The Kresling origami spring: a review and assessment

Structures inspired by the Kresling origami pattern have recently emerged as a foundation for building functional engineering systems with versatile characteristics that target niche applications spanning different technological fields. Their light weight, deployability, modularity, and customizability are a few of the key characteristics that continue to drive their implementation in robotics, aerospace structures, metamaterial and sensor design, switching, actuation, energy harvesting and absorption, and wireless communications, among many other examples. This work aims to perform a systematic review of the literature to assess the potential of the Kresling origami springs as a structural component for engineering design keeping three objectives in mind: (i) facilitating future research by summarizing and categorizing the current literature, (ii) identifying the current shortcomings and voids, and (iii) proposing directions for future research to fill those voids.


Introduction
The word Origami originated from combining two separate Japanese words: ori meaning 'to fold', and kami meaning 'paper'.It refers to the ancient Japanese art of folding paper to create aesthetically appealing structures in the form of flowers, birds, butterflies, etc [1,2].Long before becoming an art form, remarkable and intricate forms of origami had already been adopted in nature by plants and insects through millions of years of evolution [3,4].Examples can be seen in the Earwig's wings [4], the Hawkmoth's air sac [5,6], the Tortoise shell * Author to whom any correspondence should be addressed.bamboo [7], the Chestnut leaves [3,5,8], and the Potato flowers [9].Slowly, over the past two decades, origami has been transforming from an art into a science, which inspired the creation of various engineering structures with interesting functionalities that targeted niche applications spanning different technological fields [10][11][12][13].Examples of such designs include, but are not limited to, reconfigurable structures for solar panels and arrays [14], solar sails and inflatable booms for outer space applications [11], vascular stents [15], metamaterials and wave guides [16,17], and soft robotic manipulators [18].
Origami-inspired structures are typically constructed by folding paper or flat sheets of foldable materials following certain patterns that result in different shapes and functionalities.These patterns can be generally classified into two major categories: rigid and non-rigid (a.k.a.volumetric).Rigid origami results in three-dimensional structures in which only the creases between the panels undergo deformation by the folding motion during deployment.An example of rigid origami is the Miura-ori origami pattern, shown in figure 1(a), which has been used to construct three-dimensional deployable structures for space exploration [14,[19][20][21][22], deformable electronics [23,24], artificial muscles [25], and reprogrammable mechanical metamaterials [12,26,27], to name a few.Other rigid origami patterns have been utilized to create threedimensional deployable cylindrical structures called bellows [28][29][30].A successful demonstration is depicted in figure 1(b), where the Yoshimura origami pattern is used to design bellows with flexible triangular facets [31][32][33], that were used to create worm-like robots with peristaltic locomotion [34].By replacing the triangular panels with trapezoids, the flexibility of the Yoshimura deployable cylinders can be further increased, resulting in the Accordion pattern displayed in figure 1(c).This pattern has been utilized to design flexible frequency-tunable antennas [35].
The Kresling origami pattern, which constitutes the core of this review, was arguably discovered in 1993 by a student in one of Biruta Kresling's courses on bionics in Paris, France [6,7].The discovery was made, rather accidentally, by observing the pattern generated from twist buckling of foldable cylinders, and was documented in 1995 by Kresling herself [52], where she briefly described the pattern formed when cylindrical columns undergo torsional buckling akin to figure 2(a).During the same time frame, a series of investigations were conducted by Guest and Pellegrino [53][54][55] on a deployable structure, which highly resembles the Kresling pattern, but was inspired by a model of triangulated cylindrical shells used to investigate the mechanics of certain biological systems (figure 2(b)).
While B. Kresling [58] was the first to regard this natural pattern as a model that holds potential for comprehension and intentional design in the fields of science and engineering, the first recorded scientific observation of the Kresling pattern dates back to 1933 during studies performed on the behavior of thin walled tubes under axial and torsional loading, such as the one undertaken by Donnell [56].In figure 2(c), the Kresling pattern can be observed manifesting on such tested samples as they are failing under the applied axial-torsional loads.In 2005, Hunt and Ario [57] took a closer look at the twist buckling behavior of a sheet of paper rolled around two plastic mandrels (figure 2(d)) and studied, in more detail, the associated folding pattern.It was not until 2013 when the first utilization of the Kresling pattern in engineering applications was realized.Subsequent years have witnessed increased research output on using the Kresling origami pattern in different engineering fields, as depicted in figure 3.
Transformation of the Kresling pattern, from a three dimensional shape that is formed during the buckling of a cylinder under loading into an origami pattern that can be used for functional design, requires creating a repeatable process to fabricate those cylindrical bellows from a sheet of paper as typically done in origami art.Such a process is briefly described in figure 4 and starts with segmenting a flat sheet of paper into a 2n number of triangles whose edges either represent valley creases or mountain creases, as shown in figure 4(a).The faces that define the bellow are formed by these triangles, referred to as panels.Each panel is typically characterized by three geometrical parameters; namely, its two side lengths, a 0 and b 0 , and an angle, α 0 .The panels are then folded along their edges  [53].(c) Failure of thin walled tube under axial-torsional loading.Reproduced with permission from [56].(d) Folding pattern of twist buckled sheet of paper rolled around two plastic mandrels.Reproduced from [57]. in a specific manner: the shorter edge, b 0 , is folded outward as a mountain fold, while the other edge, c 0 , is folded inward as a valley fold.When the two opposite end panels are then joined together along edges AB and A ′ B ′ , they result in a cylindrical bellow-type structure consisting of similar triangles arranged in cyclic symmetry, as depicted in figure 4(b).The third sides of the panels, a 0 , form two n−sided parallel polygonal end planes each subscribing a circle of radius, R, at the top and bottom faces of the bellow.Finally, two rigid plates are attached to the top and bottom ends forming a Kresling origami spring (KOS), as shown in figure 4(d).
When an external axial load or a torque is applied to the KOS, it stretches or compresses depending on the direction of the applied load.In the process, the two parallel polygon planes, while remaining rigid as shown in figure 4(c), translate and rotate relative to each other along and about a common centroidal axis.This causes the triangular panels to deform (a 0 → a, b 0 → b, c 0 → c) and store the applied work in the form of strain energy.The stored energy is released upon the removal of the external load, which forces the bellow to spring back to its initial configuration, therewith providing a restoring element that forms the basis for the design of many interesting engineering structures.The nature of the restoring force can be qualitatively modified by tuning the geometric design parameters of the panels, resulting in springs exhibiting mono-stability, bi-stability, softening/hardening non-linearity, and quasi-zero-stiffness (QZS) behavior.Those Kresling origami springs (KOSs) are the foundational unit cells of almost every engineering design that will be reviewed and discussed in this paper.
In this paper, we aim to perform a systematic review of the literature on the implementation of KOSs in the engineering realm while keeping three objectives in mind: (i) facilitating future research by summarizing and categorizing the current literature, (ii) identifying the current shortcomings and voids, and (iii) proposing directions for future research to fill those voids.A diagram illustrating the structure of the review is shown in figure 5.The review starts by discussing and summarizing the various fabrication methods that have been proposed to construct the KOS (section 2).Subsequently, it describes the available models used to capture the quasi-static and dynamic behavior of the KOS (section 3).The underlying assumptions of the available models, and their drawbacks and benefits will also be highlighted.The review then classifies the current engineering applications of KOSs and highlights the main research performed under the following categories: Bit-memory switching, energy absorption, vibration isolation, metamaterials, aerospace structures, robotics and actuators, antennas design, energy harvesting, and force sensing (section 4).For each section, we provide a summary of the main findings followed by possible directions for future work.Finally, we provide our final remarks on the direction of KOS related research (section 5).

Fabrication
The transition of engineering designs based on the concept of the KOS from an idea that is tested in a controlled laboratory environment to an actual product that can be implemented in real life depends on the ability to fabricate durable and robust springs that can be mass manufactured through repeatable and automated processes.Manual folding of paper sheets as described in section 1 is the traditional way of building the KOS.Other, more direct methods that bypass the manual folding process and eliminate the need for a flat base sheet were also proposed and implemented.Such methods have been created with the following objectives in mind: (i) reducing stresses at the creases to prevent fatigue and increase durability, (ii) reducing variability and increasing repeatability in the response behavior of the different samples, which is key for engineering implementation, (iii) increasing the load bearing capacity of the springs, and (iv) incorporating non-paper based materials that are more suitable for implementation in realistic and harsh environments.
In general, fabrication methods can be divided into two main categories; namely, sheet-based fabrication and nonsheet fabrication.In this section, we summarize those methods and highlight their major advantages and drawbacks, if any.

Sheet-based fabrication
All sheet-based fabrication methods start from a flat sheet of material, but differ in the way the creases are created and the folds are initiated.Creating functional creases is probably the most challenging aspect of origami, as the outcome often dictates whether we can achieve the desired performance characteristics.This is assessed by the ability of the fabricated structure to create all of the possible restoring force behaviors including bi-stability in specific geometric combinations.
The most primitive sheet-based fabrication method involves the simultaneous buckling and twisting of a cylindrical paper or cardboard as shown in figure 6(a), [52,57].As aforementioned in section 1, this fabrication method is also how the Kresling pattern was initially uncovered [31,56,58].In this method, the relative angle between the distal and proximal ends of the KOS change by increasing the level of twisting, thereby altering the overall shape, class, and behavior of the Kresling structure.For instance, when minimal twist is combined with compression, it yields a Yoshimura-like bellow (figure 1(b)) [6,31,36,52].Since this method involves the application of force, it leaves the structure strained in its idle state.Furthermore, the qualitative nature of the twisting and compressing actions leads to inconsistencies in the behavior of the resulting structure.
To create creases on a flat sheet of material, different methods have been proposed.Non-subtractive methods include manual folding, embossing, or laminating as shown in figure 6(a).The process of lamination involves laying two or more dissimilar polymeric or composite sheets and bonding them together with a compatible adhesive.However, this layer-by-layer lamination is susceptible to delamination under severe loading [45].A more durable sheet can be created by wrapping the entire facets with a tough flexible film, which provides protection and acts as a flexible hinge [63].To accomplish this effectively, rotating dies, as shown in figure 6(a), may ensure uniform overlay of dissimilar sheets.Similarly, a hard sheet, which may serve as the base sheet, can be partitioned in place into the characteristic triangular facets of the KOS [64].Thereafter, a thin tough compliant film is wrapped and glued around the hard facets.We call this procedure a fullwrap lamination approach, which essentially ensures that the creases are free of hard material, enabling reliable hinge foldability during service.
Subtractive methods including laser etching, computer numerical control (CNC) machining [65], or a scraper (origami knife) were also used to create the creases.A spiral version of the Kresling pattern shown in figure 6(b) is usually utilized when applying subtractive approaches (CNC cutting) because it permits forming clean cuts to achieve the intended design.Among the crease designs made by laser etching, lowpower and short exposure time remove some material along the traced outline, creating trenches on the base sheet without cutting through it (figure 6(b)).Whereas using high power and long exposure time, the material can be completely pulverized, resulting in a full cut [66].When using dashed lines in the traced outline and high laser power, it is also possible to create a single line of evenly spaced perforations.A more effective crease design involves the use of double perforations, where two dashed outlines are positioned at a distance from the center line of the crease [40,67,68].This design results in compliant hinges that exhibit deep potential energy wells and high stiffness, along with enhanced resistance to fatigue compared to single-line perforations [40].The density of the dashed lines, the shape of the slots (subtracted holes), and the spacing between the two dashed lines can be adjusted as design variables to regulate the compliance of the creases [67].
To create the final KOS, the creased sheet is rolled over into a cylinder then preconditioned by applying several cycles of uniaxial compression.During this process, it is observed that mountain folds often experience high stresses and are prone to buckling or fracturing [60,69].Various remedies have been proposed by researchers, including incorporating large cuts at the center of the mountain fold [60] (figure 6(d)), adding circular holes at the nodes [70,71] (figure 6(e)), or completely eliminating the mountain fold [40] (figure 6(f)).KOSs without the mountain fold are typically created by tracing the 'spiral Kresling outline' shown in figure 6(b) (CNC cutting), through a 'Kirigami' approach, which is also regarded as 'Flexigami' [72].Despite the absence of the mountain fold, these Kirigami springs are as functional as conventional KOSs while being more durable.Kirigami springs made from spiral cuts do not require the final rolling of the sheet.Instead, the entire cut is pulled upright, and the overhanging facets are glued to the base.A notable difference from springs made using the traditional pattern, is that Kirigami-based structures are not strained since they are not forced into shape by rolling.This results in the absence of elastic energy buildup in the idle position of the structure.To enhance the functionality of paperbased Kirigami springs (Flexigami), stiff facets can be inserted between two paper sheets before cutting out the spiral pattern.These additional facets increase panel stiffness and deepen the energy barrier for folding, making it advantageous for achieving peristaltic locomotion [51].
2.1.1.Sheet materials.Several materials were proposed for sheet-based fabrication.The type of material used depends on the fabrication method, the application at hand, and the implementation environment.Typical materials used are: • Thick papers, such as Strathmore 500 Series 3-PLY BRISTOL and Tant paper or cardboard were utilized for creating impact mitigation devices [40], stretchable robotic arms [47], switches [74], and crawling microrobots [76].
In molding methods, a mold resembling the Kresling column (a stack of KOSs) is first prepared using 3D printing (figure 7(a)).Next, uncured silicon resin or thermoplastic Polyurethane (TPU) is poured into the mold [83,96].To ensure that a thin layer is formed, instead of a solid block of Kresling column, Li et al [83] employed spin casting, where the mold is spun at a rate of 10 rad s −1 , resulting in a homogeneous thin layer of silicon coating on the inner surfaces of the mold.As the deposited layer gradually cures (its viscosity increases), it takes the shape of the final Kresling mold.Finally, the mold and its content are cured (∼45 • C) for 30 min, and the mold is peeled off.Alternatively, a static, nonspinning mold, typically made from Steel, may be used for extrusion blow molding with materials such as high-density Polyethylene (HDPE) [102] or hydroforming of Steel sheets [103].
The mold can also be used as a rigid mandrill over which base material filaments like carbon fiber reinforced plastic (CFRP) can be wound [86] (figure 7(a)).In this case, the mold must possess sufficient radial stiffness to withstand the winding stresses.Acrylonitrile Butadiene Styrene (ABS), PET, and Polylactic acid (PLA) are deemed viable options [86].A 3D printed mold can also be used in vacuum molding via thermoformers, especially for imparting complex impression on ultra-thin base films, such as PVDF and PET.The outcome is the perfectly air-tight bellow shown under figure 7(c), which is suited for cryogenic space application as in fuel bladders [84].
Additive manufacturing, particularly 3D printing, has also been used extensively to fabricate the KOS.Fabrication methods involve both the use of single material (mono-lithic) or multiple materials (figure 7
Since a monolithic KOS is incapable of exhibiting the full features of the traditional paper-based KOSs such as multi-stability, Khazaaleh et al [94,95] proposed a composite design that comprises two materials with contrasting elastic moduli (figure 7(d)).Recognizing the 3D printing capability, particularly PolyJet technology [109] in fabricating multiphase structures (composites) and its ability to create intricate designs free of human errors with repeatable performance, they constructed the functional two-phase KOS shown in figure 7(d).The resulting structure is durable (∼ 5000 cycles), fully functional, and capable of generating multi-stability.This approach permits the design of architectured cushion material using Kresling unit cells, repeated in 3D space [48].Adding a two-phase material also enables bi-stability in the actuation of pneumatic systems, where the relative elastic moduli of the two materials serve as a tuning parameter [89].
Several researchers independently proposed truss-based models where the facets and creases are replaced with either elastic or rigid bars (i.e.linkages) accompanied by linear springs at either or both of their ends [16,38,49,93,110,111].This truss-based structure shown in figure 7(e) is composed of a number of linkages, each of which corresponds to the creases of the KOS.The ends of the linkages rest on one of the following joints: a frictionless ball socket joint (see the joints on figure 7(e)), universal joints [110], a universal spring joint [16,38], or an end rod bearing [49] (shown in figure 7(e)), which are attached to a rigid top and base plates.The addition of spring joints makes the rigid linkages flexible, allowing axial deformation of the linkage during the rotation of the top and bottom plates.Alternatively, the linkages can be made flexible by placing tensile springs in the middle of the linkage [110].The absence of folds or creases, along with the associated manufacturing challenges, eliminates many limitations on the choice of base materials which improves durability [65,87,93,112].The structure can also be actuated by attaching a direct electric drive to the linkages [93].However, such structures are generally bulky and not easily scalable, and the process of manufacturing is rather complex requiring the assembly of multiple elements.The interlocking of linkages at small deployment heights also prevents full deployment of the spring [73].

Assessment of fabrication methods
Regardless of the intended application, fabricating durable, robust, and cost-effective KOSs, that have adequate load bearing capacity without sacrificing some of the key features including multi-stability, is key towards further utilization of the KOSs in actual engineering applications.Sheet based fabrication results in KOSs that can achieve all the desired restoring characteristics (mono-, bi-stable, and QZS).They are generally low cost and can be designed to withstand the environmental elements when the proper folding material is used.However, these fabrication methods are not ideal for mass manufacturing as they require manual folding and creasing which results in variabilities among the different samples.Automated embossing, laser cutting, and the implementation of CNC to create creases improve repeatability, but the process still requires manual folding.In addition, plastic deformations that form at the creases during folding reduces durability especially under dynamic loading.The use of perforations to create the creases reduces stresses and increases durability.Laminating different materials to create the creases also reduces stresses at the folds, but makes the structure Non-sheet based fabrication methods on the other hand are either automated or semi-automated, thus they are more reliable, and more suitable for mass manufacturing.Due to the absence of creasing and folding, these methods also result in more durable structures as compared to sheet-based fabrication methods.Molding and winding are typically low cost manufacturing processes, but can only produce mono-stable KOSs, which is a major drawback.Monolithic 3D printing methods are medium cost, but their current implementation to design KOSs only resulted in mono-stable designs.
Linkage design and assembly is also medium cost.The resulting structure is generally sturdy, and can withstand the environmental elements making it suitable for engineering applications.However, linkage assembly is generally complex and not easy to automate which is a drawback for mass manufacturing.The resulting structure has limited deployment range, and scalability to the small scale can be difficult.
Finally, biphase 3D printing of two material is currently a costly process.However, unlike monolithic methods, it has been successfully used to design springs that can achieve mono-and bi-stable restoring characteristics.It is also ideal for mass manufacturing, but the resulting structure may have lower durability because of possible delamination at the interface between the two materials.
Table 1 summarizes our assessment of the main fabrication methods.

Future research directions
We believe that advanced manufacturing technologies, namely, 3D printing, is the most suitable approach to adopt for the fabrication of KOS design.In particular, monolithic 3D printing is, on one hand, not very costly, and is, on the other hand, an automatic process that is ideal for mass manufacturing.The issue remains in finding the 3D printing materials and geometric topographies that enable the design of springs that can achieve all the desired restoring force characteristics, while minimizing stresses and stress concentrations during deployment.This, in our opinion, should constitute one direction for future research.
Another area for further exploration is the use of additive manufacturing technologies to embed active elements (e.g.piezoelectric, magnetostrictive) inside the KOS to allow for actuation and sensing capabilities.Some recent research has already been performed on injecting magnetorheological fluid inside the creases to actuate the springs [115], but more systematic approaches should be devised and pursued.

Modeling and mechanics
In order to fully utilize the advantages of the KOS and optimize its response behavior for different applications, it is imperative to develop models that help researchers understand how they respond to external loading conditions.In the literature, we find two types of models: those that study the quasistatic behavior and static equilibria of the springs, and those that address their response behavior under dynamic loading, whether it is transient or steady under harmonic excitations.In what follows, we review and summarize the available literature on the modeling of KOSs with the goal of identifying any voids that can be filled through future work.

Quasi-static modeling
Most of the available literature on modeling the behavior of the KOS focuses on its behavior under quasi-static loading (axial or torsional).In such a scenario, the applied load is assumed to be slowly varying such that any dynamic effects can be safely neglected.The goal of such models is to predict the restoring force behavior of the KOS and identify the combination of design parameters that yield qualitatively different equilibrium characteristics.In general, those models can be classified into truss-based and computational.
3.1.1.The truss model.One of the earliest studies addressing the quasi-static modeling of KOSs appears to be that of Jianguo et al [37], who developed what is known as the conventional truss model.In the proposed model, each triangular panel is replaced by three truss elements of initial lengths, a 0 , b 0 , and c 0 as shown in figure 8.The position, u, and rotation angle, ϕ, of the two end planes of the KOS can then be described by the lengths, a, b, and c, of the three trusses during deployment.The forces developed in each of the truss elements during deployment can be obtained by using the proper constitutive relations.
In their work, Jianguo et al [37] assume that the sides a = a 0 , and b = b 0 remain rigid during deployment and that side c is the only side that deforms.Using this assumption, they developed an expression for the potential energy of the spring by summing the potential energy of all c trusses as following: where E is the elastic modulus, and A is the cross-sectional area of the 'c' truss element.Here, c = c(ϕ, u) is the length of the truss element during deployment, c 0 is its initial length, and n is the number of polygonal sides of the KOS.By setting the gradient of equation ( 1) with respect to u and ϕ to zero, the respective restoring force/torque behavior of the KOS and its different equilibria can be obtained.Different design parameters of the KOS result in qualitatively different restoring force behavior as shown in figure 9. Figures 9(a)-(c) represent, respectively, KOSs exhibiting mono-stable linear, nonlinear hardening, and nonlinear softening behavior.Figure 9(d), on the other hand, depicts the behavior of a bi-stable KOS which exhibits two stable equilibrium points occurring at the minima of the potential energy function (green circular markers).Those minima are separated by a potential energy barrier, ∆Π, that must be overcome by supplying enough external energy to force the KOS from one of its equilibrium states to the other.Typically, the bi-stable potential energy function of a single KOS is asymmetric; that is, higher energy levels are required to force the KOS from its trivial equilibrium point at S 0 to its non-trivial equilibrium point, S 1 , than the opposite (∆Π 0 > ∆Π 1 ).Finally, figure 9(e) represents the behavior of a QZS spring, where the KOS has nearly zero stiffness (slope of the restoring force curve) over a wide range of deflections around the u QZS point.
It was later shown by various researchers [38,42] that the assumption invoked by Jianguo et al [37] on link b being rigid is inaccurate and leads to an overprediction of the height of the potential barrier, ∆Π, for bi-stable KOSs.A modification to the truss model was initially proposed by Yasuda et al [38] and later studied in more detail by Masana and Daqaq [42].In the modified model, only truss a is assumed to be rigid.In this case, the length of the different trusses can be described in terms of the deployment height, u, and the twist angle ϕ using (see figure 4(c): Using equation ( 2), the total strain energy stored during deployment can be written as The restoring force of the KOS and its equilibrium states (u e , ϕ e ) are determined by minimizing the strain energy, equation ( 3), with respect to u and ϕ.Specifically, by enforcing Π u |(u e , ϕ e ) = Π ϕ |(u e , ϕ e ) = 0 at any equilibrium state, where Π u and Π ϕ represent ∂Π/∂u and ∂Π/∂ϕ, respectively.An equilibrium configuration is considered physically stable only if it corresponds to a minimum in the strain energy, which is satisfied when Masana and Daqaq [42] created maps in the design space of b 0 /R versus the design angle, α 0 (see figure 4(a) for the definition of α 0 ), to demarcate regions of qualitatively different spring behavior.Figure 10(a) shows such a map for a KOS with n = 6.The unshaded region, which occurs when b 0 /R > 2 sin (π/n)/ sin α 0 , represents combinations of  (α 0 , b 0 /R) where no fundamental triangle can be constructed, and hence, a KOS cannot be designed.On the other hand, the shaded region represents the geometrical parameters with which a fundamental triangle, and hence a spring, can be created.Within this shaded region, the KOSs can be developed only when b 0 /R < 2 sin (π/n)/ tan α 0 .When 2 sin (π/n)/ tan α 0 < b 0 /R < 2 sin (π/n)/ sin α 0 , the triangles can be tessellated only to develop Yoshimura pattern bellows [33].Based on the number of stable equilibria of the KOS, the shaded region can be divided into two main types; mono-and bi-stable, referred to as the M-and B-types, respectively.An M-type KOS can settle at only one equilibrium in the absence of the external load, while a B-type KOS can settle at two equilibrium positions in the absence of external loads.As shown in figures 10(b)-(f), each of the M-and B-type KOSs can be further sub-classified based on their specific potential characteristics as discussed below: 1. M1 type: A KOS of the M1-type is characterized by a mono-stable behavior, with its equilibrium position located at u/R = 0 as depicted in figure 10(b).Due to this configuration, the M1-type KOS can only be deployed when subjected to a tensile load.Once the load is removed, the KOS returns to its original state at u/R = 0. Additionally, because the potential energy at the equilibrium point is nonzero, the truss elements or creases of the KOS are under strain at this position.2. M2 type: A KOS of the M2-type is also characterized by a mono-stable behavior, but its potential energy function exhibits a qualitative difference compared to the M1-type as shown in figure 10(c).In this case, the stable equilibrium point occurs at a non-zero value of u/R, while an unstable equilibrium point is present at u/R = 0. Consequently, the M2-type KOS will settle at a non-zero deployment height at its stable equilibrium point.This unique configuration allows the spring to deform under both tensile and compressive loads.Additionally, since the potential energy at the stable equilibrium position is zero, the creases of the KOS remain completely unstrained at this point.As the b 0 /R ratio or the angle α 0 are increased within the M2 region, the equilibrium position moves farther away from u/R = 0 and the potential energy at the unstable equilibrium increases [42], implying the spring becomes stiffer.It should be noted that M2 region extends upto α 0 = 5π/n = 150 • when n = 6.3. M3 type: In contrast to an M2-type KOS, the M3-type springs exhibit strained creases at their stable equilibrium point, as indicated by the presence of non-zero potential energy at those positions as shown in figure 10(d).By increasing the ratio b 0 /R, the equilibrium position shifts towards larger values of u/R, resulting in a deeper potential well.Conversely, when α 0 is increased while keeping the b 0 /R ratio constant, the equilibrium moves to smaller values of u/R, causing the potential well to become shallower [42].
4. B1 type: A KOS of the B1-type is bi-stable with two unstrained non-zero stable equilibria separated by an unstable equilibrium point as shown in figure 10(e).As the b 0 /R ratio or α 0 is increased independently within the B1 region, the potential energy at the unstable equilibrium increases leading to deeper potential wells.This entails the need for a larger static load to initiate transition between the two equilibria.5. B2 type: A KOS of the B2-type is bi-stable with one strained stable equilibrium at u/R = 0 and one unstrained equilibrium at a non-zero value of u/R as shown in figure 10(f).As the b 0 /R ratio or α 0 is increased within the B2 region, the height of the potential barrier increases making it harder to transit from one of the stable equilibria to the other.
Table 2 summarizes the effect of the design parameters on the aforedescribed classifications.
Using the truss model, it is possible to study the influence of the number of sides, n, of the polygonal cylindrical structure on the design maps as shown in figures 11(a)-(e).Upon initial observation, it is noted that as n increases, the boundary between the regions M1-M2 and B1-B2 moves towards lower values of α 0 , while the boundary between the M3 and M1 region shifts towards higher values of b 0 /R4 .As a result, the size of the M2 region clearly increases, while the  size of the M3 region decreases.The size of the B1 and B2 regions also decreases as the number of sides of the polygon is increased, while the size of the M1 region remains almost constant.Figure 11(f) depicts variation of the relative area size of a given region within the design space as the number of sides of the polygon, n, is increased.Figure 12 shows an example of the influence of n on the potential energy function and the actual longitudinal-rotational deployment of the bistable KOSs designed using the same values of u 0 /R, ϕ 0 , and R. It is observed that, as n increases, the lower equilibrium position moves to smaller values of u/R and the potential energy barrier, ∆Π, decreases, implying a larger operation range and reduced strain levels.As such, one can correctly conclude the following: 1.The size of the geometrical design space that permits building bi-stable springs decreases as n increases [42,122,123].2. The size of the geometrical design space that permits building springs with unstrained equilibria increases as n increases, [42].3. The KOSs become softer and have increased deployability as n increases.In particular, keeping the other design parameters fixed, the potential barrier of the bi-stable springs decreases with n (figure 12).
To understand how the relative rigidity of the b and c links influence the quasi-static behavior following the truss model, we study their effect on the shape of the potential energy  function, the deployment path, and the equilibria (figure 13).It can be observed that, according to the truss model, the number of equilibria and their position in the deployment path do not change with the relative axial rigidity of the links.However, away from the equilibria, the stiffness and the coupling between the longitudinal and rotational degrees of freedom change.More specifically, assuming that the b-link is much more rigid than the c-link causes an increase in the height of the potential barrier, thereby requiring larger levels of energy to switch between the two stable equilibria of the bi-stable KOS.On the other hand, assuming that the b-link is softer than the c-link, as in Kirigami cuts, causes the height of the potential barrier to decrease, which makes it easier to switch between the different equilibria of the bi-stable KOS.
While the truss model can be used to predict the location, number, and stability of the equilibrium points of the KOS , it cannot accurately predict the quantitative quasistatic behavior away from the equilibrium points [42].This is because of the inaccuracy of many of the following underlying assumptions: 1.The triangular panels/facets do not warp.2. Deformation of the truss elements follow an ideal linear stress-strain curve which remains purely elastic irrespective of the strain level.3. The truss elements do not buckle under the applied load and their mutual interaction is ignored.4. The thickness of the panels is negligible. 5. Rotary stiffness of the creases is negligible.6.The self-contact of the panels is to be ignored; i.e. the resistance offered by panel interaction is negligible.In fact, it was observed experimentally and proven theoretically that the triangular panels do warp during deformation [42,124], and that the deformation of the truss elements exhibits a nonlinear stress-strain behavior especially under larger deformations [125].Moreover, the creases were observed to buckle under loading, and self avoidance due to panel contact was found to be significant at small deployment heights.
To capture some of the behaviors missing from the truss model, Filipov et al [126] introduced the bar and hinge model (figure 14(a)).The model captures the in-plane deformation of the panels, their bending, and the folding stiffness along the fold lines.In the proposed model, the in-plane stiffness of the panels and the creases is replaced by a bar element with axial stiffness, while the folding stiffness is replaced by linear elastic torsional hinges.The model does not take into account the buckling that can occur at the creases.
The model was improved further in a later study by Liu and Paulino [125] by assuming nonlinear constitutive relationships for both the bar and hinge mechanisms.In their seminal general formulation, which works for any non-rigid origami pattern, geometric nonlinearities due to large deformations are accounted for in a way that singularities arising from trigonometric functions are mitigated, which is key for the stability of numerical simulations.In particular, when describing the orientation of the panels, Liu and Paulino [125] used a formula that involves the cross product of the vectors normal to the planes prescribing the angle.Thus, the description of the angles and their derivatives were free from any trigonometric expressions that could yield singularities.In their formulation, the bar element is modeled as a general hyperelastic material by adopting the Ogden constitutive model resulting in either linear, softening, or hardening behavior depending on the type of material used.The hinge (rotational spring) is modeled as a nonlinear spring which is linear from large angles, but experiences a rapid increase in stiffness as the angle between the panels decreases during folding.This accounts for panels selfavoidance.The model was implemented on several origami patterns including a multi-stack KOS, and the deployment of the stack under prescribed loading was analyzed.
Masana and Daqaq [42] studied the quasi-static response experimentally and showed that there is a clear discrepancy between the experimental findings and the results of the truss model as shown in figure 14(b).In particular, it was shown that the truss model predicts the equilibria quite well, but not the actual behavior of the restoring force curves away from the equilibria and at small deployment heights.Thus, they proposed a more comprehensive truss model that takes into account both of the added stiffness due to panel contact and the buckling of the truss elements.In the new model, the rotation of the panels about the creases is modeled using torsional springs of hardening nonlinear behavior.To account for the contact of the panel at small angles, the torsional springs have an additional hardening component which is activated only when the angle of the panels is less than a critical angle.To account for the buckling behavior of the truss elements, a nonlinear transversal cubic spring is also added in the axial direction.Results of the new model were compared to experimental data showing excellent agreement in predicting the quasi-static curves even at small contact angles.Figure 14(b) depicts the modified truss model and its response compared with the truss model and the experimental results.One drawback of the modified truss model is that the stiffness of the springs need to be determined experimentally and adjusted for every type of materials used.
Yasuda et al [40] also used torsional springs (hinges) in the truss model to capture the folding stiffness around the a-links.In addition to the torsional springs around the a-links, Jin et al [99] added torsional springs between the mountain creases; i.e. between the b-links and the transverse plane, to capture the folding stiffness of the triangular panels.
Literature on the modeling of the KOS also involved a few additional deviations from the conventional truss model.For example, Pagano et al [50,78] modeled the deployment of the KOS using only torsional springs along the creases (a and clinks).To account for the out-of-plane panel bending, the panels are assumed to be bending along virtual folds.The position of the virtual folds is empirically determined to emulate the conditions observed in the experimental models.
In a series of articles, Huang et al [96,127] assumed that the axial strains in the creases and the associated strain energy is insignificant when the deployment range is limited to low values.They modeled the stiffness of the creases by using torsional springs at the mountain and valley creases (b and clinks).In a follow up article [127], Huang et al considered nonlinear torsional springs along the creases and theoretically studied the effect of the ratio between the linear and the nonlinear stiffness of the hinges on the restoring force.They reported that, for some combinations of the design parameters, the KOS can exhibit a tri-stable potential; a phenomenon which was never observed experimentally by the authors themselves or in the literature.This raises questions regarding their initial assumption that the contribution of the axial strain in the trusses is negligible, which also contradicts with earlier findings that multi-stability arises due to panel stretching, and not panel bending [37,125].KOSs with tri-stable potential can be achieved by outward popping of the valley creases [80,[128][129][130].In [80], Wang et al modified the truss model to include the outward popping of the valley creases by considering a mid-node along each valley crease.This mid-node results in two truss members along the valley creases (c-links), which, in turn, enables the truss model to capture out-of-plane bending of the panels, and therewith tri-stability, see figure 14(c).The high compressive stiffness around the new, third equilibrium position, permits carrying loads exceeding 1200 times weight of the KOS.
3.1.2.Computational models.In addition to the simplified truss models, which offer an excellent qualitative understanding of the quasi-static response behavior of the KOS, some researchers also developed computational models to assess the response behavior quantitatively.Such models are computationally more expensive, but are generally more accurate than the truss model.Computational models developed in the literature are generally based on commercial softwares such as ANSYS or ABAQUS [95,131,132].
Regardless of the software used, certain assumptions need to be invoked on the computational domain.A common assumption is the use of linear constitutive equations to model the material [95,131].Since the thickness of the panels is extremely small compared to other dimensions of the KOS (typically less than <1/15 of the characteristic length), another, rather safe assumption, involves reducing the threedimensional continuum theory to 3D planar elements, which offer better computational efficiency.Modeling self-contact between the panels at small deployment heights is an issue that requires special attention and can be addressed by using the Augmented Lagrangian method, as discussed in reference [132,133].The relative tangential shear between the rubbing panels is often neglected.
During uniaxial compression or tension, the boundary conditions at the bottom of the KOS, y = 0, can be prescribed as (figure 8): while the boundary conditions at the top surface, y = u, are where u x , u y , u z are the displacements in the (x, y, z) directions, respectively, and ω x , ω y , ω z , are the associated rotations.
Computationally, this vertical displacement, u y is applied incrementally through a ramped down step from 0. Zero rotations, ω x = ω z = 0, at the top polygon prevent out-of-plane tilting during deformation.The displacements u x , u z and the rotation, ω y = ϕ, around the y-axis are the unknowns to be determined from the computational model.For torsional loads, the rotation is prescribed at the top end, leading to the following boundary conditions: and the displacements u x , u y , u z become the unknowns to be determined from the computational model.
A mesh convergence study must be performed to determine the number of elements necessary for the stresses and reaction forces within the model to become mesh independent.The restoring force, F, is computed by summing the reaction forces over the nodes comprising the top surface.
Dalaq and Daqaq [95] used ABAQUS to develop a computational model for the 3D-printed multi-material KOS shown in figure 15(a).They validated the quasi-static behavior for different combinations of the design parameters and showed that the computational model can accurately predict the response for all qualitatively different behaviors (e.g.linear, nonlinear, QZS, and bi-stable) (figures 15(b) and (c)).Using the validated model, they generated design maps in the parameters space that demarcate the regions of different spring behavior (figure 15(d)).They also studied the effect of the number of  sides, n, on the restoring force and showed that, in general, KOSs with lower values of n are stiffer, and that no significant changes in the restoring force behavior is observed when n > 8.
Moshtaghzadeh et al [131] used finite element ANSYS modeling to study the buckling behavior of a Kresling column consisting of three KOS modules and showed that the buckling load increases as the number of sides, n, of the polygons increases; that is, columns with larger number of polygon sides are harder to collapse under loading.They also studied the effect of the b 0 /a 0 ratio and the radius, R, of the KOS, on the stability of the column and found that the buckling load decreases when b 0 /a 0 and/or R increases.

Experimental testing.
To verify the theoretical results, experimental quasi-static testing is performed using typical compressive/tensile/torsional testing machines (e.g.Instron machines).However, because of the axial-twist coupling, special setups must be devised to permit the free motion of one of the coordinates; either u or ϕ, when the restoring behavior associated with the other coordinate is being characterized.For instance, when measuring the restoring force in the axial direction, a controlled fixed-rate displacement is applied to one end of the KOS, while the other end is placed on a platform that can freely rotate about a common centroidal axis, as shown in figure 16.The restoring force can then be measured using a load cell, and the rotation of the other end can be tracked using any of the available digital image correlation (DIC) tools.On the other hand, when the restoring torque is to be measured, one end of the KOS is subjected to a controlled rate of angular displacement through a torsion testing machine, as depicted in figure 16, while the other end is placed on a sliding frictionless bearing that allows longitudinal motion to be free while the rotary motion is restrained.A torque cell is used to measure the applied torque at the restrained end, and the relative longitudinal displacement is monitored using any DIC tool or a laser position sensor.Each sample must be conditioned by performing several testing cycles before recording any data.Tests have to be performed at a very slow rate in both directions to guarantee quasi-stationarity and to assess hysteretic losses.The effect of rate dependence must also be assessed by performing the tests at different rates.If there is a clear dependence of the force/torque restoring curves on the rate of the prescribed displacement/rotation, the quasi-static response curves cannot be used directly in the dynamic models.
3.1.4.Effect of perforations.As stated in section 2.1.1,the addition of perforations to the creases decreases their effective stiffness and mitigates the damage that occurs due to kink formation or buckling of the creases.This improves the durability of the KOS without compromising much of its overall qualitative behavior.The addition of perforations has significant influence on the restoring force behavior of the KOS and has been studied comprehensively in the literature.
Hwang [67] studied the effect of perforations made at the creases on the quasi-static behavior and stability of the equilibria of a KOS.The analysis was performed using finite element methods and results were compared to experimental data.For the modeling and experiments, the authors used Polyoxymetheylene, which has high stiffness, strength, and ductility, and studied the effect of the length, L, of the slot hole, its width, W, and the horizontal and vertical distance between the holes, I and H, respectively on the quasi-static response (figure 17).The case when L = 10 mm, I = 0.7 mm, and H = 1 mm results in the smallest peak reaction force and stiffness during deployment, which is the closest behavior to the conventional truss model.The maximum reaction force of the mono-stable springs occurs when L = 5 mm, I = 1.3 mm, and H = 2 mm for a spring with n = 6, R = 36 mm, u 0 = 35 mm, and ϕ 0 = 48 • .The effect of W is deemed negligible.It was also observed that larger values of L and smaller values of H aid in the formation of bi-stable springs when the other parameters are kept unchanged.
Berre et al [60] discussed the effect of additional cuts (like in Kirigami) that are introduced at mid length of the creases.They experimentally studied the influence of the cuts by varying their lengths and measured the restoring force and the relationship between the deployment height and the rotation angle.They found that the position of the equilibria does not change with the cuts, but that the cuts modulate the stiffness of the structure.In another study, Yang et al [69] studied the impact of vertex cuts in non-rigid foldable origami bellows including Kresling and Miura-ori pattern bellows.Unlike in the work of Berre et al the cuts were introduced along the four mountain creases that meet at the vertex (see figure 6(e) in section 2).Using computational modeling and experiments, they observed that the response of the KOS is highly sensitive to the size of the cuts and that the stiffness drops by over 50% when the cuts are about 2.5% of the crease length.They also observed that, for certain designs, the second equilibrium of a bi-stable unit can completely vanish.

Quasi-static behavior of KOS pairs.
KOSs exhibit unique restoring properties with axial and torsional functionalities, making them suitable for various engineering applications.However, a single KOS always results in coupled translational-rotational motion at the free end under loading, limiting its versatility.To address this kinematic constraint, Li et al [134] exploited the modularity of the KOS by joining two KOSs in series to form a Kresling Origami Spring Pair (KOSP).Depending on the relative slope of the creases (chirality) in the stack, the free end of the KOSP can either have coupled or decoupled motion.When the creases of the constituent KOSs have opposite sign slopes (achiral), the motion at the free end becomes decoupled.Conversely, when the creases have similar sign slopes (chiral), the motion at the free end remains coupled but with an extended range of operation.We refer to KOSs with decoupled motion as d-KOSPs, while those with coupled motion are denoted as c-KOSPs.
Figure 18(a) shows the response of two KOSPs, chiral (orange) and achiral (green), when a translational motion, u T , is prescribed at the top polygon of the KOSP, while the bottom polygon is fixed.The free end of the d−KOSP (green) moves up or down without rotating under the prescribed axial motion.This rotation-free motion can only be realized because of the rotation and translation of the center polygon.In contrast, the c-KOSP (orange) undergoes coupled rotational-translational motion when a translational motion, u T , is applied to the top polygon.Similarly, as shown in figure 18(b), a prescribed rotational motion, ϕ T , applied to the top polygon of the d-KOSP results in coupled translational-rotational motion of the central polygon, while maintaining the total height of the KOSP constant.
In a theoretical study, Li et al [134] used the conventional truss model to identify the regions in the design space where  the KOSP exhibits qualitatively different features including mono-stability, and bi-stability.Masana and Daqaq [135] studied those behaviors theoretically and experimentally for d-KOSPs consisting of similar and different KOSs.They studied how the equilibria of the d-KOSP bifurcate as the precompression height of the KOS is varied.Figure 19(a) shows how the potential energy function changes shape as the prescribed height, u T , is changed.It is evident that the d−KOSP transits from being mono-stable at large values of u T to bistable at medium values, to mono-stable again at small values.
An experimental bifurcation diagram of the equilibria of the d−KOSP as u T is varied can also be seen in figure 19(b).

Dynamic modeling
While the quasi-static analysis provides valuable insights into the fundamental behavior of the KOS, real-world applications demand a deeper understanding of its dynamic characteristics.This is specially the case under dynamic loading conditions (vibration and energy absorption, metamaterials), or when the bi-stable KOS undergoes rapid state transitions (mechanical switching), which introduces additional challenges in terms of design and control.To fully exploit the benefits of KOSs in practical applications, it is therefore crucial to model and understand their dynamic behavior.
A simple extension of the quasi-static truss model to the dynamic case can be achieved by incorporating the inertia terms resulting from the kinetic energy of the KOS.Assuming that the upper and lower polygons remain horizontal and parallel during deployment, the kinetic energy of the KOS can be written as (see figure 8): where m is the total mass of the KOS, I yy is the mass moment of inertia term around the y axis.Note that the moment of inertia is a function of ϕ and u since the location of the center of mass of each panel changes with ϕ and u.The undamped equations of motion of the KOS can be derived using Lagrangian equations as: where q ≡ [u, ϕ] are the generalized coordinates, and F i ≡ [F, T] are the generalized forces.Here, F is the applied force in the n 1 direction and T is the applied torque around the n 1axis.Note that the resulting equations are inertially and elastically coupled because of the dependence of the inertia terms on ϕ and u.The potential energy function, Π, can be adjusted depending on the type of truss model used [37,38,73,125].Equation ( 8) is valid as long as the damping terms are very small compared to the inertia and stiffness terms, and no rate-dependent effects like viscoelasticity are significant.Such effects are indeed significant for some of the materials used to construct the KOS.For instance, the restoring force curves reported by Reid et al [33] and Khazaaleh et al [94] have large hysteresis loops that are typical of the viscoelastic effect.In such cases, the restoring force/torque based on the quasi-static description will deviate from the dynamic (high-frequency) restoring force behavior.To the authors' knowledge, the effects of such rate-dependent effects on the dynamic behavior of the KOS have not been studied in the literature.
Kidambi and Wang [44] were among the first to extend the quasi-static truss model to the dynamic scenario by considering the inertia of the panels.They used a purely theoretical model to study the transient deployment behavior of the KOS by compressing the springs to a certain value then releasing it and tracking the motion of the top polygon.Their model also accounted for eccentricity (off-axis response; i.e. rotation of the top polygon around x and/or y axes, combined with displacements in the x, and/or z directions) during deployment and was extended for multiple KOSs connected in series by summing the potential and kinetic energy of all the springs in the stack.Agarwal and Wang [136] used the dynamic model developed in their previous paper [44] to numerically study the nonlinear forced response behavior of a KOS under harmonic excitations.They studied bifurcations of the response amplitude as the excitation amplitude is varied and showed complex periodic and aperiodic responses that are typical of a bi-stable oscillators.These include period-doubled responses, intra-and inter-well chaos.
Also, in their study on the use of KOSs for switching, Masana et al [73] experimentally studied the forced response of a bi-stable KOS under harmonic excitations by creating frequency-response curves.They experimentally illustrated the different bifurcations, the regions of chaos, and the intra-versus inter-well motions.They also adopted a phenomenological model of the spring to study the dynamic response theoretically.Huang et al [96] developed a coupled electro-mechanical model of the KOS for energy harvesting.Specifically, the restoring force behavior, obtained using the quasi-static model, was used to develop the governing equation of the dynamic system.They demonstrated that the results from this model compare well with the experimental results.
Yasuda et al [40] used a linearized version of equation ( 8) to study the dynamic behavior of a stack of KOSs.The linearization decouples the equations of motion inertially but retains the elastic coupling.Modal analysis was then performed to determine the modal frequencies and mode shapes of the structure.Using the modal equations, they developed a relationship between the longitudinal and torsional degrees of freedom and used it to reduce the dynamics into a single degree of freedom that characterizes the fundamental vibration mode.They used the resulting equation to study the compressive and rarefaction waves in a KOS stack.Miyazawa et al [81] followed a similar approach, but without the model reduction, to develop a dynamic model of the KOS.They used the model to develop the governing equations for a chain of KOS oscillators with alternating chirality.Han et al [137] followed a similar approach to develop a set of linearized coupled differential equations to model the dynamic behavior of a d−KOSP.

Summary and directions for future research
Modeling the behavior of KOSs during deployment allows researchers to understand how the KOS responds to external loading conditions, and permits optimizing its performance for the desired application.The reviewed literature on modeling can be broadly categorized into quasi-static and dynamic.Under quasi-static modeling, the conventional truss model is the most widely utilized due to its simplicity and ability to provide a qualitative understanding of the quasi-static behavior.In particular, the conventional truss model can predict the location of the equilibria and their stability, but cannot predict the actual restoring force behavior away from the equilibria.Furthermore, under the truss model, the relative rigidity of the truss elements is shown to only affect the restoring force behavior, but not the location or number of the equilibria.Thus, it appears that including the strain energy associated with the stretching/compression of the creases while imposing linear constitutive relations is sufficient to capture the qualitative behavior of the KOS.
For more accurate quantitative predictions, several modified truss models that take into account the nonlinear constitutive behavior of the truss elements, the possible buckling of the creases, and the self-avoidance of the sheets at small deployment heights were proposed [42,125].These models were generally capable of improving the predictions of the conventional truss model, but also added additional layers of complexity including the need for experimental system identification of the new model parameters.Computational models [95] of the quasi-static behavior were also presented and appear to be very accurate, but require more computational time and resources, and may lack the qualitative insights that the conventional truss model provides.
Dynamic models are particularly needed for applications involving vibration and impact absorption, metamaterials, and switching.Some dynamic models have been developed in the literature, but they mostly invoke several assumptions including linearity (small-amplitude responses around the equilibria, and linear elastic constitutive equations).To the authors' knowledge, none of such models have been properly experimentally validated.Also, when nonlinear models are developed and used, the analysis is purely numerical and lack the necessary insights that allow for design optimization under dynamic loading.The effect of viscoelasticity on the dynamic behavior is yet to be fully understood in the dynamic case despite the quasi-static response showing hysteresis and rate dependent characteristics.Computational models are also limited to the quasi-static cases.Such models are needed to analyze the modal behavior of the KOS and can be useful to establish discretized model of the dynamics.In summary, more research on the nonlinear modeling and analysis of KOSs including both analytical tools and experimental validations is necessary to gain deeper insights into the global behavior of the springs under dynamic loads.

Mechanical memory and switches
Memory is a fundamental cognitive process involving the encoding, storage, and retrieval of information.Various energy storage devices, including electrical, magnetic, and mechanical systems, utilize the dual states of entrenched systems to facilitate memory storage.While early mechanical memory storage devices like punch cards have been replaced by electromagnetic technologies such as hard disk drives (HDDs), and subsequently by solid-state drives (SSDs) in modern computers, the exploration of different types of mechanical storage devices continues.These devices offer potential advantages in harsh environments where extreme conditions may hinder electronic device operation.Origami, known for its design flexibility and adaptability, has recently been investigated for mechanical storage and computing applications [138].Various origami patterns have been explored, and the Kresling pattern stands out due to its programmable stiffness, and modular nature, making it a promising candidate for mechanical computing and bit memory switching.
Yasuda et al [38] were the first to demonstrate the feasibility of using a Kresling origami inspired truss structure as non volatile mechanical memory storage devices.The mechanical bit was formed by joining two similar KOSs (mono-stable units) but with opposite chirality end-to-end (like a d−KOSP).Under certain levels of prestress, the KOS exhibits two distinct stable equilibria that represent the two binary states of the bits.The bit was actuated using a controlled torsional input at one end.Using a laser vibrometer, the position of the intermediate plate was recorded for precise state measurement of the bit memory as shown in figure 20(a).Exploiting the modular nature of these bits, Yasuda et al [38] further realized a 2-bit mechanical memory by interconnecting the single bits end to end, and showed that the 2-bit can be quasi-statically actuated using single torsional input applied at one end of one bit.
Exploiting the asymmetric nature of the potential energy function of the bi-stable KOSs, Masana et al [73] devised a novel idea to actuate a memory based switch by providing harmonic mechanical excitations at its base.The switch can be activated to move from one state to the other by applying harmonic excitations of frequency and amplitude that are unique to each state of the switch and that are different for switches of different design parameters (figure 20(b)).Multiple binary switches of different geometric characteristics can be combined to create a mechanical memory board of multiple bits that can be selectively actuated using a single harmonic input.
Jules et al [120] introduced a memory architecture utilizing the bi-stable d-KOSP design, akin to the configuration illustrated in figure 18(a).Each d-KOSP represents a memory bit, with its two equilibrium states representing the ON-OFF states of the switch.The authors examined the response of the switch to a quasi-static axial loading and determined the threshold force required to transition between the two states.They observed that the threshold forces required to transit from one state to another depend on the current state of the bit switch and is different for the two equilibrium states.They also explored stacking four d-KOSPs of distinct geometric parameters in series and observed that the states and threshold forces of each bit are unique, and therefore, the structure can be driven to any of its 16 permissible states by following a distinct predetermined sequence of compression and extension (figure 20(c)).

Energy absorption
Energy-absorbing materials are critical towards safeguarding against deliberate or accidental encounters with forceful loads or impacts, whether arising from self-contact or external objects [139][140][141].The features of an excellent energy absorbing material can be understood by inspecting its force deflection characteristics (F − δ curve shown in figure 21(a)).Since the area under the F − δ curve represents the raw energy absorbed by the structure during loading, a good energy absorbing material must have a large area under this curve.Furthermore, the F − δ curve of an ideal cushion material should follow the yellow line in figure 21(a); that is, it must exhibit a sharp increase in force upon compressive load/ impact without an overshoot to avoid damaging protected subject/structure.It should maintain a constant resistance to displacement/penetration and show no sign of final densification (δ f in figure 21(a) denotes the onset of densification).Ideally, it should be damage resistant by sustaining multiple impacts with minimal deterioration in effectiveness or structural integrity.It should also be light weight, low cost, environmentally friendly, and suited for mass production [48,[141][142][143][144][145].Finally, it should incorporate an energy dissipation mechanism (such as viscous force, frictional forces, magnetic damping, solid-liquid interfacial energy, phase transformation, or chemical reactions) to dissipate the absorbed energy.
Actual performance of most cushions however follows the blue curve in figure 21(a).The curve reveals three main shortcomings.First, the elastic rise ends with an overshoot, which could damage the protected subject/structure.Second, the subject experiences fluctuating resistance to displacement/penetration.Third, the plateau regime is followed by a rapid hardening that is induced by densification.Densification compromises the integrity and performance of polymeric cushion materials making their reusability futile.
Mechanically, the ideal cushion response is a typical characteristic of elastic buckling [147].Therefore, one of the earliest and most rudimentary crash boxes (shock absorbers) consists of an array of prismatic slender tubes that buckle along their longitudinal axis upon impact [143,146,[148][149][150].These tubes have cross-sectional geometries ranging from squares to circles. Figure 21(b) shows one of the examples of prismatic tube-based energy absorbers, which were crushed under impact loading.They are typically constructed using ductile metals (e.g.Steel) to induce plastic buckling during the impact process [151,152].Unfortunately, numerical studies and experiments have often revealed extreme levels of force overshoot.These extreme forces can lead to the destruction of the protected subject.To address this issue, researchers have turned to both bio-inspired and synthetic cellular materials [153].Additionally, researchers have explored deviations from the tube architecture by applying deliberate defects such as dimples, kinks, and bends into the crosssectional outline of the tube [154].These modifications target reducing the peak force.
Potential energy absorbing structures may dissipate imparted energy through plastic deformation, damage, and crack propagation, as well as through viscoelasticity and friction.Unlike damage based mechanisms, utilizing viscoelasticity and friction allows the reusability of the structure without compromising the structural integrity.As such we group various samples, particularly those related to Kresling-inspired structures, into two groups: damage-based (figures 21(c) and (d)) and reusable prototypes (figures 21(e)-(h)).
The year 1994 marked the first introduction of KOSs as a concept for energy absorption [53,54].Two years later, experimental demonstrations using Aluminum alloy and flexible Polyurethane hinges in place of creases showed a force-deflection response resembling that of the ideal cushion material shown in the yellow line on figure 21(a), albeit with a few large fluctuations [53] (figure 2(b)).These early research works on Kresling columns were visionary in creating robust prototypes that exhibit sequential engagement of KOSs upon compression as seen in the post-compression state in (figure 2(b)).The design employs a series of KOSs stacked along the longitudinal direction with the same chirality.
Conversely, Zhao et al [103] proposed to design the column by stacking the KOSs with alternating chirality, resulting in a symmetric column (figure 21(c)).Alongside various other studies, they conducted parametric and optimization studies, as well as dynamic simulations, which revealed the following findings [103,[161][162][163]: (i) the peak force decreases with the initial twist angle between the top and bottom surfaces of the KOS; (ii) the peak force increases with the initial height of the spring; (iii) the peak force increases with the sheet thickness, and notably, there is a direct relationship between energy absorption and peak force; (iv) columns made with opposite chirality are more stable under impact and slightly more effective in energy absorption when compared with same-chirality columns [164].
For high-speed impact and heavy-duty applications, energy absorbers made from ductile metals are often needed due to their effectiveness in absorbing and dissipating energy through plasticity.As an example, metallic Kresling column-based sample (Aluminum) used as a bumper in crash tests [161].Medium-to low-speed impact applications, on the other hand, permit the use of polymeric materials.For instance, massproducible Kresling columns were constructed out of carbon fiber reinforced plastic (CFRP) filaments using the winding method shown on figure 7(a) [86].Crushing, tearing, and delimitation (i.e.damage) of densely packed fibers during impact dissipate the imparted energy (figure 21(d)).
The aforementioned examples rely on damage of the internal features of the Kresling architecture often leaving behind a permanently crushed sample.Thus, metallic or rigidpolymeric designs are most suited for sacrificial components for single-impact use.To design recoverable energy dissipating Kresling columns, Wang et al constructed a closely related springs to that of Kresling, using Yoshimura-based springs out of viscoelastic flexible high-density Polyethylene (HDPE) [102].Despite the decrease in peak force and absorbed energy when using flexible HDPE, the structure can fully recover its original form within 24 h.
Recently, Dalaq et al [48] harnessed opposite chirality Kresling-based unit cells made from two contrasting viscoelastic polymer materials (figure 21(e)) to design efficient and recoverable cushion materials.They were able to tailor the force-deflection profile by varying the geometry of the KOS parameters and their numbers, resulting in a customizable 3D lattice with specified stiffness, strength, stroke distance, and energy absorption capabilities.The Kresling unit cell ensures complete recovery of shape and height even after multiple impacts, without causing material damage.Figure 21(f) shows the sequential engagement of the Kresling based layers under high-speed impact.This origami-inspired cushion exhibits remarkable energy-absorbing efficiency, η, with low density.
Huang and Yan [101] proposed an impact energy absorption mechanism which utilizes a truss-based version of the KOS as a uniaxial-rotational coupling mechanism that transfers the impact force to rotation at the base (figure 21(g)).
The rotating base then dissipates energy through contact friction under uniaxial compression.The base contains dimples that control the induced contact friction (denoted as an orange arrow), thereby controlling the amount of energy dissipation (figure 21(h)).

Vibration isolation
Low-frequency vibration isolation is another area where the tunability and the axial-twist coupling of the KOS can offer great advantages.In general, vibration isolators can only effectively suppress external excitations when the excitation frequency is larger than √ 2 times the natural frequency of the isolator [165] as shown in figure 22(a).Thus, in order to suppress low-frequency excitations that are ubiquitous in nature, the effective stiffness of the isolator must be very small.However, this creates an additional problem.Reduction of the isolator's stiffness leads to large static deflections upon carrying even the smallest static loads, which substantially reduces the load bearing capabilities of the isolator.An ideal lowfrequency vibration isolator must, therefore, have a high static stiffness to reduce the static deflection, yet a low dynamic stiffness near the equilibrium to reduce the natural frequency of oscillations.This results in what is commonly known as highstatic and low-dynamic stiffness (HSLDS) vibration isolators [166].
Figures 9(a) and (e), demonstrate, respectively, the ideal restoring force curve necessary to create HSLDS vibration isolators, and that of a low stiffness linear spring designed to suppress low-frequency excitations.It is evident that, while the system with the ideal restoring force curve can achieve near zero stiffness at a relatively small static deflections, the linear system can only be designed to have a small stiffness when the static deflection is extremely large.Vibration isolators which have a restoring force curve similar to the ideal one are often known as QZS vibration isolators.They have been widely researched in the literature and are shown to be superior in performance when compared to conventional isolators.However, traditional QZS isolators have complex designs requiring different combinations of springs, linkages, cams, and magnets to create (see [168][169][170] for more details).
Since KOSs can be designed to have a QZS behavior as shown in figure 9(e), they were exploited by Ishida et al [41] to develop truss-based structure inspired by the Kresling pattern (figure 22(b)).The links of the structure were designed to be deformable under tension and compression, and a central spring with opposing restoring behavior was installed such that the overall response of the truss is of the QZS type.Experimental testing of the QZS structure under harmonic excitation demonstrated its ability to effectively suppress vibrations at frequencies that are as low as 6 Hz.
In their research on exploring the multifunctional capabilities of coupled Kresling modular structures, Li et al [134] conducted a numerical case study investigating the use of d−KOSPs for isolating torsional vibrations.Specifically, they identified the parameter conditions under which the d−KOSP could achieve QZS.Through simulations, they evaluated the effectiveness of the d−KOSPs as torsional vibration isolators and observed significant attenuation of ultra-low-frequency vibrations with low transmissibility (figure 22(c)).
In a 2022 study, Han et al [167] performed a theoretical investigation to design and analyze the behavior of a lightweight nonlinear vibration isolator suitable for aerospace applications.The proposed isolator consists of a set of identical d-KOSPs connected in series or parallel.By employing the truss model, the authors identified the design parameters that lead to the QZS behavior and used them to study the dynamic behavior of the isolator by implementing the method of averaging, and validated the results using a multibody dynamics software.Both series and parallel configurations were considered showing that the design can work as an ultra-low frequency vibration isolator with low transmissibility (figure 22(d)).

Metamaterial bandgaps
The term 'metamaterials' (meta Greek for beyond), have been used by researchers to refer to artificially engineered materials that possess exceptional properties and functionalities not commonly found in natural materials.Metamaterials are composed of periodically-arranged constituent elements at scales smaller than the wavelengths of the interacting waves.Their distinctive functionalities arise from the deliberate manipulation of wave interactions within sub-wavelength structures.Mechanical metamaterials focus on the manipulation of mechanical waves and the design of structures with enhanced mechanical characteristics like high strength and toughness.Among their various characteristics, bandgaps in mechanical metamaterials have been extensively studied.Bandgaps refer to regions in the frequency spectrum of a metamaterial where certain types or frequencies of waves are restricted from propagating through the material.Figure 23(a) shows a schematic of mechanical metamaterials with bandgaps.These bandgaps can be tuned by adjusting the geometry, composition, and arrangement of the constituent unit cells allowing for tailored functionalities.Their presence allows for the manipulation and control of mechanical waves, enabling applications such as vibration isolation, wave filtering, and selective wave propagation.
Since origami engineering involves the creation of structures composed of tessellations or repeated patterns with specific periodicity, these structures naturally exhibit various metamaterial functionalities, including auxeticity, shape morphing, reconfigurability , enhanced strength, and the ability to have both positive and negative Poisson's ratio [13,26].Thus, among their different potential applications, origamiinspired metamaterials have been investigated to develop reconfigurable origami sonic barriers [173][174][175].
Metamaterials inspired by the Kresling origami have been also developed for effective vibration attenuation.In one demonstration, Han et al [137] presented a KOS-inspired metamaterial with multiple states, (figure 23(b)).They studied the transmissibility of longitudinal waves through the metamaterial and observed multiple bandgaps in the frequency spectrum that vary depending on the memory states of the metamaterial.
In another demonstration, Miyazawa et al [81] explored the utilization of d−KOSP lattices for the transfer of topological boundary modes 5 .By harnessing the reconfigurability of these lattices, the researchers proposed a practical and straightforward approach to efficiently manipulate wave energy, eliminating the need for pre-configured designs or complex active actuation systems typically required for wave manipulation.They showed that, by applying a quasi-static twist to the lattice, the strain energy landscape can be modified leading to Reproduced from [81].CC BY 4.0.(d) Torsional bandgap switching with KOS resonators.Reprinted from [74], with the permission of AIP Publishing.(e) Transverse vibration attenuation with KOS resonators.Republished with permission of ASME, from [122]; permission conveyed through Copyright Clearance Center,Inc.(f) Longitudinal to torsional wave converter.Reprinted from [171], Copyright (2021), with permission from Elsevier.(g) Supratransmission in a Kresling meta structure.Reprinted from [172], Copyright (2023), with permission from Elsevier.(h) Impact mitigation via rarefaction solitary wave creation.Reproduced with permission from [40].CC BY-NC 4.0.alterations in wave dispersion and band topology and facilitating the transfer of topological edge states in the bandgap (figure 23(c)).
Yasuda et al [40] also investigated nonlinear wave propagation characteristics of a modular stack of paperbased KOSs (figure 23(h)).They demonstrated the ability to attenuate compression waves and mitigate impact or shock.Interestingly, it was observed that rarefaction solitary waves overtake initial compressive strain waves, resulting in the latter part of the origami structure experiencing tension rather than compression upon impact.The phenomenon is particularly evident in a chain of softening KOSs.This dynamic mechanism offers the potential to create a highly efficient and reusable impact mitigation system without relying on material damping, plasticity, or fracture.
Another interesting realm of metamaterials is locally resonant metamaterials [176].These metamaterials offer an intriguing avenue for controlling wave propagation by utilizing the resonance properties of attached resonators.When an incident wave near the resonant frequency interacts with the metamaterial, energy is transferred to the resonators, causing them to oscillate out of phase with the base structure.As a result, wave energy within the bandgap frequencies experiences substantial attenuation enabling effective low-frequency vibration damping and wave filtering.Zhang et al [122] developed a metamaterial beam featuring bi-stable KOSs as discrete resonators for low-frequency transverse vibration isolation (figure 23(d)).The incorporation of these discrete oscillators leads to the formation of isolation zones within the frequency bandwidth of the metamaterial beam.By utilizing the different stiffness properties of the bi-stable KOSs in their two states, the location of the band gaps in the frequency domain can be adjusted.
In a related study, Xu et al [74] demonstrated switchable frequency bandgaps of a cantilever beam using KOS resonator attachments (figure 23(e)).In their design, the inertia of the Kresling oscillator is enhanced by fixing spherical masses at eccentric locations with respect to the neutral axis of the beam.By incorporating additional lumped masses in the form of eccentric balls positioned away from the beam's neutral axis, extra torsional modes are introduced into the system leading to a complex bandgap structure, which results in broader frequency zones for vibration isolation at lower frequencies.
Xu et al [171] also presented an axial-to-torsional wave converter using KOSs.The converter allows for the transformation of longitudinal waves into combined longitudinaltorsional vibrations or near-pure torsional vibrations at the output end of a finite conversion system operating at different frequencies (figure 23(f)).
Ywang et al [172] studied the nonlinear supratransmission 6 of waves in the bi-stable Kresling origami metamaterial shown in figure 23(g).They showed that, by exciting the structure near the resonant frequency of the KOS with a forcing amplitude greater than the threshold to escape the potential well of the state, the cells undergo large interwell oscillations.This behavior enables the transmission and amplification of the incident wave through the metamaterial.

Aerospace structures
The principles of origami, such as foldability, lightweight construction, and deployability, are highly desirable in aerospace applications where weight reduction and efficient storage are critical.Origami-inspired structures have been used to design solar panels [178], inflatable booms [11], antennas [179], solar sails [180], and UAV landing gears [121].These structures allow for compact packaging during transport and easy deployment in space, reducing launch costs and maximizing mission capabilities.Origami techniques also enable the creation of intricate and complex shapes with high structural integrity.
Implementation of the Kresling origami pattern as related to space applications was first discussed in the design and development of an origami-inspired concept for a sunshield used in deployable x-ray space telescopes [71] (figure 24(a)).Multiple origami patterns were also tested to design highly compressible bellows for the harsh space environment [43].It was observed that the Kresling pattern has the highest compressibility, and, due to its fewer folds and nodal stress points, is relatively less prone to failure compared to the Accordion and Tachi-Miura patterns (figure 24(b)).
Huang et al [181] investigated the use of different origami patterns to design cylindrical sandwich structures (CSS) with foldcores 7 , which act as load-bearing structures for satellites.The requirements include efficient heat conduction across the structure to minimize temperature gradients within the satellite body, and mechanical stability that remains unaffected by thermal deformation.Results indicate that the Kresling foldcore performs less favorably compared to other origamiinspired foldcore designs that were tested.

Soft robotics and actuation
The most explored applications of Kresling-inspired structures are in the fields of robotics, actuation, and locomotion due to certain features of the Kresling pattern that are attractive to these fields, such as the axial-twist coupling and multi-stability [39, 45, 47, 50, 51, 61, 62, 64, 75-80, 82, 83, 89, 93, 99, 111, 113-118, 182-189].Initial work on robotic systems was performed in 2016 by Pagano et al [50,78,189], where a crawling robot with a pair of serially-connected stacks of paperbased bi-stable Kresling units was constructed (figure 25(a)), and each stack was individually driven using motors.The direction of rotation of the motors was continually varied to control the expansion and contraction of the Kresling units, which activates the crawling motion.Steering and turning 7 Foldcores are a type of lightweight core structure used in the design of composite materials that are ideal for aerospace applications.It consists of a series of folded sheets or panels that are interconnected to form a three-dimensional lattice structure that minimizes weight while providing mechanical stability and rigidity.[186].(d) Linear actuator designed for soft robot applications driven pneumatically and with cables.© [2020] IEEE.Reprinted, with permission, from [64].(e) Self-scalable cable-driven robot.Reproduced from [93].CC BY 4.0.(f) Soft robotic arm with reconfigurable articulation.Reproduced with permission from [45].(g) Pneumatically-actuated robot with worm-like crawling and alternate stepping locomotion.© [2022] IEEE.Reprinted, with permission, from [99].(h) A robotic arm capable of multimodal deformation using a single pneumatic-actuation point.[89] John Wiley & Sons.© 2022 Wiley-VCH GmbH of the robot were achieved by the independent actuation of the two Kresling stacks.The robot takes advantage of bistability to simplify the open-loop locomotion control and reduce power consumption.This multi-stable robot was later used by Gustafson et al [185] to create a model of the crawling locomotion by studying the system level dynamics of the Kresling robot, which can be generalized to other origamibased robotic systems.Correspondingly, Bhovad et al [51,183] modeled and demonstrated the peristaltic crawling locomotion using a stack of two bi-stable KOSs (figure 25(b)).By exploiting multi-stability of the KOS, the crawling motion was achieved using a single actuator without the need of complex controllers.
Kim et al [186] proposed an actuation mechanism based on a Kresling shape-programmable architecture that combines torsional shape memory alloy (SMA) wires with origami blocks, and takes advantage of bi-stability to maintain the blocks at different states during actuation (figure 25(c)).The architecture enables actuation capabilities that were not achievable in previous programmable origami systems, such as reversibility, stability and large rapid transformation.
In another work by Zhang et al [64], a new type of linear actuator designed for soft robot applications is presented.The actuator, which combines cable and pneumatic actuation, features rigid-flexible coupling Kresling origami chambers and a deployable mechanism (figure 25(d)).It is capable of producing bidirectional motion, generating both thrust and tensile force, which can be controlled in terms of position and velocity.By utilizing axially-soft origami chambers, only minimal stress is generated during motion with no radial expansion, while also having high weightlifting capacity.As a result, low input pressure is required, making it a soft, low-threshold pressure linear actuator.
Mena et al [93] also employed cable-driven actuation, and designed, built, and tested a self-scalable origami robot.The design consists of a truss structure based on the Kresling pattern, where the truss elements are made of 3D printed variablelength piston links augmented with a soft spring-like links (figure 25(e)).Scalability and size-changing are achieved by actuating the assembled structure using a three-cables mechanism driven by a DC motor, which can be controlled to compress and expand the origami robot to any intermediate position.Kaufmann et al [45,113] studied the possibility of using Kresling origami structures for creating soft robotic arms with reconfigurable articulation.This is achieved by exploiting the low bending stiffness of a bi-stable Kresling unit in the compressed state, and its high bending stiffness in the deployed state.They built stacks of multiple KOSs, where each compressed unit acts as a flexible joint of the robotic arm, while the deployed units act as the stiff links (figure 25(f)).The reconfigurable articulation was demonstrated by manually setting selected Kresling units to a compressed state and actuating the stack by a tendon-based driving mechanism.
Along the lines of pneumatic actuation, Jin et al [99] exploited the high compressibility of Kresling origami, and its axial-twist coupling feature to create pneumatically-actuated soft robots with several locomotion behaviors.The robot was built by combining multiple 3D printed Kresling units of thermoplastic elastomer (TPE) material, and the contraction and expansion of each unit were individually controlled by a separate vacuum pressure line (figure 25(g)).They demonstrated the crawling locomotion by worm-like creeping motion in a straight line, and through alternate stepping of two legs with steering and navigation capability.They also utilized bi-stability for obstacle and collision sensing.In a different study, a very interesting utilization of pneumatic actuation was presented by Melancon et al [89], where they exploited multistability of the KOS to create modular, inflatable structures that switch between different deformation modes with a single pressure input.They 3D printed the KOS units and stacked them to create a robot that demonstrates the different deformation modes by controlling the trajectory of the applied pressure from a single point source (figure 25(h)).A more recent work by Zhang et al [189] built upon the multi-stable pneumatic actuation approach of Melancon et al [89] to create an all-purpose origami module exhibiting seven distinct deformation modes.
All of the aforementioned actuation mechanisms require direct driving through lines connected to the KOS and operated either pneumatically, with cables/wires, by motors, or a combination.The first approach for wireless remote actuation and control was performed by Novelino et al [76], who proposed a new method for magnetically actuating origami structures.This method involves coupling a bi-stable Kresling structure with a magnetically responsive material, which, in turn, is programmatically magnetized, and excited by applying an external magnetic field (figure 26(a)).This actuation method was then used in several studies spanning different applications.For example, Wu et al [47] designed a stretchable origami robotic arm that takes inspiration from the multi-functional octopus arms.The arm was remotely actuated and can bend and twist omnidirectionally.It is composed of Kresling robotic units, each containing a set of magnetic plates with in-plane magnetizations.When a controlled magnetic field is applied, the arm can stretch, bend, and twist in different directions, replicating the motion of an octopus arm in a controlled and tunable manner (figure 26(b)).A large 18unit octopus-like robotic arm was presented, demonstrating the capability of realizing omnidirectional bending to interact with objects.
Other studies utilizing the remote actuation were performed by Ze et al [62,82], where they create a millimeter-sized robot that can perform rolling, flipping, crawling and inwater spinning-induced locomotion by taking advantage of the KOS's geometrical features and its folding/unfolding capability (figure 26(c)).The robot's ability to autonomously adapt its locomotion mode to navigate complex and unstructured terrains makes it a potential minimally-invasive device for biomedical diagnoses and treatments.
Another approach for magnetic actuation was explored by Yin et al [115].They presented the design and fabrication of variable-stiffness origami structures by filling the folding joints with magnetorheological fluid (MRF) (figure 26(d)).Varying the applied magnetic flux can change the stiffness of the MRF and, as a result, tune the stiffness of the entire structure, which can be used to adjust its load bearing capacity.
The axial-twist coupling of the Kresling pattern and its compression and expansion capability serve as a pumping mechanism for controlled delivery of fluid, which also enable the design of linear displacement pumps.In one demonstration, Westra et al [116] explored the possibility of using KOSs in cryogenic applications.They stated that a polymeric-based KOS can withstand deformations at cryogenic temperatures by limiting the strain to below the elastic limit.The springs shown in figure 27(a) were fabricated from thin biaxially-oriented polyethylene terephthalate (BoPET) films and actuated linearly for 100 cycles and rotationally for 5 cycles, at 77 K by submerging them in liquid nitrogen.Results show no defects by visual inspection and dye penetrant testing.
An actual demonstration of the operation of a KOS as a linear displacement pump was performed by Lee et al [118], where they used the pump to drive a soft pneumatic actuator (figure 27(b)).Another work carried out by Kim et al [117,182] integrated the pump operation of the KOS with a robotic system.They proposed and evaluated a novel origami-based self-supplying pneumatic quadruped robot (figure 27(c)), which has a four-leg system controlled by two motors.The forelegs and hindlegs are pneumatically coupled to operate simultaneously with a tendon-driven system.Results show that the tendon-driven pneumatic origami pump actuator system is suitable for a quadruped robot without having an external air supply system.They also demonstrated   Reproduced with permission from [201].
the implementation of this system to actuate a pneumatic gripper robot manipulator.

Antennas
Conventional antennas have limited capability in adapting to changing system requirements due to their fixed operating characteristics.Thus, reconfigurability, which refers to the ability to modify antenna properties, such as frequency, radiation pattern, polarization, or a combination thereof, can greatly enhance the performance and enable miniaturization of communication systems.Recently, reconfigurable antennas have leveraged origami structures, and their shape-changing properties, to achieve tunability in frequency, radiation pattern, and polarization, as well as compactness, easy packing, and quick deployment [179,190].Specific to the Kresling origami, several helical antennas were designed, fabricated and tested by Liu et al [191][192][193][194][195][196][197][198].The antennas were created by adding copper lines on a sheet of paper or Kapton (figure 28(a)), with mono-, bi-, and quadri-filar configurations, consisting of multiple stacked Kresling units.Frequency reconfigurability was achieved by utilizing the high compressibility of the KOS.The authors were able to tune the antenna's operating frequency to frequency bands that are suitable for different wireless communication applications, including GPS, WiMax, and satellite radio communications.Rubio et al [199] proposed a design of circularly polarized antenna by integrating the Yagi-Uda array configuration with the Kresling origami structure (figure 28(b)).Similarly, Zhang et al [200] presented a Kresling antenna with an added spherical cap that enables enhanced frequency reconfigurability (figure 28(c)).
Bentley and Harne [201] designed, constructed, and experimentally tested a Kresling-inspired reconfigurable acoustic waveguiding device suitable for satellite communication systems and antennas.It consists of a Kresling structure connected to acoustic transducer arrays through scissor-like links and arranged either linearly or in spiral form (figure 28(d)).Compressing and deploying the Kresling structure contracts or expands the scissor arms respectively and, as a result, changes the radiation pattern of the acoustic waves.Moshtaghzadeh et al [119,131,202] performed several comprehensive studies to analyze the structural stability and fatigue life of Kresling origami antennas.It was found that structure's stability under buckling loads improves by increasing the number of sides and reducing the height.

Vibratory energy harvesting
Vibratory Energy harvesting is the process of capturing and converting the energy of an oscillating body and transforming into usable electrical energy.It employs various transduction principles, including piezoelectricity, triboelectricity, and electromagnetic induction.These energy harvesting technologies have a wide range of applications, from smallscale portable and wearable devices [203] to larger infrastructure systems [204].Origami, with its design adaptability and mechanical stability, has emerged as a promising approach for energy harvesting.It has been utilized for energy harvesting using different transduction mechanisms including triboelectric [205], thermoelectric [206], piezoelectric [59], and electromagnetic devices [207].
In one demonstration, Chung et al [59] introduced an innovative hybrid Kresling-origami-inspired energy harvesting system that combines piezoelectricity and triboelectricity to harness energy from the environment.The piezoelectric material was embedded in the creases and the panels, while the triboelectric material was layered on top of the panels, see figure 29(a).When the Kresling spring deploys due to external stimuli, the piezoelectric material converts the strain energy at the creases and the panels into electricity, while the intermittent contact between the panel surfaces is leveraged to generate electricity through triboelectricity.
In another demonstration, Huang et al [208] introduced an energy harvester utilizing the KOSs, where PVDF piezoelectric sheets were strategically positioned at the mountain creases, functioning as hinges that bend along the crease axis, see figure 29(b).Similar to a bending beam, the PVDF sheets undergo strain away from their neutral axis, resulting in the creation of an electric potential difference.The authors employed their proposed Kresling origami generators to develop a sensory pedestal capable of generating electric signals in response to mechanical stimuli.In a follow-up study [96], the authors explored the use of the bi-stable configuration of the KOS for energy harvesting and successfully harvested energy from low-amplitude excitations possessing frequencies below 10 Hz.

Force sensing
The flexibility, modularity, and deployability of origami structures enable the creation of sensors with customizable shapes and functionalities, making them suitable for a wide range of applications including wearable and bio-implantable devices, robotics, and environmental monitoring.In particular, the unique coupling between the axial and rotational degrees of freedom of the Kresling stack offers unique advantages for the development of reliable force (pressure) sensor.Such concepts have already been explored for vibration sensing [210], electrophysiological sensing [209,211], and in the development of pressure sensors for robotic applications [99,107,212].
In one demonstration, Kim et al [209] introduced a unique origami design for electrophysiological sensing including the measurement of blood pressure that is inspired by the leech's suction behavior, see figure 29(c).They developed a Kresling stack with a non-auxetic end that has conductive electrodes printed.The non-auxetic end is placed on the human skin.Upon applying compressive pressure during contact, the end expands bringing the underlying electrode into contact with the skin.The KOS stack located on the back end generates suction pressure to stabilize the electrode-skin contact during measurements.
Traditional resistance-based sensors rely on the deformation of an active layer or a diaphragm when subjected to force, which can lead to a nonlinear response, lack of repeatability, and even potential permanent damage.To overcome these issues, Bao et al [100] introduced a potentiometric method that utilizes the Kresling pattern origami, see figure 29(d).In particular, they designed a potentiometric sensor that exhibits rotational motion in response to axial force (pressure).The rotation results in a change in the effective length of the potentiometer, which, in turn, alters the total resistance.By detecting the resistance change, the sensor can accurately measure variations in pressure.The unique characteristics of the Kresling pattern ensures linear and repeatable responses without complex mechanisms like gears or axial shifters.
Kim et al [107] developed a reconfigurable 3D printed origami tube for a tunable and portable self-sensing ventilator, figure 29(e).The design incorporated a stack of KOSs that serves as an airbag with adjustable air volume and stiffness.Pressure measurements were obtained using a potentiometric sensor integrated using 3D printed circuitry.The accuracy of the pressure sensor was compared to commercial sensors and found to be reasonably accurate.

Assessment and directions for future work
As can be clearly seen already, there is a diverse set of applications where the unique characteristics of the Kresling origami springs (e.g.compressability, deployability, modularity, light weight, axial-twist coupling, and customizability) can be employed to advance the capabilities of current engineering systems.In terms of their future impact on a given field, our assessment is that KOSs will have the greatest impact on the fields of soft robotics, and lightweight deployable structures.In these two fields, there is immediate need for modular, flexible, deployable, and light-weight structures with adequate load bearing capacity, which are all key features readily available in the KOS.Furthermore, bi-stability and axial-twist coupling have been shown to offer additional advantages for the design, control, and locomotion of soft robots.The fields of metamaterials for wave guidance, energy absorption, and vibration attenuation are also domains where we expect to see further growth in the implementation of KOSs.The features of bi-stability, modularity, and axial-twist coupling offer key advantages that provide opportunities to improve performance.In the fields of reconfigurable antennas, non-volatile bitmemory switching, and energy harvesting, we do not see clear advantages resulting from the implementation of the KOSs since most of the desirable characteristics can be replicated via other adaptive or bi-stable structures.Finally, in the sensing domain, the impact of KOSs seems to be limited to force sensing applications, where the axial-twist coupling can be exploited for the design of reliable pressure sensors.
Despite the many different applications of the Kresling pattern in soft robotics and actuation, we noted that there are limited parametric studies performed in this field to optimize the performance of the KOS for the intended application.These studies could target critical performance objectives like minimizing power consumption, maximizing displacement range per rotation or vice versa, maximizing strength and reliability, or maximizing/minimizing stiffness, and can be achieved by understanding the influence of the design parameters on such performance metrics.Areas of future research within the field of soft robotics include devising alternative remote actuation mechanisms beside using magnetic actuators, which are generally sensitive to electromagnetic interference and to the presence of ferromagnetic materials.The ability of the bistable KOS to rapidly release stored potential energy by slight external perturbations, makes it also a good candidate for the design of robots with (hopping) locomotion capability.
As related to energy absorption and cushion materials, it appears that fulfilling the requirements needed to design and ideal cushion material as described in section 4.2 still necessitates the advancement of their fabrication/manufacturing methodologies.In particular, the use of 3D printing for multi-material fabrication, and the inclusion of active elements within the KOS can lead to better performance and enables real-time modulation of the stiffness of the individual unit cells within the lattice, which is key towards designing active energy absorbers.Furthermore, due to its bellow-like shape, air or other hydraulic material can be intentionally trapped inside the KOS to act as a viscous element that improves energy dissipation.The control of the outflow of the hydraulic material allows for live-modulation of the stiffness and the damping behavior of the spring.This attribute is especially important when mitigating impact threats of varying intensities, where the stiffness is tuned such that the energy absorber is adapted to operate within its effective range.This approach could result in smart adaptive cushion materials and shock absorbers whose stiffness can be controlled in real time using feedback control.
Since the restoring force of the KOS can be easily customized to exhibit a QZS behavior, which is key towards designing efficient HSLDS vibration isolators, we expect further growth in this research domain.More emphasis should be placed on experimental studies that assess low-frequency vibration isolators based on the KOS and on the utilization of the axial-twist coupling of the KOS for the design of low-frequency torsional vibration isolators, which are typically difficult to design using traditional mechanisms.
Among the different potential applications of the Kresling origami inspired metamaterials, their implementation as wave guides has occupied most of the available literature [74,122].The coupling between the axial and torsional degrees of freedom allows for creating metamaterials that transform purely axial waves into torsional waves and vice versa [171], which has critical implications for the design of shock absorbers.Future research should focus on using this coupling for the design of high performance shock and energy absorbers.

Final remarks
This review highlighted a plethora of engineering applications where the KOS can be utilized as a unique design element.The large number of such applications is not surprising, and is expected to continue to grow as we better understand the response behavior of the KOS, and as we devise more efficient ways to fabricate it.After all, the KOS is a spring that has unique and tunable restoring force behavior with axial-twist coupling, which are desirable characteristics for the design of many structures and machines as has been clearly highlighted in this review.
We believe that fabricating durable and robust KOSs that can be mass manufactured through repeatable and automated processes is currently the main bottleneck that hinders the transition of engineering designs inspired by the KOS concept from a laboratory idea to an actual product.This includes finding the combination of manufacturing processes and materials that results in a product which replicates the unique characteristics of the paper-based KOS, while being able to withstand large stresses, the environmental elements, and be fatigue resistant.
We believe that additive manufacturing, namely 3D printing, offers the clearest path towards achieving this objective.We argue that 3D printing and origami principles, including the Kresling pattern, are a perfect symbiosis that offers new opportunities towards redesigning many mechanical components whose initial design was dictated by conventional manufacturing processes.Obviously, the goal here is not to 'reinvent the wheel', but rather to optimize the components in such a way that they are more suitable for niche applications.The 3D printed KOS is one example supporting this argument.
Finally, while this review clearly highlights the potential of the KOS as a unique engineering element, it is important to keep in mind that there are many applications where traditional design components are more suitable in terms of functionality and cost.Thus, we recommend employing the Kresling pattern in applications that naturally lend themselves to its relevant use, and advise against using it in applications where other traditional solutions are more applicable and effective.

Figure 2 .
Figure 2. Origin of the Kresling pattern.(a) Twist buckling of foldable cylinders.Reprinted with permission from [7], Copyright (2019) by the American Physical Society.(b) Deployable structure of triangulated cylindrical shell.Reproduced with permission from [53].(c) Failure of thin walled tube under axial-torsional loading.Reproduced with permission from [56].(d) Folding pattern of twist buckled sheet of paper rolled around two plastic mandrels.Reproduced from [57].

Figure 4 .
Figure 4.The Kresling origami spring.(a) Sheet of paper segmented with the Kresling pattern.(b) Schematic representation of the bellow with n = 6 polygon, and (c) Schematic representation of the operation of the bellow under applied load.(d) An actual Kresling origami spring (KOS) fabricated from paper.

Figure 5 .
Figure 5. Structure of the review.

Figure 7 .
Figure 7. Non-sheet based fabrication methods.(a) Depictions of molding, filament winding, and additive manufacturing.Reproduced from [83, 86].CC BY 4.0.(b) Different crease designs: Monolithic and composite designs, where dark-colored and faint material denote hard and soft materials, respectively.Feature shown on the outer surface depicts wiring, sensors, and/or active materials.(c)-(e) Different types of non-sheet based KOSs.Under each type, we show an actual prototype, respectively.(c) Reproduced with permission from [84].(d) Reproduced from [48].© The Author(s).Published by IOP Publishing Ltd.CC BY 4.0.(e) Reprinted from [49], Copyright (2023), with permission from Elsevier.

Figure 8 .
Figure 8. Equivalent truss model of the KOS.

Figure 9 .
Figure 9.Typical restoring force (normalized with respect to EA), and associated normalized potential energy, Π = Π EAR , of a KOS as a function of the displacement.(a) Linear spring, (b) mono-stable hardening spring, (c) mono-stable softening spring, (d) bi-stable spring, (e) quasi-zero stiffness (QZS) spring.The circular green markers denote the equilibrium positions.

Figure 11 .
Figure 11.Design maps demarcating the regions of qualitatively different restoring force behavior for a KOS with (a) n = 4, (b) n = 8, (c) n = 12, (d) n = 16, and (e) n = 20.(f) Influence of the number of sides of the polygon, n, of the KOS on the relative size of the area corresponding to a certain spring type within the available design space.

Figure 13 .
Figure 13.Effect of the relative axial rigidity of the creases on (a) the normalized potential energy function, Π = Π EAR , as function of u/R, and (b) path of rotation of a bi-stable KOS designed using the parameters α 0 = 28 • , b 0 /R = 1.7, and n = 6.

Figure 14 .
Figure 14.Modified truss models.(a) Nonlinear bar and hinge model proposed by Liu and Paulino [125]; Reproduced with permission from [125].(b) Schematic of the modified truss model proposed by Masana and Daqaq [42], and a comparison of its results (quasi-static restoring force) with the experimental data and the regular truss model for a bi-stable KOS designed using the parameters α 0 = 30 • , b 0 /R = 1.3, n = 6, and R = 40 mm.Reprinted with permission from[42], Copyright (2019) by the American Physical Society.(c) Three stable states attained using the truss model with mid nodes proposed by[80].Reproduced from[80].CC BY 4.0.

Figure 16 .
Figure 16.Experimental setup for axial and torsional quasi-static testing of a KOS.

Figure 17 .
Figure 17.Spiral Kresling pattern with perforations showing the nomenclature of the slot dimensions.Reprinted from [67], Copyright (2021), with permission from Elsevier.

Figure 18 .
Figure 18.Deformation of the top end of a pair of KOSPs.(a) Under uni-axial loading, and (b) under torsional loading.Reproduced with permission from [135].CC BY-NC-SA 4.0.

Figure 19 .
Figure 19.(a) Potential energy function of a d−KOSP at different precompressed heights.Solid lines represent experimental results, dotted lines represent simulated results.(b) Bifurcation diagram of the d-KOSP extrapolated using the experimental data.Solid green lines represent the stable equilibria, while the red square markers represent the unstable ones.The contour map represents the potential energy of the stack measured in N-mm.Here, Sub-crit.P ≡ Sub-critical Pitchfork Bifurcation; Sup-crit.P ≡ Super-critical Pitchfork Bifurcation.Reproduced with permission from [135].CC BY-NC-SA 4.0.

Figure 20 .
Figure 20.Kresling inspired mechanical memory switches.(a) 2-bit mechanical memory truss structure.Reproduced from [38].CC BY 4.0.(b) Paper based 2-bit origami switch board, its states and switch activation criteria.Reprinted from [73], with the permission of AIP Publishing.(c) Transition graph and states of memory structure of a Kresling origami switch.Reproduced from [120].CC BY 4.0.

Figure 21 .
Figure 21.KOS-based energy absorbers.(a) Typical force-deflection curve upon subjecting a material to uniaxial compression/impact as compared to the response of an ideal energy absorber.(b) Crushed square cross-section cylinder (duct).Reprinted from [146], Copyright (1984), with permission from Elsevier.(c) Opposite chirality arrangement of a Kresling column made from steel.Reproduced with permission from [103].(d) A composite Kresling column made from CFRP filaments .Reprinted from [86], Copyright (2023), with permission from Elsevier.(e), (f) A two-phase composite array of opposite chirality Kresling-based cushion and progressive engagement of its layers using high-speed imaging.(e) and (f) Reprinted from [48], Copyright (2023), with permission from Elsevier.(g), (h) Friction-based energy dissipation device utilizing a Kresling truss-based spring and its progressive twist upon compression, showing frictional forces resisting the rotation at the base.(g) and (h) Reprinted from [101], Copyright (2023), with permission from Elsevier.

Figure 24 .
Figure 24.Kresling origami for aerospace applications.(a) Kresling origami sunshield model mounted on 1:10 scale structural model of a telescope, From [71].Reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc.(b) Highly compressible KOS.Republished with permission of ASME, from [43]; permission conveyed through Copyright Clearance Center,Inc.

Table 1 .
Assessment of the basic fabrication methods and designs of the KOSs.

Table 2 .
Classification of KOSs based on the number of stable equilibria and their location.