Mathematical Model and Experimental Characterization of Vertically Stacked Capacitive Tactile Sensors

Capacitive sensors are widely used in robotics for their compactness, high resolution, high sensitivity, and large dynamic range. In this article, we present a design solution for the manufacturing of capacitive tactile sensors with enhanced dynamic range and sensitivity. Herein, we adopted the approach of exploiting the vertical direction of the sensors by creating stacks of capacitors. The validation of the proposed model is conducted by means of finite element simulations and the effectiveness of stacked capacitors in suboptimal configurations has been experimentally tested by using inkjet printing and spin coating-based fabrication techniques. Results show that these sensors exhibit an enhanced dynamic range and sensitivity with respect to common single capacitors, for a given sensors area budget. Sensitivity increases of 235% passing from one-stack to two-stack capacitors (from 5.75 to 19.3 fF/kPa) and a growth of 23% from two-stack to three-stack capacitors (from 19.3 to 23.7 fF/kPa). These results suggest that the proposed methodology could be adopted for designing tactile sensors with higher spatial resolution and higher transduction sensitivity and dynamic range, in the perspective of an integration over large areas.


I. INTRODUCTION
R OBOTS have gathered a lot of attention in the last two decades, and they have been playing a bigger part in the industrial scene and in everyday life.Using the human as a model, advanced technology and great progresses in mimicking sensory capabilities have been demonstrated [1], [2].Tactile information, such as the magnitude and direction of a contact force, temperature, humidity, and texture, can be obtained by robots equipped with tactile sensors, which is crucial for secure grasps, path planning, and obstacle avoidance in unstructured environments [3], [4].In addition, reliable touch sensors are required for both object manipulation activities and safe human-machine interaction [5].Capacitive sensing is one of the most common principles used in robotic tactile sensing.The capacitance of a parallel plate capacitor is expressed as C = (ϵ 0 ϵ r A/d), where A is the overlapping area of the two electrodes, ϵ 0 is the permittivity of vacuum, ϵ r is the relative permittivity of the dielectric layer, and d is the distance between the electrodes (thickness).Researchers are interested in capacitive sensors because of their great sensitivity, resolution, costs, and read-out electronics.
For a good response, capacitive sensors require a wide dynamic range, defined as the ratio between the maximum and the minimum value that the sensor can detect.The dynamic range can be increased by maximizing the sensor capacitance (C) and it is strictly correlated to the sensitivity (S) of the transducer, expressed as the ratio between the variation in capacitance for a given variation of contact pressure.The capacitance C can be increased by decreasing the sensor thickness and/or by increasing the sensor area.Unfortunately, both approaches bring to some drawbacks.In particular, the decrease in the sensor thickness is limited by the technology used to manufacture the sensor.Furthermore, the increase in the sensor area can lead to space problems on small surfaces.There are works in literature that proposed methods to enhance the dynamic range of capacitive tactile sensors [6], [7], [8], [9], [10].An interesting approach is the exploitation of the vertical direction.Results from the works of [11], [12], [13], [15], and [14] show that this design for capacitive sensors reduces the projective area taken on the x y plane for enhanced spatial resolution and sensitivity and it can be used for extreme miniaturizations.Jen et al. [16] developed and characterized a capacitive tactile sensor that simultaneously supports high detection sensitivity and high spatial resolution with a hollow dielectric structure, by vertically stacking two capacitors.Liu et al. [17] and Yin et al. [18] made significant advances in flexible capacitive pressure/tactile sensors such as low-voltage organic transistors for sensitive pressure sensing and wearable applications by combining the advantages of different dielectrics.
Building on the mentioned approaches, this work aims to maximize the dynamic range of capacitive tactile sensors and their sensitivity by exploiting the vertical direction and creating stacks of more than two capacitors.If the previously mentioned works propose a specific solution, the goal of this article is to develop a model that allows to choose the design parameters and to evaluate different stacking options to reach the desired sensitivity.This work provides the following contributions.
1) The development of a mathematical model for tactile capacitive sensors, in order to find a configuration to increase their sensitivity with respect to the common single capacitors.
2) The validation of the mathematical model with finite element simulations, by studying the sensor's responses.3) A straightforward method for manufacturing and characterizing vertically stacked capacitive tactile sensors.4) Static and dynamic experimental analyses to validate the sensitivity and dynamic range increase on inkjet-printed sensors.
The remainder of the article is organized as follows.
Section II presents the theoretical modeling of capacitive sensors.The aim is to demonstrate that the sensitivity of a single capacitance is lower with respect to a stack of capacitors.The model allows to give guidelines for the design of new layouts in order to increase the performance of this class of sensors.Section III presents the results of finite element analysis for validating the proposed mathematical model and for studying the mechanical behavior of the dielectric material, also introducing nonlinear simulations to compare the results with the experiments.In Section IV, we describe the manufacturing process adopted to build the stacked capacitors.Section V presents the static and dynamic characterization of the sensor with the respective results.Conclusion is described in Section VI.

II. MATHEMATICAL MODEL OF A CAPACITIVE TACTILE
SENSOR As anticipated in Section I, in order to increase the dynamic range and the sensitivity of the response, we propose the development of stacked parallel capacitance.The sensitivity of capacitance measurements is dependent on the thick values of the capacitance.For a given thickness, the first goal is to evaluate how thick the layers composing the stack must be to optimize sensor response, respecting design and manufacturing constraints.The second purpose of the demonstration is to indicate how many capacitors have to be stacked in order to maximize the resulting sensitivity.

A. Capacitive Sensors Principle
As shown in Fig. 1, the variation of the pressure applied on the sensor due to the force F uniformly distributed on the transducer area A can be written as follows: where the pressure variation P = P − P 0 is supposed to be uniform on the whole sensor area.P is the applied pressure and P 0 is any possible equilibrium pressure (e.g., preloading pressure or atmospheric pressure in the case of pressure-compensated sensors).We start to share how an applied pressure is uniformly distributed on the sensor surface, and we can compare the deformation of the dielectric material of the capacitive sensor with the deformation of the equivalent spring, using the following formula: where k is the elastic constant of the equivalent spring, representing the linearized effect of the applied force in terms of the sensor deformation.t = t − t 0 is the displacement, t is the resulting thickness of the dielectric due to the pressure applied on the transducer, and t 0 is the initial thickness.
Using the standard stiffness engineering model [19], [20], we have that where E is the Young's Modulus of the material.
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By substituting the value of F computed from (1) in to (2) and using the definition of k of (3) we obtain the following equality: Which can also be rewritten as follows: assuming a linear stress-strain relationship for the material used to build the sensor.Let us now consider the nominal value of the sensor capacitance in the absence of contact pressure.
The value is C 0 and it is expressed as follows: γ = ϵϵ r , where ϵ and ϵ r are the vacuum electric permeability and the dielectric constant of the material used, respectively.Once pressure is applied to the sensor, the resulting capacitance becomes under the assumption that the transducer area does not change under the effect of contact pressure.The difference between C and C 0 can then be written as follows: By using (5), we can write (8) as follows:

B. Two-Layered Stacked Capacitance
In Section II-B, we analyze the dielectric thickness requirement by considering a capacitive sensor configured as parallel capacitance (stack), composed of two layers of dielectric material, as in Fig. 2.
For capacitive sensors, in general, the dielectric greatly affects the response of the device [21].In this article, we only considered the effect of the dielectric in terms of its deformation.From (9), can define the total variation of the stack as the sum of the variation of the two capacitors where C 0 = C 1 0 + C 2 0 is now the total capacitance of the stack at rest.Suppose we want to design a stacked sensor with these characteristics such that the total thickness of the sensor (design parameter) is defined as equal to t tot .In this case, the thicknesses of the two capacitors t 1 and t 2 become the design parameters.With these considerations, we can write C 0 as follows: .
So, for a given capacitor thickness, the goal is to decide how much t 1 and t 2 should be By defining we can write (11) using ( 12) and ( 13) The sensitivity S is defined as the ratio between the variation of capacitance C and the variation of the exerted pressure P (i.e., the stress) The sensitivity is proportional to C and by (10) to C 0 .Therefore, for a given P, C 0 must be maximized.
Fig. 3 shows the trend of C 0 with respect to the layer thickness, expressed with the variable x (minimum possible thickness for the dielectric).
The formulation of the computation of the maximum of C 0 must be constrained since the minimum value of x is indeed a physical constraint that might depend on the technology used for the manufacturing of the sensor and on the characteristics of the material involved in the design.
From this result, we can say that the decreasing of the layer thicknesses improves the output and so the sensitivity of the capacitive sensor.

C. Three-Layered to N-Layered Stacked Capacitance
In order to generalize to the case of an N -layer stack (N -stack), let us consider as an example the case of a threestack.Using the same notation of Section II-B, we define as x 1 , x 2 , and x 3 the normalized thicknesses of the three layers t 1 , t 2 , and t 3 with respect to the total expected sensor thickness t tot .Then the following optimized problem must be solved: In order to find a maximum of the above cost function, consider a feasible choice for x 1 , x 2 , and x 3 and assume (without loss of generality) x 1 fixed.Then following the rationale used in Section II-B, it is straightforward to verify that maximizes C 2 + C 3 .Since C 2 cannot be increased further, x 2 can be fixed and the above rationale can be repeated for x 1 and x 3 so that it can be This result eventually leads to a feasible solution to maximize the capacitance of the three-stack.It is also quite easy to check that for a given transducer thickness t tot , the capacitance of a three-stack is higher than a two-stack one.The dimension above suggests that for a given thickness t tot , the layers should have the following distribution: The formal proof is in Appendix.

III. FINITE ELEMENT SIMULATIONS
In this section, we describe simulations performed with the finite element method to validate the mathematical model presented in Section II and to explain the mechanical behavior of the dielectric material, also modeling the nonlinear effect featuring the real system.On one hand, the linear simulated model is useful to validate the models discussed in Section II; on the other hand, the nonlinear simulated model is important to compare the finite element results with the experimental ones presented in Section IV.First, we define the sensors model, which is essential to simulate the sensor response.Then, we compute the sensor's finite element responses with their corresponding sensitivity and we compare the results between a single capacitor and stacks of capacitors to demonstrate that a stack performs better than a single capacitance.

A. FEM Sensor Model
Practical guidelines for the implementations of capacitive tactile sensors have been presented in [22].The geometry and the material properties used for the simulated models have followed those guidelines and are related to existing tactile sensors implementation [36].The simulated transducer is a single circular taxel with a diameter d = 4 mm.Tests in Section III-B, which are used to validate the mathematical demonstration, include models with the same thickness t 0 = 1 mm.Tests in Section III-C, which are used to study the mechanical behavior of the dielectric material, include models with different thicknesses, which increase every time we add a capacity in parallel to the stack.In this case, each capacitance has a thickness t = 100 µm.The choice of the dielectric and conductive layer materials is consistent with the materials used for the experimental tests described in Section IV.In particular, the dielectric is made of polydimethylsiloxane (PDMS), and silver ink is used for the conductive capacitor plates and electrodes.The mechanical characteristics of the conductive layers are assumed negligible and not modeled in the FEM simulations.For the nonlinear mechanical analysis, we used a neo-Hookean constitutive model for the PDMS.Table I summarizes the most relevant material properties of the dielectric material used, which are as follows.
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1) The Young's modulus, defined as E = (σ/ϵ), where σ and ϵ are the material's stress and strain, respectively [28], is used to compute the elastomer strain under an exerted pressure.
2) The Poisson's ratio (ν) is a measure of the Poisson effect originating from the elastomer expansion along the direction normal to pressure application.
3) The dielectric permittivity (ϵ r ), which impacts on capacitance variations [22] and so on the sensor sensitivity.

B. FEM Validation of the Mathematical Model
The computational physics suite COMSOL Multiphysics has been used to implement the models and performance simulations, and simulation results have been further analyzed with MATLAB.For the simulations, two classes of contacts have been selected, following [30] and [22].The first class is referred to as gentle touch and is characterized by contact pressures in the 0-10 KPa range, whereas the second class, manipulation-like touch, refers to pressures in the 10-100 KPa range.
Looking at the consideration of Section II, we want to validate through FEM: 1) the optimal layer distribution and 2) the maximization of the performances by increasing the number of layers.For the sake of simplicity, in all models, the sensitivity has been calculated as the angular coefficient of the linear fit curve, which can cover all the ranges, with a maximum pressure of 100 kPa, from gentle touch to manipulation-like touch.
1) Optimal Layer Distribution: Let us now consider two models, which represent a stack with the same total thickness t 0 = 1 mm composed of four layers.In the first case, the layers have the same thickness, in the second case, the layers are distributed according to the analysis of Section II, assuming a minimum thickness of 100 µm.Fig. 4(a) and (b) shows the two stack configurations in the case of uniform pressure applied on the top layer of 100 kPa.Fig. 4(c) shows the finite element response of the first configuration, while Fig. 4(d) shows the finite element response of the second configuration.For validation, the models have been considered linear.By comparing the two finite element results, the second case has a 7% increase in sensitivity with respect to the first case.This further validates the first part of the mathematical demonstration, on which we propose the optimal layer distribution to increase the sensor response.
2) Maximization of the Performances by Increasing the Number of Layers: Let us now consider the other two models, which represent a stack composed of four layers and a stack composed of six layers, with the same total thickness t 0 = 1 mm, and both with the optimized layer configuration.For the four-stack capacitor, we consider the best configuration of Section III-B.1, and so we refer to Fig. 4(b) for the model and to Fig. 4(d) for the response.Fig. 5(c) shows the six-layer stack and Fig. 5(a) shows the finite element response of the six-stack model.Also in this case, for the validation, we consider linear models.By comparing the two finite element results, the six-layer stack has a greater sensitivity with respect to the four-layer stack.This comparison tells us that by increasing the number of layers in optimal configuration by 50%, a sensitivity variation of 130% is produced.
In order to show the effectiveness of the optimal configuration, we also compared the case of four layers of equal thickness [see Fig. 4(a)] with the case of six layers of equal thickness [see Fig. 5(b)].The results of these two nonoptimal configurations are shown in Figs.4(c) and 5(d).In this case, a sensitivity variation of 120% is produced.
Through the tests and comparisons described in Section III-B, we have validated the entire mathematical demonstration.We can state that the manufacturing of sensors composed of stacked capacitance could lead to an improvement in the performance of capacitive sensors.Moreover, each layer has to be designed according to the optimal distribution explained above.

C. FEM Analysis on the Mechanical Behavior of the Dielectric Material
In Section III-B, the analysis aims to demonstrate that more optimized stacked capacitors have good performances.In this section, we focus on the most technologically tractable case, in which layers have all equal thicknesses.So, we work with nonoptimal configurations from the layout point of view.In this case, all the elements necessary to make the results comparable with the experimental analysis have been introduced into the FEM.In particular, in addition to the PDMS as a dielectric, kapton was also added in the simulations, as in the experiments the sensors present both materials to form the dielectric.For the models, a single capacitor, a two-stack capacitor, and a three-stack capacitor have been considered.The first simulation test refers to the analysis of the response of a one-stack and two-stack, and the second simulation test refers to the analysis of the response of a two-stack and three-stack, in order to understand how the dielectric is compliant under the action of a uniform pressure applied on the surface area.Figs.6(a), 7(a), and 8(a) show the mechanical responses of one-stack, two-stack, and three-stack, respectively, in terms of dielectric total displacement under the maximum pressure applied.The initial values of the single capacity (C 0 1 ), of the two-stack (C 0 2 ), and of the three-stack (C 0 3 ) are 3.65, 7.29, and 10.9 pF, respectively.
Fig. 9 shows the ideal sensors' output and their corresponding sensitivity in the linear case.More detailed information are given in Figs.6(b), 7(b), and 8(b) which show the ideal sensors output and their corresponding sensitivity in the nonlinear case.This is a more real representation of the sensor's response, and for this reason, we will consider these values as a reference to compare them with the experimental results.Results from the neo-Hookean model show that the sensitivity has an increase of the 90% passing from a single capacitor (S = 8.9 fF/kPa) to two-stack (S = 16.9 fF/kPa), and an increase of 48% passing from a two-stack to a three-stack (S = 25 fF/kPa).The sensitivity has an increase of 180% if we pass from a single capacitor to a three-stack capacitor.Also, in this case, the sensitivity will increase every time we add a capacity in parallel to the stack.

IV. EXPERIMENTAL MANUFACTURING
In this section, we first show the manufacturing procedures used to make a single capacitor and stacks of capacitors (all with a diameter of 4 mm, as for the simulation models).Then, we present the characterization procedure used to compute the parameters required to validate the finite element simulations of Section III-C.The manufacturing process is based on the use of two techniques: inkjet printing and spin coating.The peculiarity of this approach is that it can be potentially used to obtain stacks of several capacitors in parallel and the procedure can be customized to optimize the sensors covering different surfaces.In this work, three kinds of sensors have been manufactured and their performances have been analyzed.The characteristics of a one-stack capacitor, the output of a two-stack capacitor, and the output of a three-stack capacitor have been computed and the results have been compared with those obtained from the simulations.
A. Single-Capacitance Tactile Sensor 1) Dielectric Preparation: Sylgard 184 (Dow Corning) is used as the dielectric material.The PDMS base and the curing agent is mixed in a 10:1 ratio (base to catalyst) in all experiments and steered for 2 min.Then, the PDMS is placed in vacuum desiccators for degassing (10-13 min).Then, a thin layer of PDMS has been deposited on the capacitor substrate by spin coating.As discussed in [31], a thickness of 25 µm guarantees excellent electrical insulation between the conductive layers.The PDMS thickness mainly depends on spin speed and duration [32], [33].In this work, this thickness has been obtained by depositing in sequence two layers of PDM, each obtained through a spin coating with a speed of 2000 r/min and a duration of 5 min.With these parameters, each layer has a thickness of ∼12 µm and so the total dielectric thickness is ∼24 µm.
The dielectric preparation is the same also in the case of the stacked capacitors described later.2) Device Fabrication: The capacitor substrate consists of a suitably cleaned kapton sheet (75 µm thick).Two layers of silver nanoparticle ink (Ag NP ink, SunTronic 1 EMD5730) have been inkjet printed directly on the substrate with the commercial inkjet printer Dimatix DMP-2850 Fujifilm.The printed substrate is briefly placed on a hot plate to evaporate the solvent until the silver of the ink emerges.After that, the curing is terminated in an oven at 150 • C for 30 min.Subsequently, the PDMS is spin-coated (as discussed in the above paragraph Section IV-A1) on top of the conductive layer and oven cured at 120 • C for 25 min allowing the solvent to evaporate and cross-link the PDMS.At the same time, another identical conductive pattern is inkjet printed on a second kapton substrate and, after the curing phase, is placed on top of the dielectric material, placed on the first substrate, to form a single capacitance.In the dashed rectangle of Fig. 10 1 Registered trademark.it is shown that the fabrication process and Fig. 11(a) shows the vertical section of the structure.

B. Stacked-Capacitance Tactile Sensors
1) Device Fabrication: Starting from the manufacturing of the single capacitance, stacks of N capacitors can be built by following the fabrication process which is summarized in Fig. 10.This method allows one to stack one capacitance on top of the other, forming a stack of capacitors in parallel.Each capacitor is manufactured with the aforementioned fabrication process (see Section IV-A2).In order to obtain a set of capacitors in parallel, all the odd layers must be electrically connected together as well as all the even armatures.To the same, the printed pattern of each armature is formed by a circle and an extruded electrode which will be used to interconnect the remaining layers.From the final transducer, the sensor is composed of 2N right electrodes and 2N − 1 left electrodes.All the right electrodes are electrically connected with each other with a conductive resin (MG Chemicals 842UR) and the same thing for the left electrodes.For a good electrical connection between the armatures, the layers are stacked one on the other and each of them is rotated by a few grades with respect to the lower, in order to avoid short circuits.This layout allows the electrical connection vertically.The use of conductive glues has been chosen to solve a problem regarding the vias fabrication in tactile sensors, in which functional devices from different layers need to be connected by vertical interconnects [27], [34].Through other common fabrication methods, some works addressed the problem by manually creating vertical feedthrough (creating holes and metalizing them), during or even after the device fabrication [13], [35].This process is normally conducted in rigid double-layer devices.More difficulties arise in the case of multilayered sensors (more than two layers).The drilling must be done with a perfect cut to ensure that the subsequent metallization contacts all the conductive layers involved.In this way, the vias making for multilayered sensors is supposed to be easier also because the via itself is not drilled but built like a conductive pillar.Fig. 11(b) shows the vertical section of the three-stack capacitor.
The manufactured three-stacked sensor is shown in Fig. 12.

V. STATIC AND DYNAMIC CHARACTERIZATION A. Static Characterization: Pressure-Capacitance Measurements
To evaluate the pressure range and the sensitivity of the manufactured capacitive sensors, the electrodes of the sensor were connected to the alligator clips of an LCR meter (GW Instek LCR-6100).We used a mechanical hand-made test rig, shown in Fig. 13, consisting of a vertical steel rod with a plastic cylindrical tip sustained by a linear bearing embedded in the supporting frame.The pressure applied to the capacitor is changed by adjusting the load by applying different weights on the upper supporting structure.For each Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.experiment, the rod is placed on top of the capacitor and loads are placed until a predetermined pressure (force divided by the  rod surface area) is reached by stacking loads in sequence.Pressures ranging from 10 to 100 kPa were tested, using 14 loads with the same weight.Each load was maintained for 10 s (recording 30 measurements) and then the next load was applied.All the measurements were repeated at least four times.While different loads were being applied, the capacitance of the sensor was measured using the LCR meter with a bias voltage of 1 V at a 1 kHz frequency.In this way, we extracted the capacitance change C of the sensor and its sensitivity S.

B. Results
We studied the pressure response of the one-stack capacitor, two-stack capacitor, and three-stack capacitor, measuring their corresponding C while the sensors were pressed.Fig. 14(a)-(c) shows the change in capacitance with respect to the applied pressure for each sensor type.Here, the sensitivity has been extracted over two adjacent pressure ranges.The first linear region included pressures from 0 to 30 kPa, whereas the second region included the range from 35 to 100 kPa.As it is possible to notice, in all the cases the sensitivity is higher in the first linear range, and after it decreases as the dielectric layer gets compressed and becomes less compliant.Thus, focusing only on the first linear ranges, the sensitivities for the single capacitor, the two-stack capacitor, and three-stack capacitor are 5.75, 19.3, and 23.7 fF/kPa, respectively.Results show that the sensitivity has an increase of the 235% passing from a single capacitor to a stack of 2, and an increase of 23% passing from a stack of 2 to a stack of 3 capacitors.The sensitivity has an increase of 312% if we pass from a single capacitor to a stack made of 3.These outcomes are consistent with the results obtained in simulation as described in Section III-C.

C. Dynamic Characterization: Dynamic Range Response
To measure the dynamic range of the manufactured sensor, we used a materials testing machine (ZwickRoell zwickiLine Z0.5) equipped with a load cell.As shown in Fig. 15, a force is applied on the sensor with a cylindrical probe.At the same time, the sensor is connected to an LCR meter (GW Instek LCR-6100) to measure the variation of capacitance.For each experiment, the cylindrical probe tip is initially set at 2.5 mm above the sensor.During the test, the probe is lowered toward the capacitor at a speed of 0.016 mm/s, until contact is made and the load cell measurements reach a threshold of 1 N .The process has been repeated for five loading-unloading cycles.

D. Results
We analyzed the dynamic range for the one-stack capacitor, two-stack capacitor, and three-stack capacitor.Fig. 16(a)-(c) shows the normalized sensitivity response over time.
Here the maximum and minimum values are highlighted to compare the dynamic ranges of the sensors.For the single capacitor, the two-stack capacitor, and three-stack capacitor the sensitivity range is 4, 7%, 10, 4%, and 32, 4%, respectively.These results demonstrate that also the dynamic range of the sensor is definitively higher in the case of parallel capacitors stacked together.

VI. CONCLUSION
In this work, we have presented a methodology for the design and manufacturing of capacitive tactile sensors.Our experimental results show that vertically stacking multiple capacitances leads to an improvement of the dynamic range  (from 4, 7% to 32, 4%) and the sensitivity (from 5.75 to 23.7 fF/kPa).Our hypothesis is motivated by the provided simplified model, which allows us to describe mathematically an optimal configuration of the layers composing the stack in order to improve the sensor response.The characteristics of the simplified model have been validated with finite element simulations.The effectiveness of stacked capacitors in a suboptimal configuration has been tested on an inkjet-printed sensor obtained with the proposed fabrication methodology.Indeed the effective physical realization of stacked capacitors is an open research problem.Thus, we propose the experimental realization methodologies in this article as a starting point for further improvements which will be considered as future work.

APPENDIX
As shown in Section II, if we consider the case N = 2, where N is a design specification related to the number of layers of the stack, the optimal thickness allocation is t min , − t min , where is the total transducer thickness intended also as a design specification as shown in Fig. 17 and t min is a technological constraint representing the minimum possible layer thickness.
N •t ≤ is the feasibility constraint ensuring the possibility of implementing an N -stack capacitor and of course, the interesting case is when N • t < or even ≫ N • t.
Let assume is given and, by induction, let assume that for a N − 1 layers stack (as shown in Fig. 18), the maximum capacitance is obtained using the configuration explained in the following equation: In the case of N layers we can observe that the distribution of (20) maximizes the capacitance for a vertical space budget This corresponds to the problem of maximizing the capacitance of a two-layers stack for an available space ¯ = − (N − 2) • t min which leads to the following allocation: Here are illustrated the simulations of the sensor's response by increasing the number of capacitors in parallel.The realization of a greater number of layers involves process difficulties and greater encumbrances.In this article, the simulations and the experimental tests have been conducted up to three layers of dielectric.For larger layers, we conducted simulations that give us information on what to expect by increasing the number of parallel capacitors (all the sensors features remain the same in Section III-C).Fig. 19(a) and (b) shows the nonlinear response of a five-stack and ten-stack capacitors.The predictions tell us that by continuing to increase the number of stacks, there are improvements in the variation of the sensor capacitance, which is, therefore, more sensitive.
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Fig. 1 .
Fig. 1.Schematic of the working principle of a capacitive tactile sensor.An external force applied on the plate at time t 0 induces a deformation of the dielectric which results in a measured capacitance variation.

Fig. 2 .
Fig. 2. Schematic of a capacitive sensor configured as parallel capacitance.The thickness variation of the electrodes is always negligible compared to the dielectric deformation.

Fig. 3 .
Fig. 3. Plot showing the initial capacitance C 0 with respect to the layer thickness.The red dot represents qualitatively the tradeoff between achieved capacitance with respect to the minimum achievable layer thickness.

Fig. 4 .
Fig. 4. (a) Simulation for a four-stack capacitor with t tot = 1 mm and t 1 = t 2 = t 3 = t 4 = 250 µm.(b).Simulation for the four-stack capacitor with t tot = 1 mm and t 1 = t 2 = t 3 = 100 µm and t 4 = 700 µm.(c) Computed change of capancitance ∆C for a change of applied pressure ∆P for the case of (a).(d) Computed change of capancitance ∆C for a change of applied pressure ∆P for the case of (b).

Fig. 5 .
Fig. 5. (a) Simulation for the six-layer stack with t tot = 1 mm and t 1 = t 2 = t 3 = t 4 = t 5 = 100 µm and t 6 = 500 µm.(b) Simulation for the six-layer stack with t tot = 1 mm and t 1 = t 2 = t 3 = t 4 = t 5 = t 6 = 100 µm.(c) Computed change of capacitance ∆C for a change of applied pressure ∆P for the case of (a).(d) Computed change of capacitance ∆C for a change of applied pressure ∆P for the case of (b).

Fig. 6 .
Fig. 6.(a) Behavior of the one-stack capacitor in terms of dielectric total displacement under a range of pressures applied.(b) Pressure/capacitance characterization obtained from finite element simulation with the corresponding sensitivity in the case of one-stack capacitor using the neo-Hookean nonlinear elastic constitutive model for the dielectric material.

Fig. 7 .
Fig. 7. Behavior of the two-stack capacitor in terms of dielectric total displacement under a range of pressures applied.(b) Pressure/capacitance characterization obtained from finite element simulation with the corresponding sensitivity in the case of two-stack capacitor using the neo-Hookean nonlinear elastic constitutive model for the dielectric material.

Fig. 8 .
Fig. 8. (a) Behavior of the three-stack capacitor in terms of dielectric total displacement under a range of pressures applied.(b) Finite element response with the corresponding sensitivity in the case of the three-stack capacitor using the neo-Hookean nonlinear elastic constitutive model for the dielectric material.

Fig. 9 .
Fig.9.Finite element response with the corresponding sensitivity in the case of 1) one-stack capacitor, 2) two-stack capacitor, and 3) threestack capacitor, assuming a linear model for the dielectric material.

Fig. 10 .
Fig.10.Schematic illustration of the sensor fabrication process.The dashed rectangle represents the method for a single capacitor.For the three-stacked capacitor, the process must be iterated.

Fig. 11 .
Fig. 11.(a) Vertical section of the structure for single dielectric.(b) Vertical section of the structure for three dielectric layers.

Fig. 13 .
Fig. 13.Experimental setup for measuring the change in capacitance in response to applied pressure.

Fig. 14 .
Fig. 14.(a) Experimental sensor output in the case of the one-stack capacitor.Output in terms of ∆C with the corresponding sensitivity in the two regions [0 30 kPa] [35, 100 kPa].(b) Experimental sensor output in the case of the two-stack capacitor.(c) Experimental sensor output in the case of the three-stack capacitor.

Fig. 15 .
Fig. 15.Experimental setup for measuring the dynamic range of the sensor.

Fig. 16 .
Fig. 16.(a) Experimental sensor output in the case of the one-stack capacitor.Output in terms of sensitivity expressed as ∆C/C0 over the time period of the load-unload cycles.(b) Experimental sensor output in the case of the two-stack capacitor.(c) Experimental sensor output in the case of the three-stack capacitor.

Fig. 19 .
Fig. 19.(a) Prediction on the sensor response in the case of a fivestack capacitor.(b) Prediction on the sensor response in the case of a ten-stack capacitor.

TABLE I MOST
RELEVANT DIELECTRIC MATERIAL PROPERTIES