SHAPE REPRESENTATION AND A MORPHING SCHEME TO

SUPPORT FLAPPING WING RESEARCH

Mohammad Sharif Khan and Tapabrata Ray

School of Engineering and Information Technology, University of New South Wales, Canberra 2610, Australia

Keywords:

Flapping wing, Shape representation, Shape matching.

Abstract:

Wing geometry is one of the most important factors that affects the performance of a ﬂapping wing. The

shape of insect wings and their nature of ﬂapping varies across insect species. In order to gain an in-depth

understanding of ﬂapping ﬂight with an aim to identify optimal wing shapes, there is a need for an universal

and ﬂexible shape representation scheme that is amenable to optimization. The paper presents a methodology

to represent boundaries of insect wings which can be subsequently morphed via an optimization algorithm.

The shapes are represented using B-splines, wherein the control points representing the shapes are repaired

and subsequently evolved within an optimization framework. Twelve insect-wing shapes have been used to

test the performance of the proposed method in the context of shape matching.

1 BACKGROUND

Research into ﬂapping wings have gained signiﬁcant

attention in the last two decades (Donoughe et al.,

2011; Nguyen et al., 2010). The wing area, wing

span, aspect ratio, stiffness and ﬂapping frequency are

factors that are known to greatly inﬂuence the perfor-

mance in terms of thrust and lift generation (Brugge-

man, 2010).

In the context of ﬂapping wing research, most

of the papers attempt to study the effects of kine-

matics. There are only a small number of papers

that have considered wing shapes (Altshuler et al.,

2004; Phillips et al., 2010; Yuan et al., 2010; Ou

and Jameson, 2011; Ou et al., 2011). An analy-

sis of aerodynamic forces of revolving hummingbird

wings and wing models is reported in (Altshuler et al.,

2004), wherein the planform of the wing model was

based on image of the left wing of a female ruby-

throated hummingbird (Archilochus colubris) with a

wing length of 46.5 mm and an aspect ratio of 7.72.

An experimental investigation of the effects of plan-

form on the ﬂow structures generated by an insect-

like ﬂapping wing in hover is presented in (Phillips

et al., 2010). Four planform shapes with a constant

area and aspect ratio of approximately 6 were con-

sidered: reverse-ellipse, rectangle, four-ellipse and

ellipse. Numerical simulations were performed in

(Yuan et al., 2010) for a symmetrical NACA0005 air-

foil in combined pitching-plunging motions at low

Reynolds numbers. Three-dimensional simulations

of the two-dimensional airfoils with rectangular plan-

form have been carried out with an Eppler61 air-

foil and an oscillating NACA0012 airfoil at Reynolds

number 46,000 and 40,000 respectively in (Ou and

Jameson, 2011).

It is interesting to observe that in the ﬁeld of ﬂap-

ping wing research, there is a move towards modeling

insect like wing planforms with an aim to understand

their effects on propulsive characteristics. The work

outlined in this paper aims to provide a mechanism to

represent and optimize wing shapes in order to un-

derstand the effects of wing shapes on the propul-

sive characteristics of a ﬂapping ﬂight. To this effect,

we developed a ﬂexible shape representation scheme.

The ﬂexible shape representation scheme when cou-

pled with an optimization algorithm forms an useful

tool that can be used to understand and answer why

certain wing shapes have evolved within certain class

of insects. It is important to highlight that both the

efﬁciency of the optimization algorithm and the ﬂex-

ibility of shape representation scheme require serious

attention. A lack of ﬂexibility in shape representation

will limit the evolution of various shapes, while an

inefﬁcient optimization algorithm will require evalua-

tion of numerous shapes prior to its convergence, both

of which are not desirable (Khan et al., 2011).

The boundary of a shape (such as a wing) can be

represented using one of the following schemes: im-

plicit polynomials (Landa et al., 2010), active con-

494

Khan M. and Ray T. (2012).

SHAPE REPRESENTATION AND A MORPHING SCHEME TO SUPPORT FLAPPING WING RESEARCH.

In Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods, pages 494-499

DOI: 10.5220/0003840304940499

 SciTePress

tours (Xu and Prince, 1998), cubic splines (Rogers

and Adams, 1990), Bezier curves (Rogers and

Adams, 1990) and B-spline curves (Cox, 1971;

Riesenfeld, 1972). Among the above listed repre-

sentations, B-spline curves have been most widely

used as it ensures smoothness, compactness, local

shape control, and afﬁne transformation invariance

(Mongkolnama et al., 2006). A B-spline representa-

tion has been used in the present study wherein the

control points of the curve are identiﬁed through an

optimization algorithm. A novel repair scheme is in-

troduced and embedded to ensure generation of valid

shapes (one without self intersection). The non-global

behavior of B-spline curves makes it attractive in the

realm of shape representation. The B-spline basis

also allows the order of the basis function and hence

the degree of the resulting curve to be changed with-

out changing the number of deﬁning polygon vertices

(Rogers and Adams, 1990). Another advantage of the

B-spline curve is its strong convex-hull property. It

also possesses the variation diminishing and afﬁne in-

variance properties.

In the context of shape matching, the aim is to

identify a shape that is similar to a target shape.

While methods based on active contours are com-

monly used in the ﬁeld of pattern recognition (Xu and

Prince, 1998), it must be highlighted that such meth-

ods use local information to update location of the

points representing its boundary and hence not suit-

able for black-box optimization problems. In the cur-

rent black-box application however, a single scalar er-

ror value (matching error) is only available to direct

the search. In the present study, we assume that the

length and the width of a common box enclosing all

insect wings are known.

Rest of the paper is organized as follows. The

method proposed in this paper is presented in Sec-

tion 2. The results are reported in Section 3. Finally,

some concluding remarks are provided in Section 4.

2 PROPOSED METHOD

In the proposed method, the number of control points

required to represent the shape, the dimensions of the

box enclosing the target shape and the centroid of the

target shape are assumed to be known. The variables

of the optimization problem are the x and y coordi-

nates of the control points, the range of which are

the same as the dimensions of the enclosing box. Ev-

ery solution generated through the process of initial-

ization or recombination is repaired, wherein the se-

quence of control points are changed to obtain a non-

intersecting polygon net while maintaining the spe-

ciﬁc values of its coordinates. The concept is illus-

trated using Figure 1, 2 and 3, where the original ran-

domly generated control polygon net and formation

of the convex-hullare presented alongside its repaired

form. Firstly, a convex-hull is generated using a set of

control points (Figure 2). Thereafter, the points (lying

inside the convex-hull) nearest to their adjacent edges

are inserted to generate the non-intersecting control

polygon net (Figure 3).

Figure 1: Position of initial control points without repair.

Figure 2: Convex-hull formation with initial control points.

Figure 3: Position of initial control points after repair.

2.1 Initialization and Repair Strategy

The number of control points (N) used to describe a

shape is a user deﬁned parameter. Since the limits

of x and y coordinates i.e. the space enclosing the

target shape is known, a random solution is created

and repaired during the phase of initialization.

2.2 Matching Error

Since the centroid of the target shape is known, the

SHAPE REPRESENTATION AND A MORPHING SCHEME TO SUPPORT FLAPPING WING RESEARCH

495

shape generated using the repaired solution (re-

ordered set of control points) is shifted such that its

centroid matches the centroid of the target shape.

The maximum of Euclidean and Hausdorff distance

is computed using the generated shape and the target

shape.

2.3 Optimization Algorithm

2.3.1 Evolution

The pseudo-code of the proposed optimization is pre-

sented in Algorithm 1 and 2. In the proposed algo-

rithm, two different evolution strategies are used for

generating the offspring population. These are:

Algorithm 1: Proposed algorithm.

1: pop

= Initialize {Assumption of number of con-

trol points}

2: pop

= Repair (pop

) {Maintaining the sequence

of control points considered as variables}

3: CP

= Centroid Shift (s

, s

) {Shifting of con-

trol points towards a particular distance measured

with the difference between the centroids of tar-

get and generated shapes}

4: E dist = Max (Eucli dists (s

, s

))

5: H dist = Hausdorff dist (s

, s

)

6: Matching Error = Max (E dist, H dist)

7: Hybrid Memetic Algorithm presented in Algo-

rithm 2

1. EA-like evolution – This includes simulated bi-

nary (SBX) crossover and polynomial mutation

(Deb and Agarwal, 1995).

2. DE-like evolution – This includes the DE expo-

nential crossover and mutation, as described in

(Das and Suganthan, 2010).

2.3.2 Ranking and Reduction

Since the test problems studied in the paper are for-

mulated as single-objective, unconstrained minimiza-

tion problems, the ranking is done by sorting the ob-

jective values in ascending order. The best N so-

lutions from the (parent+child) population form the

population for the next generation.

2.3.3 Local Search

At each generation, in addition to generation of new

solutions using recombination and mutation, a local

search is used for further improvement. Sequential

quadratic programming (SQP) (Powell, 1978) is used

for the local search in the present study. After perfor-

Algorithm 2: Hybrid memetic algorithm.

Require: Population size (N), Number of genera-

tions (N

), Crossover and mutation parameters.

1: pop

= Initialize

2: pop

= Repair (pop

)

3: Evaluate (pop

)

4: for i = 2 to N

5: if rand (0,1) ≤ 0.5 then

6: childpop

= Evolve EA (pop

i−1

)

7: else

8: childpop

= Evolve DE (pop

i−1

)

9: end if

10: childpop

= Repair (children

)

11: Evaluate (childpop

)

12: S = Rank (pop

i−1

+ childpop

)

13: if rand (0,1) ≤ 0.2 then

14: x = Random x ∈ pop

{Select a random so-

lution}

15: else

16: x = Choose start x (pop

) {Select a solution

as described in Section 2}

17: end if

18: x

best

← Local search (x) {x

best

is the best so-

lution found using local search from x}

19: Replace worst solution in pop

with x

best

20: pop

= Rank(pop

) {Rank the solutions again

in pop

}

21: if local search doesn’t improve the objective

value for K consecutive generations then

22: pop

= Re-initialize () {Retain the best solu-

tion while re-initializing}

23: pop

= Repair (pop

)

24: end if

25: end for

ming the local search, the worst solution in the popu-

lation is replaced by the best solution found from the

local search.

2.3.4 Re-initialization

In order to prevent the algorithm from stagnating at a

local optima, the population is reinitialized if there is

no improvement observed in the objective value for

more than K (=10) generations. The best solution

found during the search is preserved in the population

during re-initialization.

3 RESULTS

The boundaries of the wings are extracted digitally

and then target shapes are obtained via direct inverse

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

496

Table 2: Matching error between target and generated dragonﬂy-wing shapes.

Dragonﬂy wing wing-1 wing-2 wing-3 wing-4 wing-5 wing-6 wing-7

Matching error (proposed method) 6.4e-05 1.8e-05 1.9e-04 2.3e-05 3.8e-05 7.1e-05 2.2e-05

Matching error (real-coded EA) 4.019 3.670 2.747 3.850 5.071 5.388 3.854

Table 3: Matching error between target and generated damselﬂy-wing shapes.

Damselﬂy wing wing-1 wing-2 wing-3 wing-4 wing-5

Matching error (proposed method) 2.1e-04 2.9e-05 7.1e-04 0.9e-05 3.3e-05

Matching error (real-coded EA) 3.466 5.849 4.545 3.833 2.562

Table 4: Results of multiple runs for dragonﬂy-wing and damselﬂy-wing shapes using proposed method.

Shapes Error Measurement Best Worst Mean Median Std.

Dragonﬂy-wing-1 Max(Eucli,HD) 0.7e-05 1.57e-04 5.0e-05 3.3e-05 4.0e-05

Dragonﬂy-wing-5 Max(Eucli,HD) 0.9e-05 2.11e-04 5.2e-05 1.8e-05 6.9e-05

Damselﬂy-wing-2 Max(Eucli,HD) 0.8e-05 3.3e-04 6.2e-05 2.7e-05 8.9e-05

Damselﬂy-wing-5 Max(Eucli,HD) 0.7e-05 1.73e-04 4.0e-05 2.9e-05 3.6e-05

Table 5: Results of multiple runs for dragonﬂy-wing and damselﬂy-wing shapes using real-coded Evolutionary Algorithm.

Shapes Error Measurement Best Worst Mean Median Std.

Dragonﬂy-wing-1 Max(Eucli,HD) 1.763 8.796 4.894 4.395 1.823

Dragonﬂy-wing-5 Max(Eucli,HD) 2.813 8.085 4.663 4.034 1.686

Damselﬂy-wing-2 Max(Eucli,HD) 3.540 6.402 5.097 5.344 0.815

Damselﬂy-wing-5 Max(Eucli,HD) 2.219 6.255 3.719 3.453 1.066

Table 1: Parameters used for the proposed algorithm.

Parameter Value

Population size 40

Max. function evaluations 10000

Crossover probability 1.0

Crossover index 10

Mutation probability 0.05

Mutation index (polynomial) 20

Scale factor F (for DE mutation) 0.9

ﬁtting in order to remove noise from their bound-

aries (Figure 4). Seven dragonﬂy-wing shapes and

ﬁve damselﬂy-wing shapes are used to test the per-

formance of the proposed approach. Dragonﬂy-wing

shapes contain 493 to 525 points while Damselﬂy-

wing shapes contain 491 to 516 points. The param-

eters used for the algorithms are the same for each

shape matching exercise, i.e., no tuning of parameters

is done across the problems. The parameters are listed

in Table 1. A maximum of 10000 function evaluations

are set for each problems.

For all the problems, we have assumed they can be

represented using 20 control points which translates

to a 40 variable optimization problem. A common

box has been chosen as the search domain represent-

ing between 0 to 58 in the x direction and between

0 to 17 in the y direction for all the problems. The

results after 10000 function evaluations for dragonﬂy-

wing and damselﬂy-wing shapes are shown in Table 2

and Table 3 respectively. For the sake of comparison,

the results of using a real-coded evolutionary algo-

rithm are also presented alongside the results of the

proposed method.

The statistics of the multiple runs of some exam-

ples are presented in Table 4, which reﬂects the con-

sistency of the proposed approach. The best and worst

values are reported as the minimum and maximum er-

rors across 20 runs, respectively. The median value

reported is the average of 10

and 11

values in the

sorted list of matching errors obtained across 20 runs.

The matching errors for all the examples are fairly low

in the order of 1e

−

05. In an attempt to observe the

performance of the proposed method, the results of

the multiple runs of using a real-coded evolutionary

algorithm are also presented in Table 5. It is clear

from Table 4 and Table 5, that a signiﬁcant improve-

ment in performance has been achieved through the

SHAPE REPRESENTATION AND A MORPHING SCHEME TO SUPPORT FLAPPING WING RESEARCH

497

A. constricta

S. vicinum

E. simplicicolis

47 mm

E. cynosura

25 mm

32 mm

29 mm

A. verticalis

52 mm

S. rubicundulum 26 mm

S. tenebrosa

37 mm

I. posita

C. angustipennis

E. divagans

I. vertcalis

L. rectangularis

17 mm

40 mm

21 mm

15 mm

22 mm

Species Wing length

Epiprocta

(dragonflies)

Zygoptera

(damselflies)

Odonata

(dragonflies and damselflies)

Figure 4: A: Dragonﬂy and damselﬂy wing species with their length (Donoughe et al., 2011). B: Original shapes’ boundaries

(digitized) of wing species. C: Shapes’ boundaries are represented via direct inverse ﬁtting.

−5 0 5 10 15 20 25 30 35 40 45

−20

−15

−10

−5

−5 0 5 10 15 20 25 30 35 40 45

−20

−15

−10

−5

−5 0 5 10 15 20 25 30 35 40 45

−20

−15

−10

−5

Figure 5: Evolutions of generated damselﬂy-wing-2 (thin) towards the target damselﬂy-wing-2 (thick) for matching.

use of the proposed method across all the problems.

Various states of evolutions (from 1 to 10000 func-

tion evaluations in 3000 interval) of generated shape

towards the target or original shape are shown in Fig-

ure 5 for damselﬂy-wing-2 example.

4 DISCUSSION AND

CONCLUSIONS

There is an increasing interest to understand ﬂapping

behavior of insect wings with an aim to identify efﬁ-

cient propulsive mechanisms for micro air vehicles.

While ﬂapping wing kinematics have been the pri-

mary focus for the last decade, there are increasing

number of attempts in recent years that tend to model

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

498

insect like wing planforms. This paper presents a

boundary shape representation scheme coupled with

a novel repair method that is amenable to optimiza-

tion. The ability of the approach to represent and

morph a class of insect wings is illustrated through

shape matching, wherein the entire class of insect

wing shapes have been identiﬁed using a common

bounding frame. The approach relies on the use of

B-splines for shape representation, in which the con-

trol points are ordered using a repair strategy. The re-

pair strategy assists in generating more viable shapes,

thereby increasing the rate of convergence in the op-

timization exercise. In an attempt to enhance the

convergence further, a memetic algorithm embedded

with a local search based on SQP is designed. In or-

der to measure the matching error between the tar-

get shape and generated shape, two popular simi-

larity measures (Euclidean and Hausdorff distance)

have been considered. The proposed method has been

tested using seven dragonﬂy-wing and ﬁve damselﬂy-

wing shapes, wherein a good performance has been

obtained. It is important to highlight that although

this study focused on shape matching, i.e. by con-

sidering the objective as a similarity measure, other

objectives such as propulsive efﬁciency, lift or drag

can be easily included in the formulation of the opti-

mization problem provided they can be computed au-

tomatically without user intervention.

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