The influence of slope and speed on locomotive power in cross-country skiing

Purpose: A model was developed for cross-country skiing where locomotive power depends on speed and slope as variables, and further depends on snow friction, gravitation, mass, air drag, wind, and air density. Model parameters were estimated experimentally. Methods: Two differential equations were developed for how locomotive power depends on speed and slope. The equations are of the logistic form with a threshold determined by the skier's technique, intensity, mass, metabolic rate, gross efficiency, and lactate threshold. Three parameters were estimated by minimizing the average summed square difference between the simulated speed, using the model, and experimental speed of an elite male skier in a 4218 m track. Distance and height along the track were measured using a measuring wheel and an inclinometer generating 52 datapoints. Research assistants measured time from start at 14 different positions. Parameter values were determined from the literature. Results: We illustrated how speed and slope impact locomotive power. The model was shown to be superior to a model where locomotive power is a function of speed only. The joint dependence of locomotive power on speed and slope is crucial in the non-stationary conditions where the skier passes high and low points along the track where both speed and slope change rapidly. Conclusion: The model is useful to predict the impact of altering a subset of the 23 variables and parameters on the remaining variables, for example the impact of changed friction or terrain slope on the skier's speed and thus time to complete a ski race.