## 1 Introduction
Further, this can also be used to assess tire wear [28; 29]. Unfortunately, the measurement of the longitudinal velocity and the sideslip angle requires the use of complex and relatively expensive devices. This equipment includes optical sensors, the Global Positioning System, and the inertial platform. Furthermore, these instruments may encounter challenges, such as reflection issues with optical sensors when pointed at reflective surfaces like snow-covered roads, or general failures in the acquisition system. In addition, in some contexts like motorsport, it is not even possible to use these sensors during the races; then it becomes essential the virtual estimation of both longitudinal velocity and sideslip angle. According to the methods present in the literature, two main categories of vehicle sideslip angle estimation have been identified: observer-based and neural network-based [30; 31]. For what concerns the first one, the most famous is the Kalman filter [32; 33]. In the automotive domain, a multitude of nonlinear Kalman filters have been developed, and they find significant applications, mainly due to the nonlinearity of vehicle dynamics models [34]. On the other hand, the main obstacle with this approach lies in finding a balance between the complexity of the vehicle model, the quality of results, accuracy, and the computational burden required to achieve real-time sideslip angle estimation. The other category refers to the estimation of the sideslip angle through the use of neural networks (NN), a paradigm of the broader machine learning techniques [35; 36; 37; 38; 39]. An artificial neural network consists of interconnected neurons communicating through layers, adjusting their structure based on training information [40]. In recent years, neural networks have gained significance in the automotive industry for estimating variables without specific measuring procedures. This approach offers cost-effective computations, easy algorithm implementation in vehicles, and accurate results, especially in complex real-time control environments. It is important to note that the output of the neural network relies heavily on the data used for training. To achieve accurate generalization, the neural network requires a large amount of reliable data. However, if the network is appropriately trained on a robust and large dataset, it can accurately predict vehicle behavior, even in strongly nonlinear conditions. Moreover, the neural network does not require physical models. On the other hand, as stated, the Kalman filter requires a dependable vehicle dynamic or kinematic model to work. Regarding the sideslip angle estimation, the first type relies on tire models, which can become inaccurate in strongly nonlinear conditions, while the latter type in the Kalman filter tends to become unobservable for low yaw rate values and provide uncertain estimates [24; 41]. For this reason, starting from previous work of the authors, in which is presented a novel methodology for estimating longitudinal vehicle velocity, this paper extends the approach to evaluate the sideslip angle and integrates the estimates of these two quantities [42]. In more detail, the proposed methodology introduces a dual-network architecture, that is, two distinct neural networks working in sequence: one is a dedicated neural network for estimating the longitudinal velocity, while the other is a separate neural network for predicting the sideslip angle, which also takes as input the longitudinal velocity estimate made by the previous network. In this way, after training the two networks appropriately, it is possible to create a virtual sensor that can replace the optical one. This virtual sensor estimates the longitudinal velocity of a vehicle using commonly found sensors. Then, a second neural network can be fed with this velocity and data from the same sensors to obtain an estimated sideslip angle at any given time.