On the design of origami structures with a continuum of equilibrium shapes

Accepted Manuscript On the design of origami structures with a continuum of equilibrium shapes Luca Magliozzi, Andrea Micheletti, Attilio Pizzigoni, Giuseppe Ruscica PII:

S1359-8368(16)32242-9

DOI:

10.1016/j.compositesb.2016.10.023

Reference:

JCOMB 4620

To appear in:

Composites Part B

Received Date: 19 August 2016 Revised Date:

7 October 2016

Accepted Date: 9 October 2016

Please cite this article as: Magliozzi L, Micheletti A, Pizzigoni A, Ruscica G, On the design of origami structures with a continuum of equilibrium shapes, Composites Part B (2016), doi: 10.1016/ j.compositesb.2016.10.023. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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On the Design of Origami Structures with a Continuum of Equilibrium Shapes Luca Magliozzia , Andrea Michelettib , Attilio Pizzigonic , Giuseppe Ruscicac,∗ a TALL

Engineers, London, United Kingdom of Civil Engineering and Computer Science Engineering, University of Rome TorVergata, Italy c Department of Engineering and Applied Sciences, University of Bergamo, Italy

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b Department

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Abstract

Here we focus on the problem of how to fold/deploy an origami structure. We consider origami tessellations which can pass from one shape to another through a continuum of equilibrium configurations. Among different tessellations, we choose the one invented by Ron Resch. This tessellation features only triangular panels, a property that allows us to simplify the model, and that gives the structure higher stiffness with respect to other types of tessellations. We present

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two distinct procedures for changing the shape of an origami. In the first one, a certain number of bars is added to the system, to make the structure isostatic and to control, by varying their length, the change of shape. In the second procedure, length changes are assigned only to a minimum number of control elements, while the remaining length changes are determined by imposing a

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sufficient and minimal number of constraints. Such constraints can impose symmetry conditions and/or particular nodal trajectories. We found that often it is not possible to arbitrarily control all degrees of freedom of Resch’s origami

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structures, since a halt in the folding/deployment path can occur, with the structure locked in a singular configuration. This problem is sidestepped by assigning fewer kinematic constraints and by using the pseudo-inverse solution ∗ Corresponding

author Email addresses: [email protected] (Luca Magliozzi), [email protected] (Andrea Micheletti), [email protected] (Attilio Pizzigoni), [email protected] (Giuseppe Ruscica)

Preprint submitted to Composites Part B: Engineering

October 7, 2016

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of the system of kinematic-compatibility equations. Our approach is general

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enough to be applied to origami structures at different scales, and of different patterns.

Keywords: Origami, Smart Materials, Computational Modelling

1. Introduction

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In recent years, origami structures have attracted great interest among en-

gineers and physicists. The first studies on the mathematics of origami have been conducted by Thomas Hull [1] and Sarah-Marie Belcastro [2]. Thereafter, several other studies have been carried out in various fields, in order to assess

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different problems about the application of origami structures (see, e.g., [3, 4] and the literature cited therein).

The objective of this work is the design of origami-inspired deployable structures. These are structures composed of panels hinged to each other, which are 10

in static equilibrium in both ‘open’ and ‘closed’ configurations. Some partic-

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ular types of origami are able to change their shape and size simply through their deployment. Some of them do not require any deformation of the panels, which can therefore be considered as rigid elements. In particular, we consider origami tessellations, with additional actuating elements, which are able to pass 15

from one shape to another through a continuum of equilibrium configurations.

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It is worth recalling that the problem of using actuators to open and close an origami has been dealt with by John Hobart Culleton and Anthony Diaz within the project ‘Responsive Kinematics’ [5]. There they realized a series of proto-

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types, showing how to transform a rigid origami while maintaining its geometric

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integrity.

Origami tessellations have first been studied by Shuzo Fujimoto [6] and Ron

Resch [7]. Further studies have been carried out by David Huffman [8] and Eric Gjerde [9]. Useful software has been developed to simulate and analyze origami tessellations, for example ‘Tess’, by Eric Gjerdi and Alex Bateman

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[10], which can generate crease patterns for origami tessellations, and ‘Inter-

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active Rings Tessellation’, which is a tool developed by Robert J. Lang about

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rings-type origami tessellation [11]. Furthermore, Tomohiro Tachi has developed several origami computational tools, such as ‘Freeform Origami’, ‘Origamizer’ and ‘Rigid Origami Simulator’, which allow the user to design origami interac30

tively. Corresponding working principles and algorithms have been explained in

[12, 13, 14, 15], while other studies on rigid folding and thick origami have been

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presented in [16, 17], and [18], respectively.

In this work, among different tessellations, we choose the one invented by Ron Resch (Fig. 1). This tessellation features only triangular panels, a property that allows us to simplify the model, and that gives the structure higher stiffness

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with respect to other types of tessellations.

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Figure 1: Different configurations of a Ron Resch’s origami

Since there are only triangular panels, the kinematical description of the

system is obtained by considering an equivalent truss model in which bars correspond to folds, and pin-joints correspond to vertices of the origami. In this

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way, simpler and lighter structures can be obtained, especially considering the

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possibility or real constructions. The resulting truss possesses internal mechanisms, the same ones which were enabling the deployment of the origami. When considering arbitrary finite tessellations, the number of mechanisms can be computed by simply counting the edges on the boundary and the number of sides 45

of the panels, as detailed in the following.

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We present two folding/deploying procedures. In the first one, a certain number of bars is added to the structure to make it isostatic, and to control, by varying their length, the change of shape. Criteria suggested by symmetry

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and simplicity guide the choice of a number of patterns for the placement of the additional control elements. In the second folding/deploying procedure, length changes are assigned only to a minimal number of control elements, while the remaining length changes are determined by imposing a minimal and sufficient number of constraints, prescribing symmetry conditions and/or certain nodal trajectories. 55

Our findings show that in general it is not possible to assign arbitrary length

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changes to all control elements, since this can often cause a halt in the folding/deployment path, with the structure locked in a singular configuration. This problem is sidestepped by reducing the number of kinematic constraints and by using the pseudo-inverse solution of the system of kinematic-compatibility equations.

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The paper is organized as follows. In Section 2, the structural model is presented together with the rule for counting the number of mechanisms of an origami tessellation. In Section 3 and 4, we describe respectively the first and

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the second folding/deploying procedure, giving examples of their application.

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Our concluding remarks are given in Section 5.

2. Truss model for triangulated origami As mentioned in the Introduction, when an origami is constituted by triangular panels only, it is possible to consider an equivalent truss model where

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bars correspond to folds, and nodes correspond to vertices of the origami. We

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recall that for a generic truss, composed by nn nodes and nb bars, the minimum number of bars necessary to making it isostatic is given by Maxwell’s rule: nb = 3nn − nc ,

where nc is the number of scalar external constraints imposed on the nodes. If we want to apply this rule to a triangulated origami, it is useful to consider a result

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originated in the studies carried out by Cauchy [19], Dehn [20], and Alexandrov [21]: a truss whose bars and nodes are built on the edges and vertices of a convex

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triangulated polyhedron is isostatic. Correspondingly, we have 3nn − nb − 6 = 0,

(1)

where the number 6 represents the number of rigid-body motions for an unconstrained system in three dimensions. Relation (1) applies also to trusses having the same connectivity of a triangulated polyhedron, even if they are not shaped like one. We now recall the extended Maxwell’s rule, which relates nn and nb

truss [22]:

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to the number of mechanisms, m, and the number of self-stress states, s, of a

3nn − nb − 6 = m − s.

A truss obtained from a triangulated origami has the same connectivity of a polyhedron composed only by triangulated faces, except for one face, which

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correspond to the boundary of the origami. If there are k elements on the boundary, then a polyhedron of corresponding connectivity has a face with k sides. It is easy to see that, if we want to triangulate this polygon in order to obtain a polyhedron with triangular faces only, we need (k−3) elements. If these

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(k − 3) elements are not present, then there are (k − 3) degrees of freedom that, when the configuration is non-singular, correspond to the internal mechanisms of the truss or the origami. Thus, we can write 3nn − 6 − nb = k − 3 = m,

where we considered that s = 0 in a non-singular configuration. In the work carried out by Finbow-Singh and Whiteley [23], this result is extended to more 5

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general cases, allowing one to obtain similar counting rules for generic origami, not necessarily triangulated.

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3. First folding procedure: morphing an isostatic system

The origami trusses we are going to deal with are kinematically indeterminate, that is, they present a certain number of mechanisms. The number

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of mechanisms can be found with the counting rule presented in the previous section, that is m = k − 3, where m is the number of mechanisms and k the number of elements on the boundary, considering the absence of holes inside the

rule can be rewritten as:

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tessellation. If we want to consider also the system’s external constraints, the

m = k − nc − 3.

(2)

A first way to control the movement of the structure is that of adding external constraints until all mechanism are eliminated, allowing for configuration changes by assigning the motion of constrained nodes. This kind of operation, as it is described, can be complicate to implement: a series of rails and slider

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elements are necessary, with their associated complexity under a constructive point of view.

Another way to control the movement of the structure is that of adding a certain number of extra bars to the origami. If two points, during a shapechange process, get closer or farther apart, then we can imagine to connect

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them with a rigid bar. In this way, a mechanism has been removed. We can then add as many elements as needed to remove all mechanisms, being careful

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not to overconstraining the origami. In this last case, there is the risk to have a certain number of self-stresses and/or residual mechanisms. By eliminating

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exactly the number of mechanisms given by (2), we can obtain a statically and kinematically determinate structure, possibly externally constrained, which we can apply loads on and compute the corresponding internal stresses. We now apply these considerations to the case of a Ron Resch’s tessellation. In this tessellation (Fig. 2) we can identify: the upper equilateral triangle 6

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(shaded in gray in Fig. 2a), which is enclosed by three mountain folds; the

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equilateral triangle (shaded in Fig. 2b) composed of six right triangles; a right triangle (shaded in Fig. 2c) rotates around the side shared with the upper trian-

gle; the minimal module (Fig. 2d), constituted by an upper triangle and three

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lower right triangles.

Figure 2: The constituting parts of Ron Resch’s tessellation

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The specific choice of the nodes to be connected with additional bars may depend on the way one wants to control the structure. For example, in case of a symmetric tessellation, it is natural to consider symmetric sets of external

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constraints and additional elements. Since a symmetric Ron Resch’s tessellation presents an axis of 3-fold symmetry, if an element is added in a certain position, 100

then two more elements need to be added symmetrically. In addition, in order to have a homogeneous distribution of the opening and closing zones, it is necessary to insert the additional control elements both on the boundary and on the

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internal regions.

Once the dimensions and geometry of a tessellation have been chosen, dif105

ferent sets of additional elements can be considered (Fig. 3). Among several trial sets of additional elements, we found that some of them can guarantee a

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more precise and smooth deployment process, while others do not allow for a complete folding or deployment. The simulation of the folding/deployment process is carried out by mean of

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a custom-written finite-element code able to perform a geometrically nonlinear large-displacement analysis. Figures 4, 5, and 6 present simulation results when the sets of additional elements in Figs. 3a, 3b, and 3d are employed to control

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Figure 3: Hexagonal structure: (a) side of 1 triangular module and 9 added elements; (b) side of 2 triangular modules and 21 added elements; (c) side of 3 triangular modules and 33 added elements; (d) side of 3 triangular modules and 33 added elements

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the configuration change.

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Figure 4: From unfolded to folded configuration for the structure in Fig.3a

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Figure 5: From unfolded to folded configuration for the structure in Fig.3b

Figure 6: From unfolded to folded configuration for the structure in Fig.3c

4. Second folding procedure: solving kinematic-compatibility equations

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This procedure is divided into two phases. The target of the first phase, which is purely kinematical, is to obtain a smooth folding sequence, with no compenetration between nodes and edges of the origami truss. The actuators are added in the second phase and their length changes are assigned in order to follow the folding path found during the first phase. More precisely, the first

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phase consists of changing the configuration of the origami truss by solving a system of differential equations. Such system of differential equations is divided

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into four sets of equations, as detailed below. The first set of equations is given by the kinematic compatibility equations

for the edges of the origami truss, e.g. (pi − pj ) · (vi − vj ) = ij ,

where pi is the position vector of node i, vi = p˙ i is the velocity vector of node 125

i, and ij = lij l˙ij is the stretching velocity of edges ij. 9

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Figure 7: Folding sequence for an origami structure with control elements placed as in Fig. 3d: from top to bottom, ‘open’, ‘intermediate’, and ‘closed’ configurations (pairs of pictures on the same row depict the same configuration)

The second set of equations is given by the symmetry conditions imposed

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on a first group of additional edges, e.g. ij = hk

for edge ij and edge hk to maintain the same length. This equation can be

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rewritten as

(pi − pj ) · (vi − vj ) − (ph − pk ) · (vh − vk ) = 0.

We notice that this first group of edges can be chosen arbitrarily and independently from the actuators. The third set of equations assigns the stretching velocities to a second group of additional edges. These edges can be chosen arbitrarily or also in the previous 10

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(pn − pm ) · (vn − vm ) = nm .

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group of additional edges (first group):

The stretching velocities to be assigned to these edges can be computed as follows: 1 2 2 (l − lin ). 2 f in

In this relation, obtained by integrating ij = lij

dlij dt

assuming ij constant, lin

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ij ∆t =

is the initial length of the edge, lf in is its final length, ∆t is the time interval 130

(duration of the simulation).

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The fourth set of equations imposes the external constraints to the origami truss.

Summarizing, in matrix form, the four sets of differential equations are written as follows. For the first set of equations, we write Bv = 0,

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which correspond to the condition of rigid foldability with no stretching of the edges of the origami truss, where B is the kinematic compatibility matrix of the origami truss with no additional edges, and v is the vector containing the velocity of all nodes. For the second set, we have

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Bs v = 0,

which imposes the symmetry conditions. For the third set, we have

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Ba v = 0,

which assigns the stretching velocities. Finally, for the fourth set, we have Bc v = 0,

which imposes the external constraints, where each row in Bc contains all zeros except in the position corresponding to the constrained dof, e.g for (vi )x = 0

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the component corresponding to the x direction of node i is equal to one. 11

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Putting together all the equations, the resulting system can be written as

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Btot (p)v = etot .

Notice that the system can be underdetermined, with more unknowns than equations. In that case, the solution is provided through the pseudo-inverse

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v = (Btot )+ etot ,

providing the solution vector v whose norm is lower than the norm of any other solution. In this way, we do not force the system to follow a fully prescribed

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path, as it would be the case for a square full rank system1 . We notice that the pseudo-inverse solution may not possess a corresponding 3-fold symmetry. The solution vector is then used to write the system of differential equations to be solved numerically,

p˙ = v(p).

Figure 8 (left) shows the origami and the set of additional elements for specifying

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symmetry constraints (second set of equations). The length changes are specified for one of the elements belonging to the center red triangle and for one of the the isolated red elements (third set of equations). External constraints are added 140

(fourth set of equations) just to remove rigid-body motions. The second phase of this folding procedure consists in taking the origami

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truss and adding a series of edges in order to make it isostatic and to control it. These edges are not necessarily the same edges used in the first phase. The controlling edges are first added to the origami truss; then, their length variations are prescribed according to the folding path computed in the first

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phase. The simulation is carried out by using the same finite-element code mentioned in the previous section. Figure 8 (right) shows the origami and the set of additional edges used in the second phase. 1 The

idea of using the pseudo-inverse in origami simulations is not new. For example, it

has been adopted in [16] within a different formulation.

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(right) of the folding procedure

5. Conclusions 150

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Figure 8: The sets of additional elements used in the first phase (left) and the second phase

We presented two distinct procedures for obtaining a folding/deploying path for origami structures. Both procedures apply to triangulated origami tessel-

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lations, which are modeled here by corresponding truss structures (the latter constituting also an alternative way of realizing origami-based metamaterials). We presented the particular case of Ron Resch’s tessellations by giving examples 155

of computations of folding/deploying paths. During our simulation campaign, we often observed that it is not possible to arbitrarily control all degrees of free-

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dom of this type of tessellation, since a halt in the folding/deployment can occur, with the structure locked in a singular configuration. The second procedure we present can overcome this problem by using the pseudo-inverse solution of a reduced system of equations. Our approach is general enough to be applied to

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triangulated origami structures at different scales, and of different patterns. It is worth noticing that, since a singular configuration can also sustain self-stress states, it is also possible to exploit our procedure for discovering new types of prestressed structures.

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Acknowledgments

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Part of this study has been carried out when L.M. was working at his MS

Thesis [24] in Building Engineering and Architecture at University of Rome Tor Vergata.

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