DOI: https://doi.org/10.1007/s42235-019-0105-5 http://www.springer.com/journal/42235
*Corresponding author: Anh Tuan Nguyen E-mail: anhtuannguyen2410@gmail.com
A Neural-network-based Approach to Study the Energy-optimal Hovering Wing Kinematics of a Bionic Hawkmoth Model
Anh Tuan Nguyen1*, Ngoc Doan Tran1, Thanh Trung Vu2, Thanh Dong Pham1, Quoc Tru Vu1, Jae-Hung Han3 1. Faculty of Aerospace Engineering, Le Quy Don Technical University, 236 Hoang Quoc Viet, Vietnam
2. Office of International Cooperation, Le Quy Don Technical University, 236 Hoang Quoc Viet, Vietnam
3. Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea Abstract
This paper presents the application of an artificial neural network to develop an approach to determine and study the energy-optimal wing kinematics of a hovering bionic hawkmoth model. A three-layered artificial neural network is used for the rapid prediction of the unsteady aerodynamic force acting on the wings and the required power. When this artificial network is integrated into genetic and simplex algorithms, the running time of the optimization process is reduced considerably. The validity of this new approach is confirmed in a comparison with a conventional method using an aerodynamic model based on an extended unsteady vortex-lattice method for a sinu- soidal wing kinematics problem. When studying the obtained results, it is found that actual hawkmoths do not hover under an energy- optimal condition. Instead, by tilting the stroke plane and lowering the wing positions, they can compromise and expend some energy to enhance their maneuverability and the stability of their flight.
Keywords: optimal hovering wing kinematics, artificial neural network, insect flight, genetic algorithm, unsteady vortex-lattice method, bionics
Copyright © 2019, Jilin University.
1 Introduction
The flapping flight of micro-air vehicles as well as insects in nature is generally costly in terms of energy[1,2]. Therefore, finding the flapping wing kinematics patterns that are most efficient is important and has attracted the attention of many researchers[3–7]. One of the main concerns regarding this type of optimization problem is the large computational time required, as a great number of evaluations are typically required. In the optimization studies referred to above, the quasi-steady assumption was used to reduce the computational cost while pre- dicting the aerodynamic force. According to this as- sumption, many phenomena related to unsteadiness and the three-dimensional characteristics of the flow are ignored[8,9]. Moreover, each quasi-steady aerodynamic model is specifically designed for a narrow range of flow conditions. Therefore, prediction errors could increase significantly when some parameters fall outside this range. Based on data published in the literature[5,9–11], quasi-steady aerodynamic models can produce errors of more than 20% in some cases, and such poor prediction
can lead to inaccurate optimization results, as indicated by Zheng et al.[5]. Moreover, Yan et al.[12] demonstrated that optimal solutions are highly sensitive to the simpli- fied assumptions used with quasi-steady models.
Therefore, different simplification techniques could lead to entirely different optimal results[12].
Compared to aerodynamic models based on the quasi-steady assumption, Unsteady Vortex-Lattice Me- thods (UVLMs) have greater fidelity given that they can model a flow with unsteady effects and three- dimensional characteristics[13,14]. Nevertheless, due to the relatively high computational cost, it is quite chal- lenging to integrate this type of model into optimization algorithms for insect wing kinematics. Senda et al.[15]
used an UVLM to determine the optimal wing kine- matics of a butterfly. However, they did not reveal any information about the running time of their optimization process. Moreover, in their work, a gradient-based op- timization method was employed, making it likely that the obtained solution is locally optimal and that it strongly depends on the initial choice of the wing ki- nematics.
It should be noted that Computational Fluid Dy- namics (CFD) modeling, which involves the direct solving of Navier Stokes equations, has the highest fi- delity level. However, the requirement of vast computer resources as well as the significant computational cost[5]
make this type of modelling extremely inefficient for optimization problems.
This paper presents an attempt to integrate an Ar- tificial Neural Network (ANN) into a Genetic Algorithm (GA) to ascertain the optimal hovering wing kinematics of a hawkmoth-like bionic insect model. The ANN is trained with data from the extended UVLM[13] in an effort rapidly to predict the unsteady aerodynamic force and the required power with a high level of accuracy while also reducing the computational cost significantly.
Using this approach, the effects of the unsteady and three-dimensional characteristics of the flow can be accurately included in the optimization process. The result from this study will help answer the question of whether hawkmoths fly under an energy-optimal condi- tion. Moreover, certain important hovering flight cha- racteristics of actual hawkmoths are revealed and pre- sented for the first time in this paper.
2 Insect model and methodologies
2.1 Insect model and wing kinematics
The bionic insect model used in this work is based on the morphology and the mass parameters of the hawkmoth Manduca sexta. The wings are connected to the body by three-degree-of-freedom revolute joints.
The stroke plane of the wings is inclined at the angle β shown in Fig. 1a. Similar to an actual hawkmoth, the angle between the stroke plane and the body axis is as- sumed to remain constant, and it is equal to 120˚[16]. Some basic parameters of the wings and bodies based on data from the literature[16,17] are provided in Table 1. In this table, m, R, c and S respectively denote the mass of the insect model, the wingspan, the mean chord length and the wing area.
The orientation of a wing with respect to the stroke plane is defined by the three Euler angles of the sweep angle , the elevation angle θ and the rotation angle α (Fig. 1b), whose detailed definitions are available in the literature[13]. As in prior wing kinematics optimization studies[3,5–7], the time variations of the Euler angles can
Stroke plane (a)
β
120˚
Stroke plane
α Feath
ering axis Wing chord (b)
θ
Fig. 1 (a) Insect model; (b) angles to define the wing orientation.
Table 1 Mass and morphological parameters of the hawk- moth[16,17]
Parameters Values
m (mg) 1578.70
R (mm) 48.50
c (mm) 4.33
S (mm2) 460
be represented by:
1 1 0
0
0
sin [ sin(2π 3π)]
sin 2
cos(4π ) ,
π tanh[ sin(2π )]
2 tanh
a
a a
K ft
K ft
C ft
C
(1)
where f is the flapping frequency, while a, 0, θa, θ0, αa, α0, K and C are parameters that define the shapes of the wing kinematic functions.
Table 2 shows a list of the kinematic parameters and their constraints. The lower and upper bounds are chosen such that the optimization program can cover all reasonable wing kinematics patterns of the hawkmoth Manduca sexta as observed in nature[16]. It should be noted that the maximum sweeping amplitude a is set to
906
Table 2 Kinematic parameters and constraints
Min Max
f (Hz) 15 40
β (˚) 0 50
ϕ0 (˚) −25 25
θ0 (˚) −25 25
α0 (˚) −25 25
ϕa (˚) 30 60
θa (˚) 0 30
αa (˚) 25 75
K 0 1
C 0 5
Fig. 2 Vortex ring panels on the insect wings.
60˚ to avoid any collisions between the left and right wings.
2.2 Unsteady aerodynamic model
The aerodynamic model used in this study is based on the extended UVLM developed by Nguyen et al.[13]. Similar to other conventional UVLMs, such as that de- veloped by Roccia et al.[14], the wings are represented by a system of vortex ring panels, as shown in Fig. 2. By applying a no-penetration boundary condition at the centers of these panels (colocation points), the circula- tion values of these vortex rings are determined. The Kutta condition is imposed at the trailing edges of the wings to allow all vortices along these edges to be shed into the surrounding environment to form a free wake.
The wake is modelled by vortex lines, the core sizes of which increase over time due to viscous diffusion[13,18]. Nguyen et al.[13] also derived a model to predict the force caused by leading-edge vortices on insect wings[19]
based on the suction analogy theory of Polhamus[20]. In the present model, the coefficient of leading-edge suc- tion efficiency, which can take a value from 0 to 1.0, is introduced to improve the accuracy of the result. A value of 1.0 corresponds to the full contribution of the vortex force, while a zero coefficient of the leading-edge suc- tion efficiency signifies the absence of this force. Na- bawy et al.[21] found that taking the full contribution of
the vortex force into account in the leading-edge suction analogy model may not produce a good prediction. As demonstrated previously[13], for the current model, a value of 0.5 is most suitable as the coefficient of the leading-edge suction efficiency during hawkmoth flight.
The unsteady Bernoulli equation is applied to calculate the pressure difference, which is then integrated to ob- tain the resultant aerodynamic force.
The validity of the extended UVLM has been de- termined in several previous studies[13,22–24]. In this work, to provide a greater understanding of the fidelity of the extended UVLM, the lift, drag and aerodynamic power coefficients (CL, CD and CP) of a hawkmoth-like wing are computed and compared with those published by Lee et al.[10] from an experiment, CFD analyses, and adap- tive and non-adaptive quasi-steady models (Figs. 3a–3c).
Fig. 3d shows the wing kinematic functions used in this computation. Here, time is nondimensionalized with the stroke period, while the lift, drag and power coefficients are defined as:
2 ref
2 ref
3 ref
0.5 0.5 .
0.5
L
D
P
C L
SU C D
SU C P
SU
(2)
In the above expressions, L, D and P correspondingly denote the lift, drag and aerodynamic power; ρ is the air density; S is the area of a wing; and Uref is the reference velocity. Uref is equal to the mean wing velocity and is given by Uref = 2p–pfr2. Here, p–p is the peak-to-peak amplitude of the sweep angle, and r2 is the second mo- ment of area of a wing.
According to Lee et al.[10], the conventional non-adaptive quasi-steady model is specifically de- signed based on experimental data for a narrow range of flow conditions, with some modifications added to the adaptive quasi-steady model to make it applicable to flows with significant changes in the Reynolds number and the wing geometry from the assumed condition.
As shown in Fig. 3, when compared with the CFD and experimental results, the predictions by the extended UVLM show much better agreement than those from the quasi-steady models.
Nondimensionaltime
Nondimensionaltime
Nondimensionaltime
CP
)
(a) (b)
(c)
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8
Extended UVLM CFD
Experiment
Non-adaptive quasi-steady model Adaptive quasi-steady model 0
−2 6 4 CL 2
0 6 4 CD 2
0 6 4 CP 2
8
0
−100 150 100 50 200
Angular position (˚) −50 0.2
0.0 0.4 0.6 0.8 1.0 (a)
Nondimensional time
0.2
0.0 0.4 0.6 0.8 1.0 (b)
Nondimensional time
0.2
0.0 0.4 0.6 0.8 1.0 (c)
Nondimensional time
0.2
0.0 0.4 0.6 0.8 1.0 (d)
Nondimensional time
−2
θ ϕ α
−2
Fig. 3 Lift (a), drag (b) and aerodynamic power (c) coefficients corresponding to hawkmoth wing kinematics (d) obtained from various different methods.
2.3 Artificial neural network for the rapid prediction of the aerodynamic force and required power
When using the extended UVLM, the running time is long due to the update of the wake geometry at each time step[13]. It will be shown later that to obtain the optimum condition, millions of function evaluations are required; hence, the direct use of the extended UVLM is inefficient. To tackle this problem, the implementation of an ANN which rapidly predicts the aerodynamic forces and power from the extended UVLM is proposed.
For the unsteady aerodynamics of insect hovering flight, due to the strong wake effect, the relationships between the aerodynamic force and the kinematic parameters are so complex that they cannot be accurately formulated by simple physical equations. Hence, the ANN is a good candidate to model this problem. In this study, a three- layered ANN is designed, as shown in Fig. 4, with sig- moid and linear transfer functions in the hidden and output layers, respectively. Here, b1 and b2 are the biases, and wi and wo are weights that need to be updated during the training process. The input variables to the ANN are normalized as:
Min
Max Min
= ,
(3)
z1
z2
z3
z4
zN
Hidden layer f+
b2
b1
β+
K+ C+ Φα+
θα+
αα+
Φ0+
θ0+
α0+
P+
¯ M¯+ D+
¯ L+
¯
Output layer
Input layer wi
wo
Fig. 4 The structure of the artificial neural network.
where ξ is an input variable, and the subscripts Max and Min respectively denote the upper and lower bounds given in Table 2.
The nondimensional data in the output layer are
908 defined as:
0
=
= ,
=
= L L
W D D
W M M
Wc P P
P
(4)
where L , D, M and P denote the mean lift, drag, pitching moment and required power, respectively; W is the weight of the insect model; c is the mean wing chord; and P0 = 84.2 mW is the required power when using the biological wing kinematics[25]. The required power is determined as the sum of its components P, Pθ and Pα corresponding to the sweeping, elevating and rotating motions of the wings. According to Casey[1], for hawkmoths, the effect of elastic storage can be neglected when computing the required power. Therefore, the negative power is simply dissipated and does not con- tribute to the overall result. The power components can then be calculated as:
d d
( 0),
d d
0 ( d 0),
d ( , , ),
j j j
j j
j j
P if
t t
P if j
t j
(5)
where τ denotes the torque at the body-wing joints.
The total required power can be broken down into the inertial and aerodynamic power levels corresponding to the inertial and aerodynamic torques at the joints.
These parameters can be expressed as:
d dd , d
a a
j j
i i
j j
P j
t P j
t
(6)
where the superscripts a and i correspondingly denote the aerodynamic and inertial components.
As a compromise between accuracy and the re- quired computational effort, 500 neurons are used in the hidden layer. The design and training of the ANN are done with the Neural Network MATLAB toolbox[26]
Mean squared errors
Fig. 5 Convergence of the mean squared errors during the ANN training process.
using the Levenberg-Marquardt optimization method[27]:
T 1 T
1 ( ) ,
k k
w w J J I J e (7) where, w is a vector that contains all of the weights and biases; J is the Jacobian matrix, which encompasses the first derivatives of network errors with respect to the weights and biases; e is a vector of the network errors; I is a unit matrix; and μ is a learning step size parameter, whose value decreases after each successful step.
To provide data to the training process of the ANN, 5000 random cases corresponding to 5000 sets of the kinematic parameters satisfying the constraints given in Table 2 are generated. The simulation with the extended UVLM is performed for each case to obtain the mean lift, drag, pitching moment and required power. Herein, it should be noted that based on observations by Willmott and Ellington[16] of actual hawkmoths, the lower and upper bounds of the kinematic parameters given in Table 2 are sufficient to cover all reasonable wing ki- nematics patterns. Using larger ranges of these parame- ters is unnecessary and may lead to the requirement of additional data and computer resources needed to train the ANN.
Fig. 5 shows the convergent trends of the mean square errors against the number of iterations during the training process. At iteration 105, where the best vali- dation performance occurs, the mean-squared error on the training set is 9.7 10−5, while those on the valida- tion and testing sets are 1.0 10−3 and 1.3 10−3, re- spectively. These very small values can guarantee high accuracy of the ANN output. For the current ANN, 80%
of the data is used for the training performance, while 10% is used for testing, and the remaining 10% is placed in the validation set. To avoid overfitting, the training process terminates when the error on the validation set
begins to rise.
Figs. 6a–6d show the validation of the ANN in a comparison between the predicted results and those from the extended UVLM for 100 new random input data sets.
The root-mean-squared errors of these predictions are 2.4%, 2.3%, 2.7% and 2.9% for L, D, M and P, respectively. It is clear that the ANN can provide good predictions of the mean aerodynamic forces, moment and power while the computational time is reduced sig- nificantly. While the UVLM program requires 50 min to generate the output of 100 input data sets, it takes the ANN only 0.03 s to complete the same task on the same computer.
2.4 Optimization procedure
Optimization is carried out to obtain the solution with the minimal power consumption under the given force and moment balance conditions (zero mean drag and pitching moment, and with the mean lift force equal to the weight). It is possible to convert the current con- strained optimization problem to an unconstrained problem by introducing the fitness function F, which involves the penalty term, as[28]:
10
1
(1 ) ,
Max Min
i
i i i
F P r L D M s
(8) where r and s are positive real parameters that specify the strength of the penalty for violating the constraints; ζi is the distance by which parameter i is outside the spe- cified range given in Table 2. The nondimensionalized mean lift, drag, pitching moment and required power (L, D, M and P) are predicted by the ANN. De- spite the fact that the mean values of the aerodynamic forces, pitching moment and required power obtained by the ANN are more accurate than those from the quasi- steady models, it should be noted that the quasi- steady-based optimization approach has been widely used previously and produced solutions with acceptable accuracy levels[3,5,7]. The most noticeable advantage of quasi-steady aerodynamic models over the ANN-based approach is that they allow transparency with regard to the effects of the input variables on the output perfor- mance.
The optimization procedure is conducted using a
Fig. 6 Validation of the ANN in comparison with the simulation results for the nondimensionalized lift (a), drag (b), pitching moment (c) and required power (d).
combination of a genetic[29] and the simplex algo- rithm[30]. First, the Genetic Algorithm (GA) solver ge- nerates a large number of parameter sets that are then evolved and approach the globally minimal basin. Next, using the simplex algorithm, the final best parameter set from the GA is taken as an initial point from which to search for the local optimum of the basin. The tolerance of the fitness function F is set to be 10−6 and 10−10 for the GA and the simplex algorithms, respectively. The size of the population used in the GA will be discussed later in the convergence analysis part, while the crossover frac- tion, which is defined as the fraction of the population at the next generation created by the crossover function, is set to 0.5. The top 5.0% of the population in each gen- eration with the best fitness values are considered elite individuals that are allowed to survive to the next gen- eration. In addition to crossover and elite children, mu- tation children are created by applying Gaussian random changes to the parents. The purpose of introducing these mutation children is to avoid potential local optimal solutions.
3 Results and discussion
3.1 Convergence analysis
It is important to choose sufficiently large values of the penalty parameters to avoid a violation of the con- straints while searching for the optimum. However, when using overly large penalty parameters, the GA
910
Table 3 Variations of fitness and penalty against r
r Fitness Penalty
2.0 0.85 2.56 × 10−4 10.0 0.88 1.45 × 10−10 20.0 0.94 3.11 × 10−11 50.0 1.10 2.40 × 10−5 100.0 1.28 1.03 × 10−10
Power (mW)
Fig. 7 Optimal mean power obtained with various values of population size.
Table 4 Optimal results in the validation case Parameters Min Max Optimal
(Conventional)
Optimal (ANN-based)
f (Hz) 20 35 27.2 26.8
ϕ0 (˚) −20 20 −4.2 −3.4
α0 (˚) −20 20 −0.6 0.8 ϕa (˚) 45 60 60.0 60.0 αa (˚) 45 75 62.6 60.5
P (mW) – – 82.7 79.6
solver may move quickly towards a feasible region that does not contain the global optimum. In this study, it is found that when r and s are respectively below 2.0 and 5.0, the constraints are violated. At the same time, the solution is mostly unaffected when varying s from 5.0 to 106. However, the fitness value grows with an increase of r, as shown by Table 3. Based on this convergence trend of the fitness function, values of 2.0 and 5.0 are respectively selected for r and s for the current problem.
In the above analyses, while varying the penalty parameters, the population size N used in the GA solver is held at 105. For the next step, N is allowed to vary from 103 to 5 105, and for each value of N, the program runs five times to assess the convergence of the solution.
Fig. 7 shows the optimal mean power corresponding to multiple runs of the program with various values of N. It is observed that at low values of the population size, the
program tends to reach local optima; thus, the solution does not converge. Considering the computational time and convergence, N = 105 is suitable in this study.
3.2 Validation of the ANN-based optimization ap- proach
In this section, results obtained by the ANN-based optimization approach are validated against those from a conventional method that directly computes the aero- dynamic output by the extended UVLM. It should be noted that performing the optimization with a population size of 105 for hundreds of generations as indicated above is impractical for the conventional method due to the extremely long computational time. For validation, the problem is simplified by setting the five kinematic parameters of β, θ0, θa, K and C to zero; thus, the stroke plane is horizontal, and the wing kinematic functions become sinusoidal without an elevation angle. At the same time, the lower and upper bounds of the remaining five parameters are altered as shown in Table 3. The convergence analysis is conducted again, and the popu- lation size and the number of generations used in the GA solver are found to be at least 300 and 50, respectively.
While the conventional method needs more than five days to run, the ANN-based method completes the task within one minute. The optimal results obtained by the two methods are shown in Table 4. The close agreement between these results thus validates the ANN-based approach proposed in this work.
3.3 Optimal solutions
Case I in Table 5 presents the optimization result considering all ten parameters with the upper and lower bounds given in Table 2. While comparing this result with the reference data based on the biological wing kinematics[25], it is observed that the power per unit lift P* is 21% lower. Therefore, it is relevant to state that actual hawkmoths do not fly under the optimal hovering condition. Zheng et al.[5] also came to a similar conclu- sion that actual hawkmoths consume 25% more energy than the optimal level. The most noticeable differences between case I and the reference data are in the stroke plane angle β, the mean elevation angle θ0 and the ele- vation amplitude θa. It is necessary to discuss these dif- ferences.
Table 5 Optimal results and reference data Parameters Optimal values
(Case I)
Optimal values
(Case II) Ref. [25]
f (Hz) 25.3 25.7 26.1
β (˚) 0.9 15.0 15.0
ϕ0 (˚) −4.2 −8.8 −9.2 θ0 (˚) 15.8 0 −0.5
α0 (˚) 1.0 7.9 9.6
ϕa (˚) 60.0 60.0 57.2
θa (˚) 0 7.0 7.1
αa (˚) 60.0 53.4 58.0
K 0 0 –
C 1.0 1.3 –
P* (W/N) 4.5 5.8 5.7
α
Φ
θ
0.0 0.2 0.4 0.6 0.8 1.0 Nondimensional time
200 150 100 50 0
−50
−100
(b) Centre of mass
(a)
Optimal case I Optimal case II Actual insect
Fig. 8 (a) Wing tip trajectory of the actual hawkmoth and those in case I and case II; (b) optimal wing kinematics in case II (solid lines) and the biological wing kinematics (markers).
First, regarding the stroke plane angle β, similar to the result obtained by Zheng et al.[5], in the optimal condition, a hawkmoth model has a horizontal stroke plane. However, according to Willmott and Ellington[16], for actual hawkmoths, when the flight speed increases, this plane becomes more vertical. This behavior is ex- plained by an attempt to decrease the angle between the body axis and the horizontal plane, and hence reduce the body parasite drag in forward flight. Therefore, despite
the fact that hovering with a tilted stroke plane requires more energy, the maneuverability can be enhanced through easier and more rapid transitions between hover and forward flight.
With respect to the difference in the mean elevation angle θ0, Luo et al.[31] revealed that due to the added- rotation effect, a slightly positive mean elevation angle θ0 as shown in case I is beneficial in terms of lift gener- ation, as well as the power requirement. Nevertheless, actual hawkmoths tend to reduce this angle to zero by lowering their wings. Ristroph et al.[32] indicated that when the wings are located well above the center of mass, an uncontrolled insect model may experience significant diverging oscillations in the body pitch.
Moreover, instability of the pitch motion is the most noticeable when studying the passive flight dynamics of insects[22,23]. Hence, by lowering their wings, hawk- moths may greatly mitigate this unstable mode and therefore improve the stability of their flight.
Finally, according to Lua et al.[33], the oscillation of the elevation angle θ, which is detrimental to the lift generation mechanism[31,33], appears to be a byproduct of the requirement to execute the flapping motion.
Therefore, this oscillation should not occur in the op- timal solution, and the optimal θa is zero, as shown in Table 5.
Holding β, θ0 and θa at 15.0˚, 0˚ and 7.0˚, respec- tively, according to the empirical data[16], a new optimal result set is obtained and presented as Case II in Table 5.
Table 5 and Figs. 8a and 8b indicate that the kinematic data and the power per unit lift P* in this case are close to those of the actual hawkmoth. Additionally, the sweep- ing amplitude a is always at the maximum allowed value of 60˚. To explain this trend, we consider the in- ertial power, which is proportional to fmax2 or f3 2a[34]. Similarly, lift L is proportional to f2 2a. Hence, to mi- nimize the power while holding the lift equal to the weight of the insect model, a is maximized. This trend is similar to a finding by Nabawy and Crowther[35] that the larger the sweep amplitude is, the lower the induced power factor becomes. The result is more efficient flight.
It is observed that the optimal sweep angle varies with a sinusoidal function, whereas the rotation angle α has a round trapezoidal form. These function shapes are similar to those in studies conducted by Berman and
912
Wang[3] and Ke et al.[7] for optimal insect hovering wing kinematics. However, Taha et al.[4] and Nabawy and Crowther[6] found that for the highest flight efficiency, insects should combine the triangular and rectangular profiles for the sweep and rotation angles, respectively.
There can be several causes of this contradiction. In addition to the unsteady effect that is not included in the aerodynamic models of Taha et al.[4] and Nabawy and Crowther[6], differences in how the effects of the rota- tional circulation mechanism are considered may be among the main reasons. Because these authors consi- dered the potential flow theory to model the contribution of the wing rotation, there is no resulting drag force and hence no aerodynamic power required for this motion.
Additionally, taking the contribution of the inertial power into consideration causes the conditions of the present optimization problem to differ from those in the aforementioned studies. Therefore, the finding of dif- ferences between the present optimal wing kinematic functions and those in the literature is understandable.
Compared with the optimal result obtained by Berman and Wang[3] for a hawkmoth-like model based on a quasi-steady model, it is found that the frequency in the present study is closer to the that of actual hawkmoth (25.7 Hz vs 26.1 Hz). The optimal frequency in the quasi-steady-based optimization study of Berman and Wang[3] is 24.0 Hz for a female hawkmoth model, which is much lower than the measurement value of 26.3 Hz for an actual hawkmoth[16]. The source of this disa- greement is likely the overestimation of the lift when using the quasi-steady assumption, as shown in Fig. 3.
Berman and Wang also indicated that the lift force ob- tained by their quasi-steady model exceeds the CFD result by 15%. Due to this overestimation, the optimal frequency is reduced to ensure the equilibrium condition.
Another disagreement, as observed in the study of Berman and Wang, is the rotation amplitude. While the actual hawkmoth has an amplitude of just above 50˚, the quasi-steady-based optimal result is close to 80˚. This large discrepancy is due to the use of the simple physical model while dealing with the rotational mechanism.
Most quasi-steady aerodynamic models employ the theoretical rotational circulation derived based on the Kutta condition for small angles of attack and the Kut- ta-Jukowski equation to calculate the rotational force[9].
0 1 2 3 4
−30
−20
−10 0 10
Number of wingbeat stroke cycles Case I Case II
Fig. 9 Body pitch deflection in case I and case II due to horizontal gust.
This treatment tends to underestimate the force during the rotational phase, as shown in Fig. 3; therefore, a larger rotation amplitude may be required to compensate for this underestimation.
As mentioned above, compared to case I, the actual hawkmoth tilts its stroke plane and move the wings closer to the center of mass; these findings are illustrated in Fig. 8a. To study the difference in the pitch stability between case I and case II, we apply a linear dynamics model developed and validated in a previous study by the authors[22] to obtain the responses of the bionic hawkmoth model to a horizontal gust with a constant magnitude of 0.27 m·s−1 (10% of the mean wing veloc- ity). It is observed that case II with the lower wing po- sitions produces less pitch angle deflection (Fig. 9). In other words, the dynamic system corresponding to this case is less unstable. Thus, it is important to note that hawkmoths sacrifice energy to improve their body pitch stability.
Figs. 10a and 10b respectively show the lift and drag forces in cases I and II together with those of the actual hawkmoth in one wingbeat stroke. The aerody- namic forces in case II and in the case of the actual insect are similar in terms of the general trend. Some differ- ences are attributed to the limited number of kinematic parameters used in this optimization study. Moreover, it is noteworthy that the wings of the actual hawkmoth are flexible; therefore, some passive deformations of the wings may cause larger variations in the kinematic functions and in the aerodynamic force time histories, as
Lift (N)Drag (N)
Nondimensional time (a)
(b)
0 0.2 0.4 0.6 0.8 1.0
−0.01 0 0.01 0.02 0.03 0.04 0.05
0 0.2 0.4 0.6 0.8 1.0
−0.03
−0.02
−0.01 0 0.01 0.02 0.03
Optimal case I Optimal case II Actual insect
Fig. 10 Optimal lift (a) and drag (b) forces and those of the actual insect.
W
β T¯s
N¯s
L¯ Stroke plane
Fig. 11 Forces acting on the insect model.
shown in Figs. 10a and 10b. Comparing the forces in these cases, we find that with an inclined stroke plane, the insect model generates more lift in the first half of the wingbeat stroke (downstroke phase). As shown in Fig. 11, the mean lift force L, which opposes the weight W, can be decomposed into the mean tangential force Ts and the mean normal force Nswith respect to the stroke
(a)
(b)
Fig. 12 Unsteady wakes in the optimal case II (a) and in the case of the actual hawkmoth (b).
plane. With the horizontal stroke plane, the mean tan- gential force Ts is zero, and the variation of the rotation angle α should be symmetric; consequently, α0 is close to zero, as shown in case I in Table 5. However, when the stroke plane has an angle toward the horizontal direction,
Ts is nonzero and the hawkmoth model must generate more drag in the downstroke. Accordingly, α0 must in- crease, as indicated in case II in Table 5, to produce a larger angle of attack during the downstroke and thus more lift in the first half of the wingbeat cycle. While analyzing the sensitivity of their optimal solutions, Berman and Wang[3] observed that the symmetric time variation of the rotation angle requires the least power.
Hence, with a larger value of α0 when flying with an inclined stroke plane (case II), more energy is consumed.
The unsteady wakes obtained by the extended UVLM in case II and in the case of the actual hawkmoth are shown in Figs. 12a and 12b, respectively. These wakes have similar forms; however, the wake in case II is slightly less tangled, which may be explained by the use of the simplified wing kinematic functions.
914
4 Conclusion
The present paper introduces an ANN-based ap- proach to obtain the optimal hovering wing kinematics of a bionic hawkmoth model. The ANN is trained with data from the extended unsteady vortex-lattice method and then integrated into genetic and simplex algorithms for optimization. The use of this approach reduces the computational effort considerably and hence for the first time enables us to overcome the challenge of considering the aerodynamic unsteady effects while optimizing the insect wing kinematics. The validity of the ANN-based approach is confirmed through a comparison with the conventional optimization method. The study also re- veals that actual hawkmoths do not fly under the optimal hovering condition; instead, they sacrifice energy to enhance their maneuverability and stability by tilting the stroke plane and lowering the positions of their wings.
Moreover, in the optimal condition, oscillation of the elevation angle, which is a byproduct of the requirement to execute the flapping motion, does not occur.
Acknowledgment
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2018.05.
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