Biomechanical researches strongly rely on accurate force measurements to provide reliable studies and outstanding developments: ground reaction forces occurring during gait analysis and gripping forces occurring during object manipulation are only some applications that demand non-invasive and accurate force measurements.

Inertia and position readings are also demanded in biomechanical studies; with the added complexity that the employed transducers must be installed with minimal interference on the human or animal under study in order to avoid discomfort during motion, consequently, the sensors in Biomechanical studies must be either low profile or such variables must be remotely tracked when possible [1, 2].

Performing accurate position and inertia readings has been practically resolved through the usage of encoders, resolvers and Inertial Measuring Units (IMUs). However, if such sensors result too bulky, high-speed cameras can be used instead [3]. However, performing accurate force readings has always been a difficult task in biomechanical studies because force, unlike position, cannot be remotely tracked.

Force Sensing Resistors (FSRs) are cost-affordable force sensors that can be easily integrated into multiple Biomechanical applications [4-6]. Nonetheless, the main reasons for its widespread usage are their low profile and low weight, which are highly desirable characteristics when attempting to perform non-invasive force measurements [6]. Another reason for their wide acceptance is the simple interface circuit required to read sensor’s output, e.g.: voltage dividers or inverting amplifiers. When using an inverting amplifier, see Fig. 1a, an estimation of sensor’s conductance ( ^_1/Rs ) is obtained through output voltage ( ^_Vo1 ). Conversely, when using a voltage divider, see Fig. 1b, sensor’s resistance ( ^_Rs ) is measured through ^_Vo2 .

Figure 1 Source: The authors.

Commercially available FSRs can be found on different shapes and nominal ranges: round (FSR400 and FSR402) and squared (FSR406 and FSR408) FSRs are manufactured by Interlink Electronics, Camarillo, CA [7]. Tekscan Inc. from South Boston, MA, offers round (A201-1, A201-25 and A201-100) and several customizable FSRs in his product catalog [8], see Fig. 1c and Fig. 1d. Unfortunately, the overall performance of FSRs is poor compared to well-established force sensing solutions such as load cells and strain gauges. Previous works from Lebosse [9], Hollinger [10] and Komi [11] present a comprehensive review on FSR limitations. Hysteresis and drift are typically one or two orders of magnitude greater in FSRs than in load cells. These conditions are the main drawbacks for the extensive usage of FSRs in industrial and research applications, but a great effort is currently placed on improving their performance.

One trend, within FSR research, is to model sensors’ response with the aim of compensating hysteresis and drift. Relevant works on this scope have been developed by Lebosse [9], Schofield [5], Dabling [12], Vecchi [13] and Urban [14]. Likewise, authors’ previous work has demonstrated that the A201 sensors, working on the piezoresistive principle, are also capable of exhibiting a piezocapacitive response. Different methods were proposed and evaluated by the authors to combine Capacitance ( ^_Cs ) and Conductance ( ^_Vo1 ) readings with the aim of increasing FSR accuracy under static loading [15, 16]. It must be noted that DC/AC voltages were alternately applied to the FSRs to read ^_Vo1 and ^_Cs respectively, followed by a feedforward neural network to optimally combine ^_Vo1 and ^_Cs readings. When compared to the purely conductance model of Fig. 1e [8], a 64% reduction in the force Mean Squared Error (MSE) was obtained. The reduction in the MSE was done at the price of increasing the complexity of the driving circuit which may result prohibitive for certain applications with power or space constraints.

By modeling FSR’s hysteresis through the Preisach operator [17], an improved algorithm for estimating applied forces is here presented. The algorithm only requires conductance readings form the sensor, and thus, the simple driving circuit of Fig. 1a is needed, or Fig. 1b when using Interlink sensors. A considerably computational effort is required to run the inverse Preisach algorithm, but such an effort is justified when the force MSE is dramatically reduced. In order to obtain a valid generalization of results, a total of sixteen A201-1 FlexiForce sensors are used; this sensor matches the required force range (4.5N) of biomechanical applications involving grip and grasp operations. Nonetheless, the methods henceforth discussed are applicable to other models of FSRs.

This paper is organized as follows: Section II describes the experimental setup for gathering sensor data. Later in Section III, experimental data from the sixteen A201-1 sensors are presented with statistics regarding hysteresis. A brief description of the Preisach Operator and its inverse are also available on Section III. Later in Section IV, grip force profiles are exerted on the A201-1 sensors and the inverse Preisach algorithm is tested and compared with the traditional conductance model. In Section V, a generalized sensor model based on the Preisach Operator is presented, followed by conclusions and future work on Section VI.

Previous work demonstrated that the A201-X sensors can be electrically modeled as a parallel ^_Rs - ^_Cs device as shown in Fig. 2a [15,16], with ^_Rs exhibiting a hyperbolical dependence on the applied force (F). Since this paper focusses on reducing the force MSE through conductance readings only, capacitance measurements are not henceforth considered. For linearization purposes, conductance variations - measured through ^_Vo1 - have been preferably used in several studies to estimate F [8, 9, 12, 15]; this is possible by inverting the model of Fig. 1e, which yields:

Where m and b are obtained from a least-squares minimization process; the whole procedure is known in literature as sensor characterization and may comprise only increasing or increasing/decreasing forces. It must be noted that the application of either pattern produces a significant effect in m and b values given the hysteresis in the device; this is later exemplified on Section IV. The test bench for sensor characterization and testing comprises electrical and mechanical sensors/actuators as described next.

Figure 2 Source: The authors.

Grip force profiles, based on grip data issued from Stolt [18] and Melnyk [19], were exerted on the A201-1 sensors using the experimental set-up previously described on Section II. The performance of the CMA to estimate F was comparatively evaluated with the linear regression model (1), which is the manufacturer recommended method. It must be noted that the linear regression model is the preferred method in research applications [4, 5, 9-13]. Two sets of m and b constants were calculated using the triangle force profiles of Section III.B. The first set of m and b constants used only the increasing data points, whereas the second fit employed the whole data.

Experimental results for a given sensor are graphed on Fig. 8, together with an error bar plot on Fig. 9 representing the force MSE for the sixteen sensors and the three models, i.e.: the CMA and the two aforesaid linear regressions. In average, the CMA yields a considerable reduction in the MSE compared to the models based on (1), 54% and 76% respectively. The reduction in the force MSE is paid by a larger computation time, which may limit the usability of the CMA in some real-time applications, but if time is not a constraint; the CMA can run off-line and get the best of a FSR. This is the case of most biomechanics researches in which data are collected and later analyzed [4, 5]. However, a fast implementation of the CMA is possible if dedicated hardware is used, i.e., employing FPGA technology [22].

Figure 8 Source: The authors.

Note that the dispersion of the MSE is narrowed when force estimation is done on the CMA (red bars on Fig. 9), this occurs because the remaining MSE is only produced by: the Repeatability Error, RE, and the quantization error derived from the discrete CMA. The contribution of the HE to the MSE is suppressed thanks to the PO modeling and compensation via the CMA. Further reduction of the MSE is possible if a continuous implementation of the CMA is performed, just as Stakvik demonstrated in [20]. This statement can be illustrated from the zoom-in plot at the top of Fig. 8. Note that both linear regressions exhibit a subtle swinging due to ^_Vo variations, nonetheless, such variations are not great enough to change the memory curve Ψ, and consequently, the CMA yields a constant output. The implementation of a continuous CMA can better adapt to subtle ^_Vo variations amid a constant memory curve Ψ, and thus, the force MSE would be further reduced.

Figure 9 Source: The authors.

Obtaining ^_μ(α,β) and ^_η0 is not a straightforward procedure; specialized hardware is required and a time consuming characterization must be performed followed by non-trivial programing. Fortunately, it is possible to simplify the characterization process if the parametrization is performed as follows: Given ^_b0 , the output voltage of a FSR at null force, and ^_Vnom , the maximum output voltage at the nominal sensor range; then, the following linear transformation can be applied to any ^_Vo reading during both: the estimation of ^_μ(α,β) and ^_η0 , and later, during sensor operation.

The scaled output voltage ( ^_Vos ) is normalized, just as the resulting Preisach-scaled parameters are, ^_μs(α,β) and ^_ηs0 ; this implies that they can be used to estimate F with hysteresis compensation for any sensor, requiring only the experimental measuring of ^_b0 and ^_Vnom on each new device.

However, it must be pointed out that during sensor operation, the CMA running on the basis of ^_μs(α,β) and ^_ηs0 can compensate for a specific amount of HE, see Fig 4, this implies that if a given sensor exhibits a larger HE than that modeled by ^_μs(α,β) and ^_ηs0 , the resulting F will still exhibit some hysteresis (underfitting case). Conversely, if a given sensor has a lower HE, the resulting F will reflect hysteresis overfitting; both cases are presented on Fig. 10. Hysteresis overfitting is characterized by a time-ahead output during sensor unloading (green data on Fig. 10), especially noticeable around the maximum ^_Vou - ^_Vol which occurs at half of the nominal sensor range, see Fig. 4. Conversely, if hysteresis is under-fitted, the CMA produces a lagged sensor output during the unloading stage (red line on Fig. 10).

Figure 10 Source: The authors.

The Preisach-scaled parameters were obtained by applying the linear transformation (6) to the data gathered for the sixteen sensors. The resulting ^_μs(α,β) and ^_ηs0 were later used to estimate F using the CMA. The blue error bars of Fig. 9 shows the resulting MSE when using the Preisach-scaled parameters during force estimation. Note that the MSE is higher than that reported on Section IV (individual Preisach parameters tuned for each sensor); this is a logical consequence of hysteresis-overfitting and underfitting. Likewise, mixed results were obtained when comparing to the linear regression methods, indicating that additional work is required on this subject.

The Preisach Operator and the Closest Match Algorithm (CMA) have been successfully implemented on sixteen A201-1 sensors to both: model hysteresis and compensate for hysteresis effects during sensor operation. In the latter case, a considerably reduction of 54% was obtained in the force MSE compared with the manufacturer-recommended method based on linear regression. This is an important contribution that narrows the gap between the highly accurate load cells and the low-profile low-cost FSRs. Further MSE reduction is possible by implementing a continuous CMA, this is left as a pending task for future work. It must be remarked that hysteresis modeling and compensation required no-additional hardware during sensor operation, which is a highly desirable characteristic in applications with power or space constraints.

Considering that hysteresis modeling is not a straightforward procedure, a generalized model for hysteresis compensation has been proposed for the A201-1 sensor. However, mixed results were obtained because of hysteresis overfitting and underfitting. Future work is required on the subject in order to develop several Preisach density function focused on compensating for a specific amount of Hysteresis Error. This task can be carried out for sensors exhibiting a linear output, such as the Tekscan A201-X, and for sensors with a nonlinear response, e.g. Interlink FSR40X and Peratech QTC sensors.