# Design of Soft Origami Mechanisms with Targeted Symmetries

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## Abstract

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## 1. Introduction

## 2. Optimization Framework for Nonlinear Origami Mechanics

#### 2.1. Optimization Overview

#### 2.2. Nonlinear Origami Truss Model

## 3. Optimization Results

#### 3.1. Problem Setup

#### 3.2. Small Displacement Optimization Results

#### 3.3. Large Displacement Analysis

## 4. Multistability of Unit Actuators

## 5. Networks of Unit Actuators

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Overview of optimization framework. Inset images in flowchart (from top to bottom) show the initialization of the ground structure with uniform fold stiffness, unit truss element formulation and boundary conditions described on the ground structure, actuated design where objective function and constraints are evaluated, and design update on the fold topology through fold stiffness changes as determined from the sensitivity analysis.

**Figure 2.**Optimization problem setup for ground structure with different symmetries: (

**a**) two vertical reflection symmetries, ${\sigma}_{v}^{x}$ and ${\sigma}_{v}^{y}$, and a rotational symmetry, ${C}_{2}$; (

**b**) single vertical reflection symmetry, ${\sigma}_{v}^{x}$, (

**c**) rotational symmetry, ${C}_{2}$. The vertical reflection symmetry, ${\sigma}_{v}^{x}$ refers to symmetry about the X–Z plane, the vertical reflection symmetry, ${\sigma}_{v}^{y}$, refers to symmetry about the Y–Z plane, and the rotational symmetry, ${C}_{2}$ refers to the ${180}^{\circ}$ rotational operation about the Y–Z plane. White triangles refer to fixed nodes in the structure, blue arrows/circles refer to actuated nodes, and green arrows/circles refer to output nodes in objective function. Actuation and output directions are in the Z dimension.

**Figure 3.**Summary of optimization results for patterns with a ${C}_{2}$ rotational and two vertical reflection symmetries (see problem setup in top right of figure). (

**a**) resulting objective function value of optimized patterns for varying fold line complexity ${v}_{0}$ and input actuation ${u}_{z}$; (

**b**) internal energy of optimized patterns. Resulting fold patterns from optimization runs for largest input displacement considered are depicted in (

**c**–

**g**), (

**c**) ${v}_{0}=0.9$, (

**d**) ${v}_{0}=0.8$, (

**e**) ${v}_{0}=0.7$, (

**f**) ${v}_{0}=0.6$, and (

**g**) ${v}_{0}=0.5$. Fold topologies shown correspond to optimization runs highlighted by gray box in (

**a**). Scale bar depicts correlation of line thickness/dash length to design variable $\alpha $, where $\alpha =0$ corresponds to a soft hinge (fold line, ${G}_{\mathrm{soft}}$) and $\alpha =1$ to a stiff hinge (${G}_{\mathrm{stiff}}$).

**Figure 4.**Summary of optimization results for patterns with single vertical reflection symmetry, ${\sigma}_{v}^{x}$, (see problem setup in top right of figure). (

**a**) Resulting objective function value of optimized patterns for varying fold line complexity ${v}_{0}$ and input actuation ${u}_{z}$. (

**b**) Internal energy of optimized patterns. Resulting fold patterns from optimization runs for largest input displacement considered are depicted in (

**c**–

**g**), (

**c**) ${v}_{0}=0.9$, (

**d**) ${v}_{0}=0.8$, (

**e**) ${v}_{0}=0.7$, (

**f**) ${v}_{0}=0.6$, and (

**g**) ${v}_{0}=0.5$. Fold topologies shown correspond to optimization runs highlighted by gray box in (

**a**). Scale bar depicts correlation of line thickness/dash length to design variable $\alpha $, where $\alpha =0$ corresponds to a soft hinge (fold line, ${G}_{\mathrm{soft}}$) and $\alpha =1$ to a stiff hinge (${G}_{\mathrm{stiff}}$).

**Figure 5.**Summary of optimization results for patterns with rotational symmetry ${C}_{2}$ (see problem setup in top right of figure). (

**a**) resulting objective function value of optimized patterns for varying fold line complexity ${v}_{0}$ and input actuation ${u}_{z}$; (

**b**) internal energy of optimized patterns. Resulting fold patterns from optimization runs for largest input displacement considered are depicted in (

**c**–

**g**), (

**c**) ${v}_{0}=0.9$, (

**d**) ${v}_{0}=0.8$, (

**e**) ${v}_{0}=0.7$, (

**f**) ${v}_{0}=0.6$, and (

**g**) ${v}_{0}=0.5$. Fold topologies shown correspond to optimization runs highlighted by gray box in (

**a**). Scale bar depicts correlation of line thickness/dash length to design variable $\alpha $, where $\alpha =0$ corresponds to a soft hinge (fold line, ${G}_{\mathrm{soft}}$) and $\alpha =1$ to a stiff hinge (${G}_{\mathrm{stiff}}$).

**Figure 6.**Summary of optimized fold pattern performance from results in Figure 3, Figure 4 and Figure 5. (

**a**) performance map of all optimized designs in actuation-energy space; (

**b**) zoomed view of (

**a**); (

**c**) actuated designs for ${v}_{0}=0.9$. The optimized designs with only rotational symmetry (green patch) have the optimal amount of actuation with lower energy cost.

**Figure 7.**Results of large displacement analysis for fold patterns optimized via small displacement analysis. (

**a**) performance map of all optimized designs in actuation-energy space; (

**b**) zoomed-in view of (

**a**); (

**c**) actuated designs for ${v}_{0}=0.9$. The rotational symmetry optimized designs (green patch) have the optimal amount of actuation with lower energy cost. The patterns with all symmetries (blue patch) have more optimal performance relative to the designs with only a single vertical reflection symmetry, which is in contrast to small displacement actuation results in Figure 6.

**Figure 8.**Multistable configurations of core actuating topologies from Section 3. (

**a**) equilibrium folded state corresponding to symmetric actuator with a rotational and two vertical reflection symmetries; (

**b**,

**c**) stable pop-through instabilities exist for each arm in the pattern; (

**d**) equilibrium folded state corresponding to asymmetric actuator with single vertical reflection symmetry and asymmetric actuation of each arm; (

**e**,

**f**) stable pop-through instabilities exist for each arm in the pattern; (

**g**) equilibrium folded state corresponding to an actuator with only rotational symmetry resulting in twist motion; (

**h**,

**i**) stable pop-through instabilities exist for each arm in the pattern.

**Figure 9.**Shape change and actuation of networks composed of symmetric actuator result from Figure 3c. (

**a**) loop network of four actuators with alternating orientations; (

**b**) expanded oval shape from (

**a**) accessed through symmetric pop-through instability (see Figure 8c) of nodes labeled I; (

**c**) expanded rectangular shape accessed through asymmetric pop-through instability (see Figure 8b) of nodes labeled II; (

**d**) actuating mechanism for 4-unit network shown in (

**a**); (

**e**) actuating mechanism for 6-unit network. The networks in (

**d**,

**e**) are accessed through actuation of the center elements.

**Figure 10.**Linear actuator networks and loops composed of twist actuator with rotational symmetry. (

**a**) linear actuating network where fold patterns are aligned but orientation of folds are alternating. This results in an offset linear motion when the network is stretched/compressed; (

**b**) linear actuator with alternating orientation of fold pattern. The rotated fold patterned results in a twisting motion when linearly stretched/compressed; (

**c**) actuation of loop network of twist actuators. This looped network is composed of alternating pairs with the same fold pattern and alternating orientation. With actuation of every other unit, the shape is contracted and reoriented (bottom) into the direction perpendicular to the equilibrium expanded state (top).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gillman, A.; Wilson, G.; Fuchi, K.; Hartl, D.; Pankonien, A.; Buskohl, P. Design of Soft Origami Mechanisms with Targeted Symmetries. *Actuators* **2019**, *8*, 3.
https://doi.org/10.3390/act8010003

**AMA Style**

Gillman A, Wilson G, Fuchi K, Hartl D, Pankonien A, Buskohl P. Design of Soft Origami Mechanisms with Targeted Symmetries. *Actuators*. 2019; 8(1):3.
https://doi.org/10.3390/act8010003

**Chicago/Turabian Style**

Gillman, Andrew, Gregory Wilson, Kazuko Fuchi, Darren Hartl, Alexander Pankonien, and Philip Buskohl. 2019. "Design of Soft Origami Mechanisms with Targeted Symmetries" *Actuators* 8, no. 1: 3.
https://doi.org/10.3390/act8010003