Optimizing pacing strategies on a hilly track in cross-country skiing

Abstract

During events involving locomotive exercise, such as cross-country skiing, it is believed that pacing strategies (i.e. power distribution) have a significant impact on performance. Therefore, a program was developed for the numerical simulation and optimization of cross-country ski racing, one that numerically computes the optimal pacing strategy for a continuous track. The track is modelled by a set of cubic splines in two dimensions and can be used to simulate a closed loop track or one with the start and finish at different locations. For an arbitrary point on the two dimensional track, equations of motion are formulated parallel and normal to the track, considering the actual slope and curvature of the track. Forces considered at the studied point are the gravitational force, the normal force between snow and skis, the drag force from the wind, the frictional force between snow and ski and the propulsive force from the skier, where the latter is expressed as the available power divided by the actual speed. The differential equations of motion are solved from start to finish using the Runge-Kutta-Fehlberg method. The optimization of the ski race is carried out with the Method of Moving Asymptotes (MMA) which minimizes the racing time by choosing the optimum distribution of available power. Constraints for minimum, maximum and average power are decided by conditions of scaling by body size. Results from a simulated ski competition with optimized power distribution on a real track are presented.