Origami structures based on rigidly foldable tessellation patterns
Keywords: rigid origami
kinematics
flat-foldable origami
origami tessellation
mechanical linkage

This dissertation details the geometry and kinematics of novel origami tessellation structures designed to possess desirable properties found in commonly studied classical origami tessellations. Research on these novel origami models consists of geometric and kinematic analysis of their crease patterns and partially folded states. Each proposed origami tessellation is flat-foldable and rigidly foldable. Multiple sheets of origami tessellations assemble to form stacked origami structures that are also rigidly foldable. The research conducted in this dissertation produced the following main achievements.
First, an investigation into the geometry and kinematics of 4C origami tessellations reveals many different stacked origami structures. The degree-4 single-vertex origami model, 4C, is the simplest unit in any 4C origami tessellation. A composite model called the Double 4C demonstrates the mechanisms that emerge when stacking two flat-foldable 4Cs. A thick-panel version of the Double 4C shows it can accommodate panel thickness. The matrix method determines the kinematics of 4C origami tessellations, which reveals some to be flat-foldable and rigidly foldable. Geometric proofs show that certain 4C origami tessellations are crease-stackable, in which sheets of the origami tessellation can stack along designated sets of creases to form stacked origami structures. Crease stacking, panel stacking, and weaving are three methods to stack 4C origami tessellations, and each method yields a distinct stacked origami structure.
Second, the origami claw tessellation, OCT, presents a novel flat-foldable, rigidly foldable, and crease-stackable origami tessellation. By combining 4Cs with 6Cs, called claw units, the OCT possesses multiple DOFs. The matrix method determines the kinematics of the 4Cs and claw units. Geometric proofs show the OCT is crease-stackable by examining the geometry of the 4Cs and claw units. The numerical method computes the exact number of DOFs in the OCT. Stacking sheets of the OCT results in a stacked structure with only one DOF. Families of OCTs are defined as sets of OCTs such that any two in the same family are crease-stackable with each other, resulting in self-locking stacked OCTs.
Third, the modular origami horn chain, MOHC, and the modular origami claw chain, MOCC, are proposed. Both are flat-foldable, rigidly foldable, and crease-stackable. These origami tessellations match the shape of a non-intersecting planar curve in a specified partially folded state. A proposed algorithm converts a planar curve into a piecewise-linear curve, which is converted into the crease pattern of a MOHC/MOCC. Stacking these origami tessellations results in the stacked MOHC/MOCC. The concept of families of MOHCs and MOCCs is used to assemble self-locking stacked origami chains.
For each of these proposed origami tessellations, physical models are constructed to demonstrate their kinematic properties. CAD software is used to assemble stacked origami structures. The research in this dissertation has the potential to unfold a new generation of origami tessellations and their stacked structures across engineering fields.