A Quad‐Unit Dielectric Elastomer Actuator for Programmable Two‐Dimensional Trajectories

Resonant actuation and multi‐degree‐of‐freedom (DoF) actuation can optimize the output matrices and greatly expand the versatility of dielectric elastomer actuators (DEAs) for practical applications. However, developing a multi‐DoF resonant DEA that benefits both from the maximized stroke outputs in resonant actuation and the versatility in multi‐DoF actuation remains a challenging task due to the slow responses of the widely adopted acrylic elastomers and the exponentially scaled complexity in the nonlinear dynamics. Herein, a multi‐DoF resonant DEA is reported by adopting four radially organized planar DE units fabricated by low‐loss silicone elastomers to output versatile programmable two‐dimensional quasi‐static and resonant trajectories. The rich nonlinear dynamics of the proposed quad‐unit DEA (QUDEA) are characterized and the critical parameters for two‐dimensional trajectory programming are uncovered through extensive modeling and experimental studies. To showcase the two‐dimensional trajectory programming capabilities of the QUDEA, demonstrations of Chinese characters writing and hummingbird flapping patterns mimicking using the proposed QUDEA are developed. The proposed QUDEA is expected to have potential in various applications such as in experimental biology, biomimetic robotics, and industrial manufacturing devices.


Introduction
Dielectric elastomer actuators (DEAs) are an emerging type of soft actuators that can convert electrical stimulations into continuous mechanical deformations via the unique electromechanical coupling mechanism.[3] Among these emerging soft actuators, DEAs exhibit the attractive features of large actuation strains, high energy and power densities, and self-sensing capabilities [4][5][6] and have been applied in numerous soft robotic applications such as an insect-like agile flapping-wing micro-aerial-vehicle, [7] a sub-centimeter pipe-inspection crawling robot, [8] and selfpowered soft robotic fish for the Mariana Trench exploration. [3]11] When stimulated by a voltage signal, an ideal DEA will experience equal biaxial expansion in plane and contraction in thickness.To convert the electroactive deformations of the DE membranes into targeted stroke outputs, many DEA configurations have been developed, such as the planar DEAs, [12,13] rolled DEAs, [14,15] DE minimum-energy structures, [16,17] and cone DEAs. [18,19]To maximize the stroke/ force output performance of the DEAs, many DEA configurations are designed to have a single-actuated degree-of-freedom (DoF) where the end-effectors of the DEAs are restricted to exert a single DoF actuation motion (note that they may have multiple passive DoFs due to the compliant nature of the materials), refer refs.[20-22] for examples.However, due to the fixed motion patterns, the DEAs with a single-actuated DoF often suffer from poor adaptability to versatile application scenarios where multiple actuated DoFs are required to fulfill different tasks.
Many researchers have exploited the inherent compliance to design DEAs with multiple actuated DoFs to generate programmable trajectories of the end-effectors in two or even three dimensions.Agonistic-antagonistic conical configuration is one of the most widely adopted DEA configurations to achieve multiple actuated DoFs, mainly due to the ease of fabrication and high stability in all DoFs.[30][31][32] Many applications based on the multi-DoF DEAs have also been developed, such as the soft printable hexapod robot, [25] a versatile soft climbing robot, [32] a biomimetic soft lens, [26] and a soft robotic liquid mixer, [27] to name a few.
Despite that the aforementioned literature has made significant advances in the multi-DoF DEA designs, these studies mainly focused on the actuation of DEAs in quasi-static to low-frequency domain (e.g., <5 Hz, while the resonant frequency can be in 10 s to 100 s Hz).This is mainly because the widely adopted acrylic elastomer (VHB, 3M) in these studies suffers from more severe viscous losses as the actuation frequency increases, [33] hence resulting in poor output power densities and low electromechanical efficiencies of these multi-DoF DEAs.Resonant actuation of the DEAs made with low-loss silicone elastomers has been extensively demonstrated to maximize the output strokes, power densities, and electromechanical efficiencies. [3,34,35]Achieved by matching the excitation voltage frequency and the natural frequency of the DEA system, the peak mass-specific power of a DEA at its resonance has been pushed to ≈2 kW kg À1 . [5]The multiple independent electrode segment concept found in multi-DoF cone and rolled DEAs has also been utilized on a resonant circular DEA to achieve high-speed/power rotary motions. [36,37]By adopting multilayer silicone membranes and single-walled carbon nanotubes, Du et al. [37] reported the fastest rotary velocities of 2850 rpm to date and substantial output power density of the DEA rotary motor.However, it should be noted that the actuated DoF of these resonant DEA motors is still limited to a single DoF rotation.Despite that the multisegment electrodes are actuated independently, the end-effector is either fixed to a central shaft or meshed gear, which restricts the resonant trajectory of the end-effector to be fixed circular.It is also worth noting that the widely adopted circular configuration with multiple electrode segments may not be readily converted to multi-DoF resonant actuation as the DE membranes in this design tend to suffer from inhomogeneous stress distributions [38] and potentially increased risks of failure in long-term operations, especially in high amplitude resonances.Furthermore, complexity in nonlinear dynamics tends to scale up exponentially against the increasing actuated DoFs in DEAs where multiple resonant modes and complex interactions in between may emerge.Therefore, designing a multi-DoF DEA that is capable of achieving rich resonant trajectories in a programmable manner to expand the versatility and adaptability of resonant DEAs remains a challenging task.
In this work, we report a multi-DoF resonant DEA by adopting four evenly organized DE units that can be independently actuated to output versatile programmable two-dimensional quasistatic and resonant trajectories.The proposed quad-unit DEA (QUDEA) features four prestretched planar DE units that ensure a homogeneous uniaxial tensile stretch in each membrane during actuation.No constraints are introduced to the end-effector, which allows it to translate in the x and y axes and rotate around z axis freely for versatile programmable output trajectories (as illustrated in Figure 1).A dynamic model is developed to characterize the responses of the QUDEA in both translational and rotational DoFs.Extensive modeling and experimental studies are performed to uncover the critical parameters for twodimensional trajectory programming.To demonstrate the feasibility of achieving programmable two-dimensional trajectories, Chinese characters writing by using the proposed QUDEA is performed and a flapping wing mechanism is developed to mimic the hummingbird flapping pattern in different forward velocity scenarios.The rest of this article is organized as follows.In Section 2, the QUDEA design is introduced and its quasi-static and dynamic responses are characterized by using a dynamic model and experiments.Two-dimensional trajectory generation characteristics of the proposed QUDEA, in both quasi-static and resonant states, are investigated using the developed system model and experiments.Demonstrations are performed to showcase the two-dimensional trajectory generation capability of the QUDEA.Finally, a conclusion is drawn in Section 3.

Design Overview and Working Principles
The proposed QUDEA design is illustrated in Figure 1a, where the outer ends of the four planar DE units are attached to the support frame, while their inner ends are connected to a moving mass.These four DE units are evenly separated by π/2 rad to form a cross pattern, as shown in Figure 1c.The working principle of the QUDEA can be explained by the schematic diagram in Figure 1c.In their initial states, the DE membranes of the four DE units are in tension due to the introduced prestretch.When actuation voltages are applied to some of the DE units, say DE units I and II illustrated in Figure 1c, the voltage introduced electrostatic stress reduces the tensions on DE membranes I and II, which causes the moving mass to be pulled toward DE units III and IV until force and moment balances are reached.By utilizing this principle, various two-dimensional trajectories can be achieved by careful programming of the four actuation voltage signals, as will be demonstrated in the following section of this work.

Dynamic Model Development
As depicted in the sketch in Figure 1c, in the assembled state of the QUDEA, each DE unit has a length L and a width W for its membrane part.The inner end of the DE membrane has a distance R from the center of the moving mass due to the bonding requirements in practice.Let the DE membranes have a uniaxial prestretch of λ p in the length direction before mounting on the support frame.When actuated, let the center of the moving mass have a displacement x and y in the two principal axes and the moving mass also experiences a rotation angle of α, as shown in Figure 1c.In this state, the lengths of the four DE membranes become L I , L II , L III , and L IV for DE units I, II, III, and IV, respectively.The stretch ratios of the DE membranes along the length direction also vary due to the change in lengths and become λ L_I , λ L_II , λ L_III , and λ L_IV for DE units I, II, III, and IV, respectively.The lengths and the corresponding stretch ratios along the length directions are given in Equations (S1) and (S2), respectively, in the Supporting Information.Note that the stretch ratios of the DE membranes along the width direction are assumed to be constant during the actuation process, i.e., λ W_i = 1 where i = I, II, III, IV for each DE unit, respectively.
The forces exerted by the four DE units along their length directions can be written as: where i = I, II, III, IV for DE unit I, II, III, IV, respectively, H is the initial thickness of the membrane, and σ DE is the stress of the DE membrane along the length direction and is developed using a Kelvin-Voigt model in Equation (S3)-(S4) in the Supporting Information.
The net forces exerted by the four DE units on the moving mass in x and y directions, F DEA_x and F DEA_y , can be given by: The moment of the four DE units applied to the center of the moving mass can be given as: where i = I, II, III, IV for DE unit I, II, III, IV, respectively, β is the angle between the DE membrane and its frame connected to the moving mass and is provided in Equation (S6) the Supporting Information.The equations of motion for the moving mass in two translational DoFs and one rotational DoF can be written as: where m is the mass and I m is the moment of inertia.
With the actuation voltages for the four DE units, Φ I , Φ II , Φ III , and Φ IV , predefined the responses of the QUDEA excited by the programmed voltage signals can be obtained by solving the equations of motion of the QUDEA (Equation ( 4)) numerically using the 4th-order Runge-Kutta method.

Model Parameter Identifications and Validations
The dynamic modeling parameter values utilized in the rest of this article were identified from the quasi-static forcedisplacement characterizations of a single DE unit and the frequency sweep responses of the QUDEA.The experimental setup for characterizing the quasi-static and dynamic responses of the QUDEA is illustrated in Figure 2a and is described in the Experimental Section.The identified parameter values of the dynamic model, as well as the design parameter values of the QUDEA, are summarized in Table 1.Note that the uniaxial prestretch of λ p was set at 1.2 because a substantially higher stretch ratio could result in the necking effect, which leads to extremely uneven stress distribution in the membranes, while a lower stretch ratio may cause loss-of-tension of the membranes  during high stroke resonance.A high width-to-length ratio of W/L = 4 was also determined for the QUDEA to further minimize the effects of necking.Comparisons of the modeled and measured quasi-static voltage responses are plotted in Figure 2b.Note that the displacement of the QUDEA increases nonlinearly with the increasing voltages due to the stronger electromechanical coupling strength.Also, note that the modeled results agree well with the experiments in all voltage ranges from 0 to 4 kV.2c).Note from Figure 2d2,e2 that the QUDEA mainly oscillated in the y axis since only two DE units that are aligned with the y axis were actuated.The oscillation in the y axis first increases with the increasing excitation frequency, peaks at ≈53 Hz (resonant frequency), and then falls rapidly as the frequency increases further.The model is capable of characterizing the frequency responses of the QUDEA accurately in terms of resonant frequency and amplitude.The measured and modeled frequency sweep responses of the QUDEA under a set of forward sweep voltage signals where all four voltages are ON (Figure 2f ) are compared in Figure 2g1-2,h1-2.As the DE unit I and II, III, and IV are actuated in phase, respectively, in this case, the oscillation amplitudes of the QUDEA in the x and y axes are approximately identical, and both peak at the resonant frequency of ≈53 Hz.As illustrated in box A in Figure 2, the moving mass exhibits a high-amplitude translational oscillation with a tilt angle of approximately π/4 rad from the x axis due to the symmetries of the QUDEA in the x and y axes.It is also worth noting, however, a second peak was observed in the experimental results in Figure 2g1-2), which was not captured in the modeled results in Figure 2h1-2.As illustrated in box B in Figure 2, in the second resonant peak, the moving mass exhibits a rotational oscillation around the center.This rotational oscillation was measured in experiments as displacements in the x and y axes because the measuring points for the two laser sensors are off-centered.This rotational resonant response near 80 Hz is believed to be due to the slight mismatches in the prestretch ratios of the four DE units resulting from the fabrication and assembly tolerances.In fact, by introducing random variations within AE5% to the stretch ratios of the four DE units (<AE0.17mm tolerance), the modified modeled results in Figure 2i1-2 successfully captured the rotational resonant response with good accuracies.

Two-Dimensional Quasi-Static Trajectory Generation
The two-dimensional quasi-static trajectory generation strategy is illustrated in Figure 3 and is introduced as follows: first, the target trajectory and its time-series data are defined.Then, the target trajectory is split into four quarters based on the quadrants of the points where different voltage programming rules will be applied.To avoid overcomplicating the voltage programming algorithm, it is stated that only DE units I and II will be actuated for any trajectory points in the first quadrant.Similar rules apply to the trajectory points in the other three quadrants, see Figure 3b1 for details.By substituting the trajectory point into Equation ( 2) and solving the force balance equations for the moving mass: F x = 0 and F y = 0, the required voltages to drive the moving mass to this trajectory point quasi-statically are obtained.Note that moment balance is neglected due to the relatively small stroke of the QUDEA in quasi-static actuation.Figure 3b2 shows the generated voltage signals for the QUDEA to achieve the target trajectory illustrated in Figure 3a1-2.With the voltage signals obtained, they are applied to the QUDEA prototype by the setup illustrated in Figure 3c and the actual trajectory was measured based on the experimental setup introduced in the Experimental Section.
To validate the effectiveness of the proposed quasi-static trajectory programming strategy, a series of two-dimensional target trajectories, including ellipses, rectangles, and triangles, were selected and the comparisons on the target and measured trajectories are plotted in Figure 4a1-3,b1-3,c1-3.Note that the period to close the trajectory (returning to the first point on the trajectory) was set at 4 s.It can be noted from Figure 4a1-3,b1-3,c1-3 that the proposed QUDEA was capable of following both curved and straight lines, sharp and round turns.Overall, the measured trajectories overlap well with the target ones in all 9 cases.To further showcase the two-dimensional quasi-static trajectory programming capabilities of the proposed QUDEA, two Chinese characters "Zhong" and "Hua" (which stand for China when together) were selected as the target trajectories.Figure 4d1-2, e1-2 plots the programmed voltage signals and the comparison for the target and measured trajectories for the Chinese characters "Zhong" and "Hua," respectively.Good accuracies were also achieved on the two Chinese characters with rather complex trajectories, which prove the effectiveness of the proposed quasistatic trajectory programming strategy.

Resonant Trajectory Generation Results Overview
In the previous subsection, the capability of generating twodimensional quasi-static trajectories with excellent accuracies was demonstrated.However, such a working principle suffers from low strokes and, therefore very narrow workspaces.Then, various sets of voltage signals with different amplitude, frequency, and phase combinations are input to the dynamic model and the corresponding steady-state responses of the QUDEA are recorded to form a lookup table (a simple example is shown in Figure 6a).These responses are also verified in experiments, see Figure 6b for example.Finally, the verified lookup table, which contains various two-dimensional resonant trajectories and their corresponding required input voltage signals, can serve as the resonant trajectory generation guideline of the QUDEA.
The modeled and measured resonant trajectories of the QUDEA actuated by voltage signals at different amplitudes and relative phases are plotted in Figure 6a,b, respectively.Note that, in both models and experiments, the peak-to-peak voltage amplitudes for DE units Φ I_p-p and Φ II_p-p were varied from 1 to 3 kV, the relative phase Δθ was varied from 0 to π rad, while the frequency was fixed at 53 Hz near the resonant frequency of the QUDEA to optimize the workspaces.It can be noted from both modeled and measured resonant trajectories that increasing the amplitudes of the voltages will expand the size of the trajectories while altering the relative phases will significantly alter the configuration of the trajectories.For instance, in the Φ I_p-p = 3 kV & Φ II_p-p = 3 kV case, when Δθ = 0, the trajectory is close to a line with ≈π/4 rad incline with respect to the x axis (sequential motion shown in Figure 6c).As Δθ is increased to π/2 rad, the trajectory becomes an approximately circular pattern (sequential motion shown in Figure 6d).The trajectory becomes a line again when Δθ is increased to π rad, but with ≈π/4 rad decline with respect to the x axis instead.It should be noted that, due to the fabrication and assembly tolerances, actual trajectories tend to be imperfect compared with the modeled ones (e.g., ellipses in experiments versus circles in simulation for Δθ = π/2 rad).Nevertheless, the modeled and experimental results agree well in principle.The effects of voltage amplitude and relative phase on the configuration and size of the resonant trajectory demand dedicated investigations in the following subsections using both models and experiments.

Effects of Excitation Signal Parameters
To better characterize the effects of voltage amplitude and relative phase on the configuration and size of the resonant trajectory, the trajectory is approximated by an ellipse shown in Figure 5b, where the major axis length l maj., minor axis length l min., and the angle between the major axis and the x axis ξ are the three parameters that fully describe the configuration and size of the resonant trajectory.
Figure 7 shows the modeled and measured l maj.and l min.values and tilt angles ξ against the voltage amplitude Φ II_p-p and relative phase Δθ.Note that Δθ was varied from 0 to 2π rad in each subplot, Φ II_p-p was varied from 1 to 3 kV in Figure 7a-e while Φ I_p-p was fixed at 3 kV.It can be noted from Figure 7a1 that, when the Φ II_p-p value is much smaller than Φ I_p-p , the resonant trajectory has an extremely low l min .Regardless of the relative phase, the trajectory is approximately π/2 rad tilted (i.e., aligned with the y axis).As Φ II_p-p value increases, l min.is increased, and for each Φ II_p-p case, l maj.and l min.show opposite trends against Δθ.The major axis length l maj.reaches its maximum near Δθ = 0 and π rad, where l min.is at its minimum and approaches zero.As Δθ is increased to near π/2 or 3π/2 rad, l maj.drops close to its minimum while l min.reaches its peak.As Φ II_p-p is increased to similar as Φ I_p-p = 3 kV (Figure 7e1), l maj.and l min.become equal at Δθ = π/2 and 3π/2 rad and the ellipse trajectory becomes a circle.Note from Figure 7a2-e2 that the Φ II_p-p value also plays a critical role in the tilt angles ξ.As Φ II_p-p increases, ξ within Àπ/2 ≤ Δθ < π/2 rad decreases from ≈ π/2 to ≈ π/4 rad while ξ within π/2 ≤ Δθ < 3π/2 rad increases from ≈π/2 to ≈3π/4 rad.For a specific Φ II_p-p case, Δθ has very minor effects on ξ within the relative phase range of Àπ/2 ≤ Δθ < π/2 rad or π/2 ≤ Δθ < 3π/2 rad.
A similar study was performed on the voltage amplitude Φ I_p-p and relative phase Δθ while fixing Φ II_p-p at 3 kV, modeled and measured l maj., l min., and ξ values are plotted in The parametric studies on the axis lengths and tilt angles of the resonant trajectories in this subsection revealed the following key findings: 1) fixing the voltage amplitude of one DE unit pair while adjusting the amplitude value for another pair mainly influences the tilt angle of resonant trajectory.2) Varying the relative phase between the voltage signals mainly influences the major and minor axis lengths of the resonant trajectory.
The parametric studies on the resonant trajectories of QUDEA revealed that voltage amplitudes and relative phases are the two critical parameters that can directly tune the configuration of the resonant trajectories.It is worth noting, however, that the resonant response of the QUDEA is realized fundamentally by the inherent elasticity in the DE membranes and the excitation forces induced by the actuation voltages via electromechanical coupling.As a result, the following simulation study focuses on characteristics of the net spring force F s and excitation force exerted by the four DE units F e (equations of which are given in Equation (S5) in the Supporting Information) against voltage amplitudes and relative phases parameters and how they influence the resonant trajectory of the QUDEA.
The resonant trajectories, net spring force F s and excitation force F e vectors, and the angle between the two force vectors γ against the relative phase are plotted in Figure 9.Note that the voltage amplitudes were fixed at Φ I_p-p = Φ II_p-p = 3 kV in this study.Also note that, because the amplitude of F e is significantly smaller than F s , the forces are normalized with respect to their peak amplitudes respectively for better visualization of the relative positions of the force vectors.In Figure 9a where Δθ = 0 and the resonant trajectory is a straight line, the two force vectors are perfectly aligned (γ = 0 or π rad, Figure 9a3).Figure 9a2 shows that F s and F e are approximately π/2 rad out-of-phase, which indicates that when the amplitude of F s is at its minimum (mass moves to the origin), F e reaches its maximum, which helps to accelerate the mass to travel a longer stroke.When the amplitude of F s is at its maximum (mass reaches the furthest point), the amplitude of F e reaches its minimum, and the stored elastic energy in the DE membranes is utilized to convert into kinetic energy of the mass to maximize the energy efficiency of the oscillation system.These features match with the typical resonant DEA systems reported in the literature. [39]s the relative phase Δθ increases to π/4 rad in Figure 9b, the two force vectors are no longer aligned and the angle γ varies within π to 2π rad range, which indicates that the excitation force not only serves to accelerate and decelerate the mass but also generates a component force to alter the motion direction of the mass, therefore an ellipse trajectory, as shown in Figure 9b1.In Figure 9c where Δθ is π/2 rad, the amplitudes of the two forces are approximately constant and their angle γ is also close to a constant of 5π/4 rad.These unique features in the two forces lead to a steady circular trajectory of the QUDEA.The force analysis results in Figure 9 for the other relative phase values echo the aforementioned findings where a continuously varying angle in F s and F e leads to an ellipse trajectory, a constant angle leads to a circular trajectory, and an angle switching between 0 and π rad leads to a line trajectory.Also note that within the 0 < Δθ < π rad range, the trajectories are anticlockwise rotating over time, while in π < Δθ < 2π rad range, the trajectory rotational direction becomes clockwise.This finding indicates that the relative phase can serve as the key parameter for controlling the rotating direction of the resonant trajectory of the QUDEA.
The resonant trajectories, net spring force F s and excitation force F e vectors, and the angle between the two force vectors γ against the voltage amplitude are plotted in Figure 10 (Note that Φ I_p-p = 3 kV and Δθ = 0 in this study).In Figure 10a where Φ II_p-p = 0, F s and F e vectors are both aligned with the y axis, and the angle γ switches between 0 and π rad in one cycle.As Φ II_p-p increases, as shown in Figure 10b-f, the two force vectors are no longer aligned and γ varies continuously within 0 to π rad range in a complete cycle.The component force of the excitation force due to misalignment alters the motion direction of the mass continuously, resulting in ellipse trajectories in Figure 10b1-f1.Also, note that the tilt angle of the ellipse also varies with the increasing Φ II_p-p from Figure 10b1-f1.In Figure 10g where Φ I_p-p and Φ II_p-p are equal, the two force vectors are aligned again and the trajectory returns to a line configuration.

Hummingbird Flapping Pattern Generation Demonstrations
To demonstrate the potential applications of the resonant trajectory programming of the QUDEA, a hummingbird flapping motion pattern mimicking mechanism is developed in this work, as illustrated in Figure 11a.The proposed mechanism aims to, in principle, mimic the flapping motion patterns of the hummingbird wing tips in different forward velocity scenarios.When in actuation, the four DE units drive the mass in the center to oscillate and therefore cause the wings to flap.A dedicated motion transmission mechanism is developed to allow for complex two-dimensional motions of the wing beam (illustrated in Figure 11b), which is different from the conventional flapping wing mechanism designs where the wing beam can only flap back and forth with a single DoF.However, it is noteworthy that, due to this dedicated motion transmission system, the flapping mechanism prototype developed in this work, in its current form,   is not intended to generate substantial lift forces for take-off.Instead, this flapping pattern-generating mechanism is expected to have applications in, e.g., assisting with the experimental aerodynamic analyses of the flying insects and hummingbirds in wind tunnel tests by mimicking their flapping patterns at different forward velocities.In such applications, the developed QUDEA-driven flapping pattern-generating mechanism can be advantageous over conventional motor-driven devices in terms of reduced system complexity and the thorax-like resonant DEA may also help in gaining more insights into interactions between the aerodynamic forces and system resonance.
Illustrations of the actual hummingbird flapping patterns at different forward velocities, estimated wing tip trajectories, and the time-series displacements of the mass are plotted in Figure 12.Note that the wing tip trajectories of the mechanism were estimated from the measured deformations of the mass and the kinematics of the flapping mechanism.The distance from the moving mass to the pivot joint is 5 mm, while the distance from the wing tip to the pivot joint is 50 mm, which results in an approximately tenfold amplification of the trajectory from the mass to the wing tip.It can be noted from Figure 12a3-e3 that, with the increasing forward velocity, the flapping pattern of the hummingbird becomes more aligned with the vertical axis and the mismatches in trajectories between the power stroke (downward) and recoil stroke (upward) become more significant.These findings indicate that, for the proposed QUDEA to generate such flapping patterns in higher velocity scenarios, the tilt angle of the trajectory should drop from ≈π to π/2 rad, while the minor axis length should increase in the meantime.The estimated wing tip trajectory results of the flapping mechanism shown in Figure 12a2-e2 agree well with the actual hummingbird flapping patterns in all forward velocity scenarios.Note that some of the unique flapping patterns found in hummingbirds as well as many other flying insects, such as the figure-of-eight shown in Figure 12a3, are impractical to mimic by the current prototype due to the limitations in resonant oscillations.Future work can modify the flapping mechanism by having the four DE units as the main "muscles" to generate large stroke motions while introducing multiple small-scale DE units to fine-tune the flapping trajectory to better match the actual pattern.In future work toward the take-off of such devices, multiple DE units can be stacked to improve the power output, and a more compact and lighter mechanism can be designed to reduce the total payload.

Conclusion
This work developed a two-dimensional trajectory programmable DEA to bridge the gap between the resonant actuation of the DEAs and versatile multi-DoF output demands in various practical applications.The rich nonlinear dynamics of the QUDEA were characterized and the critical parameters for twodimensional trajectory programming were uncovered through extensive modeling and experimental studies.To showcase the two-dimensional trajectory programming capabilities of the QUDEA, Chinese characters writing and hummingbird flapping patterns mimicking using the proposed QUDEA were demonstrated.The key findings are summarized as follows: 1) The tilt angle of the resonant trajectory is mainly determined by the peakto-peak voltage amplitudes of the two agonistic-antagonistic DE unit pairs.2) The major and minor axis lengths of the resonant trajectory are mainly controlled by the relative phase between the sinusoidal voltage signals.
3) The rotating direction of the resonant trajectory is determined by whether the relative phase is in [0 π] or [π 2π] interval.
The proposed QUDEA is expected to have potential in various applications where programmable output trajectories are desirable.For instance, in experimental biology and biomimetic robotics, the resonant QUDEA can serve as the motion generator mimicking the high-speed locomotion patterns of many terrestrial and flying animals.In industrial applications, the programmable resonant patterns of the QUDEA can be applied in vibratory screening for sorting different materials.It should be noted that the rotational resonant response of the QUDEA triggered mainly by the fabrication tolerance of the DE units can also be utilized, for instance, together with the trajectory programming technique, to achieve insect or hummingbird mimicking flapping motion patterns with actively controlled angle of attacks for more agile flight simulations.Future work can also investigate closed-loop control of the two-dimensional trajectory programming of the QUDEA by introducing external position sensors or by exploiting the dynamic self-sensing capabilities of the DEAs.

Experimental Section
QUDEA fabrication process: The fabrication process of the QUDEA is described as follows.First, a 100 μm thick silicone DE membrane (ELASTOSIL 2030, Wacker Chemie AG) was bonded to rectangular acrylic frames via a silicone transfer tape (5302A, Nitto Tape).Note that the unbonded region of the DE membrane was 16.7 mm in length and 80 mm in width in this unstretched state.A total of four identical units were prepared in this step.Second, the fabricated DE units were attached to the moving mass and the support frame via Nylon bolts and nuts.During this step, the membranes were stretched to 20 mm in length, while the width remained unchanged.Then, custom carbon grease was carefully Figure 12. a1-e1) Sketch of the hummingbird flapping motion pattern (figures are adapted and redrawn with permission. [40]Copyright 2007, The Company of Biologists), a2-e2) estimated flapping trajectory of the tip of the wings, and a3-e3) measured displacements of the QUDEA.The forward velocity v = 2, 4, 6, 8, and 10 m s À1 for a to e, respectively.hand-brushed on both sides of the DE membranes and copper tapes were attached to the frames as connections between the compliant electrodes and the high-voltage cables.
Quasi-Static Force-Displacement Test of a Single DE Unit: The acrylic frame on one end of the single DE unit was mounted to the experimental rig.At the same time, a linear rail (X-LSQ150B-E01, Zaber) pulled the frame on the other end to deform the DE membrane uniaxially at a constant rate of 0.05 mm s À1 to minimize the effects of viscoelasticity.A load cell (S/N 835827, FUTEK) was adopted to measure the reaction force of the DE membrane.An actuation voltage of 3 kV was applied to the DE unit via a high voltage amplifier (HVA) (10HVA24-P1, UltraVolt) for electromechanical coupling characterizations.The experiment was controlled via a data acquisition (DAQ) device (NI-USB 6363, National Instruments) at a sampling rate of 40 kHz.The experimental setup is shown in Figure S1a (Supporting Information) and the measured and modeled forcedisplacement functions are plotted in Figure S1b (Supporting Information).
Quasi-Static and Dynamic Response Tests of the QUDEA: A prototype of the QUDEA was fixed to the testing rig with two laser displacement sensors (LK-G152 and LKGD500, Keyence) measuring the displacements of the QUDEA in the x and y axes.Four HVAs were adopted to actuate the four DE units separately.For quasi-static tests, direct current voltage signals with different amplitudes from 0 to 4 kV were applied to these DE units for a period of 10 s.The average displacements in the last 1 s of the actuation period were recorded as the quasi-static response results.

Figure 1 .
Figure 1.a) Rendered picture of the proposed QUDEA.b) Key components of the QUDEA.c) Illustration of the actuation principle of the QUDEA.

Figure 2 .
Figure 2. a) Experimental setup for quasi-static and dynamic deformation measurements.b) Modeled (blue surface) and measured (black dots) quasistatic deformation of the QUDEA under different actuation voltage combinations.c) Example of the continuous frequency sweep voltage signals with Φ I and Φ III ON while Φ II and Φ IV OFF.d1-2) Experimental and e1-2) modeled frequency responses of the QUDEA with Φ I and Φ III ON while Φ II and Φ IV OFF.f ) Example of the continuous frequency sweep voltage signals with all four voltage signals ON. g1-2) Experimental , h1-2) modeled, and i1-2) modeled with modifications frequency responses of the QUDEA with all four voltages ON (Φ I = Φ II , Φ III = Φ IV , and Φ I and Φ III are antiphase).

Figure 2d1- 2 ,
e1-2 compares the measured and modeled frequency sweep responses of the QUDEA under a set of forward sweep voltage signals where Φ I and Φ III are ON while Φ II and Φ IV are OFF (Figure

Figure 3 .
Figure 3. Two-dimensional quasi-static trajectory programming processes.a1) Target displacement trajectory and a2) its time-series data.b1) Illustration of the displacement quadrant-based voltage programming logic.b2) Programmed voltage signals based on the target trajectory.c) Illustration of the setup for applying the programmed voltages to each DE unit.

Figure 4 .
Figure 4. a1-3,b1-3,c1-3) Comparisons of the target trajectories and the measured trajectories.d1) Generated voltage signals for the QUDEA to write the Chinese character "Zhong" and d2) comparison of the target and measured trajectories for this Chinese character.e1) Generated voltage signals for the QUDEA to write the Chinese character "Hua" and e2) comparison of the target and measured trajectories for this Chinese character.

Figure 5 .
Figure 5. a) Illustration of typical actuation signals for resonant trajectory generation.b) Demonstration of a typical periodic oscillation trajectory approximated by an ellipse.

Figure 6 .
Figure 6.a) Modeled and b) measured resonant trajectories with different actuation voltage signal combinations.c) Sequential photos of the QUDEA generate a tilted linear trajectory.d) Sequential photos of the QUDEA generating an anticlockwise circular trajectory.

Figure 11 .
Figure 11.a) Prototype of the flapping motion pattern generating mechanism.b) Illustration of the QUDEA-driven flapping motion.
For dynamic response tests, a forward frequency sweep from 0 to 100 Hz at a sweep rate of 0.5 Hz s À1 was first generated in a personal computer and sent to the HVAs via a DAQ.A typical set of frequency sweep signals is illustrated in Figure 2c.Note that, to avoid overcomplicating the responses, the actuation voltage signals for opposing DE units were set as π rad antiphase in these frequency sweep tests.

Table 1 .
Parameter values adopted in the modeling study.