Analysis of electrical resistance tomography measurements for fast force localization

1 Introduction

In recent years, human-robot collaboration and interaction have received increasing attention. Microphones and loudspeakers allow oral communication between robots and humans. Nevertheless, a safe collaboration requires recognition of the environment to anticipate and then detect a possible collision and thus avoid possible damage. Current systems allow this by using cameras to observe the workspace and processing their video sequences. However, a skin-like sensor is required for full-surface contact detection. Several approaches have already been developed [1], [2], [3], [4], [5]. The necessary specifications for collaborative robots in Europe are defined by DIN ISO/TS 15066 [6]. This specifies that safe collaboration is linked to force monitoring and initiating appropriate actions in case of contact. Therefore, continuous monitoring of an impending contact as well as measuring the resultant force on contact is necessary to meet the required specifications (the contact should be reliably detected at a contact force exceeding a threshold to be defined, which should be less than 65N).

There are many approaches to create an artificial skin or a distributed sensor for force detection and localization [7], [8]. Some authors report on achieving the necessary spatial resolution by placing arrays of sensors, where each sensor is evaluated individually. For example, arrays of strain gauges [4], [9] or piezoresistive elements [10] can be used as individual sensors. The disadvantage of using arrays is that the spatial resolution is limited by the size of an individual sensor element. The use of tomographic methods or the joint evaluation of several measured values makes it possible to increase the resolution beyond the density of evaluation points. Examples of such measurement systems are given in [2], [11], where methods based on electrical impedance tomography are applied to conductive polymers.

In this contribution, the following research questions are to be addressed. Can electrodes placed in sufficient numbers at the edge of an area to be monitored, formed by an electrically conductive, thin, elastic, flat plastic matrix, be used to determine both the location of the applied force as well as its strength with a resolution sufficient to meet the applicable standard [6]? What is the optimum geometric configuration of the electrodes and which excitation scheme is optimal to achieve sufficient resolution attaining the high measurement rate necessary? A directly related question is the uniqueness of the resulting solution. Are there different distributions of force induced local conductivity deviations that cause the same observed reactions at the electrodes?

A somewhat related, but not covered here, question is: can the same electrodes be used to detect an impending approach of an object such that at least some lead time is left for obstacle avoidance schemes? This research question will be addressed in a further paper and is also covered in [12].

In this paper, the measurement data obtained by applying electrical resistance tomography (ERT) to a layer of conductive polymer contacted via a planar, Cartesian grid of conductive silver adhesive electrodes are analyzed. From this, a fast method for localizing a force is developed.

The paper is organized as follows. In Section 2 the measuring principle of ERT is discussed. Section 3 discusses the developed measurement setup. Afterward, the change in conductivity of the used material during stretching and compression is examined, from which the requirements for the readout electronics are derived. Based on the study of the attainable change in the potential difference between every electrode and reference potential caused by a local change in the conductivity of the polymer, in Section 4 a solution method is presented to localize the contact point and the locally applied force. The developed method is characterized by a reduced, compared to the literature, computational complexity. Finally, the reconstructed conductivity change is analyzed as a function of the applied force.

2 Measurement principle

Since the aim is to achieve a sensor configuration that can be expanded over a wide area, all electrodes are attached and contacted on the underside of the sensor. This allows the number of electrodes and array size to be scaled as needed. The sensor material consists of a comparatively highly conductive polymer compared to other polymers, although even the best conductive polymers have a relatively low conductivity compared to electrical conductors such as silver, copper or gold. The conductivity of the polymer allows the use of a current source designed for a compliance voltage of 3.3 V and reduces the error caused by the output impedance of the current source. However, for capacitance tomography, which was developed in parallel, the conductivity is higher than optimal.

The theory explaining the conductivity of plastics can be described very well using the percolation theory [13], which can be used to estimate the number of possible continuous paths in an area given the density of path segments. In the case considered here, the path segments consist of the particles of carbon black, and the number of continuous paths is directly proportional to the conductivity of the material. From this, it can be deduced that for selected densities of the conductive component in the non-conductive matrix around the percolation threshold, a large change in conductivity occurs, extending over several powers of magnitude. In this work, the mixing ratio was not varied, but a suitable, commercially available material was used, which was only investigated experimentally in this work. A change in conductivity can also be triggered by applying forces on the plastic, this is analyzed in Subsection 3.2. The material can thus be modeled as a distributed strain gauge. The local change in conductivity is caused by a compressive strain in the thickness direction combined with a tensile and shear strain in the in-plane direction. Through this, it seems reasonable that the force’s magnitude can be reliably measured, and the location of the – assumed single – contact point can be coarsely estimated. The local change in conductivity is a function of the stiffness of the overall sensor’s design, which consists of the plastic sheet material of certain conductivity exhibiting an elastic modulus E attached to a supporting layer, which allows selecting the overall stiffness of the structure. If a low stiffness is chosen, the applied force will induce conductivity variations over a larger area, whereas a high stiffness guarantees conductivity changes acting very locally.

A large change in conductivity per unit of force is, of course, desirable, but its useful range is limited by the low-power electronics used, in particular by the current source that has limited compliance voltage. A good quantitative reconstruction is dependent on the sensor material, its elastic properties, and, from the signal processing point of view, the electronics used. Any sensor also needs to comply with electromagnetic compatibility (EMC) standards, so the quality of the chosen estimator’s results depends on noise and interferences coupling into the system. Here the aim is to reliably achieve a variation of at least 1 % conductivity change at a force of 65 N, which is the limiting force the system must at least resolve to meet the criteria set forth in [6] even under the influence of interfering noise.

In order to move from a distributed parameter description of the sensor system to one with lumped parameters, the conductive layer is discretized into cuboids, and resistances are assigned to each cuboid between opposite surfaces. An applied force leads to changes in the associated resistances. With the help of electrical resistance tomography, this change can be reconstructed by measuring only at the boundary of the considered volume [11]. To obtain a result proportional to absolute variations in resistance (and not relative variations), a current excitation is chosen [14]. The set of transresistances

where u_ℓ,m;i,k is the voltage measured between electrodes ℓ and m, attached to the boundary of the region of interest, when the current i_i,k is injected at electrode i and outgoing at the electrode k. From those measurements, the local, force induced change in resistivity is to be estimated.

To acquire all transresistances the following measurement sequence is applied. The single excitation current source is connected via a set of demultiplexers to two electrodes, indexed by i and m (only neighboring electrodes were used in the investigated measuring method), and the resultant voltage is, again using multiplexers, readout over another pair of electrodes indexed by ℓ and k. Using this scheme, all 4-tuples are acquired before the tomographic reconstruction is initiated, see Figure 1.

Figure 1:

Measuring principle for ERT. The arrows represent the iteration over the electrode pairs (held at fixed locations) used for excitation and measurement. As in the Van der Pauw method used to determine the surface resistivity, the transresistance is measured in the ERT. Therefore, an alternating current source is applied between the excitation electrodes, and the resultant voltage between the readout electrodes is measured. Measurements are carried out for all readout electrodes for each pair of excitation electrodes. The applied current and the measured voltage are stored in the vectors i _In and u _Out for each electrode position. The conductivity distribution σ(Ω) is then reconstructed from these vectors.

For excitation, a voltage signal, with respect to the reference potential, was generated by the DDS synthesizer AD9850 and filtered with a bandpass filter. This signal was used as the input signal for the used voltage controlled current source. Two ADG1408 and three ADG1409 multiplexers from Analog Devices® were combined to select the excitation and measurement electrodes. An AD630 lock-in amplifier was used to suppress noise and to obtain a DC voltage signal proportional to the amplitude of the AC voltage signal.

In the classical ERT, based on the measurements obtained, the change in conductivity is estimated by solving an inverse ill-posed nonlinear problem [15]. This can be done in various ways. The simplest method is to linearize the problem. The obtained linear problem is highly underdetermined and thus possesses infinitely many solutions. Through regularization, a priori knowledge can be added to the problem and thus an estimate of the solution can be obtained. After linearization, the change in conductivity, not the absolute value, is reconstructed. Therefore, two measurements are required for reconstruction: one at a known state u _Out,h, around which the problem is linearized, and one at an unknown state u _Out,o (δ u _Out = u _Out,o − u _Out,h). The change in the conductivity distribution δ σ between the two measurements is then reconstructed. The linearization of the system through the finite element method results in the following system of linear equations

were N_v is the number of voxels used for discretization and linearization and

Equation (2) can be solved, for example, with additional assumptions made about δ u and δ σ , using a maximum-a-posteriori (MAP) estimator. The equation for estimating δ σ thus becomes

A unity matrix was used for the matrix W in order to model independent, identically Gaussian distributed noise on each channel. The matrix Q was calculated from the sensitivity matrix J to account for the different influences of the conductivity changes of the different voxels on the measured voltages [12].

Figure 2 is intended to show the influence of a change in conductivity on the potential distribution for the simplest case of parallel feeding electrodes. The material is assumed to be purely resistive and a quasi-static approximation is used. It can be shown from the parameters of the material and the frequency of the excitation signal used that these assumptions are valid. If, as shown, a current is fed into the area using the left and right electrodes, a current density field is generated. The principle of least action applies to the entire area. This means that the current density lines spread out in such a way that the generated heat is minimized [16]. With homogeneous conductivity, the current density lines run parallel to each other. The equipotential surfaces are oriented perpendicular to the current density lines. If for whatever reason the conductivity increases in an area the current density lines are, attracted‘ to areas of higher conductivity, again obeying the principle of least action. Due to the resultant different drift velocities of the moving charges, they collect at the interface from a region of higher to a region of lower conductivity. This generates a superposed electric field that weakens the external field (for an infinite conductivity the external field is completely canceled out). As for the stationary case, which can be used here because of the quasi-static assumption, the equipotential surfaces are always perpendicular to the current density lines, these are also curved around the region of higher conductivity.

Figure 2:

Comparison of the distribution of the current density lines and equipotential surfaces for different conductivity distributions. The electrodes are marked in ocher. In this example, the excitation electrodes are located on the left and right side of the area. Three measuring electrodes are located on the lower side. In the upper image, the conductivity distribution is homogeneous, while in the lower image, an ideal conductor has been added in the center. Due to the higher conductivity and thus lower specific resistance, the current density lines are attracted by the ideal conductor. The equipotential surfaces, which are always perpendicular to the current density lines, are also curved. As a result, the voltage measured between the measuring electrodes and the reference potential changes. This is depicted in the plot at the bottom, the root mean square of the potential differences between the electrodes and the reference potential u_RMS(x) for the two different cases are marked with asterisks.

If the potential distribution at the boundary (here at the electrodes assumed on the lower margin) is measured in the area of Figure 2, a change due to the deviation from the mean conductivity can be detected. This is precisely the aim of electrical resistance tomography: to deduce a local change in conductivity within an observed area from the measurements of voltages between pairs of electrodes on the boundary, assuming a proper excitation is applied.

3 Measurement setup

In the setup used in this work, the electrodes were not placed around the boundary, as indicated in Figure 1. Instead, they are on the underside of the sensor. In this way, they are located somewhat closer to the force application point, and the quality of the reconstruction is increased. The measurement setup consists of an easily producible layer of conductive polymer, an electrode array on the underside of the layer realized via a printed circuit board and the measurement electronics, see Figure 3. The prototype of the polymer layer is a rectangular cuboid with dimensions 85 mm × 85 mm × 8 mm. Square electrodes with a side length of 7 mm were arranged in a checkerboard pattern. The polymer and the circuit board were glued together using conductive epoxy adhesive.

Figure 3:

Measurement setup used. The gold-plated electrodes (3.5 mm × 7 mm) on a professionally manufactured circuit board are connected using conductive epoxy adhesive to the 3D printed conductive layer (85 mm × 85 mm × 8 mm). An insulating layer (in blue) covers the sensor.

3.1 Hardware

Since the electrodes are attached to the underside in this particular sensor design, a relatively good localization of the force application position can be achieved by using an excitation scheme that only uses adjacent pairs of electrodes. Furthermore, the selected scheme can easily be extended for a larger area to be covered.

An alternating current (frequency 20 kHz) is used as excitation as it allows using lock-in amplifiers for the measurements, an evaluation method that can also cope with very poor signal-to-noise ratios [17]. This frequency was chosen because it gave good results. A higher frequency would increase the capacitive influences from the environment and worsen the functionality of the lock-in amplifier. A lower frequency would increase the measuring duration. A voltage-controlled current source was designed to allow for a dynamic adaptation of the current’s magnitude to the mean conductivity of the sensor. Through a demultiplexer, see Figure 4, the electrode i into which the current source feeds can be selected. Another demultiplexer is used to connect the reference potential to the electrode k via which the current flows out of the polymer into the signal ground. This electrode is subsequently referred to as the ground electrode. The electrode used for the measurement of the response voltage (referred to as ground) is selected by a multiplexer. The voltage u_ℓ,m;i,k between the electrode ℓ and the electrode m which is used for reconstruction, see (2), is not directly measured, as shown in Figure 1, but calculated from the measured voltage values between every electrode and the reference potential. For reasons of interference suppression a lock-in amplifier was used to obtain the in-phase component of the voltage between every electrode and the reference potential with respect to the excitation signal. The resultant DC output voltage of the lock-in amplifier was discretized via an ADC with a resolution of 7.81 µV. Through a microprocessor, the discrete voltage value was transferred to a computer.

Figure 4:

Sketch of the operating principle of the measurement setup. The electrodes are attached to the lower part of the conductive layer. Through demultiplexer the excitation and ground electrodes (i and k) are selected. A multiplexer is used to select the electrode ℓ for which the voltage will be measured with respect to ground.

Only adjacent electrodes are paired and used for supplying the excitation current, the electrodes are numbered as shown in Figure 5. The pairs of electrodes to be used for excitation and voltage measurement are chosen to reduce the cycle time for a single reconstruction while maintaining sufficient spatial and force magnitude accuracy. To define the electrode pairs, the electrodes were divided into two areas (blue and red in Figure 5). In each area, the electrodes are numbered in a clockwise direction, and pairs are formed between neighboring electrodes. This was intended to prevent the burden of the current source from being exceeded. For the 16 electrodes, this resulted in the pairs 1–2, 2–3, 3–4, 4–5, …, 12–1, 13–14, …, 16–13 (the first electrode is always the excitation electrode, the second is connected to ground).

Figure 5:

Electrode numbering. The electrodes were divided into two areas. In each area, the electrodes were numbered in a clockwise direction. Only neighboring electrodes in the same area are used as electrode pairs.

3.2 Sensor material

In order to measure and localize a force for the conductive layer, a material should be used that is not only sufficiently conductive (the nominal load of the current source must not be exceeded) but whose conductivity also changes as a result of an applied force. Because of the geometry of the sensor used and the current source designed, a conductivity of at least 2.5 × 10⁻² S m⁻¹ is required.

A conductive filament from Recreus [18] was ultimately used to produce the sensor. This consists of thermoplastic polyurethane (TPU) and lampblack (a subcategory of carbon black) and is characterized by a Shore hardness of 92A and a conductivity of 25.64 × 10⁻² S m⁻¹. 3D printing not only made it possible to produce the sensor in the desired shape, but also to determine the hardness and thus the contact pressure dependence through the infill. A gyroid infill was used, which is characterized in terms of mechanical properties by isotropy, higher shear strength and lower hysteresis and stiffness as compared to the other conventional infills [19]. Figure 6 shows a cross-section of the sensor and thus the gyroid infill.

Figure 6:

Photograph of a cross section of the sensor. A gyroid infill was used.

3.2.1 Conductivity change of the sensor material

The effect of a force on the conductivity of a polymer reinforced with carbon black is illustrated in Figure 7. Deformation of the polymer leads to a change in the average distance between the carbon black particles. Some of which approach on contact if the polymer is compressed. This allows new conductive paths to form, which increases the local conductivity of the polymer [20].

Figure 7:

Effect of an applied force on a polymer filled with carbon black particles. The blue area is the non-conductive polymer matrix and the gray spheres are the highly conductive carbon black particles. The compression of the polymer reduces the distance between the particles, some that even touch each other. This allows new conductive pathways to form, increasing the conductivity of the material [21].

In order to better understand the behavior of the material used and subsequently understand the measured values, the change in conductivity of the filament was investigated under various conditions, which are shown in Figure 8.

Figure 8:

Different scenarios under which the change in resistance of the filament was examined. The filament is shown in black, the points used for measurement are marked in ochre and the direction of force is indicated by the pink arrows. In (a) the change in resistance through stretching was measured, in (b) through transverse compression, and in (c) through compression in the longitudinal direction.

In the first case, Figure 8(a), the change in resistance of the filament when stretched was measured. The resulting relative change in resistance as a function of the relative change in length, the strain ɛ of the filament is shown in the plot in Figure 9. When the filament is stretched, there is an increase in filament length on the one hand and a decrease in filament cross section on the other. The ratio between the relative change in length and width is defined by the Poisson’s ratio ν. At the same time, the distribution of the conductive particles in the polymer matrix changes. Figure 9 shows that up to a relative change in length of 0.15, the resistance decreases and thus the conductivity of the filament increases. The effect of the increase in conductivity due to the reduction in the distance between the carbon black particles appears to be dominant over the effect of the change in geometry (which would lead to an increase in resistance).

Figure 9:

Relative change in resistance per relative change in length (this is the gauge factor for a strain gauge). The measurements were carried out according to Figure 8(a). The stretching of the filament results in a change in resistance due to the change in geometry and due to the change in the distribution of the conductive particles in the filament. The second effect dominates. As a result, the conductivity increases with the elongation of the filament in the longitudinal direction.

The influence of compression of the filament in the transverse direction, Figure 8(b), was then investigated. The relative change in resistance obtained over the force exerted is shown in the graph in Figure 10. As explained in Figure 7, through squeezing, the distance between the particles is reduced, which increases the conductivity. This can also be seen from the measurement values plotted in Figure 10.

Figure 10:

Relative change in resistance versus force applied in transverse direction. The measurements were carried out according to Figure 8(b). As depicted in Figure 7 the distance between the particles is reduced by the compression. Therefore, the conductivity of the filament increases with increasing load.

Finally, the change in resistance due to compression in the longitudinal direction was measured. The opposite effect to the first case, where the filament was stretched, was observed. By compressing the filament, an increase in resistance was measured and thus a reduction in conductivity.

3.2.2 Conductivity change of the sensor

If a force is applied to the sensor, the change in conductivity results from the combination of the effects described above. In Figure 11 this is modeled by a resistance network. If a force is applied perpendicular to the sensor’s surface, it results in a local elongation in the longitudinal direction and on the one hand in a compression in the transverse direction. The conductivity changes, which were investigated by the measurements according to Figure 8(a) and (b) and shown in Figures 9 and 10, are therefore relevant. In both cases, a decrease in resistance and an increase in conductivity was measured. Therefore, it is expected that a positive change in conductivity will be reconstructed from the later measurements.

Figure 11:

Impact of a force on the conductivity of the polymer. The polymer can be modeled by a resistance network. Each of the resistances is dependent on the local deformation and, thus, on an acting force. The two networks shown are intended to model an area of the surface of the sensor once at rest (a) and once when a force is applied (b). The applied force causes a deformation and, thus, a change in the resistance values of the network. k is the k-factor, which is defined as the ratio between the relative change in the resistance value and the length l. ν is the proportionality factor between the relative change in length in the transverse direction to the direction of the applied force Δl_Q/l_Q (due to transverse contraction) and the relative change in length in the longitudinal direction Δl_L/l_L.

4 Results

The measurement results obtained are now examined. An excitation current with an amplitude of 1 mA and a frequency of 20 kHz was used, and the tomographic reconstruction was applied. Two different color scales are used to display the results. A color scale from blue (smallest value) to yellow (largest value) is used to display the measured voltages. The conductivity changes were reconstructed by linearizing the inverse problem and using a maximum-a-posteriori estimator, as described in Section 2. A color scale centered around 0 (white) and ranging from blue (negative value) to red (positive value) was used in the figures. The orientation of the images is as defined in Figure 5. The voltage at electrode 1 is shown at the top left, and the voltage at electrode 7 is shown at the bottom right. No voltage was measured for the ground electrode and the excitation electrode, therefore, the value of zero is shown at the corresponding locations in the images.

4.1 Determination of the location of the applied force

Figure 12 shows the result of the voltages measured at the electrodes, with respect to the reference potential, when electrodes 8 and 9 (the centers of the electrodes are marked by the red and green cross in Figure 12) are excited and the conductivity is nearly homogeneous (no force applied). The black lines indicate equipotential surfaces. In the case of homogeneous conductivity, the electric field is symmetrical, as can be seen from the figure.

Figure 12:

Measured voltages in mV on the electrodes as electrodes 8 and 9 (marked by the red and green cross, the crossing points of the grid mark the centers of the electrodes) are used for excitation (value zero, blue, in this picture). The values of the measured voltages are indicated at the corresponding positions. The remaining values of the displayed area are calculated by interpolation. The gray area is intended to show where the measured voltages lie compared to the total area of the sensor. The black lines are the equipotential surfaces. In this case, the field distribution is symmetric.

In the next step, the change in the measured voltages as a function of the force applied was investigated. For this purpose, the change of the measured voltages with and without a force applied in the area between electrodes 9 and 10 (the exact location is marked by the red circle in Figure 13) was analyzed for different excitation electrodes, as for the localization all the measurements are used.

Figure 13:

Comparison of the voltages measured in mV and the resulting difference for the excitation at electrodes 14 and 15 (marked by a red and a green cross) and a force applied between electrodes 9 and 10. The black lines represent the same equipotential surfaces for (a) and (b). Through the applied force, the conductivity increases locally. This changes the field distribution, as explained by Figure 2. The equipotential lines are pushed away from the point of higher conductivity. This can be seen in (b). The red circle indicates the approximate location where the force was applied. The shape of the equipotential surfaces in case (a) is indicated in gray to draw attention to their change. The resulting change in the measured voltages is shown by (c). It can be seen that on one side of the point of conductivity change, there is an increase in voltage and on the other, a decrease.

In Figure 13 the measurements when electrodes 14 and 15 (marked by the red and green cross) are excited are shown. The equipotential surfaces for the voltages measured in the homogeneous case are also shown in this figure. The equipotential surfaces show the same effect as described in Section 2. The acting force leads to a local increase in conductivity. According to the principle of least action, the current density lines are attracted and the equipotential surfaces are pushed away. The measured voltages increase behind the area of conductivity change (from the perspective of the current) and decrease in front of it. The same effect can be observed for the other measurements, whereby the measured change in voltage is greater closer to the point where the force is applied.

The change in the measured voltages when electrodes 8 and 9 (the centers of the electrodes are marked by the red and green cross) are excited is shown in Figure 14(a). In this case, the force is applied to the top layer, approximately above the ground electrode. As a result, the change in the measured voltage between each electrode and reference is negative. The greatest change in the measured voltage in this case will also occur for electrodes close to the area where the force is applied, and therefore in the proximity of the ground electrode here.

Figure 14:

Difference of the measured voltages for a force applied somewhere between electrodes 9 and 10 but on top of the sensor with an excitation of the electrodes 8 and 9 (a) 9 and 10 (b). The red cross marks the excitation electrode and the green cross the ground electrode.

The case where the electrodes 9 and 10 (the centers of the electrodes are marked by the red and green cross) are excited is shown in Figure 14(b). Here, the change in conductivity essentially takes place between the two electrodes used for excitation. Due to the increase in conductivity, the total resistance between the electrodes also decreases. As a result, the voltage between the excitation electrode and reference also decreases. In this case, a more or less pronounced decrease in the measured voltage is observed for all electrodes with respect to ground.

Each acquired measurement vector u _Out is processed to result in an image obtained with a different excitation, thus reflecting the change in the conductivity distribution. In order to obtain an overall picture of that change, it is necessary to combine the information from all acquisitions appropriately. This is done during reconstruction, see Figure 15.

Figure 15:

Exemplary representation of the reconstruction process. All measured values of a measurement cycle, which means all measured voltages for each excitation, should be used for reconstruction. During the reconstruction, the information from all measured values is combined, and a suitable estimate for the location where a force is applied is obtained. See Figure 16 for the description of the lower image and the previous figures for the description of the upper images. The color scale used is described in the text and in the other figures.

In classical resistance tomography, the problem is typically linearized. This results in a sensitivity matrix J , see (3), which reflects the influence of a local change in conductivity within each voxel (these result from an FEM simulation) on the measured voltages. There is a simpler approach, one that is used here and which is compared to the classical method. The novel approach uses the previously obtained findings about the behavior of the electric field to get a rough estimate of the location of the conductivity change and, additionally, a coarse estimate of the applied force. These observation can be summarized as follows:

1. If the change in conductivity takes place between the excitation electrodes, there is only a decrease in the voltage values. The greatest voltage change takes place close to the excitation electrodes.

2. If the change in conductivity takes place on top of the ground electrode, there is only a decrease in all voltage values. The greatest voltage change takes place near to the ground electrode.

3. If the change in conductivity takes place on top of the excitation electrode, there is only an increase in all voltage values. In this case, the greatest voltage change takes place near to the excitation electrode.

4. If both negative and positive voltage changes occur, the conductivity change takes place between the electrode with the highest and the electrode with the lowest voltage change.

If these observations are combined, an approximate position of the change in conductivity can be estimated from the various measurements.

Figure 16 shows reconstructions obtained for different force application points. The pink circle marks the actual location where the force is applied, while the green cross marks the position that was estimated using the approach described above. To compare the result obtained with the here presented method for fast localization with the result obtained through the classical way using linearization and a maximum-a-posteriori estimator, the reconstructed change in conductivity is depicted in the background. It can be seen from the images that the much simpler proposed evaluation method can be used to reliably estimate the approximate location of the force. However, the error in the actual location of the force is greater for this simpler approach compared to the solution using linearization and the MAP estimator. This applies in particular if the force is exerted in peripheral areas. The mean error between the estimated positions and the true positions of Figure 16 obtained with the new method is one and a half times the error obtained using the MAP estimator (using the point of maximal conductivity change as position). However, the advantage of the method lies in the significantly lower computing power required as compared to methods that reconstruct the entire conductivity change in the area and do not just estimate its location. The memory usage is much lower as the sensitivity matrix J does not need to be computed and stored, and the computation time is reduced (from 69.1 ms for the reconstruction through linearization using 22,396 voxels to 5.3 ms using the method presented^[1]). The solution method presented here is also much simpler than in the case of the classic ERT, where for each estimate a nonlinear, ill-posed inverse problem has to be solved, since no linearization and regularization is necessary. This new method can cope easily with the problem when scaled to a much larger area covered by a significantly increased number of electrodes.

Figure 16:

Reconstruction of the conductivity change distribution in the sensor and estimation of the location of the acting force for different points of action. The dashed lines indicate the positions of the electrodes on the bottom of the sensor. The pink circles mark the actual location of the applied forces, and the green cross marks the estimated location. In the background, the calculated estimate of the change in conductivity using a MAP estimator is shown. An increase in conductivity is shown in red. As only the localization of the force is considered in this section for every figure the obtained values for the conductivity change are normalized with the respective maximal absolute value obtained, resulting in values between −1 and 1. The relationship between the applied force and the reconstructed conductivity change is analyzed in the next section.

4.2 Reconstructed conductivity change for different forces

The approach presented in the previous subsection for localizing an applied force does not yet allow the reconstruction of the force intensity. Therefore, in this subsection, the images obtained by a MAP estimator are used to determine the relationship between the applied force and the reconstructed change in conductivity. Results for four different forces are shown in Figure 17. Figure 18 shows the normalized mean value of the reconstructed conductivity change Δσ over the force. As already stated above, the reconstructed conductivity change increases with increasing force. The result is therefore consistent with the previous measurements. The function that describes the change in conductivity as a function of the normal force acting on the sensor is injective, thus guaranteeing consistency. This means that both the force and the contact pressure can be determined from the reconstruction. As it can be seen from Figure 18 the system allows to reliably detect forces below 65 N, which is the minimal biomechanical limit value, which must not be exceeded. The sensor therefore allows a force limit to be set, which is, however, well below the load limit value.

Figure 17:

Reconstruction of an increasing force applied to the sensor from the conductivity change distribution in the sensor obtained using a MAP estimator. A positive force is shown in red.

Figure 18:

Plot of the average normalized reconstructed conductivity change for increasing force. As expected from the previous measurements, the change in conductivity increases with increasing force. Forces below 65 N can be measured. Therefore, the system allows to set a threshold for the maximum allowable force below 65 N and detect forces above that threshold. This fulfills the specification stated at the beginning.

5 Discussion

In our contribution we analyze the data obtained from the application of electrical resistance tomography on a 3D printed layer of conductive polymer. The conductivity variation of the used material under different circumstances is analyzed. A conductive filament of thermoplastic polyurethane (TPU) with lampblack infill (a subtype of carbon black) particles was used as source material. The measurements showed that the filament has a positive change in conductivity when stretched in the longitudinal direction or compressed in the transverse direction. When a force is applied perpendicularly to the surface of the 3D printed sensor, there is a change in conductivity due to both of those effects, resulting in a local increase in conductivity. This was also observed from the measurements and reconstructions done.

To localize the region where the force was applied, the measurement data obtained were examined and analyzed. From this, an algorithm was derived that can be applied to estimate the location of any deviation from the mean in conductivity for whatever cause in a rather simple way. The quality of the estimate was compared to the classical reconstruction method of ERT on the basis of various measurements. It followed that, by using the algorithm presented here, the location of the force induction can be estimated less precisely but at a very high measurement rate. The algorithm is characterized by a reduction in the required storage capacity as well as computing time. Both parameters were reduced to less than a tenth for the examples presented.

An analysis of the change in the conductivity distribution reconstructed with a maximum a posteriori estimator showed that the system is highly sensitive to forces below the limit specified by [6]. In addition, the relationship between the applied force and the change in conductivity is injective. The biomechanical limits values specified by [6] can therefore be adhered to.