Abstract:
The problem of a sphere rolling on a plane without twisting or slipping is considered. It is required to roll the sphere from one contact configuration to another so that the length of the curve described by the contact point is minimal. A parametrization of extremal trajectories is obtained. The asymptotics of extremal trajectories and the behaviour of the Maxwell time for the rolling of a sphere over sinusoids of small amplitude are studied; for such trajectories estimates for the so-called cut time are obtained.
Bibliography: 21 titles.
Keywords:
optimal control, geometric methods, symmetries of the exponential map, rolling of surfaces, Euler elastics.
Citation:
A. P. Mashtakov, Yu. L. Sachkov, “Extremal trajectories and the asymptotics of the Maxwell time in the problem of the optimal rolling of a sphere on a plane”, Sb. Math., 202:9 (2011), 1347–1371
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\by A.~P.~Mashtakov, Yu.~L.~Sachkov
\paper Extremal trajectories and the asymptotics of the Maxwell time in the problem of the optimal rolling of a~sphere on a~plane
\jour Sb. Math.
\yr 2011
\vol 202
\issue 9
\pages 1347--1371
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Linking options:
https://www.mathnet.ru/eng/sm7762
https://doi.org/10.1070/SM2011v202n09ABEH004190
https://www.mathnet.ru/eng/sm/v202/i9/p97
This publication is cited in the following 11 articles:
Yu. L. Sachkov, “Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions”, Russian Math. Surveys, 78:1 (2023), 65–163
Seyed Amir Tafrishi, Mikhail Svinin, Motoji Yamamoto, Yasuhisa Hirata, “A geometric motion planning for a spin-rolling sphere on a plane”, Applied Mathematical Modelling, 121 (2023), 542
Yu. L. Karavaev, “Spherical Robots:
An Up-to-Date Overview of Designs and Features”, Rus. J. Nonlin. Dyn., 18:4 (2022), 709–750
Alexey Mashtakov, 2021 International Conference “Nonlinearity, Information and Robotics” (NIR), 2021, 1
E. A. Mityushov, N. E. Misyura, S. A. Berestova, “Kvaternionnaya model programmnogo upravleniya dvizheniem shara Chaplygina”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 29:3 (2019), 408–421
A. P. Mashtakov, “O mnozhestve razreza na dvukhstupennykh svobodnykh gruppakh Karno”, Programmnye sistemy: teoriya i prilozheniya, 9:4 (2018), 319–360
Lazureanu C., Binzar T., “Symmetries and Properties of the Energy-Casimir Mapping in the Ball-Plate Problem”, Adv. Math. Phys., 2017, 5164602
I. Yu. Beschastnyi, “The optimal rolling of a sphere, with twisting but without slipping”, Sb. Math., 205:2 (2014), 157–191
Yu. L. Sachkov, E. F. Sachkova, “Exponential mapping in Euler's elastic problem”, J. Dyn. Control Syst., 20:4 (2014), 443–464
A. P. Mashtakov, “Algoritmicheskoe i programmnoe obespechenie resheniya konstruktivnoi zadachi upravleniya negolonomnymi pyatimernymi sistemami”, Programmnye sistemy: teoriya i prilozheniya, 3:1 (2012), 3–29
Citation in format AMSBIB
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