Hindawi Publishing Corporation International Journal of Aerospace Engineering Volume 2015, Article ID 141906, 8 pages http://dx.doi.org/10.1155/2015/141906

Research Article Equilibrium Positions for UAV Flight by Dynamic Soaring

Bingjie Zhu,1 Zhongxi Hou,1 Shangqiu Shan,1 and Xinzhu Wang2

1 College of Aerospace Sciences and Engineering, National University of Defense Technology, Changsha, Hunan 410073, China 2Chongqing Key Laboratory of Heterogeneous Material Mechanics, Department of Engineering Mechanics, Postdoctoral Station of Mechanics, Chongqing University, Chongqing 400040, China

Correspondence should be addressed to Bingjie Zhu; [email protected]

Received 23 June 2015; Revised 22 September 2015; Accepted 30 September 2015

Academic Editor: Hyochoong Bang

Copyright © 2015 Bingjie Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Dynamic soaring is a special flying technique designed to allow UAVs (unmanned aerial vehicles) to extract energy from wind gradient field and enable UAVs to increase the endurance. In order to figure out the energy-extraction mechanisms in dynamic soaring, a noninertial wind relative reference frame of aircraft is built. In the noninertial frame, there is an inertial force which is created by gradient wind field. When the wind gradient (𝐺𝑊) and the components of airspeed (V𝑧V𝑥) are positive, inertial force (𝐹) makes positive work to the aircraft. In the meantime, an equilibrium position theory of dynamic soaring is proposed. Atthe equilibrium positions, the increased potential energy is greater than the wasted kinetic energy when the aircraft is flying upwards. The mechanical energy is increased in this way, and the aircraft can store energy for flight. According to the extreme valuetheory, contour line figures of the maximum function and the component of airspeed (V𝑧) are obtained to find out the aircraft’s lifting balance allowance in dynamic soaring. Moreover, this equilibrium position theory can also help to conduct an aircraft to acquire energy from the environment constantly.

1. Introduction flight performance are taken as a reference model against the dynamic soaring performance of a small UAV. Long endurance flight is a key design criterion for unmanned Dynamic soaring is an important flight technique de- aerialvehicle[1].TheUAVs’flightendurancecanbeimproved signed to allow aircraft to extract energy from atmosphere. bytheprogressinpowercapacityandaerodynamicand BasedonLordRayleigh’sobservationsofalbatrossesinthe engine design [2], but not all such means are equal. Another south Atlantic, more recent publications are concerned with important alternative for enhancing UAV flight endurance wind estimated and dynamic soaring mechanism. In order is to seek energy from flight environment. In nature, there to estimate wind field for autonomous dynamic soaring, are many large-sized birds which make use of environment Lawrance et al. [7, 8] provided a method for taking direct energy to enhance their flight ability, such as wandering observations of the wind during flight and creating a wind albatross who extracts energy from horizontally moving air to direct future exploration. The wind estimation is the pre- and transfers energy to itself. The achieved energy enables condition for dynamic soaring. In this paper, a mathematical wandering albatrosses to fly continuously without flapping model is built for the wind field. As for dynamic soaring their wings [3, 4]. The flight manoeuvre that wandering mechanism, Grenestedt [9] provided insight into how and albatross performs is referred to as dynamic soaring [5]. Early where energy is extracted from the wind during dynamic in 1883, Lord Rayleigh published “The Soaring of Birds” in soaring. The effects of some aircraft parameters such as mass, Nature [6], which is regarded as the first paper about dynamic drag area, and wing span were discussed. Bower [5] studied soaring. Dynamic soaring is possible when there is hori- the energy transfer mechanisms for a vehicle flying in a zontally moving air in the flight environment. Horizontally spatiallyandtemporallyvaryingwindfield.Deittertetal.[10] moving air is also called gradient wind, in which wind investigated optimal trajectories for minimal and maximal speed varies with altitude. In this paper, both the size of wind conditions, and the likelihood of favorable winds was the wandering albatrosses with its 3.3 m wing span and its predicted based on long-term weather statics and knowledge 2 International Journal of Aerospace Engineering of minimal and maximal permissible wind strengths. Cone 100 [11] analyzed the dynamic soaring flight of albatross with 80 mathematical models. Peter [12] draw a conclusion that, in an energy neutral cycle of dynamic soaring, whether the vehicle 60

returns to initial velocity and height with no power input (m) z 40 depends only upon the maximum lift-to-drag ratio of the vehicleandthewindspeedvariation. 20 Scores of research findings reveal the significant meaning 0 of dynamic soaring. Based on the achievements of these 0 3 6 9 12 15 findings, some new ideas are studied in this paper. The W (m/s) authors first extend the idea of Grenestedt [9] by investigating the equilibrium position of dynamic soaring and providing Wind speed insight into energy variation in equilibrium position. The Wind gradient same as other flight styles, they applied forces on the aircraft Figure 1: Wind speed and wind gradient. for dynamic soaring varying continuously. Then there should be some positions of forces balance in the course of flight. At these points, the aircraft stays in a certain speed, with no kinetic energy loss. Of course, if all the points could chain a 𝑊(𝑧) andheightabovethesurface𝑧 is defined as follows closed route, the permanent flying would come true when the [13, 14]: aircraft flies along the route. (𝑧/𝑧 ) The rest of the paper is organized as follows: the method- 𝑊 (𝑧) =𝑊 ln 0 , ology is described in Section 2, including the wind gradient ref (𝑧 /𝑧 ) (1) ln ref 0 and the mathematical models for the UAV. Section 3 is simulation results and associated discussion. Finally, the 𝑊(𝑧) 𝑧 𝑊 where isthewindspeedatheight and ref is the concluding remarks are made in Section 4. 𝑧 𝑧 wind speed at the reference height ref.Thevariable 0 is the aerodynamic roughness length or the roughness factor. It is 2. Mathematical Formulations of the Problem an experimentally derived constant that accounts for the kind of surface over which the wind is blowing. Typically, a higher Inspired by the forces analysis methods in dynamic soaring value of the roughness length indicates more obstructions suggested by Bower [5] and Deittert et al. [10], the problem on the surface such as trees and buildings. By differentiating of equilibrium position for dynamic soaring is analyzed (1) with respect to altitude, the following equation can be intensively in this paper. The parameters that can influence obtained: the equilibrium position include the range of airspeed, the 𝑊 size of wind gradient, and other parameters. Meanwhile, the 𝑊̇ (𝑧) = ref =𝐺 , (𝑧 /𝑧 )⋅𝑧 𝑊 (2) maximum value of energy acquired in the equilibrium point ln ref 0 is analyzed. Once the parameters of an aircraft are known, the wind gradient suited for the equilibrium position of dynamic where 𝐺𝑊 is the gradient of the wind field. The variation soaring can be determined. Similarly, in a known wind field, trends of wind profile and wind gradient profile are depicted the type of aircraft that has equilibrium point in dynamic in Figure 1. soaring can be found out. A set of computational methods for Figure1showsthewindprofileandwindgradientprofile 𝑊 =15 equilibrium position of dynamic soaring is established. The for value of the roughness coefficient with ref m/s, 𝑧 = 100 𝑧 = 0.05 purposeofthissectionistoprovideatheoreticalguidance ref m, and 0 m. for the optimal dynamic soaring trajectory by means of the analysis of equilibrium position of dynamic soaring. Due 2.2. Flying Model. According to the observed data of dynamic to no kinetic energy loss in the position of forces balance, soaring [4], the albatross climbs with headwind, and the wind the mechanical energy increases with increasing value of speed increases with the increasing of flight height. After potential energy of the aircraft in the course of upward turning in high altitude, the albatross glides with leeward, flying. and the wind speed decreases with the descending of flight height. An illustration of a typical dynamic soaring trajectory 2.1. The Wind Field Model. It is assumed that the air- is shown in Figure 2. craftfliesoveraflatsurfaceandthewindfieldissteady. Compared with Earth fixed frame, the wind relative There is a horizontal shear flow of the air in the three- reference frame of aircraft is not inertial frame in the wind dimensionalspace.Theflowoftheaircausesthewind field. So there is an inertial force 𝐹𝑖 that works on the aircraft. speed to increase with increasing height above the ground. A noninertial frame is defined, which is the wind relative The air density is assumed to be constant. The wind blows reference frame of aircraft. In the noninertial frame, there are along the negative direction of 𝑥-axis, the positive 𝑧-axis four applied forces accounted for the aircraft: lift force𝐿 ( ), points to the sky, and the direction of 𝑦-axis is dependent drag force (𝐷), gravitational force (𝑚𝑔), and inertial force on right-handed rule. The relation between wind speed (𝐹𝑖). The inertial force is created by wind, so the direction of International Journal of Aerospace Engineering 3

where 𝜌 is air density, 𝑆 is wing area, 𝐶𝐿 is lift coefficient, Wx 𝐶𝐷 is drag coefficient, V is the airspeed of the aircraft, 𝐶𝐷0 2 is the parasitic drag coefficient, and 𝐾𝐶 is the induced drag B 𝐿 C coefficient. Note that 𝐿 2 −1 A 𝐾=(4( ) 𝐶 ) , D 𝐷0 (8) E 𝐷 max

where (𝐿/𝐷)max is the maximum lift over drag ratio, which, together with 𝐶𝐷0, is the most important aircraft aerody- Figure 2: The illustration trajectory of dynamic soaring. namic efficiency parameters and fully defines the aerody- namic drag polar. Z Inthecourseofflight,thedirectionofliftiscrosscutto drag all the time, so the resultant force (𝐹) of lift and drag can F v be indicated as 󳰀 L F1 1 Y |𝐹| = √𝐿2 +𝐷2 = V2𝜌𝑆√𝐶2 +𝐶2 . 2 𝐿 𝐷 (9)

𝛿 mGW z From (9), the airspeed (V)oftheaircraftis 𝛿1 D 2 |𝐹| F1 X |V| = . mg √ (10) √ 2 2 𝜌𝑆 𝐶𝐿 +𝐶𝐷

󸀠 When the value of 𝐹 is equal to 𝐹1,therelationship Figure 3: The applied forces for the UAV. between V and V𝑧 canbeattainedby(10). As for the energy variation, first of all, considering the theorem of kinetic energy in the noninertial frame, the force is the same as the wind direction. The inertial force 2 can be expressed as [10, 15, 16] 𝑑𝐸𝑘 =−𝑃𝐷 |V| 𝑑𝑙+𝑚𝐺𝑊V𝑧𝑑𝑥 − 𝑚𝑔𝑑𝑧, (11) 𝑑𝑊⇀󳨀 𝑑𝑊 𝑑𝑧⇀󳨀 ⇀󳨀 𝑑𝐸 𝑑𝑙 𝐹 =−𝑚 𝑖 =−𝑚 ⋅ 𝑖 =−𝑚𝐺 V 𝑖 , (3) where 𝑘 is the variation of kinetic energy, is the identity 𝑖 𝑑𝑡 𝑑𝑧 𝑑𝑡 𝑊 𝑧 element of flight trajectory, and 𝑃𝐷 =1/2𝜌𝑆𝐶𝐷.Theright- where 𝑖 istheunitvectorofwindfieldandV𝑧 is the airspeed hand side of (11) is infinitesimal energy expressions caused by component in the direction of positive 𝑧-axis. So a point drag force, inertial force, and gravitational force. The lift force 𝐿 mass flight model with attached forces is adopted, and the ( ) is vertical to the direction of motion, so there is no kinetic definitions of forces, speed, and angles used in the model energy variation of UAV caused by lift force. 𝑑𝑙 = V𝑑𝑡 𝑑𝑥 = V 𝑑𝑡 V are shown in Figure 3. The UAV is represented by a three- If , 𝑥 ,and 𝑥 is the airspeed component 𝑥 dimensionalpoint-massmodelwith3DOF.FromFigure3, in the direction of positive , (11) then transforms into 𝐹 is the resultant force of lift𝐿 ( )anddrag(𝐷), 𝛿 is the angle 𝑑𝐸 =(−𝑃 |V|3 +𝑚𝐺 V V )𝑑𝑡−𝑚𝑔𝑑𝑧. between 𝐹 and 𝐷, 𝐹1 is the resultant force of gravity (𝑚𝑔)and 𝑘 𝐷 𝑊 𝑧 𝑥 (12) 󸀠 inertial force (𝑚𝐺𝑊V𝑧), and 𝐹1 is the reverse force of 𝐹1.The wind blows in the direction of position 𝑋. The integration of (12) can be expressed as The barycenter kinetics equation of aircraft in the nonin- 3 ertial frame is represented as 𝐸𝑘𝑓 −𝐸𝑘0 = ∫ (−𝑃𝐷 |V| +𝑚𝐺𝑊V𝑧V𝑥)𝑑𝑡 V(𝑡) ⇀󳨀̇ ⇀󳨀 ⇀󳨀 ⇀󳨀 ⇀󳨀 (13) 𝑚 V = 𝐿+𝐷+𝑚𝑔−𝑚𝐺𝑊V𝑧 𝑖, (4) −𝑚𝑔(𝑧𝑓 −𝑧0), ⇀󳨀̇ ⇀󳨀 where V is the acceleration vector of airspeed, 𝐿 is the vector ⇀󳨀 where 𝐸𝑘𝑓 is the kinetic energy at height of 𝑧𝑓 and 𝐸𝑘0 is the 𝐷 𝑚⇀󳨀𝑔 of lift, isthevectorofdrag,and isthevectorofgravity. kinetic energy at height of 𝑧0.Aftertheaircraftflyaperiodof The last expression in (4) is the inertial force. time, if the kinetic energy of aircraft keeps in a constant value, Using the standard formulations for lift and drag force, the energy variation of this process could be indicated by the change of altitude difference, and (13) can now be stated to be 1 2 𝐿= 𝜌𝑆𝐶 V , (5) 2 𝐿 1 Δ𝑧 = ∫ (−𝑃 |V|3 +𝑚𝐺 V V )𝑑𝑡. 1 𝑚𝑔 𝐷 𝑊 𝑧 𝑥 (14) 𝐷= 𝜌𝑆𝐶 V2, V(𝑡) 2 𝐷 (6) Δ𝑧 is the variation of flight height. From (14), it can be 2 𝐶𝐷 =𝐶𝐷0 +𝐾𝐶𝐿, (7) found that the drag (𝐷)andinertialforce(𝐹𝑖)bringabout 4 International Journal of Aerospace Engineering

the variation of altitude or the variation of potential energy. V𝑧 =0, there is no balance point, because there is no inertial Thatistosay,aslongastherewaswindgradient(𝐺𝑊)inthe forcetocompensatethedrag. wind field, the aircraft could extract energy from the wind Alternatively, (18) can be expressed in another way: field in the suitable V𝑥 and V𝑧.In(14),thedrag(𝐷)makes 𝐹 negative work all the time, but inertial force ( 𝑖)doespositive √𝐿2 +𝐷2 = √𝑚2𝑔2 +𝑚2𝐺2 V2. (19) or negative work to the aircraft, depending on the specific 𝑊 𝑧 cases. If 𝐺𝑊 >0and V𝑧V𝑥 >0,inthiscase,inertialforce(𝐹𝑖) Combining (5)-(6), (10), and (19), there is doespositiveworktotheaircraft.Forthewindfield𝐺𝑊 >0, the following relation must hold: 𝑚2𝑔2 +𝑚2𝐺2 V2 V4 = 𝑊 𝑧 , V >0, 2 (20) 𝑥 𝜂𝑎

V𝑧 >0 (15) where or V𝑥 <0, 2 1 2 2 2 2 2 2 𝜂𝑎 = (𝜌 𝑆 𝐶𝐿 +𝜌 𝑆 𝐶𝐷). (21) V𝑧 <0. 4

That is, the aircraft climbs in the positive direction of 𝑥-𝑧 Equation (20) describes the relationship between V and V𝑧 or dives in the negative direction of 𝑥-𝑧; the aircraft extracts in the balance point. Simultaneously, (20) ensured that the energy. If the maneuver is reversed, the aircraft loses energy. size of aerodynamic force is equal to the resultant force of The mechanical energy of the aircraft is denoted as 𝐸𝑚.The gravitation and inertial force. variation of mechanical energy is equal to the energy caused Then, the only restriction to airspeed is the bank angle of by drag and inertial force. The identity element of mechanical air velocity vector, which is defined as 𝜓. There is a unique energy can be expressed as bank angle to enable forces to be balanced; it is necessary to figure out that the bank angle, which made the included angle 𝑑𝐸𝑚 =𝐷⋅𝑑𝑙+𝑚𝐺𝑤V𝑧𝑑𝑥 between air velocity vector (V)and𝐹1,is𝛿1,where𝐹1 is the 𝑚𝑔 𝑚𝐺𝑊V𝑧 2 resultant force of gravity ( )andinertialforce( ). In =−𝑃𝐷 ‖V‖ ⋅𝑑𝑙+𝑚⋅𝐺𝑊 ⋅ V𝑧 ⋅𝑑𝑥 Figure 3, the included angle between airspeed vector (V)and 𝐹1 canberepressedas 𝑑𝑥 =𝑚(𝑃 ‖V‖3 ⋅𝑑𝑡+𝐺 ⋅ V ⋅ ⋅𝑑𝑡) (16) 𝐷 𝑊 𝑧 𝑑𝑡 ⇀󳨀 ⇀󳨀 𝐹 1 ⋅ V cos 𝛿1 = 󵄨 󵄨 . (22) 𝑃 3 󵄨𝐹 󵄨 ⋅ V =𝑚⋅𝑑𝑡(− 𝐷 ‖V‖ +𝐺 ⋅ V ⋅ V ). 󵄨 1󵄨 | | 𝑚 𝑊 𝑧 𝑥 When 𝛿=𝛿1,inthiscase,theaerodynamicforceappears So the unit mass power of mechanical energy is to get across the inverse direction of the resultant force of gravitation and inertial force. Hence, 1 𝑑𝐸 𝑃 3 𝑚 =− 𝐷 ‖V‖ +𝐺 ⋅ V (𝑡) ⋅ V (𝑡) , (17) 𝑚 𝑑𝑡 𝑚 𝑊 𝑧 𝑥 ⇀󳨀 𝐹 ⋅⇀󳨀V 𝐷 𝛿 = 󵄨 1󵄨 = 𝛿= . (23) where v is the airspeed. Actually, (17) and (14) mean the same cos 1 󵄨 󵄨 cos 2 2 󵄨𝐹1󵄨 ⋅ |V| √𝐿 +𝐷 things. With the geometrical relationship of forces balanced, one 2.3. The Balance Points of Dynamic Soaring. In the point of can acquire the space curve of V𝑥 × V𝑦 × V𝑧,whichisobtained balance, the composition of driving force vectors is zero, and by projecting the four-dimensional space V𝑥 × V𝑦 × V𝑧 ×𝜓into theresultantforceofaircraftiszero.Thedynamicalequation three-dimensional space. The fundamental is to find outa in noninertial system can be expressed as suitable aerodynamic force to compensate the resultant force of gravitation and inertial force. ⇀󳨀 ⇀󳨀 ⇀󳨀 ⇀󳨀 ⇀󳨀 𝐿+𝐷+𝑚𝑔+𝑚𝐺𝑤V𝑧 𝑖=0. (18) The resultant force of gravitation and inertial force ⇀󳨀 ⇀󳨀 𝐹=[𝑚𝐺𝑊V𝑧,0,−𝑚𝑔], air velocity vector V =[V𝑟 cos(𝜓), Because the ground speed of the aircraft remains constant V (𝜓), V ] V = √V2 − V2 in the balance point, that is to say, the kinetic energy of the 𝑟 sin 𝑧 ,and 𝑟 𝑧; combining (20) and (23), flight system remains constant. When V𝑧 >0,thepotential there is energy of the flight system increases with increase in flight altitude, and the mechanical energy increases at the same 𝑚 𝑚2𝑔2 +𝑚2𝐺2 V2 𝑚 V <0 𝐺 V √√ 𝑊 𝑧 − V2 𝜓− V 𝑔 time. On the other hand, when 𝑧 , the mechanical energy 𝑊 𝑧 2 𝑧 cos 𝑧 𝜂𝑎 𝜂𝑎 𝜂𝑎 of the aircraft would not always increase, in which case, some (24) studies need to be done in the future. Here, the balance point 2 2 2 2 2 3/4 V >0 V <0 𝑚 𝐺 V +𝑚 𝑔 is called lifting balance point when 𝑧 ;whilewhen 𝑧 , =( 𝑊 𝑧 ) 𝛿. 2 cos the balance point is called descending balance point. When 𝜂𝑎 International Journal of Aerospace Engineering 5

GW = 0.5 GW = 1.0 GW = 1.5 1 1 1

0.5 0.5 0.5

−30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30

z z z −0.5 −0.5 −0.5

−1 −1 −1

CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5

(a) When 𝐺𝑊 =0.5, 𝐶𝐿 =0.5,1.0,and1.5,the (b) When 𝐺𝑊 =1.0, 𝐶𝐿 =0.5,1.0,and1.5,the (c) When 𝐺𝑊 =1.5, 𝐶𝐿 =0.5, 1.0, and 1.5, the extent of V𝑧 extent of V𝑧 extent of V𝑧

Figure 4: The extent of V𝑧, three typical values of 𝐺𝑊, 𝐶𝐿 are selected, which can tell the relationship between albatross’ V𝑧, 𝐺𝑊,and𝐶𝐿.

Let Table 1: Albatross and UAV demonstrator parameter values used. 𝑚 𝑚 2 𝑘= = ⋅ , Parameter Albatross UAV demonstrator 𝜂𝑎 𝑆 𝜌√𝐶2 +𝐶2 (25) Mass [kg] 8.5 5.5 𝐿 𝐷 2 Wing area 𝑆 [m ] 0.65 0.68088 where 𝑚/𝑆 is the wing loading. Aspect ratio 16.81 16.81 From (24) and (25), Span [m] 3.306 2.61 3 𝜌 [kg/m ] 1.22 1.22 cos 𝛿=𝑓(V𝑧) 𝐶𝐷0 0.033 0.033

𝐸max 20 21 √ 2 2 2 2 2 2 (26) 𝐺𝑊 ⋅ V𝑧 √𝑘 𝑔 +𝑘 𝐺𝑊V𝑧 − V𝑧 cos 𝜓−V𝑧𝑔 = √𝑘(𝑔2 +𝐺2 V2)3/4 𝑊 𝑧 inspired by albatross, wandering albatross’ parameters used bySachs[4]areassumed.Basedonthearchitectureofa which is subjected to the constraint wandering albatross, an aircraft demonstrator model is used 4 2 2 2 2 2 V −𝑘 𝐺 V −𝑘 𝑔 ≤0. (27) as a reference. The UAV demonstrator has similarities in 𝑧 𝑊 𝑧 dimensionaswellastheperformanceswithalbatross.This Obviously, there are two roots for (27), a positive root and a will ensure that the performances of the aircraft model are negative root. Moreover, the value of positive root is greater achievable by an engineering-designed vehicle. The param- than the absolute value of negative root. Thus, there is eter values of albatross and UAV demonstrator are given in Table 1, where 𝐸max is the maximum lift-to-drag ratio. Compared with the albatross, the UAV demonstrator has 𝑘2𝐺2 + √𝑘4𝐺4 +4𝑘2𝑔2 √ 𝑊 𝑊 a lighter mass. Furthermore, the flying ability close to the − ≤ V 2 𝑧 ground surface of the UAV demonstrator is limited, while (28) albatross can fly with a wing tip touching the water. For 𝑘2𝐺2 + √𝑘4𝐺4 +4𝑘2𝑔2 the reason of security, the UAV demonstrator has a smaller √ 𝑊 𝑊 ≤ wingspan. The aerofoil of UAV demonstrator is E214, which 2 makes the vehicle have a better performance than albatross, reaching a maximum lift-to-drag ratio of 21. which is one of constraint conditions for the solution of (26). From(26),itcanbefoundthattherelationshipbetween variables cos(𝛿) and cos(𝜓) is linear and the value of cos(𝜓) 3. Simulation Results and Discussions is located in the interval [−1, 1],sotherangeofcos(𝛿) can be defined. The extent of albatross’ V𝑧 and UAV demonstrator’s Because there is no closed form solution for the balance V𝑧 is shown in Figures 4 and 5, respectively. points in dynamic soaring, numerical investigations are In Figures 4 and 5, the black horizontal dotted line carried out instead. As dynamic soaring flight is directly represents cos(𝛿).Thesolidcurverepresentsthevariationof 6 International Journal of Aerospace Engineering

G = 1.0 GW = 0.5 W G = 1.5 1 W 1 1

0.5 0.5 0.5

−30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 z z z −0.5 −0.5 −0.5

−1 −1 −1

CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5 CL = 0.5 CL = 1.0 CL = 1.5

(a) When 𝐺𝑊 =0.5, 𝐶𝐿 =0.5,1.0,and1.5,the (b) When 𝐺𝑊 =1.0, 𝐶𝐿 =0.5,1.0,and1.5,the (c) When 𝐺𝑊 =1.5, 𝐶𝐿 =0.5, 1.0, and 1.5, the extent of V𝑧 extent of V𝑧 extent of V𝑧

Figure 5: The extent of UAV demonstrator’s V𝑧, three typical values of 𝐺𝑊, 𝐶𝐿 are selected.

√ 2 2 2 2 2 2 √ 2 2 2 3/4 Equation (31) then is converted to −𝐺𝑊 ⋅ V𝑧 √𝑘 𝑔 +𝑘 𝐺𝑊V𝑧 − V𝑧 − V𝑧𝑔/ 𝑘(𝑔 +𝐺𝑊V𝑧) .

√ 2 2 2 2 2 2 {𝐺 ⋅ V √√𝑘2𝑔2 +𝑘2𝐺2 V2 − V2 𝜓−V 𝑔} The dotted curve represents 𝐺𝑊 ⋅V𝑧 √𝑘 𝑔 +𝑘 𝐺𝑊V𝑧 − V𝑧 − { 𝑊 𝑧 𝑊 𝑧 𝑧 cos 𝑧 } max , (32) 2 2 2 3/4 0≤V ≤V { 2 3/4 } √ 𝑧 𝑧 max { √𝑘(𝑔2 +𝐺 V2) } V𝑧𝑔/ 𝑘(𝑔 +𝐺𝑊V𝑧) .FromFigures4and5,itcanbefound 𝑊 𝑧 { } that, no matter what set of parameters (𝐺𝑊, 𝐶𝐿)areselected, there always exists a balance point in the course of dynamic where soaring. In the range of V𝑧 <0,themaximumvalueofthe solidlineisgreaterthanone,andsolidlineanddottedlinego 𝑘2𝐺2 + √𝑘4𝐺4 +4𝑘2𝑔2 through the original point. That is, cos(𝛿) has corresponding √ 𝑊 𝑊 (33) V𝑧 max = . value in the area of V𝑧 <0.Furthermore,intheareaofV𝑧 >0, 2 the right-hand limit of (28) includes values which are greater than zero, then the expression of right-hand limit should be In order to obtain the maximum value of (32), another func- in the phase of lifting at zero point, and the derived function tion 𝑓 is created: of right-hand limit should be greater than zero at zero point; that is, √ 2 2 2 2 2 2 𝐺𝑊 ⋅ V𝑧 √𝑘 𝑔 +𝑘 𝐺𝑊V𝑧 − V𝑧 − V𝑧𝑔 𝑓= . (34) 3/4 󸀠 𝐺𝑊√𝑘𝑔 −𝑔 √𝑘(𝑔2 +𝐺2 V2) 𝑓 (V𝑧)V =0 = >0. (29) 𝑊 𝑧 𝑧 𝑘𝑔3 The contour line figure of albatross’ max 𝑓 and V𝑧 max is Hence, expressed in Figures 6 and 7. The contour line figure of UAV demonstrator’s max 𝑓 and 𝑔 V𝑧 max is expressed in Figures 8 and 9. 𝐺 > √ . (30) 𝑊 𝑘 From Figure 6, it is obvious that max 𝑓 has many values greater than zero. That is to say, (31) has reasonable solutions. In (30), there is not enough evidence to say that there From Figure 7, the maximum value of V𝑧 for the albatross’ exists balance point in the area of V𝑧 >0.FromFigures4and lifting balance allowance in dynamic soaring can be found. 5,itcanbefoundthatthemaximumvalueoftheright-hand For example, if the lift coefficient of albatross (𝐶𝐿 albatross)is1 limit of (26) cannot be less than cos(𝛿);thatis, andthegradientofthewindfield(𝐺𝑊)is1,inFigure6,the value of max 𝑓 is 0.4. Compared with Figure 7, V𝑧 =9m/s; combining with (20), it can be found that V = 17.0 m/s. That {𝐺 ⋅ V √√𝑘2𝑔2 +𝑘2𝐺2 V2 − V2 𝜓−V 𝑔} { 𝑊 𝑧 𝑊 𝑧 𝑧 cos 𝑧 } is to say, when albatross’ airspeed reaches 17.0 m/s, there exists max { } { √ 2 2 2 3/4 } equilibrium point in this wind field. Just getting the value of 𝑘(𝑔 +𝐺𝑊V𝑧) (31) { } airspeed is not enough to make conclusions. The restriction to airspeed is the bank angle of air velocity vector, which − cos 𝛿>0. is defined as 𝜓.Withthegeometricalrelationshipofforces International Journal of Aerospace Engineering 7

5 5 5 1.2 0. 1.1 max f max f 1 1.2 0.9 1.1 0.3 4 1 4 0.8 0.9 0.7 0.4 0.8 0.7 0.2 0.6 0.6 0.5 0.3 3 0.4 0.1 3 0.2 W 0 G W

G 0.1 2 2 0.3 0 0.2 0.5 1 0.1 0 0.4 0.4 1 0.80.70.60.5 0.3 1.211.10.9 0.6 0.2 0 0.80.7 0.10 0 0.5 1 1.5 2 2.5 3 1.11 0.9 0 1.2 0 0.5 1 1.5 2 2.5 3 CL

CL Figure 8: The contour line of (UAV demonstrator) max 𝑓. Figure 6: The contour line of (albatross) max 𝑓. 5

4

5 4 3

29 30

28

27

26

25

24

23

22

21

20

30 19

18

29

28 17

27

26 16

25

15

24

23 14

22 13

21

12

20

19

4 11 18

9

17 10

16 3

15 8 2

14

13 7 12 W

11 6 1

10 5 G 9 0 3 8 7 2 4 W 6 3

G 5 1 2 2 1 0 1 0 zmax 4 3

2930232427262822521192011716151121121988347065 1 2 0 4 3 10 zmax 0 0.5 1 1.5 2 2.5 3 5 911910876 0 4 30228272526232421223201917181615141312 CL 0 0.5 1 1.5 2 2.5 3 Figure 9: The contour line of (UAV demonstrator) V𝑧 max. CL

Figure 7: The contour line of (albatross) V𝑧 max. the mechanical energy variation of the aircraft equals the energy caused by drag and inertial force. When the wind balanced, one can acquire the space curve of V𝑥 × V𝑦 × V𝑧, gradient 𝐺𝑊 >0and V𝑧V𝑥 >0(V𝑥, V𝑧 are the airspeed which is obtained by projecting the four-dimensional space component in the direction of positive 𝑥-axis and 𝑧-axis), V𝑥×V𝑦×V𝑧×𝜓into three-dimensional space. The fundamental the inertial force (𝐹)doespositiveworktotheaircraft.The istofindoutasuitableaerodynamicforcetocompensate mechanical energy variation for unit mass is influenced by the resultant force of gravitation and inertial force. From wind gradient, mass (𝑚), and drag coefficient. (24),(25),and(26),thevaluerangeofthebankangleis ∘ ∘ Because of the existence of the gradient wind field, there [30 ,60 ]. The albatross can reach the equilibrium position in aresomebalancepointsduringdynamicsoaringnomatter gradient wind under these limiting conditions. Similarly, the how parameters are chosen. In the contour line figures of equilibrium points for UAV demonstrator in dynamic soaring albatross’ max 𝑓 and V𝑧 max,itiseasytofindoutthatthemax- canbefoundinFigures8and9. imum value of V𝑧 albatross, lift coefficient𝐶 ( 𝐿 albatross), airspeed (V), and bank angle (𝜓) and the gradient of the wind field are 4. Conclusions required by the dynamic soaring balance points. By means of performances comparison between UAV demonstrator Dynamic soaring gives the albatross an ability to achieve and albatross, it is manageable to apply dynamic soaring in energy from the environment and prolong its flight endur- engineering practice. ance, which is a guide to consider using dynamic soaring as a Finally, in order to make dynamic soaring profit from propulsive energy source for small UAVs. The paper presents further research work as soon as possible, dynamic soaring an equilibrium position theory of dynamic soaring which experiments are necessary. Furthermore, an investigation is the forces balance during the flight. At the equilibrium with theory of balances for 6 DOF aircraft model would be position, the aircraft stays in a certain speed without kinetic helpful for instructing the dynamic soaring experiments. energy loss. When the flight direction is upward, the mechan- ical energy would increase, which can store energy for flight. Conflict of Interests In this paper, the wind relative reference frame of aircraft is noninertial, and the inertial force 𝐹 created by wind can The authors declare that there is no conflict of interests be used to do work for aircraft. In the noninertial frame, regarding the publication of this paper. 8 International Journal of Aerospace Engineering

Acknowledgment The authors gratefully acknowledge the financial support from the Fundamental Research Funds for the Central Universities (Project no. CDJZR14325501).

References

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