spacecraft formation control using analytical integration

Spacecraft Formation Control usingAnalytical Integration of GVEICATT 2016

Mohamed Khalil Ben Larbi, 14.-17.03.2016

Introduction Gauss’ variational equations results Conclusion and outlook

Contents

IntroductionMotivationWhy ROE

Gauss’ variational equationsGVEControl scheme

Results and evaluationFormation reconfiguration and keeping

Conclusion and outlook

14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 2Spacecraft Formation Control using Analytical Integration of GVE

Introduction Gauss’ variational equations results Conclusion and outlookMotivation Why ROE

Motivation

Idea: derive a formation geometry guaranteeingminimum collision risk (passive safety)

minimum number of correction maneuvers (passive stability).

Client Satellite

Client inertial orbit

Servicer inertial orbit

Servicer Satellite

relative orbit

Challenge for active debris removalUncertainties in along-track

=⇒ Separation in RN plane for safe formation

Motivation

Client Satellite

Servicer Satellite

relative orbit

Motivation

Client Satellite

Servicer Satellite

relative orbit

Formation description

Relative orbital elements

δα =

δaδλ

δixδiy

(a2 − a1) a−11

(u2 − u1) + (Ω2 −Ω1) cos i1ex2 − ex1

ey2 − ey1

i2 − i1(Ω2 −Ω1) sin i1

with u = M +ω, ex = e cosω and ey = e sinω.

E/I polar notation

δe = δe(cosφ sinφ)T and δi = δi(cos θ sin θ)T

Eccentricity/Inclination Separation

Relative trajectory (without Drift)

E/I separation

safe formation for δe ‖ δi

δαnom =(δanom δλnom 0 −‖δenom‖ 0 +‖δinom‖

)T14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 5Spacecraft Formation Control using Analytical Integration of GVE

Why relative orbital elements

Insight into the formation geometry (E/I separation)

client

servicer

Maintains decoupling of in-plane and out-of-plane motionMore accuracy (retaining higher order terms)Adoption of Gauss variational equations (GVE)

Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme

Gauss variational equations

dαdt = 1

na D(αosc) · γ

The subscript γ•indicates the direction of the perturbationacceleration, originated -in this case- from a maneuver

GVE for s/c formation

Gauss variational equations dδαdt = f (δα,γp)

Impulsive thrust

∆v =

∫ t+M

⇓Thrust duration∆tM ≈ m2‖∆vM‖/Fmax

Finite duration thrust

∆v =

∫ t2

Thrust duration∆tM exactly solved

AimManeuver set to reconfigure the formation into δαnom

Impulsive thrust

∆v =

∫ t+M

∆v =

∫ t2

Impulsive thrust

∆v =

∫ t+M

∆v =

∫ t2

Impulsive thrust

∆v =

∫ t+M

∆v =

∫ t2

Impulsive thrust

∆v =

∫ t+M

∆v =

∫ t2

Impulsive thrust

∆v =

∫ t+M

∆v =

∫ t2

Maneuver sequence

Maneuver sequence 4T2N

u (rad)uT1 uN1 uN2uT2 uT3 uT4

solve GVE for ∆vM =⇒ ∆vM = f (∆δα, u0,∆u)

Major challenge

Compute the intermediate alteration ∆δaI = f (∆δα, u0,∆u)

Maneuver sequence

Maneuver sequence 4T2N

u (rad)uT1 uN1 uN2uT2 uT3 uT4

solve GVE for ∆vM =⇒ ∆vM = f (∆δα, u0,∆u)

Major challenge

Compute the intermediate alteration ∆δaI = f (∆δα, u0,∆u)

Impulsive & finite-duration planning

Impulsive thrust (IT)

solve GVE for ∆vM

IT major challenge

∆δaI = f (∆δα, u0,∆u) analyticallyresolvable

Finite-duration thrust (FDT)

solve integrated GVE for ∆tM

FDT major challenge

∆δaI = f (∆δα, u0,∆u) analyticallynot resolvable

Approximation

∆δaI solutionfrom IT

Computation

∆δaI numericaliteration

solve GVE for ∆vM

IT major challenge

FDT major challenge

Approximation

Computation

solve GVE for ∆vM

IT major challenge

FDT major challenge

Approximation

Computation

Introduction Gauss’ variational equations results Conclusion and outlookFormation reconfiguration and keeping

Formation reconfiguration and keeping

Tabelle 1: Initial and final configurations

aδa aδλ aδex δey aδix aδiyInitial(m) 29 −10000 0 0 0 0Final (m) 0 −2000 0 −900 0 −500

-1000 -500 0 500 1000-1000

Cross-Track (m)

TrajectoryStartpointEndpointClient

Introduction Gauss’ variational equations results Conclusion and outlookFormation reconfiguration and keeping

Error assessment

Inserting ∆tM from IT into integrated GVE

⇒ analytical assessment of formation error induced through impulsiveplanning

0 100 200 300 400 500 600 700 800 900 1000

Target a/ ey/a/ e

Error in a/ ey

Error in a/ ey @900m

Error in a/ iy

Error in a/ iy @500m

Conclusion

SummaryGVE for finite duration maneuver derived with several possibleapplications

FDT and IT control scheme using 4T2N maneuvers

OutlookVerification via high fidelity simulation with high risk debris objects

Inclusion of perturbations and estimation uncertainties.

Assessment of required computational power and suitability ason-board solution

Conclusion

spacecraft formation control using analytical integration

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