Designing Developable Mechanisms From Flat Patterns

This paper presents tools and methods to design cylindrical and conical developable mechanisms from flat, planar patterns. Equations are presented that relate the link lengths and link angles of planar and spherical mechanisms to the dimensions in a flat configuration. These flat patterns can then be formed into curved, developable mechanisms. Guidelines are established to determine if a mechanism described by a flat pattern can exhibit intramobile or extramobile behavior. A developable mechanism can only potentially exhibit intramobile or extramobile behavior if none of the links extend beyond half of the flat pattern. The behavior of a mechanism can change depending on the location of the cut of the flat pattern. Different joint designs are discussed including lamina emergent torsional (LET) joints. Physical examples are presented.

Rigid and Flat Foldability of a Degree-Four Vertex in Origami

Rigid foldability is the property of an origami that folds continuously from an unfolded to a folded state without deformation in its facets. Although extensively researched, there exist no intrinsic conditions for the rigid foldability of a degree-four vertex, which is the simplest possible origami building block that folds nontrivially. In this paper, we derive a necessary and sufficient condition for the rigid foldability of a degree-four vertex and show that it can be reduced to a purely sufficient condition, which is equivalent to a known condition from the realm of spherical mechanisms. The implications of these conditions are discussed, which reveals the connection between rigid and flat foldability, the two most important mathematical notions in origami. In practice, this work further contributes to the design synthesis and analysis of deployable structures, in which the mechanics of degree-four vertices is omnipresent.

Special Unitary Matrices in the Analysis and Synthesis of Spherical Linkages

This work seeks to systematically model and solve the equations associated with the kinematics of spherical mechanisms. The group of special unitary matrices, SU(2), is utilized throughout. Elements of SU(2) are employed here to analyze the three-roll wrist and the spherical Watt I linkage. Additionally, the five orientation synthesis of a spherical four-bar mechanism is solved, and solutions are found for the eight orientation synthesis of the Watt I linkage. Using SU(2) readily allows for the use of a homotopy-continuation-based solver, in this case Bertini. The use of Bertini is motivated by its capacity to calculate every solution to a design problem.

Membrane-Enhanced Lamina Emergent Torsional Joints for Surrogate Folds

Lamina emergent compliant mechanisms (including origami-adapted compliant mechanisms) are mechanical devices that can be fabricated from a planar material (a lamina) and have motion that emerges out of the fabrication plane. Lamina emergent compliant mechanisms often exhibit undesirable parasitic motions due to the planar fabrication constraint. This work introduces a type of lamina emergent torsion (LET) joint that reduces parasitic motions of lamina emergent mechanisms, and presents equations for modeling parasitic motion of LET joints. The membrane joint also makes possible one-way joints that can ensure origami-based mechanisms emerge from their flat state (a change point) into the desired configuration. Membrane-enhanced LET (M-LET) joints, including one-way surrogate folds, are described here and show promise for use in a wide range of compliant mechanisms and origami-based compliant mechanisms. They are demonstrated as individual joints and in mechanisms such as a kaleidocycle (a 6R Bricard linkage), degree-4 origami vertices (spherical mechanisms), and waterbomb base mechanisms (an 8R multi-degrees-of-freedom origami-based mechanism).

Design of a Bistable Origami Reverse-Fold Using Spherical Kinematics

This paper presents a new design concept for bistability that can be implemented as a reverse-fold origami mechanism or as a spherical four-bar mechanism. The design is based on the conceptual overlap between a certain simple class of origami mechanisms (the reverse-fold) and a class of spherical change-point mechanisms. Using both a partially compliant spherical mechanism and a piece of origami made with two sheets of paper, we implement the design concept for bistable behavior. The design concept consists in adapting planar two position synthesis to spherical mechanisms and in using a formal analogy between spherical mechanisms and certain simple origami folds. The dimensional synthesis of these two mechanisms is performed using parametric CAD. The design concept was successfully prototyped both as origami and as a partially compliant spherical mechanism.