Introduction

Recent advances in flexible and stretchable electronics technologies has created various new applications of soft sensors in the area of artificial skin1,2,3,4,5,6,7,8 or wearable robots9,10,11,12,13,14, taking advantage of their mechanical compliance that facilitates physical conformation of the sensors to the surfaces of the host structures with various shapes15,16,17. A robotic system often requires multiple sensors to obtain a sufficient amount of data, and an array of sensors with compact integration not only can cover a large area of interest but also provides a high spatial resolution18,19,20,21,22,23,24. However, multiple electronic components that compose a sensor array usually require a more complex hardware configuration25,26,27. Particularly in soft sensors, multiple signal wires cause practical issues, such as mechanical fragility, physical interference between adjacent wires, and system complexity. Nevertheless, in soft sensor studies this issue has not been seriously taken into account so far in spite of its practical significance.

Specifically, wires in a soft robotic system induce physical constraints when the system undergoes large dynamic motions. In this case, wires need to be long enough to cover the configuration space while neither interfering with other components nor tangling up themselves. Moreover, in typical soft robotic systems, mechanical connections between soft sensors and regular wires are usually the most fragile areas due to the physical interfaces between rigid and soft materials with high stress concentrations28. This becomes more problematic when multiple sensors are implemented as an array which inevitably contains more complex wiring connections. Furthermore, repairing the wires in soft systems often requires a tedious manual process, and sometimes the entire array needs to be replaced from the failures in wiring.

A straight way to alleviate those issues is to use less wires so to minimize the chance of failures in wires or connections. Previously, there have been several approaches to build soft sensor arrays with a reduced number of wires. One of the most common methods is to connect all the sensing modules in series with a power line and add signal wires to the nodes between adjacent modules1,29. However, this approach requires at least the same number of signal wires as that of the sensors, and it becomes even less practical if an extensive number of sensors are used. Although it is possible to use a multiplexer to sweep the output signals through the modules30,31, the processing time degrades with an increase of the number of sensors. Another approach is to configure the sensor array in a grid pattern with multiple layers32,33. Compared to the serial connection, the grid arrangement tends to use less wires when the number of sensing modules increases. However, grid-type sensor arrays often suffer from a ghost key effect, failing to demonstrate a complete multi-touch functionality. Although a ghost key effect can be addressed by using an additional diode or multiplexer, it increases the complexity of the system as well as the form factor26,27. There has been a recent study for tracking multiple soft sensors with a single-output wire using machine learning34,35, which however requires a training process with a relatively large amount of data. Assuming each sensing module requires a single input and a single-output, Supplementary Fig. 1 numerically compares the number of wires required for each approach with the number of sensing modules.

However, we approach this problem with a different perspective and pay attention to a technology that is not necessarily highly related to soft sensors or soft robotics. Telecommunication uses an advanced system that transmits a tremendous amount of data without any physical connections. We are inspired by and particularly focused on the principle of radios, in which signals are transmitted and carried by high frequency waves, so called “carrier signals”. These waves carry the signals by modulating the amplitudes or frequencies, and the receiver can distinctively extract different signals36. In a similar way, we allocate a unique frequency to each sensing module in the sensor array so that signals from multiple sensors can be embedded in different carrier signals through a single signal wire, ultimately requiring only two wires, one for input and the other for output.

In this paper, we propose a design of a soft sensor array that requires only two external wires regardless of the number of sensing modules. The proposed sensor system is an array of resistive-type soft sensors, made of silicone elastomer embedded with microchannels filled with room-temperature liquid metal (eutectic gallium-indium, i.e., EGaIn)1,37,38,39,40,41. When the microchannel deforms by compression, it increases its electrical resistance. An inductor and a capacitor in a surface-mount type were embedded next to each microchannel sensor in the elastomer structure for building a resister-inductor-capacitor (RLC) band-pass filter (BPF). By adjusting the values of the inductance and the capacitance, a different filtering frequency can be allocated to each sensing module, enabling individual sensing while sharing the same signal wire42,43. To drive the sensor array, a customized input signal generated by a microcontroller is applied to the sensor array, and the output signal is acquired by the same device. Finally, interpretation of the output signal provides the information on the resistance change of each sensing module. To build a physical prototype of the proposed sensor system, a fabrication method was developed utilizing a direct printing technique of liquid-metal patterns44. In addition, several soft tactile sensor designs using proposed scheme were introduced to demonstrate the feasibility of the proposed method in practical applications.

Results

Sensor configuration and operation principle

A prototype was fabricated in the form of a 4-by-4 array (Fig. 1a). The inductors and the capacitors were placed on one side of the device (Fig. 1b), making the entire sensor hardware flexible and stretchable (Fig. 1c). The connecting ports for two external wires were located at the corner. The EGaIn channels were fabricated by a direct-printing process on a silicone substrate with a pneumatic dispenser. Figure 1d shows the cross-section of the channel fabricated by the direct-printing technique. A graphical interface was also constructed to visualize the operation of the sensor as demonstrated in Fig. 1e and Supplementary Video 2 for both single-touch and multi-touch sensing.

Fig. 1: Sensor configuration and visualization.
figure 1

a Prototype of a 4-by-4 soft sensor array (top view) and b a closed-up view showing the sensing module and the electronics. c Flexibility of the proposed sensor system. d Cross-section of the EGaIn microchannel with the dimension parameters. e 3D visualization of single-touch and multi-touch sensing in real-time.

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The proposed sensor array was constructed by connecting multiple RLC-BPFs in parallel, composed of an inductor, a capacitor, and a variable resistor made of an EGaIn microchannel connected in series. The series connection of L, C, and R was chosen for the BPF structure as it has the simplest hardware configuration which is very crucial for arranging all the components in a single plane while not having EGaIn channels crossing each other. As depicted in Fig. 2a, the sensor hardware contains a parallel array of BPFs, while only two wires are connected to the device. Although the BPFs seem occupying a relatively large portion of the area in the sensor hardware, which may limit the flexibility or the stretchability of the entire device, it is always possible to arrange the rigid components in the area where mechanical compliance is not a critical requirement during operation depending applications, as demonstrated with examples in “Fingertip and insole sensor applications” section. A commercial device that can both generate arbitrary waveforms and read analog input voltages was used for providing the input and reading the output signals.

Fig. 2: Evaluation of sensing performance.
figure 2

a Conceptual drawing of sensor system configuration and operation schematic. b Sensor responses plotted in a frequency domain. Red circles show the 16 peaks of all the sensing modules. c Sampling rate vs. FFT sampling size for 16 sensing modules. d Frequency error in FFT as a function of the FFT sampling size for 16 sensing modules. e Sampling rate vs. the numbers sensing modules for an FFT sampling size of 1024. The error bars indicate standard deviations (n = 5). f Sensor response for applied force to sensing module 3. Red line shows the response of the sensing module 3 and other lines are the responses of the adjacent modules (Modules 2 and 4). g Sensor responses for applied force to all the sensing modules. h Force to output voltage response of the sensing module 3 when a force is applied only to the sensing module 3 (gray), and the response of sensing module 3 when the force is applied to all the 16 modules (black). i Comparison of the theoretical prediction (red) and the experimental result (black) of the resistance change in the sensing module and the output voltage.

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Each BPF has a unique resonant frequency (i.e., filtering frequency) determined by the values of the inductance and the capacitance. These frequencies of the sensing modules are distributed with sufficient spacing so that the bandwidth of each BPF does not interfere with those of the adjacent ones. A custom waveform was then designed as a combination of sinusoidal waves with the filtering frequencies of all the BPFs, and applied to the sensor array. When this signal goes through the circuit, each BPF passes only the sinusoidal wave component with the corresponding frequency while filtering out the others. The amplitude of the passing wave is modulated based on the impedance of the BPF determined by the resistance change of the sensing module, induced by the contact force applied to the EGaIn microchannel. By measuring the voltage across the reference resistance (20 Ω) and converting it in the frequency domain, the amplitude of each sinusoidal wave component can be calculated. The value for the reference resistance was set to match the impedance (see Supplementary Fig. 2) of the sensor array and to maximize the sensitivity. Finally, the applied force can be estimated from the change in the amplitudes, and by repeating the operation with a reasonably fast speed we can achieve real-time sensing.

The proposed sensor array and the operation scheme suggest two benefits compared to conventional sensor arrays. First, the sensor array can be scanned using only two external wires regardless of the number of the sensing modules used. Also, taking advantage of simple circuitry of the sensor design (parallel connection of RLC-BPFs), we can easily scale up (or down) the number of modules as well as arrange the modules in desired patterns, while not causing any significant structural complexity. Moreover, by utilizing a compact-size commercial device capable of both generating waveforms and reading analog voltage inputs, the entire system could become portable. Second, since the sweeping process through the sensor modules involves only computation, not requiring any electronic switching, the sampling rate of the system does not drop even though more sensing modules are employed and operated. The sampling frequency can be further enhanced by improving the computation efficiency.

Evaluation of sensing performance

We first evaluated the distribution of the filtering frequencies and the bandwidths of the BPFs in the frequency domain. Figure 2b shows a snapshot of the fast Fourier transform (FFT) plot when the sensor was in a resting state. Sixteen peaks are clearly seen in the figure representing that the filtering frequencies are well-distributed. We conducted FFT using every set of 1024 output voltage measurements. By setting the size of the FFT datapoints to be power of two (i.e., N = 2n, n = 1, 2, 3, …), we can easily divide the transform into two pieces of size N/2 at each step and expedite the computation process of FFT. We now need to determine a specific number for the FFT datapoints. A large number of datapoints will slow down the process. In the proposed system, a significant drop in the sampling rate was observed when the batch size was increased from 1024 to 2048 (Fig. 2c). On the other hand, the more data samples for FFT will provide the higher precision. As the frequency resolution of FFT increases by the number of the datapoints, FFT can also extract the amplitudes from closer frequencies to the filtering frequencies of the BPFs. We quantified the gap between the BPF frequencies and the frequencies from which the amplitudes were extracted in FFT and observed that the error between those two frequencies drastically increases when we use less than 1024 datapoints (Fig. 2d). The quantification process is described in Supplementary Note 2. Therefore, we set the sampling size of 1024 for FFT in our system.

We thus decided to use, 1024 datapoints for every FFT cycle, resulting in a 725 Hz sampling rate. Also, as expected from the operation principle, the sampling rate remained consistent regardless of the number of the sensing modules used (Fig. 2e). This indicated that the sampling rate will not degrade although more sensing modules are employed in the array.

We also evaluated the frequency independence of each BPF, since it is one of the critical factors that determines the multi-touch sensing functionality. A test setup was designed to apply an external force to the sensor while collecting the force data and the sensor output data simultaneously. The external force could be applied either to a single sensing module (single-touch) or to multiple sensing modules (multi-touch) at the same time (Fig. 5c) by switching the type of the indenter.

Figure 2f shows the response of the sensor when the force was applied only to the module 3 as a representative example (The data for the entire array is available in Supplementary Fig. 3). The responses of the modules 2 and 4 among the 16 modules were also presented for comparison, as adjacent modules are most likely to suffer from crosstalk in a regular sensor array. The response of each module means the change in the amplitude of the output voltage at the corresponding filtering frequency. When the external force was applied only to the module 3, it was observed that the module 3 exclusively responded while the modules 2 and 4 stayed relatively unchanged. Also, the magnitude of the response increased as the magnitude of the applied force increased.

However, when the force was applied to all the 16 modules simultaneously, all of them responded, as shown in Fig. 2g (Responses of all the 16 modules are available in Supplementary Fig. 4). To compare the sensor response with the external force in both single-touch and multi-touch situations, force-to-voltage relations in both cases were plotted (Fig. 2h). Generally, the output voltage increased with the increase of the force and hit the peak when the maximum force was applied. When the force was removed, the output voltage returned to the initial value with hysteresis45. In both single- and multi-touch experiments, the responses of the module 3 were almost identical, indicating that the sensor performance remained consistent with different loading conditions.

By directly probing across the sensing module using a digital multimeter, the relation between the resistance of the sensing module to the output voltage was examined. Figure 2i shows a reasonable agreement between the theoretical prediction and the experimental result. From the aspect of mathematics Eq. (13), it is expected that the output voltage will saturate due to the circuit characteristic when the sensor resistance becomes extremely high. However, within the resistance range of the sensing modules in our system (2–50 Ω, Supplementary Fig. 6a), the clear 1-on-1 mapping was possible between the sensor resistance and the output voltage, demonstrating the feasibility of sensing.

Fingertip and insole sensor applications

One of the promising applications of soft sensors is wearable devices12,26,32,46,47,48. The mechanical impedance of the elastomer is similar to that of human skin which contributes to easy conformation of the device to the complex geometry of a human body during operation49,50. The proposed sensor system can be constructed with a simple hardware configuration (only two wires and a portable operator device), particularly showing a practical advantage when applied to wearable devices. This section demonstrates the feasibility of the proposed system for different wearables, such as a thimble-shaped fingertip sensor for detecting pressures during dexterous manipulation and an insole sensor for foot pressure sensing in which the motion and the activity of the wearer can be analyzed based on the information acquired by the soft sensors.

Dexterous manipulation of human hands is not only achieved by sophisticated actuation of the muscles but also enabled by the capability of acquiring abundant information from the cutaneous sensory receptors. Especially, contacts between the human hand and the object mainly occur at the fingertips. In this case, the fingertips detect the contacts as well as apply forces to the object, and the measurement of the pressure distribution on the fingertip provides the key information on the interaction between the hand and the object for dexterous manipulation.

In fact, a human hand is one of the body parts which undergoes the most diverse types of contact situations, and deformation of the fingertip also occurs in different shapes. This suggests that when we design artificial fingertip sensors, constructing it with soft material can be crucial for imitating the functionality of the fingertips. In this regard, there have been many studies that propose different designs of soft fingertip sensors for robotic applications51,52. Nevertheless, most of them made of soft materials have shown only a limited number of sensing modalities, as accommodating a number of sensing modules in such a small area has been one of the biggest challenges. However, in our approach, we can arrange multiple sensing modules with a highly efficient hardware configuration within a limited area which offers a strong advantage when the sensor is applied to robots or human hands.

A soft thimble sensor was designed by placing five sensing modules at the fingertip: two on the front, two on the sides (one on each side), and one on the top (Fig. 3a). Elastomer bumps were added on each sensing module for efficient force transmission. The sensor array was first made of a flat substrate using a direct-printing process, and then the flat sensor layer was rolled up and fixed using a silicone adhesive (AXIA-2700, AXIA) to make the shape of a thimble (Fig. 3b). Two types of the fingertip sensors with different sizes were prepared for the thumb and for the index or middle finger (Fig. 3c). We tested the fingertip sensors for different manipulation tasks to evaluate the interaction between the hand of the wearer and the object to be manipulated.

Fig. 3: Fingertip sensor experiments and results.
figure 3

a 2D configuration before being rolled up, b complete prototypes worn on thumb and an index finger, and c numbering of sensing modules. d Pen drawing experiment and e the result. f Water pouring experiment and g the result. h Push and squeeze experiment and i the result.

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The first manipulation task is pen drawing. Three fingertip sensors were prepared for the thumb, the index finger, and the middle finger. The wearer drew a 2 cm square on paper with a pen, following the predefined trajectory, shown in Fig. 3d. The output data from all three sensors were recorded during the task.

Figure 3e shows the sensor responses of the three fingertip sensors. The thumb sensor gave the highest response and the middle finger sensor the lowest, indicating the levels of contribution of the three fingers to drawing. For the first line in Fig. 3d, the tip of the pen moved vertically downwards, and the index finger applied most of the force for drawing to the pen while the thumb supported the pen, showing high output signals for the two fingers. For the second line, the thumb and the index finger applied even higher force to the pen to pull it sideways. The two fingers released the force for the third line, and finally only the middle finger applied force to draw the fourth line.

The result from the pen drawing test showed that the proposed sensor was able to measure the tactile forces on fingertips during dynamic manipulation motions. We also wanted to check the performance of our sensor in a situation where the loading conditions gradually changed. The fingertip sensors worn on the thumb and the index finger, and the wearer held an empty water jug until it is filled with water and then poured it out to empty the jug again, as shown in Fig. 3f.

During the task, the thumb pressed the top of the handle of the jug and the other four fingers wrapped the handle. It should be noted that multiple areas of the thimble of the index finger were compressed during this manipulation. The front and one side of the fingertip made contacts with the handle to hold the jug while the other side was compressed by the middle finger. Therefore, most of the sensing modules (1 2, 3, and 5) gave the outputs for the index finger while only the modules 3 and 4 showed responses for the thumb (Fig. 3g). The output voltages increased as water was poured into the jug. The output signals then fluctuated when the wearer tilted the jug to empty the water, and finally the output decreased during pouring out the water.

The final task is manipulation with a concentrated force by a fingertip. In this experiment, the wearer performed a task of pumping of a dispensing-cap bottle by pushing the press-cap to pump the liquid out with the index finger while holding the bottle with the thumb and the other fingers of the same hand, as shown in Fig. 3h. The sensors were on the index finger and the thumb.

The sensor output of the index finger linearly increased during the motion of pumping (Fig. 3i). When the pumping ended, the sensor output stopped increasing and fluctuated instead due to the force continuously applied by the wearer. It was observed that the main contact location on the index finger moved from the module 3 to the module 4 during two times of pumping. The signals from the thumb sensor were more consistent than those of the index finger sensor for the two experiments.

Another application area in wearables, in which the proposed sensing system is highly useful, is foot pressure sensing in the form of an insole. A human body is supported by soles during locomotion, and measuring the pressure distribution over a sole reflects the movement of the center of gravity (COG), which is critical in various biomechanics applications, such as gait pattern analysis, gait phase estimation, and gait assistance53,54,55. Insoles experience continuous and repeated loadings in general and require physical robustness if any electronics are embedded, and reduction of the number of signal wires will thus be highly beneficial in this application. Since the typical geometry of a sole is the complicated 3D structure composed of both convex and concave region, fabricating the insole sensor with soft material is advantageous as the sensor will conform more naturally to the sole. Also considering that the sensor will be used in locomotive situation where the portability of the device should be emphasized, the fact that we comprised our system with a compact-sized operator makes the application well-suited.

The nine sensing modules were individually designed to cover the entire pressure area of the sole, and elastomer pads with different shapes were added on top of the sensing modules to amplify the pressure signals (Fig. 4a). The inductors and the capacitors were located in the area of the medial arch where pressure is applied minimally during the entire gait period in order to prevent any unexpected signal disturbances (Fig. 4b). Only two signal wires were connected through the heel (Fig. 4c). The length of the prototyped sensing insole was 280 mm. The prototypes were placed insides of the shoes, and the pressure profiles of both feet of the wearer were measured on a commercial treadmill equipped with a pressure sensing belt (Pressure Distribution Treadmills, Zebris Medical GmbH) (Fig. 4d).

Fig. 4: Insole sensor experiment and result.
figure 4

a Bottom and top views of the insole sensor with the module numbers and the dimensions. b Experimental setup for the insole sensor consisting of a commercial treadmill and a monitoring system. c Closed-up photo of the electronics in the insole sensor. d Shoes equipped with the insole sensors for the experiment. e Insole sensor data acquired during a gait cycle. f Visualization of the pressure distribution measured by the pressure sensing belt in the treadmill and g the corresponding visual presentation of the measured pressure profiles by the insole sensor for comparison.

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Figure 4e shows the sensor data during a single gait cycle, representing the movement of the COG inside each foot area, from the back to the front, and they were visualized in 3D space, as shown in Fig. 4g. Figure 4f shows the reference pressure profiles measured by the treadmill for comparison. The insole data showed reasonable agreement with the pressure profiles acquired by the treadmill.

Discussion

A concept for soft sensor arrays with an extremely simple hardware configuration was proposed and demonstrated. The proposed system utilizes unique frequencies individually allocated to the sensing modules to acquire the corresponding sensor outputs through a single common signal line. Each sensing module is equipped with a BPF composed of an inductor, a capacitor, and a variable resistor made of a microfluidic channel filled with room-temperature liquid metal. All these components are in compact form factors embedded in a highly deformable elastomer sheet, which makes the entire sensor array soft. The resistance change in each module is reflected to the amplitude of the sinusoidal wave passing through the BPF. Designing the input signal as the compound of the waves with multiple frequencies enables simultaneous sensing of multiple modules without degrading the sampling speed. The proposed system receives the entire sensor information in real time in the frequency domain rather than sweeping all the modules through electrical or mechanical switching, which makes the sampling rate not affected by the number of the sensing modules and dependent only on the performance of the processor and the computation algorithm. We achieved a sampling rate of 725 Hz in our system regardless of the number of the modules, which is expected to be further increased by improving the algorithm or by upgrading the computing power. Furthermore, the proposed system uses only two signal lines always, making the physical configuration extremely simple regardless of the number of the sensing modules.

The proposed system was tested for practical applications in the form of wearable sensing devices, taking advantage of physical compliance along with the simple hardware configuration. The first example of applications was a fingertip sensor for estimating the force profiles applied to the fingertips during various dexterous manipulation tasks with different objects, such as pen drawing, water pouring, and pumping of a dispensing-cap bottle. In the second example, we were able to demonstrate the tracking of the dynamic foot pressure profiles from both feet during a gait cycle. There has been active research on predicting and estimating the status of physical interactions between humans and robots/machines for applications in the areas of robotics, haptics, remote control, rehabilitation, augmented or virtual reality. We believe this study will open a space in these areas by offering an innovative scheme to utilize a number of sensors with a significantly simplified hardware configuration.

Methods

Modeling

Figure 5b shows the overall architecture of the sensor system. The most crucial factor to achieve multi-touch sensing is the independence of each BPF. In this section, the practical scheme to estimate the performance of the specifications for the inductors and capacitors is presented. In order to prevent crosstalk between BPFs, for each filtering frequency, the impedance of the corresponding BPF should be minimized while remaining the impedance of the other BPFs relatively high so that the amount of the current flows through the other BPFs becomes reasonably negligible. Ideally, when an alternating current (AC) with the resonance frequency flows through a BPF, the impedance of the BPF should correspond to the resistance of the sensing module. However, in reality, there exists parasitic series resistances in the inductor and the capacitor, which make the actual impedance become higher and have negative impact on the current concentration. Thus, it is necessary to choose the inductors and the capacitors with acceptable parasitic resistances.

Fig. 5: Fabrication process, sensor circuit and test setup.
figure 5

a Fabrication process of the proposed sensor array. b Description of the sensor circuit with a data processing protocol. c Test setup for the sensor characterization with the photos of the indenters used for single- and multi-touch sensing experiments.

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An inductor is typically a winding of a long wire, which entails a certain level of resistance from the aspect ratio multiplied by the resistivity of the material. However, when an AC with a high frequency (~1 MHz) passes through the inductor, the actual resistance even increases by means of the electromagnetic behaviors, such as the skin effect. In this case, the equivalent series resistance (ESR) value is the function of the frequency of the current applied to the inductor and also varies by the inductor samples, so it is challenging to estimate the ESR of the inductor precisely based on the specifications.

However, by utilizing the quality factor (Q factor) provided in the datasheet of the inductor, we can conservatively evaluate the ESR of the inductor. The Q factor of an inductor is defined as:

$$Q^L = \frac{{\omega _ML}}{{R^L}}$$
(1)

where QL is the Q factor for the inductor, L is the inductance, RL is the ESR of the inductor, and ωM is the frequency of the current used for evaluating the Q factor, which is provided along with the Q factor in the datasheet. Also, the value of the Q factor varies by the sample, and the datasheet offers the minimum value of the Q factor. Therefore, we can estimate the maximum ESR value of the inductor as:

$$R^L = \frac{{\omega _ML}}{{Q^L}}$$
(2)
$$R_{{{{\mathrm{Max}}}}}^L = \frac{{\omega _ML}}{{Q_{{{{\mathrm{Min}}}}}^L}}.$$
(3)

Similarly, the maximum ESR values of the capacitors can be calculated as:

$$R_{{{{\mathrm{Max}}}}}^C = \frac{1}{{\omega _MCQ_{{{{\mathrm{Min}}}}}^C}}.$$
(4)

Finally, by adding up the two values, we can obtain a conservative estimation of the parasitic series resistance of the BPFs:

$$R^{{{{\mathrm{Par}}}}} = R_{{{{\mathrm{Max}}}}}^L + R_{{{{\mathrm{Max}}}}}^C$$
(5)

From the values of the parasitic ESRs of the electrodes, we can then evaluate how exclusively the current is passing through the BPF for each frequency. The resonant frequency (or the filtering frequency) of the ith BPF can be calculated as:

$$\omega _i = \frac{1}{{\sqrt {L_iC_i} }}$$
(6)

where ωi is the resonant frequency of the ith BPF, Li and Ci are the inductance and the capacitance in the ith BPF, respectively. Then, when the current component with the frequency of ωi goes through the jth BPF, the impedance of the jth BPF becomes as:

$$Z_j( {\omega _i,\,R_j^S} ) = R_j^{{{{\mathrm{Par}}}}} + j\omega _iL_j + \frac{1}{{j\omega _iC_j}} + R_j^S$$
(7)
$$Z_j( {\omega _i,\,R_j^S} ) = \left( {R_j^{{{{\mathrm{Par}}}}} + R_j^S} \right) + j\left( {\omega _iL_j - \frac{1}{{\omega _iC_j}}} \right)$$
(8)

where \(R_j^S\) is the resistance of the jth sensing module which has its nominal value around 2 Ω without any external force and shows about 50 Ω of the maximum value when the force is applied (see Supplementary Fig. 5a). Here, a new variable current concentration factor (CCF) can be defined by assuming the maximum sensor resistance for the ith BPF and the resting states for the other BPFs, to represent the distinction of the BPFs, as:

$${{{\mathrm{CCF}}}}_i = \frac{{\left| {1/Z_i\left( {\omega _i,R_{i.{{{\mathrm{Max}}}}}^S} \right)} \right|}}{{\left| {\mathop {\sum}\nolimits_{j = 1,\,j \ne i}^{n = 16} {1/Z_j\left( {\omega _i,R_{j.{{{\mathrm{Nom}}}}}^S} \right) + 1/Z_i\left( {\omega _i,R_{i.{{{\mathrm{Max}}}}}^S} \right)} } \right|}}.$$
(9)

The CCFs should be the values between 0 and 1. As the system is composed of the BPFs having the CCF values closer to 1, the sensor array will show less crosstalk between the sensing modules and improve the multi-touch sensing performance. The numerical estimation of the ESRs and the CCFs for all the BPFs are presented in Supplementary Table 1.

In practical implementation, there is a functional limit in the maximum frequency that the operator device can generate (10 MHz). It is thus better to use as low frequency ranges as possible to accommodate more filters (i.e., more sensing modules), requiring a high LC value, as can be calculated in Eq. (6). However, a high inductance or capacitance increases the size of the electrodes, making it difficult to build a system with a compact form factor. Considering the above trade-offs, the inductance and the capacitance values were set as shown in Supplementary Table 1.

The impedance of the ith BPF when the corresponding wave signal is applied can be calculated in Eq. (10), which can be approximated as:

$$Z_i\left( {\omega _i,\,R_i^S} \right) = \sqrt {\left( {R_i^S + R_i^{{{{\mathrm{Par}}}}}} \right)^2 + \left( {\omega _iL_i - \frac{1}{{\omega _iC_i}}} \right)^2} \cong R_i^S + R_i^{{{{\mathrm{Par}}}}}\,\left( {{{{\mathrm{if}}}}\,\omega _i \cong \frac{1}{{\sqrt {L_iC_i} }}} \right)$$
(10)

under the assumption that the AC impedance from the inductor and the capacitor is negligible. The output voltage of the circuit reflects the changes in the resistance of the EGaIn channel caused by the contact force applied to the sensor56,57. The relationship between the output voltage and the resistance change of the sensor module can be calculated as Eqs. (11)–(13). The resistance change from the initial value can be expressed as a function of the change in the output voltage (Eq. (13)), and their relationship is shown in Fig. 2c with red dots. An analog electronic simulation program (LT spice, Analog Devices, Inc.) was used to confirm the characteristics of the circuit before fabricating an actual prototype.

$$\left. {\frac{{V_{{{{\mathrm{out}}}}}}}{{V_{{{{\mathrm{in}}}}}}}} \right|_i = \frac{{R_{{{{\mathrm{ref}}}}}}}{{R_{i.{{{\mathrm{Nom}}}}}^S + R_i^{{{{\mathrm{Par}}}}} + R_{{{{\mathrm{ref}}}}}}}$$
(11)
$$\left. {\frac{{V_{{{{\mathrm{out}}}}} + {\Delta}V_{{{{\mathrm{out}}}}}}}{{V_{{{{\mathrm{in}}}}}}}} \right|_i \cong \frac{{R_{{{{\mathrm{ref}}}}}}}{{R_{i.{{{\mathrm{Nom}}}}}^S + {\Delta}R_i^S + R_i^{{{{\mathrm{Par}}}}} + R_{{{{\mathrm{ref}}}}}}}$$
(12)
$$\left. {\frac{{{\Delta}R^S}}{{R_{{{{\mathrm{Nom}}}}}^S + R_i^{{{{\mathrm{Par}}}}} + R_{{{{\mathrm{ref}}}}}}}} \right|_i = \left. {\frac{1}{{1 + {\Delta}V_{{{{\mathrm{out}}}}}/V_{{{{\mathrm{out}}}}}}}} \right|_i - 1$$
(13)

Fabrication

Figure 5a shows the fabrication process of the sensor array. A mixture of two types of silicone elastomer (Dragon Skin 30 and Ecoflex 30, Smooth-On) was spin-coated on a flat substrate. After curing, EGaIn pattern were directly printed on the substrate, using a pneumatic liquid dispenser (Super Sigma CM III, Musashi), for the sensing modules and interconnects. In each terminal of the EGaIn pattern, the dispenser was programmed to generate a droplet of EGaIn for a stable connection to electronics including the inductors and the capacitors. In order to achieve a high yield rate during this patterning process, the elastomer was preliminarily dyed in white, which provides consistent reflectivity for the laser distance sensor in the dispensing system (Fig. 1d). The EGaIn patterns were printed at a speed of 3 mm s−1 with a stand-off distance of the nozzle of 0.2 mm. The width of the EGaIn pattern is ~0.25 mm (Fig. 1d) and the initial resistance of the EGaIn pattern for each sensing module is ~2 Ω. Surface mounted devices-type inductors (Chip coil LQH32NH series, MURATA) and capacitors (C0G (NP0) Dielectric, AVX) were placed on the elastomer substrate with alignment with the EGaIn patterns after applying conductive epoxy to their electrodes (Fig. 5a). This not only resolves the contact resistance issue between EGaIn and the rigid electrodes, but also contributes to stable positioning by adding small adhesion before encapsulation of the EGaIn patterns and the electronics with uncured elastomer poured on top. The elastomer matrix completely cures in 1 day at room temperature.

Operation parameters

The input waveform was generated by a commercial operator device (Analog Discovery 2, Digilent) in the form of a combination of sinusoidal waves with 16 resonance frequencies in Supplementary Table 1, with the generation speed of 10 MHz. The amplitudes of all the sinusoidal waves are identical and set to match 2 V peak-to-peak after harmonized in order to stay within the maximum amplitude that the operator device can generate. The voltage across 20 Ω reference resistance was collected with the sampling frequency synchronized to the generation speed of the input wave. FFT was conducted for every set of 1024 voltage measurements (Fig. 5b), and including the computation to extract the amplitudes for each filtering frequency, the sensing speed to scan 16 sensing modules was 725 Hz. Using 1024 datapoints for FFT, the amplitude was calculated with 9.77 kHz increments in the frequency domain, which resulted in <5.5% of the frequency error from the filtering frequencies used in the proposed system. The system was operated using a customized Python code composed of the interface library for the operator device and an FFT computing process. The visual presentation was prepared by linking the code with open-source 3D rendering program (Blender).

Sensor performance test

The characterization setup was designed by installing a loadcell (RFT60-HA01, Robotous) to a motorized x-y-z stage (Fig. 5c). The indenters for single-touch and multi-touch experiments were 3D-printed (Object 30, Stratasys) with a photo-curing plastic material (VeroBlack, Stratasys). The shape of the indenter tip was designed as a hemisphere with a diameter of 5 mm, and the indenter for multi-touch experiment was prepared by arranging multiple tips in a 4 by 4 array, having the same size of the sensor prototype (Fig. 5c).

A normal force was applied by controlling the position of the motorized stage. For the single-touch experiment, the force was applied with five different magnitudes with two repetitions for each force step at a speed of 0.083 Hz which was determined by the speed limit of the motorized stage. The sensor response for a faster loading speed was also investigated using a customized setup and the result is presented in Supplementary Fig. 6b. The input force was in the range of 15–20 N, and the force varied by the samples as each sensing module has different microscopic geometry.

The loading speed was reduced in the multi-touch experiment (0.047 Hz) as a higher force (200 N) had to be applied to the loadcell. The force applied to the individual modules was calculated by dividing the force measurement with the number of the sensing modules. The resistance in Fig. 2g was measured by directly probing the wires across the sensing module 3, using a digital multimeter (Fluke 8846A, Fluke Corp).