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1. Rigid foldability is NP-hard

   Akitaya, Hugo A; Demaine, Erik D; Horiyama, Takashi; Hull, Thomas C; Ku, Jason S; Tachi, Tomohiro (January 2020, Journal of computational geometry)

   null (Ed.)

   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. 

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2. Hidden symmetries generate rigid folding mechanisms in periodic origami

   https://doi.org/10.1073/pnas.2005089117  

   McInerney, James; Chen, Bryan Gin-ge; Theran, Louis; Santangelo, Christian D.; Rocklin, D. Zeb (November 2020, Proceedings of the National Academy of Sciences)

   We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell–Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origami’s vertices. This supports the recent result by Tachi [T. Tachi,Origami6, 97–108 (2015)] which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero-energy deformations in the bulk that may be used to reconfigure the origami sheet. 

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3. Enabling Robust Self‐Folding Origami by Pre‐Biasing Vertex Buckling Direction

   https://doi.org/10.1002/adma.201903006  

   Kang, Ji‐Hwan; Kim, Hyunki; Santangelo, Christian_D; Hayward, Ryan_C (August 2019, Advanced Materials)

   Abstract Self‐folding is a powerful approach to fabricate materials with complex 3D forms and advanced properties using planar patterning steps, but suffers from intrinsic limitations in robustness due to the highly bifurcated nature of configuration space around the flat state. Here, a simple mechanism is introduced to achieve robust self‐folding of microscale origami by separating actuation into two discrete steps using different thermally responsive hydrogels. First, the vertices are pre‐biased to move in the desired direction from the flat state by selectively swelling one of the two hydrogels at high temperature. Subsequently, the creases are folded toward their target angles by activating swelling of the second hydrogel upon cooling to room temperature. Since each vertex can be individually programmed to move upward or downward, it is possible to robustly select the desired branch even in multi‐vertex structures with reasonably high complexity. This strategy provides key new principles for designing shaping‐morphing materials that avoid undesired distractor states, expanding their potential applications in areas such as soft robotics, sensors, mechanical metamaterials, and deployable devices. 

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4. Kirigami-Based Deployable Transcrease Hard Stop Models Usable in Origami Patterns

   https://doi.org/10.1115/DETC2019-98056  

   Andrews, David W.; Avila, Alex; Butler, Jared; Magleby, Spencer P.; Howell, Larry L. (January 2019, Proceedings of the 2019 International Design Engineering Conference)

   Abstract Stopping origami in arbitrary fold states can present a challenge for origami-based design. In this paper two categories of kirigami-based models are presented for stopping the fold motion of individual creases using deployable hard stops. These models are transcrease (across a crease) and deploy from a flat sheet. The first category is planar and has behavior similar to a four-bar linkage. The second category is spherical and behaves like a degree-4 origami vertex. These models are based on the zero-thickness assumption of paper and can be applied to origami patterns made from thin materials, limiting the motion of the base origami pattern through self-interference within the original facets. Model parameters are based on a desired fold or dihedral angle, as well as facet dimensions. Examples show model benefits and limitations. 

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5. Implementation of an Origami Dynamics Model Based on the Absolute Nodal Coordinate Formulation (ANCF)

   https://doi.org/10.1115/SMASIS2022-89315  

   Tao, Jiayue; Li, Suyi (September 2022, ASME 2022 Conference on Smart Materials, Adaptive Structures and Intelligent Systems)

   Origami — the ancient art of paper folding — has been widely adopted as a design and fabrication framework for many engineering applications, including multi-functional structures, deployable spacecraft, and architected materials. These applications typically involve complex and dynamic deformations in the origami facets, necessitating high-fidelity models to better simulate folding-induced mechanics and dynamics. This paper presents the formulation and validation of such a new model based on the Absolute Nodal Coordinate Formulation (ANCF), which exploits the tessellated nature of origami and describes it as an assembly of flexible panels rotating around springy creases. To estimate the crease folding, we mathematically formulate a “torsional spring connector” in the framework of ANCF and apply it to the crease nodes, where the facets meshed by ANCF plate elements are interconnected. We simulate the dynamic folding of a Miura-ori unit cell and compare the results with commercial finite element software (ABAQUS) to validate the modeling accuracy. The ANCF model can converge using significantly fewer elements than ABAQUS without sacrificing accuracy. Therefore, this high-fidelity model can help deepen our knowledge of folding-induced mechanics and dynamics, broadening the appeals of origami in science and engineering. 

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