## 5 Intelligent Tire Force Estimation Method
### RBF Neural Network  
Tires represent complex nonlinear systems, making it challenging to directly establish functional relationships between sensor measurements and tire forces. The RBF neural network is an efficient feedforward architecture that uses radial basis functions as activation functions, featuring a simple structure and fast convergence. It can accurately approximate arbitrary nonlinear functions by transforming input vectors from a low-dimensional, nonlinearly inseparable space into a higher-dimensional, linearly separable space. This enables local nonlinear mappings to be represented as linearly adjustable parameters, effectively overcoming the limitations of traditional tire force estimation methods in terms of nonlinear fitting and tire force decoupling [29]. In this study, the dimensionality of the input features (derived from tire force-sensitive response signal curves) is relatively manageable, and the training dataset is of moderate size. Compared to deep neural networks, the RBF neural network offers notable advantages in terms of parameter tuning, training speed, and computational efficiency, making it more suitable for real-time deployment and validation in vehicle control systems or embedded hardware platforms. The activation function of the RBF neural network is defined as follows:  
\[g_{i}(p)=\exp\left(-\frac{\|p-c_{i}\|^{2}}{2\sigma_{i}^{2}}\right) \tag{8}\]  
Figure 12: (**a**) Lateral displacement response signal in the left sidewall area. (**b**) Characteristic values of the lateral displacement response signal in the left sidewall area. where \(p\) represents the input layer variable, \(c_{i}\) represents the center of the hidden layer neuron, and \(\sigma_{i}\) is the spread parameter that controls the radial influence range of the basis function. The output function of the RBF neural network is defined as follows:  
\[o_{j}(p)=\sum_{i=1}^{n}w_{ij}g_{i}(p)=\sum_{i=1}^{n}w_{ij}\exp\left(-\frac{ \left\|p-c_{i}\right\|^{2}}{2\sigma_{i}^{2}}\right),\ j=1,\cdots,r \tag{9}\]  
where \(n\) represents the number of hidden layer neurons, \(w_{ij}\) represents the weight connecting the \(i\)-th hidden neuron to the \(j\)-th output neuron, and \(r\) represents the number of output neurons. To satisfy the training and testing requirements of the tire force estimation model, the training and testing samples for the RBF neural network are generated from 100 sets of finite element simulation data under pure longitudinal slip and pure slip angle conditions. The tire is set with a rated inflation pressure of 0.24 MPa, a vertical load of 4821 N, a road adhesion coefficient of 0.8, a rolling speed of 70 km/h, a maximum slip ratio of 25%, and a maximum slip angle of 15\({}^{\circ}\). The network used mean squared error (MSE) as the objective function, with a spread parameter of 0.85, an error convergence threshold of \(1\times 10^{-5}\), and 1000 training iterations. The dataset is split into training and testing sets in an 8:2 ratio, resulting in 80 training samples and 20 testing samples. The input to the RBF neural network consists of the characteristic values extracted from the response signal curves of sensitive response variables in the tire force-sensitive response areas. For each complete tire rotation, 68 characteristic values are generated for each sensitive response variable. Therefore, the RBF neural network training input forms a 68 \(\times\) 80 matrix, and the corresponding output is a 1 \(\times\) 80 matrix. The longitudinal and lateral tire forces are estimated using separate RBF neural networks, as shown in Figure 13. The finite element simulation results are used as reference values for tire forces and are compared against the RBF neural network estimations. In the crown and the left/right sidewall areas of the tire, the sensitive response signals for longitudinal and lateral forces include distinct acceleration and displacement components.