Publications

This paper is concerned with the problem of an ellipsoid of revolution rolling on a horizontal plane under the assumption that there is no slipping at the point of contact and no spinning about the vertical. A reduction of the equations of motion to a fixed level set of first integrals is performed. Permanent rotations corresponding to the rolling of an ellipsoid in a circle or in a straight line are found. A linear stability analysis of permanent rotations is carried out. A complete classification of possible trajectories of the reduced system is performed using a bifurcation analysis. A classification of the trajectories of the center of mass of the ellipsoid depending on parameter values and initial conditions is performed.

This paper treats the problem of a spherical robot with an axisymmetric pendulum drive rolling without slipping on a vibrating plane. The main purpose of the paper is to investigate the stabilization of the upper vertical rotations of the pendulum using feedback (additional control action). For the chosen type of feedback, regions of asymptotic stability of the upper vertical rotations of the pendulum are constructed and possible bifurcations are analyzed. Special attention is also given to the question of the stability of periodic solutions arising as the vertical rotations lose stability.

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 first-order differential equations. It is shown that the reduced system is reversible relative to involution of codimension one and admits a two-parameter 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 two-parameter 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.

In this paper, we address the problem of an ellipsoid with axisymmetric mass distribution rolling on a horizontal absolutely rough plane under the assumption that the supporting plane performs periodic vertical oscillations. In the general case, the problem reduces to a system with one and a half degrees of freedom. In this paper, instead of considering exact equations, we use a vibrational potential that describes approximately the dynamics of a rigid body on a vibrating plane. Since the vibrational potential is invariant under rotation about the vertical, the resulting problem with the additional potential is integrable. For this problem, we analyze the influence of vibrations on the linear stability of vertical rotations of the ellipsoid.

This paper investigates the controlled motion of a spherical robot on a plane performing horizontal periodic oscillations. The spherical robot is modeled by a balanced dynamically asymmetric sphere (the Chaplygin sphere) with three noncoplanar gyrostats placed inside it. The motion of the sphere is controlled by the controlled rotation of the internal gyrostats. The paper addresses two control problems concerning the construction of controls which generate motion along a trajectory given either on a moving plane or in a fixed frame of reference. It is shown that, using a control torque constant in the fixed frame of reference, the general problem can be reduced to the problem of control on the zero level set of the angular momentum integral. It is proved that, on the zero level set of the angular momentum integral, the system under consideration is completely controllable according to the Rashevsky – Chow theorem. Control algorithms for the motion of the sphere along an arbitrary prescribed trajectory are constructed. Examples are given of controls for the sphere rolling in a straight line in an arbitrary direction and in a circle, and for the sphere turning so that the position of the center of mass, both relative to the moving plane and relative to the fixed frame of reference, remains unchanged.

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 three-dimensional 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 space-fixed 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.

This paper is concerned with the analysis of the influence of the friction model on the existence of additional integrals of motion in a system describing the sliding of a spherical top on a plane. We consider a model in which the friction is described not only by the force applied at the point of contact, but also by an additional friction torque. It is shown that, depending on the chosen friction model, the system admits various first integrals. In particular, we give examples of friction models in which either the Jellett integral or the Lagrange integral or the area integral is preserved.

In this paper, we analyze the effect which the choice of a frictionmodel has on tippe top inversion in the case where the resulting action of all dissipative forces is described not only by the force applied at the contact point, but also by the additional rolling resistance torque. We show that the possibility or impossibility of tippe top inversion depends on the existence of specific integrals of the motion of the system. In this paper, we consider an example of the law of rolling resistance by which the area integral is preserved in the system.We examine in detail the case where the action of all dissipative forces reduces to the horizontal rolling resistance torque. This model describes fast rotations of the top between two horizontal smooth planes. For this case, we find permanent rotations of the system and analyze their linear stability. The stability analysis suggests that no tippe top inversion is possible under fast rotations between two planes.

This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the motion of the Lagrange top. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.

This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the Lagrange top motion. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.

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.

This paper presents a review of papers devoted to the creation and investigation of spherical robots. Owing to their structural features, namely, geometric symmetry and resistance of actuating mechanisms and elements of the control system against aggressive environmental conditions, robotic systems of this type have a high potential of being used in problems of monitoring, reconnaissance, and transportation on Earth and other planets. A detailed description is given of the structures of prototypes of spherical robots which use different actuation principles: a spherical robot with an internal pendulum mechanism, a spherical robot with an internal omniwheel platform, a spherical robot with internal rotors, and a spherical robot of combined type. Experimental results are presented to give an estimate of the possibility and efficiency of controlled motion. The applied use of spherical robots depending on the type of the actuating mechanism is discussed.

This paper is devoted to investigations of a spherical robot rolling on an oscillating underlying surface. The model of spherical robot of combined type is considered. Based on analysis of equations of motion taking into account the oscillation of the underlying surface, a control algorithm for stabilization of the spherical robot is proposed. The influences of oscillations in horizontal and vertical directions are evaluated. The design of the spherical robot and its control system for fabrication of a prototype are described.

In this paper we consider the control of the motion of a dynamically asymmetric unbalanced ball (Chaplygin top) by means of two perpendicular rotors. We propose a mechanism for control by periodically changing the gyrostatic momentum of the system, which leads to an unbounded speedup. We then formulate a general hypothesis of the mechanism for speeding up spherical bodies on a plane by periodically changing the system parameters.

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.

In this paper, we develop a model of a controlled spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum, with a feedback that stabilizes given partial solutions for a free system at the final stage of motion. According to the proposed approach, feedback depends on phase variables (current position, velocities) and does not depend on the specific type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulations of motion with feedback that demonstrate the efficiency of the proposed model.

In this work we consider the controlled motion of a pendulum spherical robot on an inclined plane. The algorithm for determining the control actions for the motion along an arbitrary trajectory and examples of numerical simulation of the controlled motion are given.

This paper is concerned with a model of the controlled motion of a spherical robot with an axisymmetric pendulum actuator on an inclined plane. First integrals of motion and partial solutions are presented and their stability is analyzed. It is shown that the steady solutions exist only at an inclination angle less than some critical value and only for constant control action.

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 right-hand 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.

In this paper, we develop a model of a controlled spherical robot with an axisymmetric pendulum-type actuator with a feedback system suppressing the pendulum’s oscillations at the final stage of motion. According to the proposed approach, the feedback depends on phase variables (the current position and velocities) and does not depend on the type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulation of motion using feedback to illustrate compensation of the pendulum’s oscillations.

In the paper, a study of rolling of a dynamically asymmetrical unbalanced ball (Chaplygin top) on a horizontal plane under the action of periodic gyrostatic moment is carried out. The problem is considered in the framework of the model of a rubber body, i.e., under the assumption that there is no slipping and spinning at the point of contact. It is shown that, for certain values of the parameters of the system and certain dependence of the gyrostatic moment on time, an acceleration of the system, i.e., an unbounded growth of the energy of the system, is observed. Investigations of the dependence of the presence of acceleration on the parameters of the system and on the initial conditions are carried out. On the basis of the investigations of the dynamics of the frozen system, a conjecture concerning the general mechanism of acceleration at the expense to periodic impacts in nonholonomic systems is expressed.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.