2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013

A Geometric Control Approach for Optimum Maneuverability

of Flapping Wing MAV s Near Hover

Haithem E Taha 1 , Craig A Woolsey2 and Muhammad R Hajj3

Abstract- Flapping wing micro-air vehicles are aerial robots that use biomimetic actuation for propulsion and control. De­signing such a system requires an integrated system model de­scribing the flight mechanics, propulsion, and control. Relative to conventional aircraft, the resulting model is nonlinear, high­dimensional, time-varying, and underactuated, making analysis and design challenging. Geometric control and averaging theory provide useful analysis tools for biomimetic locomotion systems that use high frequency, time-periodic inputs to generate control forces and moments. Recognizing the essential role of certain Lie bracket and symmetric product vector fields in the flight mechanics of a flapping wing micro-air vehicle, we develop analytical expressions for these vector fields in terms of system parameters. Using these expressions, we then pose and solve a design optimization problem aimed at maximizing vehicle maneuverability. The example illustrates a constructive technique for the design of biomimetic robots and their gaits.

I . INTRODUCTION

The dynamics of flapping flight is challenging to model and analyze. The propulsion and control mechanization re­sults in a fundamentally underactuated system and for even the simplest models, the governing equations are nonlin­ear and time-varying. The use of time-periodic inputs to generate motion in underactuated mechanical systems is a well-established concept and leads naturally to the use of geometric control and averaging theory for analysis of these systems [ 1 ] .

In this work, we consider the longitudinal flight o f flapping wing micro-air vehicles (FWMAVs) near hover and use averaging and tools from geometric control theory to derive analytical expressions characterizing the net control force and moment that a given configuration can generate. Specif­ically, we develop analytical expressions for the Lie bracket and symmetric product vector fields that play a fundamental role in the averaged dynamic equations [ 1 ] . These expres­sions include parameters that define the system's geometry and gait; maximizing elements of the symmetric products directly influences the vehicle 's maneuverability. The work illustrates a design optimization approach for enhancing the maneuverability of a FWMAV

II . WING KINEMATICS

To formulate the flight dynamic model for a rigid-winged FWMAV, we define several reference frames : an inertial

1 PhD Candidate, Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA he z z at @vt . edu

2 Associate Professor, Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA cwo o l s e y @ vt . edu

3 Professor, Engineering Science and Mechanics, Virginia Tech, Blacks­burg, VA mha j j @vt . edu

978-1-4799-0178-4/$31.00 ©2013 AACC 597

frame {XI , YI , ZI } , a body-fixed reference frame {Xb , Yb , Zb } , and a wing-fixed reference frame {xw , Yw , zw } for each of the flight vehicle 's wings . (We will ultimately specialize to the case of longitudinal flight, so that wing motion may be described using a single wing frame.)

Following convention, the xb-axis points forward defining the vehicle 's longitudinal axis, the Yb-axis points to star­board, and the Zb-axis completes the right-handed frame. In general, the wing-fixed frame is defined using three addi­tional angles . However, based on observations of biological flyers [2] , [3] , we constrain the wing motion to the body horizontal plane (defined by Xb and Yb) . Thus, only two wing angles are required: the flapping angle 'P, describing fore and aft motion, and the "pitching" angle 'T}, describing rotation of the wing about a chord line. (The distinction between this use of the term "pitch" and its more conventional use for base body attitude should be clear from context. )

The pitching angle, 'T}, is constrained to be piecewise constant:

0 > 0 0 < 0

where ad and au are the prescribed angles in the downstroke and upstroke, respectively. The requirement is consistent with prior studies of hovering FWMAVs [4] , [5] , [6] , [7] , [8] , [9] , [ 10] . We also consider purely symmetric motion, so that ad = au = am, where am is the mean angle of attack maintained throughout the entire stroke. Thus,

sin 'T} = sin am and cos 'T} = sign ( 0) cos am

III . AERODYNAMIC MODEL

Following Taha et al . [ 1 1 ] , the total lift, drag and pitching moment about the hinge axis can be written in the form

L(t) 1 '4pCLJ21 02 (t) sin 2am

D(t) 1 2 pCLJ21 1 0 (t) 1 0 (t) sin2 am

1 2 pCLa 122 1 0 (t) 1 0 (t) sin am ( 1 )

where the Imn = 2 foR rmcn (r) dr are weighted moments of area for the two wings and CLa is the lift-curve slope of the three-dimensional wing, which is given by

IV. EQUATIONS OF MOTION

Because the wing pitch angle is kept constant throughout each half-stroke (ry == 0), except between half-strokes where the derivative is undefined, we neglect the wing pitching dynamics . Since the two wings move symmetrically, the equations of motion are defined in terms of four generalized coordinates: q = [x , Z , e , 'PV , where x and z are "qua­sicoordinates" associated with body velocity components u and w along the Xb and Zb directions, respectively. In the following subsections, we use the principle of virtual power [ 1 2] to derive the equations of motion:

OV

o L [mi (vi + ijci) - fi ] '

Oq

'

i=b ,w J

. OWi

+ [hi + miPCi x Vi - mil . -;::;-:- (2)

uqj where j E { I , 2 , 3 , 4} and PCi is the vector pointing from the reference point of the ith rigid body to its center of gravity.

A. Body Setting the origin of the body frame at the body center of

gravity, we write the velocity of the origin of the body axis system and the corresponding angular velocity as

and

where i, j , and k are unit vectors along the x, y, and Z directions in the axis-system indicated by the subscript. The angular momentum vector of the body about the body center of gravity and its inertial derivative are given by

The body experiences gravitational forces only, with no moments. Thus, we write the body forces in the body frame as

B. Wing Locating the origin of the wing frame at the hinge root,

we write the velocity of the origin of the wing frame and its angular velocity as

The wing angular velocity vector in the wing frame is given by

if? sin 1] - B cos 1] sin 'P ) B cos 'P

-if? cos 1] - B sin 1] sin 'P The position vector pointing from the hinge root to the wing center of gravity is Pcw = -diw +r egj w where d and reg are the distances between the wing root hinge point and the wing center of gravity along the negative xw-axis and the yw-axis, respectively. Thus, we find the the inertial acceleration

.. (w) _ Pcw -

( ih ) ( ih ih

d(w� + w§ ) - reg (w3 - WIW2 ) ) -d(W3 + WIW2 ) - reg (wi + w§ ) d(W2 - WIW3 ) + reg (WI + W2W3 )

598

Assuming the wing reference frame is fixed in the wing principal axes, we write the inertial time derivative of the angular momentum vector represented in the wing frame as

h

. (w) _ ( h

h: 21 ) ( IXWI + (Iz - Iy)w2w3 )

w IyWy + (Ix - IZ)WIW3 h3 Izw3 + (Iy - IX)WIW2

The wing is subjected to aerodynamic and gravitational forces. The force vector applied on the wing is ( -D cos 'P ) ( - sin e )

f�) = 0 + mwg 0 -L cos e

The moment vector contains three contributions: aerody­namic, gravitational, and the control torque. The first is written as maw = mhj w + N kb , where mh is the hinge moment as presented in Eq. ( 1 ) and N is the opposing moment due to the drag force distribution along the wing :

N(t) = pAhl l if? (t) I if? (t) sin2 am

The second is the moment due to gravity mgw = (-diw +

regjw ) x mwgkI ' The last is the control torque mcw = -T<pkb , where T<p is the actuating torque in the flapping direction. Having prepared all the required terms to apply the principle of virtual power (2), we obtain the equations of motion (3) . Note that the roll and yaw moment contributions from the two wing-halves cancel for symmetric flapping .

V. AVERAGING FOR MECHANICAL S Y S TEMS

A. Averaging and the Variation of Constants Formula In discussing vibrational control, Bullo [ 1 3] considers a

system with a time-invariant drift vector field and with high­amplitude, high-frequency forcing

d 1 -d

x = f (x ) + -g (x , tic) , t c x (O) = Xo (4)

where 0 < c « 1. The time-varying vector field g (x , tic) is assumed to be periodic in its second argument with period T. Let (l)g T denote the flow of the vector field g (x , T) . We define the time-varying vector field

F(X , T) = ( ((I)g,Tr f) (x)

where the pull-back of the vector field f along the diffeo­morphism (l)g,T is ( ((I)g,Tr f) (x) = ( d� ((I)g,T) - l O f o (l)g'T) (x)

By the variation of constants formula [ 1 ]

(l)J+9 = (1)9 0 (I) ((I)g , T r f

0 ,7 0 ,7 0 ,7 the solution of (4) satisfies

d 1 -d

x = -g (x , tic) , t c where z (t) satisfies

d dt z = F(z , tic) ,

x (O) = z (t)

z (O) = Xo

(5)

(6)

(7)

mv (u + we + g sin e) + mw (pI cos 'f/ cos <p + P2 sin <p + P3 sin 'f/ cos <p - Xh(2 ) mv (w - ue - 9 cos e) + mw (P3 cos 'f/ - PI sin 'f/ - Xh e)

= -D cos <p = -L

mw [d sin 'f/U + (d cos 'f/ cos <p - reg sin <p - Xh ) W + Xhe (Xh - d cos 'f/ cos <p + reg sin <p) + + Xh (PI sin 'f/ - P3 cos 'f/ + ue + 9 cos e) + d sin 'f/e( w - Xh e) + - (d cos 'f/ cos <p - reg sin <p) (ue + g) 1 + + Iybe + h2 cos <p - hI cos 'f/ sin <p - h3 sin 'f/ sin <p

mw (d cos 'f/ sin <p + reg cos <p) [U + (w - Xhe)e] + (hI sin 'f/ - h3 cos 'f/) = T<p (3)

Solutions x (t) of (4) exhibit slow and fast time scales as E ---+ 0+ , as do solutions of (7). One may investigate the slow time scale behavior by considering the following T­averaged system:

d - l iT -d x = F(x) := - F(X , T)dT, t T 0 x (O) = Xo (8)

Trajectories for the averaged system (8) approximate those of (4) on a time scale of order one. Moreover, if the sys­tem (8) has an asymptotically stable equilibrium point and Xo is within an E-neighborhood of it, then the true and averaged solutions remain E-close for all time. Asymptotic stability of the equilibrium of (8) implies that the corresponding periodic orbit of (4) is asymptotically stable.

To show that the system (4) is in a form amenable to conventional averaging theorems (see [ 14] , for example) , we define a new independent variable T = tiE . Recognizing that

d dx dt dx dT X (t(T) ) =

dt dT = Edt ,

we obtain the following system d dT x = Ef(x , T) + g(x , T) ,

B. Mechanical Control Systems

x (O) = Xo .

Bullo [ 1 3] considers mechanical control systems with n degrees of freedom and m :s; n inputs, which can be written in the form

where C (q , q) represents the "Coriolis and centripetal" effects associated with the positive definite kinetic energy metric M(q) . On the right hand side, the vector Xo (q) represents configuration-dependent effects such as forces arising from a scalar potential function (e.g . , moments due to gravity) . In Bullo 's work, E(q, q) = R(q)q represents dissipation that is linear in velocity. The configuration­dependent vector fields X a , where a E { l , . . . , m} , are the input vector fields. The dynamic equations then become

q = _M-I (q)C(q , q)q + M-I (q)XO (q) + .. ' .. '

Z(q,q) Xo (q) + M-I (q)R(q)q + M-I (q)Xa (q) va (t)

'-v-'" v '

R(q)q Xa (q)

( 1 0)

599

Defining the 2n-dimensional state vector x = [qT , qT] T , the dynamic equations ( 1 0) become

x = Z(x) + Yo (x) + R(x) + Ya (x)va (t) , ( 1 1 )

where terms are defined by obvious correspondence. The vector fields on the right hand side of equations ( 1 1 )

are polynomials o f known degree in q. Bullo notes several consequences of this special structure. For example, the Lie bracket of any two input vector fields vanishes :

( 1 2)

This observation, together with the Jacobi identity for Lie brackets of vector fields, implies that [Ya , [f , Ybl l = [Yb , [f , Yal l for any smooth vector field f(x) . Bullo there­fore introduces the following symmetric product with respect to Z(x) : (Xa : Xb ) = (Xb : Xa ) is the last n­components of the 2n vector field [Ya , [Z, Ybl l . This is the coordinate representation of the intrinsic definition given in [ 1 ] . There, the symmetric product of vector fields on a smooth Riemannian manifold is given in terms of the geodesic spray for the Levi-Civita connection associated with the Riemannian metric .

In discussing locomotion methods for underwater vehicles, Vela et al [ 1 5] , [ 1 6] relaxed some of Bullo 's assumptions and considered the case of

E(q , q) = Eo (q , q) + R(q)q ,

where Eo (q , q) represent "state-feedback terms" that are quadratic in velocity. In the case of a FWMAV, the terms in Eo (q , q) might instead represent aerodynamic effects that are more complicated than linear damping. Standard models for lift and drag, for example, are quadratic in velocity. Vela and colleagues [ 1 5] make the following important observation:

lYe , [Yb , [Ya , Z + Yo + R] ] ] = 0 ( 1 3)

Equations ( 1 2) and ( 1 3) imply that the integral series repre­sentation of (<I>g,T ) * f terminates .

Bullo [ 1 3] showed that if the mechanical system structure satisfies ( 1 2) and ( 1 3) and the control inputs va (t) satisfy

loT va (T) dT = 0 and loT loT va (s) ds dT = 0 ( 1 4)

then the averaged dynamics corresponding to the system ( 10) to indicate time-averaged quantities, the averaged dynamics are written as of (3) is written as

r = Z(r , r) + Xo (r) + R(r)r m

- L Aab (Xa (r) : X b (r ) ) a , b= l

where

Aab = 2� loT (loT

Va (S ) dS) (loT Vb (S ) dS) dT

That is, there exists eo > 0 such that, for all e E (0 , eo] ,

q (t) = r (t) + O(e)

( 1 5)

q(t) = r (t) + (lot va (s) dS) Xa + O(e) ( 1 6)

VI . THE FWMAV S Y S TEM REVIS ITED

In this section, we place the FWMAV model developed in Section IV in the context of Section V. Noting that one of the system parameters depends on one of the generalized velocities 'f/ = 'f/(cp) , the FWMAV system structure may not be amenable to apply the averaging results of the previous section. However, since the dependence is on the sign of cp only, the dynamic structure of the FWMAV still satisfies the conditions ( 1 2) and ( 1 3) . In particular, we have the generalized inertia

where

M13 M14 M23

[ my

M(q) = ;}13 M14

o

mwd sin 'f/ mw ( d cos 'f/ sin cp + reg cos cp )

M33 (Xh = 0) mw ( d cos 'f/ cos cp - reg sin cp - Xh ) Iyb + Iy cos2 cp+

+ sin2 cp(Ix cos2 'f/ + Iz sin2 'f/) sin cp sin 'f/ cos 'f/(Iz - Ix ) (Ix sin2 'f/ + Iz cos2 'f/)

and the input vector field is written as

Taha et at [ 1 1 ] showed that there are two ways to achieve trim for a FWMAV in hover. Here, we consider the first of these, in which the wing hinge is aligned with the body center of gravity (Xh = 0) and the wings execute symmetric flapping motions (identical upstroke and downstroke). We write the control input as

T'P (t) = U cos wt

which satisfies ( 1 4) . The symmetric product (XT<p : XT<p ) may be easily

determined using symbolic software, resulting in (lengthy) expressions involving the generalized coordinates q and the geometric and aerodynamic parameters . Using an overbar

600

q = Z(q, q) + Xo (q) + R(q)q U2

- 4w2 (XT<p (q) : XT<p (q) )

VII . MANEUVERAB ILITY OPTIMIZATION

( 17)

One can relate the maneuverability of the FWMAV to the elements of the symmetric product defined earlier, as these components directly affect the acceleration in the averaged dynamics . Our objective is to formulate an optimization problem in which the design variables are the system pa­rameters and the objective function is a specific element of a symmetric product evaluated at the desired fixed point of the averaged dynamics . This fixed point should correspond to the desired periodic orbit of the original system.

We consider symmetric downstroke/upstroke flapping at hover; that is (qo , qo ) = (0 , 0) . For this state to be a fixed point for the averaged dynamics, the right hand side of Eq. ( 17) evaluated at the origin must vanish. The vector fields in Eq. ( 17) evaluated at the origin, with Xh = 0, have the following components

Z(O , O) + Xo (O) = [0 , g , 0 , O] T

WIG (XT<p : XT<p ) (O) = [0 , 2K21 Sin 2O:m x -- , 0 , O] T my

where K21 = �pCLa I21 and WIG represents a lengthy expression containing the wing-inertia contribution. Thus, similar to the results obtained by neglecting the wing dynam­ics, aligning the hinge line with the body center of gravity along with symmetric flapping automatically ensures zero cycle-averaged forward thrust force and pitching moment; see Doman et at [8] and Taha et at [ 1 1 ] . In addition, it ensures zero cycle-averaged flapping acceleration. Hence, we are left with one trim equation to be satisfied - the cycle­averaged lift must balance the weight:

U2 x WIG K21 sin 2O:m 2w2 = myg ( 1 8)

In the case where one neglects the wing dynamics and prescribes the flapping motion cp( t) = <I> cos wt, the lift requirement becomes

( 1 9)

Note that lim WIG = J2 . \ . In the limit of a vanish-ffiw --+0 x sin am

ingly small wing mass, the flapping amplitude is related to the torque amplitude as follows :

Having achieved balance (trim) at hover, our objective i s then to maximize the upward acceleration of the averaged dynamics from the hovering position; that is, maximize tiJ(O) due to the maximum allowable torque. This quantity is

considered as the maneuverability index in this analysis and is given by

--'-( ) U;'ax - u(f w 0 max = 4w2 X <p<p2 (20)

where Umax is the maximum allowable torque amplitude, Uo is the balancing torque amplitude (required for trim) which satisfies Eq. ( 1 8) , and X <p<p2 is the second entry of the symmetric product vector field (X <p : X <p ) . We note that for the given configuration we cannot manipulate the pitch angular acceleration; setting Xh = 0 results in a lack of pitch control authority in hover [8] . Moreover, symmetric flapping with Xh = 0 results in a lack of control over the forward thrust. We focus on vertical acceleration as a simple example to illustrate our approach to maneuverability optimization.

We consider a trapezoidal wing having a radius R, aspect ratio Al, taper ratio A, and planform area S. The vector of design variables is x [Al, A, j, cxmV. Given a fixed mass of the body mb , the mass of the vehicle is then my = mb + mw , where mw = 2 foR m'c(r) dr = 2m'S and m' is the wing mass per unit area, which is assumed to be uniform. We use the following approximations

Ix = 2 1R m'r2c(r) dr = m' hI

rR A 3 ' Iy = 2 Jo

m'dc (r) dr = m'dIo3

where d is the chord-normalized distance from the wing hinge line to the center of gravity line.

foR m'rc(r) dr II I Iz = Ix + Iy and reg = -'--"------;---mw/2 2S

The optimization problem is

max W(O)max subject to x

Uo = W 2myg ---��-- � Umax K2I WIC sin 2cxm

and XLB < x < X UB

where X LB and XUB are vectors of lower and upper bounds for the design variables, respectively.

We consider the hawkmoth as an example. The values for the wing radius, body mass, and moment of inertia about the Yb-axis [ 17] are given by

R = 5 1 .9 mm, mb = 1 . 648 g and Iyb = 2 .080 g cm2

Assuming uniform areal distribution for the wing mass, its density is deduced from Ellington [ 1 8] to be m' = 0 .005 g/cm2 . A maximum allowable torque for tiny actu­ators is considered as U max = 1 mN m. We assume the hinge to lie at the 30% chord station (�x = 0 .05) and the wing-section center of gravity to lie at the 45% chord station (d = 0 . 1 5) . The distance between the wing root hinge point and the wing center of gravity along the negative xw -axis is then given by d = dCr ( 1 - (A - l ) T;t) , where Cr is the root chord dictated from the trapezoidal chord distribution

601

as Cr = ( l��)R ' Table I shows the lower and upper bounds for the design variables.

TABLE I

D E S I GN PARAMETER B OUND S .

Variable LB UB Wing aspect ratio, 1R 1 10 Wing taper ratio, .\. 0 1 Flapping frequency, f o Hz 00 Hz Mean Angle of Attack, am 0° 90°

The optimization suggests that the solution lies on the boundary of the design variable set. A parametric study supports this optimization result. Fig . 1 shows the variation in the maximum cycle-averaged upward acceleration with a given design variable. In each plot, the remaining variables are fixed according to the following nominal values :

x = [3 , 1 , 25 Hz , 300V

Maximum Cycle-Averaged Upward Acceleration 1 0 ,------------, 4

5 2

_5 L----------� 1 2 3 4 5 0 .5

AR 'A

4 ,--------, 1 50

1 00 L 50

0

2

o \ -----.....,

_2 L--------� -50 o 500 1 000 0 50

f (Hz) a m

Fig. I . Parametric analysis results: Fixed wing radius.

1 00

Including the effects of wing mass and inertia leads to considerably different optimum results than those obtained by neglecting the wing dynamics . First, although increasing the aspect ratio for a given wing radius results in a decreased wing area, and consequently decreased aerodynamic loads, the inertial contributions lead to a favorable effect of Al on the cycle-averaged vertical acceleration W. Therefore, if the wing radius is specified, a high aspect ratio rectangular wing will result in the maximum W. It should be noted, however, that fixing the wing area instead of the radius results in an unfavorable effect of Al on W, as shown in Fig . 2 .

Second, neglecting the wing dynamics and specifying the flapping kinematics suggests choosing higher flapping frequencies to give higher aerodynamic loads (particularly lift) and thus higher accelerations. However, to achieve the required flapping amplitude with a high frequency requires a large hinge torque, which is limited to Umax . That is the kinematic analysis results in the lift being proportional to

w21>2 , while including the wing dynamics results in 1> ex ::, . Thus, if U is set to Umax and the previous two relations are combined, the net result is that lower values of f (w) give higher lift and consequently higher ill.

Third, kinematic analysis suggests a 45° mean angle of attack for the maximum lift and consequently ill because of the sin 2a in the lift expression. However, as shown previously (at least in the limit to vanishing mw) 1> =

2 I U 2 ; that is, ill approaches infinity as am approaches W x sin am zero . Therefore, the maneuverability-optimum mean angle of attack is no longer 45° but as low as possible.

4

2

a

-2 a

Maxi mum Cycle-Averaged Upward Acceleration

4 ,---------------�

2 3 4 0.5 AR A

1 50

1 00 � 50

\. a

-50 500 1 000 a 50

f (Hz) a m

Fig. 2 . Parametric analysis results: Fixed wing area.

VIII . CONCLUSION

1 00

We considered the problem of optimizing maneuverability of a flapping-wing micro-air vehicle in longitudinal flight near hover. In modeling the vehicle, we accounted for the wing-dynamics of the back and forth flapping degree of freedom, so that the flight dynamics model has four de­grees of freedom. The resulting system is an under-actuated, nonlinear mechanical system with time-periodic forcing . We used tools from geometric control and averaging theory to derive analytical expressions for vector fields, known as symmetric products, which play a crucial role in the averaged dynamics . Because these vector fields are directly related to the vehicle 's acceleration and depend on system parameters, one may formulate and solve a design optimization problem to maximize maneuverability. Using the hawkmoth data, we provide an example that illustrates this approach for design optimization of a flapping wing micro-air vehicle with a particular focus on the cycle-average upward acceleration of the body as a maneuverability index: fast vertical takeoff.

Incorporating wing inertial effects in the flight dynamic model has a notable effect on the optimization results . For a given wing radius, larger aspect ratios and taper ratios result in higher generated upward acceleration. In contrast to the kinematic-based analyses, a smaller flapping frequency and mean angle of attack results in a larger average upward acceleration.

602

In closing, we note that maximizing the elements of the input vector fields does not necessarily maximize a vehicle 's maneuverability, when using high-frequency periodic inputs. The accelerations that are generated may cancel out over long time scales, resulting in a zero cycle-average acceleration and no net motion. As described here, it is more appropriate to maximize the elements of the symmetric products vector fields that describe the averaged effect of the high frequency inputs.

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