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Passive Dynamics of High Frequency Bat Wing

Flapping with an Anisotropic Membrane

Michael Zheng, S.M.Hadi Sadati*, Pendar Ghalamchi and Thrishantha Nanayakkara

Center for Robotics Research (CoRe), Department of Informatics

King's College London

London, United Kingdom

Seyedmohammadhadi.sadati@kcl.ac.uk

Abstract—We investigate how unmanned aerial vehicles

(UAVs) with flexible wings can be designed to exploit the

aeroelasticity of wing deformation that is present in bat wings,

with a view to improve the efficiency of flight. We constructed a

robotic bat wing with fully passive elastic wing-folding properties.

The robotic wing is powered by a gearbox running two

synchronised motors, effectively providing one degree of motion:

the upstroke and down-stroke of the wing. Through numerical

simulations and setup experiments, we observed that by

integrating a span-wise elastic network into the bat wing, we were

able to achieve passive wing-folding that mimics the 8-shape wing-

folding seen in bats’ high speed flight. This way, we were able to

reduce the complexity and additional actuation associated with

wing-folding in a robotic wing.

Keywords—Bat; Flapping; Passive; Anisotropic Membrane;

Simulation; Experiment

I. INTRODUCTION

The bat wing has an anisotropic stiffness distribution of

elasticity. It exhibits high stiffness strength along the digits,

compared to high failure strain in the perpendicular direction [1].

This distribution not only prevents high shear forces from

developing between the bones and the wing membrane, but also

helps to store energy by extending the membrane during the

down-stroke. It is interesting to note that exploitation of passive

dynamics arising from special distributions of body stiffness is

seen not only in flapping flight, but also in other forms of

locomotion such as swimming [2]. In [2], it has been

demonstrated that a dead trout can passively start to swim

against a stream of water by generating productive vortexes

purely based on passive resonance of constructive body

movements in hydraulic turbulence behind a cylindrical obstacle

placed in the path of a water stream. These biological examples

suggest that the management of the distribution of body

elasticity plays a vital role in efficient locomotion. This opens

the scientific question as to how the morphology of a deformable

winged UAV should be designed to be able to tune to the flight

environment without off-line planning.

By mimicking the properties of the bat wing, we may be able

to increase the efficiency of airborne vehicles, an exciting

prospect from an engineering perspective. Recently,

advancements in engineering materials, actuators and controls

have allowed us to consider mimicking flight behaviour in the

natural world [3]. There has been analysis on a resonant drive to

reduce the average battery power consumption for DC motor-

driven flapping wing robots [4]. They developed a methodology

using a non-dimensional analysis of a motor driven compliant

system. Using this, they were able to guide the mechanical

design and predict the optimal operating point based on power

efficiency. A prototype of a biologically inspired UAV imitating

the hovering flight of humming birds and dragonflies shows

great potential in developing a high performance flapping micro-

UAV [5]. There have been previous attempts to develop a bat

inspired UAV, such as BATMAV [6], RoboBat [7], and recently

in [8][9]. Here the authors aim to develop flexible wings

actuated by artificial muscles, rather than exploiting the aero-

elastic properties of the bat wing to improve the efficiency of

flight; while similar research on insect size flyers showed

advantages of anisotropic membrane in MAV’s flight

performance and control [10].

When considering the application of flapping flight, it is

advantageous to reduce the weight of the ornithopter as much as

possible. We seek to eliminate the need for active actuation of

wing folding during flight which reduces the energetic cost on

the upstroke of flapping flight [11]. Therefore, we look to the

aeroelasticity of the bat wing membrane as inspiration to

construct a passive wing folding technique that utilises the

elasticity of the wing.

Flyers’ flapping kinematics are different depending on the

wing structure and flight speed [10][12] for which the

aeroelasticity mechanisms should be investigated individually.

Researchers at Brown University have developed a 3D printed

articulated bat wing skeleton with a silicon membrane [13]. This

study in particular highlights the advantages of wing folding, not

only on energy conservation, but also in altering the

aerodynamic qualities of the wing. Our aim is to build upon this

foundation by focusing on the aeroelasticity of the bat wing as

major factor in efficient high speed flight, in particular, how we

can derive inspiration into wing-folding kinematics of high

frequency flapping.Different species of bats have different

properties [1]. Our aim is not to mimic the flight of bats, or one

particular species. Rather we seek a solution with feasible

applications in robotics and biomimetic, as well as the potential

for further study. Here, we investigated the passive role of

anisotropic wing membrane stiffness in reducing the complexity

of the flapping motion.

978-1-4799-4598-6/14/$31.00 © 2014 IEEE

II. MODELLING

A. Kinematics

Fig. 1 shows our 3D bat wing model with three links (arm,

forearm and hand) and four degrees of freedom (DOF). The

shoulder joint is attached to the base frame origin. Base frame

has an inclination equals to the wing geometrical angle of attack

(AoA) () with gravity direction (g). The wing is approximated

by a trapezoid with variable span but parallel root (cr) and tip

chords (ct). The membrane is modelled a series of linear springs

as the elastic elements along local chord-wise sweep directions.

The model parameters are derived using a bat wing design [13]

using Solidworks 2013. The shoulder rotation about the base

frame x-axis (q1) assumes a cosinusoidal form as in (1) and other

three joints are not actuated actively and their axes of rotation

are the local frames z-axis (q2, q3, q4). The centre of mass (COM)

for each link is centred at its midpoint. For simplicity and based

on observation of a real bat flapping, the following constraints

are being used between angles to reduce the under actuated order

of the model to one.

()=−A.cos(ω.+)+

−2×

+20°−2×=0

Linear and angular velocity vectors of each element derived

using 3×3 rotation (R(qi)) and 4×4 transformation matrices

(TR(R,rf)) as in (2) where qi is the angle of rotation, rf is the row

offset vector between two consecutive joint, rc is the COM

absolute position vector (w.r.t. base frame) represented in base

frame and

ω

ω

is the absolute angular velocity vector represented

in local frame (attached to each link at its first joint with the link

along its y axis and the first joint rotates along its z axis) for each

link. a, f and h are subscripts for frames attached to, and the

properties of the arm, forearm and hand links respectively. li is

the link length.

()=10 0

0cos (

)−sin (

)

0sin (

)cos (

)

()=cos ()−sin (

)0

sin ()cos (

)0

001

(,)=

0× 1×

=().[̇00

]+[00̇]

=().+[00̇]

=().+[00̇]

=(().(),[000

])

=((),[0 0])

=((),[0

0])

=.[0 /2 0]

=..[0 /2 0]

=...[0 /2 0]

B. Dynamics

We use a matrix representation for the equation of motion

(EOM) in (3) using the TMT method in (4) to derive the dynamic

model because of simplicity and efficient numerical evaluation

as in [14]. q= [q1, q2, q3, q4] is the generalised coordinates vector,

M is the inertial matrix in the Lagrange equation, Tcn is the

constrains Jacobian matrix on the generalised coordinates, λ is

the Lagrange multipliers vector, dEOM is the vector of the other

terms in the Lagrange equation, and dλ is the vector of the other

terms in the constraint equations vector (c(q)) derivatives. c(q)

is differentiated multiple times, so its order of differentiation

equals that of the EOM, namely two (6).

0+̈

−=

Fig. 1. Schematic model of a bat model: (a) base and wing frames, and apparent air stream due to free air stream and wing flapping motion at

quarter

chord of each element; (b) wing shematic with span-wise elastic elements; (c) aerodynamic forces on each element

. b is the subscripts for the base

frame and w is for the one attached to the wing surface whose origins are at the shoulder joint. Geometrical and real AoA are showed in left

figure

while they are approximated to be equal at the right-down figure.

(a)

(b)

(c)

Here T is the Jacobian matrix of xCOM which is the

combination of links COM position (rc) and pseudo-rotation

(∫.d) 3×1 vectors in terms of the generalised coordinates,

mi is ith link mass, li is the ith link 3×3 inertia matrix, fg is the

vector of conservative forces acting on each element such as

gravity, and Qc and Qnc are the virtual work vectors of other

conservative (here Qk accounts for membrane elasticity (k)

effect) and non-conservative forces (here Qdamp accounts for

joints’ rotational viscous damping (cv) and Qaero for aerodynamic

forces (lift (fL) and drag (fD) effect) in generalized coordinate

space [14].

[×]=[∫∫∫]

[×]=()/

[×]=−((.̇)/).̇

[×]=diag[,,,[×] , …], (: a,f, h)

=−g.[.sin(),0,.cos(),0,0,0,…]×,(:a,f,h)

=+ , =

=−c.̇×

=

=[−.]++

There is no need to find pseudo-rotation vectors. Here since

we have (=.̇) as the combination of links’ absolute linear

and angular velocity vectors (vb), we simply rearrange the

absolute angular velocity vectors in (2), in the form of (=

.̇) and then place the 3×4 coefficient matrices (Tω) at their

right place in T which is a 18×4 matrix (elements in the 9-18th

rows).

Tcn is derived based on the three constraints in (1).

̈()=̈+

=()

=1000

0−210

00−2 1

=−[(+)00

]

C. Membrane Model

Membrane is modelled as a trapezoid by integrating span-

wise elastic network of linear springs with elastic coefficient k

and initial length l0 as in Fig. 1. Membrane is rigid in chord-wise

direction and does not have lateral deflection normal to the wing

surface. Qc=Qk can be found from integrating springs’ virtual

work by having each ones deflection (lk-l0), direction (rk) and

attachment position to the wing tip (rtk) as in (6). Here rtw is hand

tip position vector in wing frame, b is wing span, lk is springs

instantaneous length,

η

is chord length ratio, xlr and xlt are

leading edge root and tip offset position from shoulder joint and

hand tip along x-axis respectively,

Λ

l,

Λ

t and

Λ

x are sweep angle

at leading edge, trailing edge and any arbitrary position along x-

axis, and fk is each spring’s force vector. We use cos(

Λ

x)≈1 and

tan(

Λ

x)≈sin(

Λ

x)≈

Λ

x approximations to reduce the complexity of

integration.

=((),[000

])...[0

0]

=2.,

≈/2

=/

≈−2.( −(+)) /

≈−2.((−)+−(−))/

= .(−)/(−), ≥

.(−)/((−)−) , <

=−().[10

]

=().[−(−)+.(−(−)) /2 0]

[×]=∫.(−).(()/).

().d

D. Aerodynamic Model

Bat wing is approximated as a trapezoidal thin airfoil whose

aerodynamic properties addressed in common aerodynamic

handbooks widely using blade element theory and quasi-steady

method [15]. Due to flapping motion, local effective AoA (

r)

along wing span alters as the magnitude of normal velocity of

the wing element increases. For simplicity effective and

geometrical AoA (

a) are considered equal and cos(

r)≈1 and

sin(

r)≈

r which is feasible for low AoA, high speed flapping

and forward flight, however the change in local air stream along

the wing span is considered. Aerodynamic forces are exerted on

quarter-chord from leading edge at each element. Total normal

(fnw) and tangential (ftw) aerodynamic forces are derived based

on drag (CD) and lift coefficients (CL) of a finite 3-D trapezoidal

wing as in [15] to find the vector of centre of pressure (CoP)

(rpw), where the instantaneous total force can be considered to

be exerted, all in the wing frame. Here AR is the wing aspect

ratio, cy is element chord length,

Λ

0.25 is the local quarter-chord

sweep angle, a0 and a are lift curve slope for 2-D infinite and 3-

D finite thin airfoil, ew is the Oswalt (span) efficiency factor, dfL

and dfD are each element drag and lift force magnitudes,

ρ

is the

air density, and mx and my are momentum of aerodynamic forces

about wing frame x and y axes.

≈α,=2./(

.(1+))

=.(1−2..(1−)/)

tan(.)=tan()−(1−)/(.(1+))

=./(1+1+(./(.cos (.)))),

=

C=.,C

=C+C

/(..e)

d=.C.(.̇

+

)..d/2

d=.C.(.̇

+

)..d/2

= (d.α+d)

/

.d

= (d+d.α)

/

.d

=.(d.α+d)

.d

= (+.tan()−/4).(d.α+d)

/

.d

=[/ / 0]

y element of rpw can be find by solving momentum around z

axis due to ftw too. fnw, ftw and rpw elements are derived in (8).

=(C+C)(1+3)̇

+24

(1+)/192

=(C−C)(1+3)̇

+24

(1+)/192

=3(1+4)̇

+40(1+)

10(1+3)̇

+240(1+)

/ =((3(1+4)tan()−(6+3+1))̇

+⋯

…40(+1)tan()

−40(++1)

)…

…/(10̇

(1+3)+240

(1+))

The virtual work of aerodynamic forces in term of

generalized coordinates (Qaero) can be derived as in (9). Here faero

is aerodynamic force vector and rp is CoP position vector, both

represented in base frame.

=().[ 0−

.sign(̇)]

=().

[×]=(()/).

T, d, dEOM, Qk and Qaero are derived analytically using Maple

software and not represented here.

= , dEOM and (3)

are evaluated numerically afterward to be integrated using a

Runge-Kutta 4th order method.

E. Numerical Simulation

The assumptions and model parameters of the simulation are

shown in table 1. Fig.s 2 and 3 show the joints angle and the

absolute drag force vector for two consecutive flapping cycles

(0.8 second). Aerodynamic force (faero) is symmetric with

vertical component mean value near zero due to symmetric up-

stroke and down-stroke motion. It shows the wing passively

folds a little while it is flapping and wing tip follows an 8-shape

path due to the membrane elasticity and links’ inertia and weight

(Fig. 4). A sequenced graph of the wing is shown in Fig. 4 too.

III. DESIGN AND IMPLEMENTATION

A. The Bat Wing Design and Implimention

Our wing skeleton is based upon previous research [13]. During

the design phase, we opted to make a few adjustments to this

specification, particularly with regards to the design of the joints.

As our wing folding mechanism is not mechanically actuated,

we were able to reduce the number of actuating mechanisms to

one gearbox. This gearbox controls the upstroke and down-

stroke of the wing only, the other wing movements are actuated

via the elastic network of the wing. The completed design was

3D printed using ABS plastic. We created a network of 0.5mm

elastic threads, similar to the layout of fibre bundles of a bat

wing (Fig. 4) [1]. We then attached 0.25mm latex sheeting to

this network using cyanoacrylate glue, and polyvinyl acetate

glue as sealing. This elastic network serves as a structure for the

wing membrane to attach on to. It allows the wing membrane to

mimic the high stiffness strength along the digits of a live bat.

We attached elastic across the wing so as to allow the elastic to

maintain the constraints in (1). Our bat wing is powered by a

modified Tamiya 70097 twin-motor gearbox using a 58:1 gear

ratio. The gearbox was altered so that the output shafts operate

in unison with each other. The bat wing is controlled with an

Arduino Uno microcontroller board, using C++.

TABLE I. MODEL AND SIMULATION PARAMETERS

m

a

11e-3 [Kg]

x

lr

0.2× c

r

[m]

m

f

19.8e

-3 [Kg]

x

lt

0.5× ct [m]

m

h

21e

-3 [Kg]

l

0

5.0e

-2 [m]

l

a

34e

-3 [m]

k

2.7e1 [N/m]

l

f

62e

-3 [m]

c

v

0.0 [Nm/s]

l

h

94e

-3 [m]

a

3°

[deg]

c

r

6

.3e-3 [m]

C

D0

0.05

c

t

1.4e

-3 [m]

e

w

1

ρ

1.2041

[Kg/m3]

g

9.81

[m/s2]

v∞

5.0 [m/s]

l

COM

l

i / 2 (i: a, f, h) [m]

I

a (Arm inertia matrix)

diag[3.3, 0.017, 3.3]×1e

-5 [Kg.m2

]

I

f (Forearm int. matrix)

diag[6.2, 0.076, 6.3] ×1e

-5

[Kg.m

2

]

Ih (Hand inertia matrix)

diag[1.0, 0.079, 1.1] ×1e

-5

[Kg.m

2

]

Flapping

parameters:

A = 70°

,

= 5π,

q

= -10°

Fig. 2. Joint rotations for two flapping cycles. q1

for the arm

about the

base frame x-axis and q2, q3 and q4 are for the arm

, elbow and

wrist rotations about the local frame z-axes

Fig. 3. Aerodynamic force components (faero

) [N] for two consecutive

flapping cycles.

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

q

t [s]

q1

q2

q3

q4

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

t [s]

FD - x

FD - y

FD - z

B. Set-up Experiments Results

We applied a low pass filter to the data and plotted a time

series of both motors simultaneously (Fig. 5). We also plotted a

return map for peaks in the data (Fig. 6). In order to observe how

the limit cycle of the motor settles, the first half of the data for

each motor is plotted in one colour (magenta and cyan), and the

latter half in another (red and blue). We can see that the wing

quickly settles down to a steady state. In order to observe the

movement patterns of the wing, we attached an LED to the tip

of the wing, and took 10 second long exposure photographs of

the wing in motion at different speeds (Fig. 7). The wing extends

upon the down-stroke, and folds mid-upstroke. One interesting

observation is the extension that can be seen at the top of the

upstroke, similar to the motion of a real bat. The wing will then

extend during the down-stroke.

IV. DISCUSSION

In this paper we focused on studying the role of anisotropic

stiffness distribution in the bat wing membrane on the efficiency

of upstroke. Biological literature suggest that the elastic

membrane helps the bat to exploit passive collapsing of the wing

at the end of the downstroke leading to reduced drag force during

upstroke. Through numerical simulations and experiments we

show that the anisotropy of the membrane contributes to such

efficiency gains only at high speeds of flapping. We observed an

eight-shaped passive elongation-collapsing steady state attractor

at roughly 198 rpm speed of the motors. Though we expect the

attractor to change shape with larger loop areas at higher speeds,

we could not perform such experiments due to hardware

limitations. Meanwhile, it proved difficult to simulate and

construct a robotic wing that emulated the wing folding

kinematics of a real bat using only one degree of actuated

motion. Besides we can look at various wing alterations,

possibly through computer simulation. For that means, a more

precise dynamic model of the wing is needed that can

profoundly show the characteristics of the wing membrane.

We showed that mimicking the behaviour of nature, such as

the bat wing skin, can have practical and feasible applications in

robotics. Further work could look into optimization of the

structural design parameters to improve the passive folding

mechanism in moderate and low flapping speed. Besides,

different elastic networks can affect not only the timing and

behaviour of wing folding in relation to a wing-beat, but also the

flight properties of flapping wing UAVs. One such application

could be the role of this elastic network in the UAV’s ability to

perform energy efficient quick and precise manoeuvres.

Fig. 5.

Elastic network mimicking the layout of fibre bundles of a bat

wing. Humerus: 37.2mm, Radioulna: 68.1mm, Digit I: 10.0mm,

Digit II: Metacarpal

29.7mm, Proximal phalanx 6.9mm, Middle

phalanx 4.1mm, Distal phalanx 2.6mm. Digit III: Metacarpal

39.9mm, Proximal phalanx 29.1mm, Distal phalanx 37.9. Digit

IV: Metacarpal 42.61mm, Proximal phalanx 22.1mm, Distal

phalanx 22.9mm. Leg: Femur 26.2mm, Tibia 31.5mm.

Fig. 4. 8-shape path of the hand tip projection on y-z plane for a full cycle (left). The intersection of the 8-shape path is maximized (middle). S

equential

representation of small passive folding in upstroke in 3D space (right).

V. CONCLUSION

We designed and built a robotic bat wing with an anisotropic

membrane capable of performing the 8-shape wing-folding seen

in bats’ high speed flight passively. We can collect input power

data from the wing, and observe wing-beat patterns. We

conclude that the wing must maintain a minimum flapping

frequency in order to exhibit its wing extending and folding

behaviour effectively. The research potential of the anisotropic

stiffness distribution of the bat wing and its practical

applications is great. This robotic wing serves as a solid

foundation for further exploration into the possibility and

benefits of integrating this behaviour into flapping wing UAVs.

ACKNOWLEDGMENT

The authors would like to thank the UK Engineering and

Physical Sciences Research Council (EPSRC) project REINS,

grant no. EP/I028765/1, and the Seventh Framework Program of

the European Commission in the framework of EU project

STIFF-FLOP, grant agreement 287728.

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Fig. 6.

Time series of current data after the application of a low pass

filter for downstroke and up-stroke motors

Fig. 7. Return map for peaks in the data for upstroke and downstroke

motors, showing the steady state variability of the up-

stroke and

downstroke motor currents.

Fig. 8. Inverted black and white images depicti

ng long exposures of

robotic wing flapping at different speeds. a) Shows the wing

flapping at a speed such that no wing extension and folding can

be seen. b) shows the wing flapping at a speed high enough such

that the wing passively extends and folds. The

red arrows

correspond to the downstroke, and the blue arrows correspond to

the up-stroke.