Hyper Yoshimura: How a slight tweak on a classical folding pattern unleashes meta-stability for deployable robots

Ziyang Zhou; Yogesh Phalak; Vishrut Deshpande; Ethan O'Brien; Ian Walker; Suyi Li

cs.ROMay 15, 2025

Hyper Yoshimura: How a slight tweak on a classical folding pattern unleashes meta-stability for deployable robots

Authors: Ziyang Zhou, Yogesh Phalak, Vishrut Deshpande, Ethan O'Brien, Ian Walker, Suyi Li

Organizations: Department of Mechanical Engineering, Virginia Tech,, Blacksburg & 24060, USA. · Department of Electrical Engineering and Computer Science, University of Wyoming, Laramie & 82071, USA.

Abstract

Deployable structures inspired by origami have provided lightweight, compact, and reconfigurable solutions for various robotic and architectural applications. However, creating an integrated structural system that can effectively balance the competing requirements of high packing efficiency, simple deployment, and precise morphing into multiple load-bearing configurations remains a significant challenge. This study introduces a new class of hyper-Yoshimura origami, which exhibits a wide range of kinematically admissible and locally metastable states, including newly discovered symmetric "self-packing" and asymmetric "pop-out" states. This metastability is achieved by breaking a design rule of Yoshimura origami that has been in place for many decades. To this end, this study derives a new set of mathematically rigorous design rules and geometric formulations. Based on this, forward and inverse kinematic strategies are developed to stack hyper-Yoshimura modules into deployable booms that can approximate complex 3D shapes. Finally, this study showcases the potential of hyper-Yoshimura with a meter-scale pop-up cellphone charging station deployed at our university's bus transit station, along with a 3D-printed, scaled prototype of a space crane that can function as an object manipulator, solar tracking device, or high-load-bearing structure. These results establish hyper-Yoshimura as a promising platform for deployable and adaptable robotic systems in both terrestrial and space environments.

Figures & tables

Figure 1: Motivation and Overview. (A) As Lady Mona Lisa rolled up her sleeve, the fabric inevitably buckled into a Yoshimura-like origami pattern, a deformation that Leonardo da Vinci captured with remarkable accuracy in the painting. (B) A four-module Yoshimura pattern consisting of the classical design ( $\beta=90^{\circ}/n$ ) and the new, “hyper” design proposed in this study ( $90^{\circ}/n\leq\beta\leq 45^{\circ}$ ). (C) A hyper-Yoshimura with different $\beta$ values exhibiting sequential buckling, and the distinct D eployed, F olded, and self- P acked states. (D) Engineering demonstrations in this study include a robotic gripper with an expandable workspace and a deployable boom that can reposition and reorient solar panels.

Figure 2: The Geometry of hyper-Yoshimura and its Kinematically Admissible States. (A) 3D renderings of $n=3$ and $n=4$ modules featuring the classical Yoshimura design. Here, their F olded state are exactly flat. (B) 3D renderings of $n=3$ and $n=4$ modules featuring hyper-Yoshimura design, showing their distinct D eployed, F olded, and self- P acked states. (C) Comparison between classical Yoshimura and hyper-Yoshimura under axial compression. The self-packing behavior allows the hyper-Yoshimura to settle into a compact, discrete hyperbolic surface without “springing back”. (D) Cutting experiment revealing the residual over-curvature associated with the self-Packing. (E-F) Correlations between the sector angle $\beta$ and the module’s height at the D eployed-, F olded-, and self- P acked states ( $w$ , $d$ , and $h$ , respectively). The lower plot in (G) also shows that $\chi$ begins at $180^{\circ}$ at the classical, flat-foldability boundary and decreases with $\beta$ , corresponding to increasing hyper-fold. In these plots, square symbols correspond to the lower limit of $\beta=90^{\circ}/n$ (classic Yoshimura), whereas the circular symbols correponds to the upper limit of $\beta=45^{\circ}$ .

Figure 3: Mechancis Analysis of the hyper-Yoshimura’s Buckling. (A) Snapshots from a bar-network simulation illustrating how elastic stretching/compression of the bar elements can capture the overall potential-energy landscape. (B) Potential-energy landscape, $U$ , of the hyper-Yoshimura modules over the design–configuration space. The loci of the D eployed, F olded, and self- P acked states, obtained from kinematic analysis, coincide precisely with the simulated energy minima. The lower subplot presents representative energy-landscape slices for $\beta=30^{\circ}$ , $31.72^{\circ}$ (the golden-ratio angle), and $35^{\circ}$ . Yellow stars indicate the degenerate classical endpoint at $(\beta,c)=(30^{\circ},1)$ , where the F olded and self- P acked configurations coincide. (C) Compressive reaction force, $F$ , of the hyper-Yoshimura modules over the design–configuration space. The lower subplot shows representative force–compression curves for the same sector angles. (D) Dependence of the potential-energy barrier, $\Delta U_{\mathbf{D}\rightarrow\mathbf{F}}$ , and critical buckling force, $F_{\mathrm{cr}}$ , on the sector angle $\beta$ . Such correlation is the basis of sequential buckling discussed later. (E) Comparison between experimental measurements and bar-network simulations for three 3D-printed hyper-Yoshimura modules. Here, $i$ denotes the initial D eployed state at the beginning of the compression loading cycle; $ii$ marks the onset of buckling toward the F olded state, after which the samples lose contact with the load cell; and $iii$ denotes the transition to the self- P acked state. Overall, the simulations accurately capture the relationship between the buckling response and the underlying Yoshimura crease geometry.(Movie S2)

Figure 4: Sequence Local Buckling with Hybrid Designs of Different $\beta$ Angles. (A) Simplified fabrication procedure of the hyper-Yoshimura specimen. Note that annealing is conducted at $140^{\circ}$ C. (B) Finished prototype, highlighting the different sector angles. (C, D) Mechanical compression and extension test results, respectively. The solid lines are the averaged data from 7 successive loading cycles, and the shaded bands are the standard deviation.

Figure 5: Robotic crane demonstration with a hyper-Yoshimura of $\beta=31.72^{\circ},35^{\circ}$ , $n=3$ , and $m=6$ . (A) Demonstration setup highlighting pneumatic and tendon actuation and two hot-swappable end attachments: a gripper and a solar panel. (B) Switching sequence among all six possible configurations. Hyper-Yoshimura modules are soft in bending in the F olded states, so four configurations are maneuverable by differential tendon control. (C) Deployable manipulation demonstration with an expandable workspace. (D) Solar panel re-positioning and re-orientation demonstration. (E) Load-bearing capacity of the D eployed hyper-Yoshimura, which carries 28.5 times its structural weight.

States	$\beta$	$\theta$	$d$	$\chi$	$c=1-d/\tan\beta$
Deployed	$(0^{\circ},90^{\circ})$	$180^{\circ}$	$w$	$180^{\circ}$	$0$
Folded	$\left[\frac{90^{\circ}}{n},90^{\circ}\right)$	$\theta_{\mathbf{F}}$	$d_{\mathbf{F}}$	$180^{\circ}$	$1-\sqrt{1-\frac{\tan^{2}(90^{\circ}/n)}{\tan^{2}\beta}}$
Self-Packed	$\left(\frac{90^{\circ}}{n},45^{\circ}\right)$	$\theta_{\mathbf{F}}$	$0$	$\cos^{-1}\!\left(1-\frac{2\sin^{2}(90^{\circ}/n)}{\sin^{2}\beta}\right)$	$1$

Table 1: Summary of the geometric conditions for the D eployed, F olded, and self- P acked states, under the non-dimensional convention

l=1

. At the F olded state,

d_{\mathbf{F}}=\sqrt{\tan^{2}\beta-\tan^{2}(90^{\circ}/n)}

and

\theta_{\mathbf{F}}=2\sin^{-1}(d_{\mathbf{F}}/w)=2\cos^{-1}[\tan(90^{\circ}/n)/\tan\beta]

. All angular variables follow the included-dihedral convention:

\theta

applies to mountain and valley creases, whereas

\chi

applies to virtual creases. An unbent crease therefore has an angle of

180^{\circ}

. During hyperfolding, the original crease dihedrals remain at

\theta_{\mathbf{F}}

while

\chi

decreases below

180^{\circ}

. A distinct self- P acked state exists only for

90^{\circ}/n<\beta<45^{\circ}

; at the lower boundary it coincides with the flat-Folded state, and at the upper boundary the facets reach the self-intersection limit. The normalized compression ratio is

c=1-d/\tan\beta

, with

c=0

in the Deployed state and

c=1

in the self-Packed state; at the classical flat-foldability boundary, the Folded state also reaches

c=1

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