfunction ofthedecisionvariable

∆ ˜

V

0

:

γ

i

= −v

i,1

C

x

− v

i,2

C

¯

y

− v

i,3

C

z

C

p

(ν

0 )

˜

X(ν

0 ) + B ∆ ˜

V

0 + k

i

t

(44)

Note that in order for theexpressions (43)to bevalid theinitial statefor thepropagation of the relativemotionmustsatisfytheperiodicitycondition:

M (ν

0 )

˜

X(ν

0 ) + B ∆ ˜

V

0 = 0

(45)

Ifconstraintsmustbeimposedonthe periodictrajectoryfor anentireperiod,onemust takeinto accountthefactthatthevariablechange(12)mapsthetrigonometriccircleto

R

,aninniteinterval. ThenecessaryandsucientLMIconditionsfornonnegativityofunivariatepolynomialsoninnite intervalsareslightlydierentwithrespecttothecasewhereniteintervalsareconsidered. Reference [23]demonstratesthatapolynomial

P (w)

isnonnegativeon

R

ifandonlyifthereexistsasymmetric positivesemi-denite matrix

Y ∈ R

(m+1)×(m+1)

such that thecoecientsof thepolynomial

P (w)

verify:

p = Λ

∗

(Y )

(46) where:

Λ

∗

(Y )(j) =

tr

(Y H

m,j

), j = 1..2m + 1.

(47)

It isinterestingtonoticethat thepolynomialnonnegativityconstraints(43),alongwith the peri-odicityconstraint (45), describe theset of spacecraftrelativestates that at

ν

0

belong to periodic trajectoriesthatevolveinsidethepolytopedenedby(30). Thissetrepresentsaninvariantsetfor the spacecraftrelativemotion[25]. Thepresentedmethod allowsthe designof impulsive maneu-versleadingto spacecraftrelativetrajectories that respectaset of linearconstraintscontinuously

(17)

positivedenitematrices. Thetechniquecaneasilybeextended toaccommodatealargernumber of impulsive controls

∆ ˜

V

overa larger time horizon. This aspect is detailed in the applications section.

IV. Applications

Themethod developedin theprevioussection canbe usedto designimpulsivemaneuversfor constrained spacecraft relative motion. In this section, we present three types of space missions where our approach can be eective. The hovering mission [9] requires the design of periodic trajectories that evolveinside a specied region. Thistype oftrajectoryshould enablethe visual inspection of agiven target with an innite time horizon for the observation task and with zero fuelcost. Thepassively safe rendezvous mission [7,26] requestsrendezvoustrajectoriesthat guarantee passivecollisionavoidance in caseof anomalous systembehaviour. Lastly, for the vis-ibility constrained rendezvous mission [4, 7] visibility cone constraints must be satised all alongtherendezvouspath. Ifthersttwoscenariosbelongtotheperiodicmotionframework,for therendezvousundervisibilityconstraintsscenariotheconstraintsarecontinuouslyimposedonthe nonperiodicrelativemotion. Ineachofthese scenarios,itisassumedthatthenumberofimpulses

N

,thethrustingpositions

ν

i

andtheinitialrelativestate

X

˜

1

areknown.

A. Generatinga hoveringtrajectory

Thetermhoveringreferstotheabilityofadeputyspacecrafttoremaininaspeciedareaclose tothetargetsatellite[9],inordertoinspectortomonitorit. Thedesignoffuelecientmaneuvers leadingto proximitynaturallyperiodic relativetrajectoriesbetweentwosatellitesis animportant aspect of the on-orbit inspection and on-orbit servicing missions. Moreover, this objective must to beachievedwhile takinginto accountthenecessity ofrestrictingthe evolutionof theresulting periodic trajectory to a specied region of the space. Starting from an initial state

X

˜

1

, a nal state

X(ν

˜

N

)

mustbereached,statethatrespectstheperiodicityconditionandguaranteesthatthe resultingperiodictrajectoryremainsinsideagiventoleranceregion

R

tol

. Thismustbedonewhile minimizingthefuelcostnecessarytoattainthisnalstateandrespectingthesaturationconstraints

(18)

as:

min

∆ ˜

V

P

N

i=1

k∆ ˜

V

i

k

1

s.t.











k∆ ˜

V

i

k ≤ ∆ ˜

V

i

, ∀ i = 1...N

˜

X(ν

1 ) = ˜

X

1 M (ν

N

) ˜

X(ν

N

) = 0

(˜

x(ν), ˜

y(ν), ˜

z(ν)) ∈ ˜

R

tol

, ∀ ν ≥ ν

N

(48)

The stateat theend of the maneuversplan,

X(ν

˜

N

)

, canbeexpressed asafunction of the initial state

X(ν

˜

1 )

andtheimpulsivecontrols

∆ ˜

V

byusingtheYamanaka-Ankersentransition matrix

Φ

:

˜

X(ν

N

) = A

N

∆ ˜

V

N

+ B

N

(49) where:

A

i

= [Φ

ν

i

1 B . . . Φ

˜

ν

i

ν

i

₋

1 B ˜

˜

B] B

i

= Φ

ν

i

ν

1 X(ν

˜

1 ) ∆ ˜

V

i

_{= [∆ ˜}

_V

1 . . . ∆ ˜

V

i

]

T

, i = 1 . . . N

(50)

Forconveniencewechoosethetoleranceregion

R

tol

tobeaboxcentredaroundadesiredposition

X

f

= [x

f

y

f

z

f

]

T

, whose dimensions are dened by

X

tol

= [x

tol

y

tol

z

tol

]

T

. The tolerance box constraintscanbeeasily written in the generallinearform (30)where the

V

and

K

matrices are givenby:

V =







1

0

0 −1 0

0

1

0 0 −1 0

0

1

0 0 −1







,

K =







x

f

− x

tol

−x

f

+ x

tol

y

f

− y

tol

−y

f

+ y

tol

z

f

− z

tol

−z

f

+ z

tol







,

[ν

0 , ν

f

] = [ν

N

, ν

N

+ 2π]

(51)

Followingthepreviouslydevelopedprocedure,theinnitelymanytoleranceboxconstraintscanbe transformed into anite numberof non negativity constraints onsomepolynomialswhose coe-cientsdependlinearlyonthedecisionvariables

∆ ˜

V

:

(19)

Thecoecients

γ

i

ofthepolynomials

Γ

i

(w)

dependlinearlyonthestateattheendofthemaneuvres plan:

γ

i

= t k

i

− v

i,1

C

x

+ v

i,2

C

¯

y

+ v

i,3

C

z

A

N

∆ ˜

V

N

+ B

N

(53)

Thus,usingthepreviouslydevelopedresults,theproblemofndingtheimpulsivecontrols

∆V

i

such that thechaser satellite reaches at theend of the predictionhorizon a trajectorythat isperiodic andcontainedinaspeciedtoleranceregionistransformedinto asemi-deniteprogram:

min

∆ ˜

V ,Z

P

N

i=1

Z

i

s.t.











−Z

i

≤ ∆ ˜

V

i

≤ Z

i

Z

i

≤ ∆ ˜

V

i

, ∀ i = 1...N

˜

X(ν

1 ) = ˜

X

1 M (ν

N

) ˜

X(ν

N

) = 0

∃ Y

i

0

s.t.

γ

i

= Λ

∗

_(Y

i

), ∀i = 1 . . . 6

(54)

Toillustrate thisparticularguidanceproblem, weusetherendezvousmission summarizedintable 1. The purpose is to reach aperiodic parkingrelative orbitby applying thecomputed impulsive maneuvers. Ourmethodforimposingcontinuousconstraintsontheresultingperiodictrajectoryis comparedwithamethodbasedonconstraintsdiscretization[5]. Yalmip[27]alongwiththeSDPT3 solver[28] isfor solvingthe semi-denite programming(SDP) problem (54). Thelinear program (LP) corresponding to the method used for comparison is solved with the linprog function from Matlab.

Table1: Simulationdataforconstrainedperiodicrelativemotion

e

a

[km]

N

X

1

[m,m/s]

t

1

[s]

X

f

[m]

X

tol

[m]

t

N

[s]

∆V

[m/s]

0.023776 7011 10 [1000,50,50,0,0,0] 1282 [100,0,0] [20,10,10] 18808 0.26

The obtained rendezvous trajectory is presented in gure 2. Both methods reach aperiodic trajectory atthe end oftheplan. The dierenceis that, forthe discretization basedmethod, the toleranceregionconstraintsaresometimesviolatedbetweenthevericationpoints. Noconstraints

(20)

100

200

300

400

500

600

700

800

900 1,000 1,100 1,200

−40

−20

0

20

40

x[m] y [m]

Fig.2: Therendezvoustrajectory

violationsoccurwhenusingourmethodsinceitallowsfortheconstraintsontherelativetrajectory tobeimposedcontinuouslyin time.

75

80

85

90

95

100

105

110

115

120

125

130 −10

−5

0

5

10

x[m] y [m] LP10 LP20 LP30 SDP

114

116

118

120

122 −8

−6

−4

−2

0

2

(21)

of pointswhere theconstraintsonthe trajectoryareexplicitlychecked. Theresultingtrajectories when considering 10, 20 and 30 points respectively are depicted in gure 3. Table 2 shows the comparison betweenthefuelcost, thesolvertimeand thetime spentoutsidethetolerance region for each scenario. Increasing thenumber ofverication points reduces theamount of constraints violationsbutitalsoincreasesthesolvertimeandthefuelcost. Foralmostequalfuelcost,theSDP method hastheadvantageofguaranteeingzeroconstraintsviolation.

Table2: ComparisonbetweentheSDPandLP basedmethods

Method LP10 LP20 LP30 SDP

Fuelcost[m/s] 0.48907 0.48922 0.48927 0.48927 Solvertime[s] 0.1972 0.6499 1.6241 0.9325 Timeoutofbounds[s] 1269 737 339 0

B. Orbitalrendezvous underpassivesafety constraints

Animportantissuefortheguidancealgorithmsforspacecraftproximityoperationslikeorbital rendezvous and docking is the ability to handle abnormal system behaviour. The purpose is to ensureasafebehaviourforalargeclass ofpossiblemalfunctions. Passivesafetyimpliesthedesign of rendezvoustrajectoriessuchthat disabling thefollower'sthrusters in theeventof afailurewill cause the satellites to remain on a relative fail trajectory that is guaranteed to be collision free [26]. Securityconstraintsmustbeimposedbothontherendezvoustrajectoryandthepredictedfail trajectoriesinorderto guaranteethiskindofbehaviour.

The passive safetyconstraints can be imposed on any of the

N

stepsof the rendezvous plan but adding too many constraints will increase the total fuel cost of the mission without further improvingtheoverallprobabilityof collision[7]. That iswhythesecurityconstraintswillonlybe enforcedon the last

S

steps of theplan. In order to guaranteethat the fail trajectories

X

˜

fail

are collisionfree,theyaredesignedtobeperiodicandto evolveinsideaspeciedareainproximityof thetarget

X

safe

. This hasto beachievedwhileminimizing theoverallfuelcostofthemission and

(22)

controlproblem canbewritten as:

min

∆ ˜

V

P

N

i=1

k∆ ˜

V

i

k

1

s.t.











k∆ ˜

V

i

k ≤ ∆ ˜

V

i

, ∀i = 1 . . . N

˜

X(ν

1 ) = ˜

X

1 M (ν

i

) ˜

X(ν

i

) = 0

˜

X

i

fail

(ν) ∈ X

safe

, ∀ν ≥ ν

i

, ∀i = N − S . . . N − 1

˜

X

f

− ˜

X

tol

≤ ˜

X(ν

N

) ≤ ˜

X

f

+ ˜

X

tol

(55)

The fail trajectories

X

˜

fail

are obtained through the open loop propagation of the spacecraft au-tonomous relativemotionstarting fromthe stateson therendezvoustrajectorythat mustbe ren-dered passively safe. The safearea

X

safe

is considered to be an open polytopebehind thechaser dened by:

˜

x

i

f ail

(ν) ≤ ˜x

safe

,

∀ν ≥ ν

i

,

∀i = N −S . . . N −1

(56)

Followingthesameprocedure asbefore,constraints(56)canbewritten as:

Γ

i

(w) ≥ 0,

∀w ∈ R

with:

Γ

i

(w) = T (w) x

safe

− P

xp

i

(w),

∀i = N −S . . . N −1

(57)

where

P

i

xp

(w)

isthepolynomialforthepropagationoftheautonomousperiodicmotiononthe

x

axis

starting from each state

X(ν

˜

i

)

. Inaddition, thestate

X(ν

˜

i

)

mustverifythe periodicitycondition (23). Thecoecients

γ

i

ofthepolynomials

Γ

i

(w)

canbewrittendirectlyasalinearfunctionofthe decisionvariables

∆ ˜

V

i

:

γ

i

= −C

x

C

p

(ν

i

)

A

i

∆ ˜

V

i

+ B

i

+ x

safe

t,

∀ i = N −S . . . N −1

(58)

(23)

min

∆ ˜

V ,Z

P

N

i=1

Z

i

s.t.











−Z

i

≤ ∆ ˜

V

i

≤ Z

i

Z

i

≤ ∆ ˜

V

i

, ∀ i = 1 . . . N

˜

X(ν

1 ) = ˜

X

1 M (ν

i

)

A

i

∆ ˜

V

i

+ B

i

= 0

∃ Y

i

0

s.t.

− C

x

C

p

(ν

i

)

A

i

∆ ˜

V

i

+ B

i

+ x

safe

t = Λ

∗

(Y

i

)

, ∀ i = N −S . . . N −1

˜

X

f

− ˜

X

tol

≤ A

N

∆ ˜

V

N

+ B

N

≤ ˜

X

f

+ ˜

X

tol

(59) The data presented in table 3 corresponds to the rendezvous and docking mission with passive securityconstraintsthat isconsidered forillustration.

In order to identify asuitable valuefor the security horizon

S

, therendezvous mission without

Table3: Simulationdatafortherendezvousmission withpassivesecurityconstraints

e

a

[km]

N

X

1

[m,m/s]

t

1

[s]

X

f

[m,m/s]

v

tol

[m/s]

t

N

[s]

x

safe

[m]

0.023776 7011 15 [-30,0,-3,0,0,0] 0 [-5,0,0,0,0,0] 0.01 5843 -5

securityconstraintsissolvedrst. Thefailtrajectoriesarepropagatedstartingfromeverycontrolled stateon thesecond half ofthe rendezvousplan. Theresultingtrajectoriesare presentedin gure 4. Thestateswhich originatefail trajectoriesleadingto a collisionbetweenthe twosatellites are then included in the safetyhorizon

S

. Figure 4suggeststhat a securityhorizon of

S = 4

should removethecollisionriskincaseofsystemfailure(pleasenotethatsomefailtrajectoriesoverlapfor thestateswhere theoptimized

∆V

equalszero).

Figure 5presentsthe fail trajectoriesobtainedwhen the passivesecurity constraints areenforced in thecontrol synthesis problem for asecurity horizon of

S = 4

. The fail trajectories areindeed periodic and evolvein the security areadened by

x

safe

, removingthe risk of collision in case of systemerror.

(24)

−40

−35

−30

−25

−20

−15

−10

−5

0

5 −4

−2

0

2

4

6

8

x[m] z[m]

−10

−8

−6

−4

0

2

4

Fig.4: Therendezvoustrajectorywithoutthesecurityconstraints(the dashedlines representthe fail trajectoriesstartingfromthepointsevidencedbythe*symbol)

Table 4 shows the evolution of the mission fuel cost with respect to the choice of the security horizon

S

. Asexpectedthefuelcostincreasesasthesecurityhorizonincreasessincemoreandmore constraintsareaddedtotheproblem.

Table4: Evolutionofthemissionfuelcostwiththelengthofthesecurityhorizon

S

0 1 2 3 4 5 6 7

Fuelcost[m/s] 0.0116 0.0121 0.0135 0.0146 0.0156 0.0163 0.0168 0.0174

The security constraints considered in this example were very simple in order to highlight the principle of the method without too much formal complexity. The presented method can easily accommodatedierentdescriptionsofthesaferegion,likeforinstance avisibilityconeasin[7].

C. Orbital rendezvousundervisibility constraints

(25)

− 40

− 35

− 30

− 25

− 20

− 15

− 10

− 5

0

5

10 − 8

− 6

− 4

− 2

0

2

4

6

8 x[m]

z[m]

− 5.8

− 5.6

− 5.4

− 5.2

− 5

− 4.8

− 1

0

1

Fig.5: Therendezvoustrajectorywithsecurityconstraints(thedashedlinesrepresentthefail trajectoriesstartingfromthepointsevidencedbythe*symbol)

theuseofthenonnegativepolynomialsin imposingcontinuousconstraintsonnonperiodic space-craftrelativetrajectories. Thechosenmissionrequiresforthefollowertoremaininsidetheleader's visibilityconefortheentiredurationoftherendezvousanddockingmaneuvers.Theoptimalcontrol problemcanbeformulatedasfollows:

min

∆ ˜

V

P

N

i=1

k∆ ˜

V

i

k

1

s.t.











k∆ ˜

V

i

k ≤ ∆ ˜

V

i

, ∀ i = 1...N

˜

X(ν

1 ) = ˜

X

1 (˜

x(ν), ˜

y(ν), ˜

z(ν)) ∈ X

vis

, ∀ ν ∈ [ν

1 , ν

N

]

˜

X(ν

N

) = ˜

X

f

(60)

(26)

guaranteedforeach timesegmentbetweentwothrustpositions

[ν

i

, ν

i+1

]

:

min

∆ ˜

V

P

N

i=1

k∆ ˜

V

i

k

1

s.t.











k∆ ˜

V

i

k ≤ ∆ ˜

V

i

, ∀ i = 1...N

(˜

x(ν), ˜

y(ν), ˜

z(ν)) ∈ X

vis

, ∀ ν ∈ [ν

1 , ν

2 ]

(˜

x(ν), ˜

y(ν), ˜

z(ν)) ∈ X

vis

, ∀ ν ∈ [ν

2 , ν

3 ]

. . .

(˜

x(ν), ˜

y(ν), ˜

z(ν)) ∈ X

vis

, ∀ ν ∈ [ν

N −1

, ν

N

]

˜

X(ν

N

) = ˜

X

f

(61)

The set

X

vis

is associated to the visibility cone of the leader's camera. It is represented by an openpolytopebehindtheleadersatellite(gure6), denedbytheaperture angle

β

andtheoset distance

x

safe

betweenthedockingportoftheleadersatelliteanditscenter ofgravity.

PSfragreplacements

β

M

L

M

F

Visibility cone

x

saf e

~x

L

~z

L

~

y

L

x

z

Fig.6: Theleader spacecraftvisibilitycone

(27)

K

matricesdenedby:

V =







1 0

ρ

1 0 −ρ

1 ρ

0 1 −ρ 0

1 0

0 





K =







0

0 x

safe







.

(62) and

ρ = tan(

π

2 − β)

.

Expandingthevisibilityconeconstraintsleadstopolynomialnonnegativityconstraintssimilar to(36). Forthesakeofcompleteness,theprocedurefortherstvisibilityconstraintisfullydetailed:

−˜x(ν) − ρ ˜z(ν) ≥ 0, ∀ ν ∈ [ν

i

, ν

i+1

], ∀ i = 1 . . . N −1

(63)

Usingthevariable change(12)andtheexpressions(13)forthepropagationof therelativemotion startingfromeachcontrolledstate

X(ν

˜

i

)

ontherendezvoustrajectory,therstvisibilityconstraint canfurtherbetransformedin:

Γ

i

1 (w) ≥ 0, ∀ w ∈ [w

i

w

i+1

], ∀ i = 1 . . . N −1

(64) where:

Γ

i

1 (w) = −

P

x

i

(w) + 3 d

i

4 P

Jx

(w) J

ν

i

(w)

−ρ P

z

i

(w) + 2 d

i

4 (P

Jz

(w) J

ν

i

)

≥ 0, ∀ w ∈ [w

i

, w

i+1

], ∀ i = 1 . . . N−1

(65) Theterm

J

i

ν

(w)

in(65)isreplacedoneachsegment

[w

i

, w

i+1

]

byitsupperanditslowerpolynomial bounds,

Θ

i

u

(w)

and

Θ

i

l

(w)

respectively. This leads to the following polynomial non negativity constraints:

Designing Continuously Constrained Spacecraft Relative Trajectories for Proximity Operations

HAL Id: hal-01078528

https://hal.archives-ouvertes.fr/hal-01078528

Submitted on 29 Oct 2014

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