Amplifying the Kinematics of Origami Mechanisms With Spring Joints

TL;DR

This work addresses the limitation of a single-parameter reverse fold by introducing the spring joint, a programmable, pleat-based mechanism that couples multiple reverse folds to amplify motion. The approach is grounded in the isotropic reverse-fold relation $\phi(\xi, \phi_0) = 2\arctan\left(\cos\frac{\xi}{2}\tan\frac{\phi_0}{2}\right)$, and extended via compound forms such as $\phi^{dual}_0 = \phi_0^0 - \pi + \phi_0^1$ and $\phi^{compound}_0 = -\pi (n - 1) + \sum_{k=0}^n \phi_0^k$, enabling programmable path control. The paper also introduces a minimal-layer modular variant and tilted-axis designs, and demonstrates Miura Ori substitution with $\phi_0^{compound} = \frac{5\pi}{6}$, illustrating increased motion while maintaining rigidity. Overall, the spring joint offers a high-velocity, programmable alternative to traditional reverse folds, with potential for thick-material origami deployables and CAD-assisted generalizations for broader engineering applications.

Abstract

Due to its rigid foldability and predictable kinematics, the reverse fold is the fundamental mechanism behind some of the most well known origami kinematic structures, including the Miura Ori, Yoshimura, and waterbomb patterns. However, the reverse fold only has one parameter to control its behavior: the starting fold angle. In this paper I introduce an alternative to the traditional reverse fold, based on the spring into action pattern, called the spring joint. This novel rigidly foldable mechanism is able to couple multiple reverse folds into a compact space to amplify the kinematic output of a traditional reverse fold by up to ten times, and to add one parameter for each reverse fold, giving more programmatic control of origami structures. Methods of parameterizing both the starting angle, the path of travel, and the axis of motion are also introduced. Unfortunately, this versatility comes at the cost of a large buildup of layers, making the spring joint impractical for thick origami mechanisms. To solve this problem, I also introduce a modular alternative to the spring joint that has no additional layers, with the same kinematic properties. Both of these mechanisms are tested as replacements for the reverse fold in both traditional and custom origami structures.

Figures (10)

• Figure 1: An illustration of the spring joint mechanism when folded from a square of paper
• Figure 2: (a) Reverse fold crease pattern (b) Reverse fold folded state (c) Graph of the fold angle $\phi$ as a function of the dihedral angle $\xi$ for different values of the starting fold angle.
• Figure 3: (a) Compound reverse fold crease pattern with four reverse folds and a starting fold angle of zero. The starting fold angles of the constituent reverse folds $\phi_0$ are shown $\phi_0$. (b) Compressed crease pattern of the compound reverse fold, known as a spring joint. (c) Folded spring joint (d) Graph of the fold angle $\phi$ of this spring joint. A dotted line is drawn to indicate the hard stop of the paper hitting itself
• Figure 4: A series of reverse folds, when folded with an axis tilt (and flat folded based on the Kawasaki-Justin theorem), will spread out the pleats on the opposite side of the reverse fold.
• Figure 5: (a) An example tilted spring joint crease pattern and folded state with starting fold angle $\phi^{tilt}_0=0$ (b) Physical prototype where the mechanism actuates along a different axis.