# Data-Driven Contact-Based Thermosensation for Enhanced Tactile Recognition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Contact-Based Thermosensation Design

#### 2.1. Design of the Measurement System Structure

#### 2.2. Discrete Transient Heat Transfer Model

_{h}is the thermal conductivity of the heater; q

_{v}is the power density of the heating unit; and ρ

_{h}and c

_{h}are the density and heat capacity of the heater, respectively.

_{0}, denoted as follows:

_{s}, ρ

_{s}and c

_{s}represent the thermal conductivity, density, and heat capacity of the sensing layer, respectively. N

_{1}indicates the number of discrete units at the boundary beneath the sensing layer. For the boundary unit between the measured material and the sensing layer, the discrete equation was formulated as follows:

_{o}, ρ

_{o}and c

_{o}represent the thermal conductivity, density, and heat capacity of the measured material. Subsequently, in accordance with the principle of energy conservation, the internal unit equation for the measured object was expressed as follows:

_{s}and α

_{o}denote the thermal diffusion of the sensing layer and the tested material, respectively. The equation system was presented in an explicit differential format, allowing the computation of the temperature at each node for the moment directly subsequent from the initial temperature, thus eliminating the need for solving coupled equations. Programming the model in MATLAB facilitated the calculation of physical quantities.

#### 2.3. Finite Element Simulation

^{7}W∙m

^{−3}. Furthermore, the mesh was partitioned into highly refined triangular elements.

^{2}for heat flux, affirming the model’s effectiveness.

## 3. Data-Driven Algorithm

#### 3.1. BP Neural Network

_{i}, and the expected output Y, was expressed as follows:

_{ij}and w

_{ij}

^{n}represents the neuron’s current and updated weights, respectively, while Δw

_{ij}denotes the change value, with its updated formula being as follows:

#### 3.2. Data Set and NN Training

_{v}) was inputted to calculate the material’s boundary temperature (T

_{f}) and heat flux (q

_{f}) over a given time period, expressed as follows:

_{o}and α

_{o}are the thermal conductivity and thermal diffusivity of the tested material, and f

_{h}denotes the theoretical heat transfer model. In this study, 67 different types of metals and non-metals were selected [22,31,32], with thermal conductivity ranging from 0.06 to 405.5 W∙m

^{−1}∙K

^{−1}, covering the parameter range of common materials, as shown in Figure 5. The thermal properties of these 67 standard materials were inputted into the theoretical model program f

_{h}for calculation. The study focused on transient heat transfer over a period of 1.95 s, with a set power density range of 4.0 × 10

^{7}W∙m

^{−3}to 1.4 × 10

^{8}W∙m

^{−3}, resulting in 1407 sets of time series data.

- Feature extraction. To fully describe the characteristics of heat flux and temperature signals, the linear fitting slope of the heat flux relative to its initial value and the linear fitting slope of the temperature series were calculated, denoted as u
_{1}and u_{2}, respectively. The average heat flux and average temperature were calculated as u_{3}and u_{4}. The final time’s excess temperature; the midpoint’s excess temperature; the temperature difference between the midpoint and final time; and the difference in heat flux are also calculated, respectively noted as u_{5}, u_{6}, u_{7}, and u_{8}. - Normalization. The dataset covered materials ranging from low to high thermal conductivity, with corresponding heat flux and temperature data showing significant variations. Therefore, the data features were first natural log-transformed, then normalized and denoted as X = norm(ln(u)), resulting in X = [x
_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}]. - Principal component analysis (PCA). PCA is a data analysis technique that can retain as much of the original features as possible while reducing data dimensions [33,34]. By processing data with PCA dimensionality reduction, the principal components obtained were denoted as p
_{1}to p_{8}. The contribution rates of p_{1}and p_{2}exceeded 95%, indicating that p_{1}and p_{2}can explain over 95% of the variance in the original data, thus effectively representing the original feature. The relationship between p_{1}, p_{2}, and the original features is as follows:$$\{\begin{array}{l}{p}_{1}=-0.1961{x}_{1}+0.4242{x}_{2}-0.1974{x}_{3}+0.4242{x}_{4}+0.4244{x}_{5}\\ \hspace{1em}\hspace{1em}+0.4240{x}_{6}+0.4247{x}_{7}-0.1499{x}_{8}\\ {p}_{2}=0.5393{x}_{1}+0.1403{x}_{2}+0.5384{x}_{3}+0.1408{x}_{4}+0.1395{x}_{5}\\ \hspace{1em}\hspace{1em}+0.1417{x}_{6}+0.1378{x}_{7}+0.5669{x}_{8}\end{array}$$ - With p
_{1}and p_{2}as inputs and the thermal conductivity k_{o}as output, a double hidden layer nonlinear mapping network was trained using a BP neural network. In this network, the number of neurons in the input layer was 2, the first hidden layer contained 100 neurons, the second hidden layer contained 20 neurons, and the output layer contained one neuron. The tansig function was used as the activation function for the hidden layers.

#### 3.3. BP NN with Heat Transfer Model

_{q}represents the transformed heat transfer model with heat flux as input, q

_{f}denotes the heat flux, and T

_{f}is the surface temperature of the object. Here, q

_{f}and T

_{f}are measured by sensors, k

_{o}is predicted by the BP NN model, and the function contains only one unknown variable. Thus, by scanning α

_{o}within a certain range, calculating the temperature and solving for the error with the actual measured temperature, the α

_{o}corresponding to the minimum error is searched as the result for the thermal diffusivity.

## 4. Experiment

#### 4.1. Samples of Tested Materials

#### 4.2. Experimental Measurement System

^{−6}V/(W∙m

^{−2}), and had a measurement range of (−10~10) × 10

^{3}W∙m

^{−2}.

## 5. Results and Discussion

_{o}scan range from 1 × 10

^{−7}to 1 × 10

^{−6}m

^{2}∙s

^{−1}with a step size of 0.1 × 10

^{−7}m

^{2}∙s

^{−1}. The algorithm’s calculated optimal curves and thermal properties are shown in Figure 10. The reference values were: k

_{o}= 1.097 W∙m

^{−1}∙K

^{−1}and α

_{o}= 5.52 × 10

^{−7}m

^{2}∙s

^{−1}as shown in Table 2, with the calculated results having a relative error of less than 10%.

_{o}scanning range of 1 × 10

^{−8}to 5 × 10

^{−7}m

^{2}∙s

^{−1}, and a step size of 0.05 × 10

^{−7}m

^{2}∙s

^{−1}. The calculations yielded thermal conductivity and diffusivity, with algorithm-fitting curves presented in Figure 12. These values correspond to PMMA’s reference thermophysical properties: k

_{o}= 0.185 W∙m

^{−1}∙K

^{−1}and α

_{o}= 1.10 × 10

^{−7}m

^{2}∙s

^{−1}.

_{o}scanning range was set between 1 × 10

^{−5}and 1 × 10

^{−4}m

^{2}∙s

^{−1}, with a step increment of 0.1 × 10

^{−5}m

^{2}∙s

^{−1}. Reference values for the Al alloy are as follows: k

_{o}= 120.4 W∙m

^{−1}∙K

^{−1}and α

_{o}= 5.2 × 10

^{−5}m

^{2}∙s

^{−1}.

_{o}) and diffusion coefficient (α

_{o}) across various materials, typically with less than 15% error under different heat power conditions. The algorithm simplifies the process by measuring only the surface excess temperature, eliminating the need for absolute temperature measurement and temperature compensation. This approach innovates in contact-based quantitative thermosensation.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**Experimental measurements on tempered glass (

**a**) heat flux signal and (

**b**) excess temperature.

Object | Thermal Conductivity/W∙m^{−1}∙k^{−1} | Density/kg/m^{3} | Thermal Capacity /J/kg/K |
---|---|---|---|

Heating layer and sensing layer | 0.214 | 1951.6 | 1064.6 |

Material I | 0.1 | 500.2 | 2400 |

Material II | 1.5 | 2659.6 | 800 |

Material III | 12 | 7860 | 477.1 |

Materials | Thermal Diffusivity /mm ^{2}∙s^{−1} | Standard Deviation | Heat Capacity /J∙g ^{−1}∙K^{−1} | Standard Deviation | Density /kg∙m ^{−3} | Standard Deviation |
---|---|---|---|---|---|---|

Tempered Glass | 0.552 | 0.00350 | 0.809 | 0.0012 | 2458.4 | 5.798 |

PMMA | 0.110 | 8.17 × 10^{−4} | 1.429 | 0.0029 | 1178.35 | 1.344 |

Aluminum Alloy | 51.9 | 0.22 | 0.873 | 6.03 × 10^{−4} | 2651.5 | 17.82 |

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**MDPI and ACS Style**

Ma, T.; Zhang, M.
Data-Driven Contact-Based Thermosensation for Enhanced Tactile Recognition. *Sensors* **2024**, *24*, 369.
https://doi.org/10.3390/s24020369

**AMA Style**

Ma T, Zhang M.
Data-Driven Contact-Based Thermosensation for Enhanced Tactile Recognition. *Sensors*. 2024; 24(2):369.
https://doi.org/10.3390/s24020369

**Chicago/Turabian Style**

Ma, Tiancheng, and Min Zhang.
2024. "Data-Driven Contact-Based Thermosensation for Enhanced Tactile Recognition" *Sensors* 24, no. 2: 369.
https://doi.org/10.3390/s24020369