Elena Pivovarova
Publications:
Kilin A. A., Pivovarova E. N.
Dynamics of an Unbalanced Disk with a Single Nonholonomic Constraint
2023, vol. 28, no. 1, pp. 78106
Abstract
The problem of the rolling of a disk on a plane is considered under the assumption
that there is no slipping in the direction parallel to the horizontal diameter of the disk and
that the center of mass does not move in the horizontal direction. This problem is reduced to
investigating a system of three firstorder differential equations. It is shown that the reduced
system is reversible relative to involution of codimension one and admits a twoparameter family
of fixed points. The linear stability of these fixed points is analyzed. Using numerical simulation,
the nonintegrability of the problem is shown. It is proved that the reduced system admits, even
in the nonintegrable case, a twoparameter family of periodic solutions. A number of dynamical
effects due to the existence of involution of codimension one and to the degeneracy of the fixed
points of the reduced system are found.

Kilin A. A., Pivovarova E. N.
A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane
2021, vol. 26, no. 6, pp. 775786
Abstract
In this paper we investigate the motion of a Chaplygin sphere rolling without slipping on a plane performing horizontal periodic oscillations. We show that in the system under consideration the projections of the angular momentum onto the axes of the fixed coordinate system remain unchanged. The investigation of the reduced system on a fixed level set of first integrals reduces to analyzing a threedimensional period advance map on $SO(3)$. The analysis of this map suggests that in the general case the problem considered is nonintegrable. We find partial solutions to the system which are a generalization of permanent rotations and correspond to nonuniform rotations about a body and spacefixed axis. We also find a particular integrable case which, after time is rescaled, reduces to the classical Chaplygin sphere rolling problem on the zero level set of the area integral.

Kilin A. A., Pivovarova E. N.
Stability and Stabilization of Steady Rotations of a Spherical Robot on a Vibrating Base
2020, vol. 25, no. 6, pp. 729752
Abstract
This paper addresses the problem of a spherical robot having an axisymmetric pendulum drive and rolling without slipping on a vibrating plane. It is shown that this system admits partial solutions (steady rotations) for which the pendulum rotates about its vertical symmetry axis. Special attention is given to problems of stability and stabilization of these solutions. An analysis of the constraint reaction is performed, and parameter regions are identified in which a stabilization of the spherical robot is possible without it losing contact with the plane. It is shown that the partial solutions can be stabilized by varying the angular velocity of rotation of the pendulum about its symmetry axis, and that the rotation of the pendulum is a necessary condition for stabilization without the robot losing contact with the plane.

Kilin A. A., Pivovarova E. N.
Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges
2019, vol. 24, no. 2, pp. 212233
Abstract
This paper presents a qualitative analysis of the dynamics in a fixed reference frame of a wheel with sharp edges that rolls on a horizontal plane without slipping at the point of contact and without spinning relative to the vertical. The wheel is a ball that is symmetrically truncated on both sides and has a displaced center of mass. The dynamics of such a system is described by the model of the ball’s motion where the wheel rolls with its spherical part in contact with the supporting plane and the model of the disk’s motion where the contact point lies on the sharp edge of the wheel. A classification is given of possible motions of the wheel depending on whether there are transitions from its spherical part to sharp edges. An analysis is made of the behavior of the point of contact of the wheel with the plane for different values of the system parameters, first integrals and initial conditions. Conditions for boundedness and unboundedness of the wheel’s motion are obtained. Conditions for the fall of the wheel on the
plane of sections are presented.

Kilin A. A., Pivovarova E. N.
Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges
2018, vol. 23, no. 78, pp. 887907
Abstract
This paper is concerned with the dynamics of a wheel with sharp edges moving on a horizontal plane without slipping and rotation about the vertical (nonholonomic rubber model). The wheel is a body of revolution and has the form of a ball symmetrically truncated on both sides. This problem is described by a system of differential equations with a discontinuous
righthand side. It is shown that this system is integrable and reduces to quadratures. Partial solutions are found which correspond to fixed points of the reduced system. A bifurcation analysis and a classification of possible types of the wheel’s motion depending on the system parameters are presented.

Kilin A. A., Pivovarova E. N.
The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane
2017, vol. 22, no. 3, pp. 298317
Abstract
This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous righthand side. Equations describing
the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.

Borisov A. V., Kazakov A. O., Pivovarova E. N.
Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top
2016, vol. 21, no. 78, pp. 885901
Abstract
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of perioddoubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasiperiodic.

Kilin A. A., Pivovarova E. N., Ivanova T. B.
Spherical Robot of Combined Type: Dynamics and Control
2015, vol. 20, no. 6, pp. 716728
Abstract
This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point.

Ivanova T. B., Pivovarova E. N.
Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev "How to Control the Chaplygin Ball Using Rotors. II"
2014, vol. 19, no. 1, pp. 140143
Abstract
In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.
