## 2 Observer-Based Estimation
### Kalman Filter
Several works, e.g., [26; 27; 28; 29; 30; 31; 32], propose a basic application of the KF observer (typically an EKF) to a very complex seven DOF (Degrees of Freedom) vehicle model with a fully non-linear tyre model (Pacejka or Dugoff model). As previously mentioned, a compromise between vehicle model complexity, results, accuracy, and computational burden should be considered. Moreover, when using a dynamic model, getting all of the parameters required to feed a tyre model from the manufacturer is a real challenge, since such data are considered a critical company asset, and are hence confidential. Nonetheless, as discussed in Section 2.1, should the conditions change (e.g., due to tyre wear) with respect to the modelled ones, again, the performance of the estimation would be unsatisfactory. Some authors [13; 18; 31] tried to overcome the problem by using a simple single-track model with cornering stiffness adaption. Results show that VSA estimation is very close to the actual value except for some VSA peaks, where the estimated VSA values deviate from the actual values. That is, due to simplified tyre models that operate only in the linear zone (adapting the cornering stiffness to expand its usage range) but neglecting the non-linear and saturation zone behaviour. In a similar work [24], the tyre model adaption regards the entire non-linear range. By using a standard family of curves representing a generic Pacejka model dataset and an integrated tuning procedure for the EKF observer, the authors managed to develop a tyre model that was consistent with the lateral behaviour. The tuning is performed with respect to a single parameter that is easy to set. This allows the use of a simple vehicle model without detailed knowledge of the whole number of parameters related to the tyres fitted on the vehicle, while taking the non-linear saturation tyre behaviour into account. Results showed an error under 5% in every tested manoeuvre, as well as benefits in terms of computational burden, hence its suitability for real-time applications. [PERSON] et al. in [122] proposed a dual extended Kalman filter (DEKF) approach, where two EKFs are used in parallel. The first EKF is used to estimate the vehicles states, including VSA. The second EKF is used to estimate the parameters of a rational tyre model, which are then fed to the first EKF for the vehicle state estimation. [PERSON] et al. in [120] introduces a non-linear least squares tyre parameter estimator that is able to run online (but not in real-time). The EKF has two major drawbacks that consist of the high computational effort required in the definition of the Jacobian matrices, and its intrinsic linearization errors, which force the use of very small sampling times. An effective solution to reduce the computational burden while keeping a high accuracy in the VSA estimation is the use of the unscented Kalman filter (UKF) instead of the EKF [33; 34; 35; 36; 37; 38; 39]. As a matter of fact, many recent observer-based studies on VSA estimation mostly focused on UKF [33; 35; 37; 38]. The UKF observer is based on the approximation of random statistic variables using the definition of sigma points [127]. These points are then propagated through the (non-linear) system equation yielding some random variables of a non-linear stochastic description. Afterwards, such variables are approximated by a Gaussian random function, enabling the use of the standard equation for Kalman gain calculations. A great benefit is that a UKF does not require the computation of Jacobian matrices, which makes it particularly suitable for systems with high non-linearities. The UKF achieves higher accuracy if compared to the EKF, while requiring similar effort in terms of computational load. On the other hand, the system complexity increases significantly. Several additional parameters are introduced, such as the number of sigma points, their positioning, and the weights used to compute the final estimation [127]. For instance, [PERSON] et al.