Neutral Energy Cycles for a Vehicle in Sinusoidal and

Turbulent Vertical Gusts

Lissaman P. B. S.∗ and Patel C. K.†

Energy can be extracted from natural atmospheric motions, even those with no meanvertical speed component. This procedure is extensively exploited by birds. Natural windfluctuations can be of the order of 2 m/s. For Unmanned Air Vehicles (UAVs) with flightspeeds in the range of 10 m/s, or less than about five times natural wind fluctuations,there is potential for significant energy savings. A flight maneuver in which a vehicle“rides” atmospheric fluctuations so as to complete a cycle at the same speed and altitudeas its initial state is defined as an energy neutral cycle. Such a process can be sustainedindefinitely while the atmospheric conditions maintain. Neutral energy cycles are developedfor a vehicle flying through a vertical gust of sinusoidal shape. Control laws for the liftvector are developed for neutral energy cycles by formulating the problem as a parametricoptimization problem with realistic constraints. Results for neutral trajectories for differentgust strengths and time periods are shown. Cases are given for control defined by: a locallyvariable active random input, a sinusoidal input and a sinusoidal input with linearizedsolution. Results show that the primary parameter for achieving a neutral energy is themagnitude of the maximum lift-to-drag ratio of the vehicle, that must be large. For exampleat a lift-to-drag ratio of 20, a vehicle can sustain energy neutral cruise in a gust of about20% of the cruise speed with proper control inputs. An instrumented flight test vehicle of2.0 m span and 0.48 kg mass has been designed and flown. Flight test results (not availableto date) will be used to evaluate the estimates made here and to determine the differencesfrom the analytical model in actual atmospheric conditions with practical constraints.

Nomenclature

(U, W ) Components of non-dimensional inertial velocity(X, Z) Non-dimensional spatial co-ordinatesα Angle between the airspeed vector and the horizontal planeΦ Phase angle of a sinusoidℜ{. . .} Real part of a complex entity~x Vector of state variables~Xdes Vector of design variablesCLcr Lift coefficient of the vehicle at cruiseG Maximum lift to drag ratio of the vehicleg Acceleration due to gravityL Ratio of lift coefficient to that at cruiseM Number of equal time intervalsN Ratio of vertical resultant aerodynamic force to that at cruiseN0 Constant term in N(T )N1 Amplitude of the sinusoidal term in N(T )Q Ratio of dynamic pressure to that at cruiseT Non-dimensional timeTg Non-dimensional gust period

∗President, Da Vinci Ventures, Santa Fe, NM 87505, AIAA Fellow†Doctoral Candidate, Stanford University, CA, AIAA Student Member

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45th AIAA Aerospace Sciences Meeting and Exhibit8 - 11 January 2007, Reno, Nevada

AIAA 2007-863

Copyright © 2007 by Dr. P. B. S. Lissaman and C. K. Patel. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Vcr Cruise speed of the vehicleWg Non-dimensional vertical gust velocityWg0 Amplitude of sinusoidal vertical gust velocity

I. Introduction

Leonardo da Vinci described the maneuvers of birds in natural winds in 1502, at the time that he wasconducting experiments aimed at building a man-carrying flying machine. He noted their ingenuity in

using air movements to assist their flight [Domenico, 2004]. Lord Rayleigh [1889] described how the albatrossof the Southern Ocean can maneuver to maintain altitude without flapping in winds with only horizontalflows, provided wind shear is present. Vehicles with low cruise speeds, of the order of 10 m/s, such as birdsand small Unmanned Aerial Vehicles (UAVs), can benefit significantly from the use of atmospheric energypresent in the form of spatial wind variation or turbulence and can extract energy in natural flows with nomean vertical component.

The dynamic soaring maneuver of pelagic birds, typically the albatross in the boundary layer of theSouthern Ocean, is used to traverse thousands of kilometers with minimal energy expenditure. Their flightpath has been studied by Sachs [1994] and Lissaman [2005]. In an extended analysis, Sachs and Mayrhofer[2002] analyzed the case of non-uniform wind shear, as obtains on the lee slopes of hills. Sachs developed avariational procedure to optimize the control schedule for a vehicle operating without propulsive power in alinear wind profile (uniform gradient). He derived definitive results for the minimum wind gradient requiredto execute a flight path, called an energy neutral loop, in which the vehicle returns to the original height andspeed vector. For this case the vehicle does not return to its original position in inertial coordinates, butdrifts downwind of its origin. Lissaman has extended the analysis, showing that an energy neutral closedcycle can be flown such that the vehicle does, in fact, return to its original state and inertial starting pointbut requires a slightly more intense wind gradient than the canonical case with the downwind drift. Pateland Kroo [2006] analytically studied the response of a small UAV to the vertical component of atmosphericturbulence with an approximately Dryden spectrum and estimated the energy savings possible with the useof active control.

The primary case studied here is the response of a vehicle flying a wings level course in a sinusoidalvertical flow field, called the gust input. Such a flow is typical of one component of a turbulence field. Whena vehicle flies through a gust, even for no control input, the change in angle of attack will provide a thrusteffect. This phenomenon has been called the Katzmayr effect [1922], and has been analyzed quantitativelyfor the case of constant vehicle attitude by Phillips [1975]. For the Phillips case, the lift vector is passivelymodified according to the effective angle of attack due to both the gust and the vertical motion of the vehicle.However, if the magnitude of the lift vector can be actively controlled, the aircraft can extract significantamounts of energy from a gust, as shown by Patel and Kroo [2006] for the case of turbulent gust. Activecontrol of the lift is used by sailplane pilots to gain energy from rising air masses.

The fundamental principle is that, during the upgust portion, the lift vector has a forward componentalong the flight path tending to reduce propulsive requirements, while the opposite occurs for the downflowperiod. If the lift is constant through the gust, the net thrust due to these effects is zero. However, if thelift is increased during the upgust portion and reduced during the downflow, then there is a net inducedcomponent in the direction of flight, which will reduce the flight power requirements. If sufficiently largethis effect can eliminate any required flight power, so that the cruise cycle can be repeated indefinitely. Thisstate is defined as an energy neutral cruise. Energy in natural turbulence is generally insufficient to permitfull-scale airplanes to achieve such a cruise cycle, but small UAVs, because of their slow flight speeds and lowenergy requirements can harness enough atmospheric energy to sustain flight if active control is exercised.Birds clearly do this.

The general rule for energy extraction is that the lift must increase during the upgust period and reduceduring the down. Lissaman [2005] described this in a simple but sufficient ‘belly to the breeze’ rule. Thiscan be accomplished by some control that changes N , the vertical resultant normalized aerodynamic force,according to a sine-like schedule at the same frequency as the gust input. The method by which the N controlis accomplished is not discussed here, but in general terms it is approximated by the natural response of avehicle to the gust if the attitude of the vehicle is assumed constant. Active control of N can be provided

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by deflecting control surfaces such as elevator and flaps.

The problem solved here is an idealization of a general situation. The problem is posed as a parametricoptimization problem to determine an arbitrary N control sequence subject to the constraints of energyneutral cruise and the vehicle’s longitudinal equations of motion. The flight efficiency of the vehicle can beexpressed in a single term, the maximum lift-to-drag ratio of the vehicle, called G here. The magnitude ofthe gust for a neutral energy cruise can then be calculated. It is noted that in an upward gust, the vehicleaccelerates upwards, thus attenuating the relative vertical component of the gust, and reducing the favorablethrust-wise inclination of the lift vector. Gusts of longer period are less effective in producing thrust, sincethe vehicle has time to respond and hence experiences the gust less intensely. The procedure is then modifiedwith the assumption that the N control input also follows a sinusoidal curve with the same frequency asthat of the gust. Results for vehicles with different G and different gust periods are shown.

II. Basic Flight Equations

The following equations of motion are adapted from Lissaman [2005]. Cruise speed, Vcr, is definedhere as the speed at which the vehicle operates in minimum drag state. The dimensional quantities are non-dimensionalized using suitable combinations of Vcr and the gravitational acceleration, g. Velocity componentsare normalized by Vcr, distances by the factor V 2

cr/g, and time by Vcr/g. The drag polar is assumed to bequadratic, with a maximum lift to drag ratio of G. Components of vehicle speed, U and W , are defined aspositive for forwards and downwards motion respectively. Gust speed, Wg, is positive downwards.

The study considers only wings level flight, so that only the vertical and horizontal equations of motionare required. After non-dimensionalization they are listed below:

dU

dT= −

1

2G

(

Q +N2

Q

)

cosα − N sin α

dW

dT=

1

2G

(

Q +N2

Q

)

sin α − N cosα + 1

dX

dT= U

dZ

dT= W

The above equations of motion can be represented in a concise form as d~x(T )dT

= f (~x(T ), N(T )), where

~x(T ) is a vector of states, ~x(T ) = [U(T ), W (T ), X(T ), Z(T )]T, and N(T ) is the control input. In this

analysis, the vertical gust velocity is expressed as a pure sinusoid of the form Wg = Wg0 sin (2πT/Tg).

The non-dimensional dynamic pressure, Q, is given by Q = U2 + (W − Wg)2

and the angle α is definedas follows:

tanα = −W − Wg

U

III. Problem Formulation

The problem of finding the optimal control for the load, N , as a function of time is posed as a parametricoptimization problem that can be solved using a standard optimization package [Hull 1997, Speyer 1996].The time interval [0, Tg] is divided into M equal intervals using M + 1 nodes. In this formulation, the finaltime is taken to be equal to the period of the gust, Tg. The problem now is to find the minimum gustamplitude, Wg0, and control input, N(T ), required to perform energy neutral cycle cruise, subject to thevehicle’s performance parameters and equations of motion. A vector of 5M + 1 design variables is formedusing the state variables, control inputs and the amplitude of a sinusoidal gust, as shown below.

~Xdes = [~x(T1), ~x(T2), . . . , ~x(Tg), N(T1), N(T2), . . . , N(Tg), Wg0]T

The optimization problem can now be posed as: minimize Wg0, with respect to the above design vector,subject to the constraints shown in Table 1. Constraints on the initial and final conditions are chosen so as to

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Initial conditions ~x(0) = [free, free, 0, 0]T

N(0) = free

Initialize the flight at the origin of the(X, Z) axes.

Final conditions ~x(Tg) = [U(0), W (0), free, 0]T

N(Tg) = N(0)dNdT

∣

∣

∣

Tg= dN

dT

∣

∣

∣

0

To ensure periodicity for energy neu-tral cycle.

Path constraints Lmin ≤ L(T ) ≤ Lmax

Nmin ≤ N(T ) ≤ Nmax

αmin ≤ α(T ) ≤ αmax

Path constraints imposed on the max-imum lift coefficient, the maximumload and the flight path angle.

Equations of motion d~x(T )dT

− f (~x(T ), N(T )) = 0 Using a collocation approach based onthe Simpson’s 1/3rd rule.

Table 1. List of constraints used in the optimization problem.

ensure periodicity of the solution. Path constraints are also imposed to account for realistic limitations on thecapabilities of the vehicle. The equations of motion are required to be satisfied at all the nodes throughoutthe time interval, and collocated using the Simpson’s 1/3rd rule. Suitable bounds are also placed on thedesign variables. Once the problem is posed in the above form, SNOPT is used to find the optimal solution.This formulation allows for arbitrary variation of the load, N(T ), subject to the relevant constraints. Thisrepresents the minimum gust solution to the problem. The results were verified by changing the initial guessin SNOPT and also by integrating the state variables forward in time using the optimal control functionsobtained from the optimizer.

IV. Case for Optimal Control Function

The procedure discussed in Section III was implemented for a case of G = 20, Tg = 4. The gust period wasdivided into M = 40 equal intervals using 41 nodes. The upper and lower bounds on the non-dimensionalizedlift coefficient, L(T ), were −0.5 and 1.5 respectively. Bounds of +/−3 and +/−40o were used for the N(T )and α(T ) respectively. The results for the optimal control function and state variables are shown in Figure 1.

Minimum Wg0 = 0.129

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4

Non-dimensional time, T

No

n-d

imen

sio

nal

qu

anti

ties

Wg

U

W

N

-Z

Figure 1. Time variation of parameters for neutral energy cycle

The procedure was repeated for several different gust periods and maximum values of lift to drag ratio,using the same constraints. Results from this study, shown in Figure 2, show the influence of gust period onthe minimum gust amplitude required to maintain energy neutral cruise.

Figure 3 clearly illustrates the influence of maximum lift to drag ratio, G, on the gust amplitude required

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Lmax = 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5 6 7 8Tg

Wg

0

G = 10

G = 15

G = 20

G = 25

Figure 2. Effect of gust period on minimum gust amplitude

for energy neutral cruise. As expected, higher aerodynamic efficiency enables the vehicle to maintain itsenergy level in gusts of smaller amplitude.

Lmax = 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

10 12 14 16 18 20 22 24G

Wg

0

Tg = 2

Tg = 4

Tg = 6

Tg = 7

Figure 3. Effect of aerodynamic efficiency on gust amplitude

V. Case for Sine Control Function

This section studies the effect of representing the optimal control function as a pure sinusoidal variation.This does not lead to the minimum gust solution, but an analytic representation of the control functionis possible. For the results presented in this section, the vector of design variables was modified as shownbelow:

~Xdes = [~x(T1), ~x(T2), . . . , ~x(Tg), N0, N1, Φ, Wg0]T

The control function is written as N(T ) = 1 + N0 + N1 sin (2πT/Tg + Φ), permitting modification ofthe amplitude and phase of the control, and shifting it relative to the 1-g condition. The optimizationprocedure remains the same as in Section IV. The resulting optimal solution and variation of the statevariables is presented in Figure 4. The optimal sinusoidal control function was found to be N(T ) = 1 +0.104 + 1.009 sin (2πT/Tg + 0.114). It is seen that the vertical velocity, W , is quite close to a sinusoid, as itin fact would be exactly for a linearized solution.

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Minimum Wg0 = 0.135

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

Non-dimensional time, T

No

n-d

imen

sio

nal

qu

anti

ties

Wg

U

W

N

-Z

Figure 4. Time variation of parameters for neutral energy cycle for a sinusoidal control function

Figure 5 shows a comparison of the optimal and sinusoidal control functions for G = 20 and Tg = 4. Theassociated values of the minimum gust amplitude required for energy neutral cruise are also shown. It canbe seen that the minimum gust amplitude obtained by following the optimal control function in Section IVis not much lower than what can be achieved by using a sinusoidal control function. It is noted how thatthe optimal control tends to dwell near the higher levels more than the sine control, thus requires higher gloads than the sinusoidal control.

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4T

N

Optimal N, Min Wg0 = 0.129

Sinusoidal N, Min Wg0 = 0.135

Figure 5. Optimal control schedule compared with sine model

VI. Linearized Energy Extraction Case

It is useful to develop a linearized solution for application to cases of turbulence since this avoids havingto integrate for each turbulent trace individually, as is required for the exact non-linear response but permitsuse of the standard stochastic methods for deriving the global response properties of random input functions.This procedure has been used in a similar case by Phillips [1975].

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The equations of motion may be linearized with respect to U , W and N , and written as:

dU

dT= −

(1 + N1)

G− (1 + N1) (W − Wg)

dW

dT= −

(W − Wg) (1 + N1)

G− N1

If the vertical equation is further simplified by ignoring the second order term, (W − Wg)N1, the linearresult below is obtained

dW

dT= −

(W − Wg)

G− N1

The above linear vertical equation is solved for W for input of a sinusoidal gust, Wg and Normal forceratio, N1. For N1 = ℜ

{

N∗eiωT}

and Wg = ℜ{

W ∗eiωT}

, the solution is given by:

W = ℜ

{(

W ∗

G− N∗

)

eiωT

(iω + 1/G)

}

.

The solution for vertical speed, W , can be written as:

W =

(

W ∗

G− N∗

) {

G cosωT

(1 + G2ω2)+

G2ω sinωT

(1 + G2ω2)

}

,

where it is assumed that W ∗ and N∗ are real.

Integrating the horizontal equation over a gust cycle, Tg = 2π/ω, and defining K as the fraction ofthe gust energy extracted during a cycle compared with that required to execute the cycle with no energyextraction now gives:

−1

Tg

∫ Tg

0

(Wg − W ) (1 + N1)GdT = K

K = 1, therefore, represents a neutral energy cycle. It is noted that the energy addition due to the gustis a second order term, the product of vertical flow gust speed, Wg, and normal force 1 + N1. The energyabsorbed by induced drag is, of course, also a second order term, being the product of induced flow verticalspeed and normal force. This provides:

−W ∗ =2K

(

1 + G2ω2)

G3ω2N∗+

N∗

Gω2

Taking G = 20, and a pure sine control input with amplitude N∗ = 2, the neutral energy gust speed canbe calculated using linearized theory, and can be compared with the exact result as shown in Figure 6. It isseen that there is considerable error in using the linearized model.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8 10 12

Tg

W*

Linear

Exact

Figure 6. Neutral energy gust speed for linearized model, N∗ = 2

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The expression for the minimum gust speed contains a quadratic expression for N∗, indicating that thereis an optimum value of N∗ for which the required gust speed is minimized. This value can be determined forthe appropriate parameters by setting the partial derivative of W ∗ with respect to N∗ equal to zero. Thevalue of N∗ for optimum performance is a function of gust period. This is shown in Figure 7. It is seen thata longer periods, lower values of normal acceleration, are required. The linear analysis does not impose aconstraint on the value of N∗. It is seen that N∗ for the linear case does exceed the Nmax = 3 limit imposedduring the parametric optimization.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10 12

Tg

N*

Figure 7. Optimal normal acceleration, N∗, for linearized model, G = 20.

The associated minimum gust speed is shown in Figure 8 as function of gust period. This optimized N∗

case shows a slight benefit in the required gust intensity, than the cases drawn for fixed N∗ in Figure 6.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2 4 6 8 10 12

Tg

W*

Linear

Exact

Figure 8. Neutral energy gust speed for linearized model, optimum value of N∗, G = 20.

Results indicate that lower levels of gust intensity are required at lower gust periods and higher perfor-mance parameter, G, as is expected. However it is noted that there is a discrepancy between the level ofneutral energy gust intensity determined by the linearized analysis and the exact solution. This is contraryto expectations of results of linear and exact analyses.

VII. Turbulent Case

The turbulent case is of considerable interest for operation in natural winds. An analysis of this has beenperformed using turbulent vertical flows characteristic of natural winds at altitudes less than 200 m AGL.Patel and Kroo [2005] determined optimal control laws for energy extraction in turbulent gust generatedusing the Dryden spectrum. An important aspect of the procedure was to determine control laws that yieldhigh energy savings over a wide range of gust profiles. Results presented in their paper indicate averageenergy savings of about 40% for an actively controlled vehicle of G = 18.0 flying through a turbulent gustof non-dimensionalized RMS of 15.4%. These energy savings are in comparison to the vehicle flying throughthe same gust profiles without active control.

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VIII. Practical Constraints

The solutions determined using the parametric approach in sections IV and V include some of thefundamental constraints that a flight vehicle has to satisfy. Most real vehicles and birds experience threepractical constraints on maneuvers. The first is the N (or g-load) limit which is dictated essentially bywing strength, about 5.0 for sailplanes and probably less for birds. In the results presented here, N has beenlimited to +/−3.0. The second is a Qmax constraint, related to maximum dynamic pressure and its effect onsurface covering or feathers. For vehicles this will be directly related to the “Never Exceed” speed, defined bythe flight envelope. This constraint has been imposed through bounds on the components of vehicle speed.In the results presented here, this constraint was not found to be active at any instant. The third is an Lconstraint, related to maximum lift coefficient of the wing system. Here it is noted that CLmax of about 1.5is the order of magnitude characteristic of sailplanes, UAVs and birds at the appropriate Reynolds Number.A value of Lmax = CLmax/CLcr = 1.5 has been used for the results shown in this paper. It is found thatthe stall constraint is active for high frequency gusts.

In addition to the constraints mentioned above, a few other practical limitations need to be includedwhile applying this analysis to flight test vehicles. The vehicle has to sense, or otherwise infer, and processthe atmospheric gust levels. The control surfaces are then deflected to produce the desired lift to enablethe vehicle to gain energy from the gust. Constraints related to sensor and actuator delay and bandwidthlimitations may have an impact on the amount of energy gain. For a practical vehicle, it is necessary toinclude such constraints in the optimization procedure.

IX. Operational Significance for UAVs

A feature of this research is of considerable interest to the practical operation of modern UAVs. UAVscan always profit from reduced power requirements. Even if atmospheric disturbances are not sufficientlystrong for neutral energy cycles, a UAV can be controlled to execute a cyclic N control schedule, appropriateto the length scale of the ambient turbulence, and such a procedure has excellent prospects of significantlyreducing the cruise power requirements of the vehicle, increasing its range or endurance. It is relativelysimple with current technology to sense ambient turbulent qualities and to adjust flight control systems to”ride” this turbulence. Noting from the previous results that 100% reduction of flight power can be achievedin cases of strong turbulence, it is reasonable to suppose that 30% reductions could be achieved in typicalatmospheric conditions. Calculations will be made to determine the magnitude of this effect. It is also likelythat the optimal schedule for energy saving in powered flight will be different from that for neutral energycycles on gliding vehicles.

X. Experimental Validation

Figure 9. Instrumented vehicle for energy extraction flight tests.

A flight model of 2 m wingspan and 0.48 kg mass, with a cruise speed of 6 m/s and a maximum glide ratio

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of 16 : 1 is currently being flight tested at Stanford University to validate some of the results shown here. TheUAV, powered by an electric motor of 50.0 W rating, is of conventional configuration and is instrumentedwith a custom-built autopilot board with Global Positioning System (GPS), Inertial Measurement Unit(IMU), pressure sensors, and onboard computational capability with data telemetry. The electric motor isused only to position the UAV and retrieve it after a test flight run. Preliminary flight tests have indicatedthat the instrumentation and performance will be satisfactory. Results from the flight tests will be publishedat a later date. The main goals of the flight test vehicle are to verify the applicability of the Dryden windturbulence model to small UAVs, demonstrate the energy savings estimated using the control laws, andverify the control law design procedures. The flight vehicle prototype is shown in Figure 9.

XI. Conclusion

The results show that neutral energy cycles for a vehicle flying through a sinusoidal vertical gust can bedesigned using simple control laws, varying only N , the g load factor of the vehicle. The optimal controlschedule, to minimize the required gust amplitude has been developed. The results for a simple sine controllaw are quite close to those for the true optimal schedule. For relatively modest aerodynamic efficiency, G,of the level of say 20, the neutral gust intensities are relatively small, about 20% of the cruise speed, and canbe encountered in natural atmospheric turbulence. The effects of maximum lift to drag ratio, gust amplitudeand period are displayed. A linearized response model to the sine gust yields results that are significantlydifferent than those of the exact numerical model. This unusual result is being investigated and, if correctanalytically, will be rationalized and will provide insights into the energy extraction process.

Response to gusts of a turbulent character have been determined and compared with the simple sinegust. An extensively instrumented model has been built and will be used for flight test to investigate variousreal effects of the energy neutral cycle. Comparisons of analytical results to flight test will be reported.

Extensions of this energy reduction procedure can be applied to operational UAVs where it is possiblethat significant power reductions, of the order of 30% can be achieved by ”riding” the turbulence.

Conclusions are summarized below:

1. All vehicles obeying a parabolic polar can be described by the normalized equations of motion withthe single performance parameter, G.

2. The response to a vertical sinusoidal gust has been calculated by a number of methods: using sinusoidalcontrol accelerations; a linearized approach; and an optimized approach where the control variable isoptimized for minimum gust intensity. The linearized approach, while very simple mathematically,appears a poor approximation to the exact solution.

3. The cycle for optimal control is not far different from the sinusoidal control case, a useful result fordesign of active control systems.

4. Minimum gust levels for neutral energy cycles have been plotted for gusts of varying frequency forvehicles of different performance. Real limitations such as CLmax and maximum normal accelerationhave been used.

5. Higher frequency gusts permit neutral energy cycles at lower gust magnitudes.

6. A 2.0 m span, 0.48 kg electric powered Radio Controlled (R/C) glider has been constructed, instru-mented and flown. This will be used to conduct flight experiments in the real atmosphere.

References

1Laurenzia, Domenico, [2004] Leonardo on Flight , Giunti Press, Florence - Milan, Italy.2Lord Rayleigh, J. W. S., [1889] “The Sailing Flight of the Albatross,” Nature, Vol. 40, pp. 34.3Sachs, G., [1994] “Optimal Wind Energy Extraction for Dynamic Soaring,” Applied Mathematics in Aerospace Science

and Engineering, Plenum Press, NY and London, Vol. 44, pp. 221-237.4Lissaman, P. B. S., [2005] “Wind Energy Extraction by Birds and Flight Vehicles,” AIAA paper 2005–241, Jan. 2005.5Sachs, G., and Mayrhofer, M., [2002] “Shear Wind Strength Required for Dynamic Soaring at Ridges,” Technical Soaring ,

Vol. 25, No. 4.

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6Patel, C. K., and Kroo, I. M., [2006] “Control Law Design for Improving UAV Performance Using Wind Turbulence,”AIAA paper 2006–231, Jan. 2006.

7Katzmayr, R., [1922] “Effect of Periodic Changes in Angle of Attack on Behavior of Airfoils,” NACA TM-0147, October1922.

8Phillips, W. H., [1975] “Propulsive Effects due to Flight through Turbulence,”Journal of Aircraft , Vol. 12, No. 7, 1975,pp. 624–626.

9Hull, D. G., [1997] “Conversion of Optimal Control Problems into Parameter Optimization Problems,” Journal of Guid-

ance, Control and Dynamics, Vol. 20, No. 1, Jan.-Feb. 1997.10Speyer, J. L., [1996] “Periodic Optimal Flight,” Journal of Guidance, Control and Dynamics, Vol. 19, No. 4, July-Aug.

1996.

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