A bionic approach to mathematical modeling the fold geometry of deployable reflector antennas on satellites

A bionic approach to mathematical modeling the fold

geometry of deployable reflector antennas on satellites

C.M. Feng, T.S. Liu

n,1

Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan

a r t i c l e i n f o

Article history:

Received 7 January 2014 Received in revised form 19 June 2014

Accepted 20 June 2014 Available online 28 June 2014 Keywords:

Deployable reflector antenna Deployable membrane reflector Bionics design

Fold geometry Morphological change

a b s t r a c t

Inspired from biology, this study presents a method for designing the fold geometry of deployable reflectors. Since the space available inside rockets for transporting satellites with reflector antennas is typically cylindrical in shape, and its cross-sectional area is considerably smaller than the reflector antenna after deployment, the cross-sectional area of the folded reflector must be smaller than the available rocket interior space. Membrane reflectors in aerospace are a type of lightweight structure that can be packaged compactly. To design membrane reflectors from the perspective of deployment processes, bionic applications from morphological changes of plants are investigated. Creating biologically inspired reflectors, this paper deals with fold geometry of reflectors, which imitate flower buds. This study uses mathematical formulation to describe geometric profiles of flower buds. Based on the formulation, new designs for deployable membrane reflectors derived from bionics are proposed. Adjusting parameters in the formulation of these designs leads to decreases in reflector area before deployment.

& 2014 IAA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Construction in a low- or no-gravity environment is difficult. Thus, deployable devices are widely used in space technologies due to their low construction expenses, light weights, and high packing efficiencies [1]. Solar panels, reflectors, radars, and satellite masts are examples of such deployable devices and are generally constructed to be deployable so that they are compact during launch[2–6]. Large reflectors with 4 to 25 m diameter are considered for future satellites dedicated to earth observation, telecom-munication and science missions[7]. A rigid reflector of this diameter range will not fit in most launch vehicles for

transportation into orbit.Table 1lists fairing diameters of Delta series launch vehicles [8]. To lower the launch volume and weight of the device, the reflector can be constructed to be deployable, as has been suggested from observations of several radar missions[9–11]. Because the space inside a rocket is cylindrical in shape and has a cross-sectional area that is much smaller than the satellite antenna after deployment, the question of how to decrease the cross-sectional area of the folded reflector is an important task.

Most deployable reflectors can be divided into two types: solid surface reflectors [12–14] and membrane surface reflectors[15–17]. Membrane reflectors are com-pactly stowed before deployment and can be expanded to obtain large reflective area. Most materials used in the construction of membrane reflectors are polymers onto which thin metal films are coated to reflect electromag-netic waves. These metal layers may generate gaps or cracks due to the large curvatures experienced when the Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/actaastro

Acta Astronautica

http://dx.doi.org/10.1016/j.actaastro.2014.06.029

0094-5765/& 2014 IAA. Published by Elsevier Ltd. All rights reserved. n_{Corresponding author. Tel.:þ886 3 5712121x55123;}

fax:þ886 3 5720634.

E-mail address:tsliu@mail.nctu.edu.tw(T.S. Liu). 1

Postal address: EE405, 1001 University Road, Hsinchu 30010, Taiwan, ROC.

reflector is folded before deployment. These gaps and cracks will effect the electromagnetic wave patterns of reflector antennas[18].

To design deployable membrane reflectors, this study learns from the biological world. Nature has elegantly crafted efficient solutions to many of the same or similar problems as those faced by engineers[19]. Solutions have evolved in several biological systems including insect wings, plant leaves, and flower petals[20,21]. To design deployable antennas for use in satellite, this paper aims to develop a bionic approach to designing geometric profiles of membrane reflectors that look like the bud of a morning glory, a twining or vine-like plant when folded. This study formulates geometric profiles for folded membrane reflec-tors that are derived from bionics. These are then used as the basis for new designs of deployable membrane reflectors.

2. Fold geometry design

Nature supplies a rich variety of design references for the creation of products. Morphological changes in nature provide a source of design inspiration. This paper deals with the fold geometry of reflectors by observing mor-phological changes of flower blooms. Fig. 1(a) and (b) respectively show blooming morning glory and bindweed

[20]. Bindweed belongs to the morning-glory family

[22,23]. Accordingly, both flowers have similar geometric profiles before and after bloom. From the viewpoint of geometric profiles, this study treats both flowers to be the same. Initially, petals of each flower constitute a funnel-form corolla. The petals in turn gradually expand to full bloom. In bloom processes of both flowers transforming from closed buds to the blossomy shape, the petals do not split into several lobes. In the top view of both flower buds, the petal profile consists of several concavities and bulges in cyclic arrangement. During blooming, the profile main-tains axial symmetry and gradually becomes circular. The variation of profiles in the top view can be drawn as geometric curves, as shown in Fig. 1(c). This study con-structs mathematical models to describe the geometric profile of morning glory buds. Although, several deploy-able reflector designs[24,25]in the literature are similar to the bloom process of the morning glory, such as an example shown inFig. 2, mathematical models were not created.

2.1. Geometric profile formulation

The geometries of many membrane reflectors resemble the morning glory in full bloom. A deployed reflector can be folded in a manner similar to the folding of a morning glory bud. To facilitate the folding of the reflector, the geometric profile of the reflector cross-section is designed as a ring. Fig. 3(a) and (b) depict a reflector before deployment (i.e., after folding), treated as a morning glory bud in this study, and after deployment, respectively.

The cross-sectional profile of a folded reflector, as depicted inFig. 3(a), will be designed in this study with reference to the morning glory bud. The outermost profile curve of the reflector cross-section before deployment is depicted inFig. 4and can be formulated in polar coordi-nates r and

θ

rcð

θ

Þ ¼ A cos B sin n

θ

2

 ð1Þ where A is the maximum radius of the cross-sectional profile, B is defined as the concavity factor of the curve, and n denotes the number of petals of the bud. As B increases, notches between two adjacent petals will dee-pen and approach the curve center; i.e., B determines the depth of the notches between two adjacent petals, as shown in Fig. 4 at

θ

¼ ð

π

=5Þ, ð3

π

=5Þ,

π

, ð7

π

=5Þ, and ð9

π

=5Þ. The curve inFig. 4has five identical petals, n¼5, and a maximum radius A at

θ

¼ ð2

π

=5Þ, ð4

π

=5Þ, ð6

π

=5Þ, ð8

π

=5Þ, and 2

π

. Every two neighboring radial lines inter-sect at the angleð

π

=5Þ. These intersecting lines thus divide the curve into 10 equal parts as shown inFig. 4.

Using the same values of A and n as in Eq.(1), different B values will induce different concave shapes of the curve, as depicted in Fig. 5. If B equals zero, there will be no notch, and the curve obtained from Eq.(1)will be a circle with radius A. In contrast, when B equals

π

/2, the notches between two adjacent petals of the curve based on Eq.(1)

will be so deep that the notches will intersect at the center of the curve. Although B can be any real value, to produce geometric profiles of reflectors before deployment that resemble a morning glory and to avoid the intersection of profiles at the curve center, this study confines the range of B values to between 0.6 and 1.5. As B increases, notches between two adjacent petals deepen and approach the center of the curve.

Fig. 6depicts morning glory profiles with (a) 5, (b) 15, and (c) 25 petals, with A¼50 and B¼0.8. Because the same A and B values are used inFig. 6(a) to (c), the profile curves exhibit the same radii and notch depths between two adjacent petals.

2.2. Determining the maximum radius A

If a smooth curve C is represented by x¼ f ð

θ

Þ and y¼ gð

θ

Þ such that C does not intersect itself in the interval

α

r

θ

r

β

, the arc length of C over the interval is expressed by sc¼ Z β α ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx d

θ

2 þ _ddy

_θ

2 s d

θ

ð2Þ Table 1

Fairing diameter of Delta series launch vehicles[8].

Type Diameter (m) Delta II 2.9-m 2.540 Delta II 3-m 2.743 Delta II 3-m stretched 2.743 Delta III 4-m 3.749 Delta IV M 3.750 Delta IV Mþ 4.572 Delta IV heavy 4.572

Delta IV heavy aluminum isogrid 4.572 Delta IV heavy dual manifest 4.572

The profile curve rcð

θ

Þ on the cross-section of the folded

reflector is represented by the following parametric equations x¼ rcð

θ

Þ cos

θ

y¼ rcð

θ

Þ sin

θ

ð3Þ and dx d

θ

¼ r 0 cð

θ

Þ cos

θ

rcð

θ

Þ sin

θ

dy d

θ

¼ r 0 cð

θ

Þ sin

θ

þrcð

θ

Þ cos

θ

ð4Þ

Fig. 1. Bloom processes of (a) morning glory and (b) bindweed[20]. (c) Drawings of the geometric profile based on bloom processes of both flowers in the top view.

Fully Packaged

Fully Deployed

Eq.(4)leads to dx d

θ

2 þ dy d

θ

2 ¼ ½r0 cð

θ

Þ cos

θ

rcð

θ

Þ sin

θ

2 þ½r0 cð

θ

Þ sin

θ

þrcð

θ

Þ cos

θ

2 ¼ ½rcð

θ

Þ2þ½r0cð

θ

Þ2 ð5Þ

Using Eqs.(2) and (5), the arc length sr of the profile

curve rcin 0r

θ

r2

π

is written as sr¼ Z 2π 0 Pð

θ

Þd

θ

ð6Þ where Pð

θ

Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½rcð

θ

Þ2þ½r0cð

θ

Þ2 q ð7Þ Because the explicit integral of Eq.(6)is not available, reducing the range of

θ

to 0_r

θ

_r

π

_{=n is necessary, and} using Taylor’s theorem to expand Eq.(7)about c leads to Pqð

θ

Þ ¼ PðcÞþ ∑ q q¼ 1 PðqÞðcÞ q! ð

θ

cÞ q ð8Þ

InFig. 7, function curves are drawn based on Eqs.(7) and (8). Eq.(8)is expanded about c¼

π

=ð2nÞ as 5th-, 7th-, and 9th-order Taylor series. The range of

θ

is from 0 to

π

=n. The 9th-order series is close to the function Pð

θ

Þ. Therefore, Eq.(8)can be rewritten as the 9th-order Taylor series

P9ð

θ

Þ ¼ ½0:125A2B2n2sin2ð0:707BÞþA2cos2ð0:707BÞ1=2

þ ∑9 q¼ 1 PðqÞð

π

=2nÞ q_!

θ



π

2n  q ð9Þ P9ð

θ

Þ, instead of Pð

θ

Þ, is substituted into Eq.(6)to calculate

the arc length of the profile curve rcof a cross-section at

the top of the folded reflector shown inFig. 3(a). Since D denotes the diameter of the reflector after deployment, the arc length of the profile curve rc has to equal the

circumference

π

D of the deployed reflector, shown in

Fig. 3(b). Thus, sr¼

π

D. As

θ

ranges from 0 to

π

=n, the

arc length of this interval equals s/2n. Therefore, Eq.(6)

can be rewritten as sr n¼ Z π=n 0 P9ð

θ

Þd

θ

¼

π

D n ð10Þ

Solving Eq.(10)results in the maximum radius A. For example, assume that the diameter of a reflector after deployment is D, and the reflector is folded as five petals, n_{¼5. The value of the concavity factor B is prescribed as} 1.2. As a result, Eqs.(9) and (10) yields a maximum radius of A¼0.4D. According to Eqs.(9) and (10), after parameters B and n have been prescribed, the value of A depends on the parameters B and n. Fig. 8 presents the maximum radius A vs. the concavity factor B with respect to different values of n. The maximum fold radius A decreases with increases in the concavity factor B or petal numbers n. Thus, it is advantageous to reduce the cross-sectional area before antenna deployment by increasing B or n.

Fig. 3. Contrast between a membrane reflector with five petals (a) before and (b) after deployment.

A 2 0

5

π π π 3π 2

Fig. 4. Geometric curve, drawn based on Eq.(1), of the morning glory bud’s petals. The radial lines divide the geometric curve into 10 equal parts, and two neighboring radial lines intersect at an angle ofπ/5.

B = π/2

B = 1.5

B = 1.2

B = 0.9

B = 0.6

B = 0

Fig. 5. Comparison among curves generated from Eq. (1). A larger concavity factor B results in deeper notches between two neighboring petals.

3. Curvature and cross-sectional area ratio of pro_{file curve}

It follows fromFigs. 5and6that the curvature of the profile curve rc(

θ

) vary with the concavity factor B and the

numbers of petal n, respectively. When either B or n increases beyond a certain value, the curvature at

θ

¼ ð

π

=nÞ becomes excessive. Excessive curvature may induce the formation of a crease on the reflector surface and generate gaps or cracks in the layer of metal on the reflector. These gaps and cracks will affect the pattern of electromagnetic waves on the reflector antennas [18]. Therefore, excessive curvature should be avoided.

3.1. Determining the curvature of a curve rc(

θ

)

The curvature of any profile curve rc in polar

coordi-nates is expressed by[26]

κ

ð

θ

Þ ¼jr2cþ2r02c rcr″cj ðr2 cþr02cÞ 3=2 ð11Þ π/2 π 3π/2 0 π/2 π 3π/2 0 π/2 π 3π/2 0

Fig. 6. Morning glory profiles of petal numbers (a) n¼5, (b) n¼15, and (c) n¼25 petals, with A¼50 and B¼0.8.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

P(θ)

θ

Fig. 7. Function curves drawn based on Eqs. (7) and (8). Eq. (8) is expanded about c¼π/10. 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Maximum folded radius A( D ) Concave factor B n=5 n=10 n=15 n=20 n=25 n=30

Fig. 8. Maximum radius A vs. the concavity factor B with respect to different numbers of folded petal n.

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0 1 2 3 4 5 6 7 8 9x 10 4 Concave factor B Curvature κ (1/D ) n=5 n=10 n=15 n=20 n=25 n=30

Fig. 9. Curvatureκ vs. the concavity factor B with different numbers of petal n.

where r0c¼ 

1

2ABn sin B sin n

θ

2

 cos n

θ

2
 ð12Þ and r″c¼1 4ABn 2 _{sin B sin} n

θ

2

 sin n

θ

2
 1₄AB2n2 _{cos B sin} n

θ

2

 cos2 n

θ

2
 ð13Þ As depicted inFigs. 3and4, the maximum curvature of the profile curve rcoccurs at

θ

¼

π

=n. Therefore,

substitut-ing Eqs.(1), (12) and (13) into Eq.(11)at

θ

¼

π

=n yields

κ π

 _n ¼jA

2

cos2_B0:25A2

Bn2 _{cos B sin Bj}

ðA2_cos2_BÞ3=2 ð14Þ

According to Eqs. (10) and (14), the curvature

κ

depends on parameters B and n, whose relationship is depicted inFig. 9, where D denotes the reflector diameter after deployment.

As depicted inFig. 9, increasing the number of folded petals n or concavity factor B will increase the curvature

κ

. The curvature will increase sharply when the value of B increases beyond a certain value. For example, when n and B equal 10 and 1.5, respectively, the curvature will increase sharply. With an increasing number of petals n, the critical value of B will decrease.

3.2. Cross-section areas before and after folding

This study presents mathematical models with respect to profile curves of reflector cross sections. Hence, the smaller the enclosed area of a profile curve is, the smaller the cross-section area of the reflector is. The area enclosed by a profile curve rccan be written as[27]

Af¼ 1 2 Z 2π 0 rc2ð

θ

Þd

θ

ð15Þ

Define the integrand in Eq.(15)as R¼ r2 cð

θ

Þ ¼ A 2 cos2 _{B sin} n

θ

2

 ð16Þ Because the explicit integral of R is not available, reducing the range of

θ

to 0r

θ

r

π

=n and using the 9th-order Taylor series for the expansion of R about c¼

π

=ð2nÞ leads to R9ð

θ

Þ ¼ A2cos2ð0:707BÞþ ∑ 9 q¼ 1 RðqÞð

π

=2nÞ q!

θ

2n

π

 q ð17Þ where R9ð

θ

Þ is the 9th-order Taylor series for R in Eq.(16).

Therefore, Eq.(15)can be rewritten as Af¼ n

Z π=n 0

R9ð

θ

Þd

θ

ð18Þ

The outermost geometric profile of a reflector cross-section after deployment is a circle whose area is denoted

as Ad. The ratio of the deployed reflector area vs. the area enclosed by the profile curve rcis denoted as Ad/Af, where Ad¼ 1 2 Z 2π 0 R2dð cos2

θ

þ sin 2

_θ

Þd

θ

ð19Þ

Larger ratios represent larger gains in antenna area after deployment relative to the same antenna before deployment. Obtaining a larger gain in area is the present design goal.Fig. 10presents Ad/Afratios vs. the concavity factor B with different numbers of folded petals n. Increas-ing the concavity factor B or petal numbers n will effec-tively reduce the pre-deployment reflector area. However, as shown inFig. 9, the curvature of the profile curve rc

will increase sharply when B and n increase beyond a certain value. Increasing B and n results in smaller pre-deployment cross-sectional areas, but also larger cur-vatures.

4. Discussion

Concerning reflector diameter examples, Rao et al.[28]

have presented an antenna, used inL-band communication,

with a parabolic reflector of diameter 510 cm. However, the

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0

50

100

150

200

250

n=5

n=10

n=15

n=20

n=25

n=3

Concavity factor

B

d f

A

A

Fig. 10. Ratios of the deployed reflector areas to the areas enclosed by the profile curve rc.

Table 2

Geometric profile designs with different values of B and n. Design Concavity factor B Number of petals n Curvature atθ/n (1/cm) Fold diameter (cm) a 0.9 5 0.04 491 b 1.4 5 1.72 342 c 1.2 10 1.81 233 d 1 15 1.49 214 e 0.9 20 1.84 196 f 0.8 25 1.88 195 g 0.7 30 1.67 206 h 1.2 30 46.44 83

diameter of unfolded reflectors is too large to be placed inside the launch vehicles whose fairing diameters are listed inTable 1. Using the proposed method, as shown in

Table 2this study designs eight geometries with different values of B and n. The curvatures of designs (b) to (g) are close, ranging from 1.49 to 1.88 (1/cm). In contrast to the other designs, designs (a) and (h) have excessively small and large curvatures, respectively, and are thus inadequate designs.

Fig. 11compares the areas of the deployed reflector and the reflector designs listed inTable 2, which are calculated based on Eqs.(1) and (10).Fig. 11(a) and (b) have the same number of petals, n¼5, with different concavity factors B.

As B increases, the enclosed areas of the profile curves decrease. ComparingFig. 11(b) and (c), the curvatures at

θ

/n of the profile curves are similar. However, the enclosed area in (c) is much smaller than that in (b). Accordingly, to obtain an effective fold, the profile curve rcshould possess

more than 10 petals. Therefore, (a) and (b) are ineffective designs. The designs inFig. 11(c) to (g) all have different numbers of petal and concavity factors. These five designs are effective at shrinking the enclosed areas of the profile curves to approximately 82_{–88% of the deployed area.} In Fig. 11(c) and (h), the profile curves have different numbers of petals n but the same concavity factor. As n increases, the enclosed areas of the profile curves

Fig. 11. Comparison of cross-sectional profiles before and after deployment: (a) B¼0.9; n¼5, (b) B¼1.4; n¼5, (c) B¼1.2; n¼10, (d) B¼1; n¼15, (e) B¼0.9; n¼20, (f) B¼0.8; n¼25, (g) B¼0.7; n¼30, and (h) B¼1.2; n¼30 based on Eqs.(1) and (10). Parameters B and n in Eq.(1)are fromTable 1, and parameter A is obtained by solving Eq.(10). The outermost circles represent the cross-sectional profiles of the deployed reflectors and all have the same 510 cm radius. The star-like curves inside each graph represent the cross-sectional profiles of the folded reflectors.

decrease. Although design (h) reduces the area of the folded reflector by more than 95%, its excessive curvatures should be avoided. Accordingly, (h) is treated as an inadequate design. To demonstrate the proposed method for the geometric profile design of fold reflectors, as depicted inFig. 12, this study has fabricated a membrane reflector with five petals in the fold state. Kapton mem-branes are pasted on parabolic ribs of a deployable

mechanism [29], which can fold the membrane in a

manner similar to morning glory buds with five petals. This reflector with a 120 cm diameter after deployment was folded by the deployable mechanism to produce five petals with a concavity factor of B¼0.9.Fig. 13shows the deployment process of this reflector.

5. Conclusion

Inspired by biological patterns, this study has investi-gated fold geometry for satellite deployable antennas. The profiles of folded reflectors have been formulated in polar coordinates r and

θ

. Adjusting parameters in the formula-tion leads to decrease in reflector area after fold.Fig. 11(c) to (g) illustrate that the cross-sectional area before deploy-ment and after fold can be effectively reduced when the profile curve rchas more than 10 petals. Dealing with the

example with 510 cm unfold diameter in Discussion, this study proposes five geometric profile designs used to fold membrane reflectors, as depicted inFig. 11(c) to (g) that are applicable to and contribute to the design of deploy-able antennas.

Acknowledgments

The authors would like to thank National Space Orga-nization in Taiwan, ROC, for financial support and helpful discussions.

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