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

Authors:
  • University of Maryland

Abstract and Figures

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 approximate 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.
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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. 26). Al-
though the validity of this assumption is subject to continuing
debate (Refs. 79), 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.
5,1113) 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. 21,2325), 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
xRnis the state vector, u
u
uRmis the control in-
put vector, y
y
yRlis 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
ARn(2N+1)×n(2N+1),B
B
BRn(2N+1)×m(2M+1),C
C
C
Rl(2L+1)×n(2N+1), and D
D
DRl(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
XT
k=x
x
xT
k0x
x
xT
k1cx
x
xT
k1s... x
x
xT
kNc x
x
xT
kNs (12a)
U
U
UT
k=u
u
uT
k0u
u
uT
k1cu
u
uT
k1s... u
u
uT
kNc u
u
uT
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
XT
kU
U
UT
ki(13)
where Θ
Θ
ΘkRn(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
ekRn(2N+1)and W
W
WRn(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
AkRn(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
JkRn(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
Θkˆ
J
J
J1
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 kekkbecomes 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
AsA
A
AsfA
A
Af1A
A
Afs (31a)
ˆ
B
B
B=B
B
BsA
A
AsfA
A
Af1B
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
gkd1|˙
φ|wkL˙
φ2
kd2|˙
φ|˙
φkd3w˙
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¨
φ1cTωz1sφ1sw1s˙
φ1sT=0
0
0 (35b)
˙z1s˙
φ1s˙w1s¨
φ1sT+ωz1cφ1cw1c˙
φ1cT=0
0
0 (35c)
˙z2c˙
φ2c˙w2c¨
φ2cT2ωz2sφ2sw2s˙
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 1e7. 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
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.
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
Rα11r
Rγ1
(42)
9
where:
α=ˆr1ˆr1(1ˆr1)
ˆr2
2ˆr2
1
1(43)
γ= (1ˆr1)ˆr1(1ˆr1)
ˆr2
2ˆr2
1
1(44)
β=Z1
0
ˆrα1(1ˆr)γ1dˆr(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|sinφcosφsin2ηxhXw(46c)
Zu=2Xw(46d)
Zw=2K11
m|˙
φ|cos2η(46e)
Zq=2K21
m|˙
φ|sinφcos2ηKrot12
m
˙
φcosφxhZw(46f)
Mu=4K12ˆx
Iy
|˙
φ|cos2φsinη+m
Iy
(2XqxhZu)(46g)
Mw=2K12ˆx
Iy
|˙
φ|cosφcosη+2K21
m|˙
φ|sinφcos2ηmxh
Iy
Zw
(46h)
Mq=2ˆx
Iy
|˙
φ|cosφcosη(K12xh+K22 sinφ)
+1
Iy
˙
φcosφKrot13 ˆxcos φcos η+Krot22 sinφ
2
Iy
|˙
φ|cos2ηsinφ(K21xh+K31 sinφ)
Kvµ1ω
2πIy
cos2φmxh
Iy
Zq(46i)
where Krot =πρ 1
2ˆxImn,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
TtT
40t<T
2
Φ04Φ
Tt3T
4T
2t<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) =
αd0t<T
2
παu
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
R5.19×104m
¯c1.83×104m
S947.8×106m2
a02π1/rad
m1.648×106kg
rh0 m
ˆx0.05 -
ˆr10.44 -
ˆr20.525 -
Iy2.08×107kg-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 1e7. 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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