ControlPredictivoparaRendezvous deVehículos

EspacialesRafaelVázquezValenzuela

FranciscoGavilán,E.F.CamachoJorgeGalánVioque

IICongresodeIngenieríaEspacialMadrid,23denoviembre

Aerospace Engineering at University of Seville • AerospaceEngineeringstartedin2002,intheUniv.OfSevilleSchoolofEngineering(strongtradi=oninMechanical,ElectricalandChemicalEngineeringsincethe60s)

•  TeachesUndergraduateDegree,Master’sDegreeandalsoPhD• Mainlyfocusedinaeronau=cs(atradi=onalindustryinSeville)buttherearecoursesinOrbitalMechanics,SpacecraIDynamicsandSpacecraIPropulsion

•  TheDepartmentofAerospaceEngineeringwascreatedin2006•  Somelinesofresearchrelatedtospacehaveemerged,incollabora=onwithotherdepartments

Researchers

• RafaelVazquez,SergioEsteban,FranciscoGavilan:AerospaceEngineeringDepartment

•  EduardoF.Camacho:SystemsEngineering&Automa=onDepartment•  JorgeGalan:AppliedMathema=cs

Past and Ongoing Projects

•  Threemainareas:•  Planningandschedulingforspacemissions(R.Vazquez,J.Galan,JMMelendezandothers).Problemsrelatedtoop=miza=on.Collabora=onbetweenengineers,mathema=ciansandindustry(TaitussoIware).

•  Rendezvousandforma=onflyingofspacevehicles(R.Vazquez,E.F.Camacho,F.Gavilan,J.Sanchez,JMMon=llaandothers).Analysisandcontrolofrela=veposi=onandvelocityforclosespacecraI.Applica=onsofcontroltheory.Collabora=onbetweenaerospaceengineersandcontrolengineers.

•  Iner=a-freeandadap=veaWtudecontrollaws(S.Esteban,M.Camblorincollabora=onwithU.Michigan).Analysisandsynthesisofautoma=caWtudecontrollawsthatdonotrequireknowledgeofspacecraImassdistribu=on.

• Alltheselineshavealreadyproducedjournalandconferencepapers.

Past and Ongoing projects

• Besidesthese,otherseminallinesyettoproducepublica=ons:•  Applica=onofsta=s=caltoolstostudyuncertaintyinorbitalmechanicsproblems,suchasorbitdecay(R.Vazquez,FJCamacho)

•  Analysisofgroundtrakself-intersec=onsforgeocentricsatellites(R.Vazquez,E.Teson,J.Galan)

•  Applica=onsofthethree-bodyproblemdynamicstomissiondesign(J.Galan,R.Vazquez)

Planning and scheduling for space missions •  LineofresearchthatstartedwithcontactswiththecompanyTaitusSoIware(orbitalmechanicssimulatorSaVoir).

•  Thiscompanyhasproposedanumberofproblems(relatedtoplanningandschedulingforearthobserva=onsatellites):

•  Swathacquisi=onplanningproblemformul=-satelliteEarthobserva=onmissions.

•  DesignofanautomatedtooltotomanagetheAntenna-Satelliteassignmentproblem.

•  Planningofacquisi=onsforagilesatellitesinsnapshotmode.

Swath acquisition planning problem for multi-satellite Earth observation missions The problem: swath acquisitions with "memory"

How close is a feasible solution to the optimal one?

IWPSS, 9th June 2011

Planning of acquisitions for agile satellites in snapshot mode. • Agilesatellites:allowrota=oninalldirec=onstoacquireimages•  Snapshotmode:individualimagesforselectedcoordinates

Planning of acquisitions for agile satellites in snapshot mode.

3 ALGORITMO DE BÚSQUEDA DE PUNTOS (ABP)

xR0 50 100 150 200 250 300 350 400

yR

-150

-100

-50

0

50

100

150

V = 1.2 UV, h = 100 UD,Ωmax = 1.5 /UT, tAF = 10 UT,Ψ = 30

AP : T iempo optimo mınimo−Npa = 3−Nprof = 0− n = 40

Nube de puntos

Punto de partida

Punto actual

T raza de camara

Curva lımite actual

P osicion del veıculo

P. candidatos

P. analizados

P. observados

P. elegido

Figura 3.4.3: Ejemplo de aplicación del criterio de tiempo óptimo mínimo (problema plano)

xR0 50 100 150 200 250 300 350 400

yR

-150

-100

-50

0

50

100

150

V = 1.2 UV, h = 100 UD,Ωmax = 1.5 /UT, tAF = 10 UT,Ψ = 30

AP : Desviacion mınima−Npa = 3−Nprof = 0− n = 40

Nube de puntos

Punto de partida

Punto actual

T raza de camara

Curva lımite actual

P osicion del veıculo

P. candidatos

P. analizados

P. observados

P. elegido

Figura 3.4.4: Ejemplo de aplicación del criterio de desviación mínima (problema plano)

En las figuras 3.4.1, 3.4.2, 3.4.3 y 3.4.4 se muestran algunos ejemplos de ejecución del algoritmo para losdistintos criterios del algoritmo de prioridad explicados anteriormente.

60 / 137

Analysis of groundtrack self-intersections for geocentric satellites •  Satelliteswithrepea=nggroundtrackmayormaynothavepointsofself-intersec=ondependingonorbitalelements.

•  Self-intersec=onareofinterestbecausetheyrepresentpointsofopportunityforrepeatedcommunica=onand/orobserva=on

Solución del caso impar

5.3. Verificación de los resultados obtenidos Para verificar que el cálculo de los puntos de cruce es correcto basta con representar la longitud y latitud de éstos para una inclinación cualquiera sobre la traza correspondiente. Para ello, de nuevo se ha aplicado el método numérico descrito en el apartado 5.2.1.1.

Para los ejemplos tratados hasta ahora se obtienen los siguientes resultados:

Para k=3, m=1 y 𝑖 = 44,6°, 101,11° se obtienen, respectivamente:

Para k=5, m=1 y 𝑖 = 44,5°,101,11° se obtienen, respectivamente:

Para k=5, m=3 y 𝑖 = 44,5°,101,11° se obtienen, respectivamente:

Figura 5-13. Traza con k=3, m=1, 𝐺𝑆𝑇0 = 𝜃0 = 𝑤 = 𝛺 = 0 y 𝑖 = 44,6°, 101,11° (de izquierda a derecha).

Figura 5-14. Traza con k=5, m=1, 𝐺𝑆𝑇0 = 𝜃0 = 𝑤 = 𝛺 = 0 y 𝑖 = 44,5°, 101,11° (de izquierda a derecha).

Figura 5-15. Traza con k=5, m=3, 𝐺𝑆𝑇0 = 𝜃0 = 𝑤 = 𝛺 = 0 y 𝑖 = 44,5°, 101,11° (de izquierda a derecha).

Solución del caso impar

Además, se representa en la figura 5-32 la verificación de los puntos de cruce para altos números de m y k (m=9 y k=35) para una inclinación de 𝑖 ≅ 𝜋

2.

Finalmente, atendiendo a los ejemplos mostrados, se concluye que el método de cálculo tanto analítico como numérico desarrollado en este capítulo permite calcular con gran exactitud el número y localización (latitud y longitud) de los puntos de cruce para órbitas circulares con m y k impar.

Figura 5-31. Traza con k=9, m=7, 𝐺𝑆𝑇0 = 𝜃0 = 𝑤 = 𝛺 = 0 y 𝑖 ≅ 90° y zoom de dicha traza en torno al polo (de izquierda a derecha).

Figura 5-32. Traza con k=35, m=9, 𝐺𝑆𝑇0 = 𝜃0 = 𝑤 = 𝛺 = 0 y 𝑖 ≅ 90° y zoom de dicha traza en torno al polo (de izquierda a derecha).

Analysis of groundtrack self-intersections for geocentric satellites

•  Inthecircularcasetheal=tude(whichdeterminerepea=ngproper=es)andinclina=onarethekeyvalues.

•  Inthecircularcase,wehavedevelopedalgorithmsthatgivetheexactnumberofself-intersec=onandtheirloca=ons.Thesealgorithmsmayhelpinorbitdesign.

105

105

Estudio detallado de los puntos de cruce de las trazas de satélites geocéntricos

7.5.1.2. Resultados obtenidos

En la figura 7-5 se muestra la evolución del número de puntos de cruce para las posibles órbitas heliosíncronas que se encuentran entre 300 y 20000 Km de altitud (rango de altitudes de las órbitas bajas) con 𝑚𝑚𝑎𝑥 =15. La evolución, aunque parece caótica, puede apreciarse que sigue un cierto patrón en el comportamiento, disminuyendo el máximo número de puntos de cruce a medida que aumenta el semieje mayor de la órbita.

En cuanto a las latitudes menores a las del círculo polar ártico representadas en la figura 7-6, su evolución frente al semieje mayor también sigue un patrón de comportamiento. Esta representación resulta interesante ya que muestra que se pueden tener puntos de cruce para órbitas heliosíncronas pero no necesariamente en las proximidades del polo.

Finalmente destacar que, a medida que 𝑚𝑚𝑎𝑥 aumenta, aumentan el número de puntos de cruce pero éstos se revisitarán más tarde por lo que hay un cierto compromiso entre el número de puntos de cruce y el tiempo de revisita.

Figura 7-5. Evolución del número de puntos de cruce frente al semieje mayor para las órbitas heliosíncronas con traza repetida entre 300 y 2000 Km y 𝑚𝑚𝑎𝑥 =15.

Figura 7-6. Evolución de las latitudes menores de 66º44’36’’ de los puntos de cruce frente al semieje mayor para las órbitas heliosíncronas con traza repetida entre 300 y 2000 Km y 𝑚𝑚𝑎𝑥 =15.

Rendezvous and formation flying of space vehicles

−20−10

010

20 0

20

40

60

−20

−15

−10

−5

0

5

10

15

20

y [m]

3D chaser path

x [m]

z [m

]

Starting point

Docking

LOS constraints

Safe zone

Forbiddenarea

Figure 4: Path followed by the chaser spacecraft (δ1 : δ = [0.2592 0.8065 − 0.0533] ·

10−4,√

Σii = 1 · 10−5; δθ : δi = 0.0436,√

Σii = 0.0436). Controller parameters are:

Np = 60, γ = 1000, ka = 60, p = 0.95, λ = 0.23.

21

Examples: Space Shuttle

the target vehicle’s status. If it has active sensing capabilities (i.e.,contains a transponder), the rendezvous radar system can operate atranges from 555 km to 30 m. If the target has passive sensing wherethe radar is simply reflected off the target vehicle, the rendezvousradar has a range of 22 km to 30m [4]. There are also three additionaltools available on the space shuttle to help the astronauts navigateduring the rendezvous and docking phase. Mounted in the orbiter’spayload bay is a laser ranging device that provides range, range rate,and bearing to the target for display to the crew at distances varyingfrom1.5 km to 1.5m. There is also a centerline camera attached to thecenter of the orbiter’s dockingmechanism.When the shuttle iswithin90 m of the target, it generates images that serve as a visual aid to thecrew for docking. Also available to the crew is a hand-held laserranging device which can measure range and range rate duringapproach to complement the other navigation equipment.

The aggressive requirements for the shuttle necessitated theorbiter to have the capability to rendezvous, retrieve, deploy, andservice multiple targets that had different sizes, possessed varyingdegrees of navigational aids (transponders or lights), and in manycases were not designed (or functioning) to support these operations[29]. In addition, when the shuttle wasn’t visiting one of the differentspace stations itwas typically larger than its rendezvous target, whichoften contained sensitive payloads. With Apollo and Gemini plumeimpingement issues were not significant, but for the shuttle seriousconsiderations regarding contamination and induced dynamics onthe target had to be faced. As a consequence, the approach trajectorywas redesigned. Instead of the direct approach as performedpreviously, the shuttlewould transition to a station-keeping point andthen perform one of a variety of possible final approach trajectories.

A typical rendezvous scenario for the shuttle to the InternationalSpace Station is as follows [30]. The ground controllers compute thenecessary phasingmaneuvers to get the space shuttle within 74 kmofthe target, as shown in Fig. 4 (point A). From this moment on, eitherthe shuttle’s guidance, navigation, and control (GNC) systemautomatically calculates and executes the remaining maneuvers orthe flight crew manually guides the spacecraft. Initially the onboardGNC system has control and automatically executes the firstmaneuver that transfers the crew to a specified point about 15 kmbehind the target (point B) in preparation for the terminal-phaseinitiation maneuver. Once the shuttle executes this initiation burn toplace itself near the target, it enters a trajectory that will pass

underneath the target (point C) and place it in front of the target(point D). Shortly before the shuttle passes underneath the target, theastronauts assume control over the vehicle. Using hand controls anddisplays, the crewmembers will manually guide the shuttle until it issecurely docked.

There are two common final approach modes used by the shuttle:the v-bar or r-bar approaches. (In the local vertical local horizontalreference frame, the v bar generally points along the target’s velocityvector and is commonly known as the downrange or local horizontalaxis, whereas the r bar refers to the relative altitude or local verticalaxis and points radially upward.) If the r-bar approach is selected, animpulsive maneuver in the negative downrange direction isperformed when the shuttle crosses the r bar (point C) reducing theforward velocity. Because of orbital mechanics, the shuttle willnaturally follow a course which crosses the r bar again. At this pointanother impulsive maneuver directed up and in the negativedownrange direction causes the shuttle to slowly hop itsway up to thetarget. For a v-bar approach the shuttle transfers to a point downrangefrom the target (point D). The final approach begins with a change ofvelocity toward the target. To remain on the v bar, an upward !v isrequired causing the shuttle to slowly hop toward the target. It willgradually move toward the target at a controlled rate proportional tothe relative range distance [v! "range=1000# m=s] until thevehicles have docked.

B. Soyuz/Progress

The man-rated Soyuz vehicle and the cargo carrying Progressvehicle are Russia’s work horses for space station activity. Initiallythese vehicles were equipped with the Igla rendezvous and dockingsystem but in the mid-1980s the Soviet space program replaced theIgla systemwith the newKurs (Course) system.During the transitionto the new system, the Mir space station actually incorporated both;the Kurs system from one docking port and the Igla system fromanother. Currently the Kurs system supports the rendezvous anddocking efforts at the International Space Station. It provides all ofthe necessary relative navigation information from target acquisitionto docking which includes range, range-rate, line-of-sight angles,and relative attitude measurements. Some of the noticeable changesbetween the Igla andKurs systems are that Kurs uses a different set ofantennas, allows for acquisition and maneuvering at much greater

V-bar Altitude

Downrange

Altitude

DownrangeA B

C

D

C

D

R-bar

v v v v v

vv

v

v

v

v

v

v

vv

v

Fig. 4 Shuttle orbital rendezvous scenarios.

WOFFINDEN AND GELLER 903

Di↵erent phases according to proximity.

Close approach is manual.

Examples: Soyuz

which hemisphere the target is located in (see Fig. 5). If needed, anattitude maneuver is initiated to ensure the spacecraft is properlypointed in the direction of the target. Once the Kurs system knowswhich hemisphere the target is in and the Soyuz is orientedappropriately, the scanning antenna (A3) is activated to determinemore precisely the pointing direction to the target. Eventually thedistance and orientation of the Soyuz spacecraft with respect to thetarget is sufficient to allow the main tracking antenna (A4) tointerrogate the target to obtain range and range-rate information.With this additional information, the Kurs system updates theestimated position of both vehicles and executes a correctionmaneuver (M5).

To ensure a smooth braking velocity profile when approaching thetarget vehicle, three impulsive maneuvers are implemented (M6–M8) as shown in Fig. 7. The first one (M6) occurs when the Soyuz isabout 1 km below the target’s orbit. Following the last maneuver(M8), it is likely that the current approach trajectory is not alignedwith the target’s docking port. To position itself along the target’sdocking axis for the final approach, the Soyuz performs a fly-aroundat a relative distance between 200–400 m. Regardless of whetherdocking axis is pointed along the v bar, r bar, or some inertially fixedaxis, the Soyuz begins transferring to intersect this final approachline. During the fly-around, the scanning antenna on the Soyuz (A4)tracks the target antenna B3 to obtain range, range-rate, and line of

Altitude

Downrange

M2

M1M0

M3 M4

M5

Launch

Kurs

Fig. 6 Phasing and rendezvous sequence for Soyuz/Progress vehicles.

Altitude

Downrange

M6

M7

M8

Fig. 7 Final approach sequence for Soyuz/Progress vehicles.

WOFFINDEN AND GELLER 905

Russian approach: Kurs system.

Used for MIR.

Automatic system, does not requiretarget actions (but requires systeminstalled on-board target).

Weight of 85 kg., 270 Watts ofconsumption (both on target andchaser).

Examples: ATV

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(39km, 5km)

V-Bar

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S1S0

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ESC 1

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S-1/2

(39km, 5km)

V-Bar

R-Bar

(3500m, -100m)250 m

S1S0

S2 S3

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Approach Ellipsoid (AE)

Keep Out Zone (KOZ)

Proximity link (R~30km)

ESC 1

ESC 2

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! &'(! ^((-&%3! 5'0*(! %(\1+%()! 0-! D7P! 1-)$@A+-,! $-! &'(! N&'! $.! =(5&(/8(%! >8(.$%(! &'(!*@'()16()!V%$,%(**!"JV!)$@A+-,B4!0*!C(66!0*!&$!&0%,(&!0!%((-&%3!0%(0!+-!(@6+5*(!$5&+/+S+-,!&'(!0+%?560-(!$8*(%20&+$-_!&'+*! +/56+()!0!"!C((A*!+-?.6+,'&!)1%0&+$-!15! &$!&'(!%((-&%3!+-*&(0)!$.!$-(!$%!&C$!)03*!.$%!&'(!,(-(%+@!/+**+$-4!

! &'(!^((-&%3!5'0*(!%(\1+%()!0-!D7P!%(?5'0*+-,!C+&'!R==!066$C+-,!&'(!0*&%$-01&*!&$!$8?*(%2(!&'+*!5'0*(!&'0-A*!&$!#RD`aD!(T5(%+/(-&!b9c4!bHcK!

7'(!^(-)(S2$1*!5%$T+/+&3!$5(%0&+$-*!)1%+-,!&'(!E(/$E03*!9!0-)!H!0*!C(66!0*!&'(!/+**+$-!)(*+,-!+/56+@0&+$-!%(\1+%()!83!&'(!R==!*0.(&3!'02(!8((-!)(*@%+8()!+-!5%(2+$1*!505(%*!b9c4!bHc4!b"c4!bOcK!

ATV (Automated Transfer Vehicle) fromESA carries supplies to ISS and at theend of its lifetime is ejected elevating theorbit (and carrying waste away).

Uses relative GPS and follows apre-programmed maneuver.

Additional pre-programmed maneuversfor all possible scenarios.

5 ATV vehicles had been launched (JulesVerne, Johannes Kepler, Edoardo Amaldi,Albert Einstein, Georges Lemaıtre ).

Automatic Rendezvous

The problem of designing an automatic rendezvous system oflow weight and consumption is still open and an active field ofresearch.

This would be specially attractive for small satellites thatcould rendezvous with larger spacecraft, such as spacestations.

Rendezvous and formation flying of space vehicles •  ThiswasoneofthelinesofresearchinF.GavilanPhDThesis(2012).• Wehaveconsideredtwomainproblems:

•  Safe&op=malrendezvoussubjecttomodeluncertain=es•  Fastalgorithmsforrendezvouswithon-offthrusters

•  Theproblemcanbemodeledbylinearequa=onswhenvehiclesincloseproximitybutis=me-varying(Tschauner-Hempelmodel)

• On-offthrustersmaketheproblemnonlinear

Safe & optimal rendezvous subject to model uncertainties •  Problemclassicalysolvedassumingeverythingisknown.•  Ifthereareuncertain=es,togetarobustcontroller,onehastosolveamin-maxproblem:minimizefuelconsump=onandverifysystemconstraintsevenforworstpossibleuncertain=es

•  Uncertain=eshavetobees=matedon-line•  UseofModelPredic=veControl

−20 −10 0 10 200

2

4

6

8

10

12

14

16

18

20

x [m]

y [m

]

XY plane trajectory

0 1000 2000 3000−0.1

0

0.1

u x/um

ax

Thrust

0 1000 2000 3000−0.1

0

0.1

u y/um

ax

0 1000 2000 3000−0.2

0

0.2

u z/um

ax

Time [s]

Figure 6.2: Robust MPC with thruster disturbance (δ1 ∼ N3(5 · 10−5, 0.5 · 10−5),δθ ∼ N3(0.0436, 0.0436)). Controller parameters are set to: Np = 60, γ = 1000,ka = 60, p = 0.95, λ = 0.23. Mission cost 0.4382 m/s

−10 −5 0 5 100

2

4

6

8

10

12

14

16

18

20

x [m]

y [m

]

XY plane trajectory

0 500 1000 1500 2000 2500 3000 3500−0.1

0

0.1

u x/um

ax

Thrust

0 500 1000 1500 2000 2500 3000 3500−0.2

0

0.2u y/u

max

0 500 1000 1500 2000 2500 3000 3500−0.1

0

0.1

u z/um

ax

Time [s]

Figure 6.3: Robust MPC with unmodeled dynamics and thruster disturbance (δ1 ∼N3(5 · 10−5, 0.5 · 10−5), δθ ∼ N3(0.0436, 0.0436)). Controller parameters are set to:Np = 60, γ = 1000, ka = 60, p = 0.95, λ = 0.23. Mission cost 0.4827 m/s

21

Model Predictive Control

t t+1 t+2

t+N t t+1 t+2 !!.. t+N

u(t)

Only the first control move is applied

Errors minimized over a finite horizon

Constraints taken into account

Model of process used for predicting

Model Predictive Control

t+2 t+1

t+N

t+N+1

t t+1 t+2 !!.. t+N t+N+1

u(t)

Only the first control move is applied again

MPC vs PID

MPC

PID: u(t)=u(t-1)+g0 e(t) + g1 e(t-1) + g2 e(t-2)

vs. PID

Rendezvous with on-off thrusters

• Developmentofanalgorithm(PWM-MPC)whichconsiderrealis=cthrustersoranyothertypeof=me-varyingactua=on(e.g.Rota=ngspacecraI)

•  Twomainingredients:•  Hotstartforop=miza=oncomputedfromimpulsiveactua=on•  Improvementthroughsuccessivelineariza=onsun=lconvergenceorcomputa=on=meisup.

Rendezvous with on-off thrusters

x [km]0 0.1 0.2 0.3 0.4 0.5

y[k

m]

0

0.1

0.2

0.3

0.4

0.5Starting PointPWM-MPCImpulsive-MPCOpen loopLOS constraint

Restricted area

Safe area

Constraintviolation

Figure 4: System trajectories in the target orbital plane: open-loop PWM inputs computed

from impulsive solution (dashed), closed-loop Model Predictive Control with PWM inputs

using impulsive model (dot-dashed), and closed-loop Model Predictive Control with PWM

inputs using the PWM planning algorithm (solid).

where Rk is defined as

Rk = h(k ka)I66

(87)

where, h is the step function and ka is the desired arrival time for rendezvous.

Notice that the dimensions of each block matrices are shown for disambiguation.

The reason for choosing (87) is that it is desired to arrive at the origin at time

ka (and remain there) and at the same time minimize the control e↵ort. In the

sequel ka = Np, so the spacecraft are expected to rendezvous after 50 minutes.

In the simulations three algorithms were considered: first, an impulsive open-

loop trajectory planner, as described in Section 4.1. Next, closed-loop simula-

tions using MPC but considering impulsive instead of PWM actuation in the

model (this algorithm is denoted as impulsive MPC). Finally, closed-loop sim-

28

Rendezvous of spacecraft •  Otherprospec=velines:

•  RendezvousforasmallchaserspacecraIwithonly1or2thrusters,andaWtudecontrol(JSanchezMaster’sThesis).ChallengingiftheaWtudeactuatorsaturates.Mixesrela=vedynamicsandaWtudedynamics.

•  Rendezvouswithanasteroid(JMMon=lla’sDegreeThesis).Considersatriaxialasteroid,majoraxisspinner.Toavoidimpact,atangentplane(rota=ngwiththeasteroid)canbeusedtoestablishaconstraint.

Thanks!

Ques=ons?