Origami-based structures with

programmable properties

A Thesis Presented

By

Soroush Kamrava To

The Department of Mechanical and Industrial Engineering

in partial fulfillment of the requirements for the degree of

Master of Science

in the field of

Mechanical Engineering

Northeastern University Boston, Massachusetts

August 2017

ii

ABSTRACT

Origami-inspired foldable structures and mechanisms have demonstrated a variety of

beneficial features including geometric complexity and structural resilience, but little

attention has been paid to their applications. We present a novel cellular metamaterial

constructed from origami building blocks based on Miura-ori fold. The proposed cellular

metamaterial exhibits unusual properties some of which stemming from the inherent

properties of its origami building blocks, and others manifesting due to its unique

geometrical construction and architecture. These properties include rigid-foldability,

auxeticity (i.e., negative Poisson’s ratio), bistability, and self-locking of origami building

blocks to construct load-bearing cellular metamaterials. Also, we introduce an origami

“string”: a slender structure with a programmable trajectory. The string is composed of a

number of elements that can be individually programmed to achieve a specific folding

pattern and curvature. The mechanism has a single degree of freedom, allowing it to be

actuated from one end while maintaining precise positioning along the entire string length.

These features demonstrate capabilities of the proposed origami strings for robotics

application such as a robotic gripper and a biomimetic hand.

iii

ACKNOWLEDGMENT

In the first place, I would like to express my deepest gratitude to Professor Ashkan Vaziri,

my thesis advisor, for his guidance, inspiration and unlimited encouragement during my

graduate study at Northeastern University. His supervision and support from the

preliminary to the concluding level enabled me to develop an understanding of the subject

and made him a backbone of this research and so to this thesis. During these two years, he

gives me an extraordinary experiences throughout this work. He provided me unflinching

encouragement and support in various ways. His truly scientist intuition has made him as

a constant oasis of ideas and passions in science, which exceptionally inspire and enrich

my growth as a student, a researcher and a scientist want to be. I also gratefully

acknowledge Professor Hamid Nayeb-Hashemi for his advice, crucial contribution, and

constructive comments on this thesis.

This acknowledgement would be incomplete, if I miss to mention the support I received

from my research group, especially Davood Mousanezhad, Hamid Ebrahimi, Mohammad

Sadegh Ghiasi, and Soheil Saffari for their fruitful discussions. I would also want to thank

my family for their everlasting love and unconditional encouragement throughout my

whole life. Without their help and support, I wouldn't be able to make this happen.

I dedicate my thesis to my family for their love and support.

iv

TABLE OF CONTENTS

Chapter 1: Introduction to Origami-based Mechanical Structures ..................................... 1

1.1. Literature review and overview............................................................................ 2

Chapter 2: Origami-based Cellular Metamaterial ............................................................... 4

2.1. Overview .................................................................................................................. 5

2.2. Geometry of origami-based unit cells ...................................................................... 7

2.3. Self-locking ............................................................................................................ 11

2.4. Mechanics of unit-cell ............................................................................................ 13

2.4.1. Poison’s ratio, area and volume of cross-section ............................................ 14

2.4.2. Force-Folding relations .................................................................................... 18

2.5. Conclusion .............................................................................................................. 22

2.6. Appendix A: relations between 𝜽𝜽,𝜷𝜷 𝐚𝐚𝐚𝐚𝐚𝐚 𝝃𝝃 ........................................................... 22

2.6.1. Angle 𝜷𝜷 ............................................................................................................ 23

2.6.2. Angle 𝝃𝝃 ............................................................................................................ 24

Chapter 3: Programmable Origami-based Strings ............................................................ 26

3.1. Two-dimensional folding pattern ........................................................................... 27

3.1.1. Methodology for designing desired crease patterns ........................................ 31

3.1.2. Relative motions along the string .................................................................... 35

3.2. Three-dimensional folding pattern ......................................................................... 36

3.2.2. Out-of-plane displacement .............................................................................. 39

3.2.3. Examples of 3D string ..................................................................................... 43

3.3. Robotics applications of origami string ................................................................. 44

3.3.1. Origami robotic gripper ................................................................................... 44

3.3.2. Biomimetic origami hand ................................................................................ 48

3.4. Conclusion .............................................................................................................. 49

References ......................................................................................................................... 51

v

LIST OF FIGURES

Figure 1. (a) (left image) The Miura-ori can be described by constant angle of α and

the single degree of freedom (DOF) which can be defined in terms of dihedral

angles, θ, and ξ, and the angle between mountain and front valley folding lines, β.

(middle image) Two Miura-ori units are first positioned in a zigzag pattern, then

mirrored to form a symmetric structure. (right image) ‘First-order element’, used

in developing the origami-based cellular metamaterial. (b) First-order elements

are attached together in three different ways to make a ‘second-order element’

with internal angles, γ1, γ2, and γ3. (c) From all possible closed-loop elements,

formed by using second-order elements, only one arrangement leads to a rigid-

foldable geometry while the other are all rigid. ................................................... 6

Figure 2. (a) Rigid quadrilateral closed loop element (b) ‘Kagome’ structure made

from rigid triangular and hexagonal elements. .................................................... 9

Figure 3. All possible configurations of triangular, quadrilateral, and hexagonal

closed-loop elements (the only 2D shapes which can individually tessellate the 2D

space to form periodic geometries), formed by different types of second-order

elements. ........................................................................................................... 11

Figure 4. (a) Assembly and locking procedure for two first-order elements. (b) The

assembly and self-locking feature of the first-order elements are transferred to the

building blocks. This forms the final assembly of the origami-based cellular

metamaterial. (c) Measuring the resisting force for unlocked and locked states of

two building blocks of the origami-based cellular metamaterial, where the

vi

unlocked configuration exhibits no resisting force while in the locked state the

structure shows noticeable resisting force before locking fails. ........................ 12

Figure 5. (a) Front and side views of the closed-loop element, as well as geometrical

characteristics of the first-order element. The structural organization of the first-

order element (as well as the closed-loop element) can be defined by two constant

values related to the topology of the underlying Miura-ori unit, length 𝑎𝑎 and angle

𝛼𝛼, and one variable angle which can be chosen between 𝛽𝛽, 𝜃𝜃, and 𝜉𝜉 representing

the structure’s single degree of freedom. (b) Variations of cross-sectional area and

volume of the closed-loop element (respectively normalized by 𝑎𝑎2 and 𝑎𝑎3) with

respect to the folding ratio. (c) Plots of Poisson’s ratio versus folding ratio for in-

plane diagonal directions, 𝐷𝐷1 and 𝐷𝐷2, while the insets in (b) and (c) show the

folded configurations for 𝛼𝛼 = 75𝑜𝑜, 60𝑜𝑜, 45𝑜𝑜, 30𝑜𝑜 at the specified points. (d) Rigid-

foldability of the closed-loop element under out-of-plane and in-plane loadings

(i.e., two orthogonal directions). ....................................................................... 15

Figure 6. (a) The normalized out-of-plane and in-plane folding forces (i.e., F/k, where

F is applying force and k is torsional spring constant per unit crease length) versus

the folding ratio for different values of the angle, 𝛼𝛼, ranging from 30o to 75o,

while the torsional springs are assumed to be free at 50% folding ratio [or equally

𝜃𝜃0 = 90𝑜𝑜, and 𝜉𝜉0 can be calculated from Equation (1)]. (b) The normalized out-

of-plane and in-plane folding forces versus the folding ratio for a constant value

of 𝛼𝛼 = 60𝑜𝑜, with 𝜃𝜃0 varying between the extreme cases, 𝜃𝜃0 = 0𝑜𝑜 and 𝜃𝜃0 = 180𝑜𝑜.

(c) Comparison between out-of-plane and in-plane folding forces for an RVE with

vii

𝛼𝛼 = 60𝑜𝑜 and 𝜃𝜃0 = 90𝑜𝑜. The sub-plot presents the folding ratio versus 𝛼𝛼, for the

point at which the out-of-plane and in-plane forces are equal. .......................... 21

Figure 7. Geometrical characteristics of a Mira-ori fold at an arbitrary level of folding.

......................................................................................................................... 23

Figure 8. Folding of Miura-ori elements into zigzag and curved configurations (a)

Folding of single Miura-ori. The folding angle, 𝜃𝜃 is the angle between the flat and

folded configurations. 𝛾𝛾 is the angle between lines 𝐴𝐴 and 𝐴𝐴′, as shown in the

picture. 𝛾𝛾 versus 𝜃𝜃 is plotted for different Miura-ori angles 𝛼𝛼, ranging from 0o to

180o. The markers are simulation results for 𝛼𝛼 = 60° and 95° showing excellent

agreement with the theoretical prediction. (b) 6 × 1 inch paper strips with one,

three, and five elements with 𝛼𝛼 = 60°. The flat (𝜃𝜃 = 0°) and folded (𝜃𝜃 = 10°)

configurations for each paper strip are shown. The constant value of 𝛼𝛼 results in a

zig-zag folded configuration. (c) 6 × 1 inch paper strips with one, three, and five

elements. The left sample has 𝛼𝛼 = 120°. The other samples are made with 𝛼𝛼 =

120° and 𝛼𝛼 = 60°. The flat (𝜃𝜃 = 0°) and folded (𝜃𝜃 = −10°) configurations for

each paper strip are shown, resulting in a curved folded configuration. ............ 28

Figure 9. String that fold into circle, eight-pointed star and spiral configurations (a)

12× 1 inch paper strips with different element designs. Blue and green samples

have repeating but constant values 𝛼𝛼. In the red sample, 𝛼𝛼 varies along the sample

length as shown. (b) Samples at different folding angle 𝜃𝜃. The blue sample folds

into a circle at 𝜃𝜃 = 45°, while the green and red samples fold to star and spiral

shapes, respectively at the same 𝜃𝜃 . The scale bar is the same for all three samples

for each value of 𝜃𝜃 (each row shown). .............................................................. 31

viii Figure 10. String design to achieve a specific folding configuration (a) A sample of

desired design for string origamis. The figures shows the path of central line

consisting of 18 straight lines for the desired design and required values of 𝛾𝛾 =

90° and −120°. (b) 𝛼𝛼 versus 𝜃𝜃 plotted for the required values of 𝛾𝛾 to achieve the

desired design. (c) The designed sample at 𝜃𝜃 = ±45° folding. The top image

shows the unfolded pattern for one third of the sample (6 elements). ................ 33

Figure 11. String design to achieve a specific folding configuration at 𝜃𝜃 = ±20°. . 34

Figure 12. String design to achieve a specific folding configuration at 𝜃𝜃 = ±70°. .. 34

Figure 13. Origami string with 9 elements. Local coordinate system for each element

is shown. Position and orientation of each element is expressed by matrix 𝑅𝑅𝑅𝑅

presented by Equation (20). .............................................................................. 36

Figure 14. Folding a strip of paper based on a “Four-crease” pattern. (a) Angle 𝜃𝜃

quantifies folding level (−90° ≤ 𝜃𝜃 ≤ +90°). 𝛾𝛾 is the angle between crease line B

and plane XY. ∅ is defined as the angle between projection of line B on plane XY

and X axis. Angle ∅ represents out-of-plane displacement of origami pattern. (b)

Simulation results demonstrate relations between (𝜃𝜃) and (𝛾𝛾 and ∅) as functions

of 𝛼𝛼1 and 𝛼𝛼2. (c) Illustration for folding of a pattern with 𝛼𝛼1 = 90° and 𝛼𝛼2 = 60°.

First row shows valley folding (𝜃𝜃 > 0°) and second row shows mountain folding

(𝜃𝜃 < 0°). ........................................................................................................... 39

Figure 15. Out-of-plane displacement of origami string. (a) Applying five

interconnected “four-crease” patterns in a 𝐿𝐿 × 𝐻𝐻 paper strip with repetitive 𝛼𝛼1 =

90°, 𝛼𝛼2 = 60° and their supplementary angles. (b) Simulation results for out-of-

ix

plane displacement (OPD), normalized by dividing by total length (L). [Left]

Shows variation of normalized OPD as a function of angle 𝜃𝜃 for different numbers

of division (𝑛𝑛). [Right] Normalized OPD as a function of 𝑛𝑛 for different angles 𝛼𝛼.

(c) Three origami strings with equal length and 3, 6 and 9 number of divisions at

four levels of folding (𝜃𝜃 = 5°,15°, 30° and 45°). The folded configurations for a

string with nine divisions and 𝜃𝜃 > 21° isn’t accessible due to the self-intersecting

in string. ........................................................................................................... 42

Figure 16. Design examples that fold from flat to final unique configurations. Three

strips of paper with identical length and width are patterned by different crease

line. Three folding levels (𝜃𝜃 = 5°,17° and 45°) are shown for each design. Blue,

red and green strings fold to a helical, double-spiral and star-helical final shapes

at 𝜃𝜃 = 45°, respectively. ................................................................................... 44

Figure 17. Origami robotic gripper (a) Top view of the robotic gripper with 𝛼𝛼 = 50°

and 150° at 𝜃𝜃 = 0°. The actuation moment is produced by a servo-motor and

transferred to the gripper through an embedded gear-box (b) Side views of the

robotic gripper at folding levels, 𝜃𝜃 = 12°,29°, 56°. (c) [Left] Components of linear

velocity of the right end point, normalized with respect to the angular velocity of

the servo-motor, 𝜔𝜔, and half length of the gripper, 𝑎𝑎. [Right] components of the

reaction force applied to the right end point, normalized with respect to the input

moment, 𝑀𝑀, and half length of the gripper, 𝑎𝑎. ................................................... 46

Figure 18. Biomimetic origami hand (a) Side view of the index finger of the robotic

hand at four folding levels, 𝜃𝜃 = 0°, 7°,25°, 45°. The design of the robotic finger is

shown in the inset. The actuation moment for each finger is produced by a servo-

x

motor and transferred to the finger through an embedded gear-box (b) Front and

back views of the proposed robotic hand. (c) Demonstration of functionality of

the proposed robotic hand in manipulating sample objects. .............................. 49

1

Chapter 1: Introduction to Origami-based Mechanical Structures

2 1.1.Literature review and overview

Origami, the ancient art of paper folding originating in Japan, has recently evolved into a

new paradigm for advanced technological, biomedical and engineering applications[1, 2].

Examples of innovative origami-inspired systems include deployable solar panels[3],

tunable implants[4], electric devices[5], and robotics[6, 7]. The properties of origami

mostly stem from crease patterns that can be scaled, making it compatible with a broad

range of applications from nanoscale[8-10] devices to architectural structures[11-13].

Innovative crease patterns add fascinating features to these systems including self-

locking[14], buckling-induced pop-up[15], snapping[16, 17], and programming motion

through the arrangement of folding lines[18]. Also, origami relies on seemingly

straightforward operations of concerted folding of a flat sheet of paper to produce

incredibly complicated geometrical objects. This relatively simple control of topology

makes origami an important conceptual paradigm for deployable structures across a wide

spectrum of applications. This includes several other recent demonstrations in areas as

diverse as deployable solar panels[3, 19], fold-core sandwich panels[20, 21], three-

dimensional (3D) cell-laden microstructures[22], flexible medical stents[23], flexible

electronics[24], soft pneumatic actuators[25], and self-folding robots and structures[18, 26,

27]. Furthermore, periodic cellular metamaterials have been recently designed by

assembling foldable origami units (i.e., sheets or tubes) which tessellate to fill the 3D

space[28-33]. In addition, origami has found applications in designing mechanical

metamaterials with tunable stiffness, auxeticity, bistability, load bearing capacity and self-

folding features[13, 15, 30, 31, 34-36]. In second chapter, we studied a cellular origami-

based structure which shows fascinating properties, such as auxeticity, bistability and

3 embedded self-locking. In third chapter, a novel design for one DOF origami string is

presented, which can be implemented in different robotics application due to its

programmable kinetics and kinematics.

4

Chapter 2: Origami-based Cellular Metamaterial

5 2.1. Overview

Origami construction relies on a mechanically simple folding operation and

discovering the exact sequence of folds for a desired behavior is a combinatorically

intractable problem[12, 37, 38]. In this context, simplification is possible through an

intricate coupling of topology and mechanical compatibility to design periodic fold

sequence that can be repeated to create such origami[39, 40]. An example is the pioneering

work of Tachi and Miura[29], who introduced a type of rigid origami based on the

previously-proposed Miura-ori fold[41]. Miura-ori is a single degree of freedom (DOF)

rigid-foldable origami shown in Figure 1(a) – left image. The four crease lines of Miura-

ori which result in one mountain and three valley folds define four identical parallelograms

with adjacent sides defining an acute angle, 𝛼𝛼 [shown in Figure 1(a) – left image]. As the

flat sheet deforms, these parallelograms become inclined to each other which can be

quantified in terms of dihedral angles, 𝜃𝜃 ∈ [0𝑜𝑜 ,180𝑜𝑜], 𝜉𝜉 ∈ [0𝑜𝑜 ,180𝑜𝑜], or the angle

between the mountain and front valley folding lines, 𝛽𝛽 ∈ [180𝑜𝑜 − 2𝛼𝛼, 180𝑜𝑜]. Due to the

geometrical constraints, only one of these angles (𝜃𝜃, 𝜉𝜉, or 𝛽𝛽) is independent and can then

be used to represent the single DOF of the system in analysis. For example, 𝛽𝛽 and 𝜉𝜉 can be

expressed in terms of θ, and the constant angle, 𝛼𝛼, using the following relationships:

𝛽𝛽 = 180𝑜𝑜 − 2𝑐𝑐𝑜𝑜𝑐𝑐−1

⎝

⎛ 𝑐𝑐𝑜𝑜𝑐𝑐𝛼𝛼

�1 − 𝑐𝑐𝑅𝑅𝑛𝑛2 �𝜃𝜃2� 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼⎠

⎞

𝜉𝜉 = 𝑐𝑐𝑜𝑜𝑐𝑐−1 �−(1 + 𝑐𝑐𝑜𝑜𝑐𝑐2𝛼𝛼) 𝑐𝑐𝑅𝑅𝑛𝑛2 �𝜃𝜃2�

1 − 𝑐𝑐𝑅𝑅𝑛𝑛2 �𝜃𝜃2� 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼�

(1)

6 See Appendix A for derivation of Equations (1).

Figure 1. (a) (left image) The Miura-ori can be described by constant angle of α and the single degree of freedom (DOF) which can be defined in terms of dihedral angles, θ, and ξ, and the angle between mountain and front valley folding lines, β. (middle image) Two Miura-ori units are first positioned in a zigzag pattern, then mirrored to form a symmetric structure. (right image) ‘First-order element’, used in developing the origami-based cellular metamaterial. (b) First-order elements are attached together in three different ways to make a ‘second-order element’ with internal angles, γ1, γ2, and γ3. (c) From all possible closed-loop elements, formed by using second-order elements, only one arrangement leads to a rigid-foldable geometry while the other are all rigid.

Putting Miura-ori units next to each other results in a Miura-ori sheet construction while

retaining its single DOF properties and rigid-foldability. Stacking and bonding Miura-ori

sheets along fold lines are shown to form cellular metamaterials with a single DOF that

can be machined into any desired shape while preserving its folding motion[30, 42].

7 2.2. Geometry of origami-based unit cells

In this chapter, we propose a new class of origami-based cellular metamaterials with

a wide range of interesting properties such as auxeticity, bistability, foldability, and self-

locking. We start our design with putting together four Miura-ori folds as shown in Figure

1(a) – middle image. First, two Miura-ori units were positioned in a zigzag pattern, then

mirrored to form a symmetric structure, preserving the single DOF, inherent to the original

Miura-ori fold. Based on this design, we fold a single sheet of paper to construct a ‘first-

order element’ that will be used in developing the origami-based cellular metamaterial,

Figure 1(a) – right image. It is noteworthy that folding of the first order element, for

example by changing 𝜃𝜃, results in change in its overall length; however, the left and right

parts of the element stay aligned, independent of the folding level.

First-order elements can be attached together in three different ways, shown in Figure

1(b), to make a ‘second-order element’. From these three configurations, only the

configuration shown on the right can be made by folding a single sheet of paper, and the

other two configurations can be constructed by attaching the two first-order elements. The

angle between the two segments in each second-order element is denoted by 𝛾𝛾1, 𝛾𝛾2, and

𝛾𝛾3, which can be calculated as 180𝑜𝑜 − 𝛽𝛽, 180𝑜𝑜 − 𝛽𝛽, and 𝛽𝛽, respectively (recall from Figure

1(a) that 𝛽𝛽 is an angle varying between 180𝑜𝑜 − 2𝛼𝛼 and 180𝑜𝑜). Considering 𝛾𝛾1, 𝛾𝛾2, and 𝛾𝛾3

as internal angles, these second-order elements can be connected to generate contiguous

geometrically closed-loop elements with many different topologies with the following

geometrical constraints: 1. second-order elements with 𝛾𝛾1 and 𝛾𝛾2 cannot be adjacent, 2. the

two sides of the second-order element with 𝛾𝛾3 cannot be connected to two identical

elements with 𝛾𝛾1 or 𝛾𝛾2. Note that ignoring these geometrical constraints will result in

8 closed-loop elements with at least one external angle with γ1 or γ2 or γ3 value (i.e., closed-

loop elements with at least one internal angle not equal to γ1 or γ2 or γ3). Figure 1(c) shows

three possible quadrangular configurations that satisfy above constraints.

We now prove that from all possible close-loop elements only one arrangement leads

to a rigid-foldable geometry. For each closed-loop element with 𝑛𝑛 sides, the summation of

all internal angles must be equal to 180𝑜𝑜 × (𝑛𝑛 − 2), where 𝑛𝑛 is the number of first-order

elements used to construct the closed-loop element. Denoting 𝑚𝑚𝑖𝑖 (𝑅𝑅 = 1,2,3) as the number

of 𝛾𝛾𝑖𝑖 (𝑅𝑅 = 1,2,3) angles (i.e., 𝑛𝑛 = 𝑚𝑚1 + 𝑚𝑚2 + 𝑚𝑚3) yields the following geometrical

relationship:

(𝑚𝑚1 + 𝑚𝑚2)(180𝑜𝑜 − 𝛽𝛽) + 𝑚𝑚3𝛽𝛽 = (𝑚𝑚1 + 𝑚𝑚2 + 𝑚𝑚3 − 2) × 180𝑜𝑜 (2)

To achieve a foldable configuration, the left hand side of Equation (2) must be independent

of the folding variable, 𝛽𝛽 (note that the right hand side of the equation is a constant and

independent of 𝛽𝛽). This yields 𝑚𝑚1 + 𝑚𝑚2 = 2 and 𝑚𝑚3 = 2, meaning that the only possible

foldable configuration is a ‘quadrangle’ (𝑛𝑛 = 4). The examples provided in Figure 1(c) are

the only configurations that satisfy the Equation (2). The left and middle configurations

can only built for 𝛽𝛽 = 90𝑜𝑜, while the right configuration can be built for any value of 𝛽𝛽 ∈

[180𝑜𝑜 − 2𝛼𝛼,180𝑜𝑜]. This means that the left and middle configurations are rigid and the

only possible foldable polygon is the jigsaw-puzzle-like unit cell highlighted in green.

Here, we will prove that the quadrilateral element shown in Figure 2(a), is completely

rigid, though it does not violate the geometrical constraint on internal angles, presented by

Equation (2) of the manuscript.

9

Figure 2. (a) Rigid quadrilateral closed loop element (b) ‘Kagome’ structure made from rigid triangular and hexagonal elements.

We begin the analysis by noting that the length of the edges of this closed-loop element,

𝐴𝐴𝐴𝐴����, 𝐶𝐶𝐷𝐷����, 𝐴𝐴𝐷𝐷����, and 𝐴𝐴𝐶𝐶����, can be obtained as the following [see Figure 2(a), and Figure 5(a)]:

𝐴𝐴𝐴𝐴���� = 𝐶𝐶𝐷𝐷���� = 8𝑎𝑎 − 4𝑎𝑎 𝑐𝑐𝑜𝑜𝑐𝑐 𝛽𝛽

𝐴𝐴𝐷𝐷���� = 𝐴𝐴𝐶𝐶���� = 9𝑎𝑎 − 4𝑎𝑎 𝑐𝑐𝑜𝑜𝑐𝑐 𝛽𝛽 (3)

10 where 𝑎𝑎 and 𝛽𝛽 are defined in Figure 5(a). Furthermore, the following relation must hold

for the edges and internal angles of the quadrangle, 𝐴𝐴𝐴𝐴𝐶𝐶𝐷𝐷:

𝐴𝐴𝐴𝐴���� = 𝐶𝐶𝐷𝐷���� − 𝐴𝐴𝐷𝐷���� cos𝛽𝛽 − 𝐴𝐴𝐶𝐶���� cos𝛽𝛽 = 8𝑎𝑎 − 22𝑎𝑎 cos𝛽𝛽 + 8𝑎𝑎 cos2 𝛽𝛽 (4)

Next, 𝐴𝐴𝐴𝐴���� in Equation (3) and Equation (4) must be equal, which results in the following:

8𝑎𝑎 − 4𝑎𝑎 𝑐𝑐𝑜𝑜𝑐𝑐 𝛽𝛽 = 8𝑎𝑎 − 22𝑎𝑎 cos𝛽𝛽 + 8𝑎𝑎 cos2 𝛽𝛽 (5)

which holds true only for 𝛽𝛽 = 90𝑜𝑜. Therefore, Equation (5) is not valid for all values of 𝛽𝛽

(equally, every level of folding). In other words, the element is not foldable (equally, it is

rigid). All other possible configurations of triangular, quadrilateral, and hexagonal closed-

loop elements (i.e., the only 2D shapes which can individually tessellate the 2D space to

form periodic geometries), formed by different types of second-order elements introduced

in Figure 1(b), are given in Figure 3. Note that all these elements are rigid (i.e., non-

foldable), since they don’t satisfy Equation (2), however, they can be used as building

blocks to construct rigid tessellations such as the well-known ‘Kagome’ structure made

from triangular and hexagonal elements and shown in Figure 2(b).

11

Figure 3. All possible configurations of triangular, quadrilateral, and hexagonal closed-loop elements (the only 2D shapes which can individually tessellate the 2D space to form periodic geometries), formed by different types of second-order elements.

2.3. Self-locking

It is essential to employ a connecting mechanism to link the adjacent unit cells of a

lattice structure together, to form the final configuration of the system. An example of this

mechanism is using an adhesive material to connect the unit cells together, however, this

12 may affect the foldability of the structure by restricting degrees of freedom of the system,

which will definitely alter the geometrical and mechanical properties of the final assembly.

Here, we introduce an embedded self-locking mechanism into the proposed foldable unit,

bonding the adjacent units together, which originates from the locking of first-order

elements as shown in Figures 4(a). To ensure fitting of one first-order element into another,

each element must have a folding level corresponding to 𝛽𝛽 > 90𝑜𝑜. Once a contact is

established between the two elements, self-locking can manifest by decreasing the folding

angle to 𝛽𝛽 < 90𝑜𝑜, as for example is achieved in Figures 4(a) – right image, by applying an

out-of-plane compression.

Figure 4. (a) Assembly and locking procedure for two first-order elements. (b) The assembly and self-locking feature of the first-order elements are transferred to the building blocks. This forms the final assembly of the origami-based cellular metamaterial. (c) Measuring the resisting force for unlocked and locked states of two building blocks of the origami-based cellular metamaterial, where the unlocked configuration exhibits no resisting force while in the locked state the structure shows noticeable resisting force before locking fails.

13

The foldable closed-loop element (i.e., Figure 1(c) – right image) can be stacked in the

out-of-plane direction to create a foldable tubular topology, which then can be used as

building blocks to construct a cellular metamaterial, Figures 4(b). The self-locking feature

of the first-order elements described above gets transferred to these building blocks and

similarly gets activated for folding levels with 𝛽𝛽 < 90𝑜𝑜. Note that this locked state would

impose effective contact strength between the building blocks in addition to simple

frictional assembly. To this end we subjected a prototype, made of paper, to tension, when

in locked and unlocked states, Figure 4(c). When in the unlocked state, the structure

exhibits no force resistance [i.e., force ~ 0 (N)], while in the locked state the structure

shows noticeable resisting force [i.e., force ~ 35 (N)] before locking fails. Note that the

resisting force strongly depends on folding level as well as the mechanical properties (i.e.,

elasticity) of the parent material which the plates are made of. However, the main goal of

these experiments was to demonstrate the effect of the embedded self-locking mechanism

on the structural resistance against the applied in-plane tensile load by comparing their

resisting force in unlocked versus locked configurations. In theory, since the plates are

assumed to be rigid, the resisting force will be infinite in the locked configuration.

2.4. Mechanics of unit-cell

The behavior and properties of the cellular metamaterial, which exhibits periodicity in

both in-plane as well as out-of-plane directions can be analytically evaluated by assuming

an infinite repetition of a representative volume element (i.e., RVE; same as the closed-

loop element) of the cellular metamaterial, Figure 5(a) – left and middle images. Thus, we

14 investigate the kinematics and kinetics of the cellular metamaterial by analyzing the closed-

loop element during folding.

2.4.1. Poison’s ratio, area and volume of cross-section

Figure 5(a) shows top and side views of the closed-loop element as well as the

geometrical characteristics of the constituting first-order element introduced earlier. The

in-plane diagonals, 𝐷𝐷1 and 𝐷𝐷2, and out-of-plane height, 𝐻𝐻, of the closed-loop element at an

arbitrary level of folding, illustrated in Figure 5(a), are given in terms of the geometry of

the underlying Miura-ori unit as:

𝐷𝐷1 = √2𝑎𝑎(9 − 4 cos𝛽𝛽)�1 − cos𝛽𝛽

𝐷𝐷2 = √2𝑎𝑎(9 − 4 cos𝛽𝛽)�1 + cos𝛽𝛽

𝐻𝐻 = 4𝑎𝑎 sin𝛼𝛼 sin �𝜃𝜃2� (6)

Note that 𝐷𝐷1 and 𝐷𝐷2 are diagonals of a diamond (i.e., the closed-loop element) and therefore

always perpendicular to each other. In order to quantify the folding process, we define a

non-dimensional parameter called ‘folding ratio’ as, [(180𝑜𝑜 − 𝜃𝜃) 180𝑜𝑜⁄ ] × 100%, which

varies from 0% (i.e., 𝜃𝜃 = 180𝑜𝑜) to 100% (i.e., 𝜃𝜃 = 0𝑜𝑜). In other words, 0% and 100%

folding ratios correspond to two fully-folded configurations of the proposed construction.

15

Figure 5. (a) Front and side views of the closed-loop element, as well as geometrical characteristics of the first-order element. The structural organization of the first-order element (as well as the closed-loop element) can be defined by two constant values related to the topology of the

16 underlying Miura-ori unit, length 𝑎𝑎 and angle 𝛼𝛼, and one variable angle which can be chosen between 𝛽𝛽, 𝜃𝜃, and 𝜉𝜉 representing the structure’s single degree of freedom. (b) Variations of cross-sectional area and volume of the closed-loop element (respectively normalized by 𝑎𝑎2 and 𝑎𝑎3) with respect to the folding ratio. (c) Plots of Poisson’s ratio versus folding ratio for in-plane diagonal directions, 𝐷𝐷1 and 𝐷𝐷2, while the insets in (b) and (c) show the folded configurations for 𝛼𝛼 =75𝑜𝑜, 60𝑜𝑜, 45𝑜𝑜,30𝑜𝑜 at the specified points. (d) Rigid-foldability of the closed-loop element under out-of-plane and in-plane loadings (i.e., two orthogonal directions).

The cross-sectional area of the closed-loop element, 𝑆𝑆, defined as the area of the

polygon formed by intersecting the closed-loop element with a plane normal to its height,

is constant through the height of the closed-loop element. The volume of the closed-loop

element, 𝑉𝑉, is the volume bounded by the constituting first-order elements. Figure 5(b)

depicts the variation of the cross-sectional area and volume of the closed-loop element

(respectively normalized by 𝑎𝑎2 and 𝑎𝑎3) as functions of the folding ratio, respectively,

presented for four different values of 𝛼𝛼 ranging from 30o to 75o. As the folding ratio

increases, the normalized area rises from zero (i.e., fully-folded configuration) up to a

turning point, and then decreases due to the auxetic behavior of closed-loop element in

both diagonal directions (will be discussed later). This is then followed by a plateau regime

as the closed-loop element reaches the other fully-folded configuration. The critical folding

ratio associated with the turning point decreases significantly for higher values of 𝛼𝛼.

Similar behavior is observed for the variations of the normalized volume, except the fact

that at 100% folding ratio, the volume becomes zero due to the fully-folded configuration

of the closed-loop element.

Next, for an uniaxial out-of-plane load, we calculate the Poisson’s ratio of the

closed-loop element in 𝐷𝐷1 and 𝐷𝐷2 directions (since they are always perpendicular to each

other), defined as 𝜈𝜈𝐻𝐻𝐷𝐷𝑖𝑖 = −𝑑𝑑𝐷𝐷𝑖𝑖 𝐷𝐷𝑖𝑖⁄𝑑𝑑𝐻𝐻 𝐻𝐻⁄

, where 𝑅𝑅 = 1 or 2. Differentiating Equation (6) with

17 respect to the folding angles and plugging the results into the above equations yield the

following closed-form expressions for Poisson’s ratios:

𝜈𝜈𝐻𝐻𝐷𝐷1

=−𝑐𝑐𝑜𝑜𝑐𝑐2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ ) (24 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 + 5 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ ) − 29)

(𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ ) − 1)(𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 − 1)(8 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 + 5 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ ) − 13)

𝜈𝜈𝐻𝐻𝐷𝐷2

=−𝑐𝑐𝑜𝑜𝑐𝑐2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ ) (11 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ ) − 24 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 + 13)

(𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ ) − 1)(𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ ) − 1)(8 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 + 5 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ ) − 13) (7)

It is noteworthy that although these formulations were derived for a single closed-loop unit,

they still hold true for the infinite periodic metamaterial. This is due to the fact that the

calculations were performed on an RVE, which can be tessellated in diagonal (i.e., D1 and

D2) and out-of-plane directions [as the “lattice vectors”[43]] to form the final configuration

of the metamaterial.

Figure 5(c) shows the dependence of Poisson’s ratio on the folding ratio in two orthogonal

in-plane directions (i.e., 𝐷𝐷1 and 𝐷𝐷2), for four different values of 𝛼𝛼 ranging from 30o to 75o.

𝜈𝜈𝐻𝐻𝐷𝐷1 is negative for the entire range of folding ratio and 𝛼𝛼, with a significantly pronounced

auxetic response at greater values of 𝛼𝛼. In contrast, 𝜈𝜈𝐻𝐻𝐷𝐷2 has a positive infinity value at 0%

folding ratio [theoretically, the denominator of 𝜈𝜈𝐻𝐻𝐷𝐷2 becomes zero at 0% folding ratio, see

Equation (7)], which then reduces to 0 at 100% folding ratio. For 𝛼𝛼 ≳ 60𝑜𝑜, this involves

exhibiting a negative Poisson’s ratio after a certain folding ratio. Insets in Figure 5(b) and

(c) illustrate the effect of changing 𝛼𝛼 in the geometry and folding procedure of unit-cell.

Figure 5(d) shows folding of a sample closed-loop element demonstrated under loading in

out-of-plane compression and in-plane stretching along the direction of 𝐷𝐷1. For this sample,

18 𝛼𝛼 = 60𝑜𝑜 and the fully-folded states are achieved at 𝛽𝛽 = 180𝑜𝑜 − 2𝛼𝛼 = 60𝑜𝑜 (or 𝜃𝜃 = 0𝑜𝑜)

and 𝛽𝛽 = 180𝑜𝑜 (𝜃𝜃 = 180𝑜𝑜), as shown under out-of-plane compression and in-plane

stretching experiments, respectively. Note that the closed-loop element, shown in Figure

5(d) tessellates the 3D space regardless of folding level.

2.4.2. Force-Folding relations

Next, we investigated the force required to attain a desired level of folding for each

building block of the cellular metamaterial under two loading directions (i.e., out-of-plane

and in-plane). We assumed that each building block is made of rigid plates, connected

together at straight creases modeled as linear torsional springs[31] with spring constant per

unit crease length of 𝑘𝑘(N). Also, as mentioned earlier, we idealized a building block of the

cellular metamaterial as an infinite array of closed-loop elements stacked on top of each

other, and analyzed the RVE. Following analytical expressions represent the folding force

on the RVE under out-of-plane and in-plane loadings which is derived based on the

principle of minimum total potential energy:

𝐹𝐹𝑜𝑜𝑜𝑜𝑜𝑜−𝑜𝑜𝑜𝑜−𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑘𝑘

= −104(𝜃𝜃 − 𝜃𝜃0) + 80(𝜉𝜉 − 𝜉𝜉0) 𝑑𝑑𝜉𝜉𝑑𝑑𝜃𝜃

2 sin𝛼𝛼 cos(𝜃𝜃 2⁄ )

𝐹𝐹𝑖𝑖𝑝𝑝−𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑘𝑘

= 2�1 − cos𝛽𝛽 �104(𝜃𝜃 − 𝜃𝜃0) + 80(𝜉𝜉 − 𝜉𝜉0) 𝑑𝑑𝜉𝜉𝑑𝑑𝜃𝜃

√2 sin𝛽𝛽 (17 − 12 cos𝛽𝛽)𝑑𝑑𝛽𝛽𝑑𝑑𝜃𝜃

� (8)

where 𝐹𝐹𝑜𝑜𝑜𝑜𝑜𝑜−𝑜𝑜𝑜𝑜−𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 and 𝐹𝐹𝑖𝑖𝑝𝑝−𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 denote the folding forces for out-of-plane and in-plane

loading directions, respectively, 𝜃𝜃0 and 𝜉𝜉0 are the free angles of horizontal and inclined

torsional springs, respectively (i.e., the angles at which no potential energy is stored in the

springs), and 𝑑𝑑𝜉𝜉 𝑑𝑑𝜃𝜃⁄ and 𝑑𝑑𝛽𝛽 𝑑𝑑𝜃𝜃⁄ can be calculated using Equation (1).

19

Figure 6(a) shows the plots of normalized out-of-plane and in-plane folding forces,

versus the folding ratio for different values of 𝛼𝛼, while the free angle of the torsional springs

is kept constant as 𝜃𝜃0 = 90𝑜𝑜 (i.e., 50% folding ratio; 𝜉𝜉0 can be calculated from Equation

(1) by plugging 𝜃𝜃0 instead of 𝜃𝜃). In addition, for 𝛼𝛼 = 60𝑜𝑜, we plotted the normalized out-

of-plane and in-plane folding forces versus the folding ratio for a set of 𝜃𝜃0 varying between

the extreme cases, 𝜃𝜃0 = 0𝑜𝑜 and 𝜃𝜃0 = 180𝑜𝑜, Figure 6(b). The results show a so-called

“bistable“ behavior for θ0 ≳ 155o in out-of-plane loading, and for θ0 ≲ 40o under in-

plane loading. For example, the sample with θ0 = 170o exhibits local extremum points at

20% (local maximum) and 66% (local minimum) folding ratios when subjected to out-of-

plane loading. This reveals the two stable configurations – one at the initial state (i.e.,

F k⁄ = 0) where the folding ratio is 5.5%, and – the other one at the local minimum point

at 66% folding ratio. We should note that the structure will go to the “local minimum”

point (i.e., 66% folding ratio) only if the load is still there (i.e., a pre-load), otherwise, if

we remove the load, the structure will always go back to its stable state at zero force (i.e.,

5.5% folding ratio) after going through a “snap-through”[42]. This bistability in the

response highlights the potential of the proposed cellular metamaterials for energy

absorption, energy harvesting, and impact mitigation applications[17, 44, 45]. Next, we

compare out-of-plane and in-plane loading responses for an RVE with 𝛼𝛼 = 60𝑜𝑜 and 𝜃𝜃0 =

90𝑜𝑜, see Figure 6(c). These calculations show that except for folding ratios greater than

78%, the in-plane force associated for achieving a specific folding ratio is lower than the

out-of-plane force for the same value of folding ratio. This means that for folding ratios

smaller than 78%, it is easier to fold the structure under in-plane loading (compared to an

out-of-plane loading), while the opposite is true for folding ratios greater than 78%.

20 Additionally, the inset of the figure shows that the folding ratio corresponding to the point

at which the two curves meet [shown by a hollow circle in Figure 6(c)], decreases with

increasing 𝛼𝛼, making the out-of-plane force smaller than the in-plane force for a wider span

of the folding ratio.

21

Figure 6. (a) The normalized out-of-plane and in-plane folding forces (i.e., F/k, where F is applying force and k is torsional spring constant per unit crease length) versus the folding ratio for different values of the angle, 𝛼𝛼, ranging from 30o to 75o, while the torsional springs are assumed to be free at 50% folding ratio [or equally 𝜃𝜃0 = 90𝑜𝑜, and 𝜉𝜉0 can be calculated from Equation (1)]. (b) The normalized out-of-plane and in-plane folding forces versus the folding ratio for a constant value of

22 𝛼𝛼 = 60𝑜𝑜, with 𝜃𝜃0 varying between the extreme cases, 𝜃𝜃0 = 0𝑜𝑜 and 𝜃𝜃0 = 180𝑜𝑜. (c) Comparison between out-of-plane and in-plane folding forces for an RVE with 𝛼𝛼 = 60𝑜𝑜 and 𝜃𝜃0 = 90𝑜𝑜. The sub-plot presents the folding ratio versus 𝛼𝛼, for the point at which the out-of-plane and in-plane forces are equal.

2.5. Conclusion

In this paper we propose an origami-based paradigm of constructing cellular materials

which are capable of undergoing large reversible deformation while exhibiting highly

nonlinear auxeticity, bistability and topological locking. Particularly, the locking

phenomena is used as a platform for scaling up these structures in a systematic modular

fashion into larger cellular structures with single force activation without taking recourse

to any special structural or surface modifications. The self-locking is achieved using an

applied force on the structure. Thus, in summary, this present work sets forth an important

avenue of novel cellular metamaterial design based on both self-similar and self-locking

assembly.

2.6. Appendix A: relations between 𝜽𝜽,𝜷𝜷 𝐚𝐚𝐚𝐚𝐚𝐚 𝝃𝝃

The configuration of a Miura-ori unit at an arbitrary level of folding can be fully quantified

by an angular value defining the coordinate of the single degree-of-freedom (DOF) of the

unit. This angular value can be chosen between the two dihedral angles, 𝜃𝜃 ∈ [0𝑜𝑜 ,180𝑜𝑜],

and 𝜉𝜉 ∈ [0𝑜𝑜 ,180𝑜𝑜], or the angle between the mountain and valley folds, 𝛽𝛽 ∈

[180𝑜𝑜 − 2𝛼𝛼,180𝑜𝑜], see Figure 7(a). As mentioned in the manuscript, due to the

geometrical organization of the unit and rigidity assumption, only one of these angles is

independent which can then be used as the coordinate of the single DOF of the unit for our

analysis. To this end, considering 𝜃𝜃 as an independent parameter, we can obtain 𝛽𝛽 and 𝜉𝜉 as

functions of 𝜃𝜃 (i.e., 𝛽𝛽 and 𝜉𝜉 will be considered as dependent parameters).

23

Figure 7. Geometrical characteristics of a Mira-ori fold at an arbitrary level of folding.

2.6.1. Angle 𝜷𝜷

We now begin the analysis by calculating the vectors 𝐴𝐴𝐴𝐴�����⃗ and 𝐴𝐴𝐶𝐶�����⃗ as the following [see

Figure 7(a)]:

𝐴𝐴𝐴𝐴�����⃗ = +𝐿𝐿 𝑐𝑐𝑜𝑜𝑐𝑐 𝜙𝜙 𝚤𝚤 + 𝐿𝐿 𝑐𝑐𝑅𝑅𝑛𝑛 𝜙𝜙 𝚥𝚥

𝐴𝐴𝐶𝐶�����⃗ = −𝐿𝐿 𝑐𝑐𝑜𝑜𝑐𝑐 𝛾𝛾 𝚤𝚤 − 𝐿𝐿 𝑐𝑐𝑅𝑅𝑛𝑛 𝛾𝛾 𝑘𝑘�⃗

(9)

where 𝐿𝐿 is the edge length of the four identical parallelograms forming the Miura-ori unit,

and 𝚤𝚤, 𝚥𝚥, and 𝑘𝑘�⃗ are unit vectors along the x, y, and z directions, respectively. Now, the

following expression defines the angle between the vectors 𝐴𝐴𝐴𝐴�����⃗ and 𝐴𝐴𝐶𝐶�����⃗ :

𝑐𝑐𝑜𝑜𝑐𝑐−1 �𝐴𝐴𝐴𝐴�����⃗ .𝐴𝐴𝐶𝐶�����⃗

�𝐴𝐴𝐴𝐴�����⃗ ��𝐴𝐴𝐶𝐶�����⃗ �� = 180𝑜𝑜 − 𝛼𝛼 (10)

Next, substituting Equation (9) into Equation (10) will result in the following relation

between angles 𝜙𝜙, 𝛾𝛾, and 𝛼𝛼:

𝑐𝑐𝑜𝑜𝑐𝑐 𝜙𝜙 𝑐𝑐𝑜𝑜𝑐𝑐 𝛾𝛾 = 𝑐𝑐𝑜𝑜𝑐𝑐 𝛼𝛼 (11)

24 Now, considering the isosceles triangles, ABF and AED [see Figure 7(b)], the following

relations can be obtained for the angles 𝜃𝜃 and 𝜙𝜙:

𝑐𝑐𝑅𝑅𝑛𝑛(𝜃𝜃 2⁄ ) =𝐷𝐷𝐷𝐷����

2�𝐴𝐴𝐷𝐷����

𝑐𝑐𝑅𝑅𝑛𝑛 𝜙𝜙 =𝐴𝐴𝐹𝐹����

2�𝐴𝐴𝐴𝐴����

(12)

where 𝐷𝐷𝐷𝐷���� is the length of the edge 𝐷𝐷𝐷𝐷 (similarly for other edges). We should note that

𝐴𝐴𝐷𝐷���� = 𝐴𝐴𝐴𝐴���� 𝑐𝑐𝑅𝑅𝑛𝑛 𝛼𝛼, and 𝐴𝐴𝐹𝐹���� = 𝐷𝐷𝐷𝐷����, which by substituting into Equation (12) will result in the

following:

𝑐𝑐𝑅𝑅𝑛𝑛 𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛(𝜃𝜃 2⁄ ) = 𝑐𝑐𝑅𝑅𝑛𝑛 𝜙𝜙 (13)

Finally, Figure 7(a) shows that 𝛽𝛽 = 180𝑜𝑜 − 2𝛾𝛾, which by using Equations (11) and (13)

will result in the following equation for 𝛽𝛽:

𝛽𝛽 = 180𝑜𝑜 − 2𝑐𝑐𝑜𝑜𝑐𝑐−1 �𝑐𝑐𝑜𝑜𝑐𝑐𝛼𝛼

�1 − 𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼 𝑐𝑐𝑅𝑅𝑛𝑛2(𝜃𝜃 2⁄ )� (14)

2.6.2. Angle 𝝃𝝃

In order to obtain a closed-form expression for the angle, 𝜉𝜉, as a function of 𝜃𝜃, we first

translate (with no rotations) the coordinate system of Figure 7(a) from point A to point M,

see Figure 7(c). Note that the angle 𝜉𝜉 is basically the angle between vectors 𝐺𝐺𝐴𝐴�����⃗ and 𝐺𝐺𝐻𝐻������⃗ .

We now begin the analysis by obtaining the coordinates of points G, A, and H with respect

to the new coordinate system located at point M, as the following:

25

𝐺𝐺 = �𝑋𝑋𝐺𝐺𝑌𝑌𝐺𝐺𝑍𝑍𝐺𝐺� = �

𝐿𝐿 cos𝛼𝛼 cos𝜙𝜙𝐿𝐿 cos𝛼𝛼 sin𝜙𝜙

0�

𝐴𝐴 = �𝑋𝑋𝐴𝐴𝑌𝑌𝐴𝐴𝑍𝑍𝐴𝐴� = �

𝐿𝐿 sin(𝛽𝛽 2⁄ ) 0

𝐿𝐿 cos(𝛽𝛽 2⁄ )�

𝐻𝐻 = �𝑋𝑋𝐻𝐻𝑌𝑌𝐻𝐻𝑍𝑍𝐻𝐻� = �

−𝐿𝐿 sin(𝛽𝛽 2⁄ ) + 2𝐿𝐿 cos𝛼𝛼 cos𝜙𝜙 2𝐿𝐿 cos𝛼𝛼 sin𝜙𝜙𝐿𝐿 cos(𝛽𝛽 2⁄ )

�

(15)

Note that we employed the relation, 𝑁𝑁𝐻𝐻����� = 2𝑀𝑀𝐺𝐺����� = 2𝐿𝐿 cos𝛼𝛼, to derive the set of

coordinates presented in Equation (15). Next, using Equation (15) the vectors, 𝐺𝐺𝐴𝐴�����⃗ and 𝐺𝐺𝐻𝐻������⃗ ,

will be determined as the following:

𝐺𝐺𝐴𝐴�����⃗ = �𝐿𝐿 sin(𝛽𝛽 2⁄ ) − 𝐿𝐿 cos𝛼𝛼 cos𝜙𝜙

−𝐿𝐿 cos𝛼𝛼 sin𝜙𝜙𝐿𝐿 cos(𝛽𝛽 2⁄ )

�

𝐺𝐺𝐻𝐻������⃗ = �−𝐿𝐿 sin(𝛽𝛽 2⁄ ) + 𝐿𝐿 cos𝛼𝛼 cos𝜙𝜙

𝐿𝐿 cos𝛼𝛼 sin𝜙𝜙𝐿𝐿 cos(𝛽𝛽 2⁄ )

�

(16)

Next, we calculate the angle 𝜉𝜉 as:

cos 𝜉𝜉 = �𝐺𝐺𝐴𝐴�����⃗ .𝐺𝐺𝐻𝐻������⃗

�𝐺𝐺𝐴𝐴�����⃗ ��𝐺𝐺𝐻𝐻������⃗ �� =

cos𝛽𝛽 − cos2 𝛼𝛼 + 2 cos𝛼𝛼 cos𝜙𝜙 sin(𝛽𝛽 2⁄ )1 + cos2 𝛼𝛼 − 2 cos𝛼𝛼 cos𝜙𝜙 sin(𝛽𝛽 2⁄ ) (17)

which can further be simplified (by using Equations (13) and (14)) into the following:

𝜉𝜉 = cos−1 �1 − (1 + cos2 𝛼𝛼) sin2(𝜃𝜃 2⁄ )

1 − sin2 𝛼𝛼 sin2(𝜃𝜃 2⁄ ) � (18)

26

Chapter 3: Programmable Origami-based Strings

27 3.1. Two-dimensional folding pattern

In this chapter, we introduce a family of origami-based structures with similar concept of

chapter 1. In sake of simplicity, some of the parameters’ definition are changed in this

chapter in compare to previous chapter.

One particular and well-recognized fold pattern is the Miura-ori, shown in Figure 8(a), and

first proposed in 1985 as a way to package large-area solar arrays into a small volume[19].

The strength of this fold pattern is its single degree of freedom over multiple hinges,

making it easy to deploy. A central path (longitudinal lines 𝐴𝐴 and 𝐴𝐴′) divides the width of

the strip into two equal parts, and lines 𝐴𝐴 and 𝐶𝐶 intersect the central path at the same point

with the same angle 𝛼𝛼. Angle 𝜃𝜃 describes the tilting of plates around the longitudinal

central path measured from their flat position and it varies from 𝜃𝜃 = 0° (flat) to 𝜃𝜃 = ±90°

(fully folded). 𝛾𝛾 is the angle between lines 𝐴𝐴 and 𝐴𝐴’, and it varies from 𝛾𝛾 = 0° (flat) to 𝛾𝛾 =

±2𝛼𝛼 (fully folded). For a given fold pattern defined by 𝛼𝛼, the folded configuration can be

described based on one variable (1-DOF), so we choose 𝜃𝜃 and 𝛼𝛼 as inputs of our unit and

𝛾𝛾 as output angle. Output angle (γ) can be expressed as a function of input angles (𝜃𝜃 and

𝛼𝛼) as the following,

γ = 2 𝑘𝑘 𝑐𝑐𝑜𝑜𝑐𝑐−1 �𝑐𝑐𝑜𝑜𝑐𝑐𝛼𝛼

√1 − 𝑐𝑐𝑜𝑜𝑐𝑐2𝜃𝜃𝑐𝑐𝑅𝑅𝑛𝑛2𝛼𝛼� (19)

where, 𝑘𝑘 = 1 for 𝜃𝜃 > 0 and 𝑘𝑘 = −1 for 𝜃𝜃 < 0. Figure 8(a) shows the variation of 𝛾𝛾 as a

function of 𝜃𝜃 for different values of angle 𝛼𝛼 plotted based on Equation (19). Also,

simulation results for 𝛼𝛼 = 60° and 𝛼𝛼 = 95°, shown by markers in Figure 8(a), indicate an

excellent agreement with theoretical results. For a constant value of 𝛼𝛼, increasing the

28 absolute value of 𝜃𝜃 (folding) leads to increasing absolute value of 𝛾𝛾. Also for a constant

value of 𝜃𝜃, changing 𝛼𝛼 toward 90° leads to increasing absolute value of 𝛾𝛾.

Figure 8. Folding of Miura-ori elements into zigzag and curved configurations (a) Folding of single Miura-ori. The folding angle, 𝜃𝜃 is the angle between the flat and folded configurations. 𝛾𝛾 is the angle between lines 𝐴𝐴 and 𝐴𝐴′, as shown in the picture. 𝛾𝛾 versus 𝜃𝜃 is plotted for different Miura-ori angles 𝛼𝛼, ranging from 0o to 180o. The markers are simulation results for 𝛼𝛼 = 60° and 95° showing excellent agreement with the theoretical prediction. (b) 6 × 1 inch paper strips with one, three, and five elements with 𝛼𝛼 = 60°. The flat (𝜃𝜃 = 0°) and folded (𝜃𝜃 = 10°) configurations for each paper strip are shown. The constant value of 𝛼𝛼 results in a zig-zag folded configuration. (c) 6 × 1 inch paper strips with one, three, and five elements. The left sample has 𝛼𝛼 = 120°. The other samples

29 are made with 𝛼𝛼 = 120° and 𝛼𝛼 = 60°. The flat (𝜃𝜃 = 0°) and folded (𝜃𝜃 = −10°) configurations for each paper strip are shown, resulting in a curved folded configuration.

Here, we harness this folding pattern to introduce an origami mechanical “string”, a slender

structure with one degree of freedom and shape programmability. The string is composed

of a number of elements that can be individually programmed to achieve a specific folding

pattern and curvature. When two of these elements, same as pattern shown in Figure 8(a),

are connected in series with aligned central paths, they exhibit the same dihedral angle 𝜃𝜃

and jointly have a single degree of freedom. We can repeat this process to make a string of

n elements with a single DOF. The absolute value of angle 𝜃𝜃 is the same in each element,

but the sign is reversed from one segment to the next (repeating mountain-valley sequence).

Figure 8(b) shows three prototypes made by repeating introduced folding pattern with 𝛼𝛼 =

60° and the overall length (i.e., central path length) of six inches. Folding these prototypes

from flat configuration to 𝜃𝜃 = 10° makes repeating γ = 43° and γ = −43° along the

string. Increasing 𝑛𝑛 creates larger number of γ = ±43° and ends up with a zigzag shape.

Similarly, the samples shown in Figure 8(c) have the same length of six inches, with

𝛼𝛼2𝑖𝑖−1 = 120° and 𝛼𝛼2𝑖𝑖 = 60°, where 𝑅𝑅 refers to 𝑅𝑅th element (𝑅𝑅 = 1, . . ,𝑛𝑛). Folding these

samples to 𝜃𝜃 = −10° forms 𝑛𝑛 number of equal angles, γ = 43°, making the sample curve.

Again, increasing n generates more vertices along the string and enables us to make

smoother desired curvatures.

Furthermore, 𝛼𝛼𝑖𝑖 of each element can be varied, so that properties such as 𝛾𝛾 and folding

motion can be programmed to vary along the length of the string. All properties of the

string can be prescribed using parameters 𝜃𝜃,𝛼𝛼1,𝛼𝛼2, …𝛼𝛼𝑝𝑝. Increasing 𝑛𝑛 provides more

30 design parameters and results in a better fit to any arbitrary desired trace. Each origami

string also exhibits a folding behavior, which can be controlled by the same n+1

parameters. In Figure 9, we illustrate three more elaborate string designs and their folding

procedure. The crease patterns shown in Figure 9(a) are implemented in 12-inch strips of

paper, which then fold into circle, eight-pointed star, and spiral shapes as shown in Figure

9(b). The first design (blue sample) contains repeating acute and obtuse angles 𝛼𝛼, which

form negative angles 𝛾𝛾 along the string. This design folds to a circle at 𝜃𝜃 = 45°. The second

design (green sample) consists of two periodic acute values of angle 𝛼𝛼 and results in

repeating positive and negative angles 𝛾𝛾. The design forms a closed eight-pointed star

pattern at 𝜃𝜃 = 17°. Increasing folding force results in bending of plates and subsequent

snap-through transition to a similar eight-pointed star with sharper vertices at 𝜃𝜃 = 45°. The

last design, which folds to a spiral, is generated by gradually decreasing obtuse angles 𝛼𝛼

and gradually increasing acute angles 𝛼𝛼, as shown in Figure 9(a). This sample folds to a

spiral with semi-continuous change in its curvature.

31

Figure 9. String that fold into circle, eight-pointed star and spiral configurations (a) 12× 1 inch paper strips with different element designs. Blue and green samples have repeating but constant values 𝛼𝛼. In the red sample, 𝛼𝛼 varies along the sample length as shown. (b) Samples at different folding angle 𝜃𝜃. The blue sample folds into a circle at 𝜃𝜃 = 45°, while the green and red samples fold to star and spiral shapes, respectively at the same 𝜃𝜃 . The scale bar is the same for all three samples for each value of 𝜃𝜃 (each row shown).

3.1.1. Methodology for designing desired crease patterns

Based on the folding relationship between 𝛼𝛼, 𝜃𝜃, and γ presented in Equation (19), and 𝑛𝑛 +

1 design parameters (𝜃𝜃,𝛼𝛼1,𝛼𝛼2, …𝛼𝛼𝑝𝑝), a wide range of two-dimensional designs can be

achieved by considering a desired path line or folding motion. Although we cannot trace a

smooth curve, we can approximate it with a set of 𝑛𝑛 straight lines, and as 𝑛𝑛 increases, the

approximation becomes closer to the desired curve. Here, we present a method to find these

32 parameters based on a desired pattern. Figure 10(a) shows an example desired design with

a central path consisting of 18 straight lines. This leads us to 𝑛𝑛 = 18. There is an infinite

number of designs which result in the pattern shown in Figure 10(a), however, a unique

design can be obtained for a given value of 𝜃𝜃. Figure 10(b) shows the variation of angle 𝛼𝛼

as a function of angle 𝜃𝜃, plotted for two different values of angle γ [based on Equation

(19)]. For instance, choosing 𝜃𝜃 = ±45° (so the original desired pattern shown in Figure

10(a) will be achieved by folding a flat string to 45°) results in a specific design (sign of 𝜃𝜃

changes from one element to the adjacent one) with 𝛼𝛼 = 55° and 125° for vertices with

γ = 90°, and 𝛼𝛼 = 68° and 112° for vertices with γ = −120°, as shown in Figure 10(b).

Figure 10(c) depicts the folded pattern of this design with central path that exactly

resembles the original desired design. Two alternative designs to achieve the same desired

pattern shown in Figure 10(a) at 𝜃𝜃 = ±20° and ±70° are presented in Figures 11 and 12.

33

Figure 10. String design to achieve a specific folding configuration (a) A sample of desired design for string origamis. The figures shows the path of central line consisting of 18 straight lines for the desired design and required values of 𝛾𝛾 = 90° and −120°. (b) 𝛼𝛼 versus 𝜃𝜃 plotted for the required values of 𝛾𝛾 to achieve the desired design. (c) The designed sample at 𝜃𝜃 = ±45° folding. The top image shows the unfolded pattern for one third of the sample (6 elements).

34

Figure 11. String design to achieve a specific folding configuration at 𝜃𝜃 = ±20°.

Figure 12. String design to achieve a specific folding configuration at 𝜃𝜃 = ±70°.

35

3.1.2. Relative motions along the string

To characterize the folding geometry of the origami string, we introduce a matrix [𝑅𝑅𝑖𝑖] in

Equation (20). This matrix represents the relative position and orientation of the local

coordinate system of link 𝑅𝑅 + 1 with respect to local coordinate system of link 𝑅𝑅. The origin

of the local coordinate system 𝑅𝑅 is fixed on the start point of central path in 𝑅𝑅𝑜𝑜ℎ linkage and

the 𝑥𝑥 axis is aligned with 𝑅𝑅𝑜𝑜ℎ segment of central path[46] (see Figure 13 for more details

about local coordinate systems). [𝑅𝑅𝑖𝑖] is defined as the following:

[𝑅𝑅𝑖𝑖] = �cos 𝛾𝛾𝑖𝑖 −sin 𝛾𝛾𝑖𝑖 𝑙𝑙𝑖𝑖sin 𝛾𝛾𝑖𝑖 cos 𝛾𝛾𝑖𝑖 0

0 0 1�

(20)

where, 𝛾𝛾𝑖𝑖 is the angle between linkage 𝑅𝑅 and 𝑅𝑅 + 1 [the sign is defined based on the

definition of angle 𝛾𝛾 in Figure 8(a)] and 𝑙𝑙𝑖𝑖 is the length of central line in 𝑅𝑅𝑜𝑜ℎ linkage. The

position and orientation of local coordinate system of each linkage with respect to global

coordinate system (same as the local coordinate system for the first link) can be found by

multiplying all corresponding [𝑅𝑅] matrices from the first to the desired link (i.e., ∏ [𝑅𝑅𝑖𝑖]𝑝𝑝𝑖𝑖=1 ).

Accordingly, the position of free end of the origami string can be determined using the

following equation:

�[𝑅𝑅𝑖𝑖]𝑝𝑝

𝑖𝑖=1

=

⎣⎢⎢⎢⎢⎢⎡cos(�𝛾𝛾𝑖𝑖

𝑝𝑝

𝑖𝑖=1

) − sin(�𝛾𝛾𝑖𝑖

𝑝𝑝

𝑖𝑖=1

) 𝑋𝑋𝑝𝑝𝑝𝑝𝑑𝑑

sin(�𝛾𝛾𝑖𝑖

𝑝𝑝

𝑖𝑖=1

) cos(�𝛾𝛾𝑖𝑖

𝑝𝑝

𝑖𝑖=1

) 𝑌𝑌𝑝𝑝𝑝𝑝𝑑𝑑

0 0 1 ⎦⎥⎥⎥⎥⎥⎤

(21)

36 where 𝑋𝑋𝑝𝑝𝑝𝑝𝑑𝑑 and 𝑌𝑌𝑝𝑝𝑝𝑝𝑑𝑑 are the 𝑥𝑥 and 𝑦𝑦 coordinates of the free end, respectively, and ∑ 𝛾𝛾𝑖𝑖𝑝𝑝

𝑖𝑖=1

represents the orientation of last linkage with respect to global coordinate system. Equation

(21) gives us the required information to predict the exact configuration and folding

procedure of the origami string.

Figure 13. Origami string with 9 elements. Local coordinate system for each element is shown. Position and orientation of each element is expressed by matrix [𝑅𝑅𝑖𝑖] presented by Equation (20).

3.2. Three-dimensional folding pattern

In this section, we introduce a methodology to design an origami structure with initial fully

flat configuration that folds to create a desired 3D curvature. There have been several

conceptual methods in the literature to design 3D origamis structures like discretized rigid-

foldable curvatures[47], continues buckled curvature[48, 49] and tessellated origami

patterns to approximate a 3D geometry[42], here we focus on a ‘four-crease’ pattern and

study its geometry in details to be able to predict and program the folding behavior of

origami curvature. Four-crease folding pattern is demonstrated in Figure 14(a), which

includes two longitudinal lines 𝐴𝐴 and 𝐴𝐴 named ‘central lines’ and two other crease lines 𝐷𝐷

37 and 𝐶𝐶 intersect central lines with angles 𝛼𝛼1and 𝛼𝛼2. We choose one of the central lines as

reference and measure 𝛼𝛼1and 𝛼𝛼2 with respect to it. Reference line in Figure 14(a) is line 𝐴𝐴.

Two plates which share a central line are named adjacent plates. Folding a strip of paper

based on this crease pattern is quantified by the angle 𝜃𝜃 shown in Figure 14(a). Angle 𝜃𝜃

can be calculated by 1 2� (180° − 𝜃𝜃′), where 𝜃𝜃′ is dihedral angle between two adjacent

plates. Attaching a Cartesian coordinate system with origin on the intersection of crease

lines, x axis along the reference line (𝐴𝐴) and y axis in the plane of flat origami provides

sufficient references to fully characterize the geometry of folded origami. Angle between

line 𝐴𝐴 and xy plane is 𝛾𝛾 and varies from −90° to 90°, and the angle between projection of

line 𝐴𝐴 on xy plane and x axis is ∅ and it varies from −180° to 180°. Indicating either 𝛾𝛾 and

∅ or 𝜃𝜃 can fully define the configuration of fully or partially folded origami in any level of

folding. The relation between two parameter sets, (𝛾𝛾,∅) and (𝜃𝜃), follows highly non-linear

equations. Although, some mathematical approaches are available to identify the geometry

of multi-crease-line folding patterns[50, 51], there is no available analytical solution with

desired specifications. For example, Wu et al. used numerical method to investigate the

geometrical characteristics of four-crease pattern[52]. For a special case 𝛼𝛼1 = 𝛼𝛼2,

analytical solution is available in the literature. In this work, a CAD simulation method is

applied to study the relation between angle 𝜃𝜃 and output parameters 𝛾𝛾 and ∅. Figure 14(b)

(left) shows the value of angle 𝛾𝛾 as a function of 𝜃𝜃, 𝛼𝛼1 and 𝛼𝛼2, while 𝛼𝛼2 = 60° and 𝛼𝛼1

varies from 5° to 175°. Angle 𝛾𝛾 starts from 0° at 𝜃𝜃 = 0° and goes to an optimum point by

changing 𝜃𝜃 toward −90° or 90° and it then 0gradually goes to zero after passing the

optimum point. In case of 𝛼𝛼1 ≠ 𝛼𝛼2, folding origami pattern will be stopped at some level

of folding as a result of contact between plates. Solid lines in the Figure 14(b) show value

38 of 𝛾𝛾 and ∅ in folding levels before contact happens and dash lines demonstrates the

configuration prohibited by self-intersecting. Equation (22) shows the extreme value of

folding angle 𝜃𝜃 (i.e. the angle that self-intersecting happens at) as a function of 𝛼𝛼1and 𝛼𝛼2:

𝜃𝜃𝑝𝑝𝑒𝑒𝑜𝑜𝑒𝑒𝑝𝑝𝑒𝑒𝑝𝑝 = ±(90° −12

𝑐𝑐𝑜𝑜𝑐𝑐−1 �tan𝛼𝛼2 tan𝛼𝛼1� �)

(22)

where, |tan𝛼𝛼1| > |tan𝛼𝛼2|. Folding of a four-crease pattern with 𝛼𝛼1 = 𝛼𝛼2 covers entire

range of 𝜃𝜃 [−90°, 90°] and contact doesn’t occur before getting fully folded. As 𝛼𝛼1 goes

toward 120°, which is supplementary angle of 𝛼𝛼2 = 60°, we see a sharper and faster

change in 𝛾𝛾 and also the location of optimum point gets closer to 𝜃𝜃 = 0° (quick change in

initial levels of folding). All plots corresponding to angle 𝛾𝛾 are symmetry with respect to

the origin and this indicates reversing sign of 𝜃𝜃 (folding in opposite direction) keeps the

absolute value of 𝛾𝛾 constant and reverse its sign. Figure 14(b) (right) demonstrates angle

∅ as a function of 𝜃𝜃, 𝛼𝛼1 and 𝛼𝛼2. Folding the origami (increasing the absolute value of 𝜃𝜃)

results in change of ∅ from zero to an extremum value and eventually ends in a non-zero

number at 𝜃𝜃 = ±90° and again due to self-intersecting in plates, folding will be stopped at

a certain folding level (see Equation (22)). In contrast with 𝛾𝛾, ∅ is symmetry with respect

to vertical axis (𝜃𝜃 = 0°), which means direction of folding doesn’t have effect on ∅ value.

Figure 14(c) illustrates folding of four-crease origami pattern with 𝛼𝛼1 = 90° and 𝛼𝛼2 = 60°

in both directions (±𝜃𝜃). Value of 𝛾𝛾 is positive (downward folding) and negative (upward

folding) in first and second rows, respectively, and regardless of folding direction, line 𝐴𝐴

turns toward the larger angle 𝛼𝛼, which satisfies discussed symmetries in Figure 14(b).

39

Figure 14. Folding a strip of paper based on a “Four-crease” pattern. (a) Angle 𝜃𝜃 quantifies folding level (−90° ≤ 𝜃𝜃 ≤ +90°). 𝛾𝛾 is the angle between crease line B and plane XY. ∅ is defined as the angle between projection of line B on plane XY and X axis. Angle ∅ represents out-of-plane displacement of origami pattern. (b) Simulation results demonstrate relations between (𝜃𝜃) and (𝛾𝛾 and ∅) as functions of 𝛼𝛼1 and 𝛼𝛼2. (c) Illustration for folding of a pattern with 𝛼𝛼1 = 90° and 𝛼𝛼2 =60°. First row shows valley folding (𝜃𝜃 > 0°) and second row shows mountain folding (𝜃𝜃 < 0°).

3.2.2. Out-of-plane displacement

Repeating 𝑛𝑛 four-crease folding patterns, introduced in Figure 14, along a straight line

constructs a pattern which is able to folds to 3D shapes. Angle 𝜃𝜃 is same for two sections

of a four-crease pattern (blue and red plates in Figure 14(a)) with reversed sign, so |𝜃𝜃| is

identical for entire string and represents level of folding regardless of folding direction with

|𝜃𝜃| = 0° and 90° at flat and fully folded levels of folding, respectively. Figure 15(a) shows

40 a strip of paper with length 𝐿𝐿 and width 𝐻𝐻 , which comprises six divisions (each division

contains two adjacent plates) with length 𝐿𝐿 6� , and underlying angles 𝛼𝛼1 = 90°, 𝛼𝛼2 =

60°, 𝛼𝛼′1 = 90° and 𝛼𝛼′2 = 120°, as shown in Figure 15(a). Angles 𝛼𝛼′1 and 𝛼𝛼′2 are

supplementary angles of 𝛼𝛼1 and 𝛼𝛼2 and due to periodic change in sign of angle 𝜃𝜃, this

combination results in constant sign in angle 𝛾𝛾 and makes a constant curvature along the

origami string while folding. Designing an origami pattern to create a 3D desired curvature

is associated with out-of-plane motion during folding. Hence, we define an out-of-plane

displacement (OPD), which is the distance between end point of origami string and a plane

normal to flat origami and includes central lines in flat configuration. Number of divisions

along a string (𝑛𝑛) and angles 𝛼𝛼1 and 𝛼𝛼2 are considered as three main characteristics of

presented origami and their control on OPD is discussed in this part of paper. Figure 15(b)

(left) shows 𝑂𝑂𝑂𝑂𝐷𝐷 𝐿𝐿� , which is a dimensionless parameter, as a function of 𝜃𝜃 (folding level)

for different numbers of division (𝑛𝑛), while 𝜃𝜃 varies from initial folding angle 0° to

maximum folding angle 45°. For 𝑛𝑛 = 1, normalized OPD is always zero, but for larger 𝑛𝑛,

it always starts from zero and goes to some value between 0.25 and 0.33. For 𝑛𝑛 = 2,

normalized OPD increases almost linearly, however we can see local extremum points in

larger numbers of division, for example plot of 𝑛𝑛 = 9 shows six local extremums. Figure

15(b) (right) shows variation of normalized OPD in a folded string (𝜃𝜃 = 30°) with different

angles 𝛼𝛼1 and 𝛼𝛼2 as a function of 𝑛𝑛. In all sets of 𝛼𝛼1 and 𝛼𝛼2, an immediate increase with

roughly linear regime occurs for initial values of 𝑛𝑛 and then it follows a damped sinusoidal

behavior around an constant value. As we saw in Figure 14, the magnitude of angle ∅ is

very small for 𝛼𝛼1 close to 0° and 180° and it causes small values of normalized OPD for

same 𝛼𝛼1. Figure 15(c) illustrates change of OPD during the folding for three strings with 𝑛𝑛

41 equals to 3, 6 and 9 (angles 𝛼𝛼1 and 𝛼𝛼2 are same as Figure 15(a)) in four levels of folding.

Comparing three strings confirms immediate response of string for large values of 𝑛𝑛 as

well as almost same OPD at maximum folding. When 𝑛𝑛 is relatively large, width of string

may restrict folding before 𝜃𝜃𝑝𝑝𝑒𝑒𝑜𝑜𝑒𝑒𝑝𝑝𝑒𝑒𝑝𝑝 (see Equation (22)), as it happens at 𝜃𝜃 = 30° and 45°

in last row of Figure 15(c). Decreasing 𝐻𝐻 can remove or postpone this limitation for further

levels of folding.

42

Figure 15. Out-of-plane displacement of origami string. (a) Applying five interconnected “four-crease” patterns in a 𝐿𝐿 × 𝐻𝐻 paper strip with repetitive 𝛼𝛼1 = 90°, 𝛼𝛼2 = 60° and their supplementary angles. (b) Simulation results for out-of-plane displacement (OPD), normalized by dividing by total length (L). [Left] Shows variation of normalized OPD as a function of angle 𝜃𝜃 for different numbers of division (𝑛𝑛). [Right] Normalized OPD as a function of 𝑛𝑛 for different angles 𝛼𝛼. (c) Three origami strings with equal length and 3, 6 and 9 number of divisions at four levels of folding (𝜃𝜃 =5°,15°,30° and 45°). The folded configurations for a string with nine divisions and 𝜃𝜃 > 21° isn’t accessible due to the self-intersecting in string.

43

3.2.3. Examples of 3D string

Presented origami pattern can construct origamis, which fold to innovate shapes. In Figure

16, we illustrate three examples with identical unfolded shape and completely different

folded configurations. First row of Figure 16 demonstrates a flat string of paper which folds

to a helical shape. In this design 𝛼𝛼1, 𝛼𝛼2 and their supplementary angles are repeating along

the string and length of each division is constant (same as pattern in Figure 15(b)). In the

second row, same pattern as first row is used, but angles 𝛼𝛼1 and 𝛼𝛼2 gradually decreases

from both ends toward the middle of string and length of each division follows a reverse

trend (increasing from ends to middle). This described pattern folds to a double-spiral

shape, as shown in Figure 16. Last row shows a star-shape helical, which is consisted of

repeating two pairs of acute angles 𝛼𝛼1 and 𝛼𝛼2. Discussed geometrical features of origami

string enables us to design an origami pattern that roughly fits to a desired 3D curvature at

some level of folding.

44

Figure 16. Design examples that fold from flat to final unique configurations. Three strips of paper with identical length and width are patterned by different crease line. Three folding levels (𝜃𝜃 =5°,17° and 45°) are shown for each design. Blue, red and green strings fold to a helical, double-spiral and star-helical final shapes at 𝜃𝜃 = 45°, respectively.

3.3. Robotics applications of origami string

The proposed string design enables many potential robotic applications, since it allows a

single actuator to coordinate a relatively complex motion along a set of linkages[53]. Here,

we provide two sample demonstrations of such potential applications by developing a

robotic gripper and a biomimetic hand based on the proposed origami string concept.

3.3.1. Origami robotic gripper

Figure 17 shows a robotic gripper design with overall length of 2𝑎𝑎 = 35.5 𝑐𝑐𝑚𝑚 constructed

by connecting five linkages with 𝛼𝛼 = 50° and 150°. The length of central path of each

45 linkage is 0.4 𝑎𝑎, 0.5 𝑎𝑎, 0.2 𝑎𝑎, 0.5 𝑎𝑎 and 0.4 𝑎𝑎, respectively from left to right, with the servo

motor located in the middle of the gripper. The flat sheets used in this design were

fabricated with VeroWhitePlus© material using PolyJet 3D printing technique and

connected by pin hinges. As discussed, the string design has a single degree of freedom,

allowing us to control the entire mechanism from a single hinge. Figure 17(a) displays the

top view of gripper in the flat configuration (𝜃𝜃 = 0°), the hinge pattern (similar to crease

pattern in paper samples shown in previous figures), and the servo motor connected to the

middle plates through a gear box. The gear box transfers the angular displacement and

torque of servo motor to two middle plates equally. Figure 17(b) shows the side views of

gripper at 𝜃𝜃 = 12°, 𝜃𝜃 = 29°, and 𝜃𝜃 = 56°, while approaching and gripping a spherical

ball. Using Equation (21), the Cartesian coordinate of gripper right end point is implicitly

related to 𝜃𝜃,𝛼𝛼1, and 𝛼𝛼2 through the following equation:

𝑋𝑋𝑝𝑝𝑝𝑝𝑑𝑑 = 𝑎𝑎 (0.1 + 0.4 cos(𝛾𝛾1 + 𝛾𝛾2) + 0.5 cos 𝛾𝛾1)

𝑌𝑌𝑝𝑝𝑝𝑝𝑑𝑑 = 𝑎𝑎 (−0.4 sin(𝛾𝛾1 + 𝛾𝛾2) − 0.5 cos 𝛾𝛾1)

(23)

where, 𝛾𝛾𝑖𝑖 is determined by plugging 𝛼𝛼𝑖𝑖 into Equation (19).

46

Figure 17. Origami robotic gripper (a) Top view of the robotic gripper with 𝛼𝛼 = 50° and 150° at 𝜃𝜃 = 0°. The actuation moment is produced by a servo-motor and transferred to the gripper through an embedded gear-box (b) Side views of the robotic gripper at folding levels, 𝜃𝜃 =12°,29°, 56°. (c) [Left] Components of linear velocity of the right end point, normalized with respect to the angular velocity of the servo-motor, 𝜔𝜔, and half length of the gripper, 𝑎𝑎. [Right] components of the reaction force applied to the right end point, normalized with respect to the input moment, 𝑀𝑀, and half length of the gripper, 𝑎𝑎.

Figure 17(c) [left] shows the velocity of gripper end point during different stages of folding

obtained using analytical modeling and experimental measurements[54]. The end point

velocity is proportional to the motor’s input angular velocity (𝜔𝜔 = 𝑑𝑑𝑑𝑑𝑑𝑑𝑜𝑜

), and the ratios of

47

end point velocity in 𝑥𝑥 and 𝑦𝑦 directions to 𝜔𝜔 are 𝑑𝑑𝑑𝑑𝑑𝑑

(𝑋𝑋𝑝𝑝𝑝𝑝𝑑𝑑) and 𝑑𝑑𝑑𝑑𝑑𝑑

(𝑌𝑌𝑝𝑝𝑝𝑝𝑑𝑑), respectively. In

Figure 17(c), we plotted the variation of these ratios divided by 𝑎𝑎 (resulting in

dimensionless parameters) as a function of folding angle 𝜃𝜃. Note that in this figure, the

input angle varies from 𝜃𝜃 = 0° to 𝜃𝜃 ≈ 63° (where the gripper two end points meet).

Experimental results shown by markers are measured for an input angular velocity, 𝜔𝜔 =

2 𝑟𝑟𝑎𝑎𝑑𝑑 𝑐𝑐� , demonstrating excellent agreement with the analytical results for large values of

𝜃𝜃. However, as 𝜃𝜃 becomes closer to zero (flat configuration), we observe considerable

differences between experimental and analytical results. Close to the flat configuration

(𝜃𝜃 = 0°), the behavior becomes highly sensitive to imperfections (e.g., in hinges), resulting

in wide variation in the measured data and relatively poor agreement with the analytical

results.

Next, we studied the grasping force of the proposed gripper, which is proportional to the

input torque of servo motor, denoted here by 𝑀𝑀. Neglecting gravity, the reaction forces

applied from the object to gripper end point in 𝑥𝑥 and 𝑦𝑦 directions can be determined based

on the principal of minimum total potential energy as the following:

𝐹𝐹𝑒𝑒𝑀𝑀

=𝑑𝑑𝑑𝑑𝜃𝜃 (𝑋𝑋𝑝𝑝𝑝𝑝𝑑𝑑)

( 𝑑𝑑𝑑𝑑𝜃𝜃 (𝑋𝑋𝑝𝑝𝑝𝑝𝑑𝑑) )2 + ( 𝑑𝑑𝑑𝑑𝜃𝜃 (𝑌𝑌𝑝𝑝𝑝𝑝𝑑𝑑) )2

𝐹𝐹𝑦𝑦𝑀𝑀

=𝑑𝑑𝑑𝑑𝜃𝜃 (𝑌𝑌𝑝𝑝𝑝𝑝𝑑𝑑)

( 𝑑𝑑𝑑𝑑𝜃𝜃 (𝑌𝑌𝑝𝑝𝑝𝑝𝑑𝑑) )2 + ( 𝑑𝑑𝑑𝑑𝜃𝜃 (𝑋𝑋𝑝𝑝𝑝𝑝𝑑𝑑))2

(24)

48 In Figure 17(c) [right], we plotted the variation of these ratios multiplied by a (resulting in

dimensionless parameters). Also, the insets illustrate the gripper in the flat configuration

and for 𝜃𝜃 ≈ 63°, where the gripper two end points meet.

3.3.2. Biomimetic origami hand

Next, we present another application of the proposed origami string as a biomimetic robotic

hand, where each of the five fingers is controlled using a single actuator, Figure 18.

Compared to designs based on actuation of individual fingers knuckles[55], our design is

simpler, more efficient, and lightweight as it requires only five total actuators for creating

variety of complex movements of the biomimetic robotic hand. In our design, we selected

specific values of 𝛼𝛼 to mimic a typical human hand shape and movement characteristics.

This was done by measuring the hand’s geometry such as, knuckles lengths and angles

between knuckles during different hand gestures. Figure 18 (a) shows the hinge pattern of

a designed robotic index finger as well as its folding states at 𝜃𝜃 = 0°, 𝜃𝜃 = 7°, 𝜃𝜃 = 25° and

𝜃𝜃 = 45°. The robotic index finger is made of four linkages with 𝛼𝛼1 = 134°, 𝛼𝛼2 = 65° and

𝛼𝛼3 = 146°, respectively from right to left. The movement is controlled by a servo motor

linked to the right linkage through a gear box. The other four fingers can be designed with

a similar approach and assembled into a platform. Figure 18(b) shows front and back views

of the robotic hand. The independent motion of each finger makes the origami hand

compatible with a wide range of applications, such as manipulating objects with different

geometries, as exemplified in Figure 18(c) where an apple or a pen is being held by the

robotic hand.

49

Figure 18. Biomimetic origami hand (a) Side view of the index finger of the robotic hand at four folding levels, 𝜃𝜃 = 0°, 7°,25°, 45°. The design of the robotic finger is shown in the inset. The actuation moment for each finger is produced by a servo-motor and transferred to the finger through an embedded gear-box (b) Front and back views of the proposed robotic hand. (c) Demonstration of functionality of the proposed robotic hand in manipulating sample objects.

3.4. Conclusion

In this chapter, we introduced an origami mechanical string with programmable folding,

kinetics and kinematics. We presented a simple method to design such mechanical strings

50 to achieve desired shape and performance. Precise and arbitrary motions can be controlled

with a single actuator for efficient and lightweight positioning. This is ideal for applications

that require both accuracy and sensitivity, such as surgical robots and assembly of delicate

machines[56, 57]. Given the scalable nature of fold patterns, the design is applicable at

different length scales, including developing millimeter-sized manipulators and meter-

scale space applications[56, 58]. The current work paves the way to develop more novel

origami strings, where integration of sensors and actuators can introduce the possibility of

dynamic control, and broaden the applications of such origami-based designs.

51

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