Dynamic folding of origami by exploiting asymmetric bi-stability

Abstract

In this study, we examine a rapid and reversible origami folding method by exploiting a combination of resonance excitation, asymmetric bi-stability, and active control. The underlying idea is that, by harmonically exciting a bi-stable origami at its resonance frequencies, one can induce rapid folding between its different stable equilibria with a much smaller actuation magnitude than static folding. To this end, we use a bi-stable water-bomb base as an archetypal example to uncover the underlying principles of dynamic folding based on numerical simulation and experimental testing. If the water-bomb initially settles at its “weak” stable state, one can use a base excitation to induce the intra-well resonance. As a result, the origami would fold and remain at the other “strong” stable state even if the excitation does not stop. The origami dynamics starting from the strong state, on the other hand, is more complicated. The water-bomb origami is prone to show inter-well oscillation rather than a uni-directional switch due to a nonlinear relationship between the dynamic folding behavior, asymmetric potential energy barrier, the difference in resonance frequencies, and excitation amplitude. Therefore, we develop an active feedback control strategy, which cuts off the base excitation input at the critical moment to achieve robust and uni-directional folding from the strong stable state to the weak one. The results of this study can apply to many different kinds of origami and create a new approach for rapid and reversible (self-)folding, thus advancing the application of origami in shape morphing systems, adaptive structures, and reconfigurable robotics.

Introduction

Origami—the ancient art of paper folding—has received a surge of interests over the past decade from many research communities, such as mathematicians, material scientists, biotics researchers, and engineers. A key driving factor underneath such interests is the seemingly infinite possibilities of developing three-dimensional shapes from folding a simple flat sheet. The kinematics (or shape transformation) of origami is rich and offers many desirable characteristics for constructing deployable aerospace structures [1], kinetic architectures [2], [3], self-folding robots [4], and compact surgery devices [5], [6]. The mechanics of origami offers a framework for architecting material systems [7] with unique properties [8], [9], like auxetics [10], tunable nonlinear stiffness [11], [12], and desirable dynamic responses [13], [14], [15], [16]. Moreover, the origami principle is geometric and scale-independent, so it applies to engineering systems of vastly different sizes, ranging from nanometer-scale graphene sheets [17] all the way to meter-scale civil infrastructures [18].

For most of these growing lists of origami applications, large amplitude and autonomous folding (or self-folding) are crucial for their functionality. However, achieving a (self-)folding efficiently and rapidly remains a significant challenge [19]. To this end, we have seen extensive studies of using responsive materials to achieve folding via different external stimuli, such as heat [20], magnetic field [21], ambient humidity change [22], and even light exposure [23], [24]. In a few of these studies, bi-stability was also introduced as a mechanism to facilitate folding and maintain the folded shape without requiring a continuous supply of stimulation [25], [26]. While promising, these folding or self-folding are achieved in a slow and quasi-static fashion, and some of them are non-reversible.

To achieve rapid, reversible, and efficient folding, we examine a dynamic approach by exploiting the combination of harmonic excitation and embedded asymmetric bi-stability. Bi-stable structures possess two distinct stable equilibria (or “stable states”), and this strong non-linearity can induce complex dynamic responses from external excitation, such as super-harmonics, intra/inter-well oscillations, and chaotic behaviors [27]. These nonlinear dynamics have found applications in wave propagation control [28], energy harvesting [29], sensing [30], and shape morphing [31], [32]. Here, shape morphing is particularly relevant to folding, so we used a proof-of-concept numerical simulation to demonstrate the feasibility of using harmonic excitation to induce folding in a bistable water-bomb base origami [33] (Fig. 1(a)). The bi-stability of the water-bomb base is asymmetric [25], [26], i.e. the two stable states of the structure are asymmetric with respect to the unstable state and the energy gaps of the two stable states are different, so the resonance frequencies of its two stable configurations differ significantly. Due to this asymmetry, it is possible that when the water-bomb origami is harmonically excited at the resonance frequency of its current stable state, it can rapidly fold to and remain at the other stable state. Moreover, the required excitation magnitude by this dynamic folding method is smaller than static folding.

Building upon this proof-of-concept study, this study aims to obtain a comprehensive understanding of the harmonically-excited rapid folding via a combination of dynamic modeling, experimental validation, and controller design. First, we formulate a new and nonlinear dynamic model of a generic water-bomb origami and conduct an in-depth examination into the relationships among the dynamic folding behaviors, potential energy landscape, resonance frequencies, and excitation amplitudes. It is worth noting that this model is a significant advancement to our previous study in that it releases unnecessary boundary conditions and includes the facet rigid body motion (both translational and rotational). Since the bistability of water-bomb origami is asymmetric, we can designate its two stable states as “strong” or “weak” based on the magnitude of potential energy barriers between them. Our simulation and experiment results show dynamic folding from the weak stable state to the strong one is relatively easy, but folding in the other direction is quite challenging to achieve. That is, starting from the strong stable state, the water-bomb origami tends to exhibit inter-well oscillations under most excitation conditions, which is undesirable for rapid folding purposes. This challenge is further complicated by the fact that the nonlinear dynamics of origami are highly sensitive to design variations, fabrication errors, and excessive damping. Therefore, we then devise and experimentally validate a control strategy that ensures the robustness of dynamic folding by cutting off the excitation input at a critical configuration. This control strategy is essential for practical implementations of this dynamic folding method in the future.

It is worth highlighting that although this study uses the water-bomb origami as an example, the insights into the harmonically excited folding and the control strategy can apply to many other origami designs that exhibit asymmetric multi-stability, such as stacked Miura-ori [34], Kresling [35], and leaf-out pattern [36]. Moreover, harmonic excitation at the resonance frequency has a high actuation authority, so it can be an efficient method compared to other dynamic inputs, such as impulse [37]. Therefore, the results of this study can create a new approach for rapid and reversible (self-)folding, thus advancing the application of origami in shape morphing systems, adaptive structures, and reconfigurable robotics.

In what follows, Section 2 of this paper details the dynamic modeling of the water-bomb base origami, Section 3 discusses its dynamic folding behavior under harmonic excitation, Section 4 explains the active control strategy, Section 5 provides some insights on scaling of the dynamic folding strategy and its potential applications, and Section 6 concludes this paper with a summary and conclusion.