Optimal Design and Actuation of Multistable Origami Structures

Mohamed Becha
Semester Project — Mechanics & Materials Laboratory
ETH Zürich — June 2024

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Overview

This project investigates the design and actuation of multistable origami mechanisms using a mechanics-based bar-and-hinge model.

Objectives:

1. Design linear springs such that prescribed configurations are stable equilibria.
2. Compute Minimum Energy Paths (MEPs) between stable configurations using the Nudged Elastic Band (NEB) method.

The full mathematical derivations and numerical results are available in the accompanying report.

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Mechanical Model

We consider a 2-DoF five-bar linkage composed of:

• Inextensible rigid bars
• Frictionless hinges
• Linear springs between selected nodes

The total potential energy is

$\ E(\theta_A, \theta_E) = \sum_{i=1}^{5} \frac{1}{2} k_i \left( l_i - l_{0,i} \right)^2 $

where

• $k_i$ is the spring stiffness
• $l_i$ is the current spring length
• $l_{0,i}$ is the rest length

The design variables are

$\ u = \begin{bmatrix} k_1 \\ \vdots \\ k_{n_s} \\ \sigma_1^0 \\ \vdots \\ \sigma_{n_s}^0 \end{bmatrix} $

with

$\sigma_i^0 = k_i l_{0,i}$

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Part I — Spring Design for Stable Target Configurations

Equilibrium Conditions

For each target configuration,

$$ M_{\text{config}} u = 0. $$

Stacking all configurations gives

$$ M_{\text{total}} u = 0. $$

Feasibility requires

$$ n_{\text{targets}} \cdot n_{\text{DoF}} \le 2 n_{\text{springs}}. $$

If satisfied, solutions exist in the nullspace

$$ u^\ast \in \mathcal{N}(M_{\text{total}}). $$

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Stability Conditions

For each configuration, stability requires

$$ S_i \succeq 0. $$

Equivalently,

$$ x^T S_i x \ge 0 \quad \forall x \in \mathbb{R}^2. $$

The optimization problem is formulated as

$$ \max Trace(S) $$

subject to

• $S_i \succeq 0$
• $k_{\min} \le k_i \le k_{\max}$
• $f_{\min} \le f_i \le f_{\max}$

Maximizing the trace increases the local curvature of the energy landscape near the target configurations.

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Part II — Minimum Energy Path via NEB

To compute transition paths between stable states, we use the Nudged Elastic Band (NEB) method.

A discrete path is represented by images

$$ R_0, R_1, \dots, R_{N+1}, $$

where $R_0$ and $R_{N+1}$ are fixed stable configurations.

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NEB Force Decomposition

For each intermediate image $R_i$, the NEB force is

$$ F_i = F^{\parallel}_{s,i} - \nabla V(R_i)^{\perp}. $$

The perpendicular component of the gradient is

$$ \nabla V(R_i)^{\perp} = \nabla V(R_i) - \left( \nabla V(R_i) \cdot \hat{\tau}_i \right) \hat{\tau}_i. $$

The tangential spring force is

$$ F^{\parallel}_{s,i} = k_s \left( |R_{i+1} - R_i| - |R_i - R_{i-1}| \right) \hat{\tau}_i. $$

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Optimization Criterion

The RMS stopping criterion is

$$ F_{\mathrm{RMS}} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} |F_i|^2 }. $$

Conjugate Gradient optimization is used to iteratively reduce $F_{\mathrm{RMS}}$ and converge toward a Minimum Energy Path.

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Key Results

• Three prescribed configurations successfully stabilized
• Stability matrices positive semidefinite
• Non-trivial energy landscape with additional equilibrium states
• Smooth Minimum Energy Paths computed between stable states

Energy contour plots and NEB trajectories are provided in the report.

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Repository Contents

• report.pdf — Full technical report
• figures/ — Energy landscape and NEB visualizations

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Technical Topics

• Multistable mechanisms
• Convex optimization
• Linear Matrix Inequalities (LMI)
• Nullspace methods
• Singular Value Decomposition
• Energy landscape analysis
• Conjugate Gradient optimization

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Code Availability

The MATLAB implementation is not publicly available.

All derivations and numerical results are fully documented in the report.