# Nonlinear Tactile Estimation Model Based on Perceptibility of Mechanoreceptors Improves Quantitative Tactile Sensing

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Strategy of Tactile Estimation Modeling

#### 2.2. Target Samples

^{2}piano-wire sensor (Kato Tech Co., Ltd., Kyoto, Japan), as shown in Figure 2b. The total number of samples was determined to be eight so that the subjects could concentrate on evaluating all the samples in an appropriate amount of time.

#### 2.3. Sensory Evaluation Test

#### 2.4. Vibration Measurement System and Procedure

^{TM}184 Silicone Elastomer Base, The Dow Chemical Company, Midland, TX, USA) with the hardener (SYLGARD

^{TM}184 Silicone Elastomer Curing Agent, The Dow Chemical Company, Midland, TX, USA) at 2.5% of the blended amount and a polymerization reaction at 80 °C occurred in air for more than 30 min using a high-temperature dryer. The hardness of the silicone rubber pad was designed to be equivalent to that of a human finger. Young’s modulus of the silicone is reported to be 6.7 MPa, according to the manufacturer. Figure S1 shows a comparison of the hardness between the forefinger pad and the developed sensor measured using a durometer TYPE OO (GS-754G, Teclock Co., Ltd., Nagano, Japan). The results of the Student t-test showed no significant differences between the two. A coating material (X-93-1755-1, Shin-Etsu Chemical Co., Ltd., Tokyo, Japan) was adhered to the surface of the silicone rubber pad. The outputs from the strain gauges were acquired using a dynamic strain amplifier (DPM-913B, Kyowa Electronic Instruments Co., Ltd., Tokyo, Japan). The relationship between the output of the strain gauge, V, and the vertical deformation of the tactile sensor, d, is shown in Figure S2. From the figure, we can obtain the transformation equation with a linear regression as:

#### 2.5. Data Processing Methods

_{FAI}, L

_{FAII}, L

_{SAI}, and L

_{SAII}are the thresholds for FA I, FA II, SA I, and SAII, respectively, and f is the frequency of the vibration stimulus. Each mechanoreceptor is supposed to fire when the intensity of the mechanical stimulus surpasses the corresponding threshold line.

_{i}, as shown in Figure 4. The subscript i represents the mechanoreceptors that are supposed to fire in the corresponding frequency range. Note that D

_{i}could be zero when the vibration data are always lower than the thresholds. Detailed formulas for calculating each feature are provided in the Supplementary Materials.

#### 2.6. Tactile Estimation Models

_{i}, and the dynamic friction coefficient, μ’, using the Python library and state models. Considering that the human tactile perception nature has nonlinearity [34,35,36], we developed four types of linear/nonlinear regression models:

_{0}+ β

_{1}x

_{i}+ β

_{2}x

_{j}+ β

_{1}x

_{i}x

_{j}

_{i}and x

_{j}are the explanatory variables, that is, D

_{i}and µ’. y is the objective variable, that is, the PC score. β

_{i}represents the coefficients to be determined. In the linear and logarithmic models, regression formulas were constructed for all combinations of variables. p means the number of variables to be entered and takes values from one to the maximum number of variables that can be entered. In the interaction model, any two explanatory variables were chosen to build the model, that is,

_{6}C

_{2}= 15 types of models were built for one objective variable. In the polynomial model, only one variable was entered into a single equation. a means the maximum number of dimensions of the input variable.

## 3. Results

#### 3.1. Sensory Evaluation Results

#### 3.2. Feature Values Extracted from Vibration

_{SAISAIIFAI}was significantly different, except between samples (1 and 2, 5, 6, 7), (2 and 3, 5, 6, 7), (3 and 5), (4 and 8), (5 and 6, 7), and (6 and 7). D

_{ALL}was significantly different, except between samples (1 and 8), (2 and 3), (4 and 5), (4 and 7), and (5 and 7). A one-way analysis of variance showed that there were significant differences (p < 0.05) between samples for D

_{SAISAIIFAI}, D

_{ALL}, D

_{SAISAIIFAII}, D

_{FAII}, and D

_{SAIIFAII}. Therefore, these five features for each sample were considered as index variables in the following regression analysis. In addition, the three types of features calculated based on the feature calculation method of a previous study [27] are shown in Figure S5.

#### 3.3. Regression Analysis

_{C1PC1}= −4136 + 5.544log(D

_{SAISAIIFAI}) − 98.20 log(D

_{SAIIFAII}) + 466.3log(D

_{FAII}) + 0.9490 log(µ)

_{C1PC2}= 1.828 D

_{SAIIFAII}− 3.5 × 10

^{−4}D

^{2}

_{SAIIFAII}+ 1.69 × 10

^{−8}D

^{3}

_{SAIIFAII}

_{C2PC1}= −16.88 + 2.856 log(D

_{SAISAIIFAI}) + 1.114 log(µ)

_{C2PC2}= −94.98 + 4.824 × 10

^{−3}D

_{SAISAIIFAII}− 4.270 × 10

^{−3}D

_{ALL}+ 8.772 × 10

^{−3}D

_{SAIIFAII}

_{C2PC3}= 4911 − 2.237 D

_{SAISAIIFAII}− 0.4273 D

_{SAIIFAII}+ 2.151 × 10

^{−4}D

_{SAISAIIFAI}D

_{SAIIFAII}

_{CiPCj}is the principal component score of the jth PC of Cluster i. The coefficient of determination R

^{2}, the adjusted coefficient of determination R’

^{2}, and the p-values of each regression equation are shown in Table 4.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

_{SAI}, (b) I

_{FAII}, (c) I

_{FAI}(mean ± SD, n = 11, NS: no significant difference at 5% significance probability).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Abbreviation | Definition |

PCA | principal component analysis |

PCs | principal components |

FFT | fast Fourier-transformation |

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**Figure 2.**Information of plastic plates. The plates were as follows: #1, polystyrene; #2, unknown; #3, polypropylene; #4, polyethylene; #5, polycarbonate; #6, polymethylmethacrylate; #7, unknown; #8, polyethylene. (

**a**) The enlarged views of test samples (scale bar: 5 mm). (

**b**) Dynamic friction coefficient, µ’ (mean ± SD, n = 10).

**Figure 3.**Tactile sensing system with developed tactile sensor. (

**a**) Actual image of the tactile sensor. (

**b**) Schematic diagram of the tactile sensor structure. (

**c**) Overall view of the sensing system.

**Figure 6.**Results of feature calculation for the eight samples. (

**a**) D

_{SAI}, (

**b**) D

_{SAISAIIFAI}, (

**c**) D

_{SAISAIIFAII}, (

**d**) D

_{ALL}, (

**e**) D

_{SAIIFAII}, (

**f**) D

_{FAII}, (

**g**) D

_{SAISAII}, (

**h**) D

_{SAIFAII}(mean ± SD, n = 11, NS: no significant difference at 5% significance probability, *: p < 0.05, **: p < 0.01, †: significant differences are noted in the text).

**Figure 7.**Relationship between the actual value and the predicted value. (

**a**) Cluster 1, (

**b**) Cluster 2.

**Table 1.**Classification of the regression models based upon the feature extraction method and model type.

Model Type | Feature Extraction Method | |
---|---|---|

Previously Reported Method [27] | Proposed Method | |

Linear | A-1 | B-1 |

Logarithmic | A-2 | B-2 |

Interaction | A-3 | B-3 |

Polynomial | A-4 | B-4 |

**Table 2.**The result of principal component analysis. (Bold letters indicate evaluation words with an absolute value of PC loadings of 0.5 or higher).

Evaluation Word | Principal Component Load | ||||
---|---|---|---|---|---|

Cluster 1 | Cluster 2 | ||||

PC1 | PC2 | PC1 | PC2 | PC3 | |

Smooth | −0.933 | −0.055 | −0.646 | 0.186 | −0.517 |

Sticky | 0.913 | 0.136 | 0.695 | 0.300 | −0.257 |

Pasty | 0.872 | 0.120 | 0.724 | 0.385 | 0.088 |

Feel friction-drag | 0.877 | 0.000 | 0.741 | 0.220 | 0.099 |

Moisten | 0.840 | 0.236 | 0.466 | 0.371 | 0.276 |

Sleek | −0.845 | 0.196 | −0.617 | 0.356 | −0.033 |

Slippery | −0.561 | 0.427 | −0.200 | 0.725 | 0.075 |

Velvety | −0.215 | 0.836 | −0.603 | 0.319 | 0.318 |

Fine | −0.048 | 0.810 | −0.452 | 0.356 | 0.507 |

Rough | −0.188 | −0.772 | −0.011 | −0.673 | 0.461 |

Eigenvalue | 5.50 | 2.26 | 3.18 | 1.79 | 1.00 |

Contribution rates (%) | 50.4 | 22.8 | 26.3 | 18.8 | 14.6 |

Cumulative contribution rates (%) | 50.4 | 73.2 | 26.3 | 45.1 | 59.7 |

**Table 3.**The average error of each regression model. (Bold letters indicate the model with the lowest error for each PC).

Cluster | Principal Component | Model | |||||||
---|---|---|---|---|---|---|---|---|---|

A-1 | A-2 | A-3 | A-4 | B-1 | B-2 | B-3 | B-4 | ||

Cluster 1 | PC1 | 0.134 | 0.115 | 0.876 | 0.138 | 0.052 | 0.018 | 0.061 | 0.876 |

PC2 | 0.539 | 0.506 | 1.133 | 0.795 | 0.545 | 0.535 | 0.451 | 0.338 | |

Cluster 2 | PC1 | 0.328 | 0.231 | 0.211 | 0.426 | 0.227 | 0.209 | 0.307 | 0.542 |

PC2 | 0.268 | 0.325 | 0.303 | 0.733 | 0.046 | 0.048 | 0.138 | 0.321 | |

PC3 | 0.418 | 0.416 | 0.688 | 0.360 | 0.386 | 0.385 | 0.337 | 0.441 |

Cluster | Principal Component | Equation | R^{2} | R’^{2} | p |
---|---|---|---|---|---|

Cluster 1 | PC1 | (10) | 0.995 | 0.986 | 0.000854 |

PC2 | (11) | 0.458 | 0.241 | 0.217 | |

Cluster 2 | PC1 | (12) | 0.837 | 0.772 | 0.0107 |

PC2 | (13) | 0.935 | 0.887 | 0.0077 | |

PC3 | (14) | 0.308 | −0.211 | 0.651 |

Objective Variable | Explanatory Variable | $\beta \prime $ | p |
---|---|---|---|

y_{C1PC1} | ${\mathrm{log}(D}_{\mathrm{SAISAIIFAI}})$ | 0.797 | 0.004 |

${\mathrm{log}(D}_{\mathrm{SAIIFAII}})$ | −0.426 | 0.011 | |

${\mathrm{log}(D}_{\mathrm{FAII}})$ | 0.469 | 0.018 | |

$\mathrm{log}\left(\mu \right)$ | 0.438 | 0.013 | |

y_{C2PC1} | ${\mathrm{log}(D}_{\mathrm{SAISAIIFAI}})$ | 0.515 | 0.039 |

$\mathrm{log}\left(\mu \right)$ | 0.645 | 0.018 | |

y_{C2PC2} | ${\mathrm{log}(D}_{\mathrm{SAISAIIFAII}})$ | 0.357 | 0.072 |

${D}_{\mathrm{ALL}}$ | −0.787 | 0.005 | |

${D}_{\mathrm{SAIIFAII}}$ | 0.798 | 0.005 |

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**MDPI and ACS Style**

Sagara, M.; Nobuyama, L.; Takemura, K.
Nonlinear Tactile Estimation Model Based on Perceptibility of Mechanoreceptors Improves Quantitative Tactile Sensing. *Sensors* **2022**, *22*, 6697.
https://doi.org/10.3390/s22176697

**AMA Style**

Sagara M, Nobuyama L, Takemura K.
Nonlinear Tactile Estimation Model Based on Perceptibility of Mechanoreceptors Improves Quantitative Tactile Sensing. *Sensors*. 2022; 22(17):6697.
https://doi.org/10.3390/s22176697

**Chicago/Turabian Style**

Sagara, Momoko, Lisako Nobuyama, and Kenjiro Takemura.
2022. "Nonlinear Tactile Estimation Model Based on Perceptibility of Mechanoreceptors Improves Quantitative Tactile Sensing" *Sensors* 22, no. 17: 6697.
https://doi.org/10.3390/s22176697