Collocation Method

This is what they do for mintime Raceline Optimization.

“This python module shows the planning of time-optimal trajectories, which allows an autonomous race car to drive at the handling limits, taking into account locally changing road friction values.

For this purpose, the minimum lap time problem is described as an optimal control problem, converted to a nonlinear program using direct orthogonal Gauss-Legendre collocation and then solved by the interior-point method IPOPT. Reduced computing times are achieved using a curvilinear abscissa approach for track description, algorithmic differentiation using the software framework CasADi, and a smoothing of the track input data by approximate spline regression. The vehicles behavior is approximated as a double track model with quasi-steady state tire load simplification and nonlinear tire model.

The novelty of this work is the consideration of wheel-specific tire-road friction coefficients along the racetrack using a track friction map. It is shown that variable friction coefficients have a significant impact on the trajectory, and therefore significantly improve lap times on inhomogenous racetracks.”