# Geometric and Kinematic Analyses and Novel Characteristics of Origami-Inspired Structures

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design Method of a Crease Pattern

#### 2.1. Mathematical Theory for Origami Design

- (a)
- Given two points of P1 and P2, we can make a crease over P1 and P2;
- (b)
- Given two points of P1 and P2, we can fold P1 onto P2 along a crease to make the two points coincide;
- (c)
- Given two straight lines L1 and L2, we can fold line L1 onto L2 along a crease to make the two lines coincide;
- (d)
- Given a point P1 and a line L1, we can fold line L1 onto itself along a crease passing through point P1;
- (e)
- Given two points P1 and P2 and a line L1, we can fold point P1 onto line L1 along a crease passing through point P2;
- (f)
- Given two points P1 and P2 and two lines L1 and L2, we can fold P1 and P2 onto L1 and L2, respectively, along a crease; and
- (g)
- Given a point P and two lines L1 and L2, we can fold point P onto line L1 along a crease perpendicular to L2.

#### 2.2. Common Crease Patterns

#### 2.3. Design of Crease Patterns

_{1}and G

_{2}, which act as two independent edges of the origami pattern. That is,

_{1}□ G

_{2}.

**A**(G) for the four-fold origami pattern can be easily computed by those of the subgraphs, $A\left({G}_{1}\right)$ and $A\left({G}_{2}\right)$. That is,

_{1}, ${\mathbf{I}}_{n2}$ denotes the ${n}_{2}\times {n}_{2}$ identity matrix, and ${n}_{2}$ denotes the number of vertices of the subgraph G

_{2}. This graph-theoretic method can effectively construct the involved matrices and origami models and, thus, enhance the configuration processing for origami structures.

## 3. Kinematic Analysis of Origami-Inspired Structures

#### 3.1. Rigid Folding of Origami

#### 3.2. Large Deformations of Origami Structures

#### 3.3. Kinematic Singularity and Potential Bifurcation of Origami Structures

_{2v}symmetric four-bar linkage and the typical configurations along the motion path. $({x}_{3}-{x}_{1})/{L}_{1}$ represents the ratio of the difference between the coordinates along the x-axis of the nodes 1 and 3 to the length of link 1, while ${\theta}_{14}$ denotes the angle between the links 1 and 4.

_{3v}symmetric deployable pin-jointed structure. The solid line represents the primary path, and the dashed lines represent the bifurcation paths. Nodes 7–12 are free nodes. Kinematic bifurcations can be easily noticed from the variation of the distance from a typical free node 7 to the center of the structure. When the structure transforms along the primary path and reaches the critical point, three different motion paths appear, which reveal that the structure can go on following the original path and keep C

_{3v}symmetry, or it can transform into one of the bifurcation paths and keep C

_{v}symmetry. Gan et al. [68] studied kinematic singularity of closed-loop mechanisms. Based on bifurcation of single-vertex four-fold origami, Waitukatis et al. [69] proposed a multistable origami structure with reconfigurable stiffness, and they pointed out that more than five stable states could be obtained by adjusting energy parameters. Silverberg et al. [59] studied bifurcation characteristics of torsional origami structures, and they found that such structures can be switched from monostable to bistable by bending.

- (a)
- Set the initial iteration step size to $\left|{\beta}^{0}\right|=0.1{L}_{max}$, as the ideal structure moves along the unique motion path before singularity. ${L}_{max}$ is the maximum size of the members. Set the initial value ${\mathsf{\Sigma}}_{rr}^{0}$ of ${\mathsf{\Sigma}}_{rr}$ to ${\mathsf{\Sigma}}_{rr}^{0}=1$, and ${\mathsf{\Sigma}}_{rr}$ denotes the entry of the r-th row and r-th column in the singular matrix $\mathsf{\Sigma}$.
- (b)
- In each iteration, the generalized displacement compatibility matrix
**J**of the structure can be obtained according to the geometric configuration of the iteration step $t=t+1$, and the corresponding minimum nonzero singular value ${\mathsf{\Sigma}}_{rr}^{t}$ can be obtained. - (c)
- The size of iteration step ${\beta}^{t}$ is determined by ${\mathsf{\Sigma}}_{rr}^{t}$. When ${\mathsf{\Sigma}}_{rr}^{t}\le 0.1{\mathrm{and}\mathsf{\Sigma}}_{rr}^{t}\le {\mathsf{\Sigma}}_{rr}^{t-1}$, the minimum nonzero singular value decreases and approaches zero; take ${\beta}^{t}={\mathsf{\Sigma}}_{rr}^{t}{\beta}^{0},\left|{\beta}^{t}\right|\ge 0.1{t}_{max}^{-1}\left|{\beta}^{0}\right|$ to avoid crossing singularities. ${t}_{max}$ is the maximum number of iteration steps. When ${\mathsf{\Sigma}}_{rr}^{t}>0.1{\mathrm{or}\mathsf{\Sigma}}_{rr}^{t}{\mathsf{\Sigma}}_{rr}^{t-1}$, it moves away from singularities, take ${\beta}^{t}=5{\beta}^{t-1},\left|{\beta}^{t}\right|\le \left|{\beta}^{0}\right|$ to accelerate the solution process.
- (d)
- After determining the size of ${\beta}^{t}$, the motion path can be tracked and the configuration can be updated according to the nonlinear prediction-correction method.
- (e)
- When t is greater than the maximum number of iteration steps ${t}_{max}$, the configuration is singular and the iteration ends.

## 4. Novel Characteristics of Origami Structures

#### Recongifurable Stiffness and Configuration

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Folding of a flat-foldable origami pattern with periodic and parallel creases. (

**a**) Unfolded state; (

**b**) partially folded state; (

**c**) folded state.

**Figure 4.**Waterbomb pattern: (

**a**) first stable equilibrium configuration; (

**b**) second stable equilibrium configuration.

**Figure 5.**Diagonal pattern induced by torsional buckling of cylinder: (

**a**) typical crease pattern; (

**b**,

**c**) the development of these folds in a paper specimen twisted between two mandrels.

**Figure 6.**Variation of the generalized Miura-ori and its unfolding process. (

**a**) Folded state; (

**b**,

**c**) partially folded states.

**Figure 7.**Kinematic singularity of deployable pin-jointed structures: (

**a**) C

_{2v}symmetric structure; (

**b**) C

_{3v}symmetric structure.

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Chen, Y.; Yan, J.; Feng, J.
Geometric and Kinematic Analyses and Novel Characteristics of Origami-Inspired Structures. *Symmetry* **2019**, *11*, 1101.
https://doi.org/10.3390/sym11091101

**AMA Style**

Chen Y, Yan J, Feng J.
Geometric and Kinematic Analyses and Novel Characteristics of Origami-Inspired Structures. *Symmetry*. 2019; 11(9):1101.
https://doi.org/10.3390/sym11091101

**Chicago/Turabian Style**

Chen, Yao, Jiayi Yan, and Jian Feng.
2019. "Geometric and Kinematic Analyses and Novel Characteristics of Origami-Inspired Structures" *Symmetry* 11, no. 9: 1101.
https://doi.org/10.3390/sym11091101