Mathematical Modelling & Optimization Of Crew Module Service Module Umbilical Separation Mechanism For Manned Mission

2020·0.39 MB·English

by  International Journal of Advances in Applied Mathematics and Mechanics. 7(3) (2020) 1 – 10

IJAAMM
Int.J.Adv.Appl.Math.andMech.7(3)(2020)1–10 (ISSN:2347-2529)
Journalhomepage:www.ijaamm.com
InternationalJournalofAdvancesinAppliedMathematicsandMechanics
Mathematical Modelling & Optimization of Crew Module-Service
Module Umbilical Separation Mechanism for Manned Mission
ResearchArticle
∗
LalaSuryaPrakasha, ,ChandanSaxenab,DilipVb,KrishnanandanVa
aHumanSpaceFlightGroup,LiquidPropulsionSystemsCentre,IndianSpaceResearchOrganisation,Valiamala-695547,Kerala,India
bControlSystems&UmbilicalDivision,LiquidPropulsionSystemsCentre,IndianSpaceResearchOrganisation,Valiamala-695547,Kerala,India
Received24December2019;accepted(inrevisedversion)12February2020
Abstract: Asimplifieddynamicmodelisformulatedfortheactuator-boomconfiguration,proposedfortheseparationofum-
bilicalsysteminterconnectingCrewModuleandServiceModule. Optimizationstudiesarecarriedoutusingopen-
sourceSciPylibraryonPythonplatform.Theobjectiveofthisstudyistoestimatetheforce,tobedeliveredbyactuator,
toseparatetheumbilicalsystemfromCrewModulewithconstraintsimposedfrommissionandlaunchvehicle.This
studydoesnotcoverthestructuraloptimizationaspects. However,itprovidespromisinginputsandinsighttopro-
ceedwithconfigurationanddesignofthesystem.
MSC: 70E17 • 49K21 • 90C90
Keywords: MathematicalModeling • Optimization • Umbilical • CrewModule • ServiceModule
© 2020TheAuthor(s).ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(https://creativecommons.org/licenses/by-nc-nd/3.0/).
1. Introduction
Indian Space Research Organization (ISRO) has initiated manned mission program to achieve in-house human
spaceflightcapability[1]. Towardsthis,CrewModule(CM)andServiceModule(SM)areproposedtocarrycrewto
orbitandreturnthemsafelybacktoapre-determinedlocation. CMservesasthehabitatofcrewandSMcatersfor
lifesupportandpropulsionrequirementsduringascentandon-orbitphasesofthemission. Aumbilicalsystemis
proposedtofacilitatefluidandelectricalinterconnectionbetweenCMandSMtillthetimeofCM-SMseparation.The
CMislocatedaboveSMandumbilicalsysteminterconnectsCM-SMfromoutsidetheCM-SMstructure(Fig.1).
CMhasablativelinerasheatshieldforcombatingheatduringre-entry. AsCMre-entersatmospherefacingbot-
tom,theumbilicallinescannotberoutedthroughthebottomwithoutaffectingtheheatshield. Theonlyalternative
availableistoroutethelinesoutside(Fig.1).TheproposedumbilicalsystemconsistsofUmbilicalPlatesub-assembly,
Boom,andActuator. Umbilicalplatesub-assembly,whichhousesfluid/electricalconnectors,consistsoftwoplates
inmatedconditionduringoperation. Oneplateismountedonthefore-endoftheactuatableboomandotherhalf
isfixedtoCM.Thefluid/electricallinesareroutedfromSMtoCMthoughtheboomstructure. Theboomisarigid
armwithplanarrotationaldegreeoffreedomaboutthefixedmountingpointlocatedatSMfore-end. Theactuator
providethenecessaryforceforseparationofumbilicalsystemfromCMpriortoseparationofSMduringabortand
descentphase(around120kmaltitude)ofthemission.
∗
Correspondingauthor.
E-mailaddress(es): [email protected],[email protected](LalaSuryaPrakash),[email protected](Chandan
Saxena),[email protected](DilipV),[email protected](KrishnanandanV).
1
2 MathematicalModelling&OptimizationofCrewModule-Service...
Nomenclature
CM CrewModule
SM ServiceModule
CG Centerofgravity
b lengthoftheumbilicalplateinplaneofrotation
Fa Forcedeliveredbyactuator
Fg Gravityforceontotalboom
I MomentofInertiaoftotalboomw.r.torigin
Ic MomentofInertiaofboompartonCMw.r.torigin
Im MomentofInertiaofboompartoutsideCMw.r.torigin
Ip MomentofInertiaofumbilicalplatesub-assemblyw.r.torigin
X Horizontaldistancebetweenmountingpointsofactuatorandboom
Xmax Maximumhorizontalspaceavailabletoaccomodateumbilicalsystem
Xsp Horizontaldistancebetweenmountingpointsofactuatorandactuator-boomjoint
∗
X Matrixrepresentingoptimalsolution
Y Verticaldistancebetweenmountingpointsofactuatorandboom
Ygap Verticalgapbetweenactuator-boomjointandSMtop
Ymax Maximumverticalspaceavailabletoaccomodateumbilicalsystem
Ysp Verticaldistancebetweenmountingpointsofactuatorandactuator-boomjoint
f Functionrepresentingobjectivefunction
gi Inequalityconstraints
m Totalmassoftheboomincludingfluid/electricallines
mc MassoftheboompartwhichliesonCM
mp Massoftheumbilicalplatesub-assembly
mm Massoftheboompartwhichliesoutsidetheboomandactsasmomentarmforactuator
(cid:126)rcg CGofthetotalboom
t Time
tsep TimerequiredforfullseparationofumbilicalsystemfromCM
xcg AbscissaofCGofthetotalboom
xc,cg AbscissaofCGoftheboompartwhichliesonCM
xm,cg AbscissaofCGoftheboompartwhichliesoutsideofCM
xp,cg AbscissaofCGoftheumbilicalplate
ycg OrdinateofCGofthetotalboom
yc,cg OrdinateofCGoftheboompartwhichliesonCM
ym,cg OrdinateofCGoftheboompartwhichliesoutsideofCM
yp,cg OrdinateofCGoftheumbilicalplate
la Lengthoftheactuatorinnon-actuatedcondition
lc LengthoftheboompartonCM
lm LengthoftheboompartoutsideCM
lp DistanceofCGofumbilicalplatefromtopendoftheboom(lp=0.5bforpresentstudy)
α Angularaccelerationofboomw.r.torigin
β Anglemadebyactuatorw.r.tx-axisinnon-actuatedposition
θ AnglemadebyboompartoutsideCMw.r.tx-axis
θc AnglemadebyboompartonCMw.r.tx-axis
θ FinalanglemadebyboompartonCMw.r.tx-axisafterseparation
cf
θ FinalanglemadebyboompartoutsideCMw.r.tx-axisafterseparation
f
ω Angularvelocityofboomw.r.torigin
τ Nettorqueappliedonboomw.r.torigin
NationalAeronauticsandSpaceAdministration(NASA),USAhasdevelopedactuatorbasedumbilicalseparation
mechanismforApolloandOrionprograms. Apollo’sCSM[2]andOrionspacecraft[3]routedtheirumbilicallines
outsidefromServiceModuletoCommandModule.
Thelaunchvehicleenvelopeandmissionconstraintsaremajorfactorsdecidingthedesignoftheumbilicalsys-
tem. Thispaperpresentsthedetailsofmathematicalmodellingandoptimizationstudycarriedoutfortheproposed
actuator-boomconfiguration.
2. DevelopmentofMathematicalModel
Theactuator-boomconfigurationissimilartooneemployedinhydraulicexcavators. Ref. [4]and[5]arereferred
forthedevelopmentofthemathematicalmodelofactuator-boom.Inthepresentstudy,theboomismodelledasbent
LalaSuryaPrakashetal./Int.J.Adv.Appl.Math.andMech.7(3)(2020)1–10 3
Fig.1.OverallconfigurationofCM-SMUmbilical
(a)Simplifiedsystem (b)Parameterdetails
Fig.2.Kinematicdetailsofboom-actuatorsystem
beamstructureforgeneralization.
The boom is assumed as a rigid structure with non-uniform load distribution due to the umbilical plate and
fluid/electrical lines. In order to parametrize this non-uniform load, the loads are assumed to be concentrated at
threelocationsonthebeam(Fig.2(a))umbilicalplatesub-assembly(b)partoftheboomwhichliesonCM(c)partof
theboomoutsidetheCMwhichactsasmomentarmforactuator. Inordertomaximizethemomentarm,thepoint
ofactionoftheactuatorisproposedtocoincidewiththepointofbend. Totalmassoftheboom-actuatorsystemis
expressedas-
m=m +m +m (1)
p c m
4 MathematicalModelling&OptimizationofCrewModule-Service...
2.1. Centerofgravityoftheboom
Thecenterofgravity(CG)oftotalboomstructure, includingboom, electrical/fluidlinesandumbilicalplate, is
specifiedinvectorformw.r.tXYcartesianco-ordinatesystemwithorigin(0,0)atpoint’O’(Fig.2(b)).
(cid:126)r =x iˆ+y jˆ (2)
cg cg cg
where,
x m +x m +x m
x = m,cg m c,cg c p,cg p
cg m +m +m
m c p
= 1 (cid:163)0.5m l cosθ+m (l cosθ+0.5l cosθ )+m (cid:161)l cosθ+(cid:161)l −l (cid:162)cosθ (cid:162)(cid:164)
m m c m c c p m c p c
m
y m +y m +y m
y = m,cg m c,cg c p,cg p
cg m +m +m
m c p
= 1 (cid:163)0.5m l sinθ+m (l sinθ+0.5l sinθ )+m (cid:161)l sinθ+(cid:161)l −l (cid:162)sinθ (cid:162)(cid:164)
m m c m c c p m c p c
m
2.2. MomentofInertiaoftheBoom
Momentofinertia(MI)ofthetotalboomaboutoriginpoint’O’isexpressedas
I=I +I +I (3)
m c b
where,
m l2
I = m m
m
3
(cid:183) l2 (cid:184)
I =m l2 + c +l l cos(θ −θ)
c c m 3 m c c
I =mpb2+m (cid:104)l2 +(cid:161)l −l (cid:162)2+2l (cid:161)l −l (cid:162)cos(θ −θ)(cid:105)
p 12 p m c p m c p c
2.3. Kinematics
In order to meet the separation requirements, the boom need to rotate to a final position of θ in a time t
cf sep
imposed by mission. As the expressions for MI and CG are in terms of θ, θ is converted to θ using geometric
cf f
relationship(4).
θ =θ +(θ −θ) =⇒ θ −θ =θ −θ (4)
cf f c cf c f
Assumingconstantangularacceleration,thefollowingholdstrue.
1
θ =θ+ω t + αt2
f o sep 2 sep
Astheboomisinitiallyatrest,
2(θ −θ)
α= f
t2
sep
UsingEquation(4),theaboveequationcanbetransformedto
2(θ −θ )
α= cf c (5)
t2
sep
LalaSuryaPrakashetal./Int.J.Adv.Appl.Math.andMech.7(3)(2020)1–10 5
Fig.3.FreebodydiagramoftheboomatLaunchpadcondition
2.4. Rigidbodydynamics
Theequationofmotionfortheboomisdevelopedbasedontorquebalanceaboutpivotpointoftheboom,origin
’O’ofXYco-ordinatesystem(Fig.3),forlaunchpadabortcondition.Theadditionalinertialforces,duetoacceleration
ofthevehicle,alsoplaysaroleinthedynamicsofboomandshallbeincorporatedinparameter’g’.Asthereisnoloss
ingenerality,thepresentstudydoesnotaccountforsuchinertialforce. ConsideringrotationalanalogueofNewton’s
secondlawofmotion[6],thetorque(τ)actingontheboomisexpressedaboutorigin’O’.
(cid:88)τ=Iα
Fasin(cid:161)θ+β(cid:162)lm−Fgxcg =Iα (6)
F sin(cid:161)θ+β(cid:162)l −mgx =Iα
a m cg
Re-arrangingEquation(6),theforcerequiredfromactuatorisexpressedas
Iα+mgx
F = cg (7)
a l sin(cid:161)θ+β(cid:162)
m
3. ProblemFormulation
Theobjectiveofthepresentstudyistominimizetheforcerequiredtorotatetheboomstructureforeffectivesepa-
rationofCMandSM.Theoptimizationproblemisformalizedasfollowing
Iα+mgx
Minimize: F = cg =f (cid:161)l ,θ,β(cid:162) (8)
a l sin(cid:161)θ+β(cid:162) m
m
where,I,mandx areasperEq.(1),Eq.(2)andEq.(3)respectively.
cg
Subjectto:g =X −X −l cosθ≥0
1 max sp m
g =X tanβ−Y ≥0
2 sp gap
g3=Ymax−lcsinθc−lasinβ≥0 (9)
θ =106o
cf
t ≤0.150
sep
Theconstraints,g , g ,and g ,aregeometricinnatureandderivedfromthespacelimitationofthelaunchvehicle.
1 2 3
In contrast, θ and t are functional constraints from mission point of view and these need to be satisfied for
cf sep
ensuringcleanseparationofcrewmodulefromservicemodule. Theoptimizationproblem, definedbyEquations
(8)and(9),belongstoparameterorstaticoptimisationcase.Thevariablesinvolvedintheoptimizationproblemcan
becategorisedin"decisionvariables","preassignedparameters"and"derivedparameters".
DecisionVariables:l ,θ,β
m
PreassignedParameters:m ,m ,m ,l , X ,Y ,θ ,α
m c p c sp gap c
DerivedParameters:α,l , X,Y,Y
a sp
Using(5)andmissionconstraintsfrom(9),
2×(106−76)×π
α= =23rad/s2 (10)
0.152×180
6 MathematicalModelling&OptimizationofCrewModule-Service...
Fig.4.GeometricConstraintsduetoLaunchVehicle
Table1.Inputparametersforoptimizationstudy
m 50kg
mm 0.1m=5kg
mc 0.2m=10kg
mp 0.7m=35kg
lc 650mm
Xsp 175mm
Ygap 450mm
θc 76◦
Xmax 330mm
Ymax 1090mm
Considering uncertainity in the input variables and to be more conservative, the above value is rounded off to
30rad/s2.
TheremainingparametersinthisgroupdependsonthedecisionvariablesandarecomputedinSection5.
X
l = sp
a cosβ
X =Xsp+lmcosθ (11)
Y =l sinβ−l sinθ
a m
Y =X tanβ
sp sp
4. OptimizationStudy
Theoptimizationiscarriedoutforbothunconstrainedandconstrainedcase. Thecomputationsarecarriedout
inSciPyVer. 1.1.0scientificcomputingpackage[7]ofPythonVer. 3.7.1platform. Pythonisanopen-sourcegeneral-
purposeprogramminglanguagewithsupportforscientificcomputationanddatavisualizationtools.Theminimiza-
tionoftheobjectivefunction,forthepresentstudy,iscarriedoutusingscipy.optimizewhichprovidesfollowingsub-
routinesformultivariablenonlinearoptimizationofscalarobjectivefunctions.
1. scipy.optimize.minimize:Thisfunctionsprovidesvariousstandardalgorithmslike‘Nelder-Mead’,’Powell’,’CG’
methods,’COBYLA’etc.forbothunconstrainedandunconstrainedcases.
2. GlobalOptimization:scipy.optimizecontainsalgorithmslikeBasin-Hopping,DifferentialEvolution,SHG,Dual
Annealingandbruteforceforfindingglobalminimumofmultivariatefunction.
LalaSuryaPrakashetal./Int.J.Adv.Appl.Math.andMech.7(3)(2020)1–10 7
(a)Lossofprecision (b)Convergedwith∇f (cid:54)=0
Fig.5.Resultsofunconstrainedoptimization
Fig.6.Performanceofoptimizationalgorithmwith2decisionvariables
Case1:UnconstrainedOptimization
NecessaryandSufficientconditionsforlocalminimum[8]:
∇f(X∗)=0
(12)
∇2f(X∗)(cid:194)0
The unconstrained case is solved numerically using both direct search and descent methods implemented in
scipy.optimize.minimize module. Itis found that the above problem is ill-posed and no solution found withthree
independentdecisionvariables,l ,θ, andβ. Mostofthealgorithmsdidnotconvergetosolutioncitinglossofpre-
m
cisionandsampleresultsareshowninFig.5. AsevidentfromFig.5(a),thevalueofobjectivefunctionfallsrapidly
andgoestonegativesidebeforestoppingduetolossofprecision.InFig.5(b),thefunctionvaluedwellsatzerobefore
fallingto−6.5×1016.
Basedontheaboveresults,theminimizationapproachismodifiedbyreducingthenumberofdecisionvariableto
2Nos.,l , and θ. βisexcludedasitappearsonlyinoneplaceintheexpressionofobjectivefunction(Eq. (7))and
m
doesnoteffectphysicalparametrslikeCGandMI.Moreover,βcanbeseparatelycontrolledindependentofboom
structure. Withβasinputparameter,thealgorithmsconvergesandresultsareshowninFig.6. Theresultsclearly
showsthat’Nelder-Mead’algorithmwhichisadirectsearchmethodtakesmorenumberofiterationcomparedto
gradientbasedmethodslike’BGFS’toarriveatoptimumvalue. Boththealgortihmsareinitiatedwithsameintial
condition,l =0.7mandθ=20◦.TheTable2showstheoptimumvaluesofdecisionvariables,l andθ,forminimum
m m
actuatorforce,F ,correspondingtovariousβ. ItisconcludedthatthereisnolocalminimumofF withrespectto
a a
β. Thenegativevaluesofθarediscardedastheyrepresentun-realisticscenariosforourcase. Hence,theminimum
possiblevalueforF liesbetweenβ=60oandβ=70o.Theθinthisrangeofβisnearlyequaltozeroi.e.thebentarm
a
8 MathematicalModelling&OptimizationofCrewModule-Service...
Table2.MinimumFaforrangeofβ
β lm θ Fa
(deg.) (mm) (deg.) (kN)
10 509.07 102.90 2.64
20 509.07 82.82 2.79
30 509.07 62.74 2.86
40 509.07 42.67 2.84
50 509.07 22.59 2.73
60 509.07 2.50 2.54
70 509.07 -17.59 2.27
80 509.07 -37.74 1.93
90 509.07 -57.94 1.53
Table3.ResultsofGlobalOptimization
Paramters Bounds OptimumValue
lm 100to900mm 509.07mm
θ 0to90◦ 0◦
β 0to90◦ 90◦
Fa NotApplicable 2.20kN
isalmosthorizontal.Clearly,theoutcomeoftheunconstrainedoptimizationdoesnotmeetthehorizontalconstraint
ofX (constraintg ).
max 1
Theabovebehaviourisfurtherconfirmedby’DifferentialEvolution’alogrithmforglobaloptimization. Thebounds
fordecisionvariableswerechosenjudiciouslytoallowforpracticalsolutions.Thealgorithmconvergeswithnon-zero
jacobian,suggestingnon-existenceofanyrelativeminimuminthedesignspace.Theoutcome(Table3)isinlinewith
theunconstrainedoptimizationresult,ifthenegativevaluesofθareneglectedforthelatercase.
Case2:ConstrainedOptimization
As mentioned in Section 3, constraints are of inequality type and Kuhn-Tucker conditions need to be satisfied
[8]forobtainingminimumofF (Eq. (8)). Itisobservedfromtheunconstrainedoptimizationstudy(Case1)that
a
theproblemisill-posedwithnorelativeminimum. The’SLSQP’methodisemployedwiththreedecisionvariables
viz. l , θ, β, considering the flexibility of imposing bounds via scipy.optimize.minimize function. The inequality
m
constraintsareappliedasperEq.(9).
Bounds:l =[0mm,600mm], θ=[0◦,150◦], β=[40◦,90◦]
m
Initializationvector:l =100mm, θ=30◦, β=70◦
m
The’SLSQP’algorithmtook16iterations,referFig.7,toconvergeatthesolutiongiveninTable4.
TheabovesolutionisverifiedexpliciltyforcomplianceofconstraintsandtheresultsareshowninTable5.
Fig.7.ConvergenceofSLSQPalgorithm
LalaSuryaPrakashetal./Int.J.Adv.Appl.Math.andMech.7(3)(2020)1–10 9
Table4.Optimumsolutionwithconstraints
Parameter OptimumValue
lm 156.5mm
θ 7.95◦
β 69.14◦
Fa 3.56kN
Table5.Constraintscorrespondingtooptimalsolution
Constraints ValueatX*
g1 1.56×10−8mm
g2 9.31mm
g3 2.56×10−10mm
5. ResultsandDiscussions
Asshownbytheunconstrainedandglobaloptimizationstudyinprevioussection, theoptimizationproblemof
boom-actuatorsystem,withthreedecisionvariables,isanill-posedone.Theoutcomeoftheglobaloptimization,θ=
0◦andβ=90◦,isanintiuitiveone.Observingtheobjectivefunction(Eq.(8)),itisindicative(notobviousandneces-
sary)thatactuatorforceminimizesasθ+β=90◦. However,solvingtheproblemunderconstraintsresultedinviable
solutionwhichcanbeimplementedforhumanspaceflightprogram.
1. Basedontheresultsobtainedfromconstrainedoptimizationstudy,thederivedparametersarecomputedusing
Eq. (4)and(11). Table4andTable6providesallparameters(Fig.2(b))todefinetheactuator-boomconfigura-
tionasmodeledforthepresentstudy.
2. A single bracket structure is proposed to mount the actuator and boom. The bracket is rigidly fixed to ser-
vicemodulestructure. X =329.9mm andY =437.6mm representstheoverallenvelopeofthebrackettobe
designed. Itisobservedthatthehorizontalspanofthebracketisequaltomaximumspaceconstraintinthe
horizontaldirection. So,itisprudenttokeepsomemarginwithactualvehicleenvelopepriortooptimization
study.
3. CutoutinSMneedtobeconfiguredtoroutethefluid/electricallinesthroughbrackettotheboomandfinally
toCM.Flexiblehosesarethefirstchoiceforfluidlinesconsideringthemovementrequiredduringseparation.
However, caution shall be taken to bend the hoses properly in a smooth manner around the bracket-boom
structure. Analternativeistousecombinationofflexiblehosesandrigidtubes. Flexiblehosesshallbeused
fromSMtoboombottom,facilitatingthemovementduringseparation.Rigidtubesovertheboomispreferable
asthereisnorelativemovementrequiredinthispartduringseparation.
4. Theminimumforcetobedeliveredbyactuatorisfoundtobe3.56kN (Table4).Thegeometriclength,l may
a
notbeconsideredasintiallength(non-actuatedcondtion)oftheactuator. Theactuatorcanbesmallerthan
theoptimumvalueof491.5mm.Thismeansthattheactuatortobemountedonpassivestructuretosatisfythe
geometricconstraintof491.5mm.
5. Theminimumforceobtainedfromconstrainedoptimizationisapplicableonlyfortheinitialconditionwhen
theumbilicalsystemismatedwithCM.Thevariationofforcerequirementduetomovementofumbilicalsystem
Table6.DerivedParameterscorrespondingtoOptimumsolution
Parameter OptimumValue
la 491.5mm
X 329.9mm
Y 437.6mm
Ysp 459.3mm
θ 37.95◦
f
10 MathematicalModelling&OptimizationofCrewModule-Service...
Fig.8.ForcerequiredfromActuatorduringseparation
isnecessarytodesigntheactuator. Expressionof X andY (Eq. (9))canbemanipulatedtoleadtofollowing
geometricrelation.
Y +l sinθ
tanβ= m (13)
X−l cosθ
m
UsingEq.(7)and(13),theforcerequiredbyactautorisplottedforθ=7.95◦to37.95◦(Fig.8).
6. Conclusion
Asimplifieddynamicmodel, oftheproposedactuator-boomconfiguration, isformulatedbasedonrelevantas-
sumptions.Theactuator-boomconfigurationisoptimizedconsideringconstraintsfrommissionandlaunchvehicle.
Asthereisnogeneralruletoselectthebestalgorithm,itisrecommendedtoexplorethebehaviourofobjectivefunc-
tionusingvariousavailablealgorithm. Inthiscase, theobjectivefunctionisexploredusingunconstrained, global
andconstrainedoptimizationtechinques.Itisalsoshownthatevenincaseofill-posedenigneeringproblem,feasible
solutionsarepossiblewithinthespecifiedgeometricandfunctionalconstraints.Theimplementationoftheobtained
resultstoactualhardwareisalsodiscussedinbrief.Itisnotedthatthepresentstudydoesnottakeintoaccountiner-
tialforcesduetolaunchvehiclemotion. Thisstudycanbefurthermodifiedtoincludestructuredynamicaspectsof
boom.
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