## Abstract

Lamina emergent torsion (LET) joints for use in origami-based applications enables folding of panels. Placing LET joints in series and parallel (formulating LET arrays) opens the design space to provide for tunable stiffness characteristics in other directions while maintaining the ability to fold. Analytical equations characterizing the elastic load–displacement for general serial–parallel formulations of LET arrays for three degrees-of-freedom are presented: rotation about the desired axis, in-plane rotation, and extension/compression. These equations enable the design of LET arrays for a variety of applications, including origami-based mechanisms. These general equations are verified using finite element analysis, and to show variability of the LET array design space, several verification plots over a range of parameters are provided.

## 1 Introduction

Lamina emergent mechanisms (LEMs) are compliant mechanisms formed from a planar material and have some or all of their motion out of the plane [18]. They can be fabricated using two-dimensional processes and rely on the compliance of flexible members formed from the planar material to gain their motion [9]. Apart from common benefits of compliant mechanisms such as high precision, the absence of backlash and wear, and the limited amount of parts [9], LEMs are characterized by a low manufacturing cost, a relatively simple topology, and compactness in the initial state. Characterization of the load–displacement behavior of LEMs is an important step to appropriately design mechanisms with desirable behaviors. They have been used in a variety of applications, including microelectromechanical systems [1,10] and origami-based mechanisms [11,12]. A particularly useful LEM for origami-based mechanisms is the lamina emergent torsion (LET) joint. The LET joint achieves high rotational compliance while minimizing the required footprint of the joint, allowing for origami-like folding of panels and localized joints [1315]. Recent studies have provided many folding and modeling techniques (e.g., elastic origami models [16], truss frameworks [17], topology optimization [18], and mechanical properties of paper folds [19]) and applications (e.g., kinetogami [20], deformable structures [21], and cylinders [22]) for origami-based mechanisms. Systems of panels and low-footprint joint designs are beneficial for applications where the function of the panels depends on the panel size. This feature can be used to create networks of monolithic panels and joints to obtain desired motions and functions (e.g., deployable devices [2325], printed circuit boards [26], and actuation origami [2729]).

LEMs have been placed in series and parallel to facilitate desired behaviors of systems, and work has been done to characterize the global bending stiffness in the desired degree-of-freedom (DOF) [30]. When LEMs are placed in series and parallel, they have been termed lamina emergent arrays (LEAs) or compliant arrays [31]. LEAs also have compliance in other DOFs, and motion in these other DOFs has generally been termed parasitic and undesirable. Specific topological changes of joints, known as surrogate folds for origami-based mechanisms, have been designed to reduce the parasitic motion [32] to behave more like the kinematics of smooth folding structures [33].

This paper describes the formulation of arrays of LET joints (LET arrays, a subset of LEAs) suitable for origami-based mechanisms in which panel-area conservation is desirable and develops a load–displacement relationship set of analytical equations to describe the arrays in three DOFs in the elastic region. The behavior predicted by the theoretical approach is then verified via the finite element analysis (FEA). An integrated simulation environment [34], in which a matlab parametric script guides the computer-aided design (CAD) geometry generations as well as the structural batch simulations, is used to automatically test several LET arrays for each DOF. The rest of the paper is organized as follows: Sec. 2 gives a description of LET array formulations to be studied in this paper, Sec. 3 describes the theoretical load-deflection laws of the LET arrays for each of the considered DOF, Sec. 4 provides a comparison between theoretical and FEA results, and Sec. 5 summarizes the work and presents concluding remarks.

## 2 General Lamina Emergent Torsion Arrays

Figure 1 shows the joint frame and motions considered in this paper. We will refer to the rotation about the x-axis ($Rx(γ)$) as folding (referring to the origami-based engineering nomenclature), translation along the y-axis (Ty(y)) as extension/compression (or for brevity, extension), and rotation about the z-axis ($Rz(β)$) as in-plane rotation. LET joints were introduced as joints well-suited for high rotation in the desired folding motion and where some compliance in other DOF were also observed [13]. A single LET joint was defined as a specific formulation of four compliant torsion segments in series and parallel. Two possible configurations were presented as the Outside LET and the Inside LET which had the same four torsion segments with slight topological differences resulting in different boundary conditions of the segments. The differing boundary conditions affected the off-axis motion, but had no significant effects on the desired folding motion.

Fig. 1
Fig. 1
Close modal

Researchers, to understand these motions, have developed models describing the load-displacement relationships for folding, extension, and in-plane rotation for single joints. A summary of their work is listed in Table 1. The table also shows the gaps in the literature that are important for origami-based design and this paper fills these gaps. Specifically, we give general equations which account for the various boundary conditions for the three DOF for general LET arrays with any combination of series, parallel, and configuration. Figure 2 illustrates motivations for filling these research gaps. Shown in the figure is a system of panels and joints that uses LET arrays to fold a large area into a small volume in the form of an origami pattern [23]. On the left is a close-up of these panels and joints, in the middle is the unfolded system, and on the right are details of the folded configuration. In the side view of the folded configuration, a close-up of a LET array is shown with displacements in the y-axis and about the x and z axes. It is this type of behavior which this paper characterizes to better understand how such systems of panels and joint perform.

Fig. 2
Fig. 2
Close modal
Table 1

State-of-the-art of LET joint equations

Motion
JointFolding (Rx)Extension/compression (Ty)In-plane rotation (Rz)
Outside LETEqs. (1)–(4) [13], Eqs. (1)–(2) [35]Eqs. (24)–(26) [13], Eqs. (1)–(3) [36]Eqs. (3)–(11) [35], Eqs. (6)–(9) [36]
Inside LETEqs. (5)–(7) [13]Eqs. (4)–(5) [36]Eqs. (10)–(12) [36]
Full joint arrayEqs. (5)–(7) [31]xx
General arrayxxx
Motion
JointFolding (Rx)Extension/compression (Ty)In-plane rotation (Rz)
Outside LETEqs. (1)–(4) [13], Eqs. (1)–(2) [35]Eqs. (24)–(26) [13], Eqs. (1)–(3) [36]Eqs. (3)–(11) [35], Eqs. (6)–(9) [36]
Inside LETEqs. (5)–(7) [13]Eqs. (4)–(5) [36]Eqs. (10)–(12) [36]
Full joint arrayEqs. (5)–(7) [31]xx
General arrayxxx

Standard LET joints assume four torsion segments: two in series and two in parallel. Other formulations are possible, which include odd numbers of torsion segments. To reduce confusion when discussing LET arrays made up of torsion segments, we will drop the term LET joints when describing the topology of the arrays. Instead, we will refer to LET array topologies using the following convention: SsPpc. S is the number of torsion segments in series, P is the number of torsion segments in parallel, and c is the configuration (whether the topology resembles an inside or outside LET joint, when applicable). For example, the designation 2s2pi is equivalent to an inside LET joint and 2s2po is the equivalent to an outside LET joint. A single torsion segment has the designation 1s1p and does not have a configuration c designation. Figure 3 shows possible formulations of LET arrays. The 2s2p location in the grid shows both configurations available to the 2s2p designation, with the outside LET on top and the inside LET on bottom. The outside/inside option is only available to designations with both even S and even P. The prototype shown in Fig. 2 indicates the LET array designations used in the left detail.

Fig. 3
Fig. 3
Close modal

General LET arrays can be used to tailor stiffness values in different DOF while maintaining the ability to fold. For example, assuming the geometry of individual segments remain the same, LET arrays of the same S will have the same range of motion before failure in folding but will have different folding, extension, and in-plane rotation stiffness values, as is the case for the LET arrays indicated in Fig. 2. Note that, for example, LET arrays with a P = 2 formulation are equivalent to two separate LET arrays with a P = 1 formulation in terms of folding but are not necessarily equivalent in extension and in-plane rotation as the boundary conditions of the torsion segments can be different for each case. There are unlimited variations of formulating the compliant segments. This paper focuses on the topologies depicted in Fig. 3 for any S and P.

Load–displacement relationships (stiffness rates) for each of the three DOFs considered in this paper are given in this section. It is desirable in the design of systems of panels and joints to obtain equivalent stiffness rates, expressed as scalars, in each of the DOF for each joint. Throughout this section, fundamental dimensions are used and are labeled in Fig. 4. Torsion segments and bending segments are also indicated in the figure, about which will be discussed throughout the paper. lT and wT are the length and width dimensions of torsion segments, respectively. lB and wB are the length and width dimensions of bending segments, respectively. L and W are the length and width dimensions of overall array, respectively, and are functions of the dimensions of the segments in bending and torsion. Throughout these analyses, we assume isotropic material properties. These relationships may also be applicable to symmetric and balanced laminate composites that exhibit isotropic behavior, but it is likely that other considerations, such as for coupling behavior, would be required.

Fig. 4
Fig. 4
Close modal

While LET arrays with varying dimensions are possible, we assume that for a particular LET array, the bending and torsion segments, separately, are of equal dimensions and therefore stiffness. This assumption allows for simplification of the predictive models. However, if LET arrays of varying dimensions are to be designed, the following guidelines are suggested: (1) to avoid unnecessary stress risers, ensure that dimensions of the torsion segments are the same along any particular row and (2) treat rows of identical geometry as unique LET arrays and solve the displacement of each LET array in series.

### 3.1 Folding.

We assume that the torsional stiffness kT is the same for each torsion bar. We assume that the bending stiffness for each segment in bending on the outside of the array is kB and that those on the inside of the array are twice as wide and therefore consists of two springs. A generalized analogous spring model is built by summing elements within rows in parallel and then the rows in series (Fig. 5). The resulting load–displacement relationship in folding for LET arrays is as follows:
$Mx=Keq,rxγ$
(1)
where $γ$ is the fold angle and Keq,rx is the stiffness rate found when adding the springs in parallel and in series:
$Keq,rx=PkTkBS(kT+kB)+kT$
(2)
A symmetric polar moment of inertia JT equation for beam torsional stiffness kT is given by [37]:
$kT=fJTGlT$
(3)
where f is a compensation function [37] to ensure accuracy of a symmetric torsional stiffness JT equation:
$f=1.167z5+29.49z4+30.9z3+100.9z2+30.38z+29.41z5+25.91z4+41.58z3+90.43z2+41.74z+25.21$
(4)
and
$JT=2t3wT37t2+7wT2$
(5)
where z = wT/t and G is the modulus of rigidity of the material. The bending stiffness of the segments (Euler–Bernoulli) in bending is
$kB=EIBlB$
(6)
where E is the elastic modulus of the material and IB = wBt3/12 is the moment of inertia of the segment in bending.
Fig. 5
Fig. 5
Close modal

### 3.2 Extension/Compression.

An outside LET torsion segment has a fixed-guided boundary condition, while inside LET torsion segment has a fixed-clamped condition. Here, we consider LET arrays that have a mixture of fixed-guided and fixed-clamped torsion segments, an example of which is shown in Fig. 6. The LET array has a 3s2p designation where four segments are fixed guided and two are fixed clamped. The differences between fixed guided and fixed clamped, in terms of loading conditions and mechanical responses, are discussed in Ref. [38].

Fig. 6
Fig. 6
Close modal

To model the equivalent extension/compression spring stiffness Ky,eq of LET arrays, we describe individual segment stiffness rates along the rows first (adding in parallel) and then across the rows (adding in series). Unlike the uniform distribution of stiffness rates for LET arrays in folding, LET arrays in extension/compression have a nonuniform distribution due to the mixed boundary conditions. Thus, additional row types are introduced as follows (see Fig. 7): a boundary row is a torsion row that contains a fixed-clamped segment, and if exists, it does so only at the transition from panel to joint (maximum of two in an array). A regular row is a torsion row that does not have fixed-clamped segments along the left and right edges of the array, and if exists, it may repeat (no maximum, but dependent on S). A bending segment row is composed of only bending segments. Figure 7 shows a LET array, which has two boundary rows, two regular rows, and five bending segment rows (as another example, the array in Fig. 6 has one boundary and two regular rows).

Fig. 7
Fig. 7
Close modal
The force–displacement relationship is given as follows:
$Fy=Ky,eqy$
(7)
where y is in the direction indicated in Fig. 1, and the spring stiffness Ky,eq for a LET array in extension/compression is as follows:
$Ky,eq=kboundkregkbendnregkboundkbend+nboundkregkbend+nbendkregkbound$
(8)
where kbend is the axial stiffness of bending segments:
$kbend=2PEtwBlB$
(9)
and the number of bending segment rows is as follows:
$nbend=S+1$
(10)
The number of boundary rows nbound is as follows:
(11)
For example, nbound of the LET array in Fig. 7 is 2 because S > 1 and P is odd. The number of regular rows nreg is as follows:
$nreg=S−nbound$
(12)
The boundary row force Fbound is
$Fbound=kboundybound$
(13)
where ybound is the displacement of the boundary row and the boundary row stiffness kbound is
$kbound=nbfgkfg|yfg=ybound+nbfckfc|yfc=ybound$
(14)
where yfg and yfc are the displacements of fixed-guided and fixed-clamped segments, respectively, and the regular row force Fr is
$Fr=kregyr$
(15)
where the regular row stiffness kreg is
$kreg=nrfgkfg|yfg=yr+nrfckfc|yfg=yr$
(16)
where nbfg is the number of fixed-guided segments in a boundary row:
(17)
For example, nbfg of the LET array in Fig. 7 is 1 because S > 1 and P is odd. nbfc is the number of fixed-clamped segments in a boundary row:
$nbfc=P−nbfg$
(18)
and where nrfg is the number of fixed-guided segments in a regular row:
$nrfg={1,P=12,otherwise$
(19)
For example, nrfg of the LET array in Fig. 7 is 2 because P > 1. nrfc is the number of fixed-clamped segments in a regular row:
$nrfc=P−nrfg$
(20)
The stiffness coefficient kfg of a fixed-guided beam assuming large deflections is nonlinear. Using equations for a fixed-guided beam from Ref. [9], a force–displacement relationship is found as follows:
$Ffg=4KTarcsinyfgγfglT3γfg2lT2−yfg2−lT(γfg−1)$
(21)
where $KT=2γfgKΘEIy/lT$ (the pseudo-rigid-body model (PRBM) spring constant for a fixed-guided beam) and $γfg$ and $KΘ$ are the characteristic radius factor and the stiffness coefficient and are often approximated as 0.85 and 2.65, respectively [9]. E is the material modulus of elasticity and Iy is the area moment of inertia for the torsion segment in bending in the y-direction of the array ($twT3/12$). The stiffness kfg is the derivative of Eq. (21) with respect to yfg:
$kfg=∂Ffg∂yfg$
(22)
The stiffness coefficient kfc of a fixed-clamped beam assuming large deflections is also nonlinear. The force–displacement relationship is as follows:
$Ffc=KAΔlT∂ΔlT∂yfc+2KTΘfc∂Θfc∂yfc1+LT2∂Θfc∂yfc$
(23)
where KA is the torsion segment axial stiffness:
$KA=EATγfclT+ΔlT$
(24)
where AT is the axial cross-sectional area of the torsion segment as a function of elongation and Poisson’s ratio $ν$:
$AT=twTlT(1+ΔlTlT(1−2ν))lT+ΔlT$
(25)
where ΔLT is the change in the length of the fixed-clamped beam:
$ΔLT=γfc2lT2+yfc2−γfclT$
(26)
and Θfc is the PRBM angle:
$Θfc=arctanyfcγfclT$
(27)
The kinematic coefficients are as follows:
$∂ΔlT∂yfc=yfcγfc2lT2+yfc2$
(28)
and
$∂Θfc∂yfc=γfclTγfc2lT2+yfc2$
(29)
The stiffness kfc value is the derivative of Eq. (23) with respect to yfc:
$kfc=∂Ffc∂yfc$
(30)
If there are both regular and boundary rows, respective displacements yr and yb are unknown because the stiffness values of the rows are nonlinear. Since the rows are assumed to be springs in series, the force is constant for each row:
$Fb=Fr$
(31)
and the regular row displacement can be parameterized as follows:
$yr=y−nbybnr$
(32)
such that the displacements yr and yb can be solved for using Eq. (31) as a function of joint displacement y. The evaluation of the force–displacement relationship is not straightforward. Algorithm 1 is presented for clarity.

#### Evaluating the force-displacement relationship F y

Algorithm 1

1: Evaluate integers (Eqs. (9)(12) and (17)(20)

2: for allydo

3: Formulate boundary row force $Fb$ equation (substitute (Eqs. (22) and (30) (using Eqs. (23) and (24)(26)) into Eq. (14)) into Eq. (13))

4: Formulate regular row force $Fr$ equation (substitute (Eqs. 22 and 30 (using Eqs. (23) and (24)(26)) into Eq. (16)) into Eq. (15))

5: Equate forces $Fb$ and $Fr$ (Eq. (31)), parameterize $yr$ (substitute Eq. (32) into Eq. (31)), and solve for $yb$

6: Evaluate row stiffness values $kb$ and $kr$ (Eqs. (14) and (16)) using updated $yb$ and $yr$

7: Evaluate equivalent spring stiffness $Ky,eq$ (Eq. (8)) by using updated row stiffness values $kb$ and $kr$

8: Evaluate force $Fy$ (Eq. (7)) by using updated equivalent spring stiffness $Ky,eq$

9: return$Fy$

### 3.3 In-Plane Rotation.

Different spring models are used for in-plane rotation of LET arrays for the cases when P = 1 and P > 1. For P = 1, moment-loaded beams are added in series. For P > 1, a proposed analogous spring model is developed. These cases are discussed below.

#### 3.3.1 In-Plane Rotation When P = 1.

For the case where P = 1, a moment–displacement relationship is found using a spring analogy of beams with moments applied at the end in series [9]:
$Mz=Keq,rzβ$
(33)
The equivalent spring stiffness rate is as follows:
$Keq,rz=γKΘEIylT1Scθ$
(34)
where $γ=0.7346$, KΘ = 2.0643, and cθ = 1.5164 for this case.

#### 3.3.2 In-Plane Rotation When P > 1.

A simplified spring model that assumes column extension/compression relative to a neutral axis is used to approximate the torque/displacement relationship of LET arrays of P > 1. Figure 8 illustrates how a particular LET array can be modeled as separate columns in extension and compression. Equivalent column spring stiffness values are found by adding rows of a column in series. A rotational displacement load is applied about the neutral axis, and the rotational displacement $β$ causes extension/compression displacements in the columns resulting in column forces Fi. We assume small angles such that the columns remain vertical. These forces are multiplied by their respective distances from the neutral axis ai and are summed to equate to the resultant moment Mz:
$Mz=∑i=1PFiai$
(35)
The equivalent stiffness is as follows:
$Keq,rz=∂Mz∂β$
(36)
Each LET array formulation has a left boundary column and a right boundary column and has nrc regular columns, where
$nrc=P−2$
(37)
The column forces are composed of combinations of a left boundary column force Flbc, regular column forces Frc,i, and a right boundary column force Frbc, resulting from displacements acting on the equivalent spring rates klbc, krc, and krbc, respectively. The left boundary column force Flbc is equal to the force from a fixed-guided segment Flbcfg or the force from a fixed-clamped segment Flbcfc. They are as follows:
$Flbcfg=kfg|yfg=ylbcfgylbcfg$
(38)
$Flbcfc=kfc|yfc=ylbcfcylbcfc$
(39)
By parameterizing ylbcfg as follows:
$ylbcfg=y1−nlbcfcylbcfcnlbcfg$
(40)
and by equating Eqs. (38) and (39), ylbcfc can be solved for, similar to steps 5–6 of Algorithm 1. This process is repeated for the right column force, where
$Frbcfg=kfg|yfg=yrbcfgyrbcfg$
(41)
and
$Frbcfc=kfc|yfc=yrbcfcyrbcfc$
(42)
and where
$yrbcfg=y1−nrbcfcyrbcfcnrbcfg$
(43)
Since the regular columns are composed of entirely of fixed-clamped segments, the resulting force can be evaluated directly as follows:
$Frcfc=1Skfc|yfc=y/Sy$
(44)
where ylbcfg and ylbcfc are the y displacement of fixed-guided and fixed-clamped segments, respectively, of the left boundary column; yrbcfg and yrbcfc are the y displacement of fixed-guided and fixed-clamped segments, respectively, of the right boundary column; and yrcfc is the y displacement of fixed-clamped segments of the regular column. nlbcfc is the number of fixed-clamped segments in the left boundary column:
(45)
nlbcfg is the number of fixed-guided segments in the left boundary column:
$nlbcfg=S−nlbcfc$
(46)
where nrbcfc is the number of fixed-clamped segments in the right boundary column:
(47)
and nrbcfg is the number of fixed-guided segments in the right boundary column:
$nrbcfg=S−nrbcfc$
(48)
Fig. 8
Fig. 8
Close modal
The moment arms ai are the distances from the neutral axis Cx to the forces Fi. The neutral axis is found using
$Cx=∑i=1PFixi∑i=1PFi$
(49)
where xi is the distance from one side of the array to a column force:
$xi=lT2+wB+(i−1)(lT+2wB)$
(50)
and ai is given as follows:
$ai=|xi−Cx|$
(51)
and the yi displacement for each column is given as follows:
$yi=aisinβ$
(52)
The instantaneous spring rates (kfc and kfg) are the same as before (Eqs. (30) and (22)). Due to the nonlinearity of the spring rates, the neutral axis shifts as a function of displacement. Algorithm 2 is a proposed numerical algorithm to determine the location of the neutral axis, which is required to calculate the moment–rotation relationship. The values for $γfg$ and $γfc$ are substituted in $γfgz$ and $γfcz$, respectively, for more accurate in-plane rotation results.
##### Locating the neutral axis C x
Algorithm 2

1: for all$β$do

2:  $Cx←L2$          ▹ begin by assuming symmetry

3:  Solve for $Flbc$, $Frc$, and $Frbc$

4:  ifS is even and P is odd then

5:   Calculate $Cx,new$

6:   $δCx←|Cx−Cx,new|$

7:   $Cx←(Cx,new+Cx)/2$

8:   while$δCx>ϵ$do   ▹ $ϵ$ is a predetermined convergence (i.e., $ϵ=1e−6m$)

9:    Solve for $Flbc$, $Frc$, and $Frbc$ using $Cx$

10:    Calculate $Cx,new$

11:    $δCx←|Cx−Cx,new|$

12:    $Cx←(Cx,new+Cx)/2$

13:   return$Cx$

14:  else

15:   $Cx←Cx,new$

16:   return$Cx$

## 4 Finite Element Analysis Comparison

To verify the analytical models, several LET arrays were tested using an integrated software framework [34], in which a matlab script manages the parametric study, a solidworks macro updates the geometry of the LET arrays, and ansys apdl provides the force/torque–deflection characteristics of each configuration via batch mode. A routine, overseen by matlab, was defined to automatically export the CAD files from solidworks and, subsequently, to perform the structural batch simulations of each candidate. In line with the proposed theoretical models, three DOFs were analyzed, namely $Rx(γ)$, Ty(y), and $Rz(β)$. For each of the DOF, a free mesh with second-order tetrahedral elements ansys solid187 was defined with a maximum element size of t. This element type is particularly suitable for bending-dominated problems, where the shear locking effect is undesired due to its effect on the bending stiffness. In all simulations, the nonlinear geometry (NLGEOM) option was turned on. As for the employed material, aluminum alloy 7075 (heat treated) is considered, owing to its high strength-to-modulus ratio [9]. The Young’s modulus and Poisson’s ratio are, respectively, E = 71.7 GPa and $ν=0.33$. Concerning the boundary conditions, each LET array was fixed to the ground at one end and guided in a pure translation (along the y-axis) or rotation (about the x or z axes) on the other end. Since SOLID187 elements do not have rotational DOF, MPC184 elements were used to apply kinematic constraints (spider web of beams) between the solid model’s nodes and a master node (as shown in Fig. 9), onto which rotational displacement loads can be applied. Referring to Figs. 2 and 4, the tested array parameters were lB = 2 mm; WB = 4 mm; WT = 2 mm; t = 1 mm; lT = 10 mm, lT = 20 mm, or lT = 30 mm; S (ranging from 1 to 3); P (ranging from 1 to 5); and c (either i or o, where applicable).

Fig. 9
Fig. 9
Close modal
Figure 10 shows the force/torque–deflection relationships for arrays with S = 2, lT = 10 mm and various P with the standard $γ=0.85$. The modeled and FEA results have significant differences, with the modeled values higher than the FEA in each case, as shown in Fig. 11. For design, this may not be undesirable when considering strength. However, when considering behavior of systems that implement these types of arrays, more accurate predictions of the stiffness values are desired. A possible explanation for the overestimated stiffness is the inaccurately assumed boundary conditions. The fixed-guided and fixed-clamped boundary conditions do not account for the compliance of the bending segments or of the regions not modeled. Figure 12 shows an FEA strain plot of a 2s3p LET array, indicating the regions where forces and moments are transmitted, but whose spring models are not considered even though strain is observed. A more accurate representation of the overall deflection would include the compliance of these segments. One way to model this compliance in the Ty and Mz DOF is to modify the characteristic radius factor $γ$, effectively softening the assumed rigid boundaries of these regions (hence, the designated $γfg$ or $γfc$ variables in Eqs. (21), (23), and others). In the case of the arrays in Fig. 10, increasing the effective length of the torsion segment would decrease the array stiffness, bringing them closer to the FEA results. A way to model the compliance of these segments for the Mx DOF is to effectively lengthen the bending segments so that they “extend” into these regions, which we have done by adding 1/2wT (1/4wT to both sides), making Eq. (14) equate to kB = E IB/(lB + 1/2wT). By making these modifications, the errors between the FEA and modeled results are reduced (see Fig. 11). Table 2 lists the modified characteristic radius factors (e.g., $γfg$) used for the associated lT and S values and their corresponding figures. In each row of the table, the only geometry variation was lT. The radius factors listed in the table were found using an optimization routine defined as follows:
$minimize∑Errorwith respect toγfg,γfc,γfgz,γfczsubject to0.1≤γfg,γfc,γfgz,γfcz≤2.0for eachlT∈{1.0,2.0,3.0}cm$
where Error is the absolute relative error of the load–displacement FEA and modeled results for each increment, DOF, and value of lT:
$Error=|Ty,M−Ty,FTy,F|$
(53)
where the subscripts M and F refer to modeled and FEA results, respectively, and Rz is substituted for Ty for the in-plane rotation DOF. The trend shows that for shorter torsion segments, the radius factors were increased from the standard 0.85 and that for longer torsion segments, the factors were decreased. By using the modified characteristic radius factors, the figures listed in Table 2 (i.e., Figs. 1321, representing the load–displacement relationships for all the tested configurations) suggest that accurate stiffness estimates are achieved by using the presented model for LET arrays.
Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal
Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal
Fig. 17
Fig. 17
Close modal
Fig. 18
Fig. 18
Close modal
Fig. 19
Fig. 19
Close modal
Fig. 20
Fig. 20
Close modal
Fig. 21
Fig. 21
Close modal
Table 2

List of parameters, figures, and modified characteristic radius factors

lT (cm)$γfg$$γfc$$γfgz$$γfcz$SFigure
1.01.291.061.181.161, 2, 313, 14, 15
2.00.740.730.640.821, 2, 316, 17, 18
3.00.610.640.480.801, 2, 319, 20, 21
lT (cm)$γfg$$γfc$$γfgz$$γfcz$SFigure
1.01.291.061.181.161, 2, 313, 14, 15
2.00.740.730.640.821, 2, 316, 17, 18
3.00.610.640.480.801, 2, 319, 20, 21

One exception to the modified radius factors listed in Table 2 was the case when P = 2 for the Rz DOF. The spring model presented in Sec. 3.3.2 assumes that the columns remain vertical during loading with the global rotation occurring as the left and right columns extend in opposite directions. For P = 2, however, there are no other segments in parallel to enforce vertical displacement, and so the model overestimates the stiffness by artificially enforcing the vertical constraint. Therefore, a separate optimization was performed for the case when P = 2 for each of the lT values. The optimal radius factors were as follows: (for lT = 10 mm) $γfg=γfc=1.01$, (for lT = 20 mm) $γfg=0.53$ and $γfc=0.56$, and (for lT = 30 mm) $γfg=0.40$ and $γfc=0.49$. These values are especially useful when considering the traditional outside and inside LET joints (which are equivalent to 2s2po and 2s2pi LET arrays, respectively).

It can be seen that for the Ty and Rz DOF, the force/torque–displacement relationships exhibit nonlinear behavior when S is low and P is high (see Figs. 13, 16, and 19) for both the FEA and modeled results. This is to be expected when considering the nonlinear terms in Eqs. (21) and (23) for these high-stiffness array formulations. It can also be seen that the errors are higher for these cases, especially for the Rz DOF. A possible explanation for this error in the Rz model is that for these stiff cases, the simplifying spring models do not account for compliance at the boundaries and at the interactions between the bending and torsion segments. This inaccuracy in model assumptions is less apparent in the less-stiff and linear regimes of the arrays. Other cases (higher S) show excellent agreement between the analytical and FEA results.

The comparison between the FEA and modeled results verified that the models can accurately characterize general LET arrays for elastic load–displacement behaviors. These models represent an important tool for designers of systems that incorporate LET arrays to enable design of behaviors of such systems. The equations used assume the material to remain in the elastic region throughout the motion and are particularly useful in exploring the design space of the formulation of the LET arrays. When finalizing a design, consideration of the elastic limits is required. Equations of maximum stress of LET joints [13,39] can be applied to determine the stresses of LET arrays.

## 5 Conclusion

In this paper, LET arrays were defined as torsion segments in series and in parallel, which can be used for origami-based applications and others. The load–displacement relationships for the arrays were characterized in three DOFs enabling the design of arrays that consider off-axis motion.1 The proposed models were verified using FEA for several LET array configurations resulting from an automatic framework. By using the standard $γ=0.85$ value, conservative stiffness estimates are available. If more accurate stiffness estimated are desired, modified radius factors may be used. By using these modified values, load–displacement models had good agreement with the FEA results. Based on intuition and optimization results, shorter torsion segments may use higher $γ$ values and longer torsion segments may use lower $γ$ values. In some cases, the nonlinear behavior of the arrays were sources of error between the modeled and FEA results. Because the use of LET arrays is becoming more prevalent, the analytical models presented in this work enable their design. By using LET arrays, a designer can tailor the stiffness behavior for folding, extension, and in-plane rotation to realize folding and origami-based mechanisms. The provided stiffness values can be used for multibody dynamic or other system-modeling techniques in the design of such systems. By providing several plots of the force/torque–displacement relationships to show model verification, we have also showed the variability of the design space available to LET arrays.

## Footnote

1

To facilitate rapid implementation, a repository containing the proposed models for the LET array analysis can be downloaded from the source: http://dx.doi.org/10.17632/bvcjbcg3cs.1

## Acknowledgment

The authors gratefully acknowledge critical discussions on in-plane rotation stiffness with Jared Butler, Brandon Sargent, Philip Stevenson, and Collin Ynchausti and the finite element modeling assistance from Kenny Seymour.

## Funding Data

• NASA Space Technology Research Fellowship (Grant No. 80NSSC17K0145; Funder ID: 10.13039/100000104).

## References

1.
Jacobsen
,
J.
,
Winder
,
B.
,
Howell
,
L.
, and
Magleby
,
S.
,
2010
, “
Lamina Emergent Mechanisms and Their Basic Elements
,”
ASME J. Mech. Rob.
,
2
(
1
), pp.
1
9
. 10.1115/1.4000523
2.
Qiu
,
C.
,
Qi
,
P.
,
Liu
,
H.
,
Althoefer
,
K.
, and
Dai
,
J. S.
,
2016
, “
Six-Dimensional Compliance Analysis and Validation of Orthoplanar Springs
,”
ASME J. Mech. Des.
,
138
(
4
), p.
042301
. 10.1115/1.4032580
3.
Qiu
,
L.
,
Huang
,
G.
, and
Yin
,
S.
,
2017
, “
Design and Performance Analysis of Double C-Type Flexure Hinges
,”
ASME J. Mech. Rob.
,
9
(
4
), p.
044503
. 10.1115/1.4036609
4.
Qiu
,
L.
,
Yin
,
S.
, and
Xie
,
Z.
,
2016
, “
Failure Analysis and Performance Comparison of Triple-LET and LET Flexure Hinges
,”
Eng. Failure Anal.
,
66
, pp.
35
43
. 10.1016/j.engfailanal.2016.04.006
5.
Wang
,
R.
, and
Zhang
,
X.
,
2017
, “
Optimal Design of a Planar Parallel 3-DOF Nanopositioner With Multi-Objective
,”
Mech. Mach. Theory
,
112
, pp.
61
83
. 10.1016/j.mechmachtheory.2017.02.005
6.
Gollnick
,
P. S.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2011
, “
An Introduction to Multilayer Lamina Emergent Mechanisms
,”
ASME J. Mech. Des.
,
133
(
8
), p.
081006
. 10.1115/1.4004542
7.
Alfattani
,
R.
, and
Lusk
,
C.
,
2018
, “
A Lamina-Emergent Frustum Using a Bistable Collapsible Compliant Mechanism
,”
ASME J. Mech. Des.
,
140
(
12
), p.
125001
. 10.1115/1.4037621
8.
Lobontiu
,
N.
,
Gress
,
T.
,
Munteanu
,
M. G.
, and
Ilic
,
B.
,
2019
, “
Stiffness Design of Circular-Axis Hinge, Self-Similar Mechanism With Large Out-of-Plane Motion
,”
ASME J. Mech. Des.
,
141
(
9
), p.
092302
. 10.1115/1.4042792
9.
Howell
,
L. L.
,
2001
,
Compliant Mechanisms
,
John Wiley & Sons
,
New York
.
10.
Aten
,
Q. T.
,
Jensen
,
B. D.
, and
Howell
,
L. L.
,
2012
, “
Geometrically Non-Linear Analysis of Thin-Film Compliant Mems Via Shell and Solid Elements
,”
Finite Elements Anal. Des.
,
49
(
1
), pp.
70
77
. 10.1016/j.finel.2011.08.022
11.
Francis
,
K. C.
,
Blanch
,
J. E.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2013
, “
Origami-Like Creases in Sheet Materials for Compliant Mechanism Design
,”
Mech. Sci.
,
4
(
2
), pp.
371
380
. 10.5194/ms-4-371-2013
12.
Pehrson
,
N. A.
,
Magleby
,
S. P.
,
Lang
,
R. J.
, and
Howell
,
L. L.
,
2016
, “
Introduction of Monolithic Origami With Thick-Sheet Materials
,”
Proceedings of the International Association for Shell and Spatial Structures Annual Symposium
,
Tokyo, Japan
,
Sept. 26–30
, pp.
1
10
.
13.
Jacobsen
,
J. O.
,
Chen
,
G.
,
Howell
,
L. L.
, and
Magleby
,
S. P.
,
2009
, “
Lamina Emergent Torsional (LET) Joint
,”
Mech. Mach. Theory
,
44
(
11
), pp.
2098
2109
. 10.1016/j.mechmachtheory.2009.05.015
14.
Xie
,
Z.
,
Qiu
,
L.
, and
Yang
,
D.
,
2017
, “
Design and Analysis of Outside-Deployed Lamina Emergent Joint (OD-LEJ)
,”
Mech. Mach. Theory
,
114
, pp.
111
124
. 10.1016/j.mechmachtheory.2017.03.011
15.
Xie
,
Z.
,
Qiu
,
L.
, and
Yang
,
D.
,
2018
, “
Design and Analysis of a Variable Stiffness Inside-Deployed Lamina Emergent Joint
,”
Mech. Mach. Theory
,
120
, pp.
166
177
. 10.1016/j.mechmachtheory.2017.09.023
16.
Saito
,
K.
,
Tsukahara
,
A.
, and
Okabe
,
Y.
,
2015
, “
New Deployable Structures Based on An Elastic Origami Model
,”
ASME J. Mech. Des.
,
137
(
2
), p.
021402
. 10.1115/1.4029228
17.
Chen
,
Y.
,
Sareh
,
P.
,
Yan
,
J.
,
Fallah
,
A. S.
, and
Feng
,
J.
,
2019
, “
An Integrated Geometric-Graph-Theoretic Approach to Representing Origami Structures and Their Corresponding Truss Frameworks
,”
ASME J. Mech. Des.
,
141
(
9
), p.
091402
. 10.1115/1.4042791
18.
Gillman
,
A. S.
,
Fuchi
,
K.
, and
Buskohl
,
P. R.
,
2019
, “
Discovering Sequenced Origami Folding Through Nonlinear Mechanics and Topology Optimization
,”
ASME J. Mech. Des.
,
141
(
4
), p.
041401
. 10.1115/1.4041782
19.
,
C.
,
Cavoret
,
J.
,
Dureisseix
,
D.
,
Jean-Mistral
,
C.
, and
Ville
,
F.
,
2016
, “
An Experimental Study and Model Determination of the Mechanical Stiffness of Paper Folds
,”
ASME J. Mech. Des.
,
138
(
4
), p.
041401
. 10.1115/1.4032629
20.
Gao
,
W.
,
Ramani
,
K.
,
Cipra
,
R. J.
, and
Siegmund
,
T.
,
2013
, “
Kinetogami: A Reconfigurable, Combinatorial, and Printable Sheet Folding
,”
ASME J. Mech. Des.
,
135
(
111009
), p.
111009
. 10.1115/1.4025506
21.
Geiss
,
M. J.
,
Boddeti
,
N.
,
Weeger
,
O.
,
Maute
,
K.
, and
Dunn
,
M. L.
,
2019
, “
Combined Level-Set-XFEM-Density Topology Optimization of Four-Dimensional Printed Structures Undergoing Large Deformation
,”
ASME J. Mech. Des.
,
141
(
5
), p.
051405
. 10.1115/1.4041945
22.
Bös
,
F.
,
Wardetzky
,
M.
,
Vouga
,
E.
, and
Gottesman
,
O.
,
2016
, “
On the Incompressibility of Cylindrical Origami Patterns
,”
ASME J. Mech. Des.
,
139
(
2
), p.
021404
.
23.
Pehrson
,
N. A.
,
Smith
,
S. P.
,
Ames
,
D. C.
,
Magleby
,
S. P.
, and
Arya
,
M.
,
2019
, “
Self-Deployable, Self-Stiffening, and Retractable Origami-Based Arrays for Spacecraft
,”
AIAA Scitech 2019 Forum
,
San Diego, CA
,
Jan. 7–11
,
Paper No. AIAA 2019–0484
.
24.
Huang
,
H.
,
Li
,
B.
,
Zhang
,
T.
,
Zhang
,
Z.
,
Qi
,
X.
, and
Hu
,
Y.
,
2019
, “
Design of Large Single-Mobility Surface-Deployable Mechanism Using Irregularly Shaped Triangular Prismoid Modules
,”
ASME J. Mech. Des.
,
141
(
1
), p.
012301
. 10.1115/1.4041178
25.
Zirbel
,
S. A.
,
Lang
,
R. J.
,
Thomson
,
M. W.
,
Sigel
,
D. A.
,
Walkemeyer
,
P. E.
,
Trease
,
B. P.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2013
, “
Accommodating Thickness in Origami-Based Deployable Arrays
,”
ASME J. Mech. Des.
,
135
(
11
), p.
111005
. 10.1115/1.4025372
26.
Defigueiredo
,
B. P.
,
Zimmerman
,
T. K.
,
Russell
,
B. D.
, and
Howell
,
L. L.
,
2018
, “
Regional Stiffness Reduction Using Lamina Emergent Torsional Joints for Flexible Printed Circuit Board Design
,”
ASME J. Electron. Packag.
,
140
(
4
), p.
041001
. 10.1115/1.4040552
27.
Fuchi
,
K.
,
Buskohl
,
P. R.
,
Bazzan
,
G.
,
Durstock
,
M. F.
,
Reich
,
G. W.
,
Vaia
,
R. A.
, and
Joo
,
J. J.
,
2015
, “
Origami Actuator Design and Networking Through Crease Topology Optimization
,”
ASME J. Mech. Des.
,
137
(
9
), p.
091401
. 10.1115/1.4030876
28.
Guang
,
C.
, and
Yang
,
Y.
,
2018
, “
An Approach to Designing Deployable Mechanisms Based on Rigid Modified Origami Flashers
,”
ASME J. Mech. Des.
,
140
(
8
), p.
082301
. 10.1115/1.4040178
29.
Klett
,
Y.
,
2018
, “
PALEO: Plastically Annealed Lamina Emergent Origami
,”
ASME IDETC/CIE International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
,
Aug. 26–29
, p. V05BT07A062.
30.
Nelson
,
T. G.
,
Bruton
,
J. T.
,
Rieske
,
N. E.
,
Walton
,
M. P.
,
Fullwood
,
D. T.
, and
Howell
,
L. L.
,
2016
, “
Material Selection Shape Factors for Compliant Arrays in Bending
,”
Mater. Des.
,
110
, pp.
865
877
. 10.1016/j.matdes.2016.08.056
31.
Nelson
,
T. G.
,
Lang
,
R. J.
,
Pehrson
,
N. A.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2016
, “
Facilitating Deployable Mechanisms and Structures Via Developable Lamina Emergent Arrays
,”
ASME J. Mech. Rob.
,
8
(
3
), p.
031006
. 10.1115/1.4031901
32.
Delimont
,
I. L.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2015
, “
A Family of Dual-Segment Compliant Joints Suitable for Use as Surrogate Folds
,”
ASME J. Mech. Des.
,
137
(
9
), p.
092302
. 10.1115/1.4030875
33.
Peraza Hernandez
,
E. A.
,
Hartl
,
D. J.
, and
Lagoudas
,
D. C.
,
2016
, “
Kinematics of Origami Structures With Smooth Folds
,”
ASME J. Mech. Rob.
,
8
(
6
), p.
061019
. 10.1115/1.4034299
34.
Bilancia
,
P.
,
Berselli
,
G.
,
Bruzzone
,
L.
, and
Fanghella
,
P.
,
2019
, “
A CAD/CAE Integration Framework for Analyzing and Designing Spatial Compliant Mechanisms Via Pseudo-Rigid-Body Methods
,”
Rob. Comput. Integrated Manuf.
,
56
, pp.
287
302
. 10.1016/j.rcim.2018.07.015
35.
Boehm
,
K. J.
,
Gibson
,
C. R.
,
Hollaway
,
J. R.
, and
Espinosa-Loza
,
F.
,
2016
, “
A Flexure-Based Mechanism for Precision Adjustment of National Ignition Facility Target Shrouds in Three Rotational Degrees of Freedom
,”
Fusion Sci. Technol.
,
70
(
2
), pp.
265
273
. 10.13182/FST15-217
36.
Chen
,
G.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2018
, “
Membrane-Enhanced Lamina Emergent Torsional Joints for Surrogate Folds
,”
ASME J. Mech. Des.
,
140
(
6
), p.
062303
. 10.1115/1.4039852
37.
Chen
,
G.
, and
Howell
,
L. L.
,
2009
, “
Two General Solutions of Torsional Compliance for Variable Rectangular Cross-Section Hinges in Compliant Mechanisms
,”
Precision Eng.
,
33
(
3
), pp.
268
274
. 10.1016/j.precisioneng.2008.08.001
38.
Howell
,
L. L.
,
DiBiasio
,
C. M.
,
Cullinan
,
M. A.
,
Panas
,
R. M.
, and
Culpepper
,
M. L.
,
2010
, “
A Pseudo-Rigid-Body Model for Large Deflections of Fixed-Clamped Carbon Nanotubes
,”
ASME J. Mech. Rob.
,
2
(
3
), p.
034501
. 10.1115/1.4001726
39.
Chen
,
G.
, and
Howell
,
L. L.
,
2018
, “
Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms
,”
Chin. J. Mech. Eng.
,
31
(
14
), pp.
1
1
. 10.3901/JME.2018.01.001