Electrostatic bellow muscle actuators and energy harvesters that stack up

EBM working principle

The EBM is an electrostatic transducer that converts electrical energy into mechanical work (or vice versa), taking advantage of electrostatic forces between oppositely charged electrodes (29), and it bears the ability to work both as an actuator and a generator (16, 24, 30, 23). The EBM consists of a circular pouch made of two overlapped layers of compliant polymeric dielectric material holding opposite flexible electrodes on their external faces (Fig. 1, A to D). The dielectric layers are held together by annular frames attached along their perimeter. These frames have the role of maintaining the structural integrity of the EBM and constraining its deformation. The pouch has a single central opening on the top and contains a volume of dielectric liquid. The bottom of the pouch holds a rigid end effector for force application (Fig. 1B).

In the free state, the electrodes are flat, and the polymeric dielectric layers are in contact. Once a load is applied, the membranes deform out of plane, originating a liquid-filled pouch with a shape similar to a double-truncated cone shell.

In the actuation mode (Fig. 1C), with the application of a voltage, charges on opposite electrodes attract, causing the polymeric layers to be pulled toward each other in a “zipping” motion (22). During zipping, the polymeric layers gradually come into contact (starting from the perimeter), and liquid is evacuated through a central opening toward a reservoir.

Because of its layout, the EBM can pursue two different actuation modes: as a contractile actuator that does work against an external force or as a hydrodynamic actuator that does work against the pressure of the fluid. In the first mode, Maxwell stresses (16) are exploited to produce work against a force applied at the end effector, whereas the dielectric fluid is evacuated toward a soft-walled variable volume reservoir, hence resulting in little or no variation in pressure with respect to atmospheric conditions. In the second mode, Maxwell stress is used to induce a substantial pressurization in the fluid chamber, and the fluid is delivered to a fluid utility. In the generation mode, similar to DE generators (31), external mechanical forces drive the pouch deformation, whereas the voltage is properly varied in a such a way that electrical energy is generated and supplied to the driving electronics.

Modeling of EBM

The EBM can be modeled as a variable capacitor, using an approach similar to that presented in (29), introducing the following simplifications:

1) In a generic configuration, the EBM geometry can be ideally divided into two portions (Fig. 1D): a fully zipped outer region, in which two flat dielectric polymer portions face each other over an annular surface and a double-cone chamber in which the polymer layers enclose a volume of dielectric fluid. Further assumptions on the distribution of the strains in the dielectric polymer layers are discussed in the Supplementary Materials.

2) The electrical response is purely electrostatic; i.e., leakage currents through the dielectric layers are neglected. Similar to (29), the total capacitance of the device is about equal to that of the zipped annular portion, whereas the capacitance of the double-cone portion housing the oil is negligible.

The EBM kinematics is described through two independent parameters, i.e., the height h of the pouch and the radial distance r_c from the cone axis of the inner perimeter of the zipped region (Fig. 1D). The EBM response is governed by the following equations (see derivation and validation in the Supplementary Materials){F+p·∂Ωc∂h=∂Uel∂h+ϵpE2·Az∂t∂hp·∂Ωc∂rc=∂Uel∂rc+ϵpE2·[2πtrc+Az∂t∂rc](1)where r_o and r_i are the fixed inner and outer radii of the device; Ωc=πh(ri2+rc+rirc)/3 is the volume of fluid in the double-cone chamber; U_el = U_el(h, r_c) is the elastic energy of the deformed polymer layers; Az=π(ro2−rc2) is the area of the zipped electrode portion; t = t(h, r_c) is the thickness of a single polymer layer in the zipped region (it varies with h and r_c due to strain); ϵ_p is the dielectric constant of the polymer; and E is the electric field in the zipped region, which depends on the applied voltage V: E = V/(2t). The two terms on the righthand side of both equations in Eq. 1 account for the elastic restoring forces due to the dielectric polymer elasticity and the contribution of Coulomb electrostatic forces. The latter contribution, in particular, holds a term ϵ_pE² proportional to the Maxwell stress applied on the polymeric layers over the zipping region.

The response of the EBM in the different working modes can be described in terms of the different applied loads (namely, F or p in Eq. 1). When the EBM works as a contractile actuator, the fluid dwells at a pressure close to atmospheric value (p ≃ 0), and the force F, which makes work in the direction of h, represents the useful output load. When the EBM works as a pump, p represents the useful available pressure, and F is either zero or equal to a constant preload. As a generator, the EBM can take advantage of either the input mechanical work provided by the external force (reverse muscle) or by the fluid pressure (reverse pump) to produce an electrical energy output.

Manufacturing EBMs

The pouch consists of two layers of dielectric polymer attached together with a thin annular layer of double-sided adhesive film. The top film has a circular opening serving as the fluid outlet. Stiff reinforcing annular frames are bonded on the outer faces of the dielectric films with the aim of preventing deformations at the outer diameter of the films. A reservoir is connected to the top opening and an end effector on the bottom side (Fig. 2A). The reservoir includes a manifold that generates a radial fluid flow from the axial movement of the EBM and a slack membrane that implements a (nearly) constant-pressure variable-volume sealed reservoir. Conductive paint is applied on each side of the pouch (Fig. 2B) to form the electrodes. The device is preloaded and filled up with dielectric oil through the central opening (Fig. 2C).

EBM operated as a contractile actuator

An EBM unit can work as a contractile actuator achieving contractions of up to 20% its initial length (including the length of the fluid reservoir) against external loads in the range from 1 to 6 N (up to roughly 70 times the whole muscle and fluid storage weight) with input voltages in the range from 6 to 8 kV. At 1 Hz, the EBM unit produced strokes of 1.3 mm (Fig. 3A) corresponding to a contraction of 18.0% of the length before actuation, with a maximum applied voltage of 6 kV. Neglecting the contribution of the reservoir height, the stroke corresponds to a contraction of 44.7%. Figure 3A reveals that at 1 Hz, the device sweeps the whole available stroke and remains latched at the upper and lower positions for a finite amount of time. The EBM moves synchronously with the applied voltage: When voltage is low, concurrent polymeric layers are kept apart (unzipped) under the action of the mechanical load; as the voltage increases, electrostatic forces progressively overcome the external load and cause the electrodes to zip; at large voltage (roughly above 5 kV), the electrodes stay fully zipped against each other, and almost all of the liquid is moved into the reservoir. The EBM experiences a very limited reduction in stroke due to increased frequency (Fig. 3A), similarly to hydraulically amplified muscles (24) and in contrast with frequency-sensitive fluid-based electrostatic ribbons (25). The EBM stroke at 4 and 8 Hz is, respectively, 97.0 and 92.8% of the stroke at 1 Hz, with a load of 1.5 N. Increasing the peak voltage to 8 kV, the EBM reaches a total contraction over 20% with forces of up to 2 N. An overview of the EBM unit response under quasi-static conditions under different applied forces is shown in Fig. 3B. In the low-force range (1 to 2 N), increasing the applied load leads to an increase in the stroke as a result of the increased initial length of the muscle. A further increase in the load, which acts against the electrostatic forces, causes a decrease in the actual stroke. The EBM still generates contractions over 17% with loads of up to 3 N—i.e., an equivalent lifting capability of 35 times its total weight (including the weight of the fluid reservoir) and 50 times the EBM core weight. Increasing the peak voltage generates a modest increase in stroke at small loads, whereas it has a relevant impact at high loads. In effect, at reduced loads, a relatively low voltage is sufficient to cause a contraction close or equal to the whole geometrically available stroke. A larger voltage still produces larger strokes even at 1 to 1.5 N, because it enables better fluid evacuation and the permanence of a thinner trapped fluid film in the fully zipped configuration.

In the range from 1 to 10 Hz, the EBM contraction weakly depends on the frequency even at relatively large applied loads (Fig. 3C). At 2.5 N, the contraction is nearly constant (close to 19%) at 8 kV in the range from 1 to 10 Hz, whereas it slightly decreases with frequency at 6 kV (with the stroke at 10 Hz still being 74.6% that at 1 Hz), because the electrostatic forces do not completely compensate viscous and hydraulic losses. The EBM provides actuation up to frequencies of at least 80 Hz (Fig. 3C) with 2 N of applied load, experimenting a reduction in stroke of 50% at a frequency of 37 Hz and of 80% at 80 Hz. The reduction in stroke at larger frequencies is ascribable to viscous losses due to the dielectric fluid motion.

The blocking force of the EBM (at which the stroke falls to zero) at 8 kV is close to 6 N (i.e., 98 times the weight of the EBM core), rather independently of the actuation frequency (Fig. 3D). The stroke at 5 N is roughly 65% of the maximum stroke, and it drastically falls in the interval of 5 to 6 N. The EBM unit achieves contraction rates of up to 1140%/s at 10 Hz (fig. S8), with power densities over 21 W/kg per EBM core mass (which still corresponds to 14.8 W/kg if the weight of the fluid reservoir is also considered).

The EBM has also achieved notable performance in terms of long-term cyclic operation. An EBM prototype subject to electric fields in the order of 120 kV/mm on the dielectric polymer layers was able to accomplish more than 0.1 millions of cycles at a 5 Hz operating frequency, with no visible reduction in stroke over time (fig. S11).

Scaling the EBM response

We investigated the effect of scaling the radial dimensions of an EBM contractile actuator while keeping the thickness of the dielectric polymer layers constant. According to the proposed mathematical model, a geometrically scaled device (with geometric scaling factor of s_f) loaded with a force scaled by s_f provides a stroke that is s_f times larger at a same voltage (derivation provided in section SA.3).

The experimental data on three different devices (i.e., the base-case device to which previous results refer, a smaller scale device with s_f = 2/3, and a larger device with s_f = 4/3) confirm the trends predicted by the model (Fig. 3E and fig. S9). Observed disagreements are possibly due to the reduction in stiffness of the structural rings for the upscaled device and a higher sensitivity to imperfect manual manufacturing of the downscaled device.

Reducing the scale of an actuator leads to an increase in the achievable energy density (see section SA.3 in the Supplementary Materials) due to the substantial reduction in the dielectric fluid mass (which scales with sf3, as opposed to the actuation work, which scales with sf2). A downscaled EBM prototype with lightweight polymeric structural components was able to achieve an energy density (i.e., cyclic actuation work per unit EBM core mass) of 1.16 J/kg subject to a load of 1.6 N, i.e., 127 times its own weight (fig. S10).

EBM arrays from foldable strips for scaling up force and stroke

EBM units can be combined in series and in parallel to magnify the stroke and the force, respectively, while preserving the ease of manufacturing and assembly of single-unit AMs. In-series connection is achieved by stacking up multiple devices (32, 33) and putting their fluid chambers in communication. In-parallel connection is achieved by providing a set of EBM units/stacks with a common end effector. By connecting in parallel m stacks of n in-series EBMs, we obtain m × n arrays that provide a stroke that is n times magnified in the presence of an applied force that is m times as much.

Different array layouts can be built through the same manufacturing procedure (Fig. 4, A to D, and fig. S4). The constitutive elements of the EBM units (electrodes, central openings, and frames with adhesives) are mounted equidistantly and collinearly on a strip of dielectric material, forming a row of “half-units” on one surface of the strip. Figure 4A shows such a strip in the case of a four-unit array. This strip is then folded in half (with the frames and the electrodes on the external faces) to create EBM units (Fig. 4B). In the same way as single EBM units, each couple of half-units is attached and sealed together with thin layers of double-sided adhesive film. The resulting strip of assembled single units can be folded again to create arrays with arbitrary size. Figure 4C shows folds required to create a 2 × 2 EBM array, where a strip with two units is folded onto another one of the same size. In Fig. 4D, the same initial strip is folded in a different way so as to create a 1 × 4 array.

Starting from a strip of p EBM units, arrays of different sizes can be created, obtaining a maximum blocking force equal to p times that of the single units (in the case of a p × 1 array) or a maximum stroke equal to p times that of a unit (in the case of a 1 × p array). To create an m × n array from a strip of p units (with p = m · n), n − 1 folds are required. The simplicity of this assembly procedure makes it scalable and well suitable for mass production.

Performance of contractile EBM arrays

We built EBM arrays with different configurations, comprising up to six pouches each (Fig. 5A), with the same reference dimensions for the units as those used in Fig. 3 (A to D). By connecting six in-series EBM pouches (Fig. 5B) in a 1 × 6 arrangement, we reached strokes of up to 43% the muscle initial length (including the height of the fluid reservoir). We characterized the response of in-series arrays with loads of up to 3 N (20 to 40 times their weight). We compared the performance of 1 × 3 and 1 × 6 arrays against that of a single-unit device (Fig. 5C): At 8 kV, the stroke against a force of 1.5 N surpassed 4.1 mm for a 1 × 3 device (31.0% contraction) and 9.0 mm for a 1 × 6 array (42.8% contraction), as compared with 1.5 mm for the single-unit device (20.2% contraction). By increasing the number of in-series units, the height of the fluid reservoir (which is the same for all devices) becomes increasingly small compared with the total stroke, leading to an increase in the percentage contraction. Reducing the fluid reservoir height would generate an increase between 10 and 30% in the contraction for the different arrays. For instance, at 1 N, the estimated contraction for the 1 × 6 EBM would rise to 55.2% (as compared with 43.3%) if the reservoir height was virtually reduced to zero.

We compared the dynamic responses of different in-series EBM units (Fig. 5D and movie S1). Similar to the unit actuator, in the range from 1 to 10 Hz, the contraction achieved by EBM arrays is weakly dependent on the frequency. Compared with single and 1 × 3 devices, the 1 × 6 arrays suffer larger reductions in contraction at large frequencies (above 5 Hz). This is due to dynamical effects (e.g., the increase in inertia with the number of units) and the higher sensitivity to manufacturing/misalignment errors of large EBM arrays.

EBM arrays of in-series units were able to reach peak linear speeds close to 200 mm/s (see fig. S13) and contraction over 1250%/s (see fig. S12). We measured a peak-specific power of 31 W/kg (see fig. S12) for the 1 × 3 actuator in the same order as other fluid-based electrostatic actuators (24, 25). Shape-memory alloys and coiled nylon AMs reach higher power densities of tens of watts per gram (14, 34). Nonetheless, they suffer from poor controllability at large frequencies (24) due to their slow thermal dynamics.

We developed a 2 × 3 EBM array made of two in-parallel 1 × 3 arrays (Fig. 4E). As expected, this device produces a stroke close to that of a 1 × 3 array in the presence of a load that is twice as much (Fig. 5E, fig. S14, and movie S2). In practice, scaling the force-stroke response of a 1 × 3 EBM array by a factor of 2 along the force axis leads to a slight overestimation of the 2 × 3 array stroke. This is due to a higher sensitivity of the 2 × 3 device to bending moments generated by the total load, applied between (and roughly parallel to) the axes of the in-parallel stacks. We used the 2 × 3 EBM array to lift large weights (see movie S2). The device can lift hanged masses as high as 1000 g, i.e., over 40 times its weight. This is consistent with the results of single-unit devices (Fig. 3D), which showed blocking forces for single units (and, virtually, in-series arrays) close to 6 N.

EBM for volumetric pumping

EBM arrays can be used as volumetric pumps capable to operate FASs. We used a 1 × 6 EBM with reference dimensions to inflate a thin elastomeric membrane with a 45-mm base diameter, and we reached operating pressures of up to 6 kPa and flow rates over 0.5 standard liters per minute (SLPM).

The EBM-based volumetric pump was obtained by putting the dielectric liquid chamber in communication with the air chamber that holds the membrane and using two check valves to produce a unidirectional flow (Fig. 6A), similarly to (28). With this layout, each stroke of the dielectric liquid free surface generates a useful unidirectional air flow from the atmosphere toward the bubble. By performing tests at 7 Hz with a voltage of 8 kV and a preload of 1.5 N, we were able to inflate the balloon up to 56 cm³ in 28 s (Fig. 6B and movie S3). By performing tests at different frequencies, we observed that the pumped volume increases nearly proportionally with the frequency. The trend of the pumped volume and pressure versus the number of actuation cycles is about the same at operating frequencies of 5 and 7 Hz (Fig. 6B). However, at low frequency (i.e., 3 Hz), the pressure and air volume delivered after a same number of cycles are lower. This is possibly due to air leaks in the pump body, which are independent of the pumping frequency and are not counterbalanced by a sufficient delivered flow at low operating frequencies. The pump flow rate (averaged on the single actuation cycles) as a function of the hydrostatic pressure has the trend shown in Fig. 6C. In the low-pressure range (below 2 kPa), the EBM pump delivers a flow in the range of 0.40 to 0.63 SLPM, which increases with the frequency. Under these conditions, the EBM sweeps a stroke close to 5 mm (corresponding to a volumetric displacement of 2.1 cm³) and delivers a nearly constant volumetric displacement. By measuring the bubble volume variation within a cycle (see lens in Fig. 6B), we estimated a maximum instantaneous flow rate of 1.95 SLPM (fig. S17B). As the hydrostatic pressure increases, the EBM stroke and hence the flow rate decrease. As expected, at given pressure, the delivered flow rate increases with the actuation frequency following a nearly linear trend (Fig. 6D). The blocking pressure, above which the pump is no longer able to supply a flow, is slightly larger than 6 kPa, both at 5 and 7 Hz. At lower actuation frequency, the blocking pressure decreases due to air leaks (Fig. 6C). At the blocking pressure, the EBM still exhibits a stroke of 2 mm, even though the supplied flow is equal to zero. This is due to air leaks and the effect of air compressibility in the pump body. The EBM was able to develop a pumping power (cycle average) of up to 18 mW (corresponding to a power density over 0.9 W/kg per unit EBM core mass) and instantaneous peak values of 68 mW (fig. S17).

The performance achieved by the EBM pump is comparable to that of the best-performing smart material–based FASs with similar dimensions. Similar figures of merit have only been achieved by state-of-the-art DE-driven pumps, exploiting snap-through instability of inflatable membranes (35) or a dynamic resonant design (28, 36). Compared with the first solution (35), the EBM pump offers the potential for more compact designs. Compared with resonant DE FAS (28), the EBM pump achieves large flow rates with lower actuation frequency (below 10 Hz) and with no stringent need to tune the operating frequency. At a reference pressure of 2 kPa, our prototype generated a flow rate on the same order as that of a DE pump with similar radial dimensions (36) but with frequencies one order of magnitude lower.

EBM for energy harvesting

EBM arrays can function as generators with no need for architectural modifications compared with the actuator embodiment. To demonstrate generator operation, we supplied mechanical energy to a lightweight 1 × 3 EBM with polymeric structural rings by connecting the end effector to the slider of a linear motor via a pulling nylon cable, and we controlled the applied voltage synchronously with the motor displacement. We applied a constant voltage while the motor was pulling the EBM’s end effector downward (pushing the electrodes apart) and kept the device uncharged while the motor was moving upward (allowing the electrodes to move closer) (Fig. 7A). By doing so, an amount of electrical energy is initially spent when voltage is applied; then, as the external mechanical load forces opposite electrodes to be pushed apart, the EBM positively delivers current, leading to a net gain of electrical energy. The electrical energy balance is depicted in Fig. 7B, which relates the EBM voltage to the total charge delivered by the power supply (30). Figure 7C shows the EBM force against the end-effector position: Because of voltage application, the force required to pull the electrodes apart (forward stroke) is larger than that required to allow the fluid discharge (return stroke), i.e., a net amount of mechanical work is supplied to the EBM at each cycle. Using different values for the priming voltage, we obtained a cyclic generated energy density of up to 0.41 J/kg (per EBM core mass), average power of up to 1.5 mW, and maximum efficiency close to 20% (Fig. 7D). The peak power output (for the test in Fig. 7A) is 80 mW, corresponding to a peak power density of 11.2 mW/g. These figures of merit increase monotonically with the priming voltage. In contrast with theoretical trends for electrostatic generators (30), the power and energy density have a subquadratic trend, possibly due to an increase in the leakage currents with the voltage.

As compared with DE generators, the stored elastic energy (due to mechanical loading) for EBMs is on the same order as the maximum stored electrostatic energy, whereas in DE generators, it is 3 to 50 times the stored electrical energy (37). This is a distinguishing feature that makes it easier coupling EBM harvesters with mechanical energy sources without the need of stiffness cancellation mechanisms (e.g., negative springs) (38, 39).

The values of energy and power densities could be largely improved to get closer to the values achieved by DE generators [i.e., energy densities on the order of hundreds of joules per kilogram and power densities on the order of hundreds of watts per kilogram (40, 41)]. Because of the low elastic energy stored in EBMs, mechanical losses are lower than in DE generators; nonetheless, charge leakages are estimated to be substantial. Figure 7B shows that the charge absorbed by the EBM (which differs from the real charge on the electrodes due to leaks) is notably larger than zero by the end of the cycle (state P₄), meaning that some current has been drained. These losses appear attributable to leaks in air (because the outer surfaces of the electrodes are not shielded) and through the dielectric layers, with the former likely being more relevant. Introducing shielding of electrodes and using different dielectric liquids might be the key to enhance EBM generator performance. In addition, optimizing the applied voltage waveform (40) and the duration of the charging/discharging phases and resorting to closed-loop logics is expected to lead to further improvement.

EBM materials and components

The dielectric layers in EBM are made of 25-μm-thick LINQTAPE PIT1N polyimide (PI) film by Caplinq, with nominal relative permittivity of 3.9 and dielectric strength of 236 kV/mm (43). The dielectric fluid is Xiameter PMX 200 50 cSt, a silicone oil with a relative permittivity of 2.7 (23).

Bonding of the dielectric layers on the pouch inner surfaces is made with 5-μm double-sided adhesive tape by Nitto (no. 5600). Rigid rings are attached to the dielectric layer via 100-μm-thick double-sided adhesive tape by Tesa (68144). We produced two different versions of the EBM on the basis of different materials for the structural rings: a version with thin rigid metallic frames (to grant a reduced initial length and hence achieve larger contractions) and one with lightweight PMMA (polymethyl methacrylate) rings (to reduce the weight and increase the energy density). Metallic annular frames made of steel, with a thickness of 0.1 mm, were used for the actuator and pump tests. PMMA polymeric frames with 0.5 mm thickness were used for generator tests, although some actuation tests were also performed on PMMA-based EBMs (fig. S10).

Electrodes (16 mm by 30 mm in the base-case device) are made of DAG-T-502 carbon paint by Ted Pella. This paint shows appropriate compliance and adhesion to the substrate. To attach pouches in-series in arrays and to attach the reservoir or the end effector, we used 0.5-mm-thick 3M VHB 4905 double-sided adhesive tape.

The fluid reservoir contains three main components: a rigid layer, a manifold (which controls the direction of the flow within the reservoir), and a flexible membrane. The rigid layer consists of a PI film reinforced by a 0.5-mm-thick PMMA annular frame. The manifold is printed with a Formlabs 3D printer with Grey Pro photopolymer resin. The flexible membrane is a loosely stretched layer of 3M VHB 4905, which grants the reservoir with variable volume while producing negligible pressurization in the fluid. The total height of the reservoir is 4.3 mm, whereas its mass is 2.6 g.

EBM prototypes with different radial sizes were built. A base-case reference set of dimensions were used for most of the reported tests (Figs. 3, A to D, and 5 to 7). The reference EBM has outer electrode diameter d_o = 30 mm and was realized with either 30 mm–by–42 mm steel rings (Figs. 3, A to D, 5, and 6) or 30 mm–by–40 mm PMMA rings (Fig. 7). A complete set of reference dimensions for EBM unit prototypes are summarized in Fig. 2A, fig. S5, and table S3. Smaller- or larger-scale prototypes with thin steel rings (20 mm by 32 mm and 40 mm by 52 mm) or PMMA rings (20 mm by 27 mm) were also realized to study the scaled response of AM EBMs (Fig. 3E). The dimensions and weights of the different prototypes used for the tests are summarized in table S3.

Testing methods

A test bench was developed and adapted to characterize EBMs (unit and stacks) in different working modes (see fig. S6). High voltage was supplied by means of a four-quadrant amplifier (TREK 10/10B-HS). We used voltages of up to 8 kV, corresponding to a maximum electric field of 160 kV/mm over the total polymer layers thickness, i.e., well below the nominal break-down field of the reference PI material (236 kV/mm).

Driving, control, and data acquisition were performed with a real-time target machine by Speedgoat, running MATLAB Realtime environment. The EBM end-effector stroke was measured via a laser sensor (LK-G152 by Keyence). A protruding structure was attached to the device end effector to facilitate load application and stroke measurement.

Linear muscle testing

Each linear actuation test was performed applying a constant force on the end effector, supplying a time-varying voltage profile on the electrodes, and measuring the resulting displacement. Constant forces at the EBM end effector were generated through a preloaded soft elastic band with an initial length much larger than the actuator stroke. We used sinusoidal voltage profiles, thus reverting the electrode’s polarity at each cycle to compensate for polarization of the interfaces between different dielectric layers, as suggested in literature (24, 44). The driving voltage frequency is one-half the resulting actuation frequency. In our results, the percentage contraction (see eq. S15) is defined as the ratio of the stroke and the EBM length before voltage application at the tested load case (average over the performed actuation cycles). The latter is measured from the top surface of the reservoir to the bottom layer surface (where the end effector is attached). The thickness of the end effector is not considered, because its layout is only dictated by experimental convenience, and it can be easily redesigned and optimized.

EBM pump testing

In hydrostatic actuation tests, the EBM fluid outlet was connected to a 3D printed assembly holding custom-made check valves and a membrane bubble. The inflatable membrane was realized using three layers of 500-μm-thick commercial 3M VHB 4905, with a prestretch of about 2.5. In the tests, we preloaded the EBM with a constant force of 1.5 N, and we used sinusoidal voltage inputs with frequency in the range from 1.5 to 3.5 Hz (corresponding to one-half of the resulting actuation frequency). The pressure in the air chamber holding the elastic bubble was measured with pressure sensor MPX12 by Freescale Semiconductor. The bubble volume was estimated via the frames of a high-speed camera (Point Grey GS3-U3-23S6M-C with lens 250F6C). Standard edge detection algorithms were used to resolve the bubble profile, similar to (41), and the subtended volume was calculated by numerical integration.

Energy harvester testing

To test the EBM generator, we drove the EBM displacement with linear motor L16-50-35-12-P by Actuonix. The connection between the slider and the EBM end effector was implemented via a fluorocarbon fishing wire (300 μm diameter). Compared with a rigid connection, this ensures a smoother deformation of the EBM and rejects undesired transversal loads, although it does not grant direct drivability of the end-effector position. Similar to actuation tests, the voltage polarity was reverted at any cycle. In the generation tests, we used motor displacements with amplitude of 4 mm, bias of 7.7 mm, and frequency of 0.5 Hz. The input mechanical energy was calculated from the end-effector displacement and the force (measured via a load cell Rb-Phi-118 with RB-Onl-38 amplifier by RobotShop) (see eq. S16). The net generated energy was calculated from the voltage (measured with the amplifier’s integrated sensor) and the current (measured with isolation amplifier HCPL-7800A/HCPL-7800 in series to the EBM) (see eq. S16).