1 Introduction

As an ancient art of paper folding, origami has recently attracted the interest of researchers in various fields. Nowadays, origami is no longer limited to folding “paper”, but refers to all practices that transform two-dimensional sheets with certain crease patterns into three-dimensional structures through folding. Folding not only brings about significant shape and size changes, but also induces unique or extraordinary mechanical properties, which have attracted extensive research interest, and properties that have been uncovered including bi-stability and multi-stability [1, 2], negative Poisson's ratio [3, 4], negative and quasi-zero stiffness [5], programmable stiffness [6], programmable self-locking [7], recoverable collapse [8], etc. As a consequence, origami-based and origami-inspired structures and metamaterials have been applied in many fields, such as deployable space structures [9, 10], self-deployable origami stent graft [11], origami robots [12, 13], etc.

Among various origami structures and origami metamaterials, the multi-stability property has received wide attention. Multi-stable structures and metamaterials usually have multiple stable configurations, which correspond to different mechanical properties and can be characterized by the coexisting potential energy wells within the deformation range. Generally, multi-stable structures can remain in different configurations without external forces and exhibit snap-through behaviors when switching configurations. Compared with conventional multi-stable structures such as pre-buckled beams [14], multi-stable origami structures often have truly three-dimensional shapes, and the origin of multi-stability is unique. Specifically, multi-stability can originate from folding geometry nonlinearity. For example, by assigning different values of torsional stiffness at the creases, a stacked Miura-origami (SMO) cell can show bi-stability because of the non-unique geometric correspondence [15]; alternatively, multi-stability can be achieved in non-rigid-foldable origami structures, such as the Kresling-origami, which obtain different stable configurations by allowing deformations of the facets [1]. Therefore, multi-stable origami structures have been applied in many fields, including energy harvesting [16], wave propagation [17], earthworm-like robots [18], robotic arms [12], etc.

For multi-stable origami metamaterials, it is important to predict the configuration transitions under external loading. For a bi-stable origami cell, the prediction can be achieved by considering whether the energy barrier is reached and passed, for instance, the configuration transitions of the bi-stable Kresling-origami cell [19] and the bi-stable SMO cell [20]. For metamaterials composed of multiple bi-stable origami cells, predicting the transition behavior becomes much more complicated. For example, Fang et al. studied the asymmetric transition behavior of dual-cell SMO structures with different inter-cell connections [21]. Optimization-based simulations and experiments showed that switching from one stable state to another in extension is much more difficult than the opposite [21]. Note that determining the switching behavior by looking for the local minimum in a high-dimensional potential energy profile, although effective, is computationally intensive. We then aim to determine the transition sequence by evaluating the constitutive force–displacement curves of the component origami cells. This constitutes the first research goal of this paper.

On the other hand, having clarified the transition sequence mechanism, how to regulate the multi-stability profile and the transition sequence becomes particularly important. Usually, there are two methods to tune the transition behavior of multi-stable origami structures, namely the dynamic approach and the quasi-static approach. In detail, the dynamic approach refers to the implementation of different transition paths by external excitations of different amplitudes or frequencies [22]. Note that multi-stability is a strong global nonlinearity, which brings multi-solution and extreme sensitivity to the initial conditions [23]. Therefore, a comprehensive understanding of the response characteristics and the basin of attraction of the multi-stable origami structure is required to tune the transition path, which is cumbersome and inaccurate. The quasi-static approach involves changing the inherent potential energy landscape of the origami structure by means of the magnetic field or air pressure, etc., so as to alter the transition sequence. For example, by integrating magnets into the SMO and the Kresling-origami structures, not only the constitutive force–displacement relationships can be quantitatively modified, but also the stability profiles can be qualitatively changed [1]; by adjusting the internal air pressure, the SMO structure can also be switched between monostable and bi-stable [24]. Note that the above regulation methods are mainly oriented toward single-cell bi-stable origami structures, and there is a lack of systematic studies on regulating the stability profile and transition behavior for multi-stable origami metamaterials composed of multiple bi-stable cells. This is the second research goal of the present paper.

To advance the state of the art of multi-stable origami mechanics, this study aims to gain an in-depth understanding of the mechanism of configuration transition and to propose effective methods for tuning the transition sequence of multi-stable origami metamaterials. To this end, the SMO structure is employed as a platform. A single SMO cell is composed of two geometrically compatible Miura-origami sheets; it can exhibit monostable or bi-stable profiles by adjusting the crease-stiffness ratio. Connecting multiple SMO cells in series along the height direction would yield multi-stable origami metamaterials. In this research, to avoid overcomplicating the switching order, a dual-cell SMO chain is examined, whose quasi-static transition sequence is scrutinized in terms of the fundamental mechanism and the transition rule. Rather than relying on numerical optimization, the transition sequence can be accurately predicted by evaluating the constitutive force–displacement profiles of the component cells. Specifically, the critical forces required by the component cells to switch configurations govern the overall transition order. Following this, two approaches for customizing the quasi-static transition sequence of the multi-stable origami structure are proposed by tuning the potential energy profiles of constitutive cells via the abovementioned methods: a design approach by changing the crease stiffness [15] and a pneumatic scheme by tuning the internal pressure [5]. The effectiveness of the two methods is verified through simulations and/or experiments.

The rest of this paper is organized as follows. Section 2 introduces the multi-stability of the un-pressurized dual-cell SMO structure, with a focus on the underlying mechanism that determines the transition sequence. Subsequently, Sect. 3 introduces a design method by changing the crease stiffness, and Sect. 4 introduces the pneumatic origami structure and presents a pressure-based scheme without changing the design. The effectiveness of this novel scheme is experimentally demonstrated in Sect. 5. Section 6 summarizes and concludes this paper.

2 Multi-Stability and Transition Behavior of Dual-Cell Miura-Ori Structures

In this section, the rigid-folding kinematics and the bi-stable profile of the SMO cell are first briefly revisited [15]. Then, the potential energy profile and the transition behavior of a multi-stable dual-cell SMO structure are examined, and the underlying mechanism that determines the transition sequence is uncovered.

2.1 Rigid-Folding Kinematics and Bi-stability of an SMO Cell

Miura-origami is a special four-vertex origami with collinear creases and reflectional symmetry. The geometry of a Miura-ori unit can be characterized by three parameters: the crease lengths \(a_{k}\) and \(b\), and the sector angle \(\gamma_{k}\), as shown in Fig. 1a. The subscript ‘k’ denotes the bottom unit (\(k = {\rm I}\)) and the top unit (\(k = {\rm I}{\rm I}\)), respectively. An SMO cell is constructed by stacking two Miura-ori units together, providing that the kinematic compatibility conditions are satisfied such that the two units keep connected during the whole folding process,

$$ b_{{\text{I}}} = b_{{{\rm I}{\rm I}}} = b,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a_{{\text{I}}} \cos \gamma_{{\text{I}}} = a_{{{\rm I}{\rm I}}} \cos \gamma_{{{\rm I}{\rm I}}} $$
(1)
Fig. 1
figure 1

Geometry and bi-stability of the SMO cell: a crease patterns of two Miura-ori units; b the SMO cell and its geometry, where the folding angles, dihedral angles, and external dimensions are indicated; c the potential energy of an SMO cell with \(\mu = 1\), \(\mu = 25\), and \(\mu = 50\); and d the corresponding force–displacement curves of the SMO cell. Note that in displacement control, negative stiffness will occur; while in force control, snap-through jumps will happen. The stable and unstable configurations are denoted by solid and empty circles, respectively. For reference, the configuration with \({\kern 1pt} \theta_{{\text{I}}} = 0\) is also denoted

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Without loss of generality, the bottom Miura-ori unit is assumed to be smaller than the top unit, i.e., \({\kern 1pt} a_{{\text{I}}} < a_{{{\rm I}{\rm I}}}\).

Under the rigid-folding assumption, folding of the SMO cell remains a single degree-of-freedom mechanism. To describe the folded configuration of the SMO cell, the folding angle of the bottom unit (\({\kern 1pt} \theta_{{\text{I}}}\)), defined as the dihedral angle between the facet of the bottom unit and the horizontal plane, is selected as the variable. Folding of the top and bottom units is not independent, rather, \({\kern 1pt} \theta_{{{\text{II}}}}\) is constrained by

$$ \theta_{{{\rm I}{\rm I}}} = \arccos \left( {\cos \theta_{\rm I} \frac{{\tan \gamma_{\rm I} }}{{\tan \gamma_{{{\rm I}{\rm I}}} }}} \right) $$
(2)

Note that \({\kern 1pt} \theta_{{\text{I}}}\) ranges between \(- \pi /2\) and \(\pi /2\) while \(\theta_{{{\text{II}}}}\) stays positive. Hence, the bottom cell bulges out of the larger top cell when \({\kern 1pt} \theta_{{\text{I}}} < 0\) and nests into the larger top cell otherwise. In what follows, these two qualitatively different configurations are referred to as the “bulged-out” and the “nested-in”, respectively, or “out” and “in”, respectively, for simplicity.

The folded configuration of the SMO cell can also be described by the dihedral angles between adjacent facets (\(\varphi_{1}\) to \(\varphi_{5}\), Fig. 1b), which can be derived via spherical trigonometry:

$$ \begin{aligned} & \varphi_{1} = \pi - 2\theta_{\rm I}\\ & \!\sin \frac{{\varphi_{2} }}{2} = \frac{{\cos \theta_{\rm I} }}{{\sqrt {1 - \sin^{2} \theta_{\rm I} \sin^{2} \gamma_{\rm I} } }} \\ &\varphi_{3} = \pi - 2\theta_{{{\rm I}{\rm I}}}\\ &\!\sin \frac{{\varphi_{4} }}{2} = \frac{{\cos \theta_{{{\rm I}{\rm I}}} }}{{\sqrt {1 - \sin^{2} \theta_{{{\rm I}{\rm I}}} \sin^{2} \gamma_{{{\rm I}{\rm I}}} } }} \\ &\varphi_{5} = \theta_{{{\rm I}{\rm I}}} - \theta_{\rm I} \end{aligned} $$
(3)

The external dimensions (\(L\), \(W\), \(S\), and \(H\)) of the SMO cell can be derived via solid geometry:

$$ \begin{gathered} L = \frac{{b\sin \gamma_{\rm I} \cos \theta_{\rm I} }}{{\sqrt {1 - \sin^{2} \theta_{\rm I} \sin^{2} \gamma_{\rm I} } }} \hfill \\ S = \frac{{b\cos \gamma_{\rm I} }}{{\sqrt {1 - \sin^{2} \theta_{\rm I} \sin^{2} \gamma_{\rm I} } }} \hfill \\ W = a_{\rm I} \sqrt {1 - \sin^{2} \theta_{\rm I} \sin^{2} \gamma_{\rm I} } \hfill \\ H = a_{{{\rm I}{\rm I}}} \sin \theta_{{{\rm I}{\rm I}}} \sin \gamma_{{{\rm I}{\rm I}}} - a_{\rm I} \sin \theta_{\rm I} \sin \gamma_{\rm I} \hfill \\ \end{gathered} $$
(4)

Rigid folding of the SMO cell can be considered as the rotations of rigid facets relative to hinge-like creases. Denoting \(K_{i}\) as the torsional stiffness of the crease corresponding to the dihedral angle \(\varphi_{i}\), the total potential energy of the SMO cell can be calculated as

$$ V = \frac{1}{2}\sum\limits_{i = 1}^{5} {K_{i} \left( {\varphi_{i} - \varphi_{i}^{0} } \right)^{2} } $$
(5)

where the superscript ‘0’ denotes the dihedral angle corresponding to the stress-free state (\(\theta^{0}\)), at which all creases are undeformed. The torsional stiffness per unit length of different creases can be designed to be different. Here, we assume that the creases corresponding to the dihedral angle \(\varphi_{4}\) have a larger torsional stiffness per unit length (denoted by \(\mu k\)), while other creases have equal torsional stiffness per unit length (denoted by \(k\)), i.e.,

$$ K_{1} = K_{3} = 2kb, \, K_{2} = 2ka_{\rm I} , \, K_{4} = 2\mu ka_{{{\rm I}{\rm I}}} , \, K_{5} = 4kb $$
(6)

To demonstrate the effects of the torsional stiffness ratio \(\mu\), the geometric parameters listed in Table 1 are used. Figure 1c displays evolution of the potential energy landscape with respect to \(\mu\). When \(\mu = 1\), the potential energy curve has only one minimum within the deformation range, located at \(\theta^{0} = - 60^\circ\), indicating a monostable profile of the SMO cell. Increasing the stiffness ratio \(\mu\) to 25, two wells appear on the potential energy curve, indicating the presence of bi-stability, and the two wells correspond to a stable nested-in configuration and a stable bulged-out configuration. When the stiffness ratio is further increased to \(\mu = 50\), the bi-stability turns stronger, manifesting as the higher energy barrier between the two potential wells.

Table 1 Geometric parameters of the SMO cell
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The corresponding reaction force–displacement curves of the SMO cell can be calculated as the variation of the potential energy \(V\) with respect to the changes in cell height \(H\) in displacement control, i.e.,

$$ F = \frac{{{\text{d}}V}}{{{\text{d}}H}} = \frac{{{\text{d}}V}}{{{\text{d}}\theta }}\left( {\frac{{{\text{d}}H}}{{{\text{d}}\theta }}} \right)^{ - 1} $$
(7)

With different values of \(\mu\), the obtained curves are shown in Fig. 1d. The intersections of the curve with the horizontal axis correspond to equilibrium points, and those with positive crossing slopes are stable equilibria, while those with negative crossing slopes are unstable equilibria. Accordingly, when \(\mu = 1\), there is only one stable equilibrium, corresponding to the unique stable configuration of the SMO cell. When \(\mu\) rises to 25, two stable equilibria are observed, between which there exists a segment with negative stiffness and an unstable equilibrium. For the force–displacement profile of the SMO cell, two critical forces are important: the critical extension force \(F^{{\text{E}}}\) and the critical compression force \(F^{{\text{C}}}\) at the critical configurations where negative stiffness occurs, where the superscripts ‘E’ and ‘C’ denote the extension and compression processes, respectively. When the force–displacement curve does not have a negative stiffness segment (i.e., monotonic), the configuration with \(\theta_{{\text{I}}} = 0\) is considered as the critical configuration, at which the two critical forces coincide, i.e., \(F^{{\text{E}}} = F^{{\text{C}}}\). In a force control, if the SMO cell is subjected to an external force higher than these critical forces, a snap-through transition between the two stable configurations will occur. When \(\mu = 50\), the critical forces to overcome the energy barrier are significantly increased.

2.2 Transition Behavior of a Dual-Cell SMO Chain

Connecting two SMO cells (cell A and cell B) in series along the height direction via a rigid rod, a dual-cell SMO chain can be constructed (Fig. 2a). Note that the rigid rods are not in direct contact with the crease, which will not induce force concentration on the top or bottom creases. With the rod connection, the rigid folding of the two cells is independent of each other, and the reaction forces of the two cells keep equal when extending or compressing the dual-cell SMO chain. The geometric parameters of the two cells are identical and consistent with Table 1. In addition, the two cells share the same value of \(k\) (i.e., \(k_{{\text{A}}} = k_{{\text{B}}} = k\)); without loss of generality, the stiffness ratio of cell A is assumed to be higher than cell B (specifically, \(\mu_{{\text{A}}} = 50\) and \(\mu_{{\text{B}}} = 25\) such that they are both bi-stable). Figure 2b and c displays the potential energy curves and force–displacement curves of the two cells. With \(\mu_{{\text{A}}} > \mu_{{\text{B}}}\), the energy barrier and the critical forces for snap-through transitions of cell A are larger than those of cell B.

Fig. 2
figure 2

Transition behavior of the dual-cell SMO chain: a the dual-cell SMO chain is fabricated by connecting two SMO cells in the height direction with a rod connection; b and c show the potential energy curves and the force–displacement curves of the two component SMO cells; d energy evolution paths of the dual-cell SMO chain obtained via an optimization-based method, where the potential energy contour and the stable configuration \(M_{i} \, (i = 1,2,3,4)\) are demonstrated; e and f show the potential energy and the reaction force of the chain associated with the identified transition path in (d)

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The total potential energy of the dual-cell SMO chain can be calculated as

$$ V_{{{\text{chain}}}} = V_{{\text{A}}} + V_{{\text{B}}} $$
(8)

where \(V_{{\text{A}}}\) and \(V_{{\text{B}}}\) denote the elastic potential energy of cell A and cell B, respectively, which can be obtained via Eq. (5). Figure 2d plots the potential energy contour of the dual-cell SMO chain, which contains four potential minima (denoted by red dots) that correspond to the four stable configurations of the chain. Specifically, the potential minimum \(M_{1}\) corresponds to the “in-in” configuration; the minimum \(M_{4}\) corresponds to the “out-out” configuration; the minima \(M_{2}\) and \(M_{3}\) correspond to the “in-out” and “out-in” configuration, respectively. Here, the first “in/out” indicates the configuration of the top cell A, and the second indicates the configuration of the bottom cell B.

To understand the folding behavior of the dual-cell SMO chain under extension and compression, the evolution path on the potential energy contour needs to be identified, which has been achieved by an optimization-based approach. In detail, the evolution path is determined by identifying the energy minima corresponding to the total height of the chain (i.e., \(H_{{\text{A}}} + H_{{\text{B}}} = {\text{constant}}\)). Note that the situation where two energy minima correspond to a given total height is possible. Therefore, a small step size is needed to avoid jumping between two paths when finding the minima.

As an example, Fig. 2d displays the identified energy evolution paths during displacement-controlled extension and compression, and they are separated from each other, indicating that the dual-cell SMO chain exhibits different configuration transition behaviors when being stretched and compressed. Specifically, during the extension process (red curves), the chain starts from the “in-in” configuration and passes through the stable configurations \(M_{1}\) and \(M_{2}\) in turn (i.e., the bottom cell B will be first stretched to the bulged-out configuration), and subsequently reaches the other potential energy path via a jump and then passes through \(M_{4}\). During the compression process (blue curves), the chain starts from the “out-out” configuration and passes through the nested-in configuration, and subsequently jumps back to the other potential energy path and then passes through \(M_{1}\). Note that during these jumps (indicated by dashed vertical arrows), although the total chain length changes little, the internal configuration of the two component cells changes substantially. It is worth emphasizing that such jumps (or, internal reconfiguration) are achieved with displacement control rather than force control, so they are fundamentally different from the previously discussed ‘‘snap-through’’ jumps in force control, as shown in Fig. 1d.

Correspondingly, Fig. 2e and f, respectively, display the total potential energy and reaction force associated with the identified energy paths, from which the separation of the energy evolution paths and the reconfiguration behaviors can also be observed.

2.3 Transition Sequences and Fundamental Mechanisms

The above energy evolution paths are derived via optimization. Here, we interpret the transition behavior from another perspective. Figure 3 shows the force–displacement curves of the two component cells and the reaction force curves associated with the energy paths. Note that the critical forces of cell B are smaller than cell A, i.e., \(\left| {F_{{\text{B}}}^{{\text{E}}} } \right| < \left| {F_{{\text{A}}}^{{\text{E}}} } \right|\) and \(\left| {F_{{\text{B}}}^{{\text{C}}} } \right| < \left| {F_{{\text{A}}}^{{\text{C}}} } \right|\) (Fig. 3a), and recall that with rod connection, the reaction forces of the two cells remain equal during extension or compression (Fig. 3b).

Fig. 3
figure 3

Mechanisms that govern the transition sequence of the dual-cell SMO chain: a the constitutive force–displacement curves of the two component cells, where the critical extension/compression forces are denoted; b and c show the reaction force curves in displacement control and force control, respectively. They are associated with the identified energy evolution paths in Fig. 2d

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During the extension process, the chain is in the “in-in” configuration at the initial stage, and the reaction force rises gradually. After reaching the stable configuration \(M_{1}\), the reaction force of cell B reaches the critical value \(F_{{\text{B}}}^{{\text{E}}}\) first (point \(P_{1}\)), which makes it switch to the bulged-out configuration; while the critical extension force of cell A has not been reached, and cell A remains at the nested-in configuration. Keeping stretching the chain, the chain will pass the stable configuration \(M_{2}\), and the reaction force will be further increased until the critical extension force of cell A, \(F_{{\text{A}}}^{{\text{E}}}\), is reached (point \(P_{2}\)), making it switch to the bulged-out configuration. Finally, after experiencing a jump (internal reconfiguration), the stable configuration \(M_{4}\) is reached, and the chain arrives at the “out-out” configuration. Note that in the displacement control, the above configuration transitions are manifested as the segments with negative stiffness (Fig. 3b); and if with a force control, two snap-through jumps are expected to happen (Fig. 3c).

During the compression process, the transitions do not follow the previous sequence because the critical compression force of cell B, i.e., \(F_{{\text{B}}}^{{\text{C}}}\), is first reached (point \(P_{3}\)), making cell B first switch to the nested-in configuration, and the chain arrives at the stable configuration \(M_{3}\). Another configuration transition happens when the critical compression force of cell A, \(F_{{\text{A}}}^{{\text{C}}}\), is reached at point \(P_{4}\). Finally, through a jump (internal reconfiguration), the chain returns to the “in-in” configuration and reaches the stable configuration \(M_{1}\). Similarly, snap-through jumps are expected if with a force control (Fig. 3c).

The above description is based on the interrelationship among the force–displacement profiles of the component cells, the critical forces, and the reaction forces associated with the energy evolution paths. We find that the order of configuration transitions during extension and compression is related to the values of critical force, i.e., the cell with a smaller critical force will switch first. This is because \(F_{{\text{B}}}^{{\text{E}}} < F_{{\text{A}}}^{{\text{E}}}\) makes cell B first switch its configuration to the bulged-out during extension, and \(F_{{\text{A}}}^{{\text{C}}} < F_{{\text{B}}}^{{\text{C}}}\) makes cell B first switch its configuration to the nested-in during compression. For simple representation, such transition behavior is recorded as ‘Type BA-BA’’, where the first ‘BA’ represents that cell B undergoes configuration transition before cell A during extension, and the second ‘BA’ represents that cell B switches before cell A during compression.

From the above findings and by considering the possible monostable scenario, four possible configuration transition sequences can be expected by evaluating the critical forces of the component cells. The summarized rules and underlying mechanisms are detailed below and are demonstrated in Table 2 and Fig. 4. Please note that the signs of the critical forces need to be considered in Table 2 and Fig. 4.

  1. (1)

    Types AB-BA: \(F_{\text{A}}^{{\text{E}}} < F_{\text{B}}^{{\text{E}}}\) in extension, while \(F_{\text{A}}^{{\text{C}}} < F_{\text{B}}^{{\text{C}}}\) in compression, so that cell A switches first in extension and cell B switches first in compression. The “out-in” configuration will be passed through in both extension and compression, while the “in-out” configuration will not be experienced, as shown in Fig. 4a.

  2. (2)

    Type AB-AB: \(F_{\text{A}}^{{\text{E}}} < F_{\text{B}}^{{\text{E}}}\) in extension and \(F_{{\text{B}}}^{{\text{C}}} < F_{{\text{A}}}^{{\text{C}}}\) in compression, so that cell A always switches first in both extension and compression. All four configurations are experienced during the whole process, as shown in Fig. 4b.

  3. (3)

    Type BA-BA: \(F_{{\text{B}}}^{{\text{E}}} < F_{{\text{A}}}^{{\text{E}}}\) in extension and \(F_{\text{A}}^{{\text{C}}} < F_{\text{B}}^{{\text{C}}}\) in compression, so that cell B always switches first in both extension and compression. All four configurations are experienced during the whole process, as shown in Fig. 4c.

  4. (4)

    Type BA-AB: \(F_{{\text{B}}}^{{\text{E}}} < F_{{\text{A}}}^{{\text{E}}}\) in extension and \(F_{{\text{B}}}^{{\text{C}}} < F_{{\text{A}}}^{{\text{C}}}\) in compression, so that cell B switches first in extension and cell A switches first in compression. The “in-out” configuration will be passed through in both extension and compression, while the “out-in” configuration will not be experienced, as shown in Fig. 4d.

Table 2 Conditions for the four transition sequences of the multi-stable dual-cell SMO chain
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Fig. 4
figure 4

Schematic illustrations of the four types of transition sequence and the constitutive force–displacement profiles of the two component cells: a type AB-BA; b type AB-AB; c type BA-BA; and d type BA-AB

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The rules summarized above, although not rigorously reasoned or proven, are correct and effective in predicting the transition sequence, which will be subsequently demonstrated with simulations and experiments. Based on this, two approaches to tuning the transition sequence will be proposed and exemplified subsequently.

3 Tuning of Transition Sequence by Stiffness Design

Adjusting the crease stiffness or stiffness ratio can effectively change the constitutive profile of the component cell. As a result, through careful stiffness design, the transition sequence of the dual-cell SMO chain can also be effectively tuned. Here, we comprehensively examine how the crease stiffness difference between the two component cells affects the transition sequence. To this end, the crease torsional stiffness and stiffness ratio of cell A are fixed (\(k_{{\text{A}}} = k\), \(\mu_{{\text{A}}} = 50\)), and the crease stiffness of cell B is prescribed as follows: \(k_{{\text{B}}} /k_{{\text{A}}}\) ranges from 0 to \(2.5\), and \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }}\) ranges from 0 to 2.5. Note that with a relatively small stiffness ratio (\(\mu_{{\text{B}}} < 15.7\)), cell B would become monostable. For a given pair of parameters \(k_{{\text{B}}} /k_{{\text{A}}}\) and \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }}\), the transition sequence type can be predicted based on the summarized rules. Figure 5 shows the distribution of different transition sequence types on the \((\mu_{{\text{B}}} /\mu_{{\text{A}}} ) - (k_{{\text{B}}} /k_{{\text{A}}} )\) parameter plane.

Fig. 5
figure 5

Distribution of transition sequence types on the \((\mu_{{\text{B}}} /\mu_{{\text{A}}} ) - (k_{{\text{B}}} /k_{{\text{A}}} )\) plane

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Figure 5 reveals that within the range of parameters \(k_{{\text{B}}} /k_{{\text{A}}}\) and \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }}\), the four transition sequence types are all achievable. This suggests that prescribing the crease stiffness a priori is an effective method to design the transition sequence, although the regions for types AB-BA and BA-AB are much smaller than the other two types. To verify the correctness of the summarized rules and the effectiveness of the stiffness design method, we select one point from each region and derive the corresponding transition behavior by the optimization-based method (Fig. 6).

Fig. 6
figure 6

Four cases of transition sequence: a At point Q1 (\({{k_{{\text{B}}} } \mathord{\left/ {\vphantom {{k_{{\text{B}}} } {k_{{\text{A}}} }}} \right. \kern-0pt} {k_{{\text{A}}} }} = 0.8\), \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }} = 1.25\)), type AB-BA is achieved; b at point Q2 (\({{k_{{\text{B}}} } \mathord{\left/ {\vphantom {{k_{{\text{B}}} } {k_{{\text{A}}} }}} \right. \kern-0pt} {k_{{\text{A}}} }} = 1\), \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }} = 2\)), type AB-AB is achieved; c at point Q3 (\({{k_{{\text{B}}} } \mathord{\left/ {\vphantom {{k_{{\text{B}}} } {k_{{\text{A}}} }}} \right. \kern-0pt} {k_{{\text{A}}} }} = 1\), \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }} = 0.5\)), type BA-BA is achieved; and d at point Q4 (\({{k_{{\text{B}}} } \mathord{\left/ {\vphantom {{k_{{\text{B}}} } {k_{{\text{A}}} }}} \right. \kern-0pt} {k_{{\text{A}}} }} = 1\), \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }} = 0.5\)), type BA-AB is achieved. In each subfigure, from top to bottom, the identified energy evolution path, the energy curve associated with the energy path, and the force–displacement curves (in both displacement control and force control) are demonstrated

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Figure 6, from top to bottom, shows the identified energy evolution path, the energy curve associated with the energy path, and the force–displacement curves (in both displacement control and force control) corresponding to the four points. Figure 6a reveals that at point \(Q_{1}\) (\({{k_{{\text{B}}} } \mathord{\left/ {\vphantom {{k_{{\text{B}}} } {k_{{\text{A}}} }}} \right. \kern-0pt} {k_{{\text{A}}} }} = 0.8\), \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }} = 1.25\)), the dual-cell chain exhibits an ‘AB-BA’ transition sequence, which agrees with the prediction. At point \(Q_{4}\) (\({{k_{{\text{B}}} } \mathord{\left/ {\vphantom {{k_{{\text{B}}} } {k_{{\text{A}}} }}} \right. \kern-0pt} {k_{{\text{A}}} }} = 1.25\), \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }} = 0.8\)), the transition sequence derived via an optimization-based method is ‘BA-AB’, which also agrees with the prediction. At point \(Q_{2}\) (\({{k_{{\text{B}}} } \mathord{\left/ {\vphantom {{k_{{\text{B}}} } {k_{{\text{A}}} }}} \right. \kern-0pt} {k_{{\text{A}}} }} = 1\), \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }} = 2\)) and point \(Q_{3}\) (\({{k_{{\text{B}}} } \mathord{\left/ {\vphantom {{k_{{\text{B}}} } {k_{{\text{A}}} }}} \right. \kern-0pt} {k_{{\text{A}}} }} = 1\), \({{\mu_{{\text{B}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{B}}} } {\mu_{{\text{A}}} }}} \right. \kern-0pt} {\mu_{{\text{A}}} }} = 0.5\)), the optimization results indicate that the transition sequences are ‘AB-AB’ and ‘BA-BA’, respectively, which are the same as the prediction. The above simulations, therefore, fully demonstrate the correctness of the rules and the effectiveness of the proposed stiffness design method.

4 Tuning of Multi-stability Profile and Transition Sequence via Pressure

Generally, it is difficult to precisely adjust the crease torsional stiffness of origami structures, which makes it challenging to tune the transition sequence via crease-stiffness design. In addition, the available region for transition sequence types AB-BA and BA-AB is relatively small, which further increases the difficulty of tuning. In addition, as a design method, online tuning of the multi-stability profile and the transition sequence is not possible when the prototype is fabricated. As a result, this section explores an online pressure-based scheme for tuning the multi-stability profile and the transition sequence of the dual-cell SMO chain.

4.1 Mechanical Modeling

The SMO cell contains an internal cavity that can be used to inflate. Previous research has indicated that internal pressurization can effectively change the potential energy profile and the constitutive force–displacement relation [24]. By assuming that the mechanical work done by pressure is conservative, the potential energy generated by the internal as

$$ V_{{\text{P}}} = - P\left( {{\text{Vol}} - {\text{Vol}}_{\max } } \right) $$
(9)

where \(P\) denotes the difference between the internal fluid pressure and the external atmospheric pressure, and \({\text{Vol}}\) represents the closed volume inside the SMO cell, which is determined by geometric design and the folded configuration:

$$ {\text{Vol}} = 2a_{\rm I}^{2} b\sin^{2} \gamma_{\rm I} \cos \theta_{\rm I} \left( {\sqrt {\frac{{\tan^{2} \gamma_{{{\rm I}{\rm I}}} }}{{\tan^{2} \gamma_{\rm I} }} - \cos^{2} \theta_{\rm I} } - \sin \theta_{\rm I} } \right) $$
(10)

\({\text{Vol}}_{\max }\) is the maximum closed volume, given by

$$ \begin{gathered} {\text{Vol}}_{\max } = 2a_{\rm I}^{2} b\sin^{2} \gamma_{\rm I} \frac{{\tan \gamma_{{{\rm I}{\rm I}}} }}{{\tan \gamma_{\rm I} }} \hfill \\ \left. {\theta_{\rm I} } \right|_{{{\text{Vol}}_{\max } }} = - \arcsin \left( {\frac{{\tan \gamma_{\rm I} }}{{\sqrt {\tan^{2} \gamma_{\rm I} + \tan^{2} \gamma_{{{\rm I}{\rm I}}} } }}} \right) \hfill \\ \end{gathered} $$
(11)

If the internal pressure remains constant throughout the folding process, the reaction force in the height direction originating from the internal fluid can be expressed as

$$ \begin{gathered} {\text{Vol}} = 2{\text{LWH}} \hfill \\ F_{{\text{P}}} = \frac{{\partial V_{{\text{P}}} }}{{\partial \theta_{\rm I} }} = - 2P\left( {{{LW}}\frac{\partial H}{{\partial \theta_{\rm I} }} + {{LH}}\frac{\partial W}{{\partial \theta_{\rm I} }} + {{WH}}\frac{\partial L}{{\partial \theta_{\rm I} }}} \right) \hfill \\ \end{gathered} $$
(12)

Hence, the total potential energy \(V_{{{\text{total}}}}\) and the reaction force \(F_{{{\text{total}}}}\) of a pressurized SMO cell can be derived by

$$ \begin{gathered} V_{{{\text{total}}}} = V_{{{\text{SMO}}}} + V_{{\text{P}}} \hfill \\ F_{{{\text{total}}}} = F_{{{\text{SMO}}}} + F_{{\text{P}}} \hfill \\ \end{gathered} $$
(13)

where \(V_{{{\text{SMO}}}}\) and \(F_{{{\text{SMO}}}}\) are the potential energy and reaction force of the SMO cell without internal pressure, which have been derived in Eqs. (5) and (7).

4.2 Multi-stability and Transition Sequence Tuning

As an example, by inflating pressure (\(P/k = 10\)) into an SMO cell (the geometric parameters are listed in Table 1; \(k_{{\text{A}}} = k\), \(\mu_{{\text{A}}} = 50\)), Fig. 7a displays the total potential energy of the pressurized SMO cell and its composition. When the internal volume reaches the maximum, the potential energy of the internal pressure \(V_{{\text{P}}}\) reaches its minimum 0; in addition, \(V_{{\text{P}}}\) increases with the increase or decrease in the SMO height. Due to the contribution of \(V_{{\text{P}}}\), the two potential wells deviate in both position and depth, although the pressurized SMO cell remains bi-stable. Accordingly, the force–displacement curve also changes (Fig. 7b). As the potential well becomes shallower, i.e., the energy barrier to overcome becomes lower, the critical extension force and critical compression force decrease.

Fig. 7
figure 7

Bi-stability tuning via pressure: a the potential energy of a pneumatic SMO cell and its composition; b the reaction force of a pneumatic SMO cell and its composition; c effect of internal pressure on the potential energy profile; and d effect of internal pressure on the force–displacement relations

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We further examine the effect of the internal pressure on the multi-stability profile. By inflating different pressures to an SMO cell (the geometric parameters are listed in Table 1; \(k_{{\text{A}}} = k\), \(\mu_{{\text{A}}} = 50\)), the curves of the total potential energy and the reaction forces are displayed in Fig. 7c. The internal pressure is associated with the torsional stiffness and varies from \(P = 0\) to \({P \mathord{\left/ {\vphantom {P k}} \right. \kern-0pt} k} = 30\) (the unit of \(P\) is \({\text{kPa}}\), and the unit of \(k\) is \({\text{N} \mathord{\left/ {\vphantom {\text{N {rad}}}} \right. \kern-0pt} \text{rad}}\)). When the internal pressure is relatively small (e.g., \({P \mathord{\left/ {\vphantom {P k}} \right. \kern-0pt} k} = 10\)), although the SMO cell retains its bi-stability, the potential wells become shallower and the critical extension/compression forces become smaller, indicating an easier configuration transition. With the increase in internal pressure, the potential well corresponding to the nested-in configuration will be further weakened until the SMO cells become monostable (\({P \mathord{\left/ {\vphantom {P k}} \right. \kern-0pt} k} = 20,{P \mathord{\left/ {\vphantom {P k}} \right. \kern-0pt} k} = 30\)). Correspondingly, the critical extension force decreases further and becomes negative as the SMO cell becomes monostable, indicating that no additional external force is required in extension to switch the SMO cell from the nested-in configuration to the bulged-out configuration (Fig. 7d).

The above example suggests that pressure can be another tool to adjust the stability profile and hence, to tune the transition sequence. Compared with the crease-stiffness design approach, the regulation scheme through internal pressure is more convenient and can be employed online in case the prototype has been fabricated.

By varying \(P_{{\text{A}}} /k\) and \(P_{{\text{B}}} /k\) from 0 to 40, we then comprehensively examine how pressure affects the transition sequences of the dual-cell SMO chain. Specifically, two identical SMO cells (the geometric parameters are listed in Table 1; \(k_{{\text{A}}} = k_{{\text{B}}} = k\), \(\mu_{{\text{A}}} = \mu_{{\text{B}}} = 50\)) are used to construct the chain. The critical pressure between the bi-stable and monostable profiles of the component cell is approximate \({P \mathord{\left/ {\vphantom {P k}} \right. \kern-0pt} k} = 19.3\), i.e., the SMO cell would become monostable when \({P \mathord{\left/ {\vphantom {P k}} \right. \kern-0pt} k} \ge 19.3\). For given pressures, the transition sequence can be predicted based on the rules summarized in Sect. 2.3. Figure 8 shows the distribution of the transition sequence types on the \(P_{{\text{A}}} /k - P_{{\text{B}}} /k\) plane. It reveals that the regions for the four types of transition sequence are clearly divided by two lines. When \(P_{{\text{A}}} > P_{{\text{B}}}\), transition sequence types AB-BA and AB-AB are achieved, while the other two types appear when \(P_{{\text{A}}} < P_{{\text{B}}}\). To validate the prediction, we select one point from each region and derive the corresponding transition behavior using the optimization-based method, and the results are demonstrated in Fig. 9.

Fig. 8
figure 8

Distribution of transition sequence types on the \(P_{{\text{A}}} /k - P_{{\text{B}}} /k\) plane

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Fig. 9
figure 9

Four cases of transition sequence: a at point R1 (\({{P_{{\text{A}}} } \mathord{\left/ {\vphantom {{P_{{\text{A}}} } k}} \right. \kern-0pt} k} = 25\), \({{P_{{\text{B}}} } \mathord{\left/ {\vphantom {{P_{{\text{B}}} } k}} \right. \kern-0pt} k} = 15\)), type AB-BA is achieved; b at point R2 (\({{P_{{\text{A}}} } \mathord{\left/ {\vphantom {{P_{{\text{A}}} } k}} \right. \kern-0pt} k} = 15\), \({{P_{{\text{B}}} } \mathord{\left/ {\vphantom {{P_{{\text{B}}} } k}} \right. \kern-0pt} k} = 5\)), type AB-AB is achieved; c at point R3 (\({{P_{{\text{A}}} } \mathord{\left/ {\vphantom {{P_{{\text{A}}} } k}} \right. \kern-0pt} k} = 5\), \({{P_{{\text{B}}} } \mathord{\left/ {\vphantom {{P_{{\text{B}}} } k}} \right. \kern-0pt} k} = 15\)), type BA-BA is achieved; and d at point R4 (\({{P_{{\text{A}}} } \mathord{\left/ {\vphantom {{P_{{\text{A}}} } k}} \right. \kern-0pt} k} = 15\), \({{P_{{\text{B}}} } \mathord{\left/ {\vphantom {{P_{{\text{B}}} } k}} \right. \kern-0pt} k} = 25\)), type BA-AB is achieved. In each subfigure, from top to bottom, the identified energy evolution path, the energy curve associated with the energy path, and the force–displacement curves (in both displacement control and force control) are demonstrated

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Figure 9, from top to bottom, shows the identified energy evolution path, the energy curve associated with the energy path, and the force–displacement curves (in both displacement control and force control) corresponding to the four points. At point \(R_{1}\) (\({{P_{{\text{A}}} } \mathord{\left/ {\vphantom {{P_{{\text{A}}} } k}} \right. \kern-0pt} k} = 25\), \({{P_{{\text{B}}} } \mathord{\left/ {\vphantom {{P_{{\text{B}}} } k}} \right. \kern-0pt} k} = 15\)), cell A is monostable, while cell B is bi-stable. The optimization-based simulation indicates that cell A switches its configuration first in extension, while cell B switches its configuration first in compression, i.e., an AB-BA transition sequence is achieved (Fig. 9a), which agrees with the prediction. At points \(R_{2}\) (\({{P_{{\text{A}}} } \mathord{\left/ {\vphantom {{P_{{\text{A}}} } k}} \right. \kern-0pt} k} = 15\), \({{P_{{\text{B}}} } \mathord{\left/ {\vphantom {{P_{{\text{B}}} } k}} \right. \kern-0pt} k} = 5\)), \(R_{3}\) (\({{P_{{\text{A}}} } \mathord{\left/ {\vphantom {{P_{{\text{A}}} } k}} \right. \kern-0pt} k} = 5\), \({{P_{{\text{B}}} } \mathord{\left/ {\vphantom {{P_{{\text{B}}} } k}} \right. \kern-0pt} k} = 15\)), and \(R_{4}\) (\({{P_{{\text{A}}} } \mathord{\left/ {\vphantom {{P_{{\text{A}}} } k}} \right. \kern-0pt} k} = 15\), \({{P_{{\text{B}}} } \mathord{\left/ {\vphantom {{P_{{\text{B}}} } k}} \right. \kern-0pt} k} = 25\)), the predicted transition sequence types fully coincide with the optimization-based results (Fig. 9b–d). This, therefore, not only reaffirms the accuracy of the summarized rules based on critical forces, but also illustrates the effectiveness of the proposed pressure-based scheme in tuning the multi-stability profile and the transition sequence.

Moreover, Fig. 8 suggests that tuning the transition sequence can be achieved by adjusting the pressure of a single component cell. For example, for points \(R_{1}\) and \(R_{3}\), the internal pressure of cell B remains unchanged, and the transition sequence type can be changed between BA-BA and AB-BA by simply adjusting the internal pressure of cell A. Similarly, for points \(R_{2}\) and \(R_{4}\), the transition sequence type can be changed between AB-AB and BA-AB by simply adjusting the internal pressure of cell B, while cell A keeps constant pressure.

5 Experiments

The above-summarized rules and the pressure-based scheme for tuning the multi-stability profile and the transition sequence are experimentally verified in this section. To this end, a dual-cell pneumatic SMO chain is prototyped, and quasi-static tests are carried out.

5.1 Prototype and Experimental Setup

A pneumatic SMO cell is made up of an SMO cell and an internal plastic air pouch. The SMO prototype is fabricated with steel sheets and adhesive-back plastic films, as shown in Fig. 10a. Specifically, the origami facets are laser cut from 0.25-mm-thick stainless-steel sheets (the geometric parameters are listed in Table 1) and connected along the creases by 0.1-mm-thick adhesive-back plastic films (ultrahigh-molecular-weight (UHMW) polyethylene). To ensure strong bi-stability of the SMO cell, eight 0.1-mm-thick pre-bent spring steel stripes are pasted at prescribed creases, inducing higher torsional stiffness. Stacking the bottom and top Miura-ori sheets together by adhesive-back plastic films yields an SMO cell. A custom-made air pouch is inserted into the SMO cell to provide the required internal pressurization. The pouch is made of 0.1-mm-thick low-density polyethylene film (LDPE), which is flexible enough in bending to form good contact with the inner surface of the SMO cell. Figure 10b displays the photograph of a pneumatic SMO cell prototype.

Fig. 10
figure 10

Prototype and experimental setup: a method for fabricating an SMO cell prototype; b photo of a pneumatic SMO cell prototype; and c experimental setup

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Figure 10c shows the setup for adjusting and maintaining pressure. During testing, the internal pressure of the air pouch is maintained at a constant level by an electro-pneumatic regulator (SMC, ITV-0030). Displacement-controlled quasi-static tests of the prototype are performed on a universal testing machine (Instron 5965).

5.2 Tuning the Stability Profile of a Single SMO Cell Via Pressure

We first demonstrate how pressure can be used to change the stability profile of the SMO cell. To this end, quasi-static tests are carried out on two component SMO cells with different internal pressures. In each test, five different air pressures varying from \(P = 3{\text{ kPa}}\) to \(P = 7{\text{ kPa}}\) are applied; the case without internal air pressure is also tested. In each test, the prototype is compressed and stretched in sequence at a speed of 5 mm/min, and the curves in compression and extension are averaged. Figure 11 shows the measured force–displacement curves of the two SMO cells under different internal pressures.

Fig. 11
figure 11

Experimentally measured force–displacement curves with different internal pressures: a cell A; b cell B. The monostable and bi-stable curves are indicated

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Figure 11 reveals that when the pressure is relatively low (\(P = 3{\text{ kPa}}\)), the force–displacement curves are very close to the cases without pressure (P = 0), and the two cells remain strongly bi-stable.

As the internal pressure increases, the bi-stability gradually decreases, manifesting as the decrease in the critical extension force. When the pressure is over 5 kPa, both prototypes become monostable, with the critical extension force becoming negative. Note that with the same pressure, the two prototypes behave differently. This is because the two SMO cells and the air pouches are handmade, and production errors are inevitable. The constitutive force–displacement curves of the two pneumatic SMO cell prototypes illustrated in Fig. 11, especially their differences in critical forces, can be used for transition sequence tuning.

5.3 Tuning of Transition Sequence via Pressure

Based on the results shown in Fig. 11 and the rules summarized in Sect. 2.3, four sets of pressures are selected to achieve different transition sequence types. Specifically, set 1 (\(P_{{\text{A}}} = 7\,{\text{kPa, }}P_{{\text{B}}} = 0\)) is used to achieve transition sequence type AB-BA; set 2 (\(P_{{\text{A}}} = 4\,{\text{kPa, }}P_{{\text{B}}} = 0\)) is used to achieve type AB-AB; set 3 (\(P_{{\text{A}}} = 0{, }P_{{\text{B}}} = 4\,{\text{kPa}}\)) is used to achieve type BA-BA; and set 4 (\(P_{{\text{A}}} = 0{, }P_{{\text{B}}} = 7\,{\text{kPa}}\)) is used to achieve type BA-AB. Then quasi-static tests (displacement-controlled compression and extension in sequence, at a speed of 5 mm/min) are performed on the four cases to demonstrate the effectiveness of the pressure-based scheme and the correctness of the prediction.

Figure 12 shows the results of the quasi-static tests. For set 1 (\(P_{{\text{A}}} = 7\,{\text{k Pa, }}P_{{\text{B}}} = 0\)) (Fig. 12a), the compression process starts from the ‘out-out’ configuration of the dual-cell chain prototype; during compression, cell B first switches to the nested-in configuration, and then cell A gradually changes its configuration to the nested-in so that the prototype reaches the ‘in-in’ configuration. Then, the extension process starts from the ‘in-in’ configuration; during extension, cell A first switches to the bulged-out configuration, and then cell B switches to the bulged-out to restore the ‘out-out’ configuration of the prototype. Overall, the transition sequence type is AB-BA. If the internal pressure of cell A is relatively small (i.e., set 2 (\(P_{{\text{A}}} = 4\,{\text{kPa, }}P_{{\text{B}}} = 0\))), cell A always makes the transition first in both the compression and extension processes, making the transition sequence type be AB-AB. When the internal air pressure of cell A is 0, i.e., in set 3 (\(P_{{\text{A}}} = 0{, }P_{{\text{B}}} = 4\,{\text{kPa}}\)) and set 4 (\(P_{{\text{A}}} = 0{, }P_{{\text{B}}} = 7\,{\text{kPa}}\)), another two transition sequence types, i.e., BA-BA and BA-AB, are achieved, respectively.

Fig. 12
figure 12

Experiments of tuning the transition sequence via internal pressure: a type AB-BA; b type AB-AB; c type BA-BA; and d type BA-AB. Quasi-static compression and extension are carried out in turn. The insets show the configuration of the dual-cell SMO chain prototype

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Overall, Fig. 12 indicates that the predicted transition sequence types are all correct. This, therefore, confirms that the critical force-based rules we summarized are effective and accurate in predicting the transition sequence behavior. More importantly, the experimental results in Fig. 12 manifest that rather than changing the design or re-constructing the prototypes, we can effectively tune the stability characteristics and transition sequence of the dual-cell SMO chain by simply adjusting the internal pressure.

6 Conclusions

Multi-stable origami structures and metamaterials can exhibit a variety of stable 3D configurations, often with different mechanical properties, which has led to widespread interest and potential applications in many areas, such as energy harvesting, flexible robots, and wave propagation. Although many studies have focused on the design and application of multi-stable origami structures, few studies have been conducted on the transition behavior of multi-stable origami structures between stable states, especially on how to tune the transition sequence. To fill this gap, this paper takes the dual-cell SMO chain as a platform to investigate the underlying mechanism of configuration transition and to propose effective methods for regulating the transition sequence.

The transition behavior of the multi-stable dual-cell SMO chain is interpreted from two aspects: energy evolution and critical forces. Via an optimization-based method, the transition paths are determined by identifying the minimum energy corresponding to a given displacement. By establishing the relationship among energy evolution, transition paths, and the associated force–displacement profiles, the underlying mechanism that determines the transition sequence is uncovered. The critical extension/compression forces of the component SMO cells play a critical role in governing the transition sequence; the component cell that reaches the critical force first during quasi-static extension or compression will be the first to undergo a configuration switch. Based on these findings, four types of transition sequence are possible, and their occurrence can be predicted by the four summarized rules.

To achieve effective tuning, two approaches are proposed: a crease-stiffness design method and a pressure-based scheme. In the first approach, by tailoring the crease stiffness, the four types of transition sequence can be achieved. However, the practical implementation of this approach is challenging, given the difficulty of precisely adjusting the crease torsional stiffness of origami structures and the relatively small design regions for certain transition sequence types. In addition, such a design method does not allow online tuning of the multi-stability profile or the transition sequence once the prototype is fabricated. As a result, we further propose the concept of pneumatic origami and present a pressure-based scheme for tuning the multi-stability profile and the transition sequence type. Simulations and experiments demonstrate that adjusting the internal pressure could effectively modify the constitutive force–displacement profile of the SMO cells, including the switch between monostability and bi-stability, as well as the changes in the critical extension/compression forces. This, therefore, is exploited for transition sequence tuning. Through a proof-of-concept prototype and quasi-static tests, we find that the transition sequence types can be effectively tuned, thus verifying correctness of the critical force-based rules and effectiveness of the pressure-based scheme.

Overall, this research, through theoretical and experimental efforts, reveals the underlying mechanism that governs the transition sequence of multi-stable origami structures from the perspective of critical transition forces and summarizes useful rules to predict the transition behavior. On this basis, this study also proposes a design method based on crease-stiffness assignment and an online method based on pressure regulation to tune the multi-stability profile and the configuration transition sequence. The results presented in this paper lay a solid foundation for the online configuration tuning of multi-stable origami structures/metamaterials, thus advancing their engineering applications.