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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).
1r
Client Satellite
Client inertial orbit
Servicer inertial orbit
RTNY
RTNZ
RTNX
r
z
yx
Servicer Satellite
relative orbit
Challenge for active debris removalUncertainties in along-track
=⇒ Separation in RN plane for safe formation
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 3Spacecraft 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).
1r
Client Satellite
Client inertial orbit
Servicer inertial orbit
RTNY
RTNZ
RTNX
r
z
yx
Servicer Satellite
relative orbit
Challenge for active debris removalUncertainties in along-track
=⇒ Separation in RN plane for safe formation
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 3Spacecraft 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).
1r
Client Satellite
Client inertial orbit
Servicer inertial orbit
RTNY
RTNZ
RTNX
r
z
yx
Servicer Satellite
relative orbit
Challenge for active debris removalUncertainties in along-track
=⇒ Separation in RN plane for safe formation
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 3Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookMotivation Why ROE
Formation description
Relative orbital elements
δα =
δaδλ
δex
δey
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
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 4Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookMotivation Why ROE
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
Introduction Gauss’ variational equations results Conclusion and outlookMotivation Why ROE
Why relative orbital elements
Insight into the formation geometry (E/I separation)
acde
acde
2acde
acda
acdl
client
servicer
acdi
eT
eR eR
eN
acda
Maintains decoupling of in-plane and out-of-plane motionMore accuracy (retaining higher order terms)Adoption of Gauss variational equations (GVE)
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 6Spacecraft Formation Control using Analytical Integration of 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
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 7Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
GVE for s/c formation
Gauss variational equations dδαdt = f (δα,γp)
Impulsive thrust
∆v =
∫ t+M
t−M
γpdt
⇓Thrust duration∆tM ≈ m2‖∆vM‖/Fmax
Finite duration thrust
∆v =
∫ t2
t1
γpdt
⇓
Thrust duration∆tM exactly solved
AimManeuver set to reconfigure the formation into δαnom
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 8Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
GVE for s/c formation
Gauss variational equations dδαdt = f (δα,γp)
Impulsive thrust
∆v =
∫ t+M
t−M
γpdt
⇓Thrust duration∆tM ≈ m2‖∆vM‖/Fmax
Finite duration thrust
∆v =
∫ t2
t1
γpdt
⇓
Thrust duration∆tM exactly solved
AimManeuver set to reconfigure the formation into δαnom
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 8Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
GVE for s/c formation
Gauss variational equations dδαdt = f (δα,γp)
Impulsive thrust
∆v =
∫ t+M
t−M
γpdt
⇓Thrust duration∆tM ≈ m2‖∆vM‖/Fmax
Finite duration thrust
∆v =
∫ t2
t1
γpdt
⇓
Thrust duration∆tM exactly solved
AimManeuver set to reconfigure the formation into δαnom
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 8Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
GVE for s/c formation
Gauss variational equations dδαdt = f (δα,γp)
Impulsive thrust
∆v =
∫ t+M
t−M
γpdt
⇓Thrust duration∆tM ≈ m2‖∆vM‖/Fmax
Finite duration thrust
∆v =
∫ t2
t1
γpdt
⇓
Thrust duration∆tM exactly solved
AimManeuver set to reconfigure the formation into δαnom
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 8Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
GVE for s/c formation
Gauss variational equations dδαdt = f (δα,γp)
Impulsive thrust
∆v =
∫ t+M
t−M
γpdt
⇓Thrust duration∆tM ≈ m2‖∆vM‖/Fmax
Finite duration thrust
∆v =
∫ t2
t1
γpdt
⇓
Thrust duration∆tM exactly solved
AimManeuver set to reconfigure the formation into δαnom
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 8Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
GVE for s/c formation
Gauss variational equations dδαdt = f (δα,γp)
Impulsive thrust
∆v =
∫ t+M
t−M
γpdt
⇓Thrust duration∆tM ≈ m2‖∆vM‖/Fmax
Finite duration thrust
∆v =
∫ t2
t1
γpdt
⇓
Thrust duration∆tM exactly solved
AimManeuver set to reconfigure the formation into δαnom
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 8Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
Maneuver sequence
Maneuver sequence 4T2N
u (rad)uT1 uN1 uN2uT2 uT3 uT4
u0
Δu
π
π
πnT
solve GVE for ∆vM =⇒ ∆vM = f (∆δα, u0,∆u)
Major challenge
Compute the intermediate alteration ∆δaI = f (∆δα, u0,∆u)
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 9Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
Maneuver sequence
Maneuver sequence 4T2N
u (rad)uT1 uN1 uN2uT2 uT3 uT4
u0
Δu
π
π
πnT
solve GVE for ∆vM =⇒ ∆vM = f (∆δα, u0,∆u)
Major challenge
Compute the intermediate alteration ∆δaI = f (∆δα, u0,∆u)
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 9Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
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
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 10Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
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
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 10Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlookGVE Control scheme
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
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 10Spacecraft Formation Control using Analytical Integration of GVE
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
-800
-600
-400
-200
0
200
400
600
800
1000
Cross-Track (m)
Rad
ial (
m)
TrajectoryStartpointEndpointClient
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 11Spacecraft Formation Control using Analytical Integration of GVE
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
y (m)
0
1
2
3
4
5
6
7
8
9 E
xecu
tion
Err
or (
%)
Error in a/ ey
Error in a/ ey @900m
Error in a/ iy
Error in a/ iy @500m
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 12Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlook
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
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 13Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlook
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
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 13Spacecraft Formation Control using Analytical Integration of GVE
Introduction Gauss’ variational equations results Conclusion and outlook
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
14.-17.03.2016 Mohamed Khalil Ben Larbi Seite 13Spacecraft Formation Control using Analytical Integration of GVE