If the DC source, V_s1, is chosen as the input of the driving circuit in Figure 1(c), a DC output voltage, V_o, is obtained that changes linearly as the applied force on the sensor increases (see Figure 2) [25]. This response corresponds to the piezoresistive property of the sensor and it has been thoroughly described by the sensor manufacturer [24] and many research articles [4,7,8,17,20,22,26]. However, to the best of our knowledge, no information is available about how the output voltage changes for a fixed force when the DC voltage is varied.

In order to study such behavior, the input voltage, V_s1, is swept from −1 V up to 1 V, and the output voltage, V_o, is plotted while the applied force remains constant. It is important to remark that these measurements are carried out by using a constant-puck area of 50.3 mm², smaller than the sensor’s sensing area of 71.3 mm², this ensures that every sensor is evenly loaded in each trial and avoids the pressure falling outside the sensing area of the sensor. Figure 3 shows the output voltage for randomly chosen forces of 12 N, 45 N, 82 N and 160 N. The best function for relating the input voltage, V_s1, to the corresponding sensor response, V_o, is found to be: (7) V o = − 1 q atanh   ( V s 1 k )where k and q are constants. However, it is inadvisable to fit k and q in the form presented in Equation (7), because V_o may yield complex values since the atanh domain is restricted to (−1, 1). Thus, for fitting purposes it is better to rewrite Equation (7) in terms of the tanh function as follows: (8) V s 1 = k   tanh   ( − q V o )

The minus sign in Equations (7) and (8) comes from the negative gain in the inverting amplifier [see Figure 1(c)] used to drive the sensor. The axes in Figure 3 are intentionally switched to represent V_o on the x-axis and V_s1 on the y-axis with the aim of fitting the data points with Equation (8) instead of Equation (7). The fitting process is highly fiable, with a coefficient of determination, R², of at least R² = 0.9992 for every applied force and an average R² value of 0.9995.

Parameters k and q were set to adjust independently for every applied force, however, the independent fitting processes returned values of k that remained almost constant, regardless of the exerted force, F, whereas q was shown to be hyperbolically dependent on the exerted force. In other words, 1/q is a linear function of F. Figure 4 shows the variation of k and 1/q for different applied forces within the 0 N–250 N range resulting from independent fitting processes.

In order to get a comprehensive view of sensor behavior, a relationship between Equation (8) and F must be found. For that purpose, the fact that the sensor exhibits piezoresistive behavior is useful, as thus its conductance, 1/R_s, may be modeled in terms of the applied force, F, as: (9) 1 / R s = m F + b

Equation (9) is not explicitly stated in the PFS user manual [24], however, the sensor manufacturer does declare that a linear interpolation between the conductance values and the applied forces can be done. Also, it can be easily deduced from a look at the conductance curve in Figure 2 that Equation (9) is a valid fit for 1/R_s. Considering the inverting amplifier with feedback resistor R_g, which is used to drive the PFS, it is possible to link Equation (9) with the amplifier characteristic equation: (10) V o V s 1 = − R g R sto obtain: (11) V o V s 1 = − R g   ( mF + b )

We should clarify that Equation (11) is not explicitly stated in the PFS user manual [24], and only Equation (10) is given in [24], but the manufacturer suggests that the sensor sensitivity, m, can be changed by either replacing the feedback resistor, R_g, or changing the driving voltage, V_s1.

With the aim of demonstrating that Equation (11) is an approximate expression for fitting the data points from Figure 3, the 1/q curve from Figure 4 is taken and represented as linearly dependent on the applied force: (12) 1 / q = m ′ F + b ′

Note that the 1/q curve in Figure 4 is analogous to the conductance curve of Figure 2. Equation (12) can be substituted into Equation (7), yielding: (13) V o = − ( m ′ F + b ′ )   atanh   ( V s 1 k )

Nevertheless, Equation (13) cannot be stated the same way as Equation (11), because the input voltage, V_s1, is part of the atanh argument, but if only the first term of the atanh-Taylor series is taken, the following approximate expression is obtained: (14) V o V s 1 = − 1 k   ( m ′ F + b ′ )

Equation (14) is an approximate expression for modeling sensor response, and so is Equation (11). The 1/k factor in Equation (14) is analogous to R_g in Equation (11), in the same way that m is analogous to m′, b to b′ and 1/R_s to 1/q. Equations (11) and (14) are valid if and only if the input voltage remains constant during the measurement process; this condition matches the driving conditions recommended by the manufacturer [24].

The effect of the feedback resistor can be deduced from the fact that R_g only changes the feedback gain of the amplifier, without affecting sensor current. Thus, changing R_g will produce directly proportional changes in the output voltage, and Equation (13) can be rewritten as: (15) V o = − ( R g R ref )   ( m ′ F + b ′ )   atanh   ( V s 1 k )

Henceforth Equation (15) is referred to as “the general-sensor model” under DC sourcing, with the restriction |V_s1| < 1 V, where R_ref is the feedback resistor used during the characterization to obtain the values of m′, b′ and k. In case the feedback resistor is changed after the characterization process, the output voltage is multiplied by the ratio R_g/R_ref where R_g is the new feedback resistor. Replacing the feedback resistor produces a directly proportional change in the output voltage, because the amplifier is inherently linear, whereas sensor resistance is not. In fact, linking Equations (7), (10) and (12) produces an expression that shows the nonlinear behavior of sensor conductance in response to changes in the input voltage: (16) 1 R s = ( m ′ F + b ′ ) atanh   ( V s 1 / k ) V s 1   R g

However, it must be noted from Equation (16) that sensor conductance is always linear to force changes.

Since a general-sensor model was deduced in Equation (15), a generalized method can be presented for obtaining a specific sensitivity in the sensor. The tests discussed in this section were done using four A201-100 sensors to provide more representative results. Each PFS exhibits a considerably different sensitivity. This condition prevents the sensors from having plug-and-play capability, because a characterization must be run before using a sensor. Moreover, for characterizations run at a given input voltage, it has been impossible until now to determine the new values of m and b under the new input voltage condition, because the amplitude nonlinearity was not accounted for. Nevertheless, Equation (15) accounts for changes in both the input voltage, V_s1, and the feedback resistor, R_g. This implies that if m′, b′, k, and R_ref are given, it is possible to either determine sensor sensitivity for any input voltage and feedback resistor or design a specific driving circuit ,V_s1 and R_g, with the aim of matching a target sensitivity.

Note that the k factor in Equation (15) contains information about sensor sensitivity for any input voltage. Unfortunately, the k factor is different for each sensor, just as sensitivity is too. The range of variation of the k factor for the four sensors under study was from 1.2 V to 1.5 V, with an average value of 1.41 V. If k is not given, the only way to change sensor sensitivity is to replace the feedback resistor, R_g; but, in certain applications where several sensors are being used [1,16,21,23], it is more convenient to change the driving voltage instead of changing individual feedback resistors. In addition, the feedback resistor is increased despite an increase in the noise level, so it is better to match a desired sensitivity by making a trade-off between V_s1 and R_g.

With the aim of proposing a method for obtaining a specific PFS sensitivity, it is necessary to fit the general-sensor model in Equation (15) to the experimental data. However, the feedback resistor must not play any role in the fitting process, because R_g only affects the closed loop gain of the amplifier, not the sensor response itself. Thus, it is better to use Equation (13) for the fit instead of Equation (15). In order to avoid complex numbers resulting from Equation (13) during the fit, Equation (13) is rewritten in terms of the tanh function as shown below: (17) V s 1 = − k   tanh   ( V o m ′ F + b ′ )

Note that Equation (17) implies a three-dimensional fit in terms of the variables F, V_o and V_s1, with coefficients m′, b′ and k to be determined.

In practice, the experimental data were gathered in the same way as described in Section 3.1. Nevertheless, the fit here described embraces all variables and coefficients at once (F, V_o, V_s1, m′, b′ and k), whereas the fitting process in Section 3.1 was split into two independent threads. This is an important difference, because the fit here tries to minimize the overall error, while the process in Section 3.1 independently minimizes the error for every exerted force.

Figure 5 shows the experimental data points and the surface produced by the three-dimensional fit in Equation (17). For this case, the values of R² were in general lower than for the fits given in Section 3.1. The minimum value of R² for the four sensors under study was R² = 0.989, with an average value of R² = 0.991; this is comprehensible, because all forces and voltages were embraced in a single fit, and thus a single k value was returned.

Once the fit is done, it is possible to assemble the general-sensor equation by substituting into Equation (15) the coefficient values returned from the fit (m′, b′ and k) and the value of the feedback resistor used during the characterization process, which matches R_ref in Equation (15). If R_g or V_s1 are changed at any time, it is possible to recalculate m and b from Equation (15); or, if the feedback resistor, R_g, is fixed by the application circuit, the target sensitivity, m_t, can be obtained by choosing the appropriate V_s1 from Equation (15) as shown below: (18) V s 1 = k   tanh   ( m t m ′ R ref R g )

However, Equation (18) is valid if and only if the predicted input voltage is lower than 1 V, because only the amplitude nonlinearity for |V_s1| < 1 V has been studied. Note that the k factor may be taken as a mean of amplitude nonlinearity. To confirm this, Figure 6 plots the following function, which is drawn from the sensor-conductance expression, Equation (16): (19) G ( V s 1 ) = atanh   ( V s 1 / k ) V s 1   R g

Equation (19) is a measure of how the sensor conductance is affected by changes in the input voltage. The higher k is for a given sensor, the flatter the lines in Figure 6. This can be understood as a less-noticeable amplitude nonlinearity in the sensor.

If the PFS is sourced with voltages above 1V, the fitting curve in Equation (8) and the general-sensor model in Equation (15) provide an unsatisfactory fit according to our criteria. Figure 7 shows the output and input voltage from data taken experimentally under three different forces (47 N, 97 N and 240 N) for driving voltages within the range (−6 V, 6 V) with Equation (8) as the trendline. Note that the experimental data points move away from the trendline, especially in the middle-range voltages, |V_s1| < 3 V.

Despite the fact that the coefficient of determination remained high for all sensors under study (exhibiting an average value of R² = 0.991), it is clear that Equations (8) and (15) require some changes in order to provide a better fit. So far we have not found a suitable and simple curve for modeling sensor behavior under such driving conditions.

Forces were applied within the range of 0 N to 250 N under seven sourcing conditions, starting at 0.4 V, with increments of 0.1 V, up to 1 V. For every supply voltage, the sensor sensitivity, m, and the y-intercept, b, were found according to the classical method proposed by the manufacturer [24]. These values of m, b are used as references for calculating subsequent errors in m, b for the different methods presented in this article.

Sensors were then characterized in terms of m′, b′ and k as described in Section 3.3. Doing so enabled Equation (15) to be assembled and m, b to be estimated for the aforementioned discrete DC voltages.

With the aim of demonstrating the typical error resulting from the assumption of linear-sensor response, the values of m, b measured at 1 V and Equation (11) were used to estimate m, b for the remaining six voltages (0.4 V to 0.9 V). That is, if the m value measured at 1 V is 20 mV/N, use of Equation (11) shows that the estimated value of m at 0.4 V is equal to 8 mV/N; the same calculation applies to the y-intercept coefficient, b. Table 1 compares the errors of m and b predicted from the linear, Equation (11), and the nonlinear model, Equation (15).

Two facts may be drawn from Table 1. First, the errors in the m values predicted by Equation (15) are rather small for all sensors. This means that the nonlinear model presented in this paper produces matches in predicting the sensor sensitivity for any input voltage under 1 V. In fact, the average error in m for all sensors and all voltages is only 2.04%. Second, the assumption of linear response in sensor conductance produces a reasonably greater error when predicting m via Equation (11); an average error of 11.35% was obtained, which is almost six times the error resulting from Equation (15). Figure 9 shows the values of m for sensor number four, under all the discrete applied voltages. Also shown are the trendlines arising from the linear Equation (11) and the nonlinear model, Equation (15).

Second, the average error for all sensors resulting from the estimation of the y-intercept coefficient, b, was quite high, 10.27% and 14.5%, for the two models. Despite this fact, we declare that Equation (15) can effectively model sensor behavior, because through our various optimization processes we have effectively reduced the estimation error of b by discarding the highly noisy data obtained under forces lower than 40 N. Such data are noisy because of the low amplitude of V_o, resulting from the low-force condition. If the data taken under forces lower than 40 N are discarded, the average error for the y-intercept estimation is reduced to 6.35%.

Input amplitudes are not restricted to be under 1 V for the AC sourcing experiments, because the intention is to subject the feasibility of the measuring scheme in Figure 8 to a thorough test. Figure 10 shows two different sets of capacitance values, C_s, for the same sensor, taken under sinusoidal excitation at a fixed frequency of 4 KHz. The black values were taken according to the measuring scheme in Figure 8, whereas the red data points were taken from our previous work [20,21], when the amplitude nonlinearity was unknown and the equations in Section 2 were used to estimate C_s. This set of data was taken at a fixed input amplitude of 3 V.

Considering:

That the black data points in Figure 10 are rather close to each other for the different driving voltages, and

An attempt was made to fit a linear trendline to the black data points in Figure 10. This is a logical approach, considering that our previous work reported a linear variation of C_s in response to force changes [20,21]. However, the residuals resulting from the fitting process suggested that a linear fit was unsuitable. Thus, different trendline forms were tried. The best, but also simplest, trendline that could model the variation of C_s was a square-root model dependant on the applied force, F: (27) C s = C o + a F + bwhere a, b and C_o are constants estimated from the fit.

On the other hand, the red data points in Figure 10, taken from our previous work [20,21], were fitted with a line, and an acceptable coefficient of determination was obtained, R² = 0.9942. The same data points were fitted with Equation (27), and the value of R² was even better, R² = 0.9987; this means that in principle either trendline is valid for fitting the data.

From Figure 10, it is clear that one set of data points is incorrect, because it is impossible that different values of capacitance are measured for the same sensor, under the same applied forces and the same mechanical layout. Considering that the red data points were taken under the erroneous supposition of linear conductance, it is logical to suggest that it was the assumption that yielded the incorrect values of capacitance; but, as stated in Sections 4 and 5, there are a few steps remaining to be accomplished before this hypothesis can be demonstrated.

In other words, we have theoretically and experimentally demonstrated that the black data points from Figure 10 are correct at least for |V_s1| < 1 V, but we are unable to theoretically demonstrate that the error basis from the red data points is due to the assumption of linear sensor conductance, despite some experimental results supporting this hypothesis. For instance, a look at the measured capacitance under low applied forces in Figure 10 shows that both methods estimate the same C_s. This may be understood if one recalls the piezoresistive behavior of the sensor; so, under low applied forces and AC excitation, sensor resistance can be taken as virtually infinite (See Figure 2), and its response is dominated only by capacitive factors. Under these circumstances, the resistive term in Equation (1) may be discarded, and the measured capacitance will be the same, regardless of the conductance model or the method employed for measuring C_s. It is only when the exerted force is increased that the aforementioned conditions start to play an important role in the estimation of sensor capacitance, and consequently, as F increases, the two sets of data points gradually diverge.