# 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 vertical force acting on a single wheel is estimated using the following equation:  
\[F_{z}=k_{s}\Delta z+c_{s}\dot{\Delta z}+F_{\text{cam}}(\Delta z,\dot{\Delta z })-m_{u}g, \tag{10}\]  
where:  
* \(F_{z}\): vertical force acting on the wheel;
* \(k_{s}\): suspension spring stiffness;
* \(c_{s}\): suspension damping coefficient;
* \(\Delta z\): suspension compression;
* \(\Delta z\): suspension compression velocity;
* \(F_{\text{cam}}\): force due to camber effects;
* \(m_{u}\): unsprung mass;
* \(g\): gravitational acceleration. The accuracy of the method depends on the assumption of constant suspension parameters, the precision of suspension measurements, and the simplification of unsprung mass dynamics. The research by [24] estimates the vertical forces acting on tires based on suspension deformation and acceleration data. The vertical force is determined through the relationship between suspension stiffness, damping, and tire deformation, as represented by the dynamic equations:  
\[M_{sp}\ddot{z}_{sp}=-K_{s\text{iss}}(z_{sp}-z_{un})-C_{sus}(\dot{z}_{sp}- \dot{z}_{un})+T_{m} \tag{11}\]  
\[M_{un}\ddot{z}_{un}=K_{s\text{iss}}(z_{sp}-z_{un})+C_{sus}(\dot{z}_{sp}-\dot{ z}_{un})-T_{m} \tag{12}\]  
\[-K_{tire}(z_{un}-z_{g})-C_{tire}(\dot{z}_{un}-\dot{z}_{g}) \tag{13}\]  
Here:  
* \(M_{sp}\), \(M_{un}\): sprung and unsprung masses;
* \(K_{s\text{iss}}\), \(C_{s\text{iss}}\): suspension stiffness and damping coefficients;
* \(K_{tire}\), \(C_{tire}\): tire stiffness and damping coefficients;
* \(z_{sp}\), \(z_{un}\): vertical displacements of the sprung and unsprung masses;* \(\dot{z}_{sp},\dot{z}_{um}\): velocities of the sprung and unsprung masses;
* \(T_{m}\): actuator force (if active suspension is used);
* \(z_{g},\dot{z}_{g}\): vertical displacement and velocity of the road surface. The accuracy of the method heavily depends on precise measurements of suspension deformation and sprung mass acceleration, which can be influenced by sensor noise or calibration errors. The research by [7] estimates vertical tire forces in multi-axle trucks by combining static forces with dynamic components, where the dynamic contributions account for lateral accelerations, roll angle, and suspension characteristics. The formulation relies on accurate measurements of suspension deflections. The method assumes that static vertical forces are pre-calculated and combined with dynamic forces, while neglecting small second-order effects. Recalibration is required under varying operating conditions, such as fuel consumption, refueling, pilot weight changes, and suspension movements, which cause shifts in the vehicle's center of gravity position and rotational inertia, introducing challenges for force estimation accuracy. In [10], the approach assumes that suspension dynamics remain linear and neglects higher-order effects. Additionally, the accuracy of force estimation depends on reliable measurements of accelerations and roll dynamics, as well as precise parameterization of roll stiffness, roll damping, and geometric vehicle properties. In [11], the vertical tire forces are estimated using a half-car model, where the vertical tire forces are included as states in the system state vector. The method calculates the forces based on the relationship between tire stiffness and the relative displacement between the tire mass and the ground, expressed as follows:  
\[Y=a_{x,m}-\frac{a_{y,m}L_{r}}{L}\tan(\delta), \tag{5}\]0  
Here:  
* \(F_{f},F_{r}\): vertical forces on the front and rear tires;
* \(k_{tf},k_{tr}\): tire stiffness for front and rear tires;
* \(u_{f},u_{r}\): vertical displacements of the front and rear tires;
* \(h_{f},h_{r}\): road surface roughness at the front and rear tires;
* \(y_{f},y_{r}\): road deflections due to vehicle-bridge interaction (if applicable). The method uses vehicle body vertical acceleration and angular velocity as measurable inputs. The system relies on accurate parameterization of the half-car model, particularly the tire stiffness, suspension parameters, and vehicle geometry.