Folding kinematics of kirigami-inspired space structures

Journal Pre-proof

Antonio Pedivellano, Sergio Pellegrino

PII:

S0020-7683(24)00224-5

DOI:

https://doi.org/10.1016/j.ijsolstr.2024.112865

Reference:

SAS 112865

To appear in:

International Journal of Solids and Structures

Received date : 7 August 2023

Revised date :

10 April 2024

Accepted date : 2 May 2024

Please cite this article as: A. Pedivellano and S. Pellegrino, Folding kinematics of kirigami-inspired

space structures.

International Journal of Solids and Structures

(2024), doi:

https://doi.org/10.1016/j.ijsolstr.2024.112865.

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Folding Kinematics of Kirigami-Inspired Space

1

Structures

2

Antonio Pedivellano

a,1

, Sergio Pellegrino

∗

3

a

Graduate Aerospace Laboratories, California Institute of Technology, 1200 E California

Blvd, Pasadena, 91125, CA, USA

Abstract

4

This paper studies the folding of square, kirigami-inspired space structures

5

consisting of concentrically arranged modular elements formed by thin shells.

6

Localized elastic folds are introduced in the thin shells and different folding

7

strategies can be obtained by varying the location of the folds and the se-

8

quence of imposed rotations. Modeling each modular element with rigid

9

rods connected by revolute joints, numerical simulations of the kinematics

10

of folding are obtained, including constraints that represent folding aids and

11

a gravity offload system. These simulations are used to study two specific

12

packaging schemes, and the folding envelopes of a specific structure are an-

13

alyzed to identify the scheme that is easier to implement in practice. This

14

particular scheme is demonstrated by means of a physical prototype.

15

Keywords:

Deployable structures, Thin shells, Packaging, Kinematic

16

analysis, Loop closure, Origami

17

1. Introduction

18

Origami, the japanese art of paper folding, and kirigami, the variant of

19

origami that allows cuts as well as folds in the paper, have inspired novel

20

∗

Corresponding author

Email address:

sergiop@caltech.edu

(Sergio Pellegrino)

Current address: DCubeD, Burgweg 6, 82110 Germering, Germany

Preprint submitted to International Journal of Solids and Structures

May 14, 2024

Journal Pre-proof

deployable space structures that can be efficiently packaged for launch and

21

built at low cost. Miura-ori (

Miura

1969

;

Miura and Pellegrino

2020

), a

22

well-known example of an origami-based modular packaging concept, has

23

led to the development of a novel deployable solar array (

Miura and Natori

24

1985

). Other examples include packaging schemes for thin films in solar

25

sails (

Guest and Pellegrino

1992

;

Furuya et al.

2011

) and flexible panels in

26

solar arrays (

Zirbel et al.

2013

) based on the concept of coiling a circular

27

membrane around a central hub. Removal of localized buckles near the folds

28

of a coiled membrane was achieved by allowing slip to occur along the folds

29

(

Arya et al.

2017

30

These concepts have been extended to large deployable structures that

31

will enable new space missions. An ongoing development is the ultralight,

32

scalable structural architecture for the Caltech Space Solar Power Project

33

(SSPP) (

Arya et al.

2016

), to provide square, flat structures that are tens

34

of meters in size and can be tightly folded into a cylindrical envelope. A

35

recent study (

Brophy et al.

2022

) has proposed a novel mission architecture

36

to reach Uranus and Neptune using very large ultralight solar arrays that

37

make it viable to use electric propulsion at great distances from the sun.

38

Large scale applications of this kind pose challenges beyond the existing

39

origami research. Hence, the specific challenges that are addressed in the

40

present study are related to the practical implementation of these folding

41

schemes, achievable with only simple folding aids. In particular, it has to be

42

ensured that no damage occurs during folding and that the self-weight of the

43

structure is properly supported at all stages of folding.

44

The specific focus of the present paper is the Caltech SSPP structure

45

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schematically shown in

Fig. 1a

. It consists of bending-stiff trapezoidal mod-

46

ular elements, also denoted as “strips”, in a four-fold symmetric arrangement

47

of concentric square loops. The strips are attached to four diagonal cords

48

stretched between a central deployment mechanism and the tips of four de-

49

ployable booms. The main structural elements of the trapezoids are thin-shell

50

longerons placed along the longitudinal edges of the trapezoids, connected

51

by transverse battens. Functional membranes, e.g., photovoltaic films and/or

52

RF radiating elements, are attached to the strips. Localized elastic folds in

53

the longerons allow compact packaging of these structures.

54

The packaging concept involves two steps, which are shown in

Fig. 1b

55

The first is a folding step, in which each quadrant of the structure is z-folded

56

to reach a star configuration. This step is followed by a coiling step in which

57

the four arms of the star are coiled into a cylindrical configuration. Note

58

that the folding step in

Fig. 1b

is different from

Arya et al.

(

2016

), as this

59

initial study had demonstrated the packaging concept with only thin films,

60

without considering relatively stiff structural elements.

61

The present paper studies the folding step through general numerical

62

simulations, with the objective of considering practically important effects,

63

such as the choice of constraints that provide support against self-weight,

64

and using easily implementable folding aids. Folding is very important to

65

the maturation of the overall structural concept, particularly because the

66

scope for exploring different folding strategies experimentally is limited by

67

the extensive use of carbon-fiber reinforced composite materials, whose brittle

68

behavior restricts the trial and error exploration feasible on physical models.

69

High-fidelity simulations, along the lines of

Pedivellano and Pellegrino

70

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(a) Architecture

(b) Packaging scheme

Figure 1: Structural architecture and packaging concept, with the folding step highlighted.

(

2022

), might be thought as ideal for such studies. However, high-fidelity

71

simulations are very time consuming and are best suited to the detailed

72

analysis of specific folding problems.

73

Reduced-order finite element models offer a computationally efficient al-

74

ternative to study elastic origami. A widely used approach is the bar-and-

75

hinge method, which models a planar structure as an assembly of elastic bar

76

elements forming a triangular pattern and revolute joints. Rotational springs

77

along the bars model the behavior of the creases and fold lines (

Filipov et al.

78

2017

;

Liu and Paulino

2017

). These models do a very good job in capturing

79

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the experimentally characterized behavior of deployable structures inspired

80

by Miura-ori, including snap-through behavior.

81

Several authors have applied this method to the hypar, the origami pat-

82

tern that is closest to the kirigami folding scheme studied in this paper.

83

Filipov and Redoutey

(

2018

) investigated the bi-stable behavior of the struc-

84

ture and showed that local and global buckling effects can be captured using

85

this analytical method.

Liu et al.

(

2019

) focused on the geometric properties

86

of the hypar and the tessellation of multiple hypar units to achieve multista-

87

bility. The main weakness of the bar-and-hinge method is that, because it

88

is based on an implicit finite element formulation, it struggles to deal with

89

singularities in the tangent stiffness matrix, e.g. at bifurcation points.

Liu

90

et al.

(

2023

) proposed an approach based on group theory to decompose the

91

global formulation of a hexagonal hypar into a series of independent problems

92

within different symmetry subgroups. This approach captured three differ-

93

ent bifurcation branches of the equilibrium path during folding of the hypar,

94

and examined the sensitivity of the corresponding peak loads to changes in

95

the distance between the creases. In general, the bar-and-hinge method is

96

not best suited to studying the motion of structures with a large number

97

of kinematic paths. The complexity of this problem has been recognized in

98

the physics literature, in which undesired kinematic paths are described as

99

distractors (

Stern et al.

2017

100

The simulation approach adopted in the present study is focused on the

101

specific structures of interest, which consists of long and narrow strips with

102

localized elastic folds at specific, fixed locations, as shown in the box in

103

Fig. 1b

. The elastic folds and the strip-to-cord connections are modeled as

104

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hinges (revolute joints) and the remaining parts of the strips are modeled

105

as rigid rods, hence modeling a complete structure as a mechanical link-

106

age. With this approach, established analytical techniques for structural

107

mechanisms can be used to develop an analytical framework to identify kine-

108

matically compatible paths for the structure. This formulation is then used

109

to study different folding procedures, defined by the location of the imposed

110

elastic folds and the order in which the square loops are folded.

111

The paper is organized as follows.

Section 2

introduces a kinematic model

112

of the structure where each strip is modeled as a linkage of rigid rods and

113

revolute joints. Then, the whole structure is modeled by multiple closed

114

loops, whose kinematics are simulated with a predictor-corrector algorithm.

115

Section 3

studies the kinematics of a single square loop.

Section 4

extends

116

the single-loop solution to structures consisting of multiple, interconnected

117

square loops.

Section 5

presents an experimental demonstration of the best

118

packaging scheme identified in the kinematic study. Finally,

Section 6

dis-

119

cusses the results and concludes the paper.

120

2. Kinematic Model and Simulations

121

Consider a linkage consisting of straight rigid rods connected by revolute

122

joints. A local reference frame is assigned to link

, see

Fig. 2

, with the origin

123

at one end, the

-axis aligned with the axis of the hinge, the

124

-axis in the plane defined by the

-axis and the axis of the link, and the

125

-axis chosen such as to form a right-handed reference frame.

126

Using the

Denavit and Hartenberg

(

1955

) notation, the reference frame

127

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for the next link,

+ 1, is related to the frame for link

by:

128

(1)

where the 3D coordinates of point

are transformed to the 4

1 extended

129

form:

130

⎡

⎣

⎤

⎦

(2)

Also,

131

⎡

⎣

⎤

⎦

(3)

where

are respectively a rotation matrix and a translation vector.

132

(a) Translation

to the origin of

(b) Rotation

about

-axis, such that

lies in

−

plane

about

to align

with

(d) Rotation

about

to align

with

Figure 2: Coordinate transformations between local reference frames on adjacent links

connected by a hinge.

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The rotation matrices are defined in terms of three Euler angles, obtained

133

from a sequence of elementary transformations that align the

-th reference

134

frame with the (

+ 1)-th frame. The following convention is used:

135

1. translation

of the coordinate system from

(Fig.

);

136

2. rotation about the

axis by

, until the rotated

-axis is contained

137

in the

plane (Fig.

);

138

3. rotation about the new

axis by

, until the rotated

axis is aligned

139

with

(Fig.

);

140

4. rotation about the

axis by

, until

is aligned with

141

(Fig.

142

With this convention,

is the angle between the (

+1)-th hinge axis and

143

the normal to the

-th link axis;

is the twisting angle between the ends of

144

the

-th link, and

is the fold angle, i.e., the rotation around the (

+ 1)-th

145

hinge axis.

146

Coordinate transformations between non-consecutive links can be ob-

147

tained by multiplying the transformation matrices of the links in between:

148

...T

−

(4)

For a closed-loop kinematic chain, the initial and final coordinate frames

149

must coincide, which is equivalent to the condition:

150

...T

−

(5)

This matrix equation provides the loop-closure constraint for a kinematic

151

chain with

links. Although

Eq. (5)

corresponds to sixteen scalar equations,

152

only six equations are linearly independent, and it is shown in

Gan and

153

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Pellegrino

(

2006

) that an independent set of equations can be obtained from

154

the six off-diagonal terms above the main diagonal.

155

Finding a kinematic path for the linkage requires finding a solution of

156

Eq. (5)

. However, analytical solutions of this system of trigonometric equa-

157

tions are possible only for linkages much simpler than those considered in

158

the present paper. Therefore,

Eq. (5)

was solved numerically, using the con-

159

tinuation algorithm proposed in

Gan and Pellegrino

(

2006

). This algorithm

160

traces the motion of linkages with one or more degrees of freedom and iden-

161

tifies potential path-switching configurations (also known as kinematic bifur-

162

cations (

Kumar and Pellegrino

2000

)) with a predictor-corrector algorithm

163

that carefully monitors the approach to bifurcation points.

164

The incremental solution of

Eq. (5)

is based on a two-step algorithm.

165

In the predictor step, the loop-closure equations are linearized to find kine-

166

matically admissible tangent motions and an increment of the solution is

167

computed; in the corrector step, the linear prediction is iteratively updated

168

until the error resulting from the linearization becomes smaller than a set

169

threshold. These steps are outlined next.

170

2.1. Predictor step

171

Let

be the current configuration of the linkage, defined by the

172

variables

,...x

173

In the predictor step, the loop-closure equation,

Eq. (5)

, is linearized near

174

to obtain:

175

...

−

(6)

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where each of the coefficients

is the 4

4 matrix:

176

i

...T

(

)

...

(

)

... T

n,j

i

(7)

Here,

denotes the time increment and

∈

] refers to the components

177

of the state vector

. Partial derivatives of the transformation matrices have

178

been denoted as

i,j

∂T

/∂x

179

Eq. (6)

provides up to six linearly independent scalar equations which,

180

as previously discussed, are obtained from the six terms above the main

181

diagonal of the matrix equation:

182

⎡

⎢

⎣

...A

−

...A

−

...A

−

...A

−

...A

−

...A

−

⎤

⎥

⎦

⎡

⎢

⎣

−

⎤

⎥

⎦

⎡

⎢

⎣

⎤

⎥

⎦

(8)

This equation can be written in compact form as:

183

= 0

(9)

The matrix

is the Jacobian of the system and the subscript

stands for

184

predictor.

185

Non-trivial solutions of

Eq. (9)

belong to the null space of

, and can

186

be obtained from the Singular Value Decomposition (SVD) of this matrix

187

(

Pellegrino

1993

;

Golub and Van Loan

2013

). This decomposition computes

188

three matrices

U, V

, and

such that

189

USV

(10)

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where

and

are orthogonal matrices containing the left- and right-singular-

190

vectors, with

∈

and

∈

, whereas

isa6

matrix

191

containing the singular values.

192

For pin-jointed structures,

Pellegrino

(

1993

) has shown that the columns

193

, corresponding to the null singular values, constitute a basis for the

194

space of inextensional mechanisms of the structure. Analogously, in the

195

present case, the columns of

identify a basis for the space of infinitesimal

196

configuration changes that do not violate the loop-closure equation.

197

Denoting this basis with

, the predicted configuration of the linkage is:

198

Vα

(11)

where

is a vector of scaling parameters for the independent inextensional

199

mechanisms that define the amplitude of the increment.

200

Note that, in the case of multiple mechanisms, a specific kinematic path

201

is followed during the entire simulation. The predictor algorithm calculates,

202

at each iteration, the dot product between the old eigenvector and the new

203

eigenvectors and picks the increment closest to the path in the previous

204

iteration.

205

2.2. Corrector step

206

The above linearized prediction results in a small error in the loop-closure

207

equation, which therefore is no longer satisfied exactly. Hence, the error

208

matrix

is defined from:

209

(

)

(

)

...T

−

(

)

(

(12)

Eq. (12)

is linearized following the same approach as in Section

2.1

210

correction to the state vector is computed by defining the vector form of the

211

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error,

4)]

, and then the

212

correction is obtained from

213

−

(13)

where the subscript

stands for

corrector.

214

A least squares solution of this equation can be computed with the SVD

215

(

Pellegrino

1993

;

Kumar and Pellegrino

2000

216

−

rank(S)

(14)

where

and

are the left- and right-singular vectors of

, respectively.

217

The configuration of the linkage is updated with the correction:

218

+Δ

(15)

and the correction step is repeated until the L2 norm of the error

(or any

219

other metric of choice) becomes lower than a chosen tolerance, set to 10

−

220

in the algorithm. In a typical simulation, the error converges within two

221

iterations of the correction step.

222

3. Folding a Loop of Four Strips

223

This section studies the folding kinematics of a square loop formed by

224

strips of length 2

along the mid-line and width 2

Fig. 3

. The strips are

225

connected at the ends by hinges aligned with the diagonals of the square.

226

According to the mobility formula for closed kinematic chains (

Uicker et al.

227

2003

;

McCarthy and Soh

2010

), a closed loop requires at least seven revolute

228

joints to have an internal degree of freedom. Therefore, at least three folds

229

are needed and, in fact, four is the minimum number if folding is required to

230

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preserve the symmetry of the structure. However, the folding of nested loops

231

requires at least two folds in each strip and hence this is the case considered

232

here.

233

The non-dimensional parameter

defines the distance between the folds.

234

Note that

= 0 corresponds to a single fold at the center of the strip, and

235

= 1 corresponds to two folds at the ends. Since the folds are not allowed

236

to cross the diagonal battens, the admissible range is

∈

,λ

max

] with:

237

max

−

(16)

Figure 3: Geometry of square loop, with location of diagonal hinges and fold lines.

Figure 4

shows the model of the square loop. The joints have two ro-

238

tational degrees of freedom around perpendicular axes, as shown in more

239

detail in

Fig. 5

, and hence are equivalent to hinges with a variable direction

240

axis. Hence, both bending

and torsion

of a strip are allowed. Therefore,

241

the kinematic chain for the square loop contains

= 12 links with

=24

242

degrees of freedom.

243

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Figure 4: Kinematic chain of four strips forming a square loop.

Figure 5: Detail of the kinematic model for a strip, with relevant dimensions.

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A general strip

is shown in

Fig. 5

. The elementary transformations for

244

this three-link model include the parameters

and

, corresponding to

245

the degrees of freedom of the joints. The remaining (constant) parameters

246

correspond to the geometry of the link and are provided in

Table 1

247

The angle

is defined in

Fig. 2b

. Due to the four-fold symmetry of

248

the structure, the rotation between the hinges and the folds is 45

◦

for all of

249

the strips (hinges and folds are defined in

Fig. 3

). The components of the

250

translation vector only depend on the position of the folds.

is always zero,

251

as the

-axis is perpendicular to the axis of the link, by construction. The

252

distances in the model are non-dimensionalized by the length 2

, so that the

253

obtained results are valid for strips of any size.

−

√

−

√

−

Table 1: Denavit-Hartenberg parameters for the transformation matrices between adjacent

links of a strip.

254

Once the transformation matrices have been defined, the loop-closure

255

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equation can be written in the form:

256

(17)

and can be solved as described in

Section 2

257

Any configuration of this system is defined the vector

∈

258

... β

... ξ

(18)

In general, there are at least 24

−

6 = 18 degrees of freedom, assuming

259

that the Jacobian of

Eq. (8)

has full-rank. Solutions with specific symmetry

260

properties can be obtained by introducing additional constraints.

261

3.1. Four-Fold-Symmetric Folding

262

Given that a square loop has four-fold symmetry, folding schemes that

263

preserve this symmetry are of particular interest. This symmetry assumption

264

greatly reduces the number of independent variables, by setting

and

265

∀

∈

4]. Symmetry also requires the folds on a strip to have the

266

same angle (

) and prevents any torsion between their axes (

= 0).

267

It should be noted, however, that torsion is allowed between the hinges and

268

the folds, under the constraint

−

. Therefore, only three independent

269

variables remain:

270

•

, denoted as the fold angle;

271

•

, denoted as the hinge angle;

272

•

, denoted as the torsion between the hinges and the folds.

273

Journal Pre-proof

With these constraints, the loop-closure equation and its linearized ver-

274

sions in

Eq. (8)

Eq. (9)

Eq. (13)

, reduce to a system of 3 equations in 3

275

unknowns. The reduced Jacobian matrix,

, has rank 2 throughout the

276

folding process, so that there is only one kinematically-admissible path be-

277

tween planar and folded configuration. Snapshots of the solution for four

278

different values of the hinge angle

are shown in

Fig. 6

279

(a) Hinge angle

◦

(b) Hinge angle

=90

◦

= 135

◦

λL

(d) Hinge angle

= 180

◦

Figure 6: Four-fold symmetric folding of a loop of four strips.

Figure 7

shows the variation of the fold angle

and the strip torsion

280

with the hinge angle

. The plot shows that

monotonically decreases

281

from 0

◦

to -45

◦

when the hinge angle is increased from 0

◦

to 180

◦

. The torsion

282

of the strip

remains zero throughout the entire folding.

283

Figure 8

plots the variation of the singular values of

as a function

284

. One of the singular values is always equal to zero, indicating that

285

there is a unique kinematic path. The other two singular values are greater

286

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Figure 7: Four-fold-symmetric kinematic solution for single square loop.

Figure 8: Singular values of

along the four-fold-symmetric path. The two configurations

shown on the left correspond to two different kinematic paths at the bifurcation point.

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than zero for all values of

, except for the initial and final point (

287

and

= 180

◦

), where the rank of

drops to 1. This corresponds to a

288

bifurcation point, at the intersection of two different kinematic paths. In

289

the alternative folding path, the ends of the strips fold inwards, towards the

290

center of the square, see Mode II in

Fig. 8

. To avoid the bifurcation, a

291

small perturbation of 10

−

rad was applied to the initial configuration of the

292

kinematic simulation.

293

The results in

Fig. 7

were obtained for a fold spacing of

5, but the

294

results are identical for any other values of

. This means that, although

295

different values of

correspond to different geometries of the structure, the

296

relationship between the variables

,ξ

and

remains unchanged.

297

The four-fold symmetric path considered in this section provides a single

298

degree-of-freedom mechanism with torsion-free kinematics, which is beneficial

299

for the structural integrity of the space structure. Therefore, it will used as

300

the baseline for the packaging kinematics of a multi-square-loop structure in

301

Sec.

302

3.2. Folding with a Single Plane of Mirror Symmetry

303

The kinematic formulation presented in this paper can capture more gen-

304

eral kinematic paths, which will be demonstrated by studying an alternative

305

packaging strategy. A mirror symmetric folding scheme, in which a square

306

loop is folded in two steps by moving two corners at a time, is shown in

307

Fig.

308

One of the outermost strips is folded first, by imposing 180

◦

relative

309

rotations at the end hinges. The rest of the structure follows in a symmetric

310

fashion. Then, the opposite strip is folded in a similar way.

311

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Figure 9: Mirror symmetric folding sequence for a square loop: corners C1 and C2 are

folded first (strip 1), followed by corners (C3 and C4).

To model the kinematics of this folding scheme, the 24 variables in the

312

general model of the square loop, Eq.

are reduced to only 11 by setting

313

the following 13 symmetry conditions: for bending

314

, and for torsion

=0,

−

315

=0,

−

. Hence, the reduced

316

model includes only 11 degrees of freedom (6 bending rotations

and 5

317

torsional rotations

318

During the second folding step,

and

are set to zero to fix the folded

319

corners of the square loop.

320

Under these assumptions, the linearized kinematics in Eq.

becomes

321

Journal Pre-proof

underdetermined, as it has 6 equations and 11 unknowns. Eq.

provides a

322

5-dimensional space of solutions.

323

While arbitrary values of

would be kinematically feasible, practically

324

useful solutions are obtained by solving the following optimization problem:

325

∗

= arg min

(19)

in which

and the weights

were all set equal to one in this

326

particular case.

327

The cost function imposes the following practical requirements:

328

•

1

: planarity.

During folding, the structure remains in contact with

329

a planar surface, to offload gravity;

330

•

2

: displacement.

The largest displacements (rotations) should

331

correspond to the degrees of freedom being controlled;

332

•

3

: torsion.

Torsional rotations should be as small as possible, to

333

minimize the stress on the structure.

334

The condition

is defined as follows:

335

•

First, the coordinates of the lower longerons of the strip are evalu-

336

ated at the joint locations, and expressed in their local frames (

337

−

at the elastic fold locations,

=[0

−

√

at the

338

location of the hinges between strips, where the notation of

Eq. (2)

has

339

been used.

340

•

The coordinates

are converted to the same reference using the com-

341

pound transformations

(0)

;

342

Journal Pre-proof

•

A global frame is defined, with the origin at the centroid

of the

343

points

(0)

and the axes aligned with the 0-th local frame. The global

344

coordinates of the points were computed as

(0)

−

;

345

•

A plane passing through these points is defined, using the Princi-

346

pal Component Analysis (PCA) (

Jolliffe

2005

) on the matrix

347

[

,...P

]. The matrix contains the coordinates of the above de-

348

fined points, 5 points on each longeron for a total of 20 points on the

349

4 longerons. The PCA returns the eigenvectors of the covariance ma-

350

trix

; the eigenvector corresponding to the smallest eigenvalue of

351

defines the normal

to the fitting plane.

352

•

Finally,

is defined as:

i

i

353

The condition

states that the displacement of the controlled D.o.F.’s

354

should be larger than any other component, and hence

355

−

(20)

where

is a vector containing the components of

associated with the

356

control variables.

357

The condition

minimizes the torsional components and is stated as:

358

[

, ...ξ

]

(21)

The optimization problem was solved using a quasi-Newton algorithm in

359

MATLAB 2020, with the function

fminunc

.Fig.

shows the evolution of

360

the bending and torsional degrees of freedom for the 2-step mirror symmetric

361

folding path, on a strip with

362

Journal Pre-proof

In the first step, the control variable is

, which increases monotonically

363

from 0

◦

to 180

◦

, while the fold angle

on strip 1 decreases to approximately

364

−

◦

. The other bending angles vary by up to approximately 30

◦

. Torsion is

365

generally small, with the largest value corresponding to

, i.e. the twisting

366

between the fold angles associated with strips 2 and 4.

Figure 10: Two-step folding with a plane of symmetry.

367

These results show that it is possible to star-fold a square loop structure

368

by controlling two degrees of freedom. Note that different kinematic paths

369

controlled by two degrees of freedom are also possible.

370

4. Folding of Nested Square Loops

371

Section

3.1

has computed, for a single square loop, a four-fold-symmetric

372

folding path with a single-degree-of-freedom. When multiple square loops

373

are connected to form a nested space structure, the loops can be folded in

374

sequence by ensuring that the loop kinematics are compatible.

375