58th International Astronautical Congress , Hyderabad, India, 24 - 28 September 2007. Copyright IAF/IAA. All rights reserved.

58th International Astronautical Congress 2007

Andrew Klesh∗ and Pierre Kabamba†

SOLAR-POWERED UNMANNED AERIAL VEHICLES ON MARS: PERPETUAL ENDURANCEDepartment of Aerospace Engineering

University of MichiganAnn Arbor, Michigan, 48109-2140

United States of Americaaklesh@umich.edu, kabamba@umich.edu

Abstract

The problem of quantifying a design requirement for perpetual endurance of solar-powered unmannedaerial vehicles (UAVs) is treated. The design requirement is formulated as a threshold of the Power Ratio,a non-dimensional number that characterizes the ability of an aircraft to fly while solar-powered. This so-called Perpetuity Threshold is identified, derived and shown to be related to the duration of daylight andlength of the solar day. A lower bound on the Perpetuity Threshold is given and comparisons are madeof this threshold between Earth and Mars. To illustrate the difference of requirements for perpetual flightbetween these two planets, specific aircraft examples are characterized with respect to perpetual flight anddiscussed.

1 Introduction

Future exploration of Mars, laid out by the Visionfor Space Exploration, requires long endurance un-manned aerial vehicles (UAVs) that use resourcesthat are plentiful on Mars. One possible way ofachieving these objectives is to have solar-poweredUAVs that fly perpetually. This motivates theproblem solved in this paper. The aircraft dis-cussed here are equipped with solar cells on theupper surface of the wings as well as onboard en-ergy storage.

This paper quantifies the requirement for perpet-ual endurance in solar-powered flight. Perpetualendurance is the ability of a UAV to collect moreenergy from the sun than it loses in flying over theduration of a solar day. This problem features theinteraction between three subsystems: energy col-lection, energy loss and solar elevation. While thecurrent literature discusses methods to optimizeUAV aerodynamic design for energy usage, there isno approach that specifically quantifies the require-ment for perpetual solar-powered flight in terms ofaircraft and environmental parameters. The pur-pose of this paper is to identify this requirement

∗Ph.D. Candidate, Dept. of Aerospace Engineering.†Professor, Dept. of Aerospace Engineering.

and show its applicability to solar-powered aircraftdesign.

Although the current literature on solar-poweredUAVs does not consider perpetual flight require-ments, a substantial body of work is available onthe design and analysis of solar-powered aircraft. Abrief review of this literature is as follows. The fea-sibility of solar-powered flight is reviewed in Refs.1-2 with a reference to Dr. A. Raspet’s pioneeringproposal of solar-powered flight in 1954. Hence,solar-powered aircraft have only appeared recentlyand their history is discussed in Refs. 6, 9, and 26.The general history and methods for design andanalysis of solar-powered aircraft are discussed inRefs. 3-27. References 12, 15 and 25 are unique inthat they use an optimization procedure to designthe aircraft based upon expected maneuvers andsunlight availability.

Optimal path planning for solar-powered aircraftis qualitatively discussed in the literature. Mis-sion design is found in Refs. 27-30 with particu-lar emphasis on where and when to fly. In mostreferences, efficiency through preliminary design isemphasized. Alternative methods to increase effi-ciency for solar-powered aircraft are discussed inRefs. 43-45. Reference 45 is of particular impor-tance as it achieves a 30% increase in efficiency

1

by improving the cooling of solar cells. However,nowhere in the literature is there a study quantify-ing the requirement for perpetual flight.

Solar-powered aircraft have many potential uses inexploration and civilian applications. References31-35 propose innovative designs for the use of solarpowered aircraft on Mars and Venus. In Refs. 36-41, additional proposals are made for high altitudewireless communication platforms and other uses.

The work in Ref. 57 does compare flight on Mars toflight on Earth, but does not take into account par-asitic drag. Moreover, it does not give an analyticsolution to the perpetual flight problem

The present paper makes the following two originalcontributions. First, it presents an analytic con-dition for perpetual endurance accounting for lo-cation, environment and aircraft parameters. Thecondition is in terms of the so-caled Power Ratiointroduced in Ref. 54, and requires this dimension-less number to exceed a threshold. Second, thispaper proves that the requirements for perpetualsolar-powered flight on Mars are always signiviantlymore stringent than those on Earth, implying thatsolar-powered UAVs for Mars exploration must sat-isfy tighter design specifications than their Earth-bound counterparts.

The remainder of the paper is as follows. In Sec II,the model is presented. In Sec. III, the perpetuityproblem is formulated. In Sec. IV, the PerpetuityThreshold is proven. In Sec. V, perpetuity is com-pared between Earth and Mars. Section VI pro-vides examples of aircraft and assesses their abilityto fly perpetually. Section VII provides conclusionsand discusses future work. Derivation of the solarincidence angle, proof of the optimality of level sun-seeking flight and properties of the aircraft modelsare given in Appendices A, B, and C respectively.

2 Modeling

The model consists of three parts: the energy col-lection model, the energy loss model and the solarelevation model.

2.1 Energy Collection Model

The aircraft is equipped with solar cells, mountedon the top side of the wings, and gains solar energyfrom the sun shining on the cells. Let a and e rep-resent the azimuth and elevation angles of the sun,respectively and let φ and ψ represent the bank andheading angles of the aircraft, respectively. Assum-ing that the wing configuration has zero dihedralangle, the incidence angle of the sun rays upon thesolar cells, θ, satisfies:

cos(θ) = cos(φ) sin(e)− cos(e) sin(a− ψ) sin(φ),(1)

which is derived in Appendix A. The power col-lected by the aircraft is:

Pin = ηsolPsdS cos(θ), (2)

where ηsol is the efficiency of the solar cell, Psd isthe solar spectral density, and S is the total surfacearea of the wing. If less than a full wing is coveredby solar cells, ηsol can be adjusted to account forthis decrease in solar cell area.

We will assume φ = 0 for the remainder of this pa-per, as justified in Appendix B. Equation (2) sim-plifies as

Pin = ηsolPsdS sin(e). (3)

During a time interval [to, tf ], the energy collectedby the aircraft is:

Ein =∫ tf

to

Pin(t)dt. (4)

2.2 Energy Loss Model

Energy lost by the craft is derived from standardlift, drag and propulsion models (See Ref. 53),assuming constant altitude, thrust and velocity.The constant altitude assumption requires L = Wwhere L is the lift and W is the weight of the air-craft. The equations governing the power lost driv-ing the propeller, Pout, are:

Pout =TV

ηprop, (5)

where,

2

T = D, (6)

D = 1/2ρV 2SCD, (7)

CD = CDO +KC2L, (8)

CL =2WρV 2S

, (9)

(10)

where T is the thrust of the aircraft, V is its veloc-ity, D is the drag, ηprop is the efficiency of the pro-peller, CD is the coefficient of drag, CDO is the par-asitic drag, CL is the coefficient of lift, the aerody-namic coefficient K represents the amount wherebythe induced drag exceeds that of an elliptical liftdistribution and ρ is the atmospheric density. Thenumerical values of the aircraft parameters used inthis paper are presented in Appendix C.

During a time interval [to, tf ], the energy lost bythe aircraft is:

Eout =∫ tf

to

Pout(t)dt. (11)

2.3 Solar Elevation Model

Assume that the aircraft flies in the atmosphere of aplanet that (1) is in a circular orbit around the sun,and (2) has a spin axis that is inclined with respectto its orbital plane. Define [I] = [xI , yI , zI ]T asan inertial orthonormal vectrix fixed in the orbitalplane, where xI is a unit vector along the line ofspring equinox, yI is in the orbital plane and zI isdefined so that the orbital angular motion is along+zI . If Ω is the angular velocity of orbital motionand s is a unit vector from the planet to the sun,we have

s = −[xI cos Ωt+ yI sinΩt]= −(cos Ωt, sinΩt, 0)[I]. (12)

Define [P ] = [xP , yP , zP ]T as a planet-fixed vectrixwhere xP at t = 0 is a unit vector pointing along theline of spring equinox, yP is in the equatorial planeand zP is defined so that the planet’s spin axis isalong +zP . If ω is the spin rate of the planet and iis the constant inclination of its spin axis, then

[P ] = R3(ωt)R1(i)[I], (13)

where the elementary rotation matrices R3 and R1

are defined as

R3(α) =

cosα sinα 0− sinα cosα 0

0 0 1

, (14)

R1(α) =

1 0 00 cosα sinα0 − sinα cosα

. (15)

Now, define [l] = [xl, yl, zl]T as a vectrix fixed atthe aircraft’s location, where xl is along the unitvector towards the local north, yl is along the unitvector towards the local west and zl is along theascending vertical. Then, the vectrices [P ] and [l]are related by

[l] = R3(π)R2(π

2− λ)R3(γ)[P ], (16)

where, λ is the latitude, γ is the longitude and theelementary rotation matrix R2 is defined as

R2(α) =

cosα 0 − sinα0 1 0

sinα 0 cosα

. (17)

The elevation of the sun is defined as

e = arcsin(s · zl). (18)

From (13),

[I] = R1(−i)R3(−ωt)[P ], (19)

and from (16),

[P ] = R3(−γ)R2(λ−π

2)R3(−π)[l]. (20)

Therefore, an expansion of (12) gives

s = −(cos(Ωt), sin(Ωt), 0)R1(−i)R3(−ωt)R3(−γ)

R2(λ−π

2)R3(π)[l]

= −(cos(Ωt), sin(Ωt), 0)R1(−i)R3(−ωt− γ)

R2(λ−π

2)R3(π)[l]. (21)

Finally, from (18) and (21), the elevation of the sunat the location of the aircraft is

e = arcsin(−(cos(Ωt), sin(Ωt), 0)R1(−i)R3(−ωt− γ)

R2(λ−π

2)R3(π)[001]T ) (22)

= arcsin(sin(i) sin(λ) sin(Ωt)− cos(λ)[cos(Ωt) cos(γ + ωt)+ cos(i) sin(Ωt) sin(γ + ωt)]). (23)

3

By definition, sunrise is a time tr such that e(tr) =0 and e(tr) > 0. Sunset is a time ts such thate(ts) = 0 and e(ts) < 0. Daylight is the durationbetween a sunrise and the next sunset, i.e., ts −tr. The solar day, tsd, is the interval between twoconsecutive sunrises. The daylight duty cycle is theratio between daylight and solar day, i.e., ts−tr

tsd.

3 Problem Formulation

The problem presented in this paper is motivatedby the goal of extending the endurance of a solar-powered UAV beyond the duration of a solar day.The UAVs under consideration stay within thevicinity of their initial position; hence changes inlongitude and latitude are assumed negligible.

3.1 The Power Ratio

The Power Ratio is defined as

PR(e) =2ηpropηsolρPsdS2VPowermin sin(e)CDOρ

2S2V 4Powermin

+ 4KW 2, (24)

where

VPowermin = 4

√4KW 2

3CDoρ2S2. (25)

It is shown in Ref. 54 that this ratio determines theregime of energy-optimal flight for a solar-poweredUAV. It is also shown that energy optimal flight re-quires V = VPowermin which will be assumed hence-forth.

3.2 The Perpetuity Problem

To fly perpetually, an aircraft must collect moreenergy during daylight hours than it expendsthroughout a solar day. We hypothesize that thePower Ratio can be used as a predictor of an air-craft’s capability to fly perpetually. Specifically,we hypothesize that to achieve perpetual flight, thePower Ratio of an aircraft must exceed a thresholdvalue, the Perpetuity Threshold. Hence, the prob-lem solved in this paper is two fold:

1. Prove the existence of the Perpetuity Thresh-old, i.e., prove that if the Power Ratio is largeenough, perpetual flight is possible.

2. Evaluate analytically the Perpetuity Thresh-old and study its implications on perpetualflight on Earth and Mars.

4 Perpetuity Threshold

4.1 Proof of Existence and Derivation ofthe Perpetuity Threshold

For perpetual flight, it is required that, over theduration of a solar day, the energy collected by theaircraft exceed or be equal to the energy lost, i.e.,

EinEout

≥ 1. (26)

From (4) and (11), this is equivalent to:∫ ts

tr1

Pin(t)dt∫ tr2

tr1

Pout(t)dt≥ 1, (27)

where we need only consider the daylight hours forthe power collected. Here, tr1 is the time of sunriseon a solar day, ts is the next time of sunset, and tr2is the time of sunrise on the next solar day.

Using (3) and the time-invariance of Pout, (27) isequivalent to

ηsolPsdS

∫ tS

tR

sin(e(t))dt

Pouttsd≥ 1. (28)

Let e be the average elevation of the sun, i.e., let esatisfy:

sin(e) =1

ts − tr

∫ ts

tr

sin(e(t))dt (29)

We can simplify (28) as

ηsolPsdS sin(e(t))(ts − tr)Pouttsd

≥ 1. (30)

Comparing this to (24), this inequality is equivalentto:

PR(e) ≥ tsdts − tr

. (31)

Hence (31) solves the perpetuity problem by estab-lishing that:

4

1. There exists a threshold of Power Ratio forperpetuity,

2. This threshold of perpetuity is the reciprocalof the daylight duty cycle.

In summary, perpetual flight is possible if and onlyif the Power Ratio, evaluated at the average sunelevation, exceeds the reciprocal of the daylight dutycycle.

Remark (1) Note that the right hand side of (31)is always greater than or equal to 1. Therefore, per-petual flight always requires that the Power Ratio,evaluated at the average sun elevation, be greaterthan or equal to 1.

5 Comparison of Perpetuity onEarth and Mars

The results of Section IV provide a threshold, de-pendent upon location and time, that must be ex-ceeded by the Power Ratio for perpetual flight.Note that design parameters and environmental pa-rameters affect the Power Ratio. If we fix the designof the aircraft, we can examine the effect of envi-ronmental parameters on the Power Ratio. Fur-thermore, we can compare the design requirementsfor perpetual flight between Earth and Mars.

5.1 Comparative Analysis of the Perpe-tuity Thresholds on Earth and Mars

The Perpetuity Threshold has been shown to bethe ratio, tsd

ts−tr . We can compare this ratio, as afunction of mean anomaly Ωt and latitude, betweenEarth and Mars, as shown in Figs. 1 and 2. Notethat the Perpetuity Threshold approaches infinitywhen the daylight duty cycle approaches zero andwe have limited the plot to thresholds smaller thansix. The arctic regions are those latitudes above90 − i and below −90 + i. During part of theyear, these regions can have extended periods oftotal darkness or total light, that is, there are no lo-cal sunrises or sunsets for multiple rotations of theplanet. For areas of total light (i.e., arctic summer),the Power Ratio need only exceed 1 for perpet-ual flight. We have not considered regions of totaldarkness (i.e., arctic winter) in this paper because

requirements on battery size put this case outsidethe scope of practical solar-powered aircraft.

Table 1 compares the planetary characteristics ofEarth and Mars.

Table 1: Planetary Characteristics of Earth andMars

Earth MarsDuration of Solar Day tsd 23.93 24.62 hoursDays in Year 365.25 687 daysInclination of Axis i 23.5 25 deg

Figure 1: Map of Perpetuity Threshold on Earth

The maximum deviation between the thresholds onEarth and Mars is 6.3% with a mean deviation of−5 × 10−2%. The Perpetuity Thresholds betweenEarth and Mars are therefore similar since the max-imum and average deviations are small.

5.2 Comparative Analysis of the PowerRatios on Earth and Mars

The Power Ratio, (24), can be rewritten as

PR =[0.402Psdηsol sin(e)

√ρg

32

] 4

√η4propb

6ε3π3S3

CDom6

,(32)

highlighting the distinct roles of environmental andaircraft parameters, respectively. Furthermore, ifa constant loading and thickness is assumed acrossthe wing planform, we can setm = ρwS where ρw is

5

Figure 2: Map of Perpetuity Threshold on Mars

the mass per unit area of the wing. This simplifies(32) as

PR =[0.402Psdηsol sin(e)

√ρg

32

] 4

√η4propε

3π3AR3

CDoρ6w

,(33)

which indicates that to increase the Power Ratio,a low wing density and high aspect ratio should beused.

To compare the power ratios of a given aircraft onEarth and Mars, rewrite (33) as

PR = 0.402ηsol sin(e) 4

√η4propε

3π3AR3

ρ6

(Psd

√ρ

g32 4√CDo

),

(34)

where Psd, ρ, g and CDo are all determined by theplanet. Note that CDo depends on the Reynoldsnumber of the aircraft, which itself depends uponviscocity and atmospheric density. The contribu-tion of Psd, ρ, g and CDo to the Power Ratio (34)is the term

Psd√ρ

g32 4√CDo

, (35)

whose values on Earth and Mars can be compared.

Table 3 compares the constant environmental pa-rameters, Psd, ρ and g, on Earth and Mars.

Note that the quantity Psd√ρ

g32

is 4.9 times larger on

Earth than on Mars.

Table 2: Environmental Parameters on Earth andMars

Earth MarsPsd 1,353 589 W/m2

g 9.86 3.71 m/sec2

ρ 1.29 0.015 kg/m3

Psd√ρ

g32

49.63 10.10 ( kgm3 )

32

We must additionally consider CDo in (35). SinceCDo depends upon velocity, atmospheric densityand viscocity, a comparison of its values on Earthand Mars is not straightforward. Fig. (2) illus-trates this comparison over a range of speeds.

Figure 3: Comparison of CDo on Earth and Mars

Note that CDo is always smaller for Earth than forMars.

Combining the results of Table 3 and Fig. 3 leadsto the following conclusion: The Power Ratio of anaircraft on Earth is always at least 4.9 times largerthan the Power Ratio of the same aircraft on Mars.

5.3 Comparison of Requirements forPerpetual Flight on Earth and Mars

The results of subsections V.A. and V.B. allow us tomake the following general statement: For a givenlatitude and date, it is always easier to design anaircraft to fly perpetually on Earth than on Mars.Several items contribute to this statement:

1. The Perpetuity Thresholds for a given date

6

and latitude are almost identical on both plan-ets

2. The contribution of environmental parametersin the Power Ratio is always at least 4.9 timeslarger on Earth than on Mars.

6 Examples

The University of Michigan SolarBubbles StudentTeam has been designing, building and testingan aircraft for solar-powered flight. It is namedHuitzilopochtli, or Hui for short, after the sun godof the Aztecs, and is a glider-based aircraft used forengineering education and as an autonomous vehi-cle test platform. Its primary area of flight is nearthe University of Michigan, at a latitude of 42.22Ndeg and longitude of -83.75W deg. We will assumean arbitrary flight date of August 6th.

From the analysis in section IV, the following pa-rameters can be evaluated:

Table 3: Perpetuity ParametersMean Anomaly Ωt 136 degDuration of Solar Day tsd 24 hoursDuration of Daylight ts − tr 14.2 hoursPerpetuity Threshold PT 1.7Average Elevation e 45.95 deg

Appendix C provides the design parameters for theHui aircraft. The Power Ratio of Hui on Earth is8.86. Since this Power Ratio exceeds the PerpetuityThreshold in Table 3, Hui is capable of perpetualflight on Earth. Moreover, the Power Ratio of Huion Mars is 1.8. Since the Perpetuity Thresholds onEarth and Mars are very similar, we conclude thatthe Hui is also capable of perpetual flight on Mars.

The Gossamer Penguin was the first manned,solar-powered aircraft. Built in 1979, and basedupon the Pathfinder solar panel, this aircraft wasflown several times across the Mojave desert. Thedesign parameters and assumptions about this air-craft are shown in Appendix C. If we again comparethe Power Ratio of this aircraft between Earth andMars we find that it is 1.03 on Earth while only0.21 on Mars. Hence, the Gossamer is capable ofsolar flight on Earth but not on Mars.

The above examples illustrate that some aircraft

are quite capable of perpetual flight on both Earthand Mars, but others can only fly on Earth.

7 Conclusions

This paper has investigated the requirement that asolar-powered aircraft must meet to achieve perpet-ual flight, i.e., to achieve a positive energy balance,Ein − Eout ≥ 0, over the course of a solar day.

The requirement for perpetual solar-powered flightis formulated in terms of the power ratio, whichis a dimensionless parameter that depends on theaircraft configuration and the environment. Specif-ically, perpetual solar-powered flight is achievableif and only if the power ratio, evaluated at the av-erage sun elevation, is greater than or equal to thereciprocal of the daylight duty cycle. The iden-tification of this so-called perpetuity threshold asthe requirement for perpetual flight constitutes themain original contribution of this paper.

The analytical approach of this paper allows a com-parison of the requirements for perpetual solar-powered flight on Earth and Mars. It is foundthat, in terms of the power ratio, perpetual solar-powered flight on Mars is always at least 4.9 timesmore restrictive than it is on Earth. This result sug-gests tight design specifications for solar-poweredUAVs for Mars exploration.

In future work, the perpetuity threshold will be val-idated through flight tests of a solar-powered UAV.

Appendix A: Derivation of the So-lar Incidence Angle

Define [g] = [xg, yg, zg]T as a vectrix (See Ref. 49)fixed to the ground with zg vertical ascending. Ifa and e are the azimuth and elevation of the sun,then s, the unit vector to the sun, is given as

s = [g]T

cos(e) cos(a)cos(e) sin(a)

sin(e)

. (36)

Define [a] = [xa, ya, za]T as a vectrix fixed to theaircraft. In terms of [g],

[a] = R1(φ)R3(ψ)[g], (37)

7

where,

R1(φ) =

1 0 00 cos(φ) sin(φ)0 − sin(φ) cos(φ)

, (38)

R3(ψ) =

cos(ψ) sin(ψ) 0− sin(ψ) cos(ψ) 0

0 0 1

, (39)

represent rotation matrices about the first andthird axis, respectively. By inverting this relation-ship, we obtain

[g] = R3(ψ)TR1(φ)T [a]. (40)

Hence, s can be expressed, in the aircraft-fixed vec-trix, as

s =

cos(e) cos(a)cos(e) sin(a)

sin(e)

R3(ψ)TR1(φ)T [a]. (41)

Define the incidence angle θ as the angle betweenthe line-of-sight to the sun and the z-axis of theaircraft-fixed vectrix. Then θ = arccos(s · za).Hence, (41) yields:

cos(θ) = cos(e) cos(a) sin(ψ) sin(φ) + sin(e) cos(φ)− cos(e) sin(a) cos(ψ) sin(φ), (42)

or, after applying trigonometric identities,

cos(θ) = sin(e) cos(φ)− cos(e) sin(φ) sin(a− ψ).(43)

Appendix B: Energy-OptimalFlight Paths with Free Destina-tion

The purpose of this section is to show that φ = 0yields paths that are extremal (i.e., satisfy the firstand second order necessary conditions for optimal-ity) with respect to the problem of energy-optimalflight with free destination. To do so, we will: for-mulate a model without restrictions on φ, formu-late an energy optimal path-planning problem withfree destination, formulate the first and second or-der necessary conditions for optimality, and showthat φ = 0 yields a path that satisfies the first andsecond order necessary conditions.

B.1 Modeling with φ 6= 0

In this section we present the full model of the sys-tem without assuming that φ = 0. The model con-sists of three parts: the aircraft kinematic model,the energy collection model and the energy lossmodel. Each subsystem is presented below.

B.2 Aircraft Kinematic Model

The bank-to-turn aircraft is assumed to fly in stillair and remain at constant altitude, with zero pitchangle, according to the equations:

x = V cosψ, (44)y = V sinψ, (45)

ψ =g tanφV

, (46)

where x and y are the Cartesian coordinates of theaircraft and ψ is the heading angle.

B.3 Energy Collection Model

The energy collection model is similar to that inSection II.A, except that:

Pin = ηsolPsdS(cos(φ) sin(e)− cos(e) sin(a− ψ) sin(φ)).(47)

B.4 Energy Loss Model

The energy loss model is similar to that in SectionII.B, except that:

CL =2W

ρV 2S cosφ. (48)

B.5 Model Summary

In summary, the integrated model is as follows.The bank angle determines the heading and the po-sition of the aircraft through (44)-(46). This bankangle, combined with the sun’s position determinesthe incidence angle through (47). The incidenceangle of the sun together with the bank angle andspeed determine the energy collected and lost bythe aircraft during flight through (4) and (11), re-spectively.

8

B.6 Dynamic Optimization Problem

The dynamic optimization problem presented inthis section is motivated by the requirement tomaximize, with respect to the time-histories of thebank angle and speed, the final energy of the solar-powered aircraft, i.e.,

maxφ(·),V (·)

(Ein − Eout), (49)

subject to dynamics and boundary conditions.

In practice, a larger value of objective function (7)increases the endurance of the aircraft.

The boundary conditions for this problem are

x(to) = xo, (50)y(to) = yo, (51)ψ(to) = free, (52)x(tf ) = free, (53)y(tf ) = free, (54)ψ(tf ) = free. (55)

B.7 Optimal Path Planning

In this section, we derive the necessary conditionsfor optimality that characterize the optimal trajec-tories.

The necessary conditions for optimality for themaximization problem (7) are derived in Ref. 46.Here, these necessary conditions are applied to thecurrent problem. With states [x, y, ψ]T and controlinputs (φ, V )T the Hamiltonian is:

H(x, y, ψ, λx, λy, λψ, φ, V ) = Pin − Pout + λxV cosψ

+λyV sinψ + λψg tanφV

, (56)

where λx, λy, and λψ are the costates. In this prob-lem, the only control constraints are that |φ| < π

2and V > 0. While velocity is constrained by per-formance of the engine, altitude, etc, it is assumedthat the aircraft can at least fly at the minimum-power velocity (Ref. 53):

VPowermin = 4

√4KW 2

3CDoρ2S2 cos2(φ). (57)

In our experience, it is not necessary to impose atight constraint on the magnitude of the bank an-gle. Indeed, banking requires lifting (See Eq. (48)),

lifting induces drag (See Eq. (8)), drag requiresthrust (See Eq. (6)), which implies power loss (SeeEq. (11)). Since the path planning aims at achiev-ing optimal final energy, the magnitude of the bankangle is naturally limited by these phenomena.

The state equations, derived from (56), are:

x =∂H

∂λx= V cos(ψ), (58)

y =∂H

∂λy= V sin(ψ), (59)

ψ =∂H

∂λψ=g tan(φ)

V. (60)

The costate equations are:

λx =−∂H∂x

= 0, (61)

λy =−∂H∂y

= 0, (62)

λψ =−∂H∂ψ

= −λyV cosψ + λxV sinψ

− ηsolPsdS cos e cos (a− ψ) sinφ. (63)

The first-order optimality conditions are:

∂H

∂φ= −ηsolPsdSc(cos(e) cos(φ) sin(a− ψ) + sin(e) sin(φ))

− 4KW 2 sin(φ)ηpropρSV cos3(φ)

+gλφ

V cos2(φ)= 0, (64)

∂H

∂V= λx cos(ψ) +

8KW 2 sec(φ)2

ηpropρSV 2

−3ρSV 2(CDo + 4KW 2 sec(φ)2

ρ2S2V 4

2ηprop+ λy sin(ψ)

−gλψ tan(φ)

V 2= 0. (65)

The second order Legrendre-Clebsch condition isthat the Hessian of the Hamiltonian be negativesemi-definite, i.e.:

∂2H

∂(φ, V )2≤ 0, (66)

where, if

∂2H

∂(φ, V )2=[Hφφ HφV

HφV HV V

], (67)

(68)

9

Hφφ = ηpropηsolρPsdS2V (−(cos(φ) sin(e))+cos(e) sin(a−ψ) sin(φ))

ηpropρSV+

−4KW 2 sec(φ)4+2 sec(φ)3(gλψηpropρS cos(φ)−4KW 2 sin(φ)) tan(φ)ηpropρSV

,

HφV = −gλψ sec(φ)2

V 2 + 4KW 2 sec(φ)2 tan(φ)ηpropρSV 2 ,

HV V = 8KW 2 sec(φ)2

ηpropρSV 3 −3ρSV (CDo+

4KW2 sec(φ)2

ρ2S2V 4 )

ηprop+

2gλψ tan(φ)

V 3 .

The boundary conditions for this problem are:

x(to) = xo, (69)y(to) = yo, (70)λψ(to) = 0, (71)λx(tf ) = 0, (72)λy(tf ) = 0, (73)λψ(tf ) = 0. (74)

B.8 Satisfaction of the Conditions for Op-timality

We will show in this section that φ = 0 yields a paththat satisfies the first and second order necessaryconditions for optimality.

If φ = 0, the state equations become:

x =∂H

∂λx= V cos(ψ), (75)

y =∂H

∂λy= V sin(ψ), (76)

ψ =∂H

∂λψ= 0, (77)

showing that the heading is constant.

The first two costate equations are:

λx =−∂H∂x

= 0, (78)

λy =−∂H∂y

= 0, (79)

which, combined with the boundary conditions(50)-(55) show that λx = λy ≡ 0. This simplifiesthe last costate equation as:

λψ =−∂H∂ψ

= 0, (80)

which, combined with the boundary conditions(50)-(55) shows that λφ ≡ 0.

After the above simplifications, (64) reduces to:

∂H

∂φ= −ηsolPsdSc cos(e) sin(a− ψ) = 0, (81)

which is satisfied by letting ψ = a, i.e., by lettingthe aircraft head in the direction of the sun.

Similarly, (65) reduces to:

∂H

∂V=

8KW 2

ηpropρSV 2−

3ρSV 2(CDo + 4KW 2

ρ2S2V 4 )

2ηprop= 0.

(82)

which is only satisfied if V = VPowermin .

The second order condition reduces to:

∂2H

∂(φ, V )2≤ 0, (83)

where, if

∂2H

∂(φ, V )2=[Hφφ HφV

HφV HV V

], (84)

(85)

Hφφ = ηpropηsolρPsdS2V (− sin(e))

ηpropρSV+ −4KW 2

ηpropρSV,

HφV = 0,

HV V = 8KW 2

ηpropρSV 3 −3ρSV (CDo+

4KW2

ρ2S2V 4 )

ηprop.

It is easily checked that choosing φ = 0 and V =VPowermin yields Hφφ < 0 and HφφHV V > 0, im-plying that the second order necessary condition issatisfied.

In summary, choosing φ = 0, ψ = a and V =VPowermin yields a path that satisfies the first andsecond order conditions for optimality for problemsubject to dynamics and boundary conditions.

Appendix C: Aircraft Model Pa-rameters

Hui was developed by the University of MichiganSolarBubbles Team (See Refs. 50-52). Aerody-namic coefficients were evaluated through the useof Athena Vortex Lattice and Fluent (Ref 52).

The glider aircraft, pictured in Fig. 4, has the char-acteristics listed in Table 4.

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Table 4: Hui Model ParametersWing Area S 1.25 m2 Ref(52)Mass m 1.95 kg Ref (52)Wingspan b 3.01 m Ref (52)Oswald Eff. Factor ε 0.9139 Ref (52)Parasitic Drag CDo 0.0065 Ref (52)Propeller Eff. ηprop 0.7 (est)

Figure 4: The Huitzilopochtli Aircraft

The Gossamer Penguin was developed by AeroVi-ronment in 1979 as a manned solar-powered air-craft. Approximate aerodynamic coefficients asfound in Refs (4) and (56) are listed in Table 5and the aircraft is pictured in Fig. 5.

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Table 5: Gossamer Penguin Model ParametersWing Area S 30.85 m2 Ref (4)Mass m 67.13 kg Ref (56)Wingspan b 21.6 m Ref (56)Oswald Eff. Factor ε 0.94 (est)Parasitic Drag CDo 0.01 (est)Propeller Eff. ηprop 0.7 (est)

Figure 5: The Gossamer Penguin

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