Bistable rotating mechanism based on dielectric elastomer actuator

A growing number of dielectric elastomer actuators have been proposed to build soft robots and soft electronic devices. However, it remains a big challenge to design soft bistable actuators. Most of existing bistable actuators based on soft materials generate either limited actuation range or unrepeatable actuation. In this article, we present a new bistable rotating mechanism using dielectric elastomer actuators, which exhibits a relatively large actuation angle range and repeatable motion. The mechanism is featured by an antagonistic structure with mechanical locks to create two symmetric stable configurations. We develop an analytical model to help design and explain the electromechanical behavior of the bistable rotating mechanism. Powered by the dielectric elastomer actuators, it shows fast response and large load capability. We build a binary manipulator to further demonstrate that the bistable rotating mechanism can provide a promising route for the design of binary systems.


Introduction
As a typical high-performance electroactive polymer, dielectric elastomer (DE) has an attractive combination of high energy density [1], large strain [2][3][4] and fast response [5]. As a result, DE has been widely used in robotics such as soft actuators, bionic robots and soft electronic devices [6][7][8][9][10][11]. Dielectric elastomer actuators (DEAs) are generally composed of a DE membrane sandwiched between a pair of compliant electrodes. When voltage is applied on the surface electrodes, the resulting Maxwell force compresses the DE film in thickness and expands it in area [5]. Typical DEAs include DE bistable minimum energy structure [12][13][14], DE out of plane actuator [15,16] and fiber-constrained DE actuator (FCDEA) [17,18].
Designing bistable actuators has long been a challenge in the field of robotics. Although bistable actuators with good performance have been developed in a range of structures, including bistable elements in the Hyper-Redundant robot manipulators [13,19,20], bistable elements in the space structure [21], bistable beam [22] and bistable shape memory muscle in the robot fish [23], most of them are designed for special situations and have various limitations. For instance, the bistable beam exhibits limited range of distance output. The bistable shape memory muscle of the fish robot can only be actuated once and the transitioning temperature is required to be preprogramed to reverse the actuation. The actuation of the bistable shape memory muscle is thus hard to repeat.
To overcome the above issues, we develop a novel bistable actuator composed of two symmetrically arranged uniaxial FCDEAs working against each other through a rotating mechanism [19,20,24]. Different from previous bistable systems where stable configurations are maintained through the local energy concavities of the bistable beam [16,[21][22][23]25], the stable configurations in our bistable actuator are created by mechanical locks. To switch between the stable configurations, external energy should be applied to the rotating mechanism to overcome the energy barrier. One of the uniaxial FCDEAs becomes softer when voltage is applied. It will be pulled by the other uniaxial FCDEA, allowing the whole rotating mechanism to switch from one stable configuration to the other. Once the mechanism is flipped, no further external energy is required to maintain the output.
Unlike conventional bistable actuators containing bistable beams, the proposed rotating mechanism becomes bistable through the geometric design of the structure, which leads to a convex free energy curve. The antagonistic structure is beneficial to ensure repeatable output with exact rotation angles. This type of bistable mechanism was originally proposed by Steven Dubowsky et al [19,24] in pursuit of building large-scale binary mechatronic systems containing small, lightweight and fast bistable actuators and has been adopted in building bistable actuators using shape memory polymer and DE [20,26].
We also develop an analytical model to help design and predict the electromechanical behavior of the bistable rotating mechanism. We use the Gent model to describe the electromechanical response of a single uniaxial FCDEA and then derive the total free energy of the bistable rotating mechanism by integrating the energies of two uniaxial FCDEAs with the geometry of the rotating mechanism. We build a binary manipulator composed of three bistable rotating mechanisms to demonstrate the output ability. We perform additional simulations to quantify the workspace area and density of binary manipulators with multiple bistable rotating mechanisms.
2. Design principle of the bistable rotating mechanism 2.1. Analytic model of uniaxial FCDEA Various constitutive models can be used to describe the mechanical response of soft elastomers [27][28][29][30]. In this study, Gent model is used to describe the elastic energy of a single uniaxial FCDEA: where D is the true electric displacement and ε is the dielectric constant of the material. The relationship between D and voltage f is: is the original thickness of the uniaxial FCDEA. The uniaxial FCDEA used in our work is a bi-layer structure consisting of two layers of VHB 4905. The thickness of each VHB 4905 is´-0.5 10 m. 3 Taking the derivate of W with respect to λ 1 and λ 2 , the nominal stresses s 1 and s 2 in the two principal directions for uniaxial FCDEA subjected to electromechanical loading can be obtained as:  [32,33]. Analytical model of FCDEAs has been developed in the literature [14].
Since the fibers maintain the prestretch along the transverse direction, λ 2 is fixed as l l = . The incompressibility assumption leads to l l l = 1. Equation (4) can be rewritten as: Set P as the load along the actuation direction in the actuation state. Since s 1 is determined by the load P andẼ is determined by the applied voltage f, equation (6) becomes: where L 2 is the initial length along the transverse direction.
For given load P and voltage f, λ 1 can be solved by equation (7).

Analytic model of bistable rotating mechanism
In the bistable actuator, two uniaxial FCDEAs work against each other through a rotating mechanism (figure 1(a)). This is typical of antagonistic structures well documented in the literature on the design of bistable actuators [13,22,25]. The rotating mechanism consists of two equilateral triangular frames connected head to head by a joint ( figure 1(b)). Mechanical locks (indicated by the dashed circles in figure 1(b)) are designed to limit the rotation angle of the mechanism. The left configuration corresponds to a rotation angle of 90°, while the right configuration corresponds to a rotation angle of 150°(figure 1(c)). These limit angles are determined by the geometric design of the mechanical locks. The rotation of the mechanism is driven by the torque produced by the uniaxial FCDEAs. As shown in figure 1(d), the pull force generated by the left uniaxial actuator is denoted as F Left and the corresponding arm is l . Left Similarly, the pull force generated by the right uniaxial FCDEA is F Right and the corresponding arm is l .
Right The torque M imposed on the rotating mechanism can be obtained as: Negative stiffness is known to promote the output strain of DEAs [34,35]. In this study, negative stiffness of a single uniaxial FCDEA is achieved through the antagonistic configurations and the geometric design. To demonstrate this, we assume that the rotating actuator is currently in its left stable state and the voltage is applied on the left uniaxial FCDEA through the rotating mechanism. The mechanism rotates clockwise in this situation. Assuming that both angular velocity and acceleration are small, the pre-stretch force of the left uniaxial FCDEA P Left can be written as: As the angle increases, F Right decreases while the ratio / l l Right Left rises. Substituting F Right solved from equation (2) into equation (9), we obtained P .
Left Figure 2(a) shows that the pre-stretch force of a single uniaxial FCDEA increases as the angle of the mechanism α increases from 90°to 150°, indicating that the stiffness of the actuator is negative. Figure 2(b) shows the torque-angle diagram of the rotating mechanism for different voltages. The mechanism initially stays in the left configuration (α=90°). The torque is positive if it is clockwise. When the applied voltage is lower than the trigger voltage, the initial torque at α=90°is negative, indicating that the mechanism cannot switch.
With the assumption that the material is incompressible and the deformation of the material is uniform in the uniaxial FCDEA, the total free energy E of the uniaxial FCDEA can be written as: where =´-V 3.15 10 m 7 3 is the active volume of the uniaxial FCDEA, λ 2 is fixed as 3, and λ 1 is determined by the rotation angle α of the mechanism. The total energy of the bistable rotating mechanism E sys is given as:   is the initial length of the active part of the uniaxial FCDEA ( figure 3(a)).
According to the analytic model of the bistable rotating mechanism, the switching of the bistable rotating mechanism depends on the applied voltage. Figures 4(a)-(c) show the relation between the energy E sys of the bistable rotating mechanism and the rotation angle α at different voltages. When the applied voltage f is zero ( figure 4(a)), the energy curve of the rotating mechanism has a single energy peak at α = 120°and ends at the mechanical locks of the structure (one at α = 90°and the other at α = 150°). The energy barrier is defined as the value of the peak energy minus the system energy at the mechanical locks of the structure. If the left configuration is set as the initial configuration, the energy barrier can be written as: The profile of the system energy in figure 4(a) shows that the energy barrier under f=0kV is about 0.006 J. The configurations at the two mechanical limits (90°and 150°) are stable and have the lowest system energy. The energy profile changes as f increases. Figure 4(b) shows the energy profile under f=4.5 kV. The left uniaxial FCDEA is now charged. This leads to an increase in the energy of the left lock (90°). But there still exists an energy barrier in the mechanism, meaning that the applied energy is not sufficiently large to switch the configuration. Figure 4(c) shows the energy curve of the mechanism at f=7 kV. It is clear that the energy decreases monotonically as the rotation angle increases from 90°to 150°. This implies that the applied voltage is large enough to trigger the switch of the mechanism from the left lock to the right lock. Figure 4(d) shows that the energy barrier decreases as the applied voltage increases. The critical voltage at which the energy barrier decreases to zero is the trigger voltage for switch. It demonstrates that when the voltage applied on the left uniaxial FCDEA is sufficiently large, the rotating actuator can overcome the energy barrier and switch from the left configuration to the right configuration.
Another important parameter associated with the bistable rotating mechanism is the length of the rotating mechanism L m . In this work, if L m is larger than´-7.62 10 m, 2 the FCDEAs will break down as their stretch ratio become too large. If L m is smaller than´-3.57 10 m, 2 the FCDEAs will totally relax and cannot offer pull force. In summary, when voltage is applied on the shorter uniaxial FCDEA of the bistable rotating mechanism, it becomes 'softer'. The charged bistable rotating mechanism can overcome the energy barrier from the current stable configuration to the other. After removing the applied voltage, the bistable rotating mechanism can maintain the current configuration. The pair of uniaxial FCDEAs thus forms a bistable actuators, which has been referred to as the flip-flop dielectric device [24].

Fabrication of uniaxial FCDEA
VHB series tape (3M) is a widely used type of DE. We choose VHB 4905 in our work to fabricate uniaxial FCDEAs. Figure 3(b) shows the uniaxial FCDEA consisting of two prestretched VHB 4905 membranes and five acrylic fibers. The VHB membranes are pre-stretched 3 times in both vertical and transverse direction. Acrylic fibers with a diameter of 2 mm are sandwiched between two VHB membranes to maintain the pre-stretch, forming a fiber-constrained composite membrane. Carbon grease is then brushed on the top and bottom surfaces of the composite membrane as compliant electrodes with a passive border of 2 mm in width on each side. Copper wires are used as electrodes and four clamps are used to fix the upper boundary of each membrane.

Characterization of uniaxial FCDEA
After being relaxed for 12 h, the uniaxial FCDEA is stretched in the vertical direction under a 100 g payload for 30 min. The copper electrodes of the uniaxial FCDEA are connected to a high voltage source (Trek 610E). The applied voltage increases from 0 kV to 8 kV, during which the deformation of the uniaxial FCDEA is recorded by a camera.
To study the effect of viscoelascity, three different ramping rates (500 V min −1 , 200 V min −1 and 100 V min −1 ) are used. Figure 5(b) shows that the response of the actuator does not change at ramping rates lower than 200 V min −1 , meaning that the effect of viscoelasticity could be minimized in these circumstances. During the fabrication process of the uniaxial FCDEA, we held the edges of the actuator and brushed with carbon grease. As a result, the edges (each is 2 mm in width) are not covered with carbon grease ( figure 5(a)). The width of the active part and the passive part of actuator is 41 mm and 4 mm, respectively. By taking the passive part into account, the theoretical prediction matches well with our experimental measurements at ramping rates of 100 and 200 V min −1 ( figure 5(b)).

Fabrication of the bistable rotating mechanism
Figure 1(a) shows the components of the bistable rotating actuator which consists of a pair of uniaxial FCDEAs and a rotating mechanism. Figure 1(b) shows the schematic of the rotating mechanism. Triangular frames are acrylic made by laser cutting. Connectors are made from 3D printing. The rotation angle output is determined by the geometry of the mechanical locks. In the analytical model, the angle output is designed to be 60°. However, due to fabrication challenges, the angle output is 45°in our rotating mechanism (from 97.5°t o 142.5°). Two FCDEAs are pre-stretched in the actuation direction under a dead load of 100g for 30 min and attached to the rotating mechanism. The rotation of the system is confined in the horizontal plane by an acrylic frame.

Characterization of the bistable rotating mechanism
We test the average switch time of the bistable rotating mechanism under different applied voltages. The switch time of the configuration change from left to the right is denoted as    6(a)). The left configuration is set as the initial configuration. When voltage is applied, the mechanism switches back and forth and the average switch time is recorded as shown in figure 6(b). In particular, at a voltage of 7 kV, it takes about 7 s for the bistable rotating mechanism to switch from one stable configuration to the other. This corresponds to a reasonable frequency for real life applications. The switch between the stable configurations has been recorded in supplementary movie 1 is available online at stacks.iop.org/SMS/29/ 015008/mmedia. It is seen that the mechanism rests stably at left configuration and right configuration at zero voltage. When a sufficiently large voltage is applied, the rotating mechanism overcomes the energy barrier and switches from one configuration to the other within a few seconds. Once the mechanism is flipped, the voltage can be removed, and the stability of the mechanism still maintains. Figure 6(c) shows the repeatability of the bistable rotating mechanism under voltage of 7 kV (see also supplementary movie 2). The bistable rotating mechanism shows high repeatability with the rotation angle between 97.5°and 142.5°. Furthermore, we test the torque output of the bistable rotating mechanism with different applied voltage ( figure 6(d)). The torque of the bistable rotating mechanism at rotation angle at 97.5°changes from negative values to positive values (clockwise is positive) with increasing applied voltages from 0 kV to 8 kV. The torque is compared with the system energy model, the trigger voltage respected to the torque measured by experiment is 5.3 kV. The result agrees well with the trigger voltage of 5.1 kV predicted by the energy model of the bistable rotating mechanism.

Design and fabrication of the binary manipulator
A binary manipulator composed of three bistable rotating mechanism in series is set up. The control system is built with a high voltage source and 6 switches ( figure 7). The switches are aligned in two columns. The left column of the switched is designed to control the three uniaxial FCDEAs on the left side of the manipulator, while the right column governs the remaining uniaxial FCDEAs. Each switch changes between 'on' and 'off', corresponding to the actuated state and

Experimental results
In the experiment, we connect the manipulator to the control system and set the voltage as 7 kV. We number the bistable rotating mechanism of the 3-segments manipulator from the end-effector to the fixed end ( figure 9(a)). We characterized the time (switch time) required to switch between different configurations. There is a total of 8 individual configurations. Thus, a total of 56 individual configuration switches exist. We separated these 56 configuration switches into three categories depending on the status (on or off) of the unit rotating mechanism nearest to the fixed end of the manipulator ( figure 9(b)). It is found that the switch time of the manipulator is governed by the switch time of the mechanism located nearest to the fixed end of the manipulator. Moreover, the switch time of the mechanism is related to the moment of inertia of the components it carries. The closer the mechanism is located with respect to the fixed end, the longer it needs to finish the configuration switch.

Simulation results
We performed analysis of the workspace of the binary manipulators based on a forward kinematic model. The illustration of the simulation program is shown in figure 10. Based on the model, we characterize the area and density of the workspace. The area and density of the workspace are defined as follows: (1) area of workspace: where A is the total area of unit blocks that contain the end-effector of the manipulator and = A l m 0 2 the characteristic size of the bistable rotating mechanism. The length of the triangular frame, L m , is indicated in figure 1(d).
(2) density of workspace: r = , where n is the number of rotating mechanisms in the manipulator. Figure 11(a) shows the configurations of a binary manipulator with 5 rotating mechanisms, i.e., n=5. The position of the end-effector in each of the configurations was recorded and linked with a unit block in the space ( figure 11(b)). The area of the workspace of the manipulator can be estimated as the total area of the unit blocks that contain the end-effectors of the manipulator normalized by the characteristic size of the rotating mechanism. Note each unit block may contain multiple end-effectors. As a result, the more end-effectors a block contains, the accuracy of the manipulator in this block is higher.
The variations of the area and density of the workspace of the manipulator as a function of the number of rotating mechanisms are shown in figure 12. It is seen that the area of the workspace grows rapidly as the number of rotating mechanisms increases from 1 to 16. At n=16, there exists a configuration where the end-effector can intervene with the fixed end ( figure 13). The critical number of rotating mechanisms required for such interference, n , c can be determined by where α is range of the rotation angle (45°in this program).

Conclusion
In summary, we present a bistable rotating mechanism that integrates a pair of FCDEAs attached on a rotating mechanism with an antagonistic structure. The geometric structure of the mechanism leads to a single convex free energy curve of the FCDEAs. Mechanical locks were adopted to stop the mechanism at target angles and create two symmetric stable configurations. Since the actuation element is dielectric elastomer, negatives stiffness feature is achieved to help enlarge the actuation range of the FCDEA. The bistable actuators proposed in this work possess a combination of large actuation range and repeatable actuation. It is different from conventional bistable systems in that: (1) It achieves the bistability, repeatability and large actuation at the same time. The geometric design is beneficial to the bistability. The antagonistic is beneficial to the repeatability. Negative stiffness enlarges the actuation range. (2) It contains discrete stable configurations that are determined and maintained by the mechanical locks of the structure. (3) It does NOT require applied voltage to maintain the stable configurations. After triggering the switch to the target stable configuration, the applied voltage can be fully removed.
Moreover, our bistable rotating mechanism features capabilities of simple design, large load capacity and building binary system. The bistable rotating mechanism reveals a new route for the design of large-scale binary system and soft robotics. In the present work, constitutive equations based on continuum mechanics are used to describe the electromechanical behavior of the uniaxial FCDEA, and a geometric model is introduced to describe the rotating frame. The constitutive equations and the geometric parameters can easily be changed to describe the behavior of similar bistable rotating mechanisms. The velocity of the actuation is largely limited by the viscoelasticity of the dielectric elastomers [36]. The   viscoelastic deformation consumes the majority of the time required to complete the configuration switching. This could be solved by reducing the viscoelasticity of the materials and is left for future studies.
Experiments and simulations have been performed to prove that the advantages of the binary manipulator made from the bistable rotating mechanisms. The simulation based on forward kinematic model shows the area and density of the workspace of the binary manipulator grows rapidly as the number of the bistable rotating mechanism grows. The result of experiments and simulations implies potential applications of this type of binary manipulator in complex environment and accurate operation such as planetary exploration and surgery.