applied sciences

Article Albatross-Like Utilization of Wind Gradient for Unpowered Flight of Fixed-Wing Aircraft

Shangqiu Shan * ID , Zhongxi Hou and Bingjie Zhu

College of Aerospace Sciences and Engineering, National University of Defense Technology, Changsha 410073, China; [email protected] (Z.H.); [email protected] (B.Z.) * Correspondence: [email protected]; Tel.: +86-137-8729-3515

Received: 3 September 2017; Accepted: 12 October 2017; Published: 14 October 2017

Abstract: The endurance of an aircraft can be considerably extended by its exploitation of the hidden energy of a wind gradient, as an albatross does. The process is referred to as dynamic soaring and there are two methods for its implementation, namely, sustainable climbing and the Rayleigh cycle. In this study, the criterion for sustainable climbing was determined, and a bio-inspired method for implementing the Rayleigh cycle in a shear wind was developed. The determined sustainable climbing criterion promises to facilitate the development of an unpowered aircraft and the choice of a more appropriate soaring environment, as was demonstrated in this study. The criterion consists of three factors, namely, the environment, aerodynamics, and wing loading. We develop an intuitive explanation of the Raleigh cycle and analyze the energy mechanics of utilizing a wind gradient in unpowered flight. The energy harvest boundary and extreme power point were determined and used to design a simple bio-inspired guidance strategy for implementing the Rayleigh cycle. The proposed strategy, which involves the tuning of a single parameter, can be easily implemented in real-time applications. In the results and discussions, the effects of each factor on climbing performance are examined and the sensitivity of the aircraft factor is discussed using five examples. Experimental MATLAB simulations of the proposed strategy and the comparison of the results with those of Gauss Pseudospectral Optimization Software confirm the feasibility of the proposed strategy.

Keywords: unpowered flight; dynamic soaring; wind gradient; energy extraction; micro fixed-wing unmanned aerial vehicle

1. Introduction An aircraft usually consumes a large amount of energy during flight. There is, however, the question of whether an aircraft can fly indefinitely without carrying a power source. This might be realizable if a method could be developed for the aircraft to continuously extract energy from its environment. A wind field in which the wind gradient varies with the altitude is known to contain a large amount of energy [1–3]. In addition, albatrosses have been observed to soar for long periods without flapping their wings [4–7]. Lord Rayleigh was perhaps the first to discuss the problem relevant to albatrosses soaring in a horizontal but non-uniform wind field [8]. This type of flight involves the extraction of energy from a wind gradient and is referred to as dynamic soaring. The phenomenon of dynamic soaring has attracted the interest of many scholars. The technique utilized by an albatross to obtain extra airspeed for soaring is referred to as the Rayleigh cycle [9] and is illustrated in Figure1. Idrac [ 10] showed that wind speed increases with the altitude, which generally occurs over the sea where albatrosses fly. Different researchers have observed birds to understand their soaring mechanics [11–13], and theories regarding their utilization of wind gradient have been developed. Wood [14] used a two-dimensional model to investigate two types of complete soaring flight cycles in a logarithmic wind velocity profile. Hendriks [15]

Appl. Sci. 2017, 7, 1061; doi:10.3390/app7101061 www.mdpi.com/journal/applsci Appl. Sci. 2017, 7, 1061 2 of 25 Appl. Sci. 2017, 7, 1061 2 of 25 obtainedthree‐dimensional a three-dimensional mass point mass dynamics point dynamicsequation for equation obtaining for non obtaining‐diving non-divingflight. In addition, flight. In addition,Denny Denny[16], Taylor [16], Tayloret al. et[17], al. [and17], andLiu Liuet al. et al.[18] [ 18analyzed] analyzed the the energy energy transformation transformation in in wind-gradient-basedwind‐gradient‐based flight flight from from a a physics physics viewpoint. viewpoint.

FigureFigure 1. An1. An albatross albatross flying flying over over the the sea sea to to obtainobtain extra airspeed. airspeed. Because Because the the wind wind speed speed below below the the wavewave is nearly is nearly zero zero and and the the wind wind above above the the wavewave has a certain speed, speed, the the wind wind can can be be considered considered to to comprisecomprise two two layers layers with with a wind a wind shear shear between between them. them. Suppose Suppose that thethat initial the initial airspeed airspeed of the of albatross the is 17albatross m/s in is the 17 wind m/s in layer the belowwind layer the wave. below Ifthe the wave. wind If speed the wind above speed the waveabove isthe 7 m/swave higher is 7 m/s than thathigher below than the wave,that below when the the wave, albatross when faces the thealbatross wind andfaces crosses the wind the windand crosses shear betweenthe wind the shear lower between the lower and upper wind layers, the airspeed would suddenly increase to 24 m/s. This and upper wind layers, the airspeed would suddenly increase to 24 m/s. This sudden gain in speed is sudden gain in speed is due to the lack of time for the albatross to change its ground speed. due to the lack of time for the albatross to change its ground speed. With the development of computer science, the use of numerical algorithms has become a popularWith the approach development to determine of computer the optimal science, trajectory the use of under numerical a wind algorithms gradient. has Zhao become [19,20] a popularused approachNPSOL, to a numerical determine optimization the optimal software, trajectory to underminimize a wind the dynamic gradient. soaring Zhao cycles [19,20 of] unpowered used NPSOL, a numericalaircraft and optimization powered aircraft. software, Sachs to [21,22] minimize also the solved dynamic the optimal soaring control cycles problem of unpowered of dynamic aircraft andsoaring powered and aircraft. computed Sachs the minimum [21,22] also shear solved wind the strength optimal under control certain problem conditions. of dynamic Deittert soaring et al. [23] and computedutilized the the differential minimum flatness shear wind property strength to determine under certainthe optimal conditions. trajectory Deittert for cross et‐country al. [23] flight. utilized theKaushik differential et al. flatness [24] used property the GPOPS to determine‐II optimal the control optimal software trajectory for for trajectory cross-country computation. flight. There Kaushik et al.have [24 been] used several the GPOPS-II other studies optimal on the control optimization software of dynamic for trajectory soaring computation. [25–29]. There have been severalRecent other studies improvements on the optimization were achieved of in dynamic the practical soaring application [25–29]. of dynamic soaring. Lawrance et Recental. [30,31] improvements proposed an algorithm were achieved for determining in the practicalthe flight direction application for the of dynamicquickest energy soaring. Lawranceharvest. etLangelaan al. [30,31 et] proposedal. [32] and an Rodriguez algorithm et foral. determining[33] proposed the a method flight directionfor estimating for the the quickestwind field for trajectory design. Liu et al. [34] presented a nonlinear model of a predictive control strategy energy harvest. Langelaan et al. [32] and Rodriguez et al. [33] proposed a method for estimating for trajectory determination for energy extraction. Barate et al. [35] controlled a simulated glider the wind field for trajectory design. Liu et al. [34] presented a nonlinear model of a predictive based on a set of Takagi‐Sugeno‐Kang fuzzy rules to exploit wind gradient energy. Although these control strategy for trajectory determination for energy extraction. Barate et al. [35] controlled a contributions improved the sophistication of the concept of dynamic soaring, this was accompanied simulated glider based on a set of Takagi-Sugeno-Kang fuzzy rules to exploit wind gradient energy. by a more complicated practical implementation process. In contrast, Gao et al. [36] and Yang [37] Althoughproposed these a simple contributions method improvedfor generating the sophistication a guidance strategy of the conceptfor autonomous of dynamic dynamic soaring, soaring. this was accompaniedThis involved by the a more division complicated of the Rayleigh practical cycle implementation into four sections, process. the characteristics In contrast, Gao of which et al. [were36] and Yangdetermined [37] proposed by simple a simple equations. method Gao for and generating Liu [38] also a developed guidance strategya dynamic for soaring autonomous plan in dynamicwhich soaring.the Dubins This involved path was the used division to improve of the Rayleighthe computational cycle into fourefficiency. sections, They the showed characteristics that dynamic of which weresoaring determined could be by achieved simple using equations. modern Gao technology. and Liu However, [38] also developed their strategy a dynamic has the shortcoming soaring plan of in whichinvolving the Dubins numerous path control was used parameters, to improve which the computationalmakes practical efficiency.implementation They showedcumbersome. that dynamic soaringIn could the present be achieved study, using we followed modern technology.the simplification However, route their of Gao strategy et al. has[36] thein developing shortcoming a of involvingbio‐inspired numerous Rayleigh control cycle parameters, strategy that which is more makes efficient practical in terms implementation of computation cumbersome. time than the traditionalIn the present optimization study, method. we followed In addition, the simplification we developed route a method of Gao for utilizing et al. [36 a] wind in developing gradient a bio-inspiredfor sustainable Rayleigh climbing cycle [39] strategy in unpowered that is flight. more Simulation efficient in experiments terms of computation and analysis were time used than to the traditional optimization method. In addition, we developed a method for utilizing a wind gradient Appl. Sci. 2017, 7, 1061 3 of 25 for sustainable climbing [39] in unpowered flight. Simulation experiments and analysis were used to verify the effectiveness of the proposed flight method. The results showed that the proposed method could be used to efficiently guide a micro fixed-wing aircraft through effortless soar in a shear wind. The rest of this paper is organized as follows. In Section2, the problem of dynamic soaring is defined and the development of a dynamic model of the concept is described. In Sections3 and4, the criterion for sustainable climbing is derived, the Rayleigh cycle is explained from an energy viewpoint, and a bio-inspired strategy for implementing the cycle is proposed. In Section4, the effects of different factors in the criterion are subsequently analyzed and discussed. The results of experimental simulations involving the bio-inspired strategy for implementing the Rayleigh cycle are then presented and validated by the traditional optimized trajectory. Finally, the conclusions drawn from the study are presented. The innovative ideas of this paper are as follows: On the basis of equilibrium theory, a novel criterion for sustainable climbing that is easily implemented for engineers is derived, and this criterion decouples the effects of aerodynamics, environment and wing loading. Moreover, a simple bio-inspired strategy is proposed to generate a Rayleigh cycle, which consumes much less time than the traditional numerical optimization method.

2. Dynamic Model A wind shear is generated by the wind boundary layer phenomenon, and its mathematical model can take different forms. The following example is presented in the Military Specification MIL-F-8785C [40]:   ln z z0 W = W6   , 0.9 m < h < 300 m (1) ln 6 z0 where W denotes the mean wind velocity, W6 denotes the wind velocity at an altitude of 6 m, z is the altitude, and z0 is generally set to 0.5 m. A wind shear can also be modeled by an exponential wind velocity profile [36]:

 z p W = WR (2) ZR where WR denotes the reference wind velocity at altitude ZR, and p denotes the slope of the boundary layer. Let us assume a horizontal wind, the magnitude of which varies with the altitude [36]. The wind gradient can be expressed as follows: dW G (z) = (3) w dz There are two common approaches to the modelling of dynamic soaring, namely, those proposed by Zhao and Qi [20] and Sachs and Bussotis [21]. The latter utilizes Cartesian coordinates, while the former uses spherical coordinates. A non-inertial coordinate frame that moves with the wind at the aircraft was adopted in the present study (see Figure2). The z-axis of the frame points skyward. If the wind velocity increases with the altitude, the x-axis points in the opposite direction of the wind; otherwise, it points in the direction of the wind. The y-axis is determined by the right-hand principle. In Figure2, W is the wind velocity vector at the aircraft position, and O0-x0y0z0 represents the origin at the initial time. Appl. Sci. 2017, 7, 1061 4 of 25 Appl. Sci. 2017, 7, 1061 4 of 25

θ ψ

FigureFigure 2. 2.Coordinates Coordinates and wind field. field.

TheThe problem problem formulation formulation and and modeling modeling procedures procedures adopted adopted in inthis this study study are aresimilar similar to those to those of of somesome relevant previous works works [16,21,23,36]. [16,21,23,36 ].The The aircraft aircraft is isunder under the the action action of ofits itsweight, weight, aerodynamics, and inertial force. However, the model equations differ from ordinary equations in aerodynamics, and inertial force. However, the model equations differ from ordinary equations that the atmosphere‐fixed frame is non‐inertial. Further, the model transforms the wind gradient in that the atmosphere-fixed frame is non-inertial. Further, the model transforms the wind gradient into an inertial force. Figure 1 illustrates an intuitive model that facilitates an understanding of the into an inertial force. Figure1 illustrates an intuitive model that facilitates an understanding of the generation of the inertial force. Because the wind speed below the wave is nearly zero, and that generation of the inertial force. Because the wind speed below the wave is nearly zero, and that above above the wave has a certain magnitude, the wind can be considered to comprise two layers theseparated wave has by a certain a wind magnitude, shear, around the which wind a can large be gradient considered occurs. to comprise Let us assume two layers that the separated airspeed by a windrelative shear, to the around albatross which shown a large in Figure gradient 1 is 17 occurs. m/s in Let the us wind assume layer thatbelow the the airspeed wave, and relative the wind to the albatrossspeed shownabove the in Figurewave is1 is7 m/s 17 m/s higher in thethan wind that layerbelow below the wave. the wave, If the andalbatross the wind faces speed the wind above and the wavecrosses is 7 m/s the wind higher shear than from that belowthe lower the layer wave. to If the the upper albatross layer, faces the theairspeed wind relative and crosses to it thewould wind shearsuddenly from the increase lower to layer 24 m/s to thebecause upper there layer, is no the time airspeed for a gradual relative change to it would in the suddenlyground speed. increase The to 24 m/swind because gradient there thus causes is no time the albatross for a gradual to gain change extra airspeed, in the ground which speed.is transformed The wind into gradient an inertial thus causesforce, the referred albatross to toas gain a dynamic extra airspeed, soaring whichthrust. is The transformed dynamic model into an can inertial be directly force, referred taken from to as a dynamicDeittert’s soaring [27] or thrust. Gao’s The[36] work dynamic as follows: model can be directly taken from Deittert’s [27] or Gao’s [36] work as follows:  1 2 dW mv ρsCD v  mgsinθ m cosθsinψ  / 2 . 21 2 dt dW mv = − ρsCDv − mg sin θ − m cos θ sin(ψ − π/2) 2  1 2 dWdt mvcosθψ = ρsCL v sin m cosψ /2 . 2 1 dt dW mv cos θψ = − ρsC v2 sin ϕ − m cos(ψ − π/2)  1 2 L dW mvθρ=coscossinsinsCL v2 mg θ +m θψdt /2 and . 2 dt 1 2 dW mvθ = ρsCLv cos ϕ − mgzv =cossinθθ+ m sin θ sin(ψ − π/2) and 2 dt where v denotes the airspeed, θ denotes the .pitch angle of the flight path, ϕ denotes the bank angle z = v sin θ and ψ denotes the yaw angle measured anticlockwise from x‐axis. For the parameters, g is the wheregravitationalv denotes acceleration, the airspeed, ρ isθ denotesthe density the of pitch air, s angle the wing of the area, flight m is path, the aircraftϕ denotes mass, the and bank CL and angle andCψD aredenotes respectively the yaw the anglelift and measured drag coefficients. anticlockwise According from to xthe-axis. geometric For the relationship parameters, showng is in the Figure 2, we transform the equations above to the following equations in Cartesian coordinates by gravitational acceleration, ρ is the density of air, s the wing area, m is the aircraft mass, and CL and vx = vcosθcosψ, vy = vcosθsinψ and vz =vsinθ, where vx, vy, and vz denote the velocities of the aircraft CD are respectively the lift and drag coefficients. According to the geometric relationship shown in Figurealong2, wethe transformrespective axes. the equations above to the following equations in Cartesian coordinates by vx = vcosθcosψ, vy = vcosθsinψ and vv+vv+vvvvvz32=sinvsinθ, where sinvx, vy, 2 and sinvz denote cos the velocities of the aircraft  yxyyzxz v=xDmxLm P vv+P +G(z)v wz along the respective axes. 22 (4) v+vxy 3 2 2 . vy sin ϕ + vx vy sin ϕ + vyvz sin ϕ − vvxvz cos ϕ vx = −PDmvvx + PLm q + Gw(z)vz (4) v+vv+vv+vvv322sin sin2 2 sin cos   xxyxzyzvx + vy v=PyDmyLm vvP (5) 22 v+vxy 3 2 2 . vx sin ϕ + vxvy sin ϕ + vxvz sin ϕ + vvyvz cos ϕ vy = −PDmvvy − PLm q (5) 2 2 vx + vy Appl. Sci. 2017, 7, 1061 5 of 25

. q 2 2 vz = −PDmvvz + PLm cos ϕv vx + vy − g, and (6) . z = vz (7) q 2 2 2 v = vx + vy + vz (8) where PLm and PDm are the relevant aerodynamic parameters, given by

1  s  P = ρC and (9) Lm 2 m L 1  s  P = ρC (10) Dm 2 m D where the term s/m in parentheses represents the reciprocal of the wing loading, and the lift-to-drag ratio L/D is given by C L/D = L (11) CD

3. Method for Sustainable Climbing

3.1. Equilibrium Points and Sustainable Climbing . An equilibrium point is an important concept of a nonlinear system x = f (x). A point in the state space is an equilibrium point if, when the state of the system begins at the point, it remain . there at all future times. Therefore, at an equilibrium point, x ≡ f (x) ≡ 0 must be satisfied. In the dynamic soaring system represented by Equations (4)–(6), the equilibrium points are defined by the . . . constant velocities that can be maintained by the aircraft, namely, vx ≡ 0, vy ≡ 0, and vz ≡ 0. For a constant bank angle ϕ, Zhu et al. [39] proved the calculability of the equilibrium points of a dynamic soaring system. The equilibrium also indicates the force balance during flight through a gradient wind. If the value of vz at an equilibrium point is negative, the aircraft would continue to dive at the corresponding velocity. This type of equilibrium point is thus referred to as a descending equilibrium point. Conversely, if the value of vz at an equilibrium point is positive, the aircraft would continue climbing. This type of equilibrium point is referred to as an ascending equilibrium point. The existence of a descending equilibrium point has been proved to be trivial, while an ascending one cannot be guaranteed to exist. Should an ascending equilibrium point occur, the aircraft could utilize it for sustainable climbing. Zhu et al. [39] proved that the sufficient and necessary condition for the existence of an ascending equilibrium point was the satisfaction of the following inequality:

1 max f2(vz) ≥ (12) ≤v ≤v q 0 z zmax (L/D)2 + 1

where r q  2 2 2 2 η Rgam Rgam + Rgamη vz − vz − Rgam

f2(vz) , vz 3 (13) 2 2 2
4 Rgam + Rgamη vz v u  p  u R η2 + η4 + 4 t gam v (14) zmax , 2 where Rgam and η are abbreviated parameters defined as g Rgam and (15) , p 2 2 PLm + PDm Appl. Sci. 2017, 7, 1061 6 of 25

Gw η , q (16) p 2 2 g PLm + PDm The above equations constitute the initial criterion for sustainable climbing. It is, however, difficult to apply this criterion because it requires the determination of the maximum of f 2(vz) at every time. In Section 3.2, we propose a simplified criterion based on Equation (12). The derivation of the proposed criterion is presented in Section 3.3.

3.2. Proposed Criterion for Unpowered Climbing The simplified criterion for unpowered climbing is explicitly proposed in this subsection and derived in the next subsection. The criterion for an unpowered aircraft to continue climbing in a gradient wind field can be expressed as follows:

ΠeΠa ≤ 1 (17) where Πa and Πe are the aircraft and environmental fractions, respectively. A smaller ΠaΠe increases the climbing probability of the aircraft. The environmental fraction Πe is given by

1 r gρ Πe , (18) Gw 2 which is only relevant to the environment. The aircraft fraction Πa can also be divided into two terms as follows: Πa , ΠSΠA (19) where ΠA and ΠS are the aerodynamic and wing loading fractions, respectively. ΠA is defined as

1  2 2 4 ΠA , h(L/D) CL + CD (20) where h is a function of the lift-to-drag ratio, and can be approximated as

3 2 h(L/D) ≈ 10−0.1177[lg(L/D)] +0.5525[lg(L/D)] −0.9116lg(L/D)+0.5809 0.3 ≤ L/D ≤ 60 (21)

The aerodynamic fraction ΠA depends on only the aerodynamic characteristic of the aircraft, while the wing loading fraction ΠS is given by

r s Π (22) S , m

Hence, the unpowered climbing criterion can be divided into three parts, namely, the environmental, aerodynamic, and wing loading fractions. The derivation procedure of the criterion is presented below.

3.3. Derivation of Proposed Criterion We define K1 , max f2(vz) (23) 0

Before the derivation, vz is normalized as

Appl. Sci. 2017, 7, 1061 7 of 25 vzn = vz/vzmax (25)

Before the derivation, vz is normalized as By substituting Equation (25) into Equation (13) and canceling the common factor, the function vv/vzn= z zmax (25) f 2 becomes By substituting Equation (25) into Equation (13) and canceling the common factor, the function √  √ q √  f2 becomes 2 p 2 2 Ω −2 + 2η −Ωvzn + 2 2 + Ωη vzn f (vz) = f (vzn) vzn 222 (26) 2 2n , Ω 22+ ηΩ3 v+ 22 +3Ωη v zn2 2 4 zn  2 4 (2 + Ωη vzn ) fv22znznzn= f(v)v 3 3 (26) 4 224 22+ Ωη vzn where q where Ω = η2 + 4 + η4 (27)

24 The range of the variant vzn is [0, 1]. Thus,Ωη=4 the++ functionη f 2n of vzn was determined to depend(27) on only the unique parameter η. Therefore, K1 naturally becomes a function of η: The range of the variant vzn is [0, 1]. Thus, the function f2n of vzn was determined to depend on only the unique parameter η. Therefore, K1 naturally becomes a function of η: K1(η) = max f2n(vzn) (28) 0<vzn≤1 K12η max f nzn(v ) (28) 01vzn It is difficult to obtain an analytical expression of Equation (28). Indeed, because the function of K It is difficult to obtain an analytical expression of Equation (28). Indeed, because the function of 1 includesK a1 maximizationincludes a maximization operation, operation, an analytical an analytical expression expression is almost is almost impossible impossible to obtain, to obtain, although the valuesalthough of the the equation values of can the beequation obtained can be for obtained discrete for samples. discrete samples. Such values Such values are, however, are, however, actually obtainedactually by numerical obtained calculations. by numerical calculations. For any value For of anyη ,value the maximum of η, the maximum of f 2n(v znof) f is2n(v searchedzn) is searched for in the intervalfor [0, in 1]. the A seriesinterval of [0, discrete 1]. A series samples of discrete are thus samples obtained. are thus The obtained. numerically The numerically determined determined values of K1 are plottedvalues in Figureof K1 are3 ,plotted which in clearly Figure shows3, which a clearly monotonic shows increase. a monotonic increase. ) ) D / ( L 1 ( K h

FigureFigure 3. Plots3. Plots of ofK K11 andand hh..

The value of η should be larger than 1; otherwise, the inequality in Equation (24) would not be The value of η should be larger than 1; otherwise, the inequality in Equation (24) would not be satisfied because of the negative value that K1(η) would have. Hence, K1 guarantees its own inverse satisfiedfunction because K− of1. Based the negative on Equation value (24), that theK 1following(η) would inequality have. Hence, would Kalso1 guarantees have to be itssatisfied own inverseto − functionachieveK 1. Basedthe initial on unpowered Equation climbing (24), the criterion: following inequality would also have to be satisfied to achieve the initial unpowered climbing criterion:   −1 1 η ≥ K1  q  (29) (L/D)2 + 1 Appl. Sci. 2017, 7, 1061 8 of 25

The expression of η can thus be rewritten as follows:

s r 1 2 m  1  4 = η Gw 2 2 (30) gρ s CL + CD

By substituting Equation (30) into Equation (29), the latter becomes

 r r  1  1 gρ s  2 2 4 1 ≥ CL + CD h(L/D) (31) Gw 2 m | {z }| {z }| {z } Πe ΠS ΠA | {z } Πa where   −1 1 h(L/D) , K1  q  (32) (L/D)2 + 1

Utilizing the numerical results of K1, we sought backward for the corresponding L/D values, and thus obtained discrete samples of h. The function h can be approximated by familiar functions such as Equation (21), the numerical results of which are plotted in Figure3. The approximation is for L/D values of 0.3–60, because the lift-drag ratio for a fixed-wing aircraft is generally within 1–60 [41]. This range covers most fixed-wing aircraft in practice. The highest lift-to-drag ratio for an albatross is about 20–30 [22]. Hence, Equation (31) establishes the criterion for unpowered sustainable climbing.

4. Rayleigh Cycle Method In the real world, the wind gradient is generally not sufficiently large to enable sustainable climbing by an aircraft. The Rayleigh cycle affords another flight strategy for utilizing a gradient wind field for unpowered flight. This strategy is based on the observation that albatrosses are capable of flying for long periods without flapping their wings. Idrac [10] observed that the wind layer where the albatrosses fly increases according to the altitude. A Rayleigh cycle can be divided into four phases, as shown in Figure4. Let us suppose that the wind over the ocean surface in Figure4 is divided by the wind shear into two layers, with the wind speed above the wave being 7 m/s and that below it being 0 m/s. The first phase of a Rayleigh cycle is the upwind climb segment. At the beginning of this phase, both the airspeed and the ground speed of the albatross are 17 m/s upwind. After the albatross crosses the wind shear, the ground speed is maintained at 24 m/s, but the airspeed is changed to 24 m/s. The second phase is the high-altitude turn segment. In this phase, the albatross executes an approximately uniform circular motion in the air, with its heading changing from upwind to downwind. Thus, the airspeed is still 17 m/s, but the ground speed changes to 31 m/s. The third phase is the downwind dive segment. Similar to the upwind climb phase, in this phase, the albatross crosses the wind shear in the wind direction. The ground speed is maintained at 31 m/s, and the airspeed also changes to 31 m/s. The fourth phase is the low altitude turn segment, which completes the Raleigh cycle, linking back to the first phase. Similar to the high-altitude turn phase, in this fourth phase, the albatross executes an approximately uniform circular motion in the air, and then returns to the beginning states. The final airspeed and ground speed are both 31 m/s, indicating that the albatross almost gains double the upper layer wind speed during a Raleigh cycle. In other words, it harvests energy from the wind shear. Figure4 is only an intuitive illustration of the Rayleigh cycle. In practice, the wind shear does not produce a sudden change in the wind speed, because it has a continuous wind gradient profile. The altitude fluctuation during a Rayleigh cycle is also considerable. Thus, there is always a transformation between gravitational potential energy and kinetic energy over the entire cycle, with the drag also Appl. Sci. 2017, 7, 1061 9 of 25

Figure 4 is only an intuitive illustration of the Rayleigh cycle. In practice, the wind shear does not produce a sudden change in the wind speed, because it has a continuous wind gradient profile. Appl. Sci. 2017, 7, 1061 9 of 25 The altitude fluctuation during a Rayleigh cycle is also considerable. Thus, there is always a transformation between gravitational potential energy and kinetic energy over the entire cycle, with consumingthe drag energy.also consuming Therefore, energy. we prove Therefore, below we that prove the upwind below that climb the and upwind downwind climb and dive downwind can be used to harvestdive can extra be used energy. to harvest extra energy.

FigureFigure 4. 4.Illustration Illustration ofof aa typicaltypical Rayleigh cycle. cycle.

4.1.4.1. Energy Energy Perspective Perspective Here,Here,E denotes E denotes the the mechanical mechanical energy, energy, including including the gravitational potential potential energy energy and and kinetic kinetic energy,energy, and and the theO- Oxyz‐xyzcoordinate coordinate system system is is as as shown shown inin FigureFigure 22.. According to to the the kinetic kinetic energy energy theorem,theorem, the the change change in in the the mechanical mechanical energy energy is is equalequal to the work work done done against against the the drag drag and and inertial inertial force.force. Because Because the the lift lift is is perpendicular perpendicular to to the the movement, movement, it does no no work. work. The The following following expression expression can thus be derived: can thus be derived:

dEdE= =−Ddl Ddl+ + mGmGwzw vv dxzdx (33)(33) Here,Here,dl isdl definedis defined as as dl = vdt dl = vdt (34)(34) ThisThis implies implies that that 3 dE = −PDv dt + mGwvzdx 3 dE = PDwz v dt + mG3 v dx dx = m(−PDmv + Gwvz dt )dt (35) 3 dx ==m(m(− PPDm vv3 +G+ G vwvzv )dtx)dt (35) Dm w z dt Hence, the normalized rate of the mechanical energy3 acceleration is equal to =m( PDm v +G w vv z x )dt d Hence, the normalized rate of the(E mechanical/m) = −P energyv3 + accelerationG v v is equal to (36) dt Dm w z x d (E / m) = P v3 +G v v (36) It should be noted that the temporaldt variationDm of the w mechanical z x energy is independent of the control input ϕ. The coordinates can thus be transformed as follows: It should be noted that the temporal variation of the mechanical energy is independent of the control input ϕ. The coordinates can thus be transformed as follows: vx = r cos θ cos ψ vy = r cos θ sin ψ (37) vz = r sin θ Appl. Sci. 2017, 7, 1061 10 of 25

v=rx cosθψ cos

v=ry cosθψ sin (37) Appl. Sci. 2017, 7, 1061 10 of 25 v=rz sinθ where r is the radial velocity, θ is the pitch angle, and ψ is the heading angle. Equation (36) can wherethereforer is the be radial rewritten velocity, as follows:θ is the pitch angle, and ψ is the heading angle. Equation (36) can therefore be rewritten as follows: d 32 (E / m) = PDm r +G w r sinθθψ cos cos (38) d dt 3 2 (E/m) = −PDmr + Gwr sin θ cos θ cos ψ (38) Figure 5 is a sectionaldt drawing of a contour plot of the normalized rate of the mechanical energy accelerationFigure5 is a d sectional(E/m)/dt for drawing PDm = 0.002 of a contourand Gw = plot 0.4. ofIt clearly the normalized shows that rate Equation of the mechanical(38) is symmetrical energy about the vxOvz plane. The equation of the surface on which the energy is invariant (i.e., d(E/m)/dt = 0) is acceleration d(E/m)/dt for PDm = 0.002 and Gw = 0.4. It clearly shows that Equation (38) is symmetrical as follows: about the vxOvz plane. The equation of the surface on which the energy is invariant (i.e., d(E/m)/dt = 0) is as follows: Gw r=Gw sinθθψ cos cos (39) r = PDmsin θ cos θ cos ψ (39) PDm This corresponds to the green surface in Figure 5. It exists where vxvz > 0 and comprises two This corresponds to the green surface in Figure5. It exists where vxvz > 0 and comprises two leaves joined at the origin, having a spindle‐like shape. Energy is only harvested inside the spindle, leaves joined at the origin, having a spindle-like shape. Energy is only harvested inside the spindle, which is thus called an energy‐harvest spindle. In practice, the Rayleigh cycle is a loop across the which is thus called an energy-harvest spindle. In practice, the Rayleigh cycle is a loop across the two two leaves of the spindle. The upwind climb (vx > 0, vz > 0) and downwind dive (vx < 0, vz < 0) leaves of the spindle. The upwind climb (vx > 0, vz > 0) and downwind dive (vx < 0, vz < 0) obviously obviously occur in the area where vxvz > 0. Energy would obviously be harvested if the aircraft was v > occurpositioned in the area within where the xvztwo leaves.0. Energy would obviously be harvested if the aircraft was positioned within the two leaves.

Figure 5. Sectional drawing of the contour plot of d(E/m)/dt. Figure 5. Sectional drawing of the contour plot of d(E/m)/dt.

ThereThere is a is maximum a maximum rate rate of theof the mechanical mechanical energy energy acceleration acceleration inin thethe spindle.spindle. In In addition, addition, EquationEquation (38) (38) can can be rewritten be rewritten as as d 1 (E/m) = −P r3 + G r2 sin 2θ cos ψ (40) dt Dm 2 w To maximize Equation (40), the term sin2θcosψ should be 1, because 1 is the maximum of sin2θcosψ when θ = π/4, ψ = 0 or θ = −π/4, ψ = π. This implies that the maximum point is on the Appl. Sci. 2017, 7, 1061 11 of 25

straight line defined by vx = vz and vy = 0. The maximum point is obtained by setting the derivative of Equation (40) to zero as

d  d  d  1  (E/m) = −P r3 + G r2 = −3P r2 + G r = 0 dr dt dr Dm 2 w Dm w from which the following is obtained: G r = w (41) 3PDm Equation (41) can be expressed in the Cartesian coordinate frame as √ 2Gw vx = vz = ± , vy = 0 (42) 6PDm

The maximum power is thus given by

 d  G 3 = w E/m 2 (43) dt max 54PDm

This leads to the conclusion that a larger gradient, smaller drag coefficient, lower air density, and larger wing loading enhance energy harvesting for unpowered flight. This agrees with the results of the proposed criterion presented below in Section 5.1.

4.2. Optimization Method for Generating Rayleigh Cycle The generation of a Rayleigh cycle is generally transformed into an optimal trajectory problem, which has been solved by many numerical optimization methods such as the Gauss pseudo-spectral method of the optimal software called Gauss Pseudospectral Optimization Software (GPOPS) [24–28]. The optimized trajectory should satisfy the dynamics Equations (4)–(8), and the performance function gives the total energy harvest in a cycle, expressed as   maxJ(X) = E t f − E(t0) (44) where tf and t0 are the final and initial times, respectively. The periodic boundary condition is as follows:     vx(t0) = vx t f    vy(t0) = vy t f (45)     vz(t0) = vz t f

The constraints on the lift coefficient CL and bank angle ϕ are ( (C ) ≤ C ≤ (C ) L min L L max (46) ϕmin ≤ ϕ ≤ ϕmax

Though this method for generating a Rayleigh cycle provides an optimized solution, the numerical computation is too time-consuming for real-time applications. In the present work, GPOPS [42], which is a MATLAB-based optimal process toolbox, was used to optimize the problem. The optimized results were then used to validate the proposed bio-inspired strategy for implementing the Rayleigh cycle, described in the next subsection.

4.3. Bio-Inspired Strategy Although the optimization method affords an optimal means of generating a Rayleigh Cycle [18–28], the computation procedure, as noted above, is too time-consuming for real-time Appl. Sci. 2017, 7, 1061 12 of 25

4.3. Bio‐Inspired Strategy Appl. Sci. 2017, 7, 1061 12 of 25 Although the optimization method affords an optimal means of generating a Rayleigh Cycle [18–28], the computation procedure, as noted above, is too time‐consuming for real‐time applications.applications. Hence, Hence, we we usedused a state state machine machine to to imitate imitate the the four four phases phases of the of Rayleigh the Rayleigh cycle cycleof an of analbatross, namely, namely, the the upwind climb, climb, high high-altitude‐altitude turn, turn, downwind downwind dive, dive, and and low low-altitude‐altitude turn turn phases. The transfer logic among the phase states is illustrated in Figure 6. The switch variables are phases. The transfer logic among the phase states is illustrated in Figure6. The switch variables are vz vz and the heading angle. The state machine used simple switch rules to generate the Rayleigh cycle, and the heading angle. The state machine used simple switch rules to generate the Rayleigh cycle, and and no further optimization was thus required. no further optimization was thus required.

v z  0

v z  0

Figure 6. Transfer logic among the four phases in the bio-inspired strategy for implementing the Figure 6. Transfer logic among the four phases in the bio‐inspired strategy for implementing the Raleigh cycle. Raleigh cycle.

4.3.1.4.3.1. Upwind Upwind Climb Climb WhenWhen this this state state is is active, active, the the kinetic kinetic energyenergy of the aircraft is is transformed transformed into into gravitational gravitational potentialpotential energy. energy. When When the the vertical vertical speed speed vvzz falls below zero, zero, as as shown shown in in Figure Figure 6,6 the, the next next state state of of high-altitudehigh‐altitude turn turn becomes becomes active. active. The The control control inputinput in this state state is is   0, C=CLL opt (47) ϕ = 0, CL = (CL)opt (47)

where (CL)opt is the optimum lift coefficient in Figure 7, the choice of which enables the determination C whereof the ( Lbest)opt climbingis the optimum performance, lift coefficient as described in Figure in Section7, the 5.1.1. choice of which enables the determination of the best climbing performance, as described in Section 5.1.1. 4.3.2. High‐Altitude Turn 4.3.2. High-Altitude Turn In this state, the aircraft loses energy. To decrease the energy loss, the aircraft needs to move on toIn the this state state, of downwind the aircraft dive loses as energy. soon as To possible. decrease Because the energy the airspeed loss, the is aircraft quite low, needs the to maximum move on to thebank state angle of downwind and lift coefficient dive as soon are given as possible. by Because the airspeed is quite low, the maximum bank angle and lift coefficient are given by  , max C=CLL max (48) ϕ = ϕ C = (C ) The downwind dive state becomes activemax ,whenL theL aircraftmax heading becomes the same as the(48) wind direction. The downwind dive state becomes active when the aircraft heading becomes the same as the wind4.3.3. direction. Downwind Dive

4.3.3. DownwindSimilar to the Dive upwind climb state of energy harvest, the control input in the downwind dive state is also given by Equation (47). When the z acceleration of the aircraft begins to increase, Similar to the upwind climb state of energy harvest, the control input in the downwind dive state v> 0 is also given by Equation (47). When the z accelerationz of the aircraft begins to increase, (49)

The low‐altitude turn state then becomes active.. vz > 0 (49)

The low-altitude turn state then becomes active. Appl. Sci. 2017, 7, 1061 13 of 25

4.3.4. Low-Altitude Turn Although energy is lost in this state, like in the high-altitude turn state, the airspeed is quite high. If a large lift coefficient is assigned, the corresponding large drag coefficient would produce a relatively large drag, which will result in the consumption of a large amount of energy. Consequently, a discount is set for the maximum lift coefficient to compromise for fast turning:

ϕ = ϕmax, CL = λ(CL)max + (1 − λ)(CL)min (50) where (CL)min is the minimum lift coefficient, and λ is the discount rate, which is a tunable parameter requiring tuning in an experiment.

4.3.5. Initial State

The initial state can be defined from an energy perspective. If the initial conditions are vx > 0 and vz > 0, the upwind climb state would be active. If the initial velocity components vx and vz are both negative, the downwind dive state would be the initial active state. If vx > 0 and vz < 0, the high-altitude turn state would be the initial active state. If vx < 0 and vz > 0, the low-altitude turn state would be the initial active state. This logic is summarized in Table1.

Table 1. Initial entry logic.

νx State >0 <0 >0 Upwind Climb Low Altitude Turn νz <0 High Altitude Turn Downwind Dive

5. Simulation Results and Discussion

5.1. Discussion of Criterion for Unpowered Climbing

5.1.1. Optimum Lift Coefficient The drag coefficient is generally expressed as [43]

2 CD = CD0 + nCL (51) where n and CD0 are parameters that depend on the airfoil itself. When an aircraft is in flight, its angle of attack can be adjusted by the elevators to vary the lift coefficient CL. The drag coefficient CD varies with CL according to Equation (51), and CL can thus be considered as a control input. Substituting Equations (51) and (11) into Equation (20) yields

! 1 C   2 4 = L 2 + + 2 ΠA h 2 CL CD0 nCL (52) CD0 + nCL

Equation (52) expresses ΠA as a function of CL. Minimizing ΠA optimizes CL and thus minimizes the climbing criterion in Equation (17). In the case of an albatross, CD0 and n are respectively 0.033 and 0.019 [22]. The function ΠA(CL) for this case is plotted in Figure7. Figure7 indicates that the optimal lift coefficient CL for an albatross is approximately 0.1. The lift coefficient corresponding to the maximum lift-to-drag ratio can be calculated using [21] as p CD0/n = 1.32 Appl. Sci. 2017, 7, 1061 14 of 25

As can be observed, these two CL values significantly differ. To understand the reason for the small optimal CL, we will consider a simple calculation. For CL = 1.32, we have L/D = 20, h = 1.08, and

1   2 4 2 2 CL + CD0 + nCL = 1.15

CL=1.32

Thus, from Equation (20), ΠA = 1.24. Further, for CL = 0.1, we have L/D = 3, h = 1.8, and

1   2 4 2 2 CL + CD0 + nCL = 0.32

CL=0.1

From Equation (20), ΠA = 0.58. Although the maximum L/D minimizes h, the other item in Equation (20) has a much larger effect on the calculated value of ΠA. In the general soaring problem without a wind gradient, the maximum lift-to-drag ratio is optimal for the soaring range [44]. The situation is, however, different in the presence of a wind gradient. An intuitive explanation of the difference can be presented from an energy viewpoint. In the absence of a wind gradient, maintaining the maximum lift-to-drag ratio enables energy saving, based on traditional flight mechanics. When there is a wind gradient, the aircraft can harvest extra energy from it, and a smaller lift coefficient would cause the velocity to increase, because of the higher velocity needed to generate the same lift to offset the weight. Thus, the aircraft crosses more wind layers during the same period, acquiring energy more efficiently, although it also saves less energy compared to that at the maximum lift-to-drag ratio. It is thus necessary to makeAppl. a compromise Sci. 2017, 7, 1061 between harvesting and saving energy. Therefore, the calculated optimal14 of 25 lift coefficient in Figure4 is smaller than the corresponding maximum lift-to-drag ratio.

3.5

3

2.5

2

1.5

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 C L

FigureFigure 7. Plot 7. Plot of Π ofA ΠasA aas function a function of ofCL CforL for an an albatross, albatross, showing showing thethe optimal value value of of CCL.L .

5.1.2. WingAs Loadingcan be observed, these two CL values significantly differ. To understand the reason for the small optimal CL, we will consider a simple calculation. For CL = 1.32, we have L/D = 20, h = 1.08, andAccording to Equation (22), a large wing loading produces a small wing loading fraction ΠS. Therefore, a larger wing loading is preferred for effortless climbing. This may seem incredible 1 2 considering that traditional aeronautics22 textbooks suppose4 that a small wing loading and high aspect C+CLD0 +nC L  1.15 ratio wing are favorable to climbing, with the former inducing a low speed, which reduces the drag  and power consumption. However, the important considerationCL 1.32 here is the energy harvested from the

Thus, from Equation (20), ΠA = 1.24. Further, for CL = 0.1, we have L/D = 3, h = 1.8, and

1 2 224 C+CLD0 +nC L  0.32 

CL 0.1

From Equation (20), ΠA = 0.58. Although the maximum L/D minimizes h, the other item in Equation (20) has a much larger effect on the calculated value of ΠA. In the general soaring problem without a wind gradient, the maximum lift‐to‐drag ratio is optimal for the soaring range [44]. The situation is, however, different in the presence of a wind gradient. An intuitive explanation of the difference can be presented from an energy viewpoint. In the absence of a wind gradient, maintaining the maximum lift‐to‐drag ratio enables energy saving, based on traditional flight mechanics. When there is a wind gradient, the aircraft can harvest extra energy from it, and a smaller lift coefficient would cause the velocity to increase, because of the higher velocity needed to generate the same lift to offset the weight. Thus, the aircraft crosses more wind layers during the same period, acquiring energy more efficiently, although it also saves less energy compared to that at the maximum lift‐to‐drag ratio. It is thus necessary to make a compromise between harvesting and saving energy. Therefore, the calculated optimal lift coefficient in Figure 4 is smaller than the corresponding maximum lift‐to‐drag ratio.

5.1.2. Wing Loading

According to Equation (22), a large wing loading produces a small wing loading fraction ΠS. Therefore, a larger wing loading is preferred for effortless climbing. This may seem incredible considering that traditional aeronautics textbooks suppose that a small wing loading and high Appl. Sci. 2017, 7, 1061 15 of 25 wind gradient for unpowered flight, rather than power conservation in powered flight. The speed increases due to the larger wing loading, which results in the aircraft crossing more wind layers and collecting more energy during a given period. This is intuitive from an energy viewpoint. A large aircraft generally has a larger wing load compared to a small one. McCormick [44] proposed a model for estimating the wing loading based on a cubic scaling law. He described the relationship by defining the following upper and lower boundaries:

m 1 ≤ 18.67(9.8m) 3 − 1.01 Upper boundary and (53) s and m 1 ≥ 9.78(9.8m) 3 − 1.01 Lower boundary (54) s Noth [45] validated this estimate using an aircraft database and the Great Flight diagram. Many other models have been proposed for different types of aerial vehicles [46]. Although a large wing loading is favorable for climbing, the applicable loading is limited by the structural strength, cruising speed, and mission configuration.

5.1.3. Sensitivity Analysis of Aircraft Fraction To compare the sensitivity of the two aircraft fractions in Equation (19), we define the logarithm of the aircraft fractions as follows:

Σa , lnΠa ΣA , lnΠA ΣS , lnΠS

The aircraft fraction can thus be obtained in the form of a summation from Equation (19), as follows: Σa = ΣA + ΣS Smaller values of these three parameters are also preferable, considering the multiplication criterion in Equation (17). Five example aircrafts were used to analyze the sensitivity of the fractions. The relevant parameters are presented in Table2. The parameters for four of the aircrafts were obtained from references, while those for the National University of Defense Technology (NUDT) solar aircraft were measured and estimated.

Table 2. Aircraft parameters.

NUDT Solar Aircraft Albatross [22] Avatar P2 [23] SBXC [30] Sky Sailor [45] Aircraft S (m2) 0.65 0.473 0.957 0.78 1.56 m (kg) 8.5 4.5 5.443 2.5 6.8 CD0 0.033 0.0173 0.017 0.0191 0.03 N 0.019 0.0517 0.0192 0.0268 0.0223 (CL)max 1.6 1.1 1.0 1.3 1.0 m/S 13.08 9.51 5.69 4.49 4.36 (kg/m2)

ΣA was calculated with respect to CL for each aircraft using Equation (52). The results are plotted in Figure8. As can be observed, the plots for Avatar P2, SBXC, and Sky Sailor are very close, as well as those for the albatross and NUDT Solar Aircraft. It can also be seen from Figure8 that all of the optimal lift coefficients are quite close to zero. The practical lift coefficient during flight is generally higher than the optimal value, and we therefore consider the operational ΣA to range between the minimum value and the value corresponding to the maximum lift coefficient. On this basis, we calculated the values of ΣA, ΣS, and Σa for the different aircraft, as plotted in Figure9. Appl. Sci. 2017, 7, 1061 16 of 25 Appl. Sci. 2017, 7, 1061 16 of 25

corresponding to the maximum lift coefficient. On this basis, we calculated the values of ΣA, ΣS, and corresponding to the maximum lift coefficient. On this basis, we calculated the values of ΣA, ΣS, and Σa for the different aircraft, as plotted in Figure 9. Appl. Sci.Σa for2017 the, 7, different 1061 aircraft, as plotted in Figure 9. 16 of 25

0.5 0.5

0 0

-0.5 Albatross -0.5 AvatarAlbatross P2 SBXCAvatar P2 Sky SailorSBXC NUDTSky Solar Sailor Aircraft NUDT Solar Aircraft -1 0-1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6C 0.8 1 1.2 1.4 1.6 L C L Figure 8. Logarithm of the aerodynamic fraction ΣA with respect to CL for different aircrafts. FigureFigure 8. Logarithm 8. Logarithm of the of the aerodynamic aerodynamic fraction fractionΣA ΣwithA with respect respect to toC CLL forfor different aircrafts. aircrafts.

0.5 0.5

0 0

-0.5 -0.5

-1 -1

A -1.5 A -1.5 S S a a -2 A A S S N -2 lba va BX ky UD tro tar P C Sa T S Albss Av 2 SB Skilor NUol atr ata XC y S DTar A oss r P2 ailo Soircr r lar aAft irc raft FigureFigure 9.9. ΣA,, ΣΣSS, ,and and ΣΣa aforfor different different aircrafts aircrafts (the (the wide wide and and narrow narrow bars bars respectively respectively indicate indicate the the maximummaximumFigure andand 9. Σ minimumA, ΣS, and Σvalues). values).a for different aircrafts (the wide and narrow bars respectively indicate the maximum and minimum values). In Figure 9, the wide and narrow bars respectively indicate the maximum and minimum In Figure9, the wide and narrow bars respectively indicate the maximum and minimum values. values. TheIn Figurelift coefficient 9, the widecan be and adjusted narrow to barsassign respectively an aircraft fractionindicate betweenthe maximum the minimum and minimum and The lift coefficient can be adjusted to assign an aircraft fraction between the minimum and maximum maximumvalues. values,The lift and coefficient the aerodynamic can be adjusted configuration to assign and an aircraftwing loading fraction can between also be theredesigned minimum to and values,maximum and the values, aerodynamic and the configurationaerodynamic configuration and wing loading and wing can loading also be redesignedcan also be redesigned to modify theto modify the aircraft fraction. From Figure 9, it can be seen that the variation of ΣA among the aircraftdifferentmodify fraction. aircrafts the aircraft From is as Figurelarge fraction. as9 ,that it From can of the be Figure seenwing that loading9, it the can variationfraction. be seen This of thatΣ suggestsA theamong variation that the the different of aerodynamic ΣA among aircrafts the is asfraction largedifferent as is thatgenerally aircrafts of the as wingis important as large loading as as that the fraction. of wing the wingloading This suggestsloading fraction. fraction. that Neither the This aerodynamic of suggeststhese fractions that fraction the can aerodynamic thus is generally be asignored importantfraction in a isbid as generally the to enhance wing as loading important the sustainable fraction. as the climbingwing Neither loading performance of these fraction. fractions in Neither a gradient can of thus these wind. be fractions ignored can in thus a bid be to enhanceignored the sustainablein a bid to enhance climbing the performance sustainable climbing in a gradient performance wind. in a gradient wind. Appl. Sci. 2017, 7, 1061 17 of 25

Appl. Sci. 2017, 7, 1061 17 of 25 5.1.4. Minimum Gradient and Environmental Effects 5.1.4. Minimum Gradient and Environmental Effects According to the inequality in Equation (31), the minimum gradient for climbing (Gw)min can be According to the inequality in Equation (31), the minimum gradient for climbing (Gw)min can be readily determined as follows: readily determined as follows: r r  1  gρ s  2 1 2 4 ( ) = gρ s  22+ (  ) Gw minG= C+ChL/DCL CD4 h L/D (55) wLDmin 2 m   (55) 2 m  

ThisThis promises promises to facilitate to facilitate the choicethe choice of appropriate of appropriate parameters parameters and locations and locations for dynamic for dynamic soaring, andsoaring, indicates and the indicates dependency the dependency of the aircraft of the on aircraft the wind on the field. wind field. TheThe gravitational gravitational acceleration accelerationg gand and air air density density ρρ varyvary little near near the the earth’s earth’s surface. surface. However, However, thethe situation situation is quite is quite different different near near space space and and on on other other planets.planets. The near near space space altitude altitude is is higher higher than than thatthat at which at which traditional traditional aircrafts aircrafts fly, fly, but but lower lower than than thatthat atat whichwhich traditional traditional astrovehicles astrovehicles usually usually orbit.orbit. It ranges It ranges between between 20 20 and and 100 100 km km above above sea sea level. level. The effects of of gravity gravity gradually gradually weaken weaken and and thethe air thinsair thins with with increasing increasing altitude, altitude, as as indicated indicated in in FigureFigure 1010.. The The data data presented presented in in the the figure figure were were obtainedobtained using using the the International International Standard Standard Atmosphere Atmosphere model model [47 ].[47]. The The lower lower values values of gofand g andρ lower ρ thelower minimum the minimum gradient near gradient space, near relaxing space, the relaxing required the wind required field forwind climbing. field for An climbing. advantage An of nearadvantage space is thatof near the space wind is field that has the awind steady field gradient, has a steady especially gradient, between especially 40 and between 60 km 40 above and 60 sea levelkm (Figure above 11sea). level The (Figure horizontal 11). windThe horizontal data in Changsha wind data (28in ◦Changsha11049” N, (28°11 112◦58′49042”″ N, E)112°58 presented′42″ E) in Figurepresented 11 were in obtained Figure 11 by were the MATLAB obtained functionby the MATLAB atmoshwm07 function using atmoshwm07 the Naval Research using the Laboratory Naval HorizontalResearch Window Laboratory Model Horizontal [48]. Gao Window et al. [ Model49] attempted [48]. Gao to et use al. [49] the attempted wind shear to at use the the bottom wind shear of near at the bottom of near space to achieve high‐altitude long‐endurance flight. space to achieve high-altitude long-endurance flight.

FigureFigure 10. 10.Variations Variations in in the the air air density density and and gravitational gravitational acceleration with with the the altitude. altitude.

LetLet us considerus consider a flight a flight at at an an altitude altitude of of 50 50 km km above above seasea level. The The air air density density at at this this altitude altitude is is veryvery low low at approximately at approximately 10 −104 −kg/m4 kg/m33,, comparedcompared to to 1 kg/m3 3atat sea sea level. level. There There is, is, however, however, little little changechange in the in the gravitational gravitational acceleration. acceleration. From From Equation Equation (35),(35), G gρ w min 50km above sea levels 50km above sea level (G ) | (gρ) 0.01 w min 50 kmG above sea level g50ρ km above sea level w min at sea level = at sea level ≈ 0.01 (Gw)min|at sea level (gρ)at sea level Appl. Sci. 2017, 7, 1061 18 of 25 Appl. Sci. 2017, 7, 1061 18 of 25 Appl. Sci. 2017, 7, 1061 18 of 25 Wind Speed(m/s) Wind Wind Speed(m/s) Wind Heading(degree) Heading(degree)

FigureFigure 11. 11.Variations Variations in the in windthe wind speed speed and heading and heading with altitude with altitude in Changsha in Changsha (28◦11049” (28°11 N, 112′49″ ◦58N,0 42” E) obtainedFigure112°58′ 4211.″ by E)Variations theobtained MATLAB byin thethe function MATLABwind speed atmoshwm07 function and headingatmoshwm07 using with the Naval usingaltitude the Research inNaval Changsha LaboratoryResearch (28°11 Laboratory Horizontal′49″ N, 112°58′42″ E) obtained by the MATLAB function atmoshwm07 using the Naval Research Laboratory WindowHorizontal Model. Window Model. Horizontal Window Model. This indicates that the minimum gradient for climbing at an altitude of 50 km is 1% of that near This indicates that the minimum gradient for climbing at an altitude of 50 km is 1% of that near the the earth’sThis indicates surface. that Climbing the minimum at the highgradient altitude for climbing thus requires at an altitudea relatively of 50 weak km is wind1% of gradient,that near earth’stheimplying earth’s surface. the surface. potential Climbing Climbing for at dynamic the highat the soaring altitude high altitude over thus a requires large thus area. requires a relatively a relatively weak wind weak gradient, wind gradient, implying theimplying potentialNASA the for [50,51], potential dynamic ESA for soaring [52],dynamic and over soaringJAXA a large [53] over area. are a large considering area. using a fixed‐wing aircraft for the explorationNASANASA [50 [50,51],of,51 Mars.], ESA ESA Early [ 52[52], ],in andtheand 1970s, JAXAJAXA NASA [[53]53] are built considering and tested using usingthe Mini a afixed‐ fixed-wingSniffer‐wing [54], aircraft aircraftwhich for had for the a the explorationexplorationwingspan ofof of Mars.6.7 Mars. m. EarlyThe Early Sky in in‐Sailor the the 1970s, 1970s, [45,52] NASANASA is an built ultra ‐and andlightweight tested tested the thesolar Mini Mini-Sniffer autonomous‐Sniffer [54], [ 54model which], which aircraft had had a a wingspanwingspandesigned of by of 6.7 the6.7 m. Autonomousm. The The Sky-Sailor Sky‐Sailor System [[45,52]45 Lab,52] of isis EPFL anan ultra-lightweightultra under‐lightweight a contract solar solarwith autonomous ESA, autonomous for scientific model model missions aircraft aircraft designeddesignedto Mars. by The by the theNUDT Autonomous Autonomous also developed System System a Lab prototypeLab of of EPFL EPFL solar under under aircraft a a contract contract for Mars withwith exploration ESA,ESA, for scientific scientific (Figure 12). missions missions to Mars.to Mars. The NUDTThe NUDT also also developed developed a prototype a prototype solar solar aircraft aircraft for for Mars Mars exploration exploration (Figure (Figure 1212).).

Figure 12. Prototype solar aircraft developed by NUDT. FigureFigure 12. 12.Prototype Prototype solar aircraft developed developed by by NUDT. NUDT. The exploitation of a wind shear on Mars may be a possible approach to extending the flight The exploitation of a wind shear on Mars may be a possible approach to extending the flight enduranceThe exploitation of Mars exploration of a wind shearaircrafts. on MarsThe gravitational may be a possible acceleration approach on Mars to extendingis less than the 4/9 flightof endurancethat on Earth, of Mars and explorationthe atmosphere aircrafts. is also The much gravitational thinner (Figureacceleration 13). Theon Marsminimum is less gradient than 4/9 for of endurancethat on Earth, of Mars and exploration the atmosphere aircrafts. is also The gravitationalmuch thinner acceleration (Figure 13). on The Mars minimum is less thangradient 4/9 offor that on Earth, and the atmosphere is also much thinner (Figure 13). The minimum gradient for sustainable Appl. Sci. 2017, 7, 1061 19 of 25 Appl. Sci. 2017,, 7,, 1061 19 of 25

sustainable climbing on Mars is thus considered to be smaller than that on Earth. In addition, the climbingsustainable on Mars climbing is thus on considered Mars is thus to considered be smaller to than be smaller that on than Earth. that In on addition, Earth. In the addition, wind field the on wind fieldfield on Mars significantly varies with thethe altitude [55],[55], which isis also favorablefavorable toto climbing. Mars significantly varies with the altitude [55], which is also favorable to climbing. The data presented The data presented inin Figure 13 were obtained using thethe standard Mars‐‐GRAM 2005 [55].[55]. in Figure 13 were obtained using the standard Mars-GRAM 2005 [55].

Figure 13. Variations in the air density and gravitational acceleration with the altitude on Mars. Figure 13. Variations inin thethe air density and gravitational acceleration with thethe altitude on Mars. 5.2. Rayleigh Cycle Simulation Results 5.2. Rayleigh Cycle Simulation Results The parameters of the albatross in Table2 were used for the simulation of its Raleigh cycle. The The parameters of thethe albatross inin Table 2 were used forfor thethe simulation of itsits Raleigh cycle. The windwind shear shear model, model, which which was was in inin accordance accordance with with Equation Equation (1),(1), (1), was was obtained obtained fromfrom from MIL MIL-F-8785C‐‐F‐‐8785C [40].[40]. [ 40]. W W6 was set to 15 m/s, which is a common value on Earth [48]. A dynamic model block was built and 6Wwas6 was set set to to 15 15 m/s, m/s, which which is is a acommon common value value on onEarth Earth [48]. [ 48A ].dynamic A dynamic model model block was block built was and built andused used toto to perform perform thethe the simulation simulation with with thethe the aid aid of ofthethe the Simulink Simulink and and Stateflow Stateflow software software (MathWorks,(MathWorks, (MathWorks, Natick,Natick, MA, MA, USA). USA). Figure Figure 14 14 shows shows aa snapshotsnapshot of of thethe the Simulink Simulink model, model, while while Figure Figure 15 15shows shows thethe the StateflowStateflow chart chart corresponding corresponding to toto the thethe Mode Mode SwitchSwitch block inin in Figure Figure 14. 14 .

Figure 14. Snapshot of the Simulink model. Figure 14. Snapshot of thethe Simulink model. Appl. Sci. 2017, 7, 1061 20 of 25 Appl. Sci. 2017, 7, 1061 20 of 25

Figure 15. Snapshot of the Stateflow chart. Figure 15. Snapshot of the Stateflow chart. The Stateflow chart in Figure 15 realizes the switch logic in Figure 6, with the Controller block particularlyThe Stateflow realizing chart the in Figurealgorithm 15 realizesdescribed the in switch Section logic 4.3. in In Figure Figure6, with15, the the transition Controller logic block particularlybetween the realizing climbing the state algorithm and the describedhigh‐altitude in Section turn is v 4.3z < .1, In because Figure 1 15 is, a the margin transition that ensures logic between that thethe climbing aircraft stateenters and the high the high-altitude‐altitude turn state turn before is vz < it 1, begins because to dive. 1 is aTo margin minimize that the ensures energy thatloss, the aircraftthe discount enters therate high-altitudeλ should be tuned. turn When state before the discount it begins rate to is dive.set to 0, To the minimize low‐altitude the energyturn requires loss, the discountabout 10 rate s, andλ should the final be velocity tuned. Whenis so small the discountthat the aircraft rate is is set incapable to 0, the of low-altitude connecting to turn the requires next upwind climb phase. When the discount rate λ is set to 1, although the time required for the turn is about 10 s, and the final velocity is so small that the aircraft is incapable of connecting to the next significantly shortened, the energy consumption also increases. The bisection method was used to upwind climb phase. When the discount rate λ is set to 1, although the time required for the turn is iteratively tune the discount rate in the present work, by which a value of 0.9 was found to be significantly shortened, the energy consumption also increases. The bisection method was used to suitable. iteratively tune the discount rate in the present work, by which a value of 0.9 was found to be suitable. For a given aircraft, the lift coefficient CL varies with the angle of attack, which can be approximatedFor a given as aircraft, the angle the between lift coefficient the total velocityCL varies vector with and the horizontal angle of plane attack, of whichthe aircraft. can be approximatedThe angle of attack as the is angle thus betweenthe actual the control total input. velocity Owing vector to the and availability the horizontal of very plane limited of the data aircraft. on Thethe angle aerodynamics of attack is of thus albatrosses, the actual the control specific input. relationship Owing to thebetween availability their angle of very of limited attack dataand onlift the aerodynamicscoefficient is ofunclear. albatrosses, In the thepresent specific study, relationship an albatross between was simply their considered angle of attack as an andaircraft, lift coefficientand its is unclear.lift coefficient In the was present assumed study, to an be albatross proportional was simplyto the angle considered of attack as anfor aircraft,small values and its of lift the coefficient latter. wasAlthough assumed this to is be consistent proportional with toregular the angle aerodynamics of attack fortheory small [43], values it was of still the difficult latter. Although to determine this is consistentthe proportionality. with regular aerodynamicsNevertheless, with theory the [43 assumption,], it was still difficultthe lift tocoefficient determine could the proportionality.be directly Nevertheless,controlled by with the theangle assumption, of attack in the the lift simulation. coefficient could be directly controlled by the angle of attack in the simulation.Figures 16–18 compares the simulation results for the proposed bio‐inspired strategy with thoseFigures for the16– traditional18 compares GPOPS the simulation optimized results trajectory. for the The proposed figures cover bio-inspired the same strategy period with in the those forentire the traditional time series GPOPS for the optimizedtwo methods, trajectory. coinciding The with figures a full cover cycle the of same the GPOPS period method, in the entire and timea serieslittle for longer the than two methods,a cycle generated coinciding by the with bio‐inspired a full cycle strategy. of the The GPOPS yellow method, background and in a Figure little longer 16 exactly covers a cycle of the bio‐inspired strategy. The figure shows the variations in the energy and than a cycle generated by the bio-inspired strategy. The yellow background in Figure 16 exactly altitude with time. Figure 17 shows the variations in the control inputs with time. As can be covers a cycle of the bio-inspired strategy. The figure shows the variations in the energy and altitude observed, the computed control inputs for the GPOPS method form smooth signals that also with time. Figure 17 shows the variations in the control inputs with time. As can be observed, the comprise four phases corresponding to those of the Rayleigh cycle, although the phase switch computedboundaries control are not inputs as forobvious the GPOPS as those method of the formbio‐inspired smooth signalsstrategy. that In alsothe upwind comprise climb four and phases correspondingdownwind dive to those segments, of the Rayleigh the bank cycle, angles although for both the phase methods switch are boundaries very small, are notsometimes as obvious asapproaching those of the zero. bio-inspired Conversely, strategy. the bank In the angles upwind during climb the high and‐ downwindaltitude and divelow‐altitude segments, turns the are bank angleslarge. for In bothaddition, methods the lift are coefficients very small, for sometimesboth methods approaching are maximum zero. during Conversely, the high‐ thealtitude bank turn, angles duringand minimum the high-altitude during the and upwind low-altitude climb and turns downwind are large. dive. In addition, In the low the‐altitude lift coefficients turn segment, for both methodsboth methods are maximum visibly compromise during the on high-altitude the assigned turn, lift coefficient, and minimum opting during for medium the upwind values. climb and downwind dive. In the low-altitude turn segment, both methods visibly compromise on the assigned lift coefficient, opting for medium values. Appl. Sci. 2017, 7, 1061 21 of 25 Appl. Sci. 2017, 7, 1061 21 of 25 Appl. Sci. 2017, 7, 1061 21 of 25

104 1.4 104 1.4 Bio-inspired GPOPS 1.2 Bio-inspired 1.2 GPOPS 1 Energy(J) 1 Energy(J) 0.8 L U H D L o p ig o o w- w h w w- 0.8 Al ind -Al nw Al L titu U C H tit D in L titu o d p lim ig ud o d D o d w- e T w b h e w i w- e T Al u ind -Al Tu nw ve Al u titu rn C tit rn in titu rn d lim ud d d e b e Di e Tu Tu ve Tu rn rn rn 150 150 100 100 50 Altitude(m) 50 Altitude(m) 0 02468101214 0 02468101214Time(s) Time(s)

Figure 16. Time histories of energy and altitude for the Gauss Pseudospectral Optimization Software Figure 16. Time histories of energy and altitude for the Gauss Pseudospectral Optimization Software Figure(GPOPS) 16. Time and histories proposed of bioenergy‐inspired and altitudemethods. for the Gauss Pseudospectral Optimization Software (GPOPS) and proposed bio-inspired methods. (GPOPS) and proposed bio‐inspired methods. L C L C Lift coefficient Lift coefficient Bank angle (degree) Bank angle (degree)

FigureFigure 17. Time 17. Time histories histories of the of the control control inputs inputs for for the the GPOPS GPOPS and and proposed proposed bio-inspiredbio‐inspired methods. methods. Figure 17. Time histories of the control inputs for the GPOPS and proposed bio‐inspired methods. In Figure 16, the indicated boundaries of the four phases (upwind climb, high‐altitude turn, In Figure 16, the indicated boundaries of the four phases (upwind climb, high-altitude turn, downwind dive, and low‐altitude turn) are for the bio‐inspired strategy. The phase switches of the downwindIn Figure dive, 16, and the low-altitude indicated boundaries turn) are of for the the four bio-inspired phases (upwind strategy. climb, The high phase‐altitude switches turn, of the downwindcontinuous dive, GPOPS and low optimized‐altitude trajectoryturn) are forare the not bio obvious.‐inspired The strategy. plots for The both phase methods switches in Figureof the 16 continuous GPOPS optimized trajectory are not obvious. The plots for both methods in Figure 16 continuousvalidate GPOPSthe Rayleigh optimized cycle trajectoryenergy theory are not discussed obvious. in The Section plots 4.1. for Energy both methods is harvested in Figure during 16 the validate the Rayleigh cycle energy theory discussed in Section 4.1. Energy is harvested during the validateupwind the Rayleighclimb and cycle downwind energy theory dive, discussed whereas in energy Section is4.1. lost Energy during is harvested the low during‐altitude the and upwind climb and downwind dive, whereas energy is lost during the low-altitude and high-altitude upwindhigh‐ altitudeclimb andturns. downwind However, dive,the total whereas energy energychanges is during lost duringa cycle thefor thelow two‐altitude methods and are turns.highpositive.‐altitude However, For turns. the the GPOPSHowever, total energy method, the changes total the energyenergy during increase changes a cycle during during for thea cyclea twocycle is methods aboutfor the 1300 two are J, positive.accompaniedmethods are For by the GPOPSpositive. method, For the the GPOPS energy method, increase the during energy a increase cycle is during about 1300 a cycle J, accompanied is about 1300 byJ, accompanied an altitude increase by of about 17 m. In the case of the bio-inspired strategy, the energy and altitude increases during a cycle are about 900 J and 12 m, respectively. The GPOPS method produces an optimized solution, hence Appl. Sci. 2017, 7, 1061 22 of 25

an altitude increase of about 17 m. In the case of the bio‐inspired strategy, the energy and altitude Appl.increases Sci. 2017, 7 during, 1061 a cycle are about 900 J and 12 m, respectively. The GPOPS method produces22 an of 25 optimized solution, hence its larger cyclic energy and altitude increases. Nevertheless, the proposed its largerbio‐inspired cyclic strategy energy andis simpler altitude and increases. affords a significant Nevertheless, saving the in proposedthe computation bio-inspired time. The strategy larger is cyclic energy increase of the GPOPS method is afforded by its lower energy loss during the simpler and affords a significant saving in the computation time. The larger cyclic energy increase of high‐altitude turn. the GPOPS method is afforded by its lower energy loss during the high-altitude turn. Figure 18 shows phase portraits of the GPOPS and proposed bio‐inspired methods. The plot Figure 18 shows phase portraits of the GPOPS and proposed bio-inspired methods. The plot for each method covers a cycle in the velocity space. The altitude corresponding to each point is for eachcolor‐ methodcoded, and covers the four a cycle phases in theof the velocity Rayleigh space. cycle Theof the altitude bio‐inspired corresponding strategy are to indicated each point by is color-coded,numbers, andwhere the the four 1–2, phases 2–3, 3–4, of theand Rayleigh 4–1 segments cycle respectively of the bio-inspired correspond strategy to the areupwind indicated climb, by numbers,high‐altitude where turn, the 1–2, downwind 2–3, 3–4, dive, and and 4–1 low segments‐altitude respectivelyturn phases. correspond to the upwind climb, high-altitude turn, downwind dive, and low-altitude turn phases.

FigureFigure 18. 18.Phase Phase portraits portraits of of the the Rayleigh Rayleigh cyclescycles of of the the GPOPS GPOPS and and proposed proposed bio bio-inspired‐inspired methods methods in in thethe velocity velocity spacespace (1–2:(1–2: upwindupwind climb, 2–3: high high-altitude‐altitude turn, turn, 3–4: 3–4: downwind downwind dive, dive, and and 4–1: 4–1: low-altitudelow‐altitude turn. turn.

TableTable3 summarizes 3 summarizes the the computation computation time time for for GPOPS GPOPS and and the the bio-inspired bio‐inspired strategy. GPOPS GPOPS generallygenerally consumes consumes about about 15 s 15 to computes to compute a dynamic a dynamic soaring soaring trajectory trajectory in a personal in a personal computer computer with an with an Intel®CoreTM i3‐2100 CPU running at 3.10 GHz, while the bio‐inspired strategy consumes 2 Intel® CoreTM i3-2100 CPU running at 3.10 GHz, while the bio-inspired strategy consumes 2 ms because ms because of the simple judging logic (the computation time can be checked in the Simulink report of the simple judging logic (the computation time can be checked in the Simulink report interface). interface). This comparison shows that the bio‐inspired strategy is more efficient at real‐time This comparison shows that the bio-inspired strategy is more efficient at real-time applications. applications.

Table 3. Computation times for GPOPS and the bio-inspired strategy. Table 3. Computation times for GPOPS and the bio‐inspired strategy.

MethodMethod GPOPSGPOPS Bio‐Inspired Bio-Inspired Strategy Strategy Computation timeComputation time 15 s15 s 2 ms 2 ms

6. Conclusions 6. Conclusions A criterion for sustainable climbing of an unpowered aircraft in a gradient wind was proposed andA criterion verified. The for sustainable criterion is based climbing on the of product an unpowered of the aircraft aircraft and in environmental a gradient wind fractions, was proposed where andthe verified. aircraft The fraction criterion is composed is based of on the the aerodynamic product of the and aircraft wing loading and environmental fractions. The fractions, criterion wherewas the aircraftused to develop fraction a is method composed for calculating of the aerodynamic the optimal and lift wingcoefficient loading from fractions. the aerodynamic The criterion fraction. was usedAn to expression develop a of method the minimum for calculating wind gradient the optimal required lift for coefficient unpowered from flight the was aerodynamic also derived fraction. from An expression of the minimum wind gradient required for unpowered flight was also derived from the environmental fraction, and the feasibility of implementing the flight in near space and on Mars was analyzed. A sensitivity analysis of the aircraft fraction indicated that the aerodynamic fraction was generally as important as the wing loading fraction. Further, the energy harvesting boundaries of the Rayleigh cycle for unpowered flight was investigated in the velocity space, and the maximum Appl. Sci. 2017, 7, 1061 23 of 25 power was determined. On the basis of the Rayleigh cycle energy theory, a bio-inspired Rayleigh cycle guidance strategy for unpowered flight was proposed, involving the tuning of a single parameter. A comparison of the simulation results for the proposed strategy and the traditional optimization method confirmed the effectiveness of the proposed strategy for real-time applications.

Acknowledgments: This research was supported by the National High Technology Research and Development Program of China (Grant No. 2014AA7054035). Author Contributions: Shangqiu Shan proposed the theory and wrote this paper; Zhongxi Hou provided guidelines for the work; Bingjie Zhu provided suggestions for writing the paper. Conflicts of Interest: The authors declare no conflict of interest.

References

1. El Alimi, S.; Maatallah, T.; Dahmouni, A.W.; Nasrallah, S.B. Modeling and investigation of the wind resource in the Gulf of Tunis, Tunisia. Renew. Sustain. Energy Rev. 2012, 16, 5466–5478. [CrossRef] 2. Bonnin, V.; Benard, E.; Moschetta, J.M.; Benard, E. Energy-harvesting mechanisms for UAV flight by dynamic soaring. In Proceedings of the AIAA Atmospheric Flight Mechanics (AFM) Conference, Boston, MA, USA, 19–22 August 2013. 3. Warrick, D.R.; Hedrick, T.L.; Biewener, A.A.; Crandell, K.E.; Tobalske, B.W. Foraging at the edge of the world: Low-altitude, high-speed manoeuvering in barn swallows. Phil. Trans. R. Soc. B 2016, 371, 20150391. [CrossRef][PubMed] 4. Richardson, P.L. How do albatrosses fly around the world without flapping their wings? Prog. Oceanogr. 2011, 88, 46–58. [CrossRef] 5. Yonehara, Y.; Goto, Y.; Yoda, K.; Watanuki, Y.; Young, L.C.; Weimerskirch, H.; Bost, C.-A.; Sato, K. Flight paths of seabirds soaring over the ocean surface enable measurement of fine-scale wind speed and direction. Proc. Natl. Acad. Sci. USA 2016, 113, 9039–9044. [CrossRef][PubMed] 6. Miller, T.A.; Brooks, R.P.; Lanzone, M.J.; Brandes, D.; Cooper, J.; Tremblay, J.A.; Wilhelm, J.; Duerr, A.; Katzner, T.E. Limitations and mechanisms influencing the migratory performance of soaring birds. IBIS 2016, 158, 116–134. [CrossRef] 7. Sachs, G.; Traugott, J.; Holzapfel, F. In-flight measurement of dynamic soaring in albatrosses. In Proceedings of the AIAA Guidance, Navigation, and Control Conference, Toronto, ON, Canada, 2–5 August 2010. 8. Airy, H. The soaring of birds. Nature 1883, 28, 103. [CrossRef] 9. Boslough, M.B. Autonomous Dynamic Soaring Platform for Distributed Mobile Sensor Arrays; Technical Report SAND2002-1896; Sandia National Laboratories: Albuquerque, NM, USA, 2002. 10. Idrac, P. Experimental study of the “soaring” of albatrosses. Nature 1925, 115, 532. [CrossRef] 11. Tucker, V.A.; Parrott, G.C. Aerodynamics of gliding flight in a falcon and other birds. J. Exp. Biol. 1970, 52, 345–367. 12. Weimerskirch, H.; Guionnet, T.; Martin, J.; Shaffer, S.A.; Costa, D. Fast and fuel efficient? Optimal use of wind by flying albatrosses. Proc. R. Soc. Lond. B Biol. Sci. 2000, 267, 1869–1874. [CrossRef][PubMed] 13. Rosen, M.; Hedenstrom, A. Gliding flight in a jackdaw: A wind tunnel study. J. Exp. Biol. 2001, 204, 1153–1166. [PubMed] 14. Wood, C. The flight of albatrosses (a computer simulation). IBIS 1973, 115, 244–256. [CrossRef] 15. Hendriks, F. Dynamic soaring in shear flow (of gliders). In Proceedings of the 2nd International Symposium on the Technology and Science of Low Speed and Motorless Flight, Cambridge, MA, USA, 11–13 September 1974. 16. Denny, M. Dynamic soaring: Aerodynamics for albatrosses. Eur. J. Phys. 2008, 30, 75. [CrossRef] 17. Taylor, G.K.; Reynolds, K.V.; Thomas, A.L.R. Soaring energetics and glide performance in a moving atmosphere. Phil. Trans. R. Soc. B 2016, 371, 20150398. [CrossRef][PubMed] 18. Liu, D.-N.; Hou, Z.-X.; Guo, Z.; Yang, X.-X.; Gao, X.-Z. Bio-inspired energy-harvesting mechanisms and patterns of dynamic soaring. Bioinspir. Biomim. 2016, 12, 016014. [CrossRef][PubMed] 19. Zhao, Y.J. Optimal patterns of glider dynamic soaring. Optim. Control Appl. Methods 2004, 25, 67–89. [CrossRef] Appl. Sci. 2017, 7, 1061 24 of 25

20. Zhao, Y.J.; Qi, Y.C. Minimum fuel powered dynamic soaring of unmanned aerial vehicles utilizing wind gradients. Optim. Control Appl. Methods 2004, 25, 211–233. [CrossRef] 21. Sachs, G.; Bussotti, P. Application of optimal control theory to dynamic soaring of seabirds. In Variational Analysis and Applications; Springer: Berlin, Germany, 2005; pp. 975–994. 22. Sachs, G. Minimum shear wind strength required for dynamic soaring of albatrosses. IBIS 2005, 147, 1–10. [CrossRef] 23. Deittert, M.; Richards, A.; Toomer, C.; Pipe, A. Dynamic soaring flight in turbulence. In Proceedings of the AIAA Guidance, Navigation and Control Conference, Chicago, IL, USA, 10–13 August 2005; pp. 2–5. 24. Kaushik, H.; Mohan, R.; Prakash, K.A. Utilization of wind shear for powering unmanned aerial vehicles in surveillance application: A numerical optimization study. Energy Procedia 2016, 90, 349–359. [CrossRef] 25. Yang, H.; Zhao, Y.J. A practical strategy of autonomous dynamic soaring for unmanned aerial vehicles in loiter missions. In Proceedings of the AIAA Infotech@Aerospace, Arlington, VA, USA, 26–29 September 2005; pp. 1–12. 26. Gordon, R.J. Optimal Dynamic Soaring for Full Size Sailplanes. Master’s Thesis, Defense Technical Information Center, Fort Belvoir, VA, USA, 2006. 27. Deittert, M.; Richards, A.; Toomer, C.A.; Pipe, A. Engineless unmanned aerial vehicle propulsion by dynamic soaring. J. Guid. Control Dyn. 2009, 32, 1446–1457. [CrossRef] 28. Akhtar, N.; Whidborne, J.F.; Cooke, A.K. Wind shear energy extraction using dynamic soaring techniques. In Proceedings of the 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, USA, 5–8 January 2009. 29. Du, S.; Gao, Y. Inertial aided cycle slip detection and identification for integrated PPP GPS and INS. Sensors 2012, 12, 14344–14362. [CrossRef][PubMed] 30. Lawrance, N.R.; Sukkarieh, S. Autonomous exploration of a wind field with a gliding aircraft. J. Guid. Control Dyn. 2011, 34, 719–733. [CrossRef] 31. Nguyen, J.L.; Lawrance, N.R.J.; Fitch, R.; Sukkarieh, S. Real-time path planning for long-term information gathering with an aerial glider. Auton. Robots 2016, 40, 1017–1039. [CrossRef] 32. Langelaan, J.W.; Spletzer, J.; Montella, C.; Grenestedt, J. Wind field estimation for autonomous dynamic soaring. In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), St. Paul, MN, USA, 14–18 May 2012; pp. 16–22. 33. Rodriguez, L.; Cobano, J.A.; Ollero, A. Wind field estimation and identification having shear wind and discrete gusts features with a small UAS. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, Korea, 9–14 October 2016; pp. 5638–5644. 34. Liu, Y.; Van Schijndel, J.; Longo, S.; Kerrigan, E.C. UAV energy extraction with incomplete atmospheric data using MPC. IEEE Trans. Aerosp. Electron. Syst. 2015, 51, 1203–1215. [CrossRef] 35. Barate, R.; Doncieux, S.; Meyer, J.A. Design of a bio-inspired controller for dynamic soaring in a simulated unmanned aerial vehicle. Bioinspir. Biomim. 2006, 1, 76. [CrossRef][PubMed] 36. Gao, X.Z.; Hou, Z.X.; Guo, Z.; Fan, R.F.; Chen, X.Q. Analysis and design of guidance-strategy for dynamic soaring with UAVs. Control Eng. Pract. 2014, 32, 218–226. [CrossRef] 37. Yang, Y. Controllability of spacecraft using only magnetic torques. IEEE Trans. Aerosp. Electron. Syst. 2016, 52, 954–961. [CrossRef] 38. Gao, C.; Liu, H.H.T. Dubins path-based dynamic soaring trajectory planning and tracking control in a gradient wind field. Optim. Control Appl. Methods 2016, 38, 147–166. [CrossRef] 39. Zhu, B.; Hou, Z.; Shan, S.; Wang, X. Equilibrium positions for UAV flight by dynamic soaring. Int. J. Aerosp. Eng. 2015, 2015, 141906. [CrossRef] 40. Moorhouse, D.; Woodcock, R. US Military Specification MIL-F-8785C; US Department of Defense: Washington DC, USA, 1980. 41. Lift-to-Drag Ratio. Available online: https://en.wikipedia.org/wiki/Lift-to-drag_ratio (accessed on 11 July 2017). 42. Rao, A.V.; Benson, D.A.; Darby, C. Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudo spectral method. ACM Trans. Math. Softw. 2010, 1, 2237. [CrossRef] 43. Anderson, J.D. Fundamentals of Aerodynamics; Tata McGraw-Hill Education: New York, NY, USA, 2010. 44. McCormick, B.W. Aerodynamics, Aeronautics, and Flight Mechanics; Wiley: New York, NY, USA, 1995; Volume 2. Appl. Sci. 2017, 7, 1061 25 of 25

45. Noth, A. Design of Solar Powered Airplanes for Continuous Flight. Ph.D. Thesis, ETH Zurich, Zurich, Switzerland, 2008. 46. Tennekes, H. The Simple Science of Flight: From Insects to Jumbo Jets; MIT Press: Cambridge, MA, USA, 2009. 47. US Standard Atmosphere; National Oceanic and Atmospheric Administration: Silver Spring, MD, USA, 1976. 48. Drob, D.P.; Emmert, J.T.; Meriwether, J.W.; Makela, J.J.; Doornbos, E.; Conde, M.; Hernandez, G.; Noto, J.; Zawdie, K.A.; McDonald, S.E.; et al. An update to the Horizontal Wind Model (HWM): The quiet time thermosphere. Earth Space Sci. 2015, 2, 301–319. [CrossRef] 49. Gao, X.; Hou, Z.; Guo, Z.; Liu, J.; Chen, X. The influence of wind shear to the performance of high-altitude solar-powered aircraft. Proc. IMechE Part G J. Aerosp. Eng. 2014, 228, 1562–1573. 50. Braun, R.D.; Wright, H.S.; Croom, M.A.; Levine, J.S.; Spencer, D.A. Design of the Ares Mars airplane and mission architecture. J. Spacecr. Rocket. 2006, 43, 1026–1034. [CrossRef] 51. Guynn, M.; Croom, M.; Smith, S.; Parks, R.; Gelhausen, P. Evolution of a Mars airplane concept for the Ares Mars scout mission. In Proceedings of the 2nd AIAA Unmanned Unlimited Systems, Technologies, and Operations Conference, San Diego, CA, USA, 15–18 September 2003. 52. Noth, A.; Bouabdallah, S.; Michaud, S.; Siegwart, R.; Engel, W. Sky-Sailor design of an autonomous solar powered Martian airplane. In Proceedings of the 8th ESA Workshop on Advanced Space Technologies for Robotics, Noordwick, The Netherlands, 2–4 November 2004. 53. Anyoji, M.; Nagai, H.; Asai, K. Development of low density wind tunnel to simulate atmospheric flight on Mars. In Proceedings of the 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, USA, 5–8 January 2009. 54. Murray, J.E.; Tartabini, P.V. Development of a Mars Airplane Entry, Descent, and Flight Trajectory; NASA/TM-2001-209035; NASA Dryden Flight Research Center: Edwards, CA, USA, 2001. 55. Justus, C.G.; Duvall, A. Mars aerocapture and validation of Mars-GRAM with TES data. In Proceedings of the 53rd JANNAF Propulsion Meeting, Monterey, CA, USA, 5–8 December 2005.

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