Abstract
We investigate the dynamics of a bicycle on an uneven mountain bike track split into straight sections with small jumps (kickers) and banked corners. A basic model of the interaction of bicycle and rider proposed and used to derive equations of motion, which capture the possibilities to accelerate the bicycle without pedaling. Since this is a first approach to the problem, only corners connected by straight lines are considered to compute optimal riding strategies. The simulation is validated with experimental data obtained on a real pump track. It is demonstrated that the model effectively captures the longitudinal bike acceleration resulting from the relative vertical motion between the rider’s upper body and the bicycle. Our numerical results are in good analogy with real rider’s actions on similar tracks.
Zusammenfassung
In diesem Beitrag wird die Dynamik eines Fahrrads auf einem sogenannten Pumptrack untersucht, also einem unebenen Rundkurs mit Sprunghügeln und Steilkurven. Ein elementares Modell der Interaktion von Fahrrad und Mensch wird eingeführt, mit welchem die Bewegungsdifferentialgleichungen hergeleitet werden. Die Dynamik umfasst insbesondere das Beschleunigen des Rades ohne zu pedalieren. Da es sich um eine initiale Untersuchung handelt, werden zunächst ausschließlich Steilkurven, welche durch gerade Streckenabschnitte verbunden sind, betrachtet. Simulationsergebnisse sowie optimale Fahrstrategien werden durch reale Versuchsdaten validiert. Es wird gezeigt, dass das vorgeschlagene Modell die Beschleunigung durch Schwerpunktverlagerung des Oberkörpers eines Radfahrers darstellen kann. Zudem zeigen die Optimierungsergebnisse Analogien zum Verhalten von Radfahrern auf ähnlichen Strecken.
About the authors
Julian Golembiewski received the M.Sc. degree in electrical engineering from TU Dortmund University, Germany, in 2021. Since early 2022, he has been pursuing his Ph.D. at the Institute of Energy Systems, Energy Efficiency, and Energy Economics at TU Dortmund University. His research centers around the optimal control of non-smooth dynamical systems, as well as optimization-based control facilitated through wireless networks.
Marcus Schmidt currently works at the Institute for Sport and Sport Science, TU Dortmund University. He studied Physical Education and Mathematics and was an athletics coach. Marcus holds a phd since June, 2018 and does research in Sprint and Jumping diagnostics by means of IMUs. He also focusses on evaluating the accuracy, reliability and validity of commercially available MEMS-sensors. Additionally, Marcus does research in the field of eccentric strength training, mainly regarding the execution quality of the Nordic Hamstring Exercise with a focus on injury prevention strategies and underlying concepts of motor learning, especially in the female athlete.
Benedikt Terschluse is a Ph.D. candidate at the Institute for Sport and Sport Science, TU Dortmund University, Germany. Holding a M.Ed. degree in Sport Science and English, he applies his expertise as a strength and conditioning coach for Borussia Dortmund's women's football team. His research interests encompass movement analysis, exercise physiology and strategies for injury prevention.
Prof. Dr Thomas Jaitner is head of the Division of Movement and Training science at TU Dortmund University since 2011. Prior to this, he was with the universities of Kaiserslautern, Leipzig and Frankfurt. His research focuses on biomechanical movement and load analysis on motor learning and technique training in sport, as well as on technological support of sports training.
Dr. Thomas Liebig is assistant professor for smart city science at the artificial intelligence group at TU Dortmund. He works on graph neural networks, probabilistic modeling of spatio-temporal data, and on privacy for distributed location aware applications. He obtained his PhD from the University of Bonn in 2013, where he studied probabilistic pedestrian modeling from incomplete data. He has published papers in International Conferences and Scientific Journals related to his area of expertise (ECML/PKDD, IEEE Big Data, IEEE IE, AGILE, JAOR, ..). He evaluates articles for numerous journals and conferences (amongst them IEEE ICDE, VAST, KDD, IJGI, KAIS).
Timm Faulwasser has studied engineering cybernetics at the University of Stuttgart. From 2008 until 2012 he was a member of the International Max Planck Research School for Analysis, Design and Optimization in Chemical and Biochemical Process Engineering Magdeburg; he obtained his Ph.D. from the Department of Electrical Engineering and Information Technology, Otto-von-Guericke University Magdeburg, Germany in 2012. After postdocs at École Polytechnique Fédérale de Lausanne, Switzerland, and at Karlsruhe Institute of Technology, Germany, he joined the Department of Electrical Engineering and Information Technology at TU Dortmund University, Germany in 2019. Currently, he serves as associate editor for IEEE Transactions on Automatic Control, IEEE CSS Letters, European Journal of Control, and Mathematics of Control Systems and Signals. His research interests are optimization and control of uncertain nonlinear systems and cyber-physical networks with applications in energy, mechatronics, process control, climate economics, and beyond. Dr. Faulwasser received the 2021-2023 Automatica Paper Prize.
Acknowledgment
We would like to give a special thanks to Jan Dieckmann, who supported this work as a test rider for all of our experiments. Furthermore, we would like to thank Dominik Modenbach, Gerrit Lang, and Maximilian Sueck for their support in data collection and analysis.
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Research ethics: The informed consent was obtained.
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Author contributions: Julian Golembiewski: conceptualization, methodology, data collection, data analysis, writing; Marcus Schmidt: methodology, data collection, data analysis, review; Benedikt Terschluse: data collection, data analysis, review; Thomas Jaitner: conceptualization, methodology, data, collection, writing, review, supervision; Thomas Liebig: conceptualization, data analysis, review, supervision; Timm Faulwasser: initial idea, conceptualization, methodology, data collection, review, supervision.
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Competing interests: The authors state no conflict of interest.
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Research funding: None declared.
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Data availability: The data can be obtained on request from the corresponding author.
Appendix A: Equations of motion
Terms for the equations of motions (3). We use the short hands s(x) ≔ sin(x) and c(x) ≔ cos(x):
with
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