## 5 Results  
### Experimentation Results  
A set of tests were performed to validate the developed model and to look at the achieved estimations of the sideslip angle. The following section presents the results for each neural network type and for the explained approaches. In Figures 8 and 9 are shown the feedforward neural network predictions of the angle of interest for two circuits with very different layouts using the \"All Tracks\" approach. It is clear from the red curve that the estimated angle closely matches the angle acquired by the optical sensor represented by the black curve for both tracks. Therefore, after training, the feedforward network can quickly and accurately estimate the vehicle's sideslip angle. Figure 10 displays the training, validation, and test performance over epochs during the training operation. Initially, the error rapidly decreases but then stabilizes, reflecting the network's understanding of the input-target relationship learned from the training dataset. The optimal network performance is determined based on the epoch with the lowest validation error. Beyond this point, the validation error starts to rise, indicating the onset of overfitting. The network then reverts to the epoch preceding this phenomenon. In Figure 11, regression plots illustrate the relationship between predicted values and target values. The y-label in these plots represents the equation linking the predicted output (dependent variable) to the target (independent variable). The target coefficient, close to unity, indicates the proximity of the training output to the target from the dataset. The second term, ideally close to zero, represents the error or residue required to align the scaled target with the training output. The coefficient of regression in the title, ideally equal to unity, signifies the strong alignment between the target and output. These results affirm the successful execution of the training phase, achieving optimal working conditions. In summary, these plots highlight that the training phase was effectively executed, leading to a well-performing neural network model. Figure 10: FF training performance results—all tracks approach. Figure 9: FF estimated \(\beta\) for track “B”—all tracks approach. Figures 12 and 13, on the other hand, show the results of angle predictions made with NARX and RNN neural networks on Track \"A\". To analyze and draw conclusions in more detail, as mentioned, two statistical parameters have been calculated for each network forecast: the coefficient of determination and the RMSE. For the precision on Track \"A\", the prevision errors for each network architecture are summarized in Table 4.  
In order to verify the effective work efficiency of the networks, they were tested on a second run related to the same circuit (Table 5), but this time, the vehicle was equipped with worm tires, while in the first run, a new set of tires was used. In this way, it is possible to check that the neural network has not adapted to the data of the first run, and that it is therefore able to work with different datasets, always giving equally valid results. Figure 11: FF training regression results—all tracks approach. Figure 12: NARX estimated \(\beta\) for track “A”—all tracks approach. With the same aim, Table 6 presents the statistical indicator values for the estimation related to a different circuit (\"Track B\") that has a different layout. The findings reported in the graphs and tables presented shed light on the forecast accuracy of the considered neural networks. In particular, the results obtained for the vehicle sideslip angle estimate by means of three different typologies of artificial neural networks indicate that the feedforward neural network is the most accurate, followed closely by NARX. On the other hand, it is fair to say that the RNN is the least precise among all the models. As can be seen graphically, it falls short in comparison to the others.