Modeling of the tire-road friction using neural networks including quantification of the prediction uncertainty

Despite the great success of neural networks (NN) in many application areas, it is still not obvious how to integrate an NN in a sensor fusion framework. The reason is that the computation of the for fusion required variance of NN is still a rather immature area. Here, we apply a methodology from system identification where uncertainty of the parameters in the NN are first estimated in the training phase, and then this uncertainty is propagated to the output in the prediction phase. This local approach is based on linearization, and it implicitly assumes a good signal-to-noise ratio and persistency of excitation. We illustrate the proposed method on a fundamental problem in advanced driver assistance systems (ADAS), namely to estimate the tire-road friction. This is a single input single output static nonlinear relation that is simple enough to provide insight and it enables comparisons with other parametric approaches. We compare both to existing methods for how to assess uncertainty in NN and standard methods for this problem, and evaluate on real data. The goal is not to improve on simpler methods for this particular application, but rather to validate that our method is on par with simpler model structures, where output variance is immediately provided.