Magneto-sensitive elastomers (MSEs) offer great opportunities for technical applications that require elastic components and adjustable material properties. Typically, MSEs consist of magnetic particles embedded in an elastomer matrix. Thus, the mechanical properties can be tuned via an external applied magnetic field.

Several applications based on MSEs have been proposed, like force sensor, magnetic field and acceleration sensors (). Besides sensor applications of MSE, it is also used for actively controlled base isolators (). A recent investigation proposed a tactile sensor based on MSEs (). The authors use the inductance change in a planar coil to detect a deformation of a close MSE layer. present another sensor based on the same principle that can differentiate between shear and compression stress. Most of the applications based on MSEs can be divided due to their usage of MSE as a sensitive or active element. There are a small number of investigations in which the MSE is used simultaneously as functional, sensitive element and an actively controlled part of an application ().

Tactile sensors are widely applied in various robotic and gripping systems. In general, they are used to detect a contact or measure the force or pressure. Furthermore, there are implemented as a control system featuring tactile feedback (). As a haptic control unit with force feedback, they facilitate human–machine interactions. Some tactile sensors include a position determination of the contact (). The latter are based on different concepts, mostly consisting of a discrete number of measurement units. These are utilizing, for example, optical, piezoresistive or capacitive transducers ().

In this paper, we propose a tactile sensor in which the setup allows the determination of the position of an indentation in one direction. One key element of the proposed design is a circuit board containing a linear array of multiple coils with overlapping sections. As a first approach, we present a mathematical model which is fitted to the sensor response. The aim of these investigations is to prove the feasibility, present the capabilities of the concept and obtain a deeper understanding of the coil interactions. Additionally, this study is providing a basis for further investigations.

The sensor response in the chosen region is modeled with Eqs. () and (). Firstly, we assume symmetric sensor response values with respect to the two symmetry planes of the planar coils that are aligned perpendicularly to the x and y axes. Therefore, Eq. (), representing an ellipse, is always centered on the y axis and is not tilted. Thus, the parameters x0,i, Ai and Bi represent the center position in the x direction, the semiaxis in the x direction and the semiaxis in the y direction of an ellipse. 3x-x0,i2Ai2+y2Bi2=1 In Sect. , the parameters x0,i, Ai and Bi are fitted with a polynomial to the SRM of each coil. We chose a polynomial f(δ) of the fourth degree, due to the nonlinear trend and inflection points of the parameters (see Fig. b). Thus, the fit is a compromise between model complexity and accuracy. Additionally, an analysis of the root mean square error of the polynomial fits of higher degrees does not lead to significantly more accuracy. 4f(δ)=p4⋅δ4+p3⋅δ3+p2⋅δ2+p1⋅δ+p0. Considering two intersecting ellipses, we obtain a system of two quadratic equations for x and y. In general, those equations result in up to four solutions, including two imaginary and two real solutions for y. Here, we only evaluate the solutions of the equations that lead to real numbers. Thus, the x coordinate can be estimated by solving the system of equations for x. The quadratic polynomial for x delivers two solutions. Referring to the restricted domain x∈R|(-Ai+x0,i)≤x≤(Ai+x0,i), there is only one solution for x.

All data postprocessing is done with MATLAB R2018b. The stepwise sequence of the script is shown in Fig. a. Firstly, the resolution of the SRM is refined by cubic interpolation to obtain a higher resolution. Next, a stepwise sweep through the data of the refined SRM is done, fitting Eq. () to the values of each step . The steps are executed with a threshold of ±2×10-5 and a step size of 3×10-5 of δ. This leads to a set of ellipses for each coil, obtaining x0,i, Ai and Bi for each step. Subsequently, we use polynomials of the fourth degree (Eq. ) to fit these ellipse parameters with a dependency on δ. Here, δ is calculated from the mean value of each step. The values for pk of the functions Ai(δ), Bi(δ) and x0,i(δ) are listed in Appendix . For the coil L1, the results of the data processing are plotted in Fig. b. The markers represent the ellipse parameters of each fit of Eq. () to the refined SRM. The lines in different styles depict the fit of the polynomials A1(δ), B1(δ) and x0,1(δ) (Eq. ). For each indentation, the evaluation of Ai(δ), Bi(δ) and x0,i(δ) of two overlapping coils leads to two intersecting ellipses. As a last step, the x components of the points of intersection are calculated (see Sect. ).

In Fig. , three cases of intersecting ellipses are shown. The procedure described in the previous paragraph commonly results in the first case. Cases two and three are special cases that need to be considered separately. An indentation close to y=0 leads to a single point where the ellipses touch (case two). An indentation in the area around the coil center (case three) results in δ≥1×10-3. Consequently, the semiaxes are equal, leading to a circle with a small radius (see Figs. b and ). Due to measurement and interpolating errors, both cases possibly lead to curves with no intersection. In order to avoid such cases, a constant value of 0.2 mm is added to all polynomials for the semiaxis Ai(δ). Although the values of the points of intersection are thus changed, potential cases with no intersection are avoided. Secondly, in the area close to the coil center, the points of intersection are found by evaluating the two adjacent inductances.

The polynomial fit of each ellipse parameter has an intercept of zero for a value of δ>1.2×10-3. Since the sample grid spacing is finite, this δ value might be reached for indentations close to the center of a coil. However, the distinction of the three cases described in Sect.  means avoiding such cases (see Fig. ). Furthermore, the semiaxes appear to tend to infinity for small values of δ. Due to measurement noise, the smallest measurable value of δ is greater than zero and leads to a finite length of the semiaxis. For the measurement concept, those values are neglectable since in that range the inductance values δ of adjacent coils are significantly higher.

As a first approach, we use the initial sample grid measurements to calculate the deviation of the determined x coordinate (Sect. ) from the real position of the indenter. The absolute error of the position is less than 0.8 mm (Fig. ). Thus, errors caused by the polynomial fit and fitting ellipses to the refined SRM are determined. This deviation is also increased by the constant value we added to all semiaxes, namely Ai(δ). Furthermore, four additional measurements at random positions are done, nonrecurringly, to check the model at points that are not covered by the sampling of the sensor (see Table ). For these measurements, the maximum error is at -0.71 mm. Hence, we conclude that this model can be applied to any planar position of the indenter in the measurement range and with the equal indentation h. For the validation of the model, more measurements are necessary at positions that are not covered by the sampling of the sensor and that are assured with repeatability tests.

For the proposed model and the determination of the position, it is a crucial requirement that the indenter shape and indentation depth h are not variable. The nature of the method is to model the sensor response based on a wide data set. Considering an indentation of variable depth and the indenter shape, the sampling and modeling of the sensor is rather complex. In this case, other model approaches are preferable. However, investigating the rate of the inductances of two overlapping coils is a promising approach, for instance.

Although the trend of the semiaxes Ai(δ) and Bi(δ) are similar, the trend of x0,i differs from L1 to L3. Depending on which coil is active, the relative asymmetrical arrangement of the passive coils changes. Thus, the spatial distribution of the flux density is distorted in a different manner by the opposing magnetic fields (passive coils). This leads to different trends of x0,i. To avoid this effect, the sensor can be optimized by means of geometric variations of the coil array in order to minimize the coupling coefficient.

The inductance measurements are based on high-frequency magnetic fields (≈2 MHz) at very low amplitudes. Hence, an external quasi-static magnetic field will not influence the inductance measurements, assuming that the flux density in the MSEs does not reach saturation and that the relative magnetic permeability is constant. This external magnetic field can be used to obtain a tunable stiffness. Thus, the sensor is possibly able to perform tactile measurements and control the stiffness simultaneously.

The setup of measurement electronics allows an interconnection of multiple coils by using the analogue multiplexer. Hence, the magnetic flux density distribution can be varied by using parallel or series connections of overlapping coils. This results in a larger response area compared to a single coil. Additionally, this method may be used to improve the accuracy of the sensor.

The proposed tactile sensor concept offers versatile fields of application, ranging from gripping to tactile control systems. As a first realized approach, we present a linear coil array that is adequate for the position determination, in one direction, of an indentation of constant depth. The analysis shows that the individual measurement of every coil provides sufficient data for sampling and modeling. The model formed by ellipse equations, with three parameters each and a polynomial fit, results in an absolute position determination error of less than 0.8 mm. Furthermore, the coupling of the coil array results in a distorted flux density distribution. This distortion is modeled mainly by the ellipse parameter x0,i.

Subsequent investigation will focus on expanding the control concepts and evaluation methods in order to determine the planar position. Such experiments will be executed with variable depths of indentation and will be extended to analyze the repeatability of the performance. Additionally, we will optimize the coil array regarding the coupling coefficients. Furthermore, we will add a quasi-static external magnetic field to the setup to investigate a simultaneous control of the stiffness.