An Arch-Shaped Electrostatic Actuator for Multi-Legged Locomotion

Abstract

A simple actuator to create non-reciprocal leg motion is imperative in realizing a multi-legged micro-locomotion mechanism. This work focuses on an arch-shaped electrostatic actuator as a candidate actuator, and it proposes the operation protocol to realize a non-reciprocal trajectory. The actuator consists of two hard and flexible sheets and a leg attached to the flexible sheet. The flexible sheet is deformed through an electrostatic zipping motion that changes the height and/or angle of the attached leg. The fabricated prototype weighed 0.1 g and swung about 15 degrees with the applied voltage of 1000 V. The swinging force exceeded 5 mN, five times the gravitational force on the actuator’s weight. Large performance deviations among prototypes were found, which were due to the manual fabrication process and the varying conditions of the silicone oil injected into the gap. The trajectory measurement showed that the leg tip moved along a non-reciprocal trajectory with a vertical shift of about 0.3 mm between the forward and backward swings. The prototype locomotion mechanism using four actuators successfully demonstrated forward and backward motions with the non-reciprocal swing motion of the four legs. The observed locomotion speed was about 0.3 mm/s. Although the speed was limited, the results showed the potential of the actuator for use in multi-legged micro-locomotion systems.

1. Introduction

Miniature robots with scales from millimeters to centimeters are promising for applications in narrow or small confined spaces. Such robots could easily reach narrow gaps and inspect internal situations that typical-scale robots can hardly achieve. For example, previous studies discussed robot operations inside pipes [1,2,3] or in the crevices of rubble [4,5,6,7].

The miniature robots found in the literature can be categorized into three types from the viewpoint of locomotion mechanisms: vibration, inchworm, and multi-legged (see Figure 1). The vibration type utilizes slanted bristles arranged under the robot body [8,9,10,11]. When the robot body is vibrated, the slanted bristles provide propulsive force due to their asymmetric friction against the ground. The vibration can be excited using magnetic fields [8,9], centrifugal force [10], and electrostatic force [11]. This type has a considerably simple structure and can be easily created; however, its motion is limited to one direction.

The inchworm type typically has two legs with controllable adhesion to the ground. The body length between the two legs can extend or contract, and the locomotion is achieved by synchronizing the extension/contraction with the adhesion control for the two legs [12,13]. This type can move in two directions, forward and backward, through changes in the timing of adhesion control. Previous studies utilized electrostatic force for adhesion control [12,13]. Some studies directly utilized passive friction instead of active adhesion control [14,15,16]. For extension/contraction, piezoelectric actuators [14], magnetic actuators [15], and fluid actuators [16] have been utilized.

The multi-legged type, which this study focused on, achieves locomotion by swinging multiple legs alternately [17,18,19,20]. This type of robot can move in two directions (forward and backward), and it is expected to achieve more complex motions using multiple legs. In previous studies on multi-legged miniature robots, legs were actuated using electromagnetic motors [21,22,23], SMAs (shape-memory alloys) [24,25] and piezoelectric elements [26,27,28]. Electromagnetic motors perform well, but their dimensions and the need for motion conversion make them unsuitable for miniaturization. SMAs and piezoelectric actuators seem more suitable for miniaturization. However, their limited displacements require large and complex amplification mechanisms. Therefore, another simple actuation mechanism is required.

Multi-legged robots realize locomotion using non-reciprocal motions of their legs. Here, non-reciprocal motion means that a leg moves along different trajectories for forward and backward movements; the leg goes through a lower trajectory to kick the ground in backward movement, while the leg is lifted in forward movement to avoid contacting the ground. The creation of non-reciprocal trajectories remains challenging for miniature actuation mechanisms. In previous studies, for example, miniature mechanisms using piezoelectric actuators have been studied [29,30,31,32]. In the studies, the leg lift distance was on the order of 10 μm or less. This is much smaller than the height of the robots, which is a few mm to tens of mm. A more significant lift is expected for stable locomotion.

With this in mind, this study aimed to develop a simple actuation mechanism that can produce a more significant leg-lift distance. To realize a simple mechanism, this work focused on an arch-shaped electrostatic actuator, which has already proven its simplicity in a vibration type [11]. Although the actuator reported in [11] only generated vertical vibration, a different configuration was proposed in [33] to produce horizontal motion. This work proposes a combined motion principle to realize vertical and horizontal motions in one actuator. Then, a non-reciprocal motion trajectory was created by integrating the two motions, and it was applied to multi-legged locomotion.

In the next section, the paper describes the structure and operation principles for the leg swing and lifting, and it explains how non-reciprocal leg motion is realized. That section then explains the actuator’s fabrication process and illustrates a fabricated prototype. A theoretical analysis of the motion of the leg swing is also provided. In Section 3, the basic performance regarding the leg swing, lifting, and non-reciprocal motion is experimentally investigated. The prototypes used in the experiments were fabricated manually, and the performance variation due to the manual fabrication is discussed. Section 4 reports on a multi-legged locomotion mechanism using four actuators. The four actuators, grouped into two categories, were driven to swing the corresponding legs alternately along non-reciprocal trajectories. The experimental observation confirmed that the mechanism could locomote both forward and backward by changing the phase shift between the operation commands to the two actuator groups. Finally, Section 5 provides the conclusions of this work with some future remarks.

2. Arch-Shaped Electrostatic Actuator

2.1. Structure

Figure 2 shows the structure of the arch-shaped electrostatic actuator of this work. It mainly consists of two sheets, one of which is hard and the other is flexible. Two stator electrodes are formed on the bottom of the hard sheet, which is then covered with a thin film for surface insulation. The flexible sheet is conductive and formed into an arch shape, which is adhered to the stator sheet at both ends. In the gap between the stator and the flexible sheets, silicone oil is injected near the ends of the gap to facilitate the zipping motion described below. A thin plate, which functions as a leg, is attached at the crest of the arch. The zipping motion deforms the flexible sheet, which then swings the attached leg.

2.2. Operation Principle

The actuator operates through a zipping motion [34] caused by electrostatic force, which is illustrated in the upper inset of Figure 2. When a high voltage is applied between a stator electrode and the flexible sheet, the electrostatic force closes the gap from its end, which constitutes the zipping motion. If one end of the gap is closed, the flexible sheet deforms to shift the arch’s crest away from the activated stator electrode, as shown in Figure 2. The crest shift changes the attached leg’s orientation toward the activated electrode. Through alternate applications of voltage to the two stator electrodes while connecting the flexible sheet to the electric ground, the leg swings back and forth. On the other hand, when a high voltage is applied to the two stator electrodes simultaneously, the flexible sheet is attracted to the stator via electrostatic force on both ends. This changes the height of the arch and lifts the leg.

When the swing and lifting motions are combined, the leg moves along a non-reciprocal trajectory, which is imperative for multi-legged locomotion. For example, if the actuator is used to realize leftward locomotion, the leg should move along a lower trajectory when swinging rightward to kick the ground. On the other hand, when it swings leftward, it should move along an upper trajectory to avoid contact with the ground. Such a motion is realized by controlling the voltage on the two stator electrodes, as shown in Figure 3. For the rightward (or backward) swing, the leg is lowered by deactivating the two electrodes, as in (a) in the figure, followed by the activation of the right electrode for swinging, as in (b). On the other hand, for the leftward (forward) swing, both electrodes are activated to lift the leg, as in (c). Then, the right electrode is deactivated, and the leg swings to the left, as in (d). Through the repetition of this sequence, the leg swings along a non-reciprocal trajectory. The opposite motion can also be realized by changing the roles of the two electrodes.

2.3. Prototype Fabrication

Figure 4 shows the fabrication process employed in this work. The figure illustrates the batch fabrication of four prototypes. First, the electrodes (copper foil tape) are cut, as shown in (a), and pasted onto the stator sheet (polyimide, 80 $\mathsf{\mu }$m thick), which is then covered with an insulating film (polyester, 6 $\mathsf{\mu }$m thick). The stator sheet is then fixed onto the plastic fixing base shown in (b), and the rod, having an arch-shaped cross-section, is placed on the stator sheet. Then, a flexible sheet (stainless shim tape, 5 $\mathsf{\mu }$m thick) is arranged over the rod. The cover, whose bottom surface also has an arch-shaped cross-section, is arranged over the flexible sheet to fix the shape of the flexible sheet, as in (c). After that, the flexible sheets are glued to the stator sheet at their ends using polyimide tape (30 $\mathsf{\mu }$m thick) and an adhesive bond. After the cover is removed, the legs (polypropylene plate) are bonded to the flexible sheets, and the rod is pulled out, as in (d). Finally, the structure is cut into four actuators.

Figure 5 shows the size of the prototype actuators and the appearance of one prototype. The length of the actuator is 16 mm, and the width is 9 mm. The arch of the flexible sheet has a base length of 8 mm and a height of $1.2$ mm. The leg has a thickness of $0.7$ mm, a height of 1 mm, and a width of 6 mm. The weight of one actuator is about $0.1$ g. For operation, silicone oil (Shin-Etsu Chemical Co., Ltd., KF-96-50cs, Tokyo, Japan)was injected near both ends of the gap. The injection was done manually using a pipette dropper. The amount was not precisely controlled, but it was estimated to be about 0.5 mL. However, as the silicone oil was squeezed out from the gap during actuation, the amount in the gap could fluctuate during operation. In the experiments described later, the oil was periodically wiped off and injected again to regulate the amount.

2.4. Modeling

The electrostatic zipping motion has been analyzed in the literature [11,35,36]. Since it involves non-linearity, previous studies utilized numerical simulations, such as finite element analyses. This work proposes another simple analysis. The analysis assumes that the flexible sheet follows the linear theory of elasticity and buckles in the first buckling mode of a beam fixed at both ends. Figure 6 shows the first mode buckling with a compressive force, P, and a bending moment, M. In the actuator, the force P comes from the glue connecting the flexible sheet and the stator, and the moment M comes from the electrostatic force. Since the glue is considered strong enough, we assumed that the moment M determines the buckling shape.

In linear buckling theory, the buckling shape is expressed as

(1)$y\left(x\right)={\displaystyle \frac{M}{P}}\left(1-cos{\displaystyle \frac{2\pi }{L_b}}x\right)=A\left(1-cos{\displaystyle \frac{2\pi }{L_b}}x\right),$

where $L_b$ is the base length, and $A=M/P$ is the amplitude of the buckled shape. On the other hand, the required moment is calculated from the shape as

(2)$M=-{\left.E{I}_y{\displaystyle \frac{d^2}{d{x}^2}}\left\{A\left(1-cos{\displaystyle \frac{2\pi }{L_b}}x\right)\right\}\right|}_{x=0}=-E{I}_{y}A{\displaystyle \frac{4{\pi }^2}{L_b^2}},$

where E is the Young’s modulus of the flexible sheet, and $I_y$ is the cross-sectional secondary moment.

For simplicity, we assumed that the distributed electrostatic force produces a concentrated moment at the edge. When assuming that the electric field is perpendicular to the stator surface, the moment produced by the distributed electrostatic force, $M_e$, is calculated as

(3)$M_e=-{\displaystyle \frac{{ϵ}_m}{2}}{V}^2{\int }_0^{L_e}{\displaystyle \frac{1}{{(\frac{{ϵ}_m}{{ϵ}_{in}}{t}_{in}+y)}^2}}xdx$

where ${ϵ}_m$ and ${ϵ}_{in}$ are the absolute permittivity of the silicone oil and the insulating film, $t_{in}$ is the thickness of the insulating film, and $L_e$ is the integration limit that is set to the effective length of the stator electrode.

In the next step, we estimated the shape parameters, $L_b$ and A. Theoretically, they can be obtained via the elliptic integral of the second kind with the information of the sheet length. This work, however, approximated the relation using a linear function for simplicity. We defined the length of adhesion between the flexible sheet and the stator as $\Delta L$. Then, the shape parameters, $L_b$ and A, were defined as functions of $\Delta L$.

(4)$\begin{array}{c}\hfill L_b(\Delta L)=L_{b0}-\Delta L\end{array}$

(5)$\begin{array}{c}\hfill A(\Delta L)=A_{max}-(A_{max}-A_{min}){\displaystyle \frac{L_a}{\Delta L_{max}}}\end{array}$

where $L_{b0}$ is the original length and is 8 mm for the prototype, $\Delta L_{max}$ is the maximum adhesion length, and $A_{max}$ and $A_{min}$ are the maximum and the minimum of the amplitude, A. For the prototype, the maximum length, $\Delta L_{max}$, was set to the length of the stator electrode, 3.5 mm. The maximum amplitude, $A_{max}$, is obtained when $L_a=0$, and it was 0.6 mm for the prototype. The minimum amplitude, $A_{min}$, is obtained when $L_a=\Delta L_{max}$. This was found in an exploratory manner from the numerical integration of the buckled shape length. The resulting $A_{min}$ was about 0.4566 for the prototype. Using these relations, we can obtain the required voltage by solving $M=M_e$ for an arbitrary $\Delta L$. Then, by eliminating the parameter $\Delta L$, we obtain the relationship between the voltage and the shape.

Figure 7a shows the obtained relationship between the voltage and the change in the arch height. This relation qualitatively matches the result of a finite element analysis in a previous study [11]. Once we knew the arch shape, we could calculate the angle of the leg based on geometric relationships. The calculation result is shown in Figure 7b.

3. Experiments

For the prototype actuators, leg swing, lifting, and their combination were evaluated. As prototype actuators sometimes suffered from electrical breakdowns during high-voltage applications, different experiments utilized different prototypes.

3.1. Experimental Setup

Operation voltages applied to the stator electrodes were generated via high-voltage amplifiers (NF circuit design, HVA-4321) that amplified signals from a signal generator with a gain of 1000. The motion of the leg was recorded using a high-speed camera system (Keyence, VW-6000), and the recorded movies were processed using the camera system’s built-in motion analyzer. To measure the force, a load cell (KYOWA, LTS-50GA with an amplifier DPM-711B) was used, and its output was recorded using an oscilloscope.

3.2. Leg Swing Characteristics

The motion and the output force for the leg swing were evaluated. The setup to evaluate the leg swing is shown in Figure 8. First, the step response of the leg-swing motion was measured for one prototype by applying a step voltage of 1000 V to one stator electrode. The other stator electrode remained electrically open, and the flexible sheet was grounded. The swing angle was defined as the angle between the tip of the leg before and after the movement with respect to the center of the stator (see Figure 9b). The result and the corresponding snapshots are shown in Figure 9a,c. The measurement was repeated five times for the same prototype, and the results are plotted in the figure with different colors. As can be seen from Figure 9a, the response varied from trial to trial, and the time required for the swing was found to range between 0.07 and 0.13 s.

To investigate the reasons for the performance variations, we measured the capacitance of another actuator while repeating step-response measurements in the same manner. The measured capacitance showed no significant change despite the change in the step response with each measurement. We also tried to investigate the operation current, but the current was too small (less than 1 $\mathsf{\mu }$A in our theoretical estimation) to be detected using our experimental setup. Therefore, we could not detect variations from an electrical point of view. On the other hand, it was found that silicone oil sometimes flew out from the gap during actuation and adhered to the leg, as shown in Figure 10. Therefore, it is highly possible that varying oil conditions caused the variation in performance. Especially when the oil adhered to the leg, it could modify the leg movement via surface tension.

Next, as a static characteristic, the relationship between the applied voltage and the leg angle was evaluated. From the neutral state in which no voltage was applied to the stator electrodes, a ramp voltage changing from 0 V–1000 V with a speed of 500 V/s was applied to one stator electrode, and the change in the leg angle was measured. The results for the three prototypes are shown in Figure 11. The results show that the behaviors were considerably different among prototypes. This variation would be partly due to the differences in the silicone oil conditions described above. More importantly, however, the difference was due to the variations in their arch shapes and the bonding conditions to the stator, as the bonding was done manually. In fact, the capacitance measurement for another set of actuators showed a large variation (the actuator capacitance was 5 to 8 pF), indicating that the conditions of the flexible sheets were not uniform.

The plot also shows the theoretical result for the leg angle. Compared to the theoretical result, the experimental results showed much smaller angles at lower voltages. This would imply that there is some resistance force in the actuator that prevents the motion. Although it requires further investigation, possible causes are the effect of the oil and structural failure, such as a misalignment of the film.

These results suggest that controlling the angle in a continuous manner is almost impossible for this actuator. Therefore, the leg swing should be controlled in a discrete manner by switching between three states: the left end, the right end, and the neutral state. An angle change of about $\pm 15$ degrees is expected in such a discrete control with an applied voltage of 1000 V.

Thirdly, the output force of the leg-swing motion was evaluated. The measurement setup is shown in Figure 12. In this experiment, a bipolar pulse voltage ( $\pm 1000$ V and 20 Hz) was applied because it is known that a mono-polar DC (direct-current) voltage gradually reduces the output force due to the charging of the insulator surface [37,38,39]. The bipolar pulse voltage was expected to have the same effect as the DC voltage in terms of the output force since the negative voltage should produce the same electrostatic force as the positive one.

For the measurement, a load cell was arranged to contact the leg from the side. The load cell and the leg were moved to $0.4$ mm without applying voltage to the actuator. The position of $0.4$ mm corresponds to the leg-swing angle of about 15 degrees. Then, the bipolar voltage was applied to the stator electrode near the load cell, while the other stator electrode remained electrically open, and the flexible sheet was grounded. This created the leg swing force toward the load cell. The load cell then started the measurement and was moved to $-0.4$ mm at a speed of $0.2$ mm/s using a motorized stage (SIGMAKOKI CO., LTD., OSMS20-35(X), Saitama, Japan).

The measured forces for the three prototypes are shown in Figure 13. The results show that the force considerably differed among the prototypes. However, all the measurements show that the actuator generated a force of more than about 5 mN, which is five times the gravitational force on its weight, for the range between -0.1 and 0.4 mm. This indicates that the leg can kick the ground to generate propulsive force, even under the weight of the actuator. The difference among prototypes was due to the different fixing conditions of the flexible sheets, as discussed for the angle measurement. In addition, the difference in leg-bonding conditions among the prototypes probably affected the force measurement results.

3.3. Leg Lifting Characteristics

The characteristics of the leg lifting motion were evaluated. First, the step response was measured by applying a step voltage of 1000 V to the two stator electrodes. The measured vertical displacements of the tip of the leg are plotted in Figure 14a. The measurement was repeated five times for the same actuator. As seen in the plot, the responses were more consistent than the leg-swing motion. One thing that should be noted is that the leg did not move perpendicularly during the lifting. The flexible sheet deformed from one side as the voltage was applied, resulting in an asymmetric shape, as shown in Figure 14b. The instability of this asymmetric deformation probably caused the difference in the final values of the step responses.

Next, the load characteristics were evaluated. The actuator should endure the robot’s weight when employed in a multi-legged locomotion mechanism. To reveal how the flexible sheet deforms through such a vertical load, the displacement-load characteristics were measured using the setup shown in Figure 15. This measurement was done without applying voltage to the actuator. The actuator, flipped upside down, was pushed by a load cell from the top. The motorized stage moved the load cell at $0.2$ mm/s. The results are shown in Figure 16. The measurement was carried out for three prototypes.

As seen in the plot, two different behaviors were observed. In one case, a local peak appeared at around a $0.2$ mm displacement, as shown in red and blue plots. In this case, first, the leg sunk more or less vertically, as shown in picture A. Then, the flexible sheet was deformed asymmetrically, leading to the force drop (pictures B through D). After that, the deformed sheet gradually collapsed, resulting in a gradual increase in force. In the other case, the yellow plot, the flexible sheet was deformed asymmetrically from the beginning, and the force continuously increased from the beginning to the end. The results show that, regardless of the deformation mode, the actuator was deformed by about $0.1$ mm when the actuator’s body weight was loaded.

3.4. Non-Reciprocal Motion

Combining the swing and lifting motion allows the actuator’s leg to move along a non-reciprocal trajectory, which is imperative for multi-legged locomotion, as discussed in Section 2.2. The non-reciprocal motion was tested using a bipolar voltage of 1000 V, as shown in Figure 17. Snapshots of an obtained motion are shown in Figure 18, and trajectories of the leg’s tip at different frequencies are shown in Figure 19 (see video in the Supplemental Materials). Here, the frequency refers to the cycle of the non-reciprocal motion or $1/T$, where T is the period defined in Figure 17. Each plot shows a trajectory for five cycles.

The results indicate that the trajectory is stable for lower frequencies and clearly non-reciprocal; the trajectories for forward and backward swings have about a $0.3$ mm shift in the vertical direction. However, the motion was squashed at a higher frequency, $6.0$ Hz, and the difference between the forward and backward swings was not clearly observed. These results suggest that the response frequency of this motion is about $3.0$ Hz, under which a $0.3$ mm vertical shift is expected. Given that the vertical displacement due to the actuator’s body weight was about $0.1$ mm, the actuator can realize non-reciprocal motion even if the body weight is loaded.

3.5. Discussion

The fabricated actuators showed a maximum swing angle of about 15 degrees, which corresponds to the traveling distance of 0.4 mm for the leg tip. The response time of the leg swing was about 0.1 seconds. In the combined motion, the response frequency of 3.0 Hz was obtained. In previous studies on miniature multi-legged systems, the response time was about 0.02 ms for a system using piezoelectric actuators [30] and 0.7 s for a system using SMAs [24]. The response of the actuator of this work is located in the middle of them.

The leg lift in the non-reciprocal motion was about $0.3$ mm, and it is 14% of the actuator height ($2.2$ mm). This is significantly larger compared to the piezoelectric mechanism [29,30], whose leg lift was in the order of micro-meters.

4. Multi-Legged Walking Mechanism

A multi-legged locomotion mechanism was developed using four actuators, as shown in Figure 20. The main body was a polyimide sheet with 80 $\mathsf{\mu }$m. A thin PLA (poly-lactic acid) rod was glued to the polyimide sheet as a backbone to increase the stiffness. Then, four actuators were glued to the bottom surface of the body. The size of the mechanism was $9\mathrm{mm}\times 70\mathrm{mm}\times 3\mathrm{mm}$. The total weight was $0.7$ g. This means that each actuator should support about $1.8$ mN. According to Figure 16, this load will cause the actuator’s vertical deformation to be less than $0.2$ mm, which is acceptable when considering the non-reciprocal motion observed in Figure 19.

The four actuators were grouped into two, acting to swing the legs alternately along the non-reciprocal trajectories. The voltage waveforms shown in Figure 21 were applied to the four electrode sets in the two groups. The voltage amplitude was 1000 V, and the cycle frequency of the swing motion was $0.5$ Hz. The resulting body motion measured using a laser displacement meter (OMRON: ZX1-LD100A81) is shown in Figure 22. Figure 23 shows the motions of the mechanism and its leg.

As shown in Figure 22, the forward and the backward locomotions were realized by changing the role of the two actuator groups (see video in the Supplemental Materials). Similar speeds were obtained in both directions. However, during the experiments, it was observed that some of the legs sometimes failed to move, probably due to the weight of the mechanism. This should be improved in future work.

Table 1 compares the performance of miniature locomotion mechanisms that utilizes simple actuators.The overall performance of the mechanism developed in this work was found to be within a similar range to that using SMA. However, the ratio of the leg lift against the body size was more significant with this work. Compared to piezo-actuated mechanisms, the leg lift achieved in this work is significantly larger, but the motion speed is far slower, indicating the need to improve the motion speed. Since the motion in this actuator was possibly damped due to silicone oil, removing it might improve the speed. Future studies should investigate an operation without silicone oil.

5. Conclusions

A simple actuator to create non-reciprocal leg motion is imperative in realizing a multi-legged miniature locomotion mechanism. As a candidate actuator, this work focused on an arch-shaped electrostatic actuator and proposed the operation protocol to realize a non-reciprocal trajectory. The actuator has a considerably simple structure, and the prototype fabricated in this work weighed only $0.1$ g. The experiments revealed that the actuator could swing its attached leg for $\pm 15$ degrees at an applied voltage of 1000 V. The swinging force of more than 5 mN was obtained, which is five times the gravitational force on the weight of the actuator. On the other hand, the motion or the output force varied between trials and prototypes. This suggests that continuous position control is difficult for this actuator; discrete control between three states, namely the neutral and the full swings to left and right, would better suit it.

A multi-legged locomotion mechanism was prototyped using four actuators. Although motion failure was sometimes observed for each actuator, the mechanism successfully moved forward and backward using the legs’ non-reciprocal motions. This showed the potential of this actuator to be applied to multi-legged miniature locomotion systems. The locomotion speed was still limited, and future work should try to improve the speed by upgrading the frequency response or modifying the dimensional parameters.

The experiments of this work showed that considerable performance variations exist in the actuator prototypes. The variation is probably due to the instability in the conditions of the silicone oil and fabrication errors caused by manual work. Therefore, future work should investigate the possibility of removing the oil. Some studies reported that actuators of similar classes worked without oil [40,41], and their findings might also be applied to this actuator. The removal of the oil might also contribute to an improvement in the motion speed. To reduce fabrication errors, an improvement in the fabrication process is important, especially for fixing the flexible sheet.

Finally, although this work did not discuss this possibility, the high operation voltage might be a problem in applications, and therefore, a reduction in the voltage should be pursued. For other electrostatic actuators using a similar zipping motion, it has been reported that using a thin insulating layer with high permittivity can reduce the operation voltage [42]. That should also be investigated for this actuator.

Author Contributions

Conceptualization, Y.S. and A.Y.; methodology, Y.S.; experiments, Y.S.; writing—original draft preparation, Y.S.; writing—review and editing, A.Y.; project administration, A.Y. Both authors have read and agreed to the published version of the manuscript. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors upon request.

Conflicts of Interest

The funders had no role in the design of the study, in the collection, analyses, or interpretation of data, in the writing of the manuscript, or in the decision to publish the results.

Footnotes

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Figures and Table

Figure 1. Three types of miniature locomotion mechanisms.

Figure 2. The structure and the operation principle of the arch-shaped electrostatic actuator. A high voltage is applied to the electrode shown in a red color. The flexible sheet deforms through a zipping motion, as shown in the upper inset. The attached leg will be lifted through the application of voltage to the two electrodes. On the other hand, a voltage application to one electrode will swing the leg.

Figure 3. Realization of a non-reciprocal trajectory of the leg tip.

Figure 4. Fabrication process of the arch-shaped electrostatic actuator. (a) The components of the stator sheet. (b) The stator sheet is fixed onto the fixing base, and the flexible sheet (stainless shim tape) is arranged onto the stator with the rod. (c) The flexible sheet is formed in an arch shape by combining the rod and the cover. (d) The flexible sheets are glued, and the structure is removed from the base. The structure then is cut into individual actuators, such as in (e).

Figure 5. Dimensions and appearance of the arch-shaped actuators fabricated in this work.

Figure 6. First buckling mode of a beam fixed at both ends. The analysis of this work assumed that the flexible sheet buckles in this mode.

Figure 7. The calculation results of the analytical model. (a) Change in the height, or the vertical displacement, of the flexible sheet. (b) Leg-swing angle.

Figure 8. Setup for measuring the leg-swing characteristics. When the motion was measured, the load cell was removed, and the camera measured the motion of the actuator. On the other hand, the load cell was connected when the output force was measured.

Figure 9. Step response of the leg-swing motion. In (a), a step voltage of 1000 V was applied to one stator electrode. The measurement was repeated five times for the same actuator, and it was plotted using different colors.

Figure 10. Silicone oil sometimes flew out from the gap and adhered to the leg.

Figure 11. Relationship between the leg-tilt angle and the applied voltage, measured by applying a ramp voltage. The measurement was performed for three different prototypes plotted using different colors. The applied voltage and leg-tilt angle derived from the model are also plotted.

Figure 12. Setup for measuring the output force of the leg swing. The load cell and the leg were set to [Forumla omitted. See PDF.] mm, and a bipolar voltage was applied. After that, the load cell was moved toward −0.4 mm while measuring the force.

Figure 13. Leg-swing force of three prototypes measured using the setup in Figure 12. Negative force indicates the direction of the force toward the negative x direction. The plots show that the three prototypes behaved differently.

Figure 14. Step response of the leg-lift motion. A step voltage of 1000 V was applied to the two electrodes simultaneously. The measurement was repeated five times for the same actuator.

Figure 15. Setup for measuring the load characteristics along the vertical direction.

Figure 16. The vertical resistive force when actuators were pushed by a load cell. The measurement was done for three different prototypes plotted in different colors. (a) Relationship between the vertical force and displacement. (b) Snapshots showing the actuator’s behavior under the load. The snapshots in (b) correspond to the points in (a) with the same alphabet.

Figure 17. Voltage waveforms to produce the non-reciprocal motion of the leg.

Figure 18. Snapshots showing the non-reciprocal motion for the cycling frequency of 0.5 Hz.

Figure 19. The measured trajectories of the leg tip when the actuator was driven using the waveforms in Figure 17 with different cycling frequencies. Each plot shows the trajectories of five cycles.

Figure 20. Prototype of a multi-legged locomotion mechanism.

Figure 21. Grouping of the actuators and the voltage waveforms to realize multi-legged locomotion.

Figure 22. Horizontal motion of the main body of the multi-legged mechanism. The two plots represent two different cases in which voltage patterns were swapped.

Figure 23. Photos of the multi-legged mechanism in operation and the behavior of its leg.

Table 1

Comparison of the miniature locomotion mechanisms that utilizes simple actuators.

	Wang et al. [29]	Liu et al. [30]	Asamura et al. [24]	This Study
Actuator	Piezo	Piezo	SMA	Electrostatic
Mass (g)	2.2	42.55	15.4	0.7
Size (mm)	$20\times 20\times 4.6$	$58\times 44\times 12$	$70\times 60\times 24$	$9\times 70\times 3$
Speed (mm/s)	39.3	516	0.4	0.3
Leg lift (mm)	$4\times {10}^{-3}$	$2\times {10}^{-3}$	0.7	0.3

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/robotics13090131/s1.

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