## Introduction

Robot- Robot-based manufacturing systems, such as smart factories, require efficient and controllable grippers or actuating systems1. Among the various adhesive mechanisms, electroadhesion is the most promising approach for achieving this goal2,3. A flexible functionality can also be achieved by using an appropriate substrate material4,5. Electroadhesion is the attractive force between the electroadhesive pad and the substrate6,7,8. Applying a few kilovolts generates an electric field which is strong electroadhesive forces through electric polarization9 Compared with other adhesion mechanisms, this method offers the advantages of ultralow energy consumption, gentle handling, and enhanced adaptability10,11,12.

## Methods

### Computational method

The electric field distribution E within a general electrode system can be determined using the following equation:

$$div\,\varepsilon\,grad\varphi =0,$$
(1)

where ε, φ denote the permittivity and the scalar electric potential ($$E=-grad \varphi$$), respectively. The externally generated volume-charge density was not considered. To obtain numerical solutions to Eq. (1), it is unnecessary to solve for the entire surface where the attraction force Fa occurs. On the contrary, owing to the repetitive pattern of the electrodes, E can be computed for a single segment comprising two electrodes with opposite charges40.

The electrostatic force Fe exerted on the segment can be calculated from the E distribution as follows:

$${F}_{e}= {\oint }_{s}TdS, T= \frac{1}{2}\left(E\cdot D\right)I+E\otimes D,$$
(2)

where T is the Maxwell stress tensor, D is the dielectric flux density, I is the unit diagonal matrix, and $$\otimes$$ denotes the dyadic product. Equation (2) can be simplified by assuming parallel interdigital electrodes on attractive object surfaces. The Maxwell stress tensor is expressed as follows:

$$\text{T}= \left[\begin{array}{cc}\frac{\varepsilon }{2}\left({E}_{x}^{2}-{E}_{y}^{2}\right)+\frac{1}{2\mu }({B}_{x}^{2}-{B}_{y}^{2})& (\varepsilon {E}_{x}{E}_{y}+ \frac{1}{\mu }{B}_{x}{B}_{y}+{E}_{x}{P}_{y}+{E}_{y}{P}_{x}-{B}_{x}{M}_{y}-{B}_{y}{M}_{x})\\ (\varepsilon {E}_{x}{E}_{y}+ \frac{1}{\mu }{B}_{x}{B}_{y}+{E}_{x}{P}_{y}+{E}_{y}{P}_{x}-{B}_{x}{M}_{y}-{B}_{y}{M}_{x})& \frac{\varepsilon }{2}\left({E}_{x}^{2}-{E}_{y}^{2}\right)+\frac{1}{2\mu }({B}_{x}^{2}-{B}_{y}^{2})\end{array}\right]$$

where Ex and Ey are the electric field components, Bx and By are the magnetic field components, P is polarization, and M is magnetization. As we assume our system is electrostatic condition, the B, M is neglected. The P which the function of E and ε is calculated by following equation:

$$P = \varepsilon {\varepsilon }_{relative}E$$
(3)

where εrelative denotes the relative permittivity.

The normal direction of the electrostatic force Fey acting on the segment with electrodes of length = 1 is obtained as

$${F}_{ey}={\oint }_{S}{T}_{y}dS=\frac{1}{2}\varepsilon l{\int }_{0}^{w+g}({E}_{y}^{2}-{E}_{x}^{2})dx.$$
(4)

Fey between the two parallel capacitor plates is expressed as

$${F}_{ey}=\frac{\partial {W}_{e}}{\partial y}=-\frac{1}{2}\frac{\partial {E}_{y}{D}_{y}}{\partial y}{\int }_{V}dV=-\frac{\varepsilon }{2}{\left(\frac{U}{t}\right)}^{2}wl,$$
(5)

where We, Dy, V are the total energy, the electric displacement field component, and the capacitor volume, respectively. U is the applied voltage, and t is the distance between the electrodes41.

### ML algorithms

We developed an ML model using the RF algorithm, which is known for its ability to process complex and nonlinear data. This ensemble method combines multiple decision trees to improve the prediction accuracy and prevent overfitting42. Our dataset comprised 1000 samples calculated through finite element method simulations, covering a wide range of electroadhesive actuator designs. The input features for the model included the voltage, electrode spacing, outer and inner permittivities of the protective layers, air gap, outer and inner thicknesses of the protective layers, and electrode width. These features are critical for determining the electrostatic forces exerted by actuators. We implemented rigorous training and validation procedures, including hyperparameter tuning and cross validation, to ensure the reliability of the model for predicting the effects of these diverse design parameters. In addition, we conducted an in-depth analysis of feature importance, which provided valuable insights into the most important factors that influence electrode design optimization and facilitate the improvement of the electroadhesive actuator performance.

## Results

### Electric field and force analysis

Figure 1b,c illustrate schematics of the interdigit electrodes used for electroadhesion simulation. The object is separated from the conductive electrodes by a protective layer, which is either a monolayer or a bilayer43. These protective layers serve as insulators and have ambiguous breakdown voltage shortages44,45. A simulation was conducted under the same applied voltage of 10 kV, disregarding the breaking voltage, to directly compare the electroadhesive force with respect to changes in permittivity between bilayer and monolayer protective layers. It is well-known that insulators with lower permeabilities exhibit higher breaking voltages, and vice versa25. The voltage between 1 and 10 kV is applied to one electrode, and it is set to zero (ground) for the other electrode46,47. The electric field generated between the two electrodes exerts an adhesive force on various object materials48,49,50. We have assumed that all of the electroadhesive gripping has caused no mechanical damage or chemical reactions. The detailed parameters and material properties are listed in Table 1. Further, the design of the interdigit electrode is analyzed by numerically solving a formulated mathematical model. The electrostatic force is then derived from the electric field distribution and integrated field infectivity on the object surface.

Figure 2a illustrates the electrode potential contours within a single period of the interdigitated electrode configuration. The most distinct potential gradient is observed between the adjacent electrodes, radiating outward in a radial pattern. The permittivity variations between the monolayer and bilayer protective layers profoundly affect the contour slopes. There is not a significant change observed in the voltage gradient which means the electric field along to the x-direction for both monolayer and bilayer structures, while the electric field along to the y-direction shows slightly differences at the outer layer. This electric field change which in turn influence the electrostatic forces exerted on the adhered object.

### Random forest and sensitivity analysis

Figure 3a shows that increasing the permittivity of the monolayer configuration is an effective method for enhancing the electrostatic force. However, when selecting materials for the protective layer, it is essential to consider the operational voltage requirements, particularly the breakdown voltage51. Therefore, a bilayer design with low internal and external permittivities should be carefully considered. As the permittivity of the protective monolayer increases from 1 to 3, the electrostatic force exerted on the object surface increases steadily; however, it plateaus at higher permittivity levels. This suggests that although a higher permittivity can facilitate greater electric field penetration and increase the electrostatic force, this effect is limited. Notably, the permittivity variation within the protective layer results in a nonlinear electric field distribution, underscoring the complex relationship between material properties and electrostatic phenomena.

Recently, there has been a growing interest in using data-centric approaches with machine learning and artificial intelligence to optimize design parameters. Our electroadhesive gripper system has up to 8 parameters influencing its electroadhesive force. Conventional optimization methods, such as Levenberg–Marquardt, nonlinear least squares, and Newton–Raphson, can lead to inaccuracies due to missing local minima or other reasons, which take significant time to calculate. To optimize the design parameters that significantly affect the electroadhesive force of a bilayer protective structure, we performed a sensitivity analysis using the random forest (RF) algorithm on the results of 10,000 cases. To derive the 10,000 samples, we used a simulation with random values of parameters between their minimum and maximum values. We also tested several machine learning (ML) algorithms, including k-nearest neighbor, kernel ridge, support vector, and random forest regressions (RFR)52,53,54. The RFR algorithm most accurately fit the training data, as evidenced by the 500 sample points shown in Fig. 3b. Feature importance was automatically derived from the RF model (Fig. 3c). The results revealed that the applied voltage was the most influential design parameter for determining the electroadhesive force, underscoring the critical significance of the breakdown voltage of the protective layer. The air gap between the protective layer and the object emerged as the second most significant design parameter. This gap was maintained at 50 μm to optimize computational efficiency as it is an integral part of the actuator geometry. The protective layer permittivity, which was the primary focus of this study, exhibited the lowest sensitivity in response, with the permittivity of the inner layer being marginally more sensitive than that of the outer layer. Previous research on the protective layer of bilayer protective structures did not fully consider the influence of its dielectric constant owing to its low sensitivity30. Additionally, the possibility that the electrostatic force differs from previous findings was not considered either. This study acknowledged these omissions and aimed to address them. The findings of this study can provide valuable insights into the design parameters that significantly affect the electroadhesive force in bilayer protective structures.

### Effect of design parameters on the force to object

Figure 4a,b show results of the electrostatic forces with respect to the permittivity of the bilayer. They indicate that the general electrostatic forces strengthen as bilayer permittivity increases. The permittivity of the inner layer was kept constant at 2.5, 3.5, and 4.5, whereas that of the outer layer was varied. Figure 4c indicates that the electrostatic forces reached their maximum values at the specific permittivity levels of the outer layer. These findings demonstrate the influence of the bilayer permittivity on the behavior of the electrostatic forces in a material. When the dielectric constant of the internal layer was set to 2.5, it was observed that the highest electroadhesive force of 4505 N/m2 was obtained by using an outer layer permittivity of 3.8. However, it is important to note that the electroadhesive force may vary by up to 15.7%, resulting in a difference of 702.49 N/m2 from the minimum value. As a result, it is advisable to optimize the dielectric constant when developing the bilayer. It is worth mentioning that this trend is more evident when the permittivity of the inner layer is increased. When the permittivity of 3.5 and 4.5 for the inner layer, the differences between the maximum and minimum forces reach 811.83 N/m2 and 882.33 N/m2 with permittivity of 4.4 and 4.8, respectively. This finding contradicts the previous assumption that a higher permittivity of the outer protective layer results in greater electrostatic forces.

Figure 5 shows that the electrostatic forces vary with the fixed bilayer protective layers as a function of the geometry of the interdigit electrode and the voltage applied to the electrode. In Fig. 5a, as the applied voltage increases, the adhesive force increases exponentially, which implies that the lifting force can be enhanced if the inner protective layer can sustain the applied voltage. Furthermore, decreasing the thickness of the protective layers and reducing the air gap between the objects and the outer protective layer can increase the attractive forces generated by applying a higher voltage, as shown in Fig. 5b,c. These results agree well with those reported by Liu et al., Mao et al., Guo et al., and Cao et al.14,16,55,56. As reported by Guo et al., to increase the electrostatic force, the spacing between the gap and the thick layers should be minimized14. However, a slightly different trend can be observed in the interdigit electrode width and spacing compared with the previous results, indicating that the spacing change between the electrodes generates the maximum attraction force. Figure 5d,e show that an optimum electrode width of 220 μm results in the maximum attraction force per unit area when the pad is attached to non-conductive substrates, wherein the space between the electrodes is fixed at 70 μm (optimum width/space ratio of 3.14).

Figure 6a,b illustrate the correlation between the geometry of the electrode and the electrostatic force. The electroadhesive force is calculated by considering the thickness of the protective layer and the distance between the electrodes (spacing), while the width of the electrode varies from 75 to 1000 μm. Specifically, the change in electrode width occurs simultaneously from both sides while moving in the out-plane direction of the electrode while maintaining the electrode spacing. The results indicate that the electrostatic force increases with electrode width up to a certain point, which indicates the optimal width at which the force is maximized. For thinner protective layers, such as 25 µm thickness, a more pronounced peak force is observed with an electrode width of 100 µm, while the maximum force is observed at 420 µm with the 500 µm thickness of protective layer. This result means that the optimum electrode width is dependent on the protective layer thickness. This suggests that a smaller distance between the electrode and the object enhances the electroadhesion efficiency. Moreover, the proximity of the electrodes also plays a crucial role. At a spacing of 50 µm, the most substantial force is induced with an electrode width of 225 µm. However, this force wanes when the spacing exceeds a certain threshold. This decline could be attributed to fringing effects or the dilution of the electric field over a larger area, which reduces the effective adhesive contact. The curve levels off as the electrode spacing increases, reflecting a substantial decrease in the effectiveness of the electrostatic force due to the diminished electric field intensity between the electrodes. This finding emphasizes the need for optimizing electrode width and spacing to maximize the electroadhesion force. Therefore, while designing electroadhesive systems, the physical dimensions of the electrode and air gap as well as the electrostatic principles governing adhesion must be carefully considered. Although material permittivity is a critical parameter, the geometric configuration is equally important.