Omniwheel implementation of the Suslov problem with a rheonomic constraint: dynamic model and control

E. A. Mikishanina

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 147, pp. 296-308

Résumé

The classical Suslov problem of the motion of a rigid body with a fixed point is well known and has been studied in detail. In this paper, an omniwheel implementation of the Suslov problem is proposed. The controlled motion of a rigid body with a fixed point in the presence of scleronomic nonholonomic constraints and rheonomic artificial kinematic constraint is considered. The rigid body rotates around a fixed point, rolls around a spherical shell from the inside and contacts it by means of omniwheels with a differential actuator. We believe that the omniwheels are in contact with the spherical shell only at one point. In order to subordinate the motion of the rigid body to an artificial rheonomic constraint, a differential actuator creates control torques on omniwheels. Based on the d’Alembert–Lagrange principle, equations of motion of the mechanical system with indeterminate multipliers specifying constraint reactions are constructed. The problem is reduced to the study of a non-autonomous two-dimensional dynamical system. Using the generalized Poincar.e transformation, the study of a two-dimensional dynamical system is reduced to the study of the stability of a one-parameter family of fixed points for a system of differential equations with a degenerate linear part. We determine numerical parameters for which phase trajectories of the system are bounded and for which phase trajectories of the system are unbounded. The results of the study are illustrated graphically. Based on numerical integration, maps for the period (Poincar.e sections) and a map of dynamic regimes are constructed to confirm the Feigenbaum scenario of transition to chaotic dynamics.

Détail

Keywords: dynamic model, control, Suslov problem, omniwheel, nonholonomic constraint, rheonomic constraint, feedback

Bibliographie

[1] G. K. Suslov, Theoretical mechanics, Gostekhizdat, Moscow-Leningrad, 1946 (In Russian)

[2] V. Vagner, “Geometric interpretation of the motion of nonholonomic dynamical systems”, Proceedings of the Seminar on Vector and Tensor Analysis, 1941, no. 5, 301–327 (In Russian)

[3] L. C. Garcia–Naranjo, A. J. Maciejewsk, J. C. Marrero, M. Przybylska, “The inhomogeneous Suslov problem”, Physics Letters A, 378:32–33 (2013), 2389–2394

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

[5] A. V. Borisov, I. S. Mamaev, A. A. Kilin, “Hamiltonicity and integrability of the Suslov problem”, Regular and Chaotic Dynamics, 16:1–2 (2011), 104–116 | DOI

[6] A. D. Bilimovitch, “Sur les systemes conservatifs, non holonomes avec des liaisons dependantes du temps”, Comptes rendus de l'Académie des Sciences, 156 (1913), 12–18

[7] A. V. Borisov, A. V. Tsiganov, E. A. Mikishanina, “On inhomogeneous nonholonomic Bilimovich system”, Communications in Nonlinear Science and Numerical Simulation, 94 (2021), 105573 | DOI

[8] V. I. Kirgetov, “The motion of controlled mechanical systems with prescribed constraints (servoconstraints)”, Journal of Applied Mathematics and Mechanics, 31:3 (1967), 465–477 | DOI

[9] E. A. Mikishanina, “Rolling motion dynamics of a spherical robot with a pendulum actuator controlled by the Bilimovich servo-constraint”, Theoretical and Mathematical Physics, 211:2 (2022), 679–691 | DOI | DOI

[10] V. V. Kozlov, “The dynamics of systems with servoconstraints. I”, Regular and Chaotic Dynamics, 20:3 (2015), 205–224 | DOI

[11] N. N. Bautin, E. A. Leontovich, Methods and Techniques of Qualitative Research of Dynamical Systems on a Plane, Nauka Publ., Moscow, 1990 (In Russian)

[12] I. Bizyaev, S. Bolotin, I. Mamaev, “Normal forms and averaging in an acceleration problem in nonholonomic mechanics”, Chaos, 31:1 (2021), 013132 | DOI

[13] E. A. Mikishanina, “The problem of acceleration in the dynamics of a double-link wheeled vehicle with arbitrarily directed periodic exitation”, Theoretical and Applied Mechanics, 50:2 (2023), 205–221 | DOI

[14] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, World Scientific, Singapore, New Jersey, London, Hong Kong, 2001

[15] V. R. Barseghyan, T. A. Simonyan, A. G. Matevosyan, “On one problem of quadcopter control with given intermediate values of different parts of coordinates”, Vestnik rossiyskikh universitetov. Matematika = Russian Universities Reports. Mathematics, 29:145 (2024), 29–42 (In Russian)

[16] A. A. Kilin, Yu. L. Karavaev, “The kinematic control model for a spherical robot with an unbalanced internal omniwheel platform”, Russian Journal of Nonlinear Dynamics, 10:4 (2014), 497–511 (In Russian)

[17] A. A. Kilin, Yu. L. Karavaev, “The dynamic of a spherical robot with an internal omniwheel platform”, Russian Journal of Nonlinear Dynamics, 11:1 (2015), 187–204 (In Russian)

[18] J. Carr, Applications of Centre Manifold Theory, Springer–Verlag, New York, 1981