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# Amplifying the response of soft actuators by harnessing snap-through instabilities

Edited by John W. Hutchinson, Harvard University, Cambridge, MA, and approved July 21, 2015 (received for review March 11, 2015)

## Significance

Although instabilities have traditionally been avoided as they often represent mechanical failure, here we embrace them to amplify the response of fluidic soft actuators. Besides presenting a robust strategy to trigger snap-through instabilities at constant volume in soft fluidic actuators, we also show that the energy released at the onset of the instabilities can be harnessed to trigger instantaneous and significant changes in internal pressure, extension, shape, and exerted force. Therefore, in stark contrast to previously studied soft fluidic actuators, we demonstrate that by harnessing snap-through instabilities it is possible to design and construct systems with highly controllable nonlinear behavior, in which small amounts of fluid suffice to generate large outputs.

## Abstract

Soft, inflatable segments are the active elements responsible for the actuation of soft machines and robots. Although current designs of fluidic actuators achieve motion with large amplitudes, they require large amounts of supplied volume, limiting their speed and compactness. To circumvent these limitations, here we embrace instabilities and show that they can be exploited to amplify the response of the system. By combining experimental and numerical tools we design and construct fluidic actuators in which snap-through instabilities are harnessed to generate large motion, high forces, and fast actuation at constant volume. Our study opens avenues for the design of the next generation of soft actuators and robots in which small amounts of volume are sufficient to achieve significant ranges of motion.

The ability of elastomeric materials to undergo large deformation has recently enabled the design of actuators that are inexpensive, easy to fabricate, and only require a single source of pressure for their actuation, and still achieve complex motion (1⇓⇓⇓–5). These unique characteristics have allowed for a variety of innovative applications in areas as diverse as medical devices (6, 7), search and rescue systems (8), and adaptive robots (9⇓–11). However, existing fluidic soft actuators typically show a continuous, quasi-monotonic relation between input and output, so they rely on large amounts of fluid to generate large deformations or exert high forces.

By contrast, it is well known that a variety of elastic instabilities can be triggered in elastomeric films, resulting in sudden and significant geometric changes (12, 13). Such instabilities have traditionally been avoided as they often represent mechanical failure. However, a new trend is emerging in which instabilities are harnessed to enable new functionalities. For example, it has been reported that buckling can be instrumental in the design of stretchable soft electronics (14, 15), and tunable metamaterials (16⇓–18). Moreover, snap-through transitions have been shown to result in instantaneous giant voltage-triggered deformation (19, 20).

Here, we introduce a class of soft actuators comprised of interconnected fluidic segments, and show that snap-through instabilities in these systems can be harnessed to instantaneously trigger large changes in internal pressure, extension, shape, and exerted force. By combining experiments and numerical tools, we developed an approach that enables the design of customizable fluidic actuators for which a small increment in supplied volume (input) is sufficient to trigger large deformations or high forces (output).

Our work is inspired by the well-known two-balloon experiment, in which two identical balloons, inflated to different diameters, are connected to freely exchange air. Instead of the balloons becoming equal in size, for most cases the smaller balloon becomes even smaller and the balloon with the larger diameter further increases in volume (Movie S1). This unexpected behavior originates from the balloons’ nonlinear relation between pressure and volume, characterized by a pronounced pressure peak (21, 22). Interestingly, for certain combinations of interconnected balloons, such nonlinear response can result in snap-through instabilities at constant volume, which lead to significant and sudden changes of the membranes’ diameters (Figs. S1 and S2). It is straightforward to show analytically that these instabilities can be triggered only if the pressure–volume relation of at least one of the membranes is characterized by (*i*) a pronounced initial peak in pressure, (*Analytical Exploration: Response of Interconnected Spherical Membranes Upon Inflation*).

## Highly Nonlinear Fluidic Segments

To experimentally realize inflatable segments characterized by such a nonlinear pressure–volume relation, we initially fabricated fluidic segments that consist of a soft latex tube of initial length

Next, to construct fluidic segments with a final steep increase in pressure and a response that can be easily tuned and controlled, we enclosed the latex tube by longer and stiffer braids of length *A*). It is important to note that the effect of the stiff braids is twofold. First, as *B*), and therefore apply an axial force, *F*, to the membrane. Second, at a certain point during inflation when the membrane and the braids come into contact, the overall response of the segments stiffens.

We derived a simple analytical model to predict the effect of *Simple Analytical Model to Predict the Response of the Fluidic Segments*). It is interesting to note that our analysis indicates that for a latex tube of given length, shorter braids lower the peak pressure due to larger axial forces (Fig. S4 *C* and *E*). Moreover, it also shows that *F*). However, in this case we find that shorter tubes lower the pressure of the softening region. Finally, the analytical model also indicates that the length of the fluidic segments, *E* and *F*). However, when the tube and braids come into contact, further elongation is restrained by the braids and the segments shorten as a function of the supplied volume.

Having demonstrated analytically that fluidic segments with the desired nonlinear response can be constructed by enclosing a latex tube by longer and stiffer braids, and that their response can be controlled by changing *A*). We then measure their response experimentally by inflating them with water at a rate of 60 mL/min, ensuring quasi-static conditions (Fig. 1*B* and Movie S2).

We fabricated 36 segments with *C*, all fluidic segments are characterized by the desired nonlinear pressure—volume relation and follow the trends predicted by the analytical model (Fig. S4 *E* and *F*). In particular, we find that for the 36 tested segments the initial peak in pressure ranges between 65 and 85 kPa (Fig. 1*C*). We also monitored the length of the segments during inflation (Fig. 1*D*). As predicted by the analytical model, we find that initially the segments elongate, but then shorten when the tube and braids come into contact. It is important to note that no instabilities are triggered upon inflation of the individual segments, because the supplied volume is controlled, not the pressure.

## Combined Soft Actuator

Next, we created a new, combined soft actuator by interconnecting the two segments whose individual response is shown in Fig. 2*A*. Upon inflation of this combined actuator, very rich behavior emerges (Fig. 2*C* and Movie S3). In fact, the pressure–volume response of the combined actuator is not only characterized by two peaks, but the second peak is also accompanied by a significant and instantaneous elongation. This suggests that an instability at constant volume has been triggered.

### Numerical Algorithm.

To better understand the behavior of such combined actuators, we developed a numerical algorithm that accurately predicts the response of systems containing *n* segments, based solely on the experimental pressure–volume curves of the individual segments. By using the 36 segments from experiments as building blocks, we can construct *n* segments (i.e., 630 different combined actuators for

We start by noting that, upon inflation, the state of the *i*th segment is defined by its pressure *i*th segment in the unpressurized state. When the total volume of the system, *E*, stored in the system, which is given by the sum of the elastic energy of the individual segments**2** to express the energy in terms of

Next, we implement a numerical algorithm that finds the equilibrium path followed by the actuator upon inflation (i.e., increasing *v*). Starting from the initial configuration (i.e., *v*) and locally minimize the elastic energy (**4** already takes into account the volume constraint (Eq. **2**), we use an unconstrained optimization algorithm such as the Nelder–Mead simplex algorithm implemented in Matlab (23). Note that this algorithm looks only locally for an energy minimum, similar to what happens in the experiments, and therefore it does not identify additional minima at the same volume that may appear during inflation.

Using the aforementioned algorithm, we find that for many actuators the energy can suddenly decrease upon inflation, indicating that a snap-through instability at constant volume has been triggered. To fully unravel the response of the actuators, we also detect all equilibrium configurations and evaluate their stability. The equilibrium states for the system can be found by imposing**4** into Eq. **5**, yields**2**, can be rewritten as**7** ensures that the pressure is the same in all *n* segments connected in series.

Operationally, to determine all of the equilibrium configurations of a combined soft actuator comprising *n* fluidic segments, we first define 1,000 equispaced pressure points between 0 and 100 kPa. Then, for each of the *n* segments we find all volumes that result in those values of pressure (Fig. S5*A*). Finally, for each value of pressure, we determine the equilibrium states by making all possible combinations of those volumes (Fig. S5*B*). Note that by using Eq. **2** we can also determine the total volume in the system at each equilibrium state, and then plot the pressure–volume response for the combined actuator (Fig. S5*C*).

Finally, we check the stability of each equilibrium configuration. Because an equilibrium state is stable when it corresponds to a minimum of the elastic energy **4**, at any stable equilibrium solution the Hessian matrix**8** can be evaluated as

### Numerical Results.

To demonstrate the numerical algorithm, we focus on two segments where the experimentally measured pressure–volume and length–volume responses are highlighted in Fig. 3 *A* and *B*. In Fig. 3*C* we report the evolution of the total elastic energy of the system, *E*, as a function of the volume of the first segment, *v*, and in Fig. 3*D* we show all equilibrium configurations in the *E* emerges, so that for *E* in which the system is residing disappears, so that its equilibrium configuration becomes unstable, forcing the actuator to snap to the second equilibrium characterized by a lower value of *E*. Interestingly, this instability triggers a significant internal volume flow from the second to the first segment (Fig. 3*D*) and a sudden increase in length (Fig. 3*E*). Further inflating the system to

All transitions that take place upon inflation (i.e., at *E*), and correspond to instances at which one or more of the individual segments cross their own peak in pressure. These state transitions can either be stable or unstable (Fig. 3 *C–E*). A stable transition always leads to an increase of the elastic energy stored in the system, and an instability results in a new equilibrium configuration with lower energy. Each state transition can therefore be characterized by the elastic energy release, which we define as a normalized scalar

In Fig. 4 we report

We also characterize each state transition according to the changes induced in the individual segments, and use

### Experimental Results.

To validate the numerical predictions, we measured experimentally the response of several combined actuators. In Fig. 5*A* we show the results for the system whose predicted transitions are indicated by the diamond gray markers in Fig. 4. We compare the numerically predicted and experimentally observed mechanical response, finding an excellent agreement. In particular, for this combined actuator we find that the pressure–volume curve is characterized by two peaks, indicating that two transitions take place upon inflation. Although the *A* and Movie S4). This unstable transition is also accompanied by a moderate internal volume redistribution of 22%, resulting in the sudden inflation of the top actuator (see snapshots in Fig. 5*A* and numerical result in Fig. S6*A*).

In Fig. 5*B* we present the results for the combined actuator whose response is indicated by the square gray markers in Fig. 4. Our analysis indicates that one stable *B* and numerical result in Fig. S6*B*) and a large increase in length (Movie S5), and the second instability is a

Although the results reported in Fig. 5 *A* and *B* are for actuators free to expand, these systems can also be used to exert large forces while supplying only small volumes. To this end, in Fig. 5 *C* and *D* we show the force measured during inflation when the elongation of the actuators is completely constrained. We find that also in this case an instability is triggered, resulting in a sudden, large increase in the exerted force. Note that the volume at which the instability occurs is slightly different from that found in the case of free inflation. This discrepancy arises from the fact that the pressure–volume relation of each segment is affected by the conditions at its boundaries.

The proposed approach can be easily extended to study more complex combined actuators comprising a larger number of segments. By increasing *n*, new types of state transitions can be triggered. For example, transitions of type *A–C*), in which two segments deflate into a single one, causing all three segments to cross their peak in pressure. In Fig. 6, we focus on an actuator that undergoes an unstable

## Conclusion

In summary, by combining experimental and numerical tools we have shown that snap-through instabilities at constant volume can be triggered when multiple fluidic segments with a highly nonlinear pressure–volume relation are interconnected, and that such unstable transitions can be exploited to amplify the response of the system. In stark contrast to most of the soft fluidic actuators previously studied, we have demonstrated that by harnessing snap-through instabilities it is possible to design and construct systems in which small amounts of fluid suffice to trigger instantaneous and significant changes in pressure, length, shape, and exerted force.

To simplify the analysis, in this study we have used water to actuate the segments (due to its incompressibility). However, it is important to note that the actuation speed of the proposed actuators can be greatly increased by supplying air. In fact, we find that water introduces significant inertia during inflation, limiting the actuation speed. It typically takes more than 1 s for the changes in length, pressure, and internal volume induced by the instability to fully take place (Movie S7). However, by simply using air to actuate the system and by adding a small reservoir to increase the energy stored in the system, the actuation time can be significantly reduced (from

Our results indicate that by combining fluidic segments with designed nonlinear responses and by embracing their nonlinearities, we can construct actuators capable of large motion, high forces, and fast actuation at constant volume. Although here we have focused specifically on controlling the nonlinear response of fluidic actuators, we believe that our analysis can also be used to enhance the response of other types of actuators (e.g., thermal, electrical and mechanical) by rationally introducing strong nonlinearities. Our approach therefore enables the design of a class of nonlinear systems that is waiting to be explored.

## Materials and Methods

All individual soft fluidic segments and combined actuators investigated in this study are tested using a syringe pump (Standard Infuse/Withdraw PHD Ultra; Harvard Apparatus) equipped with two 50-mL syringes that have an accuracy of

## Analytical Exploration: Response of Interconnected Spherical Membranes Upon Inflation

To identify the key components in the design of an inflatable system that undergoes a snap-through instability upon inflation, we analytically explore the equilibrium states of two spherical membranes, connected in series. We start by describing the equations governing the response of a single hyperelastic spherical membrane upon inflation (25, 26), and then discuss the behavior of two interconnected spherical membranes (27⇓⇓–30).

### Response of a Hyperelastic Spherical Membrane upon Inflation.

We consider a spherical membrane with initial radius *R*, and initial thickness *H*, and assume that the membrane is thin (i.e., *p*, a biaxial state of stress is achieved and the principal stretches are given by*r* and *h* denoting respectively the radius and thickness of the membrane in the deformed configuration. Moreover, equilibrium requires

We consider the membrane to be made of a hyperelastic, incompressible material, the response of which is captured by the strain energy *W*. Because of incompressibility we have

Substitution of Eqs. **S1**, **S2**, and **S6** into Eq. **S5** yields**S4** to express *h* as a function of *r* (i.e., **S8** clearly shows that the initial radius *R*, initial thickness *H*, and shear modulus μ only scale the value of the pressure and that the only parameter that changes the shape of the

As an example, in Fig. S1 we show the *R* and shear modulus μ and (*a*) *b*) *c*)

It is important to note that in our analysis we assume that the deformation is homogeneous throughout the membrane, although it has been shown that this assumption can be violated by spherical membranes as they can exhibit asymmetric bifurcation modes (32, 34). However, these asymmetric modes do not affect the main features of the nonlinear pressure–volume relation typical for such membranes, and as such our analysis provides enough detail to qualitatively study the response of systems of interconnected membranes.

### Response of Two Interconnected Spherical Membranes upon Inflation.

Having determined the pressure–volume curve for a single spherical membrane, we now determine the equilibrium states for a system of two interconnected membranes. For both membranes, we use the pressure–volume relation defined by Eqs. **S8** and **S9**. Specialization of Eqs. **9** and **14** to a system comprising two spherical membranes, yields

Moreover, Eq. **17** reduces to*i*th membrane. Moreover, *i*th membrane and

In Fig. S2 we show the response of the systems obtained by interconnecting membranes *a* and *b* (Fig. S2*A*), *b* and *c* (Fig. S2*B*), and *a* and *c* (Fig. S2*C*). Note that the pressure–volume curves for the individual segments are shown in Fig. S1. When connecting membranes *a* and *b* (Fig. S2*A*), upon inflation membrane *a* inflates first. Then, when this membrane starts to stiffen, membrane *b* also increases in volume. However, for this combined system all equilibrium points are stable. More interesting behavior is observed when combining membrane *b* and *c* (Fig. S2*B*). Initially, this system behaves similarly to the system shown in Fig. S2, with membrane *c* inflating first. However, at *b*. Therefore, at the instability a significant amount of volume flows from membrane *c* to membrane *b*. We also note that this system is characterized by equilibrium configurations that are disconnected from the main curve, so that they will never be reached upon inflation. Finally, even more complex behavior can be achieved by connecting membranes *a* and *c* (Fig. S2*C*). Here, a first instability causes all volume to flow from membrane *a* to membrane *c*, and a second instability forces some volume to flow back from membrane *c* to membrane *a*.

## Simple Analytical Model to Predict the Response of the Fluidic Segments

The analytical predictions reported above reveal that snap-through instabilities at constant volume can be triggered in a system comprising interconnected spherical membranes only if the pressure–volume relation of at least one of the membranes is characterized by (*i*) a pronounced initial peak in pressure, (**S14** can be obtained from the Gent energy density in the limit as

In the following, we first derive the pressure–volume relationship for a latex tube, and then we develop a simple analytical model to capture the response of the highly nonlinear braided fluidic segments used in this study.

### Response of a Latex Tube Upon Inflation.

We start by investigating analytically the response of a cylindrical latex membrane with initial radius *R*, initial length *H*, and assume that the membrane is thin (i.e., *p*. For simplicity, we neglect end effects and local instabilities resulting from bulging (36⇓–38), and assume that the cylindrical membrane deforms uniformly. The principal stretches are then given by*r*, *l,* and *h* denoting the membrane’s radius, length, and thickness in the deformed configuration, respectively. Moreover, as in experiments, we assume that the tube has closed ends, so that it is subjected to an axial force

We consider the membrane to be made of an incompressible neo-Hookean material (Eq. **S14**) and because of incompressibility we have**S15**–**S17** and **S19** into Eqs. **S21** and **S22** yields**S19**, **S23**, and **S24** results in**S26** into Eq. **S23** provides the evolution of the pressure as a function of the axial stretch **S27**, because the volume enclosed by the membrane, *v*, is given by**S27**). Furthermore, we immediately see that no stiffening occurs, even for very high volumes. As a result, we do not expect to trigger any instability at constant volume in systems comprising interconnected latex tubes. Moreover, we also note that it is not possible to tune the pressure response of these inflatable tubes by changing their length, as the pressure does not depend on

### Response of a Braided Fluidic Segment.

To construct fluidic segments with a final steep increase in pressure and whose response can be easily tuned and controlled, we enclosed the latex tube of initial length *A*). The effect of the stiff braids is twofold: (*i*) as *F*, applied to the membrane; (

#### Effect of the axial force.

To determine the effect of the braids on the initial response of the segments, we first estimate the axial force *F* introduced by *n* braids that enclose the latex tube. For the sake of simplicity, we model each individual braid as two rigid segments of length *K* (Fig. S4*A*). As the braids are longer than the latex tube, before attaching them together, they are shortened by *B*), causing them to buckle. Balance of the work done by the axial force *F* and the elastic energy requires**S30** reduces to*F*, and introduces an error that is within 9%.

Finally, we experimentally estimate the torsional stiffness *K* by performing uniaxial compression tests on braids of different length and comparing the experimentally measured critical forces to Eq. **S31**. From the results reported in Fig. S4*B* we obtain that for the braids used in this study

Next, we investigate the effect of the axial force *F* on the latex tube. To account for the axial force, Eq. **S17** modifies as**S24** becomes**S16** and **S23** remain unaltered. Note that Eq. **S34** can also be used to estimate the axial stretch introduced into the tube by the braids in the unpressurized state (i.e.,

By combining Eqs. **S4**, **S23**, and **S34** we obtain**S37** into Eq. **S34** the pressure–axial stretch relationship is finally obtained as

In Fig. S4*C* we show the analytically predicted pressure–volume curves for a latex tube of length *g* = 9.81 m s^{−1} is the gravitational acceleration). As previously observed (37), the results show that an increase in the axial force applied to the membrane results in a lower peak pressure, indicating that the axial force resulting from the braids can be used to control the initial response of the fluidic segments. In Fig. S4*C* the analytical predictions (dashed lines) are also compared with experimental results (continuous lines) obtained inflating a latex tube of length

#### Effect of contact.

At a critical point during inflation, the braids and the membrane come into contact, stiffening the overall response of the segments. Here, we assume that contact occurs when

Because the pressure–volume relationship provided by Eqs. **S38** and **S39** is valid only for *D*). Because the beams are modeled as rigid, this pressure results in two effective forces with radial, *D*), given by*F* is defined in Eq. **S31**. Furthermore, we assume that after contact the response is fully dominated by the braids so that**S38**–**S40**. Finally, combining Eqs. **S42**–**S45** we obtain

In Fig. S4*E* we show the predicted response for six segments characterized by

However, if the braid length is kept constant at *F*). However, in this case we find that shorter tubes lower the pressure of the softening region. Finally, the analytical model also indicates that the length of the fluidic segments, *E* and *F*).

Therefore, this simple analytical model indicates that, by enclosing inflatable tubes with stiffer and longer braids, fluidic segments with the desired nonlinear response can be realized. Importantly, we have also found that by changing

Finally, it is important to note that we expect this model to predict only qualitatively and not quantitatively the response of the segments. This is mainly due to the effect of boundary conditions (i.e., the deformation is not uniform throughout the membrane) and inextensibility of the braids. Moreover, it has also been shown that local instabilities resulting in bulges (36⇓–38) are triggered during the inflated of tubes. Although our model aims to provide design guidelines, accounting for all these effects would have lead to a significantly more complicated and less intuitive model, which falls outside the scope of this study.

## Acknowledgments

This work was supported by the Materials Research Science and Engineering Center under National Science Foundation Award DMR-1420570. K.B. also acknowledges support from the National Science Foundation (CMMI-1149456-CAREER) and the Wyss institute through the Seed Grant Program.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: bertoldi{at}seas.harvard.edu.

Author contributions: J.T.B.O. and K.B. designed research; J.T.B.O., T.K., and J.J.A.D. performed research; J.T.B.O., T.K., and K.B. analyzed data; and J.T.B.O. and K.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1504947112/-/DCSupplemental.

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