Rolling motion dynamics of a spherical robot with a pendulum actuator controlled by the Bilimovich servo-constraint

E. A. Mikishanina

Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 2, pp. 281-294

Résumé

We discuss the problem of rolling without slipping for a spherical shell with a pendulum actuator (spherical robot) installed in the geometric center of the sphere. The motion of the spherical robot is controlled by the Bilimovich servo-constraint. To implement the servo-constraint, the pendulum actuator creates a control torque. because the physical implementation of the Bilimovich constraint as a nonholonomic constraint is somewhat difficult, it can be implemented as a servo-constraint. Based on the general equations of motion, the kinematic constraints, and the servo-constraint, the equations of motion for this mechanical system are obtained. When the pendulum moves in the vertical plane at fixed levels of the first integrals, the resulting system of the equations of motion reduces to a non-Hamiltonian system with one degree of freedom. We find the conditions for the implementation of the motion program specified by the servo-constraint. The dynamics analysis is based on the study of phase portraits of the system, period maps, and plots of the desired mechanical parameters.

Détail

Keywords: dynamics, spherical robot, pendulum actuator, equation of motion, nonholonomic constraint
Mots-clés : servo-constraint, Bilimovich constraint.

Bibliographie

[1] A. V. Borisov, A. V. Tsiganov, E. A. Mikishanina, “On inhomogeneous nonholonomic Bilimovich system”, Commun. Nonlinear Sci. Numer. Simul., 94 (2021), 105573, 11 pp. | DOI | MR

[2] A. D. Bilimovich, “Sur les systèmes conservatifs, non holonomes avec des liaisons dépendantes du temps”, Comptes Rendus Acad. Sci. Paris, 156 (1913), 12–18

[3] V. V. Vagner, “Geometricheskaya interpretatsiya dvizheniya negolonomnykh dinamicheskikh sistem”, Trudy seminara po vektornomu i tenzornomu analizu s ikh prilozheniyami k geometrii, mekhanike i fizike, Vyp. 5, Izd-vo Mosk. un-ta, M., 1941, 301–327 | MR

[4] O. E. Fernandez, A. M. Bloch, D. V. Zenkov, “The geometry and integrability of the Suslov problem”, J. Math. Phys., 55:11 (2014), 112704, 14 pp. | DOI | MR

[5] L. C. García-Naranjo, A. J. Maciejewski, J. C. Marrero, M. Przybylska, “The inhomogeneous Suslov problem”, Phys. Lett. A, 378:32–33 (2014), 2389–2394, arXiv: 1310.3868 | DOI | MR

[6] A. V. Borisov, E. A. Mikishanina, “Two nonholonomic chaotic systems. Part I. On the Suslov problem”, Regul. Chaotic Dyn., 25:3 (2020), 313–322 | DOI | MR

[7] I. A. Bizyaev, I. S. Mamaev, “Dynamics of the nonholonomic Suslov problem under periodic control: unbounded speedup and strange attractors”, J. Phys. A: Math. Theor., 53:18 (2020), 185701, 17 pp. | DOI | MR

[8] A. B. Borisov, I. S. Mamaev, D. V. Treschev, “Kachenie tverdogo tela bez proskalzyvaniya i vercheniya: Kinematika i dinamika”, Nelineinaya dinam., 8:4 (2012), 783–797 | DOI | MR

[9] M. H. Beghin, Étude théorique des compas gyrostatiques Anschutz et Sperry, Thèses de l'entre-deux-guerres, 34, Impr. nationale, Paris, 1922 | MR

[10] P. Appell, Traité de mécanique rationnelle, v. 2, Dynamique des systèmes. Mécanique analytique, Gauthier-Villars, Paris, 1932 | Zbl

[11] A. G. Azizov, “K dinamike sistem, stesnennykh servosvyazyami”, Nauchnye trudy TashGU, 1971, no. 397, 3–9

[12] A. G. Azizov, “Dvizhenie upravlyaemykh mekhanicheskikh sistem s servosvyazyami”, PMM, 54:3 (1990), 366–372 | DOI

[13] V. I. Kirgetov, “O dvizhenii upravlyaemykh mekhanicheskikh sistem s uslovnymi svyazyami (servosvyazyami)”, PMM, 31:3 (1967), 433–446 | DOI | MR

[14] V. V. Kozlov, “Printsipy dinamiki i servosvyazi”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1989, no. 5, 59–66 | MR | Zbl

[15] V. V. Kozlov, “Dinamika sistem s servosvyazyami. I”, Nelineinaya dinam., 11:2 (2015), 353–376 | DOI | MR | Zbl

[16] V. V. Kozlov, “Dinamika sistem s servosvyazyami. II”, Nelineinaya dinam., 11:3 (2015), 579–611 | DOI | Zbl

[17] Ya. V. Tatarinov, Uravneniya klassicheskoi mekhaniki v lakonichnykh formakh, Izd-vo Mosk. un-ta, M., 2005

[18] A. V. Borisov, E. A. Mikishanina, “Dynamics of the Chaplygin ball with variable parameters”, Russ. J. Nonlinear Dyn., 16:3 (2020), 453–462 | DOI | MR

[19] I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Different models of rolling for a robot ball on a plane as a generalization of the Chaplygin ball problem”, Regul. Chaotic Dyn., 24:5 (2019), 560–582 | DOI | MR

[20] A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Kak upravlyat sharom Chaplygina pri pomoschi rotorov”, Nelineinaya dinam., 8:2 (2012), 289–307

[21] A. V. Borisov, A. A. Kilin, I. S. Mamaev, “The problem of drift and recurrence for the rolling Chaplygin ball”, Regul. Chaotic Dyn., 18:6 (2013), 832–859 | DOI | MR | Zbl

[22] S. V. Bolotin, “The problem of optimal control of a Chaplygin ball by internal rotors”, Regul. Chaotic Dyn., 17:6 (2012), 559–570 | DOI | MR | Zbl

[23] Yu. L. Karavaev, A. A. Kilin, “The dynamics of a spherical robot of combined type by periodic control actions”, Russ. J. Nonlinear Dyn., 15:4 (2019), 497–504 | DOI | MR

[24] A. A. Kilin, E. N. Pivovarova, T. B. Ivanova, “Spherical robot of combined type: dynamics and control”, Regul. Chaotic Dyn., 20:6 (2015), 716–728 ; “Dynamics-based motion planning for a pendulum-actuated spherical rolling robot”, 23:4 (2018), 372–388 | DOI | MR | DOI | MR