# Vertical Force Monitoring of Racing Tires: A Novel Deep Neural Network-Based Estimation Method
## 2 Related Works
### Observer and Filter-Based Estimation of Vertical Force  
This subsection focuses on previous notable contributions in the field of observer and filter-based estimation techniques, particularly in the context of vertical tire force estimation. The research of [5] employs estimation modules to calculate/estimate tire forces using nonlinear observers. These observers utilize measured longitudinal acceleration, measured lateral acceleration, and measured vertical acceleration to estimate the vertical force. Additionally, an observer block is developed to compute the vertical tire forces based on the acceleration of the vehicle's sprung mass in longitudinal, lateral, and vertical directions, as well as rotation around the roll and pitch axes.In this study, the vertical force estimation model relies on several vehicle geometric parameters, including the height of the center of gravity (\(H_{\text{COG}}\)), the height of the roll center (\(H_{\text{RC}}\)), the longitudinal distances to the front and rear axles (\(L_{f}\), \(L_{r}\)), and the front and rear axle widths (\(e_{f}\), \(e_{r}\)). The model assumes constant vehicle mass and a fixed center of gravity height (\(H_{\text{COG}}\)), which may not reflect dynamic loading conditions or changes in weight distribution. Its accuracy depends on precise measurements or estimates of these parameters, as any deviation directly affects the force distribution calculations. The model requires roll (\(\phi_{v}\)) and pitch (\(\theta_{v}\)) angle estimations, which rely on additional sensor data and filtering processes. The study of [6] establishes an intelligent tire prototype based on microelectromechanical system accelerometers. The research proposes a theoretical rolling kinematics model to elucidate the mechanisms of acceleration fields resulting from the coupling effect of rigid body motion and elastic deformation. Additionally, the study formulates an analytical model for real-time estimation of vertical force.The vertical force is formulated as a function of the contact patch length and fitting coefficients, where the contact patch length is estimated using radial acceleration data. The model's fitting coefficients are derived from experimental data under specific conditions, necessitating recalibration for different scenarios. In the study by [2], the longitudinal, lateral, and vertical tire forces are individually estimated utilizing a strong tracking unscented Kalman filter and a conventional Kalman filter. These estimations utilize input parameters including yaw acceleration and lateral and longitudinal forces, while the filters make use of vehicle dynamic models.The vertical force estimation model relies on roll dynamics, requiring parameters such as roll angle (\(\phi\)) and roll acceleration (\(\bar{\phi}\)), which make the model dependent on the accuracy of the roll dynamic model. Suspension dynamics, including spring compression (\(\Delta z\)) and compression rate (\(\dot{\Delta z}\)), are incorporated, necessitating precise measurements from suspension height sensors. The model involves nonlinear equations that depend on parameters such as spring stiffness (\(k_{s}\)) and damping coefficient (\(c_{s}\)), which require accurate calibration. Additionally, the model assumes small roll angles (\(\phi\)) to apply linear approximations (\(\sin\phi\approx\phi\), \(\cos\phi\approx 1\)), potentially limiting its accuracy under conditions involving larger roll angles. The effects of unsprung mass dynamics are not explicitly included, which may influence predictions in scenarios with significant dynamic load variations. The research by [5] introduces a Tire Vertical Force Observer (TVFOB) to estimate the vertical forces acting on tires by using suspension deformation and acceleration data from both the sprung and unsprung masses. The method leverages a quarter-car model, which simplifies the suspension system dynamics for analysis and observer design.