DTIC ADA523295: Six-Degree-of-Freedom Model of a Controlled Circular Parachute

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by  Defense Technical Information Center

JOURNALOFAIRCRAFT
Vol.40,No.3,May–June2003
Six-Degree-of-Freedom Model of a Controlled Circular Parachute
VladimirN.Dobrokhodov, OlegA.Yakimenko,†andChristopherJ.Junge‡
¤
U.S.NavalPostgraduateSchool,Monterey,California93943-5106
Thepapercontinuesaseriesofpublicationsdevotedtomodernadvancesinaerodynamicdeceleratorsystem
technologystarted recently (JournalofAircraft, Vol.38,No.5,2001)andaddressesthedevelopmentof a six-
degree-of-freedommodelofaguidedcircularparachute.Thepaperreviewsexistingcircularparachutemodels
anddiscussesseveralmodelingissuesunresolvedwithintheframeofexistingapproachesorcompletelyignoredso
far.Theseissuesincludeusingdataobtainedintheaerodynamicexperimentsandcomputational-(cid:143)uid-dynamics
modelingforbothundistorted(uncontrolled)anddistorted(controlled)canopyshapes,introducingandcomputing
controlderivatives,andprovidingcomparisonwiththereal(cid:143)ightdata.Thepaperprovidesstep-by-stepdevelop-
mentofthemathematicalmodelofcircularparachutethatincludesthebasicequationsofmotion,analysisand
computationoftheaerodynamicforcesandmoments,andinvestigationwithmodelingofspecialmodesobserved
in(cid:143)ight.Itthenintroducesanewapplicationofatwo-stepaerodynamicparametersidenti(cid:142)cationalgorithmthat
isbasedoncomparisonwithtwotypesoftheair-dropdata(uncontrolledsetandcontrolledone).Thepaperends
withsummaryoftheobtainedresultsandproposesavitaldirectionforthefurtherelaborationofthedeveloped
model.
Nomenclature l = lengthofactuatedpneumatic
max
muscleactuators
A = apparentmasstensor
6 6
apl£ = dimensionofacubicpayloadcontainer lPMADlmin = nominallengthofpneumatic
b = body-(cid:142)xedcoordinateframe muscleactuators
fCDg = aerodynamicdragcoef(cid:142)cient lSL = lengthofsuspensionlines
ofundisturbedcanopy M = vectorofexternalmoment
Cm = aerodynamicmomentcoef(cid:142)cient Mcanopy = aerodynamicmomentvector
ofundisturbedcanopy ofundisturbedcanopy
Cn = yawmomentcausedbyyawacceleration MiG = momentscausedbytheweightoftheith
CCnlrN == cdoanmtproinlgmmomomenetnctoceofe(cid:142)fc(cid:142)iceinetnt ccoenmteproonfengtrawvhiteynobfetihnegwtrhaonlselastyesdtetomthe
Fn = vectorofexternalforce Mrisers = vectorofthemomentcaused
Fa=d = totalaerodynamicforcevector bythechangeinriserlength
F = aerodynamicforcevector Ma=d = totalaerodynamicmomentvector
canopy
ofundisturbedcanopy m = totalmassofthesystem
Frisers = forcevectorcausedbythechange ma = massofairtrappedbythecanopy
inriserlength mi = massoftheithcomponent
G = gravitationvectorcausedbyEarth’s n = numberofactuatorsactivated(n 1;2 /
2f g
gravityg PCP = locationofthecenterofpressureinfbg
IIIjijijj === mmacbeoonommturtaeealnnxmttsseosoomffoiifennnfeebtrrsgttiioaafaoinbfeothrutetiaaitxohefscthooemfifptbhognent PPqPOuuPMDA[xu;yu;zu]T ==== pppdrooyessnsiiasttmiiuoorinnecooipnffrettphhsneeseurmuremeaoldasetiylcsimnteufmuscginlefaucgtuators
Qjj componentinitscentroid-(cid:142)xedaxes Rp = radiusofin(cid:143)atedparachute
kkCD0;kCD®;kCm == oemptpimiriiczaalticoonepf(cid:142)arcaiemnetstefrosrsymmetrical R0 = noofmuninina(cid:143)laotredrecfearneonpcyeradius
mn
(undisturbed)canopyusedtocompute ubR = rotationmatrixfromfbgtofug
apparentmasstensor’scomponents S0 = canopy’sreferencearea
l = distancebetweentheoriginof b T = totalenergy
B andthetopofpayload f g TADS = kineticenergyoftheparachute
l = relativelengthofkthactuator(l [0 1]/ Tair = kineticenergyofsurroundingair
Nk N2 I t = (cid:143)ight(descent)time
f
u = localtangentplanecoordinateframe
Received7July2002;presentedas Paper2002-4613attheAIAAAt- fVg [u;v;w]T = inertialvelocityvector
mrTehvoiissspimhoenartrieecrciFaelliivgieshddt6eMcJleaacnrheuadanrayicw2s0oC0rok3n;ofaefcrtcehenepctUee,d.MSf.ooGrnoptevureberlyni,cmCateAino,tn6a1–n49dJAiasnunugoautrsystu22b00j00e23ct;. VVauDDD[[uVax;;vVay;;wVaz]]TT == ainiresrptieaeldvveleocctoitryvectormeasuredinfug
tocopyrightprotectionintheUnitedStates. Copiesofthispapermaybe W = windvectormeasuredin u
u f g
madeforpersonalorinternaluse,onconditionthatthecopierpaythe$10.00 z = zcoordinateofthesystem’scenter
G
per-copyfeetotheCopyrightClearanceCenter,Inc.,222RosewoodDrive, ofgravityin b
Danvers,MA01923;includethecode0021-8669/03$10.00incorrespon- z = zcoordinatefofgtheithcomponent’s
dencewiththeCCC. i
centroidin b
¤NationalResearch CouncilResearch Associate,DepartmentofAero- ®;® = angleofattafckg,spatialangleofattack
nauticsandAstronautics.MemberAIAA. sp
†ResearchAssociateProfessor,DepartmentofAeronauticsandAstronau- ®mn = apparentmasstensor’scomponents
tics.AssociateFellowAIAA. ¯ = side-slipangle
‡LieutenantCommanderoftheU.S.Navy,DepartmentofAeronautics " = canopyshaperatio
andAstronautics.MemberAIAA. ½ = airdensity
482
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Six-Degree-of-Freedom Model of a Controlled Circular Parachute
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DOBROKHODOV,YAKIMENKO,ANDJUNGE 483
¿ = actuator’stransitiontime Withregardtoaerodynamicsofthefullydeployedcanopieswith
’;#;Ã = Eulerangles a symmetric shape, variousauthorshave shown that the nonlin-
› [p;q;r]T = vectorofangularvelocity ear natureof basicaerodynamicterms is a functionof the angle
D
ofattack.Forinstance,inRef.16authorsdiscussparameteriden-
Subscripts ti(cid:142)cation using (cid:143)ight-testdata,where they point out that the de-
pendenceof aerodynamicterms on the angle of attack is highly
i = componentofanaerialdelivery
nonlinearunlikethoseofanaircraft.Many ofpapers12;14;15;17 in-
system(i 1;2;3;4 /
2f g cludesuf(cid:142)cientdatafor fullydeployedcanopiesanda varietyof
j = axesof b (j x;y;z /
f g 2f g (cid:143)ightconditions.(Similartotheaircraftaerodynamicsthemajor-
k = actuatornumber(k 1;2;3;4 /
2f g ityofavailabledatahasbeenobtainedonthebaseofwind-tunnel
m,n = rowandcolumnof6 6tensorvector
£ experiments.)However,theresultsreportedemploypredetermined
symmetriccanopyshapes.Theaerodynamicsofadistortedcanopy
Introduction hasnotbeenconsidered.
PARACHUTES have been the simplest and cheapestdevices Insummary,overthe past40yearsa consensusonthe lackof
accuratedynamicmodelingofapparent-masseffectsandnonlinear
used for the deliveryof materials,people,and vehiclesever
aerodynamicsofdistortedcanopieshasemerged.
since their (cid:142)rst recorded use by Jacques Garnering who jumped
Thecomplexityofparachutemotionhasalsobeencon(cid:142)rmedby
froma balloonoverParisin 1797.However,thisverysimplicity
theextensiveamountofrealair-dropexperiments.21 24First,ithas
makestheiraerodynamicsverydif(cid:142)culttomodel.Speci(cid:142)cally,the ¡
beenveri(cid:142)edthatstronglynon-linearnatureoftheparachuteaero-
parachutenotonlyde(cid:143)ectssurroundingair,butalsoadoptsitsshape,
dynamicsis determinedbythreemajor factors:(cid:143)ightconditions,
whichisdictatedbytheair(cid:143)owgeneratedbythecanopy.Further-
canopyshape,andgeometryof“parachute-payload”system[there-
more,lackofstreamliningduringcontrolactivationmakesturbulent
afterreferredtoasanaerialdeliverysystem(ADS)].Thus,assuming
ratherthanlaminar(cid:143)owdominatetheparachute’saerodynamics.
thatthematerialgeometrydoesnotchangeduring(cid:143)ightleadstothe
Signi(cid:142)cantresearchon(cid:143)atcircularparachutemodelinghasbeen
conclusionthataerodynamicforcesandmomentsdependoncanopy
done over the past 60 years by researchersin the U.S., Europe,
shape,spatialangleofattack,anddynamicpressure.
andRussia.However,nocompletemodelofa controlledcircular
Second,thedistortionoftheaxisymmetriccanopyshaperulesout
parachutehasbeendevelopedsofar(whereasanumberofef(cid:142)cient
employingthewell-establishedanalyticalresultsusedtodetermine
techniquesaddressingthecontrolofmaneuverableparachuteslike
theapparentmasstensorforaxisymmetriccanopy.Thisproblemhas
parafoilexists).Moreover,existinghigh-degree-of-freedommodels
notbeenresolved.Therefore,basedonpreliminaryresultsprovided
ofcircularnoncontrolledparachuteslackveri(cid:142)ednonlinearaerody-
by CFD techniques25 and our own analysis,severalassumptions
namicsandusemostlyempiricalvaluesfortheapparentmassterms
havebeenmadetomodelcanopydistortioncausedbycontrolinputs.
(seeRefs.1–22).
In particular,it was assumed that the number of canopy shapes
The main contributionof the work reportedhereis the devel-
causedbycontrolinputsare(cid:142)niteandknown.
opment of a controlledmodel of a six-degree-of-freedom (DoF)
At the same time there is an essentialyaw rotationcausedby
circularparachute.Thiswasdonebyapplyinganonlinearsystem
canopyshapedistortionwhenevenasmalldifferencebetweenthe
identi(cid:142)cationalgorithmtore(cid:142)nethecomputational-(cid:143)uid-dynamics
lengthoftwo adjacentcontrolinputsispresented.Therefore,the
(CFD)valuesoftheaerodynamiccoef(cid:142)cients.Thecriticalideawas
controlinputperformancestronglyaffectsthe whole system dy-
to(cid:142)rstidentifytheuncontrolledmodelaerodynamicsandusethe
namics. Hence, the controlled parachute model should properly
resultingestimatestoidentifythecontrolledmodelaerodynamics.
considerit.
Anotherimportantstepwas to subtractthein(cid:143)uenceofthe wind
Finally,thedescentratedependsnotonlyonairdensitybutalsoon
whendeterminingthecontrolledmodel.Thisapproachwasmade
controlinputsthroughthecanopyshapedistortionthatwasalsoob-
possiblebytherichnessofthe(cid:143)ightdataset,23;24 thatis,datafor
servedduringtherealdrops.Therefore,themodelofthecontrolled
bothuncontrolledandcontrolleddropswereavailable.
(cid:143)atcircularparachuteshouldincludesixdegreesoffreedomwith
Adetailedanalysisofthecircularparachutemodelingproblem
anappropriatedescriptionofapparentmasstensor,aerodynamics,
discussedintheliteraturehasidenti(cid:142)edtwoshortcomingsinherent
andcontrolsystemdynamics.
inexistingmethods:1)estimationofapparentmasstermsisdone
Thispaperisorganizedasfollows.Thenextsectionaddressesthe
empiricallyforaxisymmetricshapes;2)theonlyaerodynamicscon-
developmentofasix-DoFparachutemodel.Itcontainsthedetailed
sideredarethoseofafullydeployedandsymmetriccanopy.
discussionofthemathematicalissuesinvolved,beginningwithanal-
BasedonHenn’s1 work introducingapparentmasscoef(cid:142)cients
ysisofthebasicequationsofmotion,followedbycomputationof
(1944),severalauthorsdevelopedmathematicalmodelsof a dif-
aerodynamicforcesandmoments,andendingwiththemodelingof
ferentdegreeofcomplexitytoinvestigatethedynamicbehaviorof
somespecialmodesobservedinthe(cid:143)ighttest.Thefollowingsec-
parachutes(e.g.,seeRef.2).Differentempiricalvaluesfortwoap-
tiondiscussesveri(cid:142)cationoftheparachutemodelusingatwo-step
parentmasseswereusedinthesestudies.In1962Lester3rederived
parameteridenti(cid:142)cationtechnique.Thepaperendswiththeconclu-
basicequationsofmotionshowingthatHenn’sequationswereer-
sions.TheAppendixcontainsmass–geometrydatumforageneric
roneous.At the same time Ibrahim4;5 has conductedthe (cid:142)rstex-
ADSconsistingofaG-12parachuteandanA-22container,which
haustiveanalysisoftheapparentmassandmomentofinertiaterms
wasusedformodeling.
ofcup-shapedbodiesinunsteadyincompressible(cid:143)ow.Hewasalso
the(cid:142)rstonewhoshowedthatresultingparachutedynamicperfor-
ModelDevelopment
manceissensitivetothevaluesoftheseterms.Thisstudyinitiated
thescienti(cid:142)cdebateontheextentofin(cid:143)uenceoftheapparentmass De(cid:142)nitionsandAssumptions
termsonthestabilityofa descendingsystem.Manyauthors13 17 Inthissectionwediscussthedevelopmentofacircularparachute
¡
concludedthatneglectingnonlineartermsandthestochasticnature model.The speci(cid:142)c parametersandgeometryofdescendingsys-
ofparachutedynamicswouldleadtocompletelyinaccuratemodel- tem usedwerethoseofa G-12 parachute.TheG-12isa 150-m2
ingandalsostressedtheneedforexperimentaldeterminationofthe nyloncargoparachutewith64suspensionlines(SL)26(Fig.1).The
apparentmassterms.Anattempttousetheoreticalapparentmass A-22deliverycontainerwasselectedasaprototypecargobox.This
andmoments-of-inertiacoef(cid:142)cientsbasedonthoserelatedtotheair 1.82-m3 almostcubiccontainerhasapayloadcapacityofnearlya
displacedbyellipsoidsofrevolutionmovinginapotential(cid:143)uidwas tonandiscommonlyusedwiththeG-12bytheU.S.Army.
madealso.12Oneofthelatestexperimentalstudies14;15suggeststhat The following controlled ADS architectureis considered.All
apparentmassesandmomentsofinertia(hereafterreferredtojointly lines are assembledinto eight link assembles (see Fig. 1). Each
astheapparentmasstensorterms)arethefunctionsofaspatialangle pairofassemblesisattachedtooneoffourrisers.Attheotherend
ofattackanddependingonaccelerationmightsigni(cid:142)cantlyexceed therisersarecoupledtothepayloadatfourdispersedpoints(see
thetheoreticalones. Fig.2).By designtheserisersallowcontrollingtheparachuteby
484 DOBROKHODOV,YAKIMENKO,ANDJUNGE
Fig.1 G-12parachutecanopy.
Fig.2 Payloadrigging. Fig.3 Systemofaxesandde(cid:142)nitionofaerodynamicterms.
lengtheningoneortwoadjacentactuatorsandhencedisturbingthe
4)Theaerodynamicforcesgeneratedbythepayloadarenegligi-
symmetricshapeof the canopy.The pneumaticmuscleactuators
ble.
(PMA)developedbyVertigo,Inc.,27 aremodeledandusedinthis
5)Undistortedcanopyhasanaxialsymmetryaboutthezaxis.
studyasaprototypeofthecontrolrisers.
Distortionofthecanopy’sshape(causedbythelengtheningofone
TwoCartesiancoordinatesystemshavebeenchosentodescribe
ortwoadjacentrisers)introducesasymmetricforcesandmoments
ADSmotion(seeFig.3).
allowingsteeringoftheADSinacertaindirectioninahorizontal
LinearpositionoftheADS iscomputedwithrespecttoalocal
plane.Moreover,itobviouslymakesADS nonsymmetrical.(One
tangentplane u .ItspositiveydirectionisalignedwithlocalNorth,
f g planeofsymmetrystillremains,butitslocationisnotconstantwith
thepositivexdirectionpointsEast,andthepositivezdirectionpoints
respecttothebodyframe.)Inthisstudyhoweverwewilladditionally
up.
assumethattheeffectofriserslengtheningandcanopydistortion
Allothercomputationsareperformedinthebody-(cid:142)xedcoordi-
from the standpointof changing tensors of inertia and apparent
nateframe b .Itsoriginisattachedtothecenteroftheopen-end
f g massesisnegligiblysmall.Thereforeevenforacontrolledcircular
planeofthecanopy.Thex and y bodyaxeslieintheplaneparal-
parachutethesetwotensorswillbeassumedtohavethesameform
leltothecanopy’s base,andz isalignedwiththeimaginaryaxis
asforuncontrolled(symmetrical)parachute.
extendingtowardthecentroidofthepayload.
Itisworthmentioningthatindifferentaerohydrodynamicstudies
theoriginof b issometimesplacedatthecanopy’scentroid.The AerialDeliverySystemGeometry
undisturbedcfangopyisalwaysassumedtobeaplanetaryellipsoid, Figure4introducesasetofparametersde(cid:142)ningtheparachute’s
butitcanhaveadifferentratioofminorandmajoraxes.Thatmakes geometry.Numbers1:::4denotetheADS’scomponents:canopy,
thelocationoftheoriginof b tobeconditionalfromtheconcrete rigging(suspension)lines,actuators(PMAs),andpayload.Thepo-
parachutedesign.Moreoverfthgezcoordinateofthecanopycentroid sitionoftheircentroidszi aredeterminedwithrespecttothebody
canbedeterminedonlyapproximatelybecauseinthegeneralcase frameorigin(pointOonFig.4).
thereisnoanalyticalformulaforellipsoidalshells.Despitethefact Onthis(cid:142)gureRpD23R0denotestheradiusofin(cid:143)atedparachute,
thatoriginallythesix-DoFmodelpresentedinthispaperwasalso andapl denotesthedimensionofacontainer(withoutlossofgen-
developedwiththissetupoftheframe b ,itwaslateronredone eralityallthreedimensionsofacargocontainerwereconsideredto
withtheoriginatthecenteroftheopen-fengdplaneofthecanopyto bethesame).
makeequationsofmotionmoreuniversal. Asjustmentioned,theshapeofundisturbedcanopyisahalfof
For simplicitywe will skipsubscriptb in the furtheranalysis, aplanetaryellipsoid(hemispheroid),meaningthatitiscircularin
assumingthatallvariablesandaerodynamiccoef(cid:142)cientswhenap- planwhenviewedalongthezdirectionandellipticalwhenviewed
plicablearede(cid:142)nedin b . fromtheside.Theratiooftheminortomajoraxes(canopyshape
The followingassumfpgtionsmostlyadoptedfrom the Tory and ratio)willbedenotedas".ToryandAyres12reportedthatthisratio
Ayrespaper12wereusedtodevelopthemodel: fora(cid:143)atcircularcanopyistypicallyequalto0.5.Otherresearchers
1)Becauseofthepredeterminedarchitecture,theparachuteand (e.g.,seeRefs.14–17)assumedcanopytobeahemisphere(" 1).
D
payloadareconsideredtoberigidlyconnectedtoeachother(Fig.2). FortheADSathand,"isequalto0.82.
2)Duringtheairdrop,thesetworigidlyconnectedpartsareas-
sumedtoexperienceonlygravityandaerodynamicforces. BasicSix-Degree-of-FreedomEquations
3)Thecanopyexperiencesallaerodynamicforcesandmoments Followingbasicanalyticalmechanicsprinciples,thetotalenergy
aboutitscenterofpressure. T of a whole descendingsystem relative to body frame can be
DOBROKHODOV,YAKIMENKO,ANDJUNGE 485
rigginglinesm ,actuatorsm ,andpayloadm
2 3 4;
X4
K m z mz
D i iD G
i 1
D
wherez isthestaticc.g.oftheoverallADSwithrespecttothepoint
G
O(seeFig.4).Theothernotationsdenotethediagonalcomponents
ofADS’sinertiatensorI ;I ;I andapparentmasstensor’scom-
xx yy zz
ponents® .Theonlyasymmetryleftintheseequationsiscaused
mn
byapossibleasymmetryofpayload.(I ingeneralmightnotbe
xx
equaltoI :/(Computationofbothtensorsisconsideredindetails
yy
inthefollowingsubsections.)
Equations(4)and(5)areverysimilartothoseusedtomodela
rigid-bodymotion[F mV m.› V/andM I› › I› ],
D C £ D P C £
andinvectorformtheycanberewrittenasfollows:
F M V ¤ M ; M I› H I M (6)
D m P C c Om D P C cOC cr
Inthelatterexpression
M diagm ® ;m ® ;m ®
m D2 f C 11 C 11 C 333g
vr wq q rp
¡ PC
¤ 4 ur wp qr p 5
c D ¡ ¡ P
vp uq 0 .p2 q2/
¡ ¡ C
M [m ® ;m ® ;K ® ]T
Om D C 11 C 33 C 15
I diag I ® ;I ® ;I ®
D 2f xx C 44 yy C 44 zzC 366g
wp ur v qr vw
Fig.4 ADSgeometry. H 4wq¡vr ¡uP ¡pr uw5
c D ¡ C P ¡
0 0 0
givenby
T DTADSCTair (1) IOD[K C®15;Iyy C®44¡Izz¡®66;®33¡®11]T
TheexpressionsforT,TADS,andTair canbefoundinRef.16. M [0;0;.I I /pq]T
Foranideal(cid:143)uidthekineticenergycanbedeterminedintermsof cr D yy ¡ xx
velocitypotential.However,itiscommonpractice16toassumethat BeingresolvedwithrespecttoVand› ,Eqs.(6)yield
P P
thekineticenergyofreal(cid:143)uidcanbede(cid:142)nedsimilarly.Usingthe
Lagrangeapproach,thebasicequationsofmotionfortheparachute– VP DMm¡1.F¡¤cMOm/; ÄP D I¡1.M¡HcIO¡Mcr/ (7)
airsystemcanbeobtained.Themostgeneralformoftheseequations TheattitudeoftheADSisdeterminedbytheEulerangles’,#,
in b is andÃ.Therelationbetweenvector› andEuleranglesisfoundin
f g
‡ ´ ‡ ´ theusualway28;29:
d @T @T 2 3 2 3
› F (2) Á 1 sinÁtgµ cosÁtgµ
dt @V C £ @V D P
‡ ´ ‡ ´ ‡ ´ 4µ5 40 cosÁ sinÁ 5› (8)
d @T @T @T P D
› V M (3) Ã 0 sinÁ.1=cosµ/ cosÁ.1=cosµ/
dt @› C £ @› C £ @V D P
ThelocaltangentplanecoordinatesP [x ;y ;z ]T oftheori-
uD u u u
whereaftersubstitutingtheexpressionforkineticenergyofthebody ginof b canbeobtainedemployingcorrespondingrotationmatrix
andtheairintoEqs.(2)and(3)andapplyingappropriateapparent uR: f g
b
masstensorforasymmetricbody,andsomealgebra,the(cid:142)nalform
ofequationsofmotionwillappeartobeasfollows: PPu DubRV (9)
2 3
.m ® /.u vr/ .m ® /wq .K ® /.q rp/
C 11 P¡ C C 33 C C 15 PC
F 4.m ® /.v ur/ .m ® /wp .K ® /.p qr/5 (4)
D C 11 PC ¡ C 33 ¡ C 15 P¡
.m ® /w .m ® /.uq vp/ .K ® /.p2 q2/
C 33 P ¡ C 11 ¡ ¡ C 15 C
2 3
.I ® /p .K ® /.v wp ur/ .I ® I ® /qr .® ® /vw
xxC 44 P¡ C 15 P¡ C ¡ yyC 44¡ zz¡ 66 C 33¡ 11
M 4.I ® /q .K ® /.u wq vr/ .I ® I ® /pr .® ® /uw5 (5)
D yy C 44 PC C 15 PC ¡ C yy C 44¡ zz¡ 66 ¡ 33¡ 11
.I ® /r .I I /pq
zz C 66 PC yy¡ xx
(Anexampleofstep-by-stepderivationoftheseequationscanbe ComputationofMomentsofInertia
foundforexampleinRef.16.) The staticmass centerand moments of inertiaare determined
InEqs.(4)and(5), basedontheweightanddimensionsofeachcomponent.
X4 First,thez coordinateofeachcomponentcentroidwithrespect
m m tooneofitssurfacesz andindividualcentralmomentsofinertia
D iD1 i foreachcomponentinQitheaxesofcorrespondentcentroidIQijj were
denotesthemassofthesystemthatincludesthemassofcanopym , derived.(Magnitudesofz thencanbecomputedwithaccountofz .)
1 i Qi
486 DOBROKHODOV,YAKIMENKO,ANDJUNGE
Table1 Relevantformulasofmomentsofinertiaforparachutecomponents
Component Canopy(1) Suspensionlines(2)andPMAs(3) Payload(4)
zQiDj¡B!Cj ¡R2p"0:83 Lc2os° a2pl
µ ¶
Ii Ii 0:248m1R¡2pe0:52" ¢ mi L2.1Ccos2°/ R2 m4ap2l
QaaDQbb 0:246m1R2pe" Ix1xDIy1y 2 ‡ 12 C´¤ 6
IQcic 23m1R2p.1C0:143 "/ mi L2s1i2n2° CR¤2 m46ap2l
Finally,themomentsofinertiaforthewholeADSwerecomputed
asasumofinertiasofcorrespondingADScomponents
X4
I Ii
jjD jj
i 1
D
Thenumericalvaluesforthemomentsofinertiaofeachcompo-
nentofagenericADSaregiveninTableA4intheAppendix.Here
itcanbestatedthatalthoughthemajorcontributionsinto I and
xx
Fig.5a Canopy(1). Iyy momentsofinertiaareobviouslycausedbythepayload(more
than98%),moment I isformedbasicallybytheparachuteitself
zz
(canopy,SLs,andPMAs)withaminorin(cid:143)uenceofpayload(less
than15%).Thatshowswhyitisreasonabletoneglecttheeffectof
ADS asymmetrywhilelengtheningrisers.Anotherfeatureisthat
thesymmetryofthecargoboxwithrespecttotheaxiszsimpli(cid:142)es
Eqs.(7)zeroingvectorM .
cr
ApparentMassTerms
Similartotherigid-bodymasstensor,theapparent(virtual)mass
tensor A has 66 36 elements,and in a real (cid:143)uid these can all
¢ D
beunique.Foranideal(cid:143)uid,however,Aisasymmetricaltensor,
leaving a maximum of 21 distinctterms. In the case of a body
withtwoplanesofsymmetryandcoordinateframeoriginlocated
Fig.5b Suspensionlines(2)andPMAs(3). somewhereontheaxisofsymmetry,tensorAcanbefurtherreduced
tothefollowingform:
2 3
® 0 0 0 ® 0
11 15
6 0 ® 0 ® 0 0 7
6 22 24 7
66 0 0 ®33 0 0 0 77
A 6 7 (11)
D6 0 ®24 0 ®44 0 0 7
4 5
® 0 0 0 ® 0
15 55
0 0 0 0 0 ®
66
Herethe(cid:142)rstthreediagonalelementsrepresentapparentmasses
oftheairvirtuallystagnantwithinandaround(below)thecanopy,
thenextthreearecorrespondentapparentmomentsofinertia,and
Fig.5c Payload(4). theoff-diagonalairmass/inertiaelementscontributetothecoupling
motion.
Because of axial symmetry of the circular canopy, ® ® ,
Table 1 containsformulasfor each component.Figures5a–5c ® ® ,and® ® .Thatleavesonlythreedistincte2le2mDen1t1s,
showthecomponents.Althoughformulasforahemisphericalshell w5h5iDcha4r4e® ,®24,D®¡,®15,and® .[Thatiswhydynamicequations
11 33 44 15 66
andasolidcube(payload)arewellknown(forexample,seeRef.30), havetheirappearance,Eqs.(4)and(5)].
theformulasforahemispheroidalshellandforafrustumrightcir- Intheearlierstudiestorepresenta(cid:143)owaroundafullydeployed
cularconeshellformedbySLsorPMAsareoriginal.Approximate canopy,the latterwas representedas a spheroid.In this casethe
formulasfor a hemispheroidalshell were speci(cid:142)cally derivedto referenceairmassandmomentsofinertiacorrespondtothoseof
matchrealvaluesintherangeof" [0:5 1].(Exactcumbersome theairdisplacedbythebody
2 I
formulasfora hemispheroidalshellcan befoundfor instancein
Ref.31.) ma Dmsa D 43¼½R3p" (12)
In Table 1 R is a radius of the shell measured at the z co-
oard=ipna2teolf itssci¤enn°tr=o2idf.o(rRP¤MDARsp.)¡TlhSeLscionn°e=h2aflfo-ranSgLlsea°ndfoRr¤thDe Ixaixr D 15maR2p.1C"2/; Izazir D 25maR2p (13)
pl C PMA (seeconventionsofTable1).
consideredADScon(cid:142)gurationcanbecomputedfromthegeometric
Todayit is a usualpracticeto referto the airtrappedwithina
relationsin° .R a =p2/.l l / 1.
D p¡ pl SLC PMA ¡ hemisperoid.Inthiscaseairmassmakesthehalfofspheroid
Theindividualinertiacomponentswerethentransferredtothe
originof b usingtheparallelaxistheorem m 0:5ms (14)
f g a D a
andformulasforthemomentsofinertiaarethesameasEqs.(13).If
Ii Ii m z2; i 1;:::;4 (10)
jj D QjjC i i D neededtobecomputedwithrespecttocentroidaxes,thezcoordinate
DOBROKHODOV,YAKIMENKO,ANDJUNGE 487
forthehemispheroid’scentroidequaltoz 3"R needstobe withintheshownrange.EighteenyearslaterLudwig and Heins2
PD¡8 p
takenintoconsideration. increasedthebaselinevalueandextendedtherange.Attheendof
Generallyspeaking,apparentmasstermsdependonthecanopy’s the1960sbasedonexperimentalresearchbyIbrahim,4;5Whiteand
con(cid:142)guration,porosity,acceleration,andspatialangleofattack.For Wolf8solvedthe(cid:142)ve-DoFequationsofmotionusingapproximately
instance,Ibrahim4;5showedthatapparentmassescandroptheirval- thesamevaluesforapparentmassesandintroducedapparentmo-
uesmorethan20timeswiththeporosityincreasedfrom0to40%. mentofinertia® .ToryandAyres12implementedtheoreticalval-
44
Apparentmomentsofinertiaalsodecreasetheirvaluesbythefac- uescomputedforthepotential(cid:143)owaroundspheroid.Notonlydid
torof2.75.IntheirexperimentalstudyYavuzandCockrell14 and theyuseerroneousequations,buttheirmodelingdataseemedtonot
Yavuz15demonstratedstrongdependenceoftheangleofattackon matchtheresultsofexperiment.
especially® .(Itdecreases4.5timeswithincreaseoftheangleof Thebeginningofthe1980swasaneweraofcircularparachute
33
attackfrom0to40deg.)Theyalsorevealedastrongdependence modeling.Eaton13 and other researchteams14;16;17 developedthe
ofapparentmasstermsfromappropriateaccelerationsshowingthat right set of six-DoF equations and performed stability analysis.
theycanchangeasmuchasbyafactorof(cid:142)vewhileexperiencing Eaton’scoef(cid:142)cientsifreferredtothemassofahemispheroidrather
steadyacceleration.With regardtothelatterstudy,howeverit is thanaspheroidwouldgivethevaluesof0.4and0.8fortheapparent
unclearhow the dependencefrom accelerationcan be takeninto masses,whichisfairlyclosetothoseofDoherrandothers.16;17
account.First,theparachutedoesnotexperienceaconstantaccel- In spite of the results of experimental work of Yavuz and
erationduringdescent.Accordingtothe(cid:143)ight-testdataavailable, Cockrell14 andYavuz15showingcompletelydifferentscaleforap-
accelerationsduringdescentofafullydeployedparachuteoscillate parentmassterms,itlookslikeeverybodyiscontinuingusingthe
aroundzero.Buteventhevalueatzeroaccelerationcannotbeim- classicalsetofDoherr(e.g.,seeRefs.18–20).
plementedbecausealldependenceshavea(cid:142)rst-orderdiscontinuity The present study was not aimed at tuning these coef(cid:142)cients,
atthispointsothatthevaluesdifferasmuchasthreetimeswhen andsoa classicalsetofapparentmasstermswasusedherealso.
approachingto the steady-statedescent (zero acceleration)from Howevertheauthorswouldliketoaddressthisissueinthefuture.It
negativetopositiveacceleration,notmentioningthatthedatawere alsoexplainswhyelement® wasleftinequationsnotwithstanding
66
obtainedforthesmallrigidmodelsofahemispherecanopy.Second, thatforundisturbedcanopyitisalwaysneglected.Forundisturbed
thephysicalvaluesofapparentmassesobtainedatzeroacceleration canopyit contributesto the damping of yaw oscillations,and so
whenbeingscaledtothewholeADSmakenosensebecausethey authorsalsonormalizeditasshowninEqs.(15)forthefuturestudy,
exceedthoseofADSitself(seethefollowing). lettingk howevertobezerohere.
66
Thereforeinthepresentstudytheauthorsfollowedallothermajor
studiesandconsideredallapparentmasstermstodependexplicitly ComputationofForcesandMoments
ontheairdensityonly.Allotherpossibleeffectswererepresented AerodynamicForcesandMomentsforUndisturbedCanopy
byconstantcoef(cid:142)cients: Thetotalexternalforceandmomentactingonthesystem(Fig.3)
arecausedbytheaerodynamiceffectsandtheweightofeachsystem
®11 Dk11ma; ®33 Dk33ma; ®44 Dk44IQxxa component.Thus,wecanwrite
X
®66Dk66IQzza; ®15 Dk15mazP (15) FDFa=dCG; MDMa=dC MGi (16)
As opposed to other studies17 where expression l i
pz [.rleSpLrCeslePnMtAin/g2¡theRd2pi]st(asneeceFfirgo.m4)fwraamseusebdaosriagirnefteorethneceploeinngBttoDhf, where GDubRT[0;0;mg]T (the apparent (virtual)masses do not
P f g contributetotheweightofthesystem).
applicationofthetranslationalapparentmasscomponent(canopy’s
Inturn,forcontrolledADS
centerofpressure)wasusedbecauseofthesenseofEqs.(4)and
(5)F.4o;5l;l9o¡w11i;n14gRef.17,coef(cid:142)cientsinEqs.(15)weretakentobeas Fa=d DFcanopyCFrisers; Ma=d DMcanopyCMrisers (17)
follows:k 0:5,k 1:0,k 0:24,andk 0:75.[Actually Let us address further in this subsection aerodynamic terms
11D 33D 44D 15D
the valuesgivenare slightlydifferentfromthe valuesin Ref.17 forundistortedcanopy.Initialvaluesofthedimensionlessaerody-
becausetheywererecomputedtomatchnotations(13)and(14).] namiccoef(cid:142)cientsofacircularparachutewereobtainedfromtwo
FortheADSathandwithm 472kg(½ 1kgm 3),thefollow- sources.Thebasicshapeoftheaerodynamiccurvesfortheundis-
ingexpressionsarevalid:® aD236kg( 2D2%ofm¡),® 472kg tortedcanopywasadoptedfromKnacke,26whereasinitialaerody-
( 44% of m), ® 160011kDgm2 (less»than 0.5% of33ID ), and namiccoef(cid:142)cientestimateswerebasedonCFDresultsprovidedby
®» 707kgm( 434%Dof K).By lookingatthesenumberxsx,itbe- Mosseev.25
15D »
comesclearthat® and® areevidentlyofmostimportance.The The functionaldependenceof the aerodynamiccoef(cid:142)cientson
11 33
differencebetweenthemaffectsthedynamicbehaviorofthesystem theangleofattackobtainedbyMosseevisshownonFig.6.
[seeEqs.(5)]. Onthis(cid:142)gureC .® /denotesaerodynamicdragcoef(cid:142)cient,and
D sp
To giveanimpressionwhatcoef(cid:142)cientswereusedbytheother C .® / denotestotal aerodynamicmoment coef(cid:142)cient,both de-
m sp
researches,Table2containssomedataonthemajorresearchinthe pendingonthespatialangleofattack.
areaforthepast60years. This spatialangle of attack ® and its components—angle of
sp
Henn1wasthe(cid:142)rsttoperformastabilityanalysiswithaccountof attack ® and sideslipangle ¯—are shown on Fig. 7 and can be
apparentmasses.Foraspheroidasareferencebody,hetook0.5as computedusing b -framecomponentsofanairspeedvectorV as
f g a
abaselinevaluesforbothcoef(cid:142)cients® and® andvariedthem follows:
11 33
Table2 Valuesofapparentmasscoef(cid:142)cients
No. Researcher Year Model Originoffbg Ref.body " k11 k33 k44 k15
1 Henn1 1944 ThreeDoF O Spheroid 0.5 0.5[0–0.6] 0.5[0–1.0] —— ——
2 Ludwig,Heins2 1962 ThreeDoF O Spheroid 0.5 1.0[0.6–1.4] 1.0[0.6–1.4] —— ——
3 White,Wolf8 1968 FiveDoF Sys.c.g. Sphere 1 0.7 0.7 0.23 ——
4 Tory,Ayres12 1977 SixDoF Can.c.g. Spheroid 0.5 1.31 2.12 1.34 ——
5 Eaton13 1981 SixDoF Can.c.g. Spheroid 0.5 0.2[0–0.5] 0.4[0–1.0] Unclear ——
6 Yavuz,Cockrell14;15 1981 SixDoF Can.c.g. Hemisphere 1 [1.31–6] [2.12–10] 1.5–3.0 ——
7 Cockrell,Doherr,Saliaris16;17 1981 SixDoF O Hemisphere 1 0.5 1.0 0.24 0.75
8 Present 2003 SixDoF O Hemispheroid 0.82 0.5 1.0 0.24 0.75
488 DOBROKHODOV,YAKIMENKO,ANDJUNGE
Fig.8 Yawrotationscheme.
Fig.6 G-12parachuteaerodynamics.
Fig.7 Flight angles determi-
nation.
‡ ´ Fig.9 Effectofchangingriserlength.
w
®sp Dcos¡1 pu2 va2 w2 (18) InEq.(22)Cndenotestheyawmomentcausedbyyawaccelera-
aC aC a tion,andforasymmetricbodyitisequaltozero.Howeverincase
‡ ´ ‡ ´
oftransitionofoneoftherisersfromonestatetoanotherwhileone
u v
® tan¡1 a ; ¯ tan¡1 p a (19) oftheadjacentrisershasbeenalreadylengthened,thismomentis
D wa D u2aCwa2 notequaltozero.Thenextsubsectionaddressesthisissue.
TheairspeedvectoraccountsforawindvectorWuinthefollow- YawMomentCausedbyDynamicAsymmetry
ingmanner: Asjustmentioned,signi(cid:142)cantyawrotationwasobservedduring
the(cid:143)ighttestswhenthelengthoftheriserwaschangingwhileone
V V uRTW (20)
a D ¡b u oftheadjacentrisershasbeenalreadylengthened.Figure8clari(cid:142)es
thissituation.
ObviouslytheaerodynamicforcevectorF dependsonthe
canopy Here“0”standsforashortenedriser(l 0/,and“1”denotesa
spatialangleofattackanddynamicpressureandcanbepresented NkD
lengthenedriser(l 1/,typicallywhilerisertransition(4:::5s)
asfollows: NkD
yawangleof15:::20degwasaccrued(Fig.9).
Therefore,ifthekthriserundergoestransitionthefollowingre-
F C .® /qS .V = V / (21)
canopy D D sp 0 a k ak lationisvalid:
wnohWremrheoiqtfeDvaen0cd:t5oW½r;okalVnfa8dksS2h0iosDwdey¼dnRath0m2aiitscthtphereelsocsnaungroietp;ukyd¢’isknradelefanenordetenthscaeenlaaErteeuarc.allidmiaon- Cn Dsign.lNk¡1¡lNkC1/CnlN.lN/lNkCCnrr (23)
In Eq. (23) l .l l /=.l l /, and the difference
tionofaparachuteinglideplanearesuf(cid:142)cientlyuncoupled.There- NkD k¡ min max¡ min
.l l /de(cid:142)nesthesignofthemoment.
fore,thelongitudinalandlateralmotionofasymmetricparachute NkB¡y1¡anaNklCyz1ingdatafrom20(cid:143)ighttests,thefunctionaldependence
can be studiedseparately,similarto the studyoftheaircraftlin-
earizeddynamics.28;29Thisleadsustoassumeadditionallythatthe CnlN.lN/wasfoundtobeaspresentedonFig.10andthecoef(cid:142)cient
Cr tobeconstantandequalto2s.
longitudinalandthelateralmotionofacircularparachuteinaglide n
Whenlor1 lissmall,ashappensattheverybeginningandat
planeareuncoupled. N ¡N
theveryendoftheactuationprocess,themagnitudeoftherotation
This last assumptionimplies that roll and pitch motion of the
is low because the canopy symmetry distortionis minimal. The
ADShavethesamemomentcharacteristics,thatis,C C .¯/,
rollD m rotationisatitshighestatthemiddleofactuationprocesswhenthe
C C .®/.Therefore,thevectorofaerodynamicmomentcan
pitchD m shapedistortionismaximal(seeFig.9).
beexpressedasfollows:
2 3
C ActuatorForcesandMoments
roll
M 2qS R 4C 5 (22) Letusde(cid:142)nenowaneffectofriseractuation.
canopy D 0 0 pitch Vertigo’sPMA27 usedasaprototypeofcontrolriserisbraided
C
n (cid:142)bertubeswithneopreneinnersleevesthatcanbepressurized.In
DOBROKHODOV,YAKIMENKO,ANDJUNGE 489
Fig.10 DependenceC¹l (¹l).
n
Fig.13 Blockdiagramofthesystemidenti(cid:142)cationtechnique.
First,expressionsforC .® /andC .® /werepresentedinthe
d sp m sp
followingform:
£ ¤
CD.®sp/DkCD0CQD®D0 CkCD® C¤D.®sp/¡C¤D.0/ (25)
Cm.®sp/DkCmCm¤.®sp/ (26)
yieldingthree optimizationparametersk , k , and k . The
CD0 CD® Cm
initialvalueofC mg.qS / 1wasobtainedfromtheobvious
equationforastQeDa®dDy0dDescentra0te¡.
Second,anappropriatecostfunctionwaschosen:
Fig.11 Dependence¹lPMA=f(PPMA). Z
tf
J P .t/ P .t/ dt (27)
D k u ¡ Ou k
0
whereP .t/istheinertialpositionofADSobtainedin(cid:143)ighttest,
u
P .t/denotestheestimatedpositionofADSobtainedinsimulation,
Ou
andt standsforthe(cid:143)ighttime.
f
Thereforethe objectiveof identi(cid:142)cation was to minimize cost
function(27) by varyingthree parameters in Eqs. (25) and (26)
(Fig.13).
Becausefromtheanalysisofthe(cid:143)ight-testdataitbecameobvious
thattherewasasigni(cid:142)cantdifferenceintheADSdynamicsduring
controlledanduncontrolleddrops,theidenti(cid:142)cationalgorithmwas
Fig.12 Dependence F =f(¹l,n). (cid:142)rstappliedtothedataobtainedfromtheuncontrolleddrop.The
kk riserskk resultingvaluesofk ,k ,andk wereusedthentoinitializethe
CD0 CD® Cm
secondstep,wherethesametechniquewasappliedtoacontrolled
the nominalstateallPMAs arepressurized.Upon venting,PMA drop.Whereasvaluesofoptimizationparametersobtainedinthe(cid:142)rst
increasesitslengthbyapproximatelyonethirdfromlPMADlmin to stepprovidedestimatesofADS aerodynamicsaroundzeroangle
lP¤MADlmaxDlPMAC1lPMA andcompressesindiameter.[Thede- of attack,their adjustmentat the secondstep characterizedADS
pendencel f.P /whereP standsforthePMApressure dynamicsathigheranglesofattackwithnonzerocontrolinputs.As
NPMAD PMA PMA
isshownonFig.11].Thisaction“deforms”theparachutecanopy aresult,someaveragedvalueswerefoundtomatchbothcontrolled
creatingan asymmetricalshape,essentiallyshiftingthe centerof anduncontrolledreal-dropsets.
pressureandprovidingadriveorslipconditionintheoppositedi- All two- and three-dimensionaltrajectoriesin this section are
rectionof the controlaction.With four independentlycontrolled plottedinEast–North–Uplocaltangentplane(LTP)coordinates.
actuators,twoofwhichcanbeactivatedsimultaneously,eightdis-
tinctcontrolactionscanbegenerated. UncontrolledModelIdenti(cid:142)cation
ThechangeintheaerodynamicforcecausedbyPMAactivation Thissubsectiondetailsresultsobtainedbyapplyingtheidenti(cid:142)-
FriserswasmodeledasafunctionofPMA’srelativelengthlN,number cationtechniquetotheuncontrolleddropdata.Becausethemodel
ofPMAactuatedn,andinvolvedactuationsystemdynamicswith assumesafullydeployedcanopy,itwasinitializedatthepointon
transitiontime¿27: thereal-droptrajectorywherethecanopywasfullydeployed.
Figures14–16 containthe three-andtwo-dimensionalplotsof
F .t/ f.l;n;¿/ (24)
k risers kD N thetrajectories,obtainedinoneofthereal(cid:143)ightthree-DoFandsix-
DoFsimulations.(Thethree-DoFmodeloftheparachutewasused
Figure12showssteady-statevaluesofthisdependence.Inturn,
inthepreliminarystudy.21)
the actuator moment can be computed as M P F ,
whereP [0;0;z ]T. risersD CP£ risers Notethattherealtrajectoryhasthreepointswheretheparachute’s
CPD P directionofmotionchangesmorethan90deg.Thisparticulardata
setwasselectedbecauseoftherichnessofthefrequencyspectrum
ModelIdenti(cid:142)cation
oftheparachutemotioninvolved.Fig.17containsthewindpro(cid:142)le
Intheprecedingsectionweoutlinedallstepsinvolvedinthede- forthesamedrop.Itclearlyindicatesthatbothwinddirectionand
velopmentoftheSix-DoFmodelforacontrolledcircularparachute. itsmagnitudeexperiencesigni(cid:142)cantchangeswithtime.
Nextthismodelwascomparedwiththe(cid:143)ight-testdata.Thiscom- Because duringthe uncontrolleddrop the angle of attack and
parisonrevealedinadequacyofthedataobtainedbyCFDanalysis sideslipangleare closeto zerothe valueof k was set to zero.
Cm
(seeFig.6).Therefore,astandardnonlinearsystemidenti(cid:142)cation Thischoiceimpliesthattheamplitudeofconingmotioncandiffer
technique32wasappliedtotunetheCFDdependencesC .® /and from the (cid:143)ight-testdata;however,its naturalfrequencydoes not
D¤ sp
C .® /(seeFig.13). dependonk foruncontrolleddrops.Finally,theoptimalvalues
m¤ sp Cm
490 DOBROKHODOV,YAKIMENKO,ANDJUNGE
Fig.14 Three-dimensionalprojections.
Fig.17 Windpro(cid:142)le.
Fig.15 Horizontalprojection.
Fig.18 ComparisonofVxcomponents.
Figure21containspowerspectrumdensity(PSD)plotsforEuler
angles(roll,pitch,andyaw)takenfromthe(cid:143)ighttestandthesix-
DoFmodelrun.Amongdozensofmethodsavailable,theMultiple
SignalClassi(cid:142)cationmethoddevelopedbySchmidt33wasapplied.
Thismethodwaschosentodemonstrateageneralsimilaritybetween
real and simulateddataas well as to emphasizethe matchingof
principaleigenfrequenciesonthebothsetsofdata.
Numerical valuesof these principaleigenfrequenciesfor each
Fig.16 Verticalprojection.
channelaredepictedinTable3.
Similarresultsfortheprincipaleigenfrequencieswereobtained
k andk wereobtainedbydeterminingaprioritherangeoftheir independentlyforanothersetofADS(cid:143)ight-testdatausingWelch’s
Cd0 CD®
possiblevaluesfromphysicalconsiderationsandapplyingsimple methodas well.34 That shows the consistencyof ADS (cid:143)ight-test
searchtechniquesovertheresultinggridofthisparameterspace. datatakeninthepresenceofthedifferentwindpro(cid:142)les,indicating
Figures18–20includeplotsofthex;y,andzcomponentsofthe thatthedatainTable3re(cid:143)ecttheinherentmotionoftheADS,and
inertialvelocityobtainedfromthesix-DoFmodelsimulationand notthewindspectrum.
duringthe(cid:143)ighttest.Analogousdataareplottedforthethree-DoF Inturn,thewindspectrumcharacteristicsfortherealdropbeing
model.Aconspicuousfeatureofthesix-DoFmodelcanbeseenin analyzed are presentedin Fig. 22. The resultsclearly show that
theplotsoftheverticalorzcomponent.(Figure20showsabsolute thewindenergywasconcentratedaroundmuchhigherfrequency
values).Althoughtheverticalcomponentofthethree-DoFmodel (1.9Hz)ratherthan0.11-Hzparachute’sharmonicjustseen.
remainsconstantduringthe(cid:143)ight,thesamecomponentofthesix- Themainconclusiontobedrawnfromspectralanalysisisthat
DoFmodelfollowsthe(cid:143)ight-testdatareasonablywell.Therefore thesix-DoFmodeldynamicsaresuf(cid:142)cientlyclosetothoseofthe
optimizationoverasetoftwoparametersgavefairlygoodresults. actualsystem.