Theoretical study of the electroactive bistable actuator and regulation methods

ABSTRACT Dielectric elastomer actuators have attracted growing interest for soft robot due to their large deformation and fast response. However, continuous high-voltage loading tends to cause the electric breakdown of the actuator due to heat accumulation, and viscoelasticity complicates precise control. The snap-through bistability of the Venus flytrap is one of the essential inspirations for bionic structure, which can be adopted to improve the shortcoming of dielectric elastomer actuators and develop a new actuation structure with low energy consumption, variable configuration, and multi-mode actuation. Hence, in this paper, the structural design principles of electroactive bistable actuators are first presented based on the total potential energy of the structure. Following that, a feasible design parameter region is provided, the influence of crucial parameters on the actuation stroke, trigger voltage, and actuation charge are discussed. Finally, according to the coupling relationship between the bending stiffness and the bistable property of the actuator, the adjusting methods of bistable actuation are explored. A qualitative experiment was performed to verify the feasibility and correctness of the bistable design methodology and the actuation regulation strategy. This study provides significant theoretical guidance and technical support for developing and applying dielectric elastomer actuators with multi-mode, high-performance, and long-life characteristics. Graphical abstract

Although the above-discussed DEA has superior actuation performance and a wide range of application scenarios, the continuous high voltage loading during the actuation process can easily lead to electrical breakdown failure [37,38]. In addition, the viscoelasticity of the material increases the difficulty of DEA control. As a plant capable of rapid deformation, when insects stimulate the leaf villi, the Venus flytrap can undergo snapthrough deformation within 0.1 seconds to capture the insects. Then, the leaf returns to its initial shape once the insects are digested [39,40]. The bistable actuation structure based on the snap-through deformation was designed and used in microfluidic control valves [41,42], soft robots [43,44], untethered control [45], and electrocaloric heat pumps [46]. Such a structure is characterized by a fast response, low energy consumption, and variable configuration advantages. Therefore, combining snap-through deformation with a dielectric elastomer actuator effectively reduces the risk posed by the electrical breakdown of elastomer films and allows the actuator to arbitrarily switch between the two steady states without a complex control system.
The existing bistable dielectric elastomer actuators can be divided into two types: modular and integrated. The modular design combines a flexible beam with bi-stability characteristics and a dielectric elastomer actuator. The dielectric elastomer actuator triggers the bi-stability deformation when a current is induced. However, since the flexural beam performance is constant after it is selected, the bi-stability actuation behavior of this design is only related to the flexible beam [47][48][49][50].
The integrated design is a minimum energy principle structure formed by assembling a dielectric elastomer film with a flexible frame. In this design, the flexible frame itself has no bistable behavior characteristics. The bi-stability behavior of the actuator is achieved by the coupling action between the dielectric elastomer film and the flexible frame. Consequently, the bi-stability actuator of this structure is controllable, and the structure is light and compact, and the preparation process is simple. Therefore, the coupled design is the current hot spot for the research of electroactive bistable devices [51][52][53][54].
Although some experimental and exploratory studies of application scenarios have been carried out for coupled bistable actuators, fewer reports can be found on critical issues such as design methods, performance characterization, and regulation strategies. In this paper, the minimum energy structure of dielectric elastomer is taken as the research object, while the design method, performance influencing factors, and actuation mode conversion of the electroactive bistable actuator are investigated from the perspective of system energy. This study can provide theoretical guidance and practical experience for developing the design and application of dielectric elastomer actuators with multi-mode, high-performance, and long-life characteristics.

The design of an electroactive bistable actuator
The minimum energy bistable structure is designed by pasting a pre-stretched film on a flexible frame and actuated by regulating the coupling between the flexible frame and the elastomer film. However, in the previous design, the elastomer film is wholly fixed on the flexible frame. Moreover, the film presents a complex saddle shape in space, which leads to the electrical breakdown damage of the elastomer film due to non-uniform deformation and weakens the actuation capability of the dielectric elastomer. Therefore, pure shear deformation of a dielectric elastomer is adopted in this paper to achieve the minimum energy bistable structure design, as shown in Figure 1(a). The dielectric elastomer films with initial dimensions of L 1 , L 2 , and H 0 are fixed on the flexible frame after applying pre-stretching λ 1p and λ 2p . Then, fiber elements are attached to the surfaces of elastomer films at equal distances along the 2-direction to limit the deformation of the films in the 1-direction. This design approach was proposed by Huang [55] and later developed by Lu [56]. The practical dimensions of the flexible frame are L � W � T, where T, L, and W represent the flexible frame's thickness, effective length, and effective width, respectively. The elastomer film shrinks to bend the flexible frame in an arch shape once the applied constraint is removed, as shown in states A or state B in Figure 1(a).
Schematic diagrams of the bistable actuation principle under two design types are shown in Figures 1 (b) and (c). The asymmetric bistable design is shown in Figure (b). Under the voltage excitation, the actuator will gradually deform from the initial equilibrium state B to the spreading state, as shown by the dashed line in the figure. At this time, the actuator will experience a snap-through deformation to the BB position. When the excitation voltage is withdrawn, the actuator will stay in the new equilibrium state A. However, when the actuator is loaded again, it will not be able to return to the initial equilibrium state B from the new equilibrium state A. For the symmetric design, as shown in Figure 1(c), the actuator is gradually deformed from the initial equilibrium state B to the BB position under voltage excitation. After momentarily withdrawing the voltage, the actuator will jump from the BB position to the new equilibrium state A and stabilize in that position. However, when the actuator is excited again, the actuator will return to the initial equilibrium state B from the new equilibrium state A through the same path.
However, compared with the former, the symmetric design requires a more complicated loading strategy (the structure needs to precisely control the loading voltage during the actuation process to obtain the kinetic energy required for the structure to jump from state BB to state A). In the experiment, this can be achieved by utilizing the inertia, abruptly withdrawing the voltage at state BB, or adding mass to the free end of the structure [52,53]. Even though the asymmetric type can only achieve unidirectional actuation, the corresponding loading control is simple. In contrast, the symmetric design can achieve bidirectional actuation. However, it requires complex loading control to achieve the inertia required for snap-through deformation. Therefore, exploring how to achieve regulation and conversion of two actuation modes under a single structure to improve the application of these electroactive bistable devices is of great significance.
Based on the issues mentioned above, a theoretical model describing the bistable design and its actuation performance from the perspective of system energy is established in this paper. The proposed model accounts for the actuation mechanism and regulation approach of two bistable designs, neglects the energy dissipation in the entire analysis process, and adopts the ideal physical model for the dielectric elastomer actuation unit and the flexible frame.
The flexible frame only undergoes bending deformation during actuation. The strain energy of the flexible frame according to the pure bending beam theory can be expressed as [57]: where K denotes the stiffness of the flexible frame along the forward bending direction. Parameter K can be calculated according to geometry and material parameters, i.e. K ¼ EI=L. Parameter α is the symmetry factor that denotes the flexible frame stiffness ratio along the two bending directions. In other words, α represents the ratio of the negative bending stiffness of the actuator to the positive bending stiffness, Where θ is the angle of the circle center corresponding to the flexible frame bending, θ þ indicates positive bending, and θ À shows negative bending and specifies that α ¼ 1 when θ > 0. Dielectric elastomers are polymeric materials, and their viscoelasticity has an unavoidable effect on the actuator's performance, often leading to response hysteresis under dynamic excitation. Still, its impact can usually be ignored as a quasi-static analysis. Thus, using the Gent free energy model, the strain energy of the pure shear actuation unit of the dielectric elastomer can be expressed as [58]: where μ is the shear modulus of the dielectric elastomer material, P is the dielectric constant of the material, J lim characterizes the ultimate stretch of the dielectric elastomer material, and Φ is the voltage applied to the dielectric elastomer. Since elastomer film deformation in direction 1 is limited by the restraining fibers, Since the angle of the center of the flexible frame circle in the bending process always satisfies a particular geometric relationship with the chord length, the chord length is equal to the length of the elastomeric actuation unit in the direction 2. Moreover, the deformation λ 2 of the elastomeric actuation unit in direction 2 is related to the bending angle θ of the flexible frame as follows: By assuming a negligible temperature change and energy loss throughout the actuation, the total system energy of the bistable actuator can be expressed as: Dimensionless parameters are defined as follows: Then, the total energy equation of the system can be simplified as follows: It can be observed that parameters J lim , λ 1p , L � , K � , Φ � , and α constitute the set of parameters for the design and regulation of the minimum energy structure bistable actuator. Moreover, J lim is taken as a constant value in the subsequent numerical analysis, i.e. J lim ¼ 100 [59]. Figure 2(a) depicts the energy curve of the system for a bistable actuator with given design parameters. The design condition for the existence of bistable performance of the actuator is the existence of two minima in the system energy curve of the structure within the feasible actuation angle range (as shown by the black circles in the figure). The parameter region for the existence of bistable performance of the actuator under symmetric design is provided in Figure 2(b). Design parameters L � , and K � are divided into the planes consisting of type I, type II, and type III. The scattered blue points shown in type II indicate the combination of parameters for achieving the bistable actuation design. Type I indicates that the force of the elastic actuation unit is too small, and the stiffness of the flexible frame is too large. Therefore, the flexible frame cannot produce bending deformation and achieve the horizontal state of the entire structure. The type III indicates that the bending stiffness of the flexible frame is too small, but the actuation unit force is too large, which leads to the flexible frame fracture.
The effect of the pre-stretching λ 1p on the effective parameter region for bistable design is demonstrated in Figure 2(c), where A 0 represents the area of the effective parameter regions at λ 1p ¼ 2. In contrast, A represents the area of the effective parameter region under different λ 1p . The results show that the effective parameter regions under different pre-stretching are relatively similar. However, the parameter region area increases with the pre-stretching; thus, more parameter combinations can be achieved for the bistable design. The changes in the effective parameter region for the symmetric and asymmetric designs are compared in Figure 2(d). The parameter region is the largest for the symmetry factor α ¼ 1 (symmetric design). The parameter region decreases regardless of whether the symmetry factor α is increased or decreased. However, the lower boundary of the parameter region for α > 1 is the same as the symmetric design, while the upper boundary for α < 1 is the same as the symmetric design.

Electroactive bistable actuator and its performance
The essence of the bistable actuator deformation is the change of the structural system energy when excited by a voltage. The strain energy of the dielectric elastomer decreases when voltage is applied, and the minimum value of the system energy curve changes to obtain the bistable actuation. When subjected to voltage for symmetric and asymmetric designs, the deformation is shown in Figure 3. According to Figure 3 (a), when the excitation voltage is Φ � ¼ 0, the equilibrium state of the actuator is at point A. As the voltage Φ � is increased, the minimum value of the system energy curve (as shown in the box in the figure) is modified. When the excitation voltage Φ � reaches Φ � c , the equilibrium state of the actuator currently is at point AA and the bending angle of the actuator is zero. Furthermore, the actuator remains balanced at the position mentioned above when the voltage is further increased until the electrical breakdown damage occurs. However, if the voltage is rapidly removed once it is loaded to Φ � c , the actuator will snap through from the balance state AA to the balance state B (as shown by the dashed line in the figure) and remain constant at that position. Since equilibrium position B is symmetrical with respect to the initial equilibrium position A, the actuator can also be deformed from state B to state A when actuated in the reverse direction.
For the asymmetric design shown in Figure 3(b), the value of symmetry factor α is 0.8, and the actuator gradually deforms from the initial equilibrium state A as the voltage increases. Since the actuator is characterized by different energy in the initial equilibrium state A and the new equilibrium state B, it cannot be actuated from state B to state A. In other words, the actuation path is unidirectional.
The effect of pre-stretching λ 1p and symmetry factor α on the bistable trigger voltage is investigated in Figure 4. According to Figure 4(a), the trigger voltage of a bistable actuator for different pre-stretching values remains approximately the same. Moreover, the larger the L � and the smaller the K � , the larger the trigger voltage Φ � required for the bistable actuator in the feasible design parameter combination region. In addition, the larger the pre-stretching and the thinner the film thickness under the same parameter combination, the smaller the voltage required for the trigger bistable actuator. Furthermore, the larger the pre-stretch, the thinner the film thickness for the same parameters. Consequently, the smaller the critical voltage required to trigger the bistable actuator. According to Figure 4 (b), the trigger voltage distribution of feasible design parameters for the asymmetric design is more uniform than that for the symmetric design. Lastly, the trigger voltage for the symmetric design is more significant than that for the asymmetric design for the same combination of parameters.
For a bistable actuator, apart from the trigger voltage Φ � c , the actuation stroke and time are critical indicators for evaluating the performance of bistable devices. The actuation stroke is the maximum achievable actuation angle at the stable equilibrium position of the actuator under the trigger voltage compared to the initial equilibrium position, as shown in Figure 3. The actuation time is required for the actuator to load from the initial equilibrium state to the trigger voltage. Once the voltage loading device is determined, the actuation charge can be used to calculate the actuation time. Thus, the actuation charge can be used to indirectly characterize the actuation time of the actuator. The actuation charge can be determined according to the actuator's energy barrier and the trigger voltage. The energy barrier of the actuator is shown in Figure 3. Therefore, the actuation stroke θ max act and the actuation charge Q nor , which characterize the bistable actuation performance of the actuator, can be expressed as follows: The influence of design parameters on the bistable actuation stroke is shown in Figure 5. According to Figure 5(a), the larger the value of L � , the larger the actuation stroke of the actuator. However, the larger the value of K � , the smaller the actuation stroke of the actuator instead. The effect of film pre-stretching λ 1p on the actuation stroke varies approximately linearly. The larger the pre-stretching, the more outstanding the actuation performance. This can be attributed to a more significant elastomer film pre-stretching, which increases the stress in the actuation unit and the actuation deformation potential for the same design parameters. The actuation stroke under symmetric design is less significant than that under asymmetric design because the structure will produce sudden Figure 5. The influence of design parameters on the bistable actuation stroke: (a) the actuation performance of the actuator for different design parameters, and (b) the effect of pre-stretching and symmetry factor of the elastomer film on the actuation performance. structure can only stabilize under the trigger at the θ ¼ 0 position. The effect of design parameters on the bistable actuation charge is shown in Figure 6. According to Figure 6(a), the more significant the L � , the larger the energy barrier of the actuator and the larger the actuation charge. On the other hand, when L � is constant, K � is independent of the energy barrier of the actuator and the actuation charge. This is mainly due to the change of the energy barrier of the actuator that depends only on the strain energy change of the dielectric elastomer actuation unit.
According to Figure 6(b), the larger the pre-stretch of the elastomer film, the larger the actuation charge. However, the symmetry factor is independent of the actuation charge because the symmetry factor is essentially the stiffness ratio of the flexible frame. As such, it only affects the bistable actuation stroke.

Comprehensive performance evaluation and regulation of electroactive bistable actuator
Four physical parameters, the feasible parameter region, trigger voltage, actuation stroke, and actuation charge of the bistable actuator, are critical indicators to evaluate the performance of the electroactive bistable actuator. Moreover, a comprehensive analysis of these four parameters provides an essential theoretical basis and technical guarantee for the actuation control of the actuator. Therefore, the combined effect of the film prestretch λ 1p and symmetry factor α on the integrated effect of bistable actuator performance is shown in Figure 7. According to Figure 7(a), the feasible parameter area ratio (with the parameter area ratio of the pre-stretched λ 1p ¼ 2), the actuation charge, and the actuation stroke become more significant as the pre-stretch increases. However, the trigger voltage gradually decreases with an increase in the pre-stretch value. According to Figure 7(b), the symmetry factor is independent of the actuation charge. Moreover, the feasible parameter area ratio and the critical trigger voltage are the largest in the symmetric design (i.e. α ¼ 1). However, symmetry factor α in the asymmetric design decreases with the actuation stroke. Consequently, the feasible parameter area ratio and the trigger voltage decrease while the actuation stroke increases.
According to the proposed electroactive bistable design method, regulating the voltage's unloading time for the symmetric type can achieve an arbitrary drive between two equilibrium states. However, for the asymmetric type, the voltage regulator can only deform the actuator from the high-energy potential barrier side to the low-energy potential barrier side. Moreover, the actuation process is unidirectional. Therefore, achieving an arbitrary actuated state between the two steady states for the asymmetric bistable actuator is important. In addition, although the symmetric bistable design is characterized by a bi-directional actuation, the precise voltage manipulation increases the control difficulty of the actuator. In contrast, the asymmetric design can precisely compensate for this deficiency. Hence, achieving the conversion of the actuator by both symmetric and asymmetric designs is another challenging task. Therefore, the bistable actuator's regulation principle and implementation path for actuating under two design types are investigated according to Figure 8. In Figure 8(a), the influences of the actuator's initial bending angle θ 0 and the symmetry factor α on the actuation performance is shown. When θ 0 < 0 and α < 1, or θ 0 > 0 and α > 1, the bistable actuator is in a lower energy equilibrium position. At this time, the actuator can only produce monostable actuation when excited by a voltage. On the other hand, when θ 0 < 0 and α > 1, or θ 0 > 0 or θ 0 > 0 and α < 1, the actuator is in a high energy equilibrium position. Once the excitation voltage reaches the trigger voltage, the actuator will  produce a snap-through deformation. However, when α ¼ 1, regardless of the value of θ 0 , the actuator is characterized by both monostable and bistable actuation behavior.
The actuation approach of the bistable actuator from the initial equilibrium state A to the new equilibrium state B is provided in Figure 8 (b). For the symmetry factor α ¼ 1, the initial equilibrium state A 2 and the new equilibrium state B have the same energy. Hence, traversing from the initial equilibrium state A 2 to the new equilibrium state B only required loading the voltage to the critical trigger voltage. The actuator needs to withdraw the voltage to ensure snap-through deformation and reach the B state, as shown in path 1 in Figure 8 For the symmetry factor α > 1, reaching the new state B from the initial state A 3 only requires loading the excitation voltage to the trigger voltage, as shown in path 2 in Figure 8 (b). However, for the symmetry factor α < 1, the actuator cannot move from state A 1 to state B by voltage alone, and the symmetry factor α requires adjusting. The A 1 state is first deformed to A 2 or A 3 state, after which the B state is achieved via path 1 or path 2, respectively.
The actuation approach of the bistable actuator between the initial equilibrium B state and the new equilibrium state A is provided in Figure 8 (c). Like the regulation method in Figure 8(b), the corresponding actuator can be completed by only regulating the loading voltage from state B to states A 1 or A 2 . However, if the actuator is to move from state B to state A 3 via path 1 or 2, it is necessary to adjust both the symmetry factor and the excitation voltage. Therefore, for a bistable actuator of the minimum energy structure principle, it is possible to achieve monostable and bistable actuations of the actuator by co-regulating the excitation voltage and symmetry factor. Moreover, it is also possible to achieve actuation conversion between symmetric and asymmetric design types of the actuation, which is an essential theoretical guidance for improving the application potential of such electroactive bistable structures.
To verify the feasibility and correctness of the above design theory and regulation method, the following two experiments were carried out. In the experiments, commercial VHB4910 was selected as the dielectric elastomer film. Polypropylene sheet with thicknesses of 0.6 mm and 0.05 mm was chosen for preparing flexible frames and airbags, respectively. Table 1 demonstrates the spatial configuration of the electroactive actuation structure for the three design parameters. The results show that the structure can hold an effective bistable configuration only when the design parameters L � and K � are matched. As described in Figure 2, when the bending stiffness of the flexible frame is enlarged, the dielectric elastomer film cannot effectively stretch the frame, so the overall structure presents a spreading state. Similarly, when a smaller K � value is chosen, the flexible frame is folded under the action of the dielectric elastomer film, which leads to the failure of the whole structure. Therefore, a suitable combination of parameters is determinative of the specific bistable design. Figure 9 illustrates the stiffness adjustment and actuation control of the bistable structure. Figure 9(a) shows the experimental setup, through which the bending stiffness adjustment of the flexible frame is achieved by pneumatic loading. Specifically, air chambers are placed on each side of the flexible frame. Then, the air chambers are inflated by an air pump to realize the bending regulation of the flexible frame in different directions. The actuation of the dielectric elastomer is directly powered by a high voltage amplifier (Trek MODEL 20/20C) as well as a signal generator (RIGOL DG4062), and the deformation of the actuator is recorded in real-time using a camera. Figure 9(b) gives the relationship between the initial equilibrium position of the actuator and the applied air pressure. The red and blue points in the figure indicate the equilibrium position of the actuator under positive and negative bending, respectively. Moreover, the experimental snapshot gives the equilibrium state of the actuator under  Figure 9(c) investigates the actuation performance of the actuator under the combined action of pressure and voltage. When the actuator is excited from the positive bending equilibrium position, the bending angle of the actuator gradually decreases with the voltage increase. As the voltage reaches 8 kV, the actuator experiences a snap-through deformation (as shown by the black arrow in the figure). The actuator transforms from positive to negative bending. When the actuator is actuated from the negative bending equilibrium position, the bending angle of the actuator gradually decreases with increasing voltage. However, when the voltage exceeds 8kV, the actuator yields significant wrinkle deformation (shown by the pentagram in the figure), followed by electrical breakdown failure. The actuator eventually fails to achieve jump transformation since the actuator has low system energy in the negative equilibrium position, such as position A 1 in Figure 8(a). Thus, the bistable actuation cannot be achieved by voltage actuation alone.
However, when the actuator is under the combined action of pressure and voltage, the actuator can achieve snap-through deformation because the bending stiffness of the frame changes after the chamber inflation. At this time, the actuator holds a larger system energy in the negative equilibrium position compared with that in the positive equilibrium position (as in position A 3 in Figure 8(a)). Therefore, snap-through deformation can be achieved when the actuation voltage reaches a critical value. Furthermore, as the pressure value increases, the required voltage for snap-through deformation is smaller, and the actuation deformation is more significant, which is in perfect agreement with the results in Figure 7(b). Namely, the critical trigger voltage of the bistable actuation structure decreases as the symmetry factor increases and the actuation stroke increases as the symmetry factor increases.
The above two experiments confirm that the design method and regulation strategy of the electroactive bistable actuator in this paper are practical and applicable, which shall offer a theoretical guidance for developing new actuation structures and expanding the bistable structure applications in the future.

Conclusions
In this paper, the influence of pre-stretching and symmetry factors on the actuator's trigger voltage, actuation stroke, and actuation charge was studied and analyzed. Furthermore, a comprehensive effect of pre-stretching and symmetry factors on the bistable actuation performance was obtained. Accordingly, it was proposed that bistable actuators in the monostable, bistable, symmetrical, and asymmetrical actuation can be achieved by cooperatively regulating the excitation voltage and symmetry factor. Then, the feasibility and correctness of the bistable design method and the actuation regulation strategy were verified qualitatively through experiments. The research in this paper provides theoretical support and guidance for developing electroactive bistable structures with high performance and new ideas for expanding the application of dielectric elastomers in the field of intelligent devices and soft robots.