Crease Pattern Research Articles

Constructing Rigid-Foldable Generalized Miura-Ori Tessellations for Curved Surfaces

Abstract Origami has shown the potential to approximate three-dimensional curved surfaces by folding through designed crease patterns on flat materials. The Miura-ori tessellation is a widely used pattern in engineering and tiles the plane when partially folded. Based on constrained optimization, this article presents the construction of generalized Miura-ori patterns that can approximate three-dimensional parametric surfaces of varying curvatures while preserving the inherent properties of the standard Miura-ori, including developability, flat foldability, and rigid foldability. An initial configuration is constructed by tiling the target surface with triangulated Miura-like unit cells and used as the initial guess for the optimization. For approximation of a single target surface, a portion of the vertexes on the one side is attached to the target surface; for fitting of two target surfaces, a portion of vertexes on the other side is also attached to the second target surface. The parametric coordinates are adopted as the unknown variables for the vertexes on the target surfaces, while the Cartesian coordinates are the unknowns for the other vertexes. The constructed generalized Miura-ori tessellations can be rigidly folded from the flat state to the target state with a single degree-of-freedom.

Stepwise Evolution of Crease Patterns on Stimuli‐Responsive Hydrogels for the Production of Long‐Range Ordered Structures

AbstractThe creases and wrinkles that form on soft matter have been comprehensively analyzed and engineered to utilize their topological advantages in various research fields. Although the principle for the formation of such structures is found to be the inhomogeneous distribution of mechanical stress, simultaneous and omnidirectional propagation of surface patterns makes it difficult to engineer these structures. A design strategy for the reversible formation of highly uniform crease patterns on hydrogel films is proposed by driving the stepwise evolution of creases. A hydrogel film with a smooth‐ and submicron‐scale surface topology, so‐called hill and valley structures, is prepared by engineering the reaction‐diffusion‐mediated photopolymerization and viscoelastic deformation of hydrogels. As the hydrogel film undergoes a morphological change in response to pH, creases are selectively nucleated from the hill and valley structures and propagate directionally, resulting in the formation of long‐range ordered crease patterns. Different interactions are observed via the investigation of three colloidal systems, demonstrating the capacity to control colloidal particles with the highly uniform surface topology of hydrogels. Finally, the pattern‐guided alignment and reproduction of yeast cells are demonstrated, providing an opportunity for producing an artificial environment for the guided growth and analysis of biomaterials.

Counting Locally Flat-Foldable Origami Configurations Via 3-Coloring Graphs

Origami, where two-dimensional sheets are folded into complex structures, is rich with combinatorial and geometric structure, most of which remains to be fully understood. In this paper we consider flat origami, where the sheet of material is folded into a two-dimensional object, and consider the mountain (convex) and valley (concave) creases that result, called a MV assignment of the crease pattern. An open problem is to count the number locally valid MV assignments $$\mu $$ of a given flat-foldable crease pattern C, where locally valid means that each vertex will fold flat under $$\mu $$ with no self-intersections of the folded material. In this paper we solve this problem for a large family of crease patterns by creating a planar graph $$C^*$$ whose 3-colorings are in one-to-one correspondence with the locally valid MV assignments of C. This reduces the problem of enumerating locally valid MV assignments to the enumeration of 3-colorings of graphs.

Origami cubes with one-DOF rigid and flat foldability

Rigid origami is a branch of origami with great potential in engineering applications to deal with rigid-panel folding. One of the challenges is to compactly fold flat a polyhedron made from rigid facets with a single degree of freedom. In this paper, we present a new method to design origami cubes with three fundamental characteristics: rigid foldability, flat foldability and one degree of freedom (DOF). A total of four cases of crease patterns that enable origami cubes with distinct folding performances have been proposed, which covers all possible layouts of the diagonal creases on the square facets of origami cubes. The folding utilises spherical linkages formed by intersecting creases, and based on the kinematic equivalence between the rigid origami and the spherical linkages, the corresponding spherical linkage loops are introduced and analysed to reveal the motion properties of the origami cubes. The newly found method can be readily utilized to design deployable structures for various engineering applications including cube-shaped cartons, small satellites, containers, etc.

Non-Euclidean origami.

Traditional origami starts from flat surfaces, leading to crease patterns consisting of Euclidean vertices. However, Euclidean vertices are limited in their folding motions, are degenerate, and suffer from misfolding. Here we show how non-Euclidean 4-vertices overcome these limitations by lifting this degeneracy, and that when the elasticity of the hinges is taken into account, non-Euclidean 4-vertices permit higher order multistability. We harness these advantages to design an origami inverter that does not suffer from misfolding and to physically realize a tristable vertex.

Curved foldings with common creases and crease patterns

Consider a curve Γ in a domain D in the plane R2. Thinking of D as a piece of paper, one can make a curved folding P in the Euclidean space R3. The singular set C of P as a space curve is called the crease of P and the initially given plane curve Γ is called the crease pattern of P. In this paper, we show that in general there are four distinct non-congruent curved foldings with a given pair consisting of a crease and crease pattern. Two of these possibilities were already known, but it seems that the other two possibilities (i.e. four possibilities in total) are presented here for the first time.

Folding simulation of rigid origami with Lagrange multiplier method

Origami crease patterns are folding paths that transform flat sheets into spatial objects. Origami patterns with a single degree of freedom (DOF) have creases that fold simultaneously. More often, origami patterns have multiple DOFs and several substeps are required to sequentially fold such origami; at each substep, some creases fold and the rest remain fixed. In this study, we combine the loop closure constraint with the Lagrange multiplier method to account for the sequential folding of rigid origami of multiple DOFs by controlling the rotation of different sets of creases during successive substeps. This strategy is also applicable in modeling origami-inspired devices, where creases may be equipped with rotational springs and the folding process thus involves elastic energy. Several examples are presented to verify the proposed algorithms in tracing the sequential folding process as well as searching the equilibrium configurations of origami with rotational springs.

Earwig fan designing: Biomimetic and evolutionary biology applications

Technologies to fold structures into compact shapes are required in multiple engineering applications. Earwigs (Dermaptera) fold their fanlike hind wings in a unique, highly sophisticated manner, granting them the most compact wing storage among all insects. The structural and material composition, in-flight reinforcement mechanisms, and bistable property of earwig wings have been previously studied. However, the geometrical rules required to reproduce their complex crease patterns have remained uncertain. Here we show the method to design an earwig-inspired fan by considering the flat foldability in the origami model, as informed by X-ray microcomputed tomography imaging. As our dedicated designing software shows, the earwig fan can be customized into artificial deployable structures of different sizes and configurations for use in architecture, aerospace, mechanical engineering, and daily use items. Moreover, the proposed method is able to reconstruct the wing-folding mechanism of an ancient earwig relative, the 280-million-year-old Protelytron permianum This allows us to propose evolutionary patterns that explain how extant earwigs acquired their wing-folding mechanism and to project hypothetical, extinct transitional forms. Our findings can be used as the basic design guidelines in biomimetic research for harnessing the excellent engineering properties of earwig wings, and demonstrate how a geometrical designing method can reveal morphofunctional evolutionary constraints and predict plausible biological disparity in deep time.

Curved creases redistribute global bending stiffness in corrugations: theory and experimentation

Corrugations offer a convenient way to make thin, lightweight sheets into stiff structures. However, traditional, v-shaped corrugations made from straight creases result in highly anisotropic stiffness which leads to undesirable flexibility in some directions of loading. In this paper, we explore the bending stiffness of curved-crease corrugations with a planar midsurface—developable corrugations made by folding thin sheets about curves and without linerboard covers on the top or bottom. The curved-crease corrugations break symmetry in the pattern and can redistribute stiffness to resist bending deformations in multiple directions. To study these systems, we formulate a framework for predicting the bending stiffness of any planar-midsurface corrugation from its multiple geometric features at different scales. We use the framework to create two predictive methods that provide valuable insight into the global stiffness of corrugations without a detailed analysis. Results from these methods match well with experimental, three-point bending tests of five corrugation geometries made from polyester film. We found that corrugations with elliptical or parabolic curved-creases that intersect with one edge of the pattern are best at redistributing stiffness in multiple directions. While a straight-crease pattern has a stiffness of about 4 N/mm in one direction and about 0 N/mm in the other, a parabolic crease pattern has a stiffness of about 2.5 N/mm in both directions. These curved-crease corrugations can enable the self-assembly and fabrication of practical, stiff structures from simple, developable sheets.

Assigning mountain-valley fold lines of flat-foldable origami patterns based on graph theory and mixed-integer linear programming

Traditional origami design is generally based on designers’ artistic intuition and skills, mathematical calculations, and experimentations, which can involve challenges for crease patterns with a large number of vertices. To develop novel origami structures for engineering applications, systematic and easy-to-implement approaches capable of generating diverse origami patterns are desired, without requiring extensive artistic skills and experience in origami mathematics. Here, we present a computational method for automatically assigning mountain-valley fold lines to given geometric configurations of origami structures. This method is based upon a geometric-graph-theoretic representation approach combined with a graph-theoretic cycle detection algorithm, taking the subgraphs of a given structure as inputs. Then, a mixed-integer linear programming (MILP) model is established to find flat-foldable origami patterns under given constraints on the local flat-foldability and degree of vertices, leading to the identification of crease lines associated with local minimum angles. Numerical examples are presented to demonstrate the performance of the proposed approach for a range of origami structures with degree-4 or -6 vertices represented by their corresponding subgraphs.

Rigid foldability is NP-hard

In this paper, we show that the rigid-foldability of a given crease pattern using all creases is weakly NP-hard by a reduction from the partition problem, and that rigid-foldability with optional creases is NP-hard by a reduction from the 1-in-3 SAT problem. Unlike flat-foldabilty of origami or flexibility of other kinematic linkages, whose complexity originates in the complexity of the layer ordering and possible self-intersection of the material, rigid foldabilltiy from a planar state is hard even though there is no potential self-intersection. In fact, the complexity comes from the combinatorial behavior of the different possible rigid folding configurations at each vertex. The results underpin the fact that it is harder to fold from an unfolded sheet of paper than to unfold a folded state back to a plane, frequently encountered problem when realizing folding-based systems such as self-folding matters and reconfigurable robots.

Crumpling-origami transition for twisting cylindrical shells.

Origami and crumpling are two processes to reduce the size of a membrane. In the shrink-expand process, the crease pattern of the former is ordered and protected by its topological mechanism, while that of the latter is disordered and generated randomly. We observe a morphological transition between origami and crumpling states in a twisted cylindrical shell. By studying the regularity of the crease pattern, acoustic emission, and energetics from experiments and simulations, we develop a model to explain this transition from frustration of geometry that causes breaking of rotational symmetry. In contrast to solving von Kármán-Donnell equations numerically, our model allows derivations of analytic formulas that successfully describe the origami state. When generalized to truncated cones and polygonal cylinders, we explain why multiple and/or reversed crumpling-origami transitions can occur.

The designs and deformations of rigidly and flat-foldable quadrilateral mesh origami

Rigidly and flat-foldable quadrilateral mesh origami is the class of quadrilateral mesh crease patterns with one fundamental property: the patterns can be folded from flat to fully-folded flat by a continuous one-parameter family of piecewise affine deformations that do not stretch or bend the mesh-panels. In this work, we explicitly characterize the designs and deformations of all possible rigidly and flat-foldable quadrilateral mesh origami. Our key idea is a rigidity theorem (Theorem 3.1) that characterizes compatible crease patterns surrounding a single panel and enables us to march from panel to panel to compute the pattern and its corresponding deformations explicitly. The marching procedure is computationally efficient. So we use it to formulate the inverse problem: to design a crease pattern to achieve a targeted shape along the path of its rigidly and flat-foldable motion. The initial results on the inverse problem are promising and suggest a broadly useful engineering design strategy for shape-morphing with origami.

On rigid origami II: quadrilateral creased papers

Miura-ori is well known for its capability of flatly folding a sheet of paper through a tessellated crease pattern made of repeating parallelograms. Many potential applications have been based on the Miura-ori and its primary variations. Here, we are considering how to generalize the Miura-ori: what is the collection of rigid-foldable creased papers with a similar quadrilateral crease pattern as the Miura-ori? This paper reports some progress. We find some new variations of Miura-ori with less symmetry than the known rigid-foldable quadrilateral meshes. They are not necessarily developable or flat-foldable, and still only have single degree of freedom in their rigid folding motion. This article presents a classification of the new variations we discovered and explains the methods in detail.

A Programmably Compliant Origami Mechanism for Dynamically Dexterous Robots

We present an approach to overcoming challenges in dynamical dexterity for robots through programmably compliant origami mechanisms. Our work leverages a one-parameter family of flat sheet crease patterns that folds into origami bellows, whose axial compliance can be tuned to select desired stiffness. Concentrically arranged cylinder pairs reliably manifest additive stiffness, extending the programmable range by nearly an order of magnitude and achieving bulk axial stiffness spanning 200–1500 Nm $^{-1}$ using 8 mil thick polyester-coated paper. Accordingly, we design origami energy-storing springs with a stiffness of 1035 N m $^{-1}$ each and incorporate them into a three degree-of-freedom (DOF) tendon-driven spatial pointing mechanism that exhibits trajectory tracking accuracy less than 15% rms error within a ( $\sim$ 2 cm) $^3$ volume. The origami springs can sustain high power throughput, enabling the robot to achieve asymptotically stable juggling for both highly elastic (1 kg resilient shotput ball) and highly damped (“medicine ball”) collisions in the vertical direction with apex heights approaching 10 cm. The results demonstrate that “soft” robotic mechanisms are able to perform a controlled, dynamically actuated task.

Reconfigurable and programmable origami dielectric elastomer actuators with 3D shape morphing and emissive architectures

Soft actuators with the capability to generate programmable and reconfigurable motions without the use of complicated and rigid infrastructures are of great interest for the development of smart, interactive, and adaptive soft electronic systems. Here, we report a new strategy to achieve a transparent and reconfigurable actuator by using a dielectric elastomer actuator (DEA), which provides mechanical strains under electrical bias, integrated with origami ethyl cellulose (EC) paper that “instructs” the shape changes of the actuator. The actuator can be reconfigured and multiple mechanical motions can be programmed in the device by creating crease patterns that induce variations in the local stiffness to direct the actuations. With the versatile design and fabrication approach, a light emission device with dynamic shape changes was demonstrated.

On rigid origami I: piecewise-planar paper with straight-line creases

Origami (paper folding) is an effective tool for transforming two-dimensional materials into three-dimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here, we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods. In the algebraic part, we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel-hinge system. In the geometric part, we analyse corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part, we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures.

Novel origami-inspired metamaterials: Design, mechanical testing and finite element modelling

Two novel, origami-inspired, metamaterials were designed, mechanically tested, and modelled. One novel origami model was folded using a triangular based crease pattern and the other was folded using a rectangular based crease pattern. The origami-inspired metamaterial sheets were fabricated from polylactic acid using fused deposition additive manufacturing. Several configurations, parameterized by varying the fold angle, were mechanically tested under compression and impact loads. It was found that the specific elastic compression modulus of these novel designs was higher, ranging from 594 MPa/kg to 926 MPa/kg, than existing origami-inspired structures made based on the popular Ron-Resch design, which had specific elastic compression moduli between 15 MPa/kg to 365 MPa/kg. A finite element model further analysed the stress distribution of the core structures under compression loads. The impact testing results showed that the pattern of the tessellated cores affected the amount of impact force transferred through the samples, whereas the fold angle of the origami-inspired design had little impact on the results. The rectangular structure was shown to transfer approximately 50–75% of the force transferred by the triangular structure under impact loads.

Curve-pleated structures

In this paper we study pleated structures generated by folding paper along curved creases. We discuss their properties and the special case of principal pleated structures. A discrete version of pleated structures is particularly interesting because of the rich geometric properties of the principal case, where we are able to establish a series of analogies between the smooth and discrete situations, as well as several equivalent characterizations of the principal property. These include being a conical mesh, and being flat-foldable. This structure-preserving discretization is the basis of computation and design. We propose a new method for designing pleated structures and reconstructing reference shapes as pleated structures: we first gain an overview of possible crease patterns by establishing a connection to pseudogeodesics, and then initialize and optimize a quad mesh so as to become a discrete pleated structure. We conclude by showing applications in design and reconstruction, including cases with combinatorial singularities. Our work is relevant to fabrication in so far as the offset properties of principal pleated structures allow us to construct curved sculptures of finite thickness.

Modeling curved folding with freeform deformations

We present a computational framework for interactive design and exploration of curved folded surfaces. In current practice, such surfaces are typically created manually using physical paper, and hence our objective is to lay the foundations for the digitalization of curved folded surface design. Our main contribution is a discrete binary characterization for folds between discrete developable surfaces, accompanied by an algorithm to simultaneously fold creases and smoothly bend planar sheets. We complement our algorithm with essential building blocks for curved folding deformations: objectives to control dihedral angles and mountain-valley assignments. We apply our machinery to build the first interactive freeform editing tool capable of modeling bending and folding of complicated crease patterns.