## 3 Neural Network-Based Estimation  
The use of the so-called soft computing techniques such as artificial neural networks, genetic algorithms, and fuzzy logic is increasingly widespread in engineering applications e.g., [128] by [PERSON] et al. Only a few works regarding VSA estimation through ANN can be found in the literature. As mentioned in Section 1, this method has been precisely designed to overcome the need of a complex vehicle model and its complex set of parameters, especially those regarding tyres. Furthermore, errors due to integration of signals with noise are also avoided, since ANNs can estimate the VSA with their ability to identify complex relationships from input-output data pairs, using only simple math operations. A detailed mathematical description of ANNs is provided in [103] by [PERSON]. For the sake of clarity, Figure 3 reports a high-level schematic that explains how the ANN estimation process works. Most of the authors who have addressed this topic used the same general approach. It consists of a three-layered neural network composed by one input layer, one hidden layer, and the output layer, where the first and second layer's neurons use the log-sigmoid transfer function, while in the third layer, neurons use the linear functions. For the sake of clarity, inputs are considered those signals that can be easily measured on board any vehicles such as the yaw rate, lateral/longitudinal acceleration, vehicle speed, and steering angle, while the only output is the VSA. This kind of layered ANN with at least two layers is also able to memorise any kind of non-linear model [97,103]. Figure 4 shows the general layout of the above-mentioned ANN. A comprehensive training dataset is required to obtain a good estimation using an ANN. Except for [99] by [PERSON], which uses a radial basis function (RBF), they use the consolidated Levenberg-Marquardt back-propagation (BP) algorithm along with the mean squared error (MSE) performance index to train the ANN. This BP optimisation algorithm updates the weights and biases of the network, as presented in [129,130] by [PERSON] and [PERSON], while the RBF is a feed-forward propagation method that requires less calculation, with a faster learning process [100]. Moreover, two of the latest works propose a general regression neural network (GRNN) instead [105,106]. GRNN is a special form of a RBF network that features good learning ability and a high function approximation capability. The main difference between GRNN and RBF is a  
Figure 4: General layout of an ANN used for vehicle sideslip angle (VSA) estimation. Figure 3: Artificial neural networks (ANN) estimation process high-level schematic. special linear layer that computes the weighted output of the first layer. Particularly, [PERSON] et al. in [106] tested this ANN on many handling manoeuvres, obtaining a fast response with high estimation accuracy. However, they did not investigate the robustness of this method against varying speeds and changing conditions (adherence, for instance). However, some studies suggest another strategy to improve efficiency and reliability. In [101] [PERSON] et al. for example, a single special training manoeuvre has been suggested to improve the ANN estimation capability, while keeping its structure simple to avoid computational burden issues, as 10 neurons were used in the hidden layer. Such a manoeuvre consists in a 45\({}^{\circ}\) sine steer performed at increasing speed from 5 km/h to 100 km/h using a 15 km/h/min slope to be performed in at least three different friction conditions (i.e., \(\mu=1\), 0.5, 0.2). This method produced very good estimation results; however, it was only tested via a CarSim Simulation, as a real-world test would be quite complicated due to the extension of the proving ground required. Furthermore, [PERSON] et al.