The eR sensor is composed of two 0.5 mm thick urethane foam dielectric layers and three silver paste electrodes, resulting in a total thickness of 1.47 mm. The dielectric layers possess distinct densities and mechanical properties, a design that expands the measurable load range. Specifically, Dielectric Layer 1 has a density of 150 kg/m³, a compressive stress of 0.006 MPa at 25% strain, and a compression set of 1.0%, whereas Dielectric Layer 2 has a density of 240 kg/m³, a compressive stress of 0.022 MPa at 25% strain, and a compression set of 4.2%.

This configuration, utilizing dielectrics with different elastic moduli, facilitates stepwise deformation in the thickness direction. Consequently, compared to a single-dielectric sensor, the capacitance change does not saturate, enabling a detectable output over a wider input range (Zhu et al., 2022). Furthermore, the top and bottom electrodes are connected to 0V to serve as ground electrodes. These act as electrostatic shields to mitigate the influence of external noise.

The principle of the eR sensor is to estimate the external force by treating the eR as a parallel-plate capacitor and utilizing the change in capacitance. When the eR deforms owing to an external force, the electrode area A and the distance between the electrodes d change; consequently, the capacitance changes. The capacitance C stored in eR is defined by the following Equation 1: C = ε r ε 0 A d (1)

Where ϵ _r : relative permittivity of the elastomer, ϵ₀: permittivity of vacuum, A: electrode area, d: distance between electrodes. ϵ₀ = 8.854 × 10–¹² F/m.

By measuring the change in capacitance (ΔC) before and after the application of an external force, the external force can be inversely calculated as Equation 2. C was defined by considering changes in each parameter. Δ C = ε r 0 + Δ ε r ε 0 A 0 + Δ A d 0 + Δ d − ε r 0 ε 0 A 0 d 0 (2)

The relative permittivity of dielectric elastomers is generally unaffected by external forces or changes in shape, with negligible variations observed (Zhao and Suo, 2010). The dielectric layer of e-rubber uses urethane foam, which consists of a bulk material and air bubbles with different relative permittivity (Figure 1). We hypothesize that while the relative permittivity of urethane foam remains largely unchanged at small strains, it changes under load as air bubbles collapse, causing the bulk material’s properties to become dominant (O'Neill et al., 2022).

The change in capacitance is defined by the change in the elastomer permittivity, electrode area, and distance between the electrodes. However, since the changes in Δd and A are not linear but rather non-linear functions, and because the elastomer’s permittivity ϵ _r changes due to density variations caused by stress and strain, the capacitance becomes a non-linear function.

Next, we examined the relationship between ΔC and the external force F. Assuming that the electrode area of eR and the relative permittivity of the elastomer does not change, the initial capacitance C₀ and the capacitance C₁ after loading are given by Equation 1 as Equations 3, 4: C 0 = ε r ε 0 A d 0 (3) C 1 = ε r ε 0 A d 0 − Δ d (4)

Thus, the change in capacitance ΔC is calculated by Equation 5: Δ C = C 1 − C 0 = ε r ε 0 A 1 d 0 − Δ d − 1 d 0 (5)

When a normal load F is applied to an elastic dielectric (thickness d₀, area A, Young’s modulus E), the change in thickness Δd is calculated with Equation 6: Δ d = F d 0 E A (6)

Substituting (6) into (5), ΔC is calculated with Equation 7 Δ C = ε r ε 0 A · 1 d 0 1 1 − F E A − 1 = C 0 1 1 − F E A − 1 (7)

Therefore, the relationship between ΔC and F can be defined as a non-linear relationship. Furthermore, for small deformations where F≪EA, using the Maclaurin expansion, ΔC is approximated by Equation 8: Δ C ≈ C 0 · F E A (8)

This indicates that, in the small-deformation region, where the external force is considerably small, the relationship between the load and capacitance can be defined as linear. However, as the external force increases, the electrode area of the eR, the relative permittivity of the elastomer change, and the structural hysteresis of the elastomer also have an effect. Consequently, the relationship between the change in capacitance and external force applied to eR may deviate from the results obtained using this equation. Therefore, we decided to plot the external force and capacitance changes experimentally and perform regression using curve fitting.

The measurements were conducted inside a TX411N constant-temperature chamber (Kusumoto Chemicals, Ltd., Tokyo, Japan) with a force gauge (EMX-1000N, Imada, Toyohashi, Japan) installed to ensure constant humidity and temperature (Figure 2). A load was continuously applied to the sensor using a force gauge, and the change in capacitance was measured. Loads were applied from 0 to 400 N at a rate of 0.5 mm/s (Figure 3). Capacitance was measured using an LCR meter (IM 3536, Hioki EE Corp., Ueda, Japan) and a custom-developed program.

We used 30 eR samples and performed three measurements on each sample. Since EAP properties vary with temperature and humidity, measurements were taken under three conditions: Condition A (room temperature assumed: 23 °C, 50% humidity), Condition B (close to biological monitoring conditions: 40 °C, 50% humidity), and Condition C (close to conditions for clothes and shoes in a sweating state: 40 °C, 80% humidity).

Polymeric materials such as EAPs exhibit the Mullins effect, where the elasticity changes after repeated loading. In applications such as measuring pressure with insoles during walking or pinching movements of the fingers, repeated loading operations can potentially alter the measurement values. Therefore, as a durability test, we performed frequent pressure operations and measured the changes before and after the operation. The durability test setup included a constant-temperature chamber (SH-642, Espec Corp., Osaka, Japan) equipped with a pusher attached to an air cylinder (COQ2B40-40DZ, SMC, Tokyo, Japan), which was monitored using a load cell (LUR-2KNSA1, KYOWA Electronic Instruments, Chofu, Japan) (Figure 3). For the durability test, simulating actual walking, we applied 200,000 cycles of reciprocal loading from 0 to 100 N per second within a constant-temperature chamber at 40 °C and 80% humidity.

To evaluate the measurement characteristics of the eR sensor, we assessed its reliability and validity based on the Consensus-based Standards for the selection of health status Measurement Instruments (COSMIN) guideline (Mokkink et al., 2010). Reliability refers to the dispersion of sample values and its magnitude represents precision. Conversely, the closeness of the sample mean indicates that the population mean was calculated with accuracy. Because the sample values increase with external force, and consequently, the standard deviation also increases, we calculated them as coefficients of variation. Accuracy is the difference between the sample and population means, calculated using a 95% confidence interval.

To measure the measurement characteristics, we calculated the coefficient of variation and confidence interval for six conditions (Conditions A, B, and C before and after durability testing) based on the measurement values from all samples at load points of 50, 100, 150, 200, 300, and 400 N. We used t-distribution to calculate the 95% confidence interval, defining the confidence interval width as the ratio of the estimated population mean divided by the 95% confidence interval.

We investigated the hysteresis of the two dielectric layer materials used as Dielectric layers 1 and 2 (Figure 1) in the eR sensor, using a force gauge according to the procedure described in Section 2.3. The samples were 32 mm in diameter before compression. The tests were conducted at 20 °C and 45% relative humidity. Each sample was compressed from 0 N to 400 N over 12.3 s. We analyzed the relationships between nominal stress (load divided by sample area) and capacitance, and between nominal strain (deformation divided by initial thickness) and capacitance change, during both loading and unloading cycles.

To evaluate hysteresis, we calculated two metrics for each relationship: The first was the mean hysteresis error (%FS), defined as the average of the absolute differences in capacitance between the loading and unloading curves at a given input value, normalized by the full-scale output. The second was the normalized hysteresis loop area, obtained by integrating the area enclosed by the loading and unloading curves using the trapezoidal rule and dividing it by the product of the full-scale input and output ranges. These values provided quantitative measures of the reversibility and repeatability of the sensor response.

We aimed to create a smart insole for measuring plantar pressure using eR and to verify the measurement accuracy of the sensor for this purpose. The planned smart insole consists of four sensors: one at the forefoot, one each on the medial and lateral sides of the midfoot, and one on the hindfoot (Figure 4A). As shown in the figure, the insole cross-sectional feature sensors were sandwiched between the Ethylene Vinyl Acetate (EVA) foam and polyurethane foam (Figure 4B). We created six smart insoles for measurement and performed three measurements for each insole.

The measurement characteristics of the sensors inserted into the insoles were evaluated, as described in Section 2.4. The measurement and durability testing equipment were the same as those described in section 2.3. The load application speed was 10 mm/min, with loads ranging from 0 to 400 N. To simulate the plantar pressure measurement, the force gauge indenter was made of POM resin and Si rubber, rather than the metal indenter used in the 2.2’s mold (Figure 5). The indenter was used to apply a load to the insole (Figure 5). For the load test, we similarly changed the indenter and applied 200,000 cycles of reciprocal loading from 0 to 230 N per second in a constant-temperature chamber at 40 °C and 80% humidity.

The 30 eRs were tested under load conditions similar to those described for the sensor in Section 2.3. For each sample, tests were conducted three times, both before and after load application, under Conditions A, B, and C.

We plotted the relationship between the obtained ΔC and the applied load, then calculated a polynomial to represent this relationship using curve fitting. The least squares method was used for the calculation, and Python’s NumPy polyfit and poly1d libraries were used for the computation. The candidate equations for curve fitting included a 5th-order polynomial, natural exponential functions, and integer-based exponential functions. Machine learning methods can be used as an alternative to curve fitting. Nevertheless, they were not considered due to the substantial processing burden they would impose on the insole’s circuitry. The accuracy of the curve fitting was evaluated using the coefficient of determination (R2) and the Root Mean Squared Error (RMSE).

Using the system described in section 2.4, a dynamic characteristics analysis was conducted by applying continuous loads to the insole sensors to simulate walking. The loading cycles were set to simulate different walking speeds: a 2-s cycle for a slow walk (S), a 1-s cycle for a normal walk (N), and a 0.6-s cycle for a quick walk (Q). A load of 0–240 N was repeatedly applied 50 times for each condition (Figure 6). The initial 10 cycles of each loading session were excluded from the data, and the subsequent 40 cycles were used for the dynamic analysis. The capacitance of the insole was sampled every 20 ms. This loading procedure was performed under the three environmental conditions A through C, detailed in section 2.4. The same durability test as in section 2.4 was conducted to examine changes before and after the test. The investigated parameters were the difference between the loading interval and the detected waveform peak interval, and the signal drift rate. To assess whether the two groups were statistically equivalent, we conducted an equivalence test using the two one-sided test (TOST) procedure. The equivalence margin was pre-specified as 50% of the standard deviation. Equivalence was concluded if the 90% confidence interval of the mean difference fell entirely within the predefined margin, and both one-sided p-values were below 0.05. Analyses were performed using Python 3.12. The signal drift rate was calculated for all conditions from section 2. By dividing the difference in estimated load (derived from capacitance) before and after the durability test by the pre-test estimated load.

The relationship between the load value from the continuous pressure applied by the force gauge and the change in capacitance of the sensor unit alone showed a linear increase in the small deformation region up to 20 N under all conditions (A, B, and C). Beyond 20 N, it exhibits a non-linear monotonic increase, closely resembling a logarithmic curve (Figure 7). Although there were differences between individual samples, no variations were observed among the samples. After the durability test, a slight change in the capacitance output was observed; however, the shape of the curve remained consistent (Figure 7).

The coefficients of variation and confidence interval widths for all samples at the specified loads are listed in Tables 1–3. The coefficient of variation was less than 1.25%, indicating that sample variability was within acceptable precision limits. Furthermore, the confidence interval width showed a slight increasing trend after durability testing, but was generally less than 0.25%, which was deemed acceptable in terms of accuracy.

When comparing measurements from the sensor unit alone to those from sensors inserted into the insole casing, differences in the load-capacitance relationship were observed. Specifically, the sensors within the insole casing showed a gentler increase in the load-capacitance curve compared to the standalone sensor, which was influenced by the gaps within the insole casing (Figure 8).

The coefficients of variation and confidence interval widths at the specified loads are listed in Tables 4–6. The coefficient of variation was less than 8%, indicating that while the sample variability was greater than that of the standalone sensor, it was still considered within acceptable precision limits. The confidence interval width also showed a slight increasing trend after durability testing, but was generally less than 1.5%, which was deemed acceptable in terms of accuracy.

For both dielectric layers, hysteresis curves were generated by plotting capacitance against nominal stress and nominal strain. These plots revealed a disparity between the loading and unloading paths, indicating the presence of hysteresis (Figure 9). The average hysteresis error with respect to both nominal stress and nominal strain was below 10% for both dielectric layer 1 and layer 2 (Table 7). Furthermore, the normalized hysteresis loop area for both layers was 0.1 or less. While this performance is not comparable to that of state-of-the-art sensors, it demonstrates a moderate level of precision.

The load-capacitance curve for the sensors embedded within the insole casing was regressed to a polynomial. A 5th-order polynomial, as shown Equation 9, was adopted: L o a d N = a · Δ C 5 + b · Δ C 4 + c · Δ C 3 + d · Δ C 2 + e · Δ C + f (9)

For example, the plot results obtained from a forefoot sensor in one casing are shown in Figure 7. From this result, the curve fitting yielded: a = 2.26 x 10⁻³, b = −9.67 x 10⁻², c = 1.53, d = −10.4, e = 33.1, f = 1.34. The accuracy of the curve fitting was calculated as R² = 0.998839 and RMSE = 3.687313 (N). We determined that the 5th-order polynomial provided a better fit for regression than a natural exponential approximation (R² = 0.99856) or an integer-based exponential function (R² = 0.99856), and thus adopted it.

The relationship between load and capacitance differs for each sensor. Therefore, we perform an individual calibration for each sensor embedded in the insole to determine the coefficients of Equation 9.

The results for the detected pressure peak intervals are shown in Tables 8–10. No significant differences were observed in any of the conditions (p < 0.05). The signal drift rate ranged from 5% to 10%, varying with environmental and walking conditions, which suggests that correction may be necessary for long-term use (Table 11).