Investigation of the orbital stability of rectilinear motions of roller-racer on a vibrating plane

E. M. Artemova   ; A. A. Kilin   ; Yu. V. Korobeinikova

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 32 (2022) no. 4, pp. 615-629

Résumé

This paper addresses the problem of a roller-racer rolling on an oscillating plane. Equations of motion of the roller-racer in the form of a system of four nonautonomous differential equations are obtained. Two families of particular solutions are found which correspond to rectilinear motions of the roller-racer along and perpendicular to the plane's oscillations. Numerical estimates are given for the multipliers of solutions corresponding to the motion of the robot along the oscillations. Also, a special case is presented in which it is possible to obtain analytic expressions of the multipliers. In this case, it is shown that the motion along oscillations of a “folded” roller-racer is linearly orbitally stable as it moves with its joint ahead, and that all other motions are unstable. It is shown that, in a linear approximation, the family corresponding to the motion of the robot is perpendicular to the plane's oscillations, that is, it is unstable.

Détail

Mots-clés : roller-racer, monodromy matrix
Keywords: nonholonomic constraints, vibrating plane, orbital stability.

Bibliographie

[1] Chaplygin S.A., “On the theory of motion of nonholonomic systems. The reducing-multiplier theorem”, Regular and Chaotic Dynamics, 13:4 (2008), 369–376 | DOI | MR

[2] Borisov A.V., Mamaev I.S., “An inhomogeneous Chaplygin sleigh”, Regular and Chaotic Dynamics, 22:4 (2017), 435–447 | DOI | MR

[3] Bizyaev I.A., Borisov A.V., Mamaev I.S., “The Chaplygin sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration”, Regular and Chaotic Dynamics, 22:8 (2017), 955–975 | DOI | MR

[4] Osborne J.M., Zenkov D.V., “Steering the Chaplygin sleigh by a moving mass”, Proceedings of the 44th IEEE Conference on Decision and Control, 2005, 1114–1118 | DOI

[5] Kuznetsov S.P., “Regular and chaotic dynamics of a Chaplygin sleigh due to periodic switch of the nonholonomic constraint”, Regular and Chaotic Dynamics, 23:2 (2018), 178–192 | DOI | MR

[6] Bizyaev I.A., “The inertial motion of a roller racer”, Regular and Chaotic Dynamics, 22:3 (2017), 239–247 | DOI | MR

[7] Bizyaev I.A., Borisov A.V., Mamaev I.S., “Exotic dynamics of nonholonomic roller racer with periodic control”, Regular and Chaotic Dynamics, 23 (2018), 983–994 | DOI | MR

[8] Krishnaprasad P.S., Tsakiris D.P., “Oscillations, SE(2)-snakes and motion control: A study of the roller racer”, Dynamical Systems, 16:4 (2001), 347–397 | DOI | MR

[9] Mikishanina E.A., “Qualitative analysis of the dynamics of a trailed wheeled vehicle with periodic excitation”, Russian Journal of Nonlinear Dynamics, 17:4 (2021), 437–451 | DOI | MR

[10] Kilin A.A., Pivovarova E.N., “A particular integrable case in the nonautonomous problem of a Chaplygin sphere rolling on a vibrating plane”, Regular and Chaotic Dynamics, 26:6 (2021), 775–786 | DOI | MR

[11] Kilin A.A., Pivovarova E.N., “Stability and stabilization of steady rotations of a spherical robot on a vibrating base”, Regular and Chaotic Dynamics, 25:6 (2020), 729–752 | DOI | MR

[12] Kilin A.A., Pivovarova E.N., “Neintegriruemost zadachi o kachenii sfericheskogo volchka po vibriruyuschei ploskosti”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 30:4 (2020), 628–644 | DOI | MR

[13] Vetchanin E.V., Mikishanina E.A., “Vibrational stability of periodic solutions of the liouville equations”, Nelineinaya Dinamika, 15:3 (2019), 351–363 | DOI | MR

[14] Artemova E.M., Karavaev Yu.L., Mamaev I.S., Vetchanin E.V., “Dynamics of a spherical robot with variable moments of inertia and a displaced center of mass”, Regular and Chaotic Dynamics, 25:6 (2020), 689–706 | DOI | MR

[15] Yakubovich V.A., Starzhinskii V.M., Lineinye differentsialnye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya, Nauka, M., 1972 | MR

[16] Borisov A.V., Kilin A.A., Mamaev I.S., “O probleme Adamara-Gamelya i dinamike kolesnykh ekipazhei”, Nelineinaya dinamika, 12:1 (2016), 145–163 | MR

[17] Borisov A.V., Mamaev I.S., “Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems”, Regular and Chaotic Dynamics, 13:5 (2008), 443–490 | DOI | MR

[18] Demidovich B.P., Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967 | MR

[19] Yefremov K.S., Ivanova T.B., Kilin A.A., Karavaev Yu.L., “Theoretical and experimental investigations of the controlled motion of the roller racer”, International Conference Nonlinearity, Information and Robotics (NIR), 2020 | DOI

[20] Yanke E., Emde F., Lesh F., Spetsialnye funktsii. Formuly, grafiki, tablitsy, Nauka, M., 1977 | MR

[21] Krylov N.M., Bogolyubov N.N., Vvedenie v nelineinuyu mekhaniku, NITs «Regulyarnaya i khaoticheskaya dinamika», M.-Izhevsk, 2004

[22] Krasilnikov P.S., Prikladnye metody issledovaniya nelineinykh kolebanii, Institut kompyuternykh issledovanii, M.-Izhevsk, 2015