Folding of Thick Origami Through Regionally Sandwiched Compliant Sheets

Origami-based mechanisms can exhibit many useful traits [1]. Flat foldability is the trait to fold compactly with panels that are parallel to each other [2]. This behavior allows for convenient stowage and transportation of the mechanism, which can then deploy to fulfill an intended function. Shelters [3], barriers [4], antennas [5], and solar arrays [6] are a few examples of origami mechanisms performing intended functions in a deployed state. Rigid foldability is a trait of some origami folding patterns, which ensures that panels do not deform during their motion [7,8], allowing the use of engineering materials or components that are required to be rigid to perform their functions (i.e., load-bearing members, sensitive electronics).

However, when using thick materials, issues such as self-intersection arise and can be prohibitive to design. Many techniques have been developed to accommodate the use of thick materials, several of which are reviewed in Ref. [9]. These thickness accommodation techniques enable engineers to design origami-based mechanisms to retain some or all of the traits of flat foldability, deployability, and rigid foldability.

Deployment and actuation of origami offers its own challenges as well. As such, efforts have been made to simplify deployment actuation, such as using magnetic fields [10,11], heat [12–14], shape-memory composites [15,16], and strain energy [17,18], and adapting patterns to better control their motion [19]. Stored strain energy in integral joints of a mechanism is a simple way to eliminate complex internal or external deployment actuators and can be termed as a form of self-deployment. In origami, strain occurs in the creases and methods have been investigated to predict the strain energy induced by deflection [20,21]. An origami-based mechanism that can self-deploy reduces complexity and can be useful for applications ranging from consumer products to deployable space structures.

Once deployed flat, many origami-based mechanisms are susceptible to bifurcating to other possible folding branches. A folding branch is a unique set of fold assignments (either mountain or valley). At a flat configuration, all of the creases in the mechanism lie in the same plane and are no longer constrained to a single mountain-valley assignment. In this state, creases often change crease assignment [22,23] and thus folding branch. This ambiguity makes folding difficult, often requiring several inputs to assign each crease appropriately. When using thick materials, self-interference can be used advantageously to constrain each fold to its proper assignment, creating a single folding branch of the mechanism. An origami-based mechanism that has only one folding branch reduces the number of inputs, simplifying the folding process.

We propose a technique to design flat-foldable, rigid-foldable, and self-deploying thick origami-based mechanisms that offers a single branch of folding. The technique utilizes regional sandwiching of a compliant sheet (ReCS) of nonzero bending stiffness between rigid panels. The rigid panels are used to create self-interference (i.e., hard stops) in the deployed flat state to define the folding branch and to define the folds of the compliant sheet as flexures. The flexures enable flat folding and store strain energy as the source of self-deployment. We present a design framework and demonstrate the feasibility of the proposed technique through physical prototypes.

The technique discussed in this paper builds upon previous work and features derived from thick-folding origami. Foundational work in thickness accommodation and interference of thick origami is discussed in this section.

Traditional origami is often modeled using a zero-thickness assumption. Once an appreciable amount of material is being folded, consideration must be made for the movement, placement, stress, and strain of the material. Many approaches have been developed with consideration of these elements, leading to new thick origami mechanisms and techniques with unique behaviors [24,25]. Thick-folding techniques have been created to allow for self-deployment [18], flat foldability [26–32], and preservation of the kinematics of the zero-thickness model [7,27,33].

Hybrids of these techniques have shown to create new behaviors and have been demonstrated by Refs. [9,30,34]. The ReCS technique exhibits novel behaviors through the combination of several thick-folding techniques.

The membrane technique [28], as shown in Fig. 1(a), uses a flexible thin layer to enable the motion of thick origami. Rigid panels are adhered to the membrane to create rigid segments and are arranged with space between the panels to allow panels to fully fold. Because the membrane is thin, it is assumed to have no thickness and no bending stiffness, allowing the device to be stowed as shown in Fig. 1(a). Bending of the membrane is assumed to have no influence on its stress, allowing for tight bends and a minimization of the gap between rigid panels. However, the lack of bending stiffness in the membrane may require external constraints to maintain a desired configuration.

The strained joint technique [31], shown in Fig. 1(b), is a thick-folding technique that allows folding of a monolithic piece of material, much like traditional origami. Material is systematically removed from a single thick sheet to create rigid and flexible segments. These segments create the panels and folds of the mechanism. Stress and strain in the thick material require folds to have some length to prevent yielding. Strain energy created during the deformation of these flexible segments is stored and can then be utilized to deploy the mechanism into a planar state. The strained joint technique is also capable of creating flat-foldable origami.

A downside to the strained joint technique is its tendency toward parasitic motion. As material is removed to create flexible segments, in-plane rotation creates undesirable motion. This parasitic motion can be mitigated by enhancing the flexible segments with membranes [35].

A thick-folding technique that sandwiches a flexible sheet of negligible bending stiffness with rigid panels is introduced in Ref. [33] and is called axially varying volume trimming. Past volume trimming approaches have removed material uniformly along the width of the fold. This technique, shown in Fig. 1(c), variably trims the volume from the panels along the width of the fold, preserving the kinematics of the zero-thickness model. This approach allows for simple manufacture as the mechanism is composed of three layers that can be attached together.

Axially varying volume trimming provides predictable motion. It cannot provide self-deployment because of the infinitesimal length of the fold and its low-bending stiffness folding layer. As a mechanism folds using this technique, self-interference occurs between panels at a valley fold, constraining the mechanism to not fold flat (as shown in the bottom of Fig. 1(c)). While this method is unable to directly create flat-foldable mechanisms, it is possible to combine this method with other thick-folding techniques to create flat-foldable thick origami, as demonstrated in Ref. [33] and as demonstrated with the proposed ReCS technique.

Origami designed with a zero-thickness assumption can have interference only when one facet contacts another facet. A unique feature of thick origami is that panels can also interfere with adjacent panels in the planar, unfolded state. Many thickness accommodation techniques, including several of the techniques reviewed in Ref. [9] and the axially varying volume trimming [33] technique discussed above, create panel interference in this unfolded state. Several thick-folding techniques that use flexible joints with nonzero length, such as the membrane and strained joint techniques discussed above, are not capable of creating hard stops at the unfolded state through the interference of panels.

The regional sandwiching of compliant sheets (ReCS) technique presented in this paper builds upon the three techniques discussed earlier. Elements from each of them can be seen in the approach, including the attachment of rigid panels to a flexible sheet from the membrane technique, energy storage through strain energy found in the strained joint technique, and axial volume trimming of two sandwiching layers from the axially varying volume trimming technique. The feature of interference as a hard stop in thick origami is also intentionally applied in the ReCS technique.

The ReCS technique uses a single compliant sheet that is sandwiched between two arrays of rigid panels. At the simplest level, the approach can be applied to a single fold. The approach can then be expanded to include a single vertex and eventually an entire fold pattern. This section describes the components of the ReCS technique and outlines terminology requisite for its implementation.

Figure 2 illustrates the ReCS technique applied to a single fold. A compliant sheet (shown in orange) is sandwiched by panels that allow for motion on the valley side of the fold (shown in green) and that interfere on the mountain side of the fold (shown in blue).

The compliant sheet serves several purposes. First, it provides a backing to which rigid panels may be attached. Second, it creates the folds that enable motion of the mechanism. Third, it stores strain energy within the folds that allows the device to self-deploy. The resulting motion is similar to the smooth folds described in Ref. [36].

Rigid panels are sandwiched about the compliant sheet to maintain rigidity and create hard stops in the planar state. Mountain-sided panels create these hard stops, and valley-sided panels will allow for motion until the device is flat folded.

We will use the term “crease” to identify a flexible portion on a zero-thickness origami pattern where folding occurs. The term “fold” will identify the flexible portion of the thick origami mechanism that deflects during folding.

The “order” of a fold describes the number of pairs of panels that “nest” between the fold’s adjacent panels during folding. For example, the zeroth-order fold in Fig. 2(a) folds its valley side panels, shown in green, directly in contact with each other and has no other sets of panels moving between them. However, the first-order fold in Fig. 2(b) must nest a zeroth-order fold between its valley side panels, requiring the length of the folding flexure L to increase.

Figure 2 also demonstrates how panels must be sized to allow for folding to occur. The panels on the valley side of the fold need to be adequately spaced to allow for folding motion. The panels on the mountain side of the fold will be placed adjacent to each other in the planar state along the crease. This prevents bifurcation of the fold, constraining motion to a single folding branch. As the order of the fold increases, the distance between panels in the folded state must also increase to accommodate the nesting of lower order folds. Panels on the mountain side of the fold remain adjacent to prevent bifurcation.

Figure 3 demonstrates the approach applied to a degree-4 vertex. Volume in the compliant sheet (Fig. 3(a)) is trimmed to create each of the four needed folds and prevent interference of panels at the vertex. Two sets of rigid panels are attached to the sheet, each on their respective side of the compliant sheet. Volume from these panels is trimmed to create a desired mountain-valley assignment at each fold. Labels for the creases C₁–C₄, sector angles α₁–α₄, the vertex at the coordinate frame, and the valley (dashed lines) and mountain (dashed-dotted line) assignments are shown in Fig. 3(a).

Figure 3(a) also demonstrates that the compliant sheet can be created from a single sheet of material with portions removed, providing strength to the technique by reducing manufacturing costs to create a self-deployable origami mechanism.

The content in this section provides a framework that guides one through the process of utilizing the proposed technique. This framework is given in the following order:

1. Selection of panel thickness

2. Design of flexure lengths

3. Volume trimming

4. Customization of flexure stiffness

Included within these steps is discussion on the mathematical basis for key components of the steps and additional insights to aid in the customization of the approach for specific applications. It should be noted that the ReCS technique presented in this paper describes an idealized model that accounts for the unfolded and folded states of the origami mechanism. Real-life applications of the technique may require additional kinematic constraints to maintain certain assumptions made in the method, particularly the assumption of evenly distributed stresses presented in step 2.

Step 1: Selection of Panel Thickness. The thickness of the front and back panels t_p will serve as a key design input whereon much of the design is dependent, including the flexure lengths and maximum compliant sheet thickness. It is assumed that for an application, the panels will be of the most interest to the designer and will give some requisite functionality, such as solar panels or ballistic shielding. Once the panel thickness is known, the rest of the design may then evolve.

Step 3: Volume Trimming.Compliant Sheet. With the flexure lengths evaluated, we can now determine where to remove material in the sheet. To maximize the stiffness within each flexure, the width b (shown in Fig. 2) is designed to extend to the interior vertex without intersecting adjacent flexures. Material is removed at the vertex of the sheet to remove stress concentrations and to enable folding. An example cutout is shown in Fig. 4, along with crease line and flexure boundary vector definitions.

Finally, the vectors to the flexure ends are formed by substituting the coefficients λ_j,j+1 and λ_j,j−1 with the selected r_j values in Eqs. (13) and (14), constituting the flexure design at the vertex. Connecting these vectors around the vertex forms the two-dimensional polygon, which, when cut out of the compliant sheet, is the volume to trim at a vertex (see Fig. 4).

Panels. Figure 6 demonstrates how volume trimming occurs on one side of the compliant sheet. A panel is trimmed about the vertex to match the trimmed polygon in the compliant sheet. Next, the panel is trimmed along each valley fold between boundary lines C′_j and C″_j defined in Eqs. (7) and (8). Finally, cuts are made along vector C_j on mountain folds. The result is multiple trimmed panels from the original, one for each facet. This process is repeated on the opposite side of the compliant sheet (when viewed from the opposite direction so as to change the perceived parity of each crease).

Equation (25) provides an upper limit to the thickness of the compliant sheet, and therefore stiffness, when designed against stress failure, given the assumption of constant curvature is held true. Once t_s has been selected, the flexure width b can be adjusted to customize the stiffness of the device without altering the stress in the flexures. The flexure can also be divided into sets of flexures (balanced about the midpoint of the crease), as shown in Fig. 7, if less stiffness is desired.

The methods discussed earlier were used to create demonstration prototypes. Figures 8 and 9 illustrate the ReCS technique in a single zeroth-order fold and a degree-4 vertex, respectively. The prototypes shown were created using clear 0.25 in. (6.35 mm) thick acrylic for the panels and 0.004 in. (0.1 mm) thick 1095 blue temper spring steel as the compliant sheet. The assembly is held together with bolts set in counterbore holes to keep the bolt heads and nuts from interfering during folding. Panels were cut, and counterbore holes were rastered using a laser cutter. The compliant sheet was cut from a computer numerical controlled router.

The ReCS technique described in this paper allows for its application in a single vertex. When multiple vertices are combined to create a tessellation, the ReCS technique can be applied to each vertex to create a self-deploying flat-foldable tessellation. Figure 10 demonstrates the ReCS technique applied to an origami pattern. The pattern, a “Troublewit” [4,37], has six degree-4 vertices. The panels were created from 0.25 in. thick acrylic, and the compliant sheet was cut from 0.004 in. (0.1 mm) thick low-carbon steel.

The tessellation demonstrated in this work employed the design logic discussed in Eqs. (17)–(20). This particular tessellation incorporates vertices whose panels do not nest panels from neighboring vertices. As such, no consideration of the folding behavior of neighboring vertices is necessary, allowing the ReCS technique described above to be applied individually to each vertex. However, if a fold pattern is selected where panels from one vertex nest panels from another, appropriate adjustments in step 2, such as increasing flexure lengths to accommodate higher order folds, would be required to allow for flat foldability.

The prototypes successfully demonstrate self-deployment from the folded to planar state. The low-carbon steel prototype was also able to deploy but with reduced intensity due to the reduced stiffness. The device was also constrained to a single folding branch with mountain and valley assignments remaining unchanged.

The ReCS technique does not constrain the mechanism to the zero-thickness kinematic motion. The use of flexures instead of hinges results in parasitic motion in several directions. Possible undesirable motions of the panels include the panels shifting in a shearing direction or rotating relative to one another, as shown in Fig. 11. This parasitic motion can lead to yielding in the compliant sheet because portions of the sheet are forced into a tighter curvature than what would be applied by a pure moment. Both of the motions demonstrated in Fig. 11 should be considered to minimize their influence on the curvature of the compliant sheet.

Parasitic motion of the degree-4 vertex is primarily driven by the first-order fold. Its stiffness is significantly less than that of the other three folds, and its length allows for a large amount of undesired motion. This difficulty is further compounded when a tessellation is used that requires first-order folds to nest within second-order folds. To remain flat foldable, the outer set of panels would need to be separated by a flexure five times longer than the zeroth-order fold flexure length, resulting in a large amount of parasitic motion. It could be possible to replace higher order folds with another thick-folding technique, such as a rolling contact element [30]. Using such a hybrid approach would remove the major driver of parasitic motion while still providing self-deployment.

Another means of reducing the likelihood of yield is to reduce the thickness of the compliant sheet from its maximum to some lower threshold using the safety factor in Eq. (25). This in turn will reduce the stiffness of the device but may be necessary to ensure that yielding does not occur.

Other materials, including much thicker materials, could be investigated for use as the compliant sheet. For example, thicker materials could incorporate lamina emergent torsional joints [38] if the compliant sheet is to serve additional functions beyond enabling flat foldability and self-deployment. It is also possible for the compliant sheet to be yielded at the folds in the direction opposite of the desired motion before assembly, preloading the system. Doing so would allow the joints to store strain energy in the planar state, increasing the stability of the device when deployed. Varying thicknesses of panels could also prove useful in certain applications, which would demand adjustment to the process of flexure design detailed in step 2.

A technique for folding thick materials through regionally sandwiched compliant sheets (ReCS) was presented. The technique incorporates strain energy in the compliant sheet in regions not sandwiched by thick panels to enable self-deployment. The interference of deliberately arranged panels determines the mountain-valley fold assignments of origami-based mechanisms. Mathematical methods to design such mechanisms for folding and against failure of the compliant sheet were presented. Suggested volume trimming of the compliant sheet was shown for general degree-four vertices. Flexure stiffness was also shown to be tailorable. The ReCS technique was applied to the design and fabrication of two physical prototypes using acrylic panels and spring steel. Extension of the technique to a multi-vertex origami-based mechanism was presented using acrylic and low-carbon steel. These prototypes demonstrated the flat-foldability, rigid foldability, self-deployment, and assigned folding inherent in the ReCS technique. Parasitic motion was observed in the prototypes, and several mitigation approaches were offered. Other possible combinations with existing thick-folding techniques were also discussed.

The authors thank Nathan Brown and Kenny Seymour for their assistance in prototyping and photographing the demonstrated examples. This paper is based on work supported by the National Science Foundation and the Air Force Office of Scientific Research (NSF Grant No. EFRI-ODISSEI-1240417; Funder ID: 10.13039/100000181), the U.S. National Science Foundation (NSF Grant No. 1663345; Funder ID: 10.13039/501100008982), a Utah NASA Space Grant Consortium Fellowship (Funder ID: 10.13039/100006310), and a NASA Space Technology Research Fellowship (Grant No. 80NSSC17K0145).

d =

half the flexure length

r =

scalar projection onto crease vector

x =

horizontal position at end of flexure

y =

vertical position at end of flexure

v =

vector to intersections

E =

modulus of elasticity

I =

second moment of area

K =

bending stiffness

L =

fold flexure length

M =

moment

C =

vector along crease line

t_p =

panel thickness

t_s =

compliant sheet thickness

S_y =

yield strength

C′ =

vector of parallel line counterclockwise of crease line

C″ =

vector of parallel line clockwise of crease line

λ =

coefficients for intersection

σ =

bending stress

ϕ =

flexure angle

ϕ_o =

angle at end of flexure