Sculpting the Vertex: Manipulating the Configuration Space Topography and Topology of Origami Vertices to Design Mechanical Robustness

TL;DR

This paper introduces a novel method for designing origami-based mechanisms by heterogeneously combining high-degree-of-freedom vertices to control the configuration space, enhancing robustness and functional constraints in micro and nano-scale machines.

Contribution

It presents a technique to manipulate the configuration space of origami vertices, enabling robust, constrained mechanisms without tight fabrication tolerances.

Findings

Mapped the configuration space of a modified degree-6 Miura-ori vertex.

Demonstrated mechanisms that isolate deformations and constrain neighboring vertices.

Achieved sequential bistable folding in origami sheets.

Abstract

The geometric, aesthetic, and mathematical elegance of origami is being recognized as a powerful pathway to self-assembly of micro and nano-scale machines with programmable mechanical properties. The typical approach to designing the mechanical response of an ideal origami machine is to include mechanisms where mechanical constraints transform applied forces into a desired motion along a narrow set of degrees of freedom. In fact, to date, most design approaches focus on building up complex mechanisms from simple ones in ways that preserve each individual mechanism's degree of freedom (DOF), with examples ranging from simple robotic arms to homogenous arrays of identical vertices, such as the well-known Miura-ori. However, such approaches typically require tight fabrication tolerances, and often suffer from parasitic compliance. In this work, we demonstrate a technique in which high-degree-of-freedom mechanisms associated with single vertices are heterogeneously combined so that the coupled phase spaces of neighboring vertices are pared down to a controlled range of motions. This approach has the advantage that it produces mechanisms that retain the DOF at each vertex, are robust against fabrication tolerances and parasitic compliance, but nevertheless effectively constrain the range of motion of the entire machine. We demonstrate the utility of this approach by mapping out the configuration space for the modified Miura-ori vertex of degree 6, and show that when strung together, their combined configuration spaces create mechanisms that isolate deformations, constrain the configuration topology of neighboring vertices, or lead to sequential bistable folding throughout the entire origami sheet.

Full text

Sculpting the Vertex: Manipulating the Configuration Space Topography and Topology of Origami Vertices to Design Mechanical Robustness

Bin Liu

Department of Physics, Cornell University, Ithaca, NY 14850

School of Natural Sciences, University of California, Merced, Merced, CA 95343, USA

  

Arthur A. Evans

Department of Physics, University of Massachusetts Amherst, Amherst, Massachusetts 01003, USA

  

Jesse L. Silverberg

Department of Physics, Cornell University, Ithaca, NY 14850

  

Christian D. Santangelo

Department of Physics, University of Massachusetts Amherst, Amherst, Massachusetts 01003, USA

  

Robert J. Lang

Lang Origami, Alamo, California 94507

  

Thomas C. Hull

Department of Mathematics, Western New England University, Springfield, MA 01119, USA

  

Itai Cohen

Department of Physics, Cornell University, Ithaca, NY 14850

**The geometric, aesthetic, and mathematical elegance of origami is being recognized as a powerful pathway to self-assembly of micro and nano-scale machines with programmable mechanical properties Lv:2014cs ; Filipov:2015cr ; tactom:2016tu ; Greenberg:2011bs ; Hanna:2014fp ; You:2007do ; Onal:2011da ; Overvelde:2016gn ; Kuribayashi:2006gh ; Dureisseix:2012tk ; Yasuda:2015eg ; Waitukaitis:2015dk ; Evans:2015gu . The typical approach to designing the mechanical response of an ideal origami machine is to include mechanisms Latombe:1991 ; Erdman:1997wi where mechanical constraints transform applied forces into a desired motion along a narrow set of degrees of freedom Lv:2014cs ; Filipov:2015cr ; tactom:2016tu ; Greenberg:2011bs ; Hanna:2014fp . In fact, to date, most design approaches focus on building up complex mechanisms from simple ones in ways that preserve each individual mechanism’s degree of freedom (DOF), with examples ranging from simple robotic arms You:2007do ; Onal:2011da ; Overvelde:2016gn to homogenous arrays of identical vertices Kuribayashi:2006gh ; Dureisseix:2012tk ; Yasuda:2015eg ; Waitukaitis:2015dk ; Evans:2015gu , such as the well-known Miura-ori MIURA:1985tt . However, such approaches typically require tight fabrication tolerances, and often suffer from parasitic compliance Tachi:2011tx ; Francis:2014dp ; Hanna:2014fp ; Chen:2015sci . In this work, we demonstrate a technique in which high-degree-of-freedom mechanisms associated with single vertices are heterogeneously combined so that the coupled phase spaces of neighboring vertices are pared down to a controlled range of motions. This approach has the advantage that it produces mechanisms that retain the DOF at each vertex, are robust against fabrication tolerances and parasitic compliance, but nevertheless effectively constrain the range of motion of the entire machine. We demonstrate the utility of this approach by mapping out the configuration space for the modified Miura-ori vertex of degree 6, and show that when strung together, their combined configuration spaces create mechanisms that isolate deformations, constrain the configuration topology of neighboring vertices, or lead to sequential bistable folding throughout the entire origami sheet. **

The basic principles of an origami mechanism are nicely illustrated by the tessellation of the Miura-ori MIURA:1985tt . The fold pattern for this vertex is illustrated by the dashed blue and solid red lines in Fig. 1(A). Here, 4 creases converge at a single degree-4 vertex. To determine its degrees of freedom (DOF) it is necessary to account for the various constraints on the configurations. Holding a degree $N$ vertex fixed at the origin, each freely rotating crease contributes $2N$ degrees of freedom. From these DOF we subtract the $N$ constraints due to the angles between the creases in the fold pattern, and 3 rotational symmetries. Thus using the degree counting rule DOF=$2N-N-3$ gives a single degree of freedom for each degree 4 Miura-ori vertex Hull:2006 ; Evans:2015et . Moreover, a tessellation of such vertices is synchronized and characterized by a single degree of freedom so that the folding of any individual crease controls the folding of the entire sheet Tachi:2009ub . This mathematical argument however ignores real world aspects of sheets, including the bending of facets, finite sheet thickness, errors in folding angles, and even distortions of the crease lines Wheeler:2016kw . Such non-ideal behaviors lead to parasitic compliance including out of plane bending and twisting Schenk:2013kk ; Wei:2013kn , as well as inhomogeneous compression and defects Silverberg:2014dn ; Evans:2015et . As such these imperfections often complicate and even destroy the single degree of freedom mechanism designed into this and other structures.

Here we propose an orthogonal approach to creating mechanisms in origami structures. Rather than using the number of constraints to reduce the range of motion for each vertex, we employ the angles in the fold pattern to restrict the configuration space and guide vertex-vertex interactions to achieve a desired mechanism. We illustrate this approach using a degree-6 Miura-ori, where each vertex has $N=6$ creases with folding angles $t_{1}$, $t_{2}$, $\cdots$, $t_{6}$. In addition to the four predetermined mountain-valley folds in a typical Miura-ori (color lines in Fig. 1(A)), two universal creases ($t_{1}$ and $t_{3}$) along the diagonal of the parallelograms adopt either mountain or valley folds (thin gray lines Fig. 1(A)). Thus, each vertex has DOF$=12-6-3=3$ and can potentially accommodate a wide range of deformations Evans:2015et . For example when the angle between the mountain and valley creases $\alpha=\pi/3$, a vertical tessellation of such vertices can include a large amount of in-plane bending in either direction due to the asymmetric folding of the two universal creases (Fig. 1(A)).

The configuration space of this degree 6 vertex can be visualized by distinguishing between allowed (filled) and forbidden (empty) regions parametrized by the three folding angles $t_{1}$, $t_{2}$, $t_{3}$ (Fig. 1(B)) (See Methods). To illustrate the accessible structures, we dissect the 3D configuration space by marking 6 representative spots accompanied with the corresponding vertex folds (Fig 1(C)): (i), the unfolded configuration ($t_{1}=t_{2}=t_{3}=0$); (ii), the flat folded configuration ($t_{1}=t_{3}=0$ and $t_{2}=-\pi$); (iii) and (iv), two symmetrically folded structures associated with out-of-plane deformations ($t_{1}=t_{3}=\pm\pi/2$ and $t_{2}=0$); (v) and (vi), two antisymmetric folded structures associated with in-plane bending ($t_{1}=-t_{3}=\pm\pi/2$ and $t_{2}=-\pi$). The black dashed line shows the path associated with the one degree of freedom folding of an ideal Miura-ori vertex ($t_{2}=t_{3}=0$). The blue line ($t_{2}=0$ and $t_{1}=t_{3}$) illustrates symmetric folding configurations. Finally, the red line ($t_{2}=-\pi$ and $t_{1}=-t_{3}$) corresponds to antisymmetric folding configurations for a flat-folded structure that is bent either to one side or the other.

Remarkably, decreasing the fold angle to $\alpha=\pi/60$, maintains the dimension of the accessible configuration space while substantially altering its range of deformations (Fig. 1(D)). For example, a vertical tessellation of these vertices is highly resistant to in-plane bending until the sheet is completely flat folded with $t_{2}=-\pi$. The origin of this rigidity can be traced to the substantially restricted configuration space shown in Fig. 1(E). This configuration space approaches the skeletal form represented by the blue, dashed black, and red lines in Fig. 1(C). To emphasize the reduction in conformations we illustrate the areal projections of this skeletal configuration space onto two planes, one at $t_{2}=1$ (purple plane) and a second at $t_{2}\approx-\pi$ (red plane). As shown in Fig. 1(F) the configuration space at $t_{2}=1$ is reduced to a single dot (purple region), thereby excluding structures such as that shown in the inset to Fig. 1(B). In addition, we find that the only pathway available for asymmetric folding, is for $t_{2}\approx-\pi$ so that the structure is flat folded and the universal creases can fold anti-symmetrically with $t_{1}\approx-t_{3}$ (red linear region and accompanying folded structures).

Breaking the reflection symmetry of the vertex by having two different angles $\alpha_{1},\alpha_{2}$ between the mountain and valley creases creates diode like in-plane bending behavior. Similar to the case for symmetric but small $\alpha$, a vertical tessellation of these vertices is highly resistant to in-plane bending until $t_{2}=-\pi$. However, even in this extreme case the structure only bends in one direction but not the other (Fig. 1(G)). We find that the configuration space of such vertices is missing one of the antisymmetric branches (Fig. 1(H,I)). This asymmetry in the configuration space in-turn gives rise to the diodic behavior with bending only towards the side with the larger crease angle.

Our configuration space analysis immediately suggests a strategy for designing mechanical metamaterials that take advantage of these vertex properties to isolate in-plane bending deformations. To demonstrate this behavior we design an origami structure that combines two opposite diotic columnar tilings, and sandwiches them between symmetric degree-6 Miura-ori tilings (Fig. 2(A)). This fold pattern is designed to prevent in-plane bending on one side of the sheet from affecting the deformations on the other side. To assess our design, we folded the origami structure, used pins to apply in plane deformations, imaged it, and rendered the three dimensional conformation of the structure and the pins as shown in Fig. 2(B) (See Methods). To characterize the deformation at each vertex we keep track of the folding angle $t_{4}$, which is a sensitive indicator of the extent to which a given vertex is folded. We find that pinching due to the pin configuration induces large deformations on both sides of the sheet indicating that both halves are free to deform. When the pinching due to the pins is only applied to the right side, however, we find that the deformations are isolated and do not penetrate past the double diode structure, in accordance with the design criteria (Fig. 2(C,D)).

There are however a couple of disadvantages for this design. First, it is not flat foldable. Second, since strain due to in-plane bending is focused along the central axis, under even moderate strains the sheet as a whole becomes quite rigid and more susceptible to tearing as its size increases. Significantly greater in plane bending for arbitrarily large sheets can be accommodated in structures where the creases $t_{2}$ and $t_{5}$ are no longer co-linear. For example, the degree-6 Miura-ori washer geometry shown in Fig. 3(A), one of the so called spiral origami structures Nojima:2002ws , is a fully collapsible structure that forms a cylinder in its folded state. Here, the angle between the mountain and valley folds, $\alpha$, decreases with radius $r$ as $\alpha=\cos^{-1}(r_{\mathrm{in}}/r)-\pi/N_{\phi}$, where $r_{\mathrm{in}}$ is the inner radius of this structure and $N_{\phi}$ is the number of vertices along the angular direction. The corresponding opposite angles $\beta$ satisfies $\alpha+\beta=\pi$, as a necessary condition for flat-foldable degree-6 Miura-ori vertices.

Relaxing the co-linearity constraint for creases $t_{2}$ and $t_{5}$ (Fig. 3(B)) substantially alters the configuration space as shown in Fig. 3(C)). In particular, we find that the configuration space has bifurcations as indicated by the distinct conical section oriented towards the upper right. Importantly, this bifurcation can be used to topologically protect the flat folded configuration. By setting $r_{\mathrm{in}}>0$ and $N_{\phi}=32$, we can force $t_{1}=-0.3$ in the flat folded state. For this value of $t_{1}$ the configurations space is topologically disconnected (blue plane and lower image in Fig. 3(C)). For the topologically disconnected configuration subspace, the large area corresponds to a very compliant and deformable structure whereas the sliver between $0.3<t_{3}<0.5$ corresponds to a highly constrained and collapsed structure. The narrowness of the configuration space for the collapsed structure suggests that it is rigid and capable of strain isolation. To jump or snap between these configurations, the sheet needs to access higher energy deformation modes such as stretching of the facets, which are not accounted for by the configuration space analysis Silverberg:2015gb .

The bifurcated configuration space also immediately suggests a mechanism in which successive layers of the structure can be made to sequentially snap onto the collapsed inner cylinder. Specifically, we choose the radial separation of the neighboring vertex so that the bifurcation in its configuration space is present when the shared universal crease $0.3<t_{3}<0.5$ corresponding to the topologically protected state for $v^{\prime}$ (Fig. 3(D)). This design ensures that once the inner vertex is locked into the highly constrained collapsed structure, the second layer will become bistable so that it can snap onto the first. It can be shown that by growing the distance between vertices in adjacent layers linearly with radius, this sequential snapping continues until the entire sheet collapses onto a fully folded cylindrical structure that is rigid and isolates strain by maintaining the geometry of its inner boundary.

To test these design principles, we fabricate a Miura-ori washer following the requirement for a vertex separation that grows linearly with radius. The structure is placed between two plates and compressed laterally (Fig. 4(A,B)). We find that the structure is able to accommodate a large degree of compressive strain $\gamma_{x}=(l_{0}-l)/l_{0}$, where $l_{0}$ and $l$ correspond to the lateral size of the unfolded and folded sheet respectively. During compression, the innermost layer is the first to collapse. Upon further compression, successive layers collapse onto the inner layer (see SI video). At the same time the remaining uncollapsed layers retain their flexibility. Importantly, throughout the entire folding process up to where the strain reaches $\gamma_{c}$ indicating full collapse, the inner boundary maintains its shape indicating effective strain isolation.

The snapping behavior of successive layers can be quantified by the compressive modulus or derivative of the applied compressive force $F$ with respect to the strain $\gamma_{x}$ (Fig. 4(C)). We find that each snap, indicated by the number of collapsed layers $n$, corresponds to a rapid decrease of $dF/d\gamma_{x}$ as the normalized strain $\gamma_{x}/\gamma_{c}$ is increased (Fig. 4(C)). This decrease arises from the extra space that is released when a layer snaps into place, which relaxes the folds in the remaining layers and reduces their resistance to compression.

To verify that snapping proceeds from the inner boundary outwards, we use 3D reconstructions to track the folding angle $t_{4}$ and map its value using a color map onto the unfolded sheet configurations (left and right sides of Fig. 4(D)). The radially averaged value for each layer as a function of the normalized strain is shown in the main part of the figure. We find that with increasing strain, the highly folded configurations indicated by $t_{4}\approx\pi$ extend to larger number of collapsed layers $n$.

To determine the theoretical strain value associated with successive snapping of layers onto the inner cylinder, we simulate an isotropic deformation of the same origami geometry (Methods). We plot the number of collapsed layers as a function of the applied normalized strain (black line and open circles). We find that these data track the experimentally determined interface between the open versus collapsed vertex structures denoted by high values of $t_{4}$. Furthermore, we find that comparison with $dF/d\gamma_{x}$ (vertical dashed lines) identifies where the structure undergoes an abrupt drop in compressive modulus due to each sequential snap (labeled by $n$).

Collectively, these studies demonstrate a powerful approach for designing and programing specific mechanical properties into origami systems. The structures we considered here are degree-6 Miura-ori vertices. Our methodology however, is general. Thus, the application of modular designs and propagation of constraints between vertices to origami patterns with different DOF will undoubtedly shed light on other nontrivial mechanical mechanisms and design principles. Moreover, these results naturally extend to other systems. For example, the interactions between vertex structures described here is reminiscent of set-valued maps commonly adopted in studies of robotic, biological, physical, and economic systems Aubin:1990vqa . There, the evolving dynamics produce constrained states that correspond to the allowed configurations we discussed. This set of constrained states, in turn, determines the possible evolution pathways for the system to achieve its next set of viable states. It may be possible to employ the same modular and bifurcating designs described here to these other systems in order to better control their behaviors. The advantage of this approach being that it introduces a “robustness” protected by the configuration space topology under different constraints.

I Method

I.1 Three-dimensional model of origami structures

A mathematical model of each experimentally generated origami structure is formulated for numerical simulation and 3D reconstruction. The model is composed of the vertex coordinates and the constraints imposed by the crease length between neighboring vertices. Given a 2D projection of all the vertex positions ($x_{i}$, $y_{i}$), the $z_{i}$ coordinates in the third dimension are obtained by minimizing a penalty function Tachi:2013im $V(z)=\frac{k}{2}\sum_{i,j}^{\prime}\left[\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{j})^{2}}-l_{ij}\right]^{2}$, where $l_{ij}$ is the length of the crease between two connected vertices of indices $i$ and $j$ on the triangular crease lattice. The $z$ coordinates can thus be obtained by solving the associated ordinary differential equation (ODE) array as $\dot{z_{i}}=-k\sum_{j}^{\prime}\left[\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{j})^{2}}-l_{ij}\right]\hat{r}_{ij}\cdot\hat{z}$, where the unit vector $\hat{r}_{ij}=((x_{i}-x_{j})\hat{x}+(y_{i}-y_{j})\hat{y}+(z_{i}-z_{j})\hat{z})/\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{j})^{2}}$, and $k$ is the virtual spring constant for perturbing the crease lengths. The ambiguity due to mountain-valley alignments is avoided by shifting the vertices up or down with a small amount in the $z$ direction as the initial condition.

I.2 Configuration space analysis

The configuration space of each vertex is described by the accessible folding angles of the connected creases. In the case of a degree-6 vertex, the configuration space is three dimensional (since DOF=3). Any potentially foldable structure is realized by rotating the creases through those 3 independent folding angles. The foldability is validated by the criterion that the distance between any pair of transformed vertices cannot be greater than that in the flat unfolded state for any inextensible sheet. Self-interceptions of the structures are also identified and excluded from the configuration space (see SI).

I.3 Experiments

The origami structures were made from a flat paper sheet (Stardream Metallics 81lb) with measured thickness $\tau=0.16$ mm and flexural rigidity $D=4.0\times 10^{-4}$ N$\cdot$m Silverberg:2015gb . The crease lines were perforated by a laser cutter and weakened manually by flat-folding along the perforation lines in both directions. Its folding kinematics were recorded by a top-view camera, which was calibrated for extracting the vertex locations. The $x-y$ coordinates of the vertices were fed into the aforementioned mathematical model of the origami to extract the $z$ depths of the vertices and the 3D configurations of the folded origami sheets were reconstructed using Matlab.

II Acknowledgments

Acknowledgements.

The authors thank Andy Ruina, Tim Healy, James Jenkins, Uyen Nguyen, Lea Freni, and the Cohen lab for useful discussions. We also thank F. Parish for assistance with the laser cutter. This work was supported by the National Science Foundation Grant No. EFRI ODISSEI-1240441. I.C. received continuing support from DMREF-1435829. C.D.S. thanks the KITP, Santa Barbara, for hospitality.