Numerical simulation of cross-country skiing

Abstract

A program for numerical simulation of a whole ski race, from start to finish, is developed in MATLAB. The track is modelled by a set of cubical splines in two dimensions and can be used to simulate a track in a closed loop or with the start and finish at different locations. The forces considered in the simulations are gravitational force, normal force between snow and skis, drag force from the wind, frictional force between snow and ski and driving force from the skier. The differential equations of motion are solved from start to finish with the Runge-Kutta method. Different wind situations during the race can be modelled, as well as different glide conditions on different parts of the track. It is also possible to vary the available power during the race. The simulation program's output is the total time of the race, together with the forces and speed during different parts of the race and intermediate times at selected points. Some preliminary simulations are also presented.

Numerical optimization of pacing strategy in cross-country skiing

“…However, in the simulation it is beneficial to solve the motion equations in the same global directions at every point along the course (Carlsson et al 2011). Therefore, equations of motion are transformed to the global -and -coordinates.…”

“…Mathematical modeling of a ski race, combined with efficient nonlinear optimization routines, provides a tool for analyzing how these efforts should be best distributed. Carlsson et al (2011) showed how numerical simulations in cross-country skiing can be performed in order to calculate the total race time for an arbitrary athlete. The simulation model solves the equations of motion for constant time steps.…”

“…Carlsson et al (2011) came up with a model which can simulate a skier along a predetermined course profile by utilizing these motion equations. These equations are generally built up by expressions that involve parameters derived from both internal and external factors.…”

Comparison of Power Output Estimates in Treadmill Roller-Skiing

“…This acceleration was estimated from the measured speed as an average between the measured values. The OP estimate is based on the numerical optimization model of Sundström et al [11] which is based on the same mechanical model as Carlsson et al [12]. The OP estimate uses the method of moving asymptotes [13] and varies the power output to minimize the square of speed difference between the mechanical model and the actual performance of a cross-country skier on a treadmill.…”

“…The evaluation model is based on the same numerical simulation approach as the OP estimate [12]. The STT was simulated by solving the motion equation of roller skiing with the power output estimates as input data.…”