Optimal velocity in the race over variable slope trace

• Autorzy: Maroński R. , Samoraj P.

Identyfikatory

DOI

10.5277/ABB-00098-2014-02

• Języki publikacji: EN

Abstrakty

EN

The minimum-time running problem is reconsidered. The time of covering a given distance is minimized. The function that should be found is the runner’s velocity that varies with the distance. The Hill-Keller model of motion is employed. It is based on the Newton second law and an equation of power balance. The new element of the current approach is that the trace slope angle varies with the distance. The problem is formulated and solved in optimal control applying the Chebyshev direct pseudospectral method. The essential finding is that the optimal velocity during the cruise is constant regardless of the local slope of the terrain. Such result is valid if the inequality constraints imposed on the propulsive force or the energy are not active.

Słowa kluczowe

EN

minimum-time running; variable slope trace; velocity

PL

czas jazdy; trasa; rower

• Wydawca: Oficyna Wydawnicza Politechniki Wrocławskiej
• Czasopismo: Acta of Bioengineering and Biomechanics
• Rocznik: 2015
• Tom: Vol. 17, no. 2
• Strony: 149--153
• Opis fizyczny: Bibliogr. 22 poz., wykr.

Twórcy

autor

Maroński R.

• Institute of Aeronautics and Applied Mechanics, Warsaw, Poland

autor

Samoraj P.

• Faculty of Power and Aeronautical Engineering, Warsaw University of Technology, Warsaw, Poland

Bibliografia

• [1] ANDREEVA E., BEHNCKE H., Competitive running on a hilly track, [in:] W.H. Schmidt et al. (eds.), Variational calculus, optimal control and applications, Int. Series of Numerical Mathematics, 1998, 124, Birkhauser Verlag, 241–250.
• [2] ARDIGO L.P., SAIBENE F., MINETTI A.E., The optimal locomotion on gradients: walking, running or cycling? Eur. J. Appl. Physiol., 2003, 90(3–4), 365–371.
• [3] BEHNCKE H., Optimization models for the force and energy in competitive sports, Math. Method Appl. Sci., 1987, 9, 298–311.
• [4] COOPER R.A., A force/energy optimization model for wheelchair athletics, IEEE T. Syst. Man. Cyb., 1990, 20, 444–449.
• [5] EBBEN W.P., The optimal downhill slope for the acute overspeed running, Int. J. of Sports Physiology and Performance, 2008, 3(1), 88–93.
• [6] FAHROO F., ROSS M., Direct trajectory optimization by Chebyshev pseudospectral method, J. Guid. Control. Dynam., 2002, 25, 160–166.
• [7] KELLER J.B., A theory of competitive running, Phys. Today, 1973, 26, 42–47.
• [8] KELLER J.B., Optimal velocity in a race, Am. Math. Mon., 1974, 81, 474–480.
• [9] KELLER J.B., Optimal running strategy to escape from pursuers, Am. Math. Mon., 2000, 107(5), 416–421.
• [10] MAROŃSKI R., Simple algorithm for computation of optimal velocity in running and swimming, XV ISB Congress “Biomechanics ’95”, 1995, Jyväskylä, Finland, 588–589.
• [11] MAROŃSKI R., Minimum-time running and swimming: an optimal control approach, J. Biomech., 1996, 29, 245–249.
• [12] MAROŃSKI R., Optimization of cruising velocity in recreational cycling, Acta Bioeng. Biomech., 2002, 4 (suppl. 1), 552–553.
• [13] MAROŃSKI R., ROGOWSKI K., Minimum-time running: a numerical approach, Acta Bioeng. Biomech., 2011, 13 (2), 83–86.
• [14] MUREIKA J.R., A simple model for predicting sprint race times accounting for energy losses on the curve, Can. J. Phys., 1997, 75, 837–851.
• [15] PANASZ P., MAROŃSKI R., Commercial airplane trajectory optimization by Chebyshev pseudospectral method, The Archive of Mechanical Engineering, 2005, LII(1), 5–19.
• [16] PITCHER A.B., Optimal strategies for a two-runner model of middle-distance running, SIAM J. Appl. Math., 2009, 70(4), 1032–1046.
• [17] PRITCHARD W.G., Mathematical models of running, SIAM Rev., 1993, 35, 359–378.
• [18] ROGOWSKI K., MAROŃSKI R., Driving techniques for minimizing fuel consumption during record vehicle competition, The Archive of Mechanical Engineering, 2009, LVI(1), 27–35.
• [19] ROGOWSKI K., MAROŃSKI R., Optimization of glider’s trajectory for given thermal conditions, The Archive of Mechanical Engineering, 2011, LVIII(1), 11–25.
• [20] SAMORAJ P., Optimization of strategy during locomotion using pseudospectral Chebyshev’s method – competitive running as an example, Master’s Thesis, Faculty of Power and Aeronautical Engineering, Warsaw University of Technology, 2013, (in Polish).
• [21] SCHULTZ G., MOMBAUR K., Modeling and optimal control of human-like running, IEEE-ASME Trans. on Mechatronics, 2010, 15 (5), 783–792.
• [22] WOODSIDE W., The optimal strategy for running the race, Math. Comput. Model, 1991, 15, 1–12