# Modeling Electronic Skin Response to Normal Distributed Force

## Abstract

**:**

## 1. Introduction

_{33}piezoelectric coefficients, which quantify pressure into charge conversion. Obtaining expected (modeled) behavior of the electrical response of each sensor to measured mechanical force at the skin surface proves that the whole fabrication process was successful, but for an acceptable dispersion in sensor functioning that can be considered intrinsic to the manufacturing process and handled by appropriate calibration.

## 2. Materials and Methods

#### 2.1. Electronic Skin Materials and Structure

_{3}= d

_{33}T

_{3},

_{3}is the charge density on the sensor surface, T

_{3}is the normal stress component (i.e., the pressure acting on the bottom of the protective layer), and d

_{33}is the piezoelectric coefficient. As PVDF sensors directly convert mechanical stimuli into charge, electronic circuits for data acquisition are based on charge amplifiers [18]. Equation (1) does not include the electric field across the PVDF sensor as it is assumed to be negligible due to the virtual ground at the operational amplifier inverting input. Hence, measuring the charge density on the sensor surface provides a direct measure of the normal stress acting on the sensor surface. Assuming circular sensors, the total sensor charge measured by the charge amplifier is given by:

_{3}averaged over a single extended sensor.

#### 2.2. Electronic Skin Model

_{meas}) to the normal stress component T

_{3}averaged over the sensor (i.e., ${\stackrel{\u203e}{T}}_{\mathit{3}}$). Scope of the following two sections is to retrieve the average normal stress component ${\stackrel{\u203e}{T}}_{\mathit{3}}$ transmitted to a single extended tactile sensor as a function of the normal force F

_{3}applied at the surface of the skin protective layer. This would allow to estimate the sensor charge Q

_{meas}(output) as a function of the normal force F

_{3}(input), which leads to an estimation of the system transfer function FRF, defined as the ratio between the Fourier transforms of the output charge and of the input force. In this paper, we only focus on normal contact forces.

#### 2.3. Simplified Analysis

#### 2.3.1. Effect of Sensor Size on Sensor Response to a Single Normal Point Force

**n**. Assume a point force F is applied at a given position on the outer surface of the layer. The surface is coated with an elastomer layer of constant thickness h (Figure 2a). A stress field is then generated in the elastomer and transmitted to the sensor. Let

**T**denote the stress tensor. The stress vector acting on the sensor reads

**n**·

**T**. This stress response can be conveyed into an appropriate circuit and retrieved by an electronic device. For example, PVDF sensors directly convert the T

_{3}stress component into charge, as indicated by Equation (1).

**F**applied on the outer surface and the stress at a given point inside the cover layer, we may get advantage of the solution of the so-called Boussinesq’s problem [19]. This problem considers an elastic half-medium bounded by a surface on which a point force is applied. For such a configuration the stress field determined by Boussinesq reads:

**r**is the radial distance of the generic point of the medium from the application point of the load

**F**, all bold-faced symbols represent tensors or vectors,

**e**

_{k}is the unit vector in the k-direction and ⊗ is the symbol of tensor product.

**r**separation vector of the force application point from the center of the sensor area. The advantage of Equation (3) is its simplicity, as the stress is uniaxial in the radial direction, and its independence of elastic parameters.

_{3}stress component on the bottom of the elastic cover of thickness h, as received by a PVDF sensor working in thickness mode Equation (1). Letting $\widehat{r}$ be the radial distance of the point where the force is applied from the sensor center projected on the outer surface (see Figure 2a), we have r

^{2}= $\widehat{r}$

^{2}+ h

^{2}and

**e**

_{r}= sinα (

**e**

_{1}cosβ +

**e**

_{2}sinβ) −

**e**

_{3}cosα (see Figure 2b), where cosα = h/r, sinα = $\widehat{r}$/r.

_{3}(F

_{1}= F

_{2}= 0), Equation (4) reduces to

_{3}is transmitted to the sensor through the elastic layer leading to an average normal stress ${\stackrel{\u203e}{T}}_{\mathit{3}}$ acting on the sensor.

_{T}/h and sensor pitch) to achieve a desired skin resolution, which is related to the spatial concentration of the mechanical stress information around a single sensor.

#### 2.3.2. Effect of Normal Force Distributed Over Circular Contact Area on Point-Like Sensor Response

_{3}by the equation [20]:

_{3}generated on a point-like sensor located at a depth h below the center of the contact area can be calculated by:

_{3}through Equation (10)) normalized to the layer thickness. Note that this is the second important result of the analysis: indeed, δ is a measure of how the force F

_{3}(distributed with Hertzian pressure distribution $q\left(\widehat{r}\right)$) is transmitted to the point sensor through the elastic layer leading to the stress T

_{3}on the sensor. In Figure 6, $\delta \hspace{0.17em}$ is plotted as a function of $\frac{a}{h}$.

#### 2.3.3. Combination of the Two Contributions: Effect of Distributed Normal Force on the Response of an Extended Sensor

_{3}to the average normal stress T

_{3}is a function of the radial distance of the point force dF

_{3}from the projection of the sensor center on the outer surface. Recall that this distance coincides with the radial coordinate $\widehat{r}$ employed to integrate over the contact area. Hence, the function γ must be included inside the integral, as shown by Equation (13).

_{3}is transmitted to the sensor through the elastic layer leading to an average normal stress ${\stackrel{\u203e}{T}}_{\mathit{3}}$ acting on the sensor.

_{33}of each PVDF sensor, once the charge Q

_{3}and the force F

_{3}have been measured. Comparison with the expected value of d

_{33}[21] allows one to validate sensor functioning and skin technology.

#### 2.4. FEM Simulations

_{0}small enough to lead to small amplitude deformations. The lower boundary is assumed to be rigid, while the side walls are free. The solution of the problem is sought numerically, with the help of the code COMSOL Multiphysics. Figure 9 illustrates an example of the results of FEM simulations of the elastomer subject to a normal force distributed over circular contact area (black line, radius a Equation (10)) with Hertzian pressure distribution. The employed value for the elastomer modulus E is the result of experimental characterization of the elastic layer of the reference e-skin as discussed in Section 3.2.1.

_{T}/h, the value of a/h has been changed by arbitrarily playing with the two parameters F

_{3}and R in Equation (10). The output of the simulation is reported in Figure 10, where it is compared with the simplified analytical solution based on Boussinesq’s approach.

_{T}/h = 0.2 and a/h = 0.289—the relative error of the analytical solution with respect to the numerical one decreases from about 40% to about 10% at 10 mm depth and 1% at 30 mm depth. This suggests that for sensors embedded into a thick protective layer the analytical solution would be quite appropriate.

_{T}/h. This has prompted us to identify what causes these oscillations. Indeed, at a more careful examination, it turns out that, in the finite case, σ depends on an additional parameter, besides a/h and r

_{T}/h. This may be readily appreciated noting that, on physical ground, ignoring the effects of the sidewalls, the response of the system may be assumed to depend on the following dimensional quantities: h, r

_{T}, E, F and R. With the help of simple dimensional arguments, one may then envisage the following dimensionless relationship:

_{T}/h, F/(R

^{2}× E))

^{2}× E) by L. The above argument suggests that the plot of Figure 10 must be modified such that each line corresponding to given r

_{T}/h be replaced by a strip of lines each associated with the same value of r

_{T}/h but a distinct value of L. The output of the simulations is therefore organized including the dependence of σ on the parameter L. Figure 11 illustrates the results for r

_{T}/h = 0.6, which corresponds to the geometry of the real e-skin prototype employed for experimental tests presented in the following section. Note that the figure confirms that distinct curves are associated with different values of L. Similar analysis can be extended to all values of r

_{T}/h and analogous strips of lines would be obtained. However, in Figure 11 we have restricted ourselves to a range of values of the parameter L of direct physical relevance for the tactile application.

## 3. Experimental Results and Discussion

_{33}of each PVDF sensor, a quantity that was a priori unknown in the present investigation. Hence, comparison between theoretical predictions and experimental observations has been employed to infer the value of the effective piezoelectric coefficient d

_{33}that leads to best fit. In other words, at the present stage this model may be described as post-dictive. It will become pre-dictive once the effective piezoelectric coefficient d

_{33}will be known a-priori.

_{33}(ω)) is approximately flat in the frequency range of interest for the tactile application (say, 1 Hz–1000 Hz). This notwithstanding, resonances may derive from a variety of additional causes (e.g., movable contacts, contact surface asperities, motor-induced vibrations), which cannot be a-priori controlled. The ultimate solution is then to identify a flat zone in the system response function and perform experiments only in that frequency range.

#### 3.1. Experimental Setup

_{3}), which is converted into charge.

#### 3.2. Results

#### 3.2.1. Characterization of the Elastic Properties of the Elastomer

#### 3.2.2. Frequency Selection

#### 3.2.3. Indenter Selection

_{33.}Best fit is obtained assuming that d

_{33}=14 [pC/N].

#### 3.2.4. Response Function of the Sensor Array

_{3}affecting the response of the PVDF sensor area is the difference between the total stress (produced by the dynamic force amplitude added to the preload) and the stress due to the preload only. Note that—while the response of the PVDF sensor is not affected by static contacts—its piezoelectric coefficient d

_{33}(Equation (15)) is likely to be dependent on the preload and some non-linearities may be expected for high preload values.

_{33}of each PVDF sensor, a quantity that was a priori unknown in the present investigation as the PDMS elastic layer was already integrated on top. Some initial characterization of the commercial PVDF film was performed around 2010 and results are reported in [21]. However, a systematic measurement of the d

_{33}coefficient over the whole sensor array is not available. In any case, although such a characterization is done prior to sensor integration into the complex multilayer skin, long-term aging and fatigue may affect sensor behavior and degradation of sensor properties is expected over time. Being able to estimate the piezoelectric d

_{33}coefficient from the overall system response function is thus a useful tool to measure the reliability of the e-skin device over time, whenever embedded sensors are not accessible anymore for a direct characterization.

_{33}that best fits the experimental observations.

_{33}coefficient, which measures the piezoelectricity of the PVDF sensor, on the preload. Data associated with the higher preload (=3 N) are well fitted with a d

_{33}value equal to 22 [pC/N], while data corresponding to the lower preload (=1 N) yield a d

_{33}value of 14 [pC/N]. These lower d

_{33}values, associated with the latest runs, can be considered reasonable as the e-skin was subjected to a huge number of stress cycles in the past 4 years, and this has likely led to film degradation and consequent PVDF aging and fatigue. It is important to appreciate that, as already noted above, the proposed model becomes fully predictive with a preliminary measurement of the PVDF d

_{33}coefficient of each sensor, prior to integration of the elastic layer on top of the sensor array.

## 4. Conclusions

_{T}and a, respectively), both scaled by h.

_{3}and E. This parameter did not appear in Boussinesq’s analytical solution.

_{33}coefficient, which turns out to exhibit some dependence on the preload. Fairly low d

_{33}values are also found and appear to be due to PVDF aging and fatigue.

**n**·

**T**at the bottom of the elastomer layer.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The PVDF sensor array is located on the bottom of the protective layer and integrated on a rigid substrate (side view).

**Figure 2.**(

**a**) The PVDF sensor (radius ${r}_{T}$) is located on the bottom of the elastic cover of thickness h and a normal point force is applied on the outer surface (view side). (

**b**)

**e**

_{r}is represented in a spherical coordinate system.

**Figure 4.**The proportionality coefficient γ between average normal stress ${\stackrel{\u203e}{T}}_{\mathit{3}}$ on the sensor and normal point force F

_{3}(see Equation (6)) is plotted versus the distance $\widehat{r}$ from sensor center (see Figure 2 for notations). Different curves are associated with different sensor sizes as measured by the dimensionless parameter $\frac{{r}_{T}}{h}$ (yellow $\frac{{r}_{T}}{h}=0.2$; green $\frac{{r}_{T}}{h}=0.5$; black dotted $\frac{{r}_{T}}{h}=0.6$; blue $\frac{{r}_{T}}{h}=0.7$ ; magenta $\frac{{r}_{T}}{h}=\mathit{1}$; light blue $\frac{{r}_{T}}{h}=1.5$; red $\frac{{r}_{T}}{h}=\mathit{2}$).

**Figure 5.**Normal force distributed over circular contact area with Hertzian pressure distribution. (

**a**) View side, (

**b**) Top side.

**Figure 6.**The proportionality coefficient δ between normal stress T

_{3}on the sensor and overall contact force F

_{3}(see Equation (12)) is plotted versus the imprint radius a (contact size) scaled by the elastomer thickness h (see Figure 5 for notations). Note that the point-like sensor is aligned with the contact center.

**Figure 7.**The extended PVDF sensor is located on the bottom of the elastic cover of thickness h and a distributed normal force applied on the outer surface is aligned with the sensor center.

**Figure 8.**The proportionality coefficient σ between average normal stress ${\stackrel{\u203e}{T}}_{\mathit{3}}$ on the sensor and overall (Hertzian) contact force F

_{3}(see Equation (14)) is plotted versus the imprint radius a (contact size) scaled by the elastomer thickness h (see Figure 7 for notations). Note that the applied force is centered on the sensor. Different curves are associated with different sensor sizes as measured by the dimensionless parameter $\frac{{r}_{T}}{h}$ (yellow $\frac{{r}_{T}}{h}=0.2$; green $\frac{{r}_{T}}{h}=0.5$; black dotted $\frac{{r}_{T}}{h}=0.6$—

**case study reported in the experimental session**; blue $\frac{{r}_{T}}{h}=0.7$; magenta $\frac{{r}_{T}}{h}=\mathit{1}$; light blue $\frac{{r}_{T}}{h}=1.5$; red $\frac{{r}_{T}}{h}=\mathit{2}$).

**Figure 9.**An example of FEM simulation results (total displacement is shown). Parameters: F

_{3}= 2 N, R = 10 mm, E = 16 MPa, ν = 0.5, r

_{T}= 1 mm. Elastomer size: thickness h = 2.5 mm, length l = 40 mm, width b = 20 mm.

**Figure 10.**A comparison between results for σ as a function of a/h for different values of r

_{T}/h (see legend), as obtained by Boussinesq’s analytical model (see Figure 8) and the numerical COMSOL simulations for the finite case.

**Figure 11.**A comparison between Boussinesq’s analytical model for the half-space (dotted line) and the numerical COMSOL simulations for the finite case (markers) is reported for r

_{T}/h = 0.6, corresponding to the geometry of the e-skin prototype employed in the experimental section. As in Figure 8 and Figure 10, σ vs. a/h is plotted. The role of the parameter L is included and confirms that distinct curves are associated with different values of L.

**Figure 15.**Single sensor analysis. A comparison is presented between numerical and experimental results for FRF vs. Preload, for 2 different indenters.

**Left**: R = 4 mm,

**Right**: R = 10 mm.

**Figure 16.**Model validation. The dynamic force amplitude is 0.09 [N] in both cases. Dotted lines correspond to the values of the response function predicted by the numerical model, for the two different preloads.

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**MDPI and ACS Style**

Seminara, L.
Modeling Electronic Skin Response to Normal Distributed Force. *Sensors* **2018**, *18*, 459.
https://doi.org/10.3390/s18020459

**AMA Style**

Seminara L.
Modeling Electronic Skin Response to Normal Distributed Force. *Sensors*. 2018; 18(2):459.
https://doi.org/10.3390/s18020459

**Chicago/Turabian Style**

Seminara, Lucia.
2018. "Modeling Electronic Skin Response to Normal Distributed Force" *Sensors* 18, no. 2: 459.
https://doi.org/10.3390/s18020459