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Numerical optimization of pacing strategies in locomotive endurance sports
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics. (Sports Tech Research Centre)ORCID iD: 0000-0003-1324-9828
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is devoted to the optimization of pacing strategies in two locomotive endurance sports; cross-country skiing and road cycling. It has been established that constant pace and variable power distributions are optimal if purely mechanical aspects of locomotion are considered in these sports. However, there is a lack of research that theoretically investigates optimal pacing for real world athletes who are constrained in their ability to generate power output through the bioenergetics of the human body.

The aims of this thesis are to develop numerical pacing strategy optimization models and bioenergetic models for locomotive endurance sports and use these to assess objectives relevant in optimal pacing. These objectives include: Investigate the impact of hills, sharp course bends, ambient wind, and bioenergetic models on optimal pacing and assess the effect of optimal pacing strategies on performance.

This thesis presents mathematical models for optimization of pacing strategies. These models are divided into mechanical locomotion, bioenergetic, and optimization models that are connected and programmed numerically. The locomotion and bioenergetic models in this thesis consist of differential equations and the optimization model is described by an iterative gradient-based routine. The mechanical model describes the relation between the power output generated by an athlete and his/her locomotion along a course profile, giving the finishing time. The bioenergetic model strives to mimic the human ability to generate power output. Therefore, the bioenergetic model is set to constrain the power output that is used in the mechanical locomotion model. The optimization routine strives to minimize the finishing time in the mechanical locomotion model by varying the distribution of power output along the course, still satisfying the constraints in the bioenergetic model.

The studies contained within this thesis resulted in several important findings regarding the general application of pacing strategies in cross-country skiing and road cycling. It was shown that the constant pace strategy is not optimal if ambient conditions change over the course distance. However, variable power distributions were shown beneficial if they vary in parallel with course inclination and ambient winds to decrease variations in speed. Despite these power variations, speed variations were not eliminated for most variable ambient conditions. This relates to the athlete’s physiological restrictions and the effect of these are hard to predict without thorough modeling of bioenergetics and muscle fatigue. Furthermore, it vi

was shown that substantial differences in optimal power distributions were attained for various bioenergetic models.

It was also shown that optimal braking and power output distributions for cycling on courses that involve sharp bends consisted of three or four phases, depending on the length of the course and the position of the bends. The four phases distinguished for reasonably long courses were a steady-state power phase, a rolling phase, a braking phase, and an all-out acceleration phase. It was also shown that positive pacing strategies are optimal on relatively long courses in road cycling where the supply of carbohydrates are limited. Finally, results indicated that optimal pacing may overlook the effect of some ambient conditions in favor of other more influential, mechanical or physiological, aspects of locomotion.

In summary, the results showed that athletes benefit from adapting their power output with respect not only to changing course gradients and ambient winds, but also to their own physiological and biomechanical abilities, course length, and obstacles such as course bends. The results of this thesis also showed that the computed optimal pacing strategies were more beneficial for performance than a constant power distribution. In conclusion, this thesis demonstrates the feasibility of using numerical simulation and optimization to optimize pacing strategies in cross-country skiing and road cycling.

Abstract [sv]

Avhandlingen handlar om optimering av farthållningsstrategier inom längdskidåkning och landsvägscykling. Det finns ett utbrett stöd för att konstant fart och varierande effektfördelningar är optimala om endast mekaniska aspekter beaktas i dessa sporter. Ändå saknas teoretiska studier som undersöker optimal farthållning för verkliga idrottsutövare som är begränsade i sin förmåga att generera effekt genom kroppens bioenergetiska system.

Målen med den här avhandlingen är att utveckla metoder för bioenergetik och optimering av farthållningsstrategier i uthållighetsidrott. Dessutom är målet att undersöka påverkan av backar, svängar, omgivande vind och bioenergetisk modellering på den optimala farthållningsstrategin samt att utreda potentialen till prestationsförbättring med optimala farthållningsstrategier.

Avhandling presenterar matematiska modeller för optimering av farthållningsstrategier. Dessa modeller delas in i en mekanisk modell för förflyttning, en bioenergetisk modell och en optimeringsmodell. De mekaniska och bioenergetiska modellerna som presenteras i avhandlingen består av differentialekvation och optimeringsmodellen utgörs av en gradient-baserad algoritm. Den mekaniska modellen beskriver förhållandet mellan utövarens effekt och den resulterande rörelsen längs banan som ger tiden mellan start och mål. Den bioenergetiska modellen beskriver människokroppens olika energisystem och dess begränsningar att generera effekt. Den bioenergetiska modellen interagerar med optimeringsmodellen genom att utgöra dess begränsningar för vad den mänskliga kroppen klarar av. Sammanfattningsvis försöker optimeringsmodellen minimera tiden mellan start och mål i den mekaniska modellen genom att variera effekten längs banan. Samtidigt ser optimeringsmetoden till att denna effektfördelning inte kränker den bioenergetiska modellen.

Studierna som ingår i avhandlingen resulterade i flera viktiga upptäckter om generella tillämpningar av farthållningsstrategier inom längdskidåkning och landsvägscykling. Det visade sig att konstant fart inte är optimalt om omgivande betingelser varierade längs banans sträckning. Däremot var varierande effektfördelning fördelaktig om den varierar parallellt med banlutning och omgivande vindpåverkan för att minska fartens variationer. Trots denna variation, visade resultaten att fartvariationerna inte eliminerades helt. Detta har att göra med utövarens fysiologiska begränsningar, vars påverkan är svår att förutspå utan genomgående modellering av bioenergetik relaterat till muskeltrötthet. Dessutom viii

visade resultaten att olika bioenergetiska metoder gav upphov till betydande skillnader i de optimala farthållningsstrategierna.

Resultaten i avhandlingen visade också att optimal effektfördelning vid kurvtagning i landsvägscykling innehåller tre eller fyra faser. The fyra faser som var utmärkande på relativt långa banor var en tröskelfas, en rullfas, en bromsfas och en maximal accelerationsfas. Resultaten visar också att positiv farthållning är optimal på relativt långa banor i landsvägscykling där tillgången på kolhydrater är begränsad. Samtidigt visade resultaten på optimala farthållningsstrategier ibland att inverkan av omgivande betingelser förbisågs till fördel för med inflytelserika betingelser som påverkar framdrivningen.

Sammantaget visar resultaten i denna avhandling att utövare gagnas av att anpassa effekten med hänsyn till varierande terräng, omgivande vind, atletens egen fysiologiska och biomekaniska förmåga, banans längd och hinder såsom kurvor. Resultaten visar också att de optimala farthållningsstrategier med varierande effektfördelning som beräknats i denna avhandling förbättrar prestationen jämfört med konstanta effektfördelningar. Sammanfattningsvis visar denna avhandling på möjligheterna att använda numerisk simulering och optimering för att optimera farthållningsstrategier i längdskidåkning och landsvägscykling.

Place, publisher, year, edition, pages
Östersund: Mid Sweden University , 2016. , 122 p.
Series
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 237
Keyword [en]
Pacing strategy, optimization, numerical simulation, equations of motion, method of moving asymptotes, cross-country skiing, cycling
Keyword [sv]
Farthållningsstrategi, optimering, numerisk simulering, rörelseekvationer, method of moving asymptotes, längdskidåkning, cykling
National Category
Applied Mechanics
Identifiers
URN: urn:nbn:se:miun:diva-26925ISBN: 978-91-88025-51-7 (print)OAI: oai:DiVA.org:miun-26925DiVA: diva2:897522
Public defence
2016-02-25, Q 221, Akademigatan 1, Östersund, 13:00 (English)
Opponent
Supervisors
Note

Vid tidpunkten för disputationen var följande delarbeten opublicerade: delarbete 5 accepterat, delarbete 6 manuskript.

At the time of the doctoral defence the following papers were unpublished: paper 5 accepted, paper 6 manuscript.

Available from: 2016-01-26 Created: 2016-01-25 Last updated: 2016-10-31Bibliographically approved
List of papers
1. Numerical optimization of pacing strategy in cross-country skiing
Open this publication in new window or tab >>Numerical optimization of pacing strategy in cross-country skiing
2013 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 47, no 6, 943-950 p.Article in journal (Refereed) Published
Abstract [en]

When studying events involving locomotive exercise,such as cross-country skiing, one generally assumesthat pacing strategies (i.e. power distributions) have a significantimpact on performance. In order to better understandthe importance of pacing strategies, a program isdeveloped for numerical simulation and optimization of thepacing strategy in cross-country ski racing. This programcomputes the optimal pacing strategy for an arbitrary athleteskiing on a delineated course. The locomotion of theskier is described by introducing the equations of motionfor cross-country skiing. A transformation of the motionequations is carried out in order to improve the simulation. Furthermore, a nonlinear optimization routine is connectedto the simulation program. Simulation and optimization areperformed on a fictional male skier. Results show that it ispossible to attain an optimal pacing strategy by simulatingcross-country skiing while connecting nonlinear optimizationroutines to the simulation. It is also shown that an optimalpacing strategy is characterized by minor variations inspeed. In our opinion, this kind of optimization could serveas essential preparations before important competitions.

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2013
Keyword
Optimization, Numerical simulation, Cross-country skiing, Pacing strategy, Power distribution
National Category
Applied Mechanics
Identifiers
urn:nbn:se:miun:diva-17529 (URN)10.1007/s00158-012-0856-7 (DOI)000318624100012 ()2-s2.0-84879103889 (Scopus ID)
Projects
Sportstech 2
Available from: 2012-12-04 Created: 2012-11-30 Last updated: 2016-01-26Bibliographically approved
2. On Optimization of Pacing Strategy in Road Cycling
Open this publication in new window or tab >>On Optimization of Pacing Strategy in Road Cycling
2013 (English)In: 6TH ASIA-PACIFIC CONGRESS ON SPORTS TECHNOLOGY (APCST), Melbourne: Elsevier, 2013, 118-123 p.Conference paper, (Refereed)
Abstract [en]

The wide-spread use of power-meters in today’s competitive road cycling has been an incentive for optimizing the distribution of power output i.e. the pacing strategy. Therefore, the aim of this study was to examine the effects of course profile on the optimal pacing strategy in road cycling. For that reason, three course profiles, built up by cubical splines, were simulated with a numerical program for a fictional cyclist. The numerical program solves the equations of motion of the athlete and bicycle while an optimal design algorithm is connected to the simulation, aiming to minimize the time between start and finish. The optimization is constrained by a power-endurance concept named the critical power model for intermittent exercise. Three course profiles with the same total elevation but different number of hills were studied. The time gains of an optimized pacing strategy were 3.0%, 5.0%, and 2.3% and the speed variances at the optimized pacing strategy were 6.54%, 1.18%, and 0.84% for the single plateau, double hill, and quadruple hill courses respectively. Hence, the course profile has great effect on the optimal pacing strategy. In addition, the results show that the potential improvement of adopting an optimized pacing strategy is substantial at the highest level of competition.

Place, publisher, year, edition, pages
Melbourne: Elsevier, 2013
Series
Procedia Engineering, ISSN 1877-7058 ; 60
Keyword
Numerical optimization, simulation, pacing strategy, road cycling
National Category
Applied Mechanics
Identifiers
urn:nbn:se:miun:diva-19816 (URN)10.1016/j.proeng.2013.07.062 (DOI)000326266100020 ()2-s2.0-84891710737 (Scopus ID)
Conference
6th Asia-Pacific Conference on Sports Technology, APCST 2013; Hong Kong; Hong Kong; 18 September 2013 through 20 September 2013; Code 101817
Projects
Sportstech II
Funder
EU, FP7, Seventh Framework Programme
Available from: 2013-09-02 Created: 2013-09-02 Last updated: 2016-01-26Bibliographically approved
3. Comparing bioenergetic models for the optimisation of pacing strategy in road cycling
Open this publication in new window or tab >>Comparing bioenergetic models for the optimisation of pacing strategy in road cycling
2014 (English)In: Sports Engineering, ISSN 1369-7072, E-ISSN 1460-2687, Vol. 17, no 4, 207-215 p.Article in journal (Refereed) Published
Abstract [en]

Road cycling performance is dependent on race tactics and pacing strategy. To optimise the pacing strategy for any race performed with no drafting, a numerical model was introduced, one that solves equations of motion while minimising the finishing time by varying the power output along the course. The power output was constrained by two different hydraulic models: the simpler critical power model for intermittent exercise (CPIE) and the more sophisticated Margaria–Morton model (M–M). These were compared with a constant power strategy (CPS). The simulation of the three different models was carried out on a fictional 75 kg cyclist, riding a 2,000 m course. This resulted in finishing times of 162.4, 155.8 and 159.3 s and speed variances of 0.58, 0.26 and 0.29 % for the CPS, CPIE and M–M simulations, respectively. Furthermore, the average power output was 469.7, 469.7 and 469.1 W for the CPS, CPIE and M–M simulations, respectively. The M–M model takes more physiological phenomena into consideration compared to the CPIE model and, therefore, contributes to an optimised pacing strategy that is more realistic. Therefore, the M–M model might be more suitable for future studies on optimal pacing strategy, despite the relatively slower finishing time.

Place, publisher, year, edition, pages
Springer London, 2014
National Category
Other Mechanical Engineering
Identifiers
urn:nbn:se:miun:diva-21879 (URN)10.1007/s12283-014-0156-0 (DOI)2-s2.0-84920253100 (Scopus ID)
Funder
EU, FP7, Seventh Framework Programme
Available from: 2014-04-28 Created: 2014-04-28 Last updated: 2016-01-26Bibliographically approved
4. The influence of course bends on pacing strategy in road cycling
Open this publication in new window or tab >>The influence of course bends on pacing strategy in road cycling
2014 (English)In: Conference Proceedings of The Engineering of Sport 10 / [ed] David James, Simon Choppin, Tom Allen, Jon Wheat, Paul Fleming, Elsevier, 2014, 835-840 p.Conference paper, (Refereed)
Abstract [en]

Road cycling races in general, but particularly criteriums (short circuit race), have a considerable number of bends along the race course. Sharp bends force the rider to decelerate in order to retain the grip between the tires and the road. This study focused on how these course bends influence the optimal pacing strategy in road cycling. For this purpose, we used a numerical model that simulates cycling by solving the equation of motion. The optimisation was carried out with the Method of Moving Asymptotes, constrained with the Margaria-Morton model for human energetics and a separate course bend constraint. The results showed that sharp course bends greatly affect the pacing strategy and finishing time. The average power output and the average speed decreased with a decrease in the curve radius. Moreover, the kinetic energy lost due to braking in sharp course bends is likely to be the crucial mechanism affecting the finishing time. Therefore, we believe that the outcome of races that contain sharp bends may be strongly dependent on the athlete’s pacing strategy.

Place, publisher, year, edition, pages
Elsevier, 2014
Series
Procedia Engineering, ISSN 1877-7058 ; 72
Keyword
Pacing strategy; power distribution; cycling; bend; performance; optimisation
National Category
Applied Mechanics
Identifiers
urn:nbn:se:miun:diva-22843 (URN)10.1016/j.proeng.2014.06.141 (DOI)000346367700140 ()2-s2.0-84903794163 (Scopus ID)
Conference
The 2014 Conference of the International Sports Engineering Association
Available from: 2014-09-09 Created: 2014-09-09 Last updated: 2016-01-26Bibliographically approved
5. On a bioenergetic four compartment model for human exercise
Open this publication in new window or tab >>On a bioenergetic four compartment model for human exercise
2016 (English)In: Sports Engineering, ISSN 1369-7072, E-ISSN 1460-2687, Vol. 19, no 4, 251-263 p.Article in journal (Refereed) Published
Abstract [en]

Bioenergetic models for exercise performance simulations and pacing strategy optimizations currently lag behind empirical knowledge in human bioenergetics. Therefore, the objective of this study was the construction of a four compartment bioenergetic model that incorporates separate oxidative phosphorylation of lipids and carbohydrates and describes the regulation of these energy substrates’ utilization. Furthermore, the aim was also to model efficiency and the impact of muscle fatigue and the force-velocity relationship on the maximal attainable rate of energy expenditure. The model was formulated with five systems of differential equations that regulated the fluid levels in three of the compartments, while the lipid compartment energy was kept constant. Regulations had to be imposed on the system of compartments to achieve the desired carbohydrate dependent functionality and efficiency of the model. Equilibrium equations were modeled for the alactic compound composition and a constraint was modeled for the maximal energy expenditure rate, dependent on the intramuscular inorganic phosphate. A separate force-velocity relationship was modeled to constrain power output at low speeds and efficiency was modeled with a linear but off-set relationship between power output and rate of energy expenditure. The relative aerobic contribution to total energy expenditure showed good congruence with empirical results, while time to exhaustion was overestimated due to the constraint on maximal rate of energy expenditure. Therefore, further experimental studies are necessary for complete validation of the model.

Keyword
Bioenergetics, Compartment model, Differential equation, Power, Performance, Substrate utilization
National Category
Other Mathematics Other Mechanical Engineering Physiology
Identifiers
urn:nbn:se:miun:diva-26923 (URN)10.1007/s12283-016-0205-y (DOI)000387943400004 ()2-s2.0-84966388954 (Scopus ID)
Note

First Online: 06 May 2016

Available from: 2016-01-25 Created: 2016-01-25 Last updated: 2016-12-13Bibliographically approved
6. Numerical optimization of pacing strategies for variable wind conditions in road cycling
Open this publication in new window or tab >>Numerical optimization of pacing strategies for variable wind conditions in road cycling
(English)Manuscript (preprint) (Other academic)
Abstract [en]

It has been shown theoretically that performance can be enhanced by varying power in parallel with variable ambient conditions. However, no theoretical model has considered aerobic substrate utilization dynamics, limited carbohydrate stores, force-velocity relationships, and proper efficiency modelling. Furthermore, no study has investigated optimal pacing for courses with continuously variable ambient wind directions. Therefore, the aim of this study was to develop a model for the optimization of pacing strategies in road cycling with an updated bioenergetics model. The purpose of this model was to optimize pacing strategies for courses with continuously variable wind directions on both short (2 km) and long (100 km) courses. For this purpose, a numerical model consisting of three sub-models was programmed into the MATLAB software. This model consisted of one mechanical simulation model for cycling locomotion, one bioenergetics model based on the Margaria-Morton-Sundström model, and the method of moving asymptotes for optimization of the pacing strategy. Results showed that by optimizing the pacing strategies, time gains of 4.9 and 5.7% were attained for the 2 km courses with and without an ambient wind of 5 m·s-1 respectively. The corresponding time gains for the 100 km courses were 1.4 and 2.0% with and without ambient wind respectively. The theoretical model in this study further resulted in all-out strategies for the flat 2 km courses with and without ambient wind. Moreover, the 100 km course without wind was met with a positive pacing strategy and the 100 km course with ambient wind was met with a compromise of positive pacing and variable power distribution in parallel with the variable ambient wind conditions. In conclusion, the model presented in this study performed more detailed bioenergetic simulations than previous pacing strategy optimization studies and this resulted in more detailed pacing strategies for long courses.

Keyword
Pacing strategy, Differential equation, Power, Performance, Bioenergetics
National Category
Applied Mechanics
Identifiers
urn:nbn:se:miun:diva-26924 (URN)
Available from: 2016-01-25 Created: 2016-01-25 Last updated: 2016-01-27Bibliographically approved

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