Energy-Optimized Trajectory Planning for High Altitude LongEndurance (HALE) Aircraft

Hamidreza Bolandhemmat Benjamin Thomsen and Jack Marriott1

Abstractmdash This paper outlines the energy-optimized tra-jectory planning problem for high altitude long endurance(HALE) aircraft and explores both offline and online optimiza-tion techniques to address it The goal is to find the optimalstate and input trajectories for the solar-powered airplanewith input and nonlinear state constraints which maximize netbattery and gravitational potential energy storage Solutions tothe energy-optimal trajectory planning problem using a sixdegree-of-freedom model of the nonlinear HALE aircraft dy-namics are computed using both an interior point optimizationtechnique and a bounded nonlinear simplex search algorithmThe optimal trajectories computed offline are utilized to trainan adaptive neuro-fuzzy inference system (ANFIS) which canbe implemented on the flight control computer for online in-flight trajectory planning Simulation results show up to 15more energy storage compared to a baseline parametrically-optimized trajectory

I INTRODUCTION

Solar-powered high altitude long endurance (HALE) air-craft aim to remain aloft for long periods of time and havebeen investigated for diverse applications including aerialsurveillance and rural connectivity To remain airborne forlong durations the aircraft must be able to remain abovea prescribed minimum altitude without fully discharging itsbatteries overnight This constraint may be satisfied througha combination of design optimization seasonal or geographicoperational limitations and trajectory optimization

The problem considered in this article is the determinationof dynamically-feasible state and input trajectories whichmaximize a solar-powered aircraftrsquos net energy storage (inboth batteries and gravitational potential) over a period intime We consider the case where the aircraft is constrainedwithin a convex region in three-dimensional space as wellas input (bank angle airspeed altitude rate) constraints andsubject to winds in a horizontal plane

Energy-optimal path planning for solar-powered aerialvehicles has been investigated in the literature in recent years[1] [2] In comparison to these articles however we focuson long-endurance operation and consider winds changingsolar position energy storage using gravitational potentialand real-time computation

In this article we will discuss and present the resultsfrom three methods of trajectory optimization In Section IIwe introduce the six degree-of-freedom rigid-body aircraftdynamics and control model which determines aircraft dy-namic behavior in the optimization routine In Section IIIwe formulate a nonlinear trajectory optimization problem

1The authors are with Facebook Connectivity Menlo Park CA USAhbolandhemmat benthomsen marriottfbcom

and solve it via single shooting using both an interior-pointoptimization routine and the nonlinear Nelder-Mead simplexmethod In Section IV we present a learned model (anadaptive neuro-fuzzy inference system) trained using resultsof the aforementioned optimization techniques to provide amethod capable of online trajectory planning

II AIRCRAFT DYNAMIC GUIDANCE MODEL

Trajectory optimization is carried out on a simulationmodel of a HALE aircraft with numerical integration of thedynamics This major components of this simulation modelare a nonlinear rigid-body aircraft dynamics model guid-ancecontrol model wind Earth and atmospheric modelsolar model and power model which are described presently

Fig 1 Rendering of solar HALE model used in optimization studies

A Dynamics and Guidance

The nonlinear aircraft dynamics model x = f(x u) hasstate vector

x =983045pn pe h u v w ϕ θ ψ p q r

983046T(1)

corresponding to north and east positions altitude linearvelocities in a body-fixed frame roll pitch heading anglesand angular rates For brevity the full nonlinear equationsof motion are not reproduced here but are derived in [3Ch 4ndash5] and [4] and the mathematical descriptions can befound therein The forces and moments acting on the aircraftare assumed to be defined fully by gravity aerodynamics(including actuator state) and propulsion

The aircraft which we model is assumed to have fourcontrol inputs given by

983045T δa δe δr

983046 corresponding

to thrust aileron deflection elevator deflection and rudderdeflection The relationships between these control inputsand the forces and moments acting on the 12-state rigidbody model of the aircraft can be found in [3 Ch 4] Asthe primary concern of this article is trajectory optimizationand not inner-loop control design we define the dynamicresponse to outer-loop commands using a dynamic guidance

model [3 Ch 9] The vehicle behavior is thus controlled bythree independent commands given by

uc =983045V ceas h

c ϕc983046

(2)

corresponding to equivalent airspeed altitude rate and bankangle These commands are fed into shaping filters giving

u =983045V cfeas h

cf ϕcf983046

(3)

which in turn are the inputs to representative control responsemodels which determine Veas h and ϕ in the nonlinearaircraft dynamics model Note that in reality these dynamicrelationships are governed by control design actuator dy-namics and any external effects but we approximate thisbehavior using linear dynamic models

B Wind Earth and Atmosphere

The wind triangle model described in [3] is used asfollows

Vg

983093

983095cosχ cos γsinχ cos γminus sin γ

983094

983096 = Va

983093

983095cosψ cos γasinψ cos γaminus sin γa

983094

983096+

983093

983095wn

we

wd

983094

983096 (4)

where Vg and Va are ground speed and true airspeed χ is thecourse angle ψ is the heading angle γ and γa are the inertialand air-mass-reference flight path angles respectively Windspeed is given by Vw =

983155w2

n + w2e + w2

d where wn weand wd are the wind components in the north-east-down(NED) frame Latitude and longitude (ξ λ) are computedfrom (pn pe h) using a WGS84 Earth model

Air density ρ is computed according to the 1976 USStandard Atmospheric Model [5] and assuming equivalencebetween height above sea level and aircraft altitude hRelationships between the vehiclersquos angle of attack (α)altitude (h) and aerodynamic coefficients are pre-computedusing computational fluid dynamics software A descriptionof the full aeroservoelastic model from which this datais derived can be found in [6] The lift force L actingon the vehicle in translational flight is computed from therelationship

L

cosϕminus mg

cos γ= m

V 2a

rc + h(5)

where mg is the aircraft weight and rc is Earthrsquos localradius [7] Using CL = L(qS) where q is the dynamicpressure the angle of attack (α) is solved for using the pre-computed aerodynamic data as is CD = D(qS) The sumof parallel forces about the center of gravity for the aircraftin translational flight with center line thrust is given by

T minusD minusmg sin γ = mVa (6)

which provides the value of thrust T used in the energyand power model The current article considers only the caseof trajectory optimization in the absence of winds Latitudeand longitude (ξ λ) are computed from (pn pe h) using aWGS84 Earth model

C Solar Model

A solar model is used to enable computation of solar fluxat the surface of the vehiclersquos onboard photovoltaic (PV)cells For our purposes the solar flux is defined as theamount of solar power radiated through one direction of agiven surface (without subtracting any flux in the oppositedirection) given in Wm2 Solar angles of azimuth (φs)and elevation (983171s) and Julian day (jd) are computed from(ξ λ h) and datetime per the algorithm described by Redaand Andreas [8] Solar irradiance is then adjusted to accountfor annual variation (due to eccentricity of Earthrsquos orbit) andatmospheric absorption with the formula

I = I0

9830591 + 0034 cos

2πjd365

983060f(h 983171s) (7)

where f(h 983171s) is an atmospheric absorption factor [9]Solar flux through the vehiclersquos photovoltaic cells can then

be calculated based on geometric relationships between thevehicle coordinate frame PV cells and local solar vector(magnitude and direction of the solar irradiance) For a PVcell with area As

k and unit normal vector983045xk yk zk

983046Tin

the Forward ndash Right Wing ndash Down coordinate system thesolar flux through this cell area is given by

Φsk = As

k

983045minuscφsc983171s minussφsc983171s s983171s

983046Rs

b

983093

983095xk

ykzk

983094

983096 (8)

where

Rsb =

983093

983095cψ minussψ 0sψ cψ 00 0 1

983094

983096

983093

983095cθ 0 sθ0 1 0

minussθ 0 cθ

983094

983096

983093

9830951 0 00 cϕ minussϕ0 sϕ cϕ

983094

983096

(9)using shorthand cα and sα to represent cos (α) and sin (α)Note that the airframe model we use has area-weighted PVcell normal vector at a 7 pitch angle relative to the vehiclersquosForward ndash Right Wing ndash Down coordinate frame

D Power and Energy Storage

Modeling of the flow of electric power in the aircraftmodel is done as follows Electric power input is given by

Pin = ηsΦs (10)

where ηs is the solar collection system efficiency and Φs =983123k Φ

sk is solar flux through the areas of the vehiclersquos PV

cellsPower may be stored in the vehiclersquos batteries or expended

through either the electric propulsion system or ldquoaccessoryrdquopower system (modeled with a constant power draw)

Power output is given by

Pout = Pacc + ηpTVa (11)

where Pacc is accessory system power and propulsion systemefficiency ηp is given by

ηp =2ηp0

1 +983156

TNAdq

+ 1 (12)

where Ad is propeller disk area N is the number ofpropellers and ηp0 is a constant efficiency factor whichaccounts for motor controller and motor efficiency These re-lationships allow for the computation of power requirementsto produce a given thrust at a given airspeed

The battery system is assumed to have efficiencies ηin andηout lt 10 for charging and discharging respectively as wellas a maximum energy storage capacity Energy stored in thebatteries is thus

Ebatt(t) =

983133 t

t0

λPnet + Ebatt(t0)

λ =

983099983105983103

983105983101

ηin Pnet ge 0 and Ebatt lt Emax

ηout Pnet lt 0 and Ebatt gt 0

0 otherwise

(13)

Note that it is possible to store energy as gravitationalpotential by converting electric power to thrust used toincrease the vehiclersquos altitude Kinetic energy storage is notincluded in this model due to its relatively small impactwithin aircraft velocity constraints

TABLE IVEHICLE MODEL PARAMETERS

Parameter ValueStation-keeping lat lon (ξ λ) (342 minus1185)

Battery energy density 320 WhkgBattery capacity (Emax) 64 kWhMaximum thrust (Tmax) 1000 N

ηp0 ηs ηin ηout 079 022 093 097Accessory power draw (Pacc) 360 W

Solar constant (I0) 1367 Wm2

Wing planform area 13675 m2

Wing aspect ratio 41Aircraft mass (m) 571 kg

Surface area of solar cells 1433 m2

Max NndashE plane distance (dmax) 3000 mhmin hmax 18290 m 24574 m

hmax 08 msϕmax 10

Vmin (EAS) Vmax (EAS) 71 ms 115 ms

III OPTIMIZATION FRAMEWORK

In this section the mathematical programs which we use tooptimize the HALE trajectory are described The operatingscenario which we consider is a single HALE aircraft ona station-keeping mission where the vehicle must remainwithin a certain distance from a fixed vertical axis and mustremain above a minimum altitude The trajectory optimiza-tion is applied the winter solstice for this station-keepingmission defined as the day with the lowest peak solarelevation angle In particular we define our optimization goalas the maximization of net energy storage for a time period(ti tf ) for example a twenty-four hour period starting atsunrise on winter solstice Recall that energy is stored bothwithin the batteries and as gravitational potential energy

The optimization problem is expressed mathematically as

maxuc(t)

J =983045Ebatt(tf )minus Ebatt(ti)

983046

+ κ983045Ep(tf )minus Ep(ti)

983046

s t |hc(t)| le hmax

|ϕc(t)| le ϕmax

Vmin le V ceas(t) le Vmax

hmin le h(t) le hmax983155p2n(t) + p2e(t) le dmax

0 le Ebatt(t) le Emax

0 le T (t) le Tmax

(14)

where J isin R is an energy difference cost function uc(t)is defined in Eq 2 Ebatt and Ep are energies stored aselectrical energy in the batteries and as gravitational potentialenergy respectively and κ is a scaling term on the potentialenergy

In order to solve this problem the optimization timeperiod is discretized into n equally-spaced segments Theoptimization variables are thus the magnitudes of bankattitude command ϕc[i] altitude rate command hc[i] andairspeed command V c

eas[i] at knot points i isin (1 n) Thesecomprise the (3 times n) optimization variables used for theshooting method of trajectory optimization [10] which wasselected over other trajectory optimization methods due tothe complex and nonlinear dynamics and the lack of final-state constraints The discretization of command trajectoriesis in general coarse compared to the time scales of the systemdynamics and thus piecewise polynomial interpolation isused for the command trajectories In particular unlessotherwise stated our optimization programs use a first-orderpolynomial (linear) interpolation of commands between knotpoints

The nonlinear optimization problem is constrained asdescribed in Eq 14 where it is noted that the first three con-straints comprise upper and lower bounds on the optimizationvariables while the remaining constraints are nonlinear in theoptimization variable space These nonlinear constraints arehandled via quadratic penalization in which the optimizationcost function is augmented with quadratic penalties on theviolation of these constraints The discretized optimizationproblem with the quadratic penalties becomes

maxuc[middot]

J =983045Ebatt[n]minus Ebatt[1]

983046+ κ

983045Ep[n]minus Ep[1]

983046

+ Ch

983131

i

R(hmin minus h[i])2 +R(h[i]minus hmax)2

+ Cd

983131

i

R(d[i]minus d2max)

s t |hc[middot]| le hmax

|ϕc[middot]| le ϕmax

Vmin le V ceas[middot] le Vmax

0 le Ebatt[middot] le Emax

0 le T [middot] le Tmax

(15)

where [middot] represents evaluation at all knot points i isin(1 n) R(middot) represents the unit ramp (rectifier) functiond[i] =

983155pn[i]2 + pe[i]2 and Ch and Cd are penalty weights

In Section III-A we describe a solution to this nonlinearoptimization problem via single shooting using the gradient-based interior point (IP) algorithm and present results fromthis optimization In Section III-B we describe a solution tothe same problem using the Nelder-Mead Nonlinear Simplexalgorithm and present results We then present results ontrajectory optimization in the presence of wind and discussreal-time online optimization considerations

A Interior Point Optimization

Starting with a general nonlinear optimization problem

minxisinRn

f(x)

s t ci(x) ge 0 i isin (1 m)(16)

slack variables w can be introduced to transform the inequal-ity constraints ci(x) ge 0 into an equality constraint of theform c(x) minus w = 0 The constraint on the slack variableitself can be absorbed into the cost function by defininglogarithmic barrier cost function

983123mi=1 ln (wi)

minmicro w x

f(x)minus micro

m983131

i=1

ln (wi)

s t c(x)minus w = 0

(17)

where micro is a small positive scalar Introducing Largrangemultiplier vector λ the Lagrangian function is given by

L = f(x)minus micro

m983131

i=1

ln (wi)minus λT (c(x)minus w) (18)

The first-order optimality conditions are found by settingthe derivatives of the Lagrangian with respect to x w andλ equal to zero (Karush-Kuhn-Tucker conditions)

nablaf(x)minusnablac(x)Tλ = 0

WΛeminus microe = 0

c(x)minus w = 0

(19)

Search directions dx dw and dλ can then be computedusing the Newton-Raphson method as described by983093

983095H 0 minusnablac(x)T

0 Λ Wnablac(x) minusI 0

983094

983096

983093

983095dx

dw

dλ

983094

983096 =

983093

983095minusnablaf(x) +nablac(x)Tλ

microeminusWΛeminusc(x) + w

983094

983096

(20)with

H = nabla2xxL(xwλ) = nabla2

xxf(x)minusm983131

i=1

λTnabla2xxc(x)

W =

983093

983097983095w1 0 0

0 0

0 0 wm

983094

983098983096 Λ =

983093

983097983095λ1 0 0

0 0

0 0 λm

983094

983098983096(21)

The optimization can then be carried out until successiveiterations satisfy some pre-defined convergence criteria It isnoted that due to the nonlinearity of the problem it is notpossible to guarantee global optimality of the solution

1) Trajectory Optimization with the Interior PointMethod The nonlinear trajectory optimization problem issolved using the interior point method provided by MAT-LABs commercial solver fmincon As the problem dimen-sion is large and highly nonlinear initial input trajectoriesare provided to the nonlinear solver to aid in convergenceto a feasible solution The initial input trajectory (whichis also our baseline for benchmarking purposes) producesa circular ground-track of radius dmax at altitude hminand at the equivalent airspeed Veas which minimizes dragpower This baseline is the constant-command trajectorywhich minimizes power draw while satisfying all constraintsThe simulation results presented in this article are obtainedfor the worst-case scenario corresponding to operation duringthe winter solstice

Figure 2 visualizes the results of trajectory optimizationover an hour-long period (ti = 9am tf = 10am) on thewinter solstice day The time interval is split into n = 180knot points such that the knot points are spaced at 20 secondintervals and there are 540 resulting optimization variablesIn this figure the black circle represents the bottom of acylinder in which the trajectory is constrained via quadraticpenalization In the xminusy trajectory plots the vertical axis isrepresentative of the north direction and the horizontal axisis representative of the east direction The altitude axis isalso included in the x minus y minus h plots where it is noted thatthe h-axis is not to scale with the x and y axes The color ofthe trajectory represents a relative airspeed where warmercolors indicate a lower velocity The direction of the sunin the x minus y plane is illustrated by yellow arrows (movingclockwise as the day progresses)

In this hour-long example the optimized trajectory in-creases net energy storage by approximately 12 comparedto that of the baseline trajectory (constant command tra-jectory which minimize power draw) The solver reachesconvergence in approximately 19 hours on a workstationwith parallelization on 6 CPU cores which is a baseline forcomparison with other optimizations discussed in this article

Figure 3 shows the result of trajectory optimization foranother hour-long period (ti = 2pm tf = 3pm) on thewinter solstice day This optimized trajectory improves netenergy storage by 15 compared to the baseline trajectoryIn both of these examples the energy-optimized trajectorieshave some periodic behavior combined with a rotation tofollow the sun A period of this behavior can be describedqualitatively as climbs at relatively low velocity away fromthe sun rapid turn towards the sun descent towards the sunat relatively high velocity rapid turn away from the sunThe relatively low velocity when climbing away from thesun produces a higher angle of attack and therefore highersolar flux through the area of the vehicle PV cells Whiledescending the higher velocity and lower angle of attacksimilarly tilts the solar cells towards the local solar vector

B Nonlinear Simplex Optimization

In this section we will present results of trajectory op-timizations solved using the nonlinear simplex optimization

Fig 2 Result of trajectory optimization from 900ndash1000am

algorithm developed by Nelder and Mead [11] The non-linear simplex optimization algorithm is gradient-free (it is adirect search method) and is thus effective in many practicaloptimization problems in which Jacobians and Hessians maybe computationally expensive or non-existent

The basis of the method is to estimate the objectivefunction gradient by the use of simplex Simplex S isin Rn isdefined as a convex envelope of n+1 vertices x0 xn isinRn The simplex-based search algorithm is initiated with anondegenerate simplex and the associated set of cost functionvalues at its vertices fj = f(xj) For a general minimizationproblem the goal is to decrease the cost function values atthe working simplex vertices with successive transformationsof the simplex consisting of

1) Reflection of vertex xh isin S with worst cost functionvalue (fh) to new vertex xr through the line definedby xh and the simplex centroid (c)

Fig 3 Result of trajectory optimization from 200ndash300pm

2) Depending on value fra) Expansion along line (c xr) if fr is best to new

vertex xe Use expanded simplex if fe is bestb) Contraction along line (c xr) if fr is worst or

second-worst3) Shrinkage of the simplex by moving vertices towards

best vertex xl isin S

Note that unlike the interior point method the explicitcomputations of Jacobians and Hessians are not requiredyet the transformations of the simplex are designed in sucha way that the working simplex should roughly follow thedirection of steepest descent A detailed description of theprocedure is omitted in this article but can be found in [11]and [12]

1) Trajectory Optimization with Nonlinear Simplex Fig-ure 5 illustrates the trajectory optimization for the hour-longperiod (ti = 9am tf = 10am) on the winter solstice day

Fig 4 Example of reflection and expansion steps for a 2D problem

solved using the Nelder-Mead nonlinear simplex algorithmThe optimization problem is identical to that presented inFig 2 with the difference being the use of the nonlinearsimplex method instead of the interior point method Thetime to convergence is approximately 1 hour on the sameworkstation used for interior point benchmarking (a 20timesimprovement over the interior point method) Using the non-linear simplex algorithm the optimal trajectory improves netenergy storage over the baseline trajectory by approximately7 (compared to 12 with the interior point method)

Figure 6 compares the hourly net energy storage from theoptimized and baseline trajectories Note that the baselinetrajectory in these simulations undergoes a mode switchwhen the batteries are full any excess solar power col-lection is used to increase the altitude of the vehicle tostore gravitational potential energy which is then expendedafter sunset until the vehicle reaches hmin The overallimprovement in net energy storage of the energy-optimizedtrajectory planning adds up to 105 kWh of additional energystorage over the winter solstice day (59 kWh with nonlinearsimplex) which is enough to extend the maximum night (thetime with no solar power collection) by approximately 160minutes At the latitude chosen for analysis both the baselineand optimized trajectories are able to survive the wintersolstice night however the battery state-of-charge margin issignificantly larger with the energy-optimized trajectory

C Trajectory Optimization in Wind

Winds introduce both additional complexities and opportu-nities for the energy-optimized trajectory planning problemWind is modeled according to the equations given in Eq 4and trajectories are optimized using the interior point algo-rithm provided by MATLABrsquos fmincon solver Optimiza-tions have been carried out for different wind directions andmagnitudes (given by gray vectors) up to the 99th-percentilewind magnitude provided by a wind distribution model at thetested location and altitude For the latitude longitude andaltitude of the station-keeping orbit considered in this article(see Table I) this magnitude is approximately 24 ms It isassumed in all cases that wd equiv 0 producing wind vectors inthe xminus y plane

Figure 7 portrays the trajectory optimization for (ti =9am tf = 11am) with the wind vector with magnitudeVw = 22 ms and direction 260 (the solar azimuth anglevaries from 138 to 151) The optimizer is initialized withan input trajectory which will approximately follow a circularground track at the drag-power-minimizing airspeed (as long

Fig 5 Result of trajectory optimization from 900ndash1000am using theNelder-Mead nonlinear simplex algorithm

Fig 6 Comparison between optimized and baseline trajectories forsuccessive hour-long optimization intervals

as the magnitude of the airspeed is higher than that of thewind) This corresponds to the baseline initialization usedin preceding sections with commands modified to accountfor the magnitude and direction of the wind As the aircrafttries to maintain its velocity close to the optimal airspeedit will spend the majority of its time flying in the straightpaths upwind The trajectory optimization in this exampleimproves net energy storage over the baseline trajectory byless than 1 We note that the improvement in energy storageachieved by the optimizer decreases with wind speeds whichmay be due to the increased complexity involved in solvingwith a single shooting method or the baseline trajectory mayapproach optimal behavior in high winds

Fig 7 Result of trajectory optimization in the presence of winds from900ndash1100am for Vw = 22 ms

Simulation results show that compared to the no-windcase low-to-moderate speed winds (Vw le 07Va) at an angleof more than plusmn90 from the solar azimuth will increase thenet energy storage while low-to-moderate speed winds at anangle of less than plusmn90 from the solar azimuth will decreasenet energy storage

D Real-Time Implementation Considerations

Because of the highly nonlinear nature of the trajectoryoptimization problem the solutions described above provideno formal guarantees on convergence to an optimal (or evenfeasible) solution This combined with the computationalcomplexity involved particularly with the use of the interiorpoint method prevents the use of these optimization routinesin an online setting where external disturbances (such aschanging winds) may necessitate frequent replanning Addi-tionally the computational burden and associated power drawmakes it impractical to perform the trajectory optimizationusing an onboard computer The trajectory optimizationwould need to be sent to the vehicle through communicationlinks which could impose system reliability issues fromsignal integrity and availability standpoints

One alternative is to encapsulate behavior of the trajectoryoptimization algorithm into a number of rules based onthe trajectory optimization via shooting which could thenbe implemented as a state machine on the onboard flightcontrol computer For low wind speeds these rules could besummarized at a high level as

1) With the sun above the horizon and Ebatt lt Emaxperform ldquoracetrackrdquo-shaped periodic climbs and de-scents with the major axis aligned with the sun andwith periodic airspeed variation until Ebatt = Emax

2) With the sun above the horizon and Ebatt = Emaxclimb at the maximum rate possible which will notdraw power from the batteries

3) With the sun below the horizon and h gt hmindescend (using an optimal combination of electric andpotential energy) to h = hmin

4) With the sun below the horizon and h = hminperform circular orbits at the constant drag-power-minimizing airspeed

IV MACHINE LEARNING FOR TRAJECTORY PLANNING

One particularly promising approach to ldquocompressrdquo thetrajectory optimization for use online is to take advantageof reinforcement learning techniques to learn the optimalbehavior based on a set of inputs This has the effect ofshifting the computational burden from the online planningphase to an offline training and validation phase wherecomputational power is cheap and readily available One suchmethod which we consider here is the adaptive neuro-fuzzyinference system (ANFIS) [13] [14] ANFIS is a particulartype of universal approximator which combines a multi-layerfeedforward neural network with fuzzy logic The idea is to

1) Generate a large set of trajectories using the optimiza-tion methods provided above for a variety of differentconditions

2) Train and validate the ANFIS system using this offlinetrajectory optimization data

3) Use the ANFIS system for trajectory planning onlineSix input variables are provided for the trajectory planningANFIS namely(u1) The angular difference between the solar azimuth

and aircraft course angle ∆χ = (χminus φs)(u2) The aircraft distance from the station-keeping max-

imum radius dperim = dmax minus983155p2n + p2e

(u3) The rate of change of dperim(u4) The battery state of charge (EbattEmax)(u5) Direction of the wind(u6) Magnitude of the wind

Outputs of the trajectory planning ANFIS are(y1) Command for rate of change of heading (χcmd)(y2) Command for altitude rate (hcmd)

The airspeed command is taken to be constant for trajectoryplanning using ANFIS

Using ANFIS we are able to produce a closed-loopformula to compute the desired commands which can poten-tially be implemented in-flight on the aircraft flight controlcomputer

A Trajectory Planning with ANFIS

Figure 8 demonstrates the performance of the trainedANFIS in energy-optimized trajectory planning for the three-hour period (ti = 9am tf = 12pm) on the winter solsticeday The improvement in net energy storage over the baselinetrajectory is approximately 12 which is comparable to theresults of the offline optimization routine The use of thelearned ANFIS model to compute trajectories given the in-puts provided earlier in this section results in processing timereduction of roughly four orders of magnitude compared torunning the offline optimization routines (providing outputsin seconds instead of hours)

Fig 8 Result of application of learned ANFIS model to trajectoryoptimization from 900amndash1200pm

To validate the ANFIS trajectory planner and examine itsrobustness more simulations were carried out in which initialconditions ndash including initial heading of the aircraft ndash arevaried This is to ensure that the trained ANFIS providesinvariance to these variables and that it is able to handle

flight scenarios which have not been included in its trainingdata The results of these simulations indicate that the ANFISis effective in handling a variety of initial conditions andmaximize net energy storage as it is intended to do

V CONCLUSIONS

This work investigates the problem of energy-optimizedtrajectory planning for solar-powered HALE aircraft Opti-mization via shooting methods is first employed to optimizecommand trajectories with solvers using the interior pointmethod and the nonlinear simplex method Results are pre-sented and compared for these two solvers and the resultingoptimal trajectories for the winter solstice day are discussedThe problem of trajectory optimization in the presence ofconstant winds is addressed These trajectory optimizationroutines are used to train an adaptive neuro-fuzzy modelwhich is demonstrated to produce trajectories for the station-keeping mission without wind with nearly the same energysaving benefits as the nonlinear optimization procedures butapproximately four orders of magnitude faster This providesa potential avenue for online energy-optimized trajectoryplanning which can be carried out at low computational coston an onboard flight computer

VI ACKNOWLEDGMENTS

The authors would like to thank Bruce Lin NicholasRoberts David Liu Xiangfei Meng and Martin Gomez ofFacebook for their important contributions to this work

REFERENCES

[1] A T Klesh and P T Kabamba ldquoSolar-powered aircraft Energy-optimal path planning and perpetual endurancerdquo Journal of GuidanceControl and Dynamics vol 32 no 4 pp 1320ndash1329 2009

[2] S C Spangelo and E G Gilbert ldquoPower optimization of solar-powered aircraft with specified closed ground tracksrdquo Journal ofAircraft vol 50 no 1 pp 232ndash238 2012

[3] R W Beard and T W McLain Small Unmanned Aircraft Theoryand practice Princeton University Press 2012

[4] B L Stevens F L Lewis and E N Johnson Aircraft Control andSimulation Dynamics Controls Design and Autonomous SystemsJohn Wiley amp Sons 2015

[5] United States Committee on Extension to the Standard AtmosphereUS Standard Atmosphere 1976 National Oceanic and AtmosphericAdministration 1976

[6] D Colas N Roberts and V Samuel ldquoOpen sourcingfacebookrsquos solar-powered aircraft design toolsrdquo 2018 [Online]Available httpscodefbcomconnectivityopen-sourcing-facebooks-solar-powered-aircraft-design-tools

[7] J D Anderson Introduction to Flight McGraw-Hill Education 2011[8] I Reda and A Andreas ldquoSolar position algorithm for solar radiation

applicationsrdquo Solar Energy vol 76 no 5 pp 577ndash589 2004[9] G S Aglietti S Redi A R Tatnall and T Markvart ldquoHarnessing

high-altitude solar powerrdquo IEEE Transactions on Energy Conversionvol 24 no 2 pp 442ndash451 2009

[10] J T Betts ldquoSurvey of numerical methods for trajectory optimizationrdquoJournal of Guidance Control and Dynamics vol 21 no 2 pp 193ndash207 1998

[11] J A Nelder and R Mead ldquoA simplex method for function minimiza-tionrdquo The Computer Journal vol 7 no 4 pp 308ndash313 1965

[12] J H Mathews K D Fink et al Numerical Methods Using MATLABPearson Prentice Hall 2004 vol 4

[13] S Kurnaz O Cetin and O Kaynak ldquoAdaptive neuro-fuzzy inferencesystem based autonomous flight control of unmanned air vehiclesrdquoExpert Systems with Applications vol 37 no 2 pp 1229ndash1234 2010

[14] K Tanaka An Introduction to Fuzzy Logic for Practical ApplicationsSpringer 1997

model [3 Ch 9] The vehicle behavior is thus controlled bythree independent commands given by

uc =983045V ceas h

c ϕc983046

(2)

corresponding to equivalent airspeed altitude rate and bankangle These commands are fed into shaping filters giving

u =983045V cfeas h

cf ϕcf983046

(3)

which in turn are the inputs to representative control responsemodels which determine Veas h and ϕ in the nonlinearaircraft dynamics model Note that in reality these dynamicrelationships are governed by control design actuator dy-namics and any external effects but we approximate thisbehavior using linear dynamic models

B Wind Earth and Atmosphere

The wind triangle model described in [3] is used asfollows

Vg

983093

983095cosχ cos γsinχ cos γminus sin γ

983094

983096 = Va

983093

983095cosψ cos γasinψ cos γaminus sin γa

983094

983096+

983093

983095wn

we

wd

983094

983096 (4)

where Vg and Va are ground speed and true airspeed χ is thecourse angle ψ is the heading angle γ and γa are the inertialand air-mass-reference flight path angles respectively Windspeed is given by Vw =

983155w2

n + w2e + w2

d where wn weand wd are the wind components in the north-east-down(NED) frame Latitude and longitude (ξ λ) are computedfrom (pn pe h) using a WGS84 Earth model

Air density ρ is computed according to the 1976 USStandard Atmospheric Model [5] and assuming equivalencebetween height above sea level and aircraft altitude hRelationships between the vehiclersquos angle of attack (α)altitude (h) and aerodynamic coefficients are pre-computedusing computational fluid dynamics software A descriptionof the full aeroservoelastic model from which this datais derived can be found in [6] The lift force L actingon the vehicle in translational flight is computed from therelationship

L

cosϕminus mg

cos γ= m

V 2a

rc + h(5)

where mg is the aircraft weight and rc is Earthrsquos localradius [7] Using CL = L(qS) where q is the dynamicpressure the angle of attack (α) is solved for using the pre-computed aerodynamic data as is CD = D(qS) The sumof parallel forces about the center of gravity for the aircraftin translational flight with center line thrust is given by

T minusD minusmg sin γ = mVa (6)

which provides the value of thrust T used in the energyand power model The current article considers only the caseof trajectory optimization in the absence of winds Latitudeand longitude (ξ λ) are computed from (pn pe h) using aWGS84 Earth model

C Solar Model

A solar model is used to enable computation of solar fluxat the surface of the vehiclersquos onboard photovoltaic (PV)cells For our purposes the solar flux is defined as theamount of solar power radiated through one direction of agiven surface (without subtracting any flux in the oppositedirection) given in Wm2 Solar angles of azimuth (φs)and elevation (983171s) and Julian day (jd) are computed from(ξ λ h) and datetime per the algorithm described by Redaand Andreas [8] Solar irradiance is then adjusted to accountfor annual variation (due to eccentricity of Earthrsquos orbit) andatmospheric absorption with the formula

I = I0

9830591 + 0034 cos

2πjd365

983060f(h 983171s) (7)

where f(h 983171s) is an atmospheric absorption factor [9]Solar flux through the vehiclersquos photovoltaic cells can then

be calculated based on geometric relationships between thevehicle coordinate frame PV cells and local solar vector(magnitude and direction of the solar irradiance) For a PVcell with area As

k and unit normal vector983045xk yk zk

983046Tin

the Forward ndash Right Wing ndash Down coordinate system thesolar flux through this cell area is given by

Φsk = As

k

983045minuscφsc983171s minussφsc983171s s983171s

983046Rs

b

983093

983095xk

ykzk

983094

983096 (8)

where

Rsb =

983093

983095cψ minussψ 0sψ cψ 00 0 1

983094

983096

983093

983095cθ 0 sθ0 1 0

minussθ 0 cθ

983094

983096

983093

9830951 0 00 cϕ minussϕ0 sϕ cϕ

983094

983096

(9)using shorthand cα and sα to represent cos (α) and sin (α)Note that the airframe model we use has area-weighted PVcell normal vector at a 7 pitch angle relative to the vehiclersquosForward ndash Right Wing ndash Down coordinate frame

D Power and Energy Storage

Modeling of the flow of electric power in the aircraftmodel is done as follows Electric power input is given by

Pin = ηsΦs (10)

where ηs is the solar collection system efficiency and Φs =983123k Φ

sk is solar flux through the areas of the vehiclersquos PV

cellsPower may be stored in the vehiclersquos batteries or expended

through either the electric propulsion system or ldquoaccessoryrdquopower system (modeled with a constant power draw)

Power output is given by

Pout = Pacc + ηpTVa (11)

where Pacc is accessory system power and propulsion systemefficiency ηp is given by

ηp =2ηp0

1 +983156

TNAdq

+ 1 (12)

where Ad is propeller disk area N is the number ofpropellers and ηp0 is a constant efficiency factor whichaccounts for motor controller and motor efficiency These re-lationships allow for the computation of power requirementsto produce a given thrust at a given airspeed

The battery system is assumed to have efficiencies ηin andηout lt 10 for charging and discharging respectively as wellas a maximum energy storage capacity Energy stored in thebatteries is thus

Ebatt(t) =

983133 t

t0

λPnet + Ebatt(t0)

λ =

983099983105983103

983105983101

ηin Pnet ge 0 and Ebatt lt Emax

ηout Pnet lt 0 and Ebatt gt 0

0 otherwise

(13)

Note that it is possible to store energy as gravitationalpotential by converting electric power to thrust used toincrease the vehiclersquos altitude Kinetic energy storage is notincluded in this model due to its relatively small impactwithin aircraft velocity constraints

TABLE IVEHICLE MODEL PARAMETERS

Parameter ValueStation-keeping lat lon (ξ λ) (342 minus1185)

Battery energy density 320 WhkgBattery capacity (Emax) 64 kWhMaximum thrust (Tmax) 1000 N

ηp0 ηs ηin ηout 079 022 093 097Accessory power draw (Pacc) 360 W

Solar constant (I0) 1367 Wm2

Wing planform area 13675 m2

Wing aspect ratio 41Aircraft mass (m) 571 kg

Surface area of solar cells 1433 m2

Max NndashE plane distance (dmax) 3000 mhmin hmax 18290 m 24574 m

hmax 08 msϕmax 10

Vmin (EAS) Vmax (EAS) 71 ms 115 ms

III OPTIMIZATION FRAMEWORK

In this section the mathematical programs which we use tooptimize the HALE trajectory are described The operatingscenario which we consider is a single HALE aircraft ona station-keeping mission where the vehicle must remainwithin a certain distance from a fixed vertical axis and mustremain above a minimum altitude The trajectory optimiza-tion is applied the winter solstice for this station-keepingmission defined as the day with the lowest peak solarelevation angle In particular we define our optimization goalas the maximization of net energy storage for a time period(ti tf ) for example a twenty-four hour period starting atsunrise on winter solstice Recall that energy is stored bothwithin the batteries and as gravitational potential energy

The optimization problem is expressed mathematically as

maxuc(t)

J =983045Ebatt(tf )minus Ebatt(ti)

983046

+ κ983045Ep(tf )minus Ep(ti)

983046

s t |hc(t)| le hmax

|ϕc(t)| le ϕmax

Vmin le V ceas(t) le Vmax

hmin le h(t) le hmax983155p2n(t) + p2e(t) le dmax

0 le Ebatt(t) le Emax

0 le T (t) le Tmax

(14)

where J isin R is an energy difference cost function uc(t)is defined in Eq 2 Ebatt and Ep are energies stored aselectrical energy in the batteries and as gravitational potentialenergy respectively and κ is a scaling term on the potentialenergy

In order to solve this problem the optimization timeperiod is discretized into n equally-spaced segments Theoptimization variables are thus the magnitudes of bankattitude command ϕc[i] altitude rate command hc[i] andairspeed command V c

eas[i] at knot points i isin (1 n) Thesecomprise the (3 times n) optimization variables used for theshooting method of trajectory optimization [10] which wasselected over other trajectory optimization methods due tothe complex and nonlinear dynamics and the lack of final-state constraints The discretization of command trajectoriesis in general coarse compared to the time scales of the systemdynamics and thus piecewise polynomial interpolation isused for the command trajectories In particular unlessotherwise stated our optimization programs use a first-orderpolynomial (linear) interpolation of commands between knotpoints

The nonlinear optimization problem is constrained asdescribed in Eq 14 where it is noted that the first three con-straints comprise upper and lower bounds on the optimizationvariables while the remaining constraints are nonlinear in theoptimization variable space These nonlinear constraints arehandled via quadratic penalization in which the optimizationcost function is augmented with quadratic penalties on theviolation of these constraints The discretized optimizationproblem with the quadratic penalties becomes

maxuc[middot]

J =983045Ebatt[n]minus Ebatt[1]

983046+ κ

983045Ep[n]minus Ep[1]

983046

+ Ch

983131

i

R(hmin minus h[i])2 +R(h[i]minus hmax)2

+ Cd

983131

i

R(d[i]minus d2max)

s t |hc[middot]| le hmax

|ϕc[middot]| le ϕmax

Vmin le V ceas[middot] le Vmax

0 le Ebatt[middot] le Emax

0 le T [middot] le Tmax

(15)

where [middot] represents evaluation at all knot points i isin(1 n) R(middot) represents the unit ramp (rectifier) functiond[i] =

983155pn[i]2 + pe[i]2 and Ch and Cd are penalty weights

In Section III-A we describe a solution to this nonlinearoptimization problem via single shooting using the gradient-based interior point (IP) algorithm and present results fromthis optimization In Section III-B we describe a solution tothe same problem using the Nelder-Mead Nonlinear Simplexalgorithm and present results We then present results ontrajectory optimization in the presence of wind and discussreal-time online optimization considerations

A Interior Point Optimization

Starting with a general nonlinear optimization problem

minxisinRn

f(x)

s t ci(x) ge 0 i isin (1 m)(16)

slack variables w can be introduced to transform the inequal-ity constraints ci(x) ge 0 into an equality constraint of theform c(x) minus w = 0 The constraint on the slack variableitself can be absorbed into the cost function by defininglogarithmic barrier cost function

983123mi=1 ln (wi)

minmicro w x

f(x)minus micro

m983131

i=1

ln (wi)

s t c(x)minus w = 0

(17)

where micro is a small positive scalar Introducing Largrangemultiplier vector λ the Lagrangian function is given by

L = f(x)minus micro

m983131

i=1

ln (wi)minus λT (c(x)minus w) (18)

The first-order optimality conditions are found by settingthe derivatives of the Lagrangian with respect to x w andλ equal to zero (Karush-Kuhn-Tucker conditions)

nablaf(x)minusnablac(x)Tλ = 0

WΛeminus microe = 0

c(x)minus w = 0

(19)

Search directions dx dw and dλ can then be computedusing the Newton-Raphson method as described by983093

983095H 0 minusnablac(x)T

0 Λ Wnablac(x) minusI 0

983094

983096

983093

983095dx

dw

dλ

983094

983096 =

983093

983095minusnablaf(x) +nablac(x)Tλ

microeminusWΛeminusc(x) + w

983094

983096

(20)with

H = nabla2xxL(xwλ) = nabla2

xxf(x)minusm983131

i=1

λTnabla2xxc(x)

W =

983093

983097983095w1 0 0

0 0

0 0 wm

983094

983098983096 Λ =

983093

983097983095λ1 0 0

0 0

0 0 λm

983094

983098983096(21)

The optimization can then be carried out until successiveiterations satisfy some pre-defined convergence criteria It isnoted that due to the nonlinearity of the problem it is notpossible to guarantee global optimality of the solution

1) Trajectory Optimization with the Interior PointMethod The nonlinear trajectory optimization problem issolved using the interior point method provided by MAT-LABs commercial solver fmincon As the problem dimen-sion is large and highly nonlinear initial input trajectoriesare provided to the nonlinear solver to aid in convergenceto a feasible solution The initial input trajectory (whichis also our baseline for benchmarking purposes) producesa circular ground-track of radius dmax at altitude hminand at the equivalent airspeed Veas which minimizes dragpower This baseline is the constant-command trajectorywhich minimizes power draw while satisfying all constraintsThe simulation results presented in this article are obtainedfor the worst-case scenario corresponding to operation duringthe winter solstice

Figure 2 visualizes the results of trajectory optimizationover an hour-long period (ti = 9am tf = 10am) on thewinter solstice day The time interval is split into n = 180knot points such that the knot points are spaced at 20 secondintervals and there are 540 resulting optimization variablesIn this figure the black circle represents the bottom of acylinder in which the trajectory is constrained via quadraticpenalization In the xminusy trajectory plots the vertical axis isrepresentative of the north direction and the horizontal axisis representative of the east direction The altitude axis isalso included in the x minus y minus h plots where it is noted thatthe h-axis is not to scale with the x and y axes The color ofthe trajectory represents a relative airspeed where warmercolors indicate a lower velocity The direction of the sunin the x minus y plane is illustrated by yellow arrows (movingclockwise as the day progresses)

In this hour-long example the optimized trajectory in-creases net energy storage by approximately 12 comparedto that of the baseline trajectory (constant command tra-jectory which minimize power draw) The solver reachesconvergence in approximately 19 hours on a workstationwith parallelization on 6 CPU cores which is a baseline forcomparison with other optimizations discussed in this article

Figure 3 shows the result of trajectory optimization foranother hour-long period (ti = 2pm tf = 3pm) on thewinter solstice day This optimized trajectory improves netenergy storage by 15 compared to the baseline trajectoryIn both of these examples the energy-optimized trajectorieshave some periodic behavior combined with a rotation tofollow the sun A period of this behavior can be describedqualitatively as climbs at relatively low velocity away fromthe sun rapid turn towards the sun descent towards the sunat relatively high velocity rapid turn away from the sunThe relatively low velocity when climbing away from thesun produces a higher angle of attack and therefore highersolar flux through the area of the vehicle PV cells Whiledescending the higher velocity and lower angle of attacksimilarly tilts the solar cells towards the local solar vector

B Nonlinear Simplex Optimization

In this section we will present results of trajectory op-timizations solved using the nonlinear simplex optimization

Fig 2 Result of trajectory optimization from 900ndash1000am

algorithm developed by Nelder and Mead [11] The non-linear simplex optimization algorithm is gradient-free (it is adirect search method) and is thus effective in many practicaloptimization problems in which Jacobians and Hessians maybe computationally expensive or non-existent

The basis of the method is to estimate the objectivefunction gradient by the use of simplex Simplex S isin Rn isdefined as a convex envelope of n+1 vertices x0 xn isinRn The simplex-based search algorithm is initiated with anondegenerate simplex and the associated set of cost functionvalues at its vertices fj = f(xj) For a general minimizationproblem the goal is to decrease the cost function values atthe working simplex vertices with successive transformationsof the simplex consisting of

1) Reflection of vertex xh isin S with worst cost functionvalue (fh) to new vertex xr through the line definedby xh and the simplex centroid (c)

Fig 3 Result of trajectory optimization from 200ndash300pm

2) Depending on value fra) Expansion along line (c xr) if fr is best to new

vertex xe Use expanded simplex if fe is bestb) Contraction along line (c xr) if fr is worst or

second-worst3) Shrinkage of the simplex by moving vertices towards

best vertex xl isin S

Note that unlike the interior point method the explicitcomputations of Jacobians and Hessians are not requiredyet the transformations of the simplex are designed in sucha way that the working simplex should roughly follow thedirection of steepest descent A detailed description of theprocedure is omitted in this article but can be found in [11]and [12]

1) Trajectory Optimization with Nonlinear Simplex Fig-ure 5 illustrates the trajectory optimization for the hour-longperiod (ti = 9am tf = 10am) on the winter solstice day

Fig 4 Example of reflection and expansion steps for a 2D problem

solved using the Nelder-Mead nonlinear simplex algorithmThe optimization problem is identical to that presented inFig 2 with the difference being the use of the nonlinearsimplex method instead of the interior point method Thetime to convergence is approximately 1 hour on the sameworkstation used for interior point benchmarking (a 20timesimprovement over the interior point method) Using the non-linear simplex algorithm the optimal trajectory improves netenergy storage over the baseline trajectory by approximately7 (compared to 12 with the interior point method)

Figure 6 compares the hourly net energy storage from theoptimized and baseline trajectories Note that the baselinetrajectory in these simulations undergoes a mode switchwhen the batteries are full any excess solar power col-lection is used to increase the altitude of the vehicle tostore gravitational potential energy which is then expendedafter sunset until the vehicle reaches hmin The overallimprovement in net energy storage of the energy-optimizedtrajectory planning adds up to 105 kWh of additional energystorage over the winter solstice day (59 kWh with nonlinearsimplex) which is enough to extend the maximum night (thetime with no solar power collection) by approximately 160minutes At the latitude chosen for analysis both the baselineand optimized trajectories are able to survive the wintersolstice night however the battery state-of-charge margin issignificantly larger with the energy-optimized trajectory

C Trajectory Optimization in Wind

Winds introduce both additional complexities and opportu-nities for the energy-optimized trajectory planning problemWind is modeled according to the equations given in Eq 4and trajectories are optimized using the interior point algo-rithm provided by MATLABrsquos fmincon solver Optimiza-tions have been carried out for different wind directions andmagnitudes (given by gray vectors) up to the 99th-percentilewind magnitude provided by a wind distribution model at thetested location and altitude For the latitude longitude andaltitude of the station-keeping orbit considered in this article(see Table I) this magnitude is approximately 24 ms It isassumed in all cases that wd equiv 0 producing wind vectors inthe xminus y plane

Figure 7 portrays the trajectory optimization for (ti =9am tf = 11am) with the wind vector with magnitudeVw = 22 ms and direction 260 (the solar azimuth anglevaries from 138 to 151) The optimizer is initialized withan input trajectory which will approximately follow a circularground track at the drag-power-minimizing airspeed (as long

Fig 5 Result of trajectory optimization from 900ndash1000am using theNelder-Mead nonlinear simplex algorithm

Fig 6 Comparison between optimized and baseline trajectories forsuccessive hour-long optimization intervals

as the magnitude of the airspeed is higher than that of thewind) This corresponds to the baseline initialization usedin preceding sections with commands modified to accountfor the magnitude and direction of the wind As the aircrafttries to maintain its velocity close to the optimal airspeedit will spend the majority of its time flying in the straightpaths upwind The trajectory optimization in this exampleimproves net energy storage over the baseline trajectory byless than 1 We note that the improvement in energy storageachieved by the optimizer decreases with wind speeds whichmay be due to the increased complexity involved in solvingwith a single shooting method or the baseline trajectory mayapproach optimal behavior in high winds

Fig 7 Result of trajectory optimization in the presence of winds from900ndash1100am for Vw = 22 ms

Simulation results show that compared to the no-windcase low-to-moderate speed winds (Vw le 07Va) at an angleof more than plusmn90 from the solar azimuth will increase thenet energy storage while low-to-moderate speed winds at anangle of less than plusmn90 from the solar azimuth will decreasenet energy storage

D Real-Time Implementation Considerations

Because of the highly nonlinear nature of the trajectoryoptimization problem the solutions described above provideno formal guarantees on convergence to an optimal (or evenfeasible) solution This combined with the computationalcomplexity involved particularly with the use of the interiorpoint method prevents the use of these optimization routinesin an online setting where external disturbances (such aschanging winds) may necessitate frequent replanning Addi-tionally the computational burden and associated power drawmakes it impractical to perform the trajectory optimizationusing an onboard computer The trajectory optimizationwould need to be sent to the vehicle through communicationlinks which could impose system reliability issues fromsignal integrity and availability standpoints

One alternative is to encapsulate behavior of the trajectoryoptimization algorithm into a number of rules based onthe trajectory optimization via shooting which could thenbe implemented as a state machine on the onboard flightcontrol computer For low wind speeds these rules could besummarized at a high level as

1) With the sun above the horizon and Ebatt lt Emaxperform ldquoracetrackrdquo-shaped periodic climbs and de-scents with the major axis aligned with the sun andwith periodic airspeed variation until Ebatt = Emax

2) With the sun above the horizon and Ebatt = Emaxclimb at the maximum rate possible which will notdraw power from the batteries

3) With the sun below the horizon and h gt hmindescend (using an optimal combination of electric andpotential energy) to h = hmin

4) With the sun below the horizon and h = hminperform circular orbits at the constant drag-power-minimizing airspeed

IV MACHINE LEARNING FOR TRAJECTORY PLANNING

One particularly promising approach to ldquocompressrdquo thetrajectory optimization for use online is to take advantageof reinforcement learning techniques to learn the optimalbehavior based on a set of inputs This has the effect ofshifting the computational burden from the online planningphase to an offline training and validation phase wherecomputational power is cheap and readily available One suchmethod which we consider here is the adaptive neuro-fuzzyinference system (ANFIS) [13] [14] ANFIS is a particulartype of universal approximator which combines a multi-layerfeedforward neural network with fuzzy logic The idea is to

1) Generate a large set of trajectories using the optimiza-tion methods provided above for a variety of differentconditions

2) Train and validate the ANFIS system using this offlinetrajectory optimization data

3) Use the ANFIS system for trajectory planning onlineSix input variables are provided for the trajectory planningANFIS namely(u1) The angular difference between the solar azimuth

and aircraft course angle ∆χ = (χminus φs)(u2) The aircraft distance from the station-keeping max-

imum radius dperim = dmax minus983155p2n + p2e

(u3) The rate of change of dperim(u4) The battery state of charge (EbattEmax)(u5) Direction of the wind(u6) Magnitude of the wind

Outputs of the trajectory planning ANFIS are(y1) Command for rate of change of heading (χcmd)(y2) Command for altitude rate (hcmd)

The airspeed command is taken to be constant for trajectoryplanning using ANFIS

Using ANFIS we are able to produce a closed-loopformula to compute the desired commands which can poten-tially be implemented in-flight on the aircraft flight controlcomputer

A Trajectory Planning with ANFIS

Figure 8 demonstrates the performance of the trainedANFIS in energy-optimized trajectory planning for the three-hour period (ti = 9am tf = 12pm) on the winter solsticeday The improvement in net energy storage over the baselinetrajectory is approximately 12 which is comparable to theresults of the offline optimization routine The use of thelearned ANFIS model to compute trajectories given the in-puts provided earlier in this section results in processing timereduction of roughly four orders of magnitude compared torunning the offline optimization routines (providing outputsin seconds instead of hours)

Fig 8 Result of application of learned ANFIS model to trajectoryoptimization from 900amndash1200pm

To validate the ANFIS trajectory planner and examine itsrobustness more simulations were carried out in which initialconditions ndash including initial heading of the aircraft ndash arevaried This is to ensure that the trained ANFIS providesinvariance to these variables and that it is able to handle

flight scenarios which have not been included in its trainingdata The results of these simulations indicate that the ANFISis effective in handling a variety of initial conditions andmaximize net energy storage as it is intended to do

V CONCLUSIONS

This work investigates the problem of energy-optimizedtrajectory planning for solar-powered HALE aircraft Opti-mization via shooting methods is first employed to optimizecommand trajectories with solvers using the interior pointmethod and the nonlinear simplex method Results are pre-sented and compared for these two solvers and the resultingoptimal trajectories for the winter solstice day are discussedThe problem of trajectory optimization in the presence ofconstant winds is addressed These trajectory optimizationroutines are used to train an adaptive neuro-fuzzy modelwhich is demonstrated to produce trajectories for the station-keeping mission without wind with nearly the same energysaving benefits as the nonlinear optimization procedures butapproximately four orders of magnitude faster This providesa potential avenue for online energy-optimized trajectoryplanning which can be carried out at low computational coston an onboard flight computer

VI ACKNOWLEDGMENTS

The authors would like to thank Bruce Lin NicholasRoberts David Liu Xiangfei Meng and Martin Gomez ofFacebook for their important contributions to this work

REFERENCES

[1] A T Klesh and P T Kabamba ldquoSolar-powered aircraft Energy-optimal path planning and perpetual endurancerdquo Journal of GuidanceControl and Dynamics vol 32 no 4 pp 1320ndash1329 2009

[2] S C Spangelo and E G Gilbert ldquoPower optimization of solar-powered aircraft with specified closed ground tracksrdquo Journal ofAircraft vol 50 no 1 pp 232ndash238 2012

[3] R W Beard and T W McLain Small Unmanned Aircraft Theoryand practice Princeton University Press 2012

[4] B L Stevens F L Lewis and E N Johnson Aircraft Control andSimulation Dynamics Controls Design and Autonomous SystemsJohn Wiley amp Sons 2015

[5] United States Committee on Extension to the Standard AtmosphereUS Standard Atmosphere 1976 National Oceanic and AtmosphericAdministration 1976

[6] D Colas N Roberts and V Samuel ldquoOpen sourcingfacebookrsquos solar-powered aircraft design toolsrdquo 2018 [Online]Available httpscodefbcomconnectivityopen-sourcing-facebooks-solar-powered-aircraft-design-tools

[7] J D Anderson Introduction to Flight McGraw-Hill Education 2011[8] I Reda and A Andreas ldquoSolar position algorithm for solar radiation

applicationsrdquo Solar Energy vol 76 no 5 pp 577ndash589 2004[9] G S Aglietti S Redi A R Tatnall and T Markvart ldquoHarnessing

high-altitude solar powerrdquo IEEE Transactions on Energy Conversionvol 24 no 2 pp 442ndash451 2009

[10] J T Betts ldquoSurvey of numerical methods for trajectory optimizationrdquoJournal of Guidance Control and Dynamics vol 21 no 2 pp 193ndash207 1998

[11] J A Nelder and R Mead ldquoA simplex method for function minimiza-tionrdquo The Computer Journal vol 7 no 4 pp 308ndash313 1965

[12] J H Mathews K D Fink et al Numerical Methods Using MATLABPearson Prentice Hall 2004 vol 4

[13] S Kurnaz O Cetin and O Kaynak ldquoAdaptive neuro-fuzzy inferencesystem based autonomous flight control of unmanned air vehiclesrdquoExpert Systems with Applications vol 37 no 2 pp 1229ndash1234 2010

[14] K Tanaka An Introduction to Fuzzy Logic for Practical ApplicationsSpringer 1997

where Ad is propeller disk area N is the number ofpropellers and ηp0 is a constant efficiency factor whichaccounts for motor controller and motor efficiency These re-lationships allow for the computation of power requirementsto produce a given thrust at a given airspeed

The battery system is assumed to have efficiencies ηin andηout lt 10 for charging and discharging respectively as wellas a maximum energy storage capacity Energy stored in thebatteries is thus

Ebatt(t) =

983133 t

t0

λPnet + Ebatt(t0)

λ =

983099983105983103

983105983101

ηin Pnet ge 0 and Ebatt lt Emax

ηout Pnet lt 0 and Ebatt gt 0

0 otherwise

(13)

Note that it is possible to store energy as gravitationalpotential by converting electric power to thrust used toincrease the vehiclersquos altitude Kinetic energy storage is notincluded in this model due to its relatively small impactwithin aircraft velocity constraints

TABLE IVEHICLE MODEL PARAMETERS

Parameter ValueStation-keeping lat lon (ξ λ) (342 minus1185)

Battery energy density 320 WhkgBattery capacity (Emax) 64 kWhMaximum thrust (Tmax) 1000 N

ηp0 ηs ηin ηout 079 022 093 097Accessory power draw (Pacc) 360 W

Solar constant (I0) 1367 Wm2

Wing planform area 13675 m2

Wing aspect ratio 41Aircraft mass (m) 571 kg

Surface area of solar cells 1433 m2

Max NndashE plane distance (dmax) 3000 mhmin hmax 18290 m 24574 m

hmax 08 msϕmax 10

Vmin (EAS) Vmax (EAS) 71 ms 115 ms

III OPTIMIZATION FRAMEWORK

In this section the mathematical programs which we use tooptimize the HALE trajectory are described The operatingscenario which we consider is a single HALE aircraft ona station-keeping mission where the vehicle must remainwithin a certain distance from a fixed vertical axis and mustremain above a minimum altitude The trajectory optimiza-tion is applied the winter solstice for this station-keepingmission defined as the day with the lowest peak solarelevation angle In particular we define our optimization goalas the maximization of net energy storage for a time period(ti tf ) for example a twenty-four hour period starting atsunrise on winter solstice Recall that energy is stored bothwithin the batteries and as gravitational potential energy

The optimization problem is expressed mathematically as

maxuc(t)

J =983045Ebatt(tf )minus Ebatt(ti)

983046

+ κ983045Ep(tf )minus Ep(ti)

983046

s t |hc(t)| le hmax

|ϕc(t)| le ϕmax

Vmin le V ceas(t) le Vmax

hmin le h(t) le hmax983155p2n(t) + p2e(t) le dmax

0 le Ebatt(t) le Emax

0 le T (t) le Tmax

(14)

where J isin R is an energy difference cost function uc(t)is defined in Eq 2 Ebatt and Ep are energies stored aselectrical energy in the batteries and as gravitational potentialenergy respectively and κ is a scaling term on the potentialenergy

In order to solve this problem the optimization timeperiod is discretized into n equally-spaced segments Theoptimization variables are thus the magnitudes of bankattitude command ϕc[i] altitude rate command hc[i] andairspeed command V c

eas[i] at knot points i isin (1 n) Thesecomprise the (3 times n) optimization variables used for theshooting method of trajectory optimization [10] which wasselected over other trajectory optimization methods due tothe complex and nonlinear dynamics and the lack of final-state constraints The discretization of command trajectoriesis in general coarse compared to the time scales of the systemdynamics and thus piecewise polynomial interpolation isused for the command trajectories In particular unlessotherwise stated our optimization programs use a first-orderpolynomial (linear) interpolation of commands between knotpoints

The nonlinear optimization problem is constrained asdescribed in Eq 14 where it is noted that the first three con-straints comprise upper and lower bounds on the optimizationvariables while the remaining constraints are nonlinear in theoptimization variable space These nonlinear constraints arehandled via quadratic penalization in which the optimizationcost function is augmented with quadratic penalties on theviolation of these constraints The discretized optimizationproblem with the quadratic penalties becomes

maxuc[middot]

J =983045Ebatt[n]minus Ebatt[1]

983046+ κ

983045Ep[n]minus Ep[1]

983046

+ Ch

983131

i

R(hmin minus h[i])2 +R(h[i]minus hmax)2

+ Cd

983131

i

R(d[i]minus d2max)

s t |hc[middot]| le hmax

|ϕc[middot]| le ϕmax

Vmin le V ceas[middot] le Vmax

0 le Ebatt[middot] le Emax

0 le T [middot] le Tmax

(15)

where [middot] represents evaluation at all knot points i isin(1 n) R(middot) represents the unit ramp (rectifier) functiond[i] =

983155pn[i]2 + pe[i]2 and Ch and Cd are penalty weights

In Section III-A we describe a solution to this nonlinearoptimization problem via single shooting using the gradient-based interior point (IP) algorithm and present results fromthis optimization In Section III-B we describe a solution tothe same problem using the Nelder-Mead Nonlinear Simplexalgorithm and present results We then present results ontrajectory optimization in the presence of wind and discussreal-time online optimization considerations

A Interior Point Optimization

Starting with a general nonlinear optimization problem

minxisinRn

f(x)

s t ci(x) ge 0 i isin (1 m)(16)

slack variables w can be introduced to transform the inequal-ity constraints ci(x) ge 0 into an equality constraint of theform c(x) minus w = 0 The constraint on the slack variableitself can be absorbed into the cost function by defininglogarithmic barrier cost function

983123mi=1 ln (wi)

minmicro w x

f(x)minus micro

m983131

i=1

ln (wi)

s t c(x)minus w = 0

(17)

where micro is a small positive scalar Introducing Largrangemultiplier vector λ the Lagrangian function is given by

L = f(x)minus micro

m983131

i=1

ln (wi)minus λT (c(x)minus w) (18)

The first-order optimality conditions are found by settingthe derivatives of the Lagrangian with respect to x w andλ equal to zero (Karush-Kuhn-Tucker conditions)

nablaf(x)minusnablac(x)Tλ = 0

WΛeminus microe = 0

c(x)minus w = 0

(19)

Search directions dx dw and dλ can then be computedusing the Newton-Raphson method as described by983093

983095H 0 minusnablac(x)T

0 Λ Wnablac(x) minusI 0

983094

983096

983093

983095dx

dw

dλ

983094

983096 =

983093

983095minusnablaf(x) +nablac(x)Tλ

microeminusWΛeminusc(x) + w

983094

983096

(20)with

H = nabla2xxL(xwλ) = nabla2

xxf(x)minusm983131

i=1

λTnabla2xxc(x)

W =

983093

983097983095w1 0 0

0 0

0 0 wm

983094

983098983096 Λ =

983093

983097983095λ1 0 0

0 0

0 0 λm

983094

983098983096(21)

The optimization can then be carried out until successiveiterations satisfy some pre-defined convergence criteria It isnoted that due to the nonlinearity of the problem it is notpossible to guarantee global optimality of the solution

1) Trajectory Optimization with the Interior PointMethod The nonlinear trajectory optimization problem issolved using the interior point method provided by MAT-LABs commercial solver fmincon As the problem dimen-sion is large and highly nonlinear initial input trajectoriesare provided to the nonlinear solver to aid in convergenceto a feasible solution The initial input trajectory (whichis also our baseline for benchmarking purposes) producesa circular ground-track of radius dmax at altitude hminand at the equivalent airspeed Veas which minimizes dragpower This baseline is the constant-command trajectorywhich minimizes power draw while satisfying all constraintsThe simulation results presented in this article are obtainedfor the worst-case scenario corresponding to operation duringthe winter solstice

Figure 2 visualizes the results of trajectory optimizationover an hour-long period (ti = 9am tf = 10am) on thewinter solstice day The time interval is split into n = 180knot points such that the knot points are spaced at 20 secondintervals and there are 540 resulting optimization variablesIn this figure the black circle represents the bottom of acylinder in which the trajectory is constrained via quadraticpenalization In the xminusy trajectory plots the vertical axis isrepresentative of the north direction and the horizontal axisis representative of the east direction The altitude axis isalso included in the x minus y minus h plots where it is noted thatthe h-axis is not to scale with the x and y axes The color ofthe trajectory represents a relative airspeed where warmercolors indicate a lower velocity The direction of the sunin the x minus y plane is illustrated by yellow arrows (movingclockwise as the day progresses)

In this hour-long example the optimized trajectory in-creases net energy storage by approximately 12 comparedto that of the baseline trajectory (constant command tra-jectory which minimize power draw) The solver reachesconvergence in approximately 19 hours on a workstationwith parallelization on 6 CPU cores which is a baseline forcomparison with other optimizations discussed in this article

Figure 3 shows the result of trajectory optimization foranother hour-long period (ti = 2pm tf = 3pm) on thewinter solstice day This optimized trajectory improves netenergy storage by 15 compared to the baseline trajectoryIn both of these examples the energy-optimized trajectorieshave some periodic behavior combined with a rotation tofollow the sun A period of this behavior can be describedqualitatively as climbs at relatively low velocity away fromthe sun rapid turn towards the sun descent towards the sunat relatively high velocity rapid turn away from the sunThe relatively low velocity when climbing away from thesun produces a higher angle of attack and therefore highersolar flux through the area of the vehicle PV cells Whiledescending the higher velocity and lower angle of attacksimilarly tilts the solar cells towards the local solar vector

B Nonlinear Simplex Optimization

In this section we will present results of trajectory op-timizations solved using the nonlinear simplex optimization

Fig 2 Result of trajectory optimization from 900ndash1000am

algorithm developed by Nelder and Mead [11] The non-linear simplex optimization algorithm is gradient-free (it is adirect search method) and is thus effective in many practicaloptimization problems in which Jacobians and Hessians maybe computationally expensive or non-existent

The basis of the method is to estimate the objectivefunction gradient by the use of simplex Simplex S isin Rn isdefined as a convex envelope of n+1 vertices x0 xn isinRn The simplex-based search algorithm is initiated with anondegenerate simplex and the associated set of cost functionvalues at its vertices fj = f(xj) For a general minimizationproblem the goal is to decrease the cost function values atthe working simplex vertices with successive transformationsof the simplex consisting of

1) Reflection of vertex xh isin S with worst cost functionvalue (fh) to new vertex xr through the line definedby xh and the simplex centroid (c)

Fig 3 Result of trajectory optimization from 200ndash300pm

2) Depending on value fra) Expansion along line (c xr) if fr is best to new

vertex xe Use expanded simplex if fe is bestb) Contraction along line (c xr) if fr is worst or

second-worst3) Shrinkage of the simplex by moving vertices towards

best vertex xl isin S

Note that unlike the interior point method the explicitcomputations of Jacobians and Hessians are not requiredyet the transformations of the simplex are designed in sucha way that the working simplex should roughly follow thedirection of steepest descent A detailed description of theprocedure is omitted in this article but can be found in [11]and [12]

1) Trajectory Optimization with Nonlinear Simplex Fig-ure 5 illustrates the trajectory optimization for the hour-longperiod (ti = 9am tf = 10am) on the winter solstice day

Fig 4 Example of reflection and expansion steps for a 2D problem

solved using the Nelder-Mead nonlinear simplex algorithmThe optimization problem is identical to that presented inFig 2 with the difference being the use of the nonlinearsimplex method instead of the interior point method Thetime to convergence is approximately 1 hour on the sameworkstation used for interior point benchmarking (a 20timesimprovement over the interior point method) Using the non-linear simplex algorithm the optimal trajectory improves netenergy storage over the baseline trajectory by approximately7 (compared to 12 with the interior point method)

Figure 6 compares the hourly net energy storage from theoptimized and baseline trajectories Note that the baselinetrajectory in these simulations undergoes a mode switchwhen the batteries are full any excess solar power col-lection is used to increase the altitude of the vehicle tostore gravitational potential energy which is then expendedafter sunset until the vehicle reaches hmin The overallimprovement in net energy storage of the energy-optimizedtrajectory planning adds up to 105 kWh of additional energystorage over the winter solstice day (59 kWh with nonlinearsimplex) which is enough to extend the maximum night (thetime with no solar power collection) by approximately 160minutes At the latitude chosen for analysis both the baselineand optimized trajectories are able to survive the wintersolstice night however the battery state-of-charge margin issignificantly larger with the energy-optimized trajectory

C Trajectory Optimization in Wind

Winds introduce both additional complexities and opportu-nities for the energy-optimized trajectory planning problemWind is modeled according to the equations given in Eq 4and trajectories are optimized using the interior point algo-rithm provided by MATLABrsquos fmincon solver Optimiza-tions have been carried out for different wind directions andmagnitudes (given by gray vectors) up to the 99th-percentilewind magnitude provided by a wind distribution model at thetested location and altitude For the latitude longitude andaltitude of the station-keeping orbit considered in this article(see Table I) this magnitude is approximately 24 ms It isassumed in all cases that wd equiv 0 producing wind vectors inthe xminus y plane

Figure 7 portrays the trajectory optimization for (ti =9am tf = 11am) with the wind vector with magnitudeVw = 22 ms and direction 260 (the solar azimuth anglevaries from 138 to 151) The optimizer is initialized withan input trajectory which will approximately follow a circularground track at the drag-power-minimizing airspeed (as long

Fig 5 Result of trajectory optimization from 900ndash1000am using theNelder-Mead nonlinear simplex algorithm

Fig 6 Comparison between optimized and baseline trajectories forsuccessive hour-long optimization intervals

as the magnitude of the airspeed is higher than that of thewind) This corresponds to the baseline initialization usedin preceding sections with commands modified to accountfor the magnitude and direction of the wind As the aircrafttries to maintain its velocity close to the optimal airspeedit will spend the majority of its time flying in the straightpaths upwind The trajectory optimization in this exampleimproves net energy storage over the baseline trajectory byless than 1 We note that the improvement in energy storageachieved by the optimizer decreases with wind speeds whichmay be due to the increased complexity involved in solvingwith a single shooting method or the baseline trajectory mayapproach optimal behavior in high winds

Fig 7 Result of trajectory optimization in the presence of winds from900ndash1100am for Vw = 22 ms

Simulation results show that compared to the no-windcase low-to-moderate speed winds (Vw le 07Va) at an angleof more than plusmn90 from the solar azimuth will increase thenet energy storage while low-to-moderate speed winds at anangle of less than plusmn90 from the solar azimuth will decreasenet energy storage

D Real-Time Implementation Considerations

Because of the highly nonlinear nature of the trajectoryoptimization problem the solutions described above provideno formal guarantees on convergence to an optimal (or evenfeasible) solution This combined with the computationalcomplexity involved particularly with the use of the interiorpoint method prevents the use of these optimization routinesin an online setting where external disturbances (such aschanging winds) may necessitate frequent replanning Addi-tionally the computational burden and associated power drawmakes it impractical to perform the trajectory optimizationusing an onboard computer The trajectory optimizationwould need to be sent to the vehicle through communicationlinks which could impose system reliability issues fromsignal integrity and availability standpoints

One alternative is to encapsulate behavior of the trajectoryoptimization algorithm into a number of rules based onthe trajectory optimization via shooting which could thenbe implemented as a state machine on the onboard flightcontrol computer For low wind speeds these rules could besummarized at a high level as

1) With the sun above the horizon and Ebatt lt Emaxperform ldquoracetrackrdquo-shaped periodic climbs and de-scents with the major axis aligned with the sun andwith periodic airspeed variation until Ebatt = Emax

2) With the sun above the horizon and Ebatt = Emaxclimb at the maximum rate possible which will notdraw power from the batteries

3) With the sun below the horizon and h gt hmindescend (using an optimal combination of electric andpotential energy) to h = hmin

4) With the sun below the horizon and h = hminperform circular orbits at the constant drag-power-minimizing airspeed

IV MACHINE LEARNING FOR TRAJECTORY PLANNING

One particularly promising approach to ldquocompressrdquo thetrajectory optimization for use online is to take advantageof reinforcement learning techniques to learn the optimalbehavior based on a set of inputs This has the effect ofshifting the computational burden from the online planningphase to an offline training and validation phase wherecomputational power is cheap and readily available One suchmethod which we consider here is the adaptive neuro-fuzzyinference system (ANFIS) [13] [14] ANFIS is a particulartype of universal approximator which combines a multi-layerfeedforward neural network with fuzzy logic The idea is to

1) Generate a large set of trajectories using the optimiza-tion methods provided above for a variety of differentconditions

2) Train and validate the ANFIS system using this offlinetrajectory optimization data

3) Use the ANFIS system for trajectory planning onlineSix input variables are provided for the trajectory planningANFIS namely(u1) The angular difference between the solar azimuth

and aircraft course angle ∆χ = (χminus φs)(u2) The aircraft distance from the station-keeping max-

imum radius dperim = dmax minus983155p2n + p2e

(u3) The rate of change of dperim(u4) The battery state of charge (EbattEmax)(u5) Direction of the wind(u6) Magnitude of the wind

Outputs of the trajectory planning ANFIS are(y1) Command for rate of change of heading (χcmd)(y2) Command for altitude rate (hcmd)

The airspeed command is taken to be constant for trajectoryplanning using ANFIS

Using ANFIS we are able to produce a closed-loopformula to compute the desired commands which can poten-tially be implemented in-flight on the aircraft flight controlcomputer

A Trajectory Planning with ANFIS

Figure 8 demonstrates the performance of the trainedANFIS in energy-optimized trajectory planning for the three-hour period (ti = 9am tf = 12pm) on the winter solsticeday The improvement in net energy storage over the baselinetrajectory is approximately 12 which is comparable to theresults of the offline optimization routine The use of thelearned ANFIS model to compute trajectories given the in-puts provided earlier in this section results in processing timereduction of roughly four orders of magnitude compared torunning the offline optimization routines (providing outputsin seconds instead of hours)

Fig 8 Result of application of learned ANFIS model to trajectoryoptimization from 900amndash1200pm

To validate the ANFIS trajectory planner and examine itsrobustness more simulations were carried out in which initialconditions ndash including initial heading of the aircraft ndash arevaried This is to ensure that the trained ANFIS providesinvariance to these variables and that it is able to handle

flight scenarios which have not been included in its trainingdata The results of these simulations indicate that the ANFISis effective in handling a variety of initial conditions andmaximize net energy storage as it is intended to do

V CONCLUSIONS

This work investigates the problem of energy-optimizedtrajectory planning for solar-powered HALE aircraft Opti-mization via shooting methods is first employed to optimizecommand trajectories with solvers using the interior pointmethod and the nonlinear simplex method Results are pre-sented and compared for these two solvers and the resultingoptimal trajectories for the winter solstice day are discussedThe problem of trajectory optimization in the presence ofconstant winds is addressed These trajectory optimizationroutines are used to train an adaptive neuro-fuzzy modelwhich is demonstrated to produce trajectories for the station-keeping mission without wind with nearly the same energysaving benefits as the nonlinear optimization procedures butapproximately four orders of magnitude faster This providesa potential avenue for online energy-optimized trajectoryplanning which can be carried out at low computational coston an onboard flight computer

VI ACKNOWLEDGMENTS

The authors would like to thank Bruce Lin NicholasRoberts David Liu Xiangfei Meng and Martin Gomez ofFacebook for their important contributions to this work

REFERENCES

[1] A T Klesh and P T Kabamba ldquoSolar-powered aircraft Energy-optimal path planning and perpetual endurancerdquo Journal of GuidanceControl and Dynamics vol 32 no 4 pp 1320ndash1329 2009

[2] S C Spangelo and E G Gilbert ldquoPower optimization of solar-powered aircraft with specified closed ground tracksrdquo Journal ofAircraft vol 50 no 1 pp 232ndash238 2012

[3] R W Beard and T W McLain Small Unmanned Aircraft Theoryand practice Princeton University Press 2012

[4] B L Stevens F L Lewis and E N Johnson Aircraft Control andSimulation Dynamics Controls Design and Autonomous SystemsJohn Wiley amp Sons 2015

[5] United States Committee on Extension to the Standard AtmosphereUS Standard Atmosphere 1976 National Oceanic and AtmosphericAdministration 1976

[6] D Colas N Roberts and V Samuel ldquoOpen sourcingfacebookrsquos solar-powered aircraft design toolsrdquo 2018 [Online]Available httpscodefbcomconnectivityopen-sourcing-facebooks-solar-powered-aircraft-design-tools

[7] J D Anderson Introduction to Flight McGraw-Hill Education 2011[8] I Reda and A Andreas ldquoSolar position algorithm for solar radiation

applicationsrdquo Solar Energy vol 76 no 5 pp 577ndash589 2004[9] G S Aglietti S Redi A R Tatnall and T Markvart ldquoHarnessing

high-altitude solar powerrdquo IEEE Transactions on Energy Conversionvol 24 no 2 pp 442ndash451 2009

[10] J T Betts ldquoSurvey of numerical methods for trajectory optimizationrdquoJournal of Guidance Control and Dynamics vol 21 no 2 pp 193ndash207 1998

[11] J A Nelder and R Mead ldquoA simplex method for function minimiza-tionrdquo The Computer Journal vol 7 no 4 pp 308ndash313 1965

[12] J H Mathews K D Fink et al Numerical Methods Using MATLABPearson Prentice Hall 2004 vol 4

[13] S Kurnaz O Cetin and O Kaynak ldquoAdaptive neuro-fuzzy inferencesystem based autonomous flight control of unmanned air vehiclesrdquoExpert Systems with Applications vol 37 no 2 pp 1229ndash1234 2010

[14] K Tanaka An Introduction to Fuzzy Logic for Practical ApplicationsSpringer 1997

where [middot] represents evaluation at all knot points i isin(1 n) R(middot) represents the unit ramp (rectifier) functiond[i] =

983155pn[i]2 + pe[i]2 and Ch and Cd are penalty weights

In Section III-A we describe a solution to this nonlinearoptimization problem via single shooting using the gradient-based interior point (IP) algorithm and present results fromthis optimization In Section III-B we describe a solution tothe same problem using the Nelder-Mead Nonlinear Simplexalgorithm and present results We then present results ontrajectory optimization in the presence of wind and discussreal-time online optimization considerations

A Interior Point Optimization

Starting with a general nonlinear optimization problem

minxisinRn

f(x)

s t ci(x) ge 0 i isin (1 m)(16)

slack variables w can be introduced to transform the inequal-ity constraints ci(x) ge 0 into an equality constraint of theform c(x) minus w = 0 The constraint on the slack variableitself can be absorbed into the cost function by defininglogarithmic barrier cost function

983123mi=1 ln (wi)

minmicro w x

f(x)minus micro

m983131

i=1

ln (wi)

s t c(x)minus w = 0

(17)

where micro is a small positive scalar Introducing Largrangemultiplier vector λ the Lagrangian function is given by

L = f(x)minus micro

m983131

i=1

ln (wi)minus λT (c(x)minus w) (18)

The first-order optimality conditions are found by settingthe derivatives of the Lagrangian with respect to x w andλ equal to zero (Karush-Kuhn-Tucker conditions)

nablaf(x)minusnablac(x)Tλ = 0

WΛeminus microe = 0

c(x)minus w = 0

(19)

Search directions dx dw and dλ can then be computedusing the Newton-Raphson method as described by983093

983095H 0 minusnablac(x)T

0 Λ Wnablac(x) minusI 0

983094

983096

983093

983095dx

dw

dλ

983094

983096 =

983093

983095minusnablaf(x) +nablac(x)Tλ

microeminusWΛeminusc(x) + w

983094

983096

(20)with

H = nabla2xxL(xwλ) = nabla2

xxf(x)minusm983131

i=1

λTnabla2xxc(x)

W =

983093

983097983095w1 0 0

0 0

0 0 wm

983094

983098983096 Λ =

983093

983097983095λ1 0 0

0 0

0 0 λm

983094

983098983096(21)

The optimization can then be carried out until successiveiterations satisfy some pre-defined convergence criteria It isnoted that due to the nonlinearity of the problem it is notpossible to guarantee global optimality of the solution

1) Trajectory Optimization with the Interior PointMethod The nonlinear trajectory optimization problem issolved using the interior point method provided by MAT-LABs commercial solver fmincon As the problem dimen-sion is large and highly nonlinear initial input trajectoriesare provided to the nonlinear solver to aid in convergenceto a feasible solution The initial input trajectory (whichis also our baseline for benchmarking purposes) producesa circular ground-track of radius dmax at altitude hminand at the equivalent airspeed Veas which minimizes dragpower This baseline is the constant-command trajectorywhich minimizes power draw while satisfying all constraintsThe simulation results presented in this article are obtainedfor the worst-case scenario corresponding to operation duringthe winter solstice

Figure 2 visualizes the results of trajectory optimizationover an hour-long period (ti = 9am tf = 10am) on thewinter solstice day The time interval is split into n = 180knot points such that the knot points are spaced at 20 secondintervals and there are 540 resulting optimization variablesIn this figure the black circle represents the bottom of acylinder in which the trajectory is constrained via quadraticpenalization In the xminusy trajectory plots the vertical axis isrepresentative of the north direction and the horizontal axisis representative of the east direction The altitude axis isalso included in the x minus y minus h plots where it is noted thatthe h-axis is not to scale with the x and y axes The color ofthe trajectory represents a relative airspeed where warmercolors indicate a lower velocity The direction of the sunin the x minus y plane is illustrated by yellow arrows (movingclockwise as the day progresses)

In this hour-long example the optimized trajectory in-creases net energy storage by approximately 12 comparedto that of the baseline trajectory (constant command tra-jectory which minimize power draw) The solver reachesconvergence in approximately 19 hours on a workstationwith parallelization on 6 CPU cores which is a baseline forcomparison with other optimizations discussed in this article

Figure 3 shows the result of trajectory optimization foranother hour-long period (ti = 2pm tf = 3pm) on thewinter solstice day This optimized trajectory improves netenergy storage by 15 compared to the baseline trajectoryIn both of these examples the energy-optimized trajectorieshave some periodic behavior combined with a rotation tofollow the sun A period of this behavior can be describedqualitatively as climbs at relatively low velocity away fromthe sun rapid turn towards the sun descent towards the sunat relatively high velocity rapid turn away from the sunThe relatively low velocity when climbing away from thesun produces a higher angle of attack and therefore highersolar flux through the area of the vehicle PV cells Whiledescending the higher velocity and lower angle of attacksimilarly tilts the solar cells towards the local solar vector

B Nonlinear Simplex Optimization

In this section we will present results of trajectory op-timizations solved using the nonlinear simplex optimization

Fig 2 Result of trajectory optimization from 900ndash1000am

algorithm developed by Nelder and Mead [11] The non-linear simplex optimization algorithm is gradient-free (it is adirect search method) and is thus effective in many practicaloptimization problems in which Jacobians and Hessians maybe computationally expensive or non-existent

The basis of the method is to estimate the objectivefunction gradient by the use of simplex Simplex S isin Rn isdefined as a convex envelope of n+1 vertices x0 xn isinRn The simplex-based search algorithm is initiated with anondegenerate simplex and the associated set of cost functionvalues at its vertices fj = f(xj) For a general minimizationproblem the goal is to decrease the cost function values atthe working simplex vertices with successive transformationsof the simplex consisting of

1) Reflection of vertex xh isin S with worst cost functionvalue (fh) to new vertex xr through the line definedby xh and the simplex centroid (c)

Fig 3 Result of trajectory optimization from 200ndash300pm

2) Depending on value fra) Expansion along line (c xr) if fr is best to new

vertex xe Use expanded simplex if fe is bestb) Contraction along line (c xr) if fr is worst or

second-worst3) Shrinkage of the simplex by moving vertices towards

best vertex xl isin S

Note that unlike the interior point method the explicitcomputations of Jacobians and Hessians are not requiredyet the transformations of the simplex are designed in sucha way that the working simplex should roughly follow thedirection of steepest descent A detailed description of theprocedure is omitted in this article but can be found in [11]and [12]

1) Trajectory Optimization with Nonlinear Simplex Fig-ure 5 illustrates the trajectory optimization for the hour-longperiod (ti = 9am tf = 10am) on the winter solstice day

Fig 4 Example of reflection and expansion steps for a 2D problem

solved using the Nelder-Mead nonlinear simplex algorithmThe optimization problem is identical to that presented inFig 2 with the difference being the use of the nonlinearsimplex method instead of the interior point method Thetime to convergence is approximately 1 hour on the sameworkstation used for interior point benchmarking (a 20timesimprovement over the interior point method) Using the non-linear simplex algorithm the optimal trajectory improves netenergy storage over the baseline trajectory by approximately7 (compared to 12 with the interior point method)

Figure 6 compares the hourly net energy storage from theoptimized and baseline trajectories Note that the baselinetrajectory in these simulations undergoes a mode switchwhen the batteries are full any excess solar power col-lection is used to increase the altitude of the vehicle tostore gravitational potential energy which is then expendedafter sunset until the vehicle reaches hmin The overallimprovement in net energy storage of the energy-optimizedtrajectory planning adds up to 105 kWh of additional energystorage over the winter solstice day (59 kWh with nonlinearsimplex) which is enough to extend the maximum night (thetime with no solar power collection) by approximately 160minutes At the latitude chosen for analysis both the baselineand optimized trajectories are able to survive the wintersolstice night however the battery state-of-charge margin issignificantly larger with the energy-optimized trajectory

C Trajectory Optimization in Wind

Winds introduce both additional complexities and opportu-nities for the energy-optimized trajectory planning problemWind is modeled according to the equations given in Eq 4and trajectories are optimized using the interior point algo-rithm provided by MATLABrsquos fmincon solver Optimiza-tions have been carried out for different wind directions andmagnitudes (given by gray vectors) up to the 99th-percentilewind magnitude provided by a wind distribution model at thetested location and altitude For the latitude longitude andaltitude of the station-keeping orbit considered in this article(see Table I) this magnitude is approximately 24 ms It isassumed in all cases that wd equiv 0 producing wind vectors inthe xminus y plane

Figure 7 portrays the trajectory optimization for (ti =9am tf = 11am) with the wind vector with magnitudeVw = 22 ms and direction 260 (the solar azimuth anglevaries from 138 to 151) The optimizer is initialized withan input trajectory which will approximately follow a circularground track at the drag-power-minimizing airspeed (as long

Fig 5 Result of trajectory optimization from 900ndash1000am using theNelder-Mead nonlinear simplex algorithm

Fig 6 Comparison between optimized and baseline trajectories forsuccessive hour-long optimization intervals

as the magnitude of the airspeed is higher than that of thewind) This corresponds to the baseline initialization usedin preceding sections with commands modified to accountfor the magnitude and direction of the wind As the aircrafttries to maintain its velocity close to the optimal airspeedit will spend the majority of its time flying in the straightpaths upwind The trajectory optimization in this exampleimproves net energy storage over the baseline trajectory byless than 1 We note that the improvement in energy storageachieved by the optimizer decreases with wind speeds whichmay be due to the increased complexity involved in solvingwith a single shooting method or the baseline trajectory mayapproach optimal behavior in high winds

Fig 7 Result of trajectory optimization in the presence of winds from900ndash1100am for Vw = 22 ms

Simulation results show that compared to the no-windcase low-to-moderate speed winds (Vw le 07Va) at an angleof more than plusmn90 from the solar azimuth will increase thenet energy storage while low-to-moderate speed winds at anangle of less than plusmn90 from the solar azimuth will decreasenet energy storage

D Real-Time Implementation Considerations

Because of the highly nonlinear nature of the trajectoryoptimization problem the solutions described above provideno formal guarantees on convergence to an optimal (or evenfeasible) solution This combined with the computationalcomplexity involved particularly with the use of the interiorpoint method prevents the use of these optimization routinesin an online setting where external disturbances (such aschanging winds) may necessitate frequent replanning Addi-tionally the computational burden and associated power drawmakes it impractical to perform the trajectory optimizationusing an onboard computer The trajectory optimizationwould need to be sent to the vehicle through communicationlinks which could impose system reliability issues fromsignal integrity and availability standpoints

One alternative is to encapsulate behavior of the trajectoryoptimization algorithm into a number of rules based onthe trajectory optimization via shooting which could thenbe implemented as a state machine on the onboard flightcontrol computer For low wind speeds these rules could besummarized at a high level as

1) With the sun above the horizon and Ebatt lt Emaxperform ldquoracetrackrdquo-shaped periodic climbs and de-scents with the major axis aligned with the sun andwith periodic airspeed variation until Ebatt = Emax

2) With the sun above the horizon and Ebatt = Emaxclimb at the maximum rate possible which will notdraw power from the batteries

3) With the sun below the horizon and h gt hmindescend (using an optimal combination of electric andpotential energy) to h = hmin

4) With the sun below the horizon and h = hminperform circular orbits at the constant drag-power-minimizing airspeed

IV MACHINE LEARNING FOR TRAJECTORY PLANNING

One particularly promising approach to ldquocompressrdquo thetrajectory optimization for use online is to take advantageof reinforcement learning techniques to learn the optimalbehavior based on a set of inputs This has the effect ofshifting the computational burden from the online planningphase to an offline training and validation phase wherecomputational power is cheap and readily available One suchmethod which we consider here is the adaptive neuro-fuzzyinference system (ANFIS) [13] [14] ANFIS is a particulartype of universal approximator which combines a multi-layerfeedforward neural network with fuzzy logic The idea is to

1) Generate a large set of trajectories using the optimiza-tion methods provided above for a variety of differentconditions

2) Train and validate the ANFIS system using this offlinetrajectory optimization data

3) Use the ANFIS system for trajectory planning onlineSix input variables are provided for the trajectory planningANFIS namely(u1) The angular difference between the solar azimuth

and aircraft course angle ∆χ = (χminus φs)(u2) The aircraft distance from the station-keeping max-

imum radius dperim = dmax minus983155p2n + p2e

(u3) The rate of change of dperim(u4) The battery state of charge (EbattEmax)(u5) Direction of the wind(u6) Magnitude of the wind

Outputs of the trajectory planning ANFIS are(y1) Command for rate of change of heading (χcmd)(y2) Command for altitude rate (hcmd)

The airspeed command is taken to be constant for trajectoryplanning using ANFIS

Using ANFIS we are able to produce a closed-loopformula to compute the desired commands which can poten-tially be implemented in-flight on the aircraft flight controlcomputer

A Trajectory Planning with ANFIS

Figure 8 demonstrates the performance of the trainedANFIS in energy-optimized trajectory planning for the three-hour period (ti = 9am tf = 12pm) on the winter solsticeday The improvement in net energy storage over the baselinetrajectory is approximately 12 which is comparable to theresults of the offline optimization routine The use of thelearned ANFIS model to compute trajectories given the in-puts provided earlier in this section results in processing timereduction of roughly four orders of magnitude compared torunning the offline optimization routines (providing outputsin seconds instead of hours)

Fig 8 Result of application of learned ANFIS model to trajectoryoptimization from 900amndash1200pm

To validate the ANFIS trajectory planner and examine itsrobustness more simulations were carried out in which initialconditions ndash including initial heading of the aircraft ndash arevaried This is to ensure that the trained ANFIS providesinvariance to these variables and that it is able to handle

flight scenarios which have not been included in its trainingdata The results of these simulations indicate that the ANFISis effective in handling a variety of initial conditions andmaximize net energy storage as it is intended to do

V CONCLUSIONS

This work investigates the problem of energy-optimizedtrajectory planning for solar-powered HALE aircraft Opti-mization via shooting methods is first employed to optimizecommand trajectories with solvers using the interior pointmethod and the nonlinear simplex method Results are pre-sented and compared for these two solvers and the resultingoptimal trajectories for the winter solstice day are discussedThe problem of trajectory optimization in the presence ofconstant winds is addressed These trajectory optimizationroutines are used to train an adaptive neuro-fuzzy modelwhich is demonstrated to produce trajectories for the station-keeping mission without wind with nearly the same energysaving benefits as the nonlinear optimization procedures butapproximately four orders of magnitude faster This providesa potential avenue for online energy-optimized trajectoryplanning which can be carried out at low computational coston an onboard flight computer

VI ACKNOWLEDGMENTS

The authors would like to thank Bruce Lin NicholasRoberts David Liu Xiangfei Meng and Martin Gomez ofFacebook for their important contributions to this work

REFERENCES

[1] A T Klesh and P T Kabamba ldquoSolar-powered aircraft Energy-optimal path planning and perpetual endurancerdquo Journal of GuidanceControl and Dynamics vol 32 no 4 pp 1320ndash1329 2009

[2] S C Spangelo and E G Gilbert ldquoPower optimization of solar-powered aircraft with specified closed ground tracksrdquo Journal ofAircraft vol 50 no 1 pp 232ndash238 2012

[3] R W Beard and T W McLain Small Unmanned Aircraft Theoryand practice Princeton University Press 2012

[4] B L Stevens F L Lewis and E N Johnson Aircraft Control andSimulation Dynamics Controls Design and Autonomous SystemsJohn Wiley amp Sons 2015

[5] United States Committee on Extension to the Standard AtmosphereUS Standard Atmosphere 1976 National Oceanic and AtmosphericAdministration 1976

[6] D Colas N Roberts and V Samuel ldquoOpen sourcingfacebookrsquos solar-powered aircraft design toolsrdquo 2018 [Online]Available httpscodefbcomconnectivityopen-sourcing-facebooks-solar-powered-aircraft-design-tools

[7] J D Anderson Introduction to Flight McGraw-Hill Education 2011[8] I Reda and A Andreas ldquoSolar position algorithm for solar radiation

applicationsrdquo Solar Energy vol 76 no 5 pp 577ndash589 2004[9] G S Aglietti S Redi A R Tatnall and T Markvart ldquoHarnessing

high-altitude solar powerrdquo IEEE Transactions on Energy Conversionvol 24 no 2 pp 442ndash451 2009

[10] J T Betts ldquoSurvey of numerical methods for trajectory optimizationrdquoJournal of Guidance Control and Dynamics vol 21 no 2 pp 193ndash207 1998

[11] J A Nelder and R Mead ldquoA simplex method for function minimiza-tionrdquo The Computer Journal vol 7 no 4 pp 308ndash313 1965

[12] J H Mathews K D Fink et al Numerical Methods Using MATLABPearson Prentice Hall 2004 vol 4

[13] S Kurnaz O Cetin and O Kaynak ldquoAdaptive neuro-fuzzy inferencesystem based autonomous flight control of unmanned air vehiclesrdquoExpert Systems with Applications vol 37 no 2 pp 1229ndash1234 2010

[14] K Tanaka An Introduction to Fuzzy Logic for Practical ApplicationsSpringer 1997

Fig 2 Result of trajectory optimization from 900ndash1000am

algorithm developed by Nelder and Mead [11] The non-linear simplex optimization algorithm is gradient-free (it is adirect search method) and is thus effective in many practicaloptimization problems in which Jacobians and Hessians maybe computationally expensive or non-existent

The basis of the method is to estimate the objectivefunction gradient by the use of simplex Simplex S isin Rn isdefined as a convex envelope of n+1 vertices x0 xn isinRn The simplex-based search algorithm is initiated with anondegenerate simplex and the associated set of cost functionvalues at its vertices fj = f(xj) For a general minimizationproblem the goal is to decrease the cost function values atthe working simplex vertices with successive transformationsof the simplex consisting of

1) Reflection of vertex xh isin S with worst cost functionvalue (fh) to new vertex xr through the line definedby xh and the simplex centroid (c)

Fig 3 Result of trajectory optimization from 200ndash300pm

2) Depending on value fra) Expansion along line (c xr) if fr is best to new

vertex xe Use expanded simplex if fe is bestb) Contraction along line (c xr) if fr is worst or

second-worst3) Shrinkage of the simplex by moving vertices towards

best vertex xl isin S

Note that unlike the interior point method the explicitcomputations of Jacobians and Hessians are not requiredyet the transformations of the simplex are designed in sucha way that the working simplex should roughly follow thedirection of steepest descent A detailed description of theprocedure is omitted in this article but can be found in [11]and [12]

1) Trajectory Optimization with Nonlinear Simplex Fig-ure 5 illustrates the trajectory optimization for the hour-longperiod (ti = 9am tf = 10am) on the winter solstice day

Fig 4 Example of reflection and expansion steps for a 2D problem

solved using the Nelder-Mead nonlinear simplex algorithmThe optimization problem is identical to that presented inFig 2 with the difference being the use of the nonlinearsimplex method instead of the interior point method Thetime to convergence is approximately 1 hour on the sameworkstation used for interior point benchmarking (a 20timesimprovement over the interior point method) Using the non-linear simplex algorithm the optimal trajectory improves netenergy storage over the baseline trajectory by approximately7 (compared to 12 with the interior point method)

Figure 6 compares the hourly net energy storage from theoptimized and baseline trajectories Note that the baselinetrajectory in these simulations undergoes a mode switchwhen the batteries are full any excess solar power col-lection is used to increase the altitude of the vehicle tostore gravitational potential energy which is then expendedafter sunset until the vehicle reaches hmin The overallimprovement in net energy storage of the energy-optimizedtrajectory planning adds up to 105 kWh of additional energystorage over the winter solstice day (59 kWh with nonlinearsimplex) which is enough to extend the maximum night (thetime with no solar power collection) by approximately 160minutes At the latitude chosen for analysis both the baselineand optimized trajectories are able to survive the wintersolstice night however the battery state-of-charge margin issignificantly larger with the energy-optimized trajectory

C Trajectory Optimization in Wind

Winds introduce both additional complexities and opportu-nities for the energy-optimized trajectory planning problemWind is modeled according to the equations given in Eq 4and trajectories are optimized using the interior point algo-rithm provided by MATLABrsquos fmincon solver Optimiza-tions have been carried out for different wind directions andmagnitudes (given by gray vectors) up to the 99th-percentilewind magnitude provided by a wind distribution model at thetested location and altitude For the latitude longitude andaltitude of the station-keeping orbit considered in this article(see Table I) this magnitude is approximately 24 ms It isassumed in all cases that wd equiv 0 producing wind vectors inthe xminus y plane

Figure 7 portrays the trajectory optimization for (ti =9am tf = 11am) with the wind vector with magnitudeVw = 22 ms and direction 260 (the solar azimuth anglevaries from 138 to 151) The optimizer is initialized withan input trajectory which will approximately follow a circularground track at the drag-power-minimizing airspeed (as long

Fig 5 Result of trajectory optimization from 900ndash1000am using theNelder-Mead nonlinear simplex algorithm

Fig 6 Comparison between optimized and baseline trajectories forsuccessive hour-long optimization intervals

as the magnitude of the airspeed is higher than that of thewind) This corresponds to the baseline initialization usedin preceding sections with commands modified to accountfor the magnitude and direction of the wind As the aircrafttries to maintain its velocity close to the optimal airspeedit will spend the majority of its time flying in the straightpaths upwind The trajectory optimization in this exampleimproves net energy storage over the baseline trajectory byless than 1 We note that the improvement in energy storageachieved by the optimizer decreases with wind speeds whichmay be due to the increased complexity involved in solvingwith a single shooting method or the baseline trajectory mayapproach optimal behavior in high winds

Fig 7 Result of trajectory optimization in the presence of winds from900ndash1100am for Vw = 22 ms

Simulation results show that compared to the no-windcase low-to-moderate speed winds (Vw le 07Va) at an angleof more than plusmn90 from the solar azimuth will increase thenet energy storage while low-to-moderate speed winds at anangle of less than plusmn90 from the solar azimuth will decreasenet energy storage

D Real-Time Implementation Considerations

Because of the highly nonlinear nature of the trajectoryoptimization problem the solutions described above provideno formal guarantees on convergence to an optimal (or evenfeasible) solution This combined with the computationalcomplexity involved particularly with the use of the interiorpoint method prevents the use of these optimization routinesin an online setting where external disturbances (such aschanging winds) may necessitate frequent replanning Addi-tionally the computational burden and associated power drawmakes it impractical to perform the trajectory optimizationusing an onboard computer The trajectory optimizationwould need to be sent to the vehicle through communicationlinks which could impose system reliability issues fromsignal integrity and availability standpoints

One alternative is to encapsulate behavior of the trajectoryoptimization algorithm into a number of rules based onthe trajectory optimization via shooting which could thenbe implemented as a state machine on the onboard flightcontrol computer For low wind speeds these rules could besummarized at a high level as

1) With the sun above the horizon and Ebatt lt Emaxperform ldquoracetrackrdquo-shaped periodic climbs and de-scents with the major axis aligned with the sun andwith periodic airspeed variation until Ebatt = Emax

2) With the sun above the horizon and Ebatt = Emaxclimb at the maximum rate possible which will notdraw power from the batteries

3) With the sun below the horizon and h gt hmindescend (using an optimal combination of electric andpotential energy) to h = hmin

4) With the sun below the horizon and h = hminperform circular orbits at the constant drag-power-minimizing airspeed

IV MACHINE LEARNING FOR TRAJECTORY PLANNING

One particularly promising approach to ldquocompressrdquo thetrajectory optimization for use online is to take advantageof reinforcement learning techniques to learn the optimalbehavior based on a set of inputs This has the effect ofshifting the computational burden from the online planningphase to an offline training and validation phase wherecomputational power is cheap and readily available One suchmethod which we consider here is the adaptive neuro-fuzzyinference system (ANFIS) [13] [14] ANFIS is a particulartype of universal approximator which combines a multi-layerfeedforward neural network with fuzzy logic The idea is to

1) Generate a large set of trajectories using the optimiza-tion methods provided above for a variety of differentconditions

2) Train and validate the ANFIS system using this offlinetrajectory optimization data

3) Use the ANFIS system for trajectory planning onlineSix input variables are provided for the trajectory planningANFIS namely(u1) The angular difference between the solar azimuth

and aircraft course angle ∆χ = (χminus φs)(u2) The aircraft distance from the station-keeping max-

imum radius dperim = dmax minus983155p2n + p2e

(u3) The rate of change of dperim(u4) The battery state of charge (EbattEmax)(u5) Direction of the wind(u6) Magnitude of the wind

Outputs of the trajectory planning ANFIS are(y1) Command for rate of change of heading (χcmd)(y2) Command for altitude rate (hcmd)

The airspeed command is taken to be constant for trajectoryplanning using ANFIS

Using ANFIS we are able to produce a closed-loopformula to compute the desired commands which can poten-tially be implemented in-flight on the aircraft flight controlcomputer

A Trajectory Planning with ANFIS

Figure 8 demonstrates the performance of the trainedANFIS in energy-optimized trajectory planning for the three-hour period (ti = 9am tf = 12pm) on the winter solsticeday The improvement in net energy storage over the baselinetrajectory is approximately 12 which is comparable to theresults of the offline optimization routine The use of thelearned ANFIS model to compute trajectories given the in-puts provided earlier in this section results in processing timereduction of roughly four orders of magnitude compared torunning the offline optimization routines (providing outputsin seconds instead of hours)

Fig 8 Result of application of learned ANFIS model to trajectoryoptimization from 900amndash1200pm

To validate the ANFIS trajectory planner and examine itsrobustness more simulations were carried out in which initialconditions ndash including initial heading of the aircraft ndash arevaried This is to ensure that the trained ANFIS providesinvariance to these variables and that it is able to handle

flight scenarios which have not been included in its trainingdata The results of these simulations indicate that the ANFISis effective in handling a variety of initial conditions andmaximize net energy storage as it is intended to do

V CONCLUSIONS

This work investigates the problem of energy-optimizedtrajectory planning for solar-powered HALE aircraft Opti-mization via shooting methods is first employed to optimizecommand trajectories with solvers using the interior pointmethod and the nonlinear simplex method Results are pre-sented and compared for these two solvers and the resultingoptimal trajectories for the winter solstice day are discussedThe problem of trajectory optimization in the presence ofconstant winds is addressed These trajectory optimizationroutines are used to train an adaptive neuro-fuzzy modelwhich is demonstrated to produce trajectories for the station-keeping mission without wind with nearly the same energysaving benefits as the nonlinear optimization procedures butapproximately four orders of magnitude faster This providesa potential avenue for online energy-optimized trajectoryplanning which can be carried out at low computational coston an onboard flight computer

VI ACKNOWLEDGMENTS

The authors would like to thank Bruce Lin NicholasRoberts David Liu Xiangfei Meng and Martin Gomez ofFacebook for their important contributions to this work

REFERENCES

[1] A T Klesh and P T Kabamba ldquoSolar-powered aircraft Energy-optimal path planning and perpetual endurancerdquo Journal of GuidanceControl and Dynamics vol 32 no 4 pp 1320ndash1329 2009

[2] S C Spangelo and E G Gilbert ldquoPower optimization of solar-powered aircraft with specified closed ground tracksrdquo Journal ofAircraft vol 50 no 1 pp 232ndash238 2012

[3] R W Beard and T W McLain Small Unmanned Aircraft Theoryand practice Princeton University Press 2012

[4] B L Stevens F L Lewis and E N Johnson Aircraft Control andSimulation Dynamics Controls Design and Autonomous SystemsJohn Wiley amp Sons 2015

[5] United States Committee on Extension to the Standard AtmosphereUS Standard Atmosphere 1976 National Oceanic and AtmosphericAdministration 1976

[6] D Colas N Roberts and V Samuel ldquoOpen sourcingfacebookrsquos solar-powered aircraft design toolsrdquo 2018 [Online]Available httpscodefbcomconnectivityopen-sourcing-facebooks-solar-powered-aircraft-design-tools

[7] J D Anderson Introduction to Flight McGraw-Hill Education 2011[8] I Reda and A Andreas ldquoSolar position algorithm for solar radiation

applicationsrdquo Solar Energy vol 76 no 5 pp 577ndash589 2004[9] G S Aglietti S Redi A R Tatnall and T Markvart ldquoHarnessing

high-altitude solar powerrdquo IEEE Transactions on Energy Conversionvol 24 no 2 pp 442ndash451 2009

[10] J T Betts ldquoSurvey of numerical methods for trajectory optimizationrdquoJournal of Guidance Control and Dynamics vol 21 no 2 pp 193ndash207 1998

[11] J A Nelder and R Mead ldquoA simplex method for function minimiza-tionrdquo The Computer Journal vol 7 no 4 pp 308ndash313 1965

[12] J H Mathews K D Fink et al Numerical Methods Using MATLABPearson Prentice Hall 2004 vol 4

[13] S Kurnaz O Cetin and O Kaynak ldquoAdaptive neuro-fuzzy inferencesystem based autonomous flight control of unmanned air vehiclesrdquoExpert Systems with Applications vol 37 no 2 pp 1229ndash1234 2010

[14] K Tanaka An Introduction to Fuzzy Logic for Practical ApplicationsSpringer 1997

Fig 4 Example of reflection and expansion steps for a 2D problem

solved using the Nelder-Mead nonlinear simplex algorithmThe optimization problem is identical to that presented inFig 2 with the difference being the use of the nonlinearsimplex method instead of the interior point method Thetime to convergence is approximately 1 hour on the sameworkstation used for interior point benchmarking (a 20timesimprovement over the interior point method) Using the non-linear simplex algorithm the optimal trajectory improves netenergy storage over the baseline trajectory by approximately7 (compared to 12 with the interior point method)

Figure 6 compares the hourly net energy storage from theoptimized and baseline trajectories Note that the baselinetrajectory in these simulations undergoes a mode switchwhen the batteries are full any excess solar power col-lection is used to increase the altitude of the vehicle tostore gravitational potential energy which is then expendedafter sunset until the vehicle reaches hmin The overallimprovement in net energy storage of the energy-optimizedtrajectory planning adds up to 105 kWh of additional energystorage over the winter solstice day (59 kWh with nonlinearsimplex) which is enough to extend the maximum night (thetime with no solar power collection) by approximately 160minutes At the latitude chosen for analysis both the baselineand optimized trajectories are able to survive the wintersolstice night however the battery state-of-charge margin issignificantly larger with the energy-optimized trajectory

C Trajectory Optimization in Wind

Winds introduce both additional complexities and opportu-nities for the energy-optimized trajectory planning problemWind is modeled according to the equations given in Eq 4and trajectories are optimized using the interior point algo-rithm provided by MATLABrsquos fmincon solver Optimiza-tions have been carried out for different wind directions andmagnitudes (given by gray vectors) up to the 99th-percentilewind magnitude provided by a wind distribution model at thetested location and altitude For the latitude longitude andaltitude of the station-keeping orbit considered in this article(see Table I) this magnitude is approximately 24 ms It isassumed in all cases that wd equiv 0 producing wind vectors inthe xminus y plane

Figure 7 portrays the trajectory optimization for (ti =9am tf = 11am) with the wind vector with magnitudeVw = 22 ms and direction 260 (the solar azimuth anglevaries from 138 to 151) The optimizer is initialized withan input trajectory which will approximately follow a circularground track at the drag-power-minimizing airspeed (as long

Fig 5 Result of trajectory optimization from 900ndash1000am using theNelder-Mead nonlinear simplex algorithm

Fig 6 Comparison between optimized and baseline trajectories forsuccessive hour-long optimization intervals

as the magnitude of the airspeed is higher than that of thewind) This corresponds to the baseline initialization usedin preceding sections with commands modified to accountfor the magnitude and direction of the wind As the aircrafttries to maintain its velocity close to the optimal airspeedit will spend the majority of its time flying in the straightpaths upwind The trajectory optimization in this exampleimproves net energy storage over the baseline trajectory byless than 1 We note that the improvement in energy storageachieved by the optimizer decreases with wind speeds whichmay be due to the increased complexity involved in solvingwith a single shooting method or the baseline trajectory mayapproach optimal behavior in high winds

Fig 7 Result of trajectory optimization in the presence of winds from900ndash1100am for Vw = 22 ms

Simulation results show that compared to the no-windcase low-to-moderate speed winds (Vw le 07Va) at an angleof more than plusmn90 from the solar azimuth will increase thenet energy storage while low-to-moderate speed winds at anangle of less than plusmn90 from the solar azimuth will decreasenet energy storage

D Real-Time Implementation Considerations

Because of the highly nonlinear nature of the trajectoryoptimization problem the solutions described above provideno formal guarantees on convergence to an optimal (or evenfeasible) solution This combined with the computationalcomplexity involved particularly with the use of the interiorpoint method prevents the use of these optimization routinesin an online setting where external disturbances (such aschanging winds) may necessitate frequent replanning Addi-tionally the computational burden and associated power drawmakes it impractical to perform the trajectory optimizationusing an onboard computer The trajectory optimizationwould need to be sent to the vehicle through communicationlinks which could impose system reliability issues fromsignal integrity and availability standpoints

One alternative is to encapsulate behavior of the trajectoryoptimization algorithm into a number of rules based onthe trajectory optimization via shooting which could thenbe implemented as a state machine on the onboard flightcontrol computer For low wind speeds these rules could besummarized at a high level as

1) With the sun above the horizon and Ebatt lt Emaxperform ldquoracetrackrdquo-shaped periodic climbs and de-scents with the major axis aligned with the sun andwith periodic airspeed variation until Ebatt = Emax

2) With the sun above the horizon and Ebatt = Emaxclimb at the maximum rate possible which will notdraw power from the batteries

3) With the sun below the horizon and h gt hmindescend (using an optimal combination of electric andpotential energy) to h = hmin

4) With the sun below the horizon and h = hminperform circular orbits at the constant drag-power-minimizing airspeed

IV MACHINE LEARNING FOR TRAJECTORY PLANNING

One particularly promising approach to ldquocompressrdquo thetrajectory optimization for use online is to take advantageof reinforcement learning techniques to learn the optimalbehavior based on a set of inputs This has the effect ofshifting the computational burden from the online planningphase to an offline training and validation phase wherecomputational power is cheap and readily available One suchmethod which we consider here is the adaptive neuro-fuzzyinference system (ANFIS) [13] [14] ANFIS is a particulartype of universal approximator which combines a multi-layerfeedforward neural network with fuzzy logic The idea is to

1) Generate a large set of trajectories using the optimiza-tion methods provided above for a variety of differentconditions

2) Train and validate the ANFIS system using this offlinetrajectory optimization data

3) Use the ANFIS system for trajectory planning onlineSix input variables are provided for the trajectory planningANFIS namely(u1) The angular difference between the solar azimuth

and aircraft course angle ∆χ = (χminus φs)(u2) The aircraft distance from the station-keeping max-

imum radius dperim = dmax minus983155p2n + p2e

(u3) The rate of change of dperim(u4) The battery state of charge (EbattEmax)(u5) Direction of the wind(u6) Magnitude of the wind

Outputs of the trajectory planning ANFIS are(y1) Command for rate of change of heading (χcmd)(y2) Command for altitude rate (hcmd)

The airspeed command is taken to be constant for trajectoryplanning using ANFIS

Using ANFIS we are able to produce a closed-loopformula to compute the desired commands which can poten-tially be implemented in-flight on the aircraft flight controlcomputer

A Trajectory Planning with ANFIS

Figure 8 demonstrates the performance of the trainedANFIS in energy-optimized trajectory planning for the three-hour period (ti = 9am tf = 12pm) on the winter solsticeday The improvement in net energy storage over the baselinetrajectory is approximately 12 which is comparable to theresults of the offline optimization routine The use of thelearned ANFIS model to compute trajectories given the in-puts provided earlier in this section results in processing timereduction of roughly four orders of magnitude compared torunning the offline optimization routines (providing outputsin seconds instead of hours)

Fig 8 Result of application of learned ANFIS model to trajectoryoptimization from 900amndash1200pm

To validate the ANFIS trajectory planner and examine itsrobustness more simulations were carried out in which initialconditions ndash including initial heading of the aircraft ndash arevaried This is to ensure that the trained ANFIS providesinvariance to these variables and that it is able to handle

flight scenarios which have not been included in its trainingdata The results of these simulations indicate that the ANFISis effective in handling a variety of initial conditions andmaximize net energy storage as it is intended to do

V CONCLUSIONS

This work investigates the problem of energy-optimizedtrajectory planning for solar-powered HALE aircraft Opti-mization via shooting methods is first employed to optimizecommand trajectories with solvers using the interior pointmethod and the nonlinear simplex method Results are pre-sented and compared for these two solvers and the resultingoptimal trajectories for the winter solstice day are discussedThe problem of trajectory optimization in the presence ofconstant winds is addressed These trajectory optimizationroutines are used to train an adaptive neuro-fuzzy modelwhich is demonstrated to produce trajectories for the station-keeping mission without wind with nearly the same energysaving benefits as the nonlinear optimization procedures butapproximately four orders of magnitude faster This providesa potential avenue for online energy-optimized trajectoryplanning which can be carried out at low computational coston an onboard flight computer

VI ACKNOWLEDGMENTS

The authors would like to thank Bruce Lin NicholasRoberts David Liu Xiangfei Meng and Martin Gomez ofFacebook for their important contributions to this work

REFERENCES

[1] A T Klesh and P T Kabamba ldquoSolar-powered aircraft Energy-optimal path planning and perpetual endurancerdquo Journal of GuidanceControl and Dynamics vol 32 no 4 pp 1320ndash1329 2009

[2] S C Spangelo and E G Gilbert ldquoPower optimization of solar-powered aircraft with specified closed ground tracksrdquo Journal ofAircraft vol 50 no 1 pp 232ndash238 2012

[3] R W Beard and T W McLain Small Unmanned Aircraft Theoryand practice Princeton University Press 2012

[4] B L Stevens F L Lewis and E N Johnson Aircraft Control andSimulation Dynamics Controls Design and Autonomous SystemsJohn Wiley amp Sons 2015

[5] United States Committee on Extension to the Standard AtmosphereUS Standard Atmosphere 1976 National Oceanic and AtmosphericAdministration 1976

[6] D Colas N Roberts and V Samuel ldquoOpen sourcingfacebookrsquos solar-powered aircraft design toolsrdquo 2018 [Online]Available httpscodefbcomconnectivityopen-sourcing-facebooks-solar-powered-aircraft-design-tools

[7] J D Anderson Introduction to Flight McGraw-Hill Education 2011[8] I Reda and A Andreas ldquoSolar position algorithm for solar radiation

applicationsrdquo Solar Energy vol 76 no 5 pp 577ndash589 2004[9] G S Aglietti S Redi A R Tatnall and T Markvart ldquoHarnessing

high-altitude solar powerrdquo IEEE Transactions on Energy Conversionvol 24 no 2 pp 442ndash451 2009

[10] J T Betts ldquoSurvey of numerical methods for trajectory optimizationrdquoJournal of Guidance Control and Dynamics vol 21 no 2 pp 193ndash207 1998

[11] J A Nelder and R Mead ldquoA simplex method for function minimiza-tionrdquo The Computer Journal vol 7 no 4 pp 308ndash313 1965

[12] J H Mathews K D Fink et al Numerical Methods Using MATLABPearson Prentice Hall 2004 vol 4

[13] S Kurnaz O Cetin and O Kaynak ldquoAdaptive neuro-fuzzy inferencesystem based autonomous flight control of unmanned air vehiclesrdquoExpert Systems with Applications vol 37 no 2 pp 1229ndash1234 2010

[14] K Tanaka An Introduction to Fuzzy Logic for Practical ApplicationsSpringer 1997

as the magnitude of the airspeed is higher than that of thewind) This corresponds to the baseline initialization usedin preceding sections with commands modified to accountfor the magnitude and direction of the wind As the aircrafttries to maintain its velocity close to the optimal airspeedit will spend the majority of its time flying in the straightpaths upwind The trajectory optimization in this exampleimproves net energy storage over the baseline trajectory byless than 1 We note that the improvement in energy storageachieved by the optimizer decreases with wind speeds whichmay be due to the increased complexity involved in solvingwith a single shooting method or the baseline trajectory mayapproach optimal behavior in high winds

Fig 7 Result of trajectory optimization in the presence of winds from900ndash1100am for Vw = 22 ms

Simulation results show that compared to the no-windcase low-to-moderate speed winds (Vw le 07Va) at an angleof more than plusmn90 from the solar azimuth will increase thenet energy storage while low-to-moderate speed winds at anangle of less than plusmn90 from the solar azimuth will decreasenet energy storage

D Real-Time Implementation Considerations

Because of the highly nonlinear nature of the trajectoryoptimization problem the solutions described above provideno formal guarantees on convergence to an optimal (or evenfeasible) solution This combined with the computationalcomplexity involved particularly with the use of the interiorpoint method prevents the use of these optimization routinesin an online setting where external disturbances (such aschanging winds) may necessitate frequent replanning Addi-tionally the computational burden and associated power drawmakes it impractical to perform the trajectory optimizationusing an onboard computer The trajectory optimizationwould need to be sent to the vehicle through communicationlinks which could impose system reliability issues fromsignal integrity and availability standpoints

One alternative is to encapsulate behavior of the trajectoryoptimization algorithm into a number of rules based onthe trajectory optimization via shooting which could thenbe implemented as a state machine on the onboard flightcontrol computer For low wind speeds these rules could besummarized at a high level as

1) With the sun above the horizon and Ebatt lt Emaxperform ldquoracetrackrdquo-shaped periodic climbs and de-scents with the major axis aligned with the sun andwith periodic airspeed variation until Ebatt = Emax

2) With the sun above the horizon and Ebatt = Emaxclimb at the maximum rate possible which will notdraw power from the batteries

3) With the sun below the horizon and h gt hmindescend (using an optimal combination of electric andpotential energy) to h = hmin

4) With the sun below the horizon and h = hminperform circular orbits at the constant drag-power-minimizing airspeed

IV MACHINE LEARNING FOR TRAJECTORY PLANNING

One particularly promising approach to ldquocompressrdquo thetrajectory optimization for use online is to take advantageof reinforcement learning techniques to learn the optimalbehavior based on a set of inputs This has the effect ofshifting the computational burden from the online planningphase to an offline training and validation phase wherecomputational power is cheap and readily available One suchmethod which we consider here is the adaptive neuro-fuzzyinference system (ANFIS) [13] [14] ANFIS is a particulartype of universal approximator which combines a multi-layerfeedforward neural network with fuzzy logic The idea is to

1) Generate a large set of trajectories using the optimiza-tion methods provided above for a variety of differentconditions

2) Train and validate the ANFIS system using this offlinetrajectory optimization data

3) Use the ANFIS system for trajectory planning onlineSix input variables are provided for the trajectory planningANFIS namely(u1) The angular difference between the solar azimuth

and aircraft course angle ∆χ = (χminus φs)(u2) The aircraft distance from the station-keeping max-

imum radius dperim = dmax minus983155p2n + p2e

(u3) The rate of change of dperim(u4) The battery state of charge (EbattEmax)(u5) Direction of the wind(u6) Magnitude of the wind

Outputs of the trajectory planning ANFIS are(y1) Command for rate of change of heading (χcmd)(y2) Command for altitude rate (hcmd)

The airspeed command is taken to be constant for trajectoryplanning using ANFIS

Using ANFIS we are able to produce a closed-loopformula to compute the desired commands which can poten-tially be implemented in-flight on the aircraft flight controlcomputer

A Trajectory Planning with ANFIS

Figure 8 demonstrates the performance of the trainedANFIS in energy-optimized trajectory planning for the three-hour period (ti = 9am tf = 12pm) on the winter solsticeday The improvement in net energy storage over the baselinetrajectory is approximately 12 which is comparable to theresults of the offline optimization routine The use of thelearned ANFIS model to compute trajectories given the in-puts provided earlier in this section results in processing timereduction of roughly four orders of magnitude compared torunning the offline optimization routines (providing outputsin seconds instead of hours)

Fig 8 Result of application of learned ANFIS model to trajectoryoptimization from 900amndash1200pm

To validate the ANFIS trajectory planner and examine itsrobustness more simulations were carried out in which initialconditions ndash including initial heading of the aircraft ndash arevaried This is to ensure that the trained ANFIS providesinvariance to these variables and that it is able to handle

flight scenarios which have not been included in its trainingdata The results of these simulations indicate that the ANFISis effective in handling a variety of initial conditions andmaximize net energy storage as it is intended to do

V CONCLUSIONS

This work investigates the problem of energy-optimizedtrajectory planning for solar-powered HALE aircraft Opti-mization via shooting methods is first employed to optimizecommand trajectories with solvers using the interior pointmethod and the nonlinear simplex method Results are pre-sented and compared for these two solvers and the resultingoptimal trajectories for the winter solstice day are discussedThe problem of trajectory optimization in the presence ofconstant winds is addressed These trajectory optimizationroutines are used to train an adaptive neuro-fuzzy modelwhich is demonstrated to produce trajectories for the station-keeping mission without wind with nearly the same energysaving benefits as the nonlinear optimization procedures butapproximately four orders of magnitude faster This providesa potential avenue for online energy-optimized trajectoryplanning which can be carried out at low computational coston an onboard flight computer

VI ACKNOWLEDGMENTS

The authors would like to thank Bruce Lin NicholasRoberts David Liu Xiangfei Meng and Martin Gomez ofFacebook for their important contributions to this work

REFERENCES

[1] A T Klesh and P T Kabamba ldquoSolar-powered aircraft Energy-optimal path planning and perpetual endurancerdquo Journal of GuidanceControl and Dynamics vol 32 no 4 pp 1320ndash1329 2009

[2] S C Spangelo and E G Gilbert ldquoPower optimization of solar-powered aircraft with specified closed ground tracksrdquo Journal ofAircraft vol 50 no 1 pp 232ndash238 2012

[3] R W Beard and T W McLain Small Unmanned Aircraft Theoryand practice Princeton University Press 2012

[4] B L Stevens F L Lewis and E N Johnson Aircraft Control andSimulation Dynamics Controls Design and Autonomous SystemsJohn Wiley amp Sons 2015

[5] United States Committee on Extension to the Standard AtmosphereUS Standard Atmosphere 1976 National Oceanic and AtmosphericAdministration 1976

[6] D Colas N Roberts and V Samuel ldquoOpen sourcingfacebookrsquos solar-powered aircraft design toolsrdquo 2018 [Online]Available httpscodefbcomconnectivityopen-sourcing-facebooks-solar-powered-aircraft-design-tools

[7] J D Anderson Introduction to Flight McGraw-Hill Education 2011[8] I Reda and A Andreas ldquoSolar position algorithm for solar radiation

applicationsrdquo Solar Energy vol 76 no 5 pp 577ndash589 2004[9] G S Aglietti S Redi A R Tatnall and T Markvart ldquoHarnessing

high-altitude solar powerrdquo IEEE Transactions on Energy Conversionvol 24 no 2 pp 442ndash451 2009

[10] J T Betts ldquoSurvey of numerical methods for trajectory optimizationrdquoJournal of Guidance Control and Dynamics vol 21 no 2 pp 193ndash207 1998

[11] J A Nelder and R Mead ldquoA simplex method for function minimiza-tionrdquo The Computer Journal vol 7 no 4 pp 308ndash313 1965

[12] J H Mathews K D Fink et al Numerical Methods Using MATLABPearson Prentice Hall 2004 vol 4

[13] S Kurnaz O Cetin and O Kaynak ldquoAdaptive neuro-fuzzy inferencesystem based autonomous flight control of unmanned air vehiclesrdquoExpert Systems with Applications vol 37 no 2 pp 1229ndash1234 2010

[14] K Tanaka An Introduction to Fuzzy Logic for Practical ApplicationsSpringer 1997

A Trajectory Planning with ANFIS

Figure 8 demonstrates the performance of the trainedANFIS in energy-optimized trajectory planning for the three-hour period (ti = 9am tf = 12pm) on the winter solsticeday The improvement in net energy storage over the baselinetrajectory is approximately 12 which is comparable to theresults of the offline optimization routine The use of thelearned ANFIS model to compute trajectories given the in-puts provided earlier in this section results in processing timereduction of roughly four orders of magnitude compared torunning the offline optimization routines (providing outputsin seconds instead of hours)

Fig 8 Result of application of learned ANFIS model to trajectoryoptimization from 900amndash1200pm

To validate the ANFIS trajectory planner and examine itsrobustness more simulations were carried out in which initialconditions ndash including initial heading of the aircraft ndash arevaried This is to ensure that the trained ANFIS providesinvariance to these variables and that it is able to handle

flight scenarios which have not been included in its trainingdata The results of these simulations indicate that the ANFISis effective in handling a variety of initial conditions andmaximize net energy storage as it is intended to do

V CONCLUSIONS

This work investigates the problem of energy-optimizedtrajectory planning for solar-powered HALE aircraft Opti-mization via shooting methods is first employed to optimizecommand trajectories with solvers using the interior pointmethod and the nonlinear simplex method Results are pre-sented and compared for these two solvers and the resultingoptimal trajectories for the winter solstice day are discussedThe problem of trajectory optimization in the presence ofconstant winds is addressed These trajectory optimizationroutines are used to train an adaptive neuro-fuzzy modelwhich is demonstrated to produce trajectories for the station-keeping mission without wind with nearly the same energysaving benefits as the nonlinear optimization procedures butapproximately four orders of magnitude faster This providesa potential avenue for online energy-optimized trajectoryplanning which can be carried out at low computational coston an onboard flight computer

VI ACKNOWLEDGMENTS

The authors would like to thank Bruce Lin NicholasRoberts David Liu Xiangfei Meng and Martin Gomez ofFacebook for their important contributions to this work

REFERENCES

[1] A T Klesh and P T Kabamba ldquoSolar-powered aircraft Energy-optimal path planning and perpetual endurancerdquo Journal of GuidanceControl and Dynamics vol 32 no 4 pp 1320ndash1329 2009

[2] S C Spangelo and E G Gilbert ldquoPower optimization of solar-powered aircraft with specified closed ground tracksrdquo Journal ofAircraft vol 50 no 1 pp 232ndash238 2012

[3] R W Beard and T W McLain Small Unmanned Aircraft Theoryand practice Princeton University Press 2012

[4] B L Stevens F L Lewis and E N Johnson Aircraft Control andSimulation Dynamics Controls Design and Autonomous SystemsJohn Wiley amp Sons 2015

[5] United States Committee on Extension to the Standard AtmosphereUS Standard Atmosphere 1976 National Oceanic and AtmosphericAdministration 1976

[6] D Colas N Roberts and V Samuel ldquoOpen sourcingfacebookrsquos solar-powered aircraft design toolsrdquo 2018 [Online]Available httpscodefbcomconnectivityopen-sourcing-facebooks-solar-powered-aircraft-design-tools

[7] J D Anderson Introduction to Flight McGraw-Hill Education 2011[8] I Reda and A Andreas ldquoSolar position algorithm for solar radiation

applicationsrdquo Solar Energy vol 76 no 5 pp 577ndash589 2004[9] G S Aglietti S Redi A R Tatnall and T Markvart ldquoHarnessing

high-altitude solar powerrdquo IEEE Transactions on Energy Conversionvol 24 no 2 pp 442ndash451 2009

[10] J T Betts ldquoSurvey of numerical methods for trajectory optimizationrdquoJournal of Guidance Control and Dynamics vol 21 no 2 pp 193ndash207 1998

[11] J A Nelder and R Mead ldquoA simplex method for function minimiza-tionrdquo The Computer Journal vol 7 no 4 pp 308ndash313 1965

[12] J H Mathews K D Fink et al Numerical Methods Using MATLABPearson Prentice Hall 2004 vol 4

[13] S Kurnaz O Cetin and O Kaynak ldquoAdaptive neuro-fuzzy inferencesystem based autonomous flight control of unmanned air vehiclesrdquoExpert Systems with Applications vol 37 no 2 pp 1229ndash1234 2010

[14] K Tanaka An Introduction to Fuzzy Logic for Practical ApplicationsSpringer 1997