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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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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.
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- PAR ID:
- 10209599
- Date Published:
- Journal Name:
- Journal of computational geometry
- Volume:
- 11
- Issue:
- 1
- ISSN:
- 1920-180X
- Page Range / eLocation ID:
- 93–124
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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