Pop-Up Catcher: A Flat-Foldable Gripper Using Multistate Passive Actuation

Abstract

Origami techniques have significantly impacted robotics, expanding its capabilities in shape transformation. Pop-up transformations, inspired by pop-up books, offer intriguing applications in robotics fields, including deployable robots. However, designing origami-inspired robots for transitioning to a completely flat state poses unique challenges, particularly in multistate passive actuation. This article introduces the “pop-up catcher,” a gripper designed for multistate passive actuation that can be folded flat and actuated passively to grasp the object. To ensure its reliable state transition, we conduct “transition path planning” with the potential energy surface modulation. We demonstrate the pop-up deployment and passive capture of the target object using our flat-foldable catcher comprised of our pop-up gripper and self-locking modular Sarrus origami that can be folded into a profile less than 25 mm thick while capturing objects over 500 mm away.

Keywords

gripper multistate origami passive actuation path potential energy

Introduction

Origami, characterized by its inherent simplicity and reconfigurability, has influenced various scientific and engineering disciplines.^1–4 Robotics represents one such field that benefits from origami, particularly in terms of design methodology and fabrication techniques. By harnessing the principles of origami, robotics has expanded its scope in shape transformation capabilities. This use has been evident in a range of applications such as lightweight manipulators and grippers,^5–10 mobile platforms,^11–14 medical robots,^15,16 and deployable space structures.^17–20

Pop-up transformations, inspired by the mechanisms found in pop-up books, present intriguing applications of origami. These transformations involve transitioning from a wholly flat shape to a specific three-dimensional form. The ability to make a completely flat shape ideally results in the minimal achievable volume, thus maximizing the space-saving benefits of origami. Although this concept originated in pop-up books, its ability to morph from a flat state to a three-dimensional structure has found high potential in diverse applications, ranging from pop-up constructions to self-assembling robots.^21–23

Despite these advantages, designing an origami robotic system that can transition to a completely flat state presents unique challenges, particularly in relation to actuation. Two primary attributes of origami—the ultra-thin profile and high kinematic degrees of freedom (DoF), which are both compelling characteristics—necessitate actuation methods that are not commonly used or readily available. To address these challenges, various actuation techniques have been developed, including pneumatic inflation,^9,23,24 shape memory alloys,^25,26 and cable-driven systems.^12,22

Passive actuation presents an alternative way to tackle these challenges, typically using structurally stored energy within the body of the system.²⁷ Systems capable of passive actuation commonly exhibit high redundancy in DoF.^27–29 These spare DoF can create an energy potential field of the system by introducing elasticity or damping characteristics that determine the system’s behavior.^27,30 As a result, interactions with the external environment trigger changes in the internal energy levels of the system. These energy-level fluctuations, in conjunction with the elasticity or damping characteristics assigned to the redundant DoF, create predefined motion sequences. This method reduces reliance on additional components, such as actuators and batteries, which are often necessary for active actuation systems, hence maintaining the inherent lightweight and compact benefits of origami.^22,31 Simultaneously, it facilitates the execution of diverse and adaptable behaviors, even when faced with the constraints of limited actuation sources.^32,33

To take advantage of passive actuation, diverse studies based on the potential energy landscape have been conducted to develop passive origami and physically intelligent systems.^34,35 In the field of origami-based systems, the potential energy-based approach has served to improve multifaceted problems: stability, reconfigurability, and functional properties. Golden-ratio Yoshimura origami enabled the pop-out of complex shapes of booms and structures by meta-stability based on intentional defiance of constraints.³⁶ Modularized multistable origami was developed into meta-materials and robotic mobilities by integrating multiple combinations of bistability.³⁷ In addition to stability, dual-stiffness and threshold characteristics were realizable by layered elastic membranes.²⁸

While passive actuation energy offers several advantages, its reliability remains a significant challenge, limiting its application in various fields. The risk mainly occurs because passive actuation depends on DoF that do not involve active control, increasing the risk of mechanical singularities in systems with redundant kinematics.^38,39 This issue becomes particularly problematic in the implementation of multistate actuation. At the bifurcation point between local energy minima, predicting the direction in which the system will be deployed becomes challenging. Common solutions include limiting the operation of passive actuation to prevent singularity or using dynamic inertia to dictate movement direction. However, these approaches restrict the design possibilities of multistate actuation systems.

In this study, we introduce the “pop-up catcher,” a gripper that can be folded into a completely flat configuration (less than 25 mm in thickness), capable of capturing objects located over 500 mm away using multistate passive actuation. The gripper is designed to operate in three distinct states: a flat-folded state, a capture-ready state, and a captured state. To address the challenge of reliable state transition in multistability systems, we have reshaped the energy potential field to avoid any expected bifurcation for passive actuation of the gripper. The modulation can be achieved by introducing additional DoF and rearranging the elastic elements, effectively circumventing singularity issues and ensuring stable actuation. Our demonstrations validate this approach, showing that the pop-up catcher maintains an extremely low profile while capturing various objects through physical contact without external actuators or sensors. In addition, we perform an analysis of the potential energy surface to realize a flat-foldable gripper origami with passive actuation capable of taking on three distinct states. Primarily, we derive the bio-inspired folding pattern and mechanical arrangement. Subsequently, experimental validations are carried out to confirm the modeling accuracy in terms of passive actuation and grasping capabilities. Finally, we introduce a fully flat-foldable and passive-actuatable pop-up catcher incorporating self-locking Sarrus origami and our passive gripper origami.

Design Approach

Initial derivation of a flat-foldable and bistable origami

Anisotropically pre-strained structures inspired by the Venus-flytrap are widely used in developing rapid capture systems (Fig. 1A, B).^40–42 These systems achieve fast motion by the snap transition from bistable energy properties.⁴³ However, storing the curved structure in a completely flat state can pose challenges and carry the risk of permanent deformation, compromising the functionality.

FIG. 1.

The design process of MPA origami. (A) The snap transition mechanism of the Venus-flytrap capable of passively capturing insects with external stimulation. (B) The structural concept of the Venus flytrap-inspired bistable sheet formed by anisotropic pre-strains (red and green elements). (C) Derivation of the TPA origami from the bistable sheet. The left-side and right-side patterns indicate the fold line alone and spring-installed patterns, respectively. The folding state is defined by $ρ_{A}$ or $ρ_{B}$ . (D) Derivation of the MPA origami by modifying the TPA origami and releasing the singularity. The additional fold line (purple) adds one extra kinematic DoF. The left-side and right-side patterns indicate the fold line alone and spring-installed patterns, respectively. The folding state is defined by $ρ_{1}$ and $ρ_{2}$ . (E) The kinematic workspace of TPA origami, which has a bifurcation on ( $ρ_{A}$ , $ρ_{B}$ ) = (0 deg, 0 deg). (F) The kinematic workspace of MPA origami. (G) The energy field of TPA origami and its major states: flat folded (ⓐ), flat unfolded (ⓑ), capture ready (stable 1) (ⓒ), captured (stable 2) (ⓓ). (H) Surface plot of the energy field of MPA origami and its major states: flat folded (i), capture ready (stable 1) (ii), flat unfolded (iii), captured (stable 2) (iv). The deployment paths include the first deployment ( $s_{i - i i}$ ), passive actuation ( $s_{i i - iii}$ ) through the increase of $ρ_{1}$ and external load, and the final snap-through deployment ( $s_{iii - i v}$ ). The color of the energy field maps illustrates the relative levels of the potential energy ( $V$ ), normalized by the maximum values of $V$ , $V_{Max}$ . DoF, degree of freedom; MPA, multistate passive actuation; TPA, two-state passive actuation.

To overcome the challenges of curved sheets, the inherent curvature can be replaced with an origami system consisting of pre-strained springs and rigid planar facets. Our approach to making a bistable yet flat-foldable system uses single-vertex origami with orthogonally crossing fold lines (Fig. 1C). The orthogonality and singularity of the fold-line pattern prevent both fold lines from varying simultaneously, leading to a cross-like kinematic configuration space of motion with respect to folding angles (Fig. 1E):

Δ ρ_{A} \cdot Δ ρ_{B} = 0

(1)

where

Δ ρ_{A}

and

Δ ρ_{B}

represent the change in folding angle A (

ρ_{A}

) and folding angle B (

ρ_{B}

), respectively. If zero folding angle is defined as a flat-unfolded state, one folding joint maintains zero, whereas the other folding joint varies:

ρ_{A} \cdot ρ_{B} = 0 .

(2)

On the basis of the double orthogonal folding modes, bistability can be formed by installing pre-strained springs with an orthogonal arrangement—each stable state is generated at each folding mode because one folding angle is determined to be zero (see Supplementary Data for calculation of potential energy (V)):

ρ_{i}^{*} = \underset{ρ_{i}}{argmin} V (ρ_{i}, ρ_{j} = 0)

(3)

where

(i, j) \in {(A, B), (B, A)}

. In the case of this two-state passive actuation (TPA) origami, only a single folding angle can vary under the ideal assumptions, the transition between the stable states also follows the straight cross-shaped lines and necessarily visits the flat-unfolded state ([

ρ_{A}, ρ_{B}] = [0 \deg, 0 \deg]

; ⓑ in Fig. 1G).

While the singular folding pattern allows for bistability and flat foldability (Supplementary Movie S1), it also introduces the risk of random deployment at the flat-unfolded state. Specifically, the pattern encounters a deterministic bifurcation point, offering two possible deployment paths (DP#1(ⓐ→ⓑ→ⓒ) and DP#2(ⓐ→ⓑ→ⓓ) in Figure 1G, which cannot guarantee the specific transition direction (Supplementary Movie S2). This bifurcation issue cannot guarantee the specific deployment even when the TPA origami is used for a gripper prototype derived by mirroring (Fig. 3A and I–K; Supplementary Movie S3).

Modification of the potential energy field for multistep deployment path

To address the multipath issue of TPA origami and release the singularity, increasing the kinematic DoF needs to be used. This objective can be achieved by modifying the fold-line arrangement and adjusting the system’s design, cutting, or changing material properties.^10,28 In our design, a 2-DoF mechanism is derived by adding an extra fold line (Fig. 1D), maintaining the assumption of rigid origami, resulting in the multistate passive actuation (MPA) origami (Fig. 1F).

The same assumptions for TPA origami are used to model the potential energy of MPA origami: zero-thickness facets and axial springs across zero-stiffness fold-lines. The extra DoF provided by the additional fold-line allows MPA origami to be deployed in a much wider kinematic region (see subsection Kinematics analysis for kinematic equations of MPA origami), thus offering a variety of possible deployment paths according to the geometry of the pattern and properties of the springs. This flexibility enables MPA origami to avoid the bifurcation during initial deployment.

The energy field of MPA origami enables the prediction of the transition paths based on relative energy levels ( $V$ / $V_{Max}$ ). Starting the deployment from the flat-folded configuration (“i”), MPA origami transitions through a stopover state (“ii”) to the final stable state (“iv”) after passing through a local maximum (“iii”) under external load or energy input (Fig. 1H and Supplementary Movie S4).

Regarding the deployment path, we can predict two types of paths: passive actuation with external interaction and self-deployment with no load. In the case of self-deployment, paths of initial deployment (“i” to “ii”) and snap-through deployment (“iii” to “iv”; Supplementary Movie S5) can be obtained by the minimum energy path under the quasi-static assumption (see Section Potential Energy Analysis for the potential energy model of MPA origami). Self-deployment occurs as energy decreases without any mechanical input:

\nabla V (s_{Self - deployment}^{*}) / / d s_{Self - deployment}^{*}

(4)

where

s_{Self - deployment}^{*}

denotes the pathway consisting of a specific set of [

ρ_{1}^{*}, ρ_{2}^{*}

] during self-deployment (

s_{i - ii}

and

s_{iii - iv}

in Fig. 1H). In this study, we predict the self-deployment path (

ρ_{1}^{*}

ρ_{2}^{*}

) by minimizing

V

with each given state of

ρ_{2} = ρ_{2}^{*}

ρ_{1}^{*} = \underset{ρ_{1}}{argmin} V (ρ_{1}, ρ_{2} = ρ_{2}^{*})

(5)

which can derive s_{Self - deployment} = \{s_{i - ii}, s_{iii - iv}\} .

Regarding the passive capture (ii to iii), a positive rotation of $ρ_{1}$ axis is adopted as a trigger mechanism for snap-through until the flat-unfolded state (iii). In this case, the passive actuation path ( $s_{ii - iii}$ ) can be derived by minimizing $V$ with respect to the given angle $ρ_{1} = ρ_{1}^{*}$ :

ρ_{2}^{*} = \underset{ρ_{2}}{argmin} V (ρ_{1} = ρ_{1}^{*}, ρ_{2}) | {d ρ}_{1} > 0, d V > 0 .

(6)

The first constraint ( ${d ρ}_{1} > 0$ ) is defined as a kinematic condition (deployment direction) for passive actuation, and the energy constraint ( $d V > 0$ ) comes from the energy increase by work done by the external load (see Section Parametric study of trigger and grasping loads for characteristics of MPA origami). During the passive actuation, $ρ_{1}$ and $V$ increase due to kinematic and mechanical input; it arrives at $ρ_{1} = 0$ , and then it rapidly suffers snap through via the flat-unfolded state (iii). Therefore, by strategically planning kinematic DoF, stable locations, and deployment paths based on the energy landscape, we can mitigate the risk of uncertain deployment behaviors. Leveraging the MPA origami, we can design a flat-foldable, passive, and deployable gripper (Fig. 3B, L, and M; Supplementary Movie S6).

Kinematics analysis

To obtain the kinematic configuration space of MPA origami (Fig. 1F), the position of node H is obtained through two independent ways: folding ρ₃ and ρ₄ and folding ρ₂ and ρ₃ (Fig. 2A). Assuming that facet OADC is a fixed reference plane and node O is the origin [0, 0, 0]^T, two sets of positions for H can be derived:

H (ρ_{3}, ρ_{4}) = [\begin{matrix} L_{2} \cos ρ_{3} + L_{4} \sin ρ_{4} \sin ρ_{3} \\ L_{4} \cos ρ_{4} \\ L_{2} \sin ρ_{3} + L_{4} \sin ρ_{4} \cos ρ_{3} \end{matrix}]

(7)

H (ρ_{1}, ρ_{2}) = [\begin{matrix} L_{2} \cos ρ_{2} \\ L_{4} \cos ρ_{1} - L_{2} \sin ρ_{2} \sin ρ_{1} \\ L_{4} \sin ρ_{1} + L_{2} \sin ρ_{2} \cos ρ_{1} \end{matrix}]

(8)

where the x-axis refers to norm(

\vec{DC}

) starting from O, the y-axis refers to the norm(

\vec{DA}

) starting from O, and the z-axis refers to the cross product of unit x-vector and y-vector departing from O (Fig. 2A and Table 1). However, the length between D and H remains constant due to the assumption of rigid facets, requiring the elimination of the redundant constraint. Therefore, the Cartesian coordinate system can be transformed into a spherical coordinate system, and two independent constraints are derived:

[\begin{matrix} \sqrt{L_{2}^{2} + L_{4}^{2}} \\ \cos^{- 1} (\frac{L_{2} \sin ρ_{3} + L_{4} \sin ρ_{4} \cos ρ_{3}}{\sqrt{L_{2}^{2} + L_{4}^{2}}}) \\ \tan^{- 1} (\frac{L_{4} \cos ρ_{4}}{L_{2} \cos ρ_{3} + L_{4} \sin ρ_{4} \sin ρ_{3}}) \end{matrix}] = [\begin{matrix} \sqrt{L_{2}^{2} + L_{4}^{2}} \\ \cos^{- 1} (\frac{L_{4} \sin ρ_{1} + L_{2} \sin ρ_{2} \cos ρ_{1}}{\sqrt{L_{2}^{2} + L_{4}^{2}}}) \\ \tan^{- 1} (\frac{L_{4} \cos ρ_{1} - L_{2} \sin ρ_{2} \sin ρ_{1}}{L_{2} \cos ρ_{2}}) \end{matrix}]

(9)

FIG. 2.

Modeling of the performances of MPA origami. (A) Representation of kinematic and geometric variables: folding angles ( $ρ_{1}$ , $ρ_{2}$ , $ρ_{3}$ , $ρ_{4}$ , and $ρ_{5}$ ), lengths ( $L_{1}$ , $L_{2}$ , $L_{3}$ , and $L_{4}$ ), in-plane split angle ( $α$ ), and nodes. All folding angles become zero in the flat-unfolded state (see Table 1). Design parameters are chosen into $λ_{1} = x_{0} / d$ , $λ_{2} = y_{0} / L_{3}$ , and $λ_{3} = α$ , which means normalized free length of spring 2 (1), normalized free length of spring 3, and split angle, respectively. (B) Springs' configurations for potential energy analysis. The tensile bar spring assumption, whose equivalent tensile stiffness is inversely proportional to the free length, is utilized. The constant, conservative moment is assumed to operate when spring 2 (1) is fully extended and suffers bending. (C) Normalized moment splines, $T_{3} (ρ_{3})$ and $T_{5} (ρ_{5})$ induced by spring 2 and spring 3, respectively. $T_{3}$ and $T_{5}$ are normalized by $EAd$ and $E A L_{3}$ . (D) Equivalent parameters that represent the shapes of splines $T_{3} (ρ_{3})$ and $T_{5} (ρ_{5}$ ) with respect to design parameters, [ $h_{3} (λ_{1})$ , $φ_{3} (λ_{1})$ ] and [ $h_{5} (λ_{2}, λ_{3})$ , $φ_{3} (λ_{2}, λ_{3})$ ], respectively. (E) Output performances (surface plots) with respect to equivalent parameters, $T_{G} (h_{3}, φ_{3})$ and $T_{T} (h_{5}, φ_{5})$ . The square markers are marked for experimental verification in Figure 4. (F) Definition of contact-point gap ( $b$ ( $ρ_{3}$ ; s)) and the dependent variables: contact distance from $ρ_{3}$ axis ( $s$ ) and the folding angle ( $ρ_{3})$ . (G) Contact-point gap profile at the given s, with respect to $ρ_{3}$ . Geometrically available configuration or range of $ρ_{3}$ for grasping is defined [90 deg, $ρ_{3, col}$ ], where $ρ_{3, col}$ denotes the critical angle at which the collision fold occurs. (H) A schematic diagram to describe the grasping force ( $F_{G}$ ) at the given distance (s) and folding angle ( $ρ_{3}$ ). The force normal to the facet is considered. The horizontal and vertical frame is defined in ^{G}x and ^{G}z axes, respectively. (I) The profiles of the product of contact distance (s) and grasping force ( $F_{G}$ ) in each axis versus the folding angle ( $ρ_{3}$ ) ( $λ_{1}$ = 1.0). The force magnitude is normalized by EAd [equations (20) and (21)]. Grasping force at $ρ_{3} = 90$ º is chosen for the parametric study and experiment.

Table 1.

Definition of Folding Angles

Folding angles	ρ ₁	ρ ₂	ρ ₃	ρ ₄	ρ ₅
Reference	OEBA	OHIE	OADC	OCFG	OGH
Active	OADC	OEBA	OCFG	OGH	OHIE
Positive direction	OA	OE	OC	OG	OH

Furthermore, an equation regarding the folding angle 5 ( $ρ_{5}$ ) and split angle ( $α$ ) is obtained:

n_{OGH} \cdot n_{OHIE} = ‖ n_{OGH} ‖ ‖ n_{OHIE} ‖ \cos ρ_{5}

(10)

where n_OGH and n_OHIE are normal vectors of plane OGH and plane OHIE, respectively. Three equations can be derived from the position of H with respect to O in Cartesian coordinate systems, but the distance between O and H is constant; a set of four equations can be reduced to three independent equations (9) and (10). Therefore, the half pattern retains two kinematic DoF from five folding angles and three kinematic constraints.

Using the kinematic information, a kinematic configuration space described by two folding angles can be produced, as shown in Figure 1F. In this work, a set of [ $ρ_{1}$ , $ρ_{2}$ ] is chosen to draw the region since $ρ_{1}$ and $ρ_{2}$ play roles in trigger and grasping motions, respectively. Therefore, kinematic configuration spaces can be estimated by investigating whether each case produces valid solutions according to equations (9) and (10). In the case of Figure 1G, the set of [ $L_{1}$ , $L_{2}$ , $L_{3}$ , $α$ , $d$ , $x_{0}$ , $y_{0}$ ] was set to [50 mm, 50 mm, 50 mm, 20 degrees, 25 mm, 25 mm, 25 mm] for sketching the coverage. Afterward, the potential energy field can be obtained throughout this coverage.

Potential energy analysis

The configurations of the folding pattern can be predicted by minimizing the potential energy state. In this study, three springs (springs 1, 2, and 3) are installed across $ρ_{2}$ , $ρ_{3}$ , and $ρ_{5}$ axes on the half pattern, respectively (Fig. 2A). The tensile bar spring model is adopted because of two characteristics: tunability of axial stiffness by varying free length and saturation of restoring moment for negative folding angles:

k_{s} = \frac{E A}{s_{0}}

(11)

where

s

s_{0}

E

A

, and

k_{s}

denote an axial direction, free length in the s-direction, Young’s modulus, initial cross-sectional area, and the equivalent spring constant. In this analysis, Young’s modulus and cross-sectional area are also assumed to be constant in the initial state.

Spring 2 can experience bending when negative across the $ρ_{3}$ folding. For negative angles, we assume that the axial springs (zero bending stiffness) exert a restoring moment ( $T_{Restoring}$ ) on $ρ_{3}$ axis because of the thickness of the springs; the minimum moment on the $ρ_{3}$ axis is saturated to $T_{Restoring}$ if $ρ_{3}$ < 0. The conservative moment results in additional potential energy that is proportional to the magnitude of $ρ_{3}$ when $ρ_{3}$ is negative:

{T_{3}|}_{ρ_{3} < 0} = T_{Restoring} = k_{x} (2 d - x_{0}) \frac{t}{2} .

(12)

where

k_{x}

is the spring constant with a free length of x₀, d is the length between the spring anchor and

ρ_{3}

axis, and t is the thickness of spring 1 and spring 2 (Fig. 2B). The potential energy of spring 2 when

ρ_{3}

is negative,

V_{Spring 2} |_{ρ_{3} < 0}

, can be obtained by summing translational potential energy and rotational potential energy (

T_{Restoring} {\cdot (- ρ}_{3})

V_{Spring 2} |_{ρ_{3} < 0} = \frac{1}{2} k_{x} {(2 d - x_{0})}^{2} - T_{Restoring} ρ_{3}

(13)

assuming that the restoring moment is conservative (Fig. 2B). If

ρ_{3} > 0

, the potential energy of spring 2 will remain the translational term (Fig. 2B),

V_{Spring 2} |_{ρ_{3} > 0} = \frac{1}{2} k_{x} {(2 d \cos \frac{ρ_{3}}{2} - x_{0})}^{2} .

(14)

The potential energy from spring 3, whose spring constant and free length are $k_{y}$ and $y_{0}$ , respectively, $V_{Spring 3}$ , can be obtained from the translational term,

V_{Spring 3} = \frac{1}{2} k_{y} {(\bar{G I} - y_{0})}^{2}

(15)

because both ends of spring 3 are fixed at the nodes G and I (Fig. 2A). In addition, we assume that the folding angle

ρ_{5}

remains positive because of the compressive force of spring 3 (Fig. 2B). On the basis of the folding pattern geometry and energy model, transition paths can be calculated by minimizing the total potential energy

V

, which is the sum of

V_{Spring 1}

V_{Spring 2}

, and

V_{Spring 3}

, at each given folding state (equations (13)–(15)).

On the basis of equations (13)–(15), the potential energy surface can be obtained throughout the kinematic configuration space (Fig. 1H). Once the energy field is obtained, deployment paths can also be expected. The paths of self-deployment, such as $s_{i - ii}$ and $s_{iii - iv}$ , can be derived by energy minimization from initial states (equation (5)). The initial points of [ $ρ_{1}$ , $ρ_{2}$ ] for $s_{i - ii}$ and $s_{iii - iv}$ are chosen to [−4°, −171°] and [1°, 2°], and the paths are derived by obtaining $ρ_{1}$ at the specific $ρ_{2}$ which increases monotonously. The path of passive trigger process ( $s_{ii - iii}$ ) to increase potential energy can also be predicted by energy minimization (equation (6)). The path departs from the first stable (destination of $s_{i - ii}$ ) and arrives at the flat-unfolded state under the assumption that $ρ_{1}$ monotonously increases; $ρ_{1}$ performs the axis for passive trigger by external contact and positive rotation during the process. Mechanical performances such as grasping and trigger loads can be estimated using the paths.

Parametric study of trigger and grasping loads

Bistable grippers can be evaluated with two performance metrics: trigger and grasping. From the energy-level analysis, two metrics can be defined: trigger moment $T_{T}$ on $ρ_{1}$ axis, and grasping moment $T_{G}$ on $ρ_{2}$ and $ρ_{3}$ axes as

T_{T} = \max \{\frac{\partial V}{\partial ρ_{1}}\} | {(ρ_{1})}_{Stable 1} < ρ_{1} < 0, d ρ_{1} > 0, d V > 0

(16)

and

T_{G} = (T_{2} + T_{3}) |_{ρ_{3} = 90 °} \approx 2 T_{3} |_{ρ_{3} = 90 \deg}

(17)

where

T_{3}

denotes the moment on the OC axis (ρ₃) caused by spring 2 (Fig. 2B, C). The grasping performance is investigated, especially when ρ₂ is 90º. The grasping moment is approximated to

2 T_{3} because ρ_{2}

equals or is similar to 90º when

ρ_{3}

is 90º.

Looking into the arrangement of springs, each spring force can be derived under the assumption of the bar spring model (Fig. 2B):

F_{Spring 2} = k_{x} (2 d \cos \frac{ρ_{3}}{2} - x_{0}) = E A (\frac{2 d}{x_{0}} \cos \frac{ρ_{3}}{2} - 1)

(18)

F_{Spring 3} = k_{y} (\bar{G I} - y_{0}) = E A (\frac{\bar{G I}}{y_{0}} - 1)

(19)

according to equations (11), (14), and (15) and Figure 2A, B. Regarding

T_{3}

for

T_{G}

, the applied moment caused by

F_{Spring 2}

T_{3}

, can be scaled by

EAd

, and its characteristic can be determined by

\frac{x_{0}}{d}

(

λ_{1}

), which is the ratio of the free length (

x_{0}

) to the installed distance of spring 2 (

d

T_{3} = F_{Spring 2} \cdot (d \sin \frac{ρ_{3}}{2}) = EAd (\frac{2 d}{x_{0}} \cos \frac{ρ_{3}}{2} - 1) \cdot \sin (\frac{ρ_{3}}{2})

(20)

For trigger performance, the effect of $T_{5}$ on $T_{T}$ is investigated since springs 1 and 2 are fully axially extended, and extension of spring 3 is required to increase the total potential energy right before snap through. The applied moment profile of $T_{5}$ is determined by the free lengths of springs and the split angle $α$ (Fig. 2A); $α$ determines the characteristic of $\bar{G I}$ and $L$ scales facet lengths, including $L_{4}$ :

T_{5} = F_{Spring 3} \cdot \frac{L_{4} L_{5}}{\bar{G I}} \cos α = E A L_{4} \cdot (\frac{\bar{G I}}{y_{0}} - 1) \cdot \frac{L_{5}}{\bar{G I}} \cos α .

(21)

Equation (21) shows that $T_{5}$ varies complicatedly with respect to $ρ_{5}$ ; $\bar{G I}$ is described by trigonometric functions of $ρ_{5}$ . Because of the complexity, we describe the output performances in terms of geometric information of the moment splines (the maximum value and the argument) rather than actual design parameters. The shape and scale of $T_{5}$ are determined by $\frac{\bar{G I}}{y_{0}}$ and $α$ , and $E A L_{4}$ , respectively; $y_{0} / L_{3}$ and $α$ were adopted for $λ_{2}$ and $λ_{3}$ , respectively (Fig. 2D).

Figure 2C–E summarizes the relationship between design parameters and output performances. Figure 2C shows the applied moment splines, $T_{3} (ρ_{3})$ and $T_{5} (ρ_{5})$ , on $ρ_{3}$ and $ρ_{5}$ axes, at the given design parameters, respectively. We define $h_{3}$ and $φ_{3}$ as the maximum and the argument of $T_{3} (ρ_{3})$ , and $h_{5}$ and $φ_{5}$ as the sets of $T_{5} (ρ_{5})$ , respectively. We investigated the effect of design parameters $λ_{1}$ on [ $h_{3}$ , $φ_{3}$ ], and $λ_{2}$ and $λ_{3}$ on [ $h_{5}$ , $φ_{5}$ ] to represent equivalent torque splines (Fig. 2D). Afterward, the grasping moment $T_{G}$ and trigger moment $T_{T}$ are plotted with respect to [ $h_{3}$ , $φ_{3}$ ] and [ $h_{5}$ , $φ_{5}$ ], respectively (Fig. 2E). Grasping moment $T_{G}$ increases as $h_{3}$ increases and $φ_{3}$ approaches 90º. The system needs a lower $h_{5}$ and a greater $φ_{5}$ for a lower trigger moment. The scale normalization factors for trigger ( $T_{5}$ , $h_{5}$ , and $T_{T}$ ) and grasping ( $T_{3}$ , $h_{3}$ , and $T_{G}$ ) performances are $E A L_{3}$ and $EAd$ , respectively (equations (20) and (21)). In summary, a specific set of design parameters can be chosen to achieve the target output performance.

In addition to the shown output performances, the activation energy for snap-through can be obtained. Activation energy, $Δ V_{Activation}$ , is defined as the difference in energy level between the first stable (before trigger) $V_{Stable 1} = V (ρ_{1}^{Stable 1}, ρ_{2}^{Stable 1})$ and flat-unfolded state $V_{Flat - unfolded} = V (0 °, 0 °)$ :

Δ V_{Activation} = V_{Flat - unfolded} - V_{Stable 1} .

(22)

Two performance parameters are considered: contact-point gap $b (ρ_{3}; s)$ and normal grasping force ( $F_{G}$ ) for grasping characterizations. The contact-point gap is the distance between two points $s = η L_{2}$ ( $0 < η < 1$ ) from the $ρ_{3}$ axis (Fig. 2F). The lower and upper bounds for $ρ_{3}$ are derived: $ρ_{3} = 90 °$ and $ρ_{3} = ρ_{3, col}$ (Fig. 2G). At the lower bound, the grasping force starts to produce a {-k} directional force. The upper bound $ρ_{3, col}$ is defined as the angle that causes the collision-fold of symmetric facets:

L_{2} \sin (π - ρ_{3, col}) = L_{1} .

(23)

Geometric capacity can be estimated with folding configurations and contact-point distance.

Grasping force ( $F_{G}$ ) is defined as the grasping moment ( $T_{G}$ ) divided by contact-point distance ( $s$ ) (Fig. 2H):

F_{G} = T_{G} / s .

(24)

The normal grasping force is investigated to generalize the grasping performance of the origami. The normal force $F_{G}$ is projected onto horizontal and vertical axes, ^{G}x and ^{G}z: ${(F_{G})}_{x}$ and ${(F_{G})}_{z}$ (Fig. 2H, I):

s \cdot {(F_{G})}_{x} = T_{G} \cos ({\frac{π}{2} + ρ}_{3})

(25)

s \cdot {(F_{G})}_{z} = T_{G} \sin ({\frac{π}{2} + ρ}_{3}) .

(26)

The equations allow for the estimation of the mechanical capacity for grasping. By referring to geometric and mechanical behavior, the folding pattern and equivalent moment profiles can be derived.

Materials and Methods

Design of folding patterns for the gripper

To extend the half patterns (Fig. 1C, D) into gripper patterns, the half-patterns of TPA and MPA origami are symmetrically mirrored, and the dotted lines mean fold lines or creases (Fig. 3A, B). Facets on the mirror plane are united into a single facet; no fold line is in the mirror plane (Fig. 3C, D). In this work, the arrangement and properties of springs are also symmetrically mirrored. The appearances of gripper prototypes of both TPA and MPA origami are presented in Figure 3I–M. The gripper is assumed to be passively actuated by $ρ_{1}$ rotation induced by contact of a target object, and the grasping motion is achieved by $ρ_{2}$ and $ρ_{3}$ rotations of both mirrored facets (Fig. 2A).

FIG. 3.

Fabrication of origami prototypes and transition test of grippers from TPA and MPA origami. (A) Flat crease pattern of the TPA origami gripper with the arrangement of springs. The TPA crease pattern with spring characteristics is symmetrically mirrored, and the facets and mirrored facets on the mirroring plane are united. Therefore, it contains four springs, “R,” and two springs, “G,” whose spring constants and free lengths are $k_{R}$ and $l_{0, R}$ , and $k_{G}$ and $l_{0, G}$ . (B) Flat crease pattern of MPA origami gripper with an arrangement of springs. Likewise, the MPA crease pattern with springs is symmetrically mirrored and united and contains two “spring 1”s, two “spring 2”s, and two “spring 3”s. (C) Drawing of MPA origami for fabrication, which includes separated facets, a flexible membrane for folding joints, and springs. (D) Drawing of MPA origami gripper for fabrication. (E) A schematic figure of a cross-sectional view of prototypes for experiments. Facets made of PETG are attached to both sides of a layer of flexible fabric by adhesive tape. The gap between facets is assigned for folding joints. A spring or elastomer band is anchored at the penetrated hole and fixed by adhesive and staples. (F) Facets with adhesive tape cut by a carbon dioxide laser and printed fabric. (G) A closed view of the spring-anchored point. (H) Fabricated prototypes of half and gripper patterns of MPA origami. (I) Scenes of the TPA origami gripper’s initial deployment succeeding in stopover (DP#1 in Fig. 1G). (J) Scenes of the TPA origami gripper’s passive capture after the stopover. (K) Scenes of the TPA origami gripper’s initial deployment failing in stopover, which loses the function of passive capture (DP#2 in Fig. 1G). (L) Scenes of the MPA origami gripper’s initial deployment. (M) Scenes of the MPA origami gripper’s passive capture after a stopover. The modified path consists of a serial process of success of stopover (i→ii) and capture mission (ii→iii→iv). PETG, polyethylene terephthalate glycol.

Fabrication of origami prototypes

For origami prototypes and specimens (Fig. 3H), multifacet patterning and multilayer lamination are adopted (Fig. 3C–F). Each facet is wholly separated and attached to both sides of a single flexible fabric layer, which plays the role of folding joints (Fig. 3E). This work adopts polyethylene terephthalate glycol (PETG) with a thickness of 0.5 mm for facets, which is quickly laser machinable (carbon dioxide laser; VLS6.75; Universal Laser). For folding joints, the inter-facet gaps are set to 1 mm, allowing a flat-folded configuration (2·0.5 mm). Drawings of origami patterns with separated facets and gaps are printed in one layer of flexible fabric with a thickness of 0.15 mm by a plotter (HP DesignJet T650).

An individual facet with an adhesive tape layer is attached to the top and bottom sides of the printed drawing (Fig. 3F), and the laminated structure’s flexural modulus is estimated to be 0.545 GPa and withstand repeated loads without significant degradation of the flexural modulus (Supplementary Data and Supplementary Fig. S2). Both ends of springs to be marked for free lengths are joined through the laminated structures and fixed by adhesive (LOCTITE® 401; Henkel) and staples (Fig. 3E–G). Springs with a thickness of 1.5 mm and a cross-sectional area of 4.5 mm² are used for experiments, and springs with a thickness of 2 mm and a cross-sectional area of 30 mm² are used to demonstrate large-scale prototypes. Young’s moduli of the elastomer bands are estimated to be 0.70 MPa and 0.86 MPa, respectively. The estimation of spring stiffnesses is presented in Supplementary Data and Supplementary Figure S1.

Prototypes for the demonstrations are also fabricated using the same method as other materials for facets, as shown in Figure 3E. The gripper of the flat-foldable pop-up catcher is fabricated with double-sided PETG (0.5 mm), and the Sarrus origami-based manipulator (Fig. 6A, B) is fabricated with PETG (0.5 mm) and acrylic sheet (1 mm). The spacing lengths between facets are 1 mm. For the second demonstration, the large-scale origami prototype is fabricated with PETG (0.5 mm) and acrylic sheet (5 mm), and its inter-facet spacing is 10 mm.

Load experiment

To experiment and measure the grasping and trigger forces, five specimens per design case are fabricated and subject to compressive load (“half” prototype in Fig. 3H). A tensile compress test machine (MCT-2150W; A&D) is used to measure the load (Figs. 3C and 4). All fixtures are made of aluminum (AL6061) and machined. The diameter of the loading nose for grasping load is 4 mm, and the facet OADC (Fig. 2A) is mechanically fixed onto the fixture (Fig. 4A, B). The radius and length of the loading nose for trigger load are 5 mm and 70 mm, respectively. The specimen’s $ρ_{1}$ axis is initially vertically aligned with the loading nose, and horizontal motion allows for passive deployment (Figs. 2A and 4C, D).

FIG. 4.

(A) Schematic of measurement of grasping force, $F_{G}$ at $ρ_{3}$ = 90º. The prototypes are fixed by bolting into an aluminum fixture through the anchoring holes. (B) Experimental setup for measuring grasping force. (C) Schematic of the measurement of trigger force, $F_{T}$ . The aluminum loading nose (radius = 5 mm) applies a vertical force downward onto the $ρ_{1}$ axis (edge). The specimens are fabricated by multimaterial layers presented in Figure 2. (D) Experimental setup for measuring trigger load. (E) Predicted values and experimental results of the grasping moment with regard to λ₁. (F) Predicted values and experimental results of the trigger moment with regard to λ₂. (G) Predicted values and experimental results of the trigger moment with regard to λ₃. Each test was repeated five times, and box plots were used to express the distribution; whiskers were set to 1.50.

Drop test

To test the trigger performances of the gripper, the standard tennis ball with a mass and diameter of 56 g and 67 mm is used as a target object (Fig. 5). The object is subject to being dropped five times per drop condition. The incidence angles are adjusted by slope structures made of acrylic sheet with a thickness of 10 mm. The origami specimens are fabricated according to Figure 3D, E. The drop height is measured from the bottom of the object to the $ρ_{1}$ axis of the gripper.

FIG. 5.

(A) A schematic figure of a target object’s collision and escaping during the passive actuation from the first stable state to the flat-unfolded state. The velocity of the object is assumed to be constant. The figure describes the geometric variables in the plane perpendicular to the $ρ_{1}$ axis (^{G}yz plane). (B) The velocity field and the estimated capture-capable regime. The minimum speed limit ( ${(v_{obj})}_{limit}$ ) and incidence angle limit ( ${(θ_{obj})}_{limit}$ ) are presented in red and blue dashed lines, respectively. Marker “○” is used for cases with success counts of 3 or more. The marker “×” is used for the other cases. (C) A schematic figure to describe the drop test. The initially stationary standard tennis ball, whose mass ( $m_{obj}$ ) and diameter are 56 g and 67 mm, respectively, falls at the drop height presented as H, which is the distance between the gripper’s $ρ_{1}$ axis and the bottom of the ball. An incidence angle is assigned as the inclination angle ( $90 ° - θ_{obj}$ ). (D) Categorized possible results. Only the case of serial success of trigger and capture is evaluated as the whole success. There are three sorts of failure modes: “triggered but bouncing out”, “1-side triggered”, and “none triggered”.

Design of a sarrus origami-based pop-up manipulator

Previous studies have explored compact manipulators inspired by Sarrus origami.^12,22 A flat-foldable pop-up manipulator is designed by exploiting the simplicity and flat foldability of the Sarrus folding mechanism (Fig. 6 and Supplementary Fig. S3). The deployed reach can be derived from the kinematic equations:

2 p \sin ψ = h

(27)

and

\tan θ \cos ψ = \tan \frac{γ}{2}

(28)

where p and θ represent the in-plane length and angle of the isosceles trapezoids (side facets), h the deployed height of the unit Sarrus origami,

ψ

the angle between two isosceles trapezoids at locking, and γ the in-plane angle of the base plate of a regular polygon (Supplementary Fig. S3). However, this origami still risks singularity when flat folded (ν = 0 deg); therefore, a planar stopper between two Sarrus modules is inserted (Fig. 6 and Supplementary Fig. S3). The design parameters for this work are (p, θ, h, β) = (101.7 mm, 72 deg, 200 mm, 79.2 deg). The manipulator prototype (Supplementary Fig. S3F) is fabricated using the same structure presented in Figure 3.

FIG. 6.

Design and prototype of the flat-foldable pop-up catcher consisting of an MPA origami gripper-based end-effector and Sarrus origami-based manipulator. (A) Flat-folded configuration of the assembled origami, featuring sub-manipulating facets and stoppers. (B) Deployed configuration of the assembled origami (before snap through). A stopper between two Sarrus origami modules can prevent kinematic singular behavior. The end-effector is attached to the end of the manipulator. (C) Prototype of the pop-up catcher, with a stowed thickness of approximately 1 inch in the flat-folded state. (D) The deployed prototype, which is the passive gripper, reaches 522 mm from the bottom of the manipulator to the top of the end-effector.

Design of an end effector of a passive gripper with MPA origami

An end-effector of the passive origami gripper from MPA origami is constructed to be attached to the manipulator’s end. The stand-alone gripper (Fig. 3L, M) is integrated with submanipulating facets to stow the gripper facets in the flat-folded state (Supplementary Fig. S3 and Fig. 6A, B). For the gripper part, the middle facets (oo’c’dc) of the gripper are altered into a pentagon (Ω < 180 deg) to avoid the kinematic singularity of the sub-manipulator of the end effector (Supplementary Fig. S3). However, the pentagon-facet design can cause incompatibility between the flat-folded and deployed states; the line oao’ cannot become the straight line at the deployed state under the assumption of rigid facets. In this work, Ω is chosen to be 140º, and the estimated incompatibility of length $\bar{o a o^{'}}$ , ${(\bar{o a o^{'}})}_{Deployed} - {(\bar{o a o^{'}})}_{Flat}$ , is 1.8 mm (Supplementary Fig. S3). The slight compliance of facets is assigned to make the configuration practically compatible. Two springs are installed at the sub-manipulator to enable passive pop-up deployment into the longitudinal direction (Supplementary Fig. S3). The end-effector prototype is fabricated using the same method and materials presented in Figure 3.

Results

Experiment for trigger and grasping loads

The experiments were conducted to verify our modeling with three design parameters: $λ_{1}$ , $λ_{2}$ , and $λ_{3}$ (Fig. 2A). Trigger force $F_{T}$ and grasping force $F_{G}$ are measured with variation of $λ_{1}$ and $λ_{2}$ , and $λ_{3}$ , respectively. A grasping experiment was conducted to investigate performance with respect to $λ_{1}$ . The grasping moment $T_{G}$ was measured using static load:

T_{G} = F_{G} s

(29)

where

F_{G}

is the static axial load measured at

ρ_{3}

= 90°, and

s

is the distance from the

ρ_{3}

axis to the loading point (30 mm in this case) (Fig. 4A, B). Next, the trigger force on the

ρ_{1}

axis by applying a compressive load was measured. The trigger moment

T_{T}

is obtained using the principle of virtual work:

T_{T} d ρ_{1} = F_{T} d (L_{3} \sin \frac{ρ_{1}}{2})

(30)

where

F_{T}

is the axial load orthogonal to both

ρ_{1}

axis and ground (Fig. 4C, D). The prototypes were subjected to a compressive force at 50 mm/min to maintain a quasi-static condition, and the maximum moment value was evaluated just before the snap transition.

The effects of three values for each parameter were investigated: {0.8, 1.0, 1.2} for $λ_{1}$ , {0.4, 0.5, 0.6} for $λ_{2}$ , and {10 deg, 20 deg, 30 deg} for $λ_{3}$ . The configuration was set to $L_{1} = L_{2} = L_{3} = 50$ mm, $d = 25$ mm, $E = 0.702 MPa$ (Supplementary Fig. S1), $A = (3) (1.5) {mm}^{2}$ , and $t = 1.5$ mm (Fig. 2A, B). The expected, normalized loads are presented in Figure 2E: $T_{G}$ = [1.09, 0.586, 0.252] for $λ_{1}$ = [0.8, 1.0, 1.2]; $T_{T}$ = [1.12, 0.672, 0.392] for $λ_{2}$ = [0.4, 0.5, 0.6]; $T_{T}$ = [0.749, 0.672, 0.563] for $λ_{3}$ = [10 deg, 20 deg, 30 deg]. $T_{G}$ experiment varied by $λ_{1}$ were conducted with [ $λ_{2}, λ_{3}$ ] = [0.5, 20 deg], $T_{T}$ experiment varied by $λ_{2}$ were conducted with [ $λ_{1}, λ_{3}$ ] = [1.0, 20 deg], and $T_{G}$ experiment varied by $λ_{3}$ were conducted with [ $λ_{1}, λ_{2}$ ] = [1.0, 0.5].

Figure 4E–G compares the predicted values presented in Figure 2E and the experimental results. According to Figure 4E, the grasping moment tends to increase as the free lengths of “spring 1” and “spring 2” ( $λ_{1} d$ ) decrease for both prediction and experiments. However, disparity might be caused by the actual nonlinearity of springs and the effect of practical facet thickness. For the trigger moment, the experimental results tend to closely follow the tendency of the predicted model. The experimental results also show that decreasing the free length of “spring 3” ( $λ_{2} L_{3}$ ) increased the trigger moment rather than increasing the split angle under the same conditions of other parameters (Fig. 4F, G). Overall, the experimental results verify the proposed model in predicting mechanical performances.

Capture characterization

The other factor is estimating the trigger-capable range of the target object’s motion. The acceptable momentum is estimated from the object velocity and folded configuration at the first stable state (“ii” in Fig. 1H). Assuming the object contacts the gripper’s $ρ_{1}$ axis with constant velocity, the object pushes the $ρ_{1}$ axis by $L_{3} \cos \frac{{(ρ_{1})}_{Stable 1}}{2}$ (where ${(ρ_{1})}_{Stable 1}$ is $ρ_{1}$ at the first stable state) makes the origami gripper flat-unfolded and get snap through (Fig. 5A). During the duration, the target object would be displaced horizontally ( $β L_{3}$ in Figure 5A, 0 < $β$ < 1) with the horizontal speed of $v_{obj} \cos θ_{obj}$ (in {-j} direction Considering the velocity-related information, the condition under which the gripper is triggered before the object escapes can be estimated:

\frac{\cos \frac{{(ρ_{1})}_{Stable 1}}{2}}{\sin θ_{obj}} < \frac{β}{\cos θ_{obj}}

(31)

where the left-side term is the lead time to trigger passively, and the right-side term is the lead time that the target object escapes during the same duration. Considering the inequality, the limit of the relative incidence angle

{(θ_{obj})}_{limit}

can be estimated (

β = 0.5

for this study; blue dashed line in Fig. 5B).

In addition, the minimum speed limit ${(v_{obj})}_{limit}$ can be derived by considering activation energy (equation (22)) and the object’s kinematic energy (red dashed line in Fig. 5B):

2 Δ V_{Activation} = \frac{1}{2} m_{obj} {(v_{obj})}_{limit}^{2}

(32)

where

Δ V_{Activation}

denotes the activation energy needed for the snap through of one finger of the gripper (Fig. 3B). These equations allow for estimating the “capture-capable regime” in the velocity field (Fig. 5B).

Trigger experiment on incidence velocity

Drop tests with a standard tennis ball (mass and diameter of 56 g and 67 mm) were conducted. The gripper configuration was set to $L_{1} = L_{3} = 35$ mm, $L_{2} = 70$ mm, $d = 30$ mm, $λ_{1} = 5 / 6$ , $λ_{2} = 1 / 2$ , $λ_{3} = α = 10$ °, $E = 0.702 MPa$ , $A = (3) (1.5) {mm}^{2}$ , and $t = 1.5$ mm (Fig. 2). Thirteen cases were investigated, varying drop height H and incidence angle $θ_{obj}$ (Fig. 5C and Table 2). The estimated minimum speed and angle limits are 1.3 m/s and 60°, respectively (equations (31) and (32)).

Table 2.

Results of the Drop Tests with Respect to Drop Height and Incidence Angle

Case	Drop height (mm), H	Incidence angle, $θ_{obj}$	Counts (five trials per case)
			Success (○)	Failure (×)
				Triggered but bouncing out	One-side triggered	None triggered
C1	50	90°		—	—	—	5
C2	75		—	—	—	5
C3	100		2	—	2	1
C4	125		5	—	—	—
C5	150		4	1	—	—
C6	200		5	—	—	—
C7	250		3	2	—	—
C8	200	60°	3	2	—	—
C9	250	60°	5	—	—	—
C10	200	45°	1	—	1	3
C11	250	45°	1	—	4	—
C12	200	30°	—	—	1	4
C13	250	30°	—	—	—	5

To investigate the speed and incidence angle of the target object, drop height and angle of inclination are assigned, respectively. A drop height, H, produces the object’s speed ( $v_{obj}$ ) caused by gravity, assuming the mechanical energy conservation, and the inclination angle of $90 ° - θ_{obj}$ is assigned to the gripper. The gripper is fabricated in the manner of Figure 3D, E. Five trials were conducted per case.

Figure 5B,D and Table 2 show the results of the drop tests. Success was counted when both trigger and capture occurred (○), while failures included cases where the object bounced out or was only partially triggered (×) (Supplementary Movie S7). For $θ_{obj} = 90 °$ , higher drop heights tended to increase success rates. Lower speeds led to “1-side triggered” or “no trigger” results, whereas excessive speed caused bouncing out because of high energy.

In the other cases of $θ_{obj}$ , two drop heights were investigated (C8-C13). Overall, the success rate tends to decrease as $θ_{obj}$ decreases. At $θ_{obj} = 60 °$ , the probability of success increases as H increases. At $θ_{obj} = 45 °$ , only one success was counted at each height. At $θ_{obj} = 30 °$ , no success is counted in both heights, and almost all trials fail to trigger. In detail, the consequences of “1-side triggered” and “none triggered” can happen with a speed less than ${(v_{obj})}_{limit}$ , whereas the maximum speed limit also might exist because of the excessive energy, which makes the target object bounce out (Fig. 5D). In addition, a low incidence angle can make them slide and escape, which is affected by material characteristics.

Demonstration

Demonstration of a flat-foldable pop-up catcher

The integrated catcher system includes an end-effector from the MPA origami gripper and the Sarrus origami-based manipulator, and its stowed thickness is 1 inch (≈ 25 mm) (Fig. 6). The origami gripper shown in Figure 3L, M is extended into an end effector (Fig. 6A, B and Supplementary Fig. S3). For the flat-foldable manipulator origami, Sarrus origami, which has been adopted for simple deployment and actuation in various works, is adopted,^44–47 and a three-leg parallel Sarrus mechanism, with three revolute joints in each leg, is used in this demonstration.⁴⁷ The pop-up deployment is achieved through pre-strains of springs installed on both the end effector and manipulator (Supplementary Fig. S3).

The integrated origami system serves two functions: triggering and retrieving the manipulator. The trigger maintains the flat-folded state with compression, whereas the motor rewinds the wire to retract the manipulator. The pop-up system can be stowed into a 1-inch thickness at the flat folded state and reach 522 mm at the deployed state, yielding a compression-to-extension ratio of 21 (Fig. 6). This pop-up origami catcher can catch floating objects with tunable bistability. The system successfully catches a 33.6 g falling ball and retracts by folding the manipulator flat (Fig. 7). Besides upward capture, the system can also catch objects in downward and sideward directions, allowing flexibility in various scenarios (Supplementary Fig. S4 and Movie S8).

FIG. 7.

Demonstrations of capture tests of a robotic application (Fig. 6) and a large-scale origami. (A) Rapid deployment of the integrated origami from flat-folded to deployed state towards a moving object (33 g). The deployed origami system can catch the object upward. The snapshots are presented over the passage of time (t₁). (B) Passive grasping and retrieval of the grasped object using a motor to wind a wire. (C) Comparison of a target object, a flat-folded large-scale origami (stowed into 570 mm × 570 mm × 7 mm), and a standard tennis ball with a diameter of 67 mm. The target object is an empty plastic box with a size of 647 mm × 437 mm × 326 mm (for an inner volume of 60 L) and a mass of 3 kg. (D) The capture-ready (first stable) stable with a span of 1.6 m. The area scale of the facet is 66 times larger than the centimeter-level prototype (Fig. 5). (E) Passive actuation (trigger) of the large-scale origami. (F) Capturing the target object by the triggered snap through. The snapshots are presented over the passage of time (t₂).

Demonstration of a large-scale origami

Exploiting the nature of rigid origami, our flat-foldable origami gripper can be scaled up geometrically (Fig. 7C–F). The large-scale origami is fabricated using the method in Figure 3D, with a flexible fabric layer (0.15 mm thickness) and 10 mm inter-facet gap for near-zero-stiffness fold lines; acrylic facets for up sides, and PETG facets for bottom sides. The design configuration [ $L_{1}$ , $L_{2}$ , $L_{3}$ , $λ_{1}$ , $λ_{2}$ , $λ_{3}$ ] is set to [285 mm, 570 mm, 285 mm, 0.54, 0.46, 10°] for symmetry.

In this demonstration, a plastic box (an inner volume of 60 L; an outer size of 647 mm × 437 mm × 326 mm) is captured by the large-scale gripper (Fig. 7C). The span length reaches 1.6 m in the capture-ready (stable 1) state (Fig. 7D). When the box was dropped onto the gripper, it induced the gripper’s $ρ_{1}$ axis to rotate passively and get a snap through (Fig. 7E, F), successfully capturing the target object despite its larger weight (Supplementary Movie S9).

According to equation (32) and given properties, the activation energy for snap through can be derived under the rigid origami assumption. The activation energy is estimated at 4.8 J, and the estimated gravitational energy of the target box is approximately (3.0 kg) (9.81 m/s²) (0.62 m) = 17.6 J. The amount of energy applied is enough to deploy the larger gripper into a flat unfolded state and induce the snap through. However, grasping behavior could become slower or suspended because of the larger inertia from increased geometric dimensions; the degradation can be compensated by the gravity-compensation systems or environments and stiff yet lightweight facets.

Discussion

In this study, we introduced a pop-up catcher capable of transitioning through multiple states: flat folded, capture ready, and captured. By designing the potential energy landscape and using transition path planning, we successfully addressed the multipath issue: a compatibility issue between flat foldability and bistability. Our demonstration showcased the capabilities of the flat-foldable pop-up catcher, achieving a 21:1 extension-to-compression ratio and capturing objects within 0.8 s. Furthermore, scalability was validated through a meter-scale gripper capable of capturing a 60 L box object.

Another aspect worth highlighting is the design philosophy aligned with embodied intelligence. Passive actuation can be viewed as a form of structural programming in which the mechanical structure itself encodes specific responses to external stimuli. This approach enables the distribution of control responsibilities between the mechanical body and electronic systems, potentially leading to more efficient overall system architectures. Such embodied design principles could offer substantial advantages, especially in scenarios where form factor, weight, or the feasibility of electromagnetic actuation is severely constrained. Although not fully explored in this work, the proposed energy landscape-based design methodology for passive actuation holds promise for integration with structural damping, nonlinear elasticity, and dynamic response characteristics. These integrations could extend the structural programming capacity and enable the design of more sophisticated passive systems that respond in diverse ways to external interactions.

However, several points of discussion about improving our gripper for future study exist. The first is the recovery mechanism from the capture-finished to the capture-ready state. The gripper can be passively actuated once unless forced recovery is needed. Adding an active actuator, such as a cable-driven system, could handle failure risks, such as activation energy scarcity or grasping failure, but it increases the system’s weight.

The second topic is expandability with manipulators. In this study, our pop-up catcher system adopted the Sarrus origami, prioritizing the extremely thin thickness. Although the manipulator origami deploys in one direction, it can be used in mobile systems, such as aerial vehicles, which can adjust position and attitude. Using versatile manipulators with high DoF and extra actuators could expand the mission workspace and diversity.

The other topic is about the optimization of grasping performance. General performances, especially equivalent moment and normal force, were investigated because rigid-origami analysis provides the estimation of the common structural characteristics. However, grasping capability can be additionally strengthened by practical considerations: inertia, elasticity, texture of facets, spring model, and contact dynamics, which could affect the potential energy of origami.

We believe that transition path planning is beneficial for designing passively actuated origami-inspired mechanics and robots to mitigate the risk of deployment or actuation failures. This planning process can also alleviate system burdens and complexity in various passive origami systems requiring multistep passive actuations in the fields of transformable or deployable robotic applications.

Data Availability Statement

All data needed to evaluate the conclusions are included in the main text or Supplementary Data.

Authors’ Contributions

S.J.L. contributed to conceptualization, analysis, fabrication, experiments, and manuscript writing and revision. J.H.J. contributed to conceptualization, experiments, and manuscript writing and revision. J.H.R. contributed to manuscript writing and revision. D.Y.L. contributed to conceptualization, analysis, and manuscript writing and revision.

Footnotes

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2025-02213804); the NRF grant funded by the Korea government (MSIT) (No. 2022R1C1C1003718); the InnoCORE program of the Ministry of Science and ICT(No. N10250155); the BK21 FOUR Program of NRF grant funded by the Ministry of Education.

Supplemental Material

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