Geodesic
Generalized Chaplygins transformation and explicit integration of a system with a spherical support
Russian journal of nonlinear dynamics, Tome 7 (2011) no. 2, pp. 313-338.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the problem of explicit integration and bifurcation analysis for two systems of nonholonomic mechanics. The first one is the Chaplygin's problem on no-slip rolling of a balanced dynamically non-symmetrical ball on a horizontal plane. The second problem is on the motion of rigid body in a spherical support. We explicitly integrate this problem by generalizing the transformation which Chaplygin applied to the integration of the problem of the rolling ball at a non-zero constant of areas. We consider the geometric interpretation of this transformation from the viewpoint of a trajectory isomorphism between two systems at different levels of the energy integral. Generalization of this transformation for the case of dynamics in a spherical support allows us to integrate the equations of motion explicitly in quadratures and, in addition, to indicate periodic solutions and analyze their stability. We also show that adding a gyrostat does not lead to the loss of integrability.
Keywords: nonholonomic mechanics, spherical support, Chaplygin ball, explicit integration, bifurcation analysis.
Mots-clés : isomorphism 
@article{ND_2011_7_2_a7,
     author = {A. V. Borisov and A. A. Kilin and I. S. Mamaev},
     title = {Generalized {Chaplygins} transformation and explicit integration of a system with a spherical support},
     journal = {Russian journal of nonlinear dynamics},
     pages = {313--338},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2011_7_2_a7/}
}
TY  - JOUR
AU  - A. V. Borisov
AU  - A. A. Kilin
AU  - I. S. Mamaev
TI  - Generalized Chaplygins transformation and explicit integration of a system with a spherical support
JO  - Russian journal of nonlinear dynamics
PY  - 2011
SP  - 313
EP  - 338
VL  - 7
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2011_7_2_a7/
LA  - ru
ID  - ND_2011_7_2_a7
ER  - 
%0 Journal Article
%A A. V. Borisov
%A A. A. Kilin
%A I. S. Mamaev
%T Generalized Chaplygins transformation and explicit integration of a system with a spherical support
%J Russian journal of nonlinear dynamics
%D 2011
%P 313-338
%V 7
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2011_7_2_a7/
%G ru
%F ND_2011_7_2_a7
A. V. Borisov; A. A. Kilin; I. S. Mamaev. Generalized Chaplygins transformation and explicit integration of a system with a spherical support. Russian journal of nonlinear dynamics, Tome 7 (2011) no. 2, pp. 313-338. http://geodesic.mathdoc.fr/item/ND_2011_7_2_a7/

[1] Borisov A. V., Bolsinov A. V., Mamaev I. S., “Topologiya i ustoichivost integriruemykh sistem”, UMN, 65:2(392) (2010), 71–132 | MR | Zbl

[2] Borisov A. V., Mamaev I. S., Dinamika tverdogo tela: Gamiltonovy metody, integriruemost, khaos, NITs «Regulyarnaya i khaoticheskaya dinamika», M.–Izhevsk, 2005, 608 pp.

[3] Borisov A. V., Mamaev I. S., Kilin A. A., Izbrannye zadachi negolonomnoi mekhaniki, Inst. kompyutern. issled., M.–Izhevsk, 2005 elektronnyi preprint: http://ics.org.ru/doc?book=30&dir=r

[4] Veselov A. P., Veselova L. E., “Integriruemye negolonomnye sistemy na gruppakh Li”, Matem. zametki, 44:5 (1988), 604–619 | MR

[5] Veselova L. E., “Novye sluchai integriruemosti uravnenii dvizheniya tverdogo tela pri nalichii negolonomnoi svyazi”, Geometriya, differentsialnye uravneniya i mekhanika, Sb. st., MGU, M., 1986, 64–68

[6] Markeev A. P., “Ob integriruemosti zadachi o kachenii shara s mnogosvyaznoi polostyu, zapolnennoi idealnoi zhidkostyu”, MTT, 21:1 (1986), 64–65

[7] Fëdorov Yu. N., “O dvizhenii tverdogo tela v sharovom podvese”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 1988, no. 5, 91–93

[8] Fëdorov Yu. N., “O dvukh integriruemykh negolonomnykh sistemakh v klassicheskoi dinamike”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 1989, no. 4, 38–41 | MR

[9] Chaplygin S. A., “O katanii shara po gorizontalnoi ploskosti”, Matem. sb., 24:1 (1903), 139–168 | Zbl

[10] Chaplygin S. A., “O katanii shara po gorizontalnoi ploskosti”, Sobr. soch., v. 1, ed. S. A. Chaplygin, GITTL, M.–L., 1948, 76–101

[11] Borisov A. V., Mamaev I. S., “Rolling of a non-homogeneous ball over a sphere without slipping and twisting”, Regul. Chaotic Dyn., 12:2 (2007), 153–159 | DOI | MR

[12] Chung W., Nonholonomic manipulators, Springer Tr. Adv. Robot., 13, Springer, Berlin–New York, 2004, 114 pp.

[13] Duistermaat J. J., Chaplygin's sphere, arXiv: math/0409019v1

[14] Eisenhart L. P., “Separable systems of Stackel”, Ann. of Math. (2), 35:2 (1934), 284–305 | DOI | MR

[15] “Jovanovic B. $LR$ and $L+R$ systems”, J. Phys. A, 42 (2009), 225202, 18 pp. | DOI | MR | Zbl

[16] Kilin A. A., “The dynamics of Chaplygin ball: The qualitative and computer analysis”, Regul. Chaotic Dyn., 6:3 (2001), 291–306 | DOI | MR | Zbl

[17] Koiller J., Ehlers K., “Rubber rolling over a sphere”, Regul. Chaotic Dyn., 12:2 (2007), 127–152 | DOI | MR

[18] Kumagai M., Ochiai T., “Development of a robot balancing on a ball”, Intern. Conf. on Control, Automation and Systems (Seoul, Oct. 14–17, 2008), 2008, 433–438

[19] Lauwers T. B., Kantor G. A., Hollis R. L., “A dynamically stable single-wheeled mobile robot with inverse mouse-ball drive”, Proc. IEEE Intern. Conf. on Robotics and Automation (Orlando, FL, May 15–19, 2006), 2006, 2884–2889

[20] Nagarajan U., Kantor G., Hollis R. L., “Trajectory planning and control of an underactuated dynamically stable single spherical wheeled mobile robot”, IEEE Intern. Conf. on Robotics and Automation (Kobe, Japan, May 12–17, 2009), 2009, 3743–3748

[21] Shen J., Schneider D. A., Bloch A. M., “Controllability and motion planning of a multibody Chaplygin's sphere and Chaplygin's Top”, Int. J. Robust Nonlinear Control, 18:9 (2008), 905–945 | DOI | MR

[22] Wilson J. L., Mazzoleni A. P., DeJarnette F. R., Antol J., Hajos G. A., Strickland C. V., “Design, analysis, and testing of Mars tumbleweed rover concepts”, J. Spacecraft Rockets, 45:2 (2008), 370–382 | DOI

[23] Xu Ya., Ou Yo., Control of single wheel robots, Springer Tr. Adv. Robot., 20, Springer, Berlin, 2005, 188 pp.

[24] http://www.dyson.co.uk/