# 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  
In [5], the vehicle model-based vertical force estimation method utilizes longitudinal and lateral accelerations along with geometric parameters of the vehicle. The center of gravity height (\(H_{COG}\)) is assumed to be constant in this method. However, under dynamic conditions, such as fuel consumption, pit stop refueling, vehicle modifications, and changes in the vehicle model, both the vehicle mass (\(m\)) and \(H_{COG}\) can vary. Fuel level changes during a race affect the total mass and the effective position of the center of gravity, introducing challenges for accurate force estimation. Unlike the center of gravity height, the vehicle mass (\(m\)) is not treated as constant and is estimated in real time using the Recursive Least Squares (RLS) method. The estimation process is described as follows:  
\[Y=\Gamma\Psi \tag{4}\]  
where the measured signals \(Y\), unknown parameters \(\Gamma\), and known variables \(\Psi\) are defined as follows:  
\[Y=a_{x,m}-\frac{a_{y,m}L_{r}}{L}\tan(\delta), \tag{5}\]  
\[\Gamma=\bigg{[}\frac{1}{m},\gamma\cos(\alpha)+\sin(\alpha)\bigg{]}, \tag{6}\]  
\[\Psi=\bigg{[}F_{x,f}\cos(\delta)-\frac{I_{zx}\tilde{\Psi}-F_{x,f}\sin(\delta) L}{L\tan(\delta)}+F_{x,r}-\frac{1}{2}QAv_{x}^{2}-g\bigg{]}. \tag{7}\]  
Here, \(a_{x,m}\) and \(a_{y,m}\) are the longitudinal and lateral accelerations, \(\delta\) is the steering angle, \(L\) is the wheelbase, \(L_{r}\) is the distance from the center of gravity to the rear axle, \(I_{zz}\) is the moment of inertia, \(Q\) is the aerodynamic drag coefficient, \(A\) is the frontal area of the vehicle, and \(v_{x}\) is the longitudinal velocity. The vertical force acting on a single wheel, incorporating the vehicle mass (\(m\)) and center of gravity height (\(H_{COG}\)), is given as follows:  
\[F_{zfr}=-\frac{mH_{COG}}{e_{f}}\left(a_{y,m}\cos(\phi_{v})-a_{z,m}\sin(\phi_{ v})\right) \tag{8}\]  
\[-\frac{m}{e_{f}}(a_{y,m}\sin(\phi_{v})+a_{z,m}\cos(\phi_{v}))\bigg{(}\frac{e_{ f}}{2}-H_{RC}\sin(\phi_{v})\bigg{)}. \tag{9}\]In Equation (9), \(F_{z/r}\) is the vertical force on the front-right wheel, \(e_{f}\) is the front track width, \(H_{RC}\) is the roll center height, \(a_{y,m}\) and \(a_{z,m}\) are the lateral and vertical accelerations, and \(\phi_{v}\) is the roll angle. While the real-time estimation of vehicle mass improves adaptability to changing conditions, it introduces additional computational load, which can be a limiting factor in high-speed applications such as motorsport. Furthermore, the assumption of a constant \(H_{\text{COG}}\) under dynamic conditions, including load redistribution and suspension movement, restricts the accuracy of vertical force estimation. In [2], the vertical force estimation method is based on the _Strong Tracking Unscented Kalman Filter (STUKF)_, which utilizes measurements of suspension compression \((\Delta z)\) and compression velocity \((\dot{\Delta z})\). These measurements are obtained from suspension height sensors. The suspension parameters, such as the spring stiffness \((k_{s})\) and damping coefficient \((c_{s})\), are assumed to remain constant throughout the estimation process. This assumption does not consider the effects of temperature changes, tire wear, or varying external loads, which can alter these parameters over time. Additionally, the method relies on accurate suspension height measurements, making it sensitive to sensor noise, vibrations, or calibration errors. The effects of unsprung mass are simplified, as its dynamics are not directly accounted for in the estimation model.