## Ice Patch Modeling and Vehicle Simulation

Methodology

Application Specifics

Results

In attempting to model random patches of ice, we are confronted with uncertainty having spatially defined distribution characteristics. To accurately represent the ice distribution, tantamount to a spatial function for friction coefficients, we must leverage a space dependent method to capture spatial averages and covariance. Gaussian processes is the most convenient method of this kind for us to work with as it is well-developed while still flexible to allow us to fit it around our problem.

Gaussian processes, just like any other uncertainty quantification method, require some data or some information about the distribution a priori. In our procedure, we use maximum likelihood estimation to extract the relevant parameters from a set of data (i.e. spatial mean function of friction coefficients, mean patch length, noise in data, etc). Then we use these parameters to interpolate points on a high resolution grid maintaining key statistical moments inherent in the observed data. Each grid point carries a sampling mean and standard deviation for friction coefficient rather than a deterministic value (see “Gaussian Processes for Machine Learning” by Rasmussen for mathematical details).

The first step as described above is depicted
in Figure 1. Deterministic values are provided at all points (x_{1i},
x_{2i}); these values do not need to be given in the shape of a grid but it is
depicted this way for simplicity. The sophisticated Gaussian processes
are used to interpolate sampling characteristics on a finer grid at points
in the vicinity of expected travel (x^{*}_{1i},x^{*}_{1i}).

**Figure 1: Interpolation with Gaussian Processes**

Once the sampling distributions have been computed at points on a fine grid, Monte Carlo simulation is carried out to predict vehicle behavior. For each iteration, a sample is pulled from each grid point and a spline is fit the points. Since the spline has spatial continuity on the region of interest, friction coefficients can be drawn from it at any in point in time and space. Figure 2 below pictorial describes the processes for a single iteration.

**Figure 1: Monte Carlo iteration for continuous friction coefficient space**

Each iteration uses a different friction coefficient space, generated from the same spatial distribution characteristics. The trajectory outputs for each iteration are finally averaged. Further analysis is conducted to obtain representative trajectory variance for each point in time.