Foldable Origami Research Articles

Optimal Curve Fitting for Serial Chain in Six-Crease Origami Unit

Abstract Origami-inspired structures have been widely used in aerospace and robotics for three-dimensional (3D) symmetrical configurations using crease-symmetrical origami basic patterns. These patterns offer advantages in repeatable and systematic modeling and mass production. However, few studies have focused on 3D non-symmetrical structures using symmetrical origami basic patterns due to their structure complexity, limiting their application. Therefore, we aim to analyze the folding behavior in 3D non-symmetrical structures using a 6-crease symmetry origami base pattern. To achieve this goal, we first focus on behavior in a two-dimensional (2D) plane. This paper presents a scheme for the behavior of origami units with an optimal curve-fitting algorithm. The curve can be any 2D space curve. The fitting curve, constructed by numerical analysis and an optimal approaching scheme, can satisfy error requirements and retain foldable origami unit features. The paper verifies the feasibility of the curve-fitting scheme by presenting two curve examples, including a quadratic curve and a sin wave function. The results show that the fitting error is reduced by 99% when no boundary conditions are applied. This research provides valuable insights into understanding origami unit kinematic optimization through forward and inverse kinematics. It offers potential applications in the engineering design of foldable structures and precision origami-inspired mechanism, thereby opening avenues for further exploration of complex origami structures and their applications in emerging technologies.

Rigid-foldable spiral origami with compression-torsion coupled motion mode

Rigid foldable origami enables smooth and precise folding without stretching or bending its constituent panels and is promising for applications such as reprogrammable matter, self-folding machines, reconfigurable antennas, and deployable spacecraft. The diverse range of potential applications necessitates the need for the design and detailed analysis of different rigid-foldable origami structures, especially those with intricate motion modes. In this paper, we introduce a rigid-foldable spiral origami design that features a compression-torsion coupled motion mode. This design exhibits rich static and dynamic properties. Under static conditions, the compression-torsion coupled motion mode creates multiple self-locking positions and allows for the development of mechanical static diodes. Under dynamic conditions, the compression-torsion coupling effect in the spiral origami facilitates precise control of wave modes within the origami chain when impacted by a ball with a moderate initial velocity. In the case of large initial velocities of the ball, the spiral origami can function as a wave generator, producing rarefaction solitary waves or compressive solitary waves. The proposed spiral origami design provides an opportunity to explore new applications of rigid-foldable origami with compression-torsion coupling effects.

Analysis of the Rigid Foldability of Origami Patterns Based on Spatial Positions of Creases

Abstract Rigid foldability is a special property of rigid origami patterns, where each origami plane remains undeformed during continuous movement along the predetermined crease. Current research on the rigid foldability of origami patterns mainly focuses on kinematics, while less attention is paid to factors that cause deformation of the folding plane. Whether the relative spatial position of adjacent creases has been changed is a critical factor that influences the state (rigid or deformed) of the folding plane between the two adjacent creases during the folding process. This study considered two factors (linear relationship and Euclidean distance) to measure the changes in the spatial positions of creases, explored the relationship between the two factors and rigid folding, and identified deformation forms that affect rigid foldability. First, the origami pattern was regarded as a linkage mechanism, and the linear relationship between creases was determined from the single-vertex origami unit forming this origami structure. Then, the geometric parameters of the origami pattern were used to calculate the Euclidean distance between two points on adjacent creases during the folding process. If the linear relationship and Euclidean distance always remain the same, the origami pattern has rigid foldability. Based on changes in the Euclidean distance, this method can also help determine the main deformation of non-rigidly foldable origami patterns. In addition, it can be applied to origami patterns with four or five vertices and multiple loops, and it further provides a novel approach for determining the layout of the crease position and the judgment of rigid foldability during origami-inspired mechanism design.

Multi-Stability of the Extensible Origami Structures.

Multi-stable structures and metamaterials with more than two stable states are widely applied in diversified engineering applications. Non-rigid foldable origami patterns have provided an effective way of designing multi-stable structures. But most of them have only two stable states and therefore require a combination of many units to achieve multi-stability. Here, a series of extensible origami structures are proposed with generic multi-stability based on non-rigid wrapping origami. Through a kinematic analysis and experiments, it is demonstrate that a sequential folding among different layers of the structures is created to generate a continuous rigid origami range and several discrete rigid origami states, which consequently leads to the multi-stability of the extensible origami structures. Moreover, the effects of design parameters on the mechanical properties of the structures are investigated by numerical simulation, enabling properties programmability upon specific needs. This study thus paves a new pathway for the development of novel multi-stable origami structures.

Efficient reciprocating burrowing with anisotropic origami feet.

Origami folding is an ancient art which holds promise for creating compliant and adaptable mechanisms, but has yet to be extensively studied for granular environments. At the same time, biological systems exploit anisotropic body forces for locomotion, such as the frictional anisotropy of a snake's skin. In this work, we explore how foldable origami feet can be used to passively induce anisotropic force response in granular media, through varying their resistive plane. We present a reciprocating burrower which transfers pure symmetric linear motion into directed burrowing motion using a pair of deployable origami feet on either end. We also present an application of the reduced order model granular Resistive Force Theory to inform the design of deformable structures, and compare results with those from experiments and Discrete Element Method simulations. Through a single actuator, and without the use of advanced controllers or sensors, these origami feet enable burrowing locomotion. In this paper, we achieve burrowing translation ratios-net forward motion to overall linear actuation-over 46% by changing foot design without altering overall foot size. Specifically, anisotropic folding foot parameters should be tuned for optimal performance given a linear actuator's stroke length.

Large impact of small vertex cuts on the mechanics of origami bellows

For origami structures, perforating or cutting slits along creases is an effective method to define fold lines and alleviate stress concentrations at vertices. In this letter we show numerically and experimentally that for non-rigid-foldable origami bellows (e.g. Miura-ori, Kresling patterns) the introduction of small cut-outs at the vertices results in up to an order of magnitude reduction of the bellows’ stiffness under axial compression. Further, the cut-outs at vertices impact the nonlinear response, e.g. the position and magnitude of a force limit point and presence of bistable configurations. As the origami bellows are not rigid foldable, an axial compression will necessarily result in facet deformations; the small cut-outs at the vertices are found to provide an unexpectedly large stress alleviation, resulting in disproportionate changes in mechanical properties of the bellows. In order to accurately model the mechanics of origami bellows, such manufacturing details must therefore be captured accurately. Lastly, introducing vertex cut-outs can be offered as a novel approach to tailoring the stiffness of non-rigid foldable origami structures.

Semi-quantitative microfluidic paper-based analytical device for ionic silica detection.

Silicate ions (SiO32-), or ionic silica, are known to cause silica scaling in industrial water applications when excess amounts are present; hence, concentrations must be monitored and kept at a constant low level. Ionic silica is conventionally measured by spectrophotometry in the form of its silicomolybdic complex based on the molybdenum blue reaction, but the operation process is complicated and not suitable for on-site detection. To solve these issues, microfluidic paper-based analytical devices (µPADs) have been gaining attention as portable, low-cost analytical devices suitable for on-site detection. The foldable origami type device described in this work enabled silica detection based on the molybdenum blue reaction, in the range of 50-1000mg/L, with a practically detectable lowest concentration of 50mg/L. The device showed selectivity for silicate ions and stability over 21days when stored at 4°C. The semi-quantitative analytical performance makes the proposed paper-based device attractive for on-site industrial monitoring.

Origami-Inspired Deployable Space Habitats

Origami is increasingly used as a source of inspiration in a wide variety of disciplines. In this project, we explore cylindrical origami structures, referred to as “origami bellows”, as novel geometries for orbital space habitats. The dimensions of space habitats are limited by the tight mass and volume constraints imposed by launcher payload fairings. Future deployable habitats based on foldable origami bellows have the potential to achieve large volumes when deployed, while being capable of compacting to smaller stowed configurations for launch. To assess the feasibility of such habitat designs, the deployment performance of a selection of bellows was investigated. Bellows formed from Kresling and Miura-ori patterns were considered; both expand axially, but Miura-ori patterns experience an additional radial expansion. Our scope was also limited to patterns which are stable in both the stowed and deployed configurations. Habitats were judged on their internal and effective volume expansions; the latter being adjusted to account for the practicalities of operating within a complex habitat geometry. We find that significant internal and effective volume expansions are achievable, particularly for Miura-ori geometries. Nonetheless, we make the argument for Kresling patterns as a more practical option due to their simpler geometries, despite smaller volume expansions. We find our Kresling geometries to have effective volumes between 2.5 - 3.6 times greater than a conventional habitat launched in a fairing of equal volume. Our work shows that origami-based designs do indeed have potential to greatly outperform current space habitat designs. Keywords: Origami Bellows, Space Habitats, Deployable Structures

DSM assists with foldable bottle

DSM is partnering with DiFOLD to further develop that firm’s foldable Origami Bottle. The bottle can be folded to 10% of its full size, which is convenient for consumers and could save on shipping volumes. The bottles are made of DSM’s Arnitel Eco thermoplastic copolyester resins, which are partly sourced from rapeseed oil.

A machine learning approach to predicting mechanical behaviour of non-rigid foldable square-twist origami

Origami-inspired metamaterials offer unlimited possibilities and broad application prospects in science and engineering. In particular, non-rigid foldable origami structures potentially provide more favorable characteristics than rigid foldable ones owing to the flexibility of thin sheets. However, the previous rigid analytical solutions are not sufficient to reflect the actual motions of non-rigid origami because of facet deformation, which prevents the widespread promotion of non-rigid origami applications. In this study, a machine learning approach has been one of first proposed for predicting the mechanical behaviours of a non-rigid foldable square-twist origami pattern (Type 1). Decision tree and multilayer perceptron were applied to classify multi-stability and predict the transition point, maximum strain energy, and energy barrier of the square-twist origami structure. Both methods efficiently provided accurate results for various geometric and material parameters. It is worth mentioning that the dimensional analysis added to this approach can reduce the computational time and increase the accuracy of the predicted results. Due to the same folding principle and similar parameters of origami patterns, the machine learning approach proposed herein is a promising alternative for other complex origami engineering problems when analytical and empirical solutions are unavailable.

Design of a foldable origami mechanism with helical motion inspired by the Resch Triangular Tessellation

Inspired by artistic origami, researchers in engineering have explored the principles of origami folding to design novel foldable mechanisms, origami robots and metamaterials. Taking inspiration from the Resch Triangular Tessellation, an artistic origami of which the central triangular panel twists during the folding/unfolding process, this paper presents a novel origami mechanism with a foldable/deployable form. It has 3-Degrees-of-Freedom (DoFs) and is capable of helical motion (one twisting motion coupled with the translation along with the same axis). The mechanism is composed of two minimum units of the Resch Triangular Tessellation pattern connecting in stack face to face. Three origami creases intersecting at a common point, equivalent to a traditional spherical joint, are adopted for the connection. The geometric and kinematic characteristics of the origami mechanism are analyzed. The workspace of the moving platform of the mechanism is analyzed. A proof-of-concept prototype of the mechanism is fabricated and experimentally demonstrated by a cable-driven actuation system. The pure helical motion mode (when three inputs are the same) is tested using the prototype and is compared with the theoretical model. Diverse motion configurations with different settings of input angles are presented by evaluating the 3D model and prototype of the novel helical mechanism. The connection of multiple helical origami mechanisms in series leads to a novel continuum robot which is also discussed. Compared with other counterparts, the novel mechanism has both the helical motion and origami-like foldability without needing flexible panels.

Designing origami tessellations composed of quadrilateral meshes and degree-4 vertices for engineering applications

Origami-based designs have been successfully adopted to solve engineering problems at different scales of application. This article proposes a method for designing origami tessellations composed of quadrilateral meshes and degree-4 vertices. Crease patterns are generated starting from base units to which are applied tailored generation functions. The theoretical framework is further implemented in a parametric design workflow and automated with the aid of an Interactive Evolutionary Algorithm (IEA). The proposed workflow is a flexible tool that could be applied to design and analyse both rigid and non-rigid foldable origami patterns. The design domain could be investigated without pre-defined design goals and with the support of user-defined performance values.

Artificial Neural Network for the prediction of fatigue life of a flexible foldable origami antenna with Kresling pattern

In this paper, we present a comprehensive fatigue analysis of a foldable origami helical antenna with Finite Element Method (FEM) and Artificial Neural Network (ANN). We study the effect of design parameters such as total height, length ratio (b/a), height of story, thickness, length and thickness ratios of creases, and radius of the circumscribed circle of polygonal on the fatigue life. We employ ANN method to reduce the computational cost of the conventional methods for predicting the fatigue life of the origami antenna. Although ANN is trained with a limited set of data, our results reveal that the proposed approach predicts the fatigue failure of the antenna with high accuracy. This trained ANN determines the life cycle of the origami structure with less than 1% error on the training set and less than 2% error on the test set. The ANN results illustrate that the fatigue life of the structure is improved by increasing the radius of the circumscribed circle and decreasing the thickness of the structure. We discover that some values of the length ratio (b/a) magnify the effect of total height on the life cycle. In addition, we show how the creases parameters of the structure play an important role in the fatigue life of the antenna.

Prediction of fatigue life of a flexible foldable origami antenna with Kresling pattern

In this paper, we present a thorough mechanical analysis of a foldable origami helical antenna using the Kresling pattern. A precise simulation is required to explicitly capture the complex motion of an origami structure during the folding process. We perform this study by employing the Finite Element Method (FEM) available in ANSYS software. First, we analyze the folding behavior of the structure with three stories. The results indicate that after the second story is folded completely, the first and third stories begin to fold. Next, we predict the fatigue life of the structure using the mean stress with fully reversed loading. We investigate the effects of several design parameters such as length ratio (b/a), height of story, total height, thickness, length and thickness ratios of creases, and radius of the circumscribed circle of polygonal on the fatigue life. We find out that the origami structure with the length ratio (b/a) of 1.9 has the highest life cycles. The results indicate that the effects of the height of the story on the life cycle depend on the length ratio (b/a). We observe that the life cycle of the origami structure improves by decreasing the thickness or increasing the radius of the circumscribed circle of hexagonal. In addition, we discover that the creases design plays an important role in the fatigue life and folding behavior of the origami structure.

Design of an Origami Bendy Straw for Robotic Multistable Structures

Abstract This article presents a soft cylindrical multistable origami structure based on “bendy straws,” consisting of multiple conical frusta mimicking the structure of a flexible drinking straw. These frusta are connected in such a way that the whole structure is axially multistable, having a stable compressed state in which its smallest frustum is collapsed. The bendy straw structure can also be modified so that the smallest frustum collapses only partially, keeping the structure in a bent state. We studied the geometry of a similar structure consisting of polygonal frusta instead of conical ones and used this geometry to design a nonrigidly foldable origami pattern folding into a similar origami bendy straw structure. Most of the origami structures presented so far have been modeled from rigidly foldable origami patterns: These origami structures do not rely on local deformations of the sheet and cannot use it to their advantage, whereas the nonrigid origami structure presented here features multistability. We have established that this origami structure is not only axially multistable, but that it can also be kept in a bent state, thanks to the use of pop-through defects (PTDs). The origami bendy straws studied here were made from paper (with a density of 90 g/m2) bilaminated with a 42.5 μm thick plastic film. A digital dynamometer was used to study the forces required to compress and expand a single origami bendy straw, create and reverse a PTD, and bend an origami bendy straw using PTDs.

Bistability-based foldable origami mechanical logic gates

Mechanical logic gates, counterparts of electronic logic gates, are highly desirable for the development of mechanical computing and robotic materials. Kirigami and origami based mechanical metamaterials have been shown to have attractive scale-free properties, e.g., multi-stability and programmability. Here, a bistability-based foldable waterbomb origami consisting of bistable kirigami units is proposed. The geometrical parameters dependence of the responses of the kirigami units and the origami is explored. Simplified models are developed, verified by numerical simulations and experiments. Based on the proposed origami, mechanical logic gates AND, OR, and NOT are designed and are experimentally demonstrated. The obtained results illustrate a possible solution for more complex logic operations (e.g., signal processing and data storage) from a mechanical deformation point of view by combining the logic gates together in a desired manner.

Rigid foldability and mountain-valley crease assignments of square-twist origami pattern

Rigid foldability allows an origami pattern to fold about crease lines without twisting or stretching component panels. It enables folding of rigid materials, facilitating the design of foldable structures. Recent study shows that rigid foldability is affected by the mountain-valley crease (M-V) assignment of an origami pattern. In this paper, we investigate the rigid foldability of the square-twist origami pattern with diverse M-V assignments by a kinematic method based on the motion transmission path. Four types of square-twist origami patterns are analyzed, among which two are found rigidly foldable, while the other two are not. The explicit kinematic equations of the rigid cases are derived based on the kinematic equivalence between the rigid origami pattern and the closed-loop network of spherical 4R linkages. We also convert a non-rigid pattern into a rigid one by introducing an extra crease. The kinematic analysis of the modified pattern reveals an interesting bifurcation behaviour. This work not only helps to deepen our understanding on the rigid foldability of origami patterns and its relationship with the M-V assignments, but also provides us an effective way to create more rigidly foldable origami patterns from non-rigid ones.

Simulation of Dynamics During Deployment of Foldable Origami Structures

Foldable origami-based structures are a type of deployable structures that are increasingly applied in the space and building industries. When folded, the small size of such structures facilitates transportation and storage. Meanwhile, the properties of their larger deployed state may be of interest to different applications. A stable working condition is established by locking the structure in its deployed state, as in the process of deployment, the driving forces may generate a dynamic effect, thus leading to instability of the system. Hence, the study of dynamic characteristics of such structures, including trajectory, duration, velocity, and acceleration is of paramount importance. In this paper, based on the general dynamic equation and Lagrange’s equations of the first kind, the finite element method is adopted to investigate the dynamic deployment of foldable plate structures in terms of the generalized nodal coordinates. The proposed geometric description of a quadrilateral plate element is based on a folding plate composed of refined triangular elements, which are used to approximate the real shells in the structure. Subsequently, a MATLAB framework is developed on the basis of the element using the Newmark integration and the Newton–Raphson iteration method to simulate the deployment process of the structure. Comparisons between MATLAB results and ADAMS results verify the reliability of the framework in analyzing the dynamic deployment of the foldable origami-based structures with sufficient accuracy.

Dynamic modeling for foldable origami space membrane structure with contact-impact during deployment

The dynamic modeling for the foldable origami space membrane structure considering contact-impact during the deployment is studied in this paper. The membrane is discretized using the triangular elements of the Absolute Nodal Coordinate Formulation (ANCF), and the stress–strain relationship of the membrane is determined based on the Stiffness Reduction Model (SRM). A mixed method is proposed for the frictionless contact problem by combining the membrane surface-to-surface (STS) contact elements with the membrane node-to-surface (NTS) contact elements to improve precision. Compared with the traditional STS contact elements, the mixed method can effectively avoid mutual penetration of the element boundaries, especially for the foldable origami membrane structures undergoing overall motions. The penalty method is adopted to enforce the nonpenetration condition. Moreover, special constraints are built for the fold lines, and then the dynamic equations of the membrane multibody system considering the damping effect are formulated. The dynamic deployment procedure of a leaf-in origami membrane structure with contact-impact is performed employing this present mixed method. The results demonstrate the effectiveness and superiority of the presented mixed method in the solution of the complicated contact problem, and the influence of the contact-impact on the dynamic performance is analyzed.

A portable three-degrees-of-freedom force feedback origami robot for human–robot interactions

Haptic interfaces can recreate the experience of touch and are necessary to improve human–robot interactions. However, at present, haptic interfaces are either electromechanical devices eliciting very limited touch sensations or devices that may provide more comprehensive kinesthetic cues but at the cost of their large volume: there is a clear trade-off between the richness of feedback and the device size. The design and manufacturing challenges in creating complex touch sensations from compact platforms still need to be solved. To overcome the physical limitation of miniaturizing force feedback robots, we adapted origami design principles to achieve portability, accuracy and scalable manufacturing. The result is Foldaway, a foldable origami robot that can render three-degrees-of-freedom force feedback in a compact platform that can fit in a pocket. This robotic platform can track the movement of the user’s fingers, apply a force of up to 2 newtons and render stiffness up to 1.2 newtons per millimetre. We experimented with different human–machine interactions to demonstrate the broad applicability of Foldaway prototypes: a portable interface for the haptic exploration of an anatomy atlas; a handheld joystick for interacting with virtual objects; and a bimanual controller for intuitive and safe teleoperation of drones. Haptic interfaces are important for the development of immersive human–machine interactions. To create a compact design with rich touch-sensitive functions, a robotic device called Foldaway, which folds flat, has been designed that can render three-degrees-of-freedom force feedback.