Numerical optimization of pacing strategy in cross-country skiing

Abstract

This is an accepted version of a paper published in Structural and multidisciplinary optimization (Print).Citation for the published paper: Sundström, D., Carlsson, P., Ståhl, F., Tinnsten, M. (2012) "Numerical optimization of pacing strategy in cross-country skiing" Structural and multidisciplinary optimization (Print) URL: http://dx.doi.org/10.1007/s00158-012-0856-7Access to the published version may require subscription. AbstractWhen studying events involving locomotive exercise, such as cross-country skiing, one generally assumes that pacing strategies (i.e. power distributions) have a significant impact on performance. In order to better understand the importance of pacing strategies, a program is developed for numerical simulation and optimization of the pacing strategy in cross-country ski racing. This program computes the optimal pacing strategy for an arbitrary athlete skiing on a delineated course. The locomotion of the skier is described by introducing the equations of motion for cross-country skiing. A transformation of the motion equations is carried out in order to improve the simulation. Furthermore, a nonlinear optimization routine is connected to the simulation program. Simulation and optimization are performed on a fictional male skier. Results show that it is possible to attain an optimal pacing strategy by simulating cross-country skiing while connecting nonlinear optimization routines to the simulation. It is also shown that an optimal pacing strategy is characterized by minor variations in speed. In our opinion, this kind of optimization could serve as essential preparations before important competitions.

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

“…They found performance benefits from all ''all-out'' start and thereafter even pacing. Sundstrøm et al (2013) similarly found that ''an optimal pacing strategy is characterized by minor variations in speed'', determined numerically by a nonlinear optimization routine. Applied to this paper's model this suggests a constant or relatively constant intensity parameter e through varying terrain (except steep downhill where intensity inevitably decreases), and intensity inverse proportional to track length.…”

“…For speeds above 10 m/s skiers usually use techniques such as the tuck position to save energy (Andersson et al, 2010). Using these insights locomotive power has recently been analyzed as a function of speed only (Moxnes et al, 2014;Sundstrøm et al, 2013). This paper's contribution is to endogenize locomotive power more thoroughly into power production, in the sense of making locomotive power a variable in the model instead of an exogenously given parameter.…”

“…In the research on power production, de Koning, Bobbert, and Foster (1999) used an energy flow model to determine pacing in cycling, de Koning and van Ingen Schenau (1994) estimated mechanical power in endurance sports, and van Ingen Schenau and Cavanagh (1990) formulated Newtonian mechanics power equations for various endurance sports. For cross-country skiing, Carlsson, Tinnsten, and Ainegren (2011) simulated skiing numerically accounting for a variety of forces, and wind and glide situations, Moxnes and Hausken (2008), Moxnes, Sandbakk, and Hausken (2013) and Moxnes et al (2014) developed motion equations, Sandbakk et al (2011) analyzed time trials in different terrain sections while considering work rate and relationships to physiological and kinematic parameters while treadmill roller ski skating, and Sundstrøm, Carlsson, Ståhl, and Tinnsten (2013) optimized pacing strategy in skiing numerically applying skiing motion equations. References Moxnes et al (2013), Norman and Komi (1987) and Sandbakk et al (2012) found that locomotive power is high in uphill terrain.…”

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.…”

Comparing bioenergetic models for the optimisation of pacing strategy in road cycling

“…(1)- (5)] projected onto these directions. Subsequently, the equations of motion were transformed according to Sundström et al [21] to make distance ( ) the dependant variable and time the independent variable. Consequently, the equation of motion was expressed as: +tan cos +( )2 1 cos + 2 (6) where is time and prime denotes differentiation by .…”