Origami has served as the inspiration for a number of engineered systems. In most cases, they require nonpaper materials where material thickness is non-negligible. Foldable mechanisms based on origami-like forms present special challenges for preserving kinematics and assuring non-self-intersection when the thickness of the panels must be accommodated. Several design approaches for constructing thick origami mechanisms by beginning with a zero-thickness origami pattern and transforming it into a rigidly foldable mechanism with thick panels are reviewed. The review includes existing approaches and introduces new hybrid approaches. The approaches are compared and contrasted and their manufacturability analyzed.

## Introduction

The Japanese art of paper-folding, now called origami, is hundreds of years old in Japan [1]. As a decorative art form, it would seem to have little to do with the field of mechanical engineering. However, as origami became a worldwide activity during the latter part of the 20th century, a rich cross-fertilization of origami concepts and applications arose between the worlds of art and technology. Connections between origami artists advanced the complexity and realism of the art, but more significantly, there arose connections between origami and mathematics, sciences, and engineering, which have led to the adoption of origami structures and mechanisms in a wide range of technical fields [25]. Structures and mechanisms, both adapted and inspired from origami, have found applications as diverse as acoustics [68], active structures [9], air bags [1013], batteries [1417], deployable shelters [1823], DNA [2428], energy absorption [2932], foldcore sandwich panels [3336], medical devices [3742], MEMS devices [4348], robots [4953], and space applications [5462].

Several circumstances can motivate the application of origami to engineered systems:

• A three-dimensional (3D) shape is to be fabricated from sheet stock and there are significant penalties for cutting and gluing.

• A shape needs to transform between a large, flat, sheet-like state (often called the “deployed” state), and a much smaller state (the “stowed” state).

• A shape that is not flat, but is composed primarily of sheet-like panels, needs to fold into a stowed configuration.

• The motion or kinematic behavior of an existing origami pattern is desired for the application.

While traditional origami is made from paper, engineering applications of origami generally use a wide range of materials, including polymers [6365], metals [38,66], composites [33,34], silicon [67], wood, etc. The size range of origami-inspired engineering forms can run from microscopic [43,68,69] to building size [18,70]. Both sizes and materials used in applications present design challenges that have little or no parallel in the traditional art.

The use of paper in origami leads to some specific behaviors:

1. (1)

The medium (paper) is sufficiently thin that its thickness can usually be ignored in design algorithms [7175].

2. (2)

The elasticity of the medium is negligible; the fold pattern can be treated as a perfectly inelastic surface.

3. (3)

There is a strong distinction between folds and facets; folds and vertices can be treated as mathematically idealized lines and points.

4. (4)

Folding motions can readily include motions not along folds: bending of facets, blurring of vertices, and continuous movements of creases through the paper.

In addition, origami usually adheres to a fairly small set of initial shapes and construction rules: most origami forms are formed from single sheets, usually square, and cutting and gluing are generally eschewed.

In origami-inspired engineered systems, few, if any, of these conditions apply. Instead, a host of issues may require consideration:

Stress/strain: Facets deform and consequently have strain energy. Folds are not entirely free, as they also store strain energy. Interactions among strained elements can give behavior at odds with that of a simple/idealized model. Compliance can be introduced intentionally. Effects such as bistability and snap-through can be present, as parasitic effects or as desired behavior.

Curved folds: While some origami has intentionally made use of curved folds [7684], most origami, and most mathematical models of origami have assumed straight folds. Curved folds induce curvature in the facets to either side of the curved fold, thus making considerations of stress and strain unavoidable.

Distributed folds: Idealized models of origami treat folds as linear features. However, in both paper origami and engineering forms, each fold extends over a finite area. Explicitly incorporating distributed folds complicates the design and modeling of origami-like forms [85].

Kinematics of motion: In origami, the flexibility and toughness of the paper allows for a wide range of manipulations in transforming the form from one state to the other; indeed, many of these manipulations, e.g., so-called “closed sinks” [86,87], are mathematically impossible to do with a finite number of fixed folds, but are routinely incorporated into designs. In origami-inspired engineered systems, however, the (generally) much higher stiffness of the material making up facets can strongly limit the possible folding motions. Thus, considerations of rigid foldability (folding with rigid/nondeforming facets), degrees-of-freedom (DOFs), bifurcations of motions, and the like play a major role.

Dynamic behavior: Considerations must be made for time-dependent behaviors such as resonances, propagation of waves within a folded structure, and effects of inertia and stiffness on motion.

External forces: Gravity must be considered when working with large origami-inspired engineered systems. Friction and van der Waals forces must be considered when working on the microscale.

Nonflat/non-Euclidean surfaces: While origami is typically created from a single flat sheet (a developable surface), origami-inspired engineered systems can be nondevelopable, e.g., eggbox pattern [8890], tubes [9195], multisheet polyhedra [92,9698], and more. In addition, one can consider folding curved surfaces, such as sections of spheres [99] or hyperbolic surfaces [100,101]; indeed, such surfaces can give new behaviors (both parasitic and intended) that are not present in their developable surface analogs.

Significant cutting: While origami has little or no cutting, in the engineering world, substantial advantages can accrue by combining cutting, gluing, and folding. The more the origami-inspired engineered systems depart from the restrictions of traditional origami, the less they are bound to the defining characteristics of origami and their specialized techniques for design and modeling.

Thickness: Creation of folded structures from materials of significant thickness creates design complications in many ways, not least being the problem of self-avoidance, also called injectivity [102] or more prosaically, non-self-intersection [72,103].

In recent decades, the interest in folded structures and mechanisms has grown dramatically; national funding agencies in several countries have underwritten explorations of folding, and any one of the topics above could well warrant a review paper. The problem of thickness, though, seems to occupy a special place within the world of origami design and modeling and it cuts across many of the topics above. Thick mechanisms are typically rigid: thus, kinematics and rigid foldability almost immediately play a role. While many origami mechanisms can be modeled as simple combinations of spherical mechanisms [104], thick materials (usually) break that simplicity. Thick materials have volume and cannot self-intersect: thus, mechanisms made from thick materials introduce problems of motion planning, which can complicate other design considerations.

In this paper, we carry out a review of the many techniques that have been developed to address the challenges of folding mechanisms where the thickness of the facet materials is non-negligible in the mechanism design. While thick folding mechanisms have been demonstrated and used for decades [96,105107], the first two decades of the millennium have been especially productive for their study; several research groups have developed distinct techniques for handling thickness. We will review and summarize various approaches, including some from our own research group, and will add to this collection with new hybrid techniques, presented for the first time here.

The term “thick origami” is a bit ambiguous in an engineering context, since we are not referring primarily to the Japanese art. We will continue to use the word origami in a somewhat metaphorical sense, to refer to any structure or mechanism composed primarily of relatively stiff sheet-like surfaces, in which folding along compliant linear regions plays a major role in its creation or usage.

One more somewhat technical requirement will be added for purposes of this paper. The definition of the preceding paragraph is a broad one, and there are many engineering forms that might qualify by that definition. Even an ordinary book folds when you open and close it; pleated fans and many other simple shapes similarly fold, too. We will restrict our discussion of origami-inspired engineered systems to include only those shapes in which there is at least one interior vertex in the fold pattern: a point at which multiple fold lines come together and the point is completely surrounded by the relatively stiff surfaces. As we will discuss, the presence of interior vertices plays a dramatic role in the kinematics of a folded mechanism, and, ultimately, in its overall design.

## Preliminaries

Traditional origami is made from thin, flexible paper, but when it is translated into engineering applications, the facets of the crease patterns typically become rigid panels, while the folds become relatively compliant hinges. The entire structure can be modeled as a collection of rigid plates joined by revolute joints. In such cases, the origami crease pattern forms an abstraction of the engineering mechanism.

While only a subset of origami patterns are rigidly foldable, this characteristic is of particular importance in thick origami as the stiffness of a panel is directly related to its thickness. If the origami form can flex while all facets remain flat, then the pattern is said to be rigidly foldable. Much work has gone into determining rigid foldability and creating rigidly foldable patterns [108112]. While it is possible to create origami mechanisms derived from non-rigid-foldable patterns [113115], all of the design techniques discussed in this paper apply specifically to rigidly foldable patterns. To further narrow the scope of this review, we will focus on straight folds and flat Euclidean surfaces.

If a collection of panels is connected in a simple linear chain, then each joint can move independently, and, apart from constraints due to self-intersection avoidance, each joint can rotate independently of the others. However, if the panels are connected in such a way as to create a loop of connections, then there are constraints of consistency around each loop. Each interior vertex of the underlying crease pattern creates a loop constraint that must be satisfied. Origami mechanisms can be extremely complex, with tens or hundreds of revolute joints, but it is usually desirable that the mechanisms have a small but positive number of degrees-of-freedom in their motion. The number of degrees-of-freedom of the complete mechanism is determined by the interplay between the number of revolute joints and constraints created by the various loop conditions around each vertex (and/or holes, if the pattern contains them) [116118].

For paper origami patterns, to a good approximation, the axes of revolution of the folds intersect at the vertices, making each vertex a spherical mechanism [95,104,119121] and the entire origami pattern a collection of linked spherical mechanisms.

### Degree-4 Vertices.

The simplest nontrivial origami mechanism is a single vertex with four folds around it: a so-called degree-4 vertex (D4V). If the fold axes intersect at the vertex, then it is a single-DOF mechanism, and shows up as a building block of many rigidly foldable origami forms. Flexing one fold forces the other three to actuate with a one-to-one relationship between fold angles—at least, for a generic vertex (no special angles).

A generic D4V is shown in Fig. 1. The angles between consecutive folds are sector angles; the angles between consecutive facets are dihedral angles. Dihedral angles are measured between the facets, but in the analysis of origami structures, it is usually more convenient to use fold angles, which are the angles between the surface normals. The fold angle is the deviation from straightness, and lies in the range $[−π,π]$. A fold angle $γ∈(0,π]$ is a valley fold; a fold angle $γ∈[−π,0)$ is a mountain fold; and a fold angle γ = 0 is unfolded.

If all of the fold angles are 0, then we say that the pattern is unfolded; if they are all $±π$, then we say that the pattern is flat folded. For a pattern that can flex from the unfolded to the flat folded state, a configuration with at least some of the angles not $±π$ is an intermediate state.

A vertex (or crease pattern) that can be flat folded is flat foldable. A vertex (or crease pattern) that can be unfolded to a flat sheet is developable. Every flat foldable developable crease pattern must satisfy the Kawasaki–Justin theorem [122126] that, at every interior vertex
$α1−α2+α3−α4+⋯=0$
(1)

Note that it is possible for a vertex to be developable but not flat foldable or flat foldable but not developable. While most origami folds assume developability, many interesting mechanisms exist based on nondevelopable vertices [9193].

A rigidly foldable developable D4V with all four folds flexing simultaneously must have three folds of one type (mountain or valley) and one of the other. If we consider the folds in opposite pairs, e.g., ${γ1,γ3}$ and ${γ2,γ4}$, the pair that are of the same type are the major folds, and the pair of opposite type are the minor folds. In Fig. 1, ${γ2,γ4}$ is the major pair.

The names major and minor reflect an inequality relationship between the pairs; using the Gaussian sphere analysis of Huffman [80], it is straightforward to show that, at any folded state, if ${γ1,γ3}$ are minor, and ${γ2,γ4}$ are major, then for flat-foldable vertices
$|γ1,3|≤|γ2,4|$
(2)
with equality reached only at the unfolded and flat-folded states. For a general vertex
$|γ1|+|γ3|≤|γ2|+|γ4|$
(3)
For a given set of sector angles of a generic vertex, any one fold angle specifies the values of the other three. The general relationships are somewhat complex trigonometric expressions, which may be found in the Appendix of Ref. [127]. However, flat-foldable D4Vs have several highly desirable properties. As noted by Tachi (in slightly different forms) [88,108] and Evans et al. (in this form) [128], for a flat-foldable D4V with major and minor folds as in Fig. 1
$γ1=−γ3, γ2=γ4$
(4)
and
$tan 12γ2 tan 12γ1= sin 12(α1+α2) sin 12(α1−α2)$
(5)

A corollary of this result is that, in a network of D4Vs, the half-angle-tangents of all fold angles are proportional to each other with constants of proportionality that do not change as any one of them is flexed. As long as those proportionality coefficients are nonzero (and noninfinite), any network of flat-foldable developable D4Vs can fold from unfolded to flat (apart from self-intersection considerations). Tachi identified and made use of this property to construct a flexible generalized Miura-ori [108]. Our group also used this property to construct a family of rigidly foldable cut flashers [127].

The case where the two major folds are collinear is special and provides the exception to the one-to-one relationship between the four sector angles. In this case, starting with a flat vertex, the major angle pairs must fold first from unfolded to flat-folded. If the sector angles on either side of the major folds are equal, then the minor pair can then fold.

### Vertices of Higher Degree.

For vertices of higher degree, the relationships between sector and dihedral angles become more complex and are typically analyzed by solving a matrix equation constructed by requiring that the product of rotation matrices about the sector angles and folds give the identity matrix [129].

That is not to say that all degree-4 or higher vertices are flexible. In a recent analysis, Abel et al. established conditions for a single-vertex crease pattern of degree > 2 to be able to fold rigidly [130]. Every rigidly foldable single-vertex pattern must contain a bird's foot: a set of three creases of one fold assignment (e.g., mountain), separated sequentially by angles strictly between 0 and π (possibly with additional creases between them), plus one additional crease of the opposite assignment (e.g., valley).

For a general crease pattern composed of multiple vertices, assuming that each vertex satisfies the bird's-foot condition, the number of degrees-of-freedom was derived by Tachi [109], as
$DOF=B−3H+S−3−∑k≥4(k−3)Pk$
(6)

where B is the number of edges on the boundary, H is the number of holes in the pattern, S is the number of redundant constraints (typically geometry-specific), and Pk is the number of k-gon facets.

If we start with a triangulated crease pattern, the last sum term in Eq. (6) goes away. Joining triangles together into polygons with larger numbers of sides will reduce the flexibility of a pattern; conversely, the greatest flexibility is attained in a fully triangulated pattern.

Note, though, the importance of the boundary term B in what is left. If there are no holes (H = 0) and no special geometry S = 0, the number of degrees-of-freedom is $B−3$, no matter how many triangles might be in the surface. Thus, for example, a generic triangulated surface with four boundary edges will, in general, have 1DOF.

If a triangulated surface with three edges is constructed on the boundary, Eq. (6) gives $DOF=S=0$ for generic (non-special-geometry) configurations, implying rigidity. Indeed, since the polygon composed of those three edges will be a triangle and must necessarily be rigid, adding a triangular facet to that closed hole will give a closed polyhedron. Cauchy's theorem established that a closed convex polyhedron was rigid, and indeed, any generic closed polyhedron, and thus, any generic no-holes crease pattern with three edges, is rigid.

Special geometries can give S = 1 and thus one degree-of-freedom even with B = 3. For polyhedra, Connelly constructed nonconvex closed flexible polyhedra, and it was subsequently shown that all such polyhedra must have constant volume as they flex. See Ref. [72] for a full discussion.

As crease patterns consisting of ever larger numbers of facets are created, unless care is taken to wrap them in ever longer edge polygons to force rigidity, as the size of the array grows, the number of facets on the boundary will grow as well.

For crease patterns composed entirely of triangles, as they scale up to larger arrays with increasing boundary they gain progressively more degrees-of-freedom; informally, they get floppier. If, however, we build patterns from quadrilaterals, the scaling behavior changes dramatically. As they scale up to larger arrays with increasing boundary they lose degrees-of-freedom. Thus, sufficiently large arrays of generic quadrilateral (or larger) facets will be rigid.

The key term here, though, is generic—meaning no special geometric properties. By careful choice of geometry, it is possible to find patterned arrays that scale arbitrarily. In particular, if a pattern is composed of rigidly foldable repeated building blocks where the fold angle consistency is maintained from one block to the next, then arbitrarily large arrays may be created with nonzero degrees-of-freedom. An example of this is the well-known Miura-ori tessellation pattern that has only one degree-of-freedom.

These laws, however, apply to zero-thickness models, in which, for the flat folded state, all layers coincide in a common plane. In real-world designs, the thickness often cannot be neglected. To avoid self-intersection, the panels that correspond to facets must be transversely displaced in the flat-folded state. Even in patterns that never fold fully flat, self-intersection avoidance requires displacement of panels and/or hinges relative to their positions in the zero-thickness model. In particular, with nonzero thickness panels, mountain and valley folds get pushed (ideally) to opposite sides of the panel to avoid self-intersection.

Once hinges are displaced from a common plane, the self-consistency requirements around each vertex jump in complexity, because the hinges around a vertex no longer form spherical mechanisms. Instead of imposing three constraints as one travels around a vertex (pitch, yaw, and roll of the hinge axes), there are now six constraints, adding three of translational symmetry. Accordingly, for origami patterns with nonspherical orientations of the hinges, the challenges of achieving single-degree-of-freedom kinematic mechanisms increase considerably.

There are now several strategies for achieving low-degree-of-freedom kinematic motion with thick-panel origami mechanisms. Some strive to replicate the zero-thickness kinematics with thick panels, some developed within just the past few years. Some identify specific configurations where displaced hinges allow kinematic motion; still others replace simple hinges with more complex configurations (pairs, flexible membranes, and shaped rollers) that allow for the desired displacement. In the sections that follow, we describe each of these approaches, note their strengths and regimes of applicability, and show examples of each, as well as introducing new hybrid approaches that combine elements of them all.

## Tapered Panels Technique

The tapered panels technique for accommodating thickness in origami was an early technique that fully preserved the kinematics of the zero-thickness model [131]. The underlying principle governing this technique is that the same zero-thickness surface from the original origami model still exists within the mechanism and thickness is added around that surface. Hence, another descriptive name for this technique is “embedded zero-thickness surface” [132]. Figure 2 shows the technique applied to a Miura-ori pattern.

How the tapered panels technique is generally applied is illustrated in Fig. 3. As shown by the “thickness added” step, material is added equally to each side of the zero-thickness surface to thicken the panels to the required total thickness. To determine the taper necessary for each panel in the mechanism to avoid intersection with its neighbors, a thickened model can be folded and the volume of the panels trimmed along the bisection of the dihedral angle at each joint. To maintain a nonzero panel thickness, the maximum fold angle of the thick origami mechanism cannot reach $180 deg$.

The length of the taper on the zero-thickness panel, as found in Ref. [131] and detailed in Fig. 4, is given by
$t cot(δ/2)$
(7)

where t is the half thickness of the panel (i.e., the maximum distance from the zero-thickness surface to the outer face of the panel), and δ is the dihedral angle of the embedded zero-thickness facets at the maximum folded state.

There are a few notable characteristics of the tapered panels technique. By adding material to the outside of the vertices in a similarly tapered fashion, it is possible to limit the unfolded state so that the zero-thickness pattern cannot reach its “flat” unfolded position, thereby achieving a potentially desirable nonflat state and avoiding singularities in the motion space that could allow undesirable movements. Also, as the technique cannot accommodate fold angles of $180 deg$ (in the zero-thickness reference), singularities in the flat-folded state can be similarly avoided. However, this property also carries with it two potential inefficiencies. First, in a single-DOF mechanism, the relative spacing of all panel pairs is defined by a single quantity; at the maximally folded state, some panels may be closely spaced, while others may be separated by larger amounts. It is generally not possible to individually tune panel spacings for maximum packing. Second, the packing efficiency of the mechanism generally decreases as the thickness of the panels is increased because the motion must halt farther away from the flat-folded state [131].

## Offset Panel Technique

Edmondson et al. proposed a thickness-accommodation technique that also maintains the same kinematics as the zero-thickness model, but rather than trimming material away from panels to avoid interference, offsets are used to position the panels away from the zero-thickness surface [133,134]. This method, as illustrated in Fig. 5, allows for a fully collapsed position where panels lie parallel to and in contact with each other, and achieves the full range of angular motion of the zero-thickness equivalent surface. The underlying principle behind the offset panel technique is that all the rotational axes lie within the same plane as the zero-thickness model in both the unfolded and flat-folded states, creating a mechanism that is kinematically equivalent to the original origami pattern. This allows the panels themselves to be displaced individually as needed to avoid self-intersection.

This technique is implemented via a series of steps: thickening the origami pattern's panels, stacking the panels in the fully folded position, selecting a zero-thickness reference plane, creating offsets that join the stacked panels to the rotational axis locations of the zero-thickness origami model, and modifying panels to avoid self-intersection [134].

A characteristic of the offset panel technique is that, in the unfolded state, the pattern is no longer planar but contains multiple elevations, as illustrated in Fig. 6. The offset panel technique offers flexibility to the designer in the choice of where to place the zero thickness plane, how to construct the offsets, and allows the thickness of each panel to vary with respect to that of the other panels.

However, the offset panel technique brings with it its own challenges. Those include the stresses that occur at hinges due to long offset lengths, as well as accommodating the additional geometry of the offsets themselves. Because all panels connect to the zero-thickness reference model, cut-outs are also often required in some panels to allow the offsets of other panels to access the zero-thickness reference plane.

Despite the challenges that arise in the offset panel technique, the ability to work with any arbitrary origami pattern and the flexibility offered to use panels of varying thickness make it a viable method for the design of products. Morgan et al. demonstrated the offset panel technique in the development of several potential product applications including a kinetic sculpture, foldable circuit board, electrical engineers toolbox, and foldable table [135]. The same work also demonstrated how the geometry of the panels can be varied not only in thickness, but also in any other dimension, as long as the rotational axes remain in their relative positions and panels do not intersect.

## Hinge Shift Technique

The hinge shift technique as illustrated in Fig. 7 accommodates thickness and avoids interference by shifting the location of the rotational axes away from a single plane. While moving rotational joints to the faces of thick panels can be seen even in 19th century patent literature [136138], complexity with such an approach arises when dealing with interior vertices.

The simplest form of accommodating thickness, though only applicable to single line folds, is to move the joint axis to the valley side of the fold, what Tachi called “axis-shift” [131]. The method of shifting hinges at a thickened interior vertex to allow for folding was demonstrated by Hoberman [96] for a symmetric bird's foot vertex used in the Miura-ori pattern [139]. This thick folding vertex, as shown in Fig. 8, utilizes two levels of thickness (where the thicker portion is simply twice the thickness of the other), with rotational axes placed on each of the faces to achieve a 1DOF motion from a completely flat to completely folded state.

A limitation of this implementation of the hinge shift technique is that only the symmetric bird's foot vertex can be thus accommodated. Despite this limitation, there are many useful origami patterns that make use of the symmetric bird's foot vertex. We show an example in Fig. 9 of this approach applied to the Yoshimura pattern [140], also known as the arc pattern [141], or accordion pattern [19].

Furthermore, by introducing different shifts, other types of vertices can, in fact, be accommodated, as we will presently show.

Trautz and Künstler demonstrated another implementation of shifting hinges, illustrated in Fig. 10. This approach shifts the hinge axes by incorporating translation into the hinges themselves [142]. In this approach, the offset distances are constrained to the thickness of the panels and the mechanism is allowed to fold through the use of sliding hinges. The sliding hinges give the mechanism the added degrees-of-freedom to function, but consequently result in a vertex with several additional degrees-of-freedom. In addition, the accumulated sliding can result in mechanical interferences, therefore requiring a reduced range of motion (ROM) [143].

De Temmerman et al. [20,144] demonstrated an implementation similar to Hoberman's for a degree-6 thickened vertex used in the Yoshimura pattern [140] as shown in Fig. 11. This type of joint utilizes a single thickness layer with all fold axes placed on either the top or bottom face of the panels. While such a joint can be used to create a thick panel folding Yoshimura pattern as shown in Fig. 12, De Temmerman alternatively utilized the joint in a folding bar structure.

More recently, Chen et al. showed an implementation that generalizes hinge shifting to encompass the symmetric bird's foot and Yoshimura-pattern axis-shifted vertices and also includes nonsymmetric thick vertices [145]. Their work made the connection between well-studied 4, 5, and 6R spatial linkages and the hinge-shifted vertices of a thick panel origami pattern.

For a 4R spatial linkage, the only known linkage is the Bennett linkage [146]. While the vertex demonstrated by Hoberman is a Bennett linkage, it is only a particular case of the Bennett linkage. Connecting this thick folding vertex to the Bennett linkage has enabled a greater range of vertex geometries to be accommodated using the hinge shift technique. One example of how this work has expanded the applicability of the hinge shift technique is shown in Fig. 13 as it is applied to a rigidly foldable square twist pattern.

The general case of a spatial 4R linkage being applied to a thick degree-four vertex is shown in Fig. 14. In this model, four sector angles (α1, α2, α3, and α4) and four offset distances (d1, d2, d3, and d4) describe the geometry of the linkage. For the 4R spatial Bennett linkage, opposing link pairs are equal in length and have the same rotational axis twist [147]. The criteria for the mechanism to achieve mobility given in Refs. [145] and [148] are
$α1+α3=α2+α4=π$
(8)

$d1=d3,d2=d4$
(9)

$d1d2= sin(α1) sin(α2)$
(10)

It is interesting to note that the criteria listed in Eq. (8) correspond with the flat foldability criteria given in Eq. (1) which establishes that, for a degree-4 vertex, the hinge shift technique can only be applied to flat foldable patterns.

Chen et al. have also expanded this technique to encompass some higher-order vertices. The 5R Myard linkage [149] gives rise to hinge-shifted degree-5 vertices and the 6R Bricard linkage [150] was utilized for degree-6 vertices [151]. The 5R and 6R linkages implemented in these vertices are single-DOF mechanisms and consequently the thick origami mechanisms are also single-DOF.

## Doubled Hinge Technique

The general principle of the doubled hinge technique is to split fold lines into two fold lines thereby expanding the crease area to allow for the thickness of the panels. A method of splitting fold lines for the purpose of accommodating stacked layers was demonstrated by Hoberman [152] in the modification of a map fold pattern. While Hoberman's method aims to allow panels to fold efficiently, it must be noted that it is a crease modification technique, not a thickness-accommodation technique and would still require a subsequent step utilizing one of the thickness-accommodation techniques discussed in this review paper. Zirbel et al. showed how Hoberman's concept of splitting a vertex in conjunction with the membrane technique can be used in the design of solar array packing [60]. The membrane technique (discussed in Sec. 8) approaches thickness accommodation similarly by widening a crease with flexible material.

Ku and Demaine introduced the offset crease technique for thickness accommodation [153], which is an implementation of the doubled hinge technique where every crease is split in two. They further provided detailed analysis of the desired splitting widths for arbitrary patterns. This technique, as shown in Fig. 15, modifies the zero-thickness model by splitting each fold line into two fold lines. The modified zero-thickness model then remains in the middle of the thick panels and material is trimmed away from the panels to prevent self-intersection similar to the tapered panels technique. An advantage that the offset crease technique has over the tapered panels technique is that each fold line only undergoes folding of $90 deg$ rather than $180 deg$ and consequently less material needs to be trimmed to prevent self-intersection. Mechanisms using this technique also maintain the ability to move from a fully flat state to a fully flat-folded state.

A requirement of this technique is that cutouts be present at the vertices of the pattern in both the flat and folded states to prevent self-intersection during folding. A cutout disconnects the folds at the vertex and, in concert with the doubling of the number of folds around the vertex, adds additional degrees-of-freedom to the mechanism.

A simple degree-4 vertex, such as the one shown in Fig. 16, using the offset crease technique typically has a loop of eight creases, thus two degrees-of-freedom. One degree-of-freedom in the offset crease vertex is the amount by which the mechanism is folded; the other degree-of-freedom is an orthogonal independent degree-of-freedom that allows “wiggliness” or play in the vertex for a given intermediate fold state. It has been shown that a kinematic path exists between the flat and fully folded states generated by this technique [154].

In our own work, discussed more detail in Secs. 7 and 11.1, we show an implementation of the doubled hinge technique where select creases are doubled for use in conjunction with the rolling contact technique [132]. While it is possible that an entire approach can be formed on the concept of selectively doubling hinges, we leave this open for future work.

## Rolling Contacts Technique

Like many concepts relevant to thick origami, rolling contacts have 19th-century roots; a familiar example of a rolling contact joint is the Jacob's ladder toy over a hundred years old [155].2 Recent advances, however, have made it possible for this type of joint to be used in engineering applications such as prosthetic knee joints [156], spinal implants [157], and robot fingers [158]. A significant refinement came with compliant contact-aided mechanisms that utilize compliant flexures to maintain contact and to enforce a nonslip condition between the rolling faces [159,160]. Such joints can be used in thick-folding mechanisms; in fact, Cai gave a kinematic analysis of cylindrical rolling joints that allow for folding of thick plate structures as an alternative method for thickness accommodation in planar mechanisms [161].

In a further generalization of rolling contacts, Lang et al. presented a generalized technique that utilizes synchronized-offset rolling-contact elements (SORCE) to achieve kinematic single-DOF motion in thick origami-inspired mechanisms [132]. This technique, illustrated in Fig. 17, utilizes rolling contacts with profiles that have been synthesized to achieve the same dihedral angles as a zero-thickness origami model while creating dynamically varying transverse offsets that address non-self-intersection. Prior to this work, rolling contacts had only been utilized for planar mechanisms and had not dealt with thickness accommodation at interior vertices of an origami pattern. A notable aspect of the SORCE technique is that it marries a fully flat unfolded state (where all primary panel areas are coplanar, not necessarily including the joints) with a folded state incorporating arbitrary offsets between panels; furthermore, the DOF of the mechanism exactly reproduces the DOF of the zero-thickness model. Single-DOF mechanisms such as the Miura-ori or split square twist will remain single-DOF when implemented as SORCE joints. In fact, the technique can be applied to any arbitrary origami pattern containing any combination of vertices of varying degree.

The SORCE technique overcomes some of the difficulties that arise in other techniques because it is able to accommodate panels of arbitrary thickness, maintain the full range of motion, and preserves kinematic single-DOF motion (if it was present in the zero-thickness reference). The SORCE technique uses a unique way of achieving a translating zero-thickness plane for each panel. All of the zero-thickness planes begin coplanar in the unfolded state and then each panel offsets along an individual and specified path throughout the motion of the mechanism. The rate of offset of the individual panels relative to the zero thickness model is synchronized between the panels by the shapes of the rolling contacts to be pairwise consistent at each fold, hence the name “synchronized offset.”

However, like all of the techniques, SORCE joints have their own special considerations. The rolling contact surfaces are uniquely specified by the desired motion and offsets, but the curvature and convexity of the rolling contact surfaces must be taken into account during design to ensure robust joints.

Lang et al. [132] gave analytic prescriptions for computing the shapes of rolling contacts for vertices in which the panel offsets were purely perpendicular to the panels (which apply to any degree-4 vertex) and those that combine perpendicular and lateral offsets (which only apply to certain special cases). Several examples were given along with proof-of-concept prototypes such as the one shown in Fig. 18.

There is an interesting duality between the SORCE technique and the hinge shift technique of Trautz and Künstler [142]. In that latter, the panels are allowed to shift along the axes of the hinges throughout the range of motion. In the former, the panels are allowed to shift in the two perpendicular directions relative to the axes, and, in fact, that type of rolling contact cannot accommodate any shift along the axes.

## Membrane Technique

Although thin membranes have been used before for prototyping thick origami, using the membrane itself to accommodate panel thickness was fully explored by Zirbel et al. [59]. Thin membranes, such as fabric, were previously used primarily to simulate ideal hinges that have the center of rotation exactly at the seam between panels. However, a thin membrane can be used to accommodate thickness in origami as illustrated in Fig. 19.

When using this technique, the panels of the mechanism are connected by a thin membrane that can, in contrast to the panels themselves, be treated as approximately zero-thickness. By increasing the spacing between panels for valley folds, the membrane can not only serve as the hinge, but also accommodate the thickness of the panels to allow folding. For a rigidly foldable origami pattern, the maximum spacing between panels necessary for a $180 deg$ valley fold is 2t, where t is the thickness of the panels. When the final desired valley fold angle is less than $180 deg$, the spacing can be decreased accordingly; for example, fold angles of $90 deg$ and $60 deg$ would require spacings of $2t$ and t, respectively.

While conceptually simple to implement, modeling flexible membrane hinges is considerably more complicated than mechanisms with discrete hinges. Recent work by Peraza-Hernandez et al. laid out a computational model for such mechanisms [85]. Further complicating the modeling is the observation that the state space for motion is extremely high-dimensional, and the path through state space of a folded state as it deploys is typically determined by energy minimization along the path; thus, one must model strain energy within the curved hinges in order to determine actual motion and reachability.

Mechanisms using the membrane thickness-accommodation technique have extra degrees-of-freedom when compared to their ideal zero-thickness origami models. In this way, the technique is closely related to the offset-crease technique (Sec. 6) as it widens the crease area to allow for the panels to meet and does not guarantee consistent motion. As this crease widening increases the number of hinge points, it results in the mechanism having extra degrees-of-freedom. However, these extra degrees-of-freedom can be advantageous in allowing folding of a pattern that is otherwise not entirely rigid-foldable [59,60]. Also, because of the flexibility of the membrane, it may be necessary to keep tension at the edges of the mechanism in the unfolded configuration.

## Strained Joint Technique

The strained joint technique for accommodating thickness, introduced by Pehrson et al. [162], is related to the membrane technique. Instead of using a thin membrane, the thick material itself acts as an effective membrane, i.e., one in which the “fold” is distributed across a region, rather than being a discrete revolute joint. In this case, the panel material itself is dissected so as to be flexible along desired hinge lines as seen in the illustration of the strained joint technique shown in Fig. 20. This gives an entirely monolithic mechanism, as can be seen in Fig. 21.

To create flexible joint areas in the otherwise rigid panel material, surrogate folds are introduced in the design. A surrogate fold is “a localized reduction in stiffness achieved through geometry to allow nonpaper materials to achieve similar behaviors to a fold in paper” [163]. An important characteristic of these surrogate folds is that the folding motion does not cause the material to plastically yield; all deformations remain within the elastic regime despite large-angle flexing. Although putting strategic cuts in materials to reduce stiffness has been used previously to improve the accuracy and decrease the effort required in sheet metal bending where yielding is desired [164], it has not been widely used for the purpose of repeated motion, i.e., motion remaining within the elastic regime.

Surrogate folds are compliant joints [165]; examples of some geometries that can be used as surrogate folds can be found in Ref. [166]. Delimont et al. have evaluated several other potential geometries for use as surrogate folds in origami-inspired mechanisms that are suited to either resist or be predisposed to particular motions and deflections [163,167].

By arraying these surrogate fold geometries in series and parallel, the amount of deflection can be customized and tailored to meet the fold characteristics required at each crease of the origami pattern. The strain allowed in the surrogate folds without material yielding is what enables the motion of the origami, hence, the term “strained joint technique.” It should be noted that the membrane technique also involves strain, localized within the membrane hinges. In this case, however, the strain takes place within the same material that constitutes the panels.

Pehrson et al. have outlined the primary design approach for applying the strained joint technique to a fold pattern [162]. Material and origami crease pattern selection occur first. Material selection holds particular importance in this technique for thickness accommodation because the motion of the joints in the mechanism results from material deflection, which is, in turn, governed by material properties. When using arrays of lamina-emergent torsion joints for the surrogate folds, a material with a high ratio of yield strength to shear modulus of elasticity is desired [168].

After material and crease pattern selection, the creases are tagged according to how many layers of material must be accommodated by the joint at that crease; only then can the joints themselves be fully designed. Because the motion of the mechanism is a result of material deflection, fatigue must also be considered in the design of the joints.

When using the strained joint technique, not all combinations of materials, thicknesses, and mechanism sizes are feasible. Mechanisms using this technique are more constrained than the others that we have reviewed thus far since the panel material itself is used to realize the deflection that achieves motion. The area used for the distributed flexible joints takes away from the area used for the rigid panels; using a relatively thick and stiff material for the mechanism will require significantly larger joint areas to achieve the same folding angles than a mechanism using thinner, more flexible material. For any given combination of material and thickness, there is a minimum mechanism size because, as a pattern size decreases, the ratio of the joint area to the panel area increases until eventually, and the panels are entirely consumed by the joint areas.

Like the membrane and offset crease techniques, a key characteristic of mechanisms made with this technique is that they contain additional degrees-of-freedom compared to the zero-thickness reference. These extra degrees-of-freedom are a result of the motion and deflection possible in the compliant joints acting as surrogate folds. Because of the extra degrees-of-freedom, the same motion planning problems that exist with the offset crease and membrane techniques also exist with the strained joint technique. However, in another similarity to the membrane technique, the extra degrees-of-freedom and possible parasitic motion in the joints can be beneficial when used to accommodate thick folding of patterns that are not quite rigidly foldable or flat-foldable.

## Comparison and Discussion of Techniques

Each of the thickness-accommodation techniques that have been discussed here have their own advantages and disadvantages. For example, the strained joint technique has the advantages of maintaining a monolithic structure similar to paper origami, but loses its rigid single degree-of-freedom movement due to the flexibility of the joints. The offset panel technique, on the other hand, can maintain a rigid single degree-of-freedom movement but is unable to fold out to a flat planar state.

In this section, the strengths and weaknesses of each thickness-accommodation technique are summarized. We also discuss how the unique characteristics of each technique make them suitable for different applications. For this discussion and comparison, the specific implementations (where applicable) of techniques referred to are Chen et al.'s generalized implementation of the hinge shift technique, the offset crease implementation of the doubled hinge technique, and the SORCE implementation of the rolling contacts technique. In comparing each of the thickness-accommodation techniques that have been reviewed in Secs. 39, we have chosen several characteristics that are of interest to designers and that vary with each technique. These characteristics are listed in Table 1, and the definition of each of these metrics is as follows:

Equivalent kinematics: indicates if the technique is kinematically equivalent to the zero-thickness origami base model. This means that the mechanism must still contain the same spherical linkages that exist in the zero-thickness model. Therefore, the same kinematic model used for the zero-thickness model can also be used to predict the motion, including the position and orientation, of the thick origami.

Preserves motion: indicates if the thick origami preserves the dihedral angles and degrees-of-freedom that are exhibited in the zero-thickness model without requiring additional constraints. When a thick origami mechanism exhibits these characteristics of the zero-thickness model, its motion is also the same. Some techniques preserve the dihedral angles by maintaining the original zero-thickness surface in the thick origami model, whereas other methods are able to preserve the motion through means such as offset functions and spatial linkages. It is also worth noting that techniques that do not always preserve the motion may be made to do so in specific configurations or through the addition of constraints that are designed to reduce the degrees-of-freedom to match the original zero-thickness model.

Full ROM: indicates if the technique preserves the ROM of the original zero-thickness origami model. This means that any thick origami based on a flat-foldable origami pattern would be able to start and end with the panel faces parallel to each other.

Flat surface: indicates if the thick origami mechanism has a flat surface in its unfolded state. Here, we will use a working definition that a thick origami mechanism has a flat surface if it can rest firmly on a flat face, such as a table top, in its unfolded state, with all panels in areal contact with the surface. This means that the large majority, if not all, of the lower panel faces must be coplanar (i.e., would rest on the “table top”) in the unfolded state, and so there cannot be any offsets protruding on one “face” of the mechanism or any significant variation in the distance of the panel face from the table top.

Arbitrary patterns: indicates whether the technique can be applied to any arbitrary origami crease pattern. Techniques that have this characteristic do not have limiting geometrical properties that make some patterns not possible for other techniques. Techniques lacking this property are only applicable to very specific fold patterns or particular geometries.

Design complexity: indicates the relative difficulty of synthesizing a thick origami mechanism based on a zero-thickness origami model using a specific thickness-accommodation technique. A technique with a “low” design complexity rating may require the use of a simple equation or two, if any at all, to develop a working mechanism model. A “high” design complexity rating, however, means that extensive computation and/or optimization is involved in the mechanism design process.

### Matching Techniques and Applications.

The importance of each of these characteristics depends on the application. For example, the ability to have a flat unfolded state may be important in the application of a folding table but may not be as important in the application of a folding tool box. A solar array would be insensitive to small offsets perpendicular to the panels, but a reflective antenna would be highly sensitive to the same. The single degree-of-freedom motion that comes with preserving the dihedral angles is desirable in most cases; however, the ability to have multiple configurations, which often accompanies multiple degree-of-freedom mechanisms, may be desired in certain situations. So, for example, while the hinge shift technique is not applicable to arbitrary crease patterns, if the pattern to be used in the application maintains the conditions listed in Eqs. (8)(10), hinge shift can create a very rigid single degree-of-freedom mechanism.

One note of potential importance is that, even though most of the techniques are applicable to an arbitrary crease pattern, some techniques do not scale as well as others to varying panel thickness-to-size ratios. One example of this is the strained joint technique; if the pattern is scaled down to be smaller while using the same material for the mechanism, the joint areas must increase in width because of their decreased length. Such scaling leads to progressively smaller panel areas until the areas from two joints meet to swallow-up the panel and there is no discernible pattern.

Another characteristic that may be of concern for some applications is whether or not the mechanism requires holes in the unfolded state. A mechanism designed using the offset crease technique, among others, will always have holes at interior vertices when in the unfolded configuration. The offset panel technique is also likely to have holes; however, these holes occur in the panels rather than at the vertices to allow all the hinges to penetrate to the zero-thickness surface. The strained joint technique also inherently leads to mechanisms with holes at interior vertices.

## Combining Techniques

The concept of combining thickness-accommodation techniques was demonstrated by Lang et al. by selectively doubling hinges (doubled hinge technique) and implementing those doubled joints as SORCE joints [132]. They also discussed the possibility of combining SORCE joints with ordinary revolute joints by taking some offset functions to be zero and others to be nonzero. In this section, we expand upon those ideas introduced in Ref. [132] and show that it is possible to combine many of the thickness-accommodation techniques discussed here and that the resulting techniques offer their own unique strengths and limitations. While we do not cover every possible technique combination—obviously, the combinatorial possibilities are great—we have chosen to illustrate a few. The combinations presented here are various combinations of the rolling contact, doubled hinge, offset panel, and hinge shift techniques.

In particular, we cover two different ways that techniques can be combined: in a pattern, and at a vertex. We will call techniques that are combined at the pattern level as hybrids and techniques that are combined at the vertex as composites.

### Rolling Contact/Doubled Hinge Composite.

In Ref. [132], which was discussed in Sec. 7, a vertex that combines the technique of doubling hinges with rolling contacts was presented. We classify this vertex as composite, as it combines multiple techniques at the vertex. The doubled hinge implementation used has only two nonzero crease widths in a degree-4 vertex as illustrated in Fig. 22. The separation distance between the split creases is not fixed, but, via rolling contacts, varies throughout the range of motion of the vertex. Once split, the modified fold pattern is translated into a thick folding mechanism by applying SORCE joints at each fold line. The advantage of using the split pattern rather than the basic SORCE concept (perpendicular motion only) is that with the addition of a dynamic split, the joint surface profiles can be more circular, thereby avoiding complex geometries.

Based upon the same bird's foot geometry, we present another variation that makes use of fixed rotational axes in combination with SORCE joints shown in Fig. 23. By combining fixed rotational axes with rolling contact joints, we are able to produce a mechanism that is self-contained within the thickness of the panels. This is an improvement over the all-SORCE configuration, which results in some of the contact joints protruding beyond the panels, as shown in Fig. 24.

### Offset Panel/Hinge Shift Hybrid.

The offset panel/hinge shift hybrid technique utilizes the hinge shift technique to join offset panel units together as illustrated in Fig. 25. While the hinge shift technique can be employed in flat-foldable degree-4, -5, or -6 configurations as shown by Chen et al., we limit our geometry to the symmetric case. By utilizing the symmetric degree-4 vertex, the geometry of the hinge shift vertex is simplified such that hinge offsets pairs d1 and d3 are twice those of d2 and d4.

We apply the offset panel/hinge shift hybrid to a simple fold pattern consisting of two vertices, the simplest form possible for a hybrid, as shown in Fig. 26. Note that, in this pattern, one of the vertices is symmetric to enable use of the hinge shift joint demonstrated by Hoberman and the other is an arbitrary degree-4 vertex that satisfies the flat-foldability criteria listed in Eq. (1). We first apply the offset panel technique to the nonsymmetric vertex on the right. We then use the hinge shift technique at the symmetric vertex on the left. Because the offset panel vertex shares two panels with the hinge shift vertex, the lengths of the offsets in the hinge shift vertex become twice the length they would be if employed at the first vertex due to the panel offsets of the offset panel vertex. The axis locations of the hinge shift vertex are then defined by the shared panels and axes of the offset panel vertex and Eqs. (8)(10).

The resulting mechanism, as shown in Fig. 27, is one that cannot be realized using hinge shift or offset panel technique alone. For hinge shift, only patterns such as the square twist pattern shown in Fig. 13 where Eqs. (8)(10) are satisfied at each vertex are possible. For the offset panel technique, this pattern would require cut-outs in some of the panels to allow otherwise intersecting offsets to pass through.

The combined offset panel/hinge shift hybrid can also solve issues that arise in the offset panel technique when dealing with tessellations (large-scale repeating patterns). In a tessellating pattern, such as the Yoshimura pattern [140], the folding/unfolding motion can go from a long flat pattern with many sections to a zero-thickness plane where all of the panels are stacked on top of each other. If the offset panel technique were applied to a pattern such as this, the length of the offsets would continue to increase layer by layer, making the technique impractical when many stacking layers are involved. By using the offset panel technique in combination with a hinge shift, a tessellating pattern using offset panel technique, as shown in Fig. 28, can be realized, that scales to arbitrary length (at least in one direction, sometimes both).

The Yoshimura pattern used in this hybrid mechanism can be realized using the hinge shift technique alone, as shown in Fig. 9. However, this is due to the symmetric nature of the pattern. In Fig. 29, we show a prototype of a bird-base hybrid mechanism that utilizes offset panel and hinge shift, but cannot be realized using symmetric hinge shift vertices alone, and would introduce cutouts if the offset panel technique were to be used alone. It is possible that the generalized hinge shift technique demonstrated by Chen et al. could be applied to this pattern; however, the design complexity would increase due to the nonsymmetric vertices.

The bird-base pattern used in this mechanism is comprised of four nonsymmetric degree-4 vertices and one symmetric degree-6 vertex. The two halves of the pattern are comprised of degree-4 vertices and each half utilizes the offset panel technique to accommodate thickness. These two halves are connected together by a single degree-6 vertex which uses the same degree-6 hinge shift vertex presented by De Temmerman et al. [20]. It is, in total, a single-DOF mechanism.

### Offset Panel/Doubled Hinge Hybrid.

To illustrate the offset panel/doubled hinge hybrid, we use the same pattern shown in Fig. 26 but split the symmetric vertex as illustrated in Fig. 22. This pattern, shown in Fig. 30, is now comprised of three degree-4 vertices rather than two. To combine the two techniques together, we first begin with the doubled hinge vertex. While we could use a SORCE + doubled hinge composite for this vertex, we have decided instead to use panels for both intermediate sections and offset the panel of the mountain fold to avoid interference. The offset panel technique is then applied to the nonsymmetric vertex on the right where the zero-thickness plane is determined by the doubled hinge vertex on the left. Because the hinge plane of the mechanism is driven by the doubled hinge vertex and does not lie in the middle of the panel stack of the offset panel vertex, cutouts are required for the offset panel vertex on the right.

The resulting mechanism is shown in Fig. 31. In this generic form, any advantages of this mechanism over the one shown in Fig. 27 are not readily apparent; discrimination between the two is going to be application-specific. Nevertheless, the fact that these two techniques can be combined gives more options to a designer for designing a thick folding mechanism.

### Doubled Hinge/Hinge Shift Hybrid.

We can similarly combine the doubled hinge technique with the hinge shift technique. To combine these two techniques, however, we utilize a symmetric vertex at both the left and right vertices, albeit dissimilar ones. The resulting mechanism, as shown in Fig. 32, offers designers yet another option when considering combining thickness-accommodation techniques.

### Other Hybrid and Composite Techniques.

Many other combinations of techniques can also be achieved. The membrane and strained joint techniques can be easily combined with other methods due to the flexibility in their motion. For example, in a thick rigid origami mechanism, part of the mechanism may not require thickness and strength and could be replaced with a thin membrane. Surrogate hinges could also be used in combination with traditional hinges. Rather than attempting to cover every possible combination—which would be combinatorially challenging—we simply present these few examples as illustration, and note that the ability for a designer to utilize different thickness-accommodation techniques at different locations within a thick origami mechanism gives them the ability to create a mechanism tailored for their specific application.

## Manufacturing Considerations

For origami-inspired design to be widely used and practiced, how such products are to be manufactured must be considered. The thickness-accommodation technique used in the design of an origami-inspired product has a significant influence on the manufacturing approaches and costs of the product. Though all thickness-accommodation techniques make manufacturing more difficult than folding a piece of paper, some techniques make the transition from paper folding to manufacturing more challenging than others [143,153]. The chosen approach determines not only the part count and what manufacturing processes are required, but also whether or not materials in sheet stock form can be easily used.

Several of the thickness-accommodation techniques previously discussed have been put into practice in more than one way. For example, Tachi has presented the tapered panels technique implemented in two ways: as panels composed of a single tapered piece and as panels composed of two constant thickness pieces joined together [131], such as those seen in the first rows of Table 2. Although these variations in how the mechanism is built do not affect the ideal performance and motion of the mechanism using the specific technique, they do have a significant impact on how the mechanism can be manufactured. In order to systematically consider the impact on manufacturing of each technique, the techniques have been divided (where applicable) into manufacturing approaches that have been suggested or employed before. A monolithic panel approach is likely to require processes to remove material to reach the final panel shape, whereas a layered or segmented approach would require additional assembly.

Table 2 lists thickness-accommodation techniques and several manufacturing characteristics relating to how these techniques are typically considered for manufacturing. The second column of Table 2 illustrates the manufacturing approaches for each technique. The characteristics relating to manufacturing shown as the other table column headings are described as follows:

Part count: indicates the relative part count of the origami-inspired mechanism. A part count is considered “baseline” when the product has as many parts as the number of facets and creases in the paper origami model. A part count may be considered high when it is roughly twice the baseline count. The part count of a product is often an indicator of its complexity because more parts corresponds to a higher potential for problematic tolerance stack-ups that lead to poor product performance.

Conducive to sheet stock: indicates whether or not the manufacturing approach is conducive to the use of materials in sheet stock form for rigid panels without significant preprocessing. For the purposes of this discussion, a technique being conducive to sheet stock means that no three-dimensional process, such as milling, is necessary to fabricate any component of the mechanism. Therefore, using a thickness-accommodation technique that is conducive to sheet stock leads to simpler manufacturing processes and potentially lower production costs overall.

Second panel process required: indicates whether the fabrication of the panels using the specified approach would require a second process after initially cutting the nominal panel shape from the stock material. Possible second panel processes include milling and other material removal processes, joining, and assembly of each panel before assembly of the mechanism as a whole. This is based on an assumption that the initial process of cutting from stock material would use a machine with no more than two degrees-of-freedom, such as a saw or most lasers and waterjets. As these second panel processes usually require jigs and/or fixtures and significant setup/processing time, there may be a significant increase in production cost for using thickness-accommodation techniques requiring such processes.

Minimum number of processes: indicates the fewest number of processes required to fabricate a product using the specified manufacturing approach. Three possible processes were assumed: cutting panel shapes from stock material (one- or two-dimensional process), material removal to achieve final three-dimensional panel shape (three-dimensional process), and assembly of the mechanism (including any applicable panel assembly). Not all manufacturing approaches require all three processes. For example, the layered hinge shift approach requires two processes: one to cut the panel components to size from stock and one to assemble the panels and mechanism. The number of processes required to manufacture a product is important as it is an indicator as to how much labor and equipment may be needed.

### Considering Manufacturing and Performance.

When choosing a thickness-accommodation technique during product development, both performance and manufacturing should be considered. There are trade-offs involved with each technique and manufacturing approach. For example, the strained joint technique is generally highly desirable from a manufacturing standpoint, but does not have the motion characteristics of preserving the dihedral angles and a single DOF. In comparison, a design that employs the SORCE rolling contacts technique would exhibit such motion characteristics, but does not have such favorable manufacturing characteristics because, in addition to requiring forming of the rolling contact surfaces, also requires tight tolerances for the surfaces, thereby furthering the increase in manufacturing cost.

One potential method of tailoring the design of an origami-inspired product to meet both the motion and manufacturing requirements is to utilize hybrid techniques. Techniques with particular motion characteristics can be implemented at some vertices in a pattern and techniques with favorable manufacturing characteristics implemented at others. By doing so, designers can pick and choose which techniques to use at various points in the pattern to reduce the overall manufacturing cost of the design while still retaining specific motion performance at critical vertices in the pattern.

## Conclusions

Origami is a vibrant ancient art that has recently extended into the realms of mathematics, science, and engineering. In this paper, we have reviewed work that has been done specifically in the area of thick origami, the translation of origami forms into mechanisms involving moving panels with non-negligible thickness. The basic terminology and mathematics used to describe origami mechanisms is presented and then used as a basis to review published techniques—most developed within the past 10 years—to create mechanisms based on origami, but using materials that are “thick.” Techniques were put into context and generalized with examples provided for each. We showed that these various techniques can be compared and contrasted with each other on a common footing, and that each technique offers advantages and disadvantages.

In addition to reviewing existing thickness-accommodation techniques, we have also expanded upon prior work by showing that multiple thickness-accommodation techniques can be combined within a single mechanism, and considered how these techniques affect the manufacturing of thick origami mechanisms. Several ways of combining thickness-accommodation techniques within a thick origami mechanism, both at the joint level and pattern level, have been presented. These hybrid and composite techniques allow for a broader design space for thick origami mechanisms. A framework for considering design and manufacturing approaches of such thick origami mechanisms is presented. Understanding how to approach the manufacturing of origami-inspired products is key to extending the applications of thick origami to new areas.

More work can yet be done in the field of thick origami. While all of the thickness-accommodation techniques discussed in this paper are effective at the single vertex level and many have shown their effectiveness in patterns with multiple vertices, patterns where thickness can accumulate, such as the Miura-ori pattern, continue to pose challenges. This is a potential area of further work, as is the development of thick origami applications. With an understanding of the design space and capabilities of these mechanisms, there are nearly infinite possibilities for applications of thick origami.

## Funding Data

• National Science Foundation and Air Force Office of Scientific Research (Grant Nos. EFRI-ODISSEI-1240417, EFRI-ODISSEI-1240441, EFRI-ODISSEI-1332249, and EFRI-ODISSEI-1332271).

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