Characterization of a compliant multi-layer system for tactile sensing with enhanced sensitivity and range

As illustrated in figure 2(d), the multi-layer sensing system consisted of two layers of foam (figure 2(b)) having different stiffness, each surmounted by a strip-shaped sensing skin (figure 2(a)). Basic indentation tests were performed to obtain the mechanical and electrical responses. Figure 2(g) shows the pressure and force as a function of probe depth. (Unloading curves are shown in SI section 4.) Up to an indentation of 10 mm, nearly up to the d₁ = 12.5 mm thickness of the upper soft foam layer (foam 1), the pressure increased linearly, to 38 kPa (3 N). The slope then increased as the probe began to deform the lower hard foam layer (foam 2). The maximum indentation was 20 mm, which is 80% of the 25 mm original total foam thickness and more than half-way into the d₂ = 12.5 mm thickness of the lower hard foam. All four scans were similar, as evident from the overlapping curves. Thus, the membranes and foams did not exhibit mechanical first cycle memory effects. The noise (figure 2(g) inset) is due to the limited force resolution when the widest range setting is used for the force transducer (±3000 N). (With a 10 bit digitization of the signal, the force resolution was limited to approximately 3 N.)

The relative change in resistance of the two sensing skins is shown in figure 2(h) as a function of pressure. The resistance increases because electrical conduction is by percolation in the EG nanoparticle network, and some percolation paths are disrupted when the particles separate under strain. This is in contrast with materials whose resistance decreases with strain, such as conductive fabrics, in which the fibers in the weave make better electrical contact under stretching. However, it is important to note that conductive fabrics and other sensors can still be used in place of the EG sensing material, since the architectural principle does not depend upon whether the GF of the sensing material is positive or negative.

The signal was larger during the first scan (due to memory effects, section 3.3.1), and then it was similar in the remaining three scans. (See section 4 in the SI for a discussion of first cycle effects.) This behavior can be attributed to break-in during the first cycle in viscoelastic materials such as foams and elastomers, which is known as the Mullins effect [62] in filled rubbers. For skin 1 at low pressure (up to 38 kPa, 0–10 mm), the sensitivity was high and substantially linear (initial slope = 1.3%/kPa or 17.6%/N in the 2nd scan), and the relative change in resistance was an easily measurable 60%. The sensitivity then gradually decreased, and above a turnover region it became linear again (above 250 kPa, slope = 0.2%/kPa or 2.9%/N for scan 2). At 1125 kPa, the change in resistance was over 250%. The utility of a monotonic, bilinear relationship should be emphasized.

For skin 2, which was sandwiched between the foam layers, the output was almost zero at low pressure, since it remained unstrained while only the upper foam experienced compression. At the turnover point for skin 1, corresponding to the corner in the force–indentation curve, the resistance of skin 2 began to increase as the foam under it was indented. Its slope was similar to that of skin 1, since both signals followed the deformation of foam 2.

The piezoresistive latex/EG sensors respond to strain [29], so the resistance of the skins reflects the surface deformations of the foams. The higher sensitivity at low forces is due to deeply indenting the soft foam 1 without much pressure (figure 2(b)). The signal from skin 1 does not saturate after foam 1 is completely crushed because the membrane continues to stretch as indentation of foam 2 begins (figure 2(c)). Because foam 2 is stiffer, there is less strain per unit force, and thus a relatively smaller change in resistance. The similar slopes for the signals from the two skins indicates that they undergo similar net changes in strain once foam 2 starts to collapse.

Comparing sensitivity with other sensing systems is challenging because of the variety of metrics reported (V, N,%, Ω/Pa, A/Pa, %/Pa [10]). A conducting polymer array showed 0.1%/kPa up to 30 kPa [63] and a conductive network of carbon nanotubes on PDMS reported 360%/kPa to 0.1 N [64]. The difficulty of balancing sensitivity and range is demonstrated by these examples, as is the utility of our approach for achieving both (1.9%/kPa up to 31 kPa and 0.055%/kPa to 1.2 MPa). The sensitivity in our system could be further increased simply by using an even softer top foam.

The displacement resolution of the measurement setup was derived from the noise level of the signal. The resolution is commonly determined by using the root mean square (rms) noise of the signal [65]. We obtained the rms noise by taking the standard deviation of ΔR/R₀ over a 5 s period with no indentation, and it was 0.0055. From the ΔR/R₀ versus pressure and pressure versus indentation curves (figures 2(g), (h)), the displacement resolution in terms of variation in indentation was determined as 0.1 mm (i.e., the indentation level corresponds to the ΔR/R₀ noise level of 0.0055). This is not a limit of the sensing system, but of the measurement system.

The contact area for human–robot interactions may be as small as a fingertip or as large as a hand. The effect of contact area on the sensing signal is thus examined here. In this system, strip-shaped sensors were used whose width w = 0.8 cm was less than the probe diameter d. We hypothesized that the piezoresistive signal would be independent of probe area if, for d > w, the strips are only strained at the probe edges, not underneath the probe (see section 3.2.3.1).

Three flat-bottom probes were used with diameters of 1, 2 and 4 cm (areas of 0.8, 3.1, and 12.6 cm², respectively). Figure 3 shows results from the second to the fourth cycles. As expected, while larger probes generated larger forces at the same indentation (figure 3(a)), they produced the same pressures (figure 3(b)). Figure 3(c) shows the normalized change in resistance of skins 1 and 2 as a function of force; ΔR/R₀ for both were greater when the contact area was larger. When plotted as a function of pressure, the electrical responses overlapped (figure 3(d)) for all three probes. These results indicate that the architecture can be used effectively as either an indentation sensor or a pressure sensor with strip-shaped sensors.

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It may seem surprising that a strain sensor can be used as a pressure sensor. This is a consequence of w < d, making the sensing system quasi-one dimensional (1D) so that the strip is always deformed by just the two edges of the probe (see sections 3.2.2 and 3.2.3), and also of the sensor's stretchability.

The question may be raised, is it necessary to have foams of different stiffness, or can two strata with the same foam stiffness achieve a similar result? In order to determine the benefit of using a combination of foam stiffnesses (soft–hard), three other foam combinations were examined (figure 4(a)): two layers of the soft open-cell foam (soft–soft, d₁ = 12.5 mm, d₂ = 12.5 mm), two layers of the stiff closed-cell foam (hard–hard, d₁ = 12.5 mm, d₂ = 12.5 mm), and a thin layer of soft foam with no second foam (soft–rigid, d₁ = 12.5 mm).

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The pressure versus indentation curve for the soft–rigid system (figure 4(b), light blue) followed the one for the soft–hard system (red) to an indentation of almost 10 mm (the thickness of the foam), but then shot up rapidly as the probe made contact with the rigid substrate. The signal from skin 1 over soft–rigid, shown in figure 4(c), had a high initial slope, like those of the soft–soft and soft–hard systems, but it then saturated once the thin layer of foam was crushed, for a maximum ΔR/R₀ of 90%. As expected, this system demonstrated relatively high sensitivity but low range. The response of skin 2 was small across the entire range (<20%) since it rested directly on the rigid substrate and experienced negligible strain.

The pressure–indentation curve for the soft–soft system (dark blue) also followed the one for the soft–hard system to an indentation of almost 10 mm, but then instead of increasing, the pressure remained low. The signal also initially followed that of the soft–hard system, but the higher sensitivity range was extended. Skin 2 began responding when the second foam layer began to deform (above 37 kPa).

The force–indentation curve for the hard–hard system (green) was similar to that of the soft–hard system, but shifted leftward toward the origin because the indenter touched the harder foam surface immediately. The initial slope of the skin 1 signal was, as expected, substantially lower than that of soft–hard due to the much smaller strain under the same force. The signal from skin 2 was smaller than from skin 1, and also smaller than the signal from skin 2 in the soft–hard system, since the high stiffness of the overlying foam in the hard–hard system induced a smaller strain in this sandwiched skin.

Comparing the various curves, the behavior of the soft–hard system combined the sensitivity of the soft foam and the wide range of the hard one. The indentation tests were repeated on duplicate sensing skins (section 7, SI). The curve shapes and the distinct behaviors were repeated, although the amplitudes varied, mainly due to the different loading histories of the skins, whose viscoelastic behavior has been widely recognized [66, 67] (see section 3.3.1).

These experiments confirm that a soft–hard combination is required and that the force range for each layer is determined by its thickness. By adjusting the absolute and relative thicknesses and stiffnesses of the foam layers, one can adjust the slopes and corner of the bilinear curve. A combination of foams is therefore a good approach to prevent the padding thickness on a robot from becoming unwieldy. Although the concept for enhancing the performance of the sensing system has been demonstrated using two layers, it is possible to use additional layers or even employ a continuous stiffness gradient.

Section 3.1.2 showed that the sensor response was independent of probe area, suggesting that at a given indentation the net strain of the sensing skin was the same, since the GF is almost constant with strain. In order to validate this hypothesis, the deformations of latex membranes over a soft foam (from pair A) under an applied indentation were 'frozen' using epoxy. Since epoxy soaks into foam, a square latex membrane was used instead of strips; we assumed the deformation, although somewhat different, would nevertheless provide insight.

Membranes deformed around flat-bottomed probes of 1 and 4 cm diameter are shown in figure 5. (For further discussion of the deformation, see SI, section 6.) Since the membrane area in contact with the probe bottom remains flat, the curved edges were compared by overlaying the images. The shapes were found to be equivalent (figure 5(c)). This accounts for the piezoresistive signals (figure 3(d)) being the same.

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In order to design the sensing architecture for obtaining a desired performance, it is necessary to understand the signal versus pressure result in figure 3(d) and the responses of the different foam systems in figure 4. Since the resulting strain in the skin depends on how the interaction between the indenter and the foam affects the shape of the foam surface, we begin by modeling that shape.

3.2.3.1. Determination of surface shape

The shapes of the deformed foams were determined using photographs taken at a series of indentations (figure 6). For this work, foam combination B was used with the spherical probe (0.95 cm dia.).

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Once the surface was traced, these curves were fit to a function. For elastic bodies under line (knife-edge) loading, derived for bodies that are large compared with the contact area, the theoretical decrease in stress intensity with distance x from the knife-edge goes as 1/x [68]. We therefore used

$$\begin{eqnarray}&&f(x)=A\displaystyle \frac{{x}_{0}}{x+{x}_{0}},\end{eqnarray} \tag{ 1 }$$

where the outer edge of the probe is at x₀, A is the indentation at x₀, and f(x) is the indentation of the foam at a distance x distal from x₀. At the edge, x = 0 and f(x) = A. Far from the edge, f(x) = 0. Using a value of x₀ = 0.4 cm provided the best fit to the experimental curve shape for both foams: their surface shapes were similar at the same indentation. It also fit the shape of the membranes shown in figure 5, confirming that the foam deformation determines the sensor strain.

3.2.3.2. Hard foam surface strain model

The second step in the modeling was to obtain the strain associated with the shape change. For simplicity we assumed that there was no strain under the probe tip itself, as illustrated in figures 7(a), (b), i.e. the probe merely pushes the surface under it downward. The deformation is symmetric around the probe, so only a half-space was considered. The total strain ε at an indentation δ = 1 was determined by summing the strains along the curve of equation (1), resulting in an increase in arc length Δh = 0.3. The strain occurred entirely within a distance of x = 3 cm from the edge of the probe. A linear relationship was postulated between strain ε and indentation δ (figure 8(a)):

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}L}{{L}_{0}}\equiv \varepsilon =\displaystyle \frac{0.3}{{L}_{0}}\delta ,\end{eqnarray} \tag{ 2 }$$

where L₀ is the original sensor length (5 cm measured from the center of the probe).

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The third step was to relate the strain to the force. The measured displacement–force relationship for the hard foam was employed to display the model strain as a function of force. The model is compared to the experimental signal in figure 8(b), where the maximum values were matched across the two axes. Prior work [29] on the sensing skins had shown that ΔR/R₀ is proportional to uniaxial strain, as expected for a constant GF. For these strips GF = 8 (SI section 1), so a normalized resistance change of 44 corresponds to a strain of 5.5%, close to the 6% predicted by the model. The agreement of the model and experimental curve shapes was excellent given the simplicity of the model and the fact that it had one fitting parameter, x₀.

3.2.3.3. Soft foam surface strain model

Applying the hard foam model to the soft foam results, the fit was not as good, as shown by the red curves ('1/x') in figure 9: the experimental data ('exp') had a steeper initial rise and a more defined corner. (The noise in the experimental curves was due to the limited force resolution when using a large force range on the transducer.)

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A revised model was therefore explored, based on the observation (figure 6(d)) that for indentations of the probe up to half the probe radius, there was little downward motion of the surrounding surface. It appeared that the foam was conformal around the tip and only with greater indentations showed the 1/x behavior. The soft foam model is illustrated in figures 7(c), (d). For this case, a separate treatment is required for indentations less than and greater than the radius.

3.2.3.3.1. Indentation less than the probe radius, δ < r

When the displacement δ is small the deformation of the soft foam can be treated as occurring solely around the indenter, which has radius r (figure 7(c)). By symmetry we considered the half plane. The length along the surface is approximated by the sum of two pieces: a linear segment h between the probe and the edge of the strip at L, and an arc s around the indenter. The strain ε is then given by

$$\begin{eqnarray}&&\varepsilon =\displaystyle \frac{{\rm{\Delta }}L}{L}=\displaystyle \frac{s+h-L}{L},\end{eqnarray} \tag{ 3 }$$

where L is the length of the sensor. When the displacement is less than the probe radius, using the definition of the arc length,

$$\begin{eqnarray}&&s=r\theta /2.\end{eqnarray} \tag{ 4 }$$

Next find θ by relating it to the height of s and the radius r,

$$\begin{eqnarray}r-a & = & r\,\cos (\theta /2)\\ \theta & = & 2{\cos }^{-1}\left(\displaystyle \frac{r-a}{r}\right),\end{eqnarray} \tag{ 5 }$$

where a = δ. Substituting,

$$\begin{eqnarray}&&s=r{\cos }^{-1}\left(\displaystyle \frac{r-\delta }{r}\right).\end{eqnarray} \tag{ 6 }$$

Next find h.

$$\begin{eqnarray}&&h=L-b.\end{eqnarray} \tag{ 7 }$$

Again use geometry to find b:

$$\begin{eqnarray}{r}^{2} & = & {(r-a)}^{2}+{b}^{2}\\ b & = & \sqrt{{r}^{2}-{(r-\delta )}^{2}}.\end{eqnarray} \tag{ 8 }$$

Then, substituting and simplifying,

$$\begin{eqnarray}&&h=L-\sqrt{2r\delta -{\delta }^{2}}.\end{eqnarray} \tag{ 9 }$$

The strain is then given by:

$$\begin{eqnarray}\varepsilon & = & \displaystyle \frac{s+h-L}{L}\\ & = & \displaystyle \frac{r{\cos }^{-1}\left(\displaystyle \frac{r-\delta }{r}\right)+L-\sqrt{2r\delta -{\delta }^{2}}-L}{L}\\ & = & \displaystyle \frac{r{\cos }^{-1}\left(\displaystyle \frac{r-\delta }{r}\right)-\sqrt{2r\delta -{\delta }^{2}}}{L}.\end{eqnarray} \tag{ 10 }$$

3.2.3.3.2. Indentation greater than probe radius, δ ≥ r

The foam wraps around the indenter for half the radius and is pushed downward the remaining fraction, δ − r, as shown in figure 7(d). The segment s has a constant length of rπ/2. The shape of the segment h is again given by equation (1). The strain is then given by:

$$\begin{eqnarray}\begin{array}{rcl}\varepsilon & = & \displaystyle \frac{s+h-L}{L}=\displaystyle \frac{r(\pi /2)+L-r+0.3(\delta -r)-L}{L}\\ & = & \displaystyle \frac{0.27r+0.3\delta }{L}.\end{array}\end{eqnarray} \tag{ 11 }$$

This prediction of this model is shown by the blue lines ('def+1/x') in figure 9, and it provided a good shape fit. The soft and hard foam models are identical above 5 mm, but the soft–soft foam undergoes greater strain at small indentations.

3.2.3.4. Two-layer modeling

3.2.3.4.1. Analytical model

The strain for the soft–hard model is the sum of the strains in the soft and hard foams. This is equal to the strain in the soft foam when the indentation is less than the soft foam thickness, and the maximum strain in the soft foam plus the strain in the hard foam when the indentation is greater. The predicted surface strains for the three foam configurations are shown as a function of indentation in figure 10(a), together with the experimental signals. Since the strain–indentation curves were the same for the soft and hard foams for δ > r, the soft–hard model was identical to the soft–soft model, as shown by the overlayed solid red and blue curves in figure 10(a). However, this model did not account for the behavior. The soft–hard signal deviated from model above 12 mm (the location of the interface), showing an unexpected increase in sensitivity. The soft–hard system behaved (black curve) as if, after a total indentation of 10 mm, the strain at the surface of the hard foam was twice as great as usual, i.e., as if the hard foam stiffness was reduced. The SI (sections 7 and 8, SI) shows that this change in slope of the skin 1 signal also occurred at the interface of a hard–hard system, suggesting that the interface is responsible. The behaviors of the three systems are shown as a function of force in figure 10(b).

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3.2.3.4.2. Finite element analysis (FEA)

Taking another approach, an axisymmetric two dimensional FEA model (SolidWorks, Dassault Systèmes) was used to model three of the configurations. The top layer thickness was set to 10 mm and the bottom layer thickness to 20 mm. The material behavior was assumed to be hyperelastic and incompressible under biaxial compressive stress and described by the Mooney–Rivlin model.

The force versus displacement curves predicted by the model at the top surface are compared with the experimental curves in figure 10(c). Again, the behaviors matched fairly well for the soft–soft and hard–hard pairs, but the predicted force for the soft–hard combination was higher than that from the experiment. Strain on the top surface versus force was also obtained from the FEA simulation (figure 10(d)). The model again showed reasonable agreement for the soft–soft and hard–hard configurations, but the predicted slope of the signal was too low for the soft–hard configuration because of the over-prediction of the stiffness. Again, the hard foam in this configuration behaved as if it were softer. Thus, neither the FEA method nor the surface model could account for the soft–hard behavior, possibly because they did not account for interfacial effects.

In the previous sections, ΔR/R₀ was presented, where R₀ was the resistance at the beginning of each cycle. Here we directly examine the voltage drop V_s over skin 1 in the soft–hard system, which was in a voltage divider circuit (section 2.3). The sensor had never been strained before and had an initial voltage reading V₀.

Figure 11 shows the voltage in response to a first indentation of 22.5 mm followed by two smaller indentations of 17 mm. There was an immediate increase in the voltage when the probe indented the surface. After the probe left the surface in the first cycle, the voltage initially dropped rapidly, but it did not return to the original baseline V₀, instead gradually settling at a new, higher baseline V'₀. After the next indentations, the voltage again returned to the new baseline V'₀, at the same rate as for the first indentation. (We have previously fit the recovery response using a generalized Maxwell model with two time constants; see SI for [29].) After the baseline shift, the subsequent cycles were relatively stable. The higher baseline resistance after the first cycle explains the smaller signals ΔR/R₀ in later scans seen in figure 2(h).

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All three signals had comparable amplitudes above the new baseline. This result suggests that memory effects could be substantially mitigated by using the resistance baseline after an event as R₀, rather than the one before it (see SI, section 9), although such an approach may not be practical in some real-time applications. Alternatively, prior work has shown that empirical models based on sensor characterization can be developed for viscoelastic materials, allowing accurate estimates of applied pressure [67].

Because of the relaxation time (figure 11), the time interval between loading events changes the signal amplitude. Four consecutive indentation loading cycles with short to long intervals between cycles were performed on the multi-layer sensing system. To examine the response of the sensing membrane alone, tensile tests were also performed with varying time intervals.

Figure 12(a) shows pressure as a function of probe indentation for the second to the fourth cycles. The mechanical responses were independent of the interval: the results overlap. Figure 12(b) shows the corresponding signals from the two sensing skins. The signals from skin 1 for time intervals of 2–10 min were almost the same, but for the shorter time interval of 30 s the signal from skin 1 was lower. The region of the skin 1 curve that was affected by the time interval was the high-slope region (which decreased from 1.6 to 1.2%/kPa) at pressures less than 50 kPa, associated with the compression of foam 1. The slopes associated with foam 2, for both skins 1 and 2, were unaffected by the time interval, suggesting that the time interval dependence arises from foam 1.

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Figure 12(c) shows force as a function of strain in the tensile tests on the sensing skin alone. All the curves overlapped, showing that the mechanical recovery is faster than the shortest interval (15 s). Figure 12(d) shows the piezoresistive responses. These curves did not stop overlapping until 15 s. Thus, the soft foam is the main contributor to recovery effects in this system and the behavior of the sensing skin on a foam is dominated by the foam to which it is attached.

Loading rates would likewise be expected to affect the response. Loading rates were therefore compared for both the soft–hard system and the sensing skin alone. Four cycles were performed at each rate, with time intervals between cycles of 5 min.

The faster loading rate resulted in a slightly higher pressure on the multi-layer system (figure 13(a)) and a correspondingly larger signal from skin 1 (figure 13(b)). Just as in figure 12(b), the part of the curve responsible for the dependence was the low pressure region corresponding to the compression of foam 1. Above 50 kPa, associated with compression of foam 2, there was no change in slope with rate, and the slope was similar for skin 2.

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The responses of a stand-alone skin are shown in figures 13(c) and (d). The mechanical stiffness was greater at the higher rate, as was the ΔR/R₀ signal. Interestingly, the performance of a compliant sensor on a foam may be less rate dependent than it would be alone, as evidenced in figure 13(b) by the nearly identical curves at the two loading rates for skin 2.