# Vertical Force Monitoring of Racing Tires: A Novel Deep Neural Network-Based Estimation Method
## 2 Related Works
### Analysis of Methodological Constraints in Related Works
#### 2.3.3 Vehicle Model-Based Estimation Approaches
The approach assumes that the vehicle dynamics are adequately represented by a linear half-car model, and higher-order effects such as nonlinear suspension behaviour are neglected. Additionally, the accuracy of the force estimation depends heavily on precise sensor measurements and the proper calibration of vehicle parameters such as tire stiffness and suspension damping. In [12], the vertical force estimation method is modeled based on the roll and pitch motions of the suspension system. The vertical force is expressed as a combination of the static load distribution and the contributions of roll torque (\(\tau_{\phi}\)) and pitch torque (\(\tau_{\theta}\)) as follows:  
\[Y=a_{x,m}-\frac{a_{y,m}L_{r}}{L}\tan(\delta), \tag{5}\]1  
The applicability of this method is subject to several constraints. First, the accuracy of the roll and pitch angles, along with their derivatives, heavily depends on sensor measurements. Noise and inaccuracies in these measurements can significantly affect the calculation of torques, thereby reducing the precision of the estimated vertical forces. Since torques cannot be directly measured in practice, they are derived from motion data using suspension parameters, such as stiffness and damping coefficients. These factors collectively introduce limitations to the method, particularly in terms of sensor reliability, parameter accuracy, and dependency on dynamic conditions. In [13], the vertical force estimation is modeled based on weight transfer, considering vehicle accelerations and the position of the center of gravity. The vertical force acting on the tires is expressed as follows:  
\[Y=a_{x,m}-\frac{a_{y,m}L_{r}}{L}\tan(\delta), \tag{5}\]2  
Here, \(F_{z0}\) represents the static load distribution, \(a_{x}\) and \(a_{y}\) are the longitudinal and lateral accelerations, \(h\) is the height of the center of gravity, \(WB\) is the wheelbase, and \(K_{\phi_{j}}\) denotes the roll stiffness. While the model offers a simplified and computationally efficient solution, it is subject to several limitations. The omission of suspension dynamics and unsprung mass effects limits the accuracy, particularly during high-frequency oscillations and sudden load changes. Furthermore, as previously noted, factors such as fuel consumption and refueling, vehicle modifications, and changes in driver weight--particularly in motorsport applications--can cause variations in the position of the center of gravity. However, these dynamic changes are not accounted for in the model. Consequently, while the method provides rapid estimations with low computational cost, the neglect of dynamic effects and parameter variations restricts its accuracy on estimation. In [16], the vertical forces acting on the tires are modeled based on the effects of longitudinal and lateral accelerations, as well as the static load distribution. The general equation for the vertical load is given as follows:  
\[Y=a_{x,m}-\frac{a_{y,m}L_{r}}{L}\tan(\delta), \tag{5}\]3  
where \(m_{y}\) is the vehicle mass, \(l_{r}\) and \(l\) represent the rear and total axle distances, \(h\) is the height of the center of gravity, \(a_{x}\) and \(a_{y}\) denote the longitudinal and lateral accelerations, \(e\) is the track width, and \(g\) is the gravitational acceleration. This equation captures the dynamic weight transfer caused by both longitudinal and lateral accelerations while incorporating the static load distribution. However, the model assumes a constant center of gravity height and does not account for suspension dynamics or unsprung mass effects. As previously noted, such assumptions limit the model's accuracy under conditions where factors like fuel consumption, vehicle modifications, and driver weight changes--especially in motorsport applications--significantly alter the center of gravity.