# Identification of Tire Model Parameters with Artificial Neural Networks
## 3 Methodology and Results
### Data Creation  
The creation of synthetic data, from the mathematical model provided by the MF, is carried out in Steps 1 to 4 (Figure 2). First, in Step 1, the parameters of the tire model are varied randomly and independently of each other, in a sufficiently wide range of values to cover the largest possible combination of true tire curves. For this study, the limits of the values that these parameters can take are indicated in Table 1. It is important to note that the synthetic data obtained are free from errors arising from the application in which the tires are used (from test bench or vehicle dynamic system acquisition). However, it must be stressed that mathematical errors always exist in the model (either because the model chosen, in this case [PERSON]'s model, does not perfectly match reality, or because of mathematical calculation errors, rounding or truncation).  
In Step 2, the mathematical curves are obtained. It can be noted that each set of random parameters produces a very different mathematical curve \(F_{i}\). For example, in Figure 3, some different curves are shown from Equation (1) for different sets of random parameters.  
It is important to remark that the range of the parameters has been selected to be sufficiently broad, so that the MF curve can have, mathematically, very different shapes.  
In Step 3, the curves are discretized in 2% slip steps in the \(x_{i}\) slip range (e.g., Figure 4). These data points are saved in the training database together with the original set of MF parameters that produced the original curve. This information will be used later in the supervised training of the ANN.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Minimum** & **Maximum** \\ \hline \(x_{i}\) & \(-0.75\) & \(+0.75\) \\ \hline \(B\) & 5 & 30 \\ \hline \(C\) & 0.5 & 2 \\ \hline \(E\) & \(-2\) & 0 \\ \hline \(Fz\) & 1000 & 5000 \\ \hline \(D\) &  
It is important to note that the curves generated from the MF are mathematically perfect. That is, as they have been generated mathematically, there is no error in the parameters, the slip or the longitudinal/lateral force. This is very convenient at this point to be able to evaluate the ability of the ANN to represent a sufficiently complicated mathematical model without including other uncertainties or error variables.  
One of the objectives of the ANN training is that the network learns to generalize the relationships between the parameters and the actual curve. This can be achieved if the training data is big enough. Step 1 to step 3 are repeated until sufficient data is gathered to train the ANN.  
There are no fixed rules to define how many data sets are necessary to train an ANN successfully. This number depends on the complexity and non-linearity of the model under study. In order to be able to determine how big the data size needs to be, during the works in this investigation, the ANN has been trained with sets of a different number of curves and the training error was measured. Figure 5 shows the root-mean-square deviation (RMS Error) at the increasing number of curves in the training set and at a different number of training iterations (epochs). The trend shows that the higher the number of data sets the smaller the training error is. It can also be seen that the error is quite similar at 25k, 50k or 100k curves (with an RMS Error around 0.010 at 8000 epochs). In this study, to make sure a good training is achieved, a data set of 100k curves was selected (Step 4).  
Figure 4: Discretization of MF curves.  
Figure 5: RMS error at different Training Data Sizes.