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 allowed in origami (see Refs. [4244]); nevertheless, the focus of this work is on rigid origami-based structures for which curved folds are not allowed because their folding deformation induces bending of the faces joined to such folds [43].

The idealization of physically folded structures as sheets having creased folds has been useful in the modeling and design of several origami-inspired structures in the past [20,23,32,45,46]. However, this simplification may not be appropriate for structures having non-negligible fold thickness or constructed from materials that do not provide sufficient strain magnitudes to generate the high curvatures required for a creased idealization. For these structures, the obtained folded regions may not be accurately represented as creases but rather as bent sheet regions having higher-order geometric continuity. These folded regions are referred to in this work as smooth folds (an example of a sheet with smooth folds is shown in Fig. 1(b)).

There have been past efforts to model and analyze surfaces that exhibit bent and creased folding. Origami-based bent and creased surfaces have been simulated using collections of developable surface subdomains [4751]. Such advancements allow for the realistic animation and rendering of curved folds and combinations of creases and bent regions. However, none of the aforementioned works [4751] have considered constraints on the geometry and deformation of the bent folded regions that are required to preserve rigid faces as in analogous rigid origami models, which are essential when fold intersections or holes are present in the sheet. In view of this, a novel model for the kinematic response of rigid origami-based structures having smooth folds is presented in this work. The proposed model for origami with smooth folds includes rigid origami with creased folds as a special case, hence being more general.

Modeling of origami-based morphing of plate structures having significant thickness at the fold regions requires the arbitrary order of continuity of smooth folds. The present model is also useful in the kinematic analysis of sheets folded via active material actuation, where the achievable curvature at the folds is limited by the maximum strain magnitude provided by such active materials [27]. Examples of these active material-based origami structures include liquid-crystal elastomer [52,53], shape memory alloy [5457], dielectric elastomer [58,59], and optically responsive polymeric self-folding sheets [6062], among others [27,63]. The different assumptions regarding strain distributions at the fold regions associated with the implementation of these various materials require the arbitrary order of continuity considered in this work.

The basic origami concepts and a review of the model for rigid origami with creased folds extended in this work are presented in Sec. 2. Section 3 presents the newly proposed model for origami with smooth folds. It includes the geometric description of smooth folds and the fold pattern, constraints required for valid configurations, and the numerical implementation of the model allowing for kinematic simulation of sheets having arbitrary patterns of smooth folds. Section 4 presents the simulation results of sheets with diverse fold patterns that demonstrate the model capabilities and the resulting realistic kinematic structural response captured by the model. Finally, Sec. 5 provides a summarizing discussion and concludes the paper.

Concepts and Review of Origami With Creased Folds