A computational tool to improve flapping efficiency of robotic insectsYufeng Chen, Alexis Lussier Desbiens, and Robert J Wood

Abstract— We implement a 2D computational model to inves-tigate the unsteady aerodynamic effects not captured by classi-cal quasi-steady models. We compare numerical simulation re-sults, experimental measurements and quasi-steady predictionsto demonstrate the strength of the numerical tool in identifyingunsteady fluid mechanisms and improving propulsive efficiencyof flapping wing robots. In particular, this study quantifies theeffect of the relative phase between wing degrees of freedomδ on lift and drag production. The computational model alsoidentifies unsteady effects such as wake capture and downwashthat are not accounted for in classical quasi-steady models.To examine the accuracy of our computational model, wefabricate millimeter-scale wings through the SCM fabricationprocesses and measure flapping kinematics and dynamics.The experiments show 2D computational model is 44% moreaccurate than the quasi-steady model and can be further used toimprove wing morphology for better aerodynamic performance.

I. INTRODUCTION

In recent years a number of flapping-wing micro-air-vehicles have achieved stable hovering flight [1], [2], [3].Compared to traditional fixed wing or helicopter flight, flap-ping wing flight observed in nature relies on unsteady fluiddynamics principles to achieve better maneuverability andsmaller vehicle size [4]. Such advantages make flapping wingair vehicles excellent candidates for surveillance and remotesensing in hazardous locations. Meanwhile, the unsteadynature of flapping flight poses modeling and control chal-lenges to improve stability, maneuverability, and propulsiveefficiency.

The current Harvard RoboBee design, shown in Figure1, uses two bimorph piezoelectric actuators to indepen-dently control wing stroke motion, and the hinge motionis mediated by passively rotating Kapton hinges. While thisdesign reduces system complexity, power consumption andvehicle mass, it poses challenges to developing a dynamicalmodel that predicts both wing hinge kinematics and thrustgeneration.

A number of quasi-steady models have been developedfrom steady state classical aerodynamics [5] to describeflapping flight. In 1963, Von Karman et al. first proposeda formula for lift and drag coefficients based on completeseparated flow computation. In 2002, Dickinson et al. [6]observed unsteady phenomena such as rotational circulation

These authors are with the School of Engineering and Applied Sciences,Harvard University, Cambridge, MA 02138, USA, and the Wyss Institute forBiologically Inspired Engineering, Harvard University, Boston, MA, 02115,USA (email: yufengchen,desbiens,rjwood@seas.harvard.edu)

1 cm

Fig. 1. The current Harvard RoboBee is an 80mg flapping wing microrobotthat can lift an extra 50mg payload. The wing design and control algorithmare based on quasi-steady models that predict time averaged forces andtorques.

and delayed stall using a robotic wing. He proposed a quasi-steady model based on his experimental results and intro-duced additional modeling terms that account for added massand rotational damping effects. Lussier Desbiens et al. [7]adopted the quasi-steady model for a passive flapping systemand demonstrated that the quasi-steady model yields accuratekinematic and thrust predictions in the time averaged sense.However, quasi-steady models cannot yield accurate predic-tions of time varying lift and drag, and this error in turnaffects prediction of aerodynamic torques that govern hingekinematics. This modeling insufficiency restricts all controlalgorithms to rely on time-averaged predictions, which ad-versely affects maneuverability and flapping efficiency.

A number of 2D [8] or 3D [9] numerical models have beendeveloped to bridge the discrepancy between experimentsand quasi-steady models. Lentink et al. [10] show that 3Dmechanisms such as spanwise flow stabilize the leadingedge vortex and delay detachment. In hovering flight, vortexshedding only happens at stroke reversal and as a result 2Dand 3D computational models yield very similar predictions.In this paper, we implement a numerical two dimensionalNavier Stokes equation solver to study effects of parametersthat are not treated by quasi-steady models. In particular, weinvestigate the effect of relative phase between stroke andhinge rotation angles δ (Figure 2) on lift and drag production.As shown in Dickinson’s robotic wing experiments [6], mod-erate differences in δ accounts for more than 30% differencein measured lift. Current wing and hinge designs rely onquasi-steady models to optimize kinematic parameters suchas stroke and hinge amplitude, but ignore the influence of

2014 IEEE International Conference on Robotics & Automation (ICRA)Hong Kong Convention and Exhibition CenterMay 31 - June 7, 2014. Hong Kong, China

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Advanced Pitch Delayed Pitch

Pitch Rotation

Stroke Acceleration

Stroke Deceleration

Pitch Rotation

Zero Pitch

Fig. 2. The top graph shows measured stroke and hinge angles and theleast squares fit to a pure sinusoid. The recorded motion is nearly purelysinusoidal, and δ indicates the relative phase between stroke and hingemotion. If δ = 0◦, then the hinge angle is 0◦ at maximum stroke, asshown in the bottom middle figure. δ < 0◦ means wing rotation leadsstroke motion and δ > 0◦ means wing rotation lags behind stroke motion.Intervals of large stroke angle (φ) represent pitch rotation, and intervalsof slowly changing hinge angle (ψ) represents wing translation. The wingtranslational phase is further divided into stroke acceleration and strokedeceleration phase based on the curvature of the stroke function. The motiontracking method is described in Section IIIB.

δ. We study this parameter’s influence on lift and dragproduction by running simulations and experiments, therebyextracting useful physical principles that will improve futurewing and hinge designs. The simulated results using our CFDsolver are compared with experimental results to validate ourfindings. This computational model is shown to be 44% moreaccurate than traditional quasi-steady models.

In addition, our simulation shows several interesting un-steady phenomena such as vortex shedding, downwash andwake capture. These physical phenomena affect lift and dragproduction but are not accounted for in quasi-steady models.By choosing appropriate morphological parameters, we canincrease lift production by making effective use of the wakecapture process. Hence, unlike quasi-steady models that onlyyield force predictions, our numerical model allows us toimprove wing morphology and motion by making effectiveuse of unsteady phenomena. Owing to this increased ac-curacy, this numerical tool is more powerful for improvingflapping efficiency than its quasi-steady counterparts. In therest of this paper we explain the implementation of thenumerical model, describe the experiment procedure, andcompare simulation with experimental results.

II. COMPUTATIONAL METHOD

A. Flapping Kinematics

As shown in Figure 3, the kinematics of a flapping winghas 2 degrees of freedom—stroke and hinge rotations. Theexperimental set up allows us to control the frequency andamplitude of stroke motion, while the hinge rotation ispassively controlled by aerodynamic and inertial torquesand hinge compliance. As shown in Figure 2, experimentalmeasurement shows the hinge motion is very close to be a

downstroke

upstroke

Stroke angle

Hinge angle 2L

r

Fig. 3. The RoboBee wing has 2 degrees of freedom. The rotation (φ)around the vertical body axis is actively controlled by piezoelectric actuators,and the rotation (ψ) around wing hinge line is passive. In blade elementmethod, the motion of a thin rectangular segment along the wing chord(shown in blue) is projected onto a 2D plane. The leading edge of thewing segment is marked green. The angular stroke motion is transformedinto planar oscillatory motion, in which the amplitude L is given byL = φmaxr, and r is half the wing radius. The black arrows indicatethe instantaneous direction and relative amplitude of aerodynamic forces.

pure sinusoid. Mathematically, the stroke and hinge motionare given by

φ = φmax sin(wt)ψ = ψmax cos(wt+ δ),

(1)

where φmax is the stroke amplitude, ψmax is the hingeamplitude and δ is the relative phase. In quasi-steady bladeelement models and 2D computational fluid dynamics mod-els, the angular stroke motion is approximated by the trans-lational motion of a thin blade element located a distance rfrom the wing root. As shown in Figure 3, the amplitude ofthe wing chord translational motion is given by L = rφmax,where r is chosen to be the wing midspan.

Previous experiments have shown that mean lift increaseswhen δ < 0◦ and decreases when δ > 0◦ [6], [7]. As shownin Figure 2, δ < 0◦ corresponds to advanced passive rotationand δ > 0◦ corresponds to delayed passive rotation. Currentquasi-steady models cannot accurately predict the effect ofvarying the kinematic parameter δ, hence we aim to inves-tigate the influence of this parameter using computationaltools.

B. Numerical solver implementation

Our computational model assumes a 2D thin flat plate ofdimension 20µm × 3mm flapping in air with kinematicsdescribed in the previous section. The two dimensional in-compressible Navier Stokes equation and the correspondingboundary conditions that govern the flapping motion are :

ρ∂u∂t + ρ(u · ∇)u = −∇p+ µ∇2u∇ · u = 0

u|wing = (u, v)wingp|∞ = 0,

where u = (u, v) is the fluid velocity field and p is thepressure field that enforces the incompressibility condition.The fluid speed along the wing surface is equal to the wingvelocity, and the pressure at far field is set to be 0. In our

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computations, the range of Reynolds number is betweem 300to 600.

We implement a numerical solver using the nodal dis-continuous Galerkin finite element method, which allowsmore geometric flexibility than the finite difference methodand requires coarser mesh resolution than the continuousGalerkin method. The solver is implemented on a movingCartesian coordinate system, and the computational Delau-nay triangulation mesh is generated by the open sourcepackage distmesh [13]. The circular mesh used for simulationcontains 2242 elements, and its radius is chosen to be sixtimes the wing chord length to avoid unintended boundaryeffects. The solution inside each mesh element is interpolatedusing 5th order Lagrange polynomials.

The structure of this solver is based on the methoddeveloped in [11]. The temporal scheme is solved using thesecond order backward Adams Bashforth method, and thespatial scheme is separated into three steps that individuallytreat nonlinear advection, pressure field contribution, andviscous correction. In addition, the flapping motion requiresthe computational mesh to move with respect to the inertialreference frame, hence a change of coordinate system isneeded. This method is not identical to solving the NavierStokes equation in a non-inertial reference frame by addingfictitious forces; it is more general because it also allowsgeometric deformation of the mesh. The transformationbetween physical coordinates that are fixed in space andcomputational coordinates that move with the wing is definedas:

u(x, y, t) = u(ζ, η, τ)v(x, y, t) = v(ζ, η, τ)p(x, y, t) = p(ζ, η, τ )

(2)

We denote the physical coordinates by x and y and thecomputational coordinates by ζ and η. The temporal vari-ables t and τ are identical, however we use different symbolsto avoid confusion between ∂

∂t and ∂∂τ . The operators in

the inertial reference frame are replaced by operators in themoving frame:

∂∂t = ∂ζ

∂t∂∂ζ + ∂η

∂t∂∂η + ∂

∂τ∂∂x = ∂ζ

∂x∂∂ζ + ∂η

∂x∂∂η

∂∂y = ∂ζ

∂y∂∂ζ + ∂η

∂y∂∂η

(3)

In component form, the Navier Stokes equation is trans-

formed to :

∂u∂τ = −∂u∂ζ ζt −

∂u∂η ηt − u

(ζx

∂u∂ζ + ηx

∂u∂η

)−v

(ζy∂u∂ζ + ηy

∂u∂η

)− 1

ρ

(ζx

∂p∂ζ + ηx

∂p∂η

)+ν

(ζ2x∂2

∂ζ2 + η2x∂2

∂η2 + ζ2y∂2

∂ζ2 + η2y∂2

∂η2

)u

+2ν(ηxζx + ηyζy) ∂∂ζ∂∂ηu

∂v∂τ = −∂v∂ζ ζt −

∂v∂ηηt − u

(ζx

∂v∂ζ + ηx

∂v∂η

)−v

(ζy∂v∂ζ + ηy

∂v∂η

)− 1

ρ

(ζy∂p∂ζ + ηy

∂p∂η

)+ν

(ζ2x∂2

∂ζ2 + η2x∂2

∂η2 + ζ2y∂2

∂ζ2 + η2y∂2

∂η2

)v

+2ν(ηxζx + ηyζy) ∂∂ζ∂∂ηv

0 = ζx∂u∂ζ + ηx

∂u∂η + ζy

∂v∂ζ + ηy

∂v∂η

(4)

where ζt, ηt are the speed of the computational coordinateswith respect to inertial reference coodinates x and y, andζx, ηx, ζy and ηy are components of the transformationJacobian between physical and computational coordinates.The parameters ρ and ν represent fluid density and kinematicviscosity. The boundary conditions of u, v and p remainunchanged. Given the fluid velocity field and pressure fieldwe can compute the force per unit length and torque per unitlength on the wing segment by integrating the stress tensoralong the wing surface as: F =

´wing

n · ¯σdl, and T =´wing

r × n · ¯σdl, where n is the local surface normal. Wecan expand the stress tensor and arrive at equations for liftand drag forces as follows:

FD = −´wing

(−pnx + 2νρ∂u∂xnx + νρ ∂v∂xny + νρ∂u∂yny

)dl

FL = −´wing

(−pny + νρ∂u∂ynx + νρ ∂v∂xny + 2νρ∂v∂yny

)dl.

(5)Finally, we can relate the computational model to the

quasi-steady model by computing lift and drag coefficients:

CL = FL12ρu

2rmsc

CD = FD12ρu

2rmsc

,(6)

where urms is the root mean square of wing velocity and cis the wing chord length.

III. EXPERIMENT SETUP

To compare the numerical model with the actual forcesgenerated by the RoboBee, we utilize an existing set up tomeasure the kinematics and dynamics of a flapping wing[7]. As shown in Figure 4, the wing is attached to a custommade carbon fiber wing driver that is mounted on a dualaxis force sensor. The wing stroke motion is controlled bya bimorph piezoelectric actuator, and the hinge motion ispassively mediated by a Kapton hinge. The optical sensorrecords the motion of the piezoelectric actuator, and thecamera records the top view of the flapping motion. Thefollowing sections describe the details of force measurement,motion measurement, and wing fabrication processes.

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high speed video camera

capacitive sensor (lift)

capacitive sensor (drag)

invar target plate

piezoelectric bending actuator

double cantilever sensorcarbon fiber wing driver

carbon fiber wing driver transmission

piezo displacement sensor

carbon fiber wing

anterior view

carbon fiber wing

perspective view

optical sensor

capacitive sensor

wing driver

Fig. 4. This illustration shows the experimental setup. The carbon fiberwing driver consists of a piezoelectric actuator (black), transmission andstructural support. An external electric signal controls the actuator to drivethe stroke motion of the millimeter-scale wing (red). The aerodynamicand inertial forces generated by a flapping wing are transformed intodisplacements of the Invar sensor. Two capacitive sensors measure thedisplacement and the data is post-processed to obtain lift and drag. A highspeed camera records the stroke motion.

A. Force measurement

The custom sensor consists of four parallel dual cantilevermodules arranged in a series-parallel configuration. Thestructure converts a load into displacements in the verticaland horizontal directions, and the displacements in bothdirections are measured by two PISeca D-510.021 capacitivesensors. We calibrate the sensors by hanging weights, andthe sensitivity was found to be -84.6 and 85.5 V/mN forthe lift and drag axes respectively. In our experiment, thedriving frequency is chosen to be 120Hz so that lift forcehas a fundamental frequency of 240Hz and drag force has afundamental frequency of 120Hz. Since the sensors measureaerodynamic and the inertial forces from the wing and wingdriver, we only report time averaged drag measurements. Onthe other hand, we can accurately measure lift by filteringout 10Hz to 200Hz and > 500Hz harmonics to eliminateactuator inertial contributions and system resonance.Thisband pass filter may eliminate higher order harmonics ofthe actual lift signal, hence for comparison purposes wealso apply the same filter to the numerically computed lift.

The wing used in our experiment weigh 0.52mg , and themagnitude of wing inertial contribution often accounts for15%−20% of the aerodynamic contribution. Using measuredflapping kinematics and the estimated mass properties fromSolidWorks, we can substract out the effect of wing inertialcontribution. In the lift axis, the formula is given by

Faero = maz −mg − Fsensor, (7)

where az id the z-component of wing inertial acceleration.We can compute az as

az = rcom,z(cos(ψ)ψ2 + sin(ψ)ψ), (8)

where rcom,z is the wing center of mass position in the z-direction. In our mass model we neglect the center of massoffset due to wing thickness.

B. Wing kinematics measurement

The wing stroke and hinge motion are recorded at 10kHzusing a Phantom V7.3 high speed video camera with an AFMICRO Nikon 200mm f/4 lens. In this experiment, we treatthe wing as a flat plate and use the top view to extract hingeand stroke motions. The details of the extraction algorithmare described in [7].

C. Wing and wing hinge design

The wing used in the experiment is made from a carbonfiber frame and polyester membrane with 3mm mean chordand 54mm2 total area. The wing hinge is made of a com-pliant 1.25mm × 140µm × 7µm kapton layer sandwichedbetween two carbon fiber layers. Details of the design andmanufacturing methodology used for the wing, transmission,and actuators are described in [14].

IV. DISCUSSION

To examine the validity of our numerical model, wecompare simulation results with the measurements. A wingis flapped with specified driving voltage and frequency pairsand we measure the corresponding kinematics and forces.The measured wing kinematics are used as the inputs tothe numerical simulation, and lift and drag coefficients arecomputed to compare with experimental results. The numer-ical simulator is also used to explore parameter spaces thatare not covered by passive rotation experiments to furtherstudy the influence of relative phase parameter δ and identifyphenomena not accounted for in quasi-steady models.

A. Comparison between experiment and computation

We run the flapping experiment with a 120Hz drivingfrequency and sweep through different voltage amplitudes tofind a case for which the relative phase δ between stroke andhinge angle is 0◦. The kinematics and forces are measuredusing the method described in the previous section. At 190V ,δ is measured to be -0.19◦ and the flapping stroke and

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hinge amplitudes are measured to be 34◦ and 43◦ degreesrespectively. The corresponding Reynolds number is

Re =umaxc

ν= 570 (9)

We then use Φmax = 34◦, Ψmax = 43◦, and δ = 0◦ asthe input parameters and solve the 2D flow problem for thechord segment at midspan of the wing. Finally, we computethe instantaneous lift and drag coefficients and comparethat with classical quasi-steady model and experimentalmeasurements.

As shown in the top graph of Figure 5, the numericalsolution (blue) shows lift peaks in the stroke decelerationphase and this agrees well with the measurement (red). Onthe other hand, the green curve (quasi-steady model) is sym-metric with respect to its local maxima so the quasi-steadymodel does not distinguish between the stroke accelerationphase and the stroke deceleration phase. While there arequasi-steady models based on 2D inviscid flow that addressadded mass and rotational circulation effects, they usuallyinvolve extra fitting parameters and are not robust for largeoperating range. Hence, the quasi-steady model we comparewith only contains the translational term. This asymmetrycan be understood by studying the flow structures aroundthe wing and can be utilized to enhance lift production.Figure 6 shows the vorticity field around a flapping wingsegment in the fourth flapping period. Large vorticity on thewing leading edge corresponds to high lift. In the strokeacceleration phase (T=3.08 to T=3.25 and T=3.58 to T=3.75),vorticity on the upper wing surface is small. In the strokedeceleration phase (T=3.25 to T=3.42 and T=3.75 to T=3.91)we observe a large vortex on the upper wing surface thatleads to large lift force. We also observe that the leading edgevortex grows only when the angle of attack is positive. Onthe other hand, the vortex shedding process is not sensitiveto angle of attack.

This observation suggests that by varying the relativephase δ between stroke and hinge angles we can increaseor reduce lift. δ < 0◦ corresponds to advanced wing pitch sothat the leading edge vortex starts to grow immediately afterstroke reversal at a positive angle of attack. These kinematicsfavor vortex development and augment lift production. Incontrast, δ > 0◦ corresponds to delayed wing pitch andinhibits vortex generation. Figure 7 shows the negativecorrelation between time averaged lift coefficient CL andtime averaged drag coefficient CD as functions of δ. In theclassical quasi-steady model, CL and CD only depend onangle of attack α. However, using the numerical solver wehave shown that lift and drag cofficients are also strongfunctions of δ. Compared to δ = 0◦, the simulation of δ =−30◦ shows 30% increase of CL and 44% increase of CD(Figure 7B). On the other hand, at δ = 30◦ we observe 47%decrease of CL and 9.7% decrease of CD (Figure 7B). Thiscomputational result agrees with the qualitative experimentresult presented in [12]. Not only does the numerical modeldescribe the influence of the kinematic parameter δ, it alsoshows other unsteady effects that can be utilized to improve

2 2.5 3 3.5 4 4.5 5 5.5 6−1

0

1

2

Cl

Time

2 2.5 3 3.5 4 4.5 5 5.5 6−1

0

1

2

Cd

Time

Fig. 5. Lift and drag coefficients for 120Hz with δ = 0◦, φmax = 35◦,and ψmax = 43◦. The x-axis is the number of flapping periods. To avoidtransient effects, this graph shows the lift and drag coefficient from the 3rdto the 6th flapping periods. The blue curves represent the numerical solutionby solving the 2D Navier Stokes equation. The green curve shows quasi-steady estimates (based on Dickinson’s formula [6]) and the red curve showssensor measurements. To eliminate wing driver inertial contributions andsensor resonance, the 10 to 200Hz and > 500Hz harmonics are filteredout. The same filter is applied to the numerically computed forces. The winginertial contribution is also subtracted out from the sensor measurement.We do not show the time varying drag measurement because the motion ofpiezo-actuator is in the direction of the drag axis.

flapping kinematics and wing morphology.

B. Identification of unsteady effects

Downwash effect: Our simulation shows that an impul-sively started wing generates more lift than that of a wingalready flapped for several periods. Since the kinematicsin both cases are identical, the quasi-steady model predictsidentical lift and drag. However, as shown in Figure 8, thepressure on the wing leading edge at T = 0.25 is morenegative than it is at T = 2.25, meaning that the instanta-neous lift generated is larger at T = 0.25. This phenomenoncan be understood by comparing the y-component of the ve-locity field. A flapping wing continuously generates lift andtransfers downward momentum to surrounding fluid. Thisdownwash tends to reduce the translational lift generated bythe wing. In our simulation, the lift of the first period is 17%higher than the time-averaged lift.

Generation and shedding of vortices: A leading edgevortex develops in the stroke acceleration phase and shedsat the end of stroke deceleration phase. As discussed in theprevious section, this physical mechanism depends on flowfield history and cannot be modeled by classical quasi-steadymodels. Figure 9 shows the vorticity field and the corre-sponding pressure field to illustrate vortex generation andvortex shedding processes. We can increase or reduce meanlift by varying the phase lag δ between stroke translation andpitch rotation.

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T=3.00 T=3.08 T=3.25T=3.17

T=3.33 T=3.42 T=3.50 T=3.58

T=3.83T=3.75 T=3.91T=3.67

Fig. 6. Vorticity plot around a flapping wing in the fourth period. Red color represents positive vorticity (points out of the page) and blue color representsnegative vorticity (points into the page). The vorticity color bar has units of 1/s. Since the flow is incompressible, the information shown by the vorticityfield is equivalent to that of the complete velocity field. Regions of high vorticity (absolute value) correspond to regions of low pressure. A large vortexgrows on the leading edge of a translating wing and is shed during stroke reversal. A lift peak occurs near T=3.42 and T=3.91 during the stroke decelerationphase.

A B

Fig. 7. CLand CD as functions of δ. All simulations are ran with identicalmesh, stroke and hinge ampltiude, while the phase parameter δ is variedfrom −40◦to 30◦. The graph on the left (A) shows time averaged liftand drag coefficients for an impulsively started wing for a half flappingperiod. The graph on the right (B) shows the same simulations for 4 flappingperiods. In the first half stroke there is no wake capture and downwasheffects so Figure 7A quantifies the effect of δ on translational lift and dragalone. After the first half flapping period we observe interaction betweenwing and shed vortices. Figure 7B shows variations of CL and CD due toδ’s effect on both translational and rotational motion.

Wake capture: Wake capture refers to the interactionbetween a wing and its previously shed vortex in the strokeacceleration phase. As described in [4], wake capture canoften lead to a secondary lift peak. In our simulation, wakecapture is beneficial to lift generation in the first periodbut becomes detrimental to lift generation for subsequentflapping periods. As shown in Figure 5, the computed liftcoefficient is negative in the stroke acceleration phase. We

Vy Vy

Pressure PressureT=0.25

T=0.25 T=2.25

T=2.25

Fig. 8. Downwash and its adverse effect on lift production. The figures inthe first row show the y-component of the velocity field. The velocity fieldcolor bar has units of m/s. More severe downward flow (blue) is observedat T=2.25 than at T=0.25. Figures in the second row show that the pressurefield on the wing upper surface is smaller at T=0.25 than at T=2.25, whichcorrespond to higher lift at T=0.25. The pressure field color bar has unitsof N/m2.

can understand this phenomenon by studying pressure andvorticity graphs shown in Figure 10. At T = 0.5 andT = 1.5 we observe similar shed vortices. In the firstperiod, the shed vortex moves over the leading edge andconvects to the opposite side of the wing at T = 0.7. Thecorresponding pressure graphs show that the region of lowpressure convects to the wing upper surface, thus creatingmore lift. On the other hand, the vortex shed at T = 1.5moves along the lower wing surface and convects toward thetrailing edge. As a result, a low pressure region along lowersurface corresponds to lower lift. This phenomenon can be

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Pressure Pressure

Vorticity VorticityT=0.25 T=0.50

T=0.25 T=0.50

Fig. 9. Vortex generation and shedding. The figures in the first row showthe pressure field around a translating (T=0.25) or rotating (T=0.50) wing.The figures in the second row show the corresponding vorticity field. Duringwing translation (T=0.25), a leading edge vortex grows and as a result alow pressure region on the wing upper surface leads to high lift. Duringwing rotation, the vortex detaches from the wing surface (T=0.50) and liftplummets. The pressure field has units of N/m2 and the vorticity field hasunits of 1/s.

explained by the downwash effect, in which the downwardmoving fluid affects the vortex convection direction.

Through simulations, we find that wake capture propertiescan be enhanced by increasing stroke amplitude or shrinkingwing chord. Figure 11A compares vorticity plots of flappingmotions with different stroke amplitudes. The shed vortexconvects along lower wing surface for L = 4mm, and itconvects to the upper wing surface for L = 6mm. Thecorresponding lift coefficients for a half flapping period(T = 2 to 2.5) is shown in Figure 11 B. While the primarylift peaks in both cases are similar, we observed differentwake capture effects. During the stroke acceleration phase,the lift coefficient for L = 4mm is negative while the liftcoefficient for L = 6mm is positive. As discussed in theprevious section, this difference depends on whether the shedvortex convects along the lower wing surface or rolls over tothe upper surface. We find that favorable wake capture leadsto 32% increase of mean lift coefficient. This simulationresult implies that future wing design must have appropriatestroke amplitude to chord length ratio to achieve favorablewake capture effects.

V. CONCLUSION AND FUTURE WORK

This paper presents a computational tool designed specif-ically to model the aerodynamic performance of RoboBeeflapping flight. Unlike classical quasi-steady models thatcalculate aerodynamic forces based only on wing geometryand kinematics, this numerical model identifies unsteadyfluid mechanisms that are important to improving propulsiveefficiency. More specifically, we quantify the effect of thephase parameter δ on lift and drag production through simu-lations. While holding other kinematic parameters constant,the mean lift coefficient CL increases by 30% and mean dragcoefficient increases by 44% if δ is reduced to δ = −30◦. Onthe other hand, if δ is increased to δ = 30◦ then we observe47% decrease in mean lift coefficient and 9.7% decrease in

Vorticity

Pressure

T=0.5 T=0.7 T=1.5 T=1.7

Fig. 10. Wake capture effect. Figures in the first row show vorticity plotswith units of 1/s and figures in the second row show pressure field withunits of N/m2. The four figures on the left show a favorable wake captureeffect in the first flapping period in which the vortex rolls over the leadingedge and its corresponding low pressure acts to increase lift. The four figureson right show an adverse wake capture effect in the second period in whichthe vortex rolls along the lower wing surface and its corresponding lowpressure region reduces lift.

T=2.00 T=2.18

T=2.00 T=2.18

A B secondary lift peak primary lift peak

Fig. 11. Wake capture effects of different flapping amplitude to chordlength ratio. The vorticity plots (A) illustrate different directions of vortexmovement for L

c= 1.33 (top row) and L

c= 2.00 (bottom row). In the

case of Lc

= 1.33, the previously shed vortex convects along the lowerwing surface. In the case of L

c= 2.00, the vortex rolls over to the upper

wing surface. The lift coefficient graph (B) compares the time varying liftcoefficients for a half flapping period. Whereas the primary translationallift peaks (T=2.2 to T=2.5) are similar, the secondary lift peaks (T=2.0 toT=2.2) are different due to differences in the wake capture process. Thelift coefficient graph shows larger L

cratio corresponds to larger mean lift

coefficient.

mean drag coefficient. This simulation result suggests thatfuture RoboBee design should utilize a stiff hinge to advancepassive wing pitch rotation. This computational result agreeswell with previous experimental findings [7].

In addition, our simulations show that wake capture effectscan be beneficial or detrimental to lift production dependingon the movement of shed vortices. We can induce favorablewake capture effects by increasing the flapping amplitudeto chord length ratio L

c . Furthermore, we have validatedour numerical model by comparing simulation results toquasi-steady predictions and experimental measurement. Itis shown that our numerical model gives a 44% percentbetter approximation to 3D experiments than the quasi-steadymodel in the least squares sense.

Whereas the quasi-steady model requires fitting coef-ficients, this numerical model is rigorously derived fromNavier Stokes equations and does not require fitting pa-

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rameters. This property makes the numerical model morereliable for future wing kinematics optimization studies.Ensuing studies should further develop this computationaltool to optimize passive rotation kinematics. Whereas thecurrent model requires completely prescribing stroke andhinge motion, the immersed boundary method can be im-plemented to allow wing-fluid interaction [15]. By definingwing stroke kinematics alone, future computational modelshould return hinge kinematics along with force estimates. Inaddition, particle image velocimetry techniques can be usedto compare experiment measurement with the computed flowfield.

VI. ACKNOWLEDGMENT

This work was partially supported by the National ScienceFoundation (award number CCF-0926148), and the Wyss In-stitute for Biologically Inspired Engineering. Any opinions,findings, and conclusions or recommendations expressed inthis material are those of the authors and do not necessarilyreflect the views of the National Science Foundation.

REFERENCES

[1] K. Ma, P. Chirarattanon, S. Fuller, and R.J. Wood, “Controlled Flightof a Biologically Inspired, Insect-Scale Robot”, Science, vol. 340, pp.603-607, 2013.

[2] Lentink, David, Stefan R. Jongerius, and Nancy L. Bradshaw. "Thescalable design of flapping micro-air vehicles inspired by insect flight."In Flying Insects and Robots, pp. 185-205. Springer Berlin Heidelberg,2010.

[3] Keennon, Matthew, Karl Klingebiel, Henry Won, and Alexander An-driukov. "Development of the Nano Hummingbird: A Tailless flappingwing micro air vehicle." In 50th AIAA Aerospace Sciences Meetingincluding the New Horizons Forum and Aerospace Exposition, pp.1-24. 2012.

[4] Shyy, Wei. Aerodynamics of low Reynolds number flyers. Vol. 22.Cambridge University Press, 2008.

[5] Anderson, John David. Fundamentals of aerodynamics. Vol. 2. NewYork: McGraw-Hill, 2001.

[6] Dickinson, Michael H., Fritz-Olaf Lehmann, and Sanjay P. Sane."Wing rotation and the aerodynamic basis of insect flight." Science284, no. 5422 (1999): 1954-1960.

[7] A.L. Desbiens, Y. Chen, and R.J. Wood, “Wing characterizationmethod for flapping wing micro air vehicle”, IEEE/RSJ Int. Conf.on Intelligent Robots and Systems, Tokyo, Japan, Nov., 2013.

[8] Wang, Z. Jane, James M. Birch, and Michael H. Dickinson. "Unsteadyforces and flows in low Reynolds number hovering flight: two-dimensional computations vs robotic wing experiments." Journal ofExperimental Biology 207, no. 3 (2004): 449-460.

[9] Zheng, Lingxiao, Tyson L. Hedrick, and Rajat Mittal. "A multi-fidelitymodelling approach for evaluation and optimization of wing strokeaerodynamics in flapping flight." Journal of Fluid Mechanics 721(2013): 118-154.

[10] Lentink, David, and Michael H. Dickinson. "Rotational accelerationsstabilize leading edge vortices on revolving fly wings." Journal ofExperimental Biology 212, no. 16 (2009): 2705-2719.

[11] Hesthaven, Jan S., and Tim Warburton. Nodal discontinuous Galerkinmethods: algorithms, analysis, and applications. Vol. 54. Springer,2008.

[12] Sane, Sanjay P., and Michael H. Dickinson. "The control of flight forceby a flapping wing: lift and drag production." Journal of experimentalbiology 204, no. 15 (2001): 2607-2626.

[13] P. Perssonn, G.Strang, A Simple Mesh Generator in MATLAB, SIAMReview, Volume 46, Number 2, June 2004, pages 329-345.

[14] Wood, R. J., S. Avadhanula, R. Sahai, E. Steltz, and R. S. Fearing."Microrobot design using fiber reinforced composites." Journal ofMechanical Design 130 (2008): 052304.

[15] Xu, Sheng, and Z. Jane Wang. "An immersed interface method forsimulating the interaction of a fluid with moving boundaries." Journalof Computational Physics 216, no. 2 (2006): 454-493.

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