Perpetual Dynamic Soaring in Linear Wind Shear

Ri ardo Ben atel 1

, Pierre Kabamba 2

and Anou k Girard 3

Department of Aerospa e Engineering, University of Mi higan,

Ann Arbor, Mi higan, 48109, United States

1

Post-Do toral Resear her, Department of Aerospa e Engineering, University of Mi higan, 1320 Beal Ave, Ann

Arbor,Mi higan,48109,AIAAMember. (ben atelumi h.edu)

2

Professor, DepartmentofAerospa e Engineering, UniversityofMi higan,1320Beal Ave, AnnArbor, Mi higan,

48109,AIAAMember.(kabambaumi h.edu)

3

Asso iateProfessor, DepartmentofAerospa eEngineering, Universityof Mi higan, 1320Beal Ave,AnnArbor,

C

L

,

C

D

,

C

D

₀

=Lift oe ient,drag oe ientanddrag oe ientwithzerolift

DS

,

DS

i

=Dynami soaringnumberanddynami soaringnumbersample

M C

i

=UniformdistributionsampleforMonte-Carlosampling

E

,

E

min

=Air rafttotalenergyandminimumsustainabletotalenergy,

J

g

=Gravitya eleration,

m/s

2

h

,

h

min

=Altitude andsafetyaltitude,

m

K

=Dragpolarindu eddraggain

k

1

,

k

2

,

k

3

=Ne essaryandsu ient onditionsinequalityparameters

L

,

D

=Liftanddragfor es,

N

L

D

,

L

D

max

=Lifttodragratioandmaximumlifttodragratio

m

=Mass,

kg

N

=Sample size

S

=Wingarea,

m

2

V

a

,

V

min

=Airspeedandstallairspeed,

m/s

w

,

W

,

W

x,y,z

=Windvelo ityve tor,windspeed,andwind velo ity omponentsin

x

,

y

,and

z

,

m/s

∂w

LW S

∂h

=Windgradientve tor,

s

−1

X

,

X

i

=Randomvariableandrandomsample

x

,

y

,

z

=Position oordinates,

m

˙x

,

˙y

,

˙z

=Velo ity omponentsinthegroundframe,

m/s

¨

x

,

y

¨

,

z

¨

=A eleration omponentsinthegroundframe,

m/s

2

β

=Experimentalveri ation onden elevel

ǫ

=Experimentalveri ationmarginoferror

γ

a

=Flightpathanglerelativeto theair,

rad

µ

,

σ

=Randomvariablemeanandstandarddeviation

ψ

,

ψ

W

,

ψ

dW

=Air raftheadingangle, winddire tionandwindgradientdire tion,

rad

φ

=Bankangle,

rad

ρ

=Fluid(air)density,

kg/m

This paperanalyzesdynami soaringin linearwind shear. Wind shearis atypeofspa e and

time-dependentairowve toreld.

Dynami soaringisa y li ighttraje torythatenablesenergyharvestingfromthesurrounding

oweld. Thedynami soaringtraje toryis y li andnotperiodi inthesense thatinea h y le

someof thestatevariablespresentthesamevaluesandtrends fortheinitial andnal onditions.

Moreover,forthe y li traje torytobepartofasustainableight,theair raftenergy,sampledat

thebeginningofa y le,shouldbenon-de reasing.

This work dis usses the ne essary and su ient onditions to enable sustainable ight with

dynami soaring. These onditionsallowtheair rafttoharvestenoughenergyfromthewindshear

to ompensatefortheenergylostdueto drag.

A. ProblemStatement

Thisworkfo usesonthestudyofdynami soaringinwindshear. Thesystemunderanalysisis

denedbyawindshearmodelwith onstantverti alvelo itygradientandtheair raftequationsof

motion,whi hmodeltheair raftbehaviorwhensubje ttoawind gradient. Thegoalofthisstudy

is to dene inequality onditions that hara terize the feasibility onditions for dynami soaring.

The onditionsareappli abletoawidevarietyofUnmannedAerialVehi les (UAV s),allowingthe

determination of the suitability of an air raft for dynami soaring under dierent environmental

onditions.

B. Ba kground

Theequationsof ightdynami sbasedontheightpathangle andtheliftare derivedin [1℄.

There areseveralstudiesonoptimaldynami soaringtraje toriesforenergyharvestingfromwind

shear [24℄. These studiesanalyze theevolutionof several ighttraje tory variables, su h asthe

loadfa tor, limbrate,heading,and bankangle, overthedynami soaring y le.

Fromthesestudies,only[2℄and[3℄studytheminimum onditionsthatallowdynami soaringto

maintainperpetualight. Thesestudieseitherrefertospe i aerodynami models,i.e.,modelswith

These resultsallow aqualitativeevaluationof howthe onditions fordynami soaringdepend on

theaerodynami parameters. However,theydonotallowaquantitativeevaluationofthisrelation

for arbitrary air raft models. As an example, these studies suggest that more aerodynami ally

e ientair raft requireweakerwind gradients. However,fromthepresenteddata,itis di ultto

quantify the exa tminimum requiredwind gradientorthe maximumenergy gainfrom a spe i

wind gradient, foran arbitrary air raft model, with spe i lift to drag ratio and minimum drag

oe ient.

C. Original ontributions

Thisworkderivesthene essaryandsu ient onditionsforsustainabledynami soaringasan

expli itfun tionofseveralair raftandenvironmentparameters.

Expressing onditionsfordynami soaringfeasibilityasanequation thatisappli abletomost

xed-wingair raftmodelsisveryimportanttoevaluatethesuitabilityofanyair raft fordynami

soaring.

II. Models

A. WindShear

DenitionII.1. Thewindve toristhehorizontal omponentoftheairvelo ityve toratagiven

positionandtime:

w

_:=









W

x

(x, t)

W

y

(x, t)









= W (x, t)









cos ψ

W

(x, t)

sin ψ

W

(x, t)









,

(1)

where

W (x, t)

is the horizontal wind speed and

ψ

W

(x, t)

is the horizontal wind ve tordire tion relativeto theNorth(g. 1).

Windshearisanatmospheri phenomenonthato urswithinthinlayersseparatingtworegions

where the predominant airowsare dierent, either in speed, in dire tion, orin both speed and

dire tion. The air layer between these regions usually presents a onsistent gradient in the ow

N

E

S

W

ψ

_W

w

Fig.1: Wind ve tor(

w

)anddire tion(

ψ

W

).

Forsimpli ity,thisstudy onsidersonlyhorizontalow;thewindve torisreferredtoas

w

and itstotalspeedas

W

.

Thehorizontalwind shearisthevariationofthewindwithaltitude:

∂w (x, t)

∂z

6= 0.

(2)

This approa h simplies the phenomenon to uniaxial (

z

) wind ve tor variations, and uses a linearlayerwindshearmodelasdenedbySa hsanddaCosta[4℄. Themodelpresentsa onstant

windgradient,withthewindvelo ityevolvinglinearlyfromthelowertotheupperlayer:

(a)Windve toralongthewindshearlayer. (b)Winddire tion hangealongthe

windshearlayer.

w

_{(h) = w (h}

_min

_{) +}

∂w

LW S

∂h

(h − h

min

) ,

(3)

where

w

(h

min

)

isthewindvelo ityve toratthelowerboundaryaltitude,

h

min

isthelower bound-ary of the layer wind shear,

∂w

LW S

∂h

=

dW

dz

[cos ψ

dW

, sin ψ

dW

]

⊺

is a onstant wind shear verti al

gradient,

dW(x,t)

dz

istheverti alwindgradient,and

ψ

dW

(x, t)

istheverti alwindgradientdire tion (g. 2).

B. Air raft Dynami s

Thisworkusesanair raftdynami smodelthatassumesthepresen eofanautopilot ontrolling

thelowleveldynami s. Thevelo ityequationis:

~

V =

















˙x

˙y

˙z

















=

















˙x

a

˙y

a

˙z

a

















+ w = V

a

















cos ψ cos γ

a

sin ψ cos γ

a

sin γ

a

















+

















W

x

W

y

W

z

















,

(4) where

V = [ ˙x, ˙y, ˙z]

~

T

is thevelo ityve torrelativeto theground,

[ ˙x

a

, ˙y

a

, ˙z

a

]

T

is thevelo ityve tor

relativetotheair,i.e., relativetotheoweld,and

[W

x

, W

y

, W

z

]

T

isthewind velo ityve tor. All

these ve tors are expressed in the ground referen eframe, where

x

,

y

, and

z

are the northward, eastward, and downwarddire tions.

V

a

is as alar representingthe total air relative speed. The air- limbangle(

γ

a

-g.3)isdenedas

γ

a

= arctan

˙

z

V

−W

z

√

( ˙

x

V

−W

x

)

2

+( ˙

y

V

−W

y

)

2

.

Sin etheautopilotisassumedto ontrolthelowleveldynami s,theside-slipisassumedtobe

regulatedandnegligible(

β ≈ 0

). TheequationsofmotiongoverningtheUAVare(g.3):

m

















¨

x

¨

y

¨

z

















= L

















− cos φ sin γ

a

cos ψ − sin φ sin ψ

− cos φ sin γ

a

sin ψ + sin φ cos ψ

− cos γ

a

cos φ

















− D

















cos γ

a

cos ψ

cos γ

a

sin ψ

− sin γ

a

















+ mg

















0

0

1

















,

(5)

where

L

and

D

aretheaerodynami liftanddrag,

m

istheair raftmass,and

g

isthea elerationof gravity. Dierentiatingequation(4)and ombiningitwiththeequationsofmotion(5), onsidering

W

z

= 0

andsolvingfor

V

˙

a

,

ψ

˙

,and

˙γ

a

,yields,

˙

V

a

= −

D

m

− g sin γ

a

− V

a

sin γ

a

cos γ

a

cos (ψ − ψ

dW

)

dW

dz

,

(6a)

˙γ

a

=

L cos φ

mV

a

−

g cos γ

a

V

a

+ sin

2

γ

a

cos (ψ − ψ

dW

)

dW

dz

,

(6b)

˙

ψ =

L sin φ

mV

a

cos γ

a

+ tan γ

a

sin (ψ − ψ

dW

)

dW

dz

.

(6 )

Liftanddragare omputedwiththeliftand drag oe ients(

C

L

, C

D

),through

L = C

L

ρSV

a

2

/2, D = C

D

ρSV

a

2

/2,

(7)

where

ρ

is the air density and

S

is the air raft wing area. The aerodynami oe ients are as-sumedtofollowaparaboli dragpolar,i.e.,thedrag oe ientisquadrati withrespe ttothelift

oe ient

C

D

= C

D

0

+ KC

2

L

,

(8)

where

C

D

₀

is the parasiti drag oe ient and

KC

2

L

is the indu ed drag oe ient. Note that

K =



4

L

D

2

max

C

D

0



−1

, where

L

D

max

is the maximum lift over drag ratio.

L

D

max

together

with

C

D

0

arethemost importantair raftaerodynami e ien y parameters,and fullydene the

E = mgh + m

V

2

a

2

.

(9)

III. Conditionsfor Sustainable Flight

DenitionIII.1(SustainableFlight). Anitedurationighttraje toryissustainablewithrespe t

to thesafetyaltitude

h

min

and theminimumair raftenergy

E

min

ifit satisesequations(4) and (6),andsatises point-wiseintimethefollowinginequalities:

h (t) ≥ h

min

, t ∈ [t

i

, t

f

] ,

(10a)

E (t) ≥ E

min

, t ∈ [t

i

, t

f

] ,

(10b)

E

min

:= mgh

min

+ m

V

2

min

2

,

(10 )

where

t

i

and

t

f

arethe initialand naltime ofthe y le, respe tively, and

V

min

is theminimum velo ityrequiredtomaintainsustainedight,i.e., thestallspeedat

h

min

.

DenitionIII.2(FlightCy le). Aight y leisatraje toryofniteduration,satisfyingequations

(4)and(6),whereoneormoreoftheair raftstate(eq. (6) )andenergy(eq. (9))variableshavethe

samevalueattheinitial andnal times,and theirinitialandnal time-derivativeshavethesame

sign. Hen e,a y li variable

κ

inaight y lesatises:

κ (t

f

) = κ (t

i

) ,

(11a)

sgn

( ˙κ (t

f

)) = sgn ( ˙κ (t

i

)) ,

(11b)

where

t

i

and

t

f

aretheinitialandnaltimeofthe y le,respe tively.

Remark III.1. For windshear soaring y lesthe y li variablesareusually the ourse,bank, and

pit hangles.

From now on, theexe ution of a sustainableight throughdynami soaring is referredto as

just dynami soaring.

A. Su ientConditions forDynami Soaring

This work studies what the minimum verti al wind gradient (

dW

dz

) whi h enables an air raft

ofmotionandtheightisnot onstrainedtobesteady.

Todenethefeasibility onditions,theminimumwindgradient(

dW

dz

min

)issetasanexpli it

fun tion of all the other parameters (independent variables). The omputation of the minimum

windgradientfor thedomain ofthe independent variablesis omplex. Hen e,aheuristi method

wasimplemented that omputeslowerbounds in the onditions to perform dynami soaring,i.e.,

su ient onditions. Tothatend, solutionsfor y li ighttraje toriesneedto be omputed. For

dynami soaring,thesetraje toriespresentequalaltitudeandair raftenergyat thebeginningand

end. The methodimplementedto ndsu htraje toriesusesGPOPS 2[5℄, a ontrol optimization

tool,withthelinearwindgradient(eq. (3))andtheair raftmodelpresentedinse tionIIB.

Theproblem variablesareseparatedintoparametersthat enter dire tly indynami equations

andvariablesthat deneinequality onstraints. To hara terizetheaerodynami propertiesofthe

air raft,andinparti ularthedragpolar urve,maximumlifttodragratio(

L

D

max

)andminimum

drag oe ient(

C

D

0

)were hosen. Theotherparametersareair raftmass(

m

),air raftwingarea (

S

),airdensity(

ρ

),andgravitya eleration(

g

). Thevariablesthatdenetheinequality onstraints aremaneuveringlimits,su hasmaximumlift oe ient,maximumwingloading,limitbankangle,

andlimitbankangularrate.

The hara terizationoftheminimumverti alwindgradientdependen e onenvironmentaland

air raftparameterswasobtainedbyrunningmultipletraje toryoptimizations,evaluatingarangeof

valuesforseveralparameters. Theanalysisbaselineparametersare

L

D

max

and

C

D

0

(g. 5). The

variablesthatdenetheinequality onstraintsarenotasrelevant,be ausetheymaybeina tive. In

fa t,for analysis oftheoptimal traje tory,theinequality onstraintswere renderedina tive. The

other parameters,likeair raft massandgravitya eleration,donotneedto besurveyed,be ause

they are used to normalize the wind gradient through the nondimensional number

dW

dz

2 2m

gρS

, as

presentedin referen e[2℄.

Hen e, for ea h pair

L

D

max

, C

D

0

in the parameters sweep, the optimization problem is

ondi-Given

_:

 L

D



max

, C

D

0

,

(12a)

Find

_:

 dW

dz



min

=

min

C

L

(t),φ(t),ψ

dW

 dW

dz

: h (t) ≥ h (0) , ∀t



.

(12b) (12 )

Fig.4: Optimaldynami soaring"S"traje tory.

Figure 4 illustrates the optimalight y le, where theenergy gained from the windgradient

mat hes exa tlythe energylost dueto drag. Theoptimal dynami soaringtraje tory presentsan

"S" shape. The air raft starts at the lowest altitude turning towards the wind. It then limbs

against the in reasing wind. Near the top of the traje tory, the air raft approa hes stallspeed,

turnsandstartstodes endwithade reasingwind. Theair rafttheninitiatesaturntorestartthe

y le.

Figure5illustratesthevariationoftheminimumverti alwindgradientwiththeair raft

L

D

max

and

C

D

0

. Theminimumwindgradientisillustratedhereforanair raftwiththemassandwingarea

hara teristi sofaCularisUAV[6℄(

m = 2.1kg

,

S = 0.55m

2

)andfortheenvironmental onditions

g = 9.8066m/s

2

and

ρ = 1.2041kg/m

3

Fig.5: Minimumwindgradientversus

L

D

max

and

C

D

0

.

lowerboundonthe onditionsfordynami soaring.

Theorem 1. The su ient onditions for dynami soaring in a linear wind shear are met if the

aerodynami , stru tural,andmaneuveringinequality onstraintsareina tiveandif

 dW

dz



2

_2m

gρS

≥ k

1

tan

k

2

C

D

0

L

D

max

+ k

3

,

(13) where

k

1

= 48.33

,

k

2

= 0.6793

,and

k

3

= −2.66 · 10

−4

.

B. Ne essary Conditionsfor Dynami Soaring

Thene essary onditionsfordynami soaring anbedened byasimilarheuristi tothatused

for the su ient onditions. However,sin e a numeri aloptimization method is being used, one

annotproveanalyti allythattheobtainedsolutionsareinfa toptimalandrepresentthene essary

onditions. Nonetheless,it ispossibleto showthat there isaveryhigh levelof onden eon this

assertion,whi hisenoughforresultsintendedforengineeringappli ations. Tothatend,onemakes

useofthelawoflargenumbersandresultsfromaMonte-Carlosamplingoftraje tories,environment

Considertherandomvariable

X =















1, ∆h ≥ 0

0, otherwise

,

(14)

where

∆h

isthealtitudedieren ebetweenthebeginningandendofadynami soaringight y le, and

∆h < 0

meansthat theight y leis notsustainable,astheair raft islosingaltitude. Ea h randomtrial(

X

i

)isaBernoullitrial,withexpe tedvalueandvarian e:

E [X] = µ,

(15a)

var (X) = σ

2

= µ (1 − µ) .

(15b)

The laimthata ertain onditionisin fa tane essary onditionissupportedif

µ → 0

. Suppose that theMonte-Carlosampling showsthat

E





N

PX

i

N





= 0,

(16a)

var





N

PX

i

N





= 0,

(16b)

andin fa t

µ → 0

withanysamplesize

N

. Thelaw of largenumbers statesthat

P

















N

PX

i

N

− µ













≥ ǫ





≤

σ

2

N · ǫ

2

= 1 − β,

(17)

where

ǫ

isthemarginoferrorand

β

isthe onden elevel. Considerthevarian efortheworst ase s enario,assumingthatthe

N + 1

st

trialwouldyielda

X

N

+1

= 1

:

σ

2

≤

_N

1



1 −

_N

1



≈

_N

1

.

(18) As

µ → 0

:

P

















N

PX

i

N













≥ ǫ





≤

1

N

2

_{· ǫ}

2

,

(19) resultingin

1

N

2

_{· ǫ}

2

= 1 − β

⇔ N =

1

ǫ

p(1 − β)

,

(20)

whi hmeansthatforamarginoferrorof0.001anda onden elevelof98.0%,

N ≥ 7072

samples areneeded. Thatmeansthat,withahighdegreeof ertainty,nomorethanonetrialin ahundred

yields

X

N+1

= 1

.

Hypothesis III.1. The ne essary onditionsfordynami soaringin alinearwind shear areonly

metiftheaerodynami ,stru tural,andmaneuveringinequality onstraintsareina tive,andif

 dW

dz



2

_2m

gρS

≥ k

1

tan

k

2

C

D

0

L

D

max

+ k

3

,

(21) where

k

1

= 48.33

,

k

2

= 0.6231

,and

k

3

= −2.70 · 10

−4

,

L

D

max

∈ [6.6, 40]

,and

C

D

0

∈ [0.005, 0.08]

.

These onditionswereobtainedbyestablishinglowerbound onditionsforthedynami soaring.

That wasa hievedbysweepingtheair raftparameters

L

D

max

and

C

D

₀

(g. 5), similarlyto the method usedtodenethesu ient onditions.

To he k this hypothesis, a Monte-Carlo simulation was run that was set up as follows. For

ea hrun,atestdynami soaringnumber

DS

isgenerated:

DS

i

= M C

i

·

k

1

tan

k

2

C

D

0

L

D

max

+ k

3

!

, M C

i

∼ U (x) : x ∈ [0, 1) ,

(22) where

C

D

0

and

L

D

max

areuniformlysampledfromtheintervals

[0.005, 0.08]

and

[6.6, 40]

, respe -tively,and

U (x)

istheuniformdistributionovertheinterval

x ∈ [0, 1)

.

M C

i

isarandomvariable sampledfrom theuniformdistributionontheinterval

[0, 1)

,asonewantstotestforany ondition underthene essary onditions. Thedynami soaringnumberisalso:

DS

i

=

 dW

dz



2

_2m

gρS

.

(23)

m

,

g

,

ρ

,and

S

aresampledfrom uniformdistributions:

m ∼ U ([1, 30]) kg,

(24a)

g ∼ U ([9, 10]) m/s

2

,

(24b)

ρ ∼ U ([0.8, 1.4]) kg/m

3

_,

(24 )

S ∼ U ([0.5, 2]) m

2

,

(24d)

gettingthetest

dW

apoolof gure"S"shapedtraje tories. Ea hinitial traje toryisused intheGPOPS 2optimizer

[5℄to obtainamoste ienttraje toryfortheparametersobtainedfrom equations(22)and(24).

The higher the nal ight y le altitude, the moree ient atraje toryis. The initial and nal

altitudesarethen omparedto he ktherandomvariable

X

(eq. (14)),thedesiredsample. As expe ted,theresultforaMonte-Carlo samplingwith 8000samplesshowsnosampleequal

to1,meaningthatnosustainableight y lewasfoundto ontradi tthene essary onditions(24).

ExperimentalVeriedCon lusion 1. HypothesisIII.1 isveriedwith margin oferrorof 0.1%

anda onden elevel of98.4%.

IV. Con lusions

Thestudy isfo usedonsustainableair rafttraje toriesthroughwindshear withalinear

ver-ti al wind gradient. The ne essary and su ient onditions for dynami soaring are dened by

the minimumwind gradientrequired. This study omputes theminimum wind gradientrequired

to exe utesustainabledynami soaring,andanalyzeshowitdependsonseveral air raftand

envi-ronmental parameters. Theminimumwind gradientis proportionalto wingarea,airdensity, and

gravitya eleration,andisinverselyproportionaltoair raftmass. Asexpe ted,theminimumwind

gradient de reases for more e ient air raft, i.e., it grows with in reasing air raft parasiti drag

oe ient(

C

D

0

)anditde reaseswith in reasingmaximumlifttodragratio(

L

D

max

).

The su ient onditionsfor dynami soaring are omputed througha heuristi method that

optimizes the ight traje tories for dierent parameters (air raft and environment). The

ne es-sary onditionsare supported by veri ation with aMonte-Carlo sampling method. Therandom

samplingresultsinahighlevelof onden eonthene essary onditions.

V. A knowledgements

We gratefully a knowledge the support of the Aerospa e & Roboti Controls Laboratory

re-sear hersat theUniversityof Mi higan. We further a knowledgethe support ofthe AsasFgroup

partbytheUnited StatesAir For eundergrantnumberFA 8650-07-2-3744.

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