Capacitive Stretch Sensing for Robotic Skins

Abstract

Various types of artificial skins have been developed to provide robots with a sense of touch. Because of their compliance, dielectric elastomer (DE) capacitive sensors are particularly suitable for soft robots. Although the electrodes of DE sensors exhibit nonlinear effects such as transient resistance changes and resistance peaks, this does not affect the capacitance readout representing stretch, as long as the frequency of the excitation voltage used for capacitance measurement is sufficiently low. At higher frequencies, however, the approximation of a DE sensor with an ideal capacitor and a series resistor accounting for electrode resistivity leads to an underestimation of capacitance in static sensors. We demonstrate how this effect is amplified by peaks and transient changes of electrode resistance caused by periodic stretching. At high frequencies, distinctive capacitance undershoots occurred that correlated with the change of electrode resistance. The close match between a simulation of the DE sensor as an R-C transmission line and recorded data supports the hypothesis of the undershoot having been caused by dynamic electrode resistance changes and the lumped parameter approximation. Our results show that nonlinear responses in DE sensors can be avoided by appropriately adjusting the excitation frequency.

Introduction

Touch allows humans to directly and intimately interact with their environment,^1,2 and losing this ability has serious consequences, such as motor impairment and a negative impact on psychological well-being.³ Against this background, we cannot expect robots that are not able to sense touch, to closely interact with humans in a safe way. This is a vital concern with robotic devices in health care and companion robots. Researchers have therefore developed artificial skins, which are capable of detecting touch stimuli through large surface areas, similar to natural skin. Sensing skins have to be adaptable to a large variety of shapes⁴ and compliant, which is particularly important for soft robots.⁵ These requirements can be met with elastomer-based materials, for example, piezoresistive sensors^6,7 and capacitive dielectric elastomer (DE) sensors,^5,8 both of which use compounds of conductive particles and nonconductive elastomer.

In piezoresistive sensors, the resistivity change is not only proportional to stretch. Transient resistance changes and resistance peaks have been observed in sensors directly printed onto motion-sensing garments.^6,7 When stretched according to a trapezoidal profile, resistance peaks appeared during the stretch and relaxation phases, each followed by a transient resistance decrease. The height of the peaks was found to increase with increasing strain rate.⁹ Resistance peaks were also observed in response to triangular stretch profiles.¹⁰ Zhang and Wang have investigated the time dependence of the resistance of polymers filled with metal powders under compression,¹¹ and attributed it to viscoelastic creep affecting the distance between clusters of conductive particles within the compound. This distance guides the tunneling current, that is, the passing of electrons between conductive clusters through nonconductive material.^12,13 Hence, a time dependence of the distance was reflected in the resistivity of the compound. According to Wang et al.,¹⁴ transient resistance changes can also be caused by time-dependent formation and breakdown of conductive paths within a conductive elastomer. Resistance peaks were attributed to a competition between the breakdown and recovery of the conductive network during cyclic loading.¹⁰ After a breakdown of the conductive network under tension, the resistance of conductive elastomers was found to decrease because of orientation effects of nonspherical particles.¹⁵ Transient resistance changes and resistance peaks make the reconstruction of deformation from resistance rather difficult.

DE sensors are capacitive consisting of parallel conductive elastomer electrodes that are separated by a dielectric. The capacitance of a DE sensor with two electrodes is determined according to Equation 1, where ε₀ · ε_r is the permittivity of the silicone dielectric, w and l are the width and length of the electrodes, and t is the thickness of the dielectric. Equation 1 assumes a negligible fringe field on the basis of t < < w and t < < l. Stretch leads to an increase of the area w · l, and a decrease of the dielectric thickness t, which causes the capacitance C to increase: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} C = \frac { { { \epsilon _0 } \cdot { \epsilon _r } \cdot w \cdot l } } { t } . \tag { 1 } \end{align*} \end{document}

Capacitance measurement is usually based on a lumped parameter model,^16–18 where the entire sensor is represented by a parallel plate capacitor with the lumped capacitance C_L and a lumped series resistor R_L to account for electrode resistivity (Fig. 1). The key benefit of the lumped parameter approximation is its simplicity. The impedance Z of the DE sensor is determined by applying a sinusoidal excitation voltage V_E with the frequency f_E, from the amplitudes of V_E, the current I, and the phase shift ϕ between them. The lumped capacitance C_L is calculated from reactance X, using Equation 2:

FIG. 1.

Lumped parameter representation of a DE sensor and corresponding vector diagrams. DE, dielectric elastomer.

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { C_L } = \frac { 1 } { { 2 \pi { f_E } { \kern 1pt } { \kern 1pt } \cdot { \kern 1pt } { \kern 1pt } \vert X \vert } } . \tag { 2 } \end{align*} \end{document}

DE sensors have been described as more accurate and repeatable than resistive sensors,¹⁹ with capacitance being affected by geometry only,⁴ which is true if the resistance of the electrodes is sufficiently low. The electrodes of DE transducers, however, have relatively high resistances. A sheet resistance of 61 kΩ/□ has been reported for a mix of carbon black and polydimethylsiloxane.²⁰ Graf and Maas have discussed the implications of electrode resistance and contact placement on the performance of DE.²¹ One of the configurations investigated was a DE with both electrode terminals on the same end and a harmonic excitation voltage, which is common in DE sensors. By means of a continuous analytical model based on transmission line theory, it was demonstrated that the attenuation of the excitation signal increased along the length of the DE with increasing frequency. It was pointed out that this effect causes a significant reduction of the penetration of high-frequency sinusoidal voltages applied to actuators, which substantially compromises their electrodynamic behavior. Tiggelman et al.²² found that lumped capacitors with resistive electrodes became distributed R-C chains at high frequencies, that is, the lumped parameter approximation was no longer valid. Such distributed transmission line models were applied to DE actuators,^23–25 by discretizing actuators into R-C elements. Each element comprises a portion of the overall capacitance, with a parallel resistor accounting for dielectric losses and series resistors representing the resistivity of the electrodes.

Xu et al. have investigated the limitations of the lumped parameter approximation in self-sensing DE actuators.²⁶ In self sensing, the movements of an actuator are inferred from its capacitance, rather than from external measurements.^16–18 In a frequency sweep on static actuators, the measured lumped capacitance C_L decreased with increasing frequency, because of the high electrode and connector resistance, which led to readouts lower than the capacitance defined by geometry (Eq. 1). As discussed by Graf et al.,²¹ excitation voltages with low frequencies propagated across the entire actuator while high-frequency voltages were attenuated, thus covering a smaller portion of the overall capacitance. This reduced capacitance was then attributed to a lumped parameter model, which resulted in an underestimation of capacitance, and thus to an incorrect representation of deformation.

In a later study, Xu et al. utilized the effect of capacitance underestimation for sensing local deformations on a single DE sensor,^27,28 by simultaneously applying sinusoidal excitation signals with different frequencies. Similar to the self-sensing example,²⁶ the resistivity of the sensor electrodes caused an attenuation of excitation voltages with high frequencies. Hence, deforming the sensor further away from its connector end caused no capacitance increase in high-frequency measurements. In contrast, all frequencies registered an increase when the deformation occurred close to the connector end. The location was detected from different response patterns at various frequencies. However, instead of a capacitance change caused purely by stretch, a dip occurred in the measured lumped capacitance and an additional time-dependent component was present.

We hypothesize that the capacitance and transient component originate from electrode resistance changes that were caused by dynamic stretching. In artificial skins for robots, electrode resistance spikes and transient changes caused by rapid movements are very likely, and the resulting erroneous capacitance measurement subsequently leads to an incorrect representation of stretch. The main contribution of this study is the investigation of the impact of electrode resistance changes under stretching and relaxation on the readout of a capacitive stretch sensor, and how it can be avoided. By applying a trapezoidal stretch profile, we show that the electrodes respond with distinct resistance peaks and a transient component. With a low enough excitation frequency, no undershoot appeared in the measured capacitance of the periodically stretched sensor. At higher excitation frequencies, the peaks in the electrode resistance amplified the effect of capacitance underestimation, which led to noticeable dips in the capacitance readout. The measured capacitances were compared with simulation results showing the same effect. Our results show that the excitation frequency can be used as a parameter to avoid unwanted dynamic effects in the capacitance signal representing stretch in DE sensors. With this taken into consideration it then becomes possible to produce a DE robotic skin that accurately and faithfully measures stretch.

Methods

The electrodes of the DE stretch sensor in this study are a compound of silicone and carbon black. They have extending tabs for resistance and capacitance measurement (Fig. 2).

FIG. 2.

DE sensor with connectors on both ends of either electrode.

Tabs on opposing ends of the same electrode were used to measure the resistance of an individual electrode. Capacitance was measured through the tabs on the same sensor end. The tab area was reinforced with nonstretchable end pieces, whereby the sensor was clamped to the stretching rig. Clamping the reinforced area beyond the electrode tabs did not cause any resistance or capacitance changes. The sensor thickness was 0.9 mm, its width was 15 mm, and the overall length was 170 mm, with a stretchable length of ∼100 mm. The capacitance of the unstretched sensor was 175 pF.

The sensor was attached to the stretching rig given in Figure 3, and prestretched by 15 mm to keep it taut in the relaxed position. The designations A and B refer to the sensor ends, and top and bottom refer to the position of the electrodes during the fabrication process. To demonstrate the influence of the sensing frequency on the lumped capacitance of the static sensor, we carried out a frequency sweep on the prestretched sensor in its stationary position. Capacitance was measured with a Hioki IM3523 LCR meter that assumed a lumped parameter model comprising a resistor in series with a capacitor.

FIG. 3.

The sensor stretching rig. LCR, inductance capacitance resistance meter.

For the periodic stretching experiments, the sensor was stretched according to the two functions given in Figure 4, with stretch rates of 10 and 20 mm/s. The sensor was stretched and released at the same time instances within a 10-s period. The top and bottom electrode resistances and capacitance from sensor end B were measured and recorded with the LCR meter. Excitation frequencies of 500 Hz, 2 kHz, and 6 kHz were chosen. Because the LCR meter has only one channel, all measurements were carried out successively, and recorded data were synchronized in MATLAB. The position encoder of the linear motor was used to measure stretch. Its output was read and recorded with LabVIEW, and synchronized with recorded resistances and capacitance in MATLAB.

FIG. 4.

Trapezoidal stretch functions.

To confirm our hypothesis of transient electrode resistance increase causing capacitance undershoot, we have modeled the DE sensor as an R-C transmission line, and simulated the process of measuring its capacitance with a lumped parameter model. In contrast to the lumped parameter model, the R-C transmission line accounts for the distributed character of resistance and capacitance. Figure 5 shows a transmission line approximation with n R-C elements, with each one representing a portion of the overall capacitance and electrode resistance, similar to the structure presented in Kaal et al.²⁵ Because of the small excitation voltage amplitude of V_E = 7 V, the leakage current across the dielectric was neglected. The electrode resistance and sensor capacitance were assumed to be distributed uniformly across the sensor. Hence, all top and bottom electrode resistance elements R_t_{, i} = R_t/n and R_b_{, i} = R_b/n are identical, and all capacitances C_i = C/n are equal. We have furthermore assumed that all transmission line elements respond to stretch equally. In addition, no parameter variations in the direction of width and thickness were considered.

FIG. 5.

R-C transmission line approximation of a DE sensor; sensor in side view. As resistance and capacitance vary with stretch, they are depicted as variable elements. Z ₁… Z _n _–1 and Z _n are the subcircuit impedances, and total impedance of the transmission line, respectively.

The block diagram in Figure 6 provides an overview of the simulation based on the transmission line model. It accepts the recorded electrode resistances, excitation frequency, and capacitance inferred from recorded stretch, and an experimentally determined stretch–capacitance curve. The input data were processed point-by-point in time. The stretch–capacitance curve was recorded in individual static measurements in increments of 3 mm, and at a frequency of 500 Hz. This was low enough for the measured capacitance to be unaffected by dynamic changes of electrode resistance.

FIG. 6.

Block diagram of the simulation.

The overall capacitance was divided into capacitances C₁…C_n. The recorded resistances of the top and bottom electrode were added, and divided into 2n resistors, as given in Figure 5. The impedance of the transmission line Z _n was calculated according to Equation 3. All resistances and capacitances are stretch dependent. Z _n is therefore also a function of stretch:

For each recorded data point along the time axis of the simulation input functions, the transmission line impedance Z _n was determined using Equation 3. To replicate the effect of applying a lumped parameter model to a sensor with distributed parameters, the impedance Z _n was attributed to a series resistor and capacitor, as given in Figure 1 and Equation 2. The result of the simulation was the lumped capacitance C_L, as it would have been measured by the LCR meter. The following section presents the results of the static sensor frequency sweep, resistance and capacitance responses to periodic stretching, and the simulation with the sensor represented by a R-C transmission line. A discussion of these results follows in a separate section.

Results

Figure 7 shows frequency sweeps of the lumped capacitance C_L for the static sensor in the prestretched state (0 mm), and at stretches of 15 and 30 mm, recorded with the LCR meter. The LCR meter internally attributed the impedance of the sensor to a lumped parameter model to determine the lumped capacitance C_L. As the frequency increases, the discrepancy between the sensor with distributed parameters and the lumped parameter approximation becomes more significant. Hence, C_L decreased with increasing frequency. This is consistent with the frequency sweeps published by Xu et al.^26–28 and Tiggelman et al.²²

FIG. 7.

Lumped capacitance C_L of the static sensor at different stretch states, measured with the LCR meter; the LCR meter was connected to the electrode tabs on sensor end B.

At 0 mm, the resistance of the top electrode was 350 kΩ, and the resistance of the bottom electrode was 361 kΩ. At 15 mm, the static resistances of both electrodes settled at approximately the same values, and at 30 mm, the top and bottom static resistances dropped by 20% and 25%, respectively. In contrast to previous work, where stretching led to a resistance increase,^{6,7,9,10,29,30} our measurements show that the resistance at full stretch was lower than that in the relaxed state.

Having confirmed the influence of sensing frequency on capacitance measurement, we investigated the sensor under periodic stretching. In all following graphs, 0 mm refers to a prestretch of 15 mm. Representative of all resistance measurements, Figure 8 shows the resistance of the top electrode for periodic stretching and relaxing at a stretch rate of 10 mm/s. It took ∼5 min for the electrode resistance to settle at a steady state. To guarantee identical conditions for all successive resistance and capacitance measurements, we have allowed for a 15-min waiting time for the electrode resistance to settle after changing the strain rate.

FIG. 8.

Resistance of the top electrode approaching a steady state at a stretch rate of 10 mm/s.

The data in Figure 9 were recorded after the electrode resistance had reached a steady state. Both electrode resistances responded to stretching and relaxing with pronounced peaks. The peaks occurred before the maximum stretch was reached, and were higher at the higher stretch rate of 20 mm/s. The resistances still changed over time once the sensor was fully relaxed or stretched. The resistances of the electrodes at maximum stationary stretch were lower than they were in the relaxed position.

FIG. 9.

Stretch and resistances of top (R_t) and bottom (R_b) electrodes at stretch rates of 10 and 20 mm/s.

The capacitances in Figure 10 were measured with the LCR meter, based on a lumped parameter approximation. The excitation frequencies f_E were set to 500 Hz, 2 kHz, and 6 kHz. The base capacitance at 0 mm decreased with increasing frequency, which is consistent with the trend of the static frequency sweep in Figure 7. However, the capacitance of the sensor at 0 mm was noticeably lower in the dynamic experiments, when compared at the same frequencies. Substantial capacitance undershoots occurred at 2 and 6 kHz, at both stretch rates, and upon both stretch and relaxation. No capacitance undershoot was noticed at 500 Hz.

FIG. 10.

Stretch and lumped capacitances C_L measured at different frequencies and stretch rates.

The connection between transient electrode resistance and capacitance undershoot was investigated in a simulation, according to the schematic in Figure 6. The stretch–capacitance relationship required to infer the input capacitance from recorded stretch is given in Figure 11. The capacitance was measured at an excitation frequency of 500 Hz, where no capacitance underestimation occurred in the previous experiments (Fig. 10).

FIG. 11.

Measured stretch–capacitance graph; the excitation frequency was 500 Hz, and the sensor was stretched in steps of 3 mm.

The simulation inputs were adapted to the transmission line approximation illustrated in Figure 5. The inputs C_input and R_t_{, i} + R_b_{, i}, and the corresponding stretches are given in Figure 12 for a stretch rate of 10 mm/s.

FIG. 12.

Stretch, resistance, and capacitance inputs for the simulation.

Figure 13 illustrates the influence of resolution, that is, the number of transmission line elements n, on the agreement between simulation and measurement, at an excitation frequency of 6 kHz and a stretch rate of 10 mm/s. The top curve describes the stretch-dependent capacitance if the sensor was a lumped parameter system with only one capacitor and one series resistor. At n = 2, n = 4, and n = 30, the lumped parameter approximation was applied to a transmission line with 2 elements, 4 elements, and 30 elements. A higher resolution reflects a more accurate representation of the distributed character of electrode resistance and sensor capacitance, which resulted in a better match between simulation and measurement. However, there was no noticeable improvement for n > 30. The good match between simulation and measurement at n = 30 shows that capacitance undershoot at high excitation frequencies was indeed caused by dynamic electrode resistance and lumped parameter-based capacitance measurement.

FIG. 13.

Variation of the number of transmission line elements n in the simulation; the excitation frequency was 6 kHz, and the stretch rate was 10 mm/s.

Figure 14 compares simulations at a resolution of n = 30 with measurements at sensing frequencies of 500 Hz, 2 kHz, and 6 kHz. At 500 Hz, the deviation between measurements and simulation was very small. At 2 kHz, the difference between the measured and simulated capacitance was 3%, and at 6 kHz the difference increased to 5%. These differences were possibly caused by the assumption of uniformly distributed resistances and capacitances in the model.

FIG. 14.

Measured and simulated capacitances. The dashed lines designate measurements (m), and the solid lines refer to simulations (s).

Discussion

Artificial skins on robots are prone to repeated stretching and relaxing, often at high strain rates. In resistive sensors, this leads to dynamic changes that no longer accurately reflect the actual deformation. We have shown that these effects can be avoided with capacitive type DE stretch sensors, where an appropriately chosen excitation frequency ensures an accurate representation of stretch. Furthermore, we have demonstrated that a static frequency sweep of the DE sensor is not sufficient, but dynamic electrode resistance has to be taken into account.

Experiments and simulations confirmed the connection between dynamic electrode resistance changes and capacitance undershoot in DE sensors at high excitation frequencies. This work extends existing research by introducing the aspect of periodic stretching, thus explaining the capacitance undershoot at high excitation frequencies observed in location sensing presented by Xu et al.^27,28 The results show that choosing an appropriate excitation frequency can avoid incorrect stretch measurements because of erroneous capacitance detection. This study is limited to a maximum stretch of 30 mm, which equates to a strain of 30%, strain rates of 10 and 20 mm/s, and trapezoidal stretch profiles with a period of 10 s. The transmission line model used in the simulation accounts for distributed parameters in longitudinal direction, but not in the direction of thickness and width. Furthermore, the electrode material was limited to a compound composed of silicone and carbon black.

The frequency sweep on the static sensor in Figure 7 with frequencies ranging from 500 Hz to 30 kHz confirmed the findings of previous research on stationary self-sensing DE actuators²⁶ and DE sensors,^27,28 namely that the lumped capacitance of a DE decreases with increasing sensing frequency. The electrode resistances led to an attenuation of high frequency excitation voltages along the sensor. Consequently, the LCR meter detected a smaller portion of the overall sensor capacitance at high frequencies, which it attributed to a lumped parameter model, thus underestimating the capacitance defined by the geometry of the sensor. The capacitance underestimation at 6 kHz was −7.5% at 0 and 30 mm, and −10% at 15 mm, compared with the values at 500 Hz. For an electrode with a strictly increasing stretch-resistance function, the underestimation would increase with increasing stretch. However, because the static resistance in our experiment did not change between 0 and 15 mm, and decreased between 15 and 30 mm, the capacitance underestimation did not simply increase with stretch. The higher resistance of both relaxed electrodes compared with their fully stretched state can be explained by the recovery dominating over the breakdown of conductive paths in the stretch direction.¹⁵

Under periodic stretching, the electrode resistance took several minutes to settle at a steady state (Fig. 8). This behavior was also observed in previous research on elastomer and carbon nanotube compounds¹⁰ and capacitive stretch sensors.²⁹ Deng et al.³⁰ have explained this through the breakdown of clusters of conductive particles into smaller ones, which are then separated by the nonconductive matrix. This reduces the electrode resistance by providing new conductive paths. The resistance graph in Figure 8 exhibits characteristic peaks during phases of stretching and relaxation. These peaks and the subsequent transient decreases can be seen in detail in Figure 9, which contains data recorded after the resistance had reached its steady state. At a stretch rate of 20 mm/s, the peaks were much higher than at 10 mm/s. Higher resistance peaks at higher stretch rates were also reported by Tognetti et al.⁹ The shapes of the curves for both electrodes are similar, but there are quantitative differences that can be explained with variations in the sensor manufacturing process. During relaxation between 5 and 10 s, the resistance of the top electrode had a tendency to further increase, but the sudden stop at 0 mm stretch caused a sharp gradient change. Similar to the static sensor, the resistance of the relaxed electrodes was higher than that of the stretched electrodes. This can again be explained by the recovery dominating over the breakdown of conductive paths in stretch direction.¹⁵ Transient changes were likely caused by viscoelastic creep in the distance between conductive particles,¹¹ and time-dependent changes in the formation and breakdown of conductive paths.¹⁴ Exploring these effects systematically would require an investigation of the electrodes on a microscopic level, which was not part of this study.

Figure 10 compares capacitance measurements at different stretch rates and frequencies. There is little difference in shape between the stretch function, and the corresponding capacitance response at 500 Hz. At 2 and 6 kHz, the base capacitance was lower, as expected from the frequency sweep in Figure 7. A distinct undershoot occurred upon stretching and relaxing. The shape of the undershoot is very similar to the resistance response in Figure 9, especially the sharp gradient change between 5 and 10 s. In the same way as the resistance increase was higher at the higher stretch rate, the capacitance undershoot was also more pronounced. At 2 kHz, it was 16 pF at 10 mm/s and 30 pF at 20 mm/s, relative to the capacitance of the unstretched sensor in Figure 10. Because of the relatively low resistance of the stretched electrodes, the effect of capacitance underestimation in the stretched sensor was less than in the relaxed sensor with higher electrode resistance. At 2 kHz, according to the static frequency sweep in Figure 7, the resting capacitance of the sensor should be ∼205 pF. During periodic stretching at 20 mm/s, however, the capacitance of the sensor in its relaxed position was only 180 pF. This reduced base capacitance and the pronounced undershoot clearly show that the dynamics of the electrode resistance must be considered for sensing frequency selection.

The simulation results in Figure 13 show that higher resolution of the R-C transmission line model leads to better agreement with the measurements. Increasing the resolution beyond n = 30 did not further reduce the discrepancy. The remaining difference is most likely caused by the assumption of uniformly distributed resistances and capacitance along the sensor in the simulation. In addition, the model assumes all sensor areas to equally respond to stretch. Furthermore, the transmission line approximation is one-dimensional, and therefore does not account for the reaction of the sensor in the direction of width and thickness. A variation of the difference with frequency is given in Figure 14. It varied from almost 0% of the sensor's capacitance at 0 mm at 500 Hz to 3% at 2 kHz, and 5% at 6 kHz, and could potentially have been caused by a nonuniform distribution of resistance and capacitance. The difference also seems to depend on stretch, which can be explained by different responses of certain sensor areas to stretch, thus altering the overall sensor capacitance specific to the excitation frequency. Our results show that capacitive stretch sensing overcomes the nonlinear problems of resistive sensing. However, to successfully use DE sensors, one has to be aware of the implications of the dynamic responses of electrode resistance to stretch and excitation frequency, on lumped capacitance measurements.

Conclusion

The interaction between humans and robots can be greatly improved by providing them with an artificial skin that enables distributed touch sensing. DE sensors are well suited for this purpose, especially for soft robots, where compliance is essential. To maintain the advantages of capacitive DE sensors over piezoresistive sensors, the dynamics of electrode resistance have to be considered. We have demonstrated that the change of electrode resistance in DE sensors under strain, in combination with high excitation frequencies and a lumped parameter approximation of a DE sensor, leads to erroneous capacitance measurements, causing a misrepresentation of stretch.

A frequency sweep on a stationary DE sensor showed that static electrode resistance causes an underestimation of capacitance at higher frequencies, which confirmed the findings of earlier research. The underestimation was amplified by stretching and relaxing the sensor, which caused distinctive peaks in the electrode resistance, followed by a transient decrease that persisted under constant stretch. Corresponding capacitance undershoot and transient capacitance changes were noticed at excitation frequencies of 2 kHz and 6 kHz, but not at 500 Hz. These effects were reproduced in a simulation based on a transmission line model of the sensor, whose capacitance was determined by means of a lumped parameter model. The good match between the measurements and simulations confirmed our hypothesis of capacitance undershoot being caused by dynamic electrode resistance changes.

If the excitation frequency is low enough, the electrode resistance has no direct impact on stretch measurement with DE sensors. It is, however, not sufficient to select a frequency based on a frequency sweep of a static sensor. The dynamics of electrode resistance must be taken into consideration. In a soft robotic sensing skin, the highest expected stretch rate should be identified beforehand, as it leads to the highest resistance peak. The excitation frequency can then be adjusted so that no capacitance undershoot appears. In resistive stretch sensors, inherent transient resistance changes and resistance peaks can only be avoided with changes to the sensor hardware or compensated with mathematical models. Both approaches are impractical for high-volume production, for example, the distribution of large numbers of sensors to achieve high resolution. Our study has demonstrated how unwanted dynamic effects in DE sensors can be easily eliminated by adjusting the excitation frequency, and we believe it provides a useful guideline for the design of capacitive artificial skins.

Footnotes

Acknowledgments

The authors thank StretchSense Ltd. for Andreas Tairych's PhD funding support and for providing production facilities and materials for the fabrication of the sensors used in this study.

Author Disclosure Statement

No competing financial interests exist.

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