Branches of Triangulated Origami Near the Unfolded State

Abstract

Origami structures are characterized by a network of folds and vertices joining unbendable plates. For applications to mechanical design and self-folding structures, it is essential to understand the interplay between the set of folds in the unfolded origami and the possible 3D folded configurations. When deforming a structure that has been folded, one can often linearize the geometric constraints, but the degeneracy of the unfolded state makes a linear approach impossible there. We derive a theory for the second-order infinitesimal rigidity of an initially unfolded triangulated origami structure and use it to study the set of nearly unfolded configurations of origami with four boundary vertices. We find that locally, this set consists of a number of distinct "branches" which intersect at the unfolded state, and that the number of these branches is exponential in the number of vertices. We find numerical and analytical evidence that suggests that the branches are characterized by choosing each internal vertex to either "pop up" or "pop down." The large number of pathways along which one can fold an initially unfolded origami structure strongly indicates that a generic structure is likely to become trapped in a "misfolded" state. Thus, new techniques for creating self-folding origami are likely necessary; controlling the popping state of the vertices may be one possibility.

Shaping the topology of folding pathways in mechanical systems

“…Self-folding sheets (or self-folding origami) are structures programmed to have one unique low- or zero-energy mode 6 , 29 , 36 . However, self-folding sheets, even when programmed with a single zero-energy mode, have been shown to have exponentially many undesirable misfolding modes accessible from the flat state 14 , 15 . We show how crease stiffness can change the topological connectivity of these modes and leave only the desired folding mode accessible from the flat state.…”

“…When designed, multiple pathways and multistability can be exploited to create mechanical switches, shapeable sheets, and many other metamaterials 2 , 6 – 12 . However, such nonlinear features can also create problems 10 , 13 – 15 . For example, self-folding origami, despite the name, has an exponential number of misfolding pathways that meet at a “branch point” at the flat state 13 , 16 – 19 , making it nearly impossible to fold into the desired folding mode 14 , 15 , 20 , 21 .…”

“…However, such nonlinear features can also create problems 10 , 13 – 15 . For example, self-folding origami, despite the name, has an exponential number of misfolding pathways that meet at a “branch point” at the flat state 13 , 16 – 19 , making it nearly impossible to fold into the desired folding mode 14 , 15 , 20 , 21 . Similar “branch points” in mechanical linkages pose challenges in robotics and other applications 22 – 24 .…”

Jigsaw puzzle design of pluripotent origami

“…Such pursuits typically focus on rigid origami, which concerns perfectly stiff plates connected by flexible hinges that are agnostic as to their mountain-valley (MV) assignment. The absence of a presupposed MV pattern opens up the possibility of pluripotent origami-crease patterns that can fold into multiple 3D target shapes [16][17][18][19] . However, the assumption of rigid plates leads to complex compatibility conditions that make designing foldable patterns notoriously difficult.…”

Transformation Dynamics in Origami

“…Once folded, an ideally rigid origami mechanism is constrained to a limited set of trajectories, but when the mechanism is in the flat-unfolded state, it is in a kinematic singularity. From this point, the mechanism can bifurcate into two or more configurations, each with a distinct shape and kinematic behavior, by changing the directions of its folds [referred to collectively as the mountain-valley (MV) assignment] [15]. In physical specimens, the crease pattern is usually fixed during fabrication, but the MV assignment can be changed, presenting a practical approach to transformation.…”