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  1. 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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  2. Self-folding origami, structures that are engineered flat to fold into targeted, three-dimensional shapes, have many potential engineering applications. Though significant effort in recent years has been devoted to designing fold patterns that can achieve a variety of target shapes, recent work has also made clear that many origami structures exhibit multiple folding pathways, with a proliferation of geometric folding pathways as the origami structure becomes complex. The competition between these pathways can lead to structures that are programmed for one shape, yet fold incorrectly. To disentangle the features that lead to misfolding, we introduce a model of self-folding origami that accounts for the finite stretching rigidity of the origami faces and allows the computation of energy landscapes that lead to misfolding. We find that, in addition to the geometrical features of the origami, the finite elasticity of the nearly-flat origami configurations regulates the proliferation of potential misfolded states through a series of saddle-node bifurcations. We apply our model to one of the most common origami motifs, the symmetric “bird's foot,” a single vertex with four folds. We show that though even a small error in programmed fold angles induces metastability in rigid origami, elasticity allows one to tune resilience to misfolding. In a more complex design, the “Randlett flapping bird,” which has thousands of potential competing states, we further show that the number of actual observed minima is strongly determined by the structure's elasticity. In general, we show that elastic origami with both stiffer folds and less bendable faces self-folds better. 
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  3. 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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