Dynamic Zero Current Method to Reduce Measurement Error in Low Value Resistive Sensor Array for Wearable Electronics

One advantage of a resistive sensor array (RSA) with shared rows (M) and shared columns (N) is the reduced number of wires from M × N + 1 to M + N which can greatly lessen the complexity and burden on wearable electronic systems. However, the drawback is the crosstalk current effect between adjacent elements, which will lead to high measurement error. Although several solutions have been reported, they mainly focus on RSAs with high resistance (≥100 Ω). There is a lack of research that addresses RSAs with resistor values below 100 Ω. Here, we introduce a new circuit design named the dynamic zero current method (DZCM) to further decrease the measurement error. From the low value RSA test with ideal resistors, the DZCM exhibits lower error than the zero potential method (ZPM). In the case of the error variation ratio of amplifier offset voltage, the DZCM has a 4%/mV (row) to 7%/mV (column) ratio, while the ZPM has an almost 25%/mV (row) to 45%/mV (column) ratio and it increases with array size.


Introduction
Eutectic gallium indium (EGaIn), an alloy consisting of 75% gallium and 25% indium, is a low viscosity liquid metal at room temperature and has good electrical conductivity [1]. These unique characteristics of EGaIn make it an ideal active component to be embedded in soft sensors. Upon deformation of the sensor microstructure, which is pre-filled with EGaIn, the strain and pressure [2][3][4] exerted can be deduced easily from the measured varying capacitance and resistance.
By encapsulating EGaIn liquid metal with soft elastomers, our group has previously designed and fabricated EGaIn based microfluidic pressure sensors [5][6][7] with relatively low baseline resistance of 10 Ω (no load) which can go up to 200 Ω depending on the loading applied to it. Several examples employing such soft sensors have been demonstrated for healthcare and wearable electronics applications. Many of the current health sensing studies focus on the self-monitoring of personal health data. One such example is the tracking of plantar pressure of individuals with diabetes with diabetic foot ulcers via a flexible pressure sensitive insole embedded with soft sensors.
These sensors are stretchable, conformable and sensitive to mechanical loading and have great potential to be used as building blocks of electronic skins. Undoubtedly, a single sensor is not sufficient to replicate the function of human skin which has closely packed mechanoreceptors with very small receptive fields. To achieve high spatial resolution, a large number of small individual sensors are needed. These sensors are required to be merged firmly to form a compact resistive sensor array (RSA). An RSA enables a high density sensing element network to be configured with minimal wire linkage and thereby confers the least possible burden to the electrical system. One major concern of this RSA architecture is the crosstalk current effect resulting from adjacent unmeasured sensors that give rise to additional measurement error. Figure 1 shows a 3 × 3 RSA. To calculate the resistance of the target resistor 'R 22 , we need to assess the current flowing through it and the voltage difference across the element, denoted as 'I 22 and 'U 22 accordingly. 'U 22 also equals the voltage difference between the row and column wire 'V row2 -V col2 . On the other hand, 'I 22 cannot be measured directly. 'I col2 in the column wire is not equivalent to 'I 22 as 'I col2 includes crosstalk currents from the adjacent unmeasured resistors. The dashed arrow lines in Figure 1 illustrate one example of these crosstalk currents. Consequently, the actual value of 'R 22 cannot be calculated easily from 'I col2 and 'U 22 . olution, a large number of small individual sensors are needed. These sensors are required to be merged firmly to form a compact resistive sensor array (RSA). An RSA enables a high density sensing element network to be configured with minimal wire linkage and thereby confers the least possible burden to the electrical system. One major concern of this RSA architecture is the crosstalk current effect resulting from adjacent unmeasured sensors that give rise to additional measurement error. Figure 1 shows a 3 × 3 RSA. To calculate the resistance of the target resistor 'R22′, we need to assess the current flowing through it and the voltage difference across the element, denoted as 'I22′ and 'U22′ accordingly. 'U22′ also equals the voltage difference between the row and column wire 'Vrow2-Vcol2′. On the other hand, 'I22′ cannot be measured directly. 'Icol2′ in the column wire is not equivalent to 'I22′ as 'Icol2′ includes crosstalk currents from the adjacent unmeasured resistors. The dashed arrow lines in Figure 1 illustrate one example of these crosstalk currents. Consequently, the actual value of 'R22′ cannot be calculated easily from 'Icol2′ and 'U22′. A number of RSA readout systems have been proposed to eliminate the crosstalk current effect. These approaches include the inserting diode method (IDM) [8,9], inserting transistor method (ITM) [10][11][12][13], passive integrator method (PIM) [14][15][16], resistance matrix approach (RMA) [17], improved RMA [18,19], incidence matrix approach (IMA) [20], voltage feedback method (VFM) [21][22][23][24][25][26][27][28][29][30] and zero potential method (ZPM) [29,[31][32][33][34][35][36][37][38][39][40]. However, these systems only cater for RSA designs with high resistance. Electronic networks with low resistance below 100 Ω are often left unaddressed as shown in Figure  2. A number of RSA readout systems have been proposed to eliminate the crosstalk current effect. These approaches include the inserting diode method (IDM) [8,9], inserting transistor method (ITM) [10][11][12][13], passive integrator method (PIM) [14][15][16], resistance matrix approach (RMA) [17], improved RMA [18,19], incidence matrix approach (IMA) [20], voltage feedback method (VFM) [21][22][23][24][25][26][27][28][29][30] and zero potential method (ZPM) [29,[31][32][33][34][35][36][37][38][39][40]. However, these systems only cater for RSA designs with high resistance. Electronic networks with low resistance below 100 Ω are often left unaddressed as shown in Figure 2.
The lack in research progress on low resistance networks is partially attributed to the large crosstalk current effect and parasitic effect in the printed circuit board (PCB). In addition, the majority of the mechanical sensors and actuators in the market use high resistance transducers, due to the construction materials, fabrication techniques and sensing mechanisms involved.
Several methods, including the ZPM, VFM and IDM, have been developed to alleviate the crosstalk current effect. Based on a detailed comparative analysis [24], Liu concluded that the ZPM has the best performance, compared to the VFM and IDM. A detailed description of the ZPM is in Appendix A.
The lack in research progress on low resistance networks is partially attributed to the large crosstalk current effect and parasitic effect in the printed circuit board (PCB). In addition, the majority of the mechanical sensors and actuators in the market use high resistance transducers, due to the construction materials, fabrication techniques and sensing mechanisms involved.
Several methods, including the ZPM, VFM and IDM, have been developed to alleviate the crosstalk current effect. Based on a detailed comparative analysis [24], Liu concluded that the ZPM has the best performance, compared to the VFM and IDM. A detailed description of the ZPM is in Appendix A.
Following this introductory section, Section 2 describes a new dynamic zero current method (DZCM) to minimize the measurement error of low value RSAs. Section 3 presents the experiments and discussion. Section 4 provides conclusions.

Dynamical Zero Current Method
In the discussion in Appendix A, we point out that the ZPM suffers from parasitic effects. The crosstalk current effect is deteriorated due to two reasons: 1. Parasitic resistance from connection wires and PCB wires contributes a large crosstalk current effect.
2. Offset voltage of row and column driving amplifiers induces a crosstalk current effect.
These two issues are difficult to eradicate, as they are caused by the intrinsic feature of these electrical components.
To fix the abovementioned problems, we proposed the DZCM. The DZCM is originated from the ZPM which drives both ends of the adjacent unmeasured resistors to zero potential. Thus, almost no current flows through these adjacent unmeasured resistors. Then, the crosstalk current path will be cut off. In the ZPM, because of the row/column parasitic resistance and row/column amplifier offset voltage, the zero current is not really zero. This amount of current is negligible in a large value RSA, but not so in the case of a low value RSA. This non-zero current must be minimized to reduce the measurement  [14][15][16][17]19,21,24,25,[27][28][29][30]33,34,36].

Dynamical Zero Current Method
In the discussion in Appendix A, we point out that the ZPM suffers from parasitic effects. The crosstalk current effect is deteriorated due to two reasons:

1.
Parasitic resistance from connection wires and PCB wires contributes a large crosstalk current effect.

2.
Offset voltage of row and column driving amplifiers induces a crosstalk current effect.
These two issues are difficult to eradicate, as they are caused by the intrinsic feature of these electrical components.
To fix the abovementioned problems, we proposed the DZCM. The DZCM is originated from the ZPM which drives both ends of the adjacent unmeasured resistors to zero potential. Thus, almost no current flows through these adjacent unmeasured resistors. Then, the crosstalk current path will be cut off. In the ZPM, because of the row/column parasitic resistance and row/column amplifier offset voltage, the zero current is not really zero. This amount of current is negligible in a large value RSA, but not so in the case of a low value RSA. This non-zero current must be minimized to reduce the measurement error in the low value RSAs. Based on the fundamental circuit topology of the ZPM, the proposed DZCM includes a feedback network to automatically enforce zero current through each row of the adjacent unmeasured resistors. This feedback feature is also able to flexibly adjust the node potential of the array resistors to match the varying row/column parasitic resistance and amplifier offset voltage in the readout system.
The DZCM circuit design for a 4 × 4 array is shown in Figure 3. R sen is the current sensing resistor, which converts current to voltage. A feedback instrument amplifier (FBIA) magnifies the voltage across R sen . SW1 are the switches with a closed (ON) state for rows containing measuring resistors and SW are the switches with an open (OFF) state for the adjacent unmeasured rows. All amplifiers in Figure 3 are non-ideal. error in the low value RSAs. Based on the fundamental circuit topology of the ZPM, the proposed DZCM includes a feedback network to automatically enforce zero current through each row of the adjacent unmeasured resistors. This feedback feature is also able to flexibly adjust the node potential of the array resistors to match the varying row/column parasitic resistance and amplifier offset voltage in the readout system.
The DZCM circuit design for a 4 × 4 array is shown in Figure 3. is the current sensing resistor, which converts current to voltage. A feedback instrument amplifier (FBIA) magnifies the voltage across . SW1 are the switches with a closed (ON) state for rows containing measuring resistors and SW are the switches with an open (OFF) state for the adjacent unmeasured rows. All amplifiers in Figure 3 are non-ideal. To explain the characteristics of the ZPM and DZCM more clearly, we simplify the network to just one unmeasured resistor to demonstrate the parasitic effect of this single resistor. Figure 4 shows the circuit of this simplified models, all amplifiers in Fig  To explain the characteristics of the ZPM and DZCM more clearly, we simplify the network to just one unmeasured resistor R um to demonstrate the parasitic effect of this single resistor. Figure 4 shows the circuit of this simplified models, all amplifiers in Figure 4 are ideal. To explain the characteristics of the ZPM and DZCM more clearly, we simplify the network to just one unmeasured resistor to demonstrate the parasitic effect of this single resistor. Figure 4 shows the circuit of this simplified models, all amplifiers in Fig   We can derive the current of R um as: I um−ZPM and I um−DZCM are the currents of the unmeasured resistor for the ZPM and DZCM, respectively. They are expressed in Equations (1a) and (1b) accordingly. V os1 and V os2 are the offset voltages of row and column driving amplifiers, R par is the parasitic resistance of row and column wires, R sen is the resistance of the current sensing resistor, A f is the gain of FBIA r and I sen is the current of sensing resistor.
From Equation (1a) for the ZPM and Equation (1a,b) for the DZCM, we can clearly see that I um−DZCM computed from Equation (1b) is smaller than I um−ZPM in Equation (1a) due to the presence of the R sen 1 + A f term in the denominator.
We assume A f = 1000, R sen = 1.0 Ω, R um = 10.0 Ω, R par = 1.0 Ω, R a = R b = 1.0 kΩ V os1 = 1.00 mV and V os2 = −1.00 mV. We can then easily evaluate I um−ZPM and I um−DZCM from Equation (1). I um−ZPM = 0.18 mA is based on Equation (1a) for the ZPM and I um−DZCM = 0.0019 mA is based on Equation (1b) for the DZCM. Resulting from V os1 , V os2 and R par , the negative feedback in the DZCM decreases its parasitic effect current down to 1% of the parasitic effect current in the ZPM circuit. To put it simply, the DZCM can greatly minimize the measurement error in the low value RSA.
We further simplify the 4 × 4 array to a 2 × 2 array to quantify the crosstalk current effect in the DZCM, as shown in Figure 5. The reduced array network also includes a parasitic effect originated from V os1 , V os2 and R par . Based on the above discussion and Equation (1a) and Equation (1b), the currents of the adjacent unmeasured resistors R 21 and R 22 in the DZCM are only 1% of those in the ZPM. Thus, in the DZCM, the crosstalk current effect from R 21 and R 22 is minimal and negligible. Compared with Figure A3, R 21 and R 22 are shaded in Figure 5 and can be ignored in the Equation (2) formula derivation. The circuit diagram of Kirchhoff's law, as shown in Figure 6, is used to analyze the DZCM network from Figure 5.   Equation (3) is a non-homogeneous linear equation • = solving steps in Appendix B, we have: In order to simplify calculation, we hypothesize that R 11 = R x , R 12 = R um and 10·R par < R um ≈ R x . After substituting (2a, 2b, 2c) with (2d, 2e), we obtain Equation (3).
Equation (3) is a non-homogeneous linear equation R·I = V and after several solving steps in Appendix B, we have: As R par R um ≈ R x , we can simplify Equation (4) as: If we assume R par R um ≈ R x , Equation (A8) can be written as Equation (6): It is clear to see the Equation (5) is similar to Equation (6), but Equation (5) does not have the term , which exists in Equation (6). This makes Equation (5) have a lower error resulting from V os12 .

Experiments for DZCM/ZPM and Discussion
Various experiments have been designed to evaluate the performances of the ZPM and DZCM under optimum circumstances. The experimental setup is shown in Table 1.   The measurement result from a multimeter of an ideal single resistor is represented as R id . Meanwhile, the output amplifier's voltage of array resistors is measured as V out by a multimeter. Using Equation (A1), we can obtain the resistance value of the resistor of interest, R x . The measurement percentage error between them is evaluated as follows: Experiment (EXP) A analyzes the effect of R x on e% in arrays of various sizes when the unmeasured array resistors R um are fixed at their lower and upper limits (i.e., 1 Ω and 200 Ω, respectively) and R par and v os are set to zero.
EXP B analyzes the effect of unmeasured array resistors R um on e% in the simplest 2 × 2 array, as R x increases and R par and v os are zero.
EXP C analyzes the effect of parasitic resistance of column and row (R parCol and R parRow ) on e% when the unmeasured array resistors R um and R x are fixed at 1Ω (V in = 10 mV), 50 Ω (V in = 100 mV), 200 Ω (V in = 1000 mV) and offset voltage v os is zero. The reason for increasing V in with increasing R um and R x is to avoid the crosstalk current effect, which will surpass the signal current of R x if V in is fixed to 10 mV and R x increases to 50 Ω or 200 Ω. The array size is set to 6 × 6 and 12 × 12.
EXP D analyzes the effect of v os on e% when the unmeasured array resistors R um and R x are fixed at 1 Ω (V in = 10 mV), 50 Ω (V in = 100 mV), 200 Ω (V in = 1000 mV) and R par is set to zero. The array size is set to 6 × 6 and 12 × 12.

Experimental Result for DZCM with Ideal Resistors
the measurement errors of the DZCM and ZPM are found to be comparable when R x falls within the range of 1 Ω to 10 Ω. There is no significant improvement on the system performance by the additional feedback network of the DZCM in this R x range. ii. As R x goes beyond 10 Ω to 200 Ω, the e% of the DZCM is noticeably larger than that of the ZPM in all array sizes due to the underlying crosstalk current effect. The crosstalk current in the DZCM feedback network amounts to 25 µA, while the offset voltage of the feedback amplifier 'AD623 is 25 µV and the resistance value of the sensing resistor is 1 Ω. This gives rise to an undesired crosstalk current in R x , as, if V in = 10 mV and R x = 200 Ω, we will have I Rx = 10 mV/200 Ω = 50 µA. Meanwhile, the offset voltage of the ZPM is intentionally and manually compensated to zero to minimize the crosstalk current down to zero. We used an adjustable resistor to form a reference voltage divider and connect it to the positive node of row/column amplifiers, shown as V os1 and V os2 , as displayed in Figure 4a. The result shows that the ZPM outperforms the DZCM within this range of R x , i.e., 10 Ω to 200 Ω, with notably smaller measurement error. Nonetheless, this crosstalk current effect in the DZCM can be further suppressed by increasing V in or R sen in the feedback network.
ii. As goes beyond 10 Ω to 200 Ω, the e% of the DZCM is noticeably larger than that of the ZPM in all array sizes due to the underlying crosstalk current effect. The crosstalk current in the DZCM feedback network amounts to 25 µ A, while the offset voltage of the feedback amplifier 'AD623′ is 25 µ V and the resistance value of the sensing resistor is 1 Ω. This gives rise to an undesired crosstalk current in , as, if = 10 mV and = 200 Ω , we will have = 10 mV 200 Ω ⁄ = 50 μA . Meanwhile, the offset voltage of the ZPM is intentionally and manually compensated to zero to minimize the crosstalk current down to zero. We used an adjustable resistor to form a reference voltage divider and connect it to the positive node of row/column amplifiers, shown as 1 and 2 , as displayed in Figure 4a. The result shows that the ZPM outperforms the DZCM within this range of , i.e., 10 Ω to 200 Ω, with notably smaller measurement error. Nonetheless, this crosstalk current effect in the DZCM can be further suppressed by increasing or in the feedback network. In the case of = 1 Ω (Figure 7b), i. e% of the DZCM is smaller than that of the ZPM within the range of 1 Ω to 10 Ω, showing the advantage of the DZCM feedback network in bringing down the crosstalk current. ii.
As increases from 10 Ω to 200 Ω, measurement error changes from zero to a significant negative value. This unfavorable event also occurs in ZPM circuitry. Moreover, this event occurred in reference [29]   In the case of R um = 1 Ω (Figure 7b), i. e% of the DZCM is smaller than that of the ZPM within the R x range of 1 Ω to 10 Ω, showing the advantage of the DZCM feedback network in bringing down the crosstalk current. ii. As R x increases from 10 Ω to 200 Ω, measurement error changes from zero to a significant negative value. This unfavorable event also occurs in ZPM circuitry. Moreover, this event occurred in reference [29]   ⁄ Ω = order as 11 , the measurement error can add up to 100 Figure 7 also shows the improvement of the DZC in the range of 1 Ω < < 10 Ω and = 1 Ω. That DZCM is capable of decreasing the measurement erro EXP B: As shown in Figure 9, the e% of the DZCM is sm range of 1 Ω < < 10 Ω. This measurement error in ZPM as increases to the range of 10 Ω < < 2  A simple 2 × 2 array (see Figure 8) is illustrated to explain the SVE. Based on Figure 8, in the extreme case of R x = R 11 = 200 Ω, current flowing through R 11 decreases, denoted by I 11 = V in /R x = 10 mV/200 Ω = 50 µA. Meanwhile, for the adjacent resistor on the same row, R 12 = 1 Ω. The current through R 12 , represented by I 12 , is equivalent to V in /R um = 10 mV/1 Ω = 10 mA. Our measurement shows that the parasitic resistances resulting from the cable linking the sensor to PCB are R par = 0.01 Ω and the voltage at column 2 is V Col2 = 10 mA × 0.01 Ω = 0.1 mV, and the voltage at column 1 is V Col1 ≈ 0 mV. Due to this potential difference between V Col1 and V Col2 , there will be a crosstalk current through R 22 and R 21 , I 22−21 = 0.1 mV/2Ω = 50 uA. As I 22−21 are in the same order as I 11 , the measurement error can add up to 100%. Figure 7 also shows the improvement of the DZCM in reducing measurement error in the range of 1 Ω < R x < 10 Ω and R um = 1 Ω. That is to say, in low resistance RSA, the DZCM is capable of decreasing the measurement error.

EXP B:
As shown in Figure 9, the e% of the DZCM is smaller or similar to the ZPM in the range of 1 Ω < R x < 10 Ω. This measurement error in the DZCM becomes larger than the ZPM as R x increases to the range of 10 Ω < R x < 200 Ω, the increment has been explained in EXP A above as SVE.
EXP C: As shown in Figure 10a,c,e, when R parRow (the parasitic resistor of row wires) changes from 0 Ω to 3.5 Ω and R um = R x = 1 Ω, 50 Ω, 200 Ω, the R x error of the ZPM is the same as in the DZCM.

When
(the parasitic resistor of column wires) changes from 0 Ω to = = 1 Ω, 50 Ω, 200 Ω, as demonstrated in Figure 10b,d,f, the error of the is higher than that of the DZCM. The larger array size results in a greater error diffe When R parCol (the parasitic resistor of column wires) changes from 0 Ω to 3.5 Ω, R um = R x = 1 Ω, 50 Ω, 200 Ω, as demonstrated in Figure 10b,d,f, the R x error of the ZPM is higher than that of the DZCM. The larger array size results in a greater error difference between the ZPM and DZCM. This error difference increases as R parCol goes up. This is due to the presence of the negative feedback network in the DZCM that effectively reduces the parasitic effect of R par in the low resistance domain.
EXP D: To analyze the system performance with regard to offset voltage, error variation is preferred over absolute error. Absolute error, as discussed in EXP A, can be nulled by manual operation. On the other hand, the fluctuation of e% is non-zero, as v osC (offset voltage of column amplifiers) and v osR (offset voltage of row amplifiers) change. This inconstancy of measurement error often results in temperature drift and process variation of amplifier chips.
Offset voltages exist across the rows and columns in RSA. In the experiment to examine the effect of v osC , v osR is kept constant at zero and v osC changes from −3 mV to 3 mV in steps of 1 mV. Similarly, for the second study to evaluate the influence of v osR , v osC is fixed to zero and v osR is varied from −3 mV to 3 mV in steps of 1 mV.
The offset voltages v osR and v osC are applied to the positive inputs of all row and column amplifiers, labeled as v os1 and v os2 , respectively (see Figure 4a,b).
Error variation is defined as the difference in measurement error associated with experimental conditions.
As seen in Figure 11a-e, when v osC and v osR change from −3 mV to +3 mV with 1 mV steps, the DZCM (the curve with solid dots) has lower R x error variation than the ZPM (the curves with open dots) in all array sizes. This reveals that the DZCM is capable of eliminating the adverse effects arising from v osR and v osC of row/column amplifiers. Such improvements can be proven in Equation (5) and Equation (1b), respectively. The DZCM Equation (5) has one less item than ZPM Equation (6). That is because, from Equation (5), the presence of A f in the DZCM feedback network helps to reduce the effect of offset voltage on the error measurement.
As seen in Figure 11f, when v osR changes from −3 mV to +3 mV with 1 mV steps, the DZCM has larger R x error variation than the ZPM in all array sizes. This reveals that the DZCM is not suitable for high value RSA. However, the error gaps between the DZCM and ZPM decrease with array size increases. This implies the DZCM will have better performance when the high value RSA has a larger array size. Table 2 shows these performances. Table 2. R x error variation by v osR , v osC with S AR = 6 × 6, 12 × 12.

DZCM and ZPM Application on Liquid Metal EGaIn Based Flexible RSA
A liquid metal EGaIn based wearable 4 × 4 RSA was fabricated [5,7], as displayed in Figure 12.
of eliminating the adverse effects arising from and of row/column ampl Such improvements can be proven in Equation (5) and Equation (1b), respectively DZCM Equation (5) has one less item than ZPM Equation (6). That is because, Equation (5), the presence of in the DZCM feedback network helps to reduce t fect of offset voltage on the error measurement. (e) (f) As seen in Figure 11f, when changes from −3 mV to +3 mV with 1 mV the DZCM has larger error variation than the ZPM in all array sizes. This reveal the DZCM is not suitable for high value RSA. However, the error gaps betwee DZCM and ZPM decrease with array size increases. This implies the DZCM will

DZCM and ZPM Application on Liquid Metal EGaIn Based Flexible RSA
A liquid metal EGaIn based wearable 4  4 RSA was fabricated [5,7], as displayed in Figure 12.  SHIMADZU EZ-SX was used in the mechanical testing to apply 0 N to 2 N loading onto the RSA. A KEITHLEY DMM 6500 digital multimeter was used to measure the single resistor values and the output voltage of DZCM/ZPM circuits. Figure 13 shows the experimental setup. SHIMADZU EZ-SX was used in the mechanical testing to apply 0 N to 2 N loading onto the RSA. A KEITHLEY DMM 6500 digital multimeter was used to measure the single resistor values and the output voltage of DZCM/ZPM circuits. Figure 13 shows the experimental setup. We firstly indented a single independent sensor and measured its resistance directly. This sensor was then connected to a number of other sensors to form a 4 × 4 flexible RSA with DZCM and ZPM readouts. Subsequently, a force was applied onto this particular sensor. Output voltages of the DZCM and ZPM were recorded. These voltage values were converted to resistance by Equation (1).
We tested the single sensor and flexible RSA on flat and curved surfaces. The curved surface was shaped by a thumb sized semi-cylinder as shown in Figure 14. In total, we conducted six experiments to assess the performance of the DZCM and its improvement on the actual EGaIn based RSA. The experimental layouts were: i.
Single sensor on a flat surface; Figure 13. Force sensor test bench.
We firstly indented a single independent sensor and measured its resistance directly. This sensor was then connected to a number of other sensors to form a 4 × 4 flexible RSA with DZCM and ZPM readouts. Subsequently, a force was applied onto this particular sensor. Output voltages of the DZCM and ZPM were recorded. These voltage values were converted to resistance by Equation (1).
We tested the single sensor and flexible RSA on flat and curved surfaces. The curved surface was shaped by a thumb sized semi-cylinder as shown in Figure 14. SHIMADZU EZ-SX was used in the mechanical testing to apply 0 N to 2 N loading onto the RSA. A KEITHLEY DMM 6500 digital multimeter was used to measure the single resistor values and the output voltage of DZCM/ZPM circuits. Figure 13 shows the experimental setup. We firstly indented a single independent sensor and measured its resistance directly. This sensor was then connected to a number of other sensors to form a 4 × 4 flexible RSA with DZCM and ZPM readouts. Subsequently, a force was applied onto this particular sensor. Output voltages of the DZCM and ZPM were recorded. These voltage values were converted to resistance by Equation (1).
We tested the single sensor and flexible RSA on flat and curved surfaces. The curved surface was shaped by a thumb sized semi-cylinder as shown in Figure 14. In total, we conducted six experiments to assess the performance of the DZCM and its improvement on the actual EGaIn based RSA. The experimental layouts were: i.
Single sensor on a flat surface; Figure 14. Flexible sensor mounted onto bending surface with thumb-like size.
In total, we conducted six experiments to assess the performance of the DZCM and its improvement on the actual EGaIn based RSA. The experimental layouts were: i.
Single sensor on a flat surface; ii. single sensor on a curved surface; iii. RSA with ZPM on a flat surface; iv. RSA with ZPM on a curved surface; v.
RSA with DZCM on a flat surface; vi. RSA with DZCM on a curved surface.
We continuously indented the sensor for about 5 h. Figure 15 shows the 5 h measurement results of the above experiments. As shown in Figure 15, experiments (i) and (ii) tested only one independent sensor. Both experiments have resistance values that decrease with time and eventually stabilize at 6 Ω on a flat surface and 5 Ω on a curved surface. This is the characteristic of EGaIn based flexible sensors [5]. Experiments (iii) and (iv) tested the RSA with a ZPM readout. The resistance values decreased with time and stabilized at 12 Ω on the flat surface and 30 Ω on the curved surface. Experiments (v) and (vi) tested the RSA with a DZCM readout. The resistance values acquired were the most stable, amounting to 9 Ω on the flat surface and 10 Ω on the curved surface.
From Figure 15, we can conclude two benefits from the DZCM design: The DZCM has less measurement error than the ZPM on both flat and curved surfaces, especially when the array has low resistor value.
Measurement data from DZCM readout were more stable than that of the ZPM in about 4 h.

Discussion
It is a challenge to create low value RSAs for low power applications due to the lack of a compatible and stable readout design. Due to the low resistance value and the huge number of resistors involved in an RSA, driving voltage has to be minimized to achieve low power consumption, then the signal level is low. Thus, the noise of the parasitic resistance coming from the connection wires and the input offset voltage of the amplifiers will increase the measurement error greatly. Finally, it leads to a low signal to noise ratio (SNR).
To achieve low power consumption in the RSA, we used: ±2.5 V supply power and 10 mV driving voltage.
To decrease the parasitic wire resistance, we used: As shown in Figure 15, experiments (i) and (ii) tested only one independent sensor. Both experiments have resistance values that decrease with time and eventually stabilize at 6 Ω on a flat surface and 5 Ω on a curved surface. This is the characteristic of EGaIn based flexible sensors [5]. Experiments (iii) and (iv) tested the RSA with a ZPM readout. The resistance values decreased with time and stabilized at 12 Ω on the flat surface and 30 Ω on the curved surface. Experiments (v) and (vi) tested the RSA with a DZCM readout. The resistance values acquired were the most stable, amounting to 9 Ω on the flat surface and 10 Ω on the curved surface.
From Figure 15, we can conclude two benefits from the DZCM design: The DZCM has less measurement error than the ZPM on both flat and curved surfaces, especially when the array has low resistor value.
Measurement data from DZCM readout were more stable than that of the ZPM in about 4 h.

Discussion
It is a challenge to create low value RSAs for low power applications due to the lack of a compatible and stable readout design. Due to the low resistance value and the huge number of resistors involved in an RSA, driving voltage has to be minimized to achieve low power consumption, then the signal level is low. Thus, the noise of the parasitic resistance coming from the connection wires and the input offset voltage of the amplifiers will increase the measurement error greatly. Finally, it leads to a low signal to noise ratio (SNR).
To achieve low power consumption in the RSA, we used: ±2.5 V supply power and 10 mV driving voltage.
To decrease the parasitic wire resistance, we used: i. Copper wires of 3.6 mm/35 mm/70 µm (width/length/thickness) to connect the resistors to form an RSA on the PCB. ii. SMA coaxial connector/cable to link RSA PCB to the readout PCB. iii. Copper wires of 0.15 mm/75 mm/70 µm (width/length/thickness) to connect the SMA connectors and amplifiers on the PCB in the EXP.
To decrease the input offset voltage of amplifiers, we used: OPA4388 with 0.25 µV offset voltage.
As demonstrated in Figures 10 and 11, the DZCM is able to decrease measurement error arising from the parasitic resistance of wires and the input offset voltage of amplifiers. These characteristics of the DZCM, as explained below, are very important to realize steady and reliable wearable applications. i.
The larger physical dimensions of an electrical connection lead to lower parasitic resistance. Nevertheless, the use of bulky connecting wires and connectors in wearable systems is not feasible as they defeat the purpose of making an accessory, which is supposed to be comfortable and easy to wear. The DZCM helps to solve this issue as it has higher tolerance for parasitic resistance. In other words, it enables thinner wires and smaller connectors to be used in the wearable. ii. The input offset voltage of the amplifier varies with the ambient temperature. Consequently, the measurement result is greatly influenced by the operating environment.
The input offset voltage also varies in mass production; thus, the measurement result changes with different product batches. By applying the DZCM, that is less sensitive to the fluctuation of the input offset voltage, these issues can be easily resolved.
The low driving voltage required in the DZCM is another advantage which enables low power consumption. This enhances the wearable's performance.
The DZCM is cost effective and most likely to be useful in RSA with large array size and high resistance values as well. We did not examine the proposed circuitry in high value RSAs. The crosstalk and parasitic effects have to be quantified to prove its usability. Nonetheless, simulation data in the literature [41] show this tendency.
Lastly, the proposed DZCM design is still in its early stage of development. The measurement error from crosstalk and parasitic resistance is still significant in the readout system. The singular values effect of the DZCM, such as that shown in Figure 8, also requires further improvement.

Conclusions
We have discussed the ZPM and established its simplified models to derive the respective output voltage equations. Subsequently, we introduced a new circuit design, called the DZCM, and established a simplified model to deduce its output voltage equations. We also analyzed its measurement performance with different array sizes, input offset voltages of driving amplifier, unmeasured resistance values, parasitic resistance values and resistance value of the resistor of interest. The results show that the DZCM has lower e% than the ZPM. In terms of error variation ratio from amplifier offset voltage, the DZCM has a 4%/mV (row) to 7%/mV (column) ratio, while ZPM has an almost 25%/mV (row) to 45%/mV (column) ratio which increases with array size.
In short, the DZCM is very useful for low value RSAs used in wearable applications. This new circuitry helps to reduce measurement error in the readout system and bring down the material cost for mass production.

Patents
One Patent Pending: A circuit to measure low resistor value resistive sensor array with dynamical zero current function (10202204307R).

Conflicts of Interest:
The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Appendix A
One ZPM structure is shown in Figure A1. There are one row driving amplifier (rAMP) in each row and one column driving amplifier (cAMP) in each column. Resistor R 1i (i = 1, 2, 3, 4) is connected to two amplifiers on each of its nodes. The resistor is driven to V in by rAMP1 and grounded by cAMPi. V outi is generated by cAMPi which drives one node of R 1i to ground.

Conflicts of Interest:
The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Appendix A
One ZPM structure is shown in Figure A1. There are one row driving amplifier (rAMP) in each row and one column driving amplifier (cAMP) in each column. Resistor 1 (i = 1, 2, 3, 4) is connected to two amplifiers on each of its nodes. The resistor is driven to by rAMP1 and grounded by cAMPi. is generated by cAMPi which drives one node of 1 to ground.  Figure A1. Circuit based on the ZPM.
Two nodes of the other adjacent unmeasured resistors ( 21 to 44 ) are grounded through their connected rAMP and cAMP. The example is 31 in Figure A1, and these resistors have zero potential difference, thus there will be no current flow through . Then, the crosstalk current flowing through will be cut off. The ideal measurement value can be computed from Equation (A1).
1 is the measured resistor, is the input voltage to drive the measured resistor, is the current of the feedback resistor of the column amplifier, is the output voltage of the column amplifier.
In the case of an actual hardware system, the parasitic effect must be taken into consideration. The ON state switch resistor ( ) and the amplifier offset voltage ( 1 , Figure A1. Circuit based on the ZPM. Two nodes of the other adjacent unmeasured resistors R um (R 21 to R 44 ) are grounded through their connected rAMP and cAMP. The example is R 31 in Figure A1, and these resistors have zero potential difference, thus there will be no current flow through R um . Then, the crosstalk current flowing through R um will be cut off. The ideal R xi measurement value can be computed from Equation (A1).
R 1i is the measured resistor, V in is the input voltage to drive the measured resistor, I f is the current of the feedback resistor R f of the column amplifier, V outi is the output voltage of the column amplifier.
In the case of an actual hardware system, the parasitic effect must be taken into consideration. The ON state switch resistor (R ON ) and the amplifier offset voltage (V os1 , V os2 ) are included in the calculation. To include the parasitic effect and explain the characteristics of the ZPM, we simplify the network to just one resistor R x . The revised circuit is shown in Figure A2.  Figure A3 shows an example of a 2 × 2 RSA, taking into account both the crosstalk and parasitic effect. In order to include non-ideal factors (row and column parasitic resistance and amplifier offset voltage ) in this model, we use the ZPM for the analysis below. Non-ideal factors are illustrated as and in the circuit model. To calculate the intrinsic crosstalk current effect, we first extract the array network and apply Kirchhoff's law, as shown in Figure A4 and Equation (A3).  Figure A3 shows an example of a 2 × 2 RSA, taking into account both the crosstalk and parasitic effect. In order to include non-ideal factors (row and column parasitic resistance R par and amplifier offset voltage V OS ) in this model, we use the ZPM for the analysis below. Non-ideal factors are illustrated as R par and V OS in the circuit model.  Figure A3 shows an example of a 2 × 2 RSA, taking into account both the crosstalk and parasitic effect. In order to include non-ideal factors (row and column parasitic resistance and amplifier offset voltage ) in this model, we use the ZPM for the analysis below. Non-ideal factors are illustrated as and in the circuit model.  Figure A3. A 2 × 2 array circuit model including parasitic and crosstalk current effects for ZPM.

Rpar
To calculate the intrinsic crosstalk current effect, we first extract the array network and apply Kirchhoff's law, as shown in Figure A4 and Equation (A3). To calculate the intrinsic crosstalk current effect, we first extract the array network and apply Kirchhoff's law, as shown in Figure A4 and Equation (A3).

Appendix B
Equation (3) is a non-homogeneous linear equation R·I = V and can be written as Equation (A9a). R x + 2R par R par R par R um + 2R par After defining R um + 2R par = R u2 and R x + 2R par = R x2 and V os11 + V in = V i1 , we have: Rearranging row 1, we have: Rearranging row 2, we have: From Equation (2a), we have I f 1 = I 11 . From Equation (A9d) row 1 (I 11 ), we have: