2013 a Simulation and Design Tool for a Passive Rotation Flapping Wing Mechanism

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 2, APRIL 2013 787

A Simulation and Design Tool for a Passive RotationFlapping Wing Mechanism

Veaceslav Arabagi, Lindsey Hines, and Metin Sitti, Senior Member, IEEE

Abstract—This paper develops a numerical simulation tool fordesigning passive rotation flapping wing mechanisms. The simu-lation tool includes a quasi-static model of the piezoelectric ac-tuator bending, transmission kinematics, and the small Reynoldsnumber aerodynamic forces governing wing dynamics. To validatethe developed tool, two single-wing systems with distinct resonantfrequencies are manufactured and characterized. Comparison toexperimental results reveals that, although discrepancies exist, thesimulation is able to predict general trends of wing kinematics andlift behavior as functions of frequency, thus, being useful as a designtool. Finally, the complex models, ranging from actuator deflectionto wing aerodynamics presented in this paper, allow analysis ofthe complete system revealing insight into several wing trajectorycontrol methodologies, and potentially serving as a design and op-timization tool for future flapping wing robots.

Index Terms—Flapping wing, passive wing pitch reversal, sim-ulation and design tool, wing lift force modulation.

I. INTRODUCTION

A S in natural flyers, miniature flapping wing vehicles areexpected to present a significant advancement in agility

from their fixed and rotary wing counterparts. Inspiration andbasic concepts of flapping flight are taken from studying thekinematics of dragonflies [1], butterflies [2], hawkmoths [3],bats [4], and flies [5]. The low Reynolds number operationregime of small-scale fliers is dominated by nonlinear aerody-namics and temporal wing–wake interactions [6], [7] that arethe underlying cause for their, unaccounted by conventionalaerodynamics, large lift force production capacity [8], respon-sible for the acrobatic maneuverability of these insects. Onemajor milestone to overcome in creating a robotic flappingwing platform capable of controlled hover is the generationof proper and controlled wing motion. Over time, two differ-ent approaches to this problem emerged: 1) active control ofboth the flapping and rotation angles, and 2) active control of

Manuscript received February 8, 2011; revised September 2, 2011 andOctober 27, 2011; accepted December 21, 2011. Date of publication February20, 2012; date of current version January 10, 2013. Recommended by TechnicalEditor S. Fatikow. The work of L. Hines was supported by the National DefenseScience and Engineering Graduate Fellowship.

V. Arabagi was with the Department of Mechanical Engineering, CarnegieMellon University, Pittsburgh, PA 15213 USA. He is now with the Departmentof Cardiovascular Surgery, Children’s Hospital Boston, Boston, MA 02115 USA(e-mail: [email protected]).

M. Sitti is with the Department of Mechanical Engineering, Carnegie MellonUniversity, Pittsburgh, PA 15213 USA (e-mail: [email protected]).

L. Hines is with the Robotics Institute, Carnegie Mellon University,Pittsburgh, PA 15213 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMECH.2012.2185707

wing flapping only, while allowing for passive rotation of thewing.

Wood et al. proposed a flapping wing platform capableof fully controlled wing trajectory, and thus body forces andtorques [9]. However, that robot design relies on a difficult tomanufacture differential mechanism, driving both the wing’sleading and trailing edges, and a complex high-bandwidthcontroller [10]. However, recent theoretical and experimentalresults of Bergou et al. suggest that at least some of the wingrotation in insects is passively assisted by inertial and aero-dynamic forces [1]. Furthermore, the flapping wing regime isknown to possess several self-stabilizing damping properties fortranslation and rotation maneuvers due to the reciprocal natureof the wing beat [11], [12], and upright maintenance with the aidof sail type devices [13]; thus, suggesting the possibility of anunderactuated wing drive mechanism. Based on this idea, a pas-sive wing rotation approach has been undertaken by Wood, witha single actuator driving the flapping motion of the wings whileallowing for passive rotation, with the amplitude of rotationlimited by mechanical stoppers [14]. This mechanism resultsin large impact forces on the wing imparted at the end of eachstroke, which could decrease its lifetime and excite undesirablestructural vibration modes.

This paper is based upon an aerial platform design featur-ing completely passive wing pitch reversal. Without mechanicalstops and suspended from an axis close to its leading edge by aspring/damper system, the motion of the wing is governed onlyby the dynamics of the wing, aerodynamic forces, and the elas-ticity of the system. Furthermore, this design not only simplifiesmechanical complexity, but also presents opportunity for theeventual introduction of wing angle of attack control via a vari-able spring stiffness at the wing rotation joint, potentially usingsmart, morphing materials. A dynamic simulation that includesa compilation of small Reynolds number aerodynamic forces,coupled with dynamics of the driving piezoelectric bending ac-tuator and kinematics of the four-bar transmission, serves as thebasis for analysis of the system’s performance. This model isemployed to simulate the behavior of two single-flapping-wingsystems, one resonating at 30 Hz and the other at 75 Hz, and theresults are compared to experimental measurements. Althoughthe theoretical model is not able to predict the exact lift forcemagnitudes, wing rotation dynamics are predicted accuratelywith experimental flapping angles as input. In addition, the sim-ulation is able to capture behavioral trends very well, provinguseful as a simulation and design tool. Finally, three methodsto control the lift force of a passively rotating flapping wingmechanism are proposed: modulating the flapping frequency,the amplitude of the actuator oscillation, and the torsional stiff-ness of the wing rotation joint.

1083-4435/$31.00 © 2012 IEEE

788 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 2, APRIL 2013

Fig. 1. Schematic of the complete single-wing system, including the actuator,slider crank, four-bar, and wing.

II. SYSTEM DYNAMICS MODELING

A numerical simulation of the actuator-transmission-wingdynamics is developed in MATLAB to serve as a design andoptimization tool. The simulation includes three main compo-nents: the actuator, a transmission mechanism, and the wing.

A. Actuator

Piezoelectric actuators are selected to power the robotic flap-per due to their high efficiencies, power densities of up to400 W/kg [15], extensive theoretical modeling [16], [17], easyincorporation into the composite manufacturing process, andtheir use with some success by other groups, such as the Berke-ley MFI [18], [19] and the Harvard MicroFly [20]. Althoughpiezoelectric materials also have several limitations, such as aneed for a high-voltage power supply, small strain (less than0.1% for PZT-5H), and high brittleness, work has been doneto make their application feasible in miniature aerial vehiclesas presented in [21]. Reciprocating wing drive is achieved witha bending unimorph piezoelectric actuator, whose schematic isportrayed in Fig. 1. The modeled actuator consists of an ac-tive bending portion and a rigid extension that increases itsstroke. Classical lamination theory was used to model the ac-tuator quasi-statically, alike presented in [17]. The structuralelasticity of the unimorph actuator is introduced into the modelas a spring of stiffness kact attached to the end of a rigid ac-tuator that bends under an applied electric field, as shown inFig. 1. The inertial properties of the actuator have been simpli-fied to the dynamics of a point mass positioned on the tip ofthe actuator. The effective mass of the actuator, meff , calculatedbased on an energy approach presented in [22], was determinedto be 15 mg for the 30-Hz resonant system. Its kinetic energy,used for the Lagrangian expression, is Ta = 1

2 meff δ2 , where δis the displacement of the slider crank end. The free, no load,displacement of the actuator tip is modeled as

Δ = g(E, actGeom) (1)

where E is the electric field in the PZT layer and actGeom isthe geometry of the actuator.

B. Transmission

A four-bar-type transmission is employed to amplify the smalltip displacements of the piezoelectric actuator. Due to actuator

Fig. 2. Displacement (left axis) and torque transmission curves (right axis)for 75-Hz resonance single-flapping-wing system. τout is the torque output ofthe four-bar. The lengths of these four-bar’s links are {L0, L1, L2, L3}= {13.5,12.1, 10, 5} mm.

bending, a slider crank mechanism is implemented to transferthe actuator output forces into the transmission. The four-barjoints are modeled as ideal pin joints, with the exception of thejoint at the output link of the four-bar, designated by θ in Fig. 1,which has a stiffness value assigned to it, as it undergoes thelargest range of motion and hence can significantly affect systemdynamics. Based on the four-bar transmission kinematics mod-eled in [23], the output angle and torque transmission functionof the four-bar is

θ = h(linksi , config, δ) (2)

�Mdrive = f(linksi , config, �Fin ) (3)

where �Mdrive is the moment that is applied to the wing aboutits flapping axis (E3 axis from Fig. 3), linksi are the lengthsof the 4 four-bar links, config is the configuration of the four-bar, i.e., its position through the stroke, and �Fin is the forceinput from the actuator, as portrayed in Fig. 1. These generalequations are sufficient to describe any transmission systemtransforming linear input displacement into angular output mo-tion. In the case of a piezoelectric actuator, Fin = kact(Δ − δ)and represents how the actuator stiffness enters into the theo-retical model. Numerical optimization is employed to achieve aparticular transmission ratio and symmetry over the actuator dc-bias voltage, yielding transmission curves portrayed in Fig. 2.Generally, higher torque transmission ratios imply a higher res-onance frequency if everything else is kept constant. Intuitively,this arises from the fact that higher transmission ratios impartsmaller wing inertial forces on the actuator and hence the lattersenses a smaller effective wing inertia.

C. Wing Aerodynamics

Passive rotation of the wing is achieved by attaching the wingto a driver spar with an elastic joint, thus, allowing the elas-tic/damping properties of the flexure material to influence wingmotion. In fact, the interaction of these elastic and dampingtorques with the wing’s inertia and aerodynamic forces is essen-tial to the generation of proper flapping wing trajectory. Fig. 3portrays the geometric setup of the simulation model and thesets of Euler angles for coordinate transformations.

Throughout its motion, the wing is affected by aerodynamicforces stemming from various small Reynolds number effects.

ARABAGI et al.: SIMULATION AND DESIGN TOOL FOR A PASSIVE ROTATION FLAPPING WING MECHANISM 789

Fig. 3. Schematics of the passive flapping wing setup. Coordinate sets repre-sent transformations established by θ (flapping angle) and ϕ (rotation angle).The coordinate systems are shifted for clarity, while in simulation they are allcentered at the point labeled “coordinate sets origin.” The �E ′′ coordinate systemis attached to the wing.

Fig. 4. (a) Wing SolidWorks model and (b) wing profile illustration of aero-dynamic force due to rotation. The wing center of mass and rotation axis areillustrated.

Overall, the total force on the wing is

Ftot = Ft + Frot + Fair + Fwc (4)

where Ft is the translational force, Frot is the force due to wingrotation, Fair is the force due to added air mass, and Fwc is theforce created by wake capture. Translational force estimates arequasi-steady approximations adapted from thin airfoil theory.Frot is the force generated by air resistance to wing rotation asportrayed in Fig. 4(b). The added air mass forces are impartedon the wing by the surrounding air and are generated due tounsteady wing motion [24]. This is also the only force that aidsthe rotation of the wing during stroke reversal, since the inertiaof the air trapped in the vicinity of the wing tends to continueits linear motion, thereby imparting a torsional moment on thewing [25]. The wake capture effect is a nonsteady phenomenonoccurring when the wing traverses vortices and air circulationgenerated by its motion prior to the direction change. Since aclosed form expression for this force cannot be determined, it isnot included in the dynamical model. Thus, the expressions foraerodynamic forces acting on chordwise wing strips, dr, takethe form

dFt =12ρU 2c(r)[C2

l (α) + C2d (α)]1/2dr (5)

dFrot = −12Crotρ|ϕ|ϕ

∫c(r)

z′′|z′′|dz′′dr (6)

dFair = −ρπ

4(θcosϕ − θϕsinϕ)rc(r)2dr

− ρπ

4ϕ

(zRA − c(r)

2

)c(r)2dr (7)

where ρ is the air density, r is the radial position of a wing stripfrom the wing’s flapping axis, c(r) is the chord length of theparticular strip, θ and ϕ are the flapping and rotation angles,respectively (portrayed in Fig. 3), α is the angle of attack ofthe wing, or (π/2 − ϕ), and zRA is the location of the rotationaxis measured from the wing’s leading edge. The translationalforce component is obtained by vector addition of mutuallyorthogonal lift and drag forces. U is taken to be the velocity inthe �E ′

2 direction of each wing strip’s midchord point

U(r) = �vmc · �E ′2 = rθ +

c(r)2

ϕcosϕ (8)

where �vmc is the midchord velocity of each wing division. Theuse of midchord velocity in the translational lift force expres-sion is justified by the fact that the incoming air stream velocityis linearly distributed along each chord of the wing; hence, thebest approximation to a constant velocity field, which was usedto obtain the translational lift coefficients in [8], is the mean ve-locity along each wing chord, observed at the midchord point.Cl , Cd , and Crot are the translational lift, drag, and rotationalforce coefficients, respectively, where the translational coeffi-cient expressions were tabulated experimentally by Sane andDickinson for a Reynolds number (Re) of 136 [25]

Cl(α) = 0.225 + 1.58 sin(2.13α − 7.2) (9)

Cd(α) = 1.92 − 1.55 cos(2.04α − 9.82) (10)

where α is in degrees. The position of the rotation axis has beenset at ≈1/4 chord length from the leading edge of the wing, atits aerodynamic center (according to thin-plate airfoil theory).The Reynolds number for our system is Re = 2800, calculatedby averaging the Re numbers obtained for each wing strip, issimilar, in aerodynamic sense, to RoboFly’s one, Re = 136,justifying the use of Sane and Dickinson’s findings as means ofapproximating the aerodynamic forces on the wing. The valuefor the rotational drag coefficient Crot was taken to be 2, asthis is the theoretical result of rotational drag on a flat platesubjected to normal flow [26], [27]. To account for the radialvariability of wing velocity, the aforementioned equations needto be integrated numerically to obtain the aerodynamic forceson the entire wing.

D. Equations of Motion of the Passive Rotation Wing System

The wing has a carbon fiber leading edge spar with an ex-truding vein for chordwise stiffness and a Kapton membraneas the main airfoil (see Fig. 4). The inertial properties of thewing required for the dynamic equations are obtained directlyfrom the SolidWorks model. In addition, the three carbon fiberlinks of the four-bar and the slider crank mechanism were mod-eled as inertial elements. The dynamic equations are formulatedin Lagrangian form, with generalized coordinates of θ and ϕbeing the flapping and rotation angles, respectively. Thus, we


formulate the Lagrangian as follows:

L = T − V =12m�v · �v +

12J�ω · �ω

+∑ 1

2mLi �vLi · �vLi +

∑ 12JLi �ωLi · �ωLi

+12meff δ2 − 1

2krotϕ

2 (11)

where m is the wing mass augmented with the added air mass[as described in (19)], �v is the velocity at the wing’s centerof gravity (CG), J is the wing’s inertia matrix taken about thewing’s center of mass, krot is the torsional spring stiffness at therotation axis, mLi , JLi , �vLi , and �ωLi are the masses, rotationalinertias, linear and angular velocities of the four-bar links 1–3,respectively, meff is the effective mass of the actuator, δ is thehorizontal velocity of the tip of the actuator (scalar), and �ω isthe angular velocity of the wing defined as

�ω = θ �E ′3 + ϕ �E ′

1 = ϕ �E ′′1 + θ sin(ϕ) �E′′

2 + θ cos(ϕ) �E ′′3 (12)

expressed in terms of basis vectors of the coordinate frame at-tached to the wing. Furthermore, the expressions for angularand linear velocity of the four-bar links are complex in closedform and, hence, have been substituted by numerically com-puted trajectories parametrized by the output flapping angleθ (i.e., v2(t) = f(θ(t)) and hence dv2 (t)

dt = θ(t) ∂f∂θ(t) ). Defin-

ing the conventions f ′ ≡ ∂f∂θ(t) and f ≡ df

dt , and substitutingall knowns into the Lagrangian L, we obtain the followingequation:

L = 1/2(−krotϕ2 + sin(ϕ)2Jyy θ2

+ ϕ(cos(ϕ)Jxz θ+Jxxϕ)+cos(ϕ)θ(cos(ϕ)Jzz θ + Jxz ϕ)

+ 1/2m((2R2CG + β2cos(2ϕ))θ2 + 4RCGβcos(ϕ)θϕ

+ 2β2 ϕ2) +3∑

i=1

mLiθ2‖�x′

Li‖2 +2∑

i=1

JLiθ2θ

′2Li

+ JL3 θ2 + meff θ2δ

′2) (13)

where RCG and βCG are the radial and vertical positions, re-spectively, of the center of mass from the origin of the coordinatesystems, as portrayed in Fig. 4, Jii are the components of thewing’s inertia matrix, �xLi is the CG position of link i of thefour-bar, and θLi is the orientation of the links (note that thefour-bar motion is planar and equations are simplified to reflect2-D motion). Note that for a thin wing, the inertial componentsJyz and Jxy are extremely small and have been omitted fromthe equations. Thus, the Lagrange equations describing the wingflapping and rotation dynamics are as follows:

d

dt

(∂L

∂ϕ

)− ∂L

∂ϕ= �Maero · �E ′′

1 − dϕ (14)

d

dt

(∂L

∂θ

)− ∂L

∂θ= ( �Mdrive + �Maero) · �E3 − Dθ (15)

where d is the rotational damping coefficient, D is the flap-ping damping coefficient arising from the transmission flexuredamping, �Mdrive is the driving flapping torque, and �Maero is the

moment due to the aerodynamic forces, explicitly defined as

�Maero =∑

i

�Fi × �βi(α) (16)

where �Fi’s are the translational and rotational aerodynamicforces and �βi’s are the respective positions of their centers ofpressure from the wing rotation axis, explicitly defined as

|βt(α)| =(

0.82π

|α| + 0.05)

c(r) (17)

|βrot | =

∫span z′′dFrot∫

span dFrot(18)

where βt defines the position of the translational lift center ofpressure as a function of angle of attack for each wing chordstrip c(r) and is obtained from experimental results of Dicksonet al. [28]. βrot is the effective moment arm of the rotationalforce distribution of Fig. 4(b).

Given that there is no analytical solution for the added airmass force on a wing planform moving in 3-D fashion, dFairof (7) does not accurately describe the moments exerted bythe added air mass on the flapping and rotation angles. As anapproximation, the effect of added air mass forces was imple-mented in the form of virtual mass that augmented the physicalmass of the wing, similar to the inertial implementation in [26]and [29]. Stemming from the concept that the added mass effectcan be estimated as dm = πρ

4 c(r)2dr [30], the expression form emerges as follows:

m = mw +∫

span

πρ

4c(r)2dr (19)

where mw is the physical mass of the wing. This approximationallows a simplified treatment of the added air mass effects thatis sufficient for the development of a simulation tool able topredict general dynamics and lift force trends. Although thismass augmentation approach is used in the differential equationsdefining the system dynamics, the expression for added air massforce of (7) is used to estimate the aerodynamic lift generatedby this effect in all lift plots presented in this paper.

The previously presented Lagrange equations (14) and (15)are fully general, governing the dynamics of a passive rotationflapping wing driven by a torque �Mdrive around the flappingaxis. In our case of the driving mechanism consisting of a piezo-electric actuator coupled to a four-bar transmission, �Mdrive isgiven by (3). Thus, substituting all the known quantities into (14)and (15) and simplifying, we obtain the nonlinear differentialequations governing the flapping and rotation angles that can befound in the Appendix, due to their length and complexity.

III. MANUFACTURING AND EXPERIMENTAL

PARAMETER DETERMINATION

In order to best quantify the theoretical simulation’s perfor-mance, two experimental systems consisting of an actuator, afour-bar and wing were manufactured, having different reso-nant frequencies of 30 and 75 Hz. Both systems feature the


Fig. 5. Overall dimensions of the manufactured wing and actuator for the(a) 30-Hz resonant system and (b) 75-Hz system. The actuator mount is 3-Dprinted from VisiJet HR 200 material, and assembled with Cyanoacrylate glue.

same construction components, differing only in their dimen-sions and lengths.

A. Component Manufacturing

All the parts comprising the flapping wing system were man-ufactured using the “Smart Composite Microstructures” (SCM)methodology and manually assembled afterward [9]. This lastmanual step generally introduces some misalignment and uncer-tainty in the final assembly, which tends to adversely affect thedynamics of the system, generally in the form of unwanted com-pliance. The transmission members are formed of two preim-pregnated, unidirectional M60J ultrahigh modulus carbon fiberlayers of 60-μm thickness (Toray) sandwiching a 12.7-μm layerof Kapton (DuPont Kapton 50HN) and cured together. The wing,whose Kapton layer serves as a membrane, is manufactured fromthinner Kapton of merely 6.3 μm (DuPont Kapton 25HN), asthe minimization of the wing weight is of major importance.

The actuator’s active layer consists of two layers of carbonfiber bonded to a 125-μm-thick layer of PZT-5H (Piezo Sys-tems), while the passive extension has an additional layer ofs-glass, increasing stiffness. Some dimensions of the actuatorand the wing used in the experimental flapping wing systemsare portrayed in Fig. 5.

B. Actuator Experimental Characterization

Since the piezoelectric actuator is of key importance in defin-ing system dynamics, experiments were performed to obtain theactual actuator blocking force, tip stiffness, and material prop-erties. Using laminar plate theory, the actuator bending stiffnesswith a force applied at its tip, labeled kact in Fig. 1, replicatingthe operational scenario, was calculated at 160 N/m. Throughindentation experiments, the actual actuator stiffness was foundto be 169 N/m, measured by the ratio of force over given tipdisplacements.

C. Wing Rotational Flexure Characterization

The dynamic parameters of the rotational flexure were exper-imentally measured after manufacturing, to account for any im-perfections. The stiffness and damping properties of each flexurewere determined by applying an impulse impact to it, after anadditional known mass was added. The resulting displacements

were recorded with a laser scan micrometer (Keyence LS-3100,Woodcliff Lake, NJ), and, being second-order responses, al-lowed the determination of stiffness and damping parametersfrom the dominating frequency and envelope of the signal. Thesetest were performed under vacuum such as to eliminate aerody-namic damping, which would constitute the main component ofdamping otherwise.

D. Transmission Compliance Characterization

Given that the slider crank and four-bar flexures are sourcesof unwanted compliance, the stiffness and damping propertiesof the transmission are measured experimentally. Complianceof the slider crank is measured via an indentation test at the tipof the first transmission link, with the actuator rigidly fixed. Therecorded tip stiffness is transformed via the transmission ratioto stiffness of the slider crank and modeled in simulation inseries with the actuator tip stiffness kact . Throughout our exper-iments, slider crank stiffness ranges as low as ksc = 300 N/mand as high as ksc = 1650 N/m. The large span of stiffness val-ues is attributed to the easy misalignment of the slider cranklinks, resulting in flexure buckling. For all simulations in thispaper, the 30-Hz resonance flapping system was simulated witha nominal slider crank stiffness of ksc = 1400 N/m, while the75-Hz system had a value of ksc = 300 N/m, per experimentalmeasurements.

The damping coefficient of the four-bar was measured by im-parting an impulse on wing-driving mechanism, with the wingrotated 90◦, i.e., α = 0◦, so as to eliminate aerodynamic dragfrom the measurements. The simulated impulse response wasthen matched to experimentally recorded vibrations via alteringthe flapping damping parameter D from (15), converging to afinal value of D = 100 μN·mm·s, employed in simulation.

IV. EXPERIMENTAL SETUP AND WING KINEMATICS

Experimental lift was measured by mounting the single-wingassembly to a load cell through a mechanical amplification link-age to increase the sensed lift forces, and the low-friction, single-degree-of-freedom (DOF) pivot isolates the vertical forces gen-erated by the wing [see Fig. 6(b)]. A 25-g load cell (TransducerTechniques) for the 75-Hz resonant prototype (30-g load cellfor the 30-Hz system) was attached to the main shaft via acontact connection, such that no torques in the shaft would betransmitted into the load cell. The actuator, wing, and four-barmechanism were mounted directly to a 3-D printed (InvisionHR 3D Printer) plastic mount that was bolted onto an aluminumshaft, portrayed in Fig. 6(a). Given the large inertia of the bal-ance system, its resonant frequency was determined at 13 Hz.This is sufficiently distant from the operating frequencies of30 and 75 Hz to ensure lift measurements were unexaggerated;however, functioning as a low-pass filter, measurements are re-stricted to mean lift force only.

During each test, the input voltage to the actuator, a sinusoidalwave specified by the peak-to-peak voltage Vpp and a constantdc offset Vdc . Wing kinematics were captured using a strobe lightand a high-definition camcorder (Sony HDR-SR11). Video wastaken from the top view of the wing, with the camera placed


Fig. 6. Photographs of the experimental lift acquisition setup with main com-ponents labeled. (a) Actuator-four-bar-wing experimental prototype mountedonto a 3-D printed plastic part. (b) Lift balance mechanical amplificationdevice.

in an offset plane parallel to the stroke plane. Three points onthe wing were tracked, based on edge contrast, from which thewing trajectory was calculated.

A. Wing Kinematics Comparison

Given the previously described setup to obtain kinematics,a quick comparison is performed to ensure that the numericalsimulation behaves similar to the system it models: experimen-tal kinematic wing data are compared to predicted results inFig. 7(a) for the 30-Hz resonant system. The kinematics featurea discrepancy of ≈10◦ peak to peak in flapping angles, propa-gating in a discrepancy of ≈20◦ peak to peak in rotation, likelydue to the asymmetry of the torque transmission curve of thephysical four-bar transmission. The simulated lift force compo-nents shown in Fig. 7(b) portray that translational lift constitutesmost of the total lift force with a small contribution from addedair mass lift. The rotational lift component is very low, insteadthe effect of this force is manifested mostly through the torqueit imparts on the wing, thereby having a great impact on shapingthe kinematics of wing rotation and influencing total lift forcein that manner.

V. RESULTS AND DISCUSSION

Characterization of the developed theoretical model of asingle-wing-flapping system is attempted through comparisonto experimentally obtained wing kinematics and mean lift forcemeasurements. The full dynamic model can be analyzed by sep-arating it into two subsystems: 1) the actuator/transmission driveand wing-flapping dynamics; 2) the aerodynamic force modelsand wing rotation dynamics. The performance of the secondsubsystem can be quantified by using the experimentally mea-sured wing flapping trajectory as a displacement source in thesimulation of wing rotation only. Running the full simulation,

Fig. 7. (a) Overlayed experimental and simulated wing kinematics, and(b) trajectories of simulated lift forces of the 30-Hz resonance single-wingsystem. System parameters: Vpp = 180 V, Vdc = 30 V, frequency = 30 Hz,ksc = 1400 N/m, krot = 20 mN·mm, and d = 10 μN·mm·s.

with the input of a sinusoidal signal driving the actuator, enablesanalysis of the wing driving mechanism. This characterizationof the simulation performance is done for both the 30- and75-Hz resonant systems, in order to give an idea about themodel’s shortcomings in simulating systems of different reso-nances.

A. Analysis of the 30-Hz Resonant System

Since a frequency response characterizes the behavior of adynamic system well, this methodology is applied to analyzethe performance of the numerical model. For the 30-Hz reso-nant experimental system, the frequency response was obtainedby recording the wing kinematics and load cell output whilesweeping the frequency of the input voltage signal at discretesteps of 3 Hz. This relatively large value of the frequency step-size is acceptable due to the low-quality factor (Q ≈ 2) of aflapping wing system, ensuring smoothness of the frequency re-sponse curve and, thus, no important missed dynamics. In orderto validate the quasi-steady aerodynamic models and underlyingrotation dynamics, the leading edge of the simulation wing isdriven based on experimental kinematics. The simulation takesas input for the flapping angle a sinusoid signal with the ampli-tude obtained from measured wing kinematics. The kinematicsplot portrays the two fundamental resonant frequencies of thesystem manifested by peaks in the flapping amplitude, occur-ring around 30 and 36 Hz, as shown in Fig. 8. The two resonantpoints of the system correspond to an optimal balance betweenthe internal stresses of the actuator, inertial forces, and nonlin-ear aerodynamic damping effects acting on the wing. Simulatedwing rotation closely resembles the experimental observations


Fig. 8. Frequency sweep of the 30-Hz resonance system simulating only wingrotation, with flapping angle set by a sinusoid wave with experimentally obtainedamplitudes. (a) Experimental and simulated wing kinematics, and (b) lift forces.System parameters: Vpp = 180 V, Vdc = 30 V, krot = 20 mN·mm, and d =10 μN·mm·s.

except for high frequencies; this is thought to occur due to theexcitement of second resonance mode and out-of-phase rota-tion, addressed in detail in Section V-C. The lift force plots ofFig. 8(b) portray very similar magnitudes as the experimentalmean lift curve. Translational lift is the dominant aerodynamicforce throughout this frequency range, with the added air masslift comprising half of this amount. Similar to the kinematics, thesimulated lift force curve diverges from experiments at high fre-quencies, suggesting the dominance of unmodeled aerodynamiceffects present in that operating regime.

Simulating the complete flapping wing system along withactuator and transmission when driven by a sinusoid signal,we obtain the plots presented in Fig. 9. Both the simulatedkinematics and total lift force are comparable in magnitude totheir experimental measurements for frequencies below the sec-ond resonance peak. The model predicts well the occurrenceof the first resonant peak while overpredicting the occurrenceof the second peak at 50 Hz, although being highly damped.The simulated frequency sweep is performed for a larger rangeof frequency values, while experimental measurements werestopped earlier due to extreme wing rotations after the secondresonance, destructive to the prototype. Furthermore, the curveslabeled with a slider crank stiffness value, ksc , represent totallift force curves obtained from simulation runs of single-wingsystems featuring bounding values for the corresponding valuesof slider crank stiffness, namely ksc = 300 and 1650 N/m. Thechange in the magnitude of the total lift curve as well as theobserved shift in system resonance illustrates the system’s sen-sitivity to sources of compliance in the wing drive mechanism.

Fig. 9. Frequency sweep obtained with the complete system simulation, in-cluding the actuator drive and wing dynamics. System parameters: Vpp = 180 V,Vdc = 30 V, ksc = 1400 N/m, krot = 20 mN·mm, and d = 10 μN·mm·s. Curveslabeled with a slider crank stiffness are obtained with simulated systems havingcritical values ksc = 300 and 1650 N/m.

Fig. 10. Frequency sweep of the 75-Hz resonant system simulating only wingrotation, with flapping angle set by a sinusoid wave with an experimentallyobtained amplitude. (a) Experimental and simulated wing kinematics, and(b) lift forces. System parameters: Vpp = 150 V, Vdc = 50 V, krot =37.5 mN·mm, and d = 13 μN·mm·s.

B. Analysis of the 75-Hz Resonant System

To obtain the 75-Hz resonant system frequency response, liftwas measured with a balance system (see Fig. 6) when the ac-tuator was driven via a continuous frequency sweep. Fig. 10(a)portrays good matching between the experimental wing rota-tion angles and the simulation modeling only wing rotation.


Fig. 11. Frequency sweep obtained with the complete simulation. Systemparameters: Vpp = 150 V, Vdc = 50 V, ksc = 300 N/m, krot = 37.5 mN·mm,and d = 13 μN·mm·s. The lift bounding curve corresponding to a slider crankstiffness of ksc = 300 N/m coincides with the nominal frequency response,while the other bounding curve corresponds to ksc = 1650 N/m.

The experimental flapping kinematics portray two resonancepeaks spaced close to each other, similar to the low resonantfrequency system. The simulated lift forces in Fig. 10(b) areslight underestimates from experimental results. The largestlift contribution comes from added air mass lift, with transla-tional lift being half of its value. This result arises from the factthat the high-frequency low-amplitude operation regime of the75-Hz system maximizes the accelerations of the wing, and thusthe air in its vicinity, i.e., added air mass, while large flappingamplitudes and translational wing motion dominate the behaviorof the 30-Hz resonant system. The large increase in simulatedlift at 95 Hz seen in Fig. 10(b), not present in the experimentalresponse, arises from the increase in flapping amplitude at thatfrequency. This occurs due to the excitement of the second res-onance mode in the experimental system, predicted poorly bythe simulated system, due to unmodeled aerodynamic effects,discussed in detail in Section V-C. Furthermore, after examin-ing the phase shift between the flapping and rotation sinusoids,a shift of 81◦ is observed in simulation and one of 138◦ inexperiment, suggesting that only the experimental system hasbegun the transition to out-of-phase rotation, thus, reducing theproduced lift.

The full simulation of the 75-Hz resonance system, portrayedin Fig. 11, overall portrays much worse agreement of simula-tion to experiment. The twofold discrepancy in total lift forceproduced is attributed primarily to the greatly underpredictedflapping amplitude in simulation, ≈20◦ peak to peak. The largeramplitudes also cause an eightfold increase in translational liftfrom the results of Fig. 10(b), rendering it the dominant liftcomponent, alike in the 30-Hz resonance system. Similar toprevious results, the simulation predicts well the first resonancemode at 75 Hz, while overestimating the second one at 115 Hz,while experimentally it is observed at 95 Hz. The curve ofksc = 1650 N/m, of Fig. 11, represents simulated total lift of a

Fig. 12. Frequency spectrum of the system impulse response. The motion ofthe first four-bar link was recorded after an impulse has been applied. The firstand second system resonant peaks are indicated by arrows. Note that the 50-and 100-Hz peaks of the experimental system are due to electrical noise in theenvironment and do not reflect physical resonant frequencies.

system featuring a stiff slider crank. The other limiting value ofstiffness ksc = 300 N/m is the nominal magnitude for this exper-imental prototype. Again, a change in this parameter propagatesinto a resonant frequency increase by more than 10 Hz, havinggreat impact on the behavior of the system.

C. Discussion

The presented comparison between experimental measure-ments and both wing rotation only and the complete systemsimulation effectively illustrates the capabilities and limitationsof the theoretical model of the single-flapping-wing system.The overall good agreement to experimental results of bothkinematics and lift forces produced by the simulation of wingrotation only suggests that wing inertial properties, flexure stiff-ness, and damping properties, as well as the rough, quasi-steadyaerodynamic models are largely correct. The complete simu-lation, encompassing the actuator drive, yields less acceptableresults, overpredicting lift force, especially for the 75-Hz reso-nant frequency system. Given that most of this discrepancy isattributed to the underprediction of flapping amplitude and thatwing rotation dynamics are correct, as established earlier, theunderlying cause of the problem is traced to the flapping mo-tion of the wing. The decrease in wing flapping displacementcould arise from actuator depolarization, inconsistencies in themanual alignment and assembly of the four-bar transmission, aswell as from spanwise aerodynamic moments not encompassedby quasi-static 2-D approximations.

One of the limitations of the developed theoretical simulationis its inability to model accurately the behavior of the system forfrequencies higher than the first resonance. These discrepanciesare thought to be products of nonlinear aerodynamic effectscaused by large stroke and rotation amplitudes. As a test tothis hypothesis, the 75-Hz resonant system’s impulse responseis recorded, featuring small flapping and rotation stroke am-plitudes (<5◦) and thus rendering the aforementioned dynamiceffects not important. Comparing the frequency componentsof the experimental and simulated impulse responses obtainedwith the fast Fourier transform algorithm, shown in Fig. 12, weobserve overlapping peaks at 78 and 129 Hz, suggesting thatthe inertial, force transmission, and stiffness properties of sys-tem components were modeled adequately in the numerical


simulation. Furthermore, the second resonance peak is de-creased from 129 to 95 Hz in actuated flapping experiments,suggesting a large effect of added air mass and potentially otherunmodeled forces on wing dynamics. The two resonance peaksshould not be viewed as the flapping and rotation resonancesof the wing, but rather as different modes of vibration of thesystem. Given that these resonances are reflected as peaks inflapping amplitude, they portray an optimally balanced statebetween internal actuator stresses and aerodynamic forces im-parted on the wing. The first resonance achieves this balancevia approximately symmetric wing rotation, with wing pitchreversal occurring roughly at the time of stroke reversal (theexact timing of rotation is a complex function of many differ-ent parameters), thus resulting in a considerable amount of liftforce. The second resonant peak maximizes flapping amplitudebecause the drag on the wing significantly decreases, thus, al-lowing larger deflections of the actuator. The reduction of dragoccurs because the wing begins to rotate out of phase with flap-ping, such that the flapping and rotation curves of Fig. 7 are 180◦

offset in phase. This onset of second resonance, when the centerof mass of the wing performs very little translational motion, isthe cause for the sharp decrease in experimental lift in the twosystems shortly after the first resonant frequency. The reasonfor the simulation’s overprediction of the second resonant fre-quency is attributed to unmodeled transient aerodynamic forces,such as wake capture [6], [25]. These dynamic effects generatedby the wing’s past trajectory create impulse-like loading on itduring each stroke, thus, exciting the second resonance modemuch earlier than predicted.

It is generally desirable to engineer the system such as to spacethe two resonance modes far apart in the frequency domain.One may attempt to do so by utilizing a stiff wing rotationaljoint, such as not to excite the second mode; however, this issuboptimal from the performance perspective, since the wingwill rotate insufficiently in its first resonance mode, which is themain operating point. Furthermore, stiffening either the wingrotation flexure or the actuator has the effect of shifting bothpeaks of Fig. 12 to higher frequencies, as well as increasingtheir separation. The proximity of the two resonance peaks inan optimized lift system is inherent in the passive wing rotationsystem at hand, where the inertia of the wing is significant andplays a major role in pitch reversal. By reducing the overallinertia/mass of the wing and employing a correspondingly stiffwing rotation joint, it is possible to increase the frequency ofthe second resonance much higher, thus, reducing interferenceand broadening the frequency range of maximum lift.

Finally, although the model generally overpredicts total liftand overestimates the system’s second resonance frequency,its usefulness lies in capturing trends in wing kinematics as afunction of frequency, amplitude, spring stiffness, etc. Thus,the model can be used as a design tool to obtain dimensionsand configuration of the actuator, four-bar transmission, andwing that would ensure wing lift is maximized at a particularfrequency. The first resonance peak is predicted accurately insimulation, but, since it is sensitive to slider crank stiffness,it should be designed in accordance with the bounding valuesof that parameter. Although the predicted magnitude of max-

imum lift is exaggerated, its occurrence at 10–20 Hz higherthan the first resonance peak is well predicted by the simulationand enforced by experimental measurements. Hence, althoughan optimization of system parameters based on the numericalmodel might yield overestimated lift forces, the physical opti-mal point will lie in the close vicinity of the one predicted, as itis defined by trends and general behavior, captured sufficientlyby the numerical model. Furthermore, given that the theoreti-cal model captures well the aerodynamic effects present in thelarge amplitude flapping regime, it should be used as an opti-mization routine for flapping wing systems designed to operatein those conditions. Moreover, given that large flapping ampli-tudes are featured by most natural flapping wing insects, thisfeature should be sought in future system designs, rendering thesimulation tool extremely useful. The optimization of such sys-tems should be done in accordance with both resonance modesof the system (such as the ones portrayed in Fig. 12), with theirproximity dictated by optimality of lift force production at firstresonance.

VI. PROPOSED WING CONTROL METHODOLOGY

The passive wing rotation design allows proper wing motionwithout the weight of additional actuators but, incorporated intoa vehicle, results in an overall underactuated system. Deliberateunderactuated design is not uncommon (see for example un-deractuated mechanical grippers [31]), but can complicate thecontrol of the final platform. A variety of control techniquesexist for certain classes of nonlinear underactuated systems, in-cluding combinations of hybrid and switching control, passivitybased methods, coordinate transformations, and basic lineariza-tion [32]. Adaptive control and model learning techniques havebeen employed in restricted DOF scenarios [33]; however, as ofyet, no overarching control theory for these systems has beendeveloped, making the success of any particular system unsure.Nevertheless, control over the produced wing lift force is a vitalrequirement for the incorporation of a flapping wing mechanisminto any robotic platform, since future vehicle design could relyprimarily on this method for the production of body torques. Wepropose three methods of modulating wing lift: controlling theflapping frequency, the waveform amplitude driving the actua-tor, and the torsional stiffness of the passive rotation joint. Theeffectiveness of these methods is investigated experimentally,resulting in data for Figs. 13 and 14. Note that these experi-ments were performed with a different single-wing prototypethan the rest of the experiments in this paper, having a resonantfrequency of ≈70 Hz; however, the mechanical constructionwas identical to the prototypes discussed earlier.

Figs. 9 and 11 portray the system response to changing flap-ping frequency. From the control standpoint, overall lift in-creases linearly until the resonant frequency, after which pointthe behavior diverges based on system construction. Further-more, given that system efficiency is maximized at resonance,only slight deviations from the main operating frequency areenvisioned, rendering frequency a poor control input.

Alternatively, varying the amplitude of the driving waveformof the actuator modulates the flapping stroke of the wing, thus,


Fig. 13. Simulated and experimental system response to a sweep in the inputwaveform amplitude. Simulation parameters: Vdc = 50 V, frequency = 70 Hz,krot = 50 mN·mm, and d = 5 μN·mm·s. Experimental lift data are averagedover three datapoints.

Fig. 14. Simulated system response to a sweep in the rotational spring stiff-ness. Simulation parameters: Vpp = 150 V, Vdc = 50 V and frequency = 70 Hz,and d = 5 μN·mm·s. Experimental lift data are averaged over three datapoints.

affecting the lift force. The simulated system response to chang-ing peak-to-peak driving amplitude, overlayed with experimen-tal data, is portrayed in Fig. 13. A good correlation betweenexperimental and simulated data is observed, although simu-lated lift is consistently higher than in experiment. The linearbehavior of the overall lift force and large span of producedforces renders this control scenario very promising. Intuitively,an increase in the lift force could always be achieved by in-creasing the flapping amplitude; yet, this approach is limited byelectrical depoling and stress induced crack propagation of thepiezoelectric actuator.

The third envisioned method of controlling lift is modulatingthe stiffness of the rotational flexure that suspends the wing.Unlike the high-bandwidth control architecture required for theMicromechanical Flying Insect [34], the inherently stable na-

Fig. 15. Experimental lift surface. Represents variation of produced lift withinput voltage and torsional spring stiffness. Experimental parameters: Vdc =50 V, frequency = 70 Hz, and d = 5 μN·mm·s.

ture of the passive flapping wing allows slow actuation of therotational joint stiffness in flight. Five flexure joints of vary-ing thickness were manufactured and tested in the experimentalsystem in order to quantify the effect of flexure stiffness. Exper-imental and simulated results are portrayed in Fig. 14 and illus-trate the expected correlation of decreasing lift with increasingflexure joint stiffness. The increase in flapping amplitude withincreasing joint stiffness is attributed to the shifting of both res-onance modes (as illustrated in Fig. 12 to higher frequencies andan increase in Q, a fact experimentally confirmed. Control overthe stiffness of the passive rotation joint is envisioned via anactively stiffness changing or smart material, such as dielectricelastomers [35] or ionic polymer metal composite (IPMC) actu-ators [36] actuators, with design, integration being future works.

Generally, it should be noted that, although each presentedcontrol method has limited application and effect, a combinationof these mechanisms could be employed in a dual wing robot.Indeed, control over rotational flexure stiffness and actuator in-put voltage allows production of any lift force on the surfaceof Fig. 15, thus, maximizing the range of achievable lift forces.The lift force surface resembles a plane, suggesting a simplelinear controller implementation. Asymmetrical lift and, there-fore, body torques could be achieved by altering the rotationalstiffness on only one passive rotation wing joint of a dual wingaerial platform via a morphing polymer. Furthermore, due to areduced number of actuation inputs into the system, some of thesix DOF will likely be coupled and uncontrolled, requiring eitheradditional actuation mechanisms or combining control inputs toachieve full position/orientation of the robot in free space. Itshould also be noted that, although being very advantageous insome respects, the passive pitch reversal wing, proposed in thispaper, would render the aerial platform less agile and mobilethan its active control counterparts. However, the mechanicalsimplicity, reduced complexity of the controller, and naturallystable behavior of wing rotation, that benefit a passive rotationdesign, could potentially bring us closer to the goal of liftoffand stable hovering.


VII. CONCLUSION

It is of great importance for any design project to have a simu-lation tool to predict the performance of the final system. Giventhat our goal is the development of a miniature, controlled, flap-ping flight robot, a full simulation of its subsystem—a single-wing design—could greatly speed up the design process, aswell as enable optimization of its components. The numeri-cal simulation presented in this paper includes a model of thebending piezoelectric actuator, transmission kinematics, and thepassively rotating wing aerodynamics. Two experimental pro-totypes featuring different resonance frequencies were manu-factured from carbon fiber composite material, enabling thecomparison of simulation to experimental results. Overall, therewas broad agreement between the flapping and rotation trajec-tory trends between the numerical simulation and experiment.The major discrepancy between the measured and predicted liftforces is attributed to manufacturing/material imperfections ofthe actuator and/or four-bar transmission, and unmodeled aero-dynamic forces. However, the model is able to predict the mainresonance peak accurately along with general trends of wingkinematics and lift behavior as functions of frequency. Thus,given its limitations in predicting the magnitude of producedlift force, the simulation can be used as a design tool for opti-mization of wing shape, actuator and four-bar geometry, as wellas a future dual-wing flapping robotic platform. Although thepresented experimental prototype features a lift-to-weight ratioof ≈1/6 (measured on a robot employing the same construc-tion), scaled down variants of this design have demonstratedliftoff [14]. Thus, given the direct scalability of the SCM man-ufacturing technique and similarity of Reynolds number, thepresented here theoretical model and analysis is valid and use-ful for scaled down robotic platforms capable of liftoff.

Compared to other wing rotation ideologies, a completelypassive rotation design benefits from mechanical simplicity,passive stability of wing trajectory, and low controller band-width. However, the high dependence of wing dynamics onaerodynamic forces renders the system as lacking in robustnessand consistent performance, requiring an adequate controller tocompensate for these design issues. Control over the amplitudeof the actuator driving waveform, as well as, the potential use ofsmart materials in the wing rotation joint, is envisioned to enableindirect control over the trajectory of the wing. The proposedpassive design is by no means seen as a replacement for activewing control mechanisms, whose ability to generate arbitraryforces and torques are essential for execution of their tasks, butmerely as an alternative flapping platform that may be realizableon a shorter time scale for specific applications.

APPENDIX

FULL EQUATIONS OF MOTION FOR THE

PASSIVE ROTATION WING

Given the definition for the Lagrangian in (13) and perform-ing the differentiations in (14) and (15), the full equations ofmotion of wing motion are obtained. The equation governingthe rotation angle ϕ is the simplest of the two as it does not

involve inertial effects of the driving subsystem

θ (mw RCGβCGcos(ϕ) + Jxz cos(ϕ))

− θ2cos(ϕ)sin(ϕ)(mβ2CG + Jyy − Jzz )

+ ϕ(mβ2CG + Jxx) + ϕkrot = �Maero · �E ′′

1 − dϕ �E ′′1 (20)

where all the constant definitions are presented in the main bodyof this paper.

The equation governing the motion of θ is more complex as itinvolves numerous inertial terms and parameterizations in termsof θ. We need to remind ourselves that according to the notationdiscussed in the text, we have the following:

f ′ ≡ ∂f

∂θ(t), f ′′ ≡ ∂2f

∂θ(t)2 , f ≡ df

dt, f ≡ d2f

dt2. (21)

Then, the equation of motion for θ becomes

θ

(sin2(ϕ)Jyy + cos2(ϕ)Jzz +

(R2

CG +β2

CG

2

)m

− 12β2cos(2ϕ)m +

3∑i=1

mLi(x′2Li + y

′2Li) + meff δ

′2

+2∑

i=1

JLiθ′2Li + JL3

)+ ϕcos(ϕ)(mRCGβCG + Jxz )

− ϕ2sin(ϕ)(mRCGβCG + Jxz ) + θϕsin(2ϕ)(mβ2CG

+ Jyy − Jzz ) +3∑

i=1

mLiθ2(x′

Lix′′Li + y′

Liy′′Li)

+2∑

i=1

JLiθ′Liθ

′′Li θ

2 + meff δ′δ′′θ2

= ( �Mdrive + �Maero) · �E3 − Dϕ. (22)

The coupled equations for θ(t) and ϕ(t) are solved numeri-cally with MATLAB’s ode15s solver. The obtained results arepresented in the main body of the paper.

ACKNOWLEDGMENT

The authors would like to thank R. Smith for his work on themean lift measuring setup, as well as prototype manufacturingand testing.

REFERENCES

[1] A. Bergou, S. Xu, and Z. Wang, “Passive wing pitch reversal in insectflight,” J. Fluid Mech., vol. 591, pp. 321–337, 2007.

[2] R. B. Srygley, A. L. R. Thomas, “Unconventional lift-generating mecha-nism in free-flying butterflies,” Nature, vol. 420, pp. 660–664, 2002.

[3] T. L. Hedrick and T. L. Daniel, “Flight control in the hawkmoth manducasexta: The inverse problem of hovering,” J. Exp. Biol., vol. 209, pp. 3114–3130, 2006.

[4] F. T. Muijres, L. C. Johansson, R. Barfield, M. Wolf, G. R. Spedding, andA. Hedenstrom, “Leading-edge vortex improves lift in slow-flying bats,”Science, vol. 319, pp. 1250–1253, 2008.

[5] S. Fry, R. Sayaman, and M. H. Dickinson, “The aerodynamics of hoveringflight in drosophila,” J. Exp. Biol., vol. 208, pp. 2303–2318, 2005.


[6] J. Birch and M. H. Dickinson, “The influence of wing-wake interactionson the production of aerodynamic forces in flapping flight,” J. Exp. Biol.,vol. 206, pp. 2267–2272, 2003.

[7] M. Hamamoto, Y. Ohta, K. Hara, and T. Hisada, “A fundamental study ofwing actuation for a 6-in-wingspan flapping microaerial vehicle,” IEEETrans. Robot., vol. 26, no. 2, pp. 244–255, Apr. 2010.

[8] S. P. Sane and M. H. Dickinson, “The aerodynamic effects of wing rotationand a revised quasi-steady model of flapping flight,” J. Exp. Biol., vol. 205,pp. 1087–1096, 2002.

[9] R. J. Wood, S. Avadhanula, M. Menon, and R. S. Fearing, “Microroboticsusing composite materials: The micromechanical flying insect thorax,” inProc. IEEE Int. Conf. Robot. Autom., 2003, pp. 1842–1849.

[10] S. Avadhanula and R. S. Fearing, “Flexure design rules for carbon fibermicrorobotic mechanisms,” in Proc. Int. Conf. Robot. Autom., 2005,pp. 1579–1584.

[11] B. Cheng and X. Deng, “Translational and rotational damping of flappingflight and its dynamics and stability at hovering,” IEEE Trans. Robot.,vol. 27, no. 5, pp. 1–16, Oct. 2011.

[12] T. L. Hedrick, B. Cheng, and X. Deng, “Wingbeat time and the scaling ofpassive rotational damping in flapping flight,” Science, vol. 324, pp. 252–255, 2009.

[13] F. V. Breugel, W. Regan, and H. Lipson, “From insects to machines:Demonstration of a passively stable, untethered flapping-hovering micro-air vehicle,” IEEE Robot. Autom. Mag., vol. 15, no. 4, pp. 68–74, Dec.2008.

[14] R. J. Wood, “The first takeoff of a biologically inspired at-scale roboticinsect,” IEEE Trans. Robot., vol. 24, no. 2, pp. 341–347, Apr. 2008.

[15] E. Steltz and R. Fearing, “Dynamometer power output measurementsof miniature piezoelectric actuators,” IEEE/ASME Trans. Mechatronics,vol. 14, no. 1, pp. 1–10, Feb. 2009.

[16] S. N. Mahmoodi, N. Jalili, and M. Daqaq, “Modeling, nonlinear dynamics,and identification of a piezoelectrically actuated microcantilever sensor,”IEEE/ASME Trans. Mechatronics, vol. 13, no. 1, pp. 58–65, Feb. 2008.

[17] R. J. Wood, E. Steltz, and R. S. Fearing, “Optimal energy density piezo-electric bending actuators,” Sens. Actuators A, vol. 119, pp. 476–488,2005.

[18] M. Sitti, D. Campolo, J. Yan, R. S. Fearing, T. Su, D. Taylor, and T. Sands,“Development of PZT/PZN-PT unimorph actuators for micromechani-cal flapping mechanisms,” in Proc. IEEE Robot. Autom. Conf., 2001,pp. 3839–3846.

[19] M. Sitti, “Piezoelectrically actuated four-bar mechanism with two flexi-ble links for micromechanical flying insect thorax,” IEEE/ASME Trans.Mechatronics, vol. 8, no. 1, pp. 26–36, Mar. 2003.

[20] R. J. Wood, “Liftoff of a 60mg flapping-wing mav,” in Proc. IEEE/RSJInt. Conf. Intell. Robots Syst., 2007, pp. 1889–1894.

[21] M. Karpelson, G. Wei, and R. J. Wood, “Milligram-scale high-voltagepower electronics for piezoelectric microrobots,” in Proc. Int. Conf. Robot.Autom., 2009, pp. 2217–2224.

[22] R. J. Wood, E. Steltz, and R. Fearing, “Nonlinear performance limits forhigh energy density piezoelectric bending actuators,” in Proc. IEEE/RSJInt. Conf. Intell. Robots Syst., 2005, pp. 3633–3640.

[23] S. Avadhanula, “The design and fabrication of the MFI thorax based on op-timal dynamics,” Ph.D. dissertation, Dept. Mech. Eng., Univ. California,Berkeley, CA, 2005.

[24] L. I. Sedov, Two-Dimensional Problems in Hydrodynamics and Aerody-namics. New York: Interscience, 1965.

[25] M. H. Dickinson, F. Lehman, and S. P. Sane, “Wing rotation and the aero-dynamic basis of insect flight,” Science, vol. 284, pp. 1954–1600, 1999.

[26] A. Andersen, U. Pesavento, and Z. J. Wang, “Unsteady aerodynamic offluttering and tumbling plates,” J. Fluid Mech., vol. 541, pp. 65–90, 2005.

[27] J. P. Whitney and R. J. Wood, “Aeromechanics of passive rotation inflapping flight,” J. Fluid Mech., vol. 660, pp. 197–220, 2010.

[28] W. B. Dickson, A. D. Straw, C. Poelma, and M. H. Dickinson, “Anintegrative model of insect flight control,” in Proc. AIAA Aerosp. Sci.Meet. Exhib., 2006, pp. 1–19.

[29] R. J. Wood, “Design, fabrication, and analysis of a 3 DOF, 3 cm flapping-wing mav,” in Proc. 2007 IEEE/RSJ Int. Conf. Intell. Robots Syst.,pp. 1576–1581.

[30] R. Dudley, The Biomechanics of Insect Flight: Form, Function and Evo-lution. Princeton, NJ: Princeton Univ. Press, 1999.

[31] S. Montambault and C. M. Gosselin, “Analysis of underactuated mechan-ical grippers,” J. Mech. Design, vol. 123, pp. 367–375, 2001.

[32] R. Olfati-Saber, “Nonlinear control of underactuated mechanical systemswith application to robotics and aerospace vehicles,” Ph.D. dissertation,Dept. Electr. Eng. Comput. Sci., Massachussets Inst. Technol., Cambridge,MA, 2000.

[33] N. Perez-Arancibia, J. Whitney, and R. Wood, “Lift force control offlapping-wing microrobots using adaptive feedforward cancelation sche-mes,” IEEE/ASME Trans. Mechatronics, [Online]. Available:http://ieeexplore.ieee.org, DOI: 10.1109/TMECH.2011.2163317.

[34] X. Deng, L. Schenato, and S. Sastry, “Flapping flight for biomimeticrobotic insects: Part II: Flight control design,” IEEE Trans. Robot., vol. 22,no. 4, pp. 789–803, Aug. 2006.

[35] P. Lotz, M. Matysek, and H. Schlaak, “Fabrication and application ofminiaturized dielectric elastomer stack actuators,” IEEE/ASME Trans.Mechatronics, vol. 16, no. 1, pp. 58–66, Feb. 2011.

[36] C. Zheng and T. Xiaobo, “A control-oriented and physics-based model forionic polymer–metal composite actuators,” IEEE/ASME Trans. Mecha-tronics, vol. 13, no. 5, pp. 519–529, Oct. 2008.

Veaceslav Arabagi received the B.Sc. degree fromthe University of California, Berkeley, in 2006,and the M.Sc. and Ph.D. degrees from CarnegieMellon University, Pittsburgh, PA, in 2008 and 2011,respectively.

He is currently a Research Fellow at Children’sHospital Boston, Boston, MA, where he is involvedin the field of surgical robotics. His research interestsinclude flapping wing flight, small-scale design andmanufacturing techniques, and bioinspired robotics.

Dr. Arabagi placed first in the 2011 ASME Grad-uate Robot Design Competition for his work on the design of a flapping wingaerial robot.

Lindsey Hines received the B.Sc. degree in mechan-ical engineering and the B.A. degree in mathematicsfrom the University of St. Thomas, Saint Paul, MN, in2008, and the M.Sc in robotics from Carnegie MellonUniversity, Pittsburgh, PA, in 2011, where she is cur-rently working toward the Ph.D. degree in robotics.

Her interests include flapping flight and robustcontrol.

Ms. Hines has been awarded both the National Sci-ence Foundation and National Defense Science andEngineering Graduate Fellowship, and placed first in

the 2011 Graduate Robot Design Competition.

Metin Sitti (S’94–A’99–M’99–SM’08) received theB.Sc. and M.Sc. degrees in electrical and electron-ics engineering from Bogazici University, Istanbul,Turkey, in 1992 and 1994, respectively, and the Ph.D.degree in electrical engineering from The Universityof Tokyo, Tokyo, Japan, in 1999.

He was a Research Scientist at the University ofCalifornia, Berkeley, from 1999 to 2002. He is cur-rently a Professor in the Department of MechanicalEngineering and Robotics Institute, Carnegie MellonUniversity, Pittsburgh, PA, where he is also the Di-

rector of the NanoRobotics Laboratory and the Center for Bio-Robotics. Hewas the Adamson Career Faculty Fellow during 2007–2010. His research in-terests include micro/nanorobotics, bioinspired miniature mobile robots, andmicro/nanomanipulation.

Dr. Sitti received the Society of Optics and Photonics Nanoengineering Pi-oneer Award in 2011. He received a National Science Foundation CAREERAward in 2005. From 2008 to 2010, he was the Vice President of Technical Ac-tivities of the IEEE Nanotechnology Council. He was elected as a DistinguishedLecturer of the IEEE Robotics and Automation Society from 2006 to 2008. Healso received the Best Paper Award at the IEEE/RSJ International Conference onIntelligent Robots and Systems in 2009 and 1998, the Best Biomimetics PaperAward at the IEEE Robotics and Biomimetics Conference in 2004, and the BestVideo Award at the IEEE Robotics and Automation Conference in 2002. He isCo-Editor-in-Chief of the Journal of Micro/Nano-Mechatronics.

Documents

2013 a Simulation and Design Tool for a Passive Rotation Flapping Wing Mechanism