Convex relaxationsfor minimumenergy orbit transfer problem

MOPTA04 Hamilton, Canada

Dorin PREDA, JosephNOAILLESpreda@enseeiht.fr, jnoaille@enseeiht.fr

ENSEEIHT–IRIT, Toulouse,France

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Outline

● Problemdescription– Coplanartransfers,coordinates,local methods

● Convexification with CartesianCoordinates– MethodsandResults

● Convexification with GaussCoordinates– GaussEquations– Equationsin time. Results– Equationsin

�. Results

● Conclusion

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Problem: Coplanar Transfers

● FrenchSpaceAgency (CNES)

● Coplanartransfer(2D):

Geostationaryfinal orbit ( � � �)

● Low thrusts(60 to 0.3Newtons)

● Minimization of the"energy"

Simplification:● constantmassduringtransfer

● "energy":

� �

��� ��� � � ��� �Constraints:● fixedinitial, final

states

●

��� �� � � � ��� ��

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Problem: Coordinates

● Cartesian

� � � ! " �# $ % &' (

– statevariablesare � � ��)+* , � and - � . � � ��/ �* /�0 �

– linearequationswith respectto control– only onetypeof nonlinearterm

● GaussCoordinates

– new state:

1* � �* �0 (first

integralsof motion)and

�

– highly nonlinearequations– "recommended"approachin

findinga local solution

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Problem: Methods

Non convex problem(Gaussor Cartesian)● local solutions

– directmethods:discretization+ NLP solver– indirectmethods:

● PontrjaginPrinciple+ BVP solver + homotopy● drawback:algebraicconstraintson statevariables

● globalsolutions(?!)– BranchandReduce(BB andInterval Analysis)

● upperbound 2 directmethods(NLP solver)● lower bound 2 convex relaxationof theoriginal problem

Overestimate/underestimatethe global solutionof the optimal

controlproblemfor agivensetof boxconstraints

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Approach: discretization

● Discretizetheproblem:– directcollocationmethod

● System:linearequations(collocation)

! $� 3 )54 % $� 3 )54 6 7 ! 78 94 ! 78 94 6 7 ! : 94 � �

7 � )54 % 7 � )54 6 7 % 3; 94 ! 3; 94 6 7 ! :)54 � �

● Nonlinearequations(systemdynamics)

94 � 9 ��)54 * � 4 � � <� < ��)54 * � 4 * �4 �

: 94 � : 9 � :)54 * � 4 � � <� < � :)54 * � 4 * �= 6 => ?� �

● Solve theproblem(NLP Solver) @ UBD

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Approach: convexification and LBD

● Computeconvex envelopesfor every nonlinearterm @ convex

problem @ LBD

94 �BA CD A E/ � � 94 �)54 * � 4 � � � 9 FGH4 �)54 * � 4 �

94 IBA CD / �) � 94 ��)J4 * � 4 � � � 94 K L4 ��)J4 * � 4 �

● Solve therelaxedproblem @ LBD

● Drawback:theconvexification introducesnew variables

● Estimatethequality of thelower estimation– boxconstraints:) M N) OP Q ! R)+* ) OP Q % R) S

– representT P QOP Q � U � R) �

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

CartesianCoordinates(I)

V WYX Z []\[]^ _ G `ba _ c 6 G `bd _ c c < e� _ cgf h a _ ceh a _ cgf i j a k [ lk a m k [ l> d m k [ l l nm 6 ?o G a _ c

e0 _ cgf h d _ ceh d _ cf i j d k [ lk a m k [ l> d m k [ l l nm 6 ?o G d _ c

G `pa _ c 6 G `pd _ cq r ` os a_ �ut 0 t h at h d c _ f v c f _ � t 0 t h at h d c

0 20 40 60 80 100 120 140 160−100

−50

0

50

t

x

0 20 40 60 80 100 120 140 160−20

−10

0

10

20

30

t

v

) and/ � vs. Time

Problems:● UBD: difficult to solve thediscretizedproblem

● oscillatinglocal solutions(seefigure)

● nondifferentiableterms

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

CartesianCoordinates(II)

Solution:changeto polar coordinates,afterthediscretization●

��)+* , � @ � #* w �

● nonlinearterms: A 7 xyz { |` %A � # }~� w or A 7z WYX { |` %A � #� ��� w

● convexify andsolve therelaxation(NLP solver Knitro):

10

15

20

25

30 01

23

45

67

−30

−20

−10

0

10

20

30

Concave/convex envelopes( E 7 w � % E � w % E $ # % E 8 )

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Results

Lower boundis 0, nomatterthebox constraintsconsidered!!!

� � � ! " �# $ % &' (

●

� � " �# $ � � � � &' ( � �

● convexificationmakesthesecondterminfluencevanish

● anull controlis ableto satisfytheinitial andfinal constraints

Constraintsof convex relaxationmustensurethatthesatellite

remainson thesameorbit if controlis null

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Gaussequations

Gaussequations:

�� � [ f � ?o� � n � m� G `�]� �� [ f ?o � _z WYX T c� ? � ` G ? 6 ?o� � ? � ` ��� �G `�]��� � [ f i ?o� _ xyz T c� ? � ` G ? 6 ?o� � ? � ` � m �G `�� � [ f ?�� �m� n � m� f 7 6�� a xyz T 6 � dz WYX T� ? f � xyz T 6�� a _ 7 6 xyz ` T c 6�� d _z WYX T xyz T c� ` f �z WYX T 6�� a _z WYX T xyz T c 6 � d _ 7 6z WYX ` T c

1vs. Time � � vs. Time �0 vs. Time

�

vs. Time

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Model I, results

Method:● useonly

<� < , < T< and � �* �0 � bounds

● objective:

� � �� ��� � � �

(� 7 contribution is lessthan13%)● use

�� � � : (

�� � � � @ 1 � constant)������ � [ f `o� � � n � m �� G `�� � [ f � j �m� � n � m � � ����� � � � [ �q `o� � � n � m �� � G ` ��� � [ f � j �m� � n � m �

Difficulties: badestimationfor � & % }~� � � � %� ��� � �0

Results:● as

R 1 � �estimationtendsto 72%of theupperbound

● lower boundis efficient (

I   �

) for verysmall

R 1

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Impr ovements

Method:tightenup theboundson

1

● variablesubstitution

¡ � 1 i 7 ¢ �

:����� � � [ f `o � � � n � m �� G `� � � [ f � j �m� � n � m � � ����� �£ � [ �q ?o � ? �� G ` ��� � [ f � j �m£ n

Results:● slightly betterfor bigger

R 1(when

R 1 ¤ 1¦¥ �¨§ ©  

)

● LBD nobetterthan

ª«

% of UBD

Conclusion:

How to obtaintight boundsfor termscontainingthe� ��� �

and}~� �

?!

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

GaussEquations revisited

Idea:Expresstheparameters

1* � �* �0 asfunctionsof�

Why?●

}~� �

and� ��� �

becomesconstantsafterdiscretization● ,

¬ 7 , ¬ � arelinearexpressionsof � � , �0 :

� f 7 6�� a xyz T 6�� dz WYX T� ? f � xyz T 6 � a _ 7 6 xyz ` T c 6 � d _z WYX T xyz T c� ` f �z WYX T 6 � a _z WYX T xyz T c 6 � d _ 7 6z WYX ` T c

Equations: �� �� f � ?o � � n �n G `�]� ��� f ­ ®°¯ �o � � m �m G ? 6 ?o� � m �m � ? G `�]� ��� f i ±² ­ �o � � m �m G ? 6 ?o � � m �n � ` G `

Quadraticobjective: � ³ � � ´ �µ�

New constraint:� � � ´ � `� ` � � �� ��

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

A simple model● useonly

<� < T and � �* �0 � (interval analysis)

● decompose� �: � � � � 6� ! � i�* � 6�* � i� I �● substitution

¡ � 1 i 8 ¢¶

:

<· < T � ! ¶� � j 7 � ` � 6� % ¶� � j 7 � ` � i�

● quadraticobjective:

T T � �� 6� � � % �� i� � �� �● additionalvariables(

� 7* � �), convexify:

� 7 � G > `� ` � � � G ¸`� `

0

2

4

6

8

10

00.05

0.10.15

0.20.25

0.30.35

35

40

45

50

55

60

65

70

75

P variation

Model I : Results

ex variation

Per

cent

age

Percentagevs.

R 1

and

R � �

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

ComplexModel

● variablesubstitution:

¡ � 7� ,

¹� � � a� ,

¹0 � � d� :

�� � � &1 � % �� � � � � 1 � % �� � � �0 1 � � �

● new notations(keepthelinearity):

º f · 6 » a xyz T 6 » dz WYX T

¼ ? f � · xyz T i » az WYX ` T 6 » dz WYX T xyz T

¼ ` f � · z WYX T 6 » az WYX T xyz T i » d xyz ` T

● Gausssystembecomes:

<· < T � ! �� j · k½ �µ lº ` � �< » a< T � % 7� j� ��� � · k ´ �µ lº � 7 % 7� j · k ´ �µ lº ` ¾ 7� �< » d< T � ! 7� j }~� � · k ´ �µ lº � 7 % 7� j · k ´ �µ lº ` ¾ �� �

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Results

● LBD improved

● around20000variables

● quadraticmodel(CPLEX)0

2

4

6

8

10

00.05

0.10.15

0.20.25

0.30.35

20

40

60

80

100

∆P

Percentage vs ∆P and ∆E

∆ex

Per

cent

age

0 0.05 0.1 0.1555

60

65

70

75

80

Pe

rce

nta

ge

0 0.05 0.1 0.1570

75

80

85

90

95

Pe

rce

nta

ge

∆P = 1

∆P = 0.1

0 1 2 3 4 5 6 7 8 9 1020

40

50

60

70

80

∆P

Per

cent

age

0 1 2 3 4 5 6 7 8 9 1020

40

60

70

80

90

100

∆P

Per

cent

age

∆ex = 0.1

∆ex = 0.01

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Underestimationvs. ¿À Á0

2

4

6

8

10

00.05

0.10.15

0.20.25

0.3

25

35

50

60

70

80

90

100

∆P

TMAX = 3N

∆ex

Per

cent

age

Underestimationfor��� �� �  Ã

0

2

4

6

8

10

00.05

0.10.15

0.20.25

0.3

20

40

60

70

80

90

100

∆P

TMAX = 27N

∆ex

Per

cent

age

Underestimationfor��� �� � © ª Ã

● no improvementin LBD for

R 1* R � � � �

● betterto havesmall

��� �� for larger

R 1* R � �

● small�Ä� �� requiresmorediscretepointsandis timeexpensive

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

Conclusions

Convexification:● over 10differentmodelsconsidered

● thebestunderestimation:Gaussequationsin L (

7� ,

� a� ,

� d� )

Improvements:● betterconvexify elementaryfunctions(

� Å0 o )Drawback:● LBD, UBD estimationsaretimeexpensive (a few secondsper

underestimation)

�Coplanar Transfers�Coordinates: Cartesian vs Gauss�Methods�Approach� ”Approach”�Cartesian Coordinates�Cartesian Coordinates�Results�Gauss equations�Model I� Improvements�Gauss equations in L�Results. Model I� Toward a complex model�Results Model II� LBD vs

� �� ��Conclusions

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NOAILLES, andDorin PREDA. Mise aupointd’uneméthodederésolutionefficacepourles

problèmesdecontrôleoptimalà solution“ bang-bang”, applicationaucalculdetrajectoiresà

pousséefaible. TechnicalReport02/CNES/0257/00- DPI 500(Rapportdefin dephase2 et3),

I.R.I.T. - E.N.S.E.E.I.H.T., Octobre2003,Juin2004.

[2] G.P. MCCORMICK. NonlinearProgramming. TheoryandApplications. JOHN WILEY & SONS,

1983.

[3] M. TAWALARMANI andN.V. SAHINIDIS. ConvexificationandGlobalOptimizationin Continuous

andMixed-Integer NonlinearProgramming:Theory, Algorithms,Software, andApplications.

Nonconvex OptimizationAnd Its Applications.KLUWER ACADEMIC PUBLISHERS, November

2002.

[4] Olivier ZARROUATI. Trajectoiresspatiales. C.N.E.S.- TechniquesSpatiales.CEPADUES, Janvier

1987.