Research Papers

Kinematics of Origami Structures With Smooth Folds OPEN ACCESS

[+] Author and Article Information
Edwin A. Peraza Hernandez

Graduate Research Assistant
Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: eperaza@tamu.edu

Darren J. Hartl

Research Assistant Professor
Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: darren.hartl@tamu.edu

Dimitris C. Lagoudas

Department of Aerospace Engineering,
Texas A&M University,
College Station, TX 77843;
Department of Materials
Science and Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: lagoudas@tamu.edu

1Corresponding author.

Manuscript received April 24, 2016; final manuscript received July 13, 2016; published online October 11, 2016. Assoc. Editor: Robert J. Wood.

J. Mechanisms Robotics 8(6), 061019 (Oct 11, 2016) (22 pages) Paper No: JMR-16-1115; doi: 10.1115/1.4034299 History: Received April 24, 2016; Revised July 13, 2016

Origami provides both inspiration and potential solutions to the fabrication, assembly, and functionality of various structures and devices. Kinematic modeling of origami-based objects is essential to their analysis and design. Models for rigid origami, in which all planar faces of the sheet are rigid and folds are limited to straight creases having only zeroth-order geometric continuity, are available in the literature. Many of these models include constraints on the fold angles to ensure that any initially closed strip of faces is not torn during folding. However, these previous models are not intended for structures with non-negligible fold thickness or with maximum curvature at the folds restricted by material or structural limitations. Thus, for general structures, creased folds of merely zeroth-order geometric continuity are not appropriate idealizations of structural response, and a new approach is needed. In this work, a novel model analogous to those for rigid origami with creased folds is presented for sheets having realistic folds of nonzero surface area and exhibiting higher-order geometric continuity, here termed smooth folds. The geometry of smooth folds and constraints on their associated shape variables are presented. A numerical implementation of the model allowing for kinematic simulation of sheets having arbitrary fold patterns is also described. Simulation results are provided showing the capability of the model to capture realistic kinematic response of origami sheets with diverse fold patterns.

Until recently, the term origami has been associated primarily with the ancient art of folding paper [1]. In origami, a goal shape is achieved from an initially planar sheet exclusively through folding. In this context, a fold is any deformation of the sheet in which the in-surface distance between any two points in the sheet is constant and the sheet does not self-intersect [2,3].

Engineering advantages of origami-inspired structures and devices include compact deployment/storage capability [4], a reduction in manufacturing complexity [5,6], and the potential for reconfigurability [7,8]. Existing and potential applications of origami solutions to device and structural design problems include deployable structures for space exploration [912], electronic components with improved properties [1315], robotic components [16,17], foldable wings [18], cellular materials [19], metamaterials [2023], and shelters [24,25], among others [2628].

Rigid origami, the special case of origami for which the planar faces of the sheet are inflexible [29,30] has been studied in the past and remains an active subject [31]. Rigid origami has been utilized for the design of deployable structures and architectural constructions [30,3234]. Theoretical modeling and simulation of rigid origami structures permit understanding of their kinematic behavior and the development of computational tools for their design. Rigid origami has been modeled using diverse approaches [29,35]. For example, Belcastro and Hull [36,37] presented a model for rigid origami derived by representing the deformation associated with folding using affine transformations. Their model provides constraints on the fold angles allowing for valid rigid origami configurations as well as mappings between unfolded and folded configurations. Tachi developed the Rigid Origami Simulator [29,38] for the simulation of rigid origami that also considered a set of constraints on the fold angles analogous to those presented in Refs. [36,37]. Using a similar approach, Tachi also developed Freeform Origami [39] for the simulation and design of freeform rigid origami structures represented as triangulated meshes [40].

Alternatively, truss representations [41] have been used wherein the polygonal faces of the sheet are triangulated, each fold or boundary edge end-point is represented by a truss joint, and each fold and boundary edge is represented by a truss member. The configurations for which the displacements of the truss joints do not cause elongations of the truss members represent valid rigid origami configurations. Additional constraints that allow the triangulated polygonal faces to remain planar are also considered for these models.

The majority of origami modeling approaches and design tools to date are based on the assumption of creased folds (see Fig. 1(a) for an example) that are straight line segments in the sheet that, upon folding deformation, the sheet has zeroth-order geometric continuity (G0) at such lines (i.e., the sheet tangent plane may be discontinuous at these folds). Curved creased folds are also a