Optimal morphing – augmented dynamic soaring maneuvers for unmanned air vehicle capable of span and sweep morphologies

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.1 (1-20)

Aerospace Science and Technology ••• (••••) •••–•••

1

67

Contents lists available at ScienceDirect

2

68

3

69

Aerospace Science and Technology

4

70

5

71

6

72

www.elsevier.com/locate/aescte

7

73

8

74

9

75

10

76

11 12 13

Optimal morphing – augmented dynamic soaring maneuvers for unmanned air vehicle capable of span and sweep morphologies

14 15 16 17 18

a

b

b

Imran Mir , Adnan Maqsood , Sameh A. Eisa , Haitham Taha , Suhail Akhtar

a

a

23 24 25 26

79 81 82 83

National University of Sciences & Technology (NUST), Islamabad, Pakistan b University of California, Irvine, USA

84 85

20 22

78 80

a

19 21

77

86

a r t i c l e

i n f o

Article history: Received 6 January 2018 Received in revised form 15 February 2018 Accepted 15 May 2018 Available online xxxx

27 28 29 30 31 32 33 34 35 36 37 38 39 40

a b s t r a c t

87 88

This paper investigates autonomous dynamic soaring maneuvers for a small Unmanned Aerial Vehicle (sUAV) having the capability to morph. Dynamic soaring for UAVs have mostly been confined in literature to fixed configurations. In order to analyze the extent to which dynamic soaring is influenced by different morphologies, an innovative concept of integrating dynamic soaring with morphing capabilities is introduced. Moreover, optimal soaring trajectories are generated for two basic wing morphologies: variable sweep and variable span. Three-dimensional point-mass UAV equations of motion and nonlinear wind gradient profile are used to model the flight dynamics. Parametric characterization of the key performance parameters is performed to determine the optimal platform configuration during various phases of the maneuver. Results presented in this paper indicate 15% lesser required wind shear by the proposed span morphology and 14% lesser required wind shear by the proposed sweep morphology, in comparison to their respective fixed wing counterparts. This shows that the morphing UAV can perform dynamic soaring in an environment, where fixed configuration UAVs might not, because of lesser available wind shears. Apart from this, span morphology reduced drag by 15%, lift requirement by 11% and angle of attack requirement by 20%, whereas increased the maximum velocity by 6.2%, normalized energies by 9% and improved loitering parameters (approximately 10%), in comparison to fixed span configurations. Similarly, sweep morphology guaranteed 20% drag reduction, 16% lesser angle of attack requirement and improved loitering performance over the fixed sweep configurations. The results achieved from this study strongly support the idea of integrating dynamic soaring with morphing capabilities and its potential benefits. © 2018 Elsevier Masson SAS. All rights reserved.

89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106

41

107

42

108

43

109

44 45

110

1. Introduction

111

46 47 48 49 50 51 52 53 54 55 56 57 58

112

Dynamic soaring as introduced by Lord Rayleigh [1] is a flight trajectory/maneuver that exploits energy from wind. This process can be so efficient that it allows un-powered (gliding) flight for hundreds of miles as typically done by the Albatross [2]. In fact, Albatrosses are so well adapted to this maneuver that they even sleep while gliding [3–5]; they have a shoulder lock mechanism that maintains their wings outstretched with almost no effort. The dynamic soaring maneuver requires not only wind but also a wind shear (wind gradient); i.e., the wind profile has to vary with altitude for the maneuver to be effectively executed. However, this wind shear naturally exists near surfaces (e.g., terrain, sea, ocean, or mountains) if there is a wind blowing over.

113 114 115 116 117 118 119 120 121 122 123 124

59

125

Fig. 1. Dynamic soaring cycle.

60 61 62 63 64 65 66

E-mail addresses: [email protected] (I. Mir), [email protected] (A. Maqsood), [email protected] (S.A. Eisa), [email protected] (H. Taha), [email protected] (S. Akhtar). https://doi.org/10.1016/j.ast.2018.05.024 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

126 127

Fig. 1 provides an illustration for how the Albatross typically executes the dynamic soaring periodic maneuver to fly crosswind.

128 129 130 131 132

JID:AESCTE

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.2 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

2

1

67

Nomenclature

2

68

3

69

U AV sU A V V

4 5 6 7

γ

8

ψ x y z

9 10 11

αL =0

12

b

13



14

CL

15

φ

16

CD E m g Vw n nmax t0

17 18 19 20 21 22 23 24

Unmanned Aerial Vehicle Small Unmanned Aerial Vehicle True air speed Flight path angle Azimuth measured clockwise from the y-axis Position vector along east direction Position vector along north direction Altitude Angle of attack at zero lift Wing span Sweep morphing angle Lift coefficient Bank angle Drag coefficient Energy Mass of the vehicle Acceleration due to gravity Wind velocity Load factor Maximum load factor Initial time

Final time Reference wind speed Reference altitude Surface correctness factor Lift curve slope of airfoil Zero lift drag coefficient ρ Density of the air n Load factor S Wing area K Aerodynamic coefficient AR Aspect ratio of the wing m Meter L/D Lift to drag ratio EPP Extended Polypropylene Particle AU W All Up Weight e Span efficiency factor NLP Non Linear Programming I P O P T Interior Point Optimization LG Legendre–Gauss G P O P S General Purpose Optimal control Software tf V re f hre f h0 cl α C D0

70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

25 26 27

91 92

Table 1 Dynamic soaring flight parameters.

93

28 29 30 31 32 33

94

No.

Nomenclature

Value

Ref.

S No.

Nomenclature

Value

Ref.

1

Minimum wind strength required for dynamic soaring at sea level Minimum wind strength required for dynamic soaring at ridges Peak altitude in the dynamic soaring maneuver Dynamic soaring cycle

5 m/s

[6,12,13]

6

Cruise speed

28 m/s

[14]

2.44 m/s

[15]

7

Max lift coefficient

1.5

[6,16,17]

15–20 m

[5]

9

±70

9.6–10.9 s

[12]

9

Maximum speed attained in a dynamic soaring cycle

28 m/s

[6,12]

10

Range for max bank angle during the turn Maximum speed attained in a dynamic soaring cycle Maximum load factor

2 3

34 35 36 37

4 5

◦

[6] [12]

3

[16]

42 43 44 45 46 47 48

53 54 55 56 57 58 59 60 61 62 63 64 65 66

99 101 102

105

The periodic maneuver consists of four phases: (i) windward climb, (ii) upper turn, (iii) leeward descent, and (iv) lower turn. The bird reaches the final point of the cycle with the same conditions as the initial point (i.e., same speed, altitude, angle of attack, flight path angle, etc), but with a gained distance along the desired flight direction without any input power; that is, a gained energy in the amount of the work that would have otherwise been exerted to overcome drag: drag force corresponding to the flight speed times the traveled distance per cycle along the desired flight direction. 1.1. Related work

51 52

98

104

49 50

97

103

39 41

96

100

14 m/s

38 40

95

Flight dynamics of soaring birds were investigated by many researchers to determine the fundamentals of soaring flight [5, 6]. Various parameters, such as peak altitude/speeds attained during the soaring maneuvers, cycle time, and required minimum wind shear were introduced in literature (Table 1). Wharington [7] first presented a heuristic approach (pitch and bank control) for formulating dynamic soaring trajectories for a UAV. Sukumar and Michael [8] investigated dynamic soaring of a sailplane in the Earth’s atmospheric boundary layer for a range of conditions. Zhao [9] analyzed different optimal dynamic soaring patterns (loiter, travel and basic modes) of a glider, by configuring the trajectory optimization problem as an optimal control one. The problem was then converted into parameter optimization via a collocation approach, and solved numerically with the software NPSOL [10]. Similarly [11] based upon direct collocation approach developed a

general optimization method to compute all the possible patterns of dynamic soaring with a small unmanned aerial vehicle. Lawrence [18–20] presented a regression-based algorithm for autonomous dynamic soaring under feasible wind shear conditions. Zhao [21] examined minimum fuel powered dynamic soaring of UAV (both propeller-driven and jet-driven) assuming a linear wind gradient profile and a 3-Degree of Freedom (DOF) pointmass model. Zhu [22] performed trajectory optimization of an unmanned aerial vehicle for dynamic soaring through numerical analysis and validated the theoretical work through flight test. Formulation of required minimum wind shear to perform dynamic soaring posed a challenging numerical problem, because of the coupled nonlinear nature of the equations of motion. Data given by Sachs [6], Idrac [12] and Pennycuick [13] refer to a wind speed of 5 m/s (close to the sea surface) as the minimum wind shear value, below which dynamic soaring is not possible. The problem of generating optimal dynamic soaring trajectories for UAV in near real time, was not investigated by many researchers. Most of the technical studies utilized iterative numerical optimization techniques, such as Advanced Launcher Trajectory Optimization Software (ALTOS), Graphical Environment for Simulation and Optimization (GESOP), Nonlinear Programming Solver (NPSOL) and Imperial College London Optimal Control Software (ICLOCS) [23,6,15,9]. These numerical optimization techniques have high computation time (100–1000 seconds (s)). In order to reduce the computational time, Akhtar [24,25] developed trajectory tracking algorithm based on Inverse Dynamics in the Virtual Domain

106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.3 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

1 2 3

6 7 8 9 10 11 12 13 14 15

67

Table 2 Aerodynamic and mass parameters of fixed configuration UAVs utilized for dynamic soaring.

68

S No

Mass (kg)

Wing span (m)

Wing area (m2 )

Chord

AR

C Lmax

( L / D )max

Wing loading

Max. speed (m/s)

Load factor

CDO

Ref.

1 2 3 4 5 6 7 8 9 10 11 12 13

6.6 – 11.3 430 15 81.7 5.4 4.5 300 17.2 – 300 1.5

2.5 – 2.6 18 3 – – 3 – 2.92 – – 1.25

0.48 – 0.81 11.6 – 4.1 0.95 0.47 11 – – 11 0.6

0.19 – 0.38 – – – – – – – – – 0.48

12.8 – 6.9 26.6 20 – 19.5 19 – 16.9 – – 2.6

1.17 – – – – 1 1.2 1.1 – – – – 1.5

20.5 30 – – 26.6 – 50 33.4 – 34 45 – –

– 12 – – 33.3 – – – – 34 50 – 20.5

– 80 – – – – – – – – – – 30

–

– – – – 0.02 0.008 0.001 0.017 – 0.02 – – 0.004

[42] [7] [43] [44] [8] [26] [45] [46] [47] [8] [15] [47] [48]

4 5

3

<6 – – –

<5 <2 – – – – –

<5

70

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

(IDVD), with a computation time of 8 s. Similarly, Gao [26] utilizing Gauss Pseudo-spectral Method (GPM) of General Purpose OPtimal control Software (GPOPS) [27,28] developed dynamic soaring trajectories with a computational time of about 15 s. Ariff and Go [29] proposed dynamic soaring algorithm for small-scale tactical UAVs utilizing Dubin’s curves. Since the navigation algorithm was based on Dubin’s curves, it was considered optimal by definition. Similarly Gao [30] formulated Dubin’s path-based trajectory planning method and tracking control approach for dynamic soaring in a gradient wind field. Sachs [23] showed that dynamic soaring by full-size sailplanes is possible with values of wind shear found near mountain ridges. Gordan [31] explained that although the full size sailplanes could extract energy from horizontal wind shears, the utility of the energy extraction could be marginal depending on the flight conditions and type of sailplane used. The design of guidance and control strategies is a promising study trend of dynamic soaring for small unmanned aerial vehicles, for which the flight modeling and simulation specifically for soaring-capable unmanned aerial vehicles is significant and necessary. In this regard, Liu [32] proposed a flight simulation platform for dynamic soaring. Based on our review of the work done in regards to dynamic soaring and the cited papers, it can be noted that certain areas are not fully covered in literature and require further investigation. Parametric studies to investigate the effect of different sizing and design parameters on dynamic soaring needs to be performed to identify the conditions most suitable for dynamic soaring. Moreover, the existing high computational time for the generation of the dynamic soaring trajectories, also needs to be reduced for practical on-board utilization purpose. 1.2. Motivating problems for this paper

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

71 72 73 74 75 76 77 78 79 80 81 82

48 49

69

In-spite of dynamic soaring being an active area of research for over a century, dynamic soaring as a mean of energy extraction from atmospheric wind shear has not been furnished to the fullest. Realizing bird flight, it is evident that soaring birds utilize the wind shear in such an optimized way that they can fly for months, almost without flapping [33–35]. However, dynamic soaring for UAVs have been confined to maneuvers, with far less benefits, than those acquired by soaring birds. In-spite being a promising choice to supply power for the small UAVs; the application of energy extraction from wind shear in case of UAVs is immature yet. This curtails the potential benefits envisaged from dynamic soaring. An area that warrants considerable attention in this regard is the improvement in the structure materials of UAV utilized for dynamic soaring. The forces acting on the UAV during high-speed dynamic soaring maneuver is of the order of several times of the gravity, which necessitates design of materials to bear such high stresses.

Although efforts have been made in improving the material composition, finding a balance between the strength of structure and the aerodynamic efficiency, while remaining within financial constraints, is still a problem for the designer. i) Bird flight inspiring morphology: Realizing bird flight, it is observed that nature provided a suitable solution to the problem. Soaring birds that inspired the idea of dynamic soaring, such as Albatrosses, which are of the size of sUAV, make changes in the shape and size of their wings continually during the dynamic soaring process [5]. The bird, while resting on its wings with a shoulder lock, skillfully, varies wing planform and twists, all without flapping, as it performs dynamic soaring maneuver [36,37]. They alter the aerodynamic forces by changing plan-form configuration and flight profile (such as angle of attack and speed). Planform configuration is mainly altered by variation in wing shape [38] through changes in wing span, sweep angle, twist angle and dihedral angles [26, 36,39]. Through this, the bird not only acquires maximum energy from the wind shear, but also reduces the aerodynamic forces acting on its body. With higher ( L / D )max, less energy is lost due to drag, and dynamic soaring becomes more efficient in terms of a lower required wind speed at the reference height [23], or a weaker required wind gradient [6]. Soaring parameters maximized by the swifts through morphing, includes glide speed, glider range, turning rate, cycle time, peak velocity, altitude and energy gain. ii) Geometrical challenges for dynamic soaring in UAVs vs. nature: From the study of relevant literature and the cited papers (such as, but not limited to [40,20,41]), it is observed that dynamic soaring maneuvers have not been considered/studied for a morphing platform. To the best of our knowledge, this is the first study that attempts to investigate dynamic soaring for a morphing capable platform. Past studies have been mostly confined to fixed configurations, in which the model under consideration cannot change its geometrical configuration. Table 2 presents details of different fixed configuration platforms that have been utilized in literature for dynamic soaring. For optimal energy gain from atmospheric wind shear, different platform configurations are required during different phases of the maneuver [49]. This is because the performance requirements vary during different phases of the maneuver, which necessitates optimized platform configurations [50]. Few such conflicting requirements are described below: i) To achieve speedy climb with higher velocity and more rolling stability, wings with less span, higher sweep and dihedral are required during climb/descent phase. However, during the higher altitude regions (where the velocity has

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

4

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.4 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

83

18

84

19

85

20 21

86

Fig. 2. Morphing techniques [52].

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

87 88

decreased considerably), higher span and lower sweep angles are desirable for providing the additional lift. ii) To increase the forward distance, wings during climb/descent phase are required to be kept in lower span, higher sweep and higher dihedral configuration. This allows speedy climb with higher velocity and more rolling stability. However, in the high altitude region, where the UAV velocity is less, wings with higher span and lower sweep angles are desirable for providing the additional lift. iii) High directional stability is required during the climb and the descent phases of the soaring flight to avoid unnecessary control efforts. This is achieved through high wing configuration. However, high directional stability is not a desired feature during high and low altitude turn phases of the maneuver, as considerably large control effort will be required to overcome this directional stability. Such conflicting requirements during different phases of the maneuver, triggers the need for having UAV with morphing capability [51]. Aircraft morphing capability can be achieved by altering aircraft wing, fuselage, tail or engine parameters. Amongst these, altering the wing parameters have the most powerful and influential impact as they create most of the lift required for the flight, and their shape and size determine the suitability of the aircraft to a particular mission. To incorporate wing, fuselage or engine morphing, different physical parameters can be changed. Complete details of these parameters that can be modified [52] are depicted in Fig. 2. Selection of the relevant configuration/morphology suitable for the desired mission type, needs to be determined through optimization techniques. Useful studies have been conducted in this regard by researchers to parameterize and determine the optimal platform configuration. Parameterization and optimization of hypersonic glide vehicle was performed by [53]. In [53], optimal configuration was determined utilizing improved three-dimensional class/shape function transformation (CST) approach. Large number of parameters were utilized to enhance the parametric ability and to determine the aerodynamic and geometry properties of the vehicle. In another study, [54] presented a design approach of wide-speed-range vehicles based on the cone-derived theory. The parametric method employed in the ascender line design makes it possible to control the overall configuration of the vehicle. Similarly, [55] performed a study of airfoil parameterization, modeling, and optimization

based on the computational fluid dynamics method. Parameterization methods of airfoil were compared and numerical techniques were utilized to optimize the airfoil to enhance the aerodynamic performance, based on the response surface model. Like soaring birds, a biologically inspired platform [56] with morphing capability, can be proposed to optimize the energy gain from the atmosphere. Consequently, it is necessary to investigate the impact of morphing on dynamic soaring and to ascertain the benefits achieved by morphing configurations in comparison with the fixed configurations. In line with that, parametric studies can be performed to determine the most beneficial morphology at distant phases of the maneuver [57].

89 90 91 92 93 94 95 96 97 98 99 100 101 102

1.3. Main contributions of this paper

103 104

i) Identification of existing challenges and solutions: One of the main contribution of this paper is to identify the challenges faced by UAVs, which are curtailing the potential utility of dynamic soaring, and to propose suitable solution. The immaturities observed in the field of dynamic soaring were attributed to the fact that dynamic soaring has been always implemented on UAVs with fixed platform configuration. Unlike soaring birds, which maximizes the benefits of soaring by dynamically modifying wing parameters during various stages of flight, the UAVs utilized for dynamic soaring always maintain a fixed wing configuration. In section 3, the problem is discussed in detail. We propose a path of research, in which the literature seems almost empty from. This proposal, discuss, including morphing in the dynamic soaring studies. It is also shown in section 3 that a biologically inspired platform with morphing capability greatly enhance the energy gain from the atmosphere. ii) Development of an original bio-inspired platform: In this paper, a mathematical framework is developed for a biologically inspired UAV that can perform dynamic soaring under morphing conditions [58,59]. The morphing techniques considered in this research includes two fundamental morphologies of a) variable span (1.25 m to 1.75 m), and b) variable sweep (0◦ to 50◦ ) configurations, see Fig. 3. Three-dimensional point-mass UAV equations of motion and nonlinear wind gradient profile are used to model flight dynamics. Utilizing UAV states, controls, operational constraints, initial and terminal conditions that enforce a periodic flight, dynamic soaring prob-

105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.5 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

5

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

Fig. 3. Morphing configurations.

16

82

17 18 19

83

Table 3 UAV modal and flight parameters.

84 85

20

S No.

Parameter

Value

S No.

Parameter

Value

86

21

1 2 3 4 5

Mass (m) Wing area (S) Nominal wing span (b) Induced drag factor (K) Span efficiency factor (e)

1.5 kg 0.6 m2 1.25 m

6 7 8 9 10

Lift coefficient (C L ) range Bank angle φ range Nominal wing chord (c) Zero lift drag coefficient (C D 0 ) Nominal aspect ratio (AR)

0–1.5 −60◦ –60◦ 0.48 m 0.02 2.6

87

22 23 24 25

1 πeAR

0.6

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

88 89 90 91 92

lem is formulated as an optimal control problem. It is then solved numerically utilizing variable-order Gaussian quadrature methods where a the continuous-time optimal control problem is approximated as a sparse nonlinear programming problem (NLP). This NLP is then solved using either the NLP solver IPOPT or the NLP solver SNOPT. iii) Extending dynamic soaring to morphing configurations: Optimal soaring trajectories are generated for two the morphologies. The sweep morphology has been introduced by the authors of this paper in [60], and will be extended and developed as part of the contribution of this paper. Additionally, this paper introduces span morphology. These morphing techniques are considered appropriate for dynamic soaring as they are the most realistic (close to nature), suitable/viable to implement, and have a direct impact on the desired performance parameters. In order to determine the extent to which the two morphologies (span and sweep) [61] impact dynamic soaring, the optimal trajectories of both the morphologies are compared against the respective extreme fixed wing configurations. The performance of the variable span configuration UAV will be compared and evaluated against two extreme fixed wing configurations having spans of 1.25 m and 1.75 m in section 3.1. Similarly, for sweep morphology, the morphing results will be evaluated against two fixed configurations having wing sweep angles of 0◦ and 50◦ (refer section 3.2). iv) Parametric characterization for key performance variables: Characterization of the key performance parameters is performed to determine the extent to which the two morphologies influence dynamic soaring characteristics. Parameters such as lift and the drag, optimization time, cycle time, the minimum required wind shear, maximum altitude gain, energy gain and ranges along east/north direction, are evaluated to determine the best morphing configuration during various phases of the maneuver. This is explained in section 3 ahead. v) Future direction: Through simulations performed in section 3.1 and section 3.2 and summarized in Table 5 and Table 7, it is shown that proposed work has considerable positive impact. This provides the research community with a new direction, through which, the energy gain from atmospheric wind shear can be optimized to ensure sustainable powerless

flight for UAVs. More work can be done to evaluate the impact of other fundamental morphologies, such as variations in wing twist, wing dihedral and so on), on dynamic soaring flight. Stability, reachability, and controllability studies can also be performed from control prospective.

93 94 95 96 97 98

2. Problem setup

99 100

2.1. UAV geometric and mass parameters

101 102

The UAV platform selected for this paper consists of a conventional Radio Controlled (RC) aircraft model, which is commercially available. It has a standard wing-tail configuration with airframe, that is consisting of Extended Polypropylene Particle (EPP) foam construction with composite landing gears. The model has a fuselage length of 1.25 m and nominal wing span of 1.25 m. The aspect ratio of the wings is 2.6. The recommended All Up Weight (AUW) for enhanced performance is about 1.5 kg, but the vehicle can fly with an AUW of approximately 1.7–1.9 kg. The geometrical and flight model parameters of the UAV utilized in this study, are defined in Table 3.

103 104 105 106 107 108 109 110 111 112 113 114

2.2. Wind shear model

115 116

Wind shear (wind gradient) is the atmospheric phenomenon that takes place on thin layers between two regions, where the airflow vector is different. The wind speed increases with altitude (within a narrow layer) before reaching the values of the free air stream. Since wind shear is a necessary condition for dynamic soaring, a well-defined model for describing wind dynamics is needed. Typically, stable atmospheric conditions exist close to the surface with the wind velocity nearly zero at the surface and increase gradually with altitude. Friction between the surface and moving air mass causes the wind speed to be strongly dependent on the height above the surface. The mean velocity profile of actual wind gradients can, therefore, be approximated by well-established linear [21,24,31], exponential [62,46] or logarithmic [63,6,8] models. The nonlinear logarithmic wind shear profile is used in this paper. This model is selected because it is normally utilized in me-

117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.6 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

6

1 2 3

teorological studies for measurements near the surface of the earth [8]. The logarithmic relation between wind speed (V W ) and height above the surface (h) is given by Eq. (1):

67 68 69

4 5 6

70

V W = V w re f

ln(h/h0 ) ln(hre f /h0 )

71

(1)

,

72

7 8 9 10 11 12 13 14 15 16 17 18 19 20

73

where V w re f denotes reference value, which represents the strength of the wind shear at reference height hre f . h0 is the surface correctness factor that determines the distribution of the wind gradients with varying altitude, reflecting the surface properties, such as irregularity, roughness and drag. The values hre f = 10 m and h0 = 0.03 m chosen in this study, are similar to the work carried out by Sachs [6]. The minimum reference wind speed value at this height, which still permits dynamic soaring is ascertained through the optimization process. The wind shear gradient is represented by Eq. (2):

V˙ W = V w re f

21 22 23 24 25 26 27

1 h ln(hre f /h0 )

h˙ ,

(2)

where h˙ is the rate of change in the altitude. While formulating Eq. (2), it is assumed that the wind is stationary (see Eq. (3)) and is a function of altitude only (V W x = V W x (h)).

V˙ W

∂VW ∂VW ∂VW ˙ = x˙ + y˙ + h. ∂x ∂y ∂h

(3)

28 29

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

In this paper, UAV dynamics is represented by a three-dimensional point-mass model [7,9,21]. For mathematical analysis, a non-rotating flat earth will be considered since dynamic soaring trajectory, typically occurs over a very brief period of time, and over a small localized area of the Earth. Aircraft is considered to be a rigid body with constant mass. The point-mass model equations with no thrust component are chosen to provide an optimal dynamic soaring flight profile for the UAV. Neglecting the UAVs rotational dynamics, the model can be represented by the nonlinear system of differential equations below:

76 77

Fig. 4. Aerodynamic forces acting on the UAV and the aerodynamic angles.

78 79

where ρ is the air density, S is the wing area, C L and C D are the lift and drag coefficients respectively. The parabolic drag coefficient is given by Eq. (6):

(6)

1

[ L sin φ − m V˙ W cos ψ], mV cos γ 1 γ˙ = [ L cos φ − mg cos γ + m V˙ W sin ψ sin γ ], mV x˙ = V cos γ sin ψ + V W ,

ψ˙ =

(4)

h˙ = V sin γ , where V is the UAV speed, γ is the flight path angle, ψ is the heading angle, φ is the bank angle and x, y , h are the UAV position vectors. For simplicity, it is assumed that the UAV flight is over a flat surface, where the air density is constant and homogeneous. The wind blows along the positive x-axis at all times (only the horizontal wind component exists). In this model (see Fig. 4), the UAV speed and the angles are modeled in a wind relative reference, while the position (x, y , h) is modeled in an Earth fixed-frame. The aerodynamic Lift (L) and Drag (D) forces are presented in Fig. 4 and mathematically formulated in Eq. (5):

L = 0.5ρ V 2 S · C L , D = 0.5ρ V 2 S · C D ,

(5)

82 84 85

2.4. Problem formulation

94

The flight model described by Eqs. (4) with nonlinear logarithmic wind profile in Eq. (1) is configured as a nonlinear system of the form in Eq. (7):

96

x˙ = F (x(t ), u (t )).

100

(7)

The state vector x(t ) is defined in Eq. (8):

x(t ) = [ V , ψ

γ , x, y , h]T , x(t ) ∈ R6

87 88 89 90 91 92 93

3

98 99 101 103

(8)

(9) (10)

where b is the wing span, and  denotes the wing sweep angle. Dynamic soaring flight is then formulated as a nonlinear optimal control problem to find a control sequence that optimizes the performance index defined in Eq. (11):

(11)

104 105 106 107 108 109 110 111 112 113 114 115 116 117

subject to the dynamic constraints presented by equations of motion in Eqs. (4), and satisfying the path and boundary constraints. Boundary conditions [18] for implementing loiter mode of dynamic soaring require terminal states to be equal to the initial states and is mathematical formulated in Eq. (12):

[ V , γ , ψ, x, y , h]tTf = [ V , γ , ψ − 2π , x, y , h]tT0 ,

97

102

Control vector u (t ) for span morphology is defined in Eq. (9) and for sweep morphology in Eq. (10): T

86

95

J = min[ V w ,re f ],

y˙ = V cos γ cos ψ,

81

where K is the induced drag coefficient represented by K = Also A R is the wing aspect ratio and e is wing span efficiency factor. The lift and drag coefficients depends on various geometrical and atmospheric parameters, such as wing span, wing sweep angle, angle of attack, altitude and are determined through empirical techniques. The estimates through empirical relationships follow similar pattern as that obtained through Vortex Lattice Method (VLM) [64].

1 πeAR .

α , φ] , u ∈ R u (t ) = [, α , φ] T , u ∈ R3

m

80

83

C D = C D 0 + K · C L2 ,

u (t ) = [b,

1 V˙ = [− D − mg sin γ − m V˙ W cos γ sin ψ],

65 66

75

2.3. Flight dynamics model

30 31

74

118 119 120 121 122 123

(12)

where t f and t o denotes the final and initial times respectively. Path constraints for the states and control during the dynamic soaring maneuver are represented in Estimates (13):

124 125 126 127 128

V min < V < V max , ψmin < ψ < ψmax , γmin < γ < γmax

129

xmin < x < xmax , ymin < y < ymax , h ≥ 0; φmin < φ < φmax , (13)

131

bmin < b < bmax , min <  < max .

130 132

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.7 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

1 2 3 4

7 8 9 10 11 12 13

67

Table 4 Numerical bounds on states and control.

68

S No.

Parameter

Value

S No.

Parameter

Value

1

hre f = 5 m, h0 = 0.03

10

Max load factor

<6

2 3 4 5 6 7 8

NonLinear logarithmic wind model parameter Initial velocity range ( V 0 ) Initial azimuth angle (ψ0 ) range Initial flight path angle (γ0 ) range Dynamic soaring cycle time Bound on x Velocity bounds Flight path angle bound

3–30 m/s from −540◦ to −90◦ −60◦ –60◦ 1–30 s −300–300 m 3–50 m/s −60 ◦ –60◦

11 12 13 14 15 16 17

Initial x position (x0 ) Initial y position ( y 0 ) Initial z position ( z0 ) Bound on y Bound on z Bound on azimuth Final state constraints

0 0 0

9

Sweep angle () range

0–50◦

18

Span variation range

5 6

7

16 17 18

−300–300 m 0–300 m from −540◦ to −90◦ Terminal state values = Initial state values 1.25–1.75 m

n=

21 22 23 24 25

L W

≤ nmax ,

(14)

where nmax is the maximum permissible load factor is defined in Table 4. The constraint of load factor limits the maximum velocity during the dynamic soaring cycle, as elaborated in Eq. (15):

26 27 28

n=

L W

29 30 31 32 33 34 35 36 37 38 39 40 41 42

V max =

= 

q∞ SC L mg

=

0.5ρ V 2 bcC L

nmg 0.5ρ S min C L

mg

, (15)

,

where g is the acceleration due to gravity, q∞ is the free stream dynamic pressure, c is the wing chord and m is the UAV mass. In Eq. (15), in order to maximize the velocity, under given constraints, the area needs to be minimized. For span morphology, this necessitates that the span should be kept at minimum possible configuration (see Eq. (16)).

S min = bmin c .

(16)

Similarly, in case of sweep morphing, the swept area is governed by Eq. (17):

43 44 45 46 47 48 49 50 51 52 53 54 55

S sweep = S nominal − 0.5c 2 (tan ).

(17)

Minimum area under sweep morphing condition is defined by Eq. (18). 2

S sweep (min) = S nominal − 0.5c (tan )max .

(18)

Numerical bounds on the state and control variables, for implementing the optimized trajectory for loiter maneuver, are listed in Table 4. In this problem, initial airspeed, flight path angle and heading angle are unconstrained. This allows the optimization process to determine their optimal values for sustainable soaring flights.

56 57

2.5. Optimization framework

58 59 60 61 62 63 64 65 66

72 73 74 75 76 77 78 79 80

Since dynamic soaring is a high maneuvering cycle, which generates high accelerations, an additional path constraint of load factor is included. This constraint specified in Eq. (14) ensures that UAVs structural bounds are not violated.

19 20

70 71

14 15

69

The selection of methodology to identify the search space for the optimal dynamic soaring trajectory was the critical part of this work. It is not surprising that the development of the numerical methods for optimization have closely paralleled to the exploration of space and the development of the digital computer [65]. Typically optimization methods are categorized in two broad groups that include deterministic and stochastic approaches. The deterministic approaches require gradient and strictly generate unique

optimal solution for specified initial conditions. Deterministic approaches have found wide-spread applications in trajectory optimization [66,67] and engineering and product design [68–70]. Stochastic techniques are broadly nature-inspired, and more optimization work has been done in this direction. The impetus of recent shift from deterministic to stochastic approaches is driven by computational economy in handling large-scale complex problems. The evolutionary or nature-inspired computing is based on metaheuristic methods of which most popular is genetic algorithms [71,72]. Various bio-inspired approaches are also gaining popularity that include neural networks [73,74], Ant Colony Optimization [75], Cuckoo search [76,77], firefly method [78], flower pollination approach [79], krill herd algorithm [80–82] and so on. For detailed review on bio-inspired optimization techniques, readers are referred to [83]. However, the fundamental drawback of nature-inspired algorithms is generation of non-unique solutions and requirement of benchmark cases. Several optimization problems do not have benchmark cases and often randomness in optimal solution leaves the least choice with the decision makers. Therefore, an established technique, which can generate unique solution in near real time environment, is generally the governing criteria of selection for the kind of studies reported in this paper. While performing this study, which integrates two fundamental concepts of dynamic soaring and morphology, selection of optimization software that is capable of generating optimal soaring trajectories in near real time was carefully performed. Most of the technical studies utilized iterative numerical optimization techniques, such as Advanced Launcher Trajectory Optimization Software (ALTOS), Graphical Environment for Simulation and Optimization (GESOP), Nonlinear Programming Solver (NPSOL) and Imperial College London Optimal Control Software (ICLOCS) [6,9, 15,23]. These numerical optimization techniques had high computation time (100 to 1000 s). Therefore, an established technique which can generate unique solution in near real time environment is what our study is looking for. The numerical technique utilized in this paper for solving the optimal control paper is a specialized software that has already been utilized in various other trajectory optimization studies [84–89,48,90,91]. This gives us a confidence to proceed with this approach. In this research, GPOPS-II [92] along with NonLinear Programming (NLP) solver Interior Point Optimization (IPOPT) are utilized. This technique has been utilized by various researchers for numerous other applications and is now considered a pretty robust and widely acceptable technique among aircraft/spacecraft trajectory optimization community. The optimization framework is based on hp-adaptive Gaussian quadrature collocation technique. In a Gaussian quadrature collocation method, the state is approximated using a Lagrange polynomial, where the support points of the Lagrange polynomial are chosen to be points associated with a Gaussian quadrature. Originally, Gaussian quadrature collocation methods were implemented as p-methods in which the convergence within a single interval is achieved by increasing the degree of the

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.8 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

polynomial approximation. An hp-adaptive method is a hybrid between a p-method and an h-method, where both the number of mesh intervals and the degree of the approximating polynomial within each mesh interval, are varied. This is to done to achieve the specified accuracy in the numerical approximation for the solution of the continuous-time optimal control problem. As a result, in an hp-adaptive method, it is possible to take advantage of the exponential convergence of the global Gaussian quadrature method in regions, where the solution is smooth. While it introduces more mesh points near potential discontinuities or in regions where the solution changes rapidly. Starting with a global approximation for the state variables, the number of segments and the degree of the polynomial on each segment are updated until user-specified tolerance is achieved. The convergence using hp-method is achieved with a significantly smaller finite-dimensional approximation as opposed to h or p-methods. This is stated in [85]. In order to utilize GPOPS-II, the problem is configured as a multiple-phase optimal control problem. In this, p ∈ [1, . . . , P ] is the phase number with P is the total number of phases, n y is the output dimension, nu is the input dimension, nq is the integral dimension, and ns is the dimension of the static parameters. The optimal control problem then tries to determine the state, ( p) ( p) ( p) x( p ) (t ) ∈ Rn y , control, u ( p ) (t ) ∈ Rnu , integrals q( p ) ∈ Rnq , start times, t 0 ( p ) ∈ R, phase terminus times, t f ( p ) ∈ R, in all phases p ∈ [1, . . . , P ], along with the static parameters s ∈ Rns , that minimizes the objective functional in Eq. (19):



28 29 30

J=

 p

( p)

( p)



tf



32 33

+

34

L

( p)

(x

( p)

(t ), u

( p)

(t ), q

( p)

39 40

dt )] ,

43 44 45 46

x˙ ( p ) = f ( p ) (x( p ) u ( p ) t ; a( p ) ),

53 54 55 56 57 58

( p)

C min ≤ C ( p )(x

( p)

p = 1, . . . , P ,

(20)

65 66

) ≤ C max) ,

p = 1, . . . , P ,

(21)

p = 1, . . . , P ,

(22)

(s)

( p 1s )

s

s

s

( prs )

; q( p1 ) , x( pr ) (t 0 ), t 0

s

; q( pr ) ≤ Lmax(s) ) (23)

The required state and control variables are approximated using polynomial interpolation. The state variables are approximated using a basis of N + 1 Lagrange interpolating polynomials as in Eq. (24):

x(τ ) ≈ X (τ ) =

X i · L i (τ ),

τ ∈ [−1, 1],

(24)

i =1

where L i (τ ) =

N  j =0, j =i

71 72

The derivative of each Lagrange polynomial at the Legendre– Gauss (LG) points is represented in a differential approximation matrix, D ∈ R N ×( N +1) . The elements of the differential approximation matrix are given in Eq. (27):

D ki (τ ) = L˙ i (τk ) =

N  l =0

N

j =0, j =i ,k

τk − τi

j =0, j =i ,k

τi − τi

N

75 76 77 78 79 80 81

,

82 83

(27)

84

The dynamic constraint in Eq. (20) is translated into algebraic constraints via the differential approximation matrix as is given by Eq. (28):

86

k = 1 , . . . , N , i = 0, . . . , N .

N 

85 87 88 89

D ki X i − (

t f − to

i =0

2

90

) f ( X k , U k , τk ; to , t f ) = 0,

91 92

k = 1, . . . , N .

(28)

N t f − to 

2

93 94 95 96 97

w k · g ( X k , U k , τk ; t 0 , t f ),

k =1

98 99 100 101 102 103 104

φ( X 0 , t 0 , X f , t f ) = 0.

(30)

Furthermore, the path constraints of Estimates (21) is evaluated at the LG points, and is given by Estimates (31):

(31)

The cost function in Eq. (29) and the algebraic constraints in Eqs. (27), (28), (30), (31), define the transformed problem whose solution is an approximate solution to the original optimal control problem from the time t 0 to t f .

105 106 107 108 109 110 111 112 113 114 115

[ p 1 , pr ∈ [ p = 1, . . . , P ]], s = [1, . . . , L ].

N 

70

74

(26)

and linkage constraints in Estimates (23):

L min ≤ L (s) (x( p 1 ) (t f ), t f

68

73

˙ (τ ). x˙ (τ ) ≈ X

C ( X k , U k , τk ; t 0 , t f ) ≤ 0 (k = 1, . . . , N ).

( p)

62 64

, t; q

( p)

φmin ≤ φ( p )(x( p ) (t 0 ), t 0 o, x( p ) (t f ), t f , ; q( p ) ) ≤ φmax) ,

61 63

,u

( p)

boundary constraints in Estimates (22),

59 60

˙ (τ ) (see Eq. (26)): approximated by X

where w k are the Gauss weights. The boundary constraint of Estimates (22) is expressed as in Eq. (30):

inequality path constraints in Estimates (21),

50 52

j =1, j =i

τ −τ j τi −τ j , i = 1, . . . , N. From Eq. (24), x˙ (τ ) is

(29)

subject to dynamic constraints in Eq. (20),

48

51

where L ∗i (τ ) =

to

47 49

67

(25)

69

N 

(19)

41 42

τ ∈ [−1, 1],

i =1

J = ( X 0 , t 0 , X f , t f ) +

36 38

X i · L ∗i (τ ),

( p)

35 37

N 

The continuous cost function of Eq. (19) is approximated using a Gauss quadrature given by Eq. (29):

( p)

[ ( p ) (x( p ) (to ), to , x( p ) (t f ), t f , q( p ) )

p =1

31

u (τ ) ≈ U (τ ) =

τ −τ j τi −τ j , i = 0, 1, . . . , N.

The control is approximated using a basis of N Lagrange interpolating polynomials as in Eq. (25):

3. Results and discussion

116 117

In order to quantify the benefits achieved by performing dynamic soaring under the two morphologies, trajectory optimization similar to Mir [48] is performed. For analysis purpose, each of the two morphing configurations (span and sweep morphing) is compared against the respective fixed wing configuration. In case of sweep morphing (0◦ to 50◦ ), the morphing results are compared against two extreme fixed wing configurations having fixed wing sweep angles of 0◦ and 50◦ respectively. Similarly, the performance of variable span configuration UAV (which can vary span from 1.25 m to 1.75 m) are compared and evaluated against two extreme fixed wing configurations having spans of 1.25 m and 1.75 m respectively. Individual aspects of dynamic soaring parameters such as cycle time, maximum velocity, minimum required wind shear, maximum altitude gain, the lift and the drag forces and so on are evaluated for both the morphologies.

118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.9 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

9

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14 15 16 17 18 19 20 21 22 23

Fig. 7. Speed variation during the maneuver.

Fig. 5. Optimized 3D dynamic soaring trajectory.

81

3.1. Perspective analysis: span morphing vs. fixed configurations In this case, optimal trajectories of dynamic soaring are formulated for dynamic soaring loiter mode. In the loitering mode, the UAV is required to return to its initial state at the end of the maneuver. Loitering mode is used to perform long endurance missions, which require continuous monitoring of a certain geographical location. Important results are summarized below:

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

80

i) 3D optimal trajectories: 3D perspective view of the optimized trajectories generated for dynamic soaring loiter mode for both the fixed wing and morphing configurations is depicted in Fig. 5. To start with the loiter maneuver, the UAV bangs into the head wind gains height, trade off kinetic energy with potential energy and at the highest point takes a steep turn and dives down with tail wind. It continues to descend until it reaches the lowest point trading potential energy for kinetic (gain in the velocity) until it reaches the minimum possible height. At that point it takes the low altitude turn and returns to the original orientation to culminate the energy neutral maneuver cycle. It is clear that under similar environmental conditions the morphing platform exhibited enhanced loiter performance with higher altitude and area coverage (distances along north and east direction). This is primarily because of the fact that the wing span is adjusted during the maneuver in a way to maximize the state parameters. ii) 2D optimal trajectories: The soaring trajectories along the inertial east, north and vertical axis for both the morphing and the fixed configurations are depicted in Fig. 6. Approximately 40% increase in the maximum altitude, 42% increase in the distance traveled along the east and 42% along the north directions during the loiter maneuver are achieved by the morphing configuration in comparison with the fixed wing configuration (1.75 m). This, clearly demonstrates the suitability of the morphing configuration for loiter/surveillance

82

missions as it gives an enhanced area coverage compared to what is achieved by fixed configuration UAV. Under similar flight conditions, morphing UAV exhibits better ranges in comparison to the fixed counterpart. iii) Optimal trajectories for UAV speed: Variation in speed during different phases of the maneuver is depicted in Fig. 7. The speed decreases during the windward climb phase of the maneuver once the UAV climbs into the head wind, going to a bare minimum at the highest point. Then the UAV, takes a turn and start descending in the direction of the wind. It starts to gain speed until it reaches the bare minimum level near the ground. At this point the speed becomes maximum and the UAV is ready to repeat the maneuver. The maximum speed that is achievable during dynamic soaring maneuver is dependent upon aerodynamic and geometrical parameters, such as dynamic pressure, lift coefficient, wing span and mass. Incorporation of morphing feature in the UAV, resulted in approximately 20% increase in the maximum speed over the fixed 1.75 m span configuration and 7% increase over the 1.25 m span configuration. This is clear from Fig. 7, that the morphing UAV achieved a maximum velocity of 120 ft/s, in comparison with the velocities of 100 ft/s and 113 ft/s achieved by fixed 1.75 m and 1.25 m span configurations respectively. iv) Normalized energies: Similarly, normalized energies (total, potential and kinetic energies) are also higher in the case of morphing UAV (see Fig. 8). Incorporation of morphing feature in the UAV, resulted in approximately 38% increase in total normalized energy over the fixed 1.75 m span configuration and 9% increase over the 1.25 m span configuration. The kinetic energy is turned into potential energy as the UAV gains height. After reaching maximum altitude, the UAV takes the high altitude turn and start its downwind flight, where the potential energy is traded for the kinetic energy. The overall

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65 66

131

Fig. 6. UAV 2D trajectories.

132

JID:AESCTE

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.10 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

10

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

83

18

84

19 20

85

Fig. 8. Normalized energy. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30

96

31

97

32

98

33

99

34

100

35 36

101

Fig. 10. Span variations with varying velocity and altitude.

Fig. 9. Optimal trajectories for minimum required wind shear.

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

103

energy at the start and end of the maneuver remains constant which ensures an energy neutral cycle. v) Minimum required wind shear: Formulation of minimum required wind shear to perform dynamic soaring pose a challenging numerical problem because of coupled and nonlinear nature of the equations of motion. The results indicate that the minimum required wind shear for dynamic soaring is 10.8 m/s (for fixed configuration with span 1.25 m), 11.4 m/s (for fixed configuration with span 1.75 m) and 9.4 m/s (for morphing configuration) at reference altitude of 10 m (Fig. 9). The speed decreases during the windward climb phase of the maneuver once the UAV climbs into the head wind, going to a bare minimum at the highest point. Then the UAV takes a turn and starts descending in the direction of the wind. It starts to gain speed until it reaches the bare minimum level near the ground. At this point, the speed becomes maximum and the UAV is ready to repeat the maneuver. The logarithmic wind model in Eq. (1) is transformed into Eqs. (32)–(34) for the three stated configurations.

58 59

V w ,re f = 10.8

ln(h/h0 )

60

ln(10/0.03)

61

ln(h/h0 )

62

V w ,re f = 11.4

63 64 65 66

102

llV w ,re f = 9.4

ln(10/0.03)

for fixed wing (1.25 m). (32)

,

for fixed wing (1.75 m). (33)

,

for morphing configuration.

ln(10/0.03) ln(h/h0 )

,

(34)

This follows that minimum required wind speeds for dynamic soaring at sea level conditions are 6.6 m/s (for fixed 1.25 m span), 6.8 m/s (for fixed 1.75 m span) and 5.6 m/s (for morphing platform). Wind shear ascertained for both the extreme fixed span configuration UAV is of the magnitude, which normally exists over conventional environments (such as over sea, over hills, rural areas and so on), making dynamic soaring possible over these areas. Autonomous dynamic soaring over densely populated urban areas, however might not be feasible with these fixed configurations because of the presence of lesser wind shears. It is in fact these urban areas over which dynamic soaring is critically required to perform long endurance missions by a comparatively smaller platform. Operations such as area surveillance/reconnaissance, medical support, search and rescue and various types of other missions are frequently required in such urban areas. In this study, the morphing configuration requires 18% lesser wind shear than required by the fixed span configuration of 1.75 m, and 15% less than fixed span configuration of 1.25 m. With the reduction in minimum required wind shear, the morphing configurations could perform dynamic soaring over such urban environments, where wind shears are comparatively weak when compared to the conventional areas, where adequate wind shear exists to perform dynamic soaring. vi) Variation in span: Variation in the wing span during various phases of the maneuver is depicted in Fig. 10. In the flight phases associated with higher velocities (head wind climb and tailwind descent), span is kept at the bare minimum. The high altitude phase having lower velocities is kept at higher

104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.11 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

11

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15 16

81

Fig. 11. Variations in angle of attack and lift coefficient.

82

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

83

span. During the initial phase, once the altitude is less, angle of attack requirement is less and the velocity is more, lower span helps in improving aerodynamic performance (low drag). As the altitude increases, velocity decreases, and the higher span wing provides better performance (higher lift coefficient). In the windward climb phase, it maintains the low span configuration until the time, when the velocity drops beyond a threshold velocity (which in this case is 40 ft/s, as seen in Fig. 10). It then shifts to higher span configuration which can provide more lift. It maintains the high span configuration throughout the high altitude turn phase, in which the velocity reaches to a bare minimum. Then in the downwind climb phase, once the velocity increases from the threshold point (40 ft/s), the UAV goes back to the low span configuration (for reducing the drag). Wings are kept at shorter span during high speed low altitude turn, in which they are less effective at generating lift, but can bear the extreme loads. Presently, the optimization algorithm performs rapid optimization of the span during various phases within the permissible range. In real world conditions, a rate limiter can be physically incorporated to ensure that the change in the span variations is not beyond admissible rates for the wing actuators. Extended wings are better for slow glides and turns, while short wings are better for fast glides and turns. vii) Variation in lift coefficient and angle of attack: Fig. 11 reflects the optimal trajectories of the lift coefficient and the angle of attack variations. The angle of attack requirement is low during the initial climb phase of the maneuver. As the altitude increases, the velocity decreases and the angle of attack requirement increases. The lift coefficient reaches large values in the high altitude turn, where the flight direction changes from windward to tailwind. In the remaining flight phases, the lift coefficient requirement is low. The morphing platform was able to perform the maneuver at much lesser angle of attack than that required by the non-morphing configurations. Approximately, 29% improvement is achieved, as the maximum angle of attack required by morphing UAV is 3.7◦ against the 5.2◦ required by 1.25 m configuration. Similarly, 20% improvement is achieved with respect to 1.75 m configuration having the maximum angle of attack of 4.6◦ . It is clear from Fig. 11a that there are rapid and abrupt changes in the angle of attack, in the early phases of the maneuver. As a result, the lift coefficient also shows large spikes, see Fig. 11b. The reason for this phenomena is that the aerodynamic properties, discussed and utilized in this section are steady or quasi-steady in nature. Such assumptions have been made for evaluating agile maneuvers in most studies [93]. In a quasisteady approach, any change in angle of attack of the aircraft results in an instantaneous change in aerodynamic properties.

84 85 86 87 88 89 90 91 92 93 94 95 96 97

Fig. 12. Lift coefficient variation with varying span.

98 99

In reality, any instantaneous change in attitude of the aerodynamic surface induces a flow-field change. As a result, the effective angle of attack is different from geometric angle of attack. The delay in achieving the new steady aerodynamic response occurs due to the time taken for the circulation around the surface to change to the new steady flow condition [94]. The quasi-steady assumption, while attractive in its simplicity, is not sufficiently accurate. Therefore, more advanced unsteady aerodynamic techniques must be used, to determine the dependency of aerodynamic coefficient on the dynamic motions. To cater for the unsteady effects, the timelag in the buildup of aerodynamic coefficients is modeled using Wagner function ( (τ )). This is elaborated in Eqs. (35):

100

C L e f f ect = C L ∗ (τ ),

114

C D e f f ect = C D ∗ (τ ).

(35)

where

τ=

2

τ +4

c¯

103 104 105 106 107 108 109 110 111 112 113 115 117 118 119 120 121

,

(36)

122 123

τ is the normalized time given by Eq. (37):

2V ∞ t

102

116

Wagner function (as defined in Eq. (36)) is a time dependent quantity that accounts for time delays experienced with changes in the instantaneous angle of attack [94].

(τ ) = 1 −

101

124 125

.

(37)

The lift coefficient catering for the unsteady dynamics through Wagner function is depicted in Fig. 12. This shows smooth transition of the lift coefficient even in the areas where there are abrupt changes in the angle of attack (especially, at the start of the maneuver). Analysis of the lift coefficient graph,

126 127 128 129 130 131 132

JID:AESCTE

12

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.12 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16 17

82

Fig. 13. Drag coefficient variation with varying span.

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30

96

31

97

32

98

33

99

34

100

35

101

Fig. 14. Variation of drag coefficient with varying velocity, altitude and span.

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

102

reveal that the morphing platform requires 11% less lift coefficient than the fixed counterparts. The UAV takes the lower span configurations in the initial phase of the maneuver associated with high velocities and lesser altitude. It then acquires the high span configurations in the high altitude regions where velocity is less. Adopting to the configuration most suitable to the flight phase, the morphing UAV acquired the inherent benefits of both the configurations. viii) Drag analysis: The drag induced by both the morphing and fixed configurations, assuming steady and unsteady aerodynamics, is depicted in Fig. 13a and Fig. 13b respectively. It is noticeable from Fig. 13a that the drag coefficient assuming steady aerodynamics has spikes in the initial flight phase. This is because of sudden changes in the lift coefficient. It is clear from Fig. 14 that in the initial phase of the maneuver, morphing platform adopts lower span configuration (1.25 m span), which has lesser drag as compared to the higher span configuration (1.75 m span). As the velocity decreases and the angle of attack requirement significantly increases, it shifts to the higher span configuration. This is more beneficial in terms of overall aerodynamic performance (more lift). Low span wings results in low drag coefficients at low angles of attack and extended wings results in high lift coefficients at high angles of attack. Adopting to the most befitted configuration, morphing platform exhibited 34% reduction in the maximum drag coefficient as compared to the fixed span configurations of 1.25 m and 15% reduction with respect to 1.75 m configuration. Reduction of drag to a

103 104 105 106 107 108 109 110 111 112 113 114 115 116

Fig. 15. Load factor.

117

significant amount crystallizes the potential viability of morphology and its profound effects on dynamic soaring. ix) Load factor: The specified path constraint of load factor, that ensures the structural limits of the vehicle throughout the maneuver is depicted in Fig. 15. The path constraint reaches to its maximum value in the high speed regime associated with the initial/terminal phase of the flight. This is where the altitude is less and speed is at the maximum. During the remaining flight phases, the load factor remains around 1 that is well within the limits. x) Dynamic soaring force: Since the wind relative frame is not inertial [95], a fictitious force, F dyn acts on the UAV and, as shown graphically in Fig. 16 and elaborated in Eq. (38):

V˙ =

1 m

118 119 120 121 122 123 124 125 126 127 128 129 130 131

[− D − mg sin γ − F dyn ].

(38)

132

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.13 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

sin γ cos γ sin ψ < 0.

1

13

(45)

2

67 68

This can be achieved through windward climb and tailwind descend. During the windward climb phase, Estimates. (46) holds.

3 4 5 6

π < ψ < 2π or − π < ψ < 0, ⇒ sin ψ < 0, so overall, sin γ cos γ sin ψ < 0.

8 9

70 71 72

0 < γ < π /2 ⇒ sin γ & cos γ > 0,

7

69

73

(46)

74 75

10

76

11

Similarly during the decent phase, Estimates. (47) holds.

77

12

− π /2 < γ < 0 ⇒ sin γ < 0, & cos γ > 0, 0 < ψ < π or − π < ψ < −2π , ⇒ sinψ > 0, so overall, sin γ cos γ sin ψ < 0.

78

13 14

Fig. 16. Dynamic soaring force.

15

79

(47)

16 17 18 19 20 21 22 23 24 25 26 27

30 31 32 33 34 35

F dyn = m V˙ w .

(39)

Its projection along the direction of airspeed is given as Eq. (40):

F dyn = m V˙ w cos γ sin ψ.

V˙ W = V w re f

1 h ln(hre f /h0 )

h˙ .

(41)

But from Eq. (4),

h˙ = V sin γ .

(42)

So Eq. (40) becomes Eq. (43):

F dyn = mV w re f

38 39

(40)

But from Eq. (2), for logarithmic wind model is given in Eq. (41):

36 37

1 hln(hre f /h0 )

cos γ sin ψ sin γ .

(43)

And Eq. (39) becomes Eq. (44):

40 41 42 43 44 45 46 47 48 49

81 82

This dynamic soaring force which is due to wind shear is given in Eq. (39):

28 29

80

1 V˙ = [− D − mg sin γ m

− mV w re f

1 h ln(hre f /h0 )

sin γ cos γ sin ψ].

(44)

In order to have sustained powerless flight for extracting energy from wind shear, the velocity added by wind shear (third term) must be greater than or at least equal to drag (first term). This requires that Estimates. (45) must be satisfied.

UAV can therefore extract energy from the atmosphere, during both the climb and descent phases. In case the maneuver is reversed, i.e. windward descent or tailwind climb, the UAV will lose energy. Fig. ?? reflects the optimized trajectories for the dynamic soaring force. This shows that dynamic soaring force is exerted in a direction, which will compensate for the energy lost due to drag during the energy neutral dynamic soaring cycle (see Eq. (44)). The morphing platform provides higher values of dynamic soaring force, which aids in providing greater loiter performance parameters, complete details of which have been discussed earlier. xi) Variation in flight path angle and heading angle: The optimized trajectories of flight path angle and heading angles are depicted in Fig. 17. Both the trajectories, satisfy the constraints imposed for windward climb (see Estimates. (46)) and tailwind descent (see Estimates. (47)). During the climb phases, the UAV follows a trajectory which maintains a heading angle within −180◦ to 0◦ and the flight path angle 0 to 90◦ . Similarly during descent phase, heading angle is between −180◦ to −360◦ and flight path angle between −90◦ to 0◦ . This implies that the trajectory formulated for UAV ensures that the energy is extracted from wind shear during both the climb and ascend phases. Both the angles follow a trajectory to ensure the UAV movement in the loiter mode with specified constraints. It is also clear from the Fig. 17 that morphing configuration follows much smoother angles as compared to both of the fixed configurations, which require sharp maneuver angles. xii) Variation in bank angle: Similarly, variations in bank angle experienced by both morphing and fixed configurations during various phases of the maneuver are depicted in Fig. 18. The time history of the bank angle, shows that in both the

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115

50

116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65 66

131

Fig. 17. Variations in flight path and heading angles.

132

JID:AESCTE

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.14 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

14

1 2 3

67

Table 5 Comparative analysis: span morphing vs fixed span configurations. S No.

Nomenclature

4 5

Fixed span (1.25 m) configuration

68

Fixed span (1.75 m) configuration

Morphing (1.25–1.75 m) configuration

6 7 8

1 2

9 10 11 12 13

3 4 5 6

14 15 16 17

7 8 9

Maximum altitude gain Maximum distance covered in east direction Maximum distance covered in north direction Normalized Energy Maximum speed Minimum wind shear at sea level (1 m) Maximum required angle of attack Maximum CL requirement Max Drag generated

% Improvement by morphing configuration w.r.t fixed 1.25 m span

% Improvement by morphing configuration w.r.t fixed 1.75 m span

Reference figures

72

197 ft 270 ft

150 ft 210 ft

210 ft 300 ft

7% 11.2%

40% 42%

Fig. 6 Fig. 6

91 ft

70 ft

100 ft

9.9%

42%

Fig. 6

6200 113 ft/s 6.6 m/s

5200 100 ft/s 6.8 m/s

7200 120 ft/s 5.6 m/s

9% 6.2% 15%

38% 20% 18%

Fig. 8 Fig. 7 Fig. 9

5.2◦

4.65◦

3.75◦

29%

20%

Fig. 11

0.45 0.03

0.45 0.023

0.4 0.0196

11% 34%

11% 15%

Fig. 12 Fig. 24

22 23 24 25 26 27 28 29 30

3.2. Perspective analysis: sweep morphing vs. fixed configurations

31 32

Fig. 18. Variations in bank angle.

34

38 39 40 41 42 43 44 45 46

76 77 78 79 80

better than higher span (1.75 m) in the results of achieving higher velocities, greater area coverage, higher altitude gains, lower required wind shear and low drag generation at low angle of attack. While, it is other way around for the results of lesser required angle of attack and high lift generation at high angle of attacks (higher altitudes). Remarkably, the proposed span morphing configuration is superior, when compared to both of the fixed configurations in every single result. Utilizing morphing, the UAV adopted to the span configuration, which ensured optimal results with minimal drag generation.

21

37

74

81 82 83 84

20

36

73 75

19

35

70 71

18

33

69

high and low altitude turns, the bank angle takes on quite large values. The bank angle constraints are therefore, active in both the phases of the dynamic soaring flight. It is also clear from the Fig. 18 that morphing configuration follows much smoother angles as compared to both of the fixed configurations, which require sharp maneuver angles. xiii) Summary of results: The entire results of section 3.1 are summarized and collected in Table 5. In Table 5, all conducted comparisons between the span morphology and the two fixed span configurations (1.25 m and 1.75 m) are presented. Also the table references the figures, where the simulations have been depicted. It is noticeable that lower span (1.25 m) is

Variation in sweep angle affects wing aerodynamics by altering the aerodynamic forces acting on the UAV. As a result, the performance envelope with swept wings is much larger as compared to the fixed wing configuration. In this case, the optimal soaring trajectories for loiter mode, are formulated for a platform capable of sweep variations from 0◦ to 50◦ . To quantify the benefits achieved, the optimized morphing trajectories will be evaluated and compared against the two extreme fixed sweep configurations having fixed 0◦ sweep and 50◦ sweep. Individual aspects of the dynamic soaring parameters evaluated in this study are presented below:

85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108

i) Optimal soaring trajectories: 3D perspective view of the optimized trajectories generated for dynamic soaring loiter mode for both the fixed wing and morphing configurations are depicted in Fig. 19.

109 110 111 112

47

113

48

114

49

115

50

116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65 66

131

Fig. 19. Optimized 3D dynamic soaring trajectory.

132

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.15 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

15

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

Fig. 20. UAV 2D trajectories.

16

82

17

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30 31

96

Fig. 21. Speed variations. Fig. 22. Variation in wing sweep.

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Under similar conditions, the morphing UAV and the fixed wing UAV with 50◦ sweep gave enhanced loiter performance parameters compared to the fixed 0◦ sweep configuration. As a result, higher altitude gains and area coverage (distances along north and east directions) were achieved. Optimal trajectories along the inertial east, north and vertical axis for both the morphing and the fixed configurations are depicted in Fig. 20. Morphing and fixed 50◦ configurations showed 29% increase in the maximum altitude and approximately 25% increase in the distance along east and north directions, in comparison with fixed 0◦ sweep configuration. ii) Variation in airspeed: Variation in UAV airspeed during different phases of the maneuver is depicted in Fig. 21. Both the morphing and the fixed high sweep (50◦ ) configuration exhibited higher maximum velocities compared to the low sweep configuration. The morphing and the fixed high sweep configuration achieved a maximum velocity of 130 ft/s in comparison with 114 ft/s achieved by low sweep configuration. This is approximately 13% improvement. The result shows that high sweep configuration is better for achieving high velocities during dynamic soaring. iii) Sweep analysis: Variation in the wing sweep during various phases of the maneuver is depicted in Fig. 22. In the flight phases associated with lower altitude and higher velocities (head wind climb and tailwind descent phases), wings are kept at higher sweep angle. During these phases, the required angle of attack is low and the velocities are high. As a result, higher sweep configuration improves aerodynamic performance by contributing low drag. As the altitude increases, velocity decreases and the lower sweep configuration provides better performance by contributing more lift. In the windward climb phase, high sweep configuration is maintained until the air speed drops beyond a threshold point. After that, lower sweep configuration is adopted to have

97 98 99

more lift. The UAV maintains the lower sweep configuration throughout the high altitude turn phase, in which the velocity is at the bare minimum. Then in the downwind climb phase, once the velocity again increases from the threshold point, it adopts the high sweep configuration (for reducing the highspeed drag). The threshold velocity, in this case is ascertained to be around 45 ft/s (as shown in Fig. 22). The optimization algorithm performs rapid optimization of the sweep during various phases within the permissible range. In real world conditions, a rate limiter can be incorporated to ensure that the change in the sweep variations is not beyond admissible rates for the wing actuators. iv) Variation in lift coefficient and angle of attack: Fig. 23 reflects the optimal trajectories of the angle of attack variations and the lift coefficient. The required angle of attack is low during the initial climb phase of the maneuver and increases as the UAV gains altitude. Results indicate that morphing UAV, could complete the trajectory with 49% lesser required angle of attack in comparison with the fixed 50◦ configuration. Likewise, a 16% improvement was achieved in comparison with the fixed 0◦ configuration. During the climb phase, lift coefficient requirement increases. It approaches the largest values in the high altitude turn phase, where the flight direction is changed from windward to tailwind direction. In the remaining phases, the required lift coefficient is less. The fixed configuration of 50◦ sweep reaches the maximum permissible angle of attack (10◦ ) during the high altitude region. As a result, the lift coefficient is also constrained, as shown in Fig. 23. The morphing platform exhibited 10% lesser maximum required lift coefficient compared to the fixed wing configuration (0◦ sweep). v) Drag analysis: Fig. 24 depicts the drag induced by both the morphing and fixed configurations. It is clear that in the ini-

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

16

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.16 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15 16

81

Fig. 23. Variations in angle of attack and lift coefficient.

82

17

83

Table 6 Minimum required wind shear: Sweep morphing configuration vs fixed configurations.

18 19 20

S No.

21 22 23

Reference altitude (m)

24 25 26

1 2

27

10 0.5

Minimum windshear required for morphing configuration (m/s)

87

11.8 8.1

10.9 7.5

9.8 6.5

92

Fig. 24. Optimal trajectories for drag.

34

37 38 39 40 41 42 43 44 45 46 47 48

89 90 91 93 95

33

36

88

94

30

35

86

Minimum windshear required for 50◦ sweep configuration (m/s)

29

32

85

Minimum windshear required for 0◦ sweep configuration (m/s)

28

31

84

tial phase of the maneuver, morphing platform adopts the higher sweep configuration, which has lesser drag as compared to the lower sweep configuration. As the velocity decreases and the required angle of attack increases (see Fig. 25), the morphing configuration adopts the low sweep configuration. This configuration is more beneficial in this stage of flight, as it generates more lift [93]. Adopting to the most beneficial configuration, morphing platform exhibited 20% reduction in the maximum drag as compared to the fixed 0◦ configurations. Similarly, 24% reduction was achieved in comparison with the fixed 50◦ configuration. vi) Minimum required wind shear: Minimum required wind shear for sustained dynamic soaring for both the fixed and the morphing configuration is presented in Table 6.

Fig. 26 shows that the morphing, significantly reduces the wind shear required for dynamic soaring. The morphing configuration showed 20% reduction in the wind shear in comparison with 0◦ sweep configuration and 14% reduction in comparison with 50◦ sweep configuration. Lesser required wind shear shows that morphing UAV could perform dynamic soaring in an environment where fixed configurations might not, because of the lesser available atmospheric wind shears. vii) Load factor: Graphical representation of the load factor constraint imposed to ensure that the structural bounds of the UAV are not violated, is shown in Fig. 27. The path constraint reaches the maximum value in the high speed regime associated with the initial/terminal phase of the flight. During the remaining phases of the flight, the load factor remains around 1 that is well within the limits. viii) Optimal trajectories for flight path and heading angles: The optimized trajectories of flight path and the heading angles are shown in Fig. 28. Both the angles follow a trajectory that ensures that the constraints for the windward climb (see Esti-

96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114

49

115

50

116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65 66

131

Fig. 25. Drag variations with velocity, altitude and sweep.

132

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.17 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

17

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14 15

80

Fig. 26. Optimal trajectories for minimum required wind shear.

Fig. 29. Dynamic soaring force.

81

16

82

17

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30 31

Fig. 30. Variations in bank angle.

Fig. 27. Load factor.

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

mates. (46)) and downwind descent (see Estimates. (47)) are satisfied. This ensures that the energy is extracted from wind shear during both the climb and ascend phases. ix) Dynamic soaring force: Fig. 29 depicts the optimized trajectories for the dynamic soaring force (DSF). It is clear from the Fig. 29 that the dynamic soaring force is exerted in a direction, which compensates for the energy lost due to drag within the energy neutral dynamic soaring cycle (see Eq. (44)). The morphing platform optimizes the dynamic soaring force for enhanced loiter performance parameters. x) Variation in bank angle: Variations in bank angle experienced by both morphing and fixed configurations during various phases of the maneuver are depicted in Fig. 30. The time history of the bank angle (φ ), shows that in both the high and low altitude turns, the bank angle takes on quite large values. The bank angle constraint is, therefore, active in both the phases of the dynamic soaring flight. It is also clear from

Fig. 30 that morphing configuration follows much smoother angles as compared to both of the fixed configurations, which require sharp maneuver angles. xi) Normalized energies: Normalized energies (total, potential and kinetic energies) are also higher in the case of morphing and high sweep configuration (Fig. 31). Approximately, 30% increase in the total normalized energy is achieved by both the morphing and high sweep platforms in comparison with the fixed wing configuration with 0◦ sweep. xii) Summary of results: The entire results of section 3.2 are summarized and collected in Table 7. In Table 7, all conducted comparisons between the sweep morphology and the two fixed sweep configurations (0◦ and 50◦ ) are presented. Also the table references the figures, where the simulations have been depicted. It is noticeable that higher sweep (50◦ ) configuration is better then lower sweep (0◦ ) configuration in the results for achieving higher velocities, greater area coverage, higher altitude gains, lesser required minimum wind

96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65 66

131

Fig. 28. Variations in flight path and heading angles.

132

JID:AESCTE

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.18 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

18

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

83

18

84

19

85

20

86

Fig. 31. Normalized energies.

21

87

22 23 24 25

88 89

Table 7 Comparative analysis: Sweep morphing vs fixed wing configurations. S No.

Nomenclature

26 27

90

Fixed 0◦ sweep configuration

Fixed 50◦ sweep configuration

Morphing (0◦ to 50◦ ) configuration

% Improvement by morphing configuration w.r.t fixed 0◦ sweep

% Improvement by morphing configuration w.r.t fixed 50◦ sweep

Reference figures

196 ft 284 ft 106 ft 115 ft/s 6600 0.45 8.1 m/s

252 ft 364 ft 132 ft 130 ft/s 8600 0.37 7.5 m/s

252 ft 365 ft 135 ft 130 ft/s 8600 0.39 6.5 m/s

29% 28% 25% 13% 30% 10% 20%

same same same same same same 14%

Fig. Fig. Fig. Fig. Fig. Fig. Fig.

5.2◦

10◦

6.2◦

0.031

0.033

0.025

16% 20%

49% 24%

Fig. 23 Fig. 24

28 29 30 31 32 33 34 35 36

1 2 3 4 5 6 7 8 9

Maximum altitude gain Distance covered in east direction Distance covered in north direction Maximum speed Normalized energy Maximum CL requirement Minimum required wind shear at sea level (1.5 m) Maximum required AoA Max Drag reduction

94

20 20 20 21 31 23 26

41 42 43 44 45

4. Conclusion

48 49 50 51 52 53 54 55

This paper introduced optimal dynamic soaring for fixed and morphing configurations. An original bio-inspired model with morphing capability is developed. Through simulations, under conditions defined in section 2, two major studies were performed. One of them analyzes the impact of span morphology and the other analyzes sweep morphology impact on dynamic soaring parameters.

56 57 58 59 60 61 62 63 64 65 66

96 97 98 99 100 101 102 104

shear, and low drag generations at low angles of attack (lower altitudes). While, it is the other way around for the results of lesser required angle of attack, and high lift generation at high angle of attacks (higher altitudes). Remarkably, the proposed sweep morphing configuration is superior, when compared to both of the fixed configurations in every single result.

46 47

95

103

38 40

92 93

37 39

91

a) Span morphology vs fixed wing morphology: The most important results attained about span work are summarized in Table 5. Concludingly, UAV with lower span (1.25 m) is better then higher span (1.75 m) configurations in the results of achieving higher velocities, greater area coverage, higher altitude gains, lower required wind shear, and low drag generation at low angle of attack. While, it is other way around, for the results of lesser required angle of attack, and high lift generation at high angle of attacks (higher altitudes). Remarkably, the proposed span morphing configuration is superior,

when compared to both of the fixed configurations in every single result. b) Sweep morphology vs fixed wing morphology: The most important results attained about span morphology are summarized in Table 7. Concludingly, UAV with higher sweep (50◦ ) configuration is better then lower sweep (0◦ ) configuration in the results of achieving higher velocities, greater area coverage, higher altitude gains, less required minimum wind shear, and low drag generations at low angles of attack (lower altitudes). While, it is other way around, for the results of lesser required angle of attack, and high lift generation at high angle of attacks (higher altitudes). Remarkably, the proposed sweep morphing configuration is superior, when compared to both of the fixed configurations in every single result. As noted in Table 5 and Table 7, proposed work has positive impacts. This study strongly supports the idea of integrating dynamic soaring under morphing conditions and its potential benefits. This triggers the possibility of performing stability, reachability, and controllability studies from control prospective. More work is also required to determine the impact of other morphologies (such as variations in wing twist [96,38], wing dihedral [97,98] and so on) on dynamic soaring. Consequently, linear/nonlinear guidance and control laws can be designed, to implement the optimized trajectories. This work is likely to open a new era for researchers, as the distinct advantages of morphing and dynamic soaring can now be integrated into a single aerial platform.

105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.19 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

1 2 3 4 5 6 7

c) Future potential: It is imperative that the investigation performed in this study, will serve as a baseline that supports the idea that dynamic soaring can be extended to a morphing platform. This shall open a new era for research, as the distinct advantages of morphing and dynamic soaring can now be integrated into a single aerial platform. Studies can then be performed for other morphologies.

8 9

Conflict of interest statement

10 11

None declared.

12 13

References

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

[1] L. Rayleigh, The soaring of birds, Nature 27 (701) (1883) 534–535. [2] J. Wilson, Sweeping flight and soaring by albatrosses, Nature 257 (5524) (1975) 307–308. [3] M. Denny, Dynamic soaring: aerodynamics for albatrosses, Eur. J. Phys. 30 (1) (2008) 75. [4] G. Sachs, J. Traugott, A. Nesterova, F. Bonadonna, Experimental verification of dynamic soaring in albatrosses, J. Exp. Biol. 216 (22) (2013) 4222–4232. [5] C. Pennycuick, The flight of petrels and albatrosses (Procellariiformes), observed in South Georgia and its vicinity, Philos. Trans. R. Soc. Lond. B, Biol. Sci. 300 (1098) (1982) 75–106. [6] G. Sachs, Minimum shear wind strength required for dynamic soaring of albatrosses, Ibis 147 (1) (2005) 1–10. [7] J.M. Wharington, Heuristic control of dynamic soaring, in: 2004 5th Asian Control Conference, vol. 2, IEEE, 2004, pp. 714–722. [8] P.P. Sukumar, M.S. Selig, Dynamic soaring of sailplanes over open fields, J. Aircr. (2013). [9] Y.J. Zhao, Optimal patterns of glider dynamic soaring, Optim. Control Appl. Methods 25 (2) (2004) 67–89. [10] P.E. Gill, W. Murray, M.A. Saunders, M.H. Wright, User’s Guide for NPSOL (Version 4.0): A Fortran Package for Nonlinear Programming, Tech. Rep., Stanford Univ CA Systems Optimization Lab, 1986. [11] D.-N. Liu, Z.-X. Hou, Z. Guo, X.-X. Yang, X.-Z. Gao, Optimal patterns of dynamic soaring with a small unmanned aerial vehicle, Proc. Inst. Mech. Eng., G, J. Aerosp. Eng. 231 (9) (2017) 1593–1608. [12] P. Idrac, Experimentelle Untersuchungen über den Segelflug, 1932. [13] C.J. Pennycuick, Gust soaring as a basis for the flight of petrels and albatrosses (Procellariiformes), Avian Sci. 2 (1) (2002) 1–12. [14] H. Weimerskirch, F. Bonadonna, F. Bailleul, G. Mabille, G. Dell’Omo, H.-P. Lipp, GPS tracking of foraging albatrosses, Science 295 (5558) (2002) 1259. [15] G. Sachs, M. Mayrhofer, Shear wind strength required for dynamic soaring at ridges, Tech. Soaring 25 (4) (2001) 209–215. [16] C. Wood, The flight of albatrosses (a computer simulation), Ibis 115 (2) (1973) 244–256. [17] C. Pennycuick, Gliding flight of the fulmar petrel, J. Exp. Biol. 37 (2) (1960) 330–338. [18] N.R. Lawrance, S. Sukkarieh, A guidance and control strategy for dynamic soaring with a gliding UAV, in: IEEE International Conference on Robotics and Automation, ICRA’09, 2009, IEEE, 2009, pp. 3632–3637. [19] N.R. Lawrance, Autonomous Soaring Flight for Unmanned Aerial Vehicles, Ph.D. Thesis, University of Sydney, 2011. [20] N.R. Lawrance, S. Sukkarieh, Wind energy based path planning for a small gliding unmanned aerial vehicle, in: AIAA Guidance, Navigation, and Control Conference, 2009, pp. 10–13. [21] Y.J. Zhao, Y.C. Qi, Minimum fuel powered dynamic soaring of unmanned aerial vehicles utilizing wind gradients, Optim. Control Appl. Methods 25 (5) (2004) 211–233. [22] B.-J. Zhu, Z.-X. Hou, H.-J. Ouyang, Trajectory optimization of unmanned aerial vehicle in dynamic soaring, Proc. Inst. Mech. Eng., G, J. Aerosp. Eng. 231 (10) (2017) 1779–1793. [23] G. Sachs, O. da Costa, Optimization of dynamic soaring at ridges, in: AIAA Atmospheric Flight Mechanics Conference and Exhibit, 2003, pp. 11–14. [24] N. Akhtar, J.F. Whidborne, A.K. Cooke, Wind Shear Energy Extraction Using Dynamic Soaring Techniques, AIAA, 734, American Institute of Aeronautics and Astronautics, 2009. [25] N. Akhtar, J. Whidborne, A. Cooke, Real-time optimal techniques for unmanned air vehicles fuel saving, Proc. Inst. Mech. Eng., G, J. Aerosp. Eng. 226 (10) (2012) 1315–1328. [26] X.-Z. Gao, Z.-X. Hou, Z. Guo, R.-F. Fan, X.-Q. Chen, Analysis and design of guidance-strategy for dynamic soaring with UAVs, Control Eng. Pract. 32 (2014) 218–226. [27] A. Rao, D. Benson, G. Huntington, Constructing an optimal control problem in GPOPS, in: A MATLAB Software for Solving Multiple-Phase Optimal Control Problem Using Pseudospectral Methods, Gainesville, 2010, pp. 18–39.

19

[28] A.V. Rao, D.A. Benson, C. Darby, M.A. Patterson, C. Francolin, I. Sanders, G.T. Huntington, Algorithm 902: GPOPS, a MATLAB software for solving multiplephase optimal control problems using the gauss pseudospectral method, ACM Trans. Math. Softw. 37 (2) (2010) 22. [29] O. Ariff, T. Go, Waypoint navigation of small-scale UAV incorporating dynamic soaring, J. Navig. 64 (1) (2011) 29–44. [30] C. Gao, H.H. Liu, Dubins path-based dynamic soaring trajectory planning and tracking control in a gradient wind field, Optim. Control Appl. Methods 38 (2) (2017) 147–166. [31] R.J. Gordon, Optimal Dynamic Soaring for Full Size Sailplanes, Tech. Rep., Air Force Inst of Tech Wright-Patterson AFB OH Dept of Aeronautics and Astronautics, 2006. [32] D.-N. Liu, Z.-X. Hou, X.-Z. Gao, Flight modeling and simulation for dynamic soaring with small unmanned air vehicles, Proc. Inst. Mech. Eng., G, J. Aerosp. Eng. 231 (4) (2017) 589–605. [33] J. Warham, The Behaviour, Population Biology and Physiology of the Petrels, Academic Press, 1996. [34] P. Jouventin, H. Weimerskirch, Satellite tracking of wandering albatrosses, Nature 343 (6260) (1990) 746–748. [35] P. Catry, R.A. Phillips, J.P. Croxall, Sustained fast travel by a gray-headed albatross (Thalassarche chrysostoma) riding an Antarctic storm, The Auk 121 (4) (2004) 1208–1213. [36] J.P. Barnes, How Flies the Albatross – The Flight Mechanics of Dynamic Soaring, Tech. Rep., SAE Technical Paper, 2004. [37] M. Berger, W. Göhde, Zur Theorie des Segelfluges von Vögeln über dem Meere, Zool. Jb. Physiol 71 (1965) 217–224. [38] M. Hassanalian, G. Throneberry, A. Abdelkefi, Wing shape and dynamic twist design of bio-inspired nano air vehicles for forward flight purposes, Aerosp. Sci. Technol. 68 (2017) 518–529. [39] M. Rosén, A. Hedenstrom, Gliding flight in a jackdaw: a wind tunnel study, J. Exp. Biol. 204 (6) (2001) 1153–1166. [40] G. Sachs, A. Knoll, K. Lesch, Optimal utilization of wind energy for dynamic soaring, Tech. Soaring 15 (2) (1991) 48–55. [41] P.L. Richardson, Upwind dynamic soaring of albatrosses and UAVs, Prog. Oceanogr. 130 (2015) 146–156. [42] V. Bonnin, E. Bénard, J.-M. Moschetta, C. Toomer, Energy-harvesting mechanisms for UAV flight by dynamic soaring, Int. J. Micro Air Veh. 7 (3) (2015) 213–229. [43] A.M. Wickenheiser, E. Garcia, Optimization of perching maneuvers through vehicle morphing, J. Guid. Control Dyn. 31 (4) (2008) 815–823. [44] N. Akhtar, Control System Development for Autonomous Soaring, 2010. [45] N. Lawrance, J. Acevedo, J. Chung, J. Nguyen, D. Wilson, S. Sukkarieh, Long endurance autonomous flight for unmanned aerial vehicles, AerospaceLab 8 (2014) 1. [46] M. Deittert, A. Richards, C. Toomer, A. Pipe, Dynamic soaring flight in turbulence, in: AIAA Guidance, Navigation and Control Conference, Chicago, Illinois, 2009, pp. 2–5. [47] Y. Liu, S. Longo, E.C. Kerrigan, Nonlinear predictive control of autonomous soaring UAVs using 3DOF models, in: 2013 European Control Conference, ECC, IEEE, 2013, pp. 1365–1370. [48] I. Mir, A. Maqsood, S. Akhtar, Optimization of dynamic soaring maneuvers to enhance endurance of a versatile UAV, Mater. Sci. Eng. Conf. Ser. 211 (2017) 012010. [49] J. Sun, Q. Guan, Y. Liu, J. Leng, Morphing aircraft based on smart materials and structures: a state-of-the-art review, J. Intell. Mater. Syst. Struct. 27 (17) (2016) 2289–2312. [50] R.M. Ajaj, C.S. Beaverstock, M.I. Friswell, Morphing aircraft: the need for a new design philosophy, Aerosp. Sci. Technol. 49 (2016) 154–166. [51] J. Fincham, M. Friswell, Aerodynamic optimisation of a camber morphing aerofoil, Aerosp. Sci. Technol. 43 (2015) 245–255. [52] A.K. Jha, J.N. Kudva, Morphing aircraft concepts, classifications, and challenges, in: Smart Structures and Materials 2004: Industrial and Commercial Applications of Smart Structures Technologies, vol. 5388, International Society for Optics and Photonics, 2004, pp. 213–225. [53] T.-t. Zhang, Z.-g. Wang, W. Huang, L. Yan, Parameterization and optimization of hypersonic-gliding vehicle configurations during conceptual design, Aerosp. Sci. Technol. 58 (2016) 225–234. [54] T.-t. Zhang, Z.-g. Wang, W. Huang, S.-b. Li, A design approach of wide-speedrange vehicles based on the cone-derived theory, Aerosp. Sci. Technol. 71 (2017) 42–51. [55] T.-t. Zhang, W. Huang, Z.-g. Wang, L. Yan, A study of airfoil parameterization, modeling, and optimization based on the computational fluid dynamics method, J. Zhejiang Univ. Sci. A 17 (8) (2016) 632–645. [56] A. Moosavian, E.J. Chae, A.M. Pankonien, A.J. Lee, D.J. Inman, A parametric study on a bio-inspired continuously morphing trailing edge, in: Bioinspiration, Biomimetics, and Bioreplication 2017, vol. 10162, International Society for Optics and Photonics, 2017, 1016204. [57] F. Peter, E. Stumpf, The Development of Morphing Aircraft Benefit Assessment, Morphing Wing Technologies, Elsevier, 2018, pp. 103–121.

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

AID:4577 /FLA

[m5G; v1.236; Prn:21/05/2018; 10:34] P.20 (1-20)

I. Mir et al. / Aerospace Science and Technology ••• (••••) •••–•••

[58] F. Nicassio, G. Scarselli, F. Pinto, F. Ciampa, O. Iervolino, M. Meo, Low energy actuation technique of bistable composites for aircraft morphing, Aerosp. Sci. Technol. (2018). [59] F. Afonso, J. Vale, F. Lau, A. Suleman, Performance based multidisciplinary design optimization of morphing aircraft, Aerosp. Sci. Technol. 67 (2017) 1–12. [60] I. Mir, A. Maqsood, S. Akhtar, Optimization of dynamic soaring maneuvers for a morphing capable UAV, in: AIAA Information Systems – AIAA Infotech@ Aerospace, 2017, 0678. [61] A.G. Quintana, M. Hassanalian, A. Abdelkefi, Aerodynamic analysis of a morphing drone with spanning and sweeping in transition modes, in: 2018 AIAA/AHS Adaptive Structures Conference, 2018, 0796. [62] X. Shen, X. Zhu, Z. Du, Wind turbine aerodynamics and loads control in wind shear flow, Energy 36 (3) (2011) 1424–1434. [63] D.-N. Liu, Z.-X. Hou, Z. Guo, X.-X. Yang, X.-Z. Gao, Bio-inspired energyharvesting mechanisms and patterns of dynamic soaring, Bioinspir. Biomim. 12 (1) (2017) 016014. [64] C.R. dos Santos, F.D. Marques, Lift prediction including stall, using vortex lattice method with Kirchhoff-based correction, J. Aircr. (2017). [65] J.T. Betts, Survey of numerical methods for trajectory optimization, J. Guid. Control Dyn. 21 (2) (1998) 193–207. [66] A. Maqsood, T.H. Go, Optimization of hover-to-cruise transition maneuver using variable-incidence wing, J. Aircr. 47 (3) (2010) 1060–1064. [67] A. Maqsood, T.H. Go, Optimization of transition maneuvers through aerodynamic vectoring, Aerosp. Sci. Technol. 23 (1) (2012) 363–371. [68] W. Shyy, N. Papila, R. Vaidyanathan, K. Tucker, Global design optimization for aerodynamics and rocket propulsion components, Prog. Aerosp. Sci. 37 (1) (2001) 59–118. [69] H. Stone, K. Wong, Preliminary design of a tandem-wing tail-sitter UAV using multi-disciplinary design optimization, in: AUVSI-Proceedings, 1996, pp. 163–178. [70] K.-Y. Kim, S.-J. Seo, Shape optimization of forward-curved-blade centrifugal fan with Navier–Stokes analysis, J. Fluids Eng. 126 (5) (2004) 735–742. [71] D.E. Golberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison Wesley, Reading, 1989. [72] J.-F. Bonnans, J.C. Gilbert, C. Lemaréchal, C.A. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects, Springer Science & Business Media, 2006. [73] K. Rashid, R. Ahmad, A. Maqsood, F. Mazhar, Knowledge-based aerodynamic estimation of airships, Mech. Eng. Robot. Res. 5 (4) (2016) 239–245. [74] H. Chiroma, S. Abdulkareem, T. Herawan, Evolutionary neural network model for West Texas intermediate crude oil price prediction, Appl. Energy 142 (2015) 266–273. [75] M. Dorigo, T. Stützle, Ant colony optimization: overview and recent advances, in: Handbook of Metaheuristics, Springer, 2010, pp. 227–263. [76] E. Valian, S. Mohanna, S. Tavakoli, Improved cuckoo search algorithm for feedforward neural network training, Int. J. Artif. Intell. Appl. 2 (3) (2011) 36–43. [77] A.H. Gandomi, X.-S. Yang, A.H. Alavi, Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems, Eng. Comput. 29 (1) (2013) 17–35. [78] A.H. Gandomi, X.-S. Yang, A.H. Alavi, Mixed variable structural optimization using firefly algorithm, Comput. Struct. 89 (23–24) (2011) 2325–2336. [79] X.-S. Yang, M. Karamanoglu, X. He, Multi-objective flower algorithm for optimization, Proc. Comput. Sci. 18 (2013) 861–868.

[80] A.H. Gandomi, A.H. Alavi, Krill herd: a new bio-inspired optimization algorithm, Commun. Nonlinear Sci. Numer. Simul. 17 (12) (2012) 4831–4845. [81] L. Guo, G.-G. Wang, A.H. Gandomi, A.H. Alavi, H. Duan, A new improved krill herd algorithm for global numerical optimization, Neurocomputing 138 (2014) 392–402. [82] G.-G. Wang, L. Guo, A.H. Gandomi, G.-S. Hao, H. Wang, Chaotic krill herd algorithm, Inf. Sci. 274 (2014) 17–34. [83] S. Binitha, S.S. Sathya, et al., A survey of bio inspired optimization algorithms, Int. J. Soft Comput. Eng. 2 (2) (2012) 137–151. [84] M.A. Patterson, A.V. Rao, GPOPS-II Version 1.0: A General-Purpose MATLAB Toolbox for Solving Optimal Control Problems Using the Radau Pseudospectral Method, University of Florida, Gainesville, FL, 2013, pp. 32611–36250. [85] M.A. Patterson, W.W. Hager, A.V. Rao, A ph mesh refinement method for optimal control, Optim. Control Appl. Methods 36 (4) (2015) 398–421. [86] P. Zhao, S. Mohan, R. Vasudevan, Optimal control for nonlinear hybrid systems via convex relaxations, preprint, arXiv:1702.04310, 2017. [87] F. Liu, W.W. Hager, A.V. Rao, Adaptive mesh refinement method for optimal control using decay rates of Legendre polynomial coefficients, IEEE Trans. Control Syst. Technol. (2017). [88] M. Kelly, An introduction to trajectory optimization: how to do your own direct collocation, SIAM Rev. 59 (4) (2017) 849–904. [89] J.P. Udani, K. Mall, M.J. Grant, D. Sun, Optimal flight trajectory to minimize noise during landing, in: 55th AIAA Aerospace Sciences Meeting, 2017, p. 1180. [90] G. Serrancolí, J. De Schutter, F. De Groote, Analysis of optimal control problem formulations in skeletal movement predictions, in: Converging Clinical and Engineering Research on Neurorehabilitation II, Springer, 2017, pp. 1269–1273. [91] M. Febrer Nafría, F. Mouzo, U. Lugrís Armesto, B.J. Fregly, J.M. Font Llagunes, Optimal control prediction of a dynamically consistent walking motion for a spinal cord-injured subject assisted by orthoses, in: ECCOMAS 2017: 8th ECCOMAS Thematic Conference on Multibody Dynamics: Book of Abstracts, Prague, Czech Republic, June 19–22, 2017, 2017, pp. 1–2. [92] M.A. Patterson, A.V. Rao, GPOPS-II: a MATLAB software for solving multiplephase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming, ACM Trans. Math. Softw. 41 (1) (2014) 1. [93] D. Lentink, U. Müller, E. Stamhuis, R. De Kat, W. Van Gestel, L. Veldhuis, P. Henningsson, A. Hedenström, J.J. Videler, J.L. Van Leeuwen, How swifts control their glide performance with morphing wings, Nature 446 (7139) (2007) 1082. [94] J.R. Wright, J.E. Cooper, Introduction to Aircraft Aeroelasticity and Loads, vol. 20, John Wiley & Sons, 2008. [95] B. Zhu, Z. Hou, S. Shan, X. Wang, Equilibrium positions for UAV flight by dynamic soaring, Int. J. Aerosp. Eng. 2015 (2015). [96] H. Rodrigue, S. Cho, M.-W. Han, B. Bhandari, J.-E. Shim, S.-H. Ahn, Effect of twist morphing wing segment on aerodynamic performance of UAV, J. Mech. Sci. Technol. 30 (1) (2016) 229–236. [97] G.T. Kendall, D.L. Lisoski, W.R. Morgan, J.A. Griecci, Active dihedral control system for a torsionally flexible wing, US Patent 9,120,555, Sept. 1, 2015. [98] F. Xi, A.D. Finistauri, A. Moosavian, Morphing wing for an aircraft, US Patent 9,856,012, Jan. 2, 2018.

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109

44

110

45

111

46

112

47

113

48

114

49

115

50

116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65

131

66

132