Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Low Thrust Minimum-Fuel Orbital Transfer: AHomotopic Approach
3rd International Workshop on Astrodynamics Tools andTechniques
ESA, DLR, CNESESTEC, Noordwijk
Joseph Gergaud and Thomas Haberkorn
2–5 October 2006
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Orbital Transfer ProblemModelisation
Homotopy MethodShooting methodHomotopyPC methods
Numerical resultsConvergence of the methodLocal minimaExamples of solutionsLinks with impulsive case
ConclusionConclusionBibliography
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Modelisation
Orbital transfer
!40
!20
0
20
40
!40
!20
0
20
40
!5
0
5
r1
ORBITE INITIALE
ORBITE FINALE
r2
r 3
!50 0 50
!40
!20
0
20
40
r1
r 2
!50 0 50!5
0
5
r2
r 3
Supported by
! LEO–GOE Transfer:P = 11.625 Mm, e = 0.75 andi = 7!
! Initial Mass: m0 = 1500 kg
! Low Thrust: Tmax = 0.1 N
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Modelisation
Coordinates
! J2 neglected
! Earth gravitational force r = ! µ0|r |3 r + T
m
! Modified Gauss Coordinates x = (P, ex , ey , hx , hy , L):Z
YX
satellite
equatorial plan
orbit
perigee
v
!
"i
!""""""#
""""""$
P = a(1! e2)ex = e cos(! + !)ey = e sin(! + !)hx = tan(i/2) cos !hy = tan(i/2) sin!L = ! + ! + " + 2#n (cumulative)
n = Number of revolutions
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Modelisation
Control
O
q
w
Sj r
v
s
i
k
! Transverse reference frame (q, s,w)
! We normalize the controlT = TmaxuSo the constraint on the control is
|u| " 1
%|u| =
&u21 + u2
2 + u23
'
! State equation:x = f0(x) + Tmax/m
(3i=1 fi (x)ui
m = !TmaxIspg0
|u|
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Modelisation
Cost
! Preliminary results! min t f
t fminTmax # c te (T. Le, S. Ge"roy, R. Epenoy)
! min Lf
(Lfmin ! L0)Tmax # c te (T. Haberkorn)
! Maximization of the final mass: max m(t f )
$% Minimization of the consumption: min) tf
0 |u|dt
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Modelisation
Optimal Control Problem
(P)
!"""""""""#
"""""""""$
min) tf
0 |u|dtx = f0(x) + Tmax/m
(3i=1 ui fi (x)
m = !$Tmax|u||u| " 1
Initial and Final ConditionsLf freet f fixed
! We define the constant ctf > 1 such that t f = ctf t fmin
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Shooting method
Di!culties of the shooting method
! The optimal control is discontinuous: The engine is o" or onwith the maximal thrust
! We don’t know the number of switching times and theirlocations
% Di#culties to find an initial guess
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Homotopy
Main idea
Cost J!(u) =) tf
0 (1! %)|u|2 + %|u|dt or) tf
0 |u|(2"!)dt
! Problem easy to solve for % = 0
! Regular problem for % < 1
! Original problem for % = 1
Optimal ControlProblems
(P!)!
Boundary ValueProblems(BVP!)
!ShootingHomotopy
S(z ,%)
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
PC methods
Homotopy algorithms
Path following of the zeros curve of the homotopy function:
S : Rn & [0, 1] !' Rn
(z ,%) (!' S(z ,%)
”Global Newton” algorithmInitialisation z0 solution of S(z0, 0) = 00 = %0 < %1 < . . . < %n = 1for i = 1, . . . , n dosolve S(z ,%i ) = 0 by Newton method with initial point z i"1
but for our orbital transfer problem we divergeHow to choose the sequence (%i )i?=% Homotopy methods (Allgower and Georg)
! Predictor-Corrector methods! Piecewise Linear methods
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
PC methods
Predictor Corrector methods
!8 !6 !4 !2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z
lambda
S(z, λ) = 0
(zλ, λ)
!
Prediction
(ez, eλ)
Tangent vector for theprediction.
!Correction(zλ+ , λ+)
Come back on the zeros path(correction).
! Until % = 1.
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
PC methods
Predictor Corrector methods
!8 !6 !4 !2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z
lambda
S(z, λ) = 0
(zλ, λ)
!
Prediction
(ez, eλ)
Tangent vector for theprediction.
!Correction(zλ+ , λ+)
Come back on the zeros path(correction).
! Until % = 1.
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
PC methods
Predictor Corrector methods
!8 !6 !4 !2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z
lambda
S(z, λ) = 0
(zλ, λ)
!
Prediction
(ez, eλ)
Tangent vector for theprediction.
!Correction(zλ+ , λ+)
Come back on the zeros path(correction).
! Until % = 1.
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
PC methods
MfMax software
! We used the software HOMPACK90 (Watson and al.) forPC-methods
! We compute S #(c(s)) by finite di"erences and S(c(s)) bynumerical integration (rkf45)=% we must have a good adequation between the step offinite di"erences and the local errors in numerical integrationin order to have a good approximation of the derivativeS #(c(s))
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Convergence of the method
Path following
0.9 1 1.1 1.2 1.30
0.5
1
(1! ! )|u|2 + ! |u|
145 150 155 160 165 1700
0.5
1
3.4 3.6 3.8 4 4.20
0.5
1
36 37 38 39 40 410
0.5
1
!1.5 !1 !0.5 00
0.5
1
!3.2 !3 !2.8 !2.6 !2.40
0.5
1
0.052 0.054 0.056 0.058 0.060
0.5
1
0.8 0.9 1 1.1 1.2 1.30
0.5
1
|u|2!!
145 150 155 160 165 1700
0.5
1
3.4 3.6 3.8 4 4.20
0.5
1
36 37 38 39 40 410
0.5
1
!1.5 !1 !0.5 00
0.5
1
!3.2 !3 !2.8 !2.6 !2.40
0.5
1
0.052 0.054 0.056 0.058 0.060
0.5
1
Smooth path
!
S(z, )! = 0
z
Discret
Differentielle
Advantage of PC methods
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Convergence of the method
Control with respect to !
Tmax = 10N
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
Critere convexe ! = 0, 0.5 et 1
t
|u|
0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
Critere puissance ! = 0, 0.5 et 1
t
|u|
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Local minima
t f fixed and Lf free: local minima
! Lots of minima becausethe number of revolutionsis free
! ' We fixe Lf :
Lf = cLf (Lfmin ! L0) + L0
t f free
30 40 50 60 70 80 901345
1350
1355
1360
1365
1370
1375
1380
1385
1390
tf
mf
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Local minima
t f fixed and Lf free: local minima
! Lots of minima becausethe number of revolutionsis free
! ' We fixe Lf :
Lf = cLf (Lfmin ! L0) + L0
t f free30 40 50 60 70 80 90
1345
1350
1355
1360
1365
1370
1375
1380
1385
1390
tf
mf
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Local minima
mf vs cLf
! t !' L
! mf does not depend onTmax (for cLf fixed)
Lf = cLf (Lfmin ! L0) + L0
! Limit mass = mass of theimpulsive case
2 3 4 5 6 7 8 9 101360
1365
1370
1375
1380
1385
1390
1395
Tmax = 10 NTmax = 1 NTmax = 0.5 NTmax = 0.1 NImpulse
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Examples of solutions
Trajectory for Tmax = 10 N et cLf = 2
! Thrust arcs on allapogees.
! Thrust arcs on lastperigees.
!40
!20
0
20
40
!40
!20
0
20
40
!5
0
5
r1
r2
r 3
!60 !40 !20 0 20 40
!40
!20
0
20
40
r1
r 2
!40 !20 0 20 40
!2
!1
0
1
2
r2
r 3
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Examples of solutions
Other initial orbits
(P0, e0) = (11.625, 0.5)
!50 !40 !30 !20 !10 0 10 20 30 40 50
!40
!30
!20
!10
0
10
20
30
40
r1
r 2
(P0, e0) = (20, 0.75)
!80 !60 !40 !20 0 20 40
!50
!40
!30
!20
!10
0
10
20
30
40
1
r 2
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Examples of solutions
Trajectory for Tmax = 0.1 N
!40
!20
0
20
40
!40
!30
!20
!10
0
10
20
30
40
!5
0
5
r1
r2
r 3
! More than 750 revolutions! More than 1500 switching times
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Links with impulsive case
Links with impulsive case
! In the coplanar transfer case, if Tmax ' +) and [L0, Lf ]contains exactly one apogee and perigee then the controlconverges to the impulsive solution
! When cLf ' +) then mf ' final mass in the impulsive case
2 3 4 5 6 7 8 9 101360
1365
1370
1375
1380
1385
1390
1395
Tmax = 10 NTmax = 1 NTmax = 0.5 NTmax = 0.1 NImpulse
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Links with impulsive case
Links with impulsive case
0 50 100 150 200 250 300 350 400 450 500!1
0
1
t
q0 50 100 150 200 250 300 350 400 450 500
!1
0
1
t
s
0 50 100 150 200 250 300 350 400 450 500!1
0
1
t
w
0 50 100 150 200 250 300 350 400 450 500
0
0.5
1
t
|u|
Thrust in time for Tmax = 10N, cLf = 5 and i0 = 500
There are three strategies in time for the optimal control: thrust atperigees, thrust at apogee and thrust at perigee, as in theimpulsive case.
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Conclusion
Conclusion and developments
! Conclusion:We have completely solve the problem without any knowledgeon the solution (number and location of switching times, ...)
! Developments! Interplanetary transfer?! State constraints?
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Outlines Orbital Transfer Problem Homotopy Method Numerical results Conclusion
Bibliography
Bibliography
! J. Gergaud et T. Haberkorn. Homotopy method for minimum consumption orbittransfer problem. Control, Optimisation and Calculus of Variations, Vol.12(2)294:310, April 2006
! J. Gergaud, T. Haberkorn and P. Martinon. Low thrust minimum-fuel orbitaltransfer: a homotopic approach, Journal of Guidance, Control, and Dynamics,Vol. 27(6)1046:1060, Nov. 2004
! T. Haberkorn. Transfert orbital a poussee faible avec minimisation de laconsommation: resolution par homotopie di!erentielle, PhD ENSEEIHT–IRIT,www.enseeiht.fr/~haberkorn, 18 octobre 2004
! J. Gergaud, T. Haberkorn and J. Noailles. MfMax(v0 & v1): Methodexplanation manual, Technical report ENSEEIHT-IRIT, UMR CNRS 5505 ,RT/APO/04/01 , january 2004, http://www.enseeiht.fr/apo/mfmax/
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