**3. Assessment of the skin behavior**

#### **3.1. Electromechanical characterization of PVDF films**

Commercial 100μm thick PVDF sheets from Measurement Specialties Inc.4 have been purchased. PVDF samples have been cut from those sheets in a square geometry of 7x7 mm2. Purchased sheets have been already stretched and poled. The process begins with the melt extrusion of the polymer resin pellets into sheet form, followed by a stretching step that reduces the sheet to about one-fifth its extruded thickness. Stretching at temperatures well below the melting point of the polymer causes chain packing of the molecules into parallel crystal planes (beta phase). The beta phase polymer is poled by application of very high electric fields (of the order of 100 V/μm) to align the crystallites to the poling field. In such conditions the piezoelectric film exhibits a material symmetry in the orthorhombic crystal system (C2V class), corresponding to that of the so-called orthotropic materials. The samples are oriented with axis 1 along the stretching direction – axis 2 in the in-plane orthogonal direction and axis 3 along the through-thickness direction (see Figure 1).

**Figure 1.** PVDF provided of electrodes to extract the charge signal. Reference axes are shown.

Linear electro-elastic constitutive equations are commonly used to describe the coupling of dielectric, elastic, and piezoelectric properties in piezoelectric materials [24].

In the frequency domain ( ( ) <sup>ˆ</sup> *f* ω means the Fourier Transform of any function *(f(t)*) such equations are:

$$
\begin{bmatrix}
\hat{S} \\
\hat{D}
\end{bmatrix} = \begin{bmatrix}
\hat{s} & \hat{d}^T \\
\hat{d} & \hat{e}
\end{bmatrix} \begin{bmatrix}
\hat{T} \\
\hat{E}
\end{bmatrix} \tag{1}
$$

where strain ˆ *S* and stress *T*ˆ are represented by 1x6 column vectors, while the electric displacement *D*ˆ and the electric field *E*ˆ are expressed by 1x3 column vectors. *s*ˆ is the 6x6 *compliance* matrix, εˆ the 3x3 *permittivity* matrix, both assumed to be symmetric, and ˆ *d* the *piezoelectric* 3x6 matrix.

In our application, the PVDF behavior is usefully described by the second row of constitutive equations:

<sup>4</sup> http://www.meas-spec.com/default.aspx

$$
\begin{bmatrix}
\hat{D}\_1\\ \hat{D}\_2\\ \hat{D}\_3
\end{bmatrix} = \begin{bmatrix}
0 & 0 & 0 & 0 & \hat{d}\_{15} & 0\\ 0 & 0 & 0 & \hat{d}\_{24} & 0 & 0\\ \hat{d}\_{31} & \hat{d}\_{32} & \hat{d}\_{33} & 0 & 0 & 0
\end{bmatrix} \begin{bmatrix}
\hat{T}\_{11}\\ \hat{T}\_{22}\\ \hat{T}\_{33}\\ \hat{T}\_{23}\\ \hat{T}\_{13}\\ \hat{T}\_{12}
\end{bmatrix} + \begin{bmatrix}
\varepsilon\_{11} & 0 & 0 & 0\\ 0 & \varepsilon\_{22} & 0 & \left[\hat{E}\_1\\ \hat{E}\_2\\ 0 & 0 & \varepsilon\_{33}
\end{bmatrix} \begin{bmatrix}
\hat{E}\_1\\ \hat{E}\_2\\ \hat{E}\_3
\end{bmatrix}
\tag{2}
$$

The structures of the piezoelectric and the permittivity matrices are due to the reported material symmetry.

As the charge signal is measured with a charge amplifier, which converts the charge input into a voltage without supplying any electric field, the electric field can be set to null. Therefore, when the film is used in *thickness mode* the previous set of equations reduces to:

$$
\hat{D}\_3 = \hat{d}\_{33}\hat{T}\_{33} \tag{3}
$$

**Figure 2.** Compression test setup to measure the *d33* coefficient.

616 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

**3.1. Electromechanical characterization of PVDF films** 

direction and axis 3 along the through-thickness direction (see Figure 1).

**Figure 1.** PVDF provided of electrodes to extract the charge signal. Reference axes are shown.

dielectric, elastic, and piezoelectric properties in piezoelectric materials [24].

*f* ω

In the frequency domain ( ( ) <sup>ˆ</sup>

ε

equations are:

where strain ˆ

*compliance* matrix,

*piezoelectric* 3x6 matrix.

constitutive equations:

4 http://www.meas-spec.com/default.aspx

Linear electro-elastic constitutive equations are commonly used to describe the coupling of

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ *<sup>T</sup> S s d T*

<sup>=</sup>

displacement *D*ˆ and the electric field *E*ˆ are expressed by 1x3 column vectors. *s*ˆ is the 6x6

In our application, the PVDF behavior is usefully described by the second row of

ε*E*

*S* and stress *T*ˆ are represented by 1x6 column vectors, while the electric

ˆ the 3x3 *permittivity* matrix, both assumed to be symmetric, and ˆ

*D d*

means the Fourier Transform of any function *(f(t)*) such

(1)

*d* the

Commercial 100μm thick PVDF sheets from Measurement Specialties Inc.4 have been purchased. PVDF samples have been cut from those sheets in a square geometry of 7x7 mm2. Purchased sheets have been already stretched and poled. The process begins with the melt extrusion of the polymer resin pellets into sheet form, followed by a stretching step that reduces the sheet to about one-fifth its extruded thickness. Stretching at temperatures well below the melting point of the polymer causes chain packing of the molecules into parallel crystal planes (beta phase). The beta phase polymer is poled by application of very high electric fields (of the order of 100 V/μm) to align the crystallites to the poling field. In such conditions the piezoelectric film exhibits a material symmetry in the orthorhombic crystal system (C2V class), corresponding to that of the so-called orthotropic materials. The samples are oriented with axis 1 along the stretching direction – axis 2 in the in-plane orthogonal

**3. Assessment of the skin behavior** 

For the present application, characterizing the electromechanical behavior of the piezoelectric polymer means to retrieve the frequency behavior of the *d33* piezoelectric coefficient.

The employed experimental equipment has been thoroughly discussed in a previous publication [25], which reports more complete characterization results of the PVDF electromechanical behavior. Briefly, the experimental setup (see Figure 2) consists of a rigid frame with a lower fixed plate to which an electro-mechanical shaker is assembled. A piezoelectric force transducer is fixed to the upper head of the frame. Samples are mounted between the force transducer and the shaker. They are pressed between two metals heads of square cross-section with machined and polished contact surfaces. The lower head is assembled to the shaker head with an interposed PMMA block, and the upper head contrasts through a spherical joint with a similar block connected to the force transducer, for

self-alignment of the contact planes. Conductive glue creates a stiff connection between the sample and the heads thus excluding possible variations in the contact area during measurements.

The test is controlled by a computer in a completely automatic way. A swept sine signal is fed into the shaker. Frequency spacing and total duration of the test are determined by the settable frequency range and number of steps. The output charge signals (response) and force transducer (stimulus) are continuously acquired and processed in frequency to give the complex piezoelectric modulus. Normally the range between 10 and 1000 Hz can be explored without difficulties.

The frequency behavior of the *d33* piezoelectric coefficient is shown in Figure 3**.** The most relevant result in this context is the almost flat behavior of both the real and imaginary parts of the modulus in the considered frequency range, in accordance with literature [26].

**Figure 3.** Frequency behavior of the real and imaginary parts of the d33 piezoelectric coefficient.

### **3.2. Electromechanical modeling of the skin structure**

In order to associate the PVDF charge response to the effective load applied on the outer skin surface, a skin model which is based on the Boussinesq's equation has been considered.

**Figure 4.** The PVDF sensor is located at the bottom of the protective layer of thickness h and a point force is applied on the outer surface.

Approximately, the relation between a point load **F** applied on the outer surface and the stress at a given point inside the cover layer is given by the Boussinesq's equation [27]:

$$T = \frac{\Im}{2\pi} \frac{F \cdot \mathbf{e}\_r}{r^2} \mathbf{e}\_r \otimes \mathbf{e}\_r \tag{4}$$

where all bold faced symbols represent tensors or vectors, **e***k* is the unit vector in the *k*direction and ⊗ is the symbol of the tensor product. The Boussinesq's problem considers a linearly elastic half-medium on which a point force is applied. Truly, the materials employed in the present application (typically elastomers or gels) are both non-linear elastic and visco-elastic. Nevertheless, Equation (4) does not depend on the elastic modulus and deviations from linearity are only expected as a consequence of geometry changes due to large deformations. For the applicability of the model, therefore, a sufficiently rigid protective layer should be chosen. In addition, Equation (4) holds whenever the Poisson ratio of the medium is close to 0.5. This holds for an elastomer (ν = 0.48), but not for other materials (e.g. a foam). A complete solution valid for all ν is available in [28], but it is considerably more complex. The major approximation concerns the finite thickness of the layer, compared with the semi-infinite medium. The advantage of (4) is its simplicity, as the stress is uniaxial in the radial direction, and its independence of elastic parameters.

Equation (4) is applied to a sensor located on the bottom of the elastic cover of thickness *h*. The sensor works in the thickness mode, i.e. it can read the *T3* stress component ('3' is the direction normal to the bottom surface) via the *d33* piezoelectric modulus. Letting *r*ˆ be the radial distance of the point where the force is applied from the sensor center projected on the outer surface, we have <sup>222</sup> *rrh* = + <sup>ˆ</sup> and 12 3 e sin (e cos e sin ) e cos *<sup>r</sup>* = +− θϕ ϕ θ **,** where sinθ= **r**ˆ *r* .

Then we obtain:

618 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

measurements.

explored without difficulties.

self-alignment of the contact planes. Conductive glue creates a stiff connection between the sample and the heads thus excluding possible variations in the contact area during

The test is controlled by a computer in a completely automatic way. A swept sine signal is fed into the shaker. Frequency spacing and total duration of the test are determined by the settable frequency range and number of steps. The output charge signals (response) and force transducer (stimulus) are continuously acquired and processed in frequency to give the complex piezoelectric modulus. Normally the range between 10 and 1000 Hz can be

The frequency behavior of the *d33* piezoelectric coefficient is shown in Figure 3**.** The most relevant result in this context is the almost flat behavior of both the real and imaginary parts

of the modulus in the considered frequency range, in accordance with literature [26].

**Figure 3.** Frequency behavior of the real and imaginary parts of the d33 piezoelectric coefficient.

In order to associate the PVDF charge response to the effective load applied on the outer skin surface, a skin model which is based on the Boussinesq's equation has been considered.

**Figure 4.** The PVDF sensor is located at the bottom of the protective layer of thickness h and a point

**3.2. Electromechanical modeling of the skin structure** 

force is applied on the outer surface.

$$\mathbf{T}\_3 = \frac{3}{2\pi} \frac{\mathbf{h}^2}{\left(\hat{r}^2 + \mathbf{h}^2\right)^5} \left\{ \left(\mathbf{F}\_1 \cos\varphi + \mathbf{F}\_1 \cos\varphi\right) \hat{r} - \mathbf{F}\_3 \mathbf{h} \right\} \tag{5}$$

For a vertical force (*F1* = *F2* = 0) Equation (5) reduces to:

$$\mathbf{T}\_3 = -\frac{3}{2\pi} \frac{\mathbf{F}\_3 \mathbf{h}^3}{\left(\hat{r}^2 + \mathbf{h}^2\right)^{5}} \tag{6}$$

As *T3* is related to the charge density *D3* on the sensor surface by the piezoelectric constitutive equation:

$$D\_3 = \mathbf{d}\_{33} \, T\_3 \, \tag{7}$$

under the hypotheses that the sensor size is sufficiently smaller than its distance from the point force, the total charge measured by the sensor can be approximated by:

$$\mathbf{q} = -\frac{3}{2\pi} A \frac{\mathbf{d}\_{33} \mathbf{F}\_3 \mathbf{h}^3}{\left(\hat{r}^2 + \mathbf{h}^2\right)^{5}} \tag{8}$$

#### **3.3. Experimental assessment of the single taxel behavior**

In order to optimize the design of the skin prototypes we checked the pertinence/limits of the mechanical model, thus quantifying the PVDF electrical response to the external mechanical stimuli and testing the effect of the protective layer (material and thickness) on the PVDF response. We performed the tests on basic single taxel prototypes, where a single circular PVDF film is covered with an elastomeric cover to protect the sensor from physical damage by shocks or chemical contamination by oil and other materials. Attention has been focused on gel/rubber layers, in that they are more controllable and reproducible systems which also allowed for the required dynamics.

An experimental campaign has been carried out in order to quantify the sensor output charge in common tactile interaction tasks. The mechanical input stimuli range from 50 Pa to 5 MPa, as reported in section 2.

**Figure 5.** Single taxel skin prototype: comparison between model and experiments.

The "modally tuned impulse force hammer" (see Section 2) has been used to mechanically stimulate the single taxel sample. In this case, we recall that the load cell measures a contact force, which depends both on the indenter and on the protective layer materials. The measured charge over the 5 kPa - 5 MPa stress range has been compared with the model presented in the previous section and results are reported in Figure 5. In this case, on the horizontal axis the peak amplitude of the *time behavior* of the contact force is reported. Similarly, the peak amplitude of the charge response is reported on the y axis. It is important to remark here that these measurements are not at all easy to perform, as it is not yet possible to accurately control the position of the hammer impact. A good statistics would be thus required to achieve reliable results. Results are however useful to understand the "order of magnitude" of the charge response and the limits of applicability of the models.

A good accordance between model and experimental data is recorded for high loads, which allows extending the model linear behavior for lower loads, thus covering the whole stress range which is of interest for the application (50Pa - 5MPa). Therefore, both charge and force ranges cover 5 orders of magnitude, the typical output charge ranging from 0.01 pC to 1 nC. This information has been a reference point for the design of the electronics.
