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Linear Time-Invariant Models of the Dynamics of Flapping-Wing Flight

Umberto Saetti*

Postdoctoral Fellow

Guggenheim School of Aerospace Engineering

Georgia Institute of Technology

Atlanta, GA 30332

Jonathan Rogers

Lockheed Martin Associate Professor

Guggenheim School of Aerospace Engineering

Georgia Institute of Technology

Atlanta, GA 30332

ABSTRACT

This paper demonstrates the extension of the harmonic decomposition methodology, originally developed for rotor-

craft applications, to the study of the nonlinear time-periodic dynamics of ﬂapping-wing ﬂight. A harmonic balance

algorithm based on harmonic decomposition is successfully applied to ﬁnd the periodic equilibrium and approxi-

mate linear time-invariant dynamics about that equilibrium of the vertical and longitudinal dynamics of a hawk moth.

These approximate linearized models are validated through simulations against the original nonlinear time-periodic

dynamics. Dynamic stability using the linear models is assessed and compared to that predicted using the averaged

dynamics. In addition, modal participation factors are computed to quantify the inﬂuence of the higher harmonics on

the ﬂight dynamic modes of motion. The study shows that higher harmonics play a key role in the overall dynamics of

ﬂapping-wing ﬂight. The higher harmonics are shown to induce a vibrational stabilization mechanism that increases

the pitch damping and stiffness while reducing the speed stability. This mechanism results in the stabilization of the

pitch oscillation mode and thus of the longitudinal hovering cubic. As such, the ﬁndings of this study suggest that

if a hovering vehicle is excited by periodic forcing at sufﬁciently high frequency and amplitude, its hovering ﬂight

dynamics may become stable.

INTRODUCTION

The ﬂight dynamics of biological ﬂyers (insects and birds),

and their man-made counterparts, ﬂapping-wing micro aerial

vehicles (MAVs), has been an active area of research within

the aerospace engineering, zoology, and biology communities

for more than two decades. One of the pioneering studies on

the subject is due to Adrian et al. (Ref. 1). Figure 1shows

the ﬂapping frequency for several biological ﬂyers capable of

hovering ﬂight. The challenge in analyzing such dynamical

systems emanates from their multi-body, nonlinear, and time-

varying nature. Typically, the multi-body characteristics are

relaxed by assuming that the effect of the wing inertial forces

on the body dynamics are negligible (see, e.g., Refs. 2–6). Al-

though the validity of this assumption is subject to continuing

debate (Refs. 7–9), researchers tend to adopt it because (i) the

mass of the wing is negligible compared to that of the body

(ii) ignoring the multi-body effects yields equations of motion

similar to those governing the ﬂight dynamics of ﬁxed-wing

aircraft. Nevertheless, there remains a major distinction be-

tween the dynamics of ﬂapping ﬂight and that of ﬁxed-wing

aircraft: because of the time-varying aerodynamic loads due

to the wing periodic motion, the ﬂapping ﬂight dynamics is

time-periodic. In fact, the dynamics of ﬂapping ﬂight is well-

described by nonlinear time-periodic systems (NLTP).

*Incoming Assistant Professor, Department of Aerospace Engineer-

ing, Auburn University, Auburn, AL 36849.

Presented at the VFS International 77th Annual Forum &

Technology Display, Virtual, May 10–14, 2021. Copyright © 2021

by the Vertical Flight Society. All rights reserved.

101102103

Flapping frequency [Hz]

Hawk Moth

Crane Fly

Bumblebee

Orchid Bee

Fruit Fly

Parasitic Wasp

Figure 1: Flapping frequency for several biological ﬂyers

capable of hovering ﬂight.

While the stability analysis of nonlinear time-invariant (NLTI)

systems can be readily performed by checking the eigenvalues

of the linearized dynamics about an equilibrium point, or by

means of Lyapounov theory, stability analysis of NLTP sys-

tems is typically more challenging. This is because the so-

lution (or equilibrium) of NLTP systems, in contrast to NLTI

systems, may be represented by a periodic orbit rather than

by a single point. As articulated in Ref. 10, two main ap-

proaches are typically adopted for determining the stability

of NLTP systems: the ﬁrst based on Floquet theory (Refs.

5,11–13) and the second based on averaging methods (Refs.

1

10,14,15). Speciﬁcally, the ﬁrst approach requires solving the

dynamic equations to ﬁnd the periodic orbit, linearizing the

dynamic equations along the periodic orbit to obtain a linear

time-periodic (LTP) system, and transforming the LTP system

into a linear time-invariant (LTI) system via Floquet transfor-

mation/decomposition. As such, the stability of the system is

assessed by checking the eigenvalues of the LTI system. The

second approach exploits averaging methods to transform the

NLTP system into an equivalent NLTI system in which the pe-

riodic orbit of the original system collapses to a single point

in the state space. The stability of the NLTI system is then as-

sessed via eigenvalue analysis of the linearized NLTI dynam-

ics around equilibrium (Ref. 14). These two main approaches

to NLTP systems stability analysis are shown qualitatively in

Fig. 2.

The dynamics of ﬂapping ﬂight exhibits two time scales: a

fast time scale for the variation of the aerodynamic loads and

a slow time scale for the aggregate body motion. For instance,

while a ﬂying insect’s body motion is perceptible to a human’s

eye, the ﬂapping motion of its wings may not be. If the ratio

of these two time scales is sufﬁciently large, then the aver-

aging approach is particularly convenient because it avoids

direct calculation of the periodic orbit. However, for biologi-

cal ﬂyers where the ﬂapping frequency is relatively low (e.g.,

large insects, birds), the validity of the averaging approach

becomes questionable (Ref. 16,17). On the other hand, the

Floquet theory-based approach does not require a separation

of time scales for the ﬂapping and aggregate motions, which

makes it particularly apt for studying the stability of biologi-

cal ﬂyers of all sizes. Still, a number of challenges arise from

the use of the Floquet theory-based approach, namely: the de-

termination the periodic orbit about which the NLTP system is

linearized and the transformation of the resulting LTP system

into an LTI system as it requires the fundamental matrix of the

LTP system (i.e., solving the LTP system from nindependent

initial conditions).

In this paper, we use a harmonic balance algorithm (Ref. 18)

to construct a higher-order LTI approximation of the LTP

system without computing the fundamental matrix. Histori-

cally, the only methods available for computing higher-order

LTI approximations of LTP systems were the Lyapounov-

Floquet method (Ref. 19) and frequency lifting methods

(Ref. 20), which both suffered from the common disadvan-

tage of the need for state transition matrices. Computation

of state transition matrices can be computationally intensive

and/or numerically very sensitive. Recent advancements in

the rotorcraft community have led to the development of a

numerical method to obtain high-order LTI approximations

of LTP systems that do not rely on state transition matrices

(i.e., the “harmonic decomposition” method of Refs. 21,22).

These LTI reformulations of LTP systems have been used to

study the interference effects between higher-harmonic con-

trol (HHC) and the aircraft ﬂight control system (AFCS)

(Refs. 21,23–25), in the design of load alleviation control

(LAC) laws (Refs. 26,27), and in the prediction and avoid-

ance of ﬂight envelope limits (Refs. 28,29). Harmonic de-

composition not only relieves the computational challenges

Figure 2: Illustration of the two main approaches to stability

analysis of NLTP systems: averaging methods (left), Floquet

theory (right). Adapted from Ref. 10

associated with computing time-invariant approximations of

LTP systems, but also provides a means to compute trim so-

lutions for ﬂight vehicles with NLTP dynamics about a peri-

odic orbit (Ref. 18). Because harmonic decomposition relaxes

all previous limitations associated with the Floquet-based ap-

proach, it can be extended to the study of dynamic stability of

ﬂapping-wing ﬂight.

The objective of this paper is to extend the harmonic decom-

position methodology as a novel dynamic analysis tool for use

in dynamic stability and ﬂight control design of bioinspired

robots. The proposed approach to stability analysis of ﬂap-

ping ﬂight introduces three major innovations when compared

to previous techniques: (i) it does not rely on state transition

matrices; (ii) it leverages approximate higher-order LTI mod-

els; (iii) it can be applied to biological ﬂyers for which the

time scale ratio between fast and slow dynamics is not neces-

sarily large.

The paper begins with a discussion of the mathematical back-

ground behind NLTP systems and an explanation of the pro-

posed periodic trim solution and linearization method. The

second section demonstrates the application of the proposed

methodology to obtain high-order LTI approximations of the

vertical dynamics of a hawk moth. Simulations are used to

2

validate the response of the linearized models against those

from nonlinear simulations. Dynamic stability is assessed and

compared to that predicted using the averaged dynamics. In

addition, modal participation factors are computed to quan-

tify the inﬂuence of the higher harmonics on the ﬂight dy-

namic modes of motion. In the third section, the analysis is

repeated for a more complex model of the longitudinal dy-

namics of the hawk moth. In this section, it is shown how

high-frequency, high-amplitude, periodic forcing has a stabi-

lizing effect on the longitudinal dynamics of the hawk moth

at hover. Final remarks summarize the overall ﬁndings of the

study and identify areas for future work.

METHODOLOGY

Mathematical Background

Consider a nonlinear time-periodic (NLTP) system in ﬁrst-

order form representative of the ﬂight dynamics of a ﬂapping-

wing biological ﬂyer/vehicle:

˙

x

x

x=f

f

f(x

x

x,u

u

u,t)(1a)

y

y

y=g

g

g(x

x

x,u

u

u,t)(1b)

where x

x

x∈Rnis the state vector, u

u

u∈Rmis the control in-

put vector, y

y

y∈Rlis the output vector, and tis the dimen-

sional time in seconds. The nonlinear functions f

f

fand g

g

gare

T-periodic in time such that:

f

f

f(x

x

x,u

u

u,t) = f

f

f(x

x

x,u

u

u,t+T)(2a)

g

g

g(x

x

x,u

u

u,t) = g

g

g(x

x

x,u

u

u,t+T)(2b)

Note that the fundamental period of the system is T=2π

ωsec-

onds, where ωis the ﬂapping frequency in rad/s. Let x

x

x∗(t)

and u

u

u∗(t)represent a periodic solution of the system such that

x

x

x∗(t) = x

x

x∗(t+T)and u

u

u∗(t) = u

u

u∗(t+T). Then, the NLTP sys-

tem can be linearized about the periodic solution. Consider

the case of small disturbances:

x

x

x=x

x

x∗+∆x

∆x

∆x(3a)

u

u

u=u

u

u∗+∆u

∆u

∆u(3b)

where ∆x

∆x

∆xand ∆u

∆u

∆uare the state and control perturbation vectors

from the candidate periodic solution. A Taylor series expan-

sion is performed on the state derivative and output vectors.

Neglecting terms higher than ﬁrst order results in the follow-

ing equations:

f

f

f(x

x

x∗+∆x

∆x

∆x,u

u

u∗+∆u

∆u

∆u,t) = f

f

f(x

x

x∗,u

u

u∗,t) + F

F

F(t)∆x

∆x

∆x+G

G

G(t)∆u

∆u

∆u

(4a)

g

g

g(x

x

x∗+∆x

∆x

∆x,u

u

u∗+∆u

∆u

∆u,t) = g

g

g(x

x

x∗,u

u

u∗,t) + P

P

P(t)∆x

∆x

∆x+Q

Q

Q(t)∆u

∆u

∆u

(4b)

where:

F

F

F(t) = ∂f

f

f(x

x

x,u

u

u)

∂x

x

xx

x

x∗,u

u

u∗,G

G

G(t) = ∂f

f

f(x

x

x,u

u

u)

∂u

u

ux

x

x∗,u

u

u∗(5a-b)

P

P

P(t) = ∂g

g

g(x

x

x,u

u

u)

∂x

x

xx

x

x∗,u

u

u∗,Q

Q

Q(t) = ∂g

g

g(x

x

x,u

u

u)

∂u

u

ux

x

x∗,u

u

u∗(5c-d)

Note that the state-space matrices in Eq. (5) have T-periodic

coefﬁcients. Equations (4a) and (4b) yield a linear time-

periodic (LTP) approximation of the NLTP system of Eq. (1)

as follows:

∆

∆

∆˙

x

x

x=F

F

F(t)∆x

∆x

∆x+G

G

G(t)∆u

∆u

∆u(6a)

∆

∆

∆y

y

y=P

P

P(t)∆x

∆x

∆x+Q

Q

Q(t)∆u

∆u

∆u(6b)

Hereafter, the notation is simpliﬁed by dropping the ∆in

front of the linearized perturbation state and control vectors

while keeping in mind that these vectors represent perturba-

tions from a periodic equilibrium. Next, the state, input, and

output vectors of the LTP systems are decomposed into a ﬁnite

number of harmonics of the fundamental period via Fourier

analysis:

x

x

x=x

x

x0+

N

∑

i=1

x

x

xic cos2πit

T+x

x

xis sin2πit

T(7a)

u

u

u=u

u

u0+

M

∑

j=1

u

u

ujc cos 2πjt

T+u

u

ujs sin 2πjt

T(7b)

y

y

y=y

y

y0+

L

∑

k=1

y

y

ykc cos 2πkt

T+y

y

yks sin 2πkt

T(7c)

As shown in Ref. 21, the harmonic decomposition method-

ology can be used to transform the LTP model into an ap-

proximate higher-order linear time-invariant (LTI) model in

ﬁrst-order form:

˙

X

X

X=A

A

AX

X

X+B

B

BU

U

U(8a)

Y

Y

Y=C

C

CX

X

X+D

D

DU

U

U(8b)

where the augmented state, control, and output vectors are:

X

X

XT=x

x

xT

0x

x

xT

1cx

x

xT

1s... x

x

xT

Nc x

x

xT

Ns (9a)

U

U

UT=u

u

uT

0u

u

uT

1cu

u

uT

1s... u

u

uT

Mc u

u

uT

Ms(9b)

Y

Y

YT=y

y

yT

0y

y

yT

1cy

y

yT

1s... y

y

yT

Lc y

y

yT

Ls(9c)

with A

A

A∈Rn(2N+1)×n(2N+1),B

B

B∈Rn(2N+1)×m(2M+1),C

C

C∈

Rl(2L+1)×n(2N+1), and D

D

D∈Rl(2L+1)×m(2M+1).

Periodic Trim Solution Algorithm

A necessary step towards the approximation of the NLTP dy-

namics of ﬂapping-wing ﬂight with LTP systems is the deter-

mination the periodic orbit about which the NLTP system is

linearized, which involves computing the states and controls

that result in a periodic equilibrium (i.e., trimming a vehi-

cle about a periodic orbit). Assuming that a periodic solution

x

x

x∗(t) = x

x

x∗(t+T)and u

u

u∗(t) = u

u

u∗(t+T)exists for the system

in Eq. (1), then the balance problem is stated as follows: de-

termine x

x

x∗(t)and u

u

u∗(t)such that:

˙

x

x

x∗=f

f

f(x

x

x∗,u

u

u∗,t)(10)

where ˙

x

x

x∗(t) = ˙

x

x

x∗(t+T). In other words, the balance problem

consists of determining the periodic state and control vectors

such that the system dynamics are periodic.

3

Several solutions exist for trimming a vehicle about a peri-

odic orbit, namely: averaged approximate trim, time march-

ing trim, autopilot trim (Ref. 30), periodic shooting (see, e.g.,

Ref. 31), harmonic balance (see, e.g., Ref. 32), and modi-

ﬁed harmonic balance (Ref. 18). In this study, the latter is

used as it incorporates three major advantages when com-

pared to other techniques: it is based on harmonic decom-

position and thus does not rely on state transition matrices, it

simultaneously solves for the approximate higher-order LTI

dynamics about the periodic solution, and it can be used to

compute the higher-harmonic control inputs that attenuate ar-

bitrary state harmonics. In addition, modiﬁed harmonic bal-

ance can calculate unstable periodic orbits, whereas methods

such as averaged approximate trim or time marching trim can-

not (Ref. 18). Because the mechanism behind dynamic stabil-

ity in ﬂapping-wing ﬂight is still not completely understood, it

is important to select a periodic trim solution method capable

of solving for unstable periodic orbits.

The modiﬁed harmonic balance algorithm begins with as-

suming that the fundamental period Tof the nonlinear time-

periodic system is known. Note that this solution strategy is

iterative in nature, in that a candidate solution is reﬁned over

a series of computational steps until a convergence criteria is

reached. Consider the candidate periodic solution at iteration

kof the algorithm: x

x

x∗

k(t)and u

u

u∗

k(t). One iteration of the algo-

rithm begins with approximating the candidate periodic solu-

tion using a Fourier series with a ﬁnite number of harmonics:

x

x

x∗

k=x

x

x∗

k0+

N

∑

i=1

x

x

x∗

kic cos2πit

T+x

x

x∗

kis sin2πit

T(11a)

u

u

u∗

k=u

u

u∗

k0+

M

∑

j=1

u

u

u∗

kjc cos 2πjt

T+u

u

u∗

kjs sin 2πjt

T(11b)

As such, the candidate periodic solution is re-written in terms

of its respective Fourier coefﬁcients:

X

X

X∗T

k=x

x

x∗T

k0x

x

x∗T

k1cx

x

x∗T

k1s... x

x

x∗T

kNc x

x

x∗T

kNs (12a)

U

U

U∗T

k=u

u

u∗T

k0u

u

u∗T

k1cu

u

u∗T

k1s... u

u

u∗T

kNc u

u

u∗T

kNs (12b)

Since the balance problem simultaneously solves for the pe-

riodic solution and the necessary control inputs that ensure it,

the harmonic realization of the candidate periodic solution of

Eq. (12a) is augmented with the harmonic realization of the

candidate control inputs of Eq. (12b) to form the vector of

unknowns at iteration k:

Θ

Θ

ΘT

k=hX

X

X∗T

kU

U

U∗T

ki(13)

where Θ

Θ

Θk∈Rn(2N+1)+m(2M+1).

Next, the state derivative vector calculated along the candidate

periodic solution over a single periodic orbit is decomposed

into a ﬁnite number of harmonics via Fourier analysis:

˙

x

x

x∗

k=˙

x

x

x∗

k0+

N

∑

i=1

˙

x

x

x∗

kic cos2πit

T+˙

x

x

x∗

kis sin2πit

T(14)

Note that the number of state derivative harmonics that are

retained in Eq. (14) is equal to the number of state harmonics

retained in Eq. (11a) (i.e.,N). Consider differentiating the

candidate periodic solution of Eq. (11a):

˙

x

x

x∗

k=d

dt x

x

x∗

k0

| {z }

˙

x

x

x∗

k0

+

N

∑

i=1d

dt x

x

x∗

kic +2πi

Tx

x

x∗

kis

| {z }

˙

x

x

x∗

kic

cos2πit

T

+d

dt x

x

x∗

kis −2πi

Tx

x

x∗

kic

| {z }

˙

x

x

x∗

kis

sin2πit

T(15)

Since at equilibrium the Fourier coefﬁcients of the system dy-

namics are constant (i.e., their time derivative is zero), the fol-

lowing integral relations are true:

˙

x

x

x∗

0=0

0

0 (16a)

˙

x

x

x∗

ic =2πi

Tx

x

x∗

is (16b)

˙

x

x

x∗

is =−2πi

Tx

x

x∗

ic (16c)

A total of n(2N+1)constraints are formed by requiring that

the state derivative Fourier coefﬁcients in Eq. (14) and the

state Fourier coefﬁcients in Eq. (11a) satisfy the integral re-

lations in Eq. (16). This leads to the deﬁnition of the error

vector at the iteration kas:

e

e

eT

k=W

W

W"˙

x

x

x∗

k0T˙

x

x

x∗

kic −2πi

Tx

x

x∗

kis T˙

x

x

x∗

kis +2πi

Tx

x

x∗

kic T#

(17)

where e

e

ek∈Rn(2N+1)and W

W

W∈Rn(2N+1)×n(2N+1)is a diagonal

scaling matrix to make all elements of the error vector ap-

proximately the same order of magnitude (e.g., 1 deg error is

equivalent to 1 ft error).

Next, the NLTP system is linearized at incremental time steps

along the candidate periodic solution, yielding the following

LTP system:

∆

∆

∆˙

x

x

x=F

F

Fk(t)∆x

∆x

∆x+G

G

Gk(t)∆u

∆u

∆u(18)

Hereafter, the notation is simpliﬁed by dropping the ∆in front

of the linearized perturbation state and control vectors while

keeping in mind that these vectors represent perturbations

from a periodic equilibrium. Next, the state and output vec-

tors of the LTP systems are decomposed into a ﬁnite number

of harmonics via Fourier analysis:

x

x

x=x

x

x0+

N

∑

i=1

x

x

xic cos2πit

T+x

x

xis sin2πit

T(19a)

u

u

u=u

u

u0+

M

∑

j=1

u

u

ujc cos 2πjt

T+u

u

ujs sin 2πjt

T(19b)

Note that the number of state harmonics retained in Eq. (19a)

is the same as in Eqs. (11a) and (14) (i.e.,N), whereas the

number of control input harmonics retained in Eq. (19b) is the

same as in Eq. (11b). As shown in Ref. 21, the LTP model can

4

be approximated by a higher-order linear time-invariant (LTI)

model in ﬁrst-order form through the harmonic decomposition

methodology:

˙

X

X

X=A

A

AkX

X

X+B

B

BkU

U

U(20)

where the augmented state and control vectors are:

X

X

XT=x

x

xT

0x

x

xT

1cx

x

xT

1s... x

x

xT

Nc x

x

xT

Ns (21a)

U

U

UT=u

u

uT

0u

u

uT

1cu

u

uT

1s... u

u

uT

Mc u

u

uT

Ms(21b)

and where A

A

Ak∈Rn(2N+1)×n(2N+1)and B

B

Bk∈

Rn(2N+1)×m(2M+1)are the LTI system and control ma-

trices. Closed-form expressions can be found in Ref. 21. It is

worth noting that harmonic decomposition does not rely on

state transition matrices, which makes the methodology more

computationally efﬁcient and less numerically sensitive than

other approaches such as the Lyapounov-Floquet method

(Ref. 19) and frequency lifting methods (Ref. 20).

The LTI system coefﬁcient matrices are used to deﬁne the Ja-

cobian matrix of the harmonic balancing algorithm at iteration

k:

J

J

Jk= [A

A

AkB

B

Bk](22)

where J

J

Jk∈Rn(2N+1)×[n(2N+1)+m(2M+1)]. The Jacobian ma-

trix is used in each algorithm iteration to compute a candidate

periodic solution and controls update (i.e., the vector of un-

knowns) given the error vector at that iteration via a Newton-

Rhapson scheme (Ref. 33). It is clear that the Jacobian matrix

is not square because the number of constraints in Eq. (16)

is less than the number of unknowns in Eq. (13). In fact,

the number of constraints is n(2N+1)whereas the number of

unknowns is n(2N+1) + m(2M+1). This leads to an under-

determined problem which does not have a unique solution.

To make the problem square such that the solution is unique,

m(2M+1)conditions have to be speciﬁed. These are the trim

conditions. Because typical periodically-forced ﬂight vehicles

only utilize control input bandwidths signiﬁcantly lower than

the forcing frequency, one can safely assume the control input

harmonics higher than the zeroth harmonic to be zero. This

corresponds to imposing 2Mm conditions, which reduces the

number of unknowns to n(2N+1) + m. It follows that mcon-

ditions still need to be speciﬁed. Note that if the minputs are

given and the corresponding equilibrium solution is required,

then the problem is fully-determined. On the other hand, in

the case where one or more (possibly all) of the mcontrol

inputs is unknown, then each input is used to ensure some de-

sired condition (e.g., trim equation). For periodically-forced

aerospace vehicles for which the vehicle dynamics are invari-

ant with respect to position and heading (Ref. 34), the zeroth

harmonic of the position and heading can be arbitrarily as-

signed and removed from the vector of unknowns. Since these

vehicles typically employ control about four axes (i.e., roll,

pitch, yaw, and heave) leading to four control inputs, ﬁxing

the three components of the zeroth harmonic of the position

(x0,y0,z0) and heading (ψ0) at equilibrium leads to a square

problem. Hence, Newton Rhapson is used to ﬁnd a candidate

periodic solution update (in harmonic form) according to:

ˆ

Θ

Θ

Θk+1=ˆ

Θ

Θ

Θk−ˆ

J

J

J−1

ke

e

ek(23)

where ˆ

Θ

Θ

Θkand ˆ

J

J

Jkare the vector of unknowns and the Jaco-

bian matrix without the unknowns that were ﬁxed, respec-

tively. As a ﬁnal step, the new candidate periodic solution

is reconstructed in the time domain:

x

x

x∗

k+1=x

x

x∗

k+10+

N

∑

i=1

x

x

x∗

k+1ic cos2πit

T+x

x

x∗

k+1is sin2πit

T

(24a)

u

u

u∗

k+1=u

u

u∗

k0+

M

∑

j=1

u

u

u∗

k+1jc cos 2πjt

T+u

u

u∗

k+1js sin 2πjt

T

(24b)

The next iteration of the algorithm then proceeds with this

new candidate solution, starting from Eqs. (11a) and (11b).

The algorithm is stopped when kekk∞becomes less than an

arbitrary tolerance. It is worth noting that the algorithm re-

quires a ﬁrst guess of the periodic solution over one periodic

orbit (although numerical results in Ref. 18 show that conver-

gence rate is usually fairly insensitive to the initial guess). A

ﬂowchart of the algorithm is shown in Fig. 3.

An added beneﬁt of the algorithm is that, to update the solu-

tion, a higher-order LTI approximation of the NLTP system

is computed at each iteration along the candidate periodic so-

lution. Thus, the algorithm not only solves for the periodic

solution of NLTP systems, but also simultaneously constructs

a higher-order LTI approximation of the NLTP system. The

higher-order LTI system can readily be used for stability anal-

ysis or feedback control design.

Modal Participation Factors

Modal participation factors are a useful tool for quantifying

the inﬂuence of higher harmonics on the dynamics of time-

periodic systems. Modal participation factors describe the

modal participation of each state to each mode through the

relative magnitude of the harmonic components of each state.

Recently, it has been shown that modal participation factors

can be computed directly from harmonic decomposition mod-

els through the following procedure (Ref. 35):

1. Solve for the eigenvalues and eigenvectors of the high-

order LTI system matrix A

A

Ain Eq. (8a). The eigenvector

corresponding to the kth mode will be in the form:

X

X

XT

k=x

x

xT

k,0x

x

xT

k,1cx

x

xT

k,1s... x

x

xT

k,Nc x

x

xT

k,Ns (25)

where x

x

xk,0is the zeroth harmonic component, and x

x

xk,nc

and x

x

xk,ms are respectively the nth cosine and sine compo-

nents of the periodic eigenvector corresponding to the kth

mode. The eigenvalues of the A

A

Amatrix are equivalent to

the Floquet exponents of the LTP system.

2. Convert the LTI eigenvector harmonic states from

real-trigonometric Fourier coefﬁcients to complex-

5

Figure 3: Periodic trim solution algorithm ﬂowchart.

exponential Fourier coefﬁcients as follows:

cj,k,0=xj,k,0(26a)

cj,k,+n=xj,k,nc −ixj,k,ns

2(26b)

cj,k,−n=xj,k,nc +ixj,k,ns

2(26c)

where xj,k,0,xj,k,nc, and xj,k,ns are respectively the zeroth,

nth cosine, and nth sine real-trigonometric harmonic LTI

eigenvector elements corresponding to the jt h LTP sys-

tem state and kth mode.

3. Compute the modal participation factors by normalizing

the modal participation with respect to the sum of the

magnitudes of all harmonic components for each partic-

ular state and mode:

φj,k,n=|cj,k,n| N

∑

i=−N

|cj,k,i|!−1

(27)

where Nis the number of state harmonics retained when

performing harmonic decomposition.

It is worth noting that those LTI system modes corresponding

to eigenvalues with imaginary parts between ±ω/2 are re-

ferred to as the base modes. Only the base modes are needed

to completely describe the system with the understanding that

higher-frequency modes simply shift the naming of harmonics

and do not affect the actual modal participation content.

Residualization

Because the measurement or estimation of the higher-

harmonic states is usually challenging or impractical,

reduced-order models can be obtained from high-order LTI

models. Ideally, these reduced-order models do not include

the higher harmonic states but still retain part of the higher-

harmonic response characteristics. This can be achieved

through residualization, a portion of singular perturbation the-

ory that pertains to LTI systems (Ref. 36). Assuming that one

or more states of the system have stable dynamics which are

faster than that of the remaining states, the state vector in Eq.

(9a) can be partitioned into fast and slow components:

X

X

XT=X

X

XT

sX

X

XT

f(28)

Then, the system in Eq. (8a) can be re-written as:

˙

X

X

Xs

˙

X

X

Xf=A

A

AsA

A

Asf

A

A

Afs A

A

AfX

X

Xs

X

X

Xf+B

B

Bs

B

B

BfU

U

U(29)

By neglecting the dynamics of the fast states (i.e.,X

X

Xf=0) and

performing a few algebraic manipulations, the equations for

a reduced-order system with the state vector composed of the

slow states may be found:

˙

X

X

Xs=ˆ

A

A

AX

X

Xs+ˆ

B

B

BU

U

U(30)

where:

ˆ

A

A

A=A

A

As−A

A

AsfA

A

Af−1A

A

Afs (31a)

ˆ

B

B

B=B

B

Bs−A

A

AsfA

A

Af−1B

B

Bf(31b)

In this study, the slow states are chosen as the zeroth harmonic

states, whereas the fast states are taken as the higher harmon-

ics:

X

X

Xs=x

x

x0(32a)

X

X

XT

f=x

x

xT

1cx

x

xT

1s... x

x

xT

Nc x

x

xT

Ns (32b)

6

VERTICAL DYNAMICS

Simulation Model

Consider the NLTP vertical dynamics of a hovering ﬂapping-

wing micro-aerial-vehicle (FWMAV) derived in Ref. 15:

˙z

˙

φ

˙w

¨

φ

=

w

˙

φ

g−kd1|˙

φ|w−kL˙

φ2

−kd2|˙

φ|˙

φ−kd3w˙

φ

+

0

0

0

1

IFcos(ωt)

U(33)

where zis the vertical position, φis the wing ﬂapping angle,

wis the vertical speed, gis the gravitational acceleration, and

kd1,kd2,kd3, and kLare constant parameters. Additionally, IF

is the ﬂapping moment of inertia, ωis the ﬂapping frequency,

and Uis the amplitude of the ﬂapping control input torque.

The state vector is x

x

xT=zφw˙

φand the control vector is

u

u

u=U. It follows that the state vector has dimension n=4

and the control vector has dimension m=1. The numerical

values of the system parameters, which are given for a hawk

moth in Ref. 15 and are reported in Table 1.

Table 1: Vertical dynamics parameters of a FWMAV

representative of a hawk moth.

Parameter Numerical Value Units

kd10.0353739 -

kd20.333915 -

kd316.5766 1/m

kL0.000621676 m

IF0.0353739 kg-m2

ω165.2478 rad/s

g9.80665 m/s2

Periodic Trim

An approximate periodic solution at hover is found by time

marching the system over one periodic orbit (T=2π/ω) us-

ing the approximate equilibrium control input and initial wing

ﬂapping angles suggested in Ref. 15. The approximate peri-

odic solution shown in Fig. 4(dashed line) provides insight

into the number of harmonics to be retained when applying

the numerical method described in the previous section. As

shown in Fig. 4, the vertical position zand vertical veloc-

ity wvary 2-times-per-revolution (or 2/rev), whereas the ﬂap-

ping angle φand the ﬂapping angular speed ˙

φvary 1-time-

per-revolution (or 1/rev). In light of these observations, it is

decided to retain up to the second state harmonic when ap-

proximating the NLTP dynamics with a higher-order LTI sys-

tem (i.e.,N=2). Since the periodicity of the control input

is incorporated in the system dynamics, only the zeroth har-

monic is retained for the control input (i.e.,M=0) such that

U=U0. It is worth noting that the zeroth harmonic of the

control input in this case corresponds to the ﬁrst (cosine) har-

monic of the ﬂapping torque. It follows that the vector of

n(2N+1) + m(2M+1) = 21 unknowns is:

Θ

Θ

ΘT=z0φ0w0˙

φ0z1cφ1cw1c˙

φ1cz1sφ1sw1s˙

φ1s

z2cφ2cw2c˙

φ2cz2sφ2sw2s˙

φ2sU0(34)

whereas the n(2N+1) = 20 constraints are:

˙z0˙

φ0˙w0¨

φ0T=0

0

0 (35a)

˙z1c˙

φ1c˙w1c¨

φ1cT−ωz1sφ1sw1s˙

φ1sT=0

0

0 (35b)

˙z1s˙

φ1s˙w1s¨

φ1sT+ωz1cφ1cw1c˙

φ1cT=0

0

0 (35c)

˙z2c˙

φ2c˙w2c¨

φ2cT−2ωz2sφ2sw2s˙

φ2sT=0

0

0 (35d)

˙z2s˙

φ2s˙w2s¨

φ2sT+2ωz2cφ2cw2c˙

φ2cT=0

0

0 (35e)

Because there are 21 unknowns and only 20 constraints, one

unknown must be ﬁxed and removed from the problem. In

this case, the zeroth harmonic of the vertical position can be

set to an arbitrary value as it does not affect the dynamics of

the FWMAV. In this example, the desired zeroth harmonic of

the vertical position is set to zero (i.e.,z0=0). This way,

the number of unknowns is now 20 such that the problem is

square.

In practice, the candidate periodic solution is discretized at

ntevenly-spaced incremental time steps over one periodic or-

bit. In this example, the number of time steps is chosen as

nt=360 such that the simulation time step is dt =T/nt. The

ﬁrst guess for the state vector is chosen as the approximate pe-

riodic solution found via time marching, whereas the control

input is initialized to:

u

u

u∗

0(t) = 1.058s2gI2

Fω2

kL

(36)

as obtained in Ref. 15 using a third-order averaging scheme.

As such, modiﬁed harmonic balance is used to reﬁne the ap-

proximate periodic solution and to obtain the higher-order LTI

approximate dynamics about that solution. Figure 4shows

the numerical solution obtained with the algorithm of Ref. 18

(dashed line) using an error tolerance of 1e−7. The numeri-

cal periodic solution is shown to enhance the approximate pe-

riodic solution obtained via time marching. The equilibrium

input thus obtained is:

u

u

u∗(t) = 1.0468s2gI2

Fω2

kL

(37)

Linearized Dynamics

To validate the approximate LTI dynamics obtained with the

modiﬁed harmonic balance algorithm, the response of the

high-order LTI system is compared with that of the NLTP dy-

namics following a doublet in the control input. As shown in

Fig. 5, the higher-order LTI response closely matches that of

the NLTP dynamics. This result suggests that the NLTP verti-

cal dynamics of a FWMAV can successfully be approximated

by a higher-order LTI system.

7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1

0

1

z [m]

10-4 Initial Guess Solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-50

0

50

[deg]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.02

0

0.02

w [m/s]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Nondimensioanl time, t/T

-1

0

1

104

Figure 4: Comparison between the numerical solution and

initial guess of the periodic motion of a ﬂapping-wing micro

aerial vehicle.

Figure 5: Response of the NLTP vertical dynamics of a

FWMAV compared to that of its higher-order LTI

approximation following a control input doublet.

The base eigenvalues of the high-order LTI model are com-

pared with those of the averaged dynamics. The averaged dy-

namics is found by retaining only the zeroth state harmonic

when performing harmonic decomposition. As such, the av-

erage dynamics is given by a 4-state system and does not con-

tain any information about the higher-harmonics of the origi-

nal system. The eigenvalues of these systems are shown qual-

itatively in Fig. 6and quantitatively in Table 2. As shown in

these results, the averaged dynamics under-predict the base

eigenvalue representing the ﬂap mode and over-predict the

eigenvalue representing the heave subsidence mode. How-

ever, the mismatch between the base eigenvalues of the high-

order LTI and the eigenvalues of the averaged dynamics is

fairly small. Based on spectral analysis only, it is not pos-

sible to reach any conclusion on the suitability of the aver-

Table 2: Vertical dynamics base eigenvalues.

LTI System λ1λ2λ3λ4

High-Order -75.93 -4.00 0.00 0.00

Averaged -75.00 -3.53 0.00 0.00

aged dynamics for describing the overall vertical dynamics

of ﬂapping-wing ﬂight. This necessitates a comparison of the

dynamics of the two models using modal participation factors.

-80 -70 -60 -50 -40 -30 -20 -10 0 10

Real

-400

-300

-200

-100

0

100

200

300

400

Imag

High-Order LTI Averaged LTI

Heave Subsidence

Integrators

Flap Mode

Figure 6: Comparison between the eigenvalues of the

high-order approximate LTI dynamics and the averaged LTI

dynamics.

Modal Participation Factors

The modal participation factors are computed for the high-

order LTI system with the states corresponding to vertical po-

sition, ﬂapping angle, and their harmonics removed. This

is done to simplify the analysis as the vertical position and

ﬂapping angle are simply described by integral relationships

and thus do not affect the dynamics of the system. Figure 7

shows the modal participation factors for the vertical speed

and ﬂapping speed states to the ﬂap and heave modes. Specif-

ically, Fig. 7a shows that the vertical speed contributes to the

ﬂap mode almost entirely with its ﬁrst harmonic, whereas it

contributes to the heave mode almost exclusively through its

zeroth harmonic. Figure 7b suggests that the ﬂapping speed

contributes to the ﬂap mode about 86% through its zeroth har-

monic, and the remaining 14% through its second harmonic.

In addition, this ﬁgure shows that the ﬂapping speed con-

tributes to the heave mode solely through its ﬁrst harmonic.

This analysis suggests that the vertical and ﬂapping speed

states contribute to the overall vertical dynamics of the

ﬂapping-wing MAV signiﬁcantly through their higher har-

monics. Thus, it is necessary to include higher-harmonic

states in the LTI approximations of the NLTP vertical ﬂight

dynamics for these approximations to be accurate. As such,

8

the averaged dynamics alone are not suitable for describing

the NLTP vertical dynamics.

2s 1s 0 1c 2c

Harmonics [N/rev]

0

20

40

60

80

100

Modal Participation Facrors [%]

Flap Mode

Heave Mode

(a) Vertical speed.

2s 1s 0 1c 2c

Harmonics [N/rev]

0

20

40

60

80

100

Modal Participation Facrors [%]

Flap Mode

Heave Mode

(b) Flapping speed.

Figure 7: Modal participation factors for the vertical

dynamics of a hawk moth at hover.

LONGITUDINAL DYNAMICS

Simulation Model

Consider now a more complex NLTP model representing the

longitudinal dynamics of a FWMAV from Ref. 16:

˙x

˙z

˙u

˙w

˙q

˙

θ

=

ucosθ+wsinθ

−usinθ+wcosθ

−qw −gsinθ

qu +gcosθ

0

q

+

0

0

X0

Y0

M0

0

+

0 0 0 0 0 0

0 0 0 0 0 0

0 0 XuXwXq0

0 0 YuYwYq0

0 0 MuMwMq0

0 0 0 0 0 0

x

z

u

w

q

θ

(38)

were xand zare the longitudinal and vertical position in the

inertial frame, uand ware the longitudinal and vertical veloc-

ities in the body-ﬁxed frame, qis the pitch rate, and θis the

pitch attitude. Assuming a horizontal stroke plane, the forces

and moments that are independent of the system’s states are

parametrized by the back-and-forth ﬂapping motion φ(t)and

a piecewise constant variation in the wing pitch angle η(t):

X0=−2K21

m

˙

φ|˙

φ|cosφsin2η(39a)

Z0=−K21

m

˙

φ|˙

φ|sin2η(39b)

M0=2˙

φ|˙

φ|K22

Iy

∆ˆxcos φ+K21

Iy

xhcosη+K31

Iy

sinφcosη

(39c)

where xhis the distance between the vehicle center of gravity

and the root of the wing hinge line (see Fig. 8) and ∆ˆxis

the chordwise distance between the center of pressure and the

root of the wing hinge line. Additionally, Kmn =1

4ρCLαImn

where ρis air density, CLαis the three-dimensional lift curve

slope of the wing, and Imn are the moments of the wing chord

distribution. The lift curve slope of the wing is given by:

CLα=πAR

2"1+rπAR

a02+1#(40)

where AR is the wing aspect ratio and a0is the lift-curve slope

of the airfoil section (Ref. 37). By deﬁnition, the aspect ratio

is given by the ratio between the wing surface and the mean

chord S/¯c. The moments of the wing chord distribution are

given by:

Imn =ZR

0

rmcn(r)dr (41)

where Ris the wing radius and c(r)is the chord distribution.

The chord distribution is given by:

c(r) = ¯c

βr

Rα−11−r

Rγ−1

(42)

9

where:

α=ˆr1ˆr1(1−ˆr1)

ˆr2

2−ˆr2

1

−1(43)

γ= (1−ˆr1)ˆr1(1−ˆr1)

ˆr2

2−ˆr2

1

−1(44)

β=Z1

0

ˆrα−1(1−ˆr)γ−1dˆr(45)

The time-varying stability derivatives are given by:

Xu=−4K11

m|˙

φ|cos2φsin2η(46a)

Xw=−K11

m|˙

φ|cosφsin2η(46b)

Xq=K21

m|˙

φ|sinφcosφsin2η−xhXw(46c)

Zu=2Xw(46d)

Zw=−2K11

m|˙

φ|cos2η(46e)

Zq=2K21

m|˙

φ|sinφcos2η−Krot12

m

˙

φcosφ−xhZw(46f)

Mu=4K12∆ˆx

Iy

|˙

φ|cos2φsinη+m

Iy

(2Xq−xhZu)(46g)

Mw=2K12∆ˆx

Iy

|˙

φ|cosφcosη+2K21

m|˙

φ|sinφcos2η−mxh

Iy

Zw

(46h)

Mq=−2∆ˆx

Iy

|˙

φ|cosφcosη(K12xh+K22 sinφ)

+1

Iy

˙

φcosφKrot13 ∆ˆxcos φcos η+Krot22 sinφ

−2

Iy

|˙

φ|cos2ηsinφ(K21xh+K31 sinφ)

−Kvµ1ω

2πIy

cos2φ−mxh

Iy

Zq(46i)

where Krot =πρ 1

2−∆ˆxImn,Kv=π

16 ρI04, and µ1depends

on the viscosity of the ﬂuid.

As suggested in Ref. 16, a triangular waveform is used for the

ﬂapping motion:

φ(t) =

Φ0+4Φ

Tt−T

40≤t<T

2

Φ0−4Φ

Tt−3T

4T

2≤t<T

(47)

where Φ0is an offset angle and Φis the amplitude of the ﬂap-

ping motion. The wing pitching motion is assumed piecewise

constant and is given by:

η(t) =

αd0≤t<T

2

π−αu

T

2t≤<T(48)

where αdand αuare, respectively, the downstroke and up-

stroke angles of attack. The base parameters used in this

analysis are reported in Table 3. The state vector is x

x

xT=

[xzuwqθ]and the control vector is chosen as u

u

uT= [Φαm].

As such, the state vector has dimension n=6 and the control

vector has dimension m=2.

Figure 8: Schematic diagram of a hovering FWMAV

(recreated from Ref. 16).

Table 3: Longitudinal dynamics parameters of a FWMAV

representative of a hawk moth.

Parameter Numerical Value Units

R5.19×10−4m

¯c1.83×10−4m

S947.8×10−6m2

a02π1/rad

m1.648×10−6kg

rh0 m

∆ˆx0.05 -

ˆr10.44 -

ˆr20.525 -

Iy2.08×10−7kg-m2

µ10.2 -

ω165.2478 rad/s

g9.80665 m/s2

Periodic Trim

A periodic solution is sought for the case where the hinge line

is aligned with the center of mass (i.e.,xh=0) and the ﬂap-

ping motion is symmetric (i.e.,Φ=0 and αd=αu=αm).

As a ﬁrst guess to the modiﬁed harmonic balance algorithm,

the controls are chosen as Φ=60.5 deg and αm=40 deg,

whereas the states are initialized to a constant value of zero

10

across the fundamental period. The control values are taken

from Ref. 16. As was done for the vertical dynamics, the

state and control harmonics retained in the modiﬁed harmonic

balance algorithm are, respectively, up to the second and the

zeroth order (i.e.,N=2 and M=0). It follows that the vector

of n(2N+1) + m(2M+1) = 32 unknowns is:

Θ

Θ

ΘT=x

x

xT

0x

x

xT

1cx

x

xT

1sx

x

xT

2cx

x

xT

2su

u

uT

0(49)

The n(2N+1) = 30 constraints are given by Eq. (16). Note

that, if trim in forward ﬂight was sought rather than at hover,

the zeroth harmonic of the derivative of the longitudinal posi-

tion state xwould be set to the desired forward speed. Because

there are m(2M+1) = 2 unknowns more than there are con-

straints, the zeroth harmonics of the position states xand z,

denoted as x0and y0, are removed from the problem and set

to arbitrary values. This choice is justiﬁed by the fact that the

zeroth harmonic of the position do not affect the dynamics of

the FWMAV. This way, the number of unknowns decreases to

30 such that the problem is square. The modiﬁed vector of

unknowns is denoted as,

ˆ

Θ

Θ

ΘT=ˆ

x

x

xT

0x

x

xT

1cx

x

xT

1sx

x

xT

2cx

x

xT

2su

u

uT

0(50)

where ˆ

x

x

x0is the zeroth-harmonic state vector without the posi-

tion states included. In this example, the number of time steps

is chosen as nt=360. Figure 9shows the periodic angular

rates obtained with the proposed algorithm using an error tol-

erance of 1e−7. The periodic equilibrium obtained is similar

to that shown in Ref. 38 in the harmonic content of each state

and in the state oscillation magnitudes. However, some dif-

ferences in the shape of the periodic orbit, especially for the

pitch rate and longitudinal speed, are evident. These differ-

ences are likely caused by the fact that the periodic equilib-

rium shown in Ref. 38 is obtained using a higher-order model

that includes the ﬂapping dynamics. Nonetheless, the agree-

ment between the periodic equilibria computed here and that

found in Ref. 38 is quite favorable. The trim control inputs

relative to the periodic equilibrium in Fig. 9are Φ=71.83

deg and αm=47.95 deg. This trim ﬂapping amplitude is close

to the observed ﬂapping amplitude for hawk moths, which is

60.5 deg (Ref. 16).

Linearized Models

To validate the approximate LTI dynamics obtained with the

modiﬁed harmonic balance algorithm, the response of the

high-order LTI system is compared to that of the NLTP dy-

namics following a doublet in the ﬂapping amplitude. As

shown in Fig. 10, the LTI response matches closely that of the

NLTP dynamics, especially for the ﬁrst second of simulation.

This result indicates that the NLTP longitudinal dynamics of a

FWMAV can successfully be approximated by a higher-order

LTI system.

The base eigenvalues of the high-order LTI model are com-

pared with the eigenvalues of the averaged and residualized

dynamics. The residualized dynamics are obtained by ne-

glecting the dynamics of the higher harmonic states, thus

0 0.2 0.4 0.6 0.8 1

-5

0

5

x [m]

10-4

0 0.2 0.4 0.6 0.8 1

-10

-5

0

z [m]

10-6

0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1

u [m/s]

0 0.2 0.4 0.6 0.8 1

0

2

4

w [m/s]

10-3

0 0.2 0.4 0.6 0.8 1

Nondimensioanl time, t/T

-500

0

500

q [deg/s]

0 0.2 0.4 0.6 0.8 1

Nondimensioanl time, t/T

-2

0

2

[deg]

Figure 9: Periodic orbit for the longitudinal dynamics of a

hawk moth at hover.

Figure 10: Response of the NLTP longitudinal dynamics of

the hawk moth compared to that of its higher-order LTI

approximation following a ﬂapping amplitude doublet.

yielding a 6-state system via singular perturbation theory. On

the other hand, the averaged dynamics are found by trun-

cating the higher-harmonic states while retaining the zeroth-

harmonic states. It follows that the average dynamics will be

a 6-state system as well. The eigenvalues of these systems are

shown qualitatively in Fig. 11 and quantitatively in Table 4.

11

Table 4: Longitudinal dynamics base eigenvalues.

LTI System λ1λ2λ3,4

High-Order -7.25 -3.14 1.16 ±2.22i

Residualized -7.32 -3.15 −1.13 ±2.22i

Averaged -10.68 -3.28 0.62 ±5.93i

Note that the longitudinal and vertical position integrators are

omitted from the table. It is observed that the eigenvalues of

the residualized dynamics match the base eignenvalues of the

high-order LTI. Notably, both set of eigenvalues predict a sta-

ble pitch oscillation mode, which is a result that was recently

observed in the literature (Ref. 16). Conversely, in addition

to largely under-predicting the eigenvalue for the pitch subsi-

dence mode, the averaged dynamics predicts an unstable pitch

oscillation mode. As such, these results indicate that the av-

eraged dynamics is not sufﬁcient to fully describe the longi-

tudinal dynamics of hovering ﬂapping-wing ﬂyers. Addition-

ally, the analysis suggests that the higher harmonics induce a

mechanism that stabilizes the dynamics at hover.

-12 -10 -8 -6 -4 -2 0 2

Real

-6

-4

-2

0

2

4

6

Imag

High-Order LTI Residualized LTI Averaged LTI

Heave Subsidence

Integrators

Pitch Subsidence

Pitch Oscillation

Figure 11: Comparison between the eigenvalues of the

high-order LTI, residualized, and averaged longitudinal

dynamics.

To investigate in more detail this vibrational stabilization

mechanism, consider the following form for the system matrix

for the averaged and residualized dynamics with the position

states removed:

A

A

A=

Xu0 0 −g

0Zw0 0

Mu0MqMθ

0 0 1 0

(51)

The numerical values of the stability derivatives derived from

Eq. (51) corresponding to the longitudinal hovering cubic and

heave dynamics are reported in Table 5for the averaged and

residualized systems. As was observed in Ref. 16, due to the

high-amplitude, high-frequency, periodic forcing, the residu-

Table 5: Stability derivatives for the longitudinal averaged

and residualized dynamics.

Derivative Averaged Residualized Units

Xu-4.46 -4.44 kg/s

Zw-3.29 -3.15 kg/s

Mu38.74 4.66 kg-m/s

Mq-4.98 -5.14 kg-m2/s

Mθ0 0.03 kg-m2/s2

alized system gains pitch damping (Mq) and some pitch stiff-

ness (Mθ) when compared to the average model. On the other

hand, the higher harmonics cause a signiﬁcant reduction in

speed stability (Mu) and a slight reduction in the longitudinal

and heave damping (Xuand Zw). Based on this analysis, the

higher harmonics induce a stabilization mechanism that in-

creases the pitch damping and stiffness while reducing speed

stability. This results in stabilization of the pitch oscillation

mode and in a pitch subsidence mode with a lower frequency,

which overall yields a stable hovering cubic. The heave dy-

namics remain largely unaffected. This results in the hypoth-

esis that periodic forcing at a high enough frequency and am-

plitude may stablilize the ﬂight modes of a hovering vehicle.

Modal Participation Factors

The modal participation factors are computed for the high-

order LTI system with the states corresponding to vertical po-

sition, longitudinal position, and their harmonics removed.

This is done to simplify the analysis as the longitudinal and

vertical position are simply described by integral relationships

and thus do not affect the dynamics of the system. Figure 12

shows the modal participation factors for the longitudinal dy-

namics of a hawk moth in hover. The following observations

can be made:

1. Figure 12a shows that the longitudinal speed contributes

to the heave subsidence mode exclusively through its ﬁrst

harmonic, whereas it contributes to the pitch oscillation

and pitch subsidence modes solely through its zeroth har-

monic.

2. Figure 12b suggests that the vertical speed contributes to

the heave subsidence mode exclusively through its zeroth

harmonic, whereas it contributes to the pitch oscillation

and pitch subsidence modes solely through its ﬁrst har-

monic.

3. Figure 12c indicates that the pitch rate contributes to the

heave subsidence mode solely through its ﬁrst harmonic.

On the other hand, it contributes to the pitch oscillation

mode roughly 45% through its zeroth harmonic, and the

remaining 55% through its second harmonic. The oppo-

site is true for the pitch subsidence mode, to which the

pitch rate contributes 55% through its zeroth harmonic,

and 45% through its second harmonic.

4. Figure 12d shows how the pitch attitude contributes to

the heave subsidence mode almost entirely with its ﬁrst

12

harmonic, while it contributes to the pitch oscillation and

pitch subsidence modes solely through its zeroth har-

monic.

As for the vertical dynamics, the modal participation analysis

suggests that the longitudinal dynamics are heavily affected

by the higher harmonics. It follows that these harmonics must

be included in linear approximations for these approximations

to be accurate representations of the NLTP dynamics. Fur-

thermore, the modal participation analysis provides valuable

insight on how each mode of motion depends on the higher

harmonics of the system. To the best knowledge of the au-

thors, this constitutes a novel approach to the analysis of the

NLTP dynamics of ﬂapping-wing ﬂight.

CONCLUSIONS

The harmonic decomposition methodology, originally devel-

oped for rotorcraft applications, has been extended to the

study of the nonlinear time-periodic dynamics of ﬂapping-

wing ﬂight. The methodology is demonstrated through two

examples involving, respectively, the vertical and longitudi-

nal dynamics of a hawk moth. In these examples, a recently-

proposed modiﬁed harmonic balance algorithm based on har-

monic decomposition is successfully applied to ﬁnd the pe-

riodic equilibrium and approximate high-order linear time-

invariant (LTI) dynamics about that equilibrium. These ap-

proximate linearized models are validated through simula-

tions against the original nonlinear time-periodic dynamics

(NLTP). Dynamic stability is assessed and compared to that

predicted using the averaged dynamics. In addition, modal

participation factors are computed to quantify the inﬂuence of

the higher harmonics on the ﬂight dynamic modes of motion.

Based on the current work, the following conclusions can be

reached:

1. The modiﬁed harmonic balance method based on har-

monic decomposition proved successful in obtaining the

periodic trim solutions and approximate high-order lin-

ear time-invariant models of the nonlinear time-periodic

dynamics of ﬂapping-wing ﬂight. The proposed ap-

proach holds an advantage over averaging methods as it

does not require any time-scale separation between the

periodic forcing function and the aggregate body motion.

Furthermore, the proposed approach is computationally

more robust than Floquet decomposition or frequency

lifting methods as it does not rely on state transition ma-

trices.

2. The modal participation analysis suggests that, for the

vertical and longitudinal dynamics of a hawk moth, there

is a signiﬁcant contribution of the states’ higher harmon-

ics to the various modes of motion. As such, the ﬂight

dynamics is heavily affected by the higher harmonics of

the ﬂapping motion. It follows that these higher harmon-

ics must be included in LTI approximate models for these

models to be accurate representations of the NLTP dy-

namics.

3. The higher harmonics are shown to induce a stabiliza-

tion mechanism that increases pitch damping and stiff-

ness while reducing speed stability. This results in the

stabilization of the pitch oscillation mode and in a pitch

subsidence mode with a lower frequency, which overall

yields a stable hovering cubic. The heave dynamics re-

main largely unaffected. As such, if a hovering vehicle

is excited by periodic forcing at a high-enough frequency

and amplitude, its ﬂight dynamics may be stable.

Since the proposed method does not require any frequency

separation between the forcing frequency and the fastest rigid-

body mode of motion, it can be applied to virtually any biolog-

ical ﬂyer. As such, future work will concentrate on analyzing

the dynamic stability of a wide spectrum of biological ﬂyers,

and possibly assessing closed-loop control laws that stabilize

these ﬂyers or enhance their ﬂight dynamics.

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