# Vertical Force Monitoring of Racing Tires: A Novel Deep Neural Network-Based Estimation Method
## 3 Random Forest Assisted Deep Neural Network Model  
This section introduces the proposed Random Forest-Assisted Deep Learning Model, specifically designed to enhance the regression performance of Random Forest by addressing its limitations in handling sequential data and complex nonlinear relationships.In general, traditional vertical force estimation methods in the literature, particularly those relying on vehicle and tire models, suffer from parametric uncertainties and inadequacies in modeling nonlinear behaviors. These approaches often depend on onboard sensors which are susceptible to noise and vibrations from the vehicle chassis, limiting their accuracy. Our study addresses these shortcomings by employing a hybrid method based on deep neural networks (DNN), which utilizes commonly available vehicle sensors, such as linear velocity and inner tire pressure sensors, as well as a single-channel vertical acceleration sensor embedded in the tire. Unlike [1], which uses acceleration data to calculate peak radial displacement as an intermediate step, our method directly maps vertical acceleration data to vertical force, eliminating the need for additional calculations and reducing complexity. The method integrates Random Forest with deep learning components, including LSTM, MGU, 1D CNN, and Multihead Attention, to achieve improved prediction accuracy for vertical tire force estimation. Utilizing notations \(F_{i}\) to represent the vertical force exerted on the tire at any moment \(t\), and \(P_{j}\) to denote the internal pressure within the tire, we define the tire's rotational velocity as \(S_{k}\). Consequently, the ensemble of potential states attainable by the tire at any given instance is articulated as follows:  
\[S=\{(F_{1},P_{1},S_{1}),\ldots,(F_{i},P_{j},S_{k}),\ldots,(F_{10},P_{2},S_{3})\} \tag{19}\]  
Table 1 presents the indices \(i\), \(j\), and \(k\) used in Equation (19), along with their physical meanings and the measured values from the experimental setup. Measurements were limited to 32 specific cases out of the 60 possible \(F_{i}\), \(P_{j}\), \(S_{k}\) combinations. \begin{table}
\begin{tabular}{c l l} \hline
**Indices/Components** & **Physical Meaning** & **Set of Measured Values** \\ \hline \(F_{i}\) & Vertical load applied to the tire (in kgf) & 150, 180, 210, 240, 270, 300, 330, 360, 390, 420 \\ \hline \(P_{j}\) & Internal pressure of the tire (in kPa) & 200, 220 \\ \hline \(S_{k}\) & Rotational speed of the tire (in km/h) & 30, 50, 70 \\ \hline \(i\) & Index representing different vertical load levels & \(i=1,2,\ldots,10\) (corresponding to 150–420 kgf) \\ \hline \(j\) & Index representing different pressure levels & \(j=1,2\) (corresponding to 200, 220 kPa) \\ \hline \(k\) & Index representing different speed levels & \(k=1,2,3\) (corresponding to 30, 50, 70 km/h) \\ \hline \(F_{i},P_{j},S_{k}\) & Vertical force under specific load, pressure, and speed & Measured for 32 unique states selected from the 60 possible combinations of \(i,j,k\) indices \\ \hline \end{tabular}
\end{table} 
Table 1: Detailed representation of indices \(F_{i}\), \(P_{j}\), \(S_{k}\), their components, and their physical meanings with measured values. Herein, \(i\) spans from 1 to 10, \(j\) encompasses values from 1 to 2, and \(k\) extends from 1 to 3. The triad \((F_{i},P_{j},S_{k})\) represents the tire's condition within the scope of this study. It is evident that this set comprises 60 distinct elements. Throughout the data acquisition phase for both the experimental and training datasets, we collated data emblematic of a subset of \(S\), denominated as \(S_{a}\), which embodies 32 distinct states/elements. Consequently, we gathered 32 subsets symbolizing different \((F_{i},P_{j},S_{k})\) states and combined them to formulate the dataset. The dataset's unprocessed data was partitioned into segments comprising 251 data points, each subsequently annotated with a label correlating to a feasible tire state \((F_{i},P_{j},S_{k})\).