Scaling laws and similarity models for the preliminary design of multirotor drones

an author's

https://oatao.univ-toulouse.fr/26688

https://doi.org/10.1016/j.ast.2019.105658

Budinger, Marc and Reysset, Aurélien and Ochotorena, Aitor and Delbecq, Scott Scaling laws and similarity models

for the preliminary design of multirotor drones. (2020) Aerospace Science and Technology, 98. 1-15. ISSN 1270-9638

multirotor

drones

M. Budinger

a

,

∗

,

A. Reysset

a

,

A. Ochotorena

a

,

S. Delbecq

b

a_{ICA,UniversitédeToulouse,UPS,INSA,ISAE-SUPAERO,MINES-ALBI,CNRS,3rueCarolineAigle,31400Toulouse,France} b_{ISAE-SUPAERO,UniversitédeToulouse,10AvenueEdouardBelin,31400Toulouse,France}

a

b

s

t

r

a

c

t

Keywords: Multirotordrones Scalinglaws Dimensionalanalysis Surrogatemodels Propulsionsystem Landinggear

Multirotordronesmodellingandparameterestimationhavegainedgreat interestbecauseoftheirvast application for civil,industrial, militaryandagricultural purposes.Atthe preliminary design level the challenge isto developlightweight modelswhichremain representativeofthe physical laws andthe systeminterdependencies.Basedonthedimensionalanalysis,thispaperpresentsavarietyofmodelling approaches fortheestimation ofthe functionalparameters andcharacteristics ofthekeycomponents ofthesystem.Throughthiswork asolidframeworkispresentedforhelping bridgethegapsbetween optimizingidealizedmodelsandselectingexistingcomponentsfromadatabase.Specialinterestisgiven to the models in terms ofreliability and error.The results are compared for various existing drone platformswithdifferentrequirementsandtheirdifferencesdiscussed.

1. Context

During the last decade, technological innovations [1–3] have significantlycontributedtothedevelopmentofsmartand power-fulmultirotor UAVs:miniaturizationandmicroelectronicwiththe integration ofinertial sensors, increased computational power of control processors, new battery technologieswith higher energy density,permanentmagnetmotorswithhighertorquedensityand highpowerdensitieselectric-converters.

The expansion of drones market has led to the decrease of drone sizes and the increase of the availability of drone compo-nents at very competitive prices. This has facilitated the exper-imentation and optimization of drone designs based on succes-sive physical tests. Most of recent designs address very specific application-cases withparticularperformances/needs forwhich a design optimization process is needed. The design optimization process becomes mandatory when the payload scale factor dif-fers from market trend. Particularly in research projects of large multirotorUAVsdesignedforthetransportofcommercialloadsor passengers [4–7] whereexpensive prototypesare used, it is sug-gestedtoperform advanceddesign studiesbefore manufacturing, integrationandtesting.Toacceleratethedesignprocess,ageneral trendistoextendtheroleofmodellingindesignandspecification. The present paper proposes a set of efficient prediction models

*

Correspondingauthor.

E-mailaddress:marc.budinger@insa-toulouse.fr(M. Budinger).

whichcan beintegratedintoanyoptimizationtool forrapid pre-liminarydesignofmultirotordrones[8,9].

Section 2 places emphasis on the sizing scenarios that influ-encethedesignofthecomponentsandintroduces theconceptof dimensionalanalysis forthecreation ofestimationmodels. Mod-elsbasedonsimilarities,alsocalledscaling laws,aredescribed in section 3and appliedin section 4forthe electricalcomponents: motor,battery,ESCandcables.

Regression models are described in section 5 and can adopt differentforms:polynomialformsexemplifiedinsection6for pro-pellers andvariable powerlawmodels insection 7forstructural components.

Finally, in section 8 several examples of existing drone plat-formsarecomparedwiththeresultsobtainedusingthemodels.

2. Needsforpredictionmodelsduringpreliminarydesign

Withinthewholeproductdevelopmentprocess,thepurposeof preliminarydesignphaseistheevaluationofarchitecture feasibil-ity,technologyselectionandcomponentshigh-level specifications definitionbasedonproduct requirementsandoperational scenar-ios.Yet,todoso,thecriticalscenariosshouldbeidentifiedandthe associatedmaincomponentscharacteristicsidentified.

2.1. Multirotormaincomponents

Fig.1showsatypicalmassdistributionforthedifferent compo-nentsofadroneaccordingtoseveralexamples[10–12].Thispaper https://doi.org/10.1016/j.ast.2019.105658

Nomenclature

ABS AcrylonitrileButadieneStyrene

DC DirectCurrent

DoE DesignofExperiments

EMF ElectroMotiveForce

ESC ElectronicSpeedController

FEM FiniteElementMethod

IGBT InsulatedGateBipolarTransistor

LiPo Lithium-Polymer

LG LandingGear

MTOW MaximumTakeoffWeight

MOSFET MetalOxideSemiconductorFieldEffectTransistor

PBS ProductBreakdownStructure

RSM ResponseSurfaceModel

UAV UnmannedAerialVehicle

VPLM VariablePowerLawMetamodel Latin formula symbols

B Fluxdensity . . . T Br Magneticremanence . . . T C Capacity . . . A h Crate C-rate . . . h−1 CP Powercoefficient CT Thrustcoefficient D Bladediameter . . . m E No-loadvoltage . . . V h Convectioncoefficient . . . _mW2_K

Harm Armheight . . . m

IO No-loadcurrent . . . A

J Advanceratio

J Windingcurrentdensity . . . _mA2

K Aircompressibility . . . N m2 keq Equivalentstiffness . . . _mN2 KT Torqueconstant . . . N mA KV Velocityconstant . . . rpm_V L Characteristiclength . . . m

larm Armlength . . . m

M Machnumber

M Mass . . . kg

np Parallelbranches

ns Seriesbranches

nT Coilturns

Piron Ironlosses . . . W

Pon Conductionlosses . . . W

Pswitch Commutationlosses . . . W

r Diameterratio . . . m

R Resistance . . .



U Voltage . . . V

UDC DCvoltage . . . V

Tf Frictiontorque . . . N m

Tnom Nominaltorque . . . N m

V Flightspeed . . . m_s

Vimpact Impactspeed . . . ms

Greek formula symbols

β

Ratiopitch-to-diameter

δ

Rotationalspeed . . . rad_s



Averagerelativeerror . . . %

ω

Rotationalspeed . . . rad_s

π

Dimensionlessnumber

ρ

Resistivity . . .



m

σ

Standarddeviation . . . %

θ

Maximumadmissibletemperature . . . K

Fig. 1. Multirotor drone average mass distribution based on [10–12].

dealswithpropulsion,energy,powerconversionandstorage com-ponentsgenerallytreatedina non-homogeneouswayindifferent papers butalsoaddressesstructural parts.All theseparts havea significantimpact onglobalmasswhich isstronglycoupled with autonomyanddynamicperformances.Thecontrolaspects and as-sociatedsensorsarenotconsideredhere.

2.2. Designdriversandsizingscenarios

The main performance requirements for drones are (regard-lessofthecontrol-laws):thepayload mass,forwardspeed,hover autonomyorrange[13].Inadditiontothesesystemlevel power-energyconsiderations,thecomponentsmustwithstandsome tran-sitoryextreme orendurance criteriathat requestspecific

knowl-edge ofthetechnologyorphysical-domain. The so-called compo-nents design drivers are summarized in Table 1 withthe corre-sponding scenarios forwhichthey should be assessed aswell as literaturereferences.

2.3. DimensionalanalysisandBuckingham’stheorem

Some references [19] or tools [20] directly use databases to evaluatetheperformanceresultingfromtheassociationofdefined components. The explosion of combinatorial solutions which in-creasewiththedatabasessizemayleadtoveryhighcomputation time. This iswhy,most ofthedesign methodologies, despite the wide variety ofimplementations [9,16,8], arebased on analytical expressionsorcatalogue dataregressionsandbenefit,asmost con-tinuousproblems,ofnumericaloptimizationcapabilitiesandhigh convergenceperformance.

DimensionalanalysisandBuckingham’stheorem[21,22] are ex-tensively used inaerodynamicsand fluid mechanicsto provide a more physicalandunified frameworkforall thecomponentsand their corresponding characteristics. This paper showshow it can beextendedtootherdomainsbyaddressingallthecomponentsof adrone.

The fundamental step isto expressone component character-istic y as an algebraic function f ofgeometrical dimensionsand physical/materialproperties: y

=

f

(

_

L

,

d1,

_

d2, ...

_

geometric

,

p1,p2, ...

  

physical

)

(1)

Table 1

Designdriversandsizingscenarios.

Components Designdrivers Parametersor characteristics

Sizingscenario segment

Ref.

Propeller Maxthrust Diameter,tip

speed

Takeoffandhigh powermission segment

[14]

Efficiency Pitch Hoverandlong

durationmission segment [14] Motor Temperature rise Nominaltorque, Resistance

Hoverandlong durationmission segment

[15]

Maxvoltage Torqueconstant, Resistance

Takeoffandhigh powermission segment

[16]

ESC Temperature

rise

Maxpower,max current

Takeoffandhigh powermission segment

[16]

Battery Energy Voltage,capacity Hoverflight

(autonomy)and forwardmission segment(range)

[14]

Power Maxdischarge

rate

Takeoffandhigh powermission segment

[16]

Frame Stress Mechanical

strength Takeoffand landing [17] Vibration Resonance frequency

Takeoffandhigh powermission segment

[18]

with L as the characteristic length anddi and pi asthe rest of

geometricandphysicalparameters.

TheBuckingham’stheoremstatesthattheoriginalequation(1), withn parametersandu physicalunits(time,mass,length...),can berewrittenwithareducedsetofn

−

u dimensionlessparameters called

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)

Therefore,thedimensionalanalysisandtheBuckingham theo-rem can permit to reduce the number of input parameters of a numericalmodelandhenceitscomplexityandcomputationalcost. Theestimationmodelusedwilldependonthecharacterofthe dimensionlessnumber.Scalinglawsarecommonlyusedwhen

π

di

and

π

pi stayconstantintherangeofaseries,forexampleforthe

sizingoftheelectricalcomponents.Whenthecomponentsdonot complywitha geometricalormaterialsimilarity, thenalternative formswill beapplied such asregressionmodels forthestudyof thepropellerparametersorthestructuralparts.

AnoverviewoftheprocessisshowninFig.2.

3. Scalinglaws

Thescalinglaws,alsocalledsimilaritylawsorallometric mod-els, are based on the dimensional analysis. These simplification lawsarereallyinterestinginapreliminarydesignprocessbecause they limit drastically the number of design parameters for each component while having good estimation properties based on a

Fig. 2. Estimation models of drone components based on dimensional analysis.

detailedreferencecomponent. These lawsare established assum-ing:

•

Geometric similarity:allthelengthratiosbetweenthecurrent componentanditscorrespondingreferenceareconstant.With thisassumptiontheaspectratios

π

di

=

x_Li areconstant,

•

Uniqueness of design driver:onlyonemaindominantphysical phenomenondrives theevolutionofthesecondary character-isticy. With thisassumption, the numberof

π

pi is equal to

zero,

•

Material similarity:allmaterialandphysicalpropertiesare as-sumedtobeidenticaltothoseofthecomponentasthe refer-ence.Ifthenumberof

π

pi isnotequaltozero,theremaining

dimensionlessnumbers can oftenbe expressed through con-stantratiosofmaterialpropertieswithsimilarunits.

Iftheseassumptionsaresatisfied,theresulting

π

di and

π

pi

di-mensionlessnumbersareconstantandconsequentlyto

π

y:

π

y

=

yLaL m



i=1 pai i

=

constant (6)

whichgivesthetypicalpowerlawformofscalinglaws:

y

∝

La

⇔

y∗

=

L∗a

⇔

y yre f

=



L Lre f



a (7)

where a is a constant representative of the problem. This paper alsousesthenotation proposedby M.Juferin[23] where a scal-ing ratio x∗ of a given parameter is calculated as x∗

=

_xx

re f. The scalinglawspresenttheassetofrequiringonlyonereference com-ponenttoextrapolatecomponentcharacteristics(written

.

re f)ona

wide rangeofexploration.More informationontheconstruction, validationanduseofscalinglawsforengineeringpurposescanbe foundin[24].Scalinglawsareusedinthispaperforthemodelling ofelectricalcomponents:brushlessmotors,inverters- partofESCs andthebattery.

3.1. Uncertaintyofscalingmodels

This article will provide the set of standard deviations of es-timation errors foreach proposed model. In order to be able to better usethesequantities,especiallyforuncertaintypropagation

Fig. 3. Comparison of error distribution to a normal law. (For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.)

orrobustoptimization,wewillshowherethaterrorsdistributions forscalinglawscanbemodelled asindependentnormallaws.

Assumingthescalinglawisdefinedas:

y

=

k

·

xa

wherex is the definitionvariableand

y is theestimated param-eterofthestudiedcomponent.The followingrelationshipisused to predict theestimated errorbetweenthe realvariable andthe estimatedone:

y

=

y

(

1

+

ε

)

ε

ismodelled asarandomvariable.Possibleerrorsourcesmaybe relatedtomultiplefactors,suchastheassumptionofthe geomet-ric similarity for much larger dimensions, unfeasibility regarding somedesignconstraints,changingmaterialsorphysicalproperties. The normaldistribution isthe mostcommonmethod usedto modelphenomena withmultiplerandomevents.Forthismethod, theprobabilitydensityfollowsaGaussiandistribution:

f

(

ε

)

=

1

σ

√

2

π

e −1 2( ε−μ σ )2

where

μ

isthemeanand

σ

thestandarddeviation.Anexampleof such distributionisillustrated inFig.3.The histogramshowsthe distributionoftherelativeerrorfortheestimatedbatterymassfor differentreferencesofLithium-Polymerbatteries.

Agraphical technique used to assessthe adequacy ofthe ob-serveddistribution toaGaussian distributionisthenormal prob-ability plot. This method fits a set of observations to a straight lineand those deviations found fromthe straight line are varia-tions fromthe normallaw.The meanandthestandard deviation arereadontheabscissafromthecrossingoftheinterpolationline with Y

=

0 and Y

=

1 respectively. The collected data series fit properlytoanormaldistributionasdepictedinFig.4withamean value

μ

=

0

.

05% andstandarddeviationvalue

σ

=

6

.

48%.

In order to minimize the error of estimation when applying scalinglaws,itisrecommendedtotakeintoaccountsomefactors. Themainfactorsorgoodpracticesare:

•

Choice of a mid-range reference point

The choice of a good referencevalue will have a strong ef-fecton thescalinglawasit willkeepthemeanerroraslow aspossible.Scalinglawsfollow,asdiscussedbefore,a normal distribution where choosing a referenceof the extremes en-tailsadifferenceincreasefortheothervalues.Thenextfigure showstheevolutionoftherelativeerroraccordingtothe dif-ferentmotortorque(Fig.5).Itisobservedthattheerrortends to decrease for references which are close to the geometric meanofminimumandmaximumvaluesofthefullrange.

Fig. 4. Normal probability plot.

Fig. 5. Effect of the reference point choice on the mean error (motor example).

•

Correlation between errors

Uncertainties from the different estimation models may be linked tofurtherrandomvariablesorindependenttoits own random variable. The dependencyis studied using a correla-tion analysis. An example is applied forthe following motor equations: Mmot

=

Mmot,re f



Tmot Tmot,re f



3/3.5 Motor mass

Tmot,f r

=

Tmot,f r,re f



Tmot Tmot,re f



3/3.5 Motor friction torque

Fig. 6. Correlation of errors between different motor models.

Fig. 7. Standard deviation associated to geometrical parameters.

Rmax,mot

=

Rmax,mot,re f



KT KT,re f



2



_T mot Tmot,re f



₋5/3.5 Motor resistance

After a correlation analysis for different motor models, it is clear that there is no clear relationship found between the differentvariables.Hence,itisassumedthattheerrorsare in-dependent (Fig.6).

•

Form of the equation

Forscalinglawslinkeddirectlytosomegeometricparameters, it can be noted that uncertainty of a dimension may grow with increasing power. Fig.7 outlines the distribution ofthe standarddeviationforallreferencesoftheBHrangefromthe manufacturerKollmorgen.

Intheabsenceofmoreinformation,ifthescalinglawsare re-latedtogeometricparameters,theestimationofuncertainties is directly related to the exponent of the scaling law asfor thepresentexample:10%forthediameter,30%massand50% inertia.

4. Scalinglawsappliedtoelectricalcomponents

4.1. Brushlessmotor

Manypapers[9,16,8] usedirectregressionsuponmanufacturer datasheets. Here, we outline how scaling laws can represent the sameinformationfromasinglereferenceandwithsimilarfidelity while enhancing understanding of major physical and/or techno-logicaleffects.

4.1.1. Designdriversandmainparameters

Themultirotor UAVsintegrate permanentmagnetsynchronous motorswithaninternal(in-runner)orexternal(out-runner)rotor. As encountered on thecurrentdrone market, mostsolutions use out-runner type which directly drives thepropeller. Thus, only a modeladaptedtosuchtypeispresentedhereafter.

Manufacturers provide technical information in the form of equivalentDCmotorsuchas:

•

The

constant velocity Kv which is generally expressed in

rpm V



. This constant is related to the speed or the torque constant KT

=

_2π60_·Kv (equivalent units



N m A

or



V rad/s

from whichitispossible toexpressthe torque T

=

KT

·

I with

re-specttothecurrentI orthenoloadvoltageE

=

KT

·

ω

with

respecttothespeed

ω

,

•

Thecontinuouscurrentrelatedtothecontinuoustorque Tnom

whichislimitedbymotorlossesandheatdissipation,

•

Amaximumtransientcurrent(ortorqueTmax)whichislinked

tothe maximumlossesandpotentially maintainedfor afew secondsorafewtensofseconds,

•

The no-load current I0 which represents the motor no-load

losses. For highspeed andhigh poles numbermotors, these losses are mainly iron losses that increase with rotational speed,

Table 2

DimensionalanalysisonEquation (8).

Parameter [Voltage] [Current] [Length] [Temperature]

J_mA2  0 1 −2 0 ρ m  1 −1 1 0 θ[K] 0 0 0 1 h_mW2K  1 1 −2 −1 L [m] 0 0 1 0 d1[m] 0 0 1 0 d2[m] 0 0 1 0 Table 3

DimensionalanalysisonEquation (9).

Parameter [Force] [Length] [Current]

T [N m] 1 1 0 J_mA2  0 −2 1 Br[Wb] 1 −1 1 L [m] 0 1 0 d1[m] 0 1 0 d2[m] 0 1 0

•

TheresistanceR whichcausestheohmicvoltagedropandhas aneffectonthecopperlosses.

4.1.2. Dimensionalanalysisandmainrelations

Inthissub-sectiontheconstructionofscalinglawswith dimen-sionalanalysis isillustrated onbrushless motors. The considered assumptions are that the main design criterion forthe motor is themaximumwindingtemperatureandthatnaturalconvectionis the dominant thermalphenomenon. We use the following nota-tionsforthefurtherdimensionalanalysis:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

J : Winding current density Br: Remanent induction

di: other motor’s dimensions permanent magnet

L: Motor length



: Rotational speed

ρ

: Cooper Resistivity Sf: Magnetic flux motor

surface

θ

: Max. Admissible temperature

rise for winding insulation B: Flux density

h: Convection coefficient T : Electromagnetic

torque

⎫

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎭

Considering thermal properties,the current density J can be linkedtothemotordimensionsandadditionalmaterialproperties:

J

=

f

(

_{  }

L

,

d1,d2 3

,

ρ

, θ,

h

  

3

)

(8)

Consideringmagneticproperties,thetorqueT canbelinkedto di-mensionsandcurrentdensity,

T

=

f

(

_{  }

L

,

d1,d2 3

,

_{  }

J

,

Br

2

)

(9)

Table 2 and Table 3 summarize the dimensional analysis of theseequations.

TheproblemrepresentedbyEquation (8) contains7parameters and4dimensionsandcanbereducedto3dimensionless param-eters. We focus on

π

J dimensionless parameter which is of the

form:

π

J

=

JaJ

·

LaL

·

ρ

aρ

· θ

aθ

·

hah

=

J2

·

L1

·

ρ

1

· θ

−1

·

h−1

Equation (8) canthenbereduced,usingBuckingham’sTheorem to:

ρ

J2_L h

θ

=

f 



d1 L

,

d2 L



(10)

SimilaranalysisconductstoEquation (9) simplification:

T J BrL4

=

f



d1 L

,

d2 L



(11)

Considering material andgeometric similarities (di

L

=

constant

and hθ_ρ

=

constant)the followingscalinglawsderive from Equa-tions(10) and(11):

J

∝

L−0.5

⇔

J∗

=

L∗−0.5 (12)

T

∝

J L4

⇔

T∗

=

J∗

·

L∗4 (13)

Thecombinationofbothpreviousequationsenablesthetorque tobeexpressedasafunctionofthemotorsize:

T

∝

L3.5

⇔

T∗

=

L∗3.5 (14)

Stillconsideringmaterial/geometricalsimilarity,themotormass canbeestimatedwithrespecttoitstorque:

M

∝

L3

∝

T33.5

⇔

_M∗

=

_T∗33.5 ₍₁₅₎

4.1.3. Scalinglawsfortheelectricalparameters

Anotherapproach presented in[23] consists in settingup the scalinglawsdirectlyusingtheanalyticalexpressionofthe param-eter understudy. As an example, the electrical circuits including windingsshouldbedesignedtomaketheiroperatingvoltage com-patiblewiththepowersourceorthepowerelectronicsvoltage.An analysis[24] oftheno-loadvoltageE givesthebackEMFconstant

KT asafunctionofthecoilturnsnumbernt:

KT

=

E



=

k

·

nt

·  ·

B

·

Sf

ω

⇒

KT

∝

nt

·

L 2 ₍₁₆₎

KT does not vary while considering similarity and is a function

ofmotorconstants:surfacecoefficient,fillingratio,winding coeffi-cient,coilsnumberpergroup,coilsnumberpergroupandphase, totalnotchesnumber,notchesnumberpergroup.

Thenumberofturnsnt isalsoinvolvedintheresistance R

ex-pression:

R

∝

n2_t

·

L−1 (17)

thisresistancecanbeusedtoexpresscopperlosses.

Anothertype oflosses canbe observedin synchronousmotor andareduetoinductionvariation:ironlosses.Thequantityofiron losses can be expressed according to the Steinmetz-like formula [25] as:

Piron

∝

fb

·

L3 (18)

where f istheelectricalfrequencylinkedtothepolenumberand mechanical rotationalspeed



: leading to frictiontorque Tf

for-mula:

Tf

=

Piron



∝ 

b−1

_·

_L3 ₍₁₉₎

Equation (35) given thereaftershowsthatthe torqueislinked to the rotational speed and the propeller diameter. Considering thatthepropellerisoperatingatconstanttipspeedthen:



∗

=

D∗−1

=⇒

T∗

=

D∗3

= 

∗−3

Thismeans,ifanaveragevalueof1.5isconsideredforb, Equa-tion(19) canthusbesimplifiedto:

Table 4

Scalinglawsappliedtoout-runnerbrushlessmotor.

Parameter & equation AXIa _SCORPIONb _KDEc

Value σ[%] Value σ[%] Value σ[%]

Nominal torque Tnom∗ [N m] 0.102 1.84 1.794

Torque constant K∗_TN m A



0.005 0.029 0.028

Maximum torque Tmax∗ [N m] 0.137 1.88 1.823

Tmax∗ =T∗nom N/A N/A N/A

Friction torque T∗f [N m] 0.006 0.041 0.021 T∗f= b−1·L3≈Tnom∗0.69 15.41 26.72 33.4 Mass M∗[g] 57 435 305 M∗=T∗ 3 3.5 nom 11.51 10.07 9.49 Resistance R∗[] 0.045 0.031 0.044 R∗=K∗2·T∗ −5 3.5

nom 13.21 N/A N/A

a _{AXIMotors:https://www.modelmotors.cz/e-shop/.}

b _{SCORPION:https://www.scorpionsystem.com/catalog/aeroplane/.}

c _{KDEDirect:https://www.kdedirect.com/collections/uas-multi-rotor-brushless-motors.}

Fig. 8. Mass vs nominal Torque for different motor families.

4.1.4. Scalinglawssummaryandvalidation

Table4synthesisesthepreviousdevelopedscalinglawsforthe motorsandgivesanexampleofreferencedata.

InthefollowingFig.8,massdataarecomparedwiththeir cor-respondingscalinglaws.Inordertoselectthebestreferencepoint withthe lowest possibleerror withrespect to the mass,the av-erage relative error was compared by testing all available data points.

The standard deviation for the mass estimation is between 9.49%and11.51%.Afigurecorrespondingtothefrictiontorque es-timation isnot shown here, butthe standarddeviation ishigher (between15.4%and33.41%).However,itremainsacceptablefora preliminarymodel.

Since the scaling lawfor the electrical resistance depends on morethanoneparameter,theuseofa X

=

Y chartcouldpermitto visualizethe estimatedvaluesagainst thedatabase. The standard deviationrelativeerroris13%fortheAXIdatabase.

4.2.Powerelectronics

Fordrones, the ElectronicSpeed Controllers (ESC)distributing powertothesynchronousmotorsaremainlyDC/ACconverters (in-verters)madeof powerelectronic switches (MOSFET transistors). Fordrone applicationcases,powerMOSFETarepreferredtoIGBT becauseoftheir highcommutation frequencyandgreatefficiency evenunderlowvoltages.

4.2.1. Designdriversandmainparameters

TheESCmaincomponentsandcorrespondingmaindesign pa-rametersarelistedhereafter.Themaincriteriainfluencingthesize ofthe componentsarerelatedto thermalaspects. Theremainder ofthissectionaimsatexpressingtheeffectofscalingonthelosses andheattransferproperties.

•

MOSFETtransistors: maximumjunction temperature is func-tionofconduction Pon,commutation losses Pswitch and

ther-malresistance Rthjc.

•

MOSFETdriver:Peakgatedrivecurrentsinfluencethe commu-tationlossesofMOSFETtransistors.

•

DCcapacitor:stabilizesthevoltagebutthehigh-frequency cur-rentcharge/dischargegenerateslosses.

•

Case:Mechanicallyprotectsthecomponentsbutalsoactsasa heatsinkforallinternallosses.

4.2.2. Scalinglawsapproach

MOSFEThavetwomaintypesoflosses[26]:

•

Conductionlosses:Pon

=

Rds,on

·

I2R M S withIR M S theRMS

cur-rentandRds,on theONresistance,

•

Switchinglosses: Pswitch

∝

UDC

·

I

·

Q_IGG

·

fswitch with Q_IGG the

commutationtime(fractionofgatecapacitanceanddrivergate current).

Forlow-powertransistorsandrelativelowswitching-frequency (as thisisthe caseforUAVs application)switchinglosses canbe neglected comparedtoconduction losses.Inaddition,we assume thattolimitconductionlosses,thenumberofparallelelementary cellsusedtorealizeaMOSFETtransistorisproportionaltothe cur-rentratingandthus Rds,on

∝

I−1.

Thisleadsto:

Ploss,M O S F E T

≈

PO N

∝

I (21)

In turn, considering that transistor design is driven by maxi-mum thermal criterion and power dissipation is mainly due to forced convection (caused by propeller air flow, ESC being inte-gratednearby),wegetatequilibrium:

θ

=

1

h

·

S

·

Ploss,M O S F E T

=⇒ θ

∗

₌

₁

₌

_l∗−2

_·

_I∗

=⇒

M∗

=

l∗3

=

I∗32

Fig. 9. ESC voltage vs. power regression.

withh theconvectioncoefficient(consideredconstant), S the con-vectionexchangesurface,

θ

thetemperaturedropconstantfora givenmaterialortechnologyandl theconvertertypicaldimension. The idea is now to link nominal power to the mass instead of currentbut to do so,the relationship between MOSFET nom-inal power and voltage has to be determined. Fig. 9 represents thepower andvoltageof75differentdrone ESCsfromJive, Kos-mik,YGE,SpinandTurnigysuppliers.Voltage levelincreaseswith poweranda directpower-lawlinking voltagetopoweris consid-eredhere:

V∗

=

P∗a

=⇒

V

=

Vre f

Pa_{re f}

·

P

a

₌

_K

_·

_Pa

=⇒

log

(

V

)

=

a

·

log

(

P

)

+

log

(

K

)

(23)

The constant K represents a referencecomponent anda the

powertobedetermined.Alinearregressionisperformedintolog spaceandleadstotheapproximationV∗

=

P∗13_.

Equation(22) canthenbemodifiedto:

M∗

=

I∗32

=



P V



_∗3 2

=

P∗ (24)

ThisisthemainresultformodellingtheESC.Yetduringdesign, anotherrepresentativevalue extractedfrommissionprofile/sizing scenario(linked to motor/batteryperformances) isthe maximum apparentpower:

Pmax

=

Umax

·

Imax

=

Ubat

·

max



Pmot

Umot



(25)

4.2.3. Scalinglawssynthesisandvalidation

The following Table5 lists themain scaling lawsdeduced for the ESC andcompares the validity of thesemodels withseveral familiesofcomponents.Moredetailsontheuseofscalinglawsfor sizing and optimizationof staticconverters can be found in [27, 28].

4.3. Batterypack

An essential step ofthe preliminarydesign process for multi-rotor drones is to select a technology and size the battery pack considering missionsperformancesneeds.Lemonetal.[29] high-light huge differences between the electrochemical couples but alsowiththecelldesignitself(Li-Ionspecificenergycanbe mul-tipliedby4).

A high discharge rate (or high power density) means high-dynamics capabilities while high energy density stands for high autonomy(longstationaryflight).Onetechnologystandsoutforits high performance capabilities: Lithium-Ion. From this family, the Lithium-IonPolymerwithslightlysimilarperformancesbuthigher life cycleduration andsafety(betterresistance toover-charge) is takenasareference.

4.3.1. Designdriversandmainparameters

Inthissub-sectionscalinglawsbasedontheanalytical expres-sion of the main sizing parameters of a battery are constructed. Thelawsarevaliddespitethecelltechnologychosen:

•

Voltage(U inV): Thebatteriesareusually asetofcellswired in series to produce the needed voltage (cell voltage being around 3.7 V). When combining ns batteries in series,

to-tal voltage U is the sum of all cells unitary voltage Ucell

(U

=

ns

·

Ucell).

•

Capacity(C inA h): Byconnectingbatteriesinseries,capacity doesnotincrease(butstoredE energydoes:E

=

C

·

U ).When the currentcapacityneeds to be increasednp branch should

beconnectedinparallel:C

=

np

·

Ci

The batterymass beingalmostlinearly connectedto itscells number (packagingputaside), thisleads tothe main scaling lawforagivencelltechnology:

M

=

ns

·

np

·

Mcell

=⇒

M∗

=

U∗

·

C∗

=

E∗ (26)

Table 5

Scalinglawsappliedtodifferentfamiliesofelectronicspeedcontrollers(ESC).

Parameters & equation JIVEa _KOSMb _YGEc _SPINd _TURNe

Value σ[%] Value σ[%] Value σ[%] Value σ [%] Value σ[%]

ESC voltage 44.4 51.8 51.8 44.4 22.2 V∗E SC[V] Maximum power 2664 10360 6216 3419 3330 P∗max[W] Maximum current 60 200 120 77 150 I∗max[A] Mass M [kg] 84 200 119 105 169 M∗=P∗ 13.75 14.14 22.58 19.42 25.36 M∗=I∗3/2 _{46.07 20.11 43.98 30.52 21.17}

a _{JIVEESC-http://shyau.com.tw/documents/.}

b _{KOSMIK-https://www.kontronik.com/en/products/speedcontroller/speedcontroller1/kosmik.html.} c _{YGE-https://www.yge.de/en/home-2/.}

d _{SPINOPTO-https://www.modelmotors.cz/product/jeti-spin/.} e _{TURNIGY-http://www.turnigy.com/.}

Table 6

Scalinglawsappliedtobatteries.

Parameters & equation GENSa _PROLb _RAMPc _TATd _KOKe

Value σ [%] Value σ[%] Value σ[%] Value σ[%] Value σ[%]

Voltage 22.2 22.2 22.2 11.1 3.7 U∗[V] Capacity 5000 4400 1800 1550 200000 C∗[mA h] Mass M [kg] 755 601 291 128 4180 M∗=C∗·U∗ 3.05 4.61 7.07 19.71 3.91

a _{GENSACE(Lithium-Polymer)https://www.gensace.de/.} b _{PROLITEX(Lithium-Polymer)http://www.thunderpowerrc.com/.}

c _{RAMPAGE(Lithium-Polymer)http://www.thunderpowerrc.com/Home/Rampage-Series.} d _{TATTU(Lithium-Polymer)https://www.genstattu.com/.}

e _{KOKAM(Lithium-Ion)http://kokam.com/.}

•

C-rate(Crateinh−1): Discharge currentisoften expressed as

a C-rate to normalize against battery capacity. A C-rate is a measureofhowfastabatterycanbedischargedrelativetoits maximum capacitywithoutpresenting permanentdamage.If

Crate

=

1,then thedischargecurrentwilldischarge theentire

batteryinonehour.FormulalinkingmaximumcurrentImaxto

C-rateisthefollowing:

Imax

=

Crate

·

C (27)

Foragivencelltechnology,Crate staysconstantandI∗max

=

C∗.

Specificpowerandenergyareassumedtoremainconstantover acell assembly even ifthe packaging can varyon a givenseries rangewithnoparticularcorrelationtocapacity.

4.3.2. Scalinglawssynthesisandvalidation

Table6summarizesthepreviousscalinglawsproposedforthe batteriesandgivesexamplesofreferencedata.187batteries com-ingfromLi-PoandLi-ionareused.

4.4.Electriccables

Electric conductors cross-section area is selected with nomi-nalcurrent. Thedesign driverofsuch acomponentisin factthe maximalinsulatortemperature.Thetemperaturedropislinkedto powerlossesthataremainlyduetotheJouleeffect:

Ploss

≈

Ploss,J oule

=

ρ

·

J2

·

V (28)

where

ρ

istheresistivity oftheconductor, J the currentdensity andV theconductorvolume.

Itisassumedherethatbecauseofthethininsulatorthickness, thethermal conductionresistance can be neglected compared to theconvectionone.Thismeansthatthepowerlosscanbe linked to insulator exchange area S

=

2

π

·

r

·

L (r cable radius and L

length):

Ploss

=

h

·

S

· θ

(29)

withh theconvectiveheattransfercoefficientand

θ

the admis-sibletemperaturedrop(constantforagivenexternal temperature andinsulatormaterial:h∗

=

1 and

θ

∗

=

1).

Equations (28) and(29) leadtothefollowingformulationofthe nominalcurrentdensity( J ):

J∗

=

r∗−21 ₍₃₀₎

ConsideringthecurrentisI

=

J

.

S,currentcanbeexpressedas afunctionoftheradius:

Table 7

Scalinglawsappliedtoelectricconductors. Parameters & Equation Wiring cablea

Value σ[%] Current I [A] 120 Radius r [mm] 5.2 r∗=I∗2/3 _5.56 Linear mass M L kg m  0.191 M L =I∗4/3 11.08

a _{PVC-insulated single wiring cables https://www.} engineeringtoolbox.com/wire-gauges-d_419.html.

Fig. 10. Radius vs. Current load.

I

=

J

·

π

·

r2

=⇒

r∗

=

I∗23 ₍₃₁₎

Equation(31) canbeusedtofindtherelationshipbetweenthe linearresistance R_L ofawireanditsnominalcurrentcaliber:

R L

=

ρ

S

∝

r −2

_=⇒



R L



_∗

=

I∗−43 ₍₃₂₎

Atthe sametime,the linearmass M_L ofthe conductorcanbe expressedasafunctionofr:



M L



∝

r2

=⇒



M L



_∗

=

I∗43 ₍₃₃₎

4.4.1. Scalinglawssummaryandvalidation

Table7summarizesthescalinglaws:

The comparison of these laws with catalogue data yields a mean relative errorof 16% forthe linear massand about8% for theradius(Fig.10).

Table 8

APC10×4.5inchmultirotordatasheet[32].

n (rpm) V (mph) CT(−) CP (−) 2000 0.0 0.1102 0.0428 2000 5.1 0.0832 0.0436 2000 13.1 0.0057 0.0114 6000 0.0 0.1126 0.0432 6000 16.1 0.0822 0.0427 6000 33.5 0.0226 0.0196 10000 0.0 0.1154 0.0485 10000 29.2 0.0789 0.0404 10000 60.7 0.0112 0.0120

5. Regressionwithdimensionlessnumbers

In the case where it is not possible to consider as constant thedimensionlessnumbers

π

di and

π

pi,theapproximationofthe

functionf canbeachievedbyperformingdataregressions[30,31]. Thedata cancomefrommanufacturer productdata aspresented insection5forthepropellers,testmeasurementsorfiniteelement simulationsresultscomputedonDoEaspresentedinsection6for thedronestructure parts.Theutilizationof

π

numbers tosetup aresponsesurface model(RSM)ora surrogatemodelhasseveral advantages:

•

Adecreaseinthenumberofvariablestobemanipulatedand therefore a drastic decrease in physicalor numerical experi-mentstobecarriedout,

•

An increase ofthe regression robustness [30] in particular if theRSMisbuiltwithinthelogarithmicspaceasfortheVPLM methodology (Variable Power Law Metamodel) [31] which showsgood resultsin interpolation butalso inextrapolation becauseofthepowerlawform.

6. Polynomialregressionappliedtopropellerdatasheets

Thepropellerrepresentsakeycomponentinthedrone propul-sion chain. Its performance can be expressed as a function of two coefficients CT andCP respectivelyexpressing thrust

(Equa-tion (34))andmechanicalpower(Equation (35))equations:

T hrust

=

CT

·

ρ

air

·

n2

·

D4 (34)

P ower

=

CP

·

ρ

air

·

n3

·

D5 (35)

where

ρ

airrepresentstheairdensity,n therotationalspeed

[

rev_s

]

andD thepropellerdiameter.

When an engineer is facing a new drone design problem, he can recursively try differentpropeller references andtest perfor-mancesagainstdifferentmissionprofiles(takeoff,dynamic perfor-mances/verticalacceleration,range...).Inthatcase,hecan interpo-latemanufacturerdatasheetssuchastheonepresentedinTable8 tohaveanestimateoftheperformancecoefficientsforeach work-ingcondition:

CT

=

ft

(

n

,

V

)

(36)

CP

=

fp

(

n

,

V

)

(37)

whereV representstherelativeairspeed.

Nevertheless,asexplainedbefore,the performance ofthe res-olution is enhanced while defining continuous expression to es-timate the performance coefficients over the whole multirotor dataset.

Cj

=

f

(

n

,

V

,

pitch

,

D

, ...)

(38)

Thereforeinthissection,wefirstapplyBuckinghamtheoremto determinetheproblemdimensionlessparametersandthenachieve apolynomialregressiontofitthemanufacturerdatasheet.

6.1. ApplicationoftheBuckingham’stheorem

Letusconsiderthefollowingproblem:

T hrust

=

f

(

D

,

pitch

  

2

,

ρ

_

air

,

K

_

,

n

,

V

_

4

)

(39)

where K istheaircompressibility.ApplyingBuckinghamTheorem leadstofollowingderivedexpression:

CT

=

T hrust

ρ

air

·

n2

·

D4

=

f

(β,

J

,

B

)

(40)

Similar analysiscan be conductedreplacing T hrust by P ower

parameterandfollowingdimensionlesssetarises:

CP

=

P ower D5

_·

_n3

_·

_ρ

air

=

f

(β,

J

,

B

)

(41)

WhereforboththrustandpowercoefficientsCT andCP:

• β =

pitch

D istheratiopitch-diameter.

•

J

=

V

n

·

D istheadvanceratio.

•

B

=

K

ρ

air

·

n2

·

D2

istheaircompressibilityindicator(similarto MachnumberM)anditseffectisvisibleathighspeeds.1 Inthissection,surrogatemodels arebuilttodescribethe pro-peller’sperformance accordingto (41) instaticscenarios

(

V

=

0

)

, applicable atthe hovering mode and at takeoff, andin dynamic scenariosfortherestofcaseswhere V

=

0.

6.2. Staticscenarios,V

=

0

Inthestaticmodels,theeffectoftheadvanceratioisremoved. In addition, for given aerodynamic configurations, the propeller mustadaptitsperformancebylimitingthespeedtoacertainRPM limit proposed by the manufacturer. For APC MR this operating rangeremainsbelow105000 R P M_inch.

In Fig. 11, the variation of CT and CP for different APC MR

Propellers are plottedagainst

β

andcompressibilityfactor B.The blue dots correspondto an operation rangebelowthe maximum RPM limit (105000 R P M_inch), whiletheblack dotsshow abehaviour beyondthatlimit.

Itcanbehypothesized,thatthereisarelationshipbetweenCT

and

β

(the samewithCP and

β

),whileonthecontraryitcanbe

seen howtheaircompressibilityindicator B hasalittleeffecton thepropeller performancewhentheoperationrangeisbelowthe establishedRPMlimit.

Beyond that limit, air compressibility effects begin to have a negative impactontheperformance ofthepropeller witha large increase of the power coefficient (CP) in comparison with the

thrustcoefficient(CT).

Theconclusionisthat,foraRPMlimitlowerthan105000 R P M_inch, thethrustandpowercoefficientCjgenerallydependson

β

:

Cj

≈

f

(β,

B

,

J

)

→

Cj

=

f

(β)

(42)

1 _{Considering that} K

ρ =a

2 _{where a is the speed of sound, this yields} a2 tip speed2∝ 1 M2 tip .

Fig. 11. Evolution of the performance coefficients in static case.

Fig. 12. Multi variable correlation matrix using R2_{heatmap (left) and scatter matrix (right).}

Theestimatedthrustcoefficient

CT andpowercoefficient

CP fora

operatingrangebelow105000 R P M_inch followtheregressionmodels:

CT

=

4

.

27

×

10−2

+

1

.

44

×

10−1

· β

R2

=

0

.

946 (43)

CP

= −

1

.

48

×

10−3

+

9

.

72

×

10−2

· β

R2

=

0

.

891 (44)

withahighvalueofR2,thesquaredcorrelationcoefficient,fitting themodelverywelltothedata.

For high values of n

·

D, a regression based only on the an-gleisnot veryaccurateduetothe effectofaircompressibility.A modelforhigherRPMshouldconsidertheeffectofbothvariables:

β

and B.

6.3. Dynamicscenarios,V

=

0

AccordingtotheBuckingham’stheorem,arelationshipmaybe foundtolinkdimensionlessparameters:

Cj

≈

fj

(β,

J

,

B

)

(45)

Data arefilteredin ordertorespectmanufacturer optimal op-erating conditions (n

.

D

≤

105000R P M_inch). Considering such an hy-pothesis, thecompressibility coefficient indicator B effect can be neglectedandEquation (45) isapproximatedby:

Cj

≈

fj

(β,

J

)

=

Cj (46)

TheuseofacorrelationmatrixandthescattermatrixinFig.12 help us to visualize that CT and CP do not depend on a single

Fig. 13. CT model relative error=f(μ,σ)evolution with terms selection.

Computations have shown that a 3rd order polynomial form canbesuitableatearlydesignphases:

Cj

=

j−∈[t,p]

(

aj,0

+

aj,1

· β +

aj,2

·

J

+

aj,11

· β

2

₊

_a j,12

· β ·

J

+

aj,22

·

J2

+

aj,111

· β

3

+

aj,112

· β

2

·

J (47)

+

aj,122

· β ·

J2

+

aj,222

·

J3

)

Whena linearregressionisperformedon normal-centred val-ues,coefficientsabsolutevaluecanbeusedtosorttermsbyorder ofimportance and regressionmodel canbe constructedwith in-creasingcomplexity(termsnumber).

AsshowninFig.13appliedtoCT problem,errormaynotbe

re-ducedafteracertaincomplexity.WithAPCdatasheet,ifa8-terms 3rd_{orderpolynomialmodelisconsidered,therelativeerroris}

lim-ited(



¯

=

0

.

4% and

σ



=

8

.

8%)asshowninFig.14:

CT

=

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

(48)

Similar results can be obtainedconsidering the Cp parameter

(



¯

=

0

.

5% and

σ



=

7

.

5%):

CP

=

0

.

01813

−

0

.

06218

· β +

0

.

00343

·

J

+

0

.

35712

· β

2

−

0

.

23774

· β

3

+

0

.

07549

· β ·

J

−

0

.

12350

·

J2 (49)

Those

CT,

CP estimators,depictedinFig. 15, presentmultiple

advantages:

•

Tobecontinuousoverpitch and D parameters,whichleadsto fasteroptimizeddesignandmayconvergetospecificpropeller design.

•

Tohaveanalyticderivatives.

Fig. 14. CT model performance considering 1stto 8thterms.

Fig. 15. 3Dregressionmodelsofestimatedperformancecoefficientsandreference points.

Fig. 16. Simplified structure hypothesis.

•

To haveregularexpressions that canbe handledby symbolic

parser for different output/inputs alternatives (ex: pitch

=

f

(

CT

,

D

,

n

,

V

)

).

7.VPLMregressionappliedtostructuralparts

The UAV structure has a non-negligible impact on its per-formance since it represents between 25% to 35% of the Maxi-mumTakeoffWeight(MTOW)thatincludesthepayload.Therefore, it is important to have an estimation of the mass of its sub-components:landinggears,framearmsandbody.

Itisdifficulttoaddressthewiderangeofcomplexgeometries which can be found over the models market. A simple geom-etry is presented in Fig. 16 for further analysis and calculation steps.

•

Bodyisassumedtobeanaluminium orcompositecrosswith hollowrectanglesection,

•

Arms are madeof aluminium hollowbars with square cross section,

•

LandinggearsareassumedtobeprintingABSplasticparts. The structure is sized with respect to two sizing scenarios: takeoffmaximum thrust (arms) anda crash witha givenimpact speed(body,arms,landinggears).

The first sizing scenario is relatively simple and considering thatthebodyisrigidcomparedtothearms,theequivalent prob-lem is a larm length cantilever beam with an applied load F

=

Thrust. The maximum stress is estimated with safety coefficient

ksas:

σ

max

=

Harm 2 12

·

T hrust

·

larm Harm4

− (

Harm

−

2e

)

4

≤

σ

alloy ks (50)

whichcanbewrittenwithdimensionlessarmaspectratio

π

arm

=

e Harm: Harm

≥



6

·

T hrust

·

larm

·

ks

σ

alloy

(

1

− (

1

−

2

·

π

arm

)

4

)



1 3 (51)

Thesecondsizingscenarioismorecomplextoconsiderandis developedinthenextsubsection.

Fig. 17. Equivalent stiffness problem decomposition with 4 landing-gears.

7.1. Crashsizingscenario

The crashsizing scenario considers amaximum speed Vimpact

ofthedronewhenhittingtheground.Atsuchspeedthestructure should resist (i.e. the maximum stress should not be exceeded) andforhigherspeeds, thelanding gears arethe partsthat break asstructuralfuses.

Tocalculatetheequivalentmaximumloadresistedbythe land-ing gears,the energyconservationlawapplies thekinetic energy storedindronemasstopotentialenergyinstructuralparts transi-torydeformation: 1 2keq

· δ

x 2

₌

1 2Mtot

·

V 2 impact

⇒

Fmax

=

1 4

(

keq

· δ

x

+

Mtotal

·

g

)

=

1 4

(

Vimpact

·



keqMtotal

+

Mtotal

·

g

)

(52)

To calculate the maximum stress induced by the maximum load Fmax appliedto onelandinggear,theequivalentstiffness keq

should be determined. For this purpose, the problem is broken downintosimplerstructuralparts,asdepictedinFig.17,andthe equivalentstiffness keq isexpressedconsideringtheeffectofeach

stiffnessonthewholepart.

keq

=

4

·

k1

·

k2

k1

+

k2

Thecalculationofthebody/armdeformationappliesthe fixed-beam formulas, while for the landing gear complex geometry a FEMsimulationsareusedtobuildananalyticmodel.

7.2. LandinggearstiffnessandinducedstressestimationapplyingVPLM

TodesignlandinggearwhichcanbeaspecificprintedABSpart, engineer willface one problem:estimate theapplied loadinthe caseofacrashinordertodeterminetheinducedstress.But,even before, as the stress depends on the load applied to the land-ing gear, an estimation ofthe stiffness

k2 should be established.

Therefore,FEMsimulationsareperformedondifferentgeometries (H1

∈ [

5

,

20

]

cm,H2

∈ [

1

,

10

]

cm,t

∈ [

3

,

15

]

mm,

θ

∈ [

20

,

45

]

deg)

withconstantthicknessb.

Iftheproblemisexpressedinregularform:

k2 b

=

f

(

  

H1,H2,t 3

,

_

E

, θ

2

)

(54)

Adimensionalanalysisleadstothefollowingproblem formula-tion:

π

k2

=

k2 b

·

E

≈

f



π

1

=

H1 H2

,

π

2

=

t H1

,

π

3

= θ



(55) andthen:

k2

=

π

k2

·

b

·

E (56)

π

k2 isfittedbya2ndordervariablepower-lawexpression:

π

k2

=

10

a0

·

_π

a1+a11·log(π1)+a12·log(π2)+a13·log(θ )

1

·

π

a2+a22·log(π2)+a23·log(θ )

2

· θ

a3+a33·log(θ )

(57)

Inlogarithmicspacethisformulacanbeadaptedtothefollowing linearform:

log

(

π

k2

)

=

a0

+

a1

·

log

(

π

1)

+

a11

·

log

(

π

1)

2

+

a12

·

log

(

π

1)

·

log

(

π

2)

+

a13

·

log

(

π

1)

·

log

(θ )

+

a2

·

log

(

π

2)

+

a22

·

log

(

π

2)2

+

a23

·

log

(

π

2)

·

log

(θ )

+

a3

·

log

(θ )

+

a33

·

log

(θ )

2

(58)

withx1

=

log

(

π1

)

,x2

=

log

(

π1

)

2...

Afteran analysis ofthe error distribution, it is observed that a 5-terms 2nd _{order model achieves a sufficient fidelity level}

(Fig.18):

π

k2

=

10−

0.37053

_·

_π

−3.11170

1

·

π

21.10205

· θ

−6.61617−4.86580·log(θ )

(59)

Adimensional analysis conductedon constraint

σ

max leads to

followingformula:

π

σmax

=

σ

max

·

b

·

H1 F

≈

f



H1 H2

,

t H1

, θ



=

f

(

π

1,

π

2, θ ) (60)

Usinga5-terms2nd _{orderform,themodelobtainedis:}

π

σmax

=

10

0.14690

_·

_π

2.08982

1

·

π

2−0.98108

· θ

3.38363+2.66468·log(θ )

(61)

Therefore,thismodelenablesto predictthe mechanicalstress generatedonthelandinggearinthecaseofacrash.The number ofdesignparametersisreasonable andthepredictionisrelatively accurate(



¯

=

0

.

1% and

σ



=

4

.

1%)similarlytotherestofthe

mod-elsproposedinthispaper.

Fig. 18.πk2model performance considering 1

st _{to 5}th_terms.

8. Assessmentofestimationmodelsperformanceduring preliminarydesign

The estimation models are the tools that allow us to work in a continuous domain, reducing the work time and eliminat-ing the limitations related to data tables. The use of analytical modelsalsofacilitatestheimplementationofsizingand optimiza-tion procedures. Reference [33] explains how to solve the main design problems.Article [34] shows indetail theimplementation of a sizing procedure for multirotor drones applying the mod-els described in this article in an optimization routine. The fol-lowing table gives the specifications and results of the overall drone sizingandtheresultingerrorswhencomparedtotheir ac-tual industrial reference. It can be seen that the proposed mod-els allowperforming arelevantpreliminarydesign ofdrones (Ta-ble9).

9. Conclusions

Inthispaperasetofvalidestimationmodelsforthekey com-ponents of all-electricmultirotor drones is presented.One ofthe main hallmarksofthesemodels liesinitsmathematical continu-ousform,whichfacilitatestheimplementationofdesignand opti-mizationtools.Theuseofdimensionalanalysis,apartfrom provid-ing thesephysically-motivatedapproacheswithaunifiedphysical basis,givesthemgreaterconfidenceandunderstanding compared to other pure statisticalapproaches inthe literature. As a result, scalinglawsshowrobustoutcomesintermsofdeviation,although itisrecommendedtofollowcertainguidelinestoreducesuch de-viation. Based on the current available technologies, they can be appliedto thepreliminarydesignofnewsystemsasonlya refer-encevalue isneeded.Surrogate modellingtechniquesare usedin orderto obtainan analyticmodel basedondata-sheetdata (pro-pellers)andonFEMsimulationsdata(landinggear).Inthesecases, the dimensional analysis facilitates the choice ofthe main inde-pendent parameters and strengthens the prediction of the mod-els.

The proposition ofthisnew multirotor dronesizing model li-brary is importantto perform an overall sizing loop.This is em-phasized in the case of high payloads, where the mass of the componentsincreasesignificantlyinordertomaintain similar au-tonomyperformancerequirements.

Declarationofcompetinginterest

Table 9

Droneplatformsreferencesandsizingresults.

Drone platforms a) MK-Quadroa _{b) Spreading Wings}b

Specifications

Payload: 1 kg 4 kg

Arms/Prop. per arm: 4/1 8/1

Climb speed: 10 m/s 6 m/s

Autonomy: 15 min 18 min

Sizing

Total mass: 1642 g (+16.5% ref.) 8513 g (−10.3% ref.)

Battery mass: 353 g (+7.29% ref.) 2133 g (+10.4% ref.)

Motor mass: 47 g (−12.9% ref.) 160 g (+1.2% ref.)

Max. Power: 173 W (−1.14% ref.) 449 W (−10.29% ref.)

Diameter propeller: 0.256 m (+2.4% ref.) 0.40 m (+5.26% ref.)

a _{MK-Quadro:http://wiki.mikrokopter.de/en/MK-Quadro.}

b _{SpreadingWingsS1000+:https://www.dji.com/fr/spreading-wings-s1000.}

Acknowledgements

Theresearch was carried out aspartofthe DroneAppproject funded by the RTRA STAE (Sciences et Technologies pour l’Aéro-nautiqueetl’Espace,www.fondation-stae.net). The authors grate-fullyacknowledgethesupportofFrançoisDEFAY.

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