Human Movement Science 38 (2014) 281–292

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The influence of slope and speed on locomotive power in cross-country skiing Kjell Hausken ⇑ Faculty of Sciences, University of Stavanger, 4036 Stavanger, Norway

a r t i c l e

i n f o

Article history: Available online 15 November 2014 PsycINFO classification: 2330 2540 Keywords: Velocity Incline Work Air drag Efficiency Friction

a b s t r a c t 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. Ó 2014 Elsevier B.V. All rights reserved.

⇑ Tel.: +47 51 831632; fax: +47 51 831550. E-mail address: [email protected] http://dx.doi.org/10.1016/j.humov.2014.08.016 0167-9457/Ó 2014 Elsevier B.V. All rights reserved.

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1. Introduction The two purposes of this paper were to develop a model of locomotive power in cross-country skiing, and estimate parameters in the model by using an experiment to illustrate the model’s applicability. The three terms locomotive power, work rate, and mechanical power have been used interchangeably in the literature (de Koning & van Ingen Schenau, 1994; Moxnes, Sandbakk, & Hausken, 2014; Sandbakk, Ettema, & Holmberg, 2012; Sandbakk, Holmberg, Leirdal, & Ettema, 2010). In this paper we preferred to use the term locomotive power defined exclusively as power generating locomotion, i.e. power propelling the skier forward, thus excluding e.g. the resting metabolic rate or power. We modeled the skier’s locomotive power to depend on five main factors, i.e. the terrain’s slope, speed, technique, intensity, and mass. A skier’s maximum locomotive power depends on his maximum and mean metabolic rate (Sandbakk et al., 2010, 2012; Sandbakk, Ettema, Leirdal, Jakobsen, & Holmerg, 2011), gross efficiency, and lactate threshold. Whereas slope and mass are given, speed, technique, and intensity further depend on air drag, wind, gravitation, air density, and friction (Bergh & Forsberg, 1992; Sandbakk et al., 2012; Smith, 1992) which in turn depends on snow quality, temperature, and mass. The importance of power production in skiing is underscored by cross-country skiing invoking all major muscle groups which burn calories excessively. Ski legend Bjørn Dæhlie recorded a VO2max of 96 ml/kg/min (World Best VO2 max Scores, 2014). 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. References Andersson et al. (2010) and Sandbakk et al. (2011, 2012) found that locomotive power decreases marginally as speed increases from zero towards 6 m/s, and decreases as speeds increase towards 10 m/s. 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. Two differential equations were determined for how locomotive power depends on the skier’s speed and the terrain’s slope, additionally accounting for technique, intensity, and mass. The endogenized role of locomotive power was analyzed together with friction, gravitation, and air drag. Parameters in the model were estimated with an experiment.

2. Methods 2.1. Theoretical model development A skier’s locomotive power P depends complexly on many factors. We here focus on the five main factors, i.e. the terrain’s slope a = a(t), speed v = v(t), technique a = a(t), intensity e = e(t), and the skier’s mass m. The slope a is measured in radians between a tangent of the track and the horizontal level. In sufficiently steep downhill terrain, with negative a, locomotive power P = P(v, a, e, v,m) is zero since the skier’s legs and arms cannot keep up with the high speed and no locomotion is generated beyond the locomotion generated by gravity (subtracting friction and air drag). As the slope a increases to moderate downhill, legs and arms can generate some locomotion and locomotive power

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P gradually increases above zero. As slope a increases further, locomotive power P increases more distinctly. In flat terrain, with slope a = 0, the skier can generate high locomotive power P. As the slope a becomes positive, and the skier starts to move uphill, his legs and arms can more easily keep up with the decreasing speed and locomotive power P increases further. Evidently an upper limit or threshold for locomotive power P exists which we refer to as Pth = Pth(e, v,m). Hence as the slope a becomes increasingly positive, locomotive power P can increase to a lower extent which means that locomotive power P levels off towards the threshold Pth = Pth(e, v, m) which we consider to be a function of the skier’s mass m, technique v, and intensity e. The skier’s intensity e, 0 6 e 6 1, is defined as the fraction of the threshold locomotive power Pth = Pth(e, v,m) that the skier chooses to exert. This gradual increase of the locomotive power P from zero, first slowly, then more abruptly, and finally levelling off, is appropriately modeled with the simple first-order non-linear differential equation:

  @P P ¼ bP 1  @a Pth

ð1Þ

Eq. (1) is frequently used to model population growth, where a is time, b is growth rate as time elapses, and Pth is carrying capacity. Verhulst (1845) developed such an equation, further elaborated upon by Lotka (1924). For our purpose we envisioned that as the slope a increases, the locomotive power P increases with growth rate b until the skier’s threshold locomotive power Pth is reached. Thus b is the growth rate of the locomotive power P as the slope a increases. More specifically, envision that P in Eq. (1) is small and substantially lower that Pth. As the slope a increases, the right hand side is positive due to a small term P (multiplied by the growth rate b) multiplied by a large term 1  P/Pth. The logic of the left hand side in the differential Eq. (1) is that this causes P to increase. As P increases, the first term bP on the right hand side increases, and the second term 1  P/Pth, decreases, until these two terms have comparable sizes. At that point P increases rapidly. As bP becomes larger that 1  P/Pth, P increases less rapidly. Eventually P approaches Pth causing the second term to approach zero and thus the right hand side in Eq. (1) approaches 0, regardless how much a increases, thus preventing P from exceeding Pth. If locomotive power P in Eq. (1) is interpreted to be exclusively a function of slope a, i.e. P(v, a, e, v, m) = P(a), Eq. (1) must be given a stationary interpretation in the sense that the skier must have remained at a sufficiently long time at a given slope a so that the speed v has stabilized. That of course never happens in practice since the slope a changes frequently as the skier proceeds through the track. For example, the skier completes an uphill part of the track at low speed v and threshold locomotive power Pth where the slope a changes from positive to negative. At first the locomotive power P remains at the threshold while the skier uses legs and arms to accelerate downhill. This means that the locomotive power P remains at the threshold while the slope a changes distinctly from positive to negative. As speed v increases downhill, legs and arms can be used to a lower extent. Eventually, for speeds above approximately 10 m/s regardless of the decline, the skier chooses the deep tuck posture while the locomotive power P decreases to zero. If the slope at that point changes from negative to positive, the skier carries with him the high speed v into the subsequent uphill part while the skier’s locomotive power P remains at zero. The uphill part can have quite a large positive slope a for a few meters while the skier remains unable to exert positive locomotive power P due to the legs and arms not being able to keep up with the high speed. This means that the locomotive power P remains at zero while the slope a changes distinctly from negative to positive to. Eventually, gravity takes its toll, the speed v decreases, the skier regains the ability to use legs and arms for locomotion, and the locomotive power P increases. This illustration of how the skier’s locomotive power P changes through varying terrain shows how slope a and speed v interact in a complicated manner to determine the skier’s locomotive power P. To determine this interaction let us first focus exclusively on letting the speed v vary. In steep uphill terrain the skier’s speed v is low and the skier can easily exert threshold locomotive power Pth. For speeds below 6 m/s it has been shown that the locomotive power P for a given metabolic energy decreases marginally as the speed v increases (Sandbakk et al., 2011, 2012). As speed v increases above a certain threshold around 6 m/s, the skier’s ability to use legs and arms decreases, slowly at first, and thereafter substantially. This causes the skier’s locomotive power P to decrease, slowly at first, thereafter more substantially, and it eventually levels off at zero locomotive power P for speeds above around 10 m/s

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where most skiers prefer the deep tuck posture. This development of the locomotive power P as a function of the speed v is the opposite of the development as a function of slope a modeled in Eq. (1). We thus propose the simple first-order non-linear differential equation:

  @P P ¼ aP @m Pth  1

ð2Þ

Eq. (2) can be perceived as analogous to population decrease from threshold carrying capacity at Pth toward extinction at P = 0 where a corresponds to the population decrease rate and v corresponds to time. Thus a is the decrease rate of the locomotive power P as the speed v increases. More specifically, envision that P in Eq. (2) is large but slightly lower that Pth. As speed v increases, the right hand side is negative due to a large term aP multiplied by P/Pth  1 which is negative with small absolute value. Hence P decreases. As P decreases, the first term aP on the right hand side decreases, but remains positive, and the second term P/Pth  1 also decreases and becomes increasingly negative, until these two terms are comparably sized. At that point P decreases rapidly. Eventually P approaches 0 causing the first term to approach zero and thus the right hand side in Eq. (2) approaches 0, preventing P from becoming negative: Eqs. (1) and (2) jointly model logistic increase of the locomotive power P as a function of the slope a, and logistic decrease of P as a function of speed v. This joint operation means that as a skier completes an uphill part of the track, his low speed v ensures that his locomotive power P remains high despite the slope a changing from positive to negative. But the speed v does not remain low for long and the locomotive power P decreases as speed v increases and slope a decreases. Conversely, as the skier completes the downhill part, his high speed v ensures that his locomotive power P remains low despite the slope a changing from negative to positive. But the speed v does not remain high for long and the locomotive power P increases as speed v decreases and slope a increases. Solving Eqs. (1) and (2) gives the solution:

P¼

Pth 1 þ Pth eav bac

ð3Þ

where c is a parameter that arises when solving the differential equations in Eqs. (1) and (2). That is, the solution in Eq. (3) is valid for all values of c. To verify that Eq. (3) is a solution, differentiating P with respect to a, and inserting P into the right hand side of Eq. (1), shows that the left hand side equals the right hand side. Analogously, differentiating P with respect to v, and inserting P into the right hand side of Eq. (2), shows that the left hand side equals the right hand side. In Eq. (3), eba in the denominator shows how the locomotive power growth rate b with respect to slope a causes the locomotive power P to increase. Conversely, eav shows how the locomotive power decrease rate a with respect to speed v causes the locomotive power P to decrease. Finally, ec expresses that locomotive power P, due to other factors than slope a and speed v, increases when c is positive and decreases when c is negative. The physical interpretation of c is thus to capture factors which impact locomotive power P and which are not present in slope a and speed v. The skier’s threshold locomotive power Pth is partly fixed by the skier’s maximum and mean metabolic rate, gross efficiency, and lactate threshold, as specified later in this section, and is partly chosen by the skier who chooses three factors. First, we considered Pth to be proportional to the skier’s chosen intensity e, 0 6 e 6 1, which is here introduced quantitatively as a parameter for the first time. Maximum intensity e = 1 can be sustained for maximally a few hundred meters. With individual starts a skier may try to keep one certain e throughout the race. Future research may establish what this specific e is dependent on the length of the race and the skier’s characteristics. One possibility is e = 0.96 throughout the race, or for example e = 0.94 for long races. In mass start races the intensity e may fluctuate substantially. Some skiers are better equipped than others to tolerate changing e. Competitive skiers have been known to come to almost a complete standstill, with e close to zero, may boost to e = 1 to lose competitors or to gain bonus points at midpoints through the race, and the intensity e usually approaches 1 towards the finish line. One coarse estimator for a skier’s intensity e is the skier’s heart rate above his resting heart rate as a percentage of his maximum heart rate minus his resting heart rate. This gives intensity e = 0 when the skier is at rest and intensity e = 1 when the skier exerts

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maximum intensity. Since a delay exists between choosing intensity and manifestation in heart rate, this estimator seems most appropriate for stationary conditions where the chosen intensity and heart rate have stabilized. Second, we assumed that Pth changes relative to a reference technique quantified as v = vref in the manner it impacts locomotive power, v > 0, vref > 0. The technique parameter is also here introduced quantitatively for the first time. Kvamme, Jakobsen, Hetland, and Smith (2005) suggested that skiers’ gear selection depends substantially on slope, whereas Andersson et al. (2010) argued that gear selection depends on slope, speed, the length of the uphill slopes and fitness level. It additionally depends on the skier’s preferences, and it may depend on snow quality, wind, height above sea level, and temperature. See Nilsson, Tveit, and Eikrehagen (2004) and Smith (2002) for further studies of gears and technique. Cross country skiing is usually classified into ski skating and classical skiing. The most common so-called gears in skating are labeled gliding diagonal skating (G1), asymmetrical double pole push in connection with every other leg push (G2), double pole push together with every leg push (G3), symmetrical double pole push in connection with every other leg push (G4), downhill skating in a low position using only the legs (G5), side-step technique for curves in which leg work is performed with or without poling (G6), and downhill skiing in a low stance position without leg or pole push (G7). The most common gears in classical skiing are labeled running diagonal stride, diagonal stride, double poling kick, double poling, G6, and G7. The skier chooses gears, and chooses techniques for each gear, which differ for uphill, flat, and downhill terrain. The skier has different proficiencies and preferences for the various techniques compared with other skiers. Higher v is referred to as a more effective technique which gives higher locomotive power. Third, we considered the skier’s mass m to be a constant changing negligibly during skiing. We assume a reference mass mref for which a reference threshold locomotive power Pth(1, vref, mref) can be determined for a given reference technique v = vref and maximum intensity e = 1. Combining these three factors, the skier’s threshold locomotive power Pth changes according to:

Pth ðe; v; mÞ ¼ e



v vref

h 

m mref

k

P th ð1; vref ; mref Þ

ð4Þ

Eq. (4) expresses that higher intensity e, a more effective technique v, and higher mass m, cause higher threshold locomotive power Pth. The parameters h and k express the sensitivity of the dependence. When h = k = 0, technique v and mass m have no impact on Pth. When 0 < h,k < 1, Pth changes concavely dependent on v and m. When h = k = 1, Pth is proportional to v and m. When h > 1 and k > 1, Pth changes convexly dependent on v and m. As shown in the literature (Moxnes et al., 2014; van Ingen Schenau et al., 1991), we modeled the skier’s center of mass as:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dv P qC d AðV þ v Þ2 Sign½V þ v  ¼  lg 1  a2  g a  ; 2m dt mv 8 > < 1 when V þ v > 0 ds ¼ v ; Sign½V þ v  ¼ 1 when V þ v < 0 > dt : 0 when V þ v ¼ 0

ð5Þ

The skier’s center of mass is used as a positional reference point to determine position s, speed v, and acceleration dv/dt, acknowledging that for example the skier’s two legs and two arms, and even the skier’s head, move back and forth and constitute poorer reference points for position. The left hand side of Eq. (5) expresses the skier’s rate of change of kinetic energy divided by mass m and speed v. On the right hand side the first term is the locomotive power P divided by mv. The second term is the friction power divided by mv. The third term is the gravitation power divided by mv. The fourth term is the air drag power divided by mv. In Eq. (5) t is time, l is friction, g is the gravitational acceleration, q is the air density, C d is the drag coefficient, A is the skier’s projected front area, and V is the wind speed. For small slopes a we have TanðaÞ ¼ dhðsÞ=dx  SinðaÞ ¼ dhðsÞ=ds  a, where h(s) is the height of the terrain relative to the starting point as a function of the accumulated distance s = s(t) along the terrain from the beginning of the track, and x is the horizontal position. The inline a varies between p/2 and p/2. The parameters l, q, Cd and m can have arbitrarily high values and be functions of time. The parameters a, l, q, Cd and m can change continuously and abruptly, but not discontinuously. Summing

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up, the model has the following 23 variables and parameters. Speed v determined by the skier and time t determined by nature are variables. Intensity e and technique v are on the one hand variables determined by the skier, but intensity e can also be determined by a training manual and technique v can be determined by a training manual or by race regulations, and then they are parameters. They thus impact the threshold locomotive power Pth which also depend on other factors discussed above such as reference mass mref which is a parameter, reference technique vref which is a parameter, maximum and mean metabolic rate, gross efficiency, and lactate threshold. The skier’s various choices impact the locomotive power P. Slope a is a parameter when dictated by the race track, and a variable when the skier chooses his own track. Mass m and projected front area A are parameters but may to some extent be altered by the skier through time. Friction l is a parameter but can be altered by the choice of skis and ski preparation. Air density q, drag coefficient C d , wind speed V, and gravitation g are parameters. Finally, a, b, c are parameters. 2.2. Comparison with experimental data Prior to conducting the study the subject was informed about the nature of the study. The subject provided a written consent to participate. The study was pre-approved by the Norwegian Regional Ethics Committee which provides IRB approval. To test the model we considered a male skier with mass m = mref = 77.5 kg, which we considered to be the reference mass. The skier used conventional racing skis with poles. He warmed up 20 min at low intensity causing starting heart rate 120 beats per minute. Heart rate was measured throughout the track using the Polar RS800 heart rate monitor. The skier mainly used the G3 technique (double pole push together with every leg push), but chose freely among the most common gears within skating dependent on the track characteristics (uphill, flat, downhill, curvature, etc.). Associated with choice of gears, the skier chose technique according to his preference, quantified as v = vref, and we set h = k = 1 so that Eq. (4) simplifies to Pth ðe; v; mÞ ¼ eP th ð1; vref ; mref Þ, which is determined as follows: The skier’s maximum metabolic rate was assumed to be 1890 J/s. Based on roller ski testing on treadmill, the gross efficiency for an uphill slope was determined to be 0.155. The skier’s lactate threshold was a fraction 0.88 of his maximum metabolic rate. Experience has shown that the skier in uphill parts of a race track like the current one has mean metabolic rate 5–10% above his lactate threshold, i.e. 1746–1830 J/s, which implicitly accounts for the intensity parameter e which is between 0.9 and 1 for this track. Using visual curve fitting, we chose the baseline 7% above the lactate threshold, i.e. 1780 J/s, which was multiplied with

Fig. 1. The height h [m] in the terrain as a function of accumulated distance s [m] in meters. The 52 stars ⁄ are the 52 datapoints for distance s versus height h. The 14 filled squares j, which coincide with the stars, are positions for time measurements.

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0.155 to yield the threshold locomotive power Pth = 275 J/s. The skier used 815 s to ski 4218 m. The empirics consist of 52 datapoints shown as stars ⁄ in Fig. 1 including start point and end point in meters, i.e. distance s from start along the horizontal axis, and height h above sea level along the vertical axis. The track profile was made by the Norwegian Ski Federation applying recommendations for World Cup races using a measuring wheel and GPS (Garmin eTrex10, Garmin International Inc., Olatha, Kansas, USA) with Wide Area Augmentation System receivers which has an accuracy of less than three meters horizontally and vertically with the current satellite availability. Using the GPS watch on a running track revealed around 1–2 m incorrectness per kilometer. The track in the terrain was measured by GPS during the summer. Distance and height along the track are measured using a measuring wheel and an inclinometer (Suunto, Finland, www.suunto.com). Points of measurement were usually where elevation changed 10–20 m vertically. Expressing the 52 datapoints as {si,hi}, i = 1,. . .,52, where si is distance from start and hi is height above sea level, slope at datapoint i is expressed as ai ¼ ðhiþ1  hi Þ=ðsiþ1  si Þ, i = 1,. . .,51, where a52 = a51. Time measurements were available for 14 of the 52 datapoints, shown with filled squares j in Fig. 1. Measuring time for a skier along a 4218 m track is labor intensive. Research assistants measured time using synchronized stop watches when the skier passed specified points along the track. The 14 specified points were chosen based on significant changes in the track profile while enabling the assistants to observe the skier before passing the point. More than 14 time measurements would be desirable, but were not obtainable for this study. The 14 time measurements, including time 0 at start and time 815 s at finish, were positioned along the track to make the comparison between empirics and model as accurate as possible. The 14 time measurements expressed as {s,t} in {meters, seconds} are {{0, 0}, {348, 123}, {600, 172}, {881, 229}, {985, 248}, {1147, 308}, {1411, 370}, {1638, 436}, {2029, 499}, {2429, 541}, {2724, 575}, {2899, 657}, {3776, 773}, {4218, 815}}. Friction depends on mass, temperature and speed, and can vary from 0.01 during wet and cold conditions to 0.06 during racing conditions. On the testing day the temperature was minus two Celsius degrees, the snow was of the old grained type, and we estimated friction l = 0.037. Air density depends on temperature and height above sea level. Using the barometric height formula, (Barometric Formula, 2014), for 2 °C at 180–250 m above sea level and standard atmospheric pressure of 1013 mb, the density is q = 1.29 g/cm3. Wind speed was V = 0 m/s. For drag and front area we assumed CdA = 0.55 m2 when speed v 6 10 m/s and CdA = 0.35 m2 in deep tuck position

Fig. 2. Locomotive power P [J/s] as a function of speed

v [m/s] and slope a [radians].

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when v > 10 m/s. See references Leirdal, Saetran, Roeleveld, et al. (2006) and Spring, Savolainen, Erkkila, Hamalainen, and Pihkala (1988) for similar values. Eqs. (1)–(5) were solved numerically. The parameters a, b, c in Eq. (3) were determined by minimizing the average summed square difference

Min a;b;c

nða;b;cÞ X i¼0

2

ðv ðiÞ  v e ðiÞÞ nða; b; cÞ

ð6Þ

between the simulated speed v(i) and experimental speed ve(i). Eq. (5) was simulated over n(a, b, c) incremental time steps i = 0,. . ., n(a, b, c), where each time step was Dt = 0.05 s, until the finish line of 4218 m is reached. The least summed square method varied the parameter a in increments of 0.1 and varied b and c in increments of 1, in three nested DO loops wrapped around the n(a, b, c) time steps. We determined the average summed square difference since the number of time steps n(a, b, c) to simulate Eq. (5) depends on a, b, c. The experimental speed ve(i) was calculated as incremental distance divided by incremental time between two subsequent time measurements. 3. Results The least square sum method gave the optimal parameter values a = 1.0, b = 58, c = 16 causing an average least square sum 3.35. For these parameter values Fig. 2 plots locomotive power P vertically as a function of simulated speed v and slope a long the two horizontal axes. The locomotive power P decreases logistically as a function of speed v, and increases logistically as a function of slope a, consistently with the elaboration of Eq. (3) in Section 2. First, in steep uphill terrain, slope a is large and speed v is eventually low causing large locomotive power P. Second, in steep downhill terrain slope a is low and speed v is eventually high causing low locomotive power P. Third, in the transition from uphill to downhill terrain so that slope a decreases, speed v is initially low causing large locomotive power P which decreases as speed v increases. Fourth, in the transition from downhill to uphill terrain so that slope a increases, speed v is initially high causing low locomotive power P which increases as speed v decreases. Points 1 and 2 are stationary conditions where the skier is located at fixed points (with fixed speed and slope) at the curved landscape in Fig. 2 for a certain time period. Points 3 and 4 are non-stationary conditions where the skier transiently moves across the curved landscape in Fig. 2 as speed v and slope a change, and these usually change in opposite directions. Fig. 3 plots simulated

Fig. 3. Simulated speed v [m/s] as filled squares j and experimental speed ve [m/s] as filled circles d as functions of time t [seconds].

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speed v as filled squares j and experimental speed ve as filled circles d as functions of time t. The experimental speed ve is the average speed between two subsequent time measurements. Comparing speed v in Fig. 3 for the 52 height measurements in Fig. 1 shows how speed v increases in downhill terrain and decreases in uphill terrain. Distance s specified with filled squares j in Fig. 1 corresponds to time t specified with filled circles d in Fig. 3. Speed v is initially low until the second time measurement of 123 s after 348 m at height 211 m. The height decrease to 207.8 m causes a substantial speed increase in Fig. 3. Thereafter follows a speed decrease until the third time measurement of 172 s after 600 m at height 212 m. The subsequent height decrease to 197 m causes a very substantial speed increase to 11.9 m/s after 177 s. Proceeding through the track in this manner, comparing Figs. 3 and 1, while accounting for the ingredients of Eqs. (1)–(5), illustrate the logic of the model. The correlation between speed v and slope a is 0.84. When b = 0 (eliminating the dependence of locomotive power P on slope a in Eq. (3)), the optimum is a = 3.0 and c = 29 causing a higher average least square sum 3.52.

4. Discussion For the logistic dependence of locomotive power P on speed v and slope a in Fig. 2, the track profile in Fig. 1 is curved causing the skier to remain in non-stationary conditions most of the time with continuously changing speed and slope. Fig. 3 shows how simulated speed v (determined using the model in Section 2 and the least square sum method in Section 3) varies substantially along the track, reaching a maximum v = 12.5 m/s in steep downhill terrain (the steepest downhill is a = 0.130), and a minimum v = 1.8 m/s in steep uphill terrain (the steepest uphill is a = 0.159). The maximum and minimum average experimental speeds (determined by the skier in Section 2) were ve = 10.5 m/s and ve = 2.1 m/s, respectively, with slightly less extreme values since these are averages. Future research should perform more time measurements than the 14 conducted in the current study (described in Section 2), especially in portions of the track with changing curvature. Measuring time at high points and low points is especially important since that’s when both speed v and slope a change most. More time measurements will give maximum speed above ve = 10.5 m/s and minimum speed below ve = 2.1 m/s since averaging has less impact. For sprint ski skating over 1425 m, Andersson et al. (2010) measured speed variation 2.8–12.9 m/s. Multiplying the maximum simulated speed 12.5 m/s with a = 1.0 in Eq. (3) yields the factor av = 12.5. Multiplying the maximum absolute slope a = 0.159 radians with b = 58 in Eq. (3) yields the factor ba = 9.2. Inserting these similarly sized factors, av = 12.5 and ba = 9.2, into the exponent av  ba  c in the denominator in Eq. (3), illustrate that both simulated speed v and slope a impact the locomotive power P in Eq. (3). This emphasizes the joint impact of speed v and slope a on locomotive power P. Decreasing the parameter b in Eq. (3) to b = 0 means eliminating the dependence of locomotive power P on slope a. This gives a higher average least square sum 3.52 in Eq. (6), compared with 3.35 when b varies, which means poorer fit between the model and data. To compensate for b = 0, the optimal a is three times higher, i.e. a = 3.0, and the optimal c is almost twice as high, i.e. c = 29, to yield a comparably sized exponent in the denominator in Eq. (3). The model in this paper can be used in many different ways, and is applicable to test many of the analyses in the papers in the reference list. A first step is to use the least squares method to estimate the parameter a related to the role of speed v, the parameter b related to the role of slope a, and the parameter c in Eq. (3) so that Eqs. (3)–(5) match the empirics for a certain skier in a certain terrain under certain conditions. This first step can be repeated for different skiers, for the same skier in different terrains, or for the same skier in the same terrain under different conditions including different techniques. Five interesting variables and parameters are speed v, slope a, intensity e, technique v, and locomotive power P. Altering any of these, the model is applicable to predict changes in one or several of the other variables and parameters. For example, decreasing the intensity e with a certain amount causes decreased speed v and decreased locomotive power P with certain amounts when slope a and technique v remain fixed. Alternatively, increased speed v causing increased locomotive power P can be used to predict increased intensity e, or altered slope a (in alternative terrain) can be used to predict

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altered technique v. Further uses of the model are to alter any of the ten parameters (variables) threshold locomotive power Pth, reference mass mref, reference technique vref, mass m, projected front area A, friction l, air density q, drag coefficient C d , wind speed V, and hypothetically gravitation g, use the least squares method in Eq. (6) to estimate new a, b, c, and predict further. Let us illustrate use of the model by considering Sandbakk et al. (2012, Table 2) who tested seven elite skiers roller ski skating on 2% and 8% slope at low, moderate and high metabolic rate. Interpreting work rate as locomotive power, at high metabolic rate they measured (P, v) equal to (192 J/s, 6.11 m/s) and (248 J/s, 3.33 m/s) at slopes 2% and 8%, respectively. These numbers match the model in this paper for example when (a, b, c) = (0.3, 9.1, 8.1), assuming threshold locomotive power Pth(1, vref, mref) = 275 J/s at maximum intensity e = 1. Different parameter values are due to factors such as different skiers, no treadmill slope transitions as in regular terrain, different friction, and roller skating at 2% slope corresponds to regular skating at 0% slope. At low metabolic rate Sandbakk et al. (2012, Table 2) measured (P, v) equal to (127 J/s, 1.667 m/s) and (05 J/s, 3.33 m/s) at slopes 2% and 8%, respectively. Inserting these lower values of locomotive power P and speed v into Eq. (3) when (a, b, c) = (0.3, 9.1, 8.1) gives Pth(e, vref,mref) equal to 113.1 J/s and 131.1 J/s, respectively. Using Eq. (4) to solve with respect to e, thus dividing with Pth(1, vref, mref) = 275 J/s, gives the intensities e = 0.41 and e = 0.48, at slopes 2% and 8%, respectively. That is, specifying a certain low metabolic rate, the model enables determining the corresponding low intensity. Conversely, specifying the desired intensity, the model enables determining the corresponding metabolic rate, or, if the metabolic rate is specified, determining a corresponding speed at fixed slope, or determining a corresponding slope at fixed speed. Calculations such as these illustrate how the model can be used to interpret empirics of locomotive power, speed, slope, etc. in skiing. Andersson et al. (2010) found the highest speed variation at the end of every uphill section while transitioning to downhill, and the lowest speed variation at the end of the downhill sections where speed was highest. The first transition is especially non-stationary since that’s when both main factors of this paper, speed v and slope a, change maximally in opposite directions. This transition needs focus during training to save time, e.g. regulate some skiers’ tendency to rest temporarily at the top before accelerating downhill. This paper’s model throws light on such regulation at the end of every uphill section. Sandbakk et al. (2011) analyzed speed, total time and time in nine different terrain sections for 12 skiers. They determined factors such as which skiers benefit in which terrain sections, and which skiers have high speed at the various slopes in which terrain sections. These empirics can in future research be combined with the 23 variables and parameters in the model in this paper to determine various relationships. One may for example scrutinize various skiers’ weaknesses, where empirics show low speed in various terrain sections when the model predicts higher speed. This is related to de Koning et al.’s (1999) analysis of optimal pacing in cycling. 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. Norman and Komi (1987) found 2.2 times higher mechanical power in 9° = 0.159 = a uphill terrain compared with 1.6° = 0.028 = a ‘‘level’’ terrain for 11 skiers using the diagonal technique. They concluded that speed may be limited by constraints on body segment utilization. Ratios such as 2.2 are directly readable along the vertical axis in Fig. 2 expressing locomotive power P as a function of speed v and incline a, with further dependence on the other variables and parameters. For example, inserting the high uphill incline a = 0.159 into Eq. (3) gives P  Pth = 275 J/s when speed v is below 7 m/s. Inserting the low incline a = 0.028 into Eq. (3) gives P = 125 J/s when speed v = 12.2 m/s, which is multiplied with 2.2 to yield 275 J/s. Sandbakk et al. (2010) compared eight world class and eight national level skiers, concluding that the world class skiers had lower anaerobic rate and higher gross efficiency at the same speed, skied more efficiently, and proposed that they had better technique. Future research can conduct such comparisons more thoroughly focusing for example especially on the efficiency parameter e, the technique parameter v, the threshold locomotive power Pth, and locomotive power Pth, also accounting for the

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other model variables and parameters. The model can be further used to test Bergh and Forsberg’s (1992) analysis of the impact of body mass on skiing performance. Inserting different mass m in the model allows predicting different speeds v in different terrains. van Ingen Schenau and Cavanagh’s (1990) determination of relationships between metabolic power input and mechanical power output, mediated by metabolic rate, gross efficiency, and lactate threshold, etc., needs continued focus in future research as technique and other factors become more optimal. Smith’s (1992) argument that our ‘‘current understanding of skiing mechanics is not sufficiently complete to adequately assess and optimize an individual skier’s technique’’ still has some validity today. Models such as the one in the current paper which identifies relationships between 23 variables and parameters, including a technique parameter v, may gradually enable us to enhance our understanding and insight and fine-tune both optimal technique and other optimal variables, parameters and relationships. This paper has enhanced our insight into how slope a, speed v, technique v, intensity e, and mass m impact locomotive power P, which operates together with friction, gravitation, and air drag to determine speed v in cross-country skiing. Future research should scrutinize these dependencies more thoroughly. For example, the various techniques v impact locomotive power P differently in uphill and downhill terrain (i.e. dependent on slope a), and impact P differently dependent on intensity e and mass m where mass m also operates differently in uphill and downhill terrain. Future research should also model how locomotive power P depends on factors such as VO2max, BMI, height, fatigue, stress, and motivation. Funding No sources of funding. Acknowledgments No funding sources were used. No conflicts of interest exist. I thank two anonymous referees of this journal, and John F. Moxnes and Øyvind Sandbakk for useful comments. References Andersson, E., Supej, M., Sandbakk, O., Sperlich, B., Stoggl, T., & Holmberg, H. C. (2010). Analysis of sprint cross-country skiing using a differential global navigation satellite system. European Journal of Applied Physiology, 110, 585–595. Bergh, U., & Forsberg, A. (1992). Influence of body mass on cross-country ski racing performance. Medicine and Science in Sports and Exercise, 24, 1033–1039. Carlsson, P., Tinnsten, M., & Ainegren, M. (2011). Numerical simulation of cross-country skiing. Computer Methods in Biomechanics and Biomedical Engineering, 14, 741–746. de Koning, J. J., Bobbert, M. F., & Foster, C. (1999). Determination of optimal pacing strategy in track cycling with an energy flow model. Journal of Science and Medicine in Sport, 2, 266–277. de Koning, J. J., & van Ingen Schenau, G. J. (1994). On the estimation of mechanical power in endurance sports. Sports Science Review, 3, 34–54. Kvamme, B., Jakobsen, B., Hetland, S., & Smith, G. (2005). Ski skating technique and physiological responses across slopes and speeds. European Journal of Applied Physiology, 95, 205–212. Leirdal, S., Saetran, L., Roeleveld, K., et al (2006). Effects of body position on slide boarding performance by cross-country skiers. Medicine and Science in Sports and Exercise, 38, 1462–1469. Lotka, A. J. (1924). Elements of mathematical biology. London: Dover Books. Moxnes, J. F., & Hausken, K. (2008). Cross-country skiing motion equations, locomotive forces and mass scaling laws. Mathematical and Computer Modelling of Dynamical Systems, 14, 535–569. Moxnes, J. F., Sandbakk, Ø., & Hausken, K. (2013). A simulation of cross-country skiing on varying terrain by using a mathematical power balance model. Open Access Journal of Sports Medicine, 4, 127–139. Moxnes, J. F., Sandbakk, Ø., & Hausken, K. (2014). Using the power balance model to simulate cross-country skiing on varying terrain. Open Access Journal of Sports Medicine, 5, 89–98. Nilsson, J., Tveit, P., & Eikrehagen, O. (2004). Effects of speed on temporal patterns in classical style and freestyle cross-country skiing. Sports Biomechanics, 3, 85–107. Norman, R. W., & Komi, P. V. (1987). Mechanical energetics of world-class cross-country skiing. International Journal of Sport Biomechanics, 3, 353–369. Sandbakk, O., Ettema, G., & Holmberg, H. C. (2012). The influence of incline and speed on work rate, gross efficiency and kinematics of roller ski skating. European Journal of Applied Physiology, 112, 2829–2838.

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