## 4 Method  
### Neural Network Approach to the Sensor Mat  
A neural network is a computational model inspired by the structure and function of the human nervous system. It can be used to recognize complex patterns and to carry out relationships in data. It is used in many fields, for example, natural language processing, computer vision, nuclear fusion, and many other applications. The artificial neural network (ANN) consists of node collections, known as \"artificial neurons\", that are organized into blocks called layers and connected to each other by weights. Each artificial neuron receives inputs from other neurons and produces an output as an adopted feed-forward algorithm, which is then assigned to the next neurons. In particular, an appropriate mathematical function, known as an activation function, maps inputs to the corresponding outputs.  
Some activation functions help to capture strong non-linearity in the data, and for this reason, the neural networks have been explored for the experimental data from mat sensors collected in the laboratory.  
In order to define a relationship between the data, the optimal weights and biases have to be set to minimize the error between the predicted and true outputs. Technically, this method is based on a backward propagation algorithm. To achieve these, it is important to use an appropriate loss function during the training process, which is an objective function to be optimized. The smaller the objective function, the better the designed network is trained. The network is trained on an initial subset of data (training data set), and the remaining test data set is used to validate the network.  
The neural network model is developed to establish the mathematical formula for the Velostat sensitive mat; for instance, a neural network's relationship between input \(x\) and output \(y\) is:  
\[y(x)=\sigma_{n}(W_{n}\sigma_{n-1}(W_{n-1}\ldots\sigma_{0}(W_{0}x+b_{0})\cdots +b_{n-1})+b_{n})\]  
where \(\sigma_{i}\) is the function as activation function, \(x\) is the input, and \(W_{i}\) and \(b_{i}\) are weights and biases with \(i=0,\ldots n-1\).  
In our case, the different geometries of mats, the pressure and voltages applied on the mat, and the number of sensing cells are considered for the approximation method based on the neural network model.  
The supply voltages, which varied in the range 5 V, 3.3 V, and 7.5 V, were measured by the RS PRO Power Supply during the laboratory tests. In the mat sensor, the active cells H14, N14, H10, L7, and G11 are subjected to the pressure applied by the FB200 200 N force gauge. The resistances \(R\) are obtained by formula \(R=V/I\), where \(V\) is the voltage measured by the RS PRO Power Supply, and \(I\) is the current measured by the Rigol DM3058E Digital Multimeter.  
The neural network provides as additional input the position of the active cell within the matrix mat identified by grid lines and coordinates, the pressure exerted on the cell,and the geometry of the mat Velostat. In this case, two types of sensor mat geometries, presented in Figure 1a,b, were considered, with the technical characteristics given in Table 3. The positions of the active cell are expressed as matrix elements arranged in specific rows and columns.  
To increase the efficiency of the learning algorithm of the neural network model based on the relation between the pressure applied on the mat and the resistance of the Velostat material, the information about the rows and column position cells of the mat matrix are requested as input. The neural network provides a relevant configuration model suitable for both geometries of mats with a high generalization performance, as shown in Figure 13.  
Thus, all the inputs mentioned above such as power voltages, position mat matrix, pressure, and type of sensor are included in the array x. Finally, y is the resistance predicted and the output of the neural network defined as the target.