Rigid Foldability and Mountain-Valley Crease Assignments of Square-Twist Origami Pattern

TL;DR

This paper investigates the rigid foldability of square-twist origami patterns with various mountain-valley crease assignments using kinematic analysis, identifying conditions for foldability and proposing methods to modify non-rigid patterns.

Contribution

It provides a detailed kinematic analysis of the rigid foldability of square-twist origami, deriving explicit equations and introducing a crease-addition method to convert non-rigid patterns.

Findings

Two of four analyzed patterns are rigidly foldable.

Explicit kinematic equations for rigid cases are derived.

A crease-addition method can convert non-rigid patterns to rigid ones.

Abstract

Rigid foldability allows an origami pattern to fold about crease lines without twisting or stretching component panels. It enables folding of rigid materials, facilitating the design of foldable structures. Recent study shows that rigid foldability is affected by the mountain-valley crease (M-V) assignment of an origami pattern. In this paper, we investigate the rigid foldability of the square-twist origami pattern with diverse M-V assignments by a kinematic method based on the motion transmission path. Four types of square-twist origami patterns are analyzed, among which two are found rigidly foldable, while the other two are not. The explicit kinematic equations of the rigid cases are derived based on the kinematic equivalence between the rigid origami pattern and the closed-loop network of spherical 4R linkages. We also propose a crease-addition method to convert the rigid foldability of the non-rigid patterns. The motion compatibility conditions of the modified patterns are checked, which verify the rigid foldability of the modified patterns. The kinematic analysis reveals the bifurcation behaviour of the modified patterns. This work not only helps to deepen our understanding on the rigid foldability of origami patterns and its relationship with the M-V assignments, but also provides us an effective way to create more rigidly foldable origami patterns from non rigid ones.