J Intell Robot SystDOI 10.1007/s10846-012-9746-x

Modeling Synchronous Muscle Function in Insect Flight:a Bio-Inspired Approach to Force Controlin Flapping-Wing MAVs

Hosein Mahjoubi · Katie Byl

Received: 29 June 2012 / Accepted: 13 July 2012© Springer Science+Business Media B.V. 2012

Abstract Micro-aerial vehicles (MAV) and theirpromising applications—such as undetected sur-veillance or exploration of environments withlittle space for land-based maneuvers—are a well-known topic in the field of aerial robotics. Inspiredby high maneuverability and agile flight of insects,over the past two decades a significant amount ofeffort has been dedicated to research on flapping-wing MAVs, most of which aim to address uniquechallenges in morphological construction, forceproduction, and control strategy. Although re-markable solutions have been found for sufficientlift generation, effective methods for motion con-trol still remain an open problem. The focus ofthis paper is to investigate general flight controlmechanisms that are potentially used by real in-sects, thereby providing inspirations for flapping-wing MAV control. Through modeling the insectflight muscles, we show that stiffness and set point

Original Work Presented at 2012 InternationalConference on Unmanned Aircraft Systems (ICUAS’12).

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

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

of the wing’s joint can be respectively tuned toregulate the wing’s lift and thrust forces. There-fore, employing a suitable controller with vari-able impedance actuators at each wing joint isa prospective approach to agile flight control ofinsect-inspired MAVs. The results of simulatedflight experiments with one such controller areprovided and support our claim.

Keywords Bio-inspired MAVs · Aerial robotics ·Insect flight · Tunable impedance ·Variable stiffness actuator · Maneuverability

Abbreviations

AoA Angle of AttackBEM Blade Element MethodCoL Center of LiftCoM Center of MassCoP Center of PressureMAV Micro-Aerial VehicleMEMS Microelectromechanical systemsPI Proportional–IntegralPID Proportional–Integral–DerivativeSC Split CycleTI Tunable Impedance

1 Introduction

The agility of insects and hummingbirds in flightprovides inspiration for the design and control of

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small micro-aerial vehicles (MAVs). Identifyingand exploiting basic principles that can be usedin man-made MAV designs has been, correspond-ingly, an active area of research in recent years.Many high-level questions remain in design-ing small, flapping-wing devices to exploit lowReynolds flapping-wing aerodynamics. For exam-ple, it is of interest to know the extent to whichachieving both stable flight and agile maneuveringrelies on (1) active control of wing shape and/ororientation during flapping, (2) high-bandwidthactuation and control loops, and (3) accurate sens-ing of full state information. Toward addressingsuch questions, we suggest first exploring the the-oretical limits of insect muscle, from a controltheory perspective.

Specifically, given the current biological un-derstanding of the capabilities and limitations ofinsect muscle, we wish to formulate likely hy-potheses on the underlying control principlesemployed by insects. Such principles would thenprovide starting points for design and control ofagile, insect-scale robots.

In this paper, we investigate a potential strat-egy that might be used by insects for control-ling aerodynamic forces. Inspired by the structureand function of insect muscles, this strategy relieson a tunable mechanical structure that employsquadratic springs. Although it remains an openquestion whether this approach is in fact em-ployed by insects, we show that it is theoreticallycapable of controlling lift and thrust forces inflapping-wing MAVs. Furthermore, this approachallows us to decouple the lift and thrust forceson a wing to a considerable extent. Thus, simplecontrollers such as PIDs could be used to controlthe vehicle. One such controller is used to simu-late the flight of a dragonfly/hummingbird-scaleMAV. The results suggest that this control ap-proach can handle various agile maneuvers whilemaintaining its stability.

The remainder of the paper is organized as fol-lows. Section 2 reviews the types and functionalityof flight muscles in insects. Using simple musclemodels, a mechanical model of the wing joint isthen developed. Section 3 presents a simple quasi-steady-state model of the aerodynamics involvedin insect flight. In Section 4, we use these modelsto explain how rotation of the wing is governed

by the interaction between aerodynamic force andimpedance properties of the wing joint. There, wewill show how adjusting these impedance proper-ties can be used to regulate lift and thrust forces.The dynamic model of the MAV and the de-signed controller are discussed in Sections 5 and 6,respectively. The results of simulated flight con-trol experiments will be presented in Section 7.Finally, Section 8 concludes this work.

2 Flight Muscles in Insects

Many flying insects employ two types of musclesduring flight: synchronous and asynchronous [1].Here, we will first briefly review the physicalstructure and functionality of each group. Thosemuscles that have the main role in adjustment ofaerodynamic forces and maneuverability are thenused as a template to develop a mechanical modelof wing actuation.

2.1 Structure and Functionality

More ancient aerobaticists of the insect worldsuch as dragonflies use only synchronous muscles,which power the wings directly as illustrated inFig. 1. Contraction of the muscles attached tothe base of the wing inside the pivot point raisesthe wing (Fig. 1a) while it is brought down bya contraction of muscles that attach to the wingoutside of the pivot point (Fig. 1b) [1, 2].

As the naming suggests, synchronous musclescontract once for every nerve impulse [1] whichis why purely synchronous muscled insects fly atlower flapping frequencies. Therefore, to producesufficient lift force, these insects often have largewings [3]. For example, a locust that beats itswings at 16 Hz has a wing area of 6–7 cm2 [4].

Asynchronous muscles are found in more ad-vanced insects such as true flies. In evolutionaryterms, these muscles are a new design featurewhich has facilitated the evolution of smaller andmuch faster agile flyers.

Asynchronous muscles are primarily responsi-ble for power generation. As shown in Fig. 2, theyindirectly move the wings by exciting a resonantmechanical load in the body structure [1]. When

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Fig. 1 An illustration ofinsect’s wing stroke usingsynchronous muscles:a upstroke andb downstroke. Muscles inthe process of contractionare shown in a darkercolor (adapted from [2])

(a) (b)

operating against inertial load of the air moved bythe wings, these muscles act as a self-sustaining os-cillator that may execute several stroke cycles forevery electrical stimulus received [5]. This justifieshow some insects can achieve very high flappingfrequencies, e.g. 1000 Hz for a small midge [4].

From Fig. 2, it can be observed that contrac-tion of each pair of asynchronous muscles indi-rectly operates both wings and thus, may adjustthe stroke magnitude and frequency of flapping.Both of these parameters influence the velocity ofair currents across the wings. From basic aerody-namic theory, we know that the relative velocityof air flow directly affects the magnitude of overallaerodynamic force. Hence, we can see that asyn-chronous muscles play a major role in productionand adjustment of this force. However, lift andthrust components of the overall force producedby each wing depend on the angle of attack (AoA)of that wing (Fig. 3a). Being indirectly connectedto the wings (Fig. 2), asynchronous muscles areprimarily used to power the wings rather thanregulating their angles of attack. Therefore, con-tribution of these muscles to individual control oflift and thrust forces should be negligible.

The structure of synchronous muscles, by con-trast, allows them to directly influence both thewing’s stroke motion and its pitch rotation (Fig. 1).

Thus, these muscles can manipulate the angle ofattack of each wing separately in order to adjustthe lift and thrust components of its aerodynamicforce, and thereby generate thrust or steeringforces.

From this review, one can see that flight controlis primarily the role of synchronous muscles. Next,we develop a simple mechanical model based onthe interaction between these muscles and thewings. This model will be then used to help us bet-ter understand the control process of aerodynamicforce.

2.2 Mechanical Model

To adopt the proper angle of attack during eachhalf stroke, the wing has to supinate before theupstroke and pronate before the downstroke [1].This means that pitch rotation of the wing is acyclic motion which happens at a rate close tothat of the flapping frequency. However, this fre-quency is usually a lot larger than the stimula-tion rate—bandwidth limitation—of synchronousmuscles [1, 5]. Therefore, these muscles cannot bedirectly responsible for the fast rotational motionof the wings. In fact, this motion is powered by thetorque due to aerodynamic force FN with respectto the wing’s pitch rotation axis (Fig. 3a).

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

(b)(a)

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Fig. 3 a Wingcross-section (duringdownstroke) at its centerof pressure, illustratingthe pitch angle of thewing ψ , and its angle ofattack αr with respect tothe air flow ur . Normaland tangentialaerodynamic forces areshown by FN and FTn. FLand FT represent the liftand thrust components ofthe overall force.b Overhead view of thewing/body setup whichdefines the stroke angle φ

y

y'

(a)

x

(b)

x'

+ + -

z

ur

ur CoP

FN

zCoP

φ

CoP

x'

αr

Wing's PitchRotation Axis

FT

FTn

ψ -

FL

Like most muscle-joint structures, contractionof agonist/antagonist synchronous muscles canchange the stiffness of the joint [1, 5]. This stiffnessresults in a resistive torque that opposes the rota-tion of the wing. With a stiffer joint, the rotationrange of the wing will be more limited. Thus,synchronous muscles can influence the angle ofattack through adjusting the stiffness of the joint.These muscles can also slowly rotate the wingsand by doing so, bias the angle of attack. Thiswould result in a phase shift between wing strokeand wing rotation which is often observed duringsteering or thrust [6].

A simplified model for muscle consists of aforce pulling one end of a nonlinear spring [7]. Theother end is attached to the joint. Here, we modelthe joint as a pulley of radius R, whose axis ofrotation is the same as wing’s pitch rotation axis.Figure 4 illustrates the complete diagram of themodel for one wing and the pair of synchronousmuscles attached to it.

The forces that model the power of musclecontraction are represented by FM1 and FM2. Thedisplacements of the input ends of springs due tothese forces are shown with x1 and x2, respec-tively. We define:

xc = 12

(x1 + x2) (1)

xd = 12

(x1 − x2) (2)

By analogy to the action of muscles about a joint,xc corresponds to co-activation while xd corre-sponds to differential activation. By differentialactivation, the muscles are able to bias the pitchangle of the wing to a unique position, ψ0. FromFig. 4, it can easily be seen that:

xd = Rψ0 (3)

Any load on the wing—i.e., the aerodynamicforce—will cause a further deflection from thisequilibrium by an amount of �ψ . The new angularposition ψ determines the displacement of the

z

x'

ψ0

R

Δψ

ψ

FM1

x2

FM2

τext

σ1

σ2

x1

Fig. 4 A simplified mechanical model of synchronousmuscle-joint structure. Each muscle is modeled as a non-linear spring with variable stiffness σ i (i = 1, 2). A forceof FMi pulls the input side of each spring, resulting inthe displacement xi. The other sides of both springs areattached to a pulley of radius R which represents the wing’sjoint

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output end of each spring. At any given time, thisdisplacement is equal to:

Rψ = R(ψ0 + �ψ

)(4)

where spring 1 moves to the left and spring 2moves to the right. The deflection of each springis the difference between the motions of its twoends. Using Eq. 1–4, it can be shown that thesedeflections are as follow:

δ1 = xc − R�ψ (5)

δ2 = xc + R�ψ (6)

Depending on the amount of its deflection, eachspring applies a force to the pulley, i.e. Fi(δi),i = 1, 2. The stiffness of each spring is alsoa function of its deflection: σ i(δi). Note thatstiffness is the derivative of force with respect todisplacement:

σi (δi) = ddδi

Fi (δi) , i = 1, 2 (7)

The forces developed in the stretched springsmust then balance the externally applied torqueτext:

τext = R [F2 (δ2) − F1 (δ1)] (8)

The effective joint stiffness krot is defined as thederivative of this torque with respect to the angu-lar position ψ . By using the chain rule along withEqs. 5–8, we have:

krot = ddψ

τext = R2 [σ1 (δ1) + σ2 (δ2)] (9)

From Eq. 9, it can be seen that if individual springsare linear (constant stiffness σ i), the effective jointstiffness is always a constant. Therefore, to modelthe joint with variable stiffness, nonlinear springsare required.

The diagram in Fig. 4 suggests that stiffness ofthe joint is primarily influenced by xc. However,fast rotation of the wing may also affect krot [7].Therefore, in reality krot slightly depends on thedeflection of the joint from its equilibrium posi-tion, i.e. �ψ . Here, in order to come up with a

simple model for nonlinearity of the springs, weassume that krot is independent of �ψ . Thus:

dd�ψ

krot =0= R2[

dd�ψ

σ1 (δ1)+ dd�ψ

σ2 (δ2)

](10)

which, from Eqs. 5 and 6, will result in the follow-ing mathematical requirement:

ddδ1

σ1 (δ1) = ddδ2

σ2 (δ2) (11)

whose general solution is given as:

σi (δi) = Aδi + Bi, i = 1,2 (12)

where A and B are constant coefficients. Assum-ing that the joint is symmetric, i.e. the springs areidentical, the subscript i on the constant B can bedropped. The force function for both springs willthen have the following form:

Fi (δi) = 12

Aδ2i + Bδi + C, i = 1,2 (13)

in which, C is another constant coefficient. FromEq. 13, quadratic springs are the type of nonlinear-ity that can satisfy Eq. 11. This type of spring hasbeen previously used to predict the mechanicalresponse of muscles during contraction with anerror no more than 15 % [8]. Therefore, whenused in the setup of Fig. 4, such springs should pro-vide a reasonable model of synchronous muscles’performance.

By replacing Eqs. 5, 6 and 12 in Eq. 9, it canbe shown that krot is now a pure function ofco-activation:

krot = 2R2 (Axc + B) (14)

Similarly, using Eqs. 3–6, 8, 13 and 14, we can seethat:

τext = krot (ψ − ψ0) (15)

Note that from Eq. 3, ψ0 is proportional to xd.Thus, the joint can be modeled as a torsionalspring whose stiffness and set point are respec-tively controlled by co-activation and differentialactivation of the synchronous muscles.

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3 Aerodynamics

Aerodynamics of insect flight, which correspondsto low Reynolds number (less than 3,000), hasbecome a popular area of research over the pastfew decades, being strongly motivated by workssuch as [9]. In particular, scaled models of flappingwings [6, 10] have been very helpful in compre-hending the involved unsteady-state aerodynamicmechanisms both qualitatively and quantitatively.Results obtained with the apparatus developed in[6], known as Robofly, have identified three mainmechanisms: delayed stall, rotational circulation,and wake capture. Detailed descriptions of thesemechanisms are available in [6, 11, 12]. Here,we will briefly review the first-order models de-veloped to estimate the aerodynamic forces pro-duced by each mechanism. Using Blade ElementMethod (BEM) [9, 13–15], these models allow usto estimate the overall aerodynamic force for anygiven wing shape and stroke profile.

3.1 Mathematical Force Models

Reynolds number in insect flight is low enoughto assume that air flow across the wing is quasi-stationary [11–14]. A quasi-steady-state aerody-namic model assumes that the force expressionsderived for 2-D thin airfoils translating with con-stant angle of attack and constant velocity can bealso used for time-varying 3-D flapping wings.

Essentially, delayed stall is the results of wing’stranslational motion [6]. Basic aerodynamic the-ory [16] shows that in steady-state conditions, theaerodynamic force per unit length on an airfoildue to its translational motion is given by:

fN = ρ

2cCN (αr) u2

r (16)

fTn = ρ

2cCT (αr) u2

r (17)

where fN and fTn are the normal and tangentialcomponents of the force. Note that these forcesalways oppose the stroke of the wing. Parametersρ, c and αr respectively represent the density ofair, cord width of the airfoil and its angle of attackrelative to air velocity ur (Fig. 3a).

CN and CT are dimensionless force coefficientswhich are generally time-dependent. However,experimental results in [6] suggest that a goodquasi-steady-state empirical approximation forthese coefficients due to delayed stall is given by:

CN (αr) = 3.4 sin αr (18)

CT (αr) ={

0.4 cos2 (2αr) , 0 ≤ αr ≤ 45◦0, otherwise

(19)

The lift force due to rotational circulation of airaround an airfoil has been modeled in [17]:

fN,rot = ρ

2c2Croturαr (20)

where Crot is the dimensionless rotational forcecoefficient. In flapping flight, this coefficient isalso generally time-dependent. However, whenusing a constant value of Crot = π [12], quasi-steady-state modeling has provided satisfactorypredictive capabilities [18]. Note that the deriva-tive of αr in Eq. 20 is in fact, an approximation ofthe wing’s angular velocity. As a reminder, fN,rot isa pure pressure force that acts normal to the airfoilprofile and in the opposite direction of wing’spitch velocity.

Wake capture is a more complex mechanism.At the end of each half-stroke, the wing beginsto change direction. However, the fluid behindthe wing tends to maintain its velocity due to itsinertia. Hence, the difference between velocitiesof fluid and wing results in production of a largeforce at the beginning of next half-stroke [6].This phenomenon cannot be easily described by aquasi-steady-state model. However, it is observedto have an insignificant overall role when strokeprofile is sinusoidal-like [11]. Since our simula-tions are based on this type of motion, the effect ofwake capture is not considered in the rest of thispaper.

3.2 Blade Element Method: Force Exertedon the Wing

In blade-element theory, a wing is divided into aset of cross-sectional strips, each of width dr, andlocated at a mean radius r from axis of wing stroke[15], as illustrated in Fig. 5. Each one of thesestrips is treated as an airfoil whose cord width is

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roke

Axi

s

r

Rw

dr

c(r)Wing's Pitch Rotation Axis CoP

zCoP

Fig. 5 The wing shape used in all modeling/simulationsand one of its blade elements. The span of the wing isrepresented by Rw

a function of r. The aerodynamic force for anyairfoil can be then calculated using Eqs. 16–20.

The air velocity observed by an airfoil at radiusr is given by:

ur = rφ (21)

in which, φ is the stroke angle as defined inFig. 3b. As mentioned earlier, we assume thatstroke profile is sinusoidal:

φ = φ0 cos (ω t) (22)

where φ0 and ω are the magnitude of the strokeand flapping frequency, respectively.

In our simulations, we consider the wing as arigid, thin plate shaped as shown in Fig. 5. Replac-ing Eq. 21 in Eqs. 16, 17 and 20 and integratingthem over the span of this wing shape will resultin the following expressions for total aerodynamicforces:

FN,trs = 0.0442ρCN (αr) φ2 R4W (23)

FTn = 0.0442ρ CT (αr) φ2 R4W (24)

FN,rot = 0.0595ρ αrφ R4W (25)

where RW represents the span of the wing. FTn

is the tangential force while FN,trs and FN,rot arethe translational and rotational components of thenormal force, FN :

FN = FN,trs + FN,rot (26)

For the stroke profile in Eq. 22, the total aero-dynamic force on the wing can be then estimated

through Eqs. 18, 19 and 23–26. However, this esti-mation is still a function of αr and its derivative.

From Section 2, the balance between mechani-cal impedance of the joint and external load, i.e.the aerodynamic force, is what determines howthe angle of attack should evolve. In the nextsection, we will further investigate this evolutionas the potential underlying mechanism for forceregulation in insect flight.

4 Force Control in Insect Flight

In Section 2, we argued that asynchronous musclesare the reason for the high wing beat rate in manyinsects. Thus, they can be thought of as the mainpower source for production of aerodynamic forceand hence, levitation. The structure of these mus-cles allows for changes in the magnitude of strokeand perhaps the frequency of flapping, both ofwhich are parameters that influence the amount ofgenerated aerodynamic force (Section 3.2). How-ever, the indirect connection of these muscles tothe joint (Fig. 2) suggests that their contractionalways affects both wings in the same way.

Maneuverability and flight control on the otherhand, require that lift and thrust forces of eachwing are individually adjusted. To change the liftand thrust components of the aerodynamic forceproduced by one wing, the insect needs to regulatethe angle of attack of that wing. Direct attachmentof one pair of synchronous muscles to each wing’sjoint makes them a convenient mechanism forthis purpose. Evolution of angle of attack andthe potential role of synchronous muscles in flightcontrol are discussed next.

4.1 Evolution of Angle of Attack

In reality, aerodynamic force is distributed overthe surface of the wing. However, it can equiva-lently be modeled as a force applied to a singlepoint [12] called the center of pressure (CoP inFigs. 3 and 5). The location of this point, in gen-eral, depends on the wing’s shape and can be esti-mated by calculation of aerodynamic force/torquepairs through BEM [15]. The distance of CoPfrom the wing’s pitch rotation axis, i.e. zCoP, de-termines the external load on the joint due to

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aerodynamic force FN (Fig. 3a). Note that tan-gential force FTn does not influence the pitchrotation of the wing. For our choice of wing shape,estimation of zCoP results in:

zCoP = 0.0673 RW (27)

The overall load tending to change the pitch angleof the wing is given by:

τext = zCoP FN − bψψ − Jψψ (28)

where Jψ and bψ are the moment of inertia andpassive damping coefficient of the wing along itspitch rotation axis, respectively.

To maintain the balance, right hand sides ofEqs. 15 and 28 must always remain equal. FromFig. 3a, it can be seen that angle of attack andpitch angle of the wing are complementary angles,i.e. αr = π/2 − ψ . This allows us to rewrite Eq. 23–25 as functions of ψ and its derivative. Then

the following ODE can be written using Eqs. 15and 28:

Jψψ + bψψ + krot (ψ − ψ0) − zCoP FN(ψ, ψ

)=0

(29)

Using the aerodynamic force model of Section 3,Eq. 29 can be numerically solved for a givenstroke profile φ and any desired set of wing jointimpedance properties, i.e. krot and ψ0. Once thewaveform of ψ is found, aerodynamic forces FN

and FTn can be calculated for any given time. Thelift and thrust components of the overall force,i.e. FL and FT , are also defined as illustrated inFig. 3a:

FL = FN cos αr − FTn sin αr (30)

FT = −FN sin αr − FTn cos αr (31)

Figure 6a illustrates the solution to Eq. 29 for asingle sinusoidal stroke cycle with φ0 = 60◦ and

Fig. 6 a Simulated pitchand stroke angles of thewing for a largedragonfly. Stroke has amagnitude of 60◦ and arate of 25 Hz. Impedanceproperties are set tokrot = 1.4 × 10−3 N.m/radand ψ0 = 0◦. Thecorrespondingaerodynamic force isplotted both inb normal/tangentialand c lift/thrust formats

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-60

-40

-20

0

20

40

60

(deg

)

(a) Pitch and Stroke Angles of the Wing

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-5

0

5(b) Corresponding Aerodynamic Force: Normal and Tangential Components

(g-F

orce

)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-4

-2

0

2

4

(g-F

orce

)

Time (sec)

(c) Corresponding Aerodynamic Force: Lift and Thrust Components

FN

FTn

FL

FT

ψ φ

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Table 1 Physicalparameters used insimulations of Section 4

Symbol Description Value

ρ Air density at sea level 1.28 kg/m3

RW Average span of each wing in a large dragonfly 8 × 10−2 mJψ Moment of inertia of each wing (pitch rotation) 1.5 × 10−4 N.m.s2

bψ Passive damping coefficient of each wing (pitch rotation) 1 × 10−8 N.m.smbody Estimated mass of a two-winged MAV with 4 × 10−3 kg

similar dimensionsg Standard gravity at sea level 9.81 m/s2

ω = 50π rad/s. These values are based on flightcharacteristics of a large dragonfly or a medium-sized hummingbird [1]. The values of other neces-sary physical parameters are listed in Table 1. Inthis diagram, the case of no differential activationhas been considered, i.e. xd = 0 m. From Eq. 3,this means that ψ0 = 0◦. As for stiffness, we havechosen a value of krot = 1.4 × 10−3 N.m/rad whichis soon shown to be the average-lift-optimizingvalue for the considered scale.

As Fig. 6a suggests, during every half stroke,the wing smoothly rotates and adapts its angleof attack to the upcoming air flow. Through thisrotation, the balance between aerodynamic forceand impedance of the joint is maintained.

The normal/tangential and lift/thrust compo-nents of the corresponding aerodynamic force areillustrated in Fig. 6b and c, respectively. Everyforce curve has been normalized by the estimatedweight of a same-sized MAV (Table 1). Figure 6c

Fig. 7 a Simulated pitchand stroke angles of thewing for a large dragonfly.Stroke has a magnitude of60◦ and a rate of 25 Hz.Here, ψ0 = 0◦ while krotis set to differentmultiples of k* = 1.4 ×10−3 N.m/rad. Thecorresponding b lift and cthrust forces are alsoplotted for comparison

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-60

-40

-20

0

20

40

60

(deg

)

(a) Pitch and Stroke Angles of the Wing

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-0.5

0

0.5

1

1.5

2

(g-F

orce

)

(b) Lift

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-6

-4

-2

0

2

4

6

(g-F

orce

)

Time (sec)

(c) Thrust

krot

=k*

krot

=2k*

krot

=8k*

krot

=k*

krot

=2k*

krot

=8k*

krot

=k*

krot

=2k*

krot

=8k*

φ

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reveals that for ψ0 = 0◦, the profile of lift forceis even-symmetric and has an average value of0.7252 g-Force per wing. Note that this value isabout 45 % larger than what is required to levitatea two-winged MAV of target size and weight.Production of such large forces is in agreementwith the highly agile movements that are oftendemonstrated by insects/hummingbirds.

On the other hand, the thrust profile in Fig. 6c isodd-symmetric; it has an average value of 9.312 ×10−7 g-Force per wing. This practically zero aver-age thrust force verifies that insects such as fliesare able to hover, another flight capability that isseldom observed outside of the insect world.

A change in contraction level of synchronousmuscles can reshape the profile of ψ and lift/thrustforces. In the remainder of this section, we willinvestigate how insects might use these changesto adjust the aerodynamic forces, and thereby,achieve flight control.

4.2 Force Control Through Impedance Tuning

Stiffness of the joint is a limiting factor for pitchrotation of the wing. A very stiff joint tends tomove less and thus, does not allow the wing toreach low angles of attack (Fig. 7a). This leads toreduction of the lift force (Fig. 7b).

Although changing krot results in different mag-nitudes of thrust force in Fig. 7c, it is observed thatas long as ψ0 = 0◦, the profile of this force remainsodd-symmetric. To produce a net thrust force forforward/backward motion or steering, the insectneeds to somehow disturb this symmetry. Figure 8shows how an asymmetric thrust profile can beachieved through changing the set point of thejoint.

Impedance set point, i.e. ψ0, acts as a biasadded to the pitch angle of the wing. The resultingshift in the waveform of ψ (Fig. 8a) creates asym-metries in both lift (Fig. 8a) and thrust profiles

Fig. 8 a Simulated pitchand stroke angles of thewing for a largedragonfly. Stroke has amagnitude of 60◦ and arate of 25 Hz. Here, krot= 1.4 × 10−3 N.m/radwhile different values ofψ0 are examined. Thecorresponding b lift and cthrust forces are alsoplotted for comparison

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-60

-40

-20

0

20

40

60

(deg

)

(a) Pitch and Stroke Angles of the Wing

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-0.5

0

0.5

1

1.5

2

2.5

(g-F

orce

)

(b) Lift

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-5

0

5

(g-F

orce

)

Time (sec)

(c) Thrust

ψ0=-15o

ψ0=0o

ψ0=15o

ψ0=-15o

ψ0=0o

ψ0=15o

ψ0=-15o

ψ0=0o

ψ0=15o

φ

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(Fig. 8c). For ψ0 = ±15◦, the average lift force inFig. 8b is equal to 0.6366 g-Force per wing. Notethat this value is still about 27.3 % larger thanthe required force for levitation of a same-sizedtwo-winged MAV. As for thrust profiles in Fig. 8c,the induced asymmetries by negative and positiveset points create a net thrust force of 0.3098 and−0.3098 g-Force per wing, respectively.

Figure 9 illustrates how net lift and thrustforces are influenced by simultaneous variationsin stiffness and set point. From Fig. 9a, maxi-mum average lift is achievable when krot = 1.4 ×10−3 N.m/rad and ψ0 = 0◦. To provide a betterview, cross-sections of the surfaces in Fig. 9 atthese values are plotted in Fig. 10.

Figure 10a suggests that through regulation ofkrot, insects are able to manipulate their lift forceand thus ascend or descend. Interestingly, as longas ψ0 is sufficiently small, changes in krot havelittle effect on the net thrust force. From Fig. 10b,it is observed that changing ψ0 between ±40◦ canproduce significant amounts of net thrust force.These values are large enough to support agile

horizontal maneuvers such as steering and for-ward/backward motion. In addition to productionof considerable net thrust, when ψ0 is between±15◦, the observed attenuation in net lift is nolarger than 13 %. Therefore, as long as set pointvariations are limited, net values of lift and thrustforces can be controlled almost independently.This is a feature that significantly improves ma-neuverability and flight control.

Finally, we emphasize that from Eqs. 3 and 14,set point and stiffness of the wing’s joint arerespectively adjusted by differential and co-activation of synchronous muscles. Since both ofthese dependencies are linear, it is easy to see thatthe pair (xd, xc) affects net lift and thrust forces inthe same way as (ψ0, krot).

5 Dynamic Model of the MAV

Performance of the proposed force regulation ap-proach is evaluated through a set of simulatedflight experiments with a modeled two-winged

Fig. 9 a Average lift andb average thrust forces vs.set point and stiffness ofthe wing’s joint

Ave

rage

Lif

t (g

-For

ce)

Ave

rage

Thr

ust (

g-F

orce

)

krot (N.m/rad )

krot (N.m/rad )ψψ

0 (deg)

ψ0 (deg)

(a)

(b)

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Fig. 10 a Average lift andthrust forces vs. stiffnessof the joint when ψ0 = 0◦.The maximum value oflift is achieved for krot =k* = 1.4 × 10−3 N.m/rad.b Average lift and thrustforces vs. set pointof the joint when krotis kept at k*

10-5

10-4

10-3

10-2

10-1

-0.2

0

0.2

0.4

0.6

0.8

(g-F

orce

)

krot

(N.m/rad

)

(a) Average Value of Aerodynamic Forces vs. Stiffness when ψ0=

0o

-80 -60 -40 -20 0 20 40 60 80-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

(g-F

orce

)

ψ0 (deg)

(b) Average Value of Aerodynamic Forces vs. Set Point when krot

= k*

Average LiftAverage Thrust

Average LiftAverage Thrust

MAV. The free-body diagram of this flapping-wing MAV is shown in Fig. 11. This body structureis based on the vehicle in [19]. The shape of each

y

(a) Frontal View

y

x

CoP CoP

z

CoM

(b) Overhead View

Yaw

Pitch

Roll

x

z

CoP CoP

FL r

R r R l

H r H l FL l

FT r F

T l

U lU r

Fig. 11 Free-body diagram of a flapping-wing MAV:a frontal and b overhead views. H, R and U are thedistance components of center of pressure (CoP) of eachwing from center of mass (CoM) of the whole system

wing is the same as Fig. 5 and other characteristicsof the wings are assumed to be as reported inTable 1.

The six-degree-of-freedom dynamic model ofthe vehicle [12] is developed by applying Newton’sequations of motion in the body frame of Fig. 11:

mbody

(⇀

v + �ω × �v)

= �F= �Fl + �Fr − mbody �G − b v

∣∣�v∣

∣ �v(32)

Jbody �ω + �ω × Jbody �ω = �τ= �Dl × �Fl + �Dr × �Fr − bω �ω

(33)

where:

�Fl =

⎡

⎢⎢⎣

FlT cos φl

FlT sin φl

FlL

⎤

⎥⎥⎦ , �Fr =

⎡

⎢⎢⎣

FrT cos φr

−FrT sin φr

FrL

⎤

⎥⎥⎦ (34)

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Table 2 Characteristics of the modeled flapping-wing MAV

Symbol Description Value

bω Passive damping coefficient of the body 3 × 10−3 N.m.sb v Coefficient of viscous friction of the body when moving in the air 1 × 10−2 N.s2/m2

H(ψ = 0˚) Distance of CoP from transverse plane of the body (xy in Fig. 11) when ψ = 0◦ 2.89 × 10−2 mR(φ = 0◦) Distance of CoP from sagittal plane of the body (xz in Fig. 11) when φ = 0◦ 6.35 × 10−2 mU(φ = 0◦) Distance of CoP from coronal plane of the body (yz in Fig. 11) when φ = 0◦ 5.8 × 10−3 mWbody Body width at the wings’ roots 1.16 × 10−2 m

and:

�Dl =⎡

⎢⎣

Ul

Rl

Hl

⎤

⎥⎦ , �Dr =

⎡

⎢⎣

Ur

−Rr

Hr

⎤

⎥⎦ (35)

The vector G expresses gravity in the body coor-dinate frame. Euler angles in the Tait-Bryan ZXYconvention are used to keep track of the orien-tation of the body with respect to the referencecoordinates. Translational and angular velocitiesof the MAV are represented by vectors v andω, respectively. H, R and U are the distancecomponents of center of pressure of each wingfrom the overall center of mass (CoM), as shownin Fig. 11. The parameter b v is the coefficient ofviscous friction and the term with bω is used tomodel the effect of passive rotational damping, asdiscussed in [20]. The values of these parametersare listed in Table 2.

Total body mass—mbody—is assumed to be4×10−3 kg for a vehicle of desired scale (Table 1).The body shape in Fig. 11 is chosen since itscorresponding inertia matrix relative to the centerof mass (CoM), i.e. Jbody, has insignificant nondi-agonal terms and thus, can be approximated by:

Jbody =⎡

⎢⎣

30 0 00 3 00 0 3

⎤

⎥⎦ × 10−5 N.m.s2 (36)

6 Controller

Figure 12 illustrates the block diagram of theemployed control module in simulations ofSection 7. A total of three sub-controllers areresponsible for stabilization of pitch angle andregulation of lift and thrust forces, thus controlling

Fig. 12 Block diagram ofthe control module andits interaction with thesimulated MAV. In eachstroke cycle, the sinusoidwave generator creates areference strokewaveform with a fixedmagnitude of 60◦ at 25 Hzwhich will be then biasedwith β. Crossoverfrequency of eachfilter/actuator isrepresented by ωc

+

-+

-+

Hybrid LiftController

[θroll

,θyaw

]

φθpitch

VV

VH

MAV

Z

[X, Y]ω

c = 30 Hz

Sinusoid WaveGenerator

Hybrid ThrustController

Magnitude = 60o

Frequncy = 25 Hz

β LPF with ω

c = 10 Hz

LPF with ω

c = 10 Hz

LPF with ω

c = 10 Hz

θpitch ref

Z ref

[X ref , Y ref ]

StrokeActuator/link Model

[krot l , k

rot r ]

[ψ0 l , ψ

0 r ]

PI PitchController

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vertical/horizontal maneuvers. In simulations, thecontroller is implemented as a discrete-time sys-tem with a fixed time-step of 0.01 s, i.e. per eachstroke cycle, 4 sets of control commands are sentto the simulated MAV. The structure of each sub-controller is briefly discussed next.

6.1 Pitch Controller

For a sinusoidal stroke profile with no bias, theaverage position of center of lift (CoL) for theflapping-wing MAV of Fig. 11 is not directlyabove its CoM. In a real MAV, similar situationscan occur when the position of CoM changes dueto modifications or carrying a payload. In suchscenarios, the pitch torque can quickly destabilizethe vehicle and causes it to crash by over-pitching.

To stabilize the pitch angle of the vehicle, i.e.θ pitch, we employed a PI controller whose outputβ is used to bias the stroke angle of the wings inorder to keep the average position of CoL aboveCoM. Note that at all times, both wings have thesame stroke profile of the form:

φ = φ0 cos (ω t) + β (37)

The reference pitch angle in Fig. 12, i.e. θrefpitch,

may vary depending on the required maneuver,but in our simulations, we never had to use avalue larger than 4◦ in magnitude. Note that β issaturated at ±15◦ to ensure that the stroke anglesremain between −75◦ and 75◦. It is also low-passfiltered in order to simulate a smooth transition inthe amount of stroke bias (Fig. 12).

6.2 Lift Controller

The relationship between krot and average pro-duction of lift force is employed in a hybrid con-troller to regulate the altitude of the vehicle. Ifthe vehicle is supposed to change its altitude Z ,a proportional controller is responsible to stabi-lize the average vertical velocity at a referencevalue of VV = 1 m/s. This reference value is de-creased when the vehicle is within 10 cm of its de-sired altitude. Upon satisfaction of the necessaryconditions (|�Z| < 0.01 m and

∣∣Z

∣∣ < 0.1 m/s),

command is switched to a PID controller that sta-

bilizes the vehicle’s vertical position at the targetaltitude.

From Fig. 10a, the nominal value of krot is cho-sen equal to 3.7×10−3 N.m/rad to ensure sufficientlift production for levitation. To regulate theproduction of lift force, krot is only allowed tochange within the interval [1.4×10−3, 1.11×10−2]N.m/rad. Finally, to avoid sudden changes in thevalue of stiffness and to simulate its smooth transi-tion as a mechanical parameter, the outputs of thelift controller are low-pass filtered prior to beingapplied to the simulated MAV (Fig. 12).

6.3 Thrust Controller

Another hybrid controller is responsible for thrustcontrol by adjusting the set points of the wings’joints. Initially, based on current orientation andposition of the MAV along with the position ofthe target, the deviation angle is calculated. If thisangle is between 2◦ and 178◦, ψ0 is calculatedproportional to the deviation angle from the tar-get. This value determines the magnitude of setpoint angles for both wings but the signs shouldbe opposite and are determined according to therequired direction of steering (Fig. 10b).

Once the deviation angle is within the accept-able range, another sub-controller will be acti-vated to govern horizontal motion and positionstabilization. This part is structurally identical tothe lift controller and uses the same values forreference horizontal velocity—VH = 1 m/s—andinternal switching conditions. However, the out-put of the controller is now used to adjust the setpoints of the wings. Note that in this case, thenet thrusts of both wings must be in the samedirection. Thus, the set points of the wings’ jointsshould have the same sign (Fig. 10b).

The activation of thrust controller is accom-panied with production of yaw and roll torques.While steering relies on yaw torques, roll torquescan destabilize the roll angle of the vehicle, i.e.θ roll, and cause undesired lateral motion. To avoidthis, we can create a small imbalance betweenthe average lift forces of the wings through fur-ther manipulation of their joints’ set points. FromFig. 10b, a slight attenuation in the magnitudeof a non-zero set point will slightly increase the

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average production of lift force by the correspond-ing wing. By doing so for the side that vehiclehas rolled to, the extra lift force on that side willresult in a roll torque that tends to decrease themagnitude of θ roll and keep it close to 0◦. For thispurpose, the attenuation factor γ is calculated asfollows:

γ = max (0, 1 − η |θroll|) (38)

where η = 100 is a constant gain.The value of ψ0 is nominally 0◦ and the thrust

controller can change it within the interval [−15◦,15◦]. This range is chosen to avoid significantintervention with lift production (Fig. 10b). Inaddition, to prevent sudden changes in the valuesof joints’ set points and to simulate their smoothtransition as mechanical parameters, the outputsof the thrust controller are fed to a low-pass filterprior to being applied to the simulated MAV(Fig. 12).

7 Flight Simulations

To evaluate the performance of proposed force/motion control approach, sample results of sim-ulated flight experiments using the MAV modeland controller from Sections 5 and 6 will be pre-sented and discussed next. The gains and timeconstants of the controller are set to reasonablevalues through a group of preliminary tuning ex-periments. The tuned sub-controllers are listed inTable 3.

Table 3 Tuned sub-controllers

Transfer function Description

10 + 0.1/s PI pitch controller25 Proportional vertical velocity

controller100 + 1/s + 5s PID vertical position controller75 Proportional steering controller25 Proportional horizontal velocity

controller500 + 1/s + 100s PID horizontal position

controller

7.1 Takeoff and Hovering

The simulated position and orientation of theMAV during a takeoff/hovering experiment areplotted in Fig. 13. The vehicle is initially inhovering mode and at t = 1 s, it starts to increaseits altitude by 1 m. After reaching the target alti-tude, hovering mode is resumed.

Figure 13 demonstrates a smooth and agilemotion with negligible orientation/position error.Note that during takeoff, the vehicle tends tomove in the horizontal plane (Fig. 13a). This is dueto the fact that changing krot affects the instanta-neous value of thrust forces as well as lift (Fig. 7).However, the thrust controller is able to quicklyrestabilize the MAV by applying small changes inthe set point angles of both wings (Fig. 13g and h).

7.2 Pitch Tracking

Considerable changes in the pitch angle of thebody are often required when an aerial vehicleis performing an agile maneuver. Therefore, it isvery important that the controller can keep upwith such changes while maintaining the stabilityof the vehicle. To investigate the performance ofour approach under these circumstances, we haveexamined the case of a hovering MAV which issupposed to adapt the pitch angle of its body tovarious sequences of rapidly changing referencevalues. The simulated results of one such trial areplotted in Fig. 14.

The reference sequence is shown in Fig. 14calong with the corresponding body pitch angle ofthe MAV. As the vehicle pitches, the directionof average aerodynamic force in the referencecoordinates changes. Therefore, the MAV tendsto move in both horizontal and vertical direc-tions (Fig. 14a and b). However, by readjustingthe impedance properties of the wings’ joints, liftand thrust controllers manage to keep the ve-hicle close to its initial position without riskingdestabilization.

7.3 Three-dimensional Target Seeking

Reaching a specified target position, i.e. (Xref ,Yref , Z ref ), generally requires a combination of

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Fig. 13 A simulatedtakeoff/hoveringexperiment: a X and Ycomponents of horizontalposition, b altitude Z ,c body pitch angle, d bodyyaw and roll angles,e stroke bias angle,f stiffness of the wings’joints, g set point angle ofthe left and h right wings

0 1 2 3 4-10

-5

0

5x 10

-3

(m)

(a) Horizontal Position

0 1 2 3 40

0.2

0.4

0.6

0.8

1

(b) Altitude

Z

(m)

0 1 2 3 4-1

-0.5

0

0.5

1θ pi

tch (

deg)

(c) Body Pitch Angle

0 1 2 3 4-5

0

5

10

β (

deg)

(e) Stroke Bias Angle

0 1 2 3 40

0.002

0.004

0.006

0.008

0.01

0.012

k rot (

N.m

/ rad)

(f) Stiffness

0 1 2 3 4-10

-5

0

5

10(g) Set Point of the Left Wing

ψ0 l

(deg

)

Time (sec)0 1 2 3 4

-10

-5

0

5

10

ψ0 r

(deg

)

(h) Set Point of the Right Wing

Time (sec)

0 1 2 3 4

-0.04

-0.02

0

0.02(d) Body Yaw and Roll Angles

(deg

)

XY

θyaw

θroll

both steering and forward/backward motion alongwith altitude regulation. Thus, by simulating athree-dimensional motion, we can examine the si-multaneous performance of both force controllersduring movement. At the same time, we willbe able to investigate whether or not our thrustcontrol method interferes with the performance oflift controller.

Figure 15 illustrates the simulated position andorientation of the MAV during one such experi-ment. Up to t = 1 s, the MAV is hovering at the

origin while facing along the X axis. At this time,a new target is specified at (1 m, 1 m, 1 m). It takes1.4 s for the MAV to completely reorient towardsthe target (t = 2.4 s in Fig. 15d). Desired altitudeis reached and stabilized within the next second(t = 3.4 s in Fig. 15b). The maneuver is completedat t = 4 s when the vehicle reaches the target(Fig. 15a). The vehicle is then quickly stabilized atthe new location and resumes hovering. Overall,the simulated MAV demonstrates a smooth andagile motion. Note that during steering phase, the

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Fig. 14 A simulated pitchtracking experiment:a X and Y components ofhorizontal position,b altitude Z , c body pitchangle, d body yaw androll angles, e stroke biasangle, f stiffness of thewings’ joints, g set pointangle of the left andh right wings

0 5 10 15 20-0.05

0

0.05

(m)

(a) Horizontal Position

0 5 10 15 20-6

-4

-2

0

2

4x 10

-3 (b) Altitude

Z

(m)

0 5 10 15 20-10

0

10

20(d

eg)

(c) Body Pitch Angle

0 5 10 15 20-20

-10

0

10

20

β (

deg)

(e) Stroke Bias Angle

0 5 10 15 201

2

3

4

5

6x 10

-3

k rot (

N.m

/ rad)

(f) Stiffness

0 5 10 15 20-10

-5

0

5

10(g) Set Point of the Left Wing

ψ0 l

(deg

)

Time (sec)0 5 10 15 20

-10

-5

0

5

10

ψ0 r

(deg

)

(h) Set Point of the Right Wing

Time (sec)

0 5 10 15 20-0.1

-0.05

0

0.05(d) Body Yaw and Roll Angles

(deg

)

X Y

θyaw

θroll

θpitch

θpitch ref

magnitude of body roll angle begins to increase.However, through the proposed countermeasurein Section 6.3, thrust controller is able to stabi-lize this angle and eventually bring it back to 0◦(Fig. 15c).

Both experiments of Figs. 13 and 15 requirethat MAV increases its altitude by 1 m. Compar-ing Fig. 13b with Fig. 15b reveals little differencein the way altitude changes. Figures 13f and 15falso indicate small differences in the transition ofstiffness. This proves that in the chosen operat-ing ranges for mechanical impedance properties,

thrust controller does not significantly interferewith the operation of lift controller.

7.4 Performance in Presence of Noise

So far, the described experiments are simulated inabsence of measurement noise. To investigate theperformance of our control approach in presenceof such noise, we repeated the experiments ofSection 7.3, this time adding white measurementnoise to all feedback parameters. Various powerlevels of noise were examined. The results suggest

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Fig. 15 A simulatedthree-dimensional targetseeking experiment:a X and Y components ofhorizontal position,b altitude Z , c body pitchand roll angles, d bodyyaw angle, e stroke biasangle, f stiffness of thewings’ joints, g set pointangle of the left andh right wings

0 2 4 6 80

0.2

0.4

0.6

0.8

1

(m)

(a) Horizontal Position

0 2 4 6 80

0.2

0.4

0.6

0.8

1

(b) Altitude

Z

(m)

0 2 4 6 8-5

0

5(d

eg)

(c) Body Pitch and Roll Angles

0 2 4 6 8-20

-10

0

10

20

β (

deg)

(e) Stroke Bias Angle

0 2 4 6 80

0.002

0.004

0.006

0.008

0.01

0.012

k rot (

N.m

/ rad)

(f) Stiffness

0 2 4 6 8-20

-10

0

10

20(g) Set Point of the Left Wing

ψ0 l

(deg

)

Time (sec)0 2 4 6 8

-20

-10

0

10

20

ψ0 r

(deg

)

(h) Set Point of the Right Wing

Time (sec)

0 2 4 6 8-10

0

10

20

30

40(d) Body Yaw Angle

θ yaw (

deg)

XY

θpitch

θroll

that our controller can handle position and an-gle measurement noise with standard deviationsas high as 2 mm and 1◦, respectively. However,considering that our current controller is basedon simple PID control laws, we believe that amore sophisticated controller can improve thesemargins further.

For comparison, the experiment of Fig. 15 isrepeated in presence of measurement noise withstandard deviations of 2 mm and 1◦. The resultsare plotted in Fig. 16. Here, evolution of posi-tion and orientation of the MAV (Fig. 16a–d)

is slightly different form the noise-free situation(Fig. 15a–d). However, the model is still capableof steering towards the target and manages toreach it without any significant degradation inperformance.

Note that noise with limited bandwidth can beused to model small physical differences betweenthe left and right sides of an actual MAV’s body.It can also represent small undesired differencesbetween impedance actuators of both sides.Therefore, we expect that in a real system, ourcontroller can deal with such imperfections.

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Fig. 16 A simulatedthree-dimensional targetseeking experiment inpresence of measurementnoise: a X and Ycomponents of horizontalposition, b altitude Z ,c body pitch and rollangles, d body yaw angle,e stroke bias angle,f stiffness of the wings’joints, g set point angle ofthe left and h right wings

0 2 4 6 80

0.2

0.4

0.6

0.8

1

(m)

(a) Horizontal Position

0 2 4 6 80

0.2

0.4

0.6

0.8

1

(b) Altitude

Z

(m)

0 2 4 6 8-5

0

5(d

eg)

(c) Body Pitch and Roll Angles

0 2 4 6 8-20

-10

0

10

20

β (

deg)

(e) Stroke Bias Angle

0 2 4 6 80

0.002

0.004

0.006

0.008

0.01

0.012

k rot (

N.m

/ rad)

(f) Stiffness

0 2 4 6 8-20

-10

0

10

20(g) Set Point of the Left Wing

ψ0 l

(deg

)

Time (sec)0 2 4 6 8

-20

-10

0

10

20

ψ0 r

(deg

)

(h) Set Point of the Right Wing

Time (sec)

0 2 4 6 8-10

0

10

20

30

40(d) Body Yaw Angle

θ yaw (

deg)

XY

θpitch

θroll

7.5 Perturbation Recovery

The ability of the proposed force/motion controlapproach in handling perturbations and restabi-lization of the vehicle after an impact is anotherimportant performance measure that should beinvestigated. Various scenarios of this type havebeen simulated and the controller was always ableto restore the balance of the MAV and resumeits interrupted action within a very short amountof time.

The example in Fig. 17 illustrates one such trial.The vehicle is initially in hovering mode until at

t = 1 s, it is briefly (�t = 7 × 10−3 s) exposedto a large external force—modeled as additionalforce and torque terms in Eqs. 32 and 33—fromthe right-front direction. By providing an initialvelocity, the impact forces the MAV to moveaway from its hovering position (Fig. 17a and b).Furthermore, the torque produced by this forcewith respect to the CoM of the vehicle results inalmost instantaneous jumps in pitch, roll and yawangles of the vehicle (Fig. 17c and d).

After the impact, the vehicle has to resumehovering at its initial position. By t = 2 s, thepitch controller stabilizes θ pitch (Fig. 17c) through

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Fig. 17 A simulatedperturbation recoveryexperiment: a X and Ycomponents of horizontalposition, b altitude Z ,c body pitch and rollangles, d body yaw angle,e stroke bias angle,f stiffness of the wings’joints, g set point angle ofthe left and h right wings

0 1 2 3 4 5 6-1.5

-1

-0.5

0

0.5

(m)

(a) Horizontal Position

0 1 2 3 4 5 6

-0.1

-0.05

0

0.05(b) Altitude

Z

(m)

0 1 2 3 4 5 6-15

-10

-5

0

5(d

eg)

(c) Body Pitch and Roll Angles

0 1 2 3 4 5 6-20

-10

0

10

20

β (

deg)

(e) Stroke Bias Angle

0 1 2 3 4 5 60

2

4

6

8x 10

-3

k rot (

N.m

/ rad)

(f) Stiffness

0 1 2 3 4 5 6-20

-10

0

10

20(g) Set Point of the Left Wing

ψ0 l

(deg

)

Time (sec)0 1 2 3 4 5 6

-20

-10

0

10

20

ψ0 r

(deg

)

(h) Set Point of the Right Wing

Time (sec)

0 1 2 3 4 5 6-20

-10

0

10

20(d) Body Yaw Angle

θ yaw (

deg)

XY

θpitch

θroll

appropriate biasing of the stroke angles (Fig. 17e).In the same period, the lift controller is able tobring the vehicle close to its initial altitude andstabilizes it within the next second (Fig. 17b).Meanwhile, the thrust controller simultaneouslyreduces the roll angle (Fig. 17c), reorients thevehicle towards the origin (Fig. 17d) and decreasesthe induced horizontal velocity due to impact(Fig. 17a). Steering is completed at t = 2.5 s.However, while reducing the horizontal velocity,the vehicle considerably moves in the horizontalplane. Therefore, another 1.5 s is required until

the MAV reaches its target and resumes hoveringat t = 4 s. Overall, the proposed control approachdemonstrates a relatively fast response during allperturbation recovery experiments.

8 Conclusion

The combination of a simple muscle model anda quasi-steady-state aerodynamic model makesit possible to estimate how angle of attack and

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aerodynamic forces of insect wings evolve as asinusoidal stroke cycle is executed. The resultsof these simulations suggest that lift and thrustforces could be controlled through regulating thestiffness (co-activation of synchronous muscles)and set point (differential activation of synchro-nous muscles) of the wing’s joint. Hence, synchro-nous muscles may play a key role in flight controlcapabilities of insects.

This potential mechanism of flight control in in-sects has inspired us to develop a similar approachfor force/motion control in insect-like MAVs suchas [19]. We refer to this method as “Tunable Im-pedance” (TI) technique. As of yet, this methodhas been only used to control simulated MAVsduring various flight maneuvers. Simulations forone such vehicle—the size of a large dragonfly—were presented in Section 7. The results of ex-periments with a simulated fly-scale MAV usinga preliminary version of this control approachare also available in [21]. In all experiments, thecontroller has demonstrated an exceptional per-formance that results in a high degree of maneu-verability and agile motions.

One advantage of force control using TI tech-nique is that it does not need to change the strokeprofiles of the wings in order to generate lift/thrustor steering torques. Therefore, both wings havethe same stroke profile at all times and can bedriven with a single actuator in a real MAV. Incontrast, ad hoc solutions such as “Split Cycle”(SC) method [22] assume that wings can havedifferent stroke profiles and use the differencebetween corresponding forces to produce steeringtorques. This means that each wing must have itsown stroke actuator. At the investigated scale inthis paper, these actuators are usually dc motorsthat impose a considerable weight challenge inthe design stage. Furthermore, solutions such asSC need to change the shape of stroke profilein order to regulate aerodynamic forces. For ex-ample, SC requires downstroke to be faster thanupstroke during a forward thrust maneuver. Op-erating at higher frequencies increases the powerconsumption of stroke actuators which is anotherdesign problem due to limited weight for onboardbatteries. In addition, the stroke actuators musthave higher bandwidth to accommodate for suchapproaches. This further complicates the design

challenges and trade-offs that must be met in theprocess.

Implementation of the TI method on a realMAV calls for development of variable im-pedance actuators that meet certain shape/weightcriteria. The diagram in Fig. 4 serves as a basicdesign scheme for such actuators; the tendonsdeveloped in [23] are a suitable replacement forquadratic springs. However, more complex andefficient designs are available for impedance al-teration [24–26]. It is worth noting that in a MAV,impedance actuators have to deal with small loadsand thus, can be piezoelectric-based or developedthrough MEMS technology. This will significantlyreduce their weight and power requirements andgreatly benefits the overall design problem.

To investigate the performance of TI techniquein a real system, we are currently implementingthis method on a scaled bench-top flapping-wingtester. Our goal is to eventually use this approachon a 3-inch-wingspan flapping-wing MAV.

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