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Flapping Flight for Biomimetic Robotic InsectsPart II: Flight Control Design

Xinyan Deng, Luca Schenato, and Shankar Sastry,Department of Electrical Engineering and Computer Sciences

University of California at Berkeley{xinyan,lusche,sastry}@eecs.berkeley.edu

Abstract— In this paper we present the design of flight controlalgorithms for flapping wing micromechanical flying insects(MFIs). Inspired by the sensory feedback and neuromotor struc-ture of biological flying insects, we propose a similar top-downhierarchical architecture to achieve high performance despitethe MFIs’ limited computational resources. Flight stabilizationis formulated as high frequency periodic control of an under-actuated system. In particular, we provide a methodology toapproximate the time-varying body dynamics caused by theaerodynamic forces with time-invariant dynamics using averagingtheory and a biomimetic parametrization of wing trajectories.This approximation leads to a simpler dynamical model thatcan be identified using experimental data from the onboardsensors and the input voltages to the wing actuators. Moreover,the overall control law is a simple periodic proportional outputfeedback. Simulations, including sensor and actuator models,demonstrate stable flight in hovering mode.

Index Terms— flapping flight, micro aerial vehicles,biomimetic, periodic control, averaging.

I. I NTRODUCTION

The recent interest in micro aerial vehicles (MAVs) [1],largely motivated by the need for aerial reconnaissance robotsinside buildings and confined spaces, has galvanized thedevelopment of inch-size flapping wing MAVs that couldmimic at least part of the extraordinary performance of insectflight. This is a challenging endeavor for several reasons.First, aerodynamics for inch-size flapping robots differ greatlyfrom manmade fixed or rotary-winged vehicles [2]. Second,size constraints forbid the use of rotary electric motors andcommercial inertial navigation system (INS), global position-ing systems (GPS) and current cameras. Finally, a flappingfrequency beyond100Hz requires sensors and processingalgorithms with bandwidth and sensitivity at least one order ofmagnitude higher than those usually found in today’s aircrafts.Nonetheless, recent technological advances, together with bet-ter understanding of insect aerodynamics and biomechanicshave promoted projects aimed at the design of Micromechan-ical Flying Insects (MFIs) [3] [4].

The goal of this paper is to develop a general frameworkto design a control unit for MFIs which would enable themto accomplish complex autonomous tasks such as search-ing, surveillance and monitoring. This paper builds upon acompanion paper [5] where comprehensive modeling of MFIaerodynamics, body dynamics, sensors, and electromechanical

This work was funded by ONR MURI N00014-98-1-0671,ONR DURIPN00014-99-1-0720 and DARPA.

Authors are with the Robotics and Intelligent Machines Lab in the Depart-ment of Electrical Engineering and Computer Sciences at the University ofCalifornia at Berkeley

actuation is presented together with a list of references torelevant research. Here, we focus on the control aspects offlapping flight for MFIs. In particular, we propose a hi-erarchical architecture for the control unit that mimics thesensory feedback and neuromotor structure of biological flyinginsects to achieve high performance while satisfying MFIsphysical and computational constraints. One of the majorcontributions of this paper is to approximate the time-varying(TV) dynamics of insect flight caused by the flapping wingswith a simpler time-invariant (TI) system whose controllerscan then be formulated using standard control design tools.This approximation relies on two ideas that can be formalizedwithin the framework of high-frequency control theory. Thefirst key idea is that the frequency of the aerodynamic forcesacting on the insect is much larger than the bandwidth of bodydynamics, and, therefore, only the mean aerodynamic forcesand torques affect the insect dynamics. The second idea isto parameterize the wing trajectory within a wingbeat periodusing some biologically inspired wing kinematic parameters,which affect the distribution of aerodynamic forces withinthe wingbeat, thus modulating the total forces and torquesacting in the insect center of mass. These parameters appearas virtual inputs in the TI approximation of flight dynamics.Finally, we show how the parameters of the TI approximationcan be identified directly from sensors measurements andactuators input voltages obtained from experiments from thetrue TV system. This approach is particularly suitable forflapping flight since it does not require the knowledge of exactaerodynamics models, which are particularly complex. Also, itprovides a model for uncertainty caused by sensor and actuatornonlinearities and external disturbances that can be used todesign robust controllers.

The paper is organized as follows. In Section II, we brieflyreview biological literature about insect flight control mecha-nisms, focusing on the interaction between the sensory systemand the neuromotor architecture. In Section III the hierarchicalarchitecture of flight control observed in true insects, andthe helicopter attitude-based navigation are then used as amodel for the design of an equivalent control system forMFIs. In Section IV we highlight analogies and differencesbetween flapping flight and helicopter flight. In Section V wepropose a formal approach to approximate the time-varyinginsect dynamics with a time invariant dynamics based onaveraging theory and wing trajectory parametrization. SectionVI presents the design of the input voltage to the actuatorsthat is required to track a desired wing trajectory. In SectionVII we model insect dynamics as a discrete time dynam-ical system where the input are the kinematic parameters

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defined in the previous section. Closed-loop identification isthen implemented to estimate the discrete-time system. Theidentified model is then used to design LQR-based feedbacklaws for hovering. Finally, in Section VIII, conclusions andfuture research directions are presented.

II. I NSECTFLIGHT SENSORS ANDCONTROL

MECHANISMS

Flies have inhabited our planet for over 300 million years,and today they account for more than 125,000 differentspecies, so that, by now, roughly every tenth known speciesis a fly [6]. This evolutionary success might spring fromtheir insuperable maneuverability and agility to survive, whichenable them, for example, to chase mates at turning velocitiesof more than3000os−1 with delay times of less than 30 ms.

The extraordinary maneuverability exhibited by flying in-sects is the result of a sophisticated neuromotor control sys-tem combined with highly specialized sensors. These sensorscomprise of the pressure sensilla, the halteres, the ocelli, andthe compound eyes.

Pressure force sensillaare present along the wing surface,the wing base, the halteres, and other parts of the body.Although their functionality in flight control is not clear, theymight play an important role in estimating the instantaneousair flow around the wing and in controlling the wing trajectory.

Thehalteres, two oscillating club-shape appendices, are thebiological equivalence of a gyroscope, and they are used toestimate the body angular velocity.

The ocelli, a sensor system composed of three wide-anglephotoreceptors oriented in a tetrahedron configuration, canestimate insect orientation relative to the horizon by comparingthe light intensity from different regions of the sky.

The compound eyesserve the purpose of estimating large-field optical flow, small-field object fixation, and object recog-nition. The large-field optical flow estimated from the com-pound eyes can provide information about the orientation,the angular velocity, and the linear velocity which can guar-antee excellent performance when combined with ocelli andhalteres [7]. The large-field optical flow from the compoundeyes, together with ocelli and halteres, play the role of theinertial navigation system (INS) in insect flight. Furthermore,compound eyes can also perform specialized visual processingfor object fixation and landmark recognition, which is used tonavigate the environment and estimate proximity of obstaclesand targets.

A more detailed description for these sensors from a flightcontrol perspective can be found in [7] [5] and in the refer-ences therein.

At present, still little is known about the flight controlmechanisms and neuromotor physiology in true insects [8] [9][6]. Experimental evidence suggests the existence of at leasttwo levels of control, as shown in Fig. 1. At the lower levelthe halteres and the ocelli control the wings muscles directlyin order to keep stable flight orientation. This level of controlseems to be reactive, since it mediates corrective reflexes tocompensate for external disturbances and to maintain a stableflight posture. At the high level, the brain, stimulated byvisual and physiological stimuli, plays the role of a navigationplanner, which plans a trajectory based on its ultimate goal,such as foraging or chasing a mate. Differently from the

Visual motor control signals

Halteremuscles

Haltere

WING

Inertial motor control signals

Ocelli

Compoundeye

Wingmuscles

Fig. 1. Neuromotor control physiology in flying insect.

halteres-ocelli system, the visual system is not connecteddirectly to the wing muscles, but to the halteres muscles.Therefore, this level of control indirectly affects the flightbehavior by biasing the motion of the halteres, thus creatingan apparent external disturbance that the lower level of controlwould try to compensate for. This structure is similar to thatbetween the vestibular-ocular reflexes and active head rotationin vertebrates [10]. The reason for this hierarchical controlarchitecture is probably an efficient solution to resolve theconflict between flight stability reflexes and goal-orientatedmanoeuvres. This typical biological neuromotor control ar-chitecture is shown in the left side of Fig. 2. Without someappropriate inhibiting mechanism, the haltere-mediated equi-librium reflexes would always counter goal-oriented motions.To resolve this potential conflict, the nervous system mustcontain the means of attenuating equilibrium reflexes duringthe generation of controlled maneuvers.

Another sublevel, as part of the reactive control system,might be present and associated with the pressure sensorswhich innervate the wings and the haltere. This bottom levelmight adjust wing motion within a single wingbeat to improveaerodynamic efficiency and compensate for local turbulence[11].

The hierarchical structure of neuromotor control in trueinsects has been adopted as a guiding model for the designof the control unit for MFIs, as described in the next section.

III. H IERARCHICAL CONTROL ARCHITECTURE

The hierarchical architecture, partially inspired by trueflying insects and autonomous aerial robots research [12],decomposes the original flight control problem into a set ofhierarchical modules, each responsible for a specific task.This way, the controllers in each module can be designedindependently of those on higher levels, thus allowing thepossibility to incrementally build more and more articulatedcontrol structures. Fig. 2 shows the architecture proposed forthe MFI control unit. It is possible to identify three main levels:thenavigation planner, theflight mode stabilizerand thewingtrajectory controller. The top level is a voluntary one sinceplanning is determined by MFI goal, the two lower levels aremore reactive since the purpose of the flight mode stabilizers

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and the wing kinematic generator is to maintain the desiredflight posture and the desired wing trajectory in the presence ofexternal disturbances, respectively. Each of these three levelsin the control unit receive specific sensory information fromdifferent sensors.

ANIMALDYNAMICS

ADAPTIVECONTROL(Learning)

ACTIVECONTROL

REACTIVECONTROL

Navigation Planner

Wing Trajectory Controller

MFIDYNAMICS

Flow SensorsHalteresOcelliCompass

Chemical sensorsCompound eyes

Actuator inputs

BIOLOGICAL NEUROMOTOR

CONTROL ARCHITETURE

MFI FLIGHTCONTROL

ARCHITETURE

Flight Mode StabilizerHover Cruise Steer Climb

Flight mode selection

Desired wing trajectory

Force sensorsat wing base

Fig. 2. Design architecture for the control unit of the MFI comparedto the neuromotor control architecture present in most animals.

At the top level of the control unit there is thenavigationplanner. Besides sensory input from the visual system, thisunit can receive commands from a communication link andinformation from application-specific sensors such as chemicalor temperature sensors. The purpose of this module is tochoose a sequence of appropriate flight modes for the flightmode stabilizer level, which enables the MFI to safely navigatethe environment and achieve the desired task such as territoryexploration, target localization and tracking.

The middle level is theflight mode stabilizerwhich isresponsible for stabilizing different flight modes available tothe MFI, such as take off, hovering, cruising, steer left, steerright, climb, dive, and land. Each flight mode is achieved bya dedicated controller that uses as inputs the signals from thehalteres, the ocelli, the large-field optical flow estimates, anda magnetic compass. Based on this information, the controllerchooses the appropriate values for the desired torques andforces that must be applied to MFI body to compensatepossible disturbances and to maintain the desired flight mode.The desired torques and forces are then mapped directly intothe corresponding wing trajectory for the next wingbeat, asshown in Section V-C.

The bottom level is thewing trajectory controllerwhichis responsible for generating the electrical signals for theactuators in order to track the desired wing motion generatedby the flight mode stabilizer module. The set of possible wingtrajectories is parameterized according to some biokinematicparameters, as described in Section V-C. These parametersare chosen based on biomimetic principles, i.e. by changingthem it is possible to replicate most of the wing trajectoriesobserved in true insects. The most important biokinematicparameters are the stroke angle amplitude and offset, timingof rotation, mean angle of attack, and upstroke-to-downstrokewing speed ratio. The active change of these parameters bytrue insects have been observed to be directly correlated tospecific maneuvers and flight modes [13]. Then every wing

trajectory is mapped to the corresponding actuator voltages viaanother map, as described in Section VI. The wing trajectorycontroller receives input information from force sensors placedat the wing’s base. This sensory information can be directlyused to estimate instantaneously the position and velocityof the wing, thus improving wing motion control throughfeedback.

IV. I NSECT VERSUSHELICOPTERFLIGHT

Similar to aerial vehicles that are based on rotary wings,such as helicopters, flying insects control their flight by con-trolling their attitude and the magnitude of the vertical thrust[13]. Position and velocity control is achieved via attitudecontrol, in fact forces acting on a plane parallel to the groundcan be generated by tilting and banking the body. Pitchingdown, for example, would result in a forward thrust, whilerolling sideward would result in a lateral acceleration. Altitudecontrol is achieved via mean lift modulation, for example,by increasing the vertical force would result in an upwardacceleration and vice versa.

However, there are some peculiar differences that preventone from directly applying successful flight control techniquesdeveloped for helicopters [14]. The spinning of the rotor bladeinduces a reaction yaw torque on the helicopter body thatwould make the latter rotate in the opposite direction if notcompensated by the tail rotor. On the other hand, the tailrotor generates a lateral thrust that needs to be compensatedby tilting the helicopter body sideways. This problem isnot present in insect flight since the wings oscillate almostsymmetrically on the opposite side of the insect body, thereforeinertial forces cancel out. Another very important difference isthe intrinsic time varying nature of the aerodynamics in insectflight. As shown in Fig. 5 the aerodynamic forces and torquesgenerated by the wings change dramatically during a wingbeat,and cannot be assumed to be constant as in the case of thehelicopter. One additional difference is that the wing trajectorycannot change dramatically from one wingbeat to the next,since the wings need to oscillate to generate sufficient lift tosustain the insect weight. Finally, in insect flight the two wingscan follow asymmetric trajectories. This peculiar characteris-tic allows insects to generate large angular accelerations bymodulating the distribution of the aerodynamic forces withina wingbeat, without strongly affecting the mean lift generation.The dependence between wing motion and torque generationin true insects has recently been considered in [15].

These similarities and differences lead us to consider thefollowing strategy when designing a robust stabilizing hover-ing controller. First, we will model the insect dynamics as aDiscrete Time Linear Time Invariant (DTLTI) system based onthe average forces and torques over a wingbeat. This approachis based on high frequency control theory that guarantees goodapproximation error between the original time-varying systemand averaged system, assuming that the wingbeat frequency issufficiently high [16]. Moreover, the design for the controlleris based on a MFI dynamics model obtained through anidentification procedure that includes the approximation errorsdue to the time varying nature of the dynamics.

Second, we parameterize the wing kinematics with four pa-rameters, such that they can be mapped uniquely into the threemean torques (roll, pitch, yaw) and mean lift. This approach

4

B

ϕwing profile

WING

3D VIEW

o

φr

φ

CENTER OFMASS

l

TOP VIEW

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x

y

z

B

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φ

to stroke plane

Planeperpendicular

ψθ

ηStroke plane

α

Plane parallel to

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Fig. 3. Definition of wing kinematic parameters: (left) 3D view ofinsect body and left wing, (right) top view of insect stroke plane.

allows a direct control of the torques and lift generation, thussimplifying the control design for the attitude and altitudeof the MFI. The dynamics of the insect is then linearizedabout the hovering equilibrium point and we propose someapproximations that decompose the original MIMO systeminto four SISO subsystems. Finally, the controller is basedon robust state feedback based on linear-quadratic regulator(LQR) design.

V. H IGH FREQUENCY INSECT FLIGHT CONTROL

A. Insect dynamics

As shown in [5], the insect dynamics can be written as:

Θ = (IW )−1[τ ba(t)−W Θ× IW Θ− IW Θ]

p = − c

mp− g +

1m

Rf ba(t) (1)

whereτ ba ∈ R3 andf b

a ∈ R3 are the torque and force vectorsgenerated by wing aerodynamics relative to the insect centerof mass. The vectorΘ = [η θ ψ]T represents theZY X Euler’sangles (roll,pitch,yaw),W = W (Θ) is the transformationmatrix from body angular velocity,ωb, to Euler’s angularvelocity, Θ, i.e. Θ = Wωb. I is the insect moment of inertiarelative to the body frame,p is the position of the insect centerof mass relative to the fixed frame,g = [0 0 − g]T is thegravity vector,c is the linear damping coefficient, andR =ezψeyθexϕ is the rotation matrix. This notation is commonlyfound in spacecraft and helicopter dynamics textbooks such as[17] and [14]. The wrench, i.e. the forces and torques, due tothe aerodynamic forces is a nonlinear function of the positionand velocity of the wing stroke angle,φ, and the angle ofattack,α, of both wings, i.e.

f ba(t) = f b

a(φr, φl, ϕr, ϕl, φr, φl, ϕr, ϕl) = f ba(u, u)

τ ba(t) = τ b

a(φr, φl, ϕr, ϕl, φr, φl, ϕr, ϕl) = τ ba(u, u) (2)

whereu = (φr, φl, ϕr, ϕl), and the lowerscriptsr, l stand forright and left wing, respectively. The stroke angle is the anglebetween the wing radial axis and they-axis of the strokeplane. The rotation angle is defined as the angle between thevertical plane and the wing profile, which corresponds to thecomplement of the angle of attackα, i.e. α = 90o − |ϕ| (seeFig. 3). The explicit expression of aerodynamics forces andtorques as a function of wing kinematics can be found in [5].The aerodynamic forces and torque are the only time-varyingelement in Equations (1), otherwise the insect dynamics would

be very similar to the time-invariant nonlinear dynamics of ahelicopter. On the other hand, the wingbeat period is muchsmaller than the responsiveness of the insect body, therefore,intuitively speaking, only mean forces and torques are relevant.In fact, this approximation has been formalized by averagingtheory [16], and high-frequency periodic control has beensuccessfully applied to trajectory tracking and approximatestabilization of underactuated systems [18] [19] [20]. However,few tools are available to synthesize the control laws system-atically and they are mainly limited to mechanical systemswith specific geometric properties [21] [22]. In particular,these tools derive a suitable time-varying input parametrizationu = g(v, t) directly from the structure of the flow of thedynamics,x = f(x, u) = f0(x) + F (x)u. In our case, wedo not directly apply these tools because of the complexityof the aerodynamic forces and torques, and thus of the vectorflow, as function of the wing’s angles and velocityu. Onepossible approach to designing the control laws is the adoptionof a linearized model of insect aerodynamics and apply theaforementioned tools, however this approach is likely to givepoor results in practice, since insect aerodynamics is a highlynonlinear function of wing position and velocity. Instead, wepropose to parameterize the wing motion based on biomimeticprinciples to design our periodic inputs, i.e. we proposeu =g(v, t). Then, by applying averaging theory to approximatethe complex time varying dynamics with the average timeinvariant dynamics, we show that there is a direct map betweenthe proposed kinematic parameters and the mean forces andtorques. The kinematics parameters appear as virtual inputsin the averaged dynamics. The averaged dynamics is thensuitable to standard controller design similarly to those foundin helicopter control.

B. Averaging

Averaging theory and high frequency control encompassseveral results and they have been applied in different scientificareas. Recently, these results have been applied specifically toinsect flight [23]. Here we report only some of the results thatwe will use for the flight controller design.

Theorem 1 ([23]). Let us consider the following systems:

x = f(x, u, u)u = g(v, t)v = h(x)g(v, t) = g(v, t + T )

(3)

˙x = f(x, v)f(x, v) = 1

T

∫ T

0f(x, g(v, t), g(v, t))dt

v = h(x)(4)

wherex, x ∈ Rn, u ∈ Rm, v ∈ Rp, and all functions and theirpartial derivatives are continuous up to second order.

If x = 0 is an exponentially stable equilibrium point forthe averaged system (4), thenx(t) − x(t) = O(T ) for allt ∈ [0,∞). Moreover the original system (3) has a unique,exponentially stable, T-periodic orbitxT (t) with the property||xT (t)|| < kT .

In our setting,T is the wingbeat period, and the systemf(x, u) is given by Equations (1) and (2), where the vectoru = (φr, φl, ϕr, ϕl) represents the right and left wing angles.As it will be shown in the next section, the wing trajectories

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are chosen to beT -periodic functions and are parameterizedby a parameter vectorv, i.e.u = g(t, v). The parameter vectorv can be interpreted as a vector of virtual inputs. Therefore,as indicated by the theorem, rather than trying to analyze theoriginal system we will focus on its averaged dynamics givenby Equations (1) where the time-varying wrench(f b

a(t), τ ba(t))

given in Equations 2 is substituted with its average:

f ba(v) ∆=

1T

∫ T

0

f ba(g(t, v), g(t, v))dt

τ ba(v) ∆=

1T

∫ T

0

τ ba(g(t, v), g(t, v))dt (5)

which is time-independent and depends only on the virtualinput vectorv. We will then look for exponentially stabilizingcontrol feedback lawv = h(x) for the averaged systems. Ifsuch a function exists, then the original time-varying systemwill have a bounded error from the desired equilibrium pointif the wingbeat periodT is sufficiently small. Althoughthis approach does not guarantee asymptotic stability for theoriginal system, we will show that the error boundkT is verysmall for insect flight as observed in true insects, and thereforeirrelevant from a practical point of view.

C. Wing Kinematic Parametrization

Although it is currently unclear how true insects accomplishthe control of their flight and maneuvering capabilities, recentwork has found that by modulating a few kinematic parameterson each wing, such as wing rotation timing at the strokereversals and the wing blade angle of attack, the insect canreadily apply torques on the body and, therefore, control itsattitude and position [2] [13]. Based on this observation, itwas suggested that a small set of wing kinematics might besufficient to generate all possible flight modes, and the keypoint for designing any of these modes is the capability tocontrol the MFI’s attitude [24].

In particular, the research done by Dickinson and his group[2] has suggested that the following kinematic parameters maysuffice to generate any flight maneuver:timing of rotation,mean angle of attack, stroke angle amplitudeandstroke angleoffset. There is a strong evidence that if these parameters canbe controlled independently, then it is possible to control thetorque and force generation during flapping. For example,a large (small) stroke angle amplitude would generate alarge (small) lift. An advanced (delayed) timing of rotationat the end of the downstroke results in a nose-up (nose-down) pitch torque. A larger (smaller) angle of attack duringthe downstroke relative to the upstroke produces a backward(forward) thrust. Most true insects flap their wings along asymmetric trajectory with a stroke angle amplitude around120o and mean angle of attack of45o on both downstrokeand upstroke [25] [6]. However, during saccades and othermaneuvers, they modify the wing trajectory by changing thekinematic parameters described above [26].

Based on these biologically inspired arguments, the problemto solve then is how to parameterize the wing trajectory to beable to mimic the real insects byindependentlycontrollingthe biokinematic parameters described above. We will thenshow how the parameters map directly to the mean torquesand forces, thus simplifying the design of a flight stabilizer.More specifically, the wing trajectory during a wingbeat is

0 0.25T 0.5T 0.75T T −60

−40

−20

0

20

40

60

Time ( T units)

Ang

les

( de

gs)

gψ(t)

gφ(t)g

v1

(t),gv

2

(t)

Fig. 4. Wing kinematic parameterizing functions given in Equa-tions (8).

described using the stroke angle,φ and the rotation angleϕ.In particular, we parameterize the wing motion of each wingwithin a wingbeat period as follows:

φ(t; υ) = gφ(t) + v1 g1(t) (6)

ϕ(t; υ) = gϕ(t) + v2 g2(t) (7)

where the functionsgi(t) areT -periodic function, i.e.gi(t +T ) = gi(t), (v1, v2) are the kinematic parameters, andTis the wingbeat period. These functions are chosen basedon the considerations above. In particular,gϕ(t) and gφ(t)generate a symmetric motion with maximum lift production,g1(t) modifies only the stroke angle amplitude,g2(t) modifiesthe timing of rotation of the angle of attack at the end of thedownstroke. Based on observations of true insect flight wechoose the following functions:

gφ(t) =π

3cos(

2π

Tt)

gϕ(t) =π

4sin(

2π

Tt)

g1(t) =π

15sin3(

π

Tt)

g2(t) = g1(t) (8)

which are defined on the intervalt ∈ [0, T ) and extended byperiodicity so thatgi(t + T ) = gi(t). Fig. 4 shows a pictorialrepresentation of wing motion and corresponding aerodynamicforces for different choices of the kinematic parameterv1 andv2. Note how these parameters affect the distribution of forcesalong the whole wingbeat period.

The wing parametrization given by Equations (8) is notunique and might not be optimal either, however it givesrise to wing trajectories that mimic some of the trajectoriesobserved in true insects. In fact, a positive (negative) value forv1 results in a large (small) stroke angle amplitude; a positive(negative) value forβ results in a delayed (advanced) timingof rotation at the end of the downstroke. If this parametrizationabove is replicated for both wings, the wings kinematicsu = (φr, φl, ϕr, ϕl) can be written in terms of the parameters

6

60 40 20 0 −20 −40 −60

symmetric motion

t=0 t=0.5T

t=0.5T t=T

DOWN−STROKE

UP−STROKE

60 40 20 0 −20 −40 −60

advanced rotation

60 40 20 0 −20 −40 −60

large stroke amplitude

60 40 20 0 −20 −40 −60

delayed rotation

φ (degs) 60 40 20 0 −20 −40 −60

small stroke amplitude

(degs) φ

Fig. 5. Pictorial sequence of the side view of wing motions and the corresponding aerodynamic forces for different choice of kinematicparameters. Symmetric motion:v1 = 0, v2 = 0. Advanced rotation:v1 = 0, v2 = 1. Delayed rotation:v1 = 0, v2 = −1. Large strokeamplitude:v1 = 1, v2 = 0. Small stroke amplitude:v1 = −1, v2 = 0.

v = (vr1, v

l1, v

r2, v

l2) as follows:

u(v, t) = g(t) + G(t)v (9)

g =

gφ

gφ

gϕ

gϕ

, G =

g1 0 0 00 g1 0 00 0 g2 00 0 0 g2

where g(t) and G(t) are a T -periodic vector and matrix,respectively, whose entries are defined in Equations (8).

It is now possible to study the effect of the chosenparametrization on the mean wrench. In fact, if we substituteEquation (9) into Equations (5), we obtain a static mapΠ :R4 → R6 from the wings parametersv ∈ R4 to the meanwrench(f b

a, τ ba) ∈ R6:

[f b

a

τ ba

]= Π(v) (10)

This is a nonlinear map and cannot be computed analyticallysince the aerodynamic force and torque are complex functionsof wing position and velocity (see Section IV in [5]). However,one could look for an affine approximation around the originof the wings parameters:

[f b

a

τ ba

]= π0 + Πlv + δ(v) (11)

whereπ0 ∈ R6, Πl ∈ R6×4, and δ(v) is the approximationerror. Although, it is not possible to linearize analyticallyEquation (11) to obtainπ0 and Πl directly, it is possible torandomly select different values for the parameter vectorv,substitute it into the parametrization given by Equation (9), andfinally compute the true mean wrench(f b

a, τ ba) via simulations

using the exact wing aerodynamics. The approximatingπ0

and Πl can then be found by rewriting Equation (11) as aleast square (LS) problem where(π0, Πl) are the unknowns.Simulations are performed based on the aerodynamic modeldescribed in [5], and on the morphological body parameterof a typical blowfly, which is the MFI target model. Theapproximating affine map is found to be as follows:

π0 =

00

mg000

, Πl = 0.1mg

0 0 −1.0 −1.00 0 0.3 −0.3

0.9 0.9 0 00.4L −0.4L −0.1L 0.1L

−0.2L −0.2L −0.4L −0.4L0 0 −0.5L 0.5L

(12)wherem is the mass of the insect,L is the length of the wing,and the zero entries correspond to estimated values negligiblerelative to the largest entries in the matrix. This approximationis quite accurate for kinematic parameters smaller than unity,||v||∞ ≤ 1. Fig. 6 shows that the estimated mean wrench,w = π0 +Πlv, predicts quite accurately the true mean wrench

7

obtained from simulations, thus validating our approach.The particular structure of this map is a consequence of the

parameterization based on the biological insights described atthe beginning of this section. In fact, as we expect, any compo-nent of the wrench depends additively or differentially on twoparameters, depending if the wings are moving symmetricallyor anti-symmetrically. Note that along the z-component thesymmetrical wing motions generate a vertical lift sufficient tobalance insect body weight. The magnitude of the coefficientsin the map are considerable. In fact, the insect can generateforward or vertical thrust in the order offa ≈ 0.1−0.2 mg, andangular torques of orderτa = 0.1− 0.2 mLg. In other words,considering that the moment of inertia of a true insect alongone of its principal axis is on the order ofI ≈ 0.1mL2 [25]and that our target size wing isL = 10mm, this is equivalentto saying that the insect can generate linear accelerationsof about alin = fa/m = 0.2g ≈ 2m/s2 and angularaccelerations of aboutaang = τa/I = g/L ≈ 105deg/s2,which are comparable with those observed in true insects.

Inspecting the structure of this parameters-to-wrench map,it is apparent that the three mean torques and the verticalthrust can be controlled almost independently by appropriatelychoosing the values for the four wing parametersv. However,there are small but non-negligible couplings between some ofthe wrench components. For example, a positive (negative)pitch torque is always associated to a positive (negative)forward thrust. Similarly, a positive (negative) yaw torqueis associate with a small positive (negative) roll torque anda small negative (positive) lateral force. Although this isundesirable, it does not undermine the stabilizability of flightmodes, as we will show in the next section. Besides, thiscoupling is not particularly harmful. In fact, as in helicopters,forward motion in MFIs is obtained by pitching down the bodyorientation. Since a pitch down torque generation is associatedwith a positive forward thrust generation, the coupling actuallyimproves responsiveness rather than reducing it.

−0.2 0 0.2

−0.2

−0.1

0

0.1

0.2

For

ce (

mg

units

)

fx

−0.2 0 0.2

−0.2

−0.1

0

0.1

0.2

0.8 1 1.2

0.8

0.9

1

1.1

1.2

−0.1 0 0.1

−0.1

0

0.1

τx

Tor

que

(mgL

uni

ts)

−0.1 0 0.1

−0.1

−0.05

0

0.05

0.1

−0.1 0 0.1

−0.1

−0.05

0

0.05

0.1

predictedsimulated

fy

fz

τy τ

z

Fig. 6. Predicted mean wrenchw = π0 + Πlv (x-axis) versusthe exact mean wrench (y-axis) obtained from simulations for100random values of the wings parameter vectorv in the unit box,i.e. ||v||∞ ≤ 1. The spreading around diagonal lines quantifies themodeling errors.

We can summarize this section by saying that, although it

is not possible to control instantaneously the insect wrench,there exist wing motions that can control independently themeanforces along thez-axis and the torques about all threeaxes. We also showed that the affine parameterization ofwing motions given by Equations (9), based on biomimeticprinciples, gives rise to a simple affine map between the meanwrench and the kinematic parameters. Based on the averagingarguments exposed in the previous section, where the inputu and the parameter vector or virtual inputv as defined inTheorem 1, correspond to the wing anglesu = (φr, φl, ϕr, ϕl)and kinematic parametersv = (vr

1, vl1, v

r2, v

l2). Also, based

on the fact that the three mean torque components and thevertical thrust can be controlled independently, it is clear thatthis is a sufficient condition to design stabilizing controllersfor insect flight. In the next section, we will show howto design stabilizing controllers for the linearized averageddynamics that also stabilize the original nonlinear time-varyingdynamics.

VI. W ING TRAJECTORYTRACKING AND ACTUATOR

CONTROL

The previous section described how to design wing trajec-tories that can generate the desired mean forces and torqueduring a wingbeat period. However, the wing trajectory cannotbe controlled directly, and appropriate input voltages to thethorax actuators must be devised to track the desired wingtrajectory. As described in [5], the dynamics of the thorax-wing structure can be approximated as a stable 2-degree offreedom second order system. Given a desired wing trajectory(φd(t), ϕd(t)), we can calculate the corresponding steady-stateinput voltages by substitution:

[V1,d(t)V2,d(t)

]=T−1

0

(M0

[φd(t)ϕd(t)

]+B0

[φd(t)ϕd(t)

]+K0

[φd(t)ϕd(t)

])

(13)where T0,M0, B0, K0 ∈ R2×2 are constant matrices, and

V1, V2 are the input voltages to the wing actuators. LetV =(V l

1 , V r1 , V l

2 , V r2 ) be the input voltages for the two wings, and

u = (φr, φl, ϕr, ϕl), then the wing-thorax dynamics for bothwings can be rewritten as follows:

Mu + Bu + Ku = V (14)

whereM,B, K are matrices that depend onT0,M0, B0,K0,and the dynamics is stable. As we will show in the nextSection, the flight mode stabilizer is assumed to be able toselect a new wing trajectory at the beginning of every wingbeatamong those defined by the parametrization in Equations (7)and (8). This is equivalent to saying that given any sequence{vn}∞n=0, wherev = (vr

1, vl1, v

r2, v

l2) is the wing kinematics

parameter vector as defined in the previous section, the wingtrajectory controller must track the trajectory:

ud(t) = g(t) + G(t)v(t), (15)

v(t) = vn, t ∈ [nT, (n + 1)T ) (16)

whereg(t) andG(t) are defined in Equation (9). Note that thematrix G(t) defined in Equations (8) was specifically chosento have the additional property

G(0) = G(0) = G(0) = G(T ) = G(T ) = G(T ) = 0 (17)

8

0 1 2 3 4 5 6

x 10−3

−10

−5

0

5

10V

1 (µ

N−

m)

0 1 2 3 4 5 6 7

x 10−3

−10

−5

0

5

10

V2 (

µ N

−m

)

time (s)

Fig. 7. Actuator voltage profile as defined in Equation (18) for 10random values of parameter vectorv with ||v||∞ ≤ 1. The solid linecorresponds tov = 0, i.e. Vd(t) = h(t). Note that ||Vd(t)||∞ ≤10µN for all ||v||∞ ≤ 1.

and, therefore, the trajectoryud(t) ∈ C2 is continuous up toits second derivative for any sequence{vn}. If we substituteEquation (15) into Equation (14) we formally obtain:

Vd(t) = h(t) + H(t)v(t) (18)

v(t) = vn, t ∈ [nT, (n + 1)T ) (19)

h(t) = Mg(t) + Bg(t) + Kg(t)H(t) = MG(t) + BG(t) + KG(t)

where h(t) and H(t) are a T -periodic vector and matrix,respectively. SinceH(t) is simply a linear combination ofG(t) and its first and second derivatives, then it follows fromEquation (17) thatH(0) = H(T ) = 0. This implies that theinput voltage vectorVd(t) ∈ C0 is continuous for any sequence{vn}.

Let us consider a desired wing trajectory vectorud(t)defined by Equations (9) and the corresponding feasible inputvoltage vectorVd(t) defined by Equations (18). We define thewing trajectory tracking error to beeu = u − ud, and applyinput voltageVd(t), then we have:

Meu = −Beu −Keu

eu(0) = u(0)− ud(0), eu(0) = u(0)− ud(0)

where we used Equation (14) and the factud(t) = −Bud(t)−Kud(t) + Vd(t) for all t ∈ [0,∞). Since the system aboveis stable, we have thatlimt→∞ eu(t) = 0, or equivalentlylimt→∞ u(t) = ud(t) for any initial condition. The rateof decay,1/τdecay, is set by the poles of the wing-thoraxmechanical system. The time constantτdecay is approximately1 to 2 wingbeat periods for the target MFI design. Thisproperty guarantees that even if we cannot directly controlthe wing trajectory, any initial perturbation would disappearwithin a few wingbeats and the wing trajectory would convergeexponentially to the steady-state solution, as shown in Fig. 8.

The wing trajectory tracking approach developed in this sec-tion is equivalent to a feed-forward control of wing trajectory

−100

−50

0

50

100

φ

−40

−20

0

20

40

0 1 2 3 4 5 6

−5

0

5

(µ N

−m)

time (T units)

V1

V2

ψψ

d

φφ

d

(deg

s)ψ

(deg

s)V

1,V2

v1=0

v2=0

v1=0

v2=1

v1=0

v2=−1

v1=1

v2=0

v1=1

v2=1

v1=1

v2=−1

Fig. 8. Simulation of actuator control given in Equations (18) showingasymptotically tracking of desired trajectory for a random sequence ofkinematic parameters{vn}, wherev = (v1, v2), and random initialcondition of actuator state vector, for one wing. From bottom to top:actuator voltagesV1, V2 as given by Equation (18) (bottom). Rotationangle, ϕ, (center), and stroke angle,φ, (top), given by Equation(14). The error between desired and true wing trajectory decays afterapproximately 3 wingbeat periods.

during a single wingbeat. In fact, it allows trajectory changesonly at the beginning of every wingbeat in such as way thatthis transition is smooth and there is no error between desiredwing trajectory an actual wing trajectory. This is equivalent tosaying that there is no error between the desired and actualmean wrench during the following wingbeat. This approachhas two main advantages. The first advantage is that we canassume to have direct control of the wing trajectory, and wecan neglect the wing-thorax dynamics since any perturbationwould die off within a few wingbeats. The second advantageis that it naturally leads to a discrete time (DT) system, sincethe wing kinematic parametersv are updated everyT seconds,i.e. at the beginning of every wingbeat. We will exploit thesetwo properties in the next Section by modelling the insectdynamics as a discrete time system where the inputs are thewing kinematic parametersv and the state is the mean valueof the body linear and angular position and velocity within theprevious wingbeat.

VII. F LIGHT CONTROL IN HOVER

Following the guidelines described in the previous section,we can now look for a stabilizing controller for hovering bydesigning a feedback lawv = h(x) such that the origin of theaveraged system is exponentially stable.

A. Identification

The analysis in the previous section provides a torquedecoupling scheme together with a set of virtual controlinputs, i.e. the wing kinematic parametersv, which entersinto the averaged system in a affine fashion. Since we areinterested in the insect dynamics close to the hovering regimewhere angular deviations and angular velocities are small,

9

we linearize the averaged system dynamics near hover. Weapproximate the continuous-time nonlinear system, with aDTLTI model in the following form:

x(n + 1) = Ax(n) + Bv(n) + δ(n)y(n) = x(n) + η(n) (20)

where x = [ηx θy ψz ωx ωy ωz px py pz vx vy vz]Tis the vector of average roll, pitch, and yaw angles, angularvelocities, positions and linear velocities over one wingbeat,respectively; δ(n) represents the unmodelled dynamics aswell as external disturbances which appear as an externalnoise to the linear model. This term includes both processnoise as well as unmodeled non-linearities. The input vectorv = [vr

1 vl1 vr

2 vl2]

T are the wing kinematic parameters, whichappear as virtual control inputs. The measurement vectory = [yo

2 yo1 yc yh

1 yh2 yh

3 ; ye1 ye

2 ye3 ye

4 ye5 ye

6]T is the vector of

measured outputs from the ocelli, halteres, magnetic compass,and compound eyes, with additional measurement noiseη(n).As described in [5], these measurements correspond to anestimate of the insect true state, i.e.y = x.

The matrices[A, B] can be obtained analytically from MFImorphological parameters such as mass, moment of inertia,center of mass, etc. However, these parameters are difficultto obtain in practice. Moreover, this approach cannot modelthe effect of the time varying part of the aerodynamic forces.Another approach would be to substitute the parameter-to-wrench map into the original nonlinear dynamics and linearizeit. Here we adopted the system identification approach,i.e., runa large number of experiments and record the pair[y(n), v(n)]of sensor measurements and kinematic parameters, and thenfind the matrices[A,B] that best fit the data. Moreover,further investigation into the particular structure of the insectdynamics given in Equation (20) results in the followingapproximate linear system to be identified:

A =

I3×3 TI3×3 03×3 03×3

03×3 A22 03×3 03×3

03×3 03×3 I3×3 TI3×3

A41 03×3 03×3 A44

, B =

03×3

B21

03×3

B41

where T is the wingbeat period, the matricesA22 and A44

account for angular and linear damping, and the matrixA41

accounts for the linear accelerations due to tilted body orien-tation. This structure is typically used in helicopter dynamicsidentification [27] [28].

We first estimate a model in open loop where only datafor the first several wingbeats are recorded. Since the sensormeasurements provide an estimate for all the entries of thestate vector, the model identification problem can be recastinto a least square solution to an overdetermined set oflinear equations asEz = d, where z = [ai,j , ..., bk,h]T isthe vector of system parameters to be estimated, andai,j

and bk,h are the nonzero entries of the matricesA and Brespectively. The matrixE = E (y(·), u(·)) and d = d (y(·))are matrices whose entries depend on the experimental data.The least square solution which minimizes the norm of theerror ||e||2 = ||d − Az||2 is given byz = E(ET E)−1ET d.The experiments were performed on the Virtual Insect FlightSimulator (VIFS), developed by the authors to provide asoftware testbed for insect flight [5]. The experimental datawere generated with random inputs and initial conditions nearthe hovering equilibrium.

0 10 20 30 40 50−10

−5

0

5

10Position (mm)

PEM−modelexact LS−model

x

0 10 20 30 40 50−10

−5

0

5

10Orientation (degs)

roll,

η

0 10 20 30 40 50

−10

−5

0

5

10

y

0 10 20 30 40 50−10

0

10

pitc

h, θ

0 10 20 30 40 50−10

−5

0

5

10

time (T units)

z

0 10 20 30 40 50

−10

0

10

yaw

, ψ

time (T units)

Fig. 9. Comparison of the exact mean angles and angular velocities(thick solid line) with those predicted with the PEM-based DTLTImodel (dashed line), the LS-based DTLTI model (thin solid line) andthose simulated using exact model over50 consecutive wingbeats.

Based on this least-squared-based model[A,B], a stabi-lizing state feedback control based on pole placement wasdesigned and tuned first on the nominal LTI model, thenverified on the fully nonlinear continuous time dynamicsprovided by VIFS. Although least-squared identification issimple and straightforward, it does not exploit the structure ofthe dynamics present in Equation (20), nor does it provide asystematic way to estimate process and output noise. However,it does provide a stabilizing controller which can be used suc-cessively to perform closed loop system identification throughprediction error method (PEM) [29]. The prediction errormethod cannot be applied directly to the system (20), since thesystem is unstable, which is why least-square identification isperformed first. The PEM-based identified model performedbetter than the least-squared-based one in predicting insectdynamics as shown in Fig. 9. Besides, the estimated processand noise variances and biases can be used to design betterrobust controllers.

B. Controller design

In order to address the trade off between regulation perfor-mance and control effort to avoid control input saturation, andalso to take into account process disturbances and measure-ment noise in Equation (20), we employed a Linear QuadraticGaussian (LQG) optimal controller design.

As a first step, a state feedback LQR regulatorv = −Kxwas designed to minimize the following quadratic cost func-tion

J = limN→∞

E(N∑

n=1

x(n)T Qx(n) + v(n)T Rv(n)) (21)

whereQ ≥ 0 andR > 0 are the weighting matrices that definethe trade-off between regulation performance and controleffort. The controller was designed with standard discrete-time LQG software, and the diagonal entries in the weighting

10

0 20 40 60 80 100

−30

−20

−10

0

10

20

30

40Position (mm)

exactcompound eye

x

0 20 40 60 80 100

−0.2

−0.1

0

0.1

0.2

Linear velocity (mm/ms)

exactoptical flow

vx

0 20 40 60 80 100−60

−40

−20

0

20

roll,

η

exactocelli

Orientation (degs)

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

exacthalteres

Angular Velocity (deg/ms)

ωx

0 20 40 60 80 100

−30

−20

−10

0

10

20

30

40

exactcompound eye

y

0 20 40 60 80 100

−0.2

−0.1

0

0.1

0.2

exactflow sensor

vy

0 20 40 60 80 100−60

−40

−20

0

20

exactocelli

pitc

h, θ

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

exacthaltere

ωy

0 20 40 60 80 100

−30

−20

−10

0

10

20

30

40exactcompound eye

z

0 20 40 60 80 100

−0.2

−0.1

0

0.1

0.2 exactflow sensor

vz

0 20 40 60 80 100−60

−40

−20

0

20

exactcompass

yaw

, ψ

0 20 40 60 80 100−1.5

−1

−0.5

0

0.5

1

1.5

exacthalteres

ωz

20 40 60 80 100−1

−0.5

0

0.5

1Kinematic parameter

υ1l

20 40 60 80 100−1

−0.5

0

0.5

1Kinematic parameter

υ2l

20 40 60 80 100−1

−0.5

0

0.5

1Kinematic parameter

υ1r

20 40 60 80 100−1

−0.5

0

0.5

1Kinematic parameter

υ1r

0 5 10 15 20 25−10

−5

0

5

10Actuator voltage (µN−m)

time (T units)

V1l

0 5 10 15 20 25−10

−5

0

5

10

time (T units)

Actuator voltage (µN−m)

V2l

0 5 10 15 20 25−10

−5

0

5

10Actuator voltage (µN−m)

time (T units)

V1r

0 5 10 15 20 25−10

−5

0

5

10

time (T units)

Actuator voltage (µN−m)

V2r

Fig. 10. Simulation of hovering control with sensor feedback and actuators dynamics. From top to bottom: insect true state and sensorsmeasurements (row 1-3); kinematics parameters given by Equation (22) (row 4); actuators voltage given by Equation (23)(row 5) during thefirst 25 wingbeats.

matrices are iteratively tuned to ensure a good transientresponse without saturating the control inputs. The above LQRoptimal state feedbackv = −Kx is then substituted with amore realistic output feedback:

v(n) = −Ky(n), (22)

where the outputy is given by the sensors measurements. Asdescribed earlier, the simplified DTLTI system (20) providesa feedback scheme to select the wing kinematic parameterfor the next wingbeat period. The true feedback controlfrom sensor measurements to actuator voltages is obtained bycombining Equation (22) with Equation (18) to give:

Vd(t) = h(t) + H(t)v(t) = h(t) + H(t)Ky(t)= h(t) + K(t)y(t) (23)

y(t) = y(nT ), t ∈ [nT, (n + 1)T ) (24)

where the sensors measurements are sampled at the beginningof each wingbeat, andK(t) is simply a proportionalT -periodic matrix gain. It is remarkable that a simple propor-tional T -periodic feedback scheme is sufficient to stabilize

the complex time-varying nonlinear insect dynamics includ-ing nonlinear sensor measurements, actuator dynamics, andprocess and output noise. More importantly, this gain can becomputed off-line and easily stored on the computation unitof the MFI.

The LQR controller was finally tested on the fully non-linear time-varying model which includes the MFI dynam-ics of Equation (1), the wing-thorax dynamics of Equations(13), and the sensor models described in [5]. The simu-lations are based on a target MFI of100mg and 10mm-wingspan with wingbeat frequencyf = 150Hz. Fig. 10shows a simulation for hovering stabilization from the initialcondition x = (ηx, θ, ψ, ωx, ωy, ωz, px, py, pz, vx, vy, vz) =(25o,−25o, 20o, 0, 0, 0, 35mm,−25mm, 25mm, 0, 0, 0), andwing state(u, u) = (φr, φl, ϕr, ϕl, φr, φl, ϕr, ϕl) = 0. Ourproposed controller design successfully achieved stabilizationdespite sensor and process noise. The initial condition corre-sponds to an offset from the desired position of about 3 body-lengths. The steady state error during hovering is< 1/10 ofthe body-length for the position and< 5o for the orientation.The MFI requires about 50 wingbeat periodsT to reach the

11

final configuration, which corresponds to about2/3rds of asecond for a wingbeat frequency of150Hz.

C. Single channel identification and control design

Based on the particular structure of the mean wrench mapgiven in Equation (12), where it appears that the mean torqueand the vertical thrust can be controlled almost independentlyby combining symmetrically or anti-symmetrically the kine-matic parametersv = (vr

1, vl1, v

r2, v

l2), we can reformulate the

flight control problem of the 6 DOF system similar to thatof helicopter control, where we have decoupled the systemdynamics into longitudinal, lateral, heave, and yaw dynamics[14] [28]. In fact, if we redefine the kinematic parameters asfollows:

v = (v1, v2, v3, v4) = (vr1−vl

1, vr2+vl

2, vr2−vl

2, vr1+vl

1) = Fv (25)

and we use these parameters as inputs for the system (20) andrepeat the identification process, then we obtain the followingmatrices:

A41 =

[0 a4 0

−a4 0 00 0 0

],

A22 = diag{a1, a2, a3},

A44 = 03×3,

B21 =

[b1 0 ∗ 00 b2 0 ∗0 0 b3 0

], B41 =

[ 0 ∗ 0 00 0 ∗ 00 0 0 b4

](26)

where the zeros entries are entries that were much smallerthan the other entries in the same row, and the asterisks,∗’s,indicate non-negligible entries. If the∗’s are neglected, it isclear that each virtual parametervi controls independently oneof the three angular accelerations and the vertical acceleration,thus justifying the single channel controller design scheme astypically done for a helicopter. The advantage of this approachis that the feedback matrix gain is given by:

Kv = diag{Klong,Klat,Kheav,Kyaw} (27)

where the matricesKlong,Klat, Kheav, Kyaw are the smallersize proportional gain matrices obtained from the decoupledinsect flight dynamics, thus reducing the computational burdenwhen computing the feedbackv = Kvy. Fig. 11 shows acomparison between the full channel controller design and thesingle channel design. The performance using single channeldesign degrades somewhat, but it is less computationallydemanding than the full channel design, which is a clearadvantage for the limited computational unit available to MFIs.

VIII. C ONCLUSION

In this paper we presented an framework for flappingflight control and navigation in biomimetic robotic insects.We started by reviewing the neuromotor architecture presentin true flying insects, and highlighting analogies and differ-ences between insect flapping flight and helicopter flight. In-spired by true insect neuromotor organization of flight controlmechanisms, we proposed a three-layered hierarchical controlstructure that simplify flight control design while preservingthe high maneuverability and the agile navigation capabilityexhibited by true insects. The first major contribution ofthis paper was to propose a suitable parametrization of wing

0 20 40 60 80 100

−20

0

20

40position (mm)

0 20 40 60 80 100

−20

0

20

40

0 20 40 60 80 100

−20

0

20

40

time (T units)

single channelfull channel

x

y

z

Fig. 11. Comparison of single channel design vs full channel design.

motion during the course of a full wingbeat and to combineit with averaging theory arguments, thus showing that theinsect time-varying dynamics can be well approximated bya discrete-time linear time-invariant (DTLTI) system wherethe wing kinematic parameters appear as virtual inputs. Thesecond major contribution was to propose an identification-based LQR controller design which does not required theknowledge of an accurate model for the insect morphologicalparameters, such as moment of inertia and mechanical part’ssizes, nor an accurate model of the aerodynamics. As a result,hovering flight mode can be stabilized with a simple affineT -periodic proportional feedback from sensor measurementsto actuator voltages. This is very important considering thelimited computational resources available to MFI’s. Althoughin this paper we focused on hovering, it has been shown thatother flight modes like cruising and steering can be stabilizedusing a affineT -periodic proportional feedback [30].

Several research directions can be explored. The mostimportant one is probably in regard to the wing parametriza-tion, which in this paper was based on the observations oftrue insect wing motions. However, different wing kinematicparameters could be chosen. Therefore, a more systematicmethodology to optimize the wing trajectory parametrizationwith respect to some metrics, such as aerodynamic power ormaximum torque production, is sought.

Another direction emerges from wing trajectory tracking.One of the major assumptions in our approach was the linearityof the actuator dynamics, so that wing trajectory tracking couldbe easily solved using a pseudo-inverse to compute the controlinput to the actuators. This assumption is true only to the firstorder, as shown in [31], and nonlinearities become particularlyimportant when rapid wing rotations at the end of the half-strokes are necessary for aggressive flight maneuvers. Novelapproaches need to be considered in this scenario.

Finally, navigation and searching algorithms in unknownenvironments have not been explored in this paper and it ispart of our current research.

12

REFERENCES

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