Efficient sizing and optimization of multirotor drones based on scaling laws and similarity models

an author's

https://oatao.univ-toulouse.fr/26691

https://doi.org/10.1016/j.ast.2020.105873

Delbecq, Scott and Budinger, Marc and Ochotorena, Aitor and Reysset, Aurélien and Defaÿ, François Efficient sizing

and optimization of multirotor drones based on scaling laws and similarity models. (2020) Aerospace Science and

Technology, 102. 1-23. ISSN 1270-9638

laws

and

similarity

models

S. Delbecq

a

,

∗

,

M. Budinger

b

,

A. Ochotorena

b

,

A. Reysset

b

,

F. Defay

a a_{ISAE-SUPAERO,UniversitédeToulouse,10AvenueEdouardBelin,31400Toulouse,France}

b_{ICA,UniversitédeToulouse,UPS,INSA,ISAE-SUPAERO,MINES-ALBI,CNRS,3rueCarolineAigle,31400Toulouse,France}

o

a

b

s

t

r

a

c

t

Keywords: Multirotordrones UAV Designmethodology Sizing Monotonicityanalysis

MultidisciplinaryDesignOptimization

Incontrasttothecurrentoverallaircraftdesigntechniques,thedesign ofmultirotorvehiclesgenerally consistsofskill-basedselectionproceduresorisbasedonpureempiricalapproaches.Theapplicationof asystemicapproachprovidesbetterdesignperformanceandthepossibilitytorapidlyassesstheeffectof changesintherequirements.Thispaperproposesagenericandefficientsizingmethodologyforelectric multirotor vehicles whichallowstooptimize aconfiguration fordifferent missions and requirements. Startingfromasetofalgebraicequationsbasedonscalinglawsandsimilaritymodels,theoptimization problemrepresentingthesizingcanbeformulatedinmanymanners.Theproposedmethodologyshowsa significantreductioninthenumberoffunctionevaluationsintheoptimizationprocessduetoathorough suppression of inequalityconstraints when compared to initial problem formulation. The results are validatedbycomparisontocharacteristics ofexistingmultirotors.Inaddition,performance predictions oftheseconfigurationsareperformedfordifferentflightscenariosandpayloads.

1. Introduction

Multirotordroneshavebecomepopulartest-bedsforUnmannedAerialVehicle(UAV)researchandapplicationspartlybecauseoftheir greatmaneuverability and abilityto execute complex movements. Thesecapabilitieshave enabledthem to be implemented ina wide rangeof both civil andmilitary applications including surveillance, agriculture and other unprecedented purposes, such as automated packagedelivery orPersonal AirVehicles (PAVs).The competitiveness andsearchfornewapplications haveencouragedindustrials and academicstodesignlighterdroneswithimprovedflighttimesandefficiency.

BecauseUAVs aresimple-to-usesystemswitha highlevelofsupportalongwithagoodperformanceformostapplications,agrowth ofcustomizedandopensolutionshastakenplace.Toovercomesomeoftheirlimitations,suchastheirreducedflighttime,asetoftools mustbedevelopedtoaidthedesignofcustomizedsolutionsthatcanbetailoredtoaparticularapplication.

Severalapproachescanbefoundintheliteraturetodeterminethemostsuitable designforthedesiredapplication.Popularly,among hobbyists,thereisapracticeoftesting“motor+propeller” combinationsandselectingthemostappropriateoneforagivenpurpose[1].This approachisnotgenerallyadoptedataresearchorcommerciallevel,duetoitshighcostorthehighnumberofcombinationsrequiredfor agivenapplication.

Thisprocess can be sped up by calculators such as eCalc [2] or DriveCalculator [3], based on the use of datatables, although the selectionofcomponentsmustbedoneindividually,whichisimpracticalwhenseveralmodelsaretobesized,andwhenthepayloadscale factordiffers fromthe typical market trend.Furthermore,these toolsdo not paythat much attentionto the interrelationsestablished betweentherestofthecomponentsandconsequently,donot givean optimaldesignofaMultirotorAerial Vehicle(MRAV)asawhole andoptimizeitsperformancewithrespecttomultidisciplinaryobjectivesandconstraints.

Holisticdesignapproachesaresubjecttomuchinterestforapplicationscompelledbycomplexinter-relationships.Suchapproachesare mandatoryforaerospacevehiclesforinstancefixed-wings[4],rotorcraft[5] ortilted-rotors[6].Toassistthedesigner,numericalmethods havetobeusedtodeterminethebestfeasibleconfigurationandcanusedifferentmethodssuchasstatistical-based[7] or optimization-based[8,9] techniques.

*

Correspondingauthor.

E-mailaddress:scott.delbecq@isae-supaero.fr(S. Delbecq).

In the literature, rotorcraft design andoptimization methodologiesare developedfor differentsub-systems to optimizethe battery capacity[10],toselecttheoptimumpropulsionsystemcomponents[11],ortodesignthecontrollersofminiaturerotorcrafts[12].There areconceptuallayoutoptimizationapplicabletomultiplemultirotordronesystems[13] buttheirrangeofapplicationandtheirsurrogate modelsarebasedonsmall-scaledrones.

Inthissense,weconsidernecessaryinthispaper:

•

Todevelopasystemicmethodologytooptimizetheglobaldesignoftheairvehicle,andforthat,wewilluseMultidisciplinaryDesign Optimization(MDO)techniques.

•

Toformulate areusable optimization codevalidfora wide rangeofMRAVs basedon scalinglawsandsurrogate models extracted fromthedimensionalanalysisorthestudyofphysicallaws,asproposedin[14].

•

Todiscussaboutthemathematicalandconflictresolutiontechniquestosolvethepossiblesingularitiesandcomplexitypresentedin anoptimizationcode.

Theglobalobjectiveistoformulateinthemostefficientwaythebasicarchitectureofthesizingofmultirotordroneswhichwouldbe extensiblewiththeadditionofhigherfidelitymodels.

Forthispurpose,threemajortasksareaddressedinthispaper:theimplementationofasizingprocedurewithspecialattentiontothe possiblesingularitiesfound,astudyofthedesignparametersfordifferentmissioncharacteristicsandperformancerequirements,andthe validationofobtainedresultsoverexistingcommercialMRAVs.

2. Preliminarydesignandsizingofmultirotordrones

2.1. Designdriversandsizingscenarios

OneofthebiggestchallengesinUAVindustrialdesignismeetingrequiredrangeandpayloadcapacityforthevarioussizingscenarios. Thesizingscenariosarepartofthespecificationprocessandmayincludebothtransientandfailureoperatingcases,suchasamaximum rateofclimb orresistance toa crash ata givenheight. Thesescenarios determine themain drone characteristicsandthe selection of mechanicalandelectroniccomponents.Thefinaloveralldesignhastoconsidersimultaneouslytransientandendurancecriteriathatmay haveimpactonthecomponents’designdriversandsystemlevelparameters.

Thediagram givenin Fig.1 showsthemain systemcomponents,their maindesign drivers,andthe mainsizing equationsforeach scenario considered. Endurance scenarios, like hovering flight (referred to as HOV in the text) need to be taken into account for the selection ofthe batteryendurance andthe evaluationofthe motortemperaturerise. Extreme performance criteria, such asmaximum transitory vertical acceleration (referred to asTO for Take-Off in the text) ormaximum climb rate (referred to asCL in the text) are fundamental to determine themaximum rotationalspeed, maximum propeller torque, ElectronicSpeed Controller(ESC) corner power, batteryvoltageandpower,andmechanicalstress.

AlistoftheequationsandmodelsusedinthispapercanbefoundinAppendixA.

2.2. Componentssizingmodels

Mostofthedesignmethodologies,despitethewidevarietyofimplementations[20–22],arebasedonanalyticalexpressionsorcatalog dataregressionsandbenefit,asmostcontinuousproblems,ofnumericaloptimizationcapabilitiesandhighconvergenceperformance.The paper[14] hasshownhowdimensionalanalysisandtheBuckinghamtheoremcanbeusedtoestimateanalyticallythemaincharacteristics ofthecomponentsofadrone.Acomponentcharacteristic y canbeexpressedasanalgebraicfunction f ofgeometricaldimensionsand physical/materialproperties: y

=

f

(

_

L

,

d1

_

,

d2

, ...

_

geometric

,

_{  }

p1

,

p2

, ...

physical

)

(1)

TheapplicationoftheBuckinghamtheoremtotheoriginalEquation(1),withn parametersandu physicalunits(time,mass,length...), theequationcanberewrittenwithareducedsetofn

−

u dimensionlessparameterscalled

π

numbers:

π

y

=

f

(

π

d1

,

π

d2

, ...,

π

p1

,

π

p2

, ...)

(2) where:

π

y

=

yay

·

LaL



i pai i (3)

π

di

=

di L (4)

π

pi

=

Lapi,0



j pa_ipi,j (5)

Inthe casewhere the dimensionlessnumbers

π

y and

π

di canbe assumed constant, such asrespectivelythe one representing the magneticsaturationandtheonerepresentingthediameter-lengthratioforthemotor,thenthefunction f canbeapproximatedbyusing scalinglawsorsimilaritymodels[23,24].Otherwise,whendataisavailable,regressionmodels[25,26] areusedsuchasformodeling the propeller performance orlanding gear structuralanalysis. A detaileddescriptionof thehypotheses andmethods associated withthese sizingmodelsofdronecomponentscanbefoundin[14].

Fig. 1. Designdriversandsizingscenariosbasedonthefollowingreferences:(propellers) [15];(motors,batteryandESC) [16,17];(frame) [18,19].(Forinterpretationofthe colorsinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

2.3.Sizingproceduredevelopment

Withprogressoftime,designershaveestablishedasequenceofwell-definedcalculationsformultidisciplinaryproblemssuchasaircraft designandhaveappliedoptimizationtechniquestosuchengineeringproblem.Thesetechniques,alsoknownasMultidisciplinaryDesign Optimization(MDO),areusefultostudythecompleteaircraftthatinvolvescouplingsbetweendifferentsubsystems.

TheresolutionofaMDOproblemrequirestoaddresstwomainaspects:

•

structuring the architecture ofthe MDO problemby choosing the computational sequence andsolving thecoupling variables. For thispurpose,the computationalprocess can berepresented bygraphs suchasthe N2 diagram [23] or theeXtended Design Struc-tureMatrix(XDSM)[27].Theresolutionofthemultidisciplinary couplingscanbeachievedusingdifferentdistributedormonolithic MDOformulationssuchastheMultiDisciplinaryFeasible(MDF)oftheIndividualDisciplineFeasible(IDF).Hereamonolithic Normal-ized VariableHybrid (NVH) formulationis usedasit isfast androbust toscale-change dueto theusage of inequalityconstraints, a smalldimensionconsistencydesignvariableandareducednumberofconstraintsanddesignvariables[28].However, itdoesnot guaranteesystemconsistencywhentheoptimizerfailswhencomparedtotheMDF.Hybridapproaches,derivedfromIDF,are partic-ularlyinteresting herebecauseofthereasonablenumberofvariablesandconstraintstobemanagedasthey keepthefeed-forward computational sequence[29],

•

implementing optimizationalgorithms to compute the best final result with far-fewer caseevaluations. The design ofa MRAVis considered hereas a standard constrainedoptimization problem[30], wherea single objective (e.g.maximum take-off weight) is minimizedwithrespectdesignvariables(e.g.batterycapacity)andsubjecttoconstraints(e.g.motormaximumtorque).

Thefollowingsectionswillshowhowtostructurethisoptimizationproblemthroughasystemicsizingprocedureinordertofacilitate theworkoftheoptimizationalgorithms.

3. Sizingproceduredefinition

Thesizing procedureis thesequenceof calculationsteps tobe carriedout to definethe sizeandfeature ofthedrone components. Thisprocedure can berepresented asan algorithm orflowchart asshownin Fig.2. The XDSMdiagram provides a visual summary of

Fig. 2. Initial XDSM diagram for multirotor drone optimization.

Table 1

Methodologysummary.

Process Methods&tools References

Step1:DefineasystemofalgebraicequationsS basedonestimation modelsandsimulationmodelssubjecttoinequalityconstraints

Analyticalmodels,Scalinglaws,Datasheetregression [24,34,14]

Step2:Reduce(whenpossible)thenumberofinequalityconstraints FirstMonotonicityPrinciple(MP1) Section3.2.1, [35]

Step3:Buildundirectedbipartitegraph/designgraphfromS,identify

StronglyConnectedComponents(SCC)andsolvethemby

implementingtheNVHformulation

Designgraphs,Symboliccomputation,Graphalgorithms Section3.3, Section3.4, [36,37,34,28]

Step4:Performamatchingandorderingofthemodifiedsystemand

generatethesizingprocedure

Graphalgorithms,Symboliccomputation [36,37,34]

theefficiencyoftheorderingandisusefultovisualizethemodelsinvolvedandthesharedvariables.Oncetherequiredthrust forevery missionisestimatedfromaninitialguessofthetotaldronemass,propellertorqueandspeedcharacteristicsareestimated.Basedonthese valuesandtheapplicationofscalinglaws,motorparametersarecalculatedandsubsequently,theESC,batteryandframeparameters.This enablesdeterminethecontributionofcomponentsmasstothetotalmassofthedrone.However,thistotalmassisrequiredtodetermine hoverandtake-offthrustthusleadingtoanalgebraicloopwhichhastobesolvedusinganMDOstrategy[31].

Feed-backloopshavetobeavoidedasmuchaspossibleinthecalculationprocedure,sincetheyhavesignificant impactonthe com-putationaleffectiveness[32].Fortheremainingloops,theNVHformulationapproachisimplementedbyintroducingdedicatedadditional constraintsand designvariables in an optimizationprocess in orderto solve the multidisciplinary couplingssuch asthemass loop in Fig.2[33,28].

3.1. Sizingmethodology

Solving a design optimization problemrequires to follow an orderly methodology. Definingthe steps ina structured wayhelps to incorporate newdesign variables in the optimization routine and serves to orientate the equations towards a solution. The proposed designoptimizationmethodologycanbesummarizedinTable1.

Oncetheequationsaredefined,therearethreemannersoftreatingtheinequalityconstraints:

A) consideralltheinequalityconstraintsfoundintheproblemasinactive1 andrelyentirelyonthesolvertofindasolution.Inthiscase theprocessskipssteps2and3.

B) identify active andinactive constraintsafterapplicationofMP1,transformthecritical2 _active3 _{inequalities toequalitiesandlet the} remaining inequalities, which boundedness behavior is undetermined, be solved by the solver. This is the approach proposed by PapalambrosandWildein[35].Inthiscasetheprocessskipsstep3.

1 _{Aninequalityconstraintg}

i(x)≤0 issaidtobeinactiveatadesignpointxkifithasnegativevalueatthatpoint(i.e.,gi(xk)<0)[38]. 2 _{Whenthereisonlyoneconstraintboundingacertainvariablepositivelyandfinitely[35].}

3 _{Aninequalityconstraintg}

C) considertheinequalities aspotentiallyactive witha specifictreatment asReysset,Budingeret al. propose in [36].In thiscasethe processusesallthe4proposedsteps.

OptionAisthesimplestway,wherethebiggesteffortisachievedbythesolverandtherefore,itrequiresalongercalculationtime.In thenextpart,OptionBisappliedbymeansoftheMonotonicityTable.

3.2.Monotonicityanalysis 3.2.1. Methodology

Themononotonicityanalysisenablestoreducetheproblemcomplexitybyreducingthenumberofconstraints.Nevertheless, simplify-ingtheproblemwillnotonlyleadtoareductionofcomputationalcost,butalsotocheckoutpossibleincorrectformulations.Itenables todecidetosetexpressionsintheformofequalitiesinsteadofinequalitiesandavoidtolimitthesolutionsearchspace.

Foranoptimizationproblemexpressedinthefollowingform:

minimize x f

(

x

)

subject to gj

(

x

)

≤

0

,

j

=

1

, . . . ,

m x

∈

X

⊆ R

n X

= {(

xi

)

∈ R

n

:

0

≤

xi

<

∞}

(6)

where f is theobjectivefunction, X thedesignvariablevector, gj theset ofinequality constraints.Thevariablesxi arealwayspositive andfinitenumbers.Toreducethenumberofconstraints gj,itisnecessarytostudytheconstraintsactivity.

The FirstMonotonicityPrinciple (MP1) developed by Wilde and Papalambros [35] is used to find active constraints and reduce the problemto the necessary conditions. It states that in a well-constrained minimization problem, every increasing variable is bounded belowbyatleastonenon-increasingactive constraint.Mathematically,itisequivalent tosaythatinastrictlyincreasingobjectiveanda decreasingconstraint,theinequalityconstraintisactiveif:



∂

f

∂

xi



∗

+

j

μ

j



∂

gj

∂

xi



∗

=

0 (7)

Since

μ

j

≥

0 (accordingtotheKarush-Kuhn-Tuckerconditions(KKT)[39]),condition(7) issatisfiedif

∂

gj

∂

xi

presentsan oppositesign to

∂

f

∂

xi

.Whenthereisexactlyoneboundingconstraint,itiscalledacriticalconstraint.

Howeverinsomecases,severalconfigurationspresentmultiplepotentiallyactiveconstraints.Ifthishappens, noclearaffirmationcan begivenaboutwhichconditionactsascriticalandthereforetheequationsmustbekeptintheformofaninequality.

Although thisprinciple could seem simple, the process of applying them many times for all the constraints of a complex system becomesconfusing when there are a large number ofdesign variables involved.Since the resultof one inference often becomesthe premise ofanother, a systematicapproach is essential to keep trackof the steps inthe reasoning toavoid mistakes. Such a record is providedbytheMonotonicityTable [35,39].Inthenextsection,theuseofthistoolisdecisiveforareductionoftheoptimizationproblem inourcasestudy:theelectricmultirotordrone.

3.2.2. Casestudy

Inthisresearch,ourfocusissetonthesizingtechniquesofaMRAVoptimizationproblem.Inafirststep,thevehiclemissionprofiles aredefined,whichwillinfluencethesizingrequirementsoftheairvehicle.Itishere,wherethecriteriaof“degradation” ordesigndrivers willbemanifested, conditioningtheoptimalperformance ofthesystem. Thesecriteriahavebeenselectedby carryingout anintensive studyoftheexistingtechnology,inwhichregressionmodelsandscalinglawsareinvolvedandwheremoreinformationcanbefoundin thearticleofBudinger, Reyssetet al. [14]. Withthepurposeofnot exceedingthescopeofthisresearch,we havesimplydetailedthem inAppendixAandFig.1.Thesesizingcriteriaarebasedonenduranceandextremeperformancescenariosandareapplicabletoawide rangeofmultirotorvehicleswithdifferentpayloadspecifications,accelerations,range,anddifferentarchitectures.

Designvariablesandspecifications

In the case of product development, a new original design is represented by its model. Design alternatives can be generated by manipulatingthevaluesofthedesignparameters.Changesinthesevariablescanshowtheeffectofexternalfactoronaparticulardesign. Inanoptimizationproblem,theobjectiveistofindthebestsetofalldesignalternativeswhichgivesanimprovementinthefinaldesign. Forthisgoal,severaliterationsaremade.

Afterthesizingequationsandmodelsforthesystemsareestablished,weassembleadesignvector x belongingtoasubset X of the n-dimensionalrealspace

R

n_,thatisx

_∈

_X

_{⊆ R}

n_{,andthattheobjectiveandconstraintsofthesystemcanbequantifiedintermsofthis} vector.Forourcasestudy,wegatherinTable2thefollowingdesignvariables.

Finally,on a morepractical level, we findthe specifications,which are usually imposed by theclients. A setof requirementsmust besatisfiedbyanyacceptable designandits choiceisintimatelyrelatedtothechoiceoftheobjectiveanddesignvariables.Wemainly highlightthepayloadcapacity,therange,theestimatedMTOW,theverticalacceleration,thedronearchitecture, andthehighestrateof climb,amongothers.

Sizingconstraints

Inorderto satisfy thelift andpowerrequirements ofhoveringflight, maximumvertical accelerationandmaximumclimb rate,the followingconstraintsareset:

Table 2

Model-dependentdesignvariables.

Vars. Description System Units

β Ratio pitch-to-diameter Propeller [–]

nD Rotational speed per diameter for take-off Propeller [Hz·m]

J Advance ratio Propeller [–]

Ubat Battery voltage Battery [V]

Cbat Battery capacity Battery [A·s]

Tmot Required nominal motor torque Motor [N·m]

KT Motor constant Motor

N·m A



PE SC ESC Power ESC [W]

Larm Arm length Frame [m]

Dout Outer arm diameter Frame [m]

kf rame Ratio inner-to-outer arm diameter Frame [–]

Mtotal Initial guess for MTOW Global [kg]

•

Hoveringflight:

– Nominaltorque,Tmot,willbeequaltoorgreaterthanthemotortorqueinhoveringflight,Tmot,hov (g1).

•

Transitoryverticalacceleration(Take-Off)

– Maximumtorque,Tmax,mot,willbeequalto orhigherthanthe motortorqueundertheconditionsofmaximumtransient vertical acceleration,Tmot,to (g2).

– Voltageprovided bythebattery, Ubat,willsatisfy themotorvoltagedemandundertheconditionsofmaximumtransient vertical acceleration,Umot,to (g3).

– Voltageprovidedbythebattery,Ubat,willsatisfytheESCvoltagedemandunderconditionsofmaximumtransientvertical acceler-ation,UE SC (g4).

– Maximumbatterypower, Pbat,willsatisfythemotorspowerdemandundertheconditionsofmaximumtransientvertical acceler-ation, Pmot,to,total (g5).

– PowerprovidedbytheESC, PE SC,willsatisfythepowerdemandundertheconditionsofmaximumtransientverticalacceleration,

PE SC,to (g6).

– Productofspeedanddiameterunderthe conditionsofmaximumtransientverticalacceleration,nD,willnot overpassthespeed limitprovidedbythepropellermanufacturer,nDmax(g7).

– Nomechanicalloadgreaterthantheonepermittedbythearmmaterial,

σ

max,shallbeexceededundertheconditionsofmaximum transientverticalacceleration(g8).

•

Maximumclimbrate

– Maximumtorque,Tmax,mot,willbeequaltoorhigherthanthepropellertorqueundertheconditionsofhighestclimbingrate,Tmot,cl (g9).

– Voltageprovidedbythebattery,Ubat,willsatisfythemotorvoltagedemandundertheconditionsofhighestclimbingrate,Umot,cl (g10).

– Voltageprovidedbythebattery,Ubat,willsatisfytheESCvoltagedemandundertheconditionsofhighestclimbingrate,UE SC (g3).

– Maximumbattery power, Pbat, willsatisfy the motorspower demand underthe conditions of highestclimbing rate, Pmot,cl,total (g11).

– PowerprovidedbytheESC,PE SC,willsatisfythepowerdemandundertheconditionsofhighestclimbingrate, PE SC,cl(g12).

– Productofspeedanddiameterundertheconditionsofhighestclimbingrate,npro,clDpro,willnotoverpassthespeedlimitprovided bythepropellermanufacturer,nDmax (g13).

– Sincethe implementationofequalityconstraintsmayengendernumericaldifficulties fortheoptimizationalgorithm [35], an ap-proachtosolvethealgebraicloopfortheequationofverticalclimbrate:

Vcl

−

Jnpro,clDpro

=

0 (8)

istobreakouttheexpressionintotwoinequalities(g14

,

g15)resultingintoanabsoluteerrorof5%.

•

Geometry

– Thearmswillhaveaminimumlength,Larm,suchthatthepropellersdonotoverlapforaregularmotorlayout(g16).

•

Globalsystemconstraints

Tothepreviousconstraints,twonecessaryconditionsareincludedtobringcoherencetotheproblem.Dependingontheobjective: – Tominimizethemass,thecalculatedendurance(thov)willconvergetotheestimatedflighttime(th).

– Tomaximizetherange,theestimatedMaximumTake-OffWeight(MTOW)willconvergetothecalculatedfinalmass(Mtotal,f inal). Inaddition,forbothcases,thefinalmass(Mtotal,f inal)willconvergewiththeinitialguess(Mtotal).

Objectivefunction

OnthebasisoftheXDSMdiagramdevelopedinFig.2,iftheproposeddesignobjectiveistheminimizationofthefinaltotalmass,the totalmasscanbeextressedasthesumofthedifferentcomponentmasses:

f

:

Mtotal,f inal

=

Mpro

+

Mmot

+

Mbat

+

ME SC

+

Mf rame (9)

Ifthis expression is broken down into minimum design variables, we can see how we obtain a complex mathematical expression dependentonthedifferentdesignvariables:

Table 3

ApplicationofFirstMonotonicityPrincipletoidentifyactiveconstraints.Theexpressionhighlighted inyellowisacriticalexpression andcanbe transformedintoequalitywithrespecttothevariableofthecolumnwitharedcircle,becausethesignofitsfirstderivativeistheonlyoppositeto thesignofthefirstderivativeoftheobjective.

Mtotal nD Tmot K tmot PE SC Ubat Cbat β J kf rame Larm Dout

Objective f + − + + − − + +

H g1:Tmot,hov−Tmot≤0 + − −

TAKE-OFF g2:Tmot,to−Tmot,max≤0 + − −

g3:Umot,to−Ubat≤0 − + − +

g4:UE SC−Ubat≤0 + −

g5:Pmot,to,total−Pbat≤0 + − −

g6:PE SC,to−PE SC≤0 − +

g7:nD−nDmax≤0 +

g8: F_πmax_·,arm·Larm

D4out−D4_in

32·Dout

−σmax≤0 + + +

−

MAX CLIMB g9:Tmot,cl−Tmot,max≤0 + − − + +

g10:Umot,cl−Ubat≤0 − + − + +

g11:Pmot,cl,total−Pbat≤0 + − + +

g12:PE SC,cl−PE SC≤0 − +

g13:npro,clDpro−nDmax≤0 − + +

g14:Vcl−Jnpro,clDpro≤0 + − − g15:Jnpro,clDpro−Vcl≤0.05 − + + GEO g16: Dpro 2·sin(Narmπ ) −Larm≤0 + − −

−

Mtotal

=

f

(

Mtotal

, β,

nD

,

Ubat

,

Tmot

,

KT

,

Cbat

,

PE SC

,

Larm

,

Dout

,

kf rame

,

J

)

(10) Next,amonotonicityanalysisoftheobjectiveandconstraintsfunctionsiscarriedout,basedonthemodelsgivenintheappendixes, overthedifferentdesignvariablesdisplayedinTable2.Inageneralway,theMonotonicityTableincludesacolumnforeachvariableanda rowfortheobjectivefunctionandeachconstraint.Theglobalsystemconstraintsareexceptedofthisanalysisbecausetheyrepresentthe necessaryconditionsfortheproperperformanceoftheoptimizationproblem.Theelementsoftheresultingmatrixindicatetheincidence andmonotonicityofeachvariableineachconstraint.

Eachcolumnrecordsavariableandthe constraintscanbound whetherfromabove(also knownasstrictly monotoneincreasing) or frombelow(strictlymonotonedecreasing).

Table 3 shows the arrangement of columns and rows. The top row contains the monotonicity of the objective function, with the variablesheadingthecolumnsindicating‘

+

’forstrictlymonotoneincreasingand‘

−

’forstrictlymonotonedecreasing.Monotonicitiesfor theconstraintsaregiveninthematrixtotherightoftheconstraintscolumn.Fortheconstraints,iftheconditionofcriticalconstraintis met,suchexpressioncanbewrittenasequality.

Theresultsshowamultipleactivityoftheconstraintswithrespecttotheobjectivefunction.Someoftheseconstraintsarecriticaland othersarepotentiallyactive.Tocitesomeexamplesofpotentiallyactiveconstraints:theobjectivefunctionisboundedabovewithrespect to N D, by g7

,

g13and g15; andbelowwithrespect to Tmot by g1

,

g2

,

g3

.

g5

,

g9

,

g10 and g11.Sincethe signofthe variableisopposite

tothatoftheobjectivefunctioninseveralconstraints,noevidentstatement canbegivenandinthat case,theseconstraintspotentially active,willbelefthereasinequalities.Thesecriticalconstraintsaredetected:

g8

(

M_total+

,

k+_{f rame}

,

Larm+

,

D−out

)

critical w.r.t Dout

→

Dout

=

3



32

·

Fmax,arm

·

Larm

π

·

σ

max

· (

1

−

Din

/

Dout

)

(11)

g16

(

M+_total

,

nD−

, β

−

,

L−arm

)

critical w.r.t Larm

→

Larm

=

Dpro

2

·

sin

(

_Narmπ

)

(12)

Thenumberofinequalityconstraintshasbeensuccessfullyreducedfromsixteentofourteen.Inthefollowingsub-section,AlgorithmC addressestherestofthepotentiallyactiveinequalities.ThroughtheconstructionofDesignGraphs [36] itispossibletoestablish relation-shipsbetweenparametersandtoproposealternativeresolutionstrategies.

3.3.Under- andover-constrainedsingularities

Thecomplexity ofa design problemcanbe significantly reducedthrough theapplicationof twomethods. Tostart, theFirst Mono-tonicityPrinciple,ageneralistapproachbasedonamathematicalstudy ofthemonotonicitysystem’sequations(algorithmBandC)can beused.Then,thecomputationalprocesscanberepresentedbydesigngraphsandtheuseofgraphalgorithmsenabletodetectpossible singularitieswhichmayrequireadditionalcomputationaleffort.Themethodforsolvingthesearepresentedandusedonlybyalgorithm C.

Theobjectiveofdesigngraphs istographicallyhighlightthemain singularitieswhenbuildingasizing code.Theultimategoalisto sequenceexplicitlyasetofequations.Theinitialsetofequationscanbe:under-constrained,over-constrained orcontainalgebraicloops.The graphedges areorientedwithan arrowwhentheverticeareknownandnon-orientedwhentheorientation oftheproblemremainsto bedetermined.Toestablishthedesigngraphs,twolayers ofknowledgearedistinguished. Asystemlayer,whichlinkstheparametersof multiplecomponentswiththespecificationsforanysizingscenario.Acomponentlayer,whichlinksparameterswithequationsbasedon theowncharacteristicsofthecomponentandareonlyappliedforaspecificcomponentusedinthedesign.Models,designequationsand designvariablesareframedinarectangle,whilethespecificparametersandcharacteristicsareincircles.

Fig. 3. Design Graph for a propeller selection (before strategy).

Fig. 4. Design Graph for a propeller selection (after strategy).

3.3.1. Under-constrainedsingularity

Sizingissue: Equationsubsethasaninfinityofsolutionsduetotheexcessofparameters.

Example: Foragivenpayloadcapacity,themaximumthrustiscalculatedaccordingtoEquation(C.1).Butthefirstproblemfacedhere is:

CT

?

=

f

(

Fmax

,

ρ

air

,

n

,

Dpro

)

forwhichonly Fmaxand

ρ

airareknown(Fig.3).

Method: Substitutionofunknownparametersbynormalizedsizingcoefficients.Theuseofstandardized coefficientsasdesignvariables willensurethevalidityofadesignrangeforawidespectrumofspecifications.

OncetheregressionmodelforthethrustcoefficientCT isobtainedasafunctionof pitch_Dpro (newdesignvariable),therangeofthethrust coefficientcanbeestimated.Ontheotherhand,theCT equationcanbereformulatediftheRPMlimitistakenintoaccountthisway:

(

nD

)

≤ (

nD

)

max

→ (

nD

)

·

kN D

= (

nD

)

max

with

(

nD

)

max dependentonthepropellertechnologyandkN D asadegreeoffreedomwithin therange

[

0

,

1

]

.Thishelpstocontinuethe computationprocessbyboundingavariablewithareducedrange.4

CT

=

Fmax

ρ

· (

nD

)

2

·

D2pro

=

Fmax

ρ

·



nDmax kN D

2

·

D2pro

Once themodification is done inside the equation, thepropeller diameter Dpro can be calculated. InFig. 4, the orientation of the problemisdepicted.Thisdiagramshouldbeunderstoodbytheendoftheexplanation.Afteracertainnumberofiterations,theoptimizer willfindtherightvalueofkN D thatsatisfiestheconstraintandthusthesystemrequirements.

AsimilarcaseisfoundintheframesizingasshowninFig.5.

Equation(11) dependsontheknownparameterFmaxandonthealreadycalculatedLarm. Dout andDinremainunknowns. Awaytosolvethisunder-determinedsituationconsistsofexpressing Din asafunctionofDout:

Din

=

kf rame

·

Dout (13)

wheretheaspectratiokf rame remainsbetween

(

0

,

1

)

.

4 _{Similarly,n orD couldhavebeenimposedinsteadofk}

N Dbutinthiscasetherangestoexplorearemuchwideranddependstronglyonthespecificationsofthedrone (smallorlargecarrier).

Fig. 5. Design Graph for the frame selection (in red underconstraint).

Fig. 6. Overconstrained singularity for the motor selection. 3.3.2. Over-constrainedsingularity

Sizingissue: Differentequationsyieldthesameoutput.

Example: Themotorsizingsetsoutthreeequationswithtwoundeterminedparameters(Tmot andTmax,mot):

Tmax,mot

≥

Tpro,to (14)

Tmot

≥

Tpro,hov (15)

Tmax,mot

=

Tmax,mot,re f

·

Tmot Tmot,re f

(16)

Thereferencecomponentdata Tmax,mot,re f andTmot,re f are availableaswellasthepropeller scenarioTpro,hov and Tpro,to.Itis high-lightedthroughtheDesignGraphshownatFig.6,thattheproblemhasinfactthreeequationswithtwoover-constrainedparameters.

Method: Shiftsomeoftheequationstotheinequalityconstraintsandintroducedegreesoffreedomtorepresenttheequalities. Toobtainanon-singularproblem,oneofthethreeequationsmustberemoved,byplacingoneoftheinequalitiesintheoptimization constraints,likeEquation (14) forexample.Then,Equation (15) istransformedtoanequalitybyintroducingadegreeoffreedomkmot:

Tmot

=

kmot

·

Tpro,hov (17)

withkmot asavariableinputwithintherange[1

,

∞)

.Thisformulationallowstohaveadesignvariablewhoselowerboundisindependent oftheapplicationspecification.

Anotherexampleis foundforthecalculation ofmaximum powerprovidedby thebattery (Fig.7). Thebatterymustprovideenough powertooperateforthemaximumtransient modes(climbandtake-off). Here,a problemwiththree equationsandone undetermined parameter( Pmax,bat)canbefound:

Pmax,bat

=

Ubat

·

Imax (18)

Pmax,bat

≥

Umot,toImot,toNpro

η

esc

(19)

Pmax,bat

≥

Umot,clImot,clNpro

η

esc

(20)

Sincethreeequationsyieldthesameoutput,thetwoinequalitiesareplacedintheoptimizationconstraints.

3.4.Resolutionofalgebraicloops

Sizingissue: Circulardependencybetweeninputsandoutputs. Example: Thisproblemisfoundduringthemotorselection:

Umot,to

=

Rmot

·

Ito

+

KT,mot

· 

mot,to (21)

Fig. 7. Overconstrained singularity for the battery selection.

Fig. 8. Design Graph for a motor selection (in red algebraic loop).

Rmot

=

Rmot,re f



KT,mot KT,mot,re f



2



_T mot Tmot,re f



₋5/3.5 (22)

Method: Byremoving thetermcorresponding totheimperfection intheEquation (21),i.e.theproduct R

·

Ito,the algebraicloop is broken.Aminimumoversizingcoefficient,kspeeedmot

≥

1,isintroducedtocompensatethemissingtermintheinitialequation.Theinitial inequalityisreintroducedintheconstraintstoensuretheconsistency.

Umot,to

=

kspeedmot

·

KT

· 

mot,to (23)

Fromthisnewexpression,thevalueofKT isobtainedandwiththis,wecalculatethemotorresistanceR substitutingthevalueofthe motorconstantinEquation (22).

Finally,aconstraintissetuptovalidatetheabovesimplification:

Ubat

≥

Umot,to (24)

ThisprocessisillustratedinFig.8wherethealgebraicloopismarkedwithredarrows.

Anotherexample is found forthe initial mass estimation.To launchthe procedure, it is assumedthat the initial guess forMTOW,

Mtotal,isamultipleoftheknownpayloadMload.

Mtotal

=

kM

·

Mload (25)

After severaliterations,the final mass,Mtotal,f inal calculatedassumofthedifferentcomponents mustconvergetothisinitial guess

Mtotal.Thisisexpressedbythefollowinginequality:

Mtotal

≥

Mtotal,f inal (26)

Thisconditionisaddedtotheglobalconstraints,sothatbothresultstendtoauniquesolution. 4. Overallsizingoptimization

4.1. Summaryofthesizingcodes

Oncethedifferentsizingequationsandexpressionshavebeenformulatedandoriented,theyareintegratedintoanoptimizationcode tocalculatetheoptimaldesignofthesystemsforaspecificobjective.Oneofthegoalsofthispaperistodemonstratehowtheapplication ofthepreviousstrategiesmayleadtoareductionoftheproblemcomplexityandtheresolutiontimeforthesameora bettersolution. Thus,threealgorithmsaredevelopedwhicharebasedonthedifferentmodificationsproposed:

Fig. 9. N2_{diagram for solving the optimization problem using algorithm A. Modifications carried out by algorithms B and C over algorithm A are highlighted with colors.}

•

AlgorithmA basedontheinitialconstraintsanddesignvariablesproposed inTable2andsection 3.2.2.ThisalgorithmappliesStep1 and4ofthemethodologyshowninTable1.

•

AlgorithmB afterhavingreducedthenumberofinequalityconstraintsthroughapplicationofMP1proposedinTable3.Thisalgorithm appliesStep1,2and4ofthemethodologyshowninTable1.

•

AlgorithmC withtheapplicationofMP1 andsubstitution ofthemodel-dependentdesign variablesby sizingcoefficientsvalidfora widerangeofmodelsassuggestedinsection3.3andsection3.4.ThisalgorithmappliesStep1,2,3and4ofthemethodologyshown inTable1.

TheoptimizerusedisaheuristicmethodfromtheSciPy Pythonlibrary:thedifferentialevolution.Thissizingprocess isillustrated in Fig.9throughtheXDSMtooltodescribetheinterfacesamongcomponentsofacomplexsystem.

This diagram, which follows the same sizing procedure as explained in section 3, highlights the different modifications made on thebasis ofthe initial algorithmA. Forthealgorithm B,asa resultof amonotonicity analysis, theinequality constraintsof theframe are eliminated, while foralgorithm C, design variables are replaced by standardized coefficients andfurther inequality constraints are removed.

Asummary ofthe different design approaches ofeach algorithm is shownin Table 4. Ifwe recap the standard formulationof an optimizationprobleminEquation (6),wecanseethatthetotalnumberofdesignconstraints(definedasm)anddesignvariables(defined asn)corresponds to theframed value of thefirst andsecond row ofTable 4foreach algorithm, respectively. Withthe applicationof theAlgorithmC wemanagetoexpresstheproblemingeneralnormalizedcoefficients.Therangeanddescriptionofthesecoefficientsare showninTable5.

Theuseofnormalizedsizingcoefficients(around1oroneboundequalto1)asoptimizationvariablesenablestohaveagenericsizing codesincethe optimizationlimitsdo notlonger dependonthe sizeofthe vehicle.Althoughtheimplementationis somehowcomplex andneedsa good understandingofthe systemtoreplace equations,theuse ofdesigngraphs andthealgorithms presented in[36,34] makesthisstepeasier.Theinteresting performances,highlighted bythenumberofiterations,andreducedresolutiontime inTable6of thisNormalizedVariableHybridformulationisalsobenchmarkedintermsofcomputationalcostandrobustnessversusothermonolithic MDOformulationsin[28].

Table 4

Algorithmsapproaches.

Algorithm A Algorithm B Algorithm C

Set of constraints gi 18

1:Tmot≥Tmot,hover

2:Tmax,mot≥Tmot,to

3:Tmax,mot≥Tmot,cl

4:Ubat≥Umot,to

5:Ubat≥Umot,cl

6:Ubat≥UE SC

7:Pbat,max≥NproPmot,to

8:Pbat,max≥NproPmot,cl

9:PE SC≥PE SC,to

10:PE SC≥PE SC,cl

11, 12:0.05≥Vcl−Jnpro,clDpro≥0

13:nD≤nDmax

14:npro,cl·Dpro≤N Dmax

15:Larm≥2·sin(π /DproNarm)

16:Fmax,arm·Larm

π·D4_out−D4 in 32·Dout ≤σmax 16

1:Tmot≥Tmot,hover

2:Tmax,mot≥Tmot,to

3:Tmax,mot≥Tmot,cl

4:Ubat≥Umot,to

5:Ubat≥Umot,cl

6:Ubat≥UE SC

7:Pbat,max≥NproPmot,to

8:Pbat,max≥NproPmot,cl

9:PE SC≥PE SC,to

10:PE SC≥PE SC,cl

11, 12:0.05≥Vcl−Jnpro,clDpro≥0

13:nD≤nDmax

14:npro,cl·Dpro≤N Dmax

13

1:Tmax,mot≥Tmot,to

2:Tmax,mot≥Tmot,cl

3:Ubat≥Umot,to

4:Ubat≥Umot,cl

5:Ubat≥UE SC

6:Pbat,max≥NproPmot,to

7:Pbat,max≥NproPmot,cl

8:PE SC≥PE SC,cl

9, 10:0.05≥Vcl−Jnpro,clDpro≥0

11:npro,cl·Dpro≤N Dmax

g:thov≥th(min mass) OR g:M T O W≥Mtotal,f inal(max time) AND g:Mtotal≥Mtotal,f inal D.Vars. 12 β,Tmot,nD, Ubat,KT,Cbat, PE SC,Larm,Dout, kf rame,Mtotal,J 10 β,Tmot,nD, Ubat,KT,Cbat,PE SC, kf rame,Mtotal,J 10 kM,kmot, kspeed,mot,kvb,kN D, kf rame,kMb, β,J, kE SC Table 5

Generalunder-/oversizingdesigncoefficients.

Coefs. Description Range

kM Oversizing coefficient on the load mass [1,+∞)

kmot Oversizing coefficient on the motor torque [1,+∞)

kspeed,mot Oversizing coefficient on the motor speed [1,+∞)

kvb Oversizing coefficient for the battery voltage [1,+∞)

kN D Slowdown propeller coefficient (0,1]

kMb Sizing coefficient on the battery load (0,+∞)

kE SC Oversizing coefficient on the ESC Power [1,+∞)

Table 6

ComparisonofalgorithmperformancesforanoctocoptermodelofSpreadingWingsS1000+[40] usingatypical

quadcorelaptop.

Spreading wings S1000+ A B C

Number of function evaluations 3.74×106 ₁

.51×106₍₁ /2.5 w.r.t. A) 7.69×105₍₁ /4.9 w.r.t. A) Number of iterations 19191 9090 (1/2.1 w.r.t. A) 4659 (1/4.1 w.r.t. A) Resolution time (s) 637.87 399.22 (1/1.6 w.r.t. A) 181.87 (1/3.5 w.r.t. A) 5. Validation

5.1. Comparisonwithexistingdronescharacteristics

Tovalidate theresultsobtainedbythethreemethods,threeMRAVmodelsofdifferentscaleandconfigurations areused:MK-Quadro

[41],designedfora verticalaccelerationof2 g(kmax,thrust

=

3)anda payloadof1 kg;theDJISpreadingWingsS1000+ [40] with similar verticalacceleration anda payload of4kg andthetaxidrone, EHANG184 [42], withtwo propellersper arm,a verticalaccelerationof 0.6 g(kmax,thrust

=

1

.

6)and100 kgofpayload.

Tables7–9showthe resultsforthedifferentcomponents’masses,propeller diameter,batteryenergyandmaximumelectricalpower forthedifferentMRAV.Thethreealgorithmsconvergetoasimilarresultbutwithatdifferentcomputationalcost.AlgorithmBenablesto decreasebyalmost2theresolutiontimewhencomparedtoAlgorithmAwhereasAlgorithmCdecreasesitbyalmost4.

Thecomputedcharacteristicsarerelativelyclosetotheonesofreferences.Deviations arebelow17%intheworstcase,whichcanbe consideredasacceptableforapreliminarydesigntool.Suchtoolenablestoassessanddeterminethemostrelevantaspectsofthesystem inaholisticmanner.

Possiblereasonsfordeviationsfromthecalculationmaybeasfollows:

•

Referencesusedinthescalinglawsarecenteredonamid-scalemodel,asrecommendedin[14],whichmayleadtoalargerdeviation forthecalculationofthemotormassontheMikrokopterMK-Quadro comparedtothemid-rangeoctocopter.

•

Aspectsofaerodynamiclosses[43,44] (notcontemplatedinthisoptimizationcode)comeintoplaythemorecomplexandlargerthe vehicleis.Tocompensatesuchlosses,apropelleroversizinginlaterstagesofdesignmayhavebeencarriedoutasforEHANG184.

•

InthereferencecatalogueofDJISpreadingWingsS1000+ [40] thebatterycapacitylieswithinarangeof10000and20000 mAh,which explainsthemassdivergencefromthereferenceandcalculatedvalues.

•

Onepossibleexplanationtoexplaintheoveralldivergencefromthereferenceandcalculatedvaluesofthebatteryproperties,maybe thatbatteriescouldbedesignedtoprovidelongerflighttimesthanindicatedforsafetyreasons.

•

Additionalweights(antenna,GPS,receivers,flightcontroller,...)maynotbeincludedinthereferencecatalogues. Quadcopter: Mikrokopter MK-Quadro

MKQUADRO.Source:[41].

⎧

⎪

⎨

⎪

⎩

Payload: 1.0 kg kmax,thrust: 3

Number of arms: 4 Number of propeller per arm 1

Climb speed: 10 m/s Drag coefficient: 1.3

Endurance: 15 min Top surface: 0

.

045 m2

⎫

⎪

⎬

⎪

⎭

Table 7

ComparisonofresultsforthedifferentalgorithmsappliedtoMK-Quadro.

Alg. A Alg. B Alg. C Reference Rel. Error (%)

Total mass (g) 1642 1409 +16.53 Battery Mass (g) 353 329 +7.29 Motor Mass (g) 47 54 −12.96 Diameter Prop (m) 0.25 0.25 0.00 Energy (W.h) 53.95 48.84 +10.46 Max Power (W) 172.99 175 −1.14

Octocopter: DJI Spreading Wings S1000+

SpreadingWingsS1000+.Source:[40].

⎧

⎪

⎨

⎪

⎩

Payload: 4 kg kmax,thrust: 3

Number of arms: 8 Number of propeller per arm 1

Climb speed: 6 m/s Drag coefficient: 1.3

Endurance: 18 min Top surface: 0

.

09 m2

⎫

⎪

⎬

⎪

⎭

Table 8

Comparison of results for the different algorithms applied to Spreading Wings

S1000+.

Alg. A Alg. B Alg. C Reference Rel. Error (%)

Total mass (g) 8513 9500 −10.38 Battery Mass (g) 2133 1932 +10.41 Motor Mass (g) 160 158 +1.26 Diameter Prop (m) 0.40 0.38 +5.26 Energy (W.h) 326.20 333 −2.04 Max Power (W) 448.54 500 −10.29

Coaxial propellers setup: EHANG 184

Ehang184autonomousaerialvehicle:Source:[42].

⎧

⎪

⎪

⎨

⎪

⎪

⎩

Payload: 100 kg kmax,thrust: 1.6

Number of arms: 4 Number of propeller per arm 2

Climb speed: 5 m/s Drag coefficient: 1.3

Endurance: 22 min Top surface: 2

.

09 m2

⎫

⎪

⎪

⎬

⎪

⎪

⎭

Table 9

ComparisonofresultsforthedifferentalgorithmsappliedtoEHANG184.

Alg. A Alg. B Alg. C Reference Rel. Error (%)

Total mass (kg) 404.18 360 +12.5

Diameter Prop (m) 2.00 1.7 +17.65

Max Power (W) 11358.27 13250 −14.28

6. Designexploration

Withinthemultidisciplinarydesign,components adapttheirperformance andcharacteristicstomeet thedifferentdesignobjectives. Inthispart,wewillfocusonsomeofthesevariationstosatisfytheobjectivesofmaximizingthemissiontimeandminimizingthemass.

6.1. Effectoftheaccelerationrequirement

InFig.10,we showanevolutionoftheflighttime ofan octocopterasafunctionoftheverticalaccelerationandits take-offweight. Thediagramshowshowahigheraccelerationimpliesareductionoftheflighttime.Thisreductionisduetotheincreaseofmotorweight inorderto providetherequiredtorque.Thistrendshowntobemorepronounced asacertain accelerationvalue isexceeded,ascanbe seenforalargedroneforvaluesbeyond1.5.

6.2. Effectofincreasedrangeoncomponentsize

AsdepictedinFig.11,theairvehiclesizeincreasesexponentiallyasalongerflighttimeisrequired.Anincreaseinrangeimpliesmore electrical powerrequired by the differentcomponentsand alsoa larger design,which can be reflected inthe growthof thepropeller diameter.

Fig. 10. Evolution of flight times for an octocopter with single motor setup with different vertical accelerations and loads.

Fig. 11. Evolution of the components’masses and propeller diameter growth curve with respect to the flight time for a quadcopter of MTOW of 30 kg.

Fig. 12. Effect of the gearbox on the evolution of the components’ masses (for a given endurance of 15 min).

6.3.Effectofagearboxonsystemmassreduction

Inthe following part, it will be shown how the motormass can be greatly optimizedwith the implementation ofa gearreducer [45].AsFig.12illustrates,thereisapositiveeffectonthesystem,leadingtoasignificantdecreaseofthetotalmass.Thismasssavingis particularlyinterestingfordroneswithlarge-scalemotors.Forsmalldrones,thesmallweightgaincannotjustifytheincreaseincomplexity provokedbytheimplementationofgearreducers.

7.Conclusions

Thispaperhasshownthatthepreliminarydesignofmultirotordrones canbe madeefficientandgenericby usingthe monotonicity analysisandtheNVHMDOformulation.Thedifferentdisciplines(aerodynamic,propulsivepowertrain,energystorage,structure)andtheir

designdriverswerepresentedaswellashowtheyarelinkedtothesizingscenariosofthemultirotor(hovering,take-off,verticalclimb).A methodologywaspresentedtoestablishthesizingprocedureofthemultirotorandtheassociatedoptimizationproblem.Threeapproaches were presented forthe resolution ofthe sizing problem. The first relies fullyon the optimizer asit doesnot modify the structure of the problem. The second used the monotonicity analysis to detect andremoved inactive constraints. The last used the monotonicity analysis andthe NVHformulation to further reduce the number of constraints in order to make the sizing procedure asefficient as possible.Theefficiencyofresolutiondependsontheproblemformulationandcantakebenefitsofgraphicrepresentationsandstrategies tosolvepotentialdifficulties:singularities(over orunderconstraints)oralgebraic loops.However,ithasbeenshownthat thelast sizing formulation significantly reduced the number of function evaluations by a factor 5, thus computational cost, making it adequate for preliminarydesign.Thesystemicandgenericsizing formulationhasthenbeenvalidatedondronesofdifferentsizes. Theexplorationof thedesignspacehasshowninparticular,thattheuseofgearreducerspermitstoimprovetheendurancebyreducingthemotorweight. Thisdesignbasecouldbeextendedbyfurtherlinkswith3Dsimulationsforpartdesign(ex.landinggear)orwith0D/1Dsimulationsfor co-design(controllerandphysicalparts)ortrajectoryoptimization.

Declarationofcompetinginterest

The authors declare that they have no known competingfinancial interests or personal relationships that could have appeared to influencetheworkreportedinthispaper.

Appendix A. Nomenclature

A.1. Powervariables

Symbol Description Units

F force, thrust [N]

T torque [N.m]

,n rotational speed, frequency [rad/s], [rev/s]

P power [W]

U voltage [V]

I current [A]

V drone speed [m/s]

KT Kt motor [N.m/A or V/(rad.s−1)]

J Inertia [kg.m2_]

Eb Energy battery [J]

A.2.Geometricaldimensions

Symbol Description Units

r radius [m]

l length [m]

D diameter [m]

H outer height of a rectangular cross section [m]

h inner height of a rectangular cross section [m]

p pitch [m]

S surface [m2_]

A.3.Othervariables

Symbol Description Units

M Mass [kg]

g Gravity constant [N/kg]

Nj number of “j” component [–]

ρair air volume density [kg/m3]

ρs structural material volume density [kg/m3]

B Air compressibility indicator [–]

Cx Drag coefficient [–]

CT Thrust coefficient [–]

CP Power coefficient [–]

A.4.Indexi (values,variabledefinition)

Symbol Description

max maximum

min minimum

f r friction

total all components

out outer

in inner

A.5.Indexj (components)

Symbol Description pro propeller mot motor esc ESC bat battery arm arm lg landing gear pay payload

body core frame

A.6.Indexk (sizingscenarios,referencevalues)

Symbol Description

hov Hover

to Take-off

cl Climb state

stat static scenario V=0 (hover or take-off scenario)

dyn dynamic scenario (climb scenario)

re f reference value

Appendix B.Systemanalysisfunctions

B.1. Thrust

Hover:

Ftotal,hov

=

Mtotal

·

g (B.1)

MaximumThrustMode: Ftotal,to

=

Ftotal,hov

·



kmax,thrust



(B.2) Climbingmode: Ftotal,cl

=

Mtotal

·

g

+

1 2

ρ

air

·

Cx

·

S

·

V 2 cl (B.3) B.2. Maximumspeeds Climbingmode: Vcl

=



2

· (

Fpro,cl

−

Mtotal

·

g0

)

ρ

air

·

CX

·

S (B.4) B.3. RPMlimit

npro,k

·

Dpro

=

kN D

· (

nD

)

max (B.5)

B.4. Thrustcoefficient[14] Statics(Vcl

=

0): CT,pro,k,stat

=

4

.

27

·

10−03

+

1

.

44

·

10−01

· β

(B.6) Dynamics(Vcl

=

0): CT,pro,k,dyn

=

0

.

02791

−

0

.

06543 J

+

0

.

11867

β

+

0

.

27334

β

2

−

0

.

28852

β

3

+

0

.

02104 J3

−

0

.

23504 J2

+

0

.

18677

β

·

J2 (B.7) B.5. Powercoefficient[14] Statics(Vcl

=

0): CP,pro,k,stat

= −

1

.

48

·

10−03

+

9

.

72

·

10−02

β

(B.8) Dynamics(Vcl

=

0): CP,pro,k,dyn

=

0

.

01813

−

0

.

06218

β

+

0

.

00343 J

+

0

.

35712

β

2

−

0

.

23774

β

3

+

0

.

07549

β

·

J

−

0

.

1235 J2 (B.9) B.6. Totalmass

Mtot

=(

Mmot

+

Mgear

+

Mesc

+

Mpro

)

·

Npro

+ (

Marm

+

MLG

)

Narm

+

Mbody

+

Mbat

+

Mpay

(B.10)

Appendix C. Componentanalysisfunctions

C.1. Propeller[14]

Thrust:

Fpro,k

=

CT

·

ρ

·

n2pro,to

·

D4pro (C.1)

Power:

Ppro,k

=

Cp

·

ρ

·

n3pro,k

·

D5pro,k (C.2)

Torque: Tpro,k

=

Ppro,k



pro,k (C.3) Propellermass: Mpro

=

Mpro,re f



Dpro Dpro,re f



3 (C.4) C.2. Motor[14] Motortorque:

Tmot,k

=

KT,mot

·

Imot,j (C.5)

Tmot,k

=

Tpro,k

+

Tf r,mot,k (C.6)

Motorvoltage:

Umot,k

=

KT,mot

· 

pro,k

+

Rmot

·

Imot,k (C.7)

Motorpower:

Pmot,k

=

Umot,k

·

Imot,k (C.8)

Requirednominaltorque:

Tmot

=

kmot

·

Tpro,hov (C.9)

Motormass: Mmot

=

Mmot,re f



Tmot Tmot,re f



3/3.5 (C.10)

Motormaximumtorque: Tmax,mot

=

Tmax,mot,re f

Tmot Tmot,re f

(C.11)

Motorfrictiontorque: Tmot,f r

=

Tmot,f r,re f



Tmot Tmot,re f



3/3.5 (C.12) Motorresistance: Rmot

=

Rmot,re f



KT,mot KT,mot,re f



2



_T mot Tmot,re f



₋5/3.5 (C.13) Motorinertia: Jmot

=

Jmot,re f



Tmot Tmot,re f



5 3.5 (C.14) C.3. Gearsystems[45]

Motortorquewithreduction: Tmot,k

=

Tpro,k Nred

(C.15)

Motorspeedwithreduction:



mot,k

= 

pro,k

·

Nred (C.16)

Ratioinputpiniontomatinggear:

mg

=

0

.

0309Nred2

+

0

.

1944Nred

+

0

.

6389 (C.17) Weightfactor:

F d2

/

C

=

1

+

1 mg

+

mg

+

m2g

+

N2_red mg

+

N_red2 (C.18) Centerdistance: C

.

D

.

=

1 2



dp

+

dg



=

dp 2



mg

+

1



(C.19) FactorC: C

=

2T K (C.20)

Applicationfactorforweightestimations:Solidrotorvolume:

Application Weight factor,μ K factor [lb/in^2]

Aircraft 0.25–0.30 600–1000 Hydrofoil 0.30–0.35 – Commercial 0.60–0.625 50–75 Industrial drive – 500–1000

F d2_{of f}

=

F d 2 Cof f

·

2T Kof f (C.21) Weightestimation: W eight

=

F d2

·

μ

(C.22)

C.4. Electronicspeedcontroller[14,46]

Cornerpowerorapparentpower: Pesc,to

=

Pmot,max

Ubat Umot,max (C.23) Scenariocondition: Pesc

≥

Pesc,to (C.24) Pesc

≥

Pesc,cl (C.25) ESCVoltage: Vesc

=

Vesc,re f



Pesc Pesc,re f



1/3 (C.26) ESCMass: Mesc

=

Mesc,re f Pesc Pesc,re f (C.27) C.5. Battery[14] Batteryvoltage: Ubat

=

3

.

7Ns,bat (C.28) Batterycapacity: Cbat

=

Ebat Vbat (C.29)

Batteryendurancecondition: thov

<

0

.

8Cbat Ibat

(C.30)

Batteryvoltagecondition:

Ubat

>

Umot,k (C.31)

Batterymaximumcurrent: Imax,bat

=

Imax,re f

Cbat Cbat,re f

(C.32)

Batterypowercondition: Ubat

·

Imax

>

Umot,k

·

Imot,k

·

Npro

η

E SC (C.33) Batterymass: Mbat

=

Mmax,re f

·

Ubat Ubat,re f

·

Cbat Cbat,re f (C.34) Batteryenergy: Ebat

=

Ebat,re f

·

Mbat Mbat,re f (C.35) C.6. Frame[47]

Minimumarmlength: Larm

≥

Dpro

/

2 sin

(

π

/

narm

)

(C.36)

Maxstress:

Circular hollow section 32Fmax,arm

·

Larm

π

D3_out

(

1

−

k4_{f rame}

)

≤

σ

max (C.37)

Square hollow section Fmax,arm

·

Larm

H3

6

−

h4 6·H

Totalmassbeams:

Circular hollow section Marm

=

π

4

·



D2_out

− (

kf rameDout

)

2

·

ρ

s

·

Larm

.

Narm (C.39)

Square hollow section Marm

=



H2_out

−

H2_in

·

ρ

s

·

Larm

·

Narm (C.40)

Massofglobalframe:

Mf rame

=

Mf rame,re f



Marm Marm,re f



(C.41)

Appendix D.Optimization(AlgorithmC)

D.1.Formulation

Theoptimizationisthefollowing:

minimize Mtot

with respect to kM

,

kmot

,

kspeed

,

kvb

,

kN D

,

kf rame

,

kMb

, β,

J

,

kE SC

subject to Tmot,to

−

Tmax,mot

≤

0

Tmot,cl

−

Tmax,mot

≤

0 Umot,to

−

Ubat

≤

0 Umot,cl

−

Ubat

≤

0 Uesc

−

Ubat

≤

0

Umot,toImot,toNpro

η

esc

−

UbatImax

≤

0 Umot,clImot,clNpro

η

esc

−

UbatImax

≤

0

0

.

05

≥

Vcl

−

Jnpro,clDpro

≥

0 npro,cl

·

Dpro

−

nDmax

≤

0

PE SC,cl

−

PE SC

≤

0

(D.1)

D.2. Inputs

Parameter Unit Description

Specs. Mload [kg] Payload mass

Vcl [m/s] Maximum climb speed

th [min] Estimated hover flight time

kmax,thrust [–] Maximum thrust coefficient

Config. Narm [–] Number arms

Np,arm [–] Number propellers per arm

Atop [m2] Top area

M T O W [kg] Estimated guess for MTOW

Air ρair [kg/m3] Air density

CD [–] Drag coefficient

D.3. Referencevalues

Parameter Value Unit Description

Motor Tmot,re f 2.32 [N m] Referencenominaltorque

Tmax,re f 2.82 [N m] Referencemaxtorque

Rmot,re f 0.03 [] Referenceresistance

Mmot,re f 0.575 [kg] Referencemotormass

KT,mot,re f 0.03 [N m/A] Referencetorquecoefficient

Tf,mot,re f 0.03 [N m] Referencefrictiontorque

Props N Dmax 105000 [RPM/in] MaxRPMspeedAPCMR

Dpro,re f 0.28 [m] ReferenceAPCpropdiameter

Mpro,re f 0.015 [kg] ReferenceAPCpropmass

Table 9 (continued)

Parameter Value Unit Description

Battery Mbat,re f 0.329 [kg] Referencebatterymass

Cbat,re f 12240 [A s] Referencebatterycapacity

Vbat,re f 14.8 [V] Referencebatteryvoltage

Ibat,max,re f 170 [A] Referencebatterycurrent

ESC PE SC,re f 3108 [W] ReferenceESCpower

VE SC,re f 44.4 [V] ReferenceESCvoltage

ME SC,re f 0.115 [kg] ReferenceESCmass

Frame Mf ra,re f 0.347 [kg] Referenceframemass

Marm,re f 0.14 [kg] Referencearmmass

σmax,re f 280×106/4 [N/m2] Compositemaxstress(dividedbyfactor2

dynamics,factor2stressconcentration)

Reference values are based from: (Motor)AXI 5325/16 GOLDLINE, (Propeller) APC Propellers 11 x 4.5, (Battery)ProLiteX TP3400-4SPX25(ESC)TurnigyK._{Force70HVand(Frame)QuadriMK7.}

D.4. Designvariablesandoutputs

Parameter Unit Description

Design variables kM [–] Oversizing coefficient load mass

kmot [–] Oversizing coefficient motor torque

kspeed,mot [–] Oversizing coefficient motor speed

kvb [–] Oversizing coefficient battery voltage

kN D [–] Undersizing coefficient propeller speed

kf rame [–] Ratio inner/outer diameter beam

kmb [–] Oversizing coefficient battery load mass

β [–] Angle pitch/diameter

J [–] Advance ratio

System Masses Mtot [kg] Drone total mass

Mpro [kg] Propeller mass (1x)

Mmot [kg] Motor mass (1x)

Mbat [kg] Battery mass (1x)

ME SC [kg] ESC mass (1x)

Marm,total [kg] Total arms mass

Mf ra,total [kg] Total frame mass

Mgear [kg] Gear mass for one motor

Frame Dout [m] Outer frame diameter

Din [m] Inner frame diameter

Lb [m] Length arms

Propeller npro,k·Dpro [RPM·inch] Speed-diameter product

Dpro [m] Propeller diameter

pitch [m] Propeller pitch

Cp,sta [–] Power coefficient in statics

Cp,dyn [–] Power coefficient in dynamics

Ct,sta [–] Thrust coefficient in statics

Ct,dyn [–] Thrust coefficient in dynamics

npro,hov [RPM] Hover speed

npro,to [RPM] Take-Off speed

npro,cl [RPM] Climbing speed

Thrust Fpro,hov [N] Thrust force for hover

Fpro,to [N] Thrust force for take-off

Fpro,cl [N] Thrust force for climbing

Power Ppro,hov [W] Power for hover

Ppro,to [W] Power for take-off

Ppro,cl [W] Power for climbing

Torque Tpro,hov [W] Torque for hover

Tpro,to [W] Torque for take-off

Tpro,cl [W] Torque for climbing

Voltage Upro,hov [W] Voltage for hover

Upro,to [W] Voltage for take-off

Upro,cl [W] Voltage for climbing

Kt,mot [N·m/A] Kt constant

Current Ipro,hov [W] Current for hover

Ipro,to [W] Transient current for take-off

Ipro,cl [W] Transient current for climbing

Battery Cbat [A·h] Battery capacity

Vbat [V] Battery voltage

Ibat [A] Battery current