lo w thrus t m inim um- f u el orbital t rans fer: a homot...

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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Orbital Transfer ProblemModelisation

Homotopy MethodShooting methodHomotopyPC methods

Numerical resultsConvergence of the methodLocal minimaExamples of solutionsLinks with impulsive case

ConclusionConclusionBibliography

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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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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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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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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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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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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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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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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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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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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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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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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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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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Local minima

t f fixed and Lf free: local minima

! Lots of minima becausethe number of revolutionsis free

! ' We fixe Lf :


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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Local minima

mf vs cLf

! t !' L

! mf does not depend onTmax (for cLf fixed)


! 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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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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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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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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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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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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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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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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