Rigidity control of general origami structures

Origami, the traditional paper-folding art, has inspired the modern design of numerous flexible structures in science and engineering. In particular, origami structures with different physical properties have been studied and utilized for various applications. More recently, several deterministic and stochastic approaches have been developed for controlling the rigidity or softness of the Miura-ori structures. However, the rigidity control of other origami structures is much less understood. In this work, we study the rigidity control of general origami structures via enforcing or relaxing the planarity condition of their polygonal facets. Specifically, by performing numerical simulations on a large variety of origami structures with different facet selection rules, we systematically analyze how the geometry and topology of different origami structures affect their degrees of freedom (DOF). We also propose a hypergeometric model based on the selection process to derive theoretical bounds for the probabilistic properties of the rigidity change, which allows us to identify key origami structural variables that theoretically govern the DOF evolution and thereby the critical rigidity percolation transition in general origami structures. Moreover, we develop a simple unified model that describes the relationship between the critical percolation density, the origami facet geometry, and the facet selection rules, which enables efficient prediction of the critical transition density for high-resolution origami structures. Altogether, our work highlights the intricate similarities and differences in the rigidity control of general origami structures, shedding light on the design of flexible mechanical metamaterials for practical applications.