Second, we integrate the DEA into a lightweight, flapping-wing mechanism and utilize system resonance to remove higher harmonics induced by the nonlinear transduction. In combination, we design a 155-mg flapping-wing module that can be assembled into several configurations. Using these modules, we are able to construct vehicles that not only demonstrate passively stable ascending flight but also controlled hovering flight.

Our robot is driven by a multi-layered DEA rolled into a cylindrical shell to generate linear actuation (see Methods section 'Fabrication of robot components' and Extended Data Figs. 1 and2 for details on DEA fabrication). The DEA is mounted in a light-weight airframe (Fig. 1a), with the two ends of the DEA attached to planar four-bar transmissions. This design allows one DEA to simultaneously actuate two wings in a manner analogous to what the indirect flight muscles do in neopteran flying insects 28 . By using the planar four-bar transmissions, the DEA's axial extension and contraction are converted into the wing's rotational stroke motion (Fig. 1b). In quasi-static operation, the actuation is unidirectional because DEA strain is proportional to the square of the applied electric field. In dynamic operation, the DEA extends and contracts as a result of its intrinsic inertia and stiffness, yet its elongation amplitude is larger than the retraction amplitude. To ensure the mean wing stroke (α) motion is symmetric with respect to the robot body, the resting wing stroke plane is offset by approximately 15° (Fig. 1b) during robot assembly. The DEA is pre-strained by 2% when it is attached to the robot transmissions, and this pre-strain loads the elastic four-bar transmissions to introduce the wing stroke bias. This small pre-strain does not noticeably change the DEA performance, and the design is advantageous compared to artificial flight muscles with a large pre-strain 11 (>100%) because it does not require a rigid and heavy supporting structure. In this way, the robot wing stroke (α) motion is fully controlled by the actuator, whereas the wing pitch (β) rotation is passively mediated by the compliant wing hinge (Fig. 1c). Figure 1d and Supplementary Video 1 show a half flapping period actuated at 280 Hz. The tracked wing stroke and pitch motion for the same experiment are shown in Fig. 1e. On the basis of an aerodynamic model developed in a previous study 29 , we estimate that this flapping motion will generate a net lift force of approximately 1.8 mN, corresponding to 1.2 times the robot weight. This modular robot can be assembled into several configurations to demonstrate different flight capabilities. For instance, Fig. 1f shows micro-aerial vehicles (MAVs) driven by one (centre panel), two (left panel) and four DEAs (right panel). These vehicles exhibit open loop lift-off (one DEA), stable ascending flight (two DEAs) and hovering flight through feedback control (four DEAs), respectively.

To achieve flight of a soft-actuated robot, the DEA must have sufficient power density and the robot transmissions and wings must be designed around the actuator's output force, displacement and bandwidth. In contrast to previous studies 11 that developed pre-strained acrylic DEAs to achieve large deformation (>30%) and high energy density (>4 J kg -1 ) but low bandwidth (<30 Hz), we use a silicone elastomer as the dielectric material for the flight muscles to achieve higher bandwidth (>400 Hz), combined with moderate strain (10-15%) and energy density (1.13 J kg -1 ). For driving frequencies lower than 600 Hz, our DEA's blocked force (Fig. 2a) is independent of frequency because its electrical properties are tuned to have a small RC time constant of 0.18 ms. The DEA's free displacement (Fig. 2b) peaks at 15% strain when it is driven at 500 Hz. The free displacement amplitude includes the

contribution from the first and higher-order harmonics in response to a sinusoidal driving signal. We observe a secondary peak of free displacement (Fig. 2b) when the driving frequency is 250 Hz, owing to exciting the second-order harmonic that is near the resonant frequency (500 Hz). Our robot design utilizes the first harmonic to drive the flapping-wing motion. By computing the fast Fourier transform (FFT) of the DEA's response to white noise, we quantify the magnitude (Fig. 2c) and phase (Fig. 2d) of the linear part of its response. When operated under the takeoff conditions (300 Hz, 1,300 V), the DEA has a power density of 300 W kg -1 and a lifetime of over 600,000 cycles (see Methods section 'DEA performance characterization' for details on the actuator characterization).

Powering MAVs using soft actuators has an advantage over state-ofthe-art flapping wing microrobots (<10 cm, <5 g) driven by rigid actuators such as piezoelectric bimorphs 3 and electromagnetic motors 5 . Although microrobotic components-such as the airframe, transmissions and wings-are robust to collisions (because inertial contributions diminish at the millimetre scale), rigid micro-actuators are fragileparticularly the piezoceramic actuators (fracture strength and failure strain are 120 MPa and 0.3%, respectively) used in many similarly sized devices 3,4 . In contrast, this DEA-driven microrobot is robust to collisions. For instance, when one wing collides with an obstacle (Fig. 2e and Supplementary Video 2), the impact is absorbed by the DEA because of its high compliance and resilience. In addition, the DEA can detect collisions (Fig. 2f) through concomitant actuation and sensing under similar principles to that of electromagnetic motors 30 and piezoelectric actuators 31 . Similarly, if an obstacle directly hits the DEA during its actuation (Fig. 2g and Supplementary Video 2), the DEA deformation can also be detected by monitoring the current (Fig. 2h). These experiments show that DEA is not only robust to collisions, but is also capable of sensing collisions with the environment (see Supplementary Information section S1 and Extended Data Fig. 3 for more experimental results on collision sensing). Despite these favourable properties (such as robustness and self-sensing), DEAs present challenges for achieving flight owing to their inherent nonlinearity. The strain in a DEA is proportional to the square of the applied electric field 7 . Consequently, a sinusoidal driving signal does not result in symmetric up stroke and down stroke motion (Fig. 3a and Supplementary Video 3) because of the influence of higher-order harmonics (see Supplementary Information section S2 and Extended Data Fig. 4 for details on nonlinear actuation and higher harmonics). For example, when operated at 100 Hz, the wing down stroke exhibits a slow reversal from T = 0.5 to T = 0.7 (Fig. 3a and Supplementary Video 3). According to a previous aerodynamic study 29 , this slow wing reversal can result in a substantial reduction in lift force. To mitigate the up stroke and down stroke asymmetry, we drive the DEA near the resonant frequency of the DEA-transmission-wing system to amplify the fundamental harmonic and attenuate higher harmonics. This asymmetry is substantially reduced when the DEA is driven at a frequency that is higher than half its resonance. Compared to flapping motion at 1 Hz or 100 Hz, the slow wing reversal is negligible when the driving frequency increases to 280 Hz (Fig. 3b and Supplementary Video 3).

In addition to exhibiting nonlinear transduction, the DEA can undergo dynamic buckling that substantially affects flapping motion and reduces the lift force. When operated near the system resonance, the DEA experiences a large compressive load that is due to the drag force from the robot wing. This normal load causes the DEA to buckle along a direction perpendicular to its actuation axis. The actuator returns to its nominal configuration as this compressive load is reduced during wing reversal. In the next flapping period, the DEA buckles in the opposite direction owing to the momentum of the restoring motion. Dynamic buckling substantially reduces the wing stroke amplitude (Fig. 3c, d and Supplementary Video 3), and it occurs at half the flapping frequency (Fig. 3d and Supplementary Video 3). Further, the large DEA deformation causes excessive electrode self-clearing and substantially reduces DEA performance and lifetime. Dynamic buckling can be inhibited by using circumferential constraints (in this case, strings) to limit the DEA's off-axis motion at its mid-plane (Fig. 3e). Figure 3f shows the left wing's stroke amplitude as a function of driving frequency and voltage. The kinks of the green lines indicate stroke amplitude reduction due to dynamic buckling. Constraining the DEA's off-axis motion enables higher driving voltages and frequencies, which correspond to higher wing stroke amplitudes. The red-shaded region indicates operating conditions that are inaccessible without constraining the DEA. Adding constraints increases the wing stroke peak-to-peak amplitude by approximately 25°, leading to a 1.6-fold increase in lift force.

Addressing the challenges of nonlinear actuation enables flight demonstrations of the DEA-driven, flapping-wing microrobots. All flight demonstrations are unconstrained, but the robots carry a thin tether for offboard power supply and control. Driven by a single DEA, the 155-mg robot demonstrates open-loop lift-off. The net lift generated by this MAV is approximately 1.8 mN, and it reaches a maximum height of 1.5 cm in 90 ms (Fig. 4a and Supplementary Video 4). To mitigate aerodynamic torque imbalances caused by fabrication and assembly imprecision, a carbon fibre rod with a point mass is attached to the robot's airframe to adjust its centre of mass position. However, without attitude (orientation of flying vehicle) and position control authority, this intrinsically unstable robot flips over within 110 ms of lift-off.

To demonstrate stable ascending flight, we build a two-actuator, four-winged robot (Fig. 1e) that uses precession around the body z-axis to achieve passive stability. We bias the resting wing's pitch angle during robot assembly to induce a net yaw torque around the robot's body z-axis. The body z-component of the angular momentum induced by precession rejects the robot's pitch and roll torque imbalances. In an open-loop takeoff experiment, we demonstrate that the robot reaches a height of 23.5 cm within 0.83 s of open-loop takeoff (Fig. 4b and Supplementary Video 5). We also construct a dynamical model and use numerical simulation to confirm the experimental observation on passive upright stability. Our simulation (Fig. 4c) shows that the robot ascends 22.7 cm in 0.83 s with a yaw rate of 17.2 revolutions per second. This passive stability property further enables us to operate more than one robot in a confined space without the need of motion tracking and feedback control. We demonstrate simultaneous takeoff flights of two robots (Supplementary Video 6) and show that they are robust against collisions with the surroundings and each other. In addition, passive stability and collision robustness can provide the ability to recover from in-flight collisions or disturbances. Figure 4d and Supplementary Video 6 show a collision recovery flight in which the robot takes off from the centre of a cylindrical shell, collides with the shell wall during its ascent, and continues to fly upward after the collision. However, passive in-flight collision recovery is a probabilistic event that depends on the robot's flight speed and the collision impact. Without any robot attitude sensing and feedback control, the robot may be destabilized after experiencing one or multiple collisions (see Supplementary Information section S3, Extended Data Figs. 56789, and Extended Data Table 2 for a detailed discussion on passive stability, collision recovery, and additional flight results).

To demonstrate controlled hovering flight, we design a four-actuator, eight-winged robot (Fig. 1e) and use a motion tracking system 3 and offboard computation for sensing and control (see Supplementary Information section S4, Extended Data Fig. 10, and Extended Data Table 3 for details on the controller design, implementation, experimental validation and repeatability). Figure 4e shows composite images of a 16-s hovering flight, and the red dot indicates the setpoint (desired hovering position). Figure 4f shows the corresponding trajectory of the same flight (Supplementary Video 7), and the colour scale represents the distance from the current position to the setpoint. For this 16-s flight, the maximum deviation of altitude, lateral position, and body angles are 12 mm (0.2 body length (BL)), 36 mm (0.6 BL) and 9°, respectively (Fig. 4g-i).

To summarize, these flights demonstrate the use of soft artificial muscles as wing actuators having sufficient power density to enable lift-off and having adequate bandwidth for flight control. Compared to the state-of-the-art MAVs driven by microscale rigid actuators (<500 mg), these soft-actuator robots have advantages such as inflight robustness to collisions and self-sensing. A feature of the DEA's fabrication scalability is that it enables efficient production of robotic modules that can be assembled in different configurations for various functions. These properties will be important for enabling swarm flight of MAVs in highly cluttered environments where collisions are difficult to avoid. However, compared to a recent piezoelectricactuator-driven MAV 32 that can demonstrate power-autonomous takeoff flights, this robot consumes 15 times more input power and requires a drive voltage 6.5 times higher. The robot's weight and net lift are 170% and 75% that of the state-of-the-art piezoelectricdriven vehicle.

To enable power-autonomous flight in soft aerial robots, future studies need to reduce a soft actuator's operating voltage, improve its power efficiency and further increase its power density. Reducing actuation voltage is crucial because up to 75% of the input electrical power can be dissipated by compact high-voltage boost and drive circuitry (as in a recent power-autonomous MAV 32 ). This challenge of lowering driving voltage can be tackled by refining DEA multi-layering techniques to further reduce the elastomer layer's thickness. Towards improving transduction efficiency, future studies could incorporate new architectures of electrically actuated soft actuators such as the electrohydraulic Peano-HASEL 33 actuators, which can use flexible metallic electrodes to reduce resistive losses. To increase power density, electroactive polymers with higher dielectric strengths and lower viscoelasticity should be explored and incorporated into future soft artificial flight muscles. From a robot design perspective, scaling up the vehicle size can substantially mitigate the challenges associated with achieving power autonomy. A larger vehicle size can provide a larger net payload, which allows the robot to carry a larger and more efficient boost circuit. In addition, scaling up the wing size corresponds to a reduction of operating frequency and leads to a linear increase in the DEA's power efficiency (see Methods section 'DEA performance characterization' and Supplementary Information S5 for a detailed discussion on future directions to achieve power autonomous flights). More broadly, our work demonstrates that soft-actuated robots can be agile, robust and controllable. These characteristics are important for developing future generations of soft robots for diverse applications such as environmental exploration and manipulation.

Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-019-1737-7.

The DEA-powered robot consists of five major components: an actuator, an airframe, transmission, two wing hinges and two wings. The two ends of the DEA are connected to the robot transmission, and the DEA's linear actuation is converted to the flapping motion of both wings. The structural design of this robot is similar to that of a microrobot powered by piezoelectric actuators that was presented in a previous study 34 . However, we needed to redesign each component to accommodate the soft actuator. In the following, we describe the design process to determine key robot parameters and present the requirements on DEA performance.

To achieve takeoff, the DEA must satisfy requirements for blocked force, resonant frequency, free displacement and power density. Specifically, the actuator needs to meet two conditions. First, the robot wings need to flap at sufficient frequency with adequate amplitude to generate a lift force that balances the robot weight. This condition places requirements on the DEA's operating frequency and displacement. Rearranging the equation that imposes the lift force and robot weight balance leads to the relationship:

where f is the robot's operating frequency, δ is half of the DEA's free displacement at the frequency f, r ˆ2 is the wing's second area moment, R is the wing span, T is the transmission ratio, AR is the wing's aspect ratio, W is the robot's weight, C ¯L is the mean lift coefficient, ρ is the air density, and f m is a mass scaling ratio such that the extra lift force can be used for flight control. In addition to satisfying this kinematic condition, the DEA needs to overcome the aerodynamic drag force during flight, and this imposes a requirement on the DEA's blocked force:

where F B is the DEA's blocked force, r cp is the wing's spanwise centre of pressure, and C ¯D is the time-averaged drag coefficient. The derivation of equations ( 1) and (2) follows closely from equations (1) to ( 14) in a previous work 34 . In equations ( 1) and ( 2), we assume that the DEA's blocked force is independent of its actuation frequency. This assumption is validated in the next section on DEA characterization. Multiplying equations ( 1) and (2) gives a requirement for the DEA's output mechanical power.

The design of a DEA-powered aerial robot also needs to satisfy an additional condition because the DEA's actuation is nonlinear with respect to input voltage. With a sinusoidal input, the DEA's actuation contains higher-order harmonics that can adversely affect flappingwing kinematics. As discussed in the main text, we attenuate higherorder harmonics by setting the robot operating frequency close to the natural frequency of the DEA-transmission-wing system. A previous study 34 shows that the actuator-transmission-wing system can be described by a lumped-parameter model. The system resonant frequency is given by:

where k m is the DEA's intrinsic stiffness, m a is the DEA's mass, k t is the transmission's torsional stiffness, and I zz is the wing's moment of inertia relative to the stroke rotational axis. For our robot, the transmission stiffness is much lower than the DEA's effective stiffness. To obtain a higher operating frequency, this condition requires a smaller wing moment of inertia. The wing moment of inertia can be decreased by reducing wing size. Using equations ( 1) to (3), we select values for the transmission ratio and the wing size while satisfying constraints imposed by our fabrication methods (that is, minimum feature size, wing inertia and so on). The values of these design parameters are reported in Extended Data Table 1. Using these parameters, we obtain the following requirements for a 100-mg DEA: F B = 0.2 N, f = 290 Hz, and δ = 0.3 mm. Multiplying these parameters shows that the DEA needs to have a minimum output power density of 200 W kg -1 . This requirement is similar to that for the MAVs powered by piezoelectric actuators 3 and to the power density values estimated for flying insects.

The robot airframe, transmission, wings and wing hinges are made using an existing multi-scale, multi-material fabrication method 35 .

The airframe consists of eight pieces of 160-µm carbon fibre laminates assembled manually and reinforced with glue (Loctite 495) (Extended Data Fig. 1a). The robot transmission is a planar four-bar mechanism.

The transmission ratio is approximated as

, where the link length l 3 is marked in Extended Data Fig. 1b. The robot transmission is attached to the DEA via a fibreglass connector, which insulates the robot structure from the DEA's driving signals. Further, the transmission connects the airframe and the wing hinge. A wing is attached onto the robot's wing hinge. The wing hinge and wing are designed according to an existing method 36 , and their geometries are illustrated in Extended Data Fig. 1c andd.

The DEA takes the form of a cylindrical shell, whose height and radius determine the actuation frequency, blocked force and free displacement. The DEA is made of a multi-layering process 9 , and it is rolled from a rectangular elastomer sheet that has embedded electrodes. Since the DEA drives two wings simultaneously, its free displacement needs to be larger than 600 µm (twice the value of the design parameter δ). Based on the values of DEA free displacement, peak loading, and elastomer stiffness, we set the actuator length to 8 mm. To obtain a blocked force over 0.2 N, the elastomer sheet (before roll-up) width is set to 5 cm. This elastomer sheet is approximately 220 µm thick, and it is manually rolled into a cylindrical shell whose inner and outer diameters are 1.5 mm and 4.5 mm, respectively.

The elastomer is a 5:4 mixture of Ecoflex 0030 (Smooth-On) and Sylgard 184 (Dow Corning). The ratio of crosslinker in Sylgard 184 is 1:40. We put a thin layer of carbon nanotube (CNT) (from Nano-C) on the elastomer and use it as the DEA's compliant electrode. For coating the electrode, we use 150 µl of CNT solution over a 90-mm-diameter polytetrafluoroethylene (PTFE) filter (Satorius 7022P). The procedures for elastomer preparation, spin coating and electrode patterning are adopted from a previous study 9 .

We made several modifications to the fabrication process to increase DEA power density and endurance. First, DEA power density can be increased by having an even number of CNT layers. Extended Data Fig. 1e shows the rolling process of a multi-layered DEA. We use greycoloured regions to denote the elastomer layers. The positive and negative electrodes are represented by red and black lines, respectively. We represent the bottom elastomer layer with a darker grey colour. When the elastomer sheet is rolled into a cylindrical shell, the DEA's bottom layer is put into contact with its top layer. This is illustrated by the inset shown in Extended Data Fig. 1f. The region highlighted by blue lines further shows that a new layer is formed by the DEA's top and bottom elastomer layers and electrodes. If the top and bottom electrodes are oppositely charged (as illustrated in Extended Data Fig. 1f), then this effective layer develops an electric field and contributes to actuation. We must have an even number of electrode layers to ensure that the bottom and top electrodes are oppositely charged.

In this work, our DEA design has six CNT and seven elastomer layers. Further, if the top and bottom elastomer layers have the same thickness as all other layers, then the electric field in this new layer is only half that of other layers because the effective layer thickness is t top + t bottom (Extended Data Fig. 1g). Hence, reducing the top-and bottomlayer thickness increases the electric field in the additional layer, and this results in an increase in DEA output power. We use a faster spincoating speed (2,700 revolutions per minute) for the top and bottom layer and a slower speed (1,700 revolutions per minute) for the middle layers. Through reducing the top and bottom elastomer layer thickness by approximately 35% (Extended Data Fig. 1h), we obtain an 11% mass reduction and a 9% increase in output power relative to a DEA with constant elastomer layer thickness. After making the elastomer layers and transferring the electrodes, we cut out the DEA from the elastomer substrate and roll it into a cylindrical shell. In the previous study 9 , the DEA is cut out manually with a razor blade. Our application requires higher accuracy, so we program a digital cutter (Silhouette Cameo) to cut out the DEA. The DEA's length is set to 8.6 mm including the exposed CNT tabs for electrical connection. With this modification, variation in the DEA length is reduced to within 150 µm. Having a precise DEA length is crucial for attaching the DEA to the robot transmission during assembly.

In addition, the DEA's bandwidth depends on several factors such as elastomer mechanical viscoelasticity (tanδ), DEA geometry, and electrode conductivity. Here, we improve the fabrication process relative to a previous study 9 to ensure good conductivity during DEA actuation (Extended Data Fig. 1i). After the DEA is rolled into a shell, carbon conductive adhesive (Electron Microscopy Sciences) is applied to the exposed electrodes and carbon fibre endcaps are glued to each end. For driving our flapping wing robot, the DEA needs to overcome aerodynamic drag during both elongation and retraction phases. During DEA retraction, aerodynamic drag opposes the DEA motion and applies a tensile stress on the DEA connections. At peak loading, this tensile stress weakens the bonding between the elastomer and the endcap, and it can create local tears and further lead to delamination. This delamination reduces electrical conductivity, which increases the DEA's time constant and reduces its bandwidth. We overcome this problem by modifying the fabrication process to increase the endcap adhesion strength. During fabrication, Loctite 416 is applied to the outer perimeter of the elastomer shell and the endcaps. The DEA is compressed with a mass of 18 g and then baked at 72 °C for 4 h. The glue cures in this process and holds the electrical connections in compression. The preload is removed after the glue cures, and other regions of the DEA return to a neutral state. A photograph of the DEA is shown in Extended Data Fig. 1j. With this procedure, we obtain an approximately four-fold increase in DEA conductivity compared to those made using previous methods 9 .

Here we describe the experimental characterization of the DEA's blocked force, free displacement, bandwidth, power consumption and efficiency. To measure the DEA's blocked force, we place the DEA under a force sensor (Nano 17 Titanium). The sensor is mounted on a twoaxis stage and is lowered until it touches the DEA's top cap (Extended Data Fig. 2a). To ensure the DEA remains securely affixed under the sensor during its retraction phase, we continue lowering the sensor to induce a preload of approximately 0.05 N. The sensor resolution and the resonant frequency are 1.5 mN and 3,000 Hz, respectively. We sample the sensor reading at 10 kHz and apply a 1,500-Hz noncausal low-pass filter to post-process the data. To measure the DEA's free displacement, we place a DEA under a laser vibrometer (Polytec PSV-500). The vibrometer measures the instantaneous velocity of the DEA's oscillatory motion (Extended Data Fig. 2b) approximately 40 times per period. For time sequence measurements, the vibrometer averages over five cycles to reduce measurement noise. The measured velocity is integrated numerically to calculate the DEA displacement. In addition, the vibrometer can measure the DEA's frequency response by driving the DEA with white noise and computing the FFT of the displacement. This measurement gives a linear approximation of the device frequency response. It quantifies the DEA's resonant modes and phase shift (Fig. 2c,d). This information is useful for robot design because the DEA's motion is approximated as linear around system resonance at flight conditions.

Sample experimental measurements of blocked force and free displacement are shown in Extended Data Fig. 2c andd, respectively. In these experiments, the DEA is driven at 350 Hz and 1,300 V. The amplitude of the DEA's blocked force is calculated as the maximum value of the measured force and it does not include the preload force (the range is labelled by the red arrows in Extended Data Fig. 2c). In our experiments, we vary the preload in the range of 0.025 N to 0.1 N and find that the magnitude of preload has a negligible effect on the blocked force measurement. The amplitude of the DEA's free displacement is calculated as the difference between the maximum and the minimum value (as indicated by the red arrows in Extended Data Fig. 2d). We report the peak-to-peak displacement value because the DEA does mechanical work during both elongation and retraction. To characterize DEA performance for different operating conditions (Fig. 2a,b), we vary input voltage amplitudes and driving frequencies from 800 V to 1,300 V, and from 1 Hz to 600 Hz. Based on the force and displacement measurements, the actuator energy and power density are calculated as:

Equations ( 4) and 5 assume the elastomer's stress-strain relationship is approximately linear. Through conducting tensile tests using an Instron materials testing machine, we find that the elastomer exhibits a linear response for a strain less than 20%. The elastomer Young's modulus is measured to be 140 kPa. The maximum measured energy density (Extended Data Fig. 2e) and power density (Extended Data Fig. 2f) are 1.13 J kg -1 and 563 W kg -1 , respectively (at 500 Hz and 1,300 V). These values satisfy the criteria for robot takeoff (Supplementary Information section S1). The DEA's driving voltage can be further increased to 1,500 V in controlled hovering flight demonstrations, so the DEA's peak power density is estimated to be 15% higher than the reported value. The DEA experiences dielectric breakdown for a driving voltage higher than 1,500 V. In our flight experiments, the robot is driven by an external power source through a thin tether. Here we quantify the DEA's resistance, capacitance, power consumption and efficiency. These parameters are important for achieving power autonomous flights in future studies. To quantify the DEA's power consumption, we measure the DEA's input voltage (V) and corresponding current (I) at flight conditions. The average electrical power input is:

T in 0

A sample measurement of instantaneous power is shown in Extended Data Fig. 2g, in which the average power consumption is 450 mW. We further measure the DEA's resistance and capacitance by sending a step input and measuring the corresponding current response. The system is modelled as a RC circuit, and parameters such as series resistance, capacitance and time constant can be obtained by fitting a first-order system to the current response. The DEA's resistance, capacitance, and time constant are 170 kΩ, 1.04 nF and 178 µs, respectively. Having calculated the DEA's resistance, we further compute the average power dissipated due to electrical resistance:

T ele 0

The DEA electrode dissipates 330 mW of power at flight conditions. The mechanical power output at this operating condition is calculated as P Fδf = 1 2 B , where the values of F B , δ and f are 0.19 N, 0.89 mm and 300 Hz, respectively. The estimated power output is 25 mW, which implies that the DEA efficiency is 5.6%. Over 73% of the power is dissipated by the electrode resistance, and the rest of the power dissipation is contributed by the elastomer's viscoelastic damping. This power dissipation leads to substantial heating of the DEA. The system can be described by a first-order conduction model:

a where T is the DEA temperature, T a is the ambient temperature, K is the dissipation rate, Q is the heat inflow, and C is the DEA's heat capacity. This first-order differential equation has a closed form solution. The solutions for the rising and the cooling phases are:

where T i is the initial temperature at the onset of cool down. The dissipation coefficients (K 1 and K 2 ) in the heating and the cooling phases are different because the flapping motion during the heating phase induces an airflow that facilitates convective cooling. The values of these modelling parameters are reported in Extended Data Table 1.

We use a FLIR T440 thermal camera to measure the DEA temperature when the robot operates under takeoff conditions (Extended Data Fig. 2h). The DEA temperature increases from 28 °C to 70 °C in 90 s. An analytical fit is superimposed on the same graph (Extended Data Fig. 2h). Snapshots of a thermal video are shown in Extended Data Fig. 2i. The maximum DEA temperature reaches 70.0 °C before cool down. This experiment shows most of the input electrical power is dissipated in the form of heat. Generating excessive amount of heat can lead to thermal failure and reduce actuator lifetime. Through our experiments, we find that our DEA can operate for over 600,000 cycles under takeoff conditions, equivalent to 33 min of flight time.

In this study, our DEA has a low transduction efficiency of 5.6%. This low transduction efficiency would not be conducive to power-autonomous flights. In addition, it requires a 1,300V driving signal to achieve takeoff, which creates challenges for developing high-efficiency boost circuitry. Although this study does not aim to achieve power-autonomous flight, it is important to identify major challenges and potential solutions. Future studies should focus on increasing the DEA electrode's conductivity, reducing elastomer layer thickness to reduce the driving voltage, and redesigning the DEA geometry and robot wings to reduce the flapping frequency. First, increasing electrode conductivity will lead to a reduction of resistive power loss. This can be done by exploring new electrode materials such as a hybrid network of carbon nanotubes, graphene and silver nanowires 37 or intrinsically stretchable electrodes such as conductive hydrogels 38 or liquid metal. Second, reducing elastomer thickness will reduce the operating voltage. We can achieve this by increasing the spin-coating speed or exploring alternative method such as using an automatic thin film applicator. Further, the spin coating and the electrode transfer process can be done in a clean room environment to reduce the number of particulates in the elastomer and on the electrodes. Third, new electroactive materials such as bottlebrush elastomers 39 can be explored to further increase the actuator's energy density. In addition, our experiments show that DEA power consumption is linearly proportional to its operating frequency. To reduce power expenditure, future studies can redesign the DEA geometry and robot transmission to reduce system resonant frequency. Alternatively, nonlinear controllers can be developed so that the DEA motion does not need to be linearized around its resonance. Beyond improving the DEAs, we can apply a new class of electrostatic actuators named Peano-HASEL 33,40 that have shown promise for achieving very high energy density and moderate bandwidth. For that class of actuators, it would be important to work on device miniaturization to reduce the driving voltage.