# Modeling and Calibration of Pressure-Sensing Insoles via a New Plenum-Based Chamber

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## Abstract

**:**

## 1. Introduction

#### 1.1. Existing Insoles

^{®}features seven sensors per foot [7], and PODOSmart

^{®}contains an inertial platform [8]. This brief list, though not exhaustive, highlights the low number of sensors that are usually integrated in such devices (to the best of our knowledge, only XSENSOR

^{®}currently offers an insole using a much higher number of sensing units). With respect to published results, [9] presented insoles that were based on 75 resistive sensors each, but the measurement frequency was limited to just 13 Hz. In [10], interesting results were achieved, however, only 24 sensors were used and the range of pressure considered (0–2 bar) did not cover typical working conditions experienced by the sole in many situations (e.g., when running). From the point of view of gait analysis, many other investigations have used a limited number of sensors that, while enabling recognition of discrete gait events and extraction of meaningful features, do not provide information that would enable a full understanding of foot-ground interactions and their continuous real-time variations to be obtained [11,12,13,14]. To address this concern, ref. [6] presented an insole carrying as many as 280 capacitive pressure sensors. This hardware, and a more refined calibration pipeline for it, is the focus of this investigation.

#### 1.2. Existing Calibration Setups

#### 1.3. Overview of Models Used for Calibration

#### 1.4. Contribution

- by introducing a novel, pressure-based, calibration setup;
- by proposing a refined calibration procedure for our insole, using a polynomial model of each sensor that also considers information coming from its past outputs;
- by validating the model found with static and dynamic trials performed using our setup.

## 2. Background

#### 2.1. Sensing Technology

^{®}from BASF (1 cm thick), whose properties of very low electrical conductivity and high elasticity make it ideal for our application. When the insole is pressed, the distance between the two conductive plates decreases, leading to a change in the measured capacitance value. The more pressure (or force) is applied to the top layer of the insole, the greater the capacitance variation is, but how to relate these is not straightforward. Determining this relationship experimentally and validating it are primary purposes of this paper.

#### 2.2. Physics-Based Approach

^{2}), it is possible to assume uniform pressure P acting on it, so that:

- k changes with the pressure and with time, since the linear model is valid only for small deformations;
- ${d}_{0}$ is not constant due to mechanical hysteresis;
- A can vary slightly from one taxel to another.

#### 2.3. Problem Statement

## 3. Calibration Setup

- a steel pressure tank, capable of withstanding a maximum pressure of 10 bars;
- a custom support, specially designed to house the insole during calibration;
- a 24-L air compressor;
- a closed-loop pressure regulator (QB4 from ProportionAir);
- a high-precision pressure sensor (PN2014 from IFM);
- several high-pressure elements (pipes and valves) to connect the various parts of the setup together.

## 4. Modeling and Identification

#### 4.1. Polynomial Model

- ${\mathit{C}}_{i}$ is the vector containing all of the capacitance measurements recorded by taxel i;
- ${\widehat{P}}_{i}({\mathit{C}}_{i},t)$ is the pressure estimated by taxel i at time t based on its measurements;
- $\pi \left({C}_{i,t}\right)={a}_{0,0}+{a}_{0,1}{C}_{i,t}+...+{a}_{0,{n}_{p}}{C}_{i,t}^{{n}_{p}}$ is a polynomial of order ${n}_{p}$ that links the instantaneous value of the capacitance ${C}_{i,t}$ to the pressure estimate;
- ${h}_{k}\left({C}_{i,t-k}\right)={a}_{k,1}{C}_{i,t-k}+...+{a}_{k,{n}_{ps}}{C}_{i,t-k}^{{n}_{ps}}$ is a polynomial of order ${n}_{ps}$ contributing to enhance the pressure estimate with information coming from the past state of the sensor (at time $t-k$);
- ${n}_{s}$ is a parameter selecting the number of past samples to be considered for the current estimate.

#### 4.2. Data Collection

^{2}is the area of a single triangular patch and ${n}_{t}$ is the number of patches on which the force is exerted. A subject of 80 kg balancing on one foot (28 triangles) would produce an average pressure $P\approx 0.85$ bar, while standing on the toes would result in $P\approx 3.4$ bar (considering just seven triangles activated).

#### 4.3. Mathematical Methods

#### 4.4. Data Preprocessing

- implement a Tikhonov regularization [32] of our problem to favour solutions with a lower norm;
- normalize the regressor ${\mathbf{\Phi}}_{i}$ (i.e., each column is scaled by the value of its maximum element). As a result, all the elements in the new regressor will belong to the range $[0,1]$. This is necessary, since, from its definition in (9), it is evident that different columns of ${\mathbf{\Phi}}_{i}$ have very different orders of magnitude. The solution of the scaled problem is also to be scaled similarly in order to counter-balance the smaller value range considered;
- subsample the original dataset to scale down the dimension of the problem by retaining a similar level of information.

#### 4.5. Calibration of the Base Instantaneous Polynomial

#### 4.6. Expansion of the Base Model

## 5. Results and Discussion

**Model A**, found by setting $\{{n}_{p},{n}_{s},{n}_{ps}\}=\{3,60,4\}$, that achieves a lower RMSE on the validation scenarios considered;**Model B**, found by setting $\{{n}_{p},{n}_{s},{n}_{ps}\}=\{3,40,4\}$, that, even if slightly less accurate, requires a lower dimensionality of the optimization vector.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The novel calibration setup that we propose is visualized: (

**a**) cross-section of the pressure tank, enabling visualization of the support holding the insole during the calibration phase; (

**b**) CAD image of the parts composing the support, safely housing the insole and its electronics.

**Figure 3.**Calibration of the base instantaneous polynomial, showing the RMSE on the validation dataset as a function of the order ${n}_{p}$ of the polynomial. A very large range of values for ${n}_{p}$ is examined; however, from the plot it is evident that a local minimum can be found at ${n}_{p}=7$. Therefore, a restricted yet meaningful range for the polynomial orders (up to ${n}_{p}=8$) is considered in the second step of the calibration.

**Figure 4.**RMSE resulting from the pressure estimates achieved with the optimal parameter sets ${\mathbf{\lambda}}_{i}$, depending on various values of ${n}_{p}$, ${n}_{s}$ and ${n}_{ps}$. Each surface represents a different value of ${n}_{p}$, as described in Section 4.6.

**Figure 5.**Comparison of the different performances, in terms of RMSE, for some chosen models. Models with just 20, 40 and 60 past samples are analyzed, characterized by base instantaneous polynomials of order 3, 4, 6 or 7.

**Figure 6.**Analysis of the best results achieved. Of the different models represented here, two are selected by virtue of their accuracy and reduced complexity, and indicated with a star-shape on the graph.

**Model A**represents the best trade-off in the validation scenarios considered, since it achieves a smaller RMSE when compared to other models which are still easily treatable (i.e., where ${n}_{ps}\le 7)$.

**Model B**retains good performances (as well as the property of being locally optimal given the number of past samples considered) and is more reactive to dynamic variations of the pressure signal as it considers a reduced range of the dynamic history of the sensor’s state.

**Figure 7.**Pressure estimates retrieved from Model A on the validation dataset (comprised of 3 different experiments). The true pressure level recorded during the experiments is shown, together with the best-fitting estimate (orange), and the worst-fitting estimate (green). Both curves show high similarity to the target of the estimation.

RMSE [bar] | ${\mathit{n}}_{\mathit{p}}$ | ${\mathit{n}}_{\mathit{s}}$ | ${\mathit{n}}_{\mathbf{ps}}$ | Dimension of ${\mathbf{\lambda}}_{\mathit{i}}$ | Error % | |
---|---|---|---|---|---|---|

Model A | 0.158 | 3 | 60 | 4 | 244 | 4.4 |

Model B | 0.167 | 3 | 40 | 4 | 164 | 4.6 |

Model C | 0.147 | 7 | 60 | 8 | 488 | 4.1 |

from [6] | 0.322 | 3 | 0 | 0 | 3 | 8.9 |

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## Share and Cite

**MDPI and ACS Style**

Belli, I.; Sorrentino, I.; Dussoni, S.; Milani, G.; Rapetti, L.; Tirupachuri, Y.; Valli, E.; Vanteddu, P.R.; Maggiali, M.; Pucci, D.
Modeling and Calibration of Pressure-Sensing Insoles via a New Plenum-Based Chamber. *Sensors* **2023**, *23*, 4501.
https://doi.org/10.3390/s23094501

**AMA Style**

Belli I, Sorrentino I, Dussoni S, Milani G, Rapetti L, Tirupachuri Y, Valli E, Vanteddu PR, Maggiali M, Pucci D.
Modeling and Calibration of Pressure-Sensing Insoles via a New Plenum-Based Chamber. *Sensors*. 2023; 23(9):4501.
https://doi.org/10.3390/s23094501

**Chicago/Turabian Style**

Belli, Italo, Ines Sorrentino, Simeone Dussoni, Gianluca Milani, Lorenzo Rapetti, Yeshasvi Tirupachuri, Enrico Valli, Punith Reddy Vanteddu, Marco Maggiali, and Daniele Pucci.
2023. "Modeling and Calibration of Pressure-Sensing Insoles via a New Plenum-Based Chamber" *Sensors* 23, no. 9: 4501.
https://doi.org/10.3390/s23094501