1 Introduction

Tactile robotics is a fast-growing field attempting to utilize tactile information in robotics, such as contact-based decision-making, manipulation, and physical human–robot interaction [1,2,3,4]. Unlike vision or auditory systems, tactile sensing has yet stagnated due to its diverse and multidisciplinary requirements; one of the challenging requirements is applying a tactile sensing system on a robot’s flexible and soft body surfaces [5].

For many years, researchers have primarily investigated flexible and stretchable electronics to create soft tactile sensors. Highly conductive and stretchable materials have demonstrated electronic skins that are thin and stretchable [6]. Although these materials have achieved many extraordinary characteristics, practical usage of these approaches in soft robotics is challenging because these materials are typically fabricated on a plane.

Tomographic, or super-resolution, tactile sensing approac-hes have recently emerged as a promising approach to create soft tactile skins using reconstruction [7,8,9,10]. The tomographic approach generally uses sparsely distributed sensing elements with wide, overlapping sensitive fields. Although the number of sensing elements is small, tactile information of higher spatial resolution can be computed by decoupling the convoluted raw signals.

The overlapping sensitive fields can be achieved by either resistive-type transducers [11] or capacitive-type transducers [12, 13]. Among these transducing methods, tactile sensors based on resistance tomography have shown practical advantages in soft skins, since the sensor design can be simply applied to three-dimensional geometries compared to the capacitance transducer that typically requires two, thin parallel stretchable electrodes. Thus, the resistance tomography method has been successfully applied to both soft materials [8, 14] and three-dimensional geometry [15].

Fig. 1
figure 1

Overview of the resistance tomographic electronics for flexible tactile sensing

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This paper introduces an electronic design strategy dividing frontend and backend electronics, to achieve flexible and scalable tactile skin. Figure 1 illustrates the advantage of this electronic design. Soft robots require different tactile skins, according to the body locations. The frontend is for flexible transducers that can have different geometry and scale. The backend is unified tomographic electronics for current injection and voltage measurement that can be applied to every soft robot’s body surfaces, regardless of the frontend’s geometry and scale.

This strategy is useful for tomographic tactile sensors because the change of frontend can be easily updated on the simulation model used in the resistance reconstruction. The use of a simulation model for sensing has been known to decouple hardware dependencies [16].

2 Fundamentals of the resistance tomography

Resistance tomography is a reconstruction method that the resistance distribution of the inside of a conductive medium using a low number of electrodes on the surface [17].

This approach has been applied to tactile sensing because only a few electrodes can estimate pressure distribution over a large and soft piezoresistive material [11]. This feature enabled many tactile sensors to detect multiple physical contacts on soft and stretchable materials [8, 14, 18].

2.1 Mathematical formulation

When electric current (j) is injected into a conductive medium (\(\Omega \)), the voltage distribution (\(\phi \)) is formed, as follows:

$$\begin{aligned} \begin{aligned}&\nabla \cdot (\sigma \nabla \phi ) = 0\,\,\,\,\,\,\,\,\,\text {in}\,\,\,\Omega \\&j = \sigma \nabla \phi \, \cdot \,\mathbf{{n}}\,\,\,\,\,\,\,\,\,\,\,\text {in}\,\,\partial \,\Omega \ \end{aligned} \end{aligned}$$
(1)

where \(\sigma \) denotes the conductivity and \(\textbf{n}\) is the unit normal vector on the boundary \(\partial \Omega \).

Equation (1) can be approximated into a linear equation form using the voltages (V) and current (I) on the electrodes [19].

$$\begin{aligned} {{\textbf {V}}} = {{\textbf {R}}}(\sigma ) \cdot {{\textbf {I}}}\ \end{aligned}$$
(2)

where R(\(\sigma \)) is the transfer resistance matrix, which is a function of the conductivity \(\sigma \) of the piezoresistive material.

In resistance tomography, we usually inject an electric current by selecting an electrode pair and measure voltage from another electrode pair. The voltage measurement obtained from the pair selections (\(E_i\) \(_j\)) can be defined as below:

$$\begin{aligned} {{E}_{ij}} = {{\textbf {M}}}_i^{\intercal }{{\textbf {R}}}(\sigma ){{{\textbf {P}}}_j}\ \end{aligned}$$
(3)

where P\(_j\) is the j-th column vector of the current injection pattern matrix and M\(_i\) is the i-th column vector of the voltage measurement pattern matrix.

These pair selections are sequentially changed to scan the entire conductive medium. In case of a piezoresistive material, the conductivity changes when it is deformed, causing a voltage measurement change \(\Delta \)E at the electrodes, as follows:

$$\begin{aligned} \Delta {{E}_{ij}} = {{\textbf {M}}}_i^{\intercal }\big ({{\textbf {R}}}(\sigma + \Delta \sigma ) - {{\textbf {R}}}(\sigma )\big ){{{\textbf {P}}}_j}\ \end{aligned}$$
(4)

where \(\Delta \sigma \) is the conductivity change of the piezoresistive material.

When the current injection and voltage measurement are fixed, we can transform Eq. (4), as follows:

$$\begin{aligned} {\Delta }{{\textbf {E}}} \approx {{\textbf {J}}}\Delta {\sigma } + {{\textbf {w}}}\ \end{aligned}$$
(5)

where \({\textbf {J}}\) is the Jacobian matrix calculated from the initial conductivity with a given pattern and \({\textbf {w}}\) is measurement noise.

Using the Jacobian matrix, the conductivity distribution (\(\sigma \)) can be reconstructed when a sequence of the voltage measurement change \(\Delta \) E is acquired. This procedure is often formulated with a regularization matrix to stabilize the solution:

$$\begin{aligned} \Delta \sigma = {({{{\textbf {J}}}^{\intercal }}{{\textbf {J}}} + {\alpha ^2}{{\varvec{\Gamma }}^{\intercal }}{\varvec{\Gamma }})^{ - 1}}{{{\textbf {J}}}^{\intercal }}\Delta {{\textbf {E}}} \end{aligned}$$
(6)

where \(\alpha \) is a scalar hyperparameter that adjusts the strength of the regularization, and \(\varvec{\Gamma }\) is a regularization matrix established from a priori knowledge.

The hyperparameter \(\alpha \) is heuristically determined from the real data with the simulation model in this study for simplicity. For the systematic hyperparameter selection, Graham and Adler well summarized possible approaches [20].

This mapping matrix in Eq. (6) can be precomputed from the finite element model of the sensor.

2.2 Current injection and voltage measurement patterns

Fig. 2
figure 2

Illustration of the measurement protocol. Despite that the first and the last electrodes are not literally adjacent in the last measurement, the term is still used in this study because the format of the current injection and voltage measurement pattern matrices is the same as the conventional matrices format of the adjacent protocol

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The choice of the current injection and voltage measurement patterns is not a trivial problem as it can influence the optimality of the measurement [21]. The adjacent pattern is widely used due to its simple implementation with moderate performance [17].

Practically, analog multiplexers are often used to control all the electrodes, instead of using multiple current sources and voltage readouts [17]. In the multiplexed electrode configuration, three electrodes are typically controlled for the voltage measurement: an electrode for the current source (voltage input), an electrode for the current sink (ground connection), and an electrode for our measurement readout (voltage output).

As illustrated in Fig. 2, for the first measurement set, the current source and sink are connected to the first and second electrodes, respectively, while the voltage of every electrode in the 64 array is sampled once in a sequence. For the second set, the current source and sink are shifted to the second and third electrodes, respectively, while the voltage sampling sequence is repeated. This shifting process and its corresponding sampling sequence are repeated 64 times to cover all the electrodes of the sensor and generate a total of 4096 voltage measurements.

In the last shifting, the current source and sink are placed at the first and last electrodes, so the next shifting will return them to their initial configuration. For this reason and since this pattern is constantly repeated to keep the measurements of the sensor updated, the current source and sink are always considered adjacent, even if they appeared to be separated in the last configuration.

3 Electronic components for resistance tomography

To implement resistance tomography-based tactile sensing, dedicated electronics are essential. Figure 1 shows an overview of the system, showing the flexible transducer frontend and the analog signal acquisition backend.

3.1 Flexible transducer frontend

The flexible transducer frontend is made of a piezoresistive layer and an underneath layer with sparsely distributed electrodes. The piezoresistive layer can be any material or structure that can change resistance when it is pressed or deformed. In this article, a piezoresistive structure consists of two conductive layers (Fig. 3): a carbon-sprayed layer (838AR, MG-Chemicals, Canada) and 5.5 mm thickness of low-resistance fabric patches (Medtex P70, Statex, Germany) with a neoprene sheet, which is introduced in the literature [15].

Fig. 3
figure 3

Schematic of the backend circuit

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The layer with electrodes is a flexible-printed circuit board (fPCB) with 64 copper electrodes exposed on the surface as shown in Fig. 4. With a conductive adhesive, the conductive fabric sheet is connected to the electrodes. The electrodes are made of copper and are essentially conductors with no resistance. The carbon-sprayed layer is made on top of the respective electrodes of the fPCB. The low-resistance fabric patches are loosely placed upon the carbon-sprayed layer, creating a slight air gap. Only the edges are pasted using glue to prevent the two layers to stop contacting each other when pressure is applied. In this design, the entire frontend can maintain mechanical flexibility while keeping the electrical connection.

3.2 Analog signal acquisition backend

The key concept of the analog signal acquisition backend is to reuse the same electronics for various transducer frontends of different shapes and scales.

The core components of the circuit are an analog multiplexer (AMUX), a current source, an analog-to-digital converter (ADC), and a microcontroller. A detailed schematic is shown in Fig. 5.

Fig. 4
figure 4

Flexible transducer frontend (top) and analog signal acquisition backend (bottom)

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Fig. 5
figure 5

Schematic of the backend circuit

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The electrodes of the transducer frontend are directly connected to a 64-channel dedicated analog multiplexer (AMUX) with a current source on chip (swmux, IMS CHIPS, Germany). The schematic of the AMUX is shown in Fig. 6.

The AMUX chip consists of two analog outputs connected to 64 analog inputs. Four of these analog inputs can be connected to one pair of electrodes for the current injection and another pair of electrodes for the voltage measurement, explained in Fig. 2. The current generator is connected to a reference ground and a programmable bias current source (ranging from 0 to 40 \(\upmu \)A with a stepsize of 2.6 \(\upmu \)A).

The voltage measurement pair is connected to the two analog outputs of the AMUX, which are later connected to the ADC. The on-resistance between these input and output connections corresponds to the AMUX internal resistance that is approximately 400\(\Omega \). This resistance can be neglected if compared to the much larger resistance of the piezoresistive material.

The chip has the size of 2700 \(\times \) 2600 \(\upmu \)m\(^2\), which is compact enough for various on-board applications. The control of the chip is done through a 4-wire SPI interface. The maximum multiplexing frequency of the AMUX is designed to be 10 MHz, which is sufficient for the resistance tomography.

When the electrodes are selected from the AMUX, a voltage measurement is conducted via the ADC (ADS115, Texas Instruments, USA). This 16-bit high-resolution ADC is specifically chosen for its superior performance in signal-to-noise ratio (SNR). The ADC is connected to the microcontroller through an I2C protocol.

Fig. 6
figure 6

Schematic of the 64-channel analog multiplexer

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A microcontroller is required to control AMUX and ADC. Any microcontroller can be used if it has SPI and I2C interfaces. In this study, a microcontroller with the Arduino architecture (MKR1010, Smart Projects, Italy) was chosen for the ease of programming and open-source development.

4 Contact sensing experiments and results

To evaluate the frontend and backend strategy, we prepared two frontend transducers, having different geometry and scale: one with 150 mm by 150 mm square geometry and another with 120 mm diameter circular geometry (see Fig. 7).

Fig. 7
figure 7

The square- and circular-shaped frontends

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The experiments aim to show that the same backend module can be used for different frontend transducers while achieving a similar level of contact sensing performance. The contact sensing performance is evaluated with two metrics: minimum detectable contact force (limit of detection) and minimum discernible contact distance.

To carry out the experiments, an indenter is placed on the frontend as shown in Fig. 8. The indenter is composed by a circular base of 22 mm diameter and 1.8 g that ensures the homogeneity of the contact area in all experiments and an interchangeable metallic weight.

4.1 Minimum detectable contact force

Fig. 8
figure 8

a Indenter placed on the frontend, b circular base of the indenter and c interchangeable metallic weight and base of the indenter

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Fig. 9
figure 9

The RMS of the measured voltages corresponding to the contact pressure for a square frontend and b circular frontend

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The minimum contact force detection limit is calculated using the limit of detection [22].

A sequence of the voltage is a vector of 4096 dimensions according to the current injection and voltage measurement patterns. Typically, the root mean square is used to estimate the power of the voltage measurements [23]. The limit of detection is defined by the contact force producing the voltage measurement power that is 3.3 times larger than the standard deviation of the noise signal.

As an experiment, a known weight is placed on the sensor’s central location as shown in Fig. 8. The contact area is kept using a 22 mm diameter indenter when the weight is placed, which results in an effective pressure area of 380.13 mm\(^2\).

The weight applied by the indenter starts at 1.8 g (base alone), and then, it is increased to 2.8, 3.8, 6.8, 8.8, 11.8, 16.8, 21.8, and 31.8 g. This corresponds to a force variation from 0.01766 to 0.3119 N and a pressure variation from \(0.4645\times 10^{-4}\) to \(8.2066\times 10^{-4}\) N/mm\(^2\). The variation in applied pressure modifies the \(V_{\textrm{rms}}\) value which is acquired and saved as a sequence of data points.

Figure 9 shows the power of the signal when the contact pressure increases, the first point corresponds to a measurement without the indenter. The standard deviation of the noise signal is calculated from repetitive signals (\(n=30\)) when no contact force is applied.

The minimum detectable contact force is estimated to be \(0.83\times 10^{-4}\) N/mm\(^2\) and \(1.19\times 10^{-4}\) N/mm\(^2\) for square specimen and circular specimen, respectively. The experiment was replicated five times under similar conditions, this data is illustrated in the form of box plots in Fig. 9. The compress shape of the boxes and the low presence of outliers indicate the repeatability of the measurements.

4.2 Minimum discernible contact distance

The minimum discernible contact distance is calculated using the Pearson correlation coefficient (PCC) between a voltage sequence from one contact position and another voltage sequence from a slightly shifted contact position.

As an experiment, the voltage sequence of the sensor’s central location is first obtained. A 10 g weight is used for this experiment, as this weight is sufficiently heavy to generate significant voltages. Secondly, the contact location of the weight is shifted, and another voltage sequence is measured. The PCC is calculated between the voltage from the origin and the voltage from the shifted location. This procedure is repeated until the shift distance reaches 40 mm.

Figure 10 shows that PCC decreases as the shift distance increases. In this article, the minimum discernible distance (MDD) is defined as the distance where PCC is 0.9 \(({\hbox {MDD}}_{0.9})\) and 0.5 \(({\hbox {MDD}}_{0.5})\). For the square-shaped frontend, the \({\hbox {MDD}}_{0.9}\) was 2.51 mm and \({\hbox {MDD}}_{0.5}\) was 16.92 mm. For the circular-shaped frontend, the \({\hbox {MDD}}_{0.9}\) was 3.42 mm and \({\hbox {MDD}}_{0.5}\) was 15.53 mm. Similarly to the minimum detectable contact force, the box plots in Fig. 10 were construct using the data of five experiments and show the repeatability of the measurements.

Fig. 10
figure 10

Localization performance for a square frontend and b circular frontend

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4.3 Demonstration

The frontend was mounted on a curved surface and resistance reconstruction is conducted when various contacts were applied. The curved surface was a cylindrical shape with a diameter of 190 mm and both the printed circuit board and the low-resistance fabric patches were attached, as shown in Fig. 11.

Fig. 11
figure 11

Front and side perspectives of the flexible frontend mounted on a curved surface

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Fig. 12
figure 12

Tactile sensing demonstration of a single-contact with lower pressure, b single-contact with high pressure, and c multi-contact

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The sensor was exposed to three types of stimuli as shown in Fig. 12. First, a moderate pressure was applied to a point located in the upper-left corner. The corresponding resistance distribution is reconstructed and shown on the screen. As a second test, high pressure was applied on the same point, causing higher resistance change. Finally, two contact points were pressed. This multi-contact has successfully reconstructed from the sensor, showing two peaks on the screen.

5 Discussion

This article introduces the transducer frontend and signal acquisition backend for tomographic tactile sensing. The scheme of frontend and backend design has been widely used in tactile sensing systems; however, a specific design for resistance tomography-based tactile sensors has been less focused.

The result in Sect. 4 indicates that the analog multiplexer with a programmable current source can be reused for frontends of different geometry and scale, achieving similar sensing performance: The circular-shaped frontend has 143.37% for the minimum detectable contact force (LOD), 136.26% for the minimum discernible contact distance \(({\hbox {MDD}}_{0.9})\), compared to the square-shaped frontend. Considering that the 11,310.73 mm\(^2\) area of the circular-shaped frontend (60 mm radius) is the 50.27% of the square-shaped frontend (150 mm square) 22,500 mm\(^2\) area, this performance difference appears to be acceptable.

The piezoresistive layer of the frontend is made of carbon spray, which may be inhomogeneous. This inhomogeneous conductivity results in simulation inaccuracy, causing overall performance reduction. The data-driven approaches have shown promising performance in handling this fabrication inhomogeneity [15]. Although the flexible-printed circuit boards are used as transducer frontend in this study, they can be extended to stretchable transducers if soft and stretchable electronics are used. The use of a stretchable transducer can be useful to soft robot systems, such as soft grippers.

Regarding the backend, the current AMUX chip in Fig. 4 is installed on a much larger package due to a supply problem. The entire backend circuit can be tremendously miniaturized when a proper package is used. The ADC can also be integrated as an application-specific integrated circuit in the future. Since the AMUX selection can be modified by software, the backend developed in this study is able to handle the connection to a frontend with a different number of electrodes (up to 64). This is an advantage of the frontend and backend scheme as it only requires changes in the simulation model.

It should be noted that a small computer systems interface (SCSI) cable was used to connect electrodes from the frontend to the backend. This analog interface may be limited when the frontend is distanced from the backend or high-speed multiplexing is required.

Although there are more metrics to evaluate resistance tomography-based tactile sensors [18], two essential metrics were chosen to show the unified backend can achieve similar sensing performance for different frontends. Since the data obtained from the experiments does not fall exactly on the LOD and the MDD, these values were estimated by finding a rough intersection point. As a future study, the effect of the geometrical change in terms of electrode arrangement can be further analyzed.

In Sect. 4.3, the curved surface is used to demonstrate the single- and multi-contact sensing capability of the proposed system. Although a simple cylindrical geometry is used in this study, this approach can be applied to even more complex-shaped surfaces.

6 Conclusion

The resistance tomography-based tactile sensors are promising to create soft tactile skins that may have various scales, geometries, and flexibility. To enhance the adaptability of the tactile skin, an electronic design strategy dividing frontend and backend electronics is investigated. The frontend is dedicated to the flexible transducer part with the piezoresistive structure. The backend is the tomographic electronics comprising an analog multiplexer, a current source, an analog-to-digital converter, and a microcontroller. The use of a unified backend is validated from the experiments of two frontends that have different scales and geometries. In the future, this electronic scheme can be potentially used in soft robots, which require diverse geometries with challenging mechanical compliance.