stabilization of flapping-wing micro-air vehicles in gust environmentspaper single space

8/11/2019 Stabilization of Flapping-Wing Micro-Air Vehicles in Gust EnvironmentsPaper Single Space

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LQR Controller for Stabilization of Flapping Wing

MAVs in Gust Environments

Manav Bhatia∗, Mayuresh Patil†, Craig Woolsey‡,

Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

and

Bret Stanford§, Philip Beran¶

U.S. Air Force Research Laboratory, Wright-Patterson AFB, OH, 45433

This study presents an approach to develop a controller for stabilization of a flappingwing micro-air vehicle (MAV) operating in gusty environments. The rigid-wing MAVis modeled as a nonlinear periodic system and the periodic-shooting method is used tofind a trimmed periodic orbit. A linearized discrete-time representation of the system

is created about this trimmed periodic orbit. This linearized representation is used forcontrol synthesis based on linear quadratic regulator (LQR) theory. The kinematic variablesdefining the wing motion are used as control inputs. The controller is implemented on thenonlinear system model to stabilize the system in the presence of external disturbances,modeled as discrete gusts in this study. The performance of the controller, in terms of the gust speed tolerance of the nonlinear, closed-loop system, is compared for variouscontroller designs. The LQR based controller is capable of stabilizing the system underboth longitudinal and lateral gust disturbances, however the maximum gust speed that canbe tolerated by a given controller is influenced by a variety of parameters, as discussed inthe paper. Numerical simulations show that tolerance of longitudinal gusts is far higherthan lateral gust tolerance. The study also shows that lateral control of the MAV can beachieved using only the wing-stroke magnitude and wing-stroke offset as the control inputsfor each wing.

Nomenclature

ω Angular velocity (rad/sec)Φ Vector of Euler angles (rad)δ Reduction in wing-stroke frequency during first half of stroke-cycle (rad/sec)η Wing feathering angle (rad)ω Wing-stroke frequency (rad/sec)φ Wing stroke angle (rad)ρ Scaling factor for control-cost in LQR synthesisθ Wing deviation angle (rad)ω Wing-stroke frequency during second half of stroke-cycle (rad/sec)

ω Skew-symmetric angular rotation operator

ξ Phase-shift with frequency ω during second half of stroke-cycle (rad)B Superscript, indicates quantity defined for the bodyW Superscript, indicates quantity defined for the wing

0 Subscript, indicates offset of wing motion dofs (rad), eg. φ0, θ0, η0

∗PostDoctoral Associate, Department of Aerospace and Ocean Engineering, Presently: Research Engineer, Universal Tech-nology Corporation, [email protected], Senior Member AIAA

†Associate Professor, Department of Aerospace and Ocean Engineering, [email protected], Associate Fellow AIAA‡Associate Professor, Department of Aerospace and Ocean Engineering, [email protected], Associate Fellow AIAA§Research Engineer, Universal Technology Corporation, [email protected], Senior Member AIAA¶Principal Research Aerospace Engineer, AFRL, [email protected], Associate Fellow AIAA

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American Institute of Aeronautics and Astronautics



B Subscript, indicates quantity measured in the body frame

I Subscript, indicates quantity measured in the inertial frame

m Subscript, indicates magnitude of angle describing wing motion dofs (rad), eg. φm, θm, ηms Subscript, indicates phase-shift of angle with reference to stroke-plane (rad), eg. θs, ηsW Subscript, indicates quantity measured in the wing frameAφ, Aθ Amplitude scaling factors to ensure C 0 continuity of wing motion across consecutive cyclesJ Quadratic cost for LQR synthesisK η Parameter controlling the nature of feathering motion, a low value (<< 1) describes sinusoidal wave,

and a higher value describes square waveK φ Parameter controlling the nature of stroke-plane motion, a low value (<< 1) describes sinusoidal

wave, and a value close to 1 describes triangular waveN θ Multiplier of wing flapping frequency to obtain the wing deviation motion frequency. Value of 1 gives

an elliptical wing-tip motion, while value of 2 gives a figure-of-eight wing tip motion.A System matrix of discrete linear time-invariant system modelB Control coefficient matrix of discrete linear time-invariant system modelD Disturbance coefficient matrix of discrete linear time-invariant system modelEBI Transformation matrix from body Euler angular rates to angular velocity with respect to inertial

frameEWB Transformation matrix from wing Euler angular rates to angular velocity with respect to body frameFforce, Fmoment Aerodynamic and gravitational force and moment vectors, respectively, in body frameK Optimum gain matrixP Solution of Lyapunov Equation or Algebraic Riccati EquationPforce, Pmoment Inertial force and moment vectors, respectively, in body frameQ State-weighting matrix used for LQR synthesisq Vector of kinematic variablesqcontrol Subset of kinematic variables used as by LQR controller, qcontrol and qtrim need not be exclusiveqtrim Subset of kinematic variables used for trim solutionR Control-weighting matrix used for LQR synthesisr Position vector (m)TBI Transformation matrix from body frame to inertial frameTWB Transformation matrix from wing frame to body framev Velocity vector (m/s)

I. Introduction

Recent years have seen a significant amount of research in order to better understand and enhance theperformance of flapping wing micro-air vehicles (MAVs). These studies have collectively established thatmultidisciplinary interactions can significantly influence the performance of such vehicles.1,2 This realizationhas important implications for design of flapping wing MAVs. Significant multidisciplinary interactions implythat a multidisciplinary design optimization (MDO) procedure be used, requiring models that adequatelycapture the cross-disciplinary couplings. Several recent studies have addressed cross-disciplinary coupling inflapping wing MAV models,3,4,5,6,7,8 with a view toward MDO.

An important requirement, often ignored in design studies, is the ability of the vehicle to withstandexternal disturbances, such as gusts. In conceptual design of conventional airplanes, little consideration isgiven to control authority and disturbance rejection, since these are not the primary design drivers for larger

aircraft. For smaller airplanes, however, the sensitivity of the vehicle to external disturbances increases, andthe ability to withstand them becomes more critical. Gust tolerance is far more important for flapping wingMAVs as they are inherently open-loop unstable during hovering flight and frequently operate under strictenergy budgets. And while a typical gust encountered by a large airplane is a small fraction of the flightspeed, a gust encountered by a MAV may be as fast as the vehicle. Hence, it is important to be able todesign a vehicle with optimum mission performance under external disturbances.

The review article by Orlowski and Girard9 notes a general consensus among stability studies of flappingwing MAVs: in the absence of active control, the MAV system representations are unstable. Although theydid not qualify this statement for a flight condition (hover, forward flight, climbing flight, etc.), our studieshave found both forward flight and hover to be unstable. Of relevance here, is the work by Stanford et al.10

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where open-loop stability was considered for optimization of flapping wing kinematics. The authors were ableto obtain open-loop stable designs, but concluded that closed-loop control is still needed for station-keepingin the presence of a disturbance, or for maintaining stability in the presence of large disturbances. Anotherimportant consideration is that the flight vehicle being studied is tailless, requiring that all control forcesand moments be generated by variations in wing kinematics, according to some appropriate control law.

With regard to dynamic modeling of flapping wing MAVs Orlowski and Girard,9 and several of thereferences which they cite, highlight the importance of wing-mass inertia. The presence of flapping wingscan exert significant inertial forces on the flight vehicle to which the wings are attached. These inertial forces

influence the dynamics of a vehicle that is free to translate and rotate in space. The motion of the body, inturn, influences the wing motion and the resulting aerodynamic forces. This coupled flight dynamic behavioris nonlinear and time-varying for the system under consideration and can have a significant influence on theoverall system performance. In the present study, we model the flight dynamic system including the wing-mass inertia for the purpose of closed-loop control. Recent derivations of nonlinear flight dynamic modelsfor flapping wing MAVs can be found in papers by Sun, Wang and Xiong 11 and Orlowski and Girard12

Orlowski and Girard9 note that none of the existing work on control of MAVs integrates a 6-degrees-of-freedom (6DOF) flight dynamic model that includes wing-mass inertia effects. Some of the recent work oncontrol of this class of vehicles includes that of Khan and Agrawal,13 who used a very simple time-averagedmodel to study the implementation of a nonlinear controller for longitudinal flight. Doman, Oppenheimerand Sigthorsson14 used analytical expressions of cycle-average force and moments for 6DOF control of aflapping wing MAV. They later extended this work to obtain 6DOF control using wing-beat bias, as opposedto a bob-weight that had been included in their previous work.15 In their work, however, the system flightdynamics and wing inertial influence were neglected. More references on control of MAVs are available inthe paper by Orlowski and Girard.9 A more recent work not found in this review article is that of Dietland Garcia.16,17 They studied the dynamics of longitudinal flight of an ornithopter with a fuselage and ahorizontal and vertical tail. They focused on development of control laws to stabilize forward flight16 andlater extended this work to enable transition from forward flight to hovering.17

This study focuses on the design of a controller for a flapping wing MAV with rigid wings, based onexisting work by Stanford, et al.10 and Sun, Wang and Xiong.11 A generalized wing kinematic descriptionis developed, combining the features of Berman and Wang’s description18 and that of Oppenheimer, Domanand Sigthorsson.15 The nonlinear flight dynamic equations are solved for periodic trim conditions using aperiodic-shooting method.19 The flight dynamic equations are linearized about this trim motion and areused to construct a model-based controller using linear quadratic regulator (LQR) theory. This controller isthen implemented in a numerical simulation of the nonlinear model in order to study the closed-loop system’s

response to gust disturbances. The emphasis is on understanding the influence of kinematic parameters usedfor control and on the performance of the controller under disturbances of increasing magnitude.The wing kinematic description is discussed in Section II. The details of aerodynamic modeling and flight

dynamics are discussed in Section III. The details of linearization and control synthesis are discussed inSection IV. The results from numerical studies are discussed in Section V. Finally, the conclusions from thisstudy are discussed in the Section VI.

II. Kinematic Parameterization

A. Parametric Definition

Here, we develop a kinematic parameterization that incorporates the split-cycle approach proposed by Do-man, Oppenheimer and Sigthorsson15 and the earlier parameterization by Berman and Wang.18 This gener-

alized parameterization allows one to smoothly vary the shape of a flapping cycle between two very differentcharacteristic motions: triangular versus sinusoidal stroke-plane motion and square versus sinusoidal feath-ering motion. The parameterization is developed with a requirement to maintain C 0 continuity in wingangular motion, which is consistent with any physically realizable motion. The flexibility of parametrizationdeveloped here allows for several choices of control inputs, however, the control study presented in this paperdoes not make use of split-cycle control.

Doman, Oppenheimer and Sigthorsson15 proposed the use of three independent kinematic parameters forthe wing: the split-cycle control parameter (δ ), the wing-beat bias (φW

0 ) and the flapping frequency (ω). Thesplit-cycle control parameter changes the flapping frequency during the up- and down-strokes while keepingthe overall flapping period constant at 2π

ω . The wing-beat bias shifts the flapping angle for an entire cycle

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up or down by φW 0 .

Our formulation includes a phase shift in the deviation (θW s ) and feathering (ηW

s ) angles. We assume,however, that split-cycle control does not change these values from the baseline kinematics. The controlparameters include frequencies (ω and δ ), amplitudes (φW

m , θW m and ηW

m ) and offsets (φW 0 , θW

0 and ηW 0 ). In

a manner consistent with split-cycle control, these are defined at the beginning of each cycle and remaininvariant during the flapping period.

Kinematic parameters are defined independently for the left and right wings. Continuity of flappingangles across consecutive flapping cycles is ensured for each angle by modifying the first quarter of each

flapping cycle. This is in contrast to the work by Doman, Oppenheimer and Sigthorsson,15 who applied thebias correction after a delay of one cycle. In their formulation, the control parameters for the ith cycle areδ i, ωi and φW

0,i−1 (as opposed to φW 0,i). In the present work, a factor Aφ is calculated using φW

0,i−1 and φW 0,i

and is applied during the first one-fourth of the flapping cycle to ensure continuity. The calculation of thetime derivatives of flapping angles assumes that the kinematic parameters are independent of time within aflapping cycle.

K φ and K η provide control of stroke and feathering motion waveforms, respectively. A low value of bothparameters ( 1) creates a sinusoidal wave, while K φ = 1 creates a triangular wave and K η > 1 tendstowards a square wave. The parameter N θ is a multiplier of wing flapping frequency to obtain the wingdeviation motion frequency, where a value of 1 gives an elliptical wing-tip motion, and a value of 2 gives afigure-of-eight wing tip motion.

The wing has three degrees of freedom: the azimuthal stroke plane angle (φ), the wing-deviation angle(θ) and the feathering, or wing-rotation angle (η). Before stating the kinematic parameterization, we definefour non-dimensional parameters:

Aφ =φW m,old + φW

0,old − φW 0

φW m

− 1 (1)

Aθ =θW m,old cos θW

s + θW 0,old

θW m cos θW

s + θW 0

− 1 (2)

ω = ω(ω − δ )

ω − 2δ (3)

ξ = −2πδ

ω − 2δ (4)

Here, the subscript old indicates a value from the previous flapping cycle. With these parameters defined,

we present the following kinematic parameterization for flapping wing motion.

φW (t) = φW m (1 + Aφ)

sin−1(K φ sin((ω − δ )t + π2

))

sin−1 K φ+ φW

0 0 ≤ t < π

2(ω − δ ) (5)

φW (t) = φW m

sin−1(K φ sin((ω − δ )t + π2

))

sin−1 K φ+ φW

0

π

2(ω − δ ) ≤ t <

π

ω − δ (6)

φW (t) = φW m

sin−1(K φ sin(ωt + ξ + π2

))

sin−1 K φ+ φW

0

π

ω − δ ≤ t <

2π

ω (7)

θW (t) = (1 + Aθ cos((ω − δ )t))(θW

m cos(N

θ(ω − δ )t + θW

s ) + θW

0 ) 0 ≤ t <

π

2(ω − δ ) (8)

θW (t) = θW m cos(N θ(ω − δ )t + θW

s ) + θW 0

π

2(ω − δ ) ≤ t <

π

ω − δ (9)

θW (t) = θW m cos(N θωt + N θξ + θW

s ) + θW 0

π

ω − δ ≤ t <

2π

ω (10)

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ηW (t) =

ηW m

(ω − δ )t

π/2 + ηW

m,old

1 −

(ω − δ )t

π/2

tanh(K η sin((ω − δ )t + ηW

s ))

tanh K η

+

ηW 0

(ω − δ )t

π/2 + ηW

0,old

1 −

(ω − δ )t

π/2

0 ≤ t <

π

2(ω − δ ) (11)

ηW (t) = ηW m

tanh(K η sin((ω − δ )t + ηW s ))

tanh K η+ ηW

0

π

2(ω − δ )

≤ t < π

ω − δ

(12)

ηW (t) = ηW m

tanh(K η sin(ωt + ξ + ηW s ))

tanh K η+ ηW

0

π

ω − δ ≤ t <

2π

ω (13)

An example of wing kinematic description with varying split-cycle control and wing-beat across consec-utive flapping cycles is provided in Table 1. The wing angles obtained from the kinematic parameterizationpresented in this section are plotted in Fig. 1. Note that C 0 continuity is maintained across consecutivecycles for each case.

Table 1. Kinematic parameters describing wing motion with varying wing-beat bias and split-cycle frequencyacross two consecutive cycles

Param Val Param Val Param Val

ω 125 N θ 2 K η 4δ 30 θW

m 1 ηW m 1

K φ 0.001 θW 0 0.2 ηW

0 0.2

φW m 1 θW

s 0.785 ηW s -0.785

φW 0 0.2 θW

m,old 0.75 ηW m,old 0.75

φW m,old 0.75 θW

0,old -0.2 ηW 0,old -0.2

φW 0,old -0.2

0 2 4 6 8 10

·10−2

−2

−1

0

1

2

time (s)

A n g l e ( r a d )

φW θW ηW

Figure 1. Wing angles with split-cycle frequency, wing-beat bias and variation of flapping magnitude and biasbetween two cycles

B. A Note on Terminology

In order to facilitate discussion in later sections, it is clarified that a set of kinematic parameters is representedas q. The complete definition of wing kinematic motion during a given cycle is defined with a total of 12parameters: ω, δ , φW

m , φW 0 , K φ, θW

m , θW 0 , θW

s , ηW m , ηW

0 , ηW s and K W

η . The numerical procedure to obtain

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a trim orbit (discussed later in Sec. IV.A) uses two kinematic parameters, and this subset is referred toas qtrim. The numerical values of qtrim are known for a trimmed vehicle. For such a vehicle the controlsynthesis procedure (discussed in Sec. IV.C) defines the set of kinematic variables, qcontrol, that will be usedby a given controller. This set of kinematic variables can contain one or all of the kinematic parameterslisted above, and is independent of the kinematic parameters used for trim.

III. Flight Dynamics

The system under consideration for this work is that of a rigid tailless body connected to a series of rigidflapping wings (two for this work, though this is not restricted in the derivation). The wings are pinned tothe body. The orientation of each wing with respect to the body is defined by the three Euler angles φW , θW ,and ηW corresponding to that particular wing. For clarity and economy of notation, we make no distinctionbetween the port and starboard wing in this presentation. The wing motion (i.e., the time histories of φW ,θW , and ηW ) is prescribed, but the overall motion of the body is computed using the flight dynamics modeldeveloped below.

A. Base Body Kinematics and Dynamics

The system under consideration is depicted in Fig. 2. There exists a fixed inertial coordinate system(xI , yI , zI ), a coordinate system attached to the center of gravity of the fuselage body ( xB, yB, zB), lo-cated at point B, and a third coordinate system (xW , yW , zW ) located at the hinge point H, which rotateswith the rigid wing. The center of gravity of the wing is located at point W. The attitude and position of the body is defined by

ΦB =

φB

θB

ηB

(14)

rI =

xI yI zI

(15)

The angular velocity of the body, written in the body frame, is

ωBB =

cos θB cos ηB sin ηB 0

− cos θB sin ηB cos ηB 0

sin θB 0 1

φB

θB

ηB

= EBI ΦB

(16)

The transformation matrix from the body frame to the inertial frame is

TBI =

1 0 0

0 cos φB − sin φB

0 sin φB cos φB

cos θB 0 sin θB

0 1 0

− sin θB 0 cos θB

cos ηB − sin ηB 0

sin ηB cos ηB 0

0 0 1

(17)

The velocity of the body is

rBI = xBI

yBI zBI

= vBI = TBI vBB (18)

where vBB

is the velocity of point B written in the body-attached coordinate system. The acceleration of thebody is

vBI = TBI ωB

BvBB + TBI v

BB (19)

where the tilde indicates the skew-symmetric matrix satisfying ab = a × b for vectors a and b.

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! "

$ "

% "

! &

$ &

% &

! '

$ '

% '

&

(

)

Figure 2. MAV and flapping wing coordinate systems

The prescribed kinematics of the wing (i.e., the attitude of the wing with respect to the body) aregoverned by three Euler angles

ΦW =

φW

θW

ηW

(20)

The angular velocity of the wing with respect to the body is

ωW W =

− sin θW 0 1

cos θW sin ηW cos ηW 0

cos θW cos ηW − sin ηW 0

φW

θW

ηW

= EWB ΦW

(21)

The transformation matrix from the wing frame to the body frame is

TWB =

cos φW − sin φW 0

sin φW cos φW 0

0 0 1

cos θW 0 sin θW

0 1 0

− sin θW 0 cos θW

1 0 0

0 cos ηW − sin ηW

0 sin ηW cos ηW

(22)

The equations of motion have been derived by Sun, Wang and Xiaong;11 only the final form is givenhere:

I

I

Imtot

IB

rBI

ΦB

vBB

ωBB

+

0

0

mtotωBB

ωBBIB

rBI ΦB

vBB

ωBB

+

0

0

Pforce

Pmoment

=

TBI vBB

E−1BI ωBB

Fforce

Fmoment

(23)

where I is the identity matrix, mtot is the complete mass of the vehicle (body plus wings), and IB is theinertia tensor of the body, written in the body coordinate system at point B. The vectors Pforce and Pmoment

are inertial forces and moments imposed on the body due to the wing motion, while Fforce and Fmoment aregravitational and aerodynamic terms.

B. Aerodynamics

The aerodynamic terms are computed with the two-dimensional quasi-steady blade element model discussedby Berman and Wang18 and used in the subsequent study by Stanford et. al.10 The wing is discretized into

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a series of blade elements for which two forces (lift and drag) and a pitching moment are calculated basedon quasi-steady assumptions.

It should be noted that this aerodynamic model cannot capture many potentially important physicalphenomena: aerodynamic interactions between the two wings, interactions between the wings and the body,flow over the body itself, or unsteady inflow from the shed wake. Despite these shortcomings, favorablecomparisons of this model with higher-fidelity Navier-Stokes solvers have been demonstrated for a flappingwing pinned to a stationary point.18

IV. Trim, Linearization and Control Synthesis

A. Trim

The physical system under consideration is modeled by a system of nonlinear, periodic equations. In orderto proceed with control synthesis, the system is linearized about a trim state: a periodic orbit of the dynamicequations with the same period as the wing flapping cycle.

Following the approach developed by Stanford, et al ,10 a periodic-shooting method is used to obtain trimstates for this system. The procedure defines two kinematic parameters (qtrim) and the body pitch-angleand angular-rate at the beginning of a cycle as the unknowns. A Newton iteration is used to calculate thesolution, which requires sensitivity of the states with respect to the kinematic parameters.

B. Linearized Representation

The present work uses linear quadratic regulator (LQR) theory for the synthesis of a controller to stabilizethe system in the presence of disturbances. The system state at time t0 is defined by the linear and angularposition and their time derivatives:

xt0 =

rBI ΦB

vBB

ωBB

t=t0

(24)

Having numerically determined a periodic trim state, the system is linearized about this periodic orbit bycalculating the sensitivity of the end-of-period states with respect to change in the states and kinematicparameters at the beginning of the period. For the purpose of this study it is assumed that the stateperturbations can be calculated and are available to the controller.

If the flight condition is trimmed, and no disturbances are present, the system will follow the periodicorbit during one flapping cycle and the state at the end of this cycle will be identical to the initial state:xT +t0 = xt0 . This will not be the case, however, if the initial state is perturbed by a small amountdxt0 . Depending on the nature of the system, the result may be a growing (unstable) or decaying (stable)discrepancy between the true state and the trimmed state. In any case, however, an initial perturbation willresult in some nonzero perturbation at the end of the period, represented as dxT +t0 .

The nonlinear system shown in Eq. (23) is now represented in a generic form as (25).

x = R(x, x, q) (25)

x(0) = xt0 (26)

In order to calculate dxT +t0 , given the initial disturbance dxt0 , we compute the sensitivity of the system

equations to small changes in the initial state: dx

dxt0

=

∂ R

∂ x

dx

dxt0

+

∂ R

∂ x

dx

dxt0

(27)

dx(0)

dxt0

= I12×12 (28)

The system (27) is a linear ordinary matrix differential equation with dxdxt0

as the unknown. The quantities

∂ R∂ x

and

∂ R∂ x

are available from the solution of Eq. (25). Integrating Eq. (27) in time for one period, one

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obtains the sensitivity of the state at the end of one flapping cycle (t = t0 + T ) with respect to the state atthe beginning of the flapping cycle (t = t0). This sensitivity matrix may then be used to compute

dxT +t0 =

dx

dxt0

t=T +t0

dxt0 (29)

The sensitivity matrix in Eq. (29) is also called the monodromy matrix 19 or the state-transition matrix .20

This matrix is calculated as a part of the periodic-shooting method used to obtain a trimmed solution. The

reader is referred to the preceding work by Stanford et al.10 for more details on this process.The evolution of the system state is affected by changes in initial conditions, as discussed above, changes in

the wing kinematic variables, dqt0 and dqt0−T , and any other external disturbances, termed ∆t0 . Althoughthe controller will have access to the subset qcontrol, the formulation presented here is applicable to anysubset of kinematic variables. Hence, to maintain generality, the vector of kinematic variables is representedas q. The sensitivities of xT +t0 with respect to the latter two parameters can be calculated using the sameprocedure that led us to Eq. (29). For perturbations due to the kinematic parameters, q, the equation is

dx

dqt0

=

∂ R

∂ x

dx

dqt0

+

∂ R

∂ x

dx

dqt0

+

∂ R

∂ qt0

(30)

dx(0)

dqt0

= 0 (31)

dxdqt0−T

= ∂ R∂ x dx

dqt0−T + ∂ R

∂ x dx

dqt0−T + ∂ R

∂ qt0−T (32)

dx(0)

dqt0−T

= 0 (33)

dxT +t0 =

dx

dqt0

t=T +t0

dqt0 +

dx

dqt0−T

t=T +t0

dqt0−T (34)

It is important to note that the kinematic variables from the previous cycle also influence the current cycledue to the C 0 continuity requirement of wing kinematics; see § II. Similarly the perturbation equation dueto external disturbances, ∆t0 , is

dx

d∆t0 = ∂ R

∂ x dx

d∆t0 + ∂ R

∂ x dx

d∆t0 + ∂ R

∂ ∆t0 (35)dx(0)

d∆t0

= 0 (36)

dxT +t0 =

dx

d∆t0

t=T +t0

∆t0 (37)

Eqs. (29), (34) and (37) can be combined to give

dxT +t0 =

dx

dxt0

t=T +t0

dxt0 +

dx

dqt0

t=T +t0

dqt0 +

dx

dqt0−T

t=T +t0

dqt0−T

+

dx

d∆t0

t=T +t0

∆t0 (38)

In simpler notation, we have

dx1 = Adx0 + Bdq0 + B−1dq−1 + D∆0 (39)

where the superscripts represent the following short-hand for time stamps: 0 = T , 1 = t0+T and −1 = t0−T ,

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and where

A =

dx

dxt0

t=T +t0

(40)

B =

dx

dqt0

t=T +t0

(41)

B−1 = dx

dqt0−

T t=T +t0 (42)

D =

dx

d∆t0

t=T +t0

(43)

Eq. (39) is a linear, discrete-time system which maps an initial state, at the beginning of a flapping cycle,to a final state at the end of the flapping cycle. As a linearization, this map is only approximate, but theapproximation is quite good when the flapping period is small relative to the time constants of the basebody’s motion.

The linear system (39) provides several opportunities with regard to the nonlinear, time-varying sys-tem (25). First, it provides a mechanism to calculate the change in system state due to a change in param-eters, provided the parameter change is small, without need for solving the governing nonlinear equation.Second, the eigenvalues of A characterize the stability of the trim state. Eigenvalues with magnitude greaterthan one are associated with unstable modes; eigenvalues with magnitude less than one correspond to stablemodes. Complex eigenvalues, which can occur only in conjugate pairs, are associated with oscillatory modeswhile real-valued eigenvalues are associated with non-oscillatory modes. Third, the linearized system lendsitself to control synthesis using linear optimal control approaches, such as LQR synthesis.

C. LQR Control Synthesis

LQR theory provides a method for determining a linear, static state feedback control law

dq0 = −Kdx0 (44)

for which the gain matrix K minimizes a quadratic cost function for a control system with linear dynamics.To develop an LQR controller for the flapping wing MAV model described earlier, Eq. (39) is first

converted to state-space form

dx1 = Adx0 + Bdq0 + D∆0 (45)

dx0 ≡

dx0

dq−1

(46)

dx1 ≡

dx1

dq0

(47)

A ≡

A B−1

0 0

(48)

B ≡

B

B0→−1

(49)

D ≡

D

0

(50)

In the state vector dx0, the term dq−1 is the vector of kinematic parameters from the previous cycle thatinfluence the state variables during the current cycle. These include the magnitude and bias of the threewing degrees-of-freedom. The matrix B0→−1 is used to map the dq0 to dq0, and depends on the choice of kinematic parameters used for control. As an example, if the wing stroke magnitude (φm), bias (φ0) andsharpness parameter (K φ), the wing deviation magnitude (θm), and the wing feathering magnitude (ηm)

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and sharpness parameter (K η) are used as control parameters, then we have

dq0 = B0→−1dq0 (51)

dq0 =

φm φ0 K φ θm ηm K η

(52)

dq0 =

φm φ0 θm ηm

(53)

B0→−1 ≡

1 0 0 0 0 0

0 1 0 0 0 00 0 0 1 0 0

0 0 0 0 1 0

(54)

LQR control synthesis attempts to minimize a quadratic cost metric

J = 1

2 limN →∞

1

N E

N n=1

dxnT Qdxn + ρdqnT Rdqn

(55)

for the system (45), where xn and un are the state and control input at the nth discrete time instant,respectively, and E[] is the expected value of its argument. The choice of penalty parameters Q, R and ρare important considerations in LQR control synthesis; the role of these penalty terms is briefly discussed in§ V.

The system dynamics (45), along with the linear quadratic cost (55), define the LQR control designproblem. The solution, in the form of the gain matrix K which minimizes the cost, is obtained by solving adiscrete time Riccati equation.21 The gain matrix that minimizes the quadratic cost function is21

K =

ρR + BT PB−1BT PA

(56)

where P is obtained from solution of the discrete algebraic Riccati equation

P = AT PA + Q − AT PB(ρR + BTPB)−1BTPA (57)

In the present study, the Riccati equation is solved using MATLAB. 22

V. Results

The control synthesis procedure described in the previous section is tested for performance throughnumerical simulation of the nonlinear model under gust disturbances. The state perturbations obtainedfrom the nonlinear model are used to calculate the control inputs using the gain matrix. This procedureallows evaluation of the gust tolerance of the linearized controller on a nonlinear model.

This set of numerical simulation cases are outlined in §A. The closed-loop numerical simulations arepresented for longitudinal gust disturbances in §B and lateral gust disturbances in §C.

A. Outline of numerical simulations

Configuration details of the MAV studied here are presented in Table 2. The MAV is trimmed for hover ata flapping frequency of 125 rad/sec. With each wing’s length of 0.1 m the wing-tip speed is 12.5 m/s. Thetrim values of kinematic parameters, qtrim, are obtained from periodic-shooting method are listed in Table3.

The gust disturbances are modeled as (1 − cos) discrete gusts, which are widely used in aircraft gustresponse studies.23 The discrete gust used in this study extend for 75 flapping cycles, and the system issimulated for a total of 150 flapping cycles. Cases with both longitudinal and lateral gust disturbanceswith different gust speeds are simulated and presented here. Three sets of control inputs are consideredand are listed in Table 4. All simulations assume independent control inputs for the kinematic parametersgoverning the left and right wing motion. Hence, the number of control inputs, N control, is twice the numberof parameters specified here.

Three separate Q matrices, listed in Table 5, are considered for control synthesis. The table also liststhe control weighting parameters, R matrices and ρ. The choice of these parameters significantly influencesthe controller behavior. A short discussion of the different State Penaltying matrices is warranted here:

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• The state penalty matrix Q1 penalizes all states and the resulting controller gives equal priority todrive all perturbations to zero, including the physical displacement of the vehicle and its orientation.

• The state penalty matrix Q2 equally penalizes all states except displacement (xyz−location). Theresulting controller tries to drive all perturbations in attitude and/or velocity to zero, while allowingthe vehicle to slowly return to its original undisturbed location.

• The state penalty matrix Q3 strongly penalizes any perturbations in the attitude and angular rate of theMAV. The low weights associated with displacement result in a relatively low priority for maintainingposition.

It is important to note that all response quantities presented in the following results are “normalized”by the maximum gust speed V g. For a linear system, these normalized responses would coincide. (Doublingthe disturbance doubles the response, so dividing by the disturbance magnitude yields a single normalizedresponse.) Deviations among the normalized responses therefore indicate nonlinear behavior.

Table 2. Baseline MAV configuration

Parameter Value

Body mass 0.01 kg

Wing-radius (half-span) 0.1 m

Wing material density 1000 kg/m3

Wing-chord 0.025 m

Wing-thickness 0.6 mm

Flapping frequency 125 rad/sec

CG to right-wing hinge (0.015, 0.03, 0.0) m

Table 3. Baseline MAV kinematic parameters, parameters in bold were obtained from periodic shootingmethod for trim

Parameter Value

φm 1.22 radφ0 0.0 rad

K φ 0.01

θm 0.39 rad

θ0 0.0 rad

θs π/2 rad

ηm π/4 rad

η0 π/2 rad

ηs 0.0 rad

K η 0.1

B. Longitudinal gust disturbances

This section presents the closed-loop simulation of a MAV under a strictly longitudinal gust along the inertialz−axis. The gust profile is shown in Fig. 3. The gust spans 75 flapping cycles and attains a maximumspeed, V g, at 37 cycles. The following simulations use the same gust profile, but vary the maximum gustspeed to test the controller performance on a nonlinear system under increasingly higher gust speeds.

The open-loop poles (i.e., the eigenvalues of the state matrix for the linearized system) of the MAVtrimmed for hover are shown in Fig. 4. The plot shows three rigid body modes on the unit circle at (1.0,0.0). The two complex-conjugate poles outside the unit circle correspond to an unstable longitudinal mode.

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Fig. 5 shows the open-loop response of a hovering MAV under a longitudinal gust disturbance of maximumspeed, V g = 0.01 m/s. The perturbations in yI and zI locations and the pitch angle, φB , are plotted.Simulation of 100 cycles shows that the vehicle is unstable in its trim condition and any disturbances uponencountering a small gust are amplified at each cycle. This diverging mode corresponds to the unstable polesthat appear outside the unit circle in Fig. 4.

−1.5 −1 −0.5 0 0.5 1 1.5

−1

0

1

Real

I m a g

Open-loop Poles

Figure 4. Linearized discrete-time invariant open-loop poles of the trimmed MAV; Region enclosed by unitcircle is stable

0 20 40 60 80 100−6

−4

−2

0

2

N cycles

∆ d o f / V g

yI , m·s/m zI , m·s/m φB , rad·s/m

Figure 5. Unstable open-loop discrete-time response of a hovering MAV under a longitudinal discrete gust

disturbance along z-axis (V g = 0.01 m/s)

1. Parameter Set 1, State Penalty Matrix Q1

The closed-loop poles for this configuration are shown in Fig. 6. The open circles are the open-loop polesrepeated from Fig. 4, while the filled circles are the closed-loop poles. It can be seen that the LQR controllerpushes all the poles to lie within the unit circle. Thus, for the linearized dynamics, the controller can stabilizethe system under all perturbations. The performance of this controller for the nonlinear system, however,must still be verified.

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The scaled response and control inputs of a hovering MAV at four gust speeds is shown in Fig. 7. Uponencountering the gust the vehicle pitches up and starts to climb. In each case the physical displacement isreduced to zero by the end of 150 flapping cycles. The responses in Fig. 7(a) show that the vehicle respondslinearly to yI −perturbations. The system response to zI − and φB perturbations is nonlinear, however, asindicted by the disparate normalized responses. The maximum gust velocity that the controller was able tostabilize for the nonlinear system model is 0.75 m/s.

The control inputs are shown in Fig. 7(b). The simulation was performed on a 6DOF model withindependent control inputs to the left and right wing. As expected, the controller used the same inputs for

each wing. The scaled control inputs show nonlinear behavior of wing stroke magnitude, φW m , while both the

wave shaping parameters K φ and ηs saw little use by the controller.The maximum gust tolerance of 0.75 m/s is only 6% of the wing tip speed; one might expect the controller

to exhibit greater robustness to disturbances. It is likely that the nonlinearities arising from the vehicle pitchmotion move the system state beyond the region within which linear approximation is valid. If so, improvingdisturbance rejection performance by the linear controller would require suppressing responses that showstrong nonlinearity. This observation is investigated in the following two simulations where the Q matrix ischanged to reduce the penalty on perturbations in position and pitch response.

−1.5 −1 −0.5 0 0.5 1 1.5

−1

0

1

Real

I m a g

Closed-loop Poles: Set 1: Q1

Figure 6. Linearized discrete-time invariant closed-loop poles for Control input Set 1 and State Penalty Q1;Open circle are open-loop poles and filled circles are closed-loop poles and region enclosed by unit circle isstable


The rationale for relaxing the penalty on position in the state penalty matrix Q2 is to reduce the resultingcontrol inputs, thereby limiting nonlinear behavior that may compromise performance of the linear controller.

The closed-loop poles of the linearized discrete-time system are shown in Fig. 8. In comparison withthe previous case (shown in Fig. 6), the modification to the State Penaltying matrix Q2 pushes the threeeigenvalues corresponding to rigid-body translation modes closer to the unit circle, but stability of all modesis maintained.

The scaled responses are shown in Fig. 9(a). As prescribed to the controller by the choice of Q = Q2,

both yI − and zI − vehicle location perturbations are slowly reduced to zero. The advantage of this is that themaximum gust speed sustained by the controller is increased to 1.5 m/s, albeit at a higher zI displacementof 0.07 units. It is also evident from the response that the pitch rotations show a strongly nonlinear behaviorwith increasing gust velocity. Hence, a change in the Q matrix to suppress these rotations may improve thecontroller gust tolerance.

Out of the set of scaled control inputs shown in Fig. 9(a), φW m shows a strongly nonlinear behavior. The

changes in K φ and ηs are again small in comparison with the other control inputs. Once the gust disturbancehas died down after 75 flapping cycles, the controller uses perturbations in φW

m , φW 0 and ηW

m to return thevehicle to its original unperturbed location.

15 of 29




yI , m·s/m zI , m·s/m φB, rad·s/m

0 50 100 150

0

1

2

3·10−2

N cycles

∆ d o

f / V g

0.001m/s

0 50 100 150

0

1

2

3·10−2

N cycles

0.05m/s

0 50 100 150

0

1

2

3·10−2

N cycles

0.5m/s

0 50 100 150

0

1

2

3·10−2

N cycles

0.75m/s

(a) State Perturbations−

−

−

− −

−

− − −

−

−

−

−

−

−

−

φW m

, rad·s/m φW 0

, rad·s/m K φ,s/m ηW m

, rad·s/m ηW 0

, rad·s/m ηW s

, rad·s/m

0 50 100 150

−0.2

−0.1

0

N cycles

∆

k i n e m a t i c / V g

0.001m/s

0 50 100 150

−0.2

−0.1

0

N cycles

0.05m/s

0 50 100 150

−0.2

−0.1

0

N cycles

0.5m/s

0 50 100 150

−0.2

−0.1

0

N cycles

0.75m/s

(b) Wing Kinematic Parameter Perturbations

Figure 7. Closed-loop response of a hovering MAV under longitudinal discrete gust disturbance; Control inputSet 1; State Penalty Q1; Simulated gust speeds are 0.001 m/s, 0.05 m/s, 0.5 m/s and 0.75 m/s

−1.5 −1 −0.5 0 0.5 1 1.5

−1

0

1

Real

I m a g



16 of 29





0 50 100 150

0

5

·10−2

N cycles

∆ d o f / V g

0.001m/s

0 50 100 150

0

5

·10−2

N cycles

0.05m/s

0 50 100 150

0

5

·10−2

N cycles

0.5m/s

0 50 100 150

0

5

·10−2

N cycles

1.5m/s


−

−

− −

−

− − −

−

−

−

−

−

−

−

φW m

, rad·s/m φW 0


, rad·s/m ηW 0

, rad·s/m ηW s

, rad·s/m

0 50 100 150

−4

−2

0

2

·10−2

N cycles

∆

k i n e

m a t i c / V g

0.001m/s

0 50 100 150

−4

−2

0

2

·10−2

N cycles

0.05m/s

0 50 100 150

−4

−2

0

2

·10−2

N cycles

0.5m/s

0 50 100 150

−4

−2

0

2

·10−2

N cycles

1.5m/s


Figure 9. Closed-loop response of a hovering MAV under longitudinal discrete gust disturbance; Control inputSet 1; State Penalty Q2; Simulated gust speeds are 0.001 m/s, 0.05 m/s, 0.5 m/s and 1.5 m/s

17 of 29





The previous simulations suggest that pitching motion results in strongly nonlinear behavior, possibly push-ing the controller out of the region in which the linear controller is effective. Here, we choose a state penaltymatrix that strongly penalizes perturbations in attitude and angular rate.

Closed-loop pole locations for this configuration are shown in Fig. 10. As in the previous case, all polesare stable, with three that lie just within the unit circle.

The resulting scaled responses in Fig. 11(a) show that the controller is able to tolerate a maximum gust

speed of 2.5 m/s. This is higher than the previous two cases of 0.75 m/s and 1.5 m/s. The maximum scaledzI displacement is virtually the same as the previous case (0.07 units), but higher than the first case (0.03units).

−1.5 −1 −0.5 0 0.5 1 1.5

−1

0

1

Real

I m a g




The closed-loop pole locations are shown in Fig. 12 and the closed-loop response for control Set 2, withfour control kinematic parameters for each wing, are presented in Fig. 13. The closed-loop pole locationsshow a similar behavior as in the previous case, with three poles just within the unit circle. The response(Fig. 13(a)) shows that a low priority to maintaining the position allows the system to tolerate significantperturbations under gusts. The system was tested and found to recover from gusts up to a maximum speedof 6.5 m/s, which is 54% of the wing tip speed. Such a gust results in a considerable deviation from thetrim orbit, about which the system was linearized. We also note that the maximum scaled displacement isan order of magnitude larger than in the previous three cases.

The closed-loop simulation for a gust speed of 6.5 m/s (last plot in Fig. 13(b)) shows growing oscillationsin ηW

m after 100 s. Although it is not show in the responses, this is a result of the unstable oscillations in thevehicle pitch rate, φB, that start growing around 75 s.

The primary difference between these and the previous simulations is the set of control inputs, where

K φ and ηW

s are excluded from the present simulations. The significant improvement in gust tolerance showsthat these two control parameters have a strongly nonlinear influence on the flight mechanics. Excludingthese inputs maintains the validity of the linearization for a much larger range of perturbations from thetrim condition, thereby improving gust tolerance at the expense of larger perturbations from the referenceposition.


This set of simulations uses even fewer control variables (Set 3) that include only wing stroke-motion control.The closed-loop poles are shown in Fig. 14 and the response and control inputs are shown in Fig. 15. The

18 of 29





0 50 100 1500

0.2

0.4

0.6

N cycles

∆ d o f / V g

1.0m/s

0 50 100 1500

0.2

0.4

0.6

N cycles

3.0m/s

0 50 100 1500

0.2

0.4

0.6

N cycles

4.0m/s

0 50 100 1500

0.2

0.4

0.6

N cycles

6.5m/s

(a) State Perturbations

−

−

φW m , rad·s/m φW

0 , rad·s/m ηW

m ,rad·s/m ηW 0

, rad·s/m

0 50 100 150−0.1

0

0.1

N cycles

∆

k i n e m a t i c / V g

1.0m/s

0 50 100 150−0.1

0

0.1

N cycles

3.0m/s

0 50 100 150−0.1

0

0.1

N cycles

4.0m/s

0 50 100 150−0.1

0

0.1

N cycles

6.5m/s


Figure 13. Closed-loop response of a hovering MAV under longitudinal discrete gust disturbance; Controlinput Set 2; State Penalty Q3; Simulated gust speeds are 1 m/s, 3 m/s, 4 m/s and 6.5 m/s

20 of 29




state penalty used for this study is the same as those used in the previous case (Fig. 13). Although thecontroller is able to recover from gusts up to 12 m/s, the MAV sees much larger perturbation in the z−location with little coupling between the y− and z− perturbations. Note that the system is able to maintainstability with only two control inputs for each wing (Fig. 13(b)) – a simpler flapping motion.

Additionally, from the study of these simulations, it is not possible to infer the performance of the linearcontroller on a nonlinear system using the location of the closed-loop poles. The control parameter set 1 withQ1 has the most stable locations of the closed-loop poles, but has the worst performance. While, the presentcase with most of the poles lying closer to the unit circle has the best performance. Hence, it suggests that

a highly nonlinear system like this is sensitive to perturbations in the system dynamics that come from thecontroller.

−1.5 −1 −0.5 0 0.5 1 1.5

−1

0

1

Real

I m a g



C. Lateral gust disturbances

This section discusses the simulation results for a flapping wing MAV under a purely lateral discrete gustdisturbance along the inertial x−axis. Like the previous section, different gust velocities are simulated withthe same gust profile as shown in Fig. 3. The gust spans 75 flapping cycles and achieves the maximum gustspeed at the end of 37 cycles.


The response of the MAV under a lateral gust disturbance is shown in Fig. 16(a). Although the responsesshow a strongly linear scaling (with respect to the gust speed) for both gust speeds, the controller is ineffectiveat rejecting gust disturbances larger than 0.25 m/s. This is lower than the corresponding limit of 0.75 m/sfor a hovering MAV under a purely longitudinal gust disturbance.

It is interesting to note that the controller prevents longitudinal motion in response to the disturbance;the response is purely lateral. This happens due to the choice of Q1 matrix which penalizes perturbations

in all degrees of freedom, including those associated with longitudinal motion. The significantly highercontrol responsiveness in the longitudinal direction results in negligible motion in comparison with lateralperturbations. System response with other choices of Q matrix presented later exhibit coupling betweenlateral and longitudinal degrees of freedom in response to a lateral gust.

The control inputs in Figs. 16(b) and 16(c) show that the controller makes strong use of K φ to achievelateral control, which is unlike any of the longitudinal cases where the controller provided very small inputsto K φ in comparison with the other control variables. All other control inputs in Figs. 16(b) and 16(c) showa primarily anti-symmetric variation between the left and right wings.

21 of 29





0 50 100 150

0

1

2

N cycles

∆ d o f / V g

1m/s

0 50 100 150

0

1

2

N cycles

3m/s

0 50 100 150

0

1

2

N cycles

5m/s

0 50 100 150

0

1

2

N cycles

12m/s

(a) State Perturbations

−

−


0 , rad·s/m

0 50 100 150−2

0

2

·10−2

N cycles

∆

k i n e m

a t i c / V g

1m/s

0 50 100 150−2

0

2

·10−2

N cycles

3m/s

0 50 100 150−2

0

2

·10−2

N cycles

5m/s

0 50 100 150−2

0

2

·10−2

N cycles

12m/s


Figure 15. Closed-loop response of a hovering MAV under longitudinal discrete gust disturbance along z−axis; Control input Set 3; State Penalty Q3; Simulated gust speeds are 1 m/s, 3 m/s, 5 m/s and 12 m/s

22 of 29




−

−

−

−

−

− −

−

xI , m·s/m yI , m·s/m zI , m·s/m φB, rad·s/m θB, rad·s/m ηB, rad·s/m

−

0 50 100 150−0.2

0

0.2

N cycles

∆ d o f / V g

0.001m/s

0 50 100 150−0.2

0

0.2

N cycles

0.01m/s

0 50 100 150−0.2

0

0.2

N cycles

0.05m/s

0 50 100 150−0.2

0

0.2

N cycles

0.25m/s


−

−

− −

−

− − −

−

−

−

−

−

−

−

φW m

, rad·s/m φW 0


, rad·s/m ηW 0

, rad·s/m ηW s

, rad·s/m

0 50 100 150−0.2

0

0.2

0.4

0.6

N cycles

∆ k i n e m a t i c / V g

0.001m/s

0 50 100 150−0.2

0

0.2

0.4

0.6

N cycles

0.01m/s

0 50 100 150−0.2

0

0.2

0.4

0.6

N cycles

0.05m/s

0 50 100 150−0.2

0

0.2

0.4

0.6

N cycles

0.25m/s

(b) Left-Wing Kinematic Perturbations−

−

−

− −

−

− − −

−

−

−

−

−

−

−

φW m

, rad·s/m φW 0


, rad·s/m ηW 0

, rad·s/m ηW s

, rad·s/m

0 50 100 150−0.6

−0.4

−0.2

0

0.2

N cycles


0.001m/s

0 50 100 150−0.6

−0.4

−0.2

0

0.2

N cycles

0.01m/s

0 50 100 150−0.6

−0.4

−0.2

0

0.2

N cycles

0.05m/s

0 50 100 150−0.6

−0.4

−0.2

0

0.2

N cycles

0.25m/s

(c) Right-Wing Kinematic Perturbations

Figure 16. Closed-loop response of a hovering MAV under lateral discrete gust disturbance along x− axis;Control input Set 1; State Penaltys Q1; Simulated gust speeds are 0.001 m/s, 0.01 m/s, 0.05 m/s and 0.25 m/s

23 of 29





Changing the State Penaltying matrix to Q3 with the same set of control inputs doubles the gust toleranceto 0.5 m/s. The responses and control inputs are shown in Figs. 17. As expected, the vehicle is displacedin the direction of the gust and the controller slowly reduces this perturbation to zero. The maximumscaled displacement increased to approximately 1.0 as opposed to 0.3 in the previous case. Unlike theprevious case, the however, response from the present controller exhibits coupling between the lateral andlongitudinal degrees of freedom with the vehicle showing (small) displacements in the yI − and zI −locations.

The control inputs (Figs. 17(b) and 17(c)) are antisymmetric for the left and right wings and the controllerprovides significant inputs to K φ. However, the control law results in very small perturbations in thefeathering phase shift, ηW

s .


The response and control inputs for the case with four control variables for each wing and Q3 as the stateweighting matrix are shown in Fig. 18. The system was found to be unstable for speeds greater than 2 m/s.Thus, exclusion of the K φ and ηW

s from control inputs significantly improves the gust tolerance of the system,but also increases the scaled displacement to approximately 1.45.

The responses (Fig. 18(a)) show that under the influence of the gust, the vehicle sees a coupled rotationin the roll and yaw angles, which are effectively damped out by the controller. At the same time, however,very low displacement in the longitudinal plane is seen. This is in contrast with a similar study with Q = Q1

where the system did not show any considerable displacement in the longitudinal plane and the vehicle simplywas displaced along the xI − axis and returned to its original position at the end of the gust disturbance.


When the number of control inputs is reduced to only two for controlling wing stroke motion, the gusttolerance limit is still the same as the previous case with four control inputs, i.e. 2 m/s. The response andcontrol inputs are shown in Fig. 19. The scaled displacement of 1.5 is close to the value seen in the previouscase. Thus, similar performance is achieved using half the number of control inputs, which may allow for asignificant simplification in the physical realization of the flapping flight control system.

VI. Conclusions

This study describes the development of a linearized, discrete-time model for the motion of a flappingwing MAV that is used in an LQR formulation to obtain a flight control law. The controller is implementedon the nonlinear, time-varying model to evaluate its performance under external disturbances, modeled asdiscrete gusts. Longitudinal and lateral gusts are considered. Controller performance is compared based onthe ability to recover trimmed flight following gusts of increasing speed.

The simulation results show that the linear controller is able to recover stable trimmed flight in responseto a gust, although the maximum gust velocity that can be withstood by each controller depends on the LQRparameters used in the control synthesis, the type of wing motion, and the direction of the gust disturbance.In all cases, the controller had significantly higher tolerance for longitudinal gusts than for lateral gusts.

Experimentation with different state-weighting matrices for LQR synthesis showed that a higher penaltyon attitude and angular rate perturbations, and a lower penalty on position increases gust tolerance. At thesame time, a lower penalty on position allows more substantial and longer lived displacements in responseto a gust. This observation suggests a tradeoff between maintaining position and tolerating disturbances. It

is important to note, however, that the choice of state-weighting is often impacted by the vehicle mission,though the designer may have some latitude to optimize performance while meeting mission requirements.

The performance of the controller is influenced by the choice of the controlled kinematic parameters.Excluding the wing stroke sharpness parameter, K φ, and the wing feathering phase-shift, ηW

s , for example,had a favorable influence on the controller performance. These two parameters appear to result in morestrongly nonlinear vehicle motion.

The results also demonstrate that the vehicle can recover from both longitudinal and lateral gust distur-bances, in both hovering and forward flight, using only the wing-stroke amplitude and offset for each wing(four control parameters in all). This observation may suggest simpler control mechanizations. Using fewer

24 of 29




−

−

−

−

− −

−


−

0 50 100 150

0

0.2

0.4

0.6

0.8

N cycles

∆ d o f / V g

0.01m/s

0 50 100 150

0

0.2

0.4

0.6

0.8

N cycles

0.05m/s

0 50 100 150

0

0.2

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0.8

N cycles

0.1m/s

0 50 100 150

0

0.2

0.4

0.6

0.8

N cycles

0.5m/s


−

−

− −

−

− − −

−

−

−

−

−

−

−

φW m

, rad·s/m φW 0


, rad·s/m ηW 0

, rad·s/m ηW s

, rad·s/m

0 50 100 150−0.2

0

0.2

0.4

0.6

N cycles


0.01m/s

0 50 100 150−0.2

0

0.2

0.4

0.6

N cycles

0.05m/s

0 50 100 150−0.2

0

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0.6

N cycles

0.1m/s

0 50 100 150−0.2

0

0.2

0.4

0.6

N cycles

0.5m/s

(b) Wing Kinematic Perturbations−

−

−

− −

−

− − −

−

−

−

−

−

−

−

φW m , rad

·s/m φ

W 0 , rad

·s/m K φ,s/m η

W m , rad

·s/m η

W 0 , rad

·s/m η

W s , rad

·s/m

0 50 100 150

−0.6

−0.4

−0.2

0

0.2

N cycles


0.01m/s

0 50 100 150

−0.6

−0.4

−0.2

0

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N cycles

0.05m/s

0 50 100 150

−0.6

−0.4

−0.2

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N cycles

0.1m/s

0 50 100 150

−0.6

−0.4

−0.2

0

0.2

N cycles

0.5m/s

(c) Wing Kinematic Perturbations

Figure 17. Closed-loop response of a hovering MAV under lateral discrete gust disturbance along x− axis;Control input Set 1; State Penaltys Q3; Simulated gust speed are 0.01 m/s, 0.05 m/s, 0.1 m/s and 0.5 m/s

25 of 29




−

−

−

−

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− −

−


−

0 50 100 150

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1

2

N cycles

∆ d o f / V g

0.1m/s

0 50 100 150

0

1

2

N cycles

0.5m/s

0 50 100 150

0

1

2

N cycles

1.0m/s

0 50 100 150

0

1

2

N cycles

2.0m/s

(a) Responses

−

−


0 , rad·s/m

0 50 100 150−0.1

0

0.1

N cycles


0.1m/s

0 50 100 150−0.1

0

0.1

N cycles

0.5m/s

0 50 100 150−0.1

0

0.1

N cycles

1.0m/s

0 50 100 150−0.1

0

0.1

N cycles

2.0m/s

(b) Wing Kinematic Perturbations

−

−

φW m

, rad·s/m φW 0

, rad·s/m

0 50 100 150

−0.1

0

0.1

N cycles


0.1m/s

0 50 100 150

−0.1

0

0.1

N cycles

0.5m/s

0 50 100 150

−0.1

0

0.1

N cycles

1.0m/s

0 50 100 150

−0.1

0

0.1

N cycles

2.0m/s

(c) Wing Kinematic Perturbations

Figure 19. Closed-loop response of a hovering MAV under lateral discrete gust disturbance along x− axis;Control input Set 3; State Penalty Q3; Simulated gust speeds are 0.1 m/s, 0.5 m/s, 1 m/s and 2 m/s

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control parameters results in larger perturbations, however, suggesting a tradeoff between design simplicityand control authority.

This paper describes a control design procedure for a linearized, simplified model of a flapping wingMAV and provides comparisons of controller performance for various design and disturbance parameters.The simulations also uncover important nonlinear effects that can limit control system performance, evenfor this simplified, rigid-wing model. In practice, a flapping wing MAV experiences additional internal andexternal disturbance effects due, for example, to wing flexibility and unsteady aerodynamics. Such effectswarrant more detailed analysis and perhaps the development of more sophisticated control algorithms, but

the approach and observations described here can provide valuable insight concerning flapping wing MAVflight control and vehicle design.

Acknowledgements

This material is based on research sponsored by Air Force Research Laboratory under agreement numberFA8650-09-2-3938. The U.S. Government is authorized to reproduce and distribute reprints for Governmentalpurposes notwithstanding any copyright notation thereon. The authors would like to thank Dr. RaymondKolonay and Dr. Rakesh K. Kapania for their leadership in the Collaborative Center on MultidisciplinarySciences. The views and conclusions contained herein are those of the authors and should not be interpretedas necessarily representing the official policies or endorsements, either expressed or implied, of Air ForceResearch Laboratory or the U.S. Government.

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