The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane
Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 298-317
Abstract
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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 right-hand 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.

Keywords:

integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, absolute dynamics

Citation:

Kilin A. A., Pivovarova E. N., The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 298-317

Stability analysis of steady motions of a spherical robot of combined type
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 611–623
Abstract
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The dynamics of a spherical robot of combined type consisting of a spherical shell and a pendulum attached at the center of the shell is considered. At the end of the pendulum a rotor is installed. For this system we carry out a stability analysis for a partial solution which in absolute space corresponds to motion along a circle with constant velocity. Regions of stability of a partial solution are found depending on the orientation of the spherical robot during the motion, its velocity and the radius of the circle traced out by the point of contact.

Pivovarova E. N., Stability analysis of steady motions of a spherical robot of combined type, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 611–623

Regular and chaotic dynamics in the rubber model of a Chaplygin top
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 277-297
Abstract
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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 quasi-periodic.

Keywords:

Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact

Citation:

Borisov A. V., Kazakov A. O., Pivovarova E. N., Regular and chaotic dynamics in the rubber model of a Chaplygin top, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 2, pp. 277-297

Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top
Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901
Abstract
pdf (2.21 Mb)

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 period-doubling 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 quasi-periodic.

Keywords:

Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact

Citation:

Borisov A. V., Kazakov A. O., Pivovarova E. N., Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901

Influence of rolling friction on the controlled motion of a robot wheel
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 583-592
Abstract
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This paper presents an experimental investigation of the influence of rolling friction on the dynamics of a robot wheel. The robot is set in motion by changing the proper gyrostatic momentum using the controlled rotation of a rotor installed in the robot. The problem is considered under the assumption that the center of mass of the system does not coincide with its geometric center. In this paper we derive equations describing the dynamics of the system and give an example of the controlled motion of a wheel by specifying a constant angular acceleration of the rotor. A description of the design of the robot wheel is given and a method for experimentally determining the rolling friction coefficient is proposed. For the verification of the proposed mathematical model, experimental studies of the controlled motion of the robot wheel are carried out. We show that the theoretical results qualitatively agree with the experimental ones, but are quantitatively different.

Keywords:

robot-wheel, rolling friction, displacement of the center of mass

Citation:

Pivovarova E. N., Klekovkin A. V., Influence of rolling friction on the controlled motion of a robot wheel, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 583-592

A model of a screwless underwater robot
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 544-553
Abstract
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The paper is devoted to the development of a model of an underwater robot actuated by inner rotors. This design has no moving elements interacting with an environment, which minimizes a negative impact on it, and increases noiselessness of the robot motion in a liquid. Despite numerous discussions on the possibility and efficiency of motion by means of internal masses' movement, a large number of works published in recent years confirms a relevance of the research. The paper presents an overview of works aimed at studying the motion by moving internal masses. A design of a screwless underwater robot that moves by the rotation of inner rotors to conduct theoretical and experimental investigations is proposed. In the context of theoretical research a robot model is considered as a hollow ellipsoid with three rotors located inside so that the axes of their rotation are mutually orthogonal. For the proposed model of a screwless underwater robot equations of motion in the form of classical Kirchhoff equations are obtained.

Keywords:

mobile robot, screwless underwater robot, movement in ideal fluid

Citation:

Vetchanin E. V., Karavaev Y. L., Kalinkin A. A., Klekovkin A. V., Pivovarova E. N., A model of a screwless underwater robot, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 544-553

Spherical Robot of Combined Type: Dynamics and Control
Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728
Abstract
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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.

Kilin A. A., Pivovarova E. N., Ivanova T. B., Spherical Robot of Combined Type: Dynamics and Control, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 716-728

Comments on the Paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev "How to Control the Chaplygin Ball Using Rotors. II"
Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 140-143
Abstract
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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.

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", Regular and Chaotic Dynamics, 2014, vol. 19, no. 1, pp. 140-143

Comment on the paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev “How to control the Chaplygin ball using rotors. II”
Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 127-131
Abstract
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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.

Ivanova T. B., Pivovarova E. N., Comment on the paper by A.V. Borisov, A.A. Kilin, I.S. Mamaev “How to control the Chaplygin ball using rotors. II”, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 1, pp. 127-131

Determination of moments of inertia and the position of the center of mass of robotic devices
Bulletin of Udmurt University. Physics and Chemistry, 2014, no. 4, pp. 79-86
Abstract
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In this paper we describe an inertiameter, which is an experimental facility for determining the inertia tensor components and the position of the center of mass of compound bodies. An algorithm for determining these dynamical properties is presented. Using the algorithm obtained, the displacement of the center of mass and the tensor of inertia are determined experimentally for a spherical robot of combined type.

Keywords:

inertiameter, spherical robot, moment of inertia, center of mass

Citation:

Alalykin S. S., Bogatyrev A. V., Ivanova T. B., Pivovarova E. N., Determination of moments of inertia and the position of the center of mass of robotic devices, Bulletin of Udmurt University. Physics and Chemistry, 2014, no. 4, pp. 79-86

Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator
Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 507-520
Abstract
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This paper investigates the possibility of the motion control of a ball with a pendulum mechanism with non-holonomic constraints using gaits — the simplest motions such as acceleration and deceleration during the motion in a straight line, rotation through a given angle and their combination. Also, the controlled motion of the system along a straight line with a constant acceleration is considered. For this problem the algorithm for calculating the control torques is given and it is shown that the resulting reduced system has the first integral of motion.

Keywords:

non-holonomic constraint, control, spherical shell, integral of motion

Citation:

Ivanova T. B., Pivovarova E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, Russian Journal of Nonlinear Dynamics, 2013, vol. 9, no. 3, pp. 507-520

Stability analysis of periodic solutions in the problem of the rolling of a ball with a pendulum
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 4, pp. 146-155
Abstract
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In the paper we study the stability of a spherical shell rolling on a horizontal plane with Lagrange’s gyroscope inside. A linear stability analysis is made for the upper and lower position of a top. A bifurcation diagram of the system is constructed. The trajectories of the contact point for diﬀerent values of the integrals of motion are constructed and analyzed.

Keywords:

rolling motion, stability, Lagrange’s gyroscope, bifurcational diagram

Citation:

Pivovarova E. N., Ivanova T. B., Stability analysis of periodic solutions in the problem of the rolling of a ball with a pendulum, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2012, no. 4, pp. 146-155