Abstract
This paper deals with the brachistochronic motion of the Chaplygin sleigh on a horizontal plane surface. Two cases are considered, such as: the case when the magnitude of the horizontal reaction force at the contact point of the knife edge with the surface is unbounded, and the case when the magnitude of this force is bounded. The problem is solved by applying Pontryagin’s maximum principle and singular optimal control theory. The angular acceleration of the sleigh is taken for a control variable. The problem considered is reduced to solving the corresponding two-point boundary value problem. In order to solve the obtained boundary value problem, an appropriate numerical procedure based on the shooting method is presented. Also, the paper analyzes the influence of the bounded magnitude of the reaction force on the structure of the controller sequence. It is shown that in the case of unbounded magnitude of the reaction force, the control is singular, and in the case of bounded magnitude, the control is, in a general case, a combination of singular and bang–bang controls.
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Chaplygin, S.A.: On the theory of motion of nonholonomic systems. The reducing-multiplier theorem. Math. Collect. 28(1), 303–314 (1911). [English Translation by A. V. Getling. Regular and Chaotic Dynamics. 13(4), 369–376 (2008)]
Caratheodory C.: Der Schlitten. ZAMM-Z. Angew. Math. Me. 13, 71–76 (1933)
Papastavridis J.G.: Time-integral theorems for nonholonomic systems. Int. J. Eng. Sci. 25(7), 833–854 (1987)
Borisov A.V., Mamayev I.S.: The dynamics of a Chaplygin sleigh. PMM-J. Appl. Math. Mech. 73, 156–161 (2009)
Fedorov, Y.N., Garca-Naranjo, L.C.: The hydrodynamic Chaplygin sleigh. J. Phys. A: Math. Theor. 43, 434013 (2010)
Antunes A.C.B., Sigaud C.: Controlling nonholonomic Chaplygin systems. Braz. J. Phys. 40(2), 131–140 (2010)
Neimark Ju. I., Fufaev N.A.: Dynamics of Nonholonomic Systems. Nauka, Moscow (1967)
Dobronravov V.V.: Foundations of Mechanics of Nonholonomic Systems. Vischaya Schkola, Moscow (1970)
Djukić, Dj.: On the brachistochronic motion of the nonholonomic mechanical systems. In: Proceedings of the 14th Yugoslavia Congress of Rational and Applied Mechanics, Portorož, pp. 73–80 (1978)
Obradović A., Čović V., Vesković M., Drazić M.: Brachistochronic motion of a nonholonomic rheonomic mechanical system. Acta Mech. 214, 291–304 (2010)
Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F.: The Mathematical Theory of Optimal Processes. Wiley, New Jersey (1962)
Bryson A.E., Ho Y.C.: Applied Optimal Control. Hemisphere, New York (1975)
Jeremić O., Šalinić S., Obradović A., Mitrović Z.: On the brachistochrone of a variable mass particle in general force fields. Math. Comput. Model. 54(11–12), 2900–2912 (2011)
Kelley H., Kopp R.E., Moyer G.H.: Singular extremals. In: G., Leitmann (ed.) Mathematics in Science and Engineering, vol. 31. Topics in Optimization, pp. 63–101. Academic Press, New York (1967)
McDanell J.P., Powers W.F.: Necessary conditions for joining optimal singular and nonsingular subarcs. SIAM J. Control 9(2), 161–173 (1971)
Stoer J., Bulirsch J.: Introduction to Numerical Analysis. Springer, Berlin (1993)
Chen Y.: Existence and structure of minimum-time control for multiple robot arms handling a common object. Int. J. Control 53(4), 855–869 (1991)
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Šalinić, S., Obradović, A., Mitrović, Z. et al. On the brachistochronic motion of the Chaplygin sleigh. Acta Mech 224, 2127–2141 (2013). https://doi.org/10.1007/s00707-013-0878-2
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DOI: https://doi.org/10.1007/s00707-013-0878-2