High‐Speed Rotary Motor for Multidomain Operations Driven by Resonant Dielectric Elastomer Actuators

The motion of rotation is ubiquitous in daily life and industry. Compared with electromagnet motors, smart‐material‐based rotary motors may exhibit more flexibility, robustness, and adaptability. Herein, a lightweight, compact dielectric elastomer actuator (DEA) rotary motor with remarkable performance is reported. The motor is driven by a multilayered, slightly‐prestretched (<10%) dielectric elastomer (DE) composite membrane which consists of low‐loss silicone as the DE layers and single‐walled carbon nanotube as electrodes. When operating at frequencies around its resonance, it achieves a maximum rotating speed of 2,850 rpm (to the best of the authors' knowledge, the fastest among all reported DEA rotary motors so far), torque of 0.655 mN m, and power of 34.7 mW with optimized transmission parameters. To demonstrate the motor's practical application, an off‐the‐shelf air vehicle propeller is driven by the motor, and a lift of 26 mN (1.54 times the DEA membrane's own weight) is obtained. An underwater robot driven by a water propeller is designed and a high speed of 175 mm s−1 (1.58 bl s−1) is obtained. The DEA rotary motor of this work paves a new way for driving agile robots in multidomain operations.


Introduction
The motion of rotation is ubiquitous in daily life and industry, such as fans, pumps, propellers, wheels, robotic arm joints, and many household appliances.While electromagnetic motors have been the most widely adopted devices to generate rotational motions and torques, they have limitations, [1] such as the large weight due to necessary metal components (e.g., copper wire windings, permanent magnets or electromagnets, and the iron cores), inadequate robustness to collisions, limited environmental adaptability (e.g., waterproof sealing is essential for electric motor's underwater usages due to the deadly corrosion to the key materials and water's conductivity [2] ), and cooling considerations.These characteristics make electromagnetic motors less than the first choice for building multidomain autonomous systems such as agile robots. [3]In recent years, compliant actuators constructed from smart flexible, and soft materials have attracted increasing attention, for they exhibit better flexibility, robustness, adaptability, and overloading resistance, [4,5] especially for usages in confined spaces and extreme environments. [6,7]Various types of smart materials for generating rotational motions and torques have been proposed, such as fluidic actuators, [8,9] shape memory alloys (SMAs), [10,11] liquid crystal elastomer actuators, [12] and dielectric elastomer actuators (DEAs). [13]Among them, DEAs stand out most for their electrically controlled motions (albeit usually at high voltage), fast response [14] and high energy/power density, [15] as well as high efficiency.Therefore, DEA is an ideal candidate for high-performance rotary motors.
Different from more commonly used DEA configurations for generating linear [16][17][18] or bending [19][20][21] motions, which have been successfully applied in developing various types of multidomain robots including walkers, [22,23] climbers, [24] swimmers, [25] air vehicles, [26] and robots for navigating in thin tubes, [27] the rotary configurations for DEAs are far from being developed or adopted.Challenges associated with rotary DEAs include mechanism design for generating full (360°) rotation, developing simple and efficient transmission structures, and most crucially, producing desired output speeds and torques.A few groundbreaking rotary configurations based on DEAs were proposed to address the above issues.Kornbluh et al. [28] designed a rotary motor from rolled DEAs and a one-way clutch.Similarly, Jung et al. [29] utilized stacked DEA and ratchet gear to mimic the human elbow joint and muscles.Rémi Waché et al. [30] introduced a torsional DEA where the elastomeric film could generate a rotational angle of 10°without any transmission mechanism.Though promising, none of the work above achieved full, DOI: 10.1002/aisy.202300243 The motion of rotation is ubiquitous in daily life and industry.Compared with electromagnet motors, smart-material-based rotary motors may exhibit more flexibility, robustness, and adaptability.Herein, a lightweight, compact dielectric elastomer actuator (DEA) rotary motor with remarkable performance is reported.The motor is driven by a multilayered, slightly-prestretched (<10%) dielectric elastomer (DE) composite membrane which consists of low-loss silicone as the DE layers and single-walled carbon nanotube as electrodes.When operating at frequencies around its resonance, it achieves a maximum rotating speed of 2,850 rpm (to the best of the authors' knowledge, the fastest among all reported DEA rotary motors so far), torque of 0.655 mN m, and power of 34.7 mW with optimized transmission parameters.To demonstrate the motor's practical application, an off-the-shelf air vehicle propeller is driven by the motor, and a lift of 26 mN (1.54 times the DEA membrane's own weight) is obtained.An underwater robot driven by a water propeller is designed and a high speed of 175 mm s À1 (1.58 bl s À1 ) is obtained.The DEA rotary motor of this work paves a new way for driving agile robots in multidomain operations.continuous rotational motions.A landmark configuration was proposed by Anderson et al. [31] as a membrane DEA with multiple sequentially activated sector electrodes.This configuration had the advantage of immunity to an intense magnetic field, compactness, and large output torque density for full and continuous rotation.Building on this work, researchers further developed rotary DEA motors that could self-commutate, [32] withstand large deformation, [33] and achieve untethered operation. [34]owever, the widely-used acrylic material as DEA's dielectric layer limited these motors' output speed to only a few hundred revolutions per minute (rpm), [35] far lower than that of an electromagnetic motor.Rosset and Shea [36] built a smaller version of the rotary motor combining silicone dielectric elastomer and pad-printed electrodes.An exceptional rotational speed of 1,500 rpm was achieved thanks to the low mechanical loss of silicone elastomers, yet the friction transmission adopted in this DEA motor, though lightweight and easy to manufacture, may lead to poor output torque due to slippage.A rolling robot that can roll along a track was developed upon the above DEA motor and achieved a speed of up to 15 cm s À1 and a prefailure distance of 25.8 km.In such a demonstration, the motor was seen to output more pure rotational motion rather than torque and power.Designing DEA rotary motors that combine ideal output characteristics of both rotary motion and torque, robust and lightweight transmission, and adaptability to various domains still remains a key challenge.
In this work, we report a lightweight, compact DEA rotary motor with remarkable performance in both its output speed and torque and at the same time can drive off-the-shelf propellers both in the air and underwater.The motor is driven by a multilayered, slightly-prestretched (<10%) DE composite membrane running at frequencies around its resonance, consisting of low-loss silicone as the dielectric elastomer layers and single-walled carbon nanotube (SWCNT) as electrodes.Besides, smart composite microstructure (SCM) technology [37] is adopted to precisely fabricate the transmission gears (module of gear %300 μm, weight <100 mg), achieving stable and precise transmission of motion and torques at high speed.By theoretical analysis and experimental testing with multiple setups, we systematically investigated the output characteristics of the DEA rotary motor with respect to the driving voltages, waveforms, frequencies, as well as transmission ratios, achieving a maximum rotating speed of 2,850 rpm (to the best of our knowledge, the fastest among all reported DEA rotary motors so far), torque of 0.655 mN m, and power of 34.7 mW (power density %20.5 W kg À1 if only considering the weight of the DEA, efficiency 1.7%) with optimized transmission parameters.To demonstrate the motor's practical application, an off-the-shelf air vehicle propeller was driven by the motor, and a thrust of 26 mN (1.54 times the DEA membrane's own weight) was obtained.Finally, to demonstrate the motor's environmental adaptability, we designed an underwater robot driven by a water propeller.Without any further waterproofing to the DEA membrane, the motor could work in water and drive the robot to reach a high speed of 175 mm s À1 (1.58 body length/second).The DEA rotary motor in our work has paved a new way for driving agile robots in multidomain operations.

Design of the DEA Rotary Motor
The DEA rotary motor adopted the membrane actuation and center transmission configuration similar to the work presented by Anderson et al. [38] The motor mainly consisted of a multilayered DEA membrane slightly prestretched and fixed on a rigid ring frame (as shown in Figure 1A, diameter of the membrane %66 mm), a transmission gear set made from lightweight carbon fibers, an output shaft, and other auxiliary structures, weighing 15.9 g in total (DEA membrane 1.69 g).Its assembly diagram is shown in Figure 1B.When the output shaft was connected to a rotary device such as a drone propeller, as shown in Figure 1C, the rotary motor drove the propeller to rotate and resist the force from the surrounding fluids, therefore generating thrust force.So is the situation in the water, as shown in Figure 1D, where the DEA rotary motor is driving an underwater vehicle to move forward in water through a water propeller.
The basic driving principle is shown in Figure 1E.Four sections of sectored electrodes were actuated sequentially with a phase difference of 90°, generating deformation and pushing the center part of the DEA membrane to have a two-dimensional translation (no rotation).Then the orbit gear fixed at the center of the DEA membrane transformed this motion into the rotation of the rotor gear which was meshed with the orbit gear and the output shaft.In this design, we chose the four-phase DEA configure instead of three as it exhibits superior performance in output power. [38]Compared with crank-rotor transmission, using meshed gears as the transmission led to minimum weight and inertia of the moving part, for no bearings inside the center structure was required.Besides, the gears could optimize speed and torque output as a reducer, with no need of extra external reducer, which further reduced the weight of the motor.Moreover, different gear sets could be used to achieve various transmission ratios, and thus the motor could have adjustable output characteristics to meet different application requirements.The output shaft was supported by two bearings (one on top of the membrane and the other below it).Finally, a retainer was fixed on the shaft to constrain the motion of the orbit gear on the horizontal plane and eliminate its undesired wobble induced by DEA membrane's out-of-plane vibrations. [39]ased on the transmission design, we could analyze the output speed and torque of the motor.The transmission ratio was where z 1 and z 2 are tooth numbers of the orbit gear and the rotor gear, respectively.The minus sign indicates opposite motion directions between gears.Thus the output rotating speed in rpm could be calculated as where f is the driving frequency of the DEA.The output torque can also be calculated analytically based on the meshing of the gears and the output forces of the DEA.Due to the relatively complex geometries of the sectored electrodes, we conducted finite element analysis in COMSOL and simulated the motion and force output of the DEA upon applied voltages (details in Experimental Section).The relationship among driving voltage, free radial deformation, and radial blocked force with a single activated electrode was obtained (Figure S1, Supporting Information).Combining the simulation results and the transmission principle, we could calculate the stall torque under quasi-static conditions as where m and α are the module and working pressure angle of the gears, F max and s max are the maximum radial blocked force and center's free radial displacement of the DEA membrane, and θ is the angle between the direction of the DEA output force and the direction from meshing point to the rotor gear center (Figure S2, Supporting Information).Details for the calculation process are in Supporting Information.Equation (2) indicates that simply by adjusting gear tooth numbers and driving frequency, we could obtain a desired output speed, and Equation (3) indicates that by increasing DEA deformation and force output, adopting a larger rotor gear, or reducing tooth number difference, we could increase the torque output.The above equations describe the ideal static performance (i.e., rotational speed and stall torque) of the motor and can help us to determine the optimal design parameters that fit specific application requirements.

Structure and Fabrication of the DEA Membrane and the Gears
A previous study has demonstrated that stacking DEAs can amplify the torque and power of a DEA rotary motor.Here, instead of using a stack of separated DEA monolayers, [31] we have adopted the configuration of a single membrane consisting of multilayered DEA composites, to save space and reduce the connecting components.The DEA membrane was made with eight layers of 50 μm thick silicone elastomer and seven layers of electrodes (Experimental Section), each between two elastomer layers.Each of the four electrode sections had two separate connectors for positive and negative inputs from the power supply (Figure 2A and S3, Supporting Information).Figure 2B shows the cross-section of DEA membrane.The top and bottom elastomer layers served as the protecting layers, conserving the electrodes during the self-cleaning process, and improving DEA electric strength. [40]The method of tape casting was adopted to fabricate the stacked membrane.To prevent buckling in the idle part of the membrane during actuation, the membrane was prestretched to 110% of its original diameter, at 66 mm.This 10% prestrain exceeds the maximum strain induced by voltage during the actuation of the membrane.
The design parameters of transmission gears were determined from DEA's characteristics and the transmission model derived earlier.To ensure the gears stay correctly meshed, the center distance of two meshing gears should not exceed DEA's maximum free radial displacement, which was approximately 0.8 mm in the quasi-static state.We set the module of the gears to be 0.3, and designed three tooth number combinations to achieve different motor output characteristics.To avoid teeth interference during internal meshing, the minimum tooth number difference was set to three and tooth profile modifications were applied.The three transmission gear sets had transmission ratio i = 4, 6, and 10.The main design parameters of the gear sets are summarized in Table 1, and more modification parameters are presented in Table S1, Supporting Information.Ideally, a DEA radial displacement larger than meshed gears center distance should ensure the gears stay correctly meshed.As the gear tooth profile was at a submillimeter scale, SCM technology was used to manufacture the gears.Multiple carbon fiber composite sheets and adhesive sheets were precisely laser-cut into the designed shape and hot pressed into a single piece of gear, with a thickness of 0.6 mm (rotor gear) or 1.2 mm (orbit gear), as illustrated in Figure 2C.As a result, the gears realized smooth meshing, with the weight of the largest orbit gear and rotor gear at merely 91 mg and 35 mg, respectively.The three sets of gears are displayed in Figure 2D, with a zoomed-in view of the rotor gear teeth.Details of the fabrication process can be found in the Experimental Section.

Characterization of DEA Membrane
Before testing the assembled rotary motor, we first conducted experiments on the characteristics of the DEA membrane, including both static and dynamic tests.
The experiment setups are shown in Figure 3A.Under static conditions, we measured the output free radial displacement and radial blocked force of the center part on the DEA membrane, when a single sector of the electrode was activated.As displayed in Figure 3A upper left, a KEYENCE LK-H050 laser displacement sensor was used to measure the displacement of the DEA center part.Figure 3B shows the relationship between the free displacement and driving voltage, comparing simulation and experiment results.The measured free displacement showed a quadratic relationship with the voltage, reaching 0.78 mm at 2,000 V.There was a 10% difference between experiment and simulation results at voltages over 1,000 V which may be due to the hyperelasticity of the elastomer, yet the tendencies and quadratic relationship of both results were consistent.At 2,000 V, an electrode section could generate a blocked force of 0.35 N. A quadratic relationship was also observed from the results.Here, the experiment was highly consistent with the simulation result.Under dynamic working conditions, we drove the membrane's all four electrodes, and each pair of nearby electrodes had a phase difference of 90°.Here, we used LabVIEW to generate the four-channel control signals, which were fed into four Trek 2220 high-voltage amplifiers through a NI USB-6363 (shown in Figure S4, Supporting Information).The free displacement here was defined as the displacement of a point on the center frame in horizontal (x) direction, and was measured with laser sensor, as shown at bottom of Figure 3A.We investigated the relationship between this free displacement and the driving frequency.As shown in Figure 3D, a frequency sweep was conducted in the range of 5 to 230 Hz, at the interval of 5 Hz.The driving signals were in the form of sinusoidal wave and square wave of 50% duty cycle, both with an amplitude of 800 V and a bias of 800 V, or U = 1,600 V according to the driving voltage signal, as shown in Figure 3D.The data showed that the DEA resonated at 160 Hz, with the displacement reaching 3.90 mm for square wave and 3.18 mm for sinusoidal wave.It could be seen that the square wave had a larger displacement than sinusoidal wave throughout the frequency range.At around 55 Hz, a peak of displacement appeared when the DEA was under square wave actuation, but not under sinusoidal wave.This will be further investigated in the following sections, by studying the trajectories of DEA's center part.
We further measured the input voltage and current curves when the membrane was actuated at high frequencies.Here, a Trek 615-3 high-voltage amplifier was used to generate the driving signal.The voltage and current were measured using the contained output monitor functions of the amplifier.During the test, an electrode section was driven at 100 Hz and 1,600 V (with an amplitude of 800 V and a bias of 800 V).The center part of the DEA was set free. Figure 3E shows the curves of input voltage and current under the square wave.At the rising edge of the square wave, the current rapidly rose to a peak of 1.5 mA, and then gradually decreased to zero.The rise time was far less than 1 ms, indicating a negligible electrical delay of the system.Figure 3F shows the input voltage and current of the DEA under the sinusoidal wave.The current was in a sinusoidal form as well, but it had a phase lead of 61°compared to the input voltage, caused by the typical RC circuit characteristics.The capacitance of a single electrode section on a DEA is 2.6 nF, and its internal serial resistance is 120 kΩ.In this way, the input power of the motor from the power supply at various driving conditions could be calculated from these measurements.
We then studied the full 2D trajectories of DEA membrane center part.The experiment setup was the same as the dynamic displacement measurement (Figure 3A bottom), but utilizing two laser sensors altogether to monitor the center displacements in two orthogonal directions (x direction and y direction).To suppress any undesired out-of-plane vibrations around the resonate frequency, the motion of the DEA center part in its z direction was constrained, in the same way as the DEA motor after assembly (Figure 1B). Figure 4A shows the measured trajectories of the DEA center part under 10, 120, 160, and 180 Hz sinusoidal wave actuation at 1,600 V. Multiple cycles of trajectories were recorded, exhibiting remarkable consistency.The trajectory was roughly circular at low frequency, and became more circular as frequency increased.The dimension of the trajectories peaked at a resonate frequency of around 160 Hz, with 3.18 mm in its x direction.In Figure 4B and Movie S1, Supporting Information, we made the trajectories directly visible with a tiny LED attached to the DEA center part.The circular shapes and changes in sizes under various frequencies were clearly demonstrated in the pictures and video.However, to ensure better visibility of the trajectories, the motion in z-direction was not constrained.As a result, the pattern at resonate frequency (160 Hz) was distorted by the out-of-plane vibrations, although its size was still considerably enlarged, showing the influence of resonance.Figure 4C shows the recorded trajectories under square wave actuation, as well as video captures and the simulation results.The bars in video captures indicate a length of 1 mm.Under different driving frequencies, the trajectories varied both in shape and in size.At 10 Hz, the inertia of the DEA membrane and center part led to overshoot under step excitation, i.e., at the rising and falling edges of the square wave.As the frequency increased, the overshoot and shake appeared not only along, but also perpendicular to the main motion direction at 30 Hz.At 55 Hz, the trajectory changed into a pattern similar to four rings combined.The overshoot caused by the impulse of square wave led to a resonation at around 55 Hz, therefore, the first peak of maximum displacement under square wave actuation appeared in Figure 3D.As there was no impulse under sinusoidal wave actuation, only one peak existed in Figure 3D for sinusoidal wave.Despite relatively high maximum displacement around the first peak, the overall displacement (or mean displacement) of the trajectory is not always large under these driving conditions.As a result, the driving ability of the DEA around the first peak would not be as ideal as the maximum displacement seemed.Then, as the driving frequency increased, the trajectory shifted into a square at around 120 Hz, and finally developed into a circular shape from 140 Hz onward.Movie S2, Supporting Information, shows the same trajectory evolution recorded by a camera.Since the effective transmission requires constant meshing of two gears, a circular trajectory with larger diameter is preferable, and inadequate displacement and complex trajectories at low frequency would lead to degradation in performance.
To better understand the complex motion behaviors, two models were built to explain the effects of both types of actuation waveform.As for sinusoidal wave at low frequency, based on the results under static conditions, the blocked force can be considered to be quadratic of the driving voltage, and the direction is along the electrode central line.The driving voltage of each electrode section is where i, U 0, and t represent electrode section numbers in clockwise order from 1 to 4, maximum voltage and time, respectively.Thus the magnitudes of blocked forces can be expressed as where c F is a constant coefficient in the force-voltage relationship.As a result, the total blocked force is which indicates that the total blocked force is a rotating force with a constant magnitude and angular speed.As the center displacement is proportional to DEA output force if linear material is assumed, the motion trajectory under low-frequency sinusoidal wave should be circular.At high frequencies, the damping and inertia might affect the size of trajectory, but the shape is always circular.
As for square wave actuation, we assumed a mass-springdamper model to describe the system.The DEA center part is represented by the mass, and it is connected with two sets of spring and damper, one in x direction and the other in y direction.The motions in x and y directions are independent.The system's equations of motion are where m, F x , F y, b, and k are the mass, input forces in x, y direction, damper dissipation coefficient, and spring stiffness, respectively.We used the step response data of one sector of the actuator to fit the coefficients such as m, b, and k.Then we used MATLAB to simulate the motion of the system under square wave actuation, in the same way as we actuated the four electrode sectors.The simulation results are displayed in Figure 4C, which are very consistent with experiment data in aspects of driving frequency, shape, and dimension, meaning that this simple model is accurate in describing-or even predicting-the performance of an actuator.

Maximum Speed and Stall Torque of the Assembled DEA Rotary Motor
We assembled the motor and tested its driving characteristics.First, the rotating speed without any load was measured to verify the DEA's actuation ability and transmission design.The motor was driven under various signal inputs, and with different transmission setups as well.A laser tachometer (VICTOR 6234P) was used to measure the rotational speed.As it offered noncontact measuring, no extra load was applied to the motor.
Figure 5A shows the output speed of the motor under square wave signals and with a transmission ratio of 10 (gear set iii in Table 1).The yellow dashed line indicated the target speed, which was calculated from Equation (2).According to the test result, the motor was able to match the target speed under relatively high voltage and at a certain frequency range.Here we define the frequency range where the motor reached its theoretical target speed as "fully meshing band", meaning the transmission gears meshed completely.When the driving voltage was 1,200 V, the motor was unable to reach target speed across all tested frequencies.At 1,400 V, a fully meshing bandwidth of 45 Hz was achieved.With increasing driving voltages, the fully meshing bandwidth expanded in both directions, leading to a broader ideal operating range.At 1,800 V, the motor outputs a maximum speed of 1,290 revolutions per minute with a fully meshing bandwidth of 100 Hz wide.The center of the fully meshing band was almost at the resonant frequency of the DEA membrane (Figure 3D), where the free displacement was remarkably amplified.Outside the fully meshing bandwidth, the gears could not mesh as designed, mainly because of insufficient DEA displacement.Therefore, the output speed was compromised.Interestingly, a number of experiment data appeared at exactly 2/3 of the target speed (indicated by the brown dashed line in Figure 5A,B).As the tooth number difference of the gear set was three, the 2/3 speed output might be associated with periodic inadequate meshing of skipping one tooth out of three per cycle.This phenomenon will be discussed further in a later section.When a sinusoidal driving signal was applied, we could get similar results as displayed in Figure 5B.The top speed of the motor was 1,260 rpm at 1,800 V and 210 Hz.However, due to lower DEA displacement output compared with square waveform (Figure 3D), the motor had a narrower fully meshing bandwidth, especially at 1,400 V.However, the center of the fully meshing bandwidth was still close to the resonant frequency.
Besides, the motor tended to produce lower speed under insufficient meshing conditions, which was most evident when comparing speeds with square wave under 1,200 V actuation.
The investigation of transmission gear sets was conducted as well.As shown in Figure 5C, the motor speeds with three transmission ratios i = 4, 6, and 10 were tested in the motor under 1,800 V square wave actuation.The yellow dashed lines indicated the target speed of each gear set.With the increase of the transmission ratio, the output speed decreased accordingly, while the fully meshing bandwidth expanded from 55 to 100 Hz.The motor reached a maximum speed output of 2,850 rpm with i = 4 at 190 Hz actuation (Movie S3, Supporting Information), and speeds of 2,100 and 1,290 rpm were attained with i = 6 and 10 sets. Figure 5D contains video captures of the slowmotion video in which the motor produced 2,850 rpm rotating speed.The video was captured with a Phantom VEO high-speed camera.At 2,850 rpm, the shaft rotated a full revolution in 21 ms.Here one-third of a revolution was shown.
The speed tests with no load demonstrated the effectiveness of the motor design.The transmission gear setups were verified, achieving a wide output speed range from hundreds to nearly 3,000 rpm.To understand its torque characteristics, we further conducted motor stall torque tests, measuring the maximum torque at different driving conditions.The experiment setup is illustrated in Figure 6A.When the motor was able to lift a certain amount of weight, we could calculate the torque output from the arm length and the weight.The motor was placed horizontally, driving the output shaft in vertical direction.A reel was mounted on the shaft and was connected to the weight with thin cable.A pulley was used to change the force direction from horizontal to vertical.Thus, the motor's torque equals the product of the gravity of the weight and reel radius.The gravity of the thin cable was ignored, as the cable of 0.053 mm diameter had minor influence on the total force.Figure 6B displays the relationship of the torque of the motor over the actuation frequency, under various driving voltages and two different waveforms.The transmission ratio of i = 10 was used during this test.The stall torque was defined by the gravity of the maximum weight the motor could lift.The experiment results indicated that the motor always produced a peak stall torque at 160 Hz whatever the voltage or the waveform was.Increasing the driving voltage and applying square wave actuation increased the stall torque.Peak stall torques of 0.655, 0.443, and 0.213 mN m were observed under 1,800, 1,600, and 1,400 V square wave actuation, respectively.By comparison, sinusoidal actuation signal generated smaller peak torque outputs as 0.370, 0.236, and 0.139 mN m under identical voltage magnitudes.The trend of the stall torque over frequency was similar to the results of speed test.For example, the motor exhibited an elevation in both its free speed and stall torque within the 130 to 210 Hz frequency range and reached the maximum value at a frequency close to the resonant frequency of the DEA membrane.However, the enhanced actuation ability of the square wave appeared more evident in torque test than in speed test.This might indicate that at insufficient meshing conditions, especially under heavy load, square waveform had greater actuation advantage over sinusoidal waveform.
Then, we tested the stall torque output of different gear sets.According to Equation (3), the torque was closely related to the size of the rotor gear.So among the three gear sets, higher transmission ratio represented larger rotor gear diameter, thus leading to higher torque output.Figure 6C shows the torque output with three gear sets.The curves had a similar tendency of increasing first then decreasing and peaking at 160 Hz.The gear set of i = 10 produced significantly larger torque as expected.Besides a 0.655 mN m peak torque output, the i = 10 gear set could maintain at least 80% of peak output within 130 to 195 Hz frequency range.In comparison, with i = 6 and 4 gear sets, the motor could produce 0.303 and 0.200 mN m peak output, respectively, merely a half and one-third of i = 10 gear set.Combining the results above and the speed test results in Figure 5C, the i = 10 gear set demonstrated superior transmission efficiency and therefore could potentially generate more output power.
We compared the no-load speed and stall torque density (against mass) of our motor with the previously mentioned DEA rotary motors in Figure 6D.Even though our stall torque density did not exceed a few other DEA rotary motors made from VHB, the speed of our DEA motor was, to the best of our knowledge, the fastest so far.6,38,44] Note: *The torque density in work [36] is estimated according to the article and video.

Torque-Speed Characteristics of the DEA Rotary Motor
The torque-speed characteristic is important for evaluating and using a motor.When the motor was driven at a specific voltage and frequency, the external load will affect its rotating speed.At maximum load condition, the output speed was close to zero, while reducing the load could accelerate the rotating speed.To identify the motor's torque-speed characteristics and maximum power output, we fixed the actuation condition as 1,800 V square wave at 180 Hz with i = 10 transmission, which displayed the largest product of no-load speed and stall torque.By changing the load applied to the motor, we measured the output speed and the output torque.The result is shown in Figure 7A, where the blue curve represents output torque at various speeds, and the yellow curve is the calculated power of the motor.At a target speed of 1,080 rpm, which is shown with the dashed line in the graph, the motor could generate a torque of 0.296 mN m.Therefore, for any load less than that, the motor could rotate at its theoretical speed, maintaining a fully meshing state for the gears (Movie S4, Supporting Information).Figure 7B shows the video captures of the meshing gears at perfectly meshing condition.During this 2 ms, the orbit gear was pushed by the DEA to translate to the left, driving the rotor gear to rotate clockwise in the process.All the gears meshed correctly and thoroughly in the meantime, and as a result, the motor rotated at target speed of 1,080 rpm.
If the load (or torque) exceeded the threshold value for target speed (e.g., 0.296 mN m in this case in Figure 7A), however, the output speed would be reduced.In other words, to generate sufficient torque for the increasing load, the motor had to operate at a slower speed.Interestingly though, at around 720 rpm-twothirds of the target speed-the output torque of the motor rose unusually rapidly with a slight decrease in speed, compared with the less steep curves at both sides.We further investigated the meshing states in this situation.As shown in Figure 7C, at a torque output of 0.40 mN m, the gears could not mesh correctly.Instead, slippage occurred between contacting teeth, which is shown in the zoomed-in views in Figure 7C.However, the rotor gear could still rotate at a constant speed even though the slippage happened.Part 2 in Movie S4, Supporting Information, shows the relatively steady state of transmission at 720 rpm speed.Thus, this unexpected transmission state boosted the motor's output by a fraction.The rotor was pushed forward by two teeth (three teeth for fully meshing state) in every actuation cycle and the speed was stable in such state.The phenomenon also explained the 2/3 output speed tendency that happened in no load speed tests at insufficient actuation state (Figure 5).As for the meshing condition in other parts of the curve, the third part in Movie S4, Supporting Information, showed inadequate meshing state at 0.32 mN m torque.The gears meshed correctly for most of the time.However, occasionally slippage happened and a tooth in the rotor gear was skipped.This reduced the speed to 1,041 rpm, 3.6% off the target speed.As the load increased, slippage appeared more frequently, until a new steady state at 2/3 speed was established.When the load exceeded 0.45 mN m, more slippage arose inevitably, further reducing the speed.
The output power of the motor is calculated as the yellow curve in Figure 7A.The maximum power of 34.7 mW was generated at 0.32 mN m load and 1,041 rpm.When only considering the weight of the DEA membrane, the power density is 20.5 W kg À1 .Another power peak was at 2/3 speed zone, where it reached 30.0 to 34.3 mW.However, considering the system stability, speed uniformity, and gear abrasive wear, the motor is preferred to operate at fully meshing state.We also calculated the power efficiency of the current motor by dividing the input power from the power source and the result is approximately 1.7%.
To practically demonstrate our rotary DEA motor's capability to output power, we installed an off-the-shelf drone propeller (Mavic Air 2 7238 propeller, from DJI) on the motor's shaft (Figure 7D) and drove the propeller with a driving voltage of 1,800 V. We measured its lift from the propeller through a scale.From Movie S5, Supporting Information, we could see that, as the driving frequency gradually increased from 120 to 185 Hz, the lift generated by the device grew to 26 mN (2.6 g force) in 20 s of continuous driving, exceeding the gravity of the DEA membrane (1.69g) itself, while the rotor operated at 810 rpm.To eliminate the ground effect of the rotor, we lifted the whole device away from the scale, so that the distance from the propeller to the scale was more than 1.5 times the propeller's diameter.According to the experiment in Sanchez-Cuevas et al.'s work, [41] for single rotor system, at such distance the influence of ground effect was less than 5% of the total thrust.
As the lifetime is an important aspect of the DEA motor's performance, we carried out an endurance test, where the DEA motor operated continuously for over 100 min and 1 million cycles (which is ideally 100 000 revolutions at full meshing, see Movie S6, Supporting Information).Here, the motor was driven under square wave at 1,600 V and 175 Hz, and with a transmission ratio i = 10.By monitoring its charging current, we found that after a million working cycles, its capacitance dropped by 30% (Figure S5, Supporting Information).This drop in capacitance would lead to reduced performance, as investigated by Jiang et al. [40] The result indicated that the actuator can work for longer without electric breakdown but at the cost of performance.

Design and Test of an Underwater Robot Driven by the DEA Motor
To further demonstrate the motor's driving ability and environmental adaptability, we designed an underwater robot driven by the DEA rotary motor and a water propeller.Compared with conventional electromagnetic motors which have to be sealed underwater, our DEA motor requires minimum waterproof measures, thanks to the top and bottom protective elastomer layers in the stacked DEA.Only the connectors of the cables were sealed with Smooth-On Sil-Poxy adhesive.The robot is shown in Figure 8A and S6, Supporting Information.The DEA motor was placed horizontally, and bevel gears were used to transmit the vertical rotation into a horizontal one.This design of orienting the DEA motor horizontally significantly reduced the drag force from water during forward motion, compared with direct drive with the motor being vertically mounted, the largest surface facing the front.The robot had a floating body to balance its buoyancy and weight.By driving the propeller at the rear, the robot was able to go forward and backward underwater.The backward motion was achieved simply by switching the driving sequence of the DEA membrane's electrodes.The DEA rotary motor was fully submerged in water, demonstrating excellent domain adaptability and robustness.
As the density of water is considerably larger than air, the motion of DEA center part would be affected by water resistance under high-frequency actuation.For underwater experiments, we drove the motor with 1,500 V square wave at 120 Hz to reach a balance between high performance and low chance of electric breakdown.To optimize the robot's speed, the thrust forces with different propellers (from TFL Hobby) were characterized.The experiment setup is displayed in Figure 8B, where the robot pulled the cable which was connected to a force gauge.The propeller configures and test results are shown in Figure 8C.As the propeller dimensions increased, the rotating speed of the shaft decreased due to larger resistance.The thrust peaked at 29.9 mN for the propeller of 48 mm diameter, with an operating speed of 286 rpm.So the propeller setup C was chosen.The propulsion speed test was then conducted.The experiment setup is displayed in Figure S7, Supporting Information.As shown in Figure 8D, the robot achieved a maximum speed of 175 mm s À1 (1.58 bl s À1 ), which was relatively fast compared with similar DEA-driven underwater robots (Table 2).Besides, the robot could reverse at a speed of 75 mm s À1 .This underwater robot's good locomotion ability is due to not only the DEA motor's relatively high power output, but also the advantage of rotary motion in propelling in water.The environmental adaptability of the robot is attributed to DEA's own capability of surviving and functioning well in water, and even the deep sea. [42]However, the performance of the robot could still be further improved.As there is only a single propeller, the random reactive force from the surroundings would make the robot tilt to its left side, as shown in Movie S7, Supporting Information.Although adding additional counterweight could alleviate the problem to some extent, equipping two or more propellers with different rotation directions might be a better option.

Conclusion
In this study, we developed a high-performance rotary motor based on a multilayered DEA membrane.By applying low-loss DE materials that could work at hundreds of hertz and highprecision and lightweight transmission gears fabricated with SCM technology, the rotary motor obtained a maximum rotating speed of up to 2850 rpm and a maximum stall torque of 0.655 mN m with optimized gear sets.The peak power output was 34.7 mW.We systematically characterized the speed-torque curve for our motor and analyzed the special phenomena such as fully meshing and transmission with slippage.
To demonstrate the motor's driving capability and environmental adaptability, we used the motor to drive propellers both in the air and underwater.An aquatic locomotion robot was designed and achieved 175 mm s À1 (1.58 bl s À1 ) velocity, while the unprotected DEA membrane surface was completely submerged in water.Compared with other DEA rotary actuators, our motor exhibited both excellent speed and high torque output.Besides, the different transmission setups expanded the operation range of the motor, providing more flexibility for future applications.Similarly, our underwater locomotion robot outpaced other underwater robots actuated by DEA, demonstrating the advantages of rotation actuation driven by artificial muscles.
The potential future applications of the DEA motor include miniature robots that require actuators with a very thin profile, and deep sea explorations where actuators that are free from waterproof and pressure hull exhibit advantages over traditional electromagnetic motors.However, along with those very promising results, there still remain several aspects that could be further optimized for the DEA rotary motor.First, the transmission gears are not able to maintain ideal fully meshing conditions across the whole functioning frequency range.The actuation performance of DEAs should be further improved, and a new mechanism is required to keep the gears fully meshing in all situations.Next, the DEA motor requires high voltage power supply and portable high-voltage power supply and control circuits were essential to develop an untethered robot driven by DEA motors. [43]Finally, the overall power efficiency of the DEA motor was merely 1.7% at maximum output power conditions.To substantially increase the efficiency, we should turn to softer elastomer material to generate larger strains and energy recovery methods during DEA charging and discharging processes.Anyway, thanks to the robustness and environmental adaptability of DEA, the motor's future usage in extreme environments and  Shintake et al. [45] 37.2 0.25 5 kV, 0.75 Hz Berlinger et al. [20] 55 0.55 2 kV, 2.2 Hz Wang et al. [46] 76.7 0.77 4.8 kV, 1.5 Hz Li et al. [25] 135 1.45 10 kV, 5 Hz (tethered) 64 0.69 9.5 kV, 5 Hz (untethered) Li et al. [42] 38.9 0.34 8 kV, 1 Hz Tang et al. [47] 33 0.

Experimental Section
Finite Element Analysis of the DEA Membrane's Static Performance: We conducted finite element analysis to study the characteristics of DEA.COMSOL multiphysics was used for the electromechanical analysis.We built the DEA model according to the design and real actuator, including the material properties, stacked structure, and prestretch of the membrane.Figure S1A, Supporting Information, displays the strain ε zz of the DEA after prestretch and under 2,000 V actuation voltage.The activated electrode section in the left part is compressed in z-direction, with a strain of around À0.2.Meanwhile, at the opposite side to the activated area, ε zz remains negative, indicating that the membrane stays in a stretched state and no buckle happens to compromise the movement of the DEA center part.There are certain areas that have positive ε zz at the contact interface of the membrane and the center part (rigid).However, the positive strain in this small area has negligible influence on the principle deformation of the DEA.The red arrow in Figure S1A, Supporting Information, shows the displacement field of DEA membrane under 2,000 V actuation voltage compared with zero input.The activated area expands in xy plane and pushes the center part to the direction of increasing x.The results validate the DEA design and prestretch treatment.Then, we studied the free displacement of the DEA when one electrode section was driven at various voltages and its blocked force when the center displacement was constrained to zero.The results are shown in Figure S1B,C, Supporting Information, demonstrating a quadratic relationship with driving voltage.Next, to verify the linear relationship between output displacement and force under the same voltage, we calculated the force output at different displacement conditions.The results are shown in Figure S1D, Supporting Information.It can be observed that with the increase in displacement, the available output force drops in a linear relationship.Thus, the assumptions we utilized during the modeling and analysis of the transmission characteristics were verified.
DEA Materials and Fabrication: The DEA dielectric was composed of silicone Dow Corning Sylgard 184 (1:10) and Wacker Elastosil 7670 (1:1), at a mixture ratio of 1:5.We used single-walled CNT (Nanjing XFNANO Materials Tech.) as electrode material.We adopted the tape casting method with a doctor blade to fabricate the elastomer thin film.An automatic film applicator (Elcometer 4340) was used to produce thin elastomer film.The major steps include: 1) Apply a sacrificial layer of polyvinyl pyrrolidone (4.5%) in isopropanol on a 0.3 mm thick polyethylene terephthalate (PET) sheet with the doctor blade and the film applicator.The sacrificial layer eases the detaching of the fabricated DEA from the PET sheet afterward.2) Apply a 50 μm thick silicone film onto the sacrificial layer, and heat it at 75 °C for 20 min.3) Fabricate the CNT electrode.Suction filtration was used to extract the CNTs from the solution onto a nylon filter membrane.The resulting areal density of the CNTs on the filter membrane was 2.9 Â 10 À9 g mm À2 .Then, a precut mask was covered on the elastomer film.The mask was made with 25 μm thick PET film, and was hollow in the place of the electrode.With the mask, the CNTs were transferred from the filter membrane to the elastomer film in the exact shape that we designed.4) Go through multilayer fabrication process.Repeat steps 2 and 3 for another 6 times, making seven layers of elastomer and seven layers of CNT electrode with positive and negative layout staggered according to Figure S3, Supporting Information.Finally, follow step 2 and apply the 8 th elastomer layer.5) Apply further heat treatment to the stacked elastomer film, at 60 °C for 4 h.6) Cut off the DEA elastomer from the film and prestretch it to 110% of its diameter, to prevent buckle at the unactuated electrode sections.Attach the acrylic top and bottom frames to the elastomer with instant glue, then cut off the membrane outside the frame, exposing the cross-section of the electrode connectors.7) Connect the electrode and power supply cables with silver epoxy conductive adhesive (8331 from MG Chemicals) at the connectors.
Gear Fabrication: The gears were fabricated using smart composite microstructure (SCM) technology to ensure manufacturing precision and reduce weight.Multiple carbon fiber sheets of 100 μm and DuPont Pyralux FR0100 adhesive sheets of 25 μm were laser-cut into the designed patterns with 355 nm picosecond ultraviolet laser (Wuhan Hero Optoelectronics Technology Co. Ltd.), and hot pressed (190 °C, 300 kPa for 60 min) into a single part.The thickness of the orbit gear and rotor gear were 1.2 and 0.6 mm, respectively.
Driving Setup of the DEA Rotary Motor: The DEA rotary motor requires four channels of high-voltage power input up to a few hundred hertz.Four high-voltage amplifiers (Trek 2220) were used to generate four channels of driving signal.They were controlled with LabVIEW program and a multifunction I/O device (USB-6363 from National Instruments).The sensors, including the force gauge and laser displacement sensors, were connected to the USB-6363 as well.The system schematic is shown in Figure S4, Supporting Information.

Figure 1 .
Figure 1.Design, functionality, and working principle of the DEA rotary motor.A) A slightly-prestretched DEA multilayer (diameter 66 mm, thickness 330 μm after prestretch) fixed on a circular frame.B) Exploded schematic to show the motor's main components.C) The rotary DEA motor drives an offthe-shelf drone propeller.D) The motor drives a water propeller underwater.E) The driving principle of the DEA rotary motor in one actuation cycle.

Figure 2 .
Figure 2. DEA membrane and transmission gears.A) Internal structures of DEA membrane.The diagram of a single electrode section shows the stacked structure of DEA and positive/negative connectors.B) Cross-section of DEA membrane, with each layer being 50 μm thick.C) Multiple carbon fiber sheets and adhesive sheets are stacked and hot-pressed to fabricate transmission gears.D) Photograph of three sets of transmission gears.

Figure
Figure3Cshows the relationship between the blocked force and driving voltage.The test setup is shown in Figure3Aupper right.The blocked force was measured with a Transducer Techniques GSO-100 force gauge, under the condition of zero displacement.At 2,000 V, an electrode section could generate a blocked force of 0.35 N. A quadratic relationship was also observed from the results.Here, the experiment was highly consistent with the simulation result.Under dynamic working conditions, we drove the membrane's all four electrodes, and each pair of nearby electrodes had a phase difference of 90°.Here, we used LabVIEW to generate the four-channel control signals, which were fed into four Trek 2220 high-voltage amplifiers through a NI USB-6363 (shown in FigureS4, Supporting Information).The free displacement here was defined as the displacement of a point on the center frame in horizontal (x) direction, and was measured with laser sensor, as shown at bottom of Figure3A.We investigated the relationship between this free displacement and the driving frequency.As shown in Figure3D, a frequency sweep was

Figure 3 .
Figure 3. Static and dynamic characteristics of DEA.A) Experiment setups for static displacement (left), blocked force (right), and dynamic displacement and trajectory (bottom) measurements.B) Static-free displacement output of DEA with voltages.C) Static-blocked force output of DEA with voltages.D) Sweep frequency of DEA dynamic maximum displacement with U = 1,600 V. E) Input voltage and current of the DEA under square wave actuation.F) Input voltage and current of the DEA under sinusoidal wave actuation.

Figure 4 .
Figure 4. Trajectories of DEA center part during dynamic actuation.A) Trajectories under sinusoidal wave actuation.B) Video captures of DEA center part's motion with the help of an LED, under sinusoidal wave actuation.C) Trajectories under square wave actuation, compared with video captures and simulation results.The bars in video captures indicate a length of 1 mm.

Figure 5 .
Figure 5. DEA motor's rotary speed with no load.A) Speed over frequency under square wave of different voltages.B) Speed over frequency under sinusoidal wave of different voltages.C) Speed over frequency with different transmission ratios.D) Video captures of the motor rotating at 2,850 rpm.

Figure 6 .
Figure 6.Stall torque characterization.A) Experimental setup for torque measurement.B) Maximum torque output of the motor under various driving voltages over frequency.C) Maximum torque output with different gear sets over frequency.D) A comparison among DEA rotary motors.[28,[31][32][33][34]36,38,44] Note: *Thtorque density in work[36] is estimated according to the article and video.

Figure 7 .
Figure 7. Torque-speed characteristics of the motor.A) The relationship between output torque and speed of the motor, and its power output.B) The gears at correctly meshing condition.C) The gears meshed with slippage, at 2/3 target speed.D) The DEA motor driving an off-the-shelf drone propeller and the lift being measured by a table scale.

Figure 8 .
Figure 8. Underwater robot driven by the proposed DEA rotary motor and a water propeller.A) Photographs of the underwater robot.The sole waterproof measures are the sealed connectors to the powering cables.All the DEA membrane is directly exposed to water.B) Thrust measurement setup.C) The thrust and shaft speed with different propellers.D) The robot achieved 175 mm s À1 speed underwater.

Table 1 .
Design parameters of transmission gears.

Table 2 .
Summary of underwater locomotion robot driven by DEAs.