This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

This content was downloaded by: jaehunghan

IP Address: 143.248.69.126

This content was downloaded on 23/01/2014 at 06:43

Please note that terms and conditions apply.

A multibody approach for 6-DOF flight dynamics and stability analysis of the hawkmoth

Manduca sexta

View the table of contents for this issue, or go to the journal homepage for more

2014 Bioinspir. Biomim. 9 016011

(http://iopscience.iop.org/1748-3190/9/1/016011)

Home Search Collections Journals About Contact us My IOPscience

Bioinspiration & Biomimetics

Bioinspir. Biomim. 9 (2014) 016011 (21pp) doi:10.1088/1748-3182/9/1/016011

A multibody approach for 6-DOF flightdynamics and stability analysis of thehawkmoth Manduca sexta

Joong-Kwan Kim and Jae-Hung Han

Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology,291 Daehak-ro, Yuseong-gu, Daejeon, Republic of Korea

E-mail: jaehunghan@kaist.ac.kr

Received 8 July 2013, revised 18 December 2013Accepted for publication 20 December 2013Published 22 January 2014

AbstractThis paper investigates the six degrees of freedom (6-DOF) flight dynamics and stability of thehawkmoth Manduca sexta using a multibody dynamics approach that encompasses the effectsof the time varying inertia tensor of all the body segments including two wings. Thequasi-steady translational and unsteady rotational aerodynamics of the flapping wings aremodeled with the blade element theory with aerodynamic coefficients derived from relevantexperimental studies. The aerodynamics is given instantaneously at each integration time stepwithout wingbeat-cycle-averaging. With the multibody dynamic model and the aerodynamicmodel for the hawkmoth, a direct time integration of the fully coupled 6-DOF nonlinearmultibody dynamics equations of motion is performed. First, the passive damping magnitudeof each single DOF is quantitatively examined with the measure of the time taken to half theinitial velocity (thalf). The results show that the sideslip translation is less dampedapproximately three times than the other two translational DOFs, and the pitch rotation is lessdamped approximately five times than the other two rotational DOFs; each DOF has the valueof (unit in wingbeat strokes): thalf,forward/backward = 7.10, thalf,sideslip = 17.95, thalf,ascending = 7.13,thalf,descending = 5.77, thalf,roll = 0.68, thalf,pitch = 2.39, and thalf,yaw = 0.25. Second, the naturalmodes of motion, with the hovering flight as a reference equilibrium condition, are examinedby analyzing fully coupled 6-DOF dynamic responses induced by multiple sets of force andmoment disturbance combinations. The given disturbance combinations are set to excite thedynamic modes identified in relevant eigenmode analysis studies. The 6-DOF dynamicresponses obtained from this study are compared with eigenmode analysis results in therelevant studies. The longitudinal modes of motion showed dynamic modal characteristicssimilar to the eigenmode analysis results from the relevant literature. However, the lateralmodes of motion revealed more complex behavior, which is mainly due to the coupling effectin the lateral flight states and also between the lateral and longitudinal planes of motion. Themain sources of the flight instability of the hovering hawkmoth are examined as either thelongitudinal instability grown from the coupled forward/backward velocity and the pitch rate,or the lateral instability grown from the coupled sideslip velocity and the roll rate.

Keywords: hawkmoth, 6-DOF, flight stability, multibody dynamics, dynamic modes of motion

(Some figures may appear in colour only in the online journal)

1748-3182/14/016011+21$33.00 1 © 2014 IOP Publishing Ltd Printed in the UK

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

1. Introduction

Flying insects possess unsurpassable manoeuvrability andstability over any micro air vehicles (MAVs) developed byhumankind. This superiority in their flight performance hasstimulated many researchers to investigate the underlyingmechanics of their flight, ultimately to utilize the knowledgefor our own platforms to perform in the same way. Based ona sound understanding of the unsteady low-Reynolds numberflapping wing aerodynamics (for a review, see Sane 2003 andShyy et al 2010), among the fundamental flight mechanics,the dynamic flight stability is the most vital factor that canenhance our insight of insect flight. A clear understanding ofthe dynamic flight stability can provide us with informationon the natural tendencies of the flight motion under externaldisturbances, i.e., which dynamic degrees of freedom (DOFs)are stable and how stable they are? These open-loop dynamicstability characteristics of an insect are equivalent to thequantified system’s plant characteristics, hence it can bedirectly used for the control law identification. For example, byunderstanding an instability growth timescale of a particularunstable flight mode, we can reversely estimate the requiredperformance of the insect’s flight control system. Therefore,the understandings on the insect flight dynamics and stabilityenable us to design MAVs with a biologically inspired controlstrategy that ensures well managed trade-off between themanoeuvrability and stability.

The dynamic stability of a system can be examinedby the well-established eigenmode analysis technique, andit has been widely used for conventional aircraft. However,the dynamics of insect flight is normally characterized asa nonlinear time periodic (NLTP) system due to its rapidwingbeat motions and its consequent periodic aerodynamicand inertial forces; hence the eigenmode analysis techniquecannot be directly used because it is only applicable to a lineartime invariant (LTI) system. Therefore, several assumptionsfor the simplification of the NLTP system to an LTI systemhave been used in the insect flight dynamic stability analysis:(1) neglecting time varying inertia of the wings, (2) wingbeat-cycle-averaging of the aerodynamics, and (3) linearization ofthe dynamic equations of motion about a reference condition,i.e., the hovering flight condition. Under these assumptions, theeigenmode analysis technique with simplified insect dynamicmodels has been employed in various studies to examine thedynamic flight stability of several insect species. From an earlyinvestigation on the longitudinal flight stability of the desertlocust (Taylor and Thomas 2003), the natural modes of motionof the hover fly, crane fly, drone fly, fruit fly, stalk-eyed fly,bumblebee, and hawkmoth have been identified (longitudinal:Sun et al 2007, Gao et al 2009, Faruque and Humbert 2010a,2010b; lateral: Zhang and Sun 2010a, Zhang et al 2012; 6-DOF: Cheng and Deng 2011).

On the other hand, a more direct approach, which regardsthe insect flight dynamics as a periodic system, has alsobeen taken for the insect flight dynamic stability analysis.An NLTP model of the desert locust was constructed withexperiment-based instantaneous aerodynamics represented bya eight order Fourier series, and its dynamic stability was

examined (Taylor and Zbikowski 2005). That study suggestedthat the proper definition of stability in flapping flight is thatof the asymptotic orbital stability. Gao et al (2010) conducteda perturbation analysis to investigate the dynamic stabilityof the fruit fly by numerically solving 6-DOF single rigidbody nonlinear equations of motion with computational fluiddynamics (CFD)-based aerodynamics represented by a tenthorder Fourier series. Sun and colleagues conducted a directlycoupled analysis between CFD and rigid body equations ofmotion and found a periodic solution of the hovering flight(Wu et al 2009). In addition, the validity of the wingbeat-cycle-averaged model was examined by comparing the resultswith CFD coupled dynamic simulations (longitudinal, Zhangand Sun 2010b; lateral, Zhang et al 2012). Later, Wu andSun (2012) reported that insects with relatively large bodyoscillation, such as the hawkmoth, require a cyclic-motionstability analysis rather than a fixed-point equilibrium stabilityanalysis. These studies suggest that the simplifications madeduring the modeling process can possibly omit importantnatural dynamic characteristics of insect flight (Orlowski andGirard 2011a, 2011b), particularly for insects with a relativelylow wingbeat frequency and large wing to body mass ratio,e.g., the hawkmoth.

The main objective of this study is to investigate thedynamic stability characteristics of the hawkmoth flightwithout the need of any assumptions for simplification toan LTI system. To this end, we first quantitatively comparedthe passive aerodynamic damping in each isolated singleDOF of the hawkmoth flight dynamics. The hawkmothmodel was perturbed with an initial velocity (translational orrotational) along a free DOF whereas the other five DOFswere constrained, and the consequent damped dynamics wascompared among all the DOFs with the measure of the half-life(thalf) of the initial velocity. This quantitative information canserve as a clue for anticipating dynamic stability of free flightsituation where multiple DOFs are coupled. Then we examinedthe fully coupled 6-DOF dynamic responses of the hoveringhawkmoth to the external disturbances. The disturbances weregiven as multiple sets of combined force and moment initialconditions and were set to excite the dynamic modes identifiedin relevant eigenmode analysis studies. Then we compared theperturbed 6-DOF dynamic responses from our study with theeigenmode analysis results.

This study takes a multibody dynamics approach (Leeet al 2011, Kim et al 2012) to consider the time varying inertiatensor of all the body components: head, thorax, abdomen, andtwo wings with three angular DOFs for the wing kinematics.As Orlowski and Girard (2011a) reported, the time varyinginertia of the wings has a significant effect on the overall flightdynamics because ignoring the wing inertia may lead to adifferent flight trajectory under the same initial conditions andthe wing kinematics. Particularly, this wing inertia effect canbe more noticeable in a situation where the wing kinematicsis not symmetric and varies with respect to control inputs.

In terms of the aerodynamic model, we used instantaneoustranslational (Usherwood and Ellington 2002) and rotational(Sane and Dickinson 2002) aerodynamic models for theflapping wings based on the blade element theory without

2

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

Table 1. Morphological parameters of the hawkmoth wing (single wing).

Parameter (Unit) mw (mg) m/mtotal (%) R (mm) c (mm) S (mm2) t (mm) r2/R (–)

Value 48.33 3.32 48.30 18.09 883.75 3.67 × 10−2 0.51

the wingbeat-cycle-averaging. Also, the effect of the arbitrary6-DOF movement of the hawkmoth body on the inflow velocityand effective angle of attack of the wings was considered. Tahaet al (2012) noted in their review paper that the averaging of thedynamic forcing from the flapping wings may omit importantaspects, such as the possible energy transfer from the high-frequency modes to the low-frequency modes of the bodymotions. Further, Orlowski and Girard (2011b) showed that thewingbeat-cycle-averaging of the flapping wing aerodynamicsresulted in a flight trajectory different from the numericalsolution of the full equations.

For the numerical flight simulations in this study, weperformed a direct time integration of the fully coupled 6-DOFnonlinear multibody dynamics equations of motion. Thus,there is no need for the linearization of the dynamic equationsof motion about a reference condition, so the dynamic analysisis not confined within a small perturbation boundary. Inaddition, there is no elimination of the coupled dynamicsbetween the longitudinal and lateral planes of motion, whichoccurs during the linearization process; hence, the analysis isnot confined to a single plane of motion. Fully coupled 6-DOFdynamic responses of the hawkmoth model to various externaldisturbances are simulated.

2. Insect modeling and simulation methodology

2.1. Insect model

The model insect for this study is the hawkmoth Manducasexta. This insect is in the order of Lepidoptera, which mothsand butterflies belong to. It has two sets of wings and eachset consists of fore- and hind-wings. Normally, both wingsflap in a synchronized manner connected by a hook structurecalled retinaculum and frenulum, and the flight is primarilygoverned by the motion of the fore-wings only (Jantzen andEisner 2008). For the dynamic analysis in this study, thefore- and hind-wings are assumed to be joined together attheir interface. All three body components (head, thorax, andabdomen) and two wings are independently modeled for amultibody dynamic modeling of the hawkmoth: the head,thorax, and abdomen are held together with fixed joints; thewings are attached to the thorax with a 3-DOF revolute jointfor the wing kinematics inputs. Biologically, however, all thebody components are connected each other with compliantstructures that allows small passive translational and rotationalmovements due to the inertia of the components. In the case ofactive controlled flight, the effect of these relative movementof the body components, particularly the abdomen which has amass fraction of about 50% (O’Hara and Palazotto 2012), has aprominent effect on the overall flight dynamics as reported byDyhr et al (2013) in their recent experimental study. However,for the open-loop dynamics and stability analysis in this study,we assumed that the passive deflection of the abdomen is small

Table 2. Morphological parameters of the hawkmoth body.

Parameter mtotal mb L l/L l1/L(Unit) (mg) (mg) (mm) (–) (–)

Value 1456.33 1359.67 40.16 0.51 0.27

and its consequent effect to the overall dynamics is negligiblecompared to the wing aerodynamics, hence we used a fixedjoint as the boundary condition between the body components.Other body components, such as six legs and two antennae,are not independently modeled because their mass fraction isalmost negligible. However, the mass of the head includes themass of two antennae, and the mass of the thorax includes themass of six legs.

2.1.1. Morphological parameters. Ellington (1984), O’Haraand Palazotto (2012), and Hedrick and Daniel (2006) are themain sources of the morphological parameters we used for thehawkmoth multibody model in this study. As noted, fore- andhind-wings were modeled as a single wing structure. Referto table 1 and figure 1(a) for the detailed wing morphology,where mw denotes the wing mass; R denotes the wing length;c denotes the mean chord; S denotes the wing area; t denotesthe wing thickness; r2 denotes the radius of second moment ofarea of the wing. The total mass of the two wings forms 6%of the whole body.

Refer to tables 2, 3, and figure 1(b) for the body’smorphological data. In table 2, mb denotes the body mass;L denotes the body length (the anterior tip to the posteriortip); l denotes the length between the anterior tip to the centerof mass; l1 denotes the length between the wing-base pivotand the center of mass. In table 3, rx denotes the major axisof the ellipsoid; ry denotes the minor axis of the ellipsoid;lh, t, a denote the length between the anterior tip to the head,thorax, and abdomen, respectively; mh, t, a denote the mass ofthe head, thorax, and abdomen, respectively; ρh, t, a denote thedensity of the head, thorax, and abdomen, respectively.

The geometry of the head, thorax, and abdomen weresimplified to an ellipsoid of revolution (see figure 1). Themass distribution of the body components was adjusted tolocate the center of mass to the measured position (Ellington1984) by varying the density of each body component. Thedensity of the constituting materials of the insect’s body isnormally 1200 kg m−3 for solid cuticle, and 1000 kg m−3 forsoft tissue (Ellington 1984). Considering cavities inside thebody and other internal organs, the density in table 3 used tomodel the multibody hawkmoth model seem to be reasonable.The mass fraction of each body component is also comparedwith O’Hara and Palazotto (2012) and these values showedgood agreement with the averaged measurement data from atotal 30 real hawkmoths.

3

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

bx

by

R

bx

bz

βχ spz

spx

wx

wy

GX0

BG r

( ), ( ), ( )t t tΦ Θ Ψ

GX

GY GZ

GX

GZ

l

1l

strx

stryi = 1i = 2i = 3i = 4i = 5

(a) (b)

Figure 1. A schematic for the wing and body morphology and the coordinate systems.

Table 3. Morphological parameters of the hawkmoth’s body components.

Parameter (Unit) rx (mm) ry (mm) lh, t, a /L (–) mh, t, a (mg) mh, t, a/mtotal (%) ρh, t, a (kg m−3)

Head 3.79 3.38 0.094 34.01 2.34 187.45Thorax 7.03 5.46 0.27 481.98 33.10 550.00Abdomen 13.55 4.96 0.66 843.67 57.93 605.00

2.1.2. Coordinate systems. To describe the 6-DOF flightdynamics, a total four major coordinate systems were usedfor the flight dynamic analysis of the hawkmoth model: (1)wing-fixed [xw yw zw], (2) stroke-plane [xsp ysp zsp], (3) body-fixed [xb yb zb], and (4) inertial [XG YG ZG] coordinate systems.These coordinate systems are illustrated in figure 1.

The origin of the wing-fixed frame is located at thewing-base pivot, and the yw-axis points to the wing span-wise direction. This frame is used to define the wingkinematics with respect to the stroke-plane frame, and all theaerodynamic forces are described with respect to this frame. Inaddition, each aerodynamic strip for the blade element theoryapplication has its own strip coordinate system aligned alongthe same direction to the wing-fixed frame.

The stroke-plane frame is mainly used for the inclusionof 6-DOF body flight states to the aerodynamic modelformulation as explained in section 2.2.2. The flight statevector of the center of mass described by the body-fixedframe (denoted by subscript b, e.g. ub, qb) is transformedto corresponding values at the stroke-plane frame, and thesetransformed flight states (denoted by subscript sp, e.g. usp,qsp) are used to compute instantaneous aerodynamic forcesinduced by the body movements. Note that the xsp-axis makesan angle of β with respect to the XG–YG plane, which is calledthe stroke-plane angle.

The body-fixed frame is attached to the center of the mass.The angle χ is called the body angle and it is defined asthe angle between the XG–YG plane and the body long axiswhen the hawkmoth is in hovering. The xb-axis points forwardmaking an angle of χ with respect to the body long axis, andit makes the xb-axis be parallel with the XG–YG plane whenin the hovering flight state. The yb-axis points to the directionof the right wing tip when it is fully stretched out. This framedescribes the 6-DOF body flight states of the hawkmoth model.

The inertial frame is used to observe the global location,attitude and corresponding velocities of the hawkmoth model.

The orientation of the body-fixed frame with respect to theinertial frame is parameterized by the Euler angles: � (yaw),� (pitch), and � (roll) in that order.

2.1.3. Wing kinematics. The wing kinematics for the insectsis defined with three rotational degrees of freedom at thewing-base pivot: (1) φ(t), stroke positional angle, (2) α(t),feathering angle, and (3) θ (t), deviation angle (following theconvention of Sane and Dickinson 2001). The stroke positionalangle governs the back and forth motion of the wing; thefeathering angle governs the geometric angle of attack of thewing surface; the deviation angle governs the up and downmotion of the wing with respect to the stroke-plane.

Generally, the flapping frequency of the hoveringhawkmoth is known to be approximately 26 Hz, and thisfrequency remains almost unchanged inside the normal flightregime (Willmott and Ellington 1997). In this study, however,a flapping frequency of 26 Hz could not provide enoughaerodynamic forces to lift up its own weight under the givenaerodynamic model and the associated wing kinematics. Thisis considered to be mainly due to the neglected aerodynamiccomponents such as the wake capture effect which canchange aerodynamic forces on the wings (see section 2.2.1).Therefore, we used a slightly higher flapping frequency(29.46 Hz) for all the simulations in this study. This flappingfrequency was selected on the basis of preliminary simulationswe ran to search for the flapping frequency that could generatea lift force equal to the weight of the insect.

Basically, this difference in the flapping frequency affectsthe aerodynamic damping in all the DOFs of the hawkmothflight, because the magnitude of the aerodynamic dampingdue to the flapping motion and the body dynamics is linearlydependent on (1) the body translational/angular velocities,(2) wingbeat stroke amplitude, and (3) its frequency (Hedricket al 2009, Cheng and Deng 2011). Therefore, the increasedflapping frequency used for this study will reduce the time

4

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

0.15.00.0-60

-30

0

30

60

90

120

150

180

measured (Willmott and Ellington 1997) simplified

wingbeat stroke

ang

le (

deg

)

down stroke up stroke

α (t)

φ (t)

θ (t)

Figure 2. The wing kinematics used for the dynamic analysis of thehawkmoth model: the solid line indicates the measured wingkinematics from a real hovering hawkmoth (Willmott and Ellington1997) and the dashed line indicates simplified sinusoidal wingkinematics.

taken to half the initial velocity (thalf) than a real situationwhere the flapping frequency is about 26 Hz. However,considering the linear dependency between the frequencyand the damping magnitude, we think that the difference inthe flapping frequency of around 10% will not significantlychange the qualitative trend of the stability characteristics ofthe hawkmoth flight.

As noted in the introduction, two different dynamicanalyses were conducted for the hawkmoth model: (1) thepassive damping analysis on each single DOF, and (2)the natural modes of motion analysis on fully coupled 6-DOF dynamic responses induced by external disturbances.The wing kinematics used for each analysis were different.A simplified sinusoidal wing kinematics was used for thepassive damping analysis, and measured wing kinematics fromhovering hawkmoth (Willmott and Ellington 1997) with asmall adjustment was used for the natural modes of motionanalysis (see figure 2).

The simplified sinusoidal wing kinematics for the passivedamping analysis is defined as:

φ(t) = φamp sin(2π f t), (1)

α(t) = αamp

tanh(Cα )tanh(Cα sin(2π f t + ψα)) + α0, (2)

θ (t) = 0, (3)

where f = 29.46 Hz, φamp = 55.4◦, αamp = 45.0◦, Cα = 4.5,ψα = −π/2, α0 = 90◦.

Here, the Cα is the tuning coefficient for the shape ofthe feathering angle α(t); as Cα goes to infinity, the functionbecomes a square wave function, and as Cα goes to zero, thefunction becomes a sine wave (Berman and Wang 2007). Bythis relationship, the Cα in (2) changes the stroke reversal timeof the feathering angle. To define this stroke reversal time,we referred to the original data of the outer (0.6 R to 0.9 R)feathering angle distribution along the span-wise location fromfigure 9 of Willmott and Ellington (1997), and we found that

Table 4. Fourier coefficients for the measured wing kinematics.

Kinematic variables K αk βk

φ(t) 0 0.28301 0.9238 −0.12112 0.0104 −0.04653 0.0388 0.0262

α(t) (averaged from 0.6R to 0.8R) 0 3.47 8901 −0.2616 −1.02842 0.0089 −0.00443 −0.0162 −0.1341

θ (t) 0 −0.06 7701 0.0247 0.04322 0.1082 0.01003 0.0078 −0.0125

the Cα of 4.5 that gives a stroke reversal time of 0.15 wingbeatstroke period is an appropriate choice for simplifying high-harmonic feathering angle observed in their study.

The measured wing kinematics (Willmott and Ellington1997) for the natural modes of motion analysis is representedby a third order Fourier series as:

[φ(t) α(t) θ (t)]

= ao

2+

3∑k=1

(ak cos(k2π f t) + bk sin(k2π f t)), (4)

where f = 29.46 Hz for all the kinematic variables, andthe Fourier coefficients are tabulated in table 4 (coefficientsobtained from Ellington (2011)). In Willmott and Ellington(1997), the feathering angle, α(t), was measured from sevenpoints along the span-wise direction (0.3 R to 0.9 R, incrementof 0.1 R). They divided the span-wise location of the wing intoinner (0.3 R to 0.5 R) and outer (0.6 R to 0.8 R) portions, andthis study took the averaged value of the outer portion of thewing.

2.2. Aerodynamic model

The aerodynamic model used in this study is based on theblade element theory (such as in Truong et al (2011)). Theinteraction between the wing and body, or the wing and wakewas not considered, also the span-wise flow generated fromthe wingbeat motion and the body dynamics induced span-wise flow components were ignored. Cheng and Deng (2011)proposed two parameters which are the chord-wise tip velocityratio and the span-wise tip velocity ratio to consider the span-wise flow on the flapping wing. However, they concluded that itneeds further research due to the lack of an accurate estimationof relationship between the aerodynamic coefficients versusthe defined parameters. At the moment, only the CFD andexperimental studies can capture the effect of the whole three-dimensional flow field including the span-wise flow and thewake interactions. Several CFD works of Sun and colleaguesreported that the sideslip velocity can change the formation ofthe leading edge vortex (LEV) by changing the span-wise axialflow, meaning a tip to root span-wise flow reduces the LEVintensity and the endurance time thus reduces aerodynamiccoefficient on the corresponding wing (Zhang and Sun 2010a,Zhang et al 2012). Even though the blade element approach

5

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

with quasi-steady aerodynamic coefficients cannot model theunsteady wake capture effect, the experimental studies withscale-up robotic wing models showed a reasonable accuracy inthe prediction of the aerodynamic forces (Dickinson et al 1999,Sane and Dickinson 2001, 2002, Usherwood and Ellington2002) Therefore, we assume that the use of the quasi-steadyaerodynamic model is reasonable for the purpose of this study.Each wing is divided into five aerodynamic strips for theblade element approach (see figure 1(a)); the area and meanchord of each strip were accurately calculated based on themorphological data from section 2.1.1.

2.2.1. A quasi-steady blade element approach with anunsteady effect consideration. Sane and Dickinson (2002)reported that the aerodynamics around flapping wings canbe categorized into four distinct components as in (5). Theinstantaneous aerodynamic force on each aerodynamic strip isthe sum of: (1) the force due to the inertia of the added-mass ofthe fluid around the wing, (2) the force due to the translationalmovement of the wing (mainly from φ(t), θ (t)), (3) the forcedue to the rotational movement of the wing (mainly from α(t)),and (4) the force due to the wake capture effect:

dFinstantaneous = dFadded−mass + dFtranslation

+ dFrotation + dFwake−capture. (5)

In this study, aerodynamic forces generated fromtranslational and rotational movement of the wing are modeled.The wake capture effect is neglected because the inherentunsteadiness of the phenomenon makes it hard to be modeledwith a quasi-steady approach. Although the wake captureeffect is excluded for the aerodynamic model used in thisstudy, as Sane and Dickinson (2002) reported, a quasi-steadymodel based on both translation and rotation captures the timehistory of the aerodynamic force generation with a reasonableaccuracy except for the impulsive aerodynamic peaks dueto the wake capture effect. During a wingbeat stroke, whichconsists of up- and down-stroke, the aerodynamic peak fromthe wake capture effect is generated at the start of each half-stroke (Dickinson et al 1999, Sane and Dickinson 2002).Therefore, in terms of the flight dynamics of the insect bodyitself, a pitching moment due to the aerodynamic peak fromthe start of the up-stroke roughly cancels out that of the down-stroke. Thus, we assume that the aerodynamic peaks fromthe wake capture have little impact to the overall pitchingdynamics of the center of mass of the insect. However,the missing wake capture peak can reduce the wingbeat-cycle-averaged lift and this reduction is compensated withslightly increased flapping frequency (see section 2.1.3). Theinstantaneous translational and rotational forces from eachaerodynamic strip are calculated by

dFlift, translation, i(Vi, αi) = 12ρV 2

i dSiCL, (6)

dFdrag, translation, i(Vi, αi) = 12ρV 2

i dSiCD, (7)

dFrotation, i(x0,Vi, αi) = ρViCrotαicidSi, (8)

where the subscript i denotes the designated number of eachaerodynamic strip (see figure 1(a)), Vi denotes the incident

airflow speed on each aerodynamic strip, αi denotes theeffective angle of attack of each aerodynamic strip, αi denotesthe first order time derivative of αi, ρ denotes the density ofair (1.225 kg m−3), dSi denotes the area of each aerodynamicstrip, ci denotes the mean chord of each aerodynamic strip.The detailed definition of Vi and αi is provided in section 2.2.2.Note that all the inflow velocity is defined with respect to thestroke-plane frame. As noted, each wing was divided into fiveaerodynamic strips (see figure 1(a)). These forces on eachaerodynamic strip were integrated along the entire wing inorder to obtain the instantaneous aerodynamics produced byflapping wings.

The aerodynamic coefficients (CL, CD, and Crot) in(6)–(8) are defined as follows. The lift and dragcoefficients with respect to the varying angle of attackof the hawkmoth wing for the quasi-steady translationalaerodynamic components were extracted from an experimentalwork by Usherwood and Ellington (2002). They conducteda vertical/horizontal force measurement experiment with ascaled hawkmoth wing rotating with an angular velocitycorresponding to Re = 8071. We curve-fitted the experimentaldata to obtain the lift and drag coefficients shown in (9). Theunsteady rotational aerodynamic component generated from arapid feathering motion of wing at each stroke reversal wasmodeled from an experimental study of Sane and Dickinson(2002) and it is shown in (10). Because the experiment isconducted with a fruitfly wing (Re = 115), some discrepanciesdue to the Reynolds number effect may arise in the estimationof the rotational force component:

CL(αi) = 15.84 sin(0.04167αi + 0.0262)

+14.74 sin(0.04238αi − 3.109) + 0.0654

CD(αi) = 22.55 sin(0.02481αi + 0.595)

+21.87 sin(0.02607αi + 3.761) + 0.0983 (9)

Crot,exp = f (ωi, x0,i). (10)

Here, the coefficient for the rotational aerodynamiccomponent (Crot,exp) is in tabular form, which is a function ofthe nondimensional angular velocity of the strip (ωi = αici/Vi),and the nondimensional axis of rotation of the strip (x0,i = x/ci,where x is the length from the leading edge to the featheringaxis, yw). The detailed values of the coefficients can be foundin figure 2 of Sane and Dickinson (2002).

2.2.2. Inclusion of 6-DOF body flight states. The incidentairflow to the wing aerodynamic strips has two main sources:one is the relative velocity due to the flapping motion itself,and the other is the induced velocity from 6-DOF bodyvelocity state variables: ub (forward/backward), vb (sideslip),wb (up/down), pb (roll), qb (pitch), and rb (yaw), whichare described by the body-fixed frame. The translationaland rotational motions of the hawkmoth body itself addglobal motions to the wings, which originally have theirown local flapping motion relative to the body. These addedglobal velocity components to the flapping wings alter themagnitude and direction of the incident airflow to the wings.Therefore, to properly simulate the flight dynamics under

6

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

external disturbances, the time varying 6-DOF body flightstates need to be associated with the aerodynamic model.

The resultant incident airflow speed on each aerodynamicstrip (Vi) is defined in

Vi = [(Vhori,i + Vusp + Vvsp + Vrsp )2

+ (Vvert,i + Vwsp + Vpsp + Vqsp )2]1/2. (11)

Here, the incident airflow components are described bythe stroke-plane frame, and these are categorized into two:(1) horizontal inflow components with respect to the stroke-plane, and (2) vertical inflow components with respect to thestroke-plane (xsp–ysp plane).

The horizontal inflow components are defined as:

Vhori,i = −yw,i φ(t), Vusp = usp cos(|φ(t)|),Vvsp = vsp sin(−φ(t)), Vrsp = yw,i rsp, (12)

and the vertical inflow components are defined as:

Vvert,i = yw,i θ (t), Vwsp = wsp, Vpsp = yw,i psp cos(φ(t)),

Vqsp = yw,i qsp sin(|φ(t)|), (13)

where yw,i denotes the location of each aerodynamic strip alongthe direction of the yw-axis of the wing-fixed frame. Here, allthe 6-DOF body flight state variables [ub vb wb pb qb rb] aretransformed to the velocity states in the stroke-plane frame[usp vsp wsp psp qsp rsp]. Note that all the wing kinematicvariables are originally defined with respect to the stroke-planeframe (see section 2.1.3).

Consequently, the effective angle of attack of eachaerodynamic strip (αi) can be defined from a perpendicularset of the incident airflow components, vertical and horizontalto the stroke-plane frame:

αi = α(t) + tan−1

(Vvert,i + Vwsp + Vpsp + Vqsp

Vhori,i + Vusp + Vvsp + Vrsp

), (14)

where α(t) is the geometric angle of attack governed by thegiven wing feathering kinematics (see section 2.1.3).

2.3. Multibody dynamics simulation environment

A multibody dynamics simulation environment for thehawkmoth model and the corresponding flapping wingaerodynamic model were established based on a multibodydynamics code (MSC.Adams, MSC Software Corp.). Itperforms a direct time integration of fully coupled6-DOF equations of motion of multibody dynamic systems.This multibody dynamics simulation environment has beendeveloped through the author’s previous studies on theflapping-wing flight dynamics (Pfeiffer et al 2010, Lee et al2011, 2012, Kim et al 2012, Kim and Han 2013a, 2013b).

This simulation environment first generates a setof nonlinear differential-algebraic equations (known as aDAE system), which are based on a defined multibodyconfiguration; for this study, there are five rigid bodies andfour kinematic constraints (the two connecting the bodycomponents are fixed joints, and the other two connecting thethorax and two wings are 3-DOF revolute joints). A genericform of this DAE system is:

Mq + ηTq λ − AT F(q, q) = 0

η(q, t) = 0 (15)

where M is the mass matrix of the system, q is the set ofcoordinates used to represent displacements, η is the set ofthe configuration and applied motion constraints (kinematicconstraint equations), λ is the Lagrange multipliers forhandling multiple constraints, F is the set of applied forcesand gyroscopic terms of the inertia forces, AT is the matrix thatprojects the applied forces in the direction q, ηq is the gradientof the constraints at any given state and can be thought of asnormal to the constraint surface in the configuration space.

The solution process for the governing DAEs followsa numerical approach based on a DAE integrator developedby Gear (1971). The solution process of this integratorconsists of two phases: a prediction (explicit process, basedon Taylor’s series) followed by a correction (implicit processusing the Newton–Raphson algorithm, based on a differencerelationship: yn+1 = yn + hyn+1, where yn is the solution att = tn, h is the integration step size, yn+1 is the solution att = tn+1). During the correction phase, the tolerance for thecorrector defines the level of acceptable error. The correctorrejects the solution if the estimated error is greater than thespecified integration tolerance, or accepts the solution if theestimated error is less than the specified integration tolerance.All the states defined in the equations of motion includingdisplacements, velocities, and applied forces are monitoredby this error criteria. This backward-difference formula basedstiff integrator is used with several modifications, such as indexreduction of DAE. For this study, the GSTIFF (based on theGear’s stiff integrator, Gear 1971) integrator with an index-2 formulation was used to solve the governing DAEs of thehawkmoth model.

The aerodynamics on the flapping wings in section 2.2was independently coded with FORTRAN and appended asa subroutine to the GSTIFF integrator. At each integrationtime step, the GSTIFF integrator provides the aerodynamicsubroutine with instantaneous 6-DOF flight state vectorsand the wing kinematics variables. Then the aerodynamicsubroutine computes corresponding aerodynamic forces andreturns them to compute the dynamic solution for thenext integration time step. For the efficiency and accuracyof the dynamic solution, convergence tests are conductedwith respect to the two control parameters (the correctortolerance, and the integration time step) that directly affectthe convergence of the dynamic solution. From the tests, acorrector tolerance of 0.1% ensured the solution convergence,and an integration time step of 0.001 s (=1 ms) that isequivalent to 34 time steps per a wingbeat stroke was enoughto simulate the oscillatory dynamics of the hawkmoth modelwith an acceptable computation time.

A block diagram showing the process of the hawkmothmultibody dynamic modeling and the simulation is shown infigure 3.

3. Results and discussion

Two different dynamic analyses were conducted forthe established multibody hawkmoth model using thedescribed simulation environment. First, the passive dampingcharacteristics of the hawkmoth were investigated on each

7

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

Fully coupled nonlinear equations

of motion

Kinematic constraints equations

DAE formulation

Solution of hawkmoth multibody dynamics

Wing kinematicsφ(t), α(t), θ(t)

Kinematic constraints(joint between body

components)

Multibody configuration information

Body/wing morphological

parameters Flapping wing aerodynamics

Direct time integration

DAE integrator (GSTIFF)

( , ) 0

( , ) 0

T TqMq A F q q

q t

η λη

+ − =

=

, ,

, ,

,

lift translation i

drag translation i

rotation i

dF

dF

dF

[ ]

[ ( ) ( ) ( )]

[ ( ) ( ) ( )]

spu v w p q r

t t t

t t t

φ α θφ α θ

Figure 3. A block diagram for the multibody dynamics simulation environment for the hawkmoth.

Table 5. Nondimensionalization of the flight state variables, where � is the stroke amplitude, f is the flapping frequency (29.46 Hz) and g isthe gravitational acceleration (9.80 665 m s−2).

Category Item Nondimensionalized by c.f.

Translational velocity u, v, w U U = 2�fr2 = 2.81 m s−1

Rotational velocity p, q, r U/c c = c = 18.09 mm

Global location XG, YG, ZG R R = 48.30 mm

Forces Fx, Fy, Fz W W = mtotal g = 0.014 N

Moments Mx, My, Mz Wc –

Time T T T = 1/ f

single DOF; only a single DOF was free while the otherfive were constrained. Second, the natural modes of motionunder external disturbances when all the 6-DOFs are fullycoupled were investigated. Throughout the analysis, all theinitial conditions and the flight state responses are presentedin a nondimensionalized form (refer table 5).

3.1. Independent DOF’s passive dynamic stability

For the passive damping analysis of each single DOF, thehawkmoth model was pushed (or rotated) with an initialtranslational (or angular) velocity along the free DOF whereasthe other five DOFs were constrained. In the analysis, allthe flight states and the initial condition, and the dynamicresponses are given and described in the stroke-plane frame(see figure 1), i.e., the hawkmoth can translate or rotatewith respect to the stroke-plane frame. Here, we wantedto accentuate the pure effect of the flapping wings and itsconsequent stroke-plane only, eliminating the effect from themorphological angles. The body-fixed frame is defined with amorphological parameter, the body angle (χ ). Also the stroke-plane angle (β) has a relationship with the body orientation.Hence the body-fixed frame is not an appropriate choicebecause it varies along the flight condition of the hawkmoth.Therefore, the aerodynamic damping characteristics which aredescribed in the stroke-plane frame can serve as basis forexamining the stability characteristics in different referenceframes. The wing kinematics used for this single DOF analysiswas the simplified sinusoidal form defined in section 2.1.3.The magnitude of the translational initial condition was varied

from ± 0.1 to ± 0.3 with an interval of ± 0.1 (equivalentto ± 0.28–84 m s−1); the magnitude of the rotational initialcondition was varied from ± 0.05 to ± 0.15 with an interval of± 0.05 (equivalent to ± 7.77–23.32 rad s−1). The magnitudeof the given initial conditions was in a reasonable rangewhich was determined on the basis of the measurementdata of a real hawkmoth flight (Willmott and Ellington1997, Hedrick et al 2009, Hedrick and Robinson 2010, andSpringthorpe et al 2012), and based on relevant simulationstudies (Sun et al 2007, Zhang and Sun 2010a, and Zhang et al2012). Specifically, the normal forward flight speed of thehawkmoth falls inside a boundary of 5 m s−1, and the normalrotational velocity falls inside a boundary of 15 rad s−1.

The response of each single DOF to the initial velocityconditions was quantitatively examined with a measure ofthe half-life (thalf) of the initial velocity. See figure 4 forthe results. Note that for all the longitudinal flight stateresponses [usp wsp qsp], non-perturbed responses (when theinitial conditions were zero) were subtracted from the originalresponses for more clear comparison. This is because of theoscillatory feature of the longitudinal flight state responsesoriginating from periodically varying aerodynamics from theflapping wings. By subtracting the non-perturbed responsefrom the original response, we can clearly distinguish theeffect of the given initial condition and its decaying dynamics.Overall, the results showed that the ysp-directional (sideslip)translation is less damped approximately three times than theother two translational DOFs, and the xsp-directional (pitch)rotation is less damped approximately five times than the othertwo rotational DOFs. The results are tabulated in table 6.

8

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

(ai )

(aii )

(aiii)

(bi )

(bii)

(biii )

Figure 4. Passive damping characteristics on each dynamic degree of freedom. (The shaded region indicates the half-amplitude decay timeof the response.) (a) The translational flight state responses with the range of the abscissa at 0–100 wingbeat strokes. (b) The rotational flightstate responses with the range of the abscissa at 0–10 wingbeat strokes.

Table 6. Results of the passive damping analysis of each single DOF (in the stroke-plane frame).

State usp vsp wsp psp qsp rsp

thalf (wingbeat stroke) 7.10 17.95 5.77 0.68 2.39 0.25(descending)7.13(ascending)

Major frequency component of the response ( f = 29.46 Hz) f – 2 f 2 f f , 2 f 2 f

3.1.1. Passive damping in the translational DOFs. The leftcolumn of figure 4 shows the responses of the translationalDOFs. Note that the range of the abscissa is 0–100 wingbeatstrokes. The shaded region in each graph indicates the timetaken to half the initial velocity. The results were: thalf,u =

7.10, thalf,v = 17.95, thalf,w,ascending = 7.13, and thalf,w,descending =5.77. Among three translational DOFs the ysp-directional(sideslip) translation turned out to be the least damped DOF.It took about 18 wingbeat strokes to damp out half the initialvelocity. The xsp-directional (forward/backward) translation

9

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

Table 7. Natural modes of motion of the hawkmoth from eigenmode analyses in the relevant literature.

Dominant coupled CorrespondingPlane of Sun et al (2007), flight states disturbances formotion Mode Zhang et al (2012) Cheng and Deng (2011) Gao et al (2009) in eigenvector this study

Longitudinal 1 Unstable Unstable Stable oscillatory δub, δqb Fx,b = 2oscillatory oscillatory (in-phase) My,b = 0.1

2 Stable fast Stable fast Stable fast δub, δqb Fy,b = 2subsidence subsidence subsidence (out-of-phase) Mx,b = −0.1

3 Stable slow Stable slow Stable slow δwb Fz,b = 2subsidence subsidence subsidence

Lateral 1 Unstable Stable slow – δvb, δpb Fy,b = 2slow divergence subsidence (in-phase) Mx,b = 0.1

2 Stable slow Unstable – δvb, δpb Fy,b = 2oscillatory oscillatory (out-of-phase) Mx,b = −0.1

3 Stable fast Stable fast – δpb, δrb Mx,b = 0.1subsidence subsidence (out-of-phase) Mz,b = −0.1

and the zsp-directional (up/down) translation showed similardamping characteristics, except for the descending dynamicsof the up/down translation. As Parks et al (2011) pointedout in their experimental study with translational damping ofCicada wings, the up/down translation is more complicatedbecause wing-wake interaction is presented in this case. Theaerodynamic model used in this study was based on a quasi-steady blade element approach and the wake interactionwas not considered; therefore, the up/down results may notcoincide with the experimental results. In this study, thexsp-directional (forward/backward) translation exhibited thehighest damping (thalf,u = 7.10) while the ysp-directional(sideslip) translation was the least damped DOF (thalf,v =17.95), which had a trend qualitatively similar to theexperimental study (Parks et al 2011). In conclusion, for thetranslational movements, there was a passive damping in allthe DOFs.

3.1.2. Passive damping in the rotational DOFs. The rightcolumn of figure 4 shows the responses of the rotationalDOFs. Note that the range of the abscissa is 0–10 wingbeatstrokes. The results showed that the rotational movements haverelatively large passive damping characteristics, which are:thalf,p = 0.68, thalf,q = 2.39, and thalf,r = 0.25. Particularly,zsp-directional (yaw) rotation was subjected to the highestdamping magnitude among three rotational DOFs. Only aquarter wingbeat stroke was sufficient to damp out theinitial yaw angular velocity to half. In addition, the xsp-directional (roll) rotation was under a high degree of passivedamping as well. The combination of these two axesof rotation (roll and yaw) develops a rotation along thebody long axis (see figure 1(b)), and also high degree ofpassive damping is anticipated along this axis. However, theysp-directional (pitch) rotation showed relatively smalldamping characteristics (almost five times smaller) than theother two rotational DOFs. In conclusion, for the rotationalmovements, there was a passive damping in all the DOFs aswell. However, the ysp-directional (pitch) rotation had muchsmaller damping than the other two.

3.2. 6-DOF free flight disturbance analysis

We found that all the independent DOFs of the hawkmoth flightdynamics are stable and possess inherent passive translationaland rotational damping from the previous section 3.1.However, there was a relative instability in a quantitativesense that the sideslip translation was less damped almostthree times than the other two translational DOFs, and thepitch rotation was less damped almost five times than theother two rotational DOFs. Also, in terms of the dynamicplanes of motion, the pitch rotation and the sideslip translationwere the least damped DOFs in the longitudinal and thelateral planes of motion, respectively. This imbalance in thedamping magnitude among DOFs can play an importantrole when it comes to a more realistic situation where allthe dynamic DOFs are unconstrained and coupled (6-DOFfree flight condition). Normally, eigenmode analysis is usedto investigate the natural modes of motion when dynamicdegrees of freedom are coupled. Linear eigenmode analysiscan provide dynamic characteristics around a certain referencepoint within a boundary of small perturbations, i.e., thehovering condition. However, this small perturbation approachcannot capture a full nonlinear dynamics within a long timewindow where the values of the disturbed body flight statesexceed the linearization-applicable range.

To identify the natural modes of motion through numericaldirect time integration, not through the eigenmode analysiswith a simplified linear system model, the free flyinghawkmoth multibody model was disturbed from the referencecondition (the hovering condition found in section 3.2.1) andthe resulting fully coupled 6-DOF dynamic response wasexamined. The magnitude and phase of the given disturbancewas determined to excite the natural modes of motion identifiedfrom the previous eigenmode analyses in the relevant literature.Table 7 shows: (1) the identified natural modes of motion of thehawkmoth from eigenmode analyses in the relevant literature,(2) the corresponding coupled flight states for each dynamicmode, and (3) the corresponding disturbance combinations forthe hawkmoth model for this study.

The dynamic response of the hawkmoth model to eachdisturbance combination was examined by the time taken for

10

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

the disturbed motion to damp/amplify to half/double of itsinitial magnitude. The frequency of the oscillatory dynamicmotion was also of interest. A comparison with the resultsfrom the previous eigenmode analyses in the relevant literaturewas conducted as well.

For the application of the disturbance, the nondimension-alized (see table 5) force and moment disturbance combina-tions (see table 7) were given at a time when the down-strokestarts during the hovering flight condition, and the disturbancecontinued for a wingbeat stroke (T = 1/ f ) having a smoothstep function profile which approximates the Heaviside stepfunction with a cubic polynomial defined in

h(t) =⎧⎨⎩

h0 ; t � t0h0 + a�2(3 − 2�) ; t0 < t < t1

h1 ; t � t1

⎫⎬⎭ (16)

where, a = h1 − h0, � = (t − t0)/(t1 − t0). All thedisturbance sets were applied with respect to the body-fixedframe. The magnitude of the nondimensionalized disturbanceforce and moment (Fx,y,z = 2, Mx,y,z = 0.1) were determinedto perturb the hawkmoth model with an initial velocity ofapproximately 0.25 m s−1 for the body translational DOFs, andapproximately 2.5 rad s−1 for the body rotational DOFs. Weassume that the magnitude of the disturbances can representa typical wind gust that the hawkmoth experiences duringthe flight, considering the normal flight speed (5 m s−1) andthe rotation speed (15 rad s−1) of the real hawkmoth (Willmottand Ellington 1997, Hedrick et al 2009, Hedrick and Robinson2010, Springthorpe et al 2012).

3.2.1. Reference hovering condition for the disturbanceanalysis. The dynamic stability of an aircraft can beevaluated by the dynamic responses after it has been disturbedfrom a steady condition, i.e., a reference equilibrium flightcondition. In the case of flying animals, the reference conditioncan normally be either a constant speed trimmed forwardflight or a hovering flight. The interested flight mode in thisstudy is the hovering flight of the hawkmoth, so the referenceequilibrium flight condition was chosen to be a hovering flight.

The hovering flight state is defined as (17). For thehovering flight, all the time-averaged 6-DOF velocity statesneed to be zero. If any time-averaged velocity state has non-zero value, it physically means a drift in the translational orrotational direction, and this drift causes the insect not tomaintain the hovering flight:

|mean[kb(t)]| − εk = 0

mean[kb(t)] = 1

T

∫ mT

nTkb(t) dt

k = [u v w p q r]

t ∈ [nT, mT ], n < m, ∀n, m ∈ Z (17)

where the error tolerance coefficient εk has a value of 1% ofpeak-to-peak amplitude of each flight state variable monitoredduring a time period defined in (17); T is the flapping period(T = 1/ f ).

The reference hovering condition was heuristicallysearched for by changing a few variables of the measured

Figure 5. The wing kinematics used for the natural modes of motionanalysis: the solid line indicates the measured wing kinematics froma real hovering hawkmoth (Willmott and Ellington 1997) and thedashed line indicates the tuned wing kinematics that satisfies thecriteria for a hovering trim condition.

wing kinematics (Willmott and Ellington 1997) defined insection 2.1.3 using a third order Fourier series expansion (seetable 4). By only changing the coefficient α0 of φ(t) and α(t),which alters the mean value of the stroke positional angleand feathering angle, the multibody hawkmoth model couldachieve a reference hovering condition satisfying the criteria in(17). The coefficient α0 of φ(t) was changed to 0.05 964 from0.2830 (mean value decreased by −6.40◦), and coefficientα0 of α(t) was changed to 3.8918 from 3.4789 (mean valueincreased by 11.83◦). The resulting wing kinematics is plottedin figure 5.

The 6-DOF flight states during the reference hoveringcondition with respect to the body-fixed frame and the inertialframe is depicted in figure 6. The peak-to-peak value ofeach flight state is also shown in figure 6. Oscillations inforward/backward velocity (ub) has a peak-to-peak value of0.108 U (0.12U in Wu and Sun (2012)), and oscillations inup/down velocity (wb) has a peak-to-peak value of 0.0919U(0.082U in Wu and Sun (2012)). The pitch angular velocity(qb) has a peak-to-peak value of 0.108U/c (0.093U/c in Wuand Sun (2012)). The oscillation of the global location on theXG–ZG plane has a magnitude within 0.04R (0.083R in Wuet al (2009)). The oscillation of the Euler pitch angle has apeak-to-peak value of 3.761◦ (5.1◦ in Wu and Sun (2012)).We compared the dynamic solution at hover in this study withthe CFD coupled flight dynamic simulation results of Wu et al(2009) and Wu and Sun (2012) in terms of the peak-to-peakvalues. This comparison may only show a qualitative similaritybetween two studies because of the difference in the used wingkinematics and also the fidelity of the aerodynamics used forthe simulation. Refer to table 5 for the nondimensionalization.Note that the value of U is 2.578 m s−1 and c is 18.37 mm inWu et al (2009) and Wu and Sun (2012).

Based on this reference hovering condition, naturalmodes of motion of the hawkmoth multibody model wereexamined. Each disturbance combination in table 7 for exciting

11

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

-0.2

-0.1

0.0

0.1

0.2

-0.2

-0.1

0.0

0.1

0.2

-3

-2

-1

0

1

2

3

-80

-60

-40

-20

0

20

40

60

80

wb, p-p

=0.0919U (=0.259m/s)

up stroke

[ub, v

b, w

b] /

U

ub

vb

wb

down stroke

ub, p-p

=0.108U (=0.305m/s)

qb, p-p

=0.108U/c (=16.793rad/s)

[pb, q

b, r

b]c

/U

pb

qb

rb

XG

YG

ZG

[XG, Y

G, Z

G] /

R

XG,p-p

=0.0393R (=1.9mm)Z

G,p-p=0.0248R (=1.2mm)

Θp-p

=3.761 deg

Eul

er a

ngle

s (d

eg)

roll pitch yaw

wingbeat stroke210

Figure 6. Reference hovering condition for the disturbance analysis.

a particular mode of motion was given to the hawkmothmultibody model and the corresponding fully coupled 6-DOFdynamic response was examined.

3.2.2. Longitudinal natural modes of motion. From theeigenmode analysis results in the relevant literature (seetable 7), three longitudinal dynamic modes of motion wereidentified. In most of the literature, one unstable oscillatorymode and two stable subsidence modes constitute the dynamicmodal structure in the longitudinal plane of motion.

To excite the longitudinal mode 1, an in-phase excitationof δub and δqb was given to the hawkmoth multibody model,and the corresponding dynamic responses of 6-DOF body-fixed flight states and global flight states are plotted in figure 7.As identified from the eigenmode analysis results in therelevant literature, the consequent response of the hawkmothmultibody model shows a diverging oscillatory dynamics inthe ub (tdouble = 7.5) and qb (tdouble = 7.5) states. As noted inSun et al (2007), a stability derivative Mu (pitching momentdue to the forward/backward velocity) seems to be the mainsource of the instability. Qualitatively, this mode of motion isa forward/backward sway motion with an in-phase pitchingmotion as in a falling leaf. Not only the ub and qb dynamics,but also the wb dynamics has a diverging oscillation whichaffects a ZG directional fall. All the lateral flight states areunaffected from the longitudinal disturbance which meansthat the lateral dynamics is decoupled from the longitudinaldynamics. A frequency domain analysis using the fast FourierTransform (FFT) shows that the oscillating ub–qb coupleddynamics has a dominant frequency component of 1.22 Hz,and the wb dynamics has a dominant frequency component of3.05 Hz. All three longitudinal states, ub, wb, and qb have thesame frequency component which is the flapping frequency,29.46 Hz. A graphical representation of the correspondingdynamic response of the longitudinal mode 1 is shown infigure 8, where �t between each frame was 3/4 of a wingbeatstroke (≈0.026 s).

To excite the longitudinal mode 2, an out-of-phaseexcitation of δub and δqb was given to the hawkmoth multibodymodel, and the corresponding dynamic responses of 6-DOFbody-fixed flight states and global flight states are plotted infigure 9. The eigenmode analysis in the relevant literatureexamined this longitudinal mode of motion as a stable fastsubsidence mode. The simulation result in this study showedthat this mode of motion is a forward translation with a pitchdown rotation as shown in figure 10. During approximatelyeight wingbeat strokes after the disturbance, this dynamicresponse had characteristics of the fast subsidence mode inub (thalf = 8.0) and qb (thalf = 3.0). However, the positiveMu stability derivative kept on affecting the ub–qb coupleddynamics by generating nose-up moment, hence altering theqb again in-phase with ub resulting in an unstable motion.After approximately six wingbeat strokes, the mode of motionagain changed to an unstable oscillatory mode as in the caseof longitudinal mode 1 (in-phase δub and δqb disturbance,figures 7 and 8). The lateral flight states were also unaffected bythe longitudinal disturbance, indicating a decoupled dynamics.An FFT result of this mode of motion had almost the samefrequency spectrum as the mode 1 excitation case: ub–qb

coupled dynamics had a frequency component of 1.22 Hz, wb

dynamics had a frequency component of 3.05 Hz, and all threelongitudinal states, ub, wb, and qb have a frequency componentof 29.46 Hz (the flapping frequency).

12

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

(ai ) (aii )

(bi ) (bii )

Figure 7. Longitudinal mode 1, 6-DOF dynamic response due to the disturbance: +δub, +δqb (coupled in-phase).

Figure 8. A graphical representation of the longitudinal mode 1, �tbetween frames is 3/4 of a wingbeat stroke. Thick arrows attachedto the hawkmoth body represent the initial disturbance (+δub, +δqb)given at the hovering condition.

Lastly, the longitudinal mode 3 was excited by givingthe hawkmoth multibody model a δwb disturbance alone. Thecorresponding dynamic responses of the 6-DOF body-fixedflight states and global flight states are plotted in figure 11.A graphical representation of the corresponding dynamicresponse to the disturbance combination is shown in figure 12.The eigenmode analysis in the relevant literature determinedthis longitudinal mode of motion as a stable slow subsidencemode in which the stability derivative Zw plays an importantrole (Sun et al 2007). The simulation result in this study showed

that this mode of motion started with a damped decay motionin the wb axis with a thalf of about 2.5 wingbeat strokes. Thenthe motion eventually diverged in the ub–qb coupled dynamics(figures 11 and 12) as the other two longitudinal modes ofmotion 1 and 2 exhibit. In addition, the δwb disturbance did notaffect any of lateral flight states, which indicate a decoupleddynamics. The frequency spectrum of this mode of motionwas almost the same as the cases above: wb dynamics had afrequency component of 3.05 Hz, ub–qb coupled dynamics hada frequency component of 1.22 Hz, and all three longitudinalstates, ub, wb, and qb had a frequency component the same asthe flapping frequency (29.46 Hz).

3.2.3. Lateral natural modes of motion. From the eigenmodeanalysis results in the relevant literature (see table 7),three lateral dynamic modes of motion were identified.Unlike the longitudinal dynamics, previous studies reporteddifferent stability characteristics of the lateral dynamic modes;eigenvalues near the imaginary axis in the s-plane were foundto be the cause of the difference, which implies that thesensitivity of the location of the eigenvalues near the imaginaryaxis is high with respect to the aerodynamic model used orassumptions during the simplified dynamic model formulation.

To excite the lateral mode 1, an in-phase excitation of δvb

and δpb was given to the hawkmoth multibody model, and

13

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

(ai ) (aii )

(bi ) (bii )

Figure 9. Longitudinal mode 2, 6-DOF dynamic response due to the disturbance: +δub, −δqb (coupled out-of-phase).

Figure 10. A graphical representation of the longitudinal mode 2,�t between frames is 3/4 of a wingbeat stroke. Thick arrowsattached to the hawkmoth body represent the initial disturbance(+δub, −δqb) given at the hovering condition.

the corresponding 6-DOF body-fixed flight states and globalflight states are plotted in figure 13. A graphical representationis shown in figure 14. The simulation result shows that theinitial δvb disturbance monotonically diverged with a tdouble ofapproximately six wingbeat strokes. The initial δpb disturbancesharply decayed in the beginning (thalf = 0.5) due to thepassive damping in the axis as shown in section 3.1.2, but soondiverged relatively slower than the vb dynamics. This unstabledynamic motion is in line with the results of Zhang and Sun(2010a). The divergence of the initial δvb disturbance seemsto affect not only the pb dynamics but also the rb dynamics

as shown in figure 13(aii). Unlike the results in the previoussection 3.2.2 on the longitudinal disturbance cases, here thelateral disturbance affected the longitudinal flight dynamicstates as well. As shown in figures 13 and 14, the ub, wb, andqb dynamics were all affected by the lateral initial disturbance.This coupling between planes of motion induced the excitationof the ub–qb coupled unstable longitudinal mode of motion(the main instability of the longitudinal plane of motion), andresulted in an unstable dynamic response in combination ofthe vb–pb coupled lateral instability and the ub–qb coupledlongitudinal instability. The FFT analysis showed that the mostprominent frequency component of the pb and rb dynamics wasthe flapping frequency of 29.46 Hz. A frequency componentof 1.22 Hz with a small amplitude also appeared in the pb andrb dynamics which seemed to be the effect of the couplingbetween the longitudinal and lateral planes of motion.

To excite the lateral mode 2, an out-of-phase excitation ofδvb and δpb was given to the hawkmoth multibody model,and the corresponding 6-DOF body-fixed flight states andglobal flight states are plotted in figure 15. The simulationresult shows that the initial δvb disturbance decayed to itshalf amplitude within approximately six wingbeat strokes. Theinitial δpb disturbance also damped out quickly at first (thalf =0.5) then decayed along with the vb dynamics, exhibiting astable dynamics. The rb dynamics was also influenced by the

14

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

(ai ) (aii )

(bi ) (bii )

Figure 11. Longitudinal mode 3, 6-DOF dynamic response due to the disturbance: +δwb.

Figure 12. A graphical representation of the longitudinal mode 3,�t between frames is 3/4 of a wingbeat stroke. Thick arrowsattached to the hawkmoth body represent the initial disturbance(+δwb) given at the hovering condition.

given disturbances but it converged with a rate almost thesame as pb dynamics. In terms of the coupling betweenthe planes of motion, there was a major coupling in the ub

longitudinal dynamics and little coupling in the wb dynamics.

The qb dynamics were relatively unaffected. Qualitatively,this mode of motion had a stable dynamics within a shorttime window of less than ten wingbeat strokes. However, asthe simulation proceeded the small amplitude of the coupledub dynamics prompted instability in the qb dynamics andeventually diverged in the longitudinal plane with the sameunstable mode with the longitudinal mode 1. A frequencydomain analysis showed that a major frequency componentof 29.46 Hz (flapping frequency) was found from the pb andrb dynamics. However the magnitude of the oscillation wasnot as high as that of the lateral mode 1 case. A graphicalrepresentation of the corresponding dynamic response to thedisturbance combination is shown in figure 16.

Lastly, the lateral mode 3 was excited with an out-of-phaseexcitation of δpb and δrb to the hawkmoth multibody model.This initial disturbance gave the hawkmoth multibody modela rotation around the body long axis (see figure 1(b)). Thecorresponding 6-DOF body-fixed flight states and global flightstates are plotted in figure 17, and a graphical representationof this mode of motion is depicted in figure 18. This modeof motion has been commonly examined as a stable fastsubsidence mode from the eigenmode analysis in the relevantliterature (table 7). In addition, the simulation result alsoshowed a fast decaying dynamics within a half wingbeatstroke both in the pb and rb dynamics (figure 17(aii)) as

15

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

(ai ) (aii )

(bi ) (bii )

Figure 13. Lateral mode 1, 6-DOF dynamic response due to the disturbance: +δvb, +δpb (coupled in-phase).

Figure 14. A graphical representation of the lateral mode 1, �t between frames is 3/4 of a wingbeat stroke. Thick arrows attached to thehawkmoth body represent the initial disturbance (+δvb, +δpb) given at the hovering condition.

examined in section 3.1. However, the vb dynamics was alsoexcited from the out-of-phase pure rotational excitation of δpb

and δrb (figure 17(ai)) due to the coupling effect resultingfrom a tilt of the resultant aerodynamic vector, and this vb

dynamics excitation then created an instability of pb and rb

dynamics again simultaneously in a long time window. Theub–qb coupled longitudinal instability also intervened due tothe slightly coupled ub from the lateral disturbances. The FFTresults showed frequency components similar to the mode 1case: pb and rb dynamics had a major frequency component of

29.46 Hz, and also a frequency component of 1.22 Hz with asmall amplitude due to the coupling between the longitudinaland lateral planes of motion.

3.2.4. Implication of disturbance analysis results. Based onthe understandings of the passive damping characteristicson each single DOF of the hawkmoth flight dynamics, weanalyzed the stability characteristics of the unconstrained freeflight condition, and table 8 shows a summary of the results. Inthe table, the dynamic responses of the perturbed hawkmoth

16

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

(ai ) (aii )

(bi ) (bii )

Figure 15. Lateral mode 2, 6-DOF dynamic response due to the disturbance: +δvb, −δpb (coupled out-of-phase).

Table 8. Summary of the results of the 6-DOF disturbance analysis; short time window indicates a time span (approximately ten wingbeatstrokes) right after the initial disturbance; here, thalf or tdouble indicate the decay or amplification time from the short time window results.

Plane of motion Mode Short time window thalf or tdouble Long time window

Longitudinal 1 ub–qb coupled unstable ub: tdouble = 7.5 ub–qb divergence (1.22 Hz)oscillatory (1.22 Hz) qb: tdouble = 7.5 wb divergence (3.05 Hz)

2 ub-qb coupled ub: thalf = 8.0 ub–qb divergence (1.22 Hz)subsidence qb: thalf = 3.0 wb divergence (3.05 Hz)

3 wb subsidence wb: thalf = 2.5 ub–qb divergence (1.22 Hz)wb divergence (3.05 Hz)

Lateral 1 vb divergence vb: tdouble = 6.0 vb–pb–rb divergencepb fast subsidence then slow divergence pb: thalf = 0.5 ub–qb divergence (1.22 Hz)

2 vb subsidence vb: thalf = 6.0 ub–qb slow divergence (1.22 Hz)pb fast subsidence then slow subsidence pb: thalf = 0.5

3 pb, rb fast subsidence, pb: thalf = 0.5 vb–pb–rb divergencevb divergence rb: thalf = 0.5 ub–qb slow divergence (1.22 Hz)

are viewed through two time windows: (1) a short time windowrepresenting a near-equilibrium range, and (2) a long timewindow representing a fully nonlinear range.

In comparison with the eigenmode analysis results in therelevant literature, the longitudinal modes of motion examinedin this study showed a similar dynamic modal structure whenviewed through the short time window. The three longitudinaldynamic modes which are an unstable oscillatory mode andtwo stable subsidence modes were clearly found. However,

the wb dynamics showed a prominent unstable oscillatorydynamics (3.05 Hz) coupled with the other two longitudinalvelocity states, whereas the eigenmode analysis results showeda decoupled dynamics: the effect of δwb in the eigenvectors oflongitudinal mode 1 and 2 was negligible, and reversely theeffect of δub in δqb in the eigenvectors of longitudinal mode 3was also negligible (Sun et al 2007, Cheng and Deng 2011).This coupling between wb and the other two longitudinal statesare more noticeable in the longitudinal mode 3. In the result

17

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

Figure 16. A graphical representation of the lateral mode 2, �tbetween frames is 3/4 of a wingbeat stroke. Thick arrows attachedto the hawkmoth body represent the initial disturbance (+δvb, −δpb)given at the hovering condition.

of this mode 3 (see figure 11), only δwb was excited as aninitial disturbance but the ub and qb dynamics were also excitedalmost at the same time. This result implies that the couplingbetween wb and the other two longitudinal states are muchstronger than identified through the eigenmode analysis withthe linearized system model of the hovering hawkmoth. Ifviewed through the long time window, we found that all threelongitudinal dynamic modes eventually falls into an oscillatorydivergence with coupled wb (3.05 Hz) and ub–qb (1.22 Hz)dynamics. However, there was no coupling effect between the

Figure 18. A graphical representation of the lateral mode 3, �tbetween frames is 3/4 of a wingbeat stroke. Thick arrows attachedto the hawkmoth body represent the initial disturbance (+δpb, −δrb)given at the hovering condition.

longitudinal and lateral planes of motion resulting from thedisturbances on the longitudinal plane.

The lateral modes of motion of the hawkmoth flightdynamics showed much more complex behavior than thatof examined by the eigenmode analysis technique (althoughthe main instability due to the coupled vb–pb dynamics iswell captured in this study). This was mainly due to thecoupling effect between the vb and rb dynamics that wereunderestimated by the previous eigenmode analysis with arelatively small Nv (yaw moment due to the sideslip velocity)stability derivative (Cheng and Deng 2011, Zhang et al 2012).

(ai ) (aii )

(bi ) (bii )

Figure 17. Lateral mode 3, 6-DOF dynamic response due to the disturbance: +δpb, −δrb (coupled out-of-phase).

18

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

We can see that not only in the lateral mode 3 where δpb

and δrb are simultaneously excited, but also in the other twolateral modes where only δvb and δpb were excited also showedstrongly coupled responses in rb–pb dynamics as shown infigures 13(aii), 15(aii) and 17(aii), which indicates a strongcoupling. In addition, we can infer that the δvb has a closerelationship with rb dynamics because, in the lateral mode 1and 2, the diverging (converging) vb dynamics induced thesame diverging (converging) rb dynamics in an oscillatorymanner (figures 13(a) and 15(a)).

This coupled oscillatory dynamics also can be found infigure 10 of Zhang et al (2012) that shows simulation results(from a directly coupled CFD and flight dynamics) of thehawkmoth compared with a linearized model. Here, we canfind a limitation of the linearized model with a wing-beat-cycleaveraging. Even though the actual dynamics has an oscillatingcharacteristics, the wing-beat-cycle averaging by its definitioneliminates all the within-wingbeat dynamics and only showsthe averaged values of it. Therefore, a possible dependencybetween two states is neglected and results in a relatively smallor almost zero stability derivatives such as Nv (yaw momentdue to the sideslip velocity) or Yr (sideslip force due to the yawrate). For example in this study, this underestimated couplingcaused a divergence in vb dynamics (figure 17) when therewas only an out-of-phase excitation of δpb and δrb that isknown to be the most stable fast subsidence mode from theeigenmode analysis due to the large passive damping in eachaxis of rotation.

Unlike the longitudinal mode of motion, the lateralmodes of motion exhibit a cross-coupling between planesof motion, i.e., the longitudinal and the lateral plane. Thiscross-coupling becomes prominent approximately after fiveto six wingbeat strokes (figures 13(ai), 15(ai) and 17(ai)),and affects the overall flight dynamics when viewed throughthe long time window. In all three lateral modes of motion,the initial lateral disturbances induced a slight perturbationin the longitudinal velocities that excited the ub–qb coupledlongitudinal instability, which eventually caused an unstabledynamic response in combination with the vb–pb coupledlateral instability. This cross-coupling between planes ofmotion occurs in the short time window implies that the high-order coupling terms have non-negligible effect even on thenear-equilibrium flight (hovering), which has been neglectedduring the linearization process of the 6-DOF full equationsof motion (such as equations (24) and (27) in Cheng andDeng (2011)). This particular cross-coupling might degradethe performance of the flight control designed based only onthe decoupled linearized dynamics assumption, thus furtherconsideration need to be made in the control design process.

In both the longitudinal and lateral planes of motion, wefound that all the growth rate of the instabilities has a timescaleof less than approximately seven wingbeat strokes to doublethe initial amplitude. It is known that the latency of the angularrate sensing mediated through the antennae is less than 0.5wingbeat stroke (Sane et al 2007) and the angular orientationdetection through the visual sensing has a larger latency ofone to three wingbeat strokes (Sprayberry 2009). Therefore,the open-loop dynamic analysis in this study indicates that the

hawkmoth has enough time to sense the perturbations in theattitude rate or angle, and to actuate wing motor or body partsto recover from the instability before the growth gets larger(as experimentally shown with real hawkmoth flight in Chenget al (2011)).

Among the unstable modes of motion of the hawkmothanalyzed in this study, the longitudinal instability due to thecoupled ub–qb dynamics, and the lateral instability due tothe coupled vb–pb dynamics were found to be the majorunstable modes. This result is in line with the passivedamping analysis on each isolated independent DOF insection 3.1, where the xsp-directional (pitch) rotation showedless damping approximately five times than the other tworotational DOFs, and the ysp-directional (sideslip) translationshowed less damping approximately three times than theother two translational DOFs. Therefore, in terms of the gustsusceptibility for a hovering hawkmoth, perturbations relatedto the Mu (pitching moment due to the forward/backwardvelocity) and Lv (roll moment due to the sideslip velocity)stability derivatives would more strongly excite instabilitieshence requires more control efforts than the other types ofperturbations.

4. Conclusion

This work investigated the 6-DOF flight dynamics and stabilitycharacteristics of the hawkmoth Manduca sexta. A multibodyapproach was used to consider the effects of the time varyinginertia tensor of all the body segments including two wings. Adirect time integration of the fully coupled 6-DOF nonlinearmultibody dynamics equations of motion was conductedwith an instantaneous aerodynamics without wingbeat-cycle-averaging. The study accomplished the following: (1) theinherent passive damping in each DOF was quantitativelyexamined and all the independent DOFs were found tobe passively stable; (2) while possessing passive stability,the pitch rotation and the sideslip translation were found tobe least damped DOFs; (3) the natural modes of motion of thehawkmoth multibody model were investigated by analyzingfully coupled 6-DOF dynamic responses to the force andmoment disturbance initial conditions; and (4) the longitudinalinstability due to the coupled ub–qb dynamics, and the lateralinstability due to the coupled vb–pb dynamics were found to bethe major unstable modes of the hawkmoth flight. Based on thepresent study, future works will focus more on the effect of thebody and wing compliance to the flight stability. As evidencedin experimental studies with the hawkmoth (Mountcastle andDaniel 2009, Dyhr et al 2013) and in a numerical study(Kim and Han 2013a), the effect of the flexibility in theaerodynamic surfaces and the flexible joints connecting thebody components is not negligible. By understandingthe principal dynamic characteristics of the hawkmoth, we canenhance our understanding in the stability of the flapping wingflight dynamics and utilize the knowledge to design MAVs witha biologically inspired control strategy.

19

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

Acknowledgments

The authors would like to thank the anonymous reviewersfor their constructive comments and criticisms that helped usto improve the quality of the present study. This work wassupported by the National Research Foundation grant fundedby the Ministry of Education, Science and Technology ofKorea (no. 2011-0015569). The first author thanks the projectof Global PhD Fellowship, which the National ResearchFoundation of Korea conducts from 2011. The authors thankDr Charlie Ellington (University of Cambridge) for providingthe Fourier coefficients in table 4 for the measured wingkinematics of the hovering hawkmoth.

References

Berman G J and Wang Z 2007 Energy-minimizing kinematics inhovering insect flight J. Fluid Mech. 582 153–68

Cheng B and Deng X 2011 Translational and rotational damping offlapping flight and its dynamics and stability at hovering IEEETrans. Robot. 27 849–64

Cheng B, Deng X and Hedrick T L 2011 The mechanics and controlof pitching manoeuvres in a freely flying hawkmoth (Manducasexta) J. Exp. Biol. 214 4092–106

Dickinson M H, Lehmann F-O and Sane S P 1999 Wing rotationand the aerodynamic basis of insect flight Science 284 1954–60

Dyhr J P, Morgansen K A, Daniel T L and Cowan N J 2013 Flexiblestrategies for flight control: an active role for the abdomenJ. Exp. Biol. 216 1523–36

Ellington C P 1984 The aerodynamics of hovering insect flight: II.Morphological parameters Phil. Trans. R. Soc. B 305 17–40

Ellington C P 2011 Private communication, University ofCambridge

Faruque I and Humbert J S 2010a Dipteran insect flight dynamics:part 1. Longitudinal motion about hover J. Theor. Biol.264 538–52

Faruque I and Humbert J S 2010b Dipteran insect flight dynamics:part 2. Lateral-directional motion about hover J. Theor. Biol.265 306–13

Gao N, Aono H and Liu H 2009 A numerical analysis of dynamicflight stability of Hawkmoth hovering J. Biomech. Sci. Eng.4 105–16

Gao N, Aono H and Liu H 2010 Perturbation analysis of 6DoF flightdynamics and passive dynamic stability of hovering fruit flyDrosophila melanogaster J. Theor. Biol. 270 98–111

Gear W C 1971 Simultaneous numerical solution ofdifferential-algebraic equations IEEE Trans. Circuit TheoryCT-18 89–95

Hedrick T L, Cheng B and Deng X 2009 Wingbeat time and thescaling of passive rotational damping in flapping flight Science324 252–5

Hedrick T L and Daniel T L 2006 Flight control in the hawkmothManduca sexta: the inverse problem of hovering J. Exp. Biol.209 3114–30

Hedrick T L and Robinson A K 2010 Within-wingbeat damping:dynamics of continuous free-flight yaw turns in Manduca sextaBiol. Lett. 6 422–5

Jantzen B and Eisner T 2008 Hindwings are unnecessary for flightbut essential for execution of normal evasive flight inLepidoptera Proc. Natl Acad. Sci. 105 16636–40

Kim J-K and Han J-H 2013a An investigation of 6-DOF insect flightdynamics with a flexible multibody dynamics approach Proc.SPIE 8686 868611

Kim J-K and Han J-H 2013b Control effectiveness analysis of thehawkmoth Manduca sexta: a multibody dynamics approachInt. J. Aeronaut. Space Sci. 14 152–61

Kim J-K, Lee J-S and Han J-H 2012 Passive longitudinal stability inornithopter flight J. Guid. Control Dyn. 35 669–73

Lee J-S, Kim J-K, Han J-H and Ellington C P 2012 Periodic tailmotion linked to wing motion affects the longitudinal stabilityof ornithopter flight J. Bionic Eng. 9 18–28

Lee J-S, Kim J-K, Kim D-K and Han J-H 2011 Longitudinal flightdynamics of bioinspired ornithopter considering fluid-structureinteraction J. Guid. Control Dyn. 34 667–77

Mountcastle A and Daniel T L 2009 Aerodynamic andfunctional consequences of wing compliance Exp. Fluids46 873–82

O’Hara R P and Palazotto A N 2012 The morphologicalcharacterization of the forewing of the Manduca sexta speciesfor the application of biomimetic flapping wing micro airvehicles Bioinspir. Biomim. 7 046011

Orlowski C T and Girard A R 2011a Modeling and simulation ofnonlinear dynamics of flapping wing micro air vehicles AIAA J.49 969–81

Orlowski C T and Girard A R 2011b Averaging of thenonlinear dynamics of flapping wing micro air vehiclesfor symmetrical flapping Proc. 49th AIAA AerospaceSciences Meeting including the New Horizons Forum andAerospace Exposition (Orlando, FL, USA, 4–7 Jan.) AIAA2011-1228

Parks P, Cheng B, Hu Z and Deng X 2011 Translational damping onflapping cicada wings Proc. 2011 IEEE/RSJ Int. Conf. onIntelligent Robots and Systems (San Francisco, CA, USA,25–30 Sept.) 574–9

Pfeiffer A T, Lee J-S, Han J-H and Baier H 2010 Ornithopter flightsimulation based on flexible multi-body dynamics J. BionicEng. 7 102–11

Sane S P 2003 The aerodynamics of insect flight J. Exp. Biol.206 4191–208

Sane S P and Dickinson M H 2001 The control of flight forceby a flapping wing: lift and drag productionJ. Exp. Biol. 204 2607–26http://jeb.biologists.org/content/204/15/2607.abstract

Sane S P and Dickinson M H 2002 The aerodynamic effects ofwing rotation and a revised quasi-steady model for flappingflight J. Exp. Biol. 205 1087–96http://jeb.biologists.org/content/205/8/1087.abstract

Sane S P, Dieudonne A, Willis M A and Daniel T L 2007 Antennalmechanosensors mediate flight control in moths Science315 863–6

Shyy W, Aono H, Chimakurthi S K, Trizila P, Kang C-K,Cesnik C E S and Liu H 2010 Recent progress in flappingwing aerodynamics and aeroelasticity Prog. Aerosp. Sci.46 284–327

Sprayberry J D H 2009 Responses of descending visually-sensitiveneurons in the hawkmoth, Manduca sexta, to three-dimensionalflower-like stimuli J. Insect Sci. 9 7

Springthorpe D, Fernandez M J and Hedrick T L 2012Neuromuscular control of free-flight yaw turns in thehawkmoth Manduca sexta J. Exp. Biol. 215 1766–74

Sun M, Wang J and Xiong Y 2007 Dynamic flight stability ofhovering insects Acta Mech. Sin. 23 231–46

Taha H E, Hajj M R and Nayfeh A H 2012 Flight dynamics andcontrol of flapping-wing MAVs: a review Nonlinear Dyn.70 907–39

Taylor G K and Thomas A L R 2003 Dynamic flight stability in thedesert locust Schistocerca gregaria J. Exp. Biol.206 2803–29

Taylor G K and Zbikowski R 2005 Nonlinear time-periodic modelsof the longitudinal flight dynamics of desert locustsSchistocerca gregaria J. R. Soc. Interface 2 197–221

Truong Q T, Nguyen Q V, Truong V T, Park H-C, Byun D-Yand Goo N-S 2011 A modified blade element theory forestimation of forces generated by a beetle-mimicking flappingwing system Bioinspir. Biomim. 6 036008

20

Bioinspir. Biomim. 9 (2014) 016011 J-K Kim and J-H Han

Usherwood J R and Ellington C P 2002 The aerodynamics ofrevolving wings: I. Model hawkmoth wings J. Exp.Biol. 205 1547–64http://jeb.biologists.org/content/205/11/1547.abstract

Willmott A P and Ellington C P 1997 The mechanics offlight in the hawkmoth Manduca sexta: I. Kinematicsof hovering and forward flight J. Exp. Biol.200 2705–22http://jeb.biologists.org/content/200/21/2705.abstract

Wu J H and Sun M 2012 Floquet stability analysis of thelongitudinal dynamics of two hovering model insects J. R. Soc.Interface 9 2033–46

Wu J-H, Zhang Y-L and Sun M 2009 Hovering of model insects:simulation by coupling equations of motion withNavier–Stokes equations J. Exp. Biol. 212 3313–29

Zhang Y-L and Sun M 2010a Dynamic flight stability of a hoveringmodel insect: lateral motion Acta Mech. Sin. 26 175–90

Zhang Y-L and Sun M 2010b Dynamic flight stability of hoveringmodel insects: theory versus simulation using equations ofmotion coupled with Navier–Stokes equations Acta Mech. Sin.26 509–20

Zhang Y-L, Wu J-H and Sun M 2012 Lateral dynamic flight stabilityof hovering insects: theory versus numerical simulation ActaMech. Sin. 28 221–31

21