J Intell Robot SystDOI 10.1007/s10846-013-9928-1

Efficient Flight Control via MechanicalImpedance Manipulation: Energy Analysesfor Hummingbird-Inspired MAVs

Hosein Mahjoubi · Katie Byl

Received: 31 August 2013 / Accepted: 12 September 2013© Springer Science+Business Media Dordrecht 2013

Abstract Within the growing family of unmannedaerial vehicles (UAV), flapping-wing micro aer-ial vehicles (MAV) are a relatively new fieldof research. Inspired by small size and agileflight of insects and birds, these systems offer agreat potential for applications such as reconnais-sance, surveillance, search and rescue, mapping,etc. Nevertheless, practicality of these vehiclesdepends on how we address various challengesranging from control methodology to morpho-logical construction and power supply design. Areasonable approach to resolving such problemsis to acquire further inspiration from solutionsin nature. Through modeling synchronous mus-cles in insects, we have shown that manipula-tion of mechanical impedance properties at wingjoints can be a very efficient method for control-ling lift and thrust production in flapping-wingMAVs. In the present work, we describe howthis approach can be used to decouple lift/thrustregulation, thus reducing the complexity of flightcontroller. Although of simple design, this con-

H. Mahjoubi (B) · K. BylRobotics Laboratory, Department of Electrical andComputer Engineering, University of California atSanta Barbara, Santa Barbara, CA 93106, USAe-mail: h.mahjoubi@ece.ucsb.edu

K. Byle-mail: katiebyl@ece.ucsb.edu

troller is still capable of demonstrating a highdegree of precision and maneuverability through-out various simulated flight experiments withdifferent types of trajectories. Furthermore, weuse these flight simulations to investigate thepower requirements of our control approach. Theresults indicate that these characteristics are con-siderably lower compared to when conventionalcontrol strategies—methods that often rely on ma-nipulating stroke properties such as frequency ormagnitude of the flapping motion—are employed.With less power demands, we believe our pro-posed control strategy is able to significantly im-prove flight time in future flapping-wing MAVs.

Keywords Bio-inspired MAV · Flapping flight ·Tunable impedance · Variable stiffness actuator ·Maneuverability · Energy efficiency

1 Introduction

Originally introduced during World War I [1],Unmanned Aerial Vehicles (UAVs) have experi-enced a substantial level of growth over the pastcentury; both in military and civilian applicationdomains. Surveillance, reconnaissance, search andrescue, traffic monitoring, fire detection and envi-ronment mapping are just a few examples of theirmany possible applications.

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UAVs can be categorized into three generalgroups: fixed-wing airplanes, rotary-wing vehiclesand flapping wing systems. Each type has differentadvantages and disadvantages. However, as wego smaller in size to design Micro Aerial Vehi-cles (MAVs), the limitations of fixed and rotary-wing technologies become more apparent; theReynolds number decreases as the wing surfacebecomes smaller, resulting in unsteady aerody-namic effects that are still a subject of research.On the other hand, flapping wing designs in-spired by birds and insects offer the most po-tential for high maneuverability at miniaturizedscales. Unparalleled dynamics, speed and agilityare known traits of insects and hummingbirdsthat turn them into the most efficient flyers innature. Furthermore, their ability to adapt withvarying weather and environmental conditionsis phenomenal; hence mimicking their flight dy-namics [2–7] and actuation mechanisms wouldbe a reasonable approach to designing practicalMAVs. Note that in contrast to most birds whosewing stroke is largely restricted within the coro-nal plane, insects—and hummingbirds—employ awing stroke motion that is mostly contained withinthe axial plane [5]. This feature enables the crea-ture to control its thrust force and perform taskssuch as hovering. A schematic description of wingstroke motion in insect flapping flight has beendemonstrated in Fig. 1 [8].

Past research on flapping-wing MAVs hasmainly concentrated on development of flappingmechanisms that are capable of producingsufficient lift for levitation. With such platformsavailable, e.g. Harvard’s Robotic Fly [9], Delf ly[10] and the Entomopter [11], force manipulationand motion control are the next challenges thatmust be faced. Recent work focused on vertical

acceleration and altitude control has led tosome impressive results [12, 13]. Research onforward flight control has been less developedin comparison [14]. In addition to flight controlchallenges, the small size of flapping-wing MAVsposes considerable limits on available space foractuators, power supplies and any sensory orcontrol modules. Although a major portion of theoverall mass is often allocated to batteries [9, 10],the flight time of these vehicles rarely exceedsa few minutes. Thus in terms of efficiency andflight duration, current flapping-wing MAVs arestill far inferior to their fixed and rotary-wingcounterparts.

The majority of works concentrating on forcecontrol employ modifications in wing-beat profileto achieve their objective [15, 16]. While this ap-proach proves to be successful in separate controlof lift [17] or horizontal thrust [18], simultaneouscontrol of these forces is more challenging. Bothforces are directly influenced by velocity of airflow over the wings, a factor that is heavily de-termined by the rate of wing-beat [19]. Therefore,modifying wing-beat profile will lead to couplingbetween lift and thrust forces, i.e. an inconve-nience for motion control and maneuverability ofthe vehicle.

In contrast to stroke actuation methods, onecould investigate possible mechanisms that insectsand small birds employ to regulate aerodynamicforce during various aerial maneuvers. Controlstrategies developed based on such mechanismsmay further improve controllability and energyefficiency of future MAV prototypes. The Tun-able Impedance (TI) method [20] is one suchstrategy. Inspired by the structure of antagonis-tic muscle pairs attached to wing joints—alsoknown as synchronous muscles [21]—in insects

Fig. 1 Wing stroke cyclein insect flapping flight(adapted from [8])

Upstroke

Downstroke

1

6 7

2 3

8 9

4 5

10

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and hummingbirds, we have previously shownthat mechanical impedance properties of eachwing joint can be modeled as a linear torsionalspring [22, 23]. Tunable Impedance approachstates that through manipulating the properties ofthese springs, one can modify the passive pitch re-versal of each wing. Wing pitch reversal [24] playsa significant role in production and manipulationof lift/thrust through regulating the angle of attack(AoA) of the wing relative to air flow [19]. Thus,TI method can indirectly adjust the mean pro-duction of lift and thrust in each individual wing[20, 25]. Although this approach is able to pro-vide a higher degree of controllability comparedto traditional wing-beat modification approaches[25, 26], its lift and thrust manipulation proceduresare not always completely decoupled [26].

In this work, our first goal is to identify theconditions that may reduce coupling between liftand thrust manipulation when TI approach is em-ployed. We then use these findings to developa motion controller that is capable of trackingvarious trajectories in the sagittal plane with aconsiderably high degree of precision as verifiedby our simulated experiments.

Unlike conventional approaches to flapping-wing MAV control, TI does not require anymodifications in flapping properties such as fre-quency or magnitude of the stroke. This suggeststhat a fixed stroke profile can be used duringall operational modes including takeoff, forwardacceleration and hovering. Hence, regardless ofthe maneuver, the MAV will need a relativelyconstant amount of power to flap its wings. Nat-urally, this amount should be large enough toprovide sufficient lift for levitation. Dependingon the desired maneuver, extra power may berequired to change the mechanical impedanceproperties of each wing joint via suitable actua-tors. Through simulated flight experiments, thiswork will show that such changes are often verysmall and do not demand considerable amountsof power. Thus, with a motion controller basedon Tunable Impedance approach, we expect thatan actual MAV’s power requirements during anymaneuver remain almost constant, i.e. close to theamount necessary for hovering. To appreciate thesignificance of this feature, a new set of flightexperiments were simulated. In this group of ex-

periments, wings’ mechanical impedance proper-ties were fixed and aerodynamic forces were con-trolled through changes in shape and frequencyof the flapping profile. Comparing both cases, itis observed that power requirements in the Tun-able Impedance method are considerably lower.Therefore, implementation of a similar controlapproach on an actual MAV may significantly im-prove efficiency and result in longer flight times.

The remainder of this work is organized asfollows. Section 2 briefly explains our model formechanical impedance at the wing joints. Sec-tion 3 reviews the employed aerodynamic modelfor flight simulations. Using this model, we willthen describe the underlying concept of TunableImpedance approach. Lift/thrust decoupling con-ditions are investigated in Section 4. The dynamicmodel of MAV and the structure of designedcontrollers are discussed in Sections 5 and 6, re-spectively. Results of simulated flight experimentsare presented in Section 7. Finally, Section 8 con-cludes this paper.

2 Mechanical Impedance of the Wing Joint

In [23], we reviewed the different types of musclethat most insects employ during flight. Amongthem, synchronous muscles [21] are the only kindthat is directly attached to the wing joint. Thus,they should have a key role in determining me-chanical impedance of the wing joint. This me-chanical impedance is responsible for regulationof changes in wing’s angle of attack (AoA) inresponse to aerodynamic torques [20]. Thus, mod-eling these muscles could further improve ourunderstanding of insect flight.

2.1 Muscle-Joint Connection: Mechanical Model

The antagonistic synchronous muscle pairs inFig. 2 [27] and their connection to the wing jointcan be modeled by the mechanical structure illus-trated in Fig. 3. The behavior of each muscle canbe approximately modeled by a quadratic spring[28], i.e. its output force F is equal to:

F (δi) = 1

2Aδ2

i + Bδi + C, i = 1, 2 (1)

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Fig. 2 An illustration ofinsect’s wing stroke usingsynchronous muscles:a upstroke andb downstroke. Muscles inthe process of contractionare shown in a darkercolor (adapted from [27])

(a) (b)

where A, B and C are constant coefficients andδi represent the corresponding change in length ofeach spring.

In [23], using Eq. 1 we have shown that rota-tional stiffness of the joint in Fig. 3 is derived as:

krot = 2R2 (Axc + B) (2)

Here, R is the radius of the joint and xc representsthe displacement due to co-activation of antago-nistic muscle pair, i.e.:

xc = 1

2(x1 + x2) (3)

The displacement induced by actuator force FMi

on the input end of each spring is shown by xi. Thedifferential displacement of these springs, i.e. xd, isresponsible for adjustment of wing’s equilibriumpitch angle ψ0:

xd = 1

2(x1 − x2) = Rψ0 (4)

It was also shown [23] how an external torque τext

influences the pitch angle ψ of the wing joint:

τext = krot (ψ − ψ0) (5)

Hence, the wing joint can be considered as a tor-sional spring whose stiffness krot and equilibriumpoint ψ0 are respectively controlled through co-activation and differential activation of an an-tagonistic synchronous muscle pair. In Section 3we show how these parameters influence lift andthrust production.

2.2 Active Impedance Regulation:Power Requirements

Contraction of synchronous muscle pairs maychange both stiffness and equilibrium position ofthe wing joint. Like all other muscles, these con-tractions consume energy. In Fig. 3, integration ofEq. 1 with respect to δi shows how much work Wis required to change the length of each spring:

W (δi) = 1

6Aδ3

i + 1

2Bδ2

i + C δi, i = 1, 2 (6)

Since:

δ1 = xc − R (ψ − ψ0) (7)

Fig. 3 A simplifiedmechanical model ofsynchronous muscle-jointstructure. Each muscle ismodeled by a quadraticspring. A force of FMipulls the input end ofeach spring, resulting inthe displacement xi. Theother sides of bothsprings are attached to apulley of radius R whichmodels the joint

z

x'

ψ0

R

Δψ

ψ

FM1

x2

FM2

τextx1

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δ2 = xc + R (ψ − ψ0) (8)

from Eqs. 2 to 4 the overall work is equal to:

W (δ1) + W (δ2) = 2W (xc) + 1

2krot (ψ − ψ0)

2 (9)

The required power for impedance manipulation,i.e. PTI is then calculated by differentiating Eq. 9with respect to time:

PTI = 2F (xc) xc + 1

2krot (ψ − ψ0)

2

− krot (ψ − ψ0) ψ0 (10)

Using Eqs. 1 and 2, it is easy to show that choosingC = 0.5B2/A:

F (xc) xc = 1

16A2 R6k2

rotkrot (11)

which is then replaced in Eq. 10 to yield:

PTI = 1

8A2 R6k2

rotkrot + 1

2krot (ψ − ψ0)

2 −krot (ψ − ψ0) ψ0

(12)

The last expression allows us to estimate powerrequirements for manipulation of krot and ψ0

based on instantaneous values of these parametersand their time derivatives. In Section 7, we willuse Eq. 12 for power estimation during simu-lated flight experiments with Tunable Impedancecontroller.

3 The Aerodynamic Model

With low Reynolds numbers, aerodynamics of in-sect flight has become an active field of researchover the past several decades [2–7, 29–31]. Var-ious unsteady-state mechanisms have been iden-tified that depending on the species, demonstratedifferent amounts of contribution to productionof aerodynamic forces. Among these phenomena,three mechanisms are the most significant in manyflying insects: delayed stall [4], rotational circula-tion [3, 4], and wake capture [4, 29]. Experimentaldata have been used to develop mathematicalexpressions for estimation of forces produced bythese mechanisms [4, 19, 32]. We have previouslyused these results to model the overall aerody-namic force in a hummingbird scale flapping-wingMAV [23]. Here, we will only repeat the final

equations and then proceed to explain the interac-tion between aerodynamic force and mechanicalimpedance of the wing joint. Interested readersare encouraged to review [23] for further detailson this model.

3.1 Wing Shape and Aerodynamic Forces

The components of aerodynamic force at centerof pressure (CoP) of the wing are illustrated inFig. 4a. In this diagram, pitch angle of the wingis represented by ψ . The stroke profile of thewing—i.e. the angle φ in Fig. 4b—is assumed tobe sinusoidal:

φ = φ0 cos (2π fst) + β (13)

where φ0 is the magnitude of stroke and fs rep-resents flapping frequency. β is a slowly varyingbias angle and has an insignificant effect on forceproduction [23]. More details on this parameterwill be provided in Section 6.

Using the quasi-steady-state aerodynamic mo-del of [23] along with Blade Element Method(BEM) [3, 33, 34], the aerodynamic force for thewing shape in Fig. 5 can be estimated by:

FN = 0.0442ρ(CN (ψ) φ + 1.3462ψ

)φR4

W (14)

y

y'

(a)

x

(b)

x'

+ + -

z

zCoP

x'

-

CoP

FD

FX

FY

Downstroke

Downstroke

ψ φ

FN F

Z

CoP Wing's PitchRotation Axis

Wing'sStroke Axis

FD

FT

Fig. 4 a Wing cross-section (during downstroke) at centerof pressure, illustrating the pitch angle ψ . Normal andtangential aerodynamic forces are represented by FN andFT . FZ and FD represent the lift and drag componentsof the overall force. b Overhead view of the wing/bodysetup which demonstrates the stroke angle φ. FX and FYare components of FD which represent forward and lateralthrust, respectively

J Intell Robot SystW

ing'

s St

roke

Axi

s

Rw

CoP

rCoP

Wing's Pitch Rotation Axis

zCoP

z

y'

Fig. 5 The wing shape used in aerodynamic model andflight simulations. The span of the wing is representedby Rw

FT = 0.0442ρCT (ψ) φ2 R4W (15)

where ρ is the density of air and RW representsthe span of the wing. As illustrated in Fig. 4a, FN

and FT are the normal and tangential componentsof the overall aerodynamic force with respect tothe surface of the wing. CN and CT are angle-dependent aerodynamic coefficients [4, 23]:

CN (ψ) = 3.4 cos ψ (16)

CT (ψ) ={

0.4 cos2 (2ψ) , |ψ | ≥ π

4rad

0, otherwise(17)

Lift FZ and drag FD components of the aerody-namic force are calculated as follows:

FZ = −FN sin ψ − FT cos ψ (18)

FD = −FN cos ψ + FT sin ψ (19)

Forward and lateral thrust, i.e. FX and FY , respec-tively, are also defined as illustrated in Fig. 4b:

FX = FD cos φ (20)

FY = −FD sin φ (21)

Finally, it is important to note that the center ofpressure [23] in Fig. 5 is located at:

zCoP = 0.0673RW (22)

rCoP = 0.7221RW (23)

3.2 The Tunable Impedance Approach:A Review

As it was stated in Section 2.1, mechanical im-pedance properties of the joint tend to oppose anyexternal torque on the wing according to Eq. 5.Aerodynamic force and dynamics of the winginfluence the rotation of the wing via the followingtorque expression:

τ = zCoP FN − bψψ − Jψψ (24)

in which Jψ and bψ are the moment of inertia andpassive damping coefficient of the wing along itspitch rotation axis, respectively. As illustrated inFig. 5, zCoP is the distance of CoP from this axis.

According to Newton’s Equations of Motion,the left-hand sides of Eqs. 5 and 24 must be equal,hence resulting in:

Jψψ + bψψ + krot (ψ − ψ0) − zCoP FN = 0 (25)

Using the presented aerodynamic model inSection 3.1, Eq. 25 can be numerically solved tocalculate changes in ψ and FN for any strokeprofile and mechanical impedance values. Otheraerodynamic profiles will then be available viaEq. 15 and Eqs. 18–21.

The values of wing parameters used in all futuresimulations are reported in Table 1. Since theemployed MAV model is roughly the same sizeas an average hummingbird, stroke parameters

Table 1 Physicalcharacteristics used inSection 3

Symbol Description Value

ρ Air density at sea level 1.28 kg· m−3

RW Average span of each wing in an average hummingbird 8 × 10−2 mJψ Moment of inertia of each wing (pitch rotation) 1.564 × 10−8 N· m· s2

bψ Passive damping coefficient of each wing (pitch rotation) 5 × 10−6 N· m· smbody Estimated mass of a two-winged MAV with dimensions 4 × 10−3 kg

of an average hummingbirdg Standard gravity at sea level 9.81 m· s−2

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during hovering flight are set to φ0 = 60◦ andfs = 25 Hz [21]. Similar to a hummingbird, theMAV model has two wings and throughout thiswork, it is assumed that they have always the samestroke profiles and mechanical impedance proper-ties. Under these assumptions, symmetry dictates

that overall lateral thrust and roll/yaw torques arezero; hence motion is effectively limited withinthe sagittal plane. This allows us to concentrateonly on lift FZ and forward thrust FX of a singlewing. Roll and yaw responses of the model arepreviously investigated in [23, 26].

0 0.01 0.02 0.03 0.04-60

-40

-20

0

20

40

60

80

(deg

)

(a) Pitch Angle of the Wing vs. krot

0 0.01 0.02 0.03 0.04-1

0

1

2

3

Lif

t (g

-For

ce)

(b) Instantaneous and Average Lift: FZ

0 0.01 0.02 0.03 0.04-6

-4

-2

0

2

4

6

(c) Instantaneous and Average Thrust: FX

Time (s)

Thr

ust

(g-F

orce

)

0 0.01 0.02 0.03 0.04-60

-40

-20

0

20

40

60

80

(deg

)

(d) Pitch Angle of the Wing vs. ψ0

0 0.01 0.02 0.03 0.04-1

0

1

2

3

Lif

t (g

-For

ce)

(e) Instantaneous and Average Lift: FZ

0 0.01 0.02 0.03 0.04-6

-4

-2

0

2

4

6

Time (s)

Thr

ust

(g-F

orce

)

(f) Instantaneous and Average Thrust: FX

krot

=k*

krot

=2.5k*

krot

=5k*

ψ0=0 o

ψ0= -20 o

ψ0=20 o

φ φ

Fig. 6 a Pitch angle evolution of one wing for ψ0 = 0◦and different values of krot (k∗ = 1.5 × 10−3 N· m/rad).The corresponding instantaneous (blue) and average (red)lift and forward thrust are plotted in b and c respectively.d Pitch angle evolution of one wing for krot = k∗ and

different values of ψ0. The corresponding instantaneous(blue) and average (red) lift and forward thrust are plottedin e and f, respectively. All forces are normalized by theestimated weight of the vehicle

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The tunable impedance approach suggests thatby modification of impedance properties, one canmanipulate pitch profile of either wing while itsstroke profile remains unperturbed. In fact, thevalue of krot primarily determines the range ofvariations in ψ (Fig. 6a) which in turn affectsthe magnitude of aerodynamic forces (Fig. 6b, c).Note that when ψ0 = 0◦, pitch and thrust profilesare odd-symmetric, i.e. the average thrust is closeto 0. Introducing a nonzero value of ψ0 perturbsthis symmetry (Fig. 6f) by biasing the pitch profile(Fig. 6d). In short, tunable impedance allows usto control overall lift and thrust by manipulatingkrot and ψ0, respectively [23]. From Fig. 6, al-though instantaneous forces are time-dependent,their average values remain constant as longas mechanical impedance properties and strokeparameters are unperturbed. We can use thesemean values to further investigate the influenceof impedance/stroke parameters in lift/thrustproduction.

Figure 7a, b illustrate the changes in mean FZ

and FX of a single wing as various krot and ψ0 pairsare examined. All force values are normalized bythe weight of the model. At ψ0 = 0◦, instanta-neous forward thrust has an approximately zeromean value regardless of the selected krot (Fig. 7c).On the other hand, mean value of lift significantlyvaries as krot is changed. Particularly, when krot =kop = 3.92 × 10−3 N· m/rad, two wings will be ableto produce just enough lift to levitate the model– i.e. 0.5 g-Force per wing. Therefore, we choosethis point as the nominal mechanical impedancevalues for hovering. Note that when krot = kop,small changes in ψ0 slightly influence mean lift butalso result in significant nonzero values of meanthrust (Fig. 7d).

From Fig. 7c, d, around krot = kop and ψ0 =0◦, average lift and thrust are respectively pro-portional to stiffness and equilibrium angle ψ0 ofthe joint. Using these relationships, the TunableImpedance method enables the model to regulate

Fig. 7 a Mean FZ vs. krotand ψ0. b Mean FX vs.krot and ψ0. c Meanvalues of FZ and FX vs.krot when ψ0 = 0◦. At krot= kop = 3.92 × 10−3 N·m/rad, two wings are ableto produce just enoughlift to levitate the model.d Mean values of FZ andFX vs. ψ0 when krot =kop. All force values arenormalized by the weightof the model. Strokeprofile is alwayssinusoidal with constantparameters: φ0 = 60◦ andfs = 25 Hz

-5

0 -500

50

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 (deg)log10 krot

Mea

n F

Z

(g-F

orce

)

-5

0 -500

50

-0.5

0

0.5

0 (deg)log10 krot

Mea

n F

X

(g-F

orce

)

-5 -4 -3 -2 -1 0-0.2

0

0.2

0.4

0.6

0.8

log10 krot

(g-F

orce

)

(c) Mean FZ and FX vs. krot ( 0 = 0o)

-60 -40 -20 0 20 40 60-0.4

-0.2

0

0.2

0.4

0.6

0 (deg)

(d) Mean FZ and FX vs. 0 (krot = kop)

(g-F

orce

)

LiftThrust

LiftThrust

(a) (b)

0.5

kop= 0.00392 Nm/rad

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aerodynamic forces during various agile flight ma-neuvers [23]. However, Fig. 7 also demonstratesthat lift and thrust are influenced by both mechan-ical impedance parameters. Therefore, an attemptto control one force through adjustment of thecorresponding mechanical impedance parametermay result in perturbation of the other force, thuscomplicating simultaneous control of both forces.In Section 4 we will investigate possible conditionsthat reduce the coupling between lift and thrustforces, hence allowing us to employ simpler con-trol structures.

The present work also concentrates on flightefficiency and power requirements of the TunableImpedance approach. To do so, our proposedmethod will be compared to a conventional ad hoccontrol approach that manipulates frequency andshape of the stroke profile to adjust production ofaerodynamic forces. From Eqs. 13–15, the magni-tude of overall instantaneous aerodynamic forceis approximately a quadratic function of fs. Thisrelationship can also be observed in Fig. 8a wherechanges in mean lift and thrust vs. fs are plottedfor a single wing. Note that in this case, mechani-cal impedance properties are always constant, i.e.krot = kop and ψ0 = 0◦.

From this diagram, stroke frequency can beused to control lift. However, mean thrust isunaffected by changes in fs. Note that as longas ψ0 = 0◦ and φ has a perfect sinusoidal wave-form, thrust profile over one stroke cycle remains

odd-symmetric [23]. Therefore, mean thrust willbe close to zero. One way of producing nonzerothrust values is to employ asymmetric strokeprofiles. This method is also known as Split Cycle[15]. Normally, upstroke and downstroke in onecycle take the same amount of time, hence theirindividual mean thrusts are equal and in oppositedirections. By increasing the flapping speed inone part and reducing it in the other one, it ispossible to disturb this balance and create forwardor backward thrust. To this end, we define Tds asthe ratio of time spent in downstroke to durationof the full stroke cycle. By definition, Tds = 0.5 isequivalent to a symmetric stroke profile and zeromean thrust. Figure 8b shows that a 20 % changein this value slightly increases mean lift, but atthe same time, a mean thrust of approximately0.25 g-Force per wing will be achievable. Hence,simultaneous manipulation of fs and Tds can beused as a direct approach to force/motion controlin flapping-wing MAVs.

4 Decoupling Conditions for Lift/Thrust Control

To investigate the possibility of decoupledlift/thrust control via tunable impedance ap-proach, we first revisit the diagrams in Fig. 7. FromFig. 7a and d, small changes in the value of ψ0 havelittle influence on overall lift. Similarly, Fig. 7b, cshow that small changes in krot around kop only

Fig. 8 a Mean values ofFZ and FX vs. fs whenTds = 0.5. At fs = 25 Hz,two wings are able toproduce just enough liftto levitate the model.b Mean values of FZ andFX vs. Tds when fs =25 Hz. All force valuesare normalized by theweight of the model.Mechanical impedanceparameters are alwaysconstant: krot = kop andψ0 = 0◦

10 20 30 40-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

fs (Hz)

(g-F

orce

)

(a) Average FZ and FX vs. fs

LiftThrust

0.2 0.4 0.6 0.8-1

-0.5

0

0.5

1

1.5

(g-F

orce

)

Tds

(b) Average FZ and FX vs. Tds

LiftThrust

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slightly affect overall thrust. Hence, as a trade-offbetween these characteristics and production ofsufficiently large forces, we have chosen to limitthe mechanical impedance parameters such thatkrot ∈ [0.2, 2] × 10−2 N·m/rad and |ψ0| ≤ 20◦.

The force values in Fig. 7 are calculated for con-stant impedance parameters. However, the linearrelationships between steady-state values suggestthat a linearized model around krot = kop andψ0 = 0◦ will be able to provide a reasonable ap-proximation of dynamic behavior of these forcesin response to varying impedance properties. Tofind this linear model, we employ system iden-tification techniques—computation/smoothing ofEmpirical Transfer Function Estimates followedby subspace identification method [35] for transferfunction matching—in order to estimate the com-ponents of nonlinear MIMO system that relatesimpedance parameters to aerodynamic forces:

δFZ (s) = GZ K (s) δK (s) + Z (s) (26)

δFX (s) = (1 + X K (s)) GX (s) ψ0 (s) (27)

δFZ and δFX respectively represent the changesin average lift and forward thrust due to variationsof krot and ψ0 values around krot = kop and ψ0 =0◦. Note that δK = log10(krot / kop). GZ K is thetransfer function that relates δK to average liftwhile GX represents the transfer function fromψ0 to average thrust. The impact of ψ0 varia-

tions on average lift and stiffness variations onaverage thrust are modeled by disturbance termsZ and X K, respectively. Figure 9 illustrates acloser look at Fig. 7a, b around the chosen opera-tion point. These diagrams support our particularchoice of disturbance terms in Eqs. 26 and 27.

In all system identification experiments, lim-its of |δK| < 0.05 and |ψ0| < 0.1 rad have beenenforced. The Bode diagrams of identified trans-fer functions are plotted in Fig. 10. Figure 11illustrates the frequency profile of maximum gainfrom each mechanical impedance parameter tothe corresponding disturbance term. In low fre-quencies, both GZ K and GX have steady-stategains close to 0 dB, while in comparison, themaximum gains in Fig. 11 are well below unity.Therefore, we expect that simultaneous manipu-lation of mechanical impedance parameters in thelow frequency range has a less perturbing effecton individual lift and thrust control mechanisms.

To choose a suitable upper limit for this range,three groups of numerical experiments have beenconducted. In the first group, sinusoidal wavesof magnitude 0.05 and various frequencies wereused for δK while ψ0 was set to 0 rad. In thiscase, changes in average lift and thrust are respec-tively treated as ideal FZi and disturbance FXd. Inthe second group, sinusoidal waves of magnitude0.1 rad and various frequencies were used for ψ0

while krot was kept at kop. In this case, changesin average lift and thrust are respectively treatedas disturbance FZ d and ideal FXi. Finally, the

Fig. 9 a Average lift forimpedance values aroundkrot = kop and ψ0 = 0◦.b Average thrust forimpedance values aroundkrot = kop and ψ0 = 0◦.All force values arenormalized by theestimated weight of thevehicle. Note that δK =log10(krot/kop)

-0.2

0

0.2 -10

0

100.2

0.3

0.4

0.5

0.6

0.7

0 (deg)K

Ave

rage

FZ

(g-F

orce

)

-0.2

0

0.2-10

0

10

-0.2

-0.1

0

0.1

0.2

0 (deg)K

Ave

rage

FX

(g-F

orce

)

(b)(a)

J Intell Robot Syst

Fig. 10 Bode diagrams ofthe identified transferfunctions a from δK toaverage lift, i.e. GZ K andb from ψ0 to averagethrust, i.e. GX

10-2

100

102

104

-80

-60

-40

-20

0

(a) GZK

Mag

(dB

)

10-2

100

102

104

50

100

150

200

Frequency (Hz)

Pha

se (

deg)

10-2

100

102

104

-80

-60

-40

-20

0

(b) GXΨ

Mag

(dB

)

10-2

100

102

104

50

100

150

200

Frequency (Hz)

Pha

se (

deg)

third group of experiments uses both describednonzero inputs at the same time. Lift and thrustchanges in this group are the overall results ofimpedance manipulation, i.e. FZ o and FXo. Thefrequency profiles of correlation between ideal(disturbance) and overall lift outputs are plottedin Fig. 12a, b. Similarly, frequency profiles of cor-relation between ideal (disturbance) and overallthrust outputs are plotted in Fig. 12c, d.

From Fig. 12, when the frequency content ofboth inputs is below 10 Hz, the correlation be-tween ideal and overall profile of either force isabove 0.917 while the correlation between dis-turbance and overall values remains below 0.239.Thus, by employing low-pass filters to remove thefrequency content of δK and ψ0 above 10 Hz, cou-pling between lift and thrust manipulation couldbe reduced significantly.

Fig. 11 Frequency profileof maximum gain a fromψ0 to lift disturbanceZ and b from δK tothrust disturbance X K

10-2

100

102

104

-100

-90

-80

-70

-60

-50

-40

-30

-20(a) ΔZΨ

(s)/Ψ0(s)

Frequency (Hz)

Mag

(dB

)

10-2

100

102

104

-100

-90

-80

-70

-60

-50

-40

-30

-20(b) ΔXK(s)/δK(s)

Frequency (Hz)

Mag

(dB

)

J Intell Robot Syst

Fig. 12 Correlationcoefficient vs. frequencyof mechanical impedancemanipulation for a FZiand FZ o, i.e. ideal andoverall lift, b FZ d andFZ o, i.e. disturbance andoverall lift, c FXi andFXo, i.e. ideal and overallthrust, d FXd and FXo,i.e. disturbance andoverall thrust. Here, fand fK respectivelyrepresent the frequencyof employed sinusoidalsignals for ψ0 and δK

0

10

20

0

10

20

0

0.5

1

f K (Hz)f (Hz)

(a) (FZi , FZo)

0

10

20

0

10

20

0

0.5

1

f K (Hz)f (Hz)

(c) (FXi , FXo )

0

10

20

0

10

20

0

0.5

1

f K (Hz)f (Hz)

(b) (FZd, FZo)

0

10

20

0

10

20

0

0.5

1

f K (Hz)f (Hz)

(d) (FXd, FXo )

5 Dynamic Model of the Flapping-Wing MAV

The free-body diagram of a typical two-wingedflapping-wing MAV model is illustrated in Fig. 13.

This body structure is inspired the proposed ve-hicle in [9]. The shape of each wing is as demon-strated in Fig. 5 while its other characteristics areas reported in Table 1.

Fig. 13 Free-bodydiagram of a two-wingedflapping-wing MAV:a frontal and b overheadviews. H, R and U are thedistance components ofcenter of pressure (CoP)of each wing from centerof mass (CoM) of themodel. For simulationpurposes, Euler angles inTait-Bryan XYZconvention are employedto update the orientationof the body

y

(a) Frontal View

y

x

z

x

FZ l

CoP

FX lF

X r(b) Overhead View

Roll

Yaw

Pitch

CoM

R l

H r

U rz

CoP

R r

H l

U l

CoPCoP

FZ r

FY lF

Y r

J Intell Robot Syst

Through employing Newton’s equations of mo-tion, a six degree-of-freedom (DoF) mathematicalmodel can be derived in this body frame [19]:

mbody

(v + �ω × �v

)= �Fl + �Fr − mbody �G − b v |�v| �v

(28)

Jbody �ω + �ω × Jbody �ω = �Dl × �Fl + �Dr × �Fr − bω �ω(29)

where:

�Fl =⎡

⎣Fl

X−Fl

YFl

Z

⎤

⎦ , �Fr =⎡

⎣Fr

XFr

YFr

Z

⎤

⎦ (30)

and:

�Dl =⎡

⎣Ul

Rl

Hl

⎤

⎦ , �Dr =⎡

⎣Ur

−Rr

Hr

⎤

⎦ (31)

Here, translational and angular velocities of theMAV’s body are represented by vectors v and ω,respectively. H, R and U are the distance com-ponents of center of pressure of each wing fromthe overall center of mass (CoM), as shown inFig. 13. Superscripts r and l respectively refer tothe right and left wings. The vector G expressesgravity in the body coordinate frame. To keeptrack of the orientation of the body with respectto the reference coordinates, Euler angles in theTait-Bryan XYZ convention are employed. The

coefficient of viscous friction is represented by b v

while the parameter bω is used to model the effectof passive rotational damping, as discussed in [36].

Mass of the body, i.e. mbody, is assumedto be 4 × 10−3 kg for a vehicle of desired scale(Table 1). The body shape in Fig. 13 is chosensince its corresponding inertia matrix relative tothe center of mass (CoM), i.e. Jbody, has in-significant nondiagonal terms and thus, can beapproximated by:

Jbody =⎡

⎣Jroll 0 00 Jpitch 00 0 Jyaw

⎤

⎦ (32)

where Jroll, Jpitch and Jyaw are the body’s individ-ual moments of inertia around x, y and z axes,respectively. All related physical parameters ofthe body are listed in Table 2.

6 Flight Controllers

Based on the discussions in Section 3.2, twodifferent controllers are developed to control thesimulated flight of dynamic model described inSection 5. As it was noted earlier, motion ofthe model will be restricted within sagittal plane,hence both controllers only deal with regulationof lift and forward thrust forces. In the remainderof this section, we will review the details andstructure of these controllers.

Table 2 PhysicalCharacteristics of themodeled MAV’s body

Symbol Description Value

bω Passive damping coefficient of the body (rotation) 3 × 10−3 N· m· sbv Viscous friction coefficient of the body when 4 × 10−4 N· m−2· s2

moving in the airJpitch = Jroll Pitch and roll moments of inertia of the body 4.38 × 10−6 N· m· s2

relative to CoMJyaw Yaw moment of inertia of the body relative to CoM 1.15 × 10−7 N· m· s2

H(ψ = 0◦) Distance of CoP from axial plane of the body 2.89 × 10−2 m(xy in Fig. 13) when ψ = 0◦

R(φ = 0◦) Distance of CoP from sagittal plane of the body 5.78 × 10−2 m(xz in Fig. 13) when φ = 0◦

U(φ = 0◦) Distance of CoP from coronal plane of the body 5.8 × 10−3 m(yz in Fig. 13) when φ = 0◦

Wbody Body width at the base of wings 1.16 × 10−2 m

J Intell Robot Syst

Fig. 14 Block diagram ofthe proposed TunableImpedance motioncontroller in interactionwith MAV model. Cutofffrequency f c of eachlow-pass filter is set to10 Hz. Both wings employsimilar values of β, krotand ψ0 at all times. Thestroke profile propertiesare always constant: i.e.φ0 = 60◦, fs = 25 Hz andTds = 0.5

+

+-

-

-+

PD Pitch Controller

X ref

Z ref

0

LPF withfc = 10 Hz

×

kop

pitch ref = 0o

X.

pitch

Sinusoid WaveGenerator

LQR Thrust Controller

LQR Lift ControllerZ

.Z

X

0 = 60o

fs = 25 Hz

+

LPF withfc = 10 Hz

LPF withfc = 10 Hz

K

MAVModel

f(x)=10 x

krot

6.1 Controller 1: Tunable Impedance Approach

The block diagram in Fig. 14 shows how the Tun-able Impedance controller is structured. To calcu-late appropriate LQR gains, the estimated trans-fer functions in Section 4 have been used to de-velop a linear model of the system from mechan-ical impedance parameters to vertical/horizontalposition and velocity. The final result is as follows:

δK = 25 (Z − Zref) + 2Z (33)

ψ0 = 2000 (X − Xref) + 100X (34)

where X and Z are respectively the horizontal andvertical positions of the model in sagittal plane.Xref and Zref define the reference trajectory ofmotion within this plane. To reduce coupling(Fig. 7 c, d), the outputs of these controllers arechosen to be limited such that krot ∈ [0.2, 2] × 10−2

N·m/rad and |ψ0| ≤ 20◦. As argued in Section 4,presence of low-pass filters on the controller out-puts further improves decoupling [37]. Note thatfor controller 1, the stroke profile properties ofboth wings are always constant: φ0 = 60◦, fs =25 Hz and Tds = 0.5.

Pitch angle of the body θpitch tends to be unsta-ble. To keep this angle small, i.e. to maintain the

Fig. 15 Block diagram ofcontroller 2 in interactionwith MAV model. Bothwings employ similarvalues of β, fs and Tds atall times. The magnitudeof stroke and mechanicalimpedance parametersare always constant: i.e.φ0 = 60◦, krot = kop andψ0 = 0◦

+

+-

-

-+

PD Pitch Controller

X ref

Z ref

LPF withfc = 10 Hz

pitch ref = 0o

X.

pitch

Z.

Z

X

Thrust Controller (Gain)

+

Tds

fs

0 = 60o Function

Generator

Lift Controller (Gain)

MAVModel

J Intell Robot Syst

upright orientation of the model, we chose to biasthe stroke profile with a variable amount repre-sented by β. The details of this control mechanismare discussed in [23]. The value of β is adjustedby a PD sub-controller which is tuned to 10 +0.05s through trial and error. The output of thiscontroller is limited such that |β| ≤ 15◦.

6.2 Controller 2: Split Cycle and FrequencyManipulation

Figure 15 demonstrates the block diagram of thesecond controller. In this approach, mechanicalimpedance parameters are always constant: krot =kop and ψ0 = 0◦. Here, lift and forward thrust are

respectively controlled through changes in fs andTds. The pitch sub-controller in this schematic isidentical to the one described earlier for controller1. The outputs of other sub-controllers are calcu-lated as:

fs = 25 − 187.5 (Z − Zref) − 12.5Z (35)

Tds = 0.5 − 2.5 (X − Xref) − 0.2X (36)

and are limited to fs ∈ [14.3, 28.5] Hz and Tds ∈[0.4, 0.6]. The gains in Eqs. 35 and 36 are tunedvia trial and error such that in both control meth-ods, simulated responses of the MAV to similarreferences remain close. The limits on fs andTds are imposed to restrict achievable values of

Fig. 16 Simulated resultsfor tracking a squaretrajectory using controller1: a displacement along Xand b Z axes, c pitchequilibrium of the wingsψ0, d stiffness of the jointskrot, e pitch angle of thebody θpitch, f stroke biasangle of the wings β andg overall trajectory in XZplane

0 5 10 150

0.5

1

(a) Displacement along X-axis

X (m

)

0 5 10 150

0.5

1

(b) Displacement along Z-axis

Z (m

)

0 5 10 15-10

0

10(c) Pitch Equilibrium of the Wings

0 (d

eg)

0 5 10 152

4

6x 10

-3 (d) Stiffness of the Joints

Time (s)

k rot

(N.m

/ rad)

0 5 10 15-2

0

2

pitc

h (d

eg)

(e) Pitch Angle of the Body

0 5 10 152

4

6

8

10

(d

eg)

(f) Stroke Bias Angle of the Wings

Time (s)

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

(g) Trajectory in XZ plane

X (m)

Z (m

)

XRefXActual

ZRefZActual

ReferenceActual

J Intell Robot Syst

aerodynamic forces within the same range as con-troller 1 (Fig. 8a, b).

The output of the function generator in Fig. 15is originally a sinusoid of constant magnitude φ0 =60◦ and variable frequency fs. Depending on thevalue of Tds, this waveform is further manipulatedsuch that two consecutive half-cycles have appro-priate durations.

7 Simulated Experiments

The results of several simulated flight experimentswill be discussed in this section. In the first group,we only employ controller 1 to investigate theperformance of Tunable Impedance approach and

its potential for decoupled control of lift/thrustforces. In the second group, experiments are re-peated with both controllers. Overall performanceand energy consumption in both cases can then becompared.

7.1 Decoupled Lift/Thrust Control via TunableImpedance Approach

Using controller 1, motion of the model alongvarious trajectories in the XZ plane has beensimulated. In each one of these experiments, themodel is initially hovering at X = 0 m and Z =0 m, i.e. Xref = 0 m and Zref = 0 m. Later, newreference position-time profiles are introduced tosub-controllers that guarantee a smooth velocity

Fig. 17 Simulated resultsfor tracking a lineartrajectory with a slope of1 using controller 1:a displacement along Xand b Z axes, c pitchequilibrium of the wingsψ0, d stiffness of the jointskrot, e pitch angle of thebody θpitch, f stroke biasangle of the wings β andg overall trajectory in XZplane

0 5 100

0.5

1

(a) Displacement along X-axis

X (

m)

0 5 100

0.5

1

(b) Displacement along Z-axis

Z (

m)

0 5 10-10

0

10(c) Pitch Equilibrium of the Wings

0 (d

eg)

0 5 102

4

6x 10

-3 (d) Stiffness of the Joints

Time (s)

k rot

(N.m

/ rad)

0 5 10-2

0

2

pitc

h (d

eg)

(e) Pitch Angle of the Body

0 5 102

4

6

8

10

(d

eg)

(f) Stroke Bias Angle of the Wings

Time (s)

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

(g) Trajectory in XZ plane

X (m)

Z (

m)

XRef

XActual

ZRef

ZActual

ReferenceActual

J Intell Robot Syst

profile over desired trajectory. We observed thatas long as acceleration demands of these profilesare within the capabilities of the model—primarilydue to enforced limits on δK and ψ0—trackingwill be achieved with a considerably high degreeof precision. The results of simulated experimentswith three important types of trajectories are pre-sented next.

7.1.1 Strictly Horizontal (Vertical) Maneuvers

As the most trivial group of maneuvers, the MAVmust be able to move along one axis while main-taining a constant position with respect to theother axis. Figure 16 demonstrates the results of

tracking a square trajectory that requires a combi-nation of all these maneuvers, i.e. takeoff, landingand forward/backward motion. As it is illustrated,controller 1 successfully handles such maneuversand keeps the model close to reference trajectory(Fig. 16g). Note that during each sub-maneuver,only one of the mechanical impedance parametersexperiences significant changes while the otherone remains almost constant (Fig. 16c, d).

7.1.2 Simultaneous Motion along Two Axes:A Linear Trajectory

To further investigate the effectiveness of ourproposed approach in decoupling lift and thrust

Fig. 18 Simulated resultsfor tracking a partiallycircular trajectory usingcontroller 1:a displacement along Xand b Z axes, c pitchequilibrium of the wingsψ0, d stiffness of the jointskrot, e pitch angle of thebody θpitch, f stroke biasangle of the wings β andg overall trajectory in XZplane

0 5 100

0.5

1

(a) Displacement along X-axis

X (

m)

0 5 100

0.5

1

(b) Displacement along Z-axis

Z

(m)

0 5 10-10

0

10(c) Pitch Equilibrium of the Wings

0 (d

eg)

0 5 102

4

6x 10

-3(d) Stiffness of the Joints

Time (s)

k rot

(N.m

/ rad)

0 5 10-2

0

2

pitc

h (d

eg)

(e) Pitch Angle of the Body

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

(g) Trajectory in XZ plane

X (m)

Z (

m)

XRef

XActual

ZRef

ZActual

0 5 102

4

6

8

10

(d

eg)

(f) Stroke Bias Angle of the Wings

Time (s)

ReferenceActual

J Intell Robot Syst

control, different trajectories with displacementsalong both X and Z axes were examined. A sim-ple example of such trajectories is a straight linewith a slope of 1. Simulation results for the corre-sponding experiment are illustrated in Fig. 17. Acombination of forward flight and takeoff motionstarting at t = 1 s enables the model to reach X= 1 m and Z = 1 m by t = 3 s. The simulatedMAV maintains this position until t = 4 s when thedirection of motion is reversed. Through simul-taneous descent and backward flight, the modelreturns to its original position at t = 6 s. All thewhile, the slope of its trajectory remains close to 1(Fig. 17g).

7.1.3 Movement Along Curves

Curves are a more complex case of trajectorieswith simultaneous displacement along both X andZ axes. The results of an experiment with a par-tially circular reference trajectory are plotted inFig. 18. The model still manages to follow this tra-jectory with reasonable precision (Fig. 18g). Notethat compared to other trajectories, correctionof motion over curves requires slower changesin impedance parameters (Fig. 18c, d). This ispartially due to the fact that stable motion alongsuch trajectories is rarely accompanied by highvelocities and/or acceleration rates.

Fig. 19 Simulated resultsfor hovering withcontroller 1:a displacement along Xand b Z axes, c pitchequilibrium of the wingsψ0, d stiffness of the jointskrot, e pitch angle of thebody θpitch, f stroke biasangle of the wings β andg overall energyconsumption ETI

0 0.5 1-0.01

0

0.01(a) Displacement along X-axis

X (

m)

0 0.5 1-0.01

0

0.01(b) Displacement along Z-axis

Z (

m)

0 0.5 1-2

0

2

4(c) Pitch Equilibrium of the Wings

0 (d

eg)

0 0.5 12

4

6x 10

-3 (d) Stiffness of the Joints

k rot

(N.m

/ rad)

0 0.5 1-2

0

2

pitc

h (d

eg)

(e) Pitch Angle of the Body

0 0.5 12

4

6

8

10

(d

eg)

(f) Stroke Bias Angle of the Wings

Time (s)0 0.5 1

0

0.5

1

1.5

2

2.5(g) Overall Energy Consumption

Time (s)

ET

I (J

)

XRef

XActual

ZRef

ZActual

J Intell Robot Syst

7.2 Method Comparison: Performanceand Energy Consumption

Several simulated trajectory tracking experimentsare chosen to be repeated with both controllersof Section 6. Some of these results are presentedhere to compare flight performance and powerrequirements. To estimate stroke power in bothsets of experiments, stroke torque of each wing iscalculated first:

τstroke = FDrCoP + bφφ + Jφ φ (37)

Here, bφ = 1 × 10−5 N·m·s is the passive dampingcoefficient of each wing when rotating around its

stroke axis while Jφ represents the wing’s momentof inertia along this axis. The calculated value forthis parameter is equal to 4.894 × 10−7 N·m·s2

(Fig. 5 with wing mass = 2 × 10−4 kg).The energy required for both wings to follow

a specific stroke profile, i.e. Estroke is then esti-mated by:

Estroke = 2∫ t

0

∣∣τstrokeφ

∣∣dt (38)

Since actuation in the second control approach isonly limited to changes in stroke profile, Eq. 38is in fact an estimate of this method’s overallenergy consumption. As for controller 1, changesin mechanical impedance properties require extra

Fig. 20 Simulated resultsfor hovering withcontroller 2:a displacement along Xand b Z axes,c downstroke durationratio Tds, d strokefrequency fs, e pitchangle of the body θpitch,f stroke bias angle of thewings β andg overall energyconsumption Estroke

0 0.5 1-0.01

0

0.01(a) Displacement along X-axis

X (

m)

0 0.5 1-0.01

0

0.01(b) Displacement along Z-axis

Z (

m)

0 0.5 10.4

0.5

0.6(c) Downstroke Duration Ratio

Tds

0 0.5 123

24

25

26

27(d) Stroke Frequency

f s (H

z)

0 0.5 1-2

0

2

pitc

h (d

eg)

(e) Pitch Angle of the Body

0 0.5 12

4

6

8

10

(d

eg)

(f) Stroke Bias Angle of the Wings

Time (s)0 0.5 1

0

0.5

1

1.5

2

2.5(g) Overall Energy Consumption

Time (s)

Est

roke

(J)

XRef

XActual

ZRef

ZActual

J Intell Robot Syst

power. Using Eq. 12, total energy consumption inthis approach, i.e. ETI is estimated by:

ETI = Estroke + 2∫ t

0|PTI|dt (39)

Based on dimensions of a medium-sized hum-mingbird, parameters A and R in Eq. 12 are setto appropriate values such that A2 R6 = 2.5 ×10−5 N2·m2.

7.2.1 Hovering

Perhaps, hovering is the only flight capability thatis unique to insects and hummingbirds. Flapping-wing MAVs with a similar stroke pattern arealso capable of hovering. In this part, hovering of

the MAV model over a period of 1 s has beensimulated. The results for controllers 1 and 2 areplotted in Figs. 19 and 20, respectively.

From these diagrams, it is observed that bothcontrollers are able to achieve stable hoveringwithout requiring significant changes in their out-put values. Since all control outputs remain closeto their nominal values, it can be concluded that:1) body pitch profiles in both approaches areclose and 2) the value of PTI for controller 1is insignificant. As a result, both cases shouldhave similar power requirements. In fact over ahovering period of 1 s, energy consumption withcontrollers 1 and 2 is estimated to be 2.2116 J and2.2078 J, respectively (Figs. 19g and 20g). Energy-time profiles in both cases are approximately

Fig. 21 Simulated resultsfor vertical takeoff andzigzag descent withcontroller 1:a displacement along Xand b Z axes, c pitchequilibrium of the wingsψ0, d stiffness of the jointskrot, e pitch angle of thebody θpitch, f stroke biasangle of the wings β andg overall energyconsumption ETI

0 5 10 15-0.5

0

0.5

(a) Displacement along X-axis

X (

m)

0 5 10 150

5

10

(b) Displacement along Z-axis

Z (

m)

0 5 10 15

-20

0

20

(c) Pitch Equilibrium of the Wings

0 (d

eg)

0 5 10 150

0.01

0.02

(d) Stiffness of the Joints

k rot

(N.m

/ rad)

0 5 10 15

-2

0

2

pitc

h (d

eg)

(e) Pitch Angle of the Body

0 5 10 15-5

0

5

10

15

(d

eg)

(f) Stroke Bias Angle of the Wings

Time (s)0 5 10 15

0

5

10

15

20

25

30

35(g) Overall Energy Consumption

Time (s)

ET

I (J

)

XRef

XActual

ZRef

ZActual

J Intell Robot Syst

linear, suggesting a constant energy consumptionrate of 2.2116 Watt and 2.2078 Watt, respec-tively. Note that these values are required tomaintain levitation and do not represent cost oftransportation.

It is worth mentioning that hummingbirdsspend up to 85 % of their time sitting and digesting[38]. The rest is spent for various flight activitiesthat often involve foraging. In [39], it has beenobserved that a 4.3-gram Broad-tailed humming-bird consumes about 7.31 Calories of sugar perday. If the energy consumed at rest is ignored, thisamount would be equivalent to an approximatein-flight energy consumption rate of 2.36 Wattwhich is close to the power requirements of ourMAV model during hovering.

7.2.2 Vertical Takeof f Followed by ZigzagDescent

The results of a takeoff/descent maneuver withcontrollers 1 and 2 are respectively plotted inFigs. 21 and 22. In these experiments, the hoveringmodel is initially commanded to vertically takeoffsuch that it reaches an altitude of 10 m within 4 s.After 1 s of hovering at this altitude, the modelmust descend to a height of 5 m within another4 s. However, reference displacements along Xand Z -axes are designed such that this part of thetrajectory follows a zigzag pattern (Figs. 21a, b and22a, b).

Both employed control strategies are able toprecisely navigate the model along this trajectory

Fig. 22 Simulated resultsfor vertical takeoff andzigzag descent withcontroller 2:a displacement along Xand b Z axes,c downstroke durationratio Tds, d strokefrequency fs, e pitchangle of the body θpitch,f stroke bias angle of thewings β and g overallenergy consumptionEstroke

0 5 10 15-0.5

0

0.5

(a) Displacement along X-axis

X (

m)

0 5 10 150

5

10

(b) Displacement along Z-axis

Z (

m)

0 5 10 150.4

0.5

0.6(c) Downstroke Duration Ratio

Tds

0 5 10 1515

20

25

30

(d) Stroke Frequency

f s (H

z)

0 5 10 15

-2

0

2

pitc

h (d

eg)

(e) Pitch Angle of the Body

0 5 10 15-5

0

5

10

15

(d

eg)

(f) Stroke Bias Angle of the Wings

Time (s)0 5 10 15

0

5

10

15

20

25

30

35(g) Overall Energy Consumption

Time (s)

Est

roke

(J)

ZRef

ZActual

XRef

XActual

J Intell Robot Syst

(Fig. 21a, b and 22a, b). Note that during takeoffwith controller 1, the outputs of thrust and pitchsub-controllers are almost constant (t = 1 s to t =5 s in Fig. 21c and f). The same is not true forcontroller 2 (Fig. 22c and f). Furthermore, com-pared to controller 1, it seems the thrust and pitchsub-controllers in controller 2 have to operate athigher frequencies in order to provide the sameoverall performance.

Compared to hovering experiments, the modelwith controller 1 does not demonstrate consid-erable changes in terms of energy consumption.From Fig. 21g, average power requirements dur-ing takeoff and descent maneuvers are estimatedas 2.1651 Watt and 2.1596 Watt, respectively. Notethat these values are slightly less than power re-quirements during hovering. From Eq. 12, only a

small fraction of overall power (less than 1 %)is spent for impedance manipulation (Fig. 21c,d). The rest, i.e. Estroke, is used to maintain asinusoidal stroke profile with constant magnitudeand frequency. Therefore, from Eqs. 37 and 38,β and FD are the only variables that can affectaverage stroke power requirements. Changes in β

are often small and slow (Fig. 21f). As for FD, ithas been previously shown [23] that around kop,smaller values of stiffness decrease the magnitudeof this force. Hence, the slight reduction in overallvalue of Estroke (and ETI) in this experiment isprimarily due to changes in stiffness (Fig. 21d).

Unlike controller 1, controller 2 relies onmodification of stroke properties. Thus, Eqs. 37and 38 imply that power requirements in experi-ments with this controller will depend on desired

Fig. 23 Simulated resultsfor forward flight at 10 m·s−1 with controller 1:a displacement along Xand b Z axes, c pitchequilibrium of the wingsψ0, d stiffness of the jointskrot, e pitch angle of thebody θpitch, f stroke biasangle of the wings β andg overall energyconsumption ETI

0 5 10 150

50

100

150(a) Displacement along X-axis

X (

m)

0 5 10 15-0.03

-0.02

-0.01

0

0.01(b) Displacement along Z-axis

Z (

m)

0 5 10 15-30

-20

-10

0

10(c) Pitch Equilibrium of the Wings

0 (d

eg)

0 5 10 150

2

4

6

8x 10

-3(d) Stiffness of the Joints

k rot

(N.m

/ rad)

0 5 10 15

0

20

40

pitc

h (d

eg)

(e) Pitch Angle of the Body

0 5 10 155

10

15

20

(d

eg)

(f) Stroke Bias Angle of the Wings

Time (s)0 5 10 15

0

5

10

15

20

25

30

35(g) Overall Energy Consumption

Time (s)

ET

I (J

)

XRef

XActual

ZRef

ZActual

J Intell Robot Syst

maneuver. From Fig. 22g, these requirements dur-ing takeoff and descent maneuvers are estimatedas 2.3164 Watt and 2.1876 Watt, respectively. Thepower used for takeoff is approximately 5 % morethan hovering requirements. Since necessary liftfor takeoff is achieved through faster stroke cycles(Fig. 22d), the increase in energy consumption isto be expected. As for descent, a similar argu-ment can be used to justify the slight reduction inpower requirements compared to those of hover-ing mode.

7.2.3 Forward Flight at High Velocities

Although average flight speed of hummingbirdsis about 5–7 m·s−1, they are known to be ableto reach velocities as high as 14 m·s−1 [40]. The

thrust boost necessary for such velocities is pro-vided through forward pitching of the body. Thedownside to this strategy is that it also reducesthe maximum possible value of lift; hence, verticalmaneuvers at such velocities are very challenging.

The presented flapping-wing MAV model alsodemonstrates such behavior with both controller1 and 2. Here, the model can reach velocities ashigh as 10 m·s−1 before lift sub-controllers fail tostabilize its altitude. The simulated results withcontroller 1 and 2 at 10 m·s−1 are illustrated inFigs. 23 and 24, respectively. In both cases, thestrained performance of lift sub-controller resultsin a slight altitude loss (Figs. 23b and 24b).

From Fig. 23g, the model with controller 1requires only 2.0018 Watt to maintain its targetvelocity. This value is 9.5 % lower than the power

Fig. 24 Simulated resultsfor forward flight at 10 m·s−1 with controller 2:a displacement along Xand b Z axes, cdownstroke durationratio Tds,d stroke frequency fs,e pitch angle of the bodyθpitch, f stroke bias angleof the wings β andg overall energyconsumption Estroke

0 5 10 150

50

100

150(a) Displacement along X-axis

X (

m)

0 5 10 15-0.03

-0.02

-0.01

0

0.01(b) Displacement along Z-axis

Z (

m)

0 5 10 150.45

0.5

0.55

0.6

(c) Downstroke Duration Ratio

Tds

0 5 10 1524

26

28

30(d) Stroke Frequency

f s (H

z)

0 5 10 15

0

20

40

pitc

h (d

eg)

(e) Pitch Angle of the Body

0 5 10 15

5

10

15

(d

eg)

(f) Stroke Bias Angle of the Wings

Time (s)0 5 10 15

0

5

10

15

20

25

30

35

40

45(g) Overall Energy Consumption

Time (s)

Est

roke

(J)

ZRef

ZActual

XRef

XActual

J Intell Robot Syst

required for hovering. Similar to the takeoff ex-periment in Section 7.2.2, this reduction is causedby lowered values of stiffness (Fig. 23d).

The model with controller 2 reaches target ve-locity through employing higher values of strokefrequency (Fig. 24d). From Fig. 24g, this approachrequires 2.8541 Watt which is 29.3 % larger thanthe power required for hovering. Note that for thesame maneuver with controller 1, required poweris reduced by 0.8523 Watt, i.e. almost 30 %.

8 Conclusions

Inspired by insect/hummingbird flight, tunableimpedance has been proposed as a semi-passiveapproach to force manipulation in flapping-wingMAVs. This method states that pitch rotationprofile of each wing can be modified throughchanges in mechanical impedance properties of itsjoint (Fig. 6). Consequently, as shown by Eq. 25,magnitude of variations and average value of thisprofile are influential factors in adjustment ofaerodynamic forces. Therefore, regulation of me-chanical impedance properties can be used as anindirect means of controlling these forces (Fig. 7).

The primary advantage of TI over conventionalmethods of force control is that it requires nomodification in stroke profile of either wing. Infact, both wings can use the same sinusoidal pat-tern of flapping with fixed parameters – exceptfor stroke bias angle β which is an independentcontrol parameter. This effectively simplifies thecontrol system and can reduce the amount of on-board hardware in an actual prototype. Under thisapproach, the system is also more controllable andhas a higher degree of mobility [26]. However, liftand thrust control are not completely decoupledwhich at times may result in limited maneuverabil-ity of the vehicle.

In this work, we showed that in a system withTI as its underlying strategy for force control,coupling between lift and thrust effectively oc-curs when either impedance variable carries sig-nificant components at high frequencies, i.e. closeto the wing-beat rate (Fig. 12). Removing highfrequency content from both impedance parame-ters will considerably reduce lift/thrust couplingand results in better performance compared to

our previous work [26]. In fact, a simple controlstructure such as Fig. 14 is sufficient to stabilizemotion along various trajectories.

Results of simulated experiments in Section 7verify the tracking capabilities of this method. Ithas been observed that as long as a trajectory canbe described using smooth velocity profiles, theonly concern for successful tracking is the cor-responding acceleration requirements. While im-posed limits on impedance parameters guaranteelinear relationships with forces, they also restrictacceleration capabilities of the model (Fig. 7).Thus, successful tracking also requires that time-derivatives of these velocity profiles lie withinacceleration limits of the model. In a real system,designing such profiles relies on information fromboth environment and vehicle itself. A high levelcontroller must first come up with an appropriatetrajectory based on surroundings of the vehicleand relative location of its target. The currentvelocity and acceleration limits of the MAV mustthen be treated as constraints of designing a suit-able velocity profile for tracking purposes.

The second goal of this work was to investigatethe impact of Tunable Impedance approach onpower requirements of a flapping-wing MAV. Itwas observed that the necessary amount of en-ergy for impedance manipulation is a lot smallerthan the amount required for stroke motion. Sincestroke profile is approximately constant, the to-tal power requirement in this approach is almostindependent of desired maneuver. For the sim-ulated MAV model in this paper, the requiredpower is always less than 2.21 Watt. Interestingly,this value is in agreement with average in-flightenergy consumption rate of a hummingbird withsimilar body dimensions.

In comparison to Tunable Impedance, othercontrol approaches that rely on manipulationof stroke properties often require considerablygreater amounts of power. This increase in powerconsumption is specifically observed when a ma-neuver demands more lift or thrust production(e.g. forward acceleration in Fig. 24). Faster strokemotions are usually employed to provide for thisexcess in force requirements.

The gap in energy consumption will widen fur-ther when we consider the case in which pitchrotation of the wing is not governed by a passive

J Intell Robot Syst

mechanism. Consequently, additional actuatorsare required to directly rotate the wing and modifyits angle of attack throughout each stroke cycle.From Eqs. 14 and 24, one can estimate the activetorque τψ = τext that must be applied to each wingso that it follows a predefined pitch profile ψ . Theenergy required for active pitch rotation of bothwings, i.e. Eψ is then calculated as:

Eψ = 2∫ t

0

∣∣τψψ

∣∣dt (40)

From Eq. 40, if the model with controller 2 wereto actively generate the same wing pitch profilesas the simulations in Section 7.2, its power re-quirements would increase by 0.5571 Watt duringhovering, 0.6613 Watt during takeoff and 0.9540Watt when moving forward at 10 m·s−1.

As a bio-inspired control strategy with lowpower demands, the Tunable Impedance methodis a promising solution to both problems of ma-neuverability and power efficiency in flapping-wing MAVs. Considering the limitations of avail-able batteries, implementation of this approachmay significantly improve flight time. Our futurework is focused on development of appropri-ate actuators for impedance manipulation. Aboveall, such actuators must have minimized loadingeffects so that wing’s dynamics are not greatly per-turbed. Upon achieving satisfactory results in im-pedance manipulation, our eventual goal is to in-stall the designed actuators on a 3-inch-wingspanflapping-wing MAV and conduct actual flight ex-periments.

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