Publications in the last 5 years

In this paper we consider the dynamics of a roller bicycle on a horizontal plane. For this bicycle we derive a nonlinear system of equations of motion in a form that allows us to take into account the symmetry of the system in a natural form. We analyze in detail the stability of straight-line motion depending on the parameters of the bicycle. We find numerical evidence that, in addition to stable straight-line motion, the roller bicycle can exhibit other, more complex, trajectories for which the bicycle does not fall.

This paper is concerned with the experimental development of the stabilizing regulator for a spherical pendulum-type robot moving on an oscillating base. Using a mathematical model of the motion of the spherical robot with an internal pendulum mechanism, a regulator stabilizing the lower position of the pendulum is developed. The developed regulator has been tested in practice by means of a real prototype of the spherical robot. The results of real experiments are presented to assess the stabilization of the lower position of the pendulum of the spherical robot during its motion along a straight line on a plane executing longitudinal oscillations, and during the stabilization of the lower position of the pendulum, when the spherical shell remains fixed relative to the plane.

This paper addresses the problem of the Chaplygin sleigh moving on an inclined plane under the action of periodic controls. Periodic controls are implemented by moving point masses. It is shown that, under periodic oscillations of one point mass in the direction perpendicular to that of the knife edge, for a nonzero initial velocity there exists a motion with acceleration or a uniform motion (on average per period) in the direction opposite to that of the largest descent. It is shown that adding to the system two point masses which move periodically along some circle enables a period-averaged uniform motion of the system from rest.

This paper addresses the problem of the motion of two point vortices of arbitrary strengths in an ideal incompressible fluid on a finite flat cylinder. A procedure of reduction to the level set of an additional first integral is presented. It is shown that, depending on the parameter values, three types of bifurcation diagrams are possible in the system. A complete bifurcation analysis of the system is carried out for each of them. Conditions for the orbital stability of generalizations of von Kármán streets for the problem under study are obtained.

The problem of describing the motion of a rigid body in a fluid is addressed by considering a symmetric Joukowsky foil. Within the framework of the model of an ideal fluid, the force and torque acting on an unsteady moving foil are calculated. The analytical results are compared with those obtained based on the numerical solution of the Navier – Stokes equations. It is shown that analytical expressions for the force and torque can be consistent with the results of numerical simulations using scaling and a delayed arguments.

This paper describes the design of an aquatic robot moving on the surface of a fluid and driven by two internal moving masses. The body of the aquatic robot in cross section has the shape of a symmetrical airfoil with a sharp edge. In this prototype, two internal masses move in circles and are rotated by a single DC motor and a gear mechanism that transmits torque from the motor to each mass. Angular velocities of moving masses are used as a control action, and the developed kinematic scheme for transmitting rotation from the motor to the moving masses allows the rotation of two masses with equal angular velocities in magnitude, but with a different direction of rotation. It is also possible to install additional tail fins of various shapes and sizes on the body of this robot. Also in the work for this object, the equations of motion are presented, written in the form of Kirchhoff equations for the motion of a solid body in an ideal fluid, which are supplemented by terms of viscous resistance. A mathematical description of the additional forces acting on the flexible tail fin is presented. Experimental studies on the influence of various tail fins on the speed of motion in the fluid were carried out with the developed prototype of the robot. In this work, tail fins of the same shape and size were installed on the robot, while having different stiffness. The experiments were carried out in a pool with water, over which a camera was installed, on which video recordings of all the experiments were obtained. Next processing of the video recordings made it possible to obtain the object’s movements coordinates, as well as its linear and angular velocities. The paper shows the difference in the velocities developed by the robot when moving without a tail fin, as well as with tail fins having different stiffness. The comparison of the velocities developed by the robot, obtained in experimental studies, with the results of mathematical modeling of the system is given.

Finite-dimensional equations constructed earlier to describe the motion of an aquatic drop-shaped robot due to given rotor oscillations are studied. To study the equations of motion, we use the Poincaré map method, estimates of the Lyapunov exponents, and the parameter continuation method to explore the evolution of asymptotically stable solutions. It is shown that, in addition to the so-called main periodic solution of the equations of motion for which the robot moves in a circle in a natural way, an additional asymptotically stable periodic solution can arise under the influence of highly asymmetric impulsive control. This solution corresponds to the robot’s sideways motion near the circle. It is shown that this additional periodic solution can lose stability according to the Neimark–Sacker scenario, and an attracting torus appears in its vicinity. Thus, a quasiperiodic mode of motion can exist in the phase space of the system. It is shown that quasiperiodic solutions of the equations of motion also correspond to the quasiperiodic motion of the robot in a bounded region along a trajectory of a rather complex shape. Also, strange attractors were found that correspond to the drifting motion of the robot. These modes of motion were found for the first time in the dynamics of the drop-shaped robot.

A finite-dimensional model is developed, which describes the motion of a balanced circular foil with proper circulation in the field of a fixed vortex source. The motion of the foil has been studied in two special cases: that of a fixed vortex and that of a fixed source. It is shown that in the absence of proper circulation, the fixed vortex and the fixed source have the same impact on the motion of the foil. However, adding non- zero proper circulation leads to qualitative differences in the foil’s dynamics. For a fixed vortex, there exist three types of motions: the fall on a vortex in finite time, periodic and quasiperiodic motion around the vortex. The investigation of this case reduces to analysis of a Hamiltonian system with one degree of freedom. Typical phase portraits and graphs of the effective potential of the system are plotted vs the distance between the geometric center of the foil and the vortex. For a fixed source, two types of motions are possible: the fall on the source in finite time and unbounded escape from the source. For small intensities of the source, the asymptotics of escape to infinity is constructed.

A mathematical model featuring the motion of a multilink wheeled vehicle is developed using a nonholonomic model. A detailed analysis of the inertial motion is made. Fixed points of the reduced system are identified, their stability is analyzed, and invariant manifolds are found. For the case of three platforms (links), a phase portrait for motion on an invariant manifold is shown and trajectories of the attachment points of the wheel pairs of the three-link vehicle are presented. In addition, an analysis is made of motion in the case where the leading platform has a rotor whose angular velocity is a periodic function of time. The existence of trajectories for which one of the velocity components increases without bound is established, and the asymptotics for it is found.

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 investigates the trajectories of neutral particles in the Schwarzschild-Melvin spacetime. After reduction by cyclic coordinates this problem reduces to investigating a two-degree-of-freedom Hamiltonian system that has no additional integral. A classification of regions of possible motion of a particle is performed according to the values of the momentum and energy integrals. Bifurcations of periodic solutions of the reduced system are analyzed using a Poincaré map.

A symmetric mathematical model of a wheeled robot with a trailer is considered for various types of coupling between the robot and the trailer. It is shown that for fixed coupling parameters and fixed initial position of the robot with trailer there are two symmetric abnormal extremals. In motion along these trajectories the robot and the trailer traverse normal extremal trajectories for the sub-Riemannian problem on the group of motions of the plane; the coupling point always draws an inflectional elastica or a straight line.

In this work, a model that describes the motion of point vortices in an ideal incompressible fluid on a finite flat cylinder is obtained. The case of two vortices is considered in detail. It is shown that the equations of motion of vortices can be represented in Hamiltonian form and have an additional first integral. A procedure of reduction to a fixed level of the first integral is proposed. For the reduced system, phase portraits are constructed, fixed points and singularities of the system are indicated.

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 investigate a nonholonomic system with parametric excitation, a Roller Racer with variable gyrostatic momentum. We examine in detail the problem of the existence of regimes with unbounded growth of energy (nonconservative Fermi acceleration). We find a criterion for the existence of trajectories for which one of the velocity components increases withound bound and has asymptotics $t^{1/3}$. In addition, we show that the problem under consideration reduces to analysis of a three-dimensional Poincaré map. This map exhibits both regular attractors (a fixed point, a limit cycle and a torus) and strange attractors.

The article is devoted to the motion analysis of a highly maneuverable mobile robot with four omniwheels, taking into account the conditions for the appearance of wheel detachment from the surface, and the occurrence of wheel slipping. Within the motion analysis the task of determination support reactions for a mobile robot is considered. To solve this task, the design of a mobile robot is presented in the form of the frame with rods. To disclosure the static indeterminacy of the considered system the forces method is used. Dependences of support reactions from the position of the center mass are obtained. The feature of the considered system is that the obtained dependencies of the support reactions are nonlinear. Based on the obtained dependences of the support reactions, the influence of the position of the center of mass of a mobile robot with four wheels on the occurrence of detachment and slipping of wheels of a mobile robot was considered. Investigation was carried out within the framework of the dry friction model, according to which module of the friction force proportionally depends on the support reaction acting on the wheel from the side of the motion surface. Simulation was carried out, as a result of which the conditions for the position of the center of mass of a mobile robot were determined, in which wheels of a mobile robot do not detach from the motion surface, and there is no wheel slipping.

This paper investigates the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that the sphere can slip in the direction of the projection of the symmetry axis onto the supporting plane. Equations of motion are obtained and their first integrals are found. It is shown that in the general case the system considered is nonintegrable and does not admit an invariant measure with smooth density. Some particular cases of the existence of an additional integral of motion are found and analyzed. In addition, the limiting case in which the system is integrable by the Euler – Jacobi theorem is established.

This paper presents the design of an aquatic robot actuated by one internal rotor. The robot body has a cylindrical form with a base in the form of a symmetric airfoil with a sharp edge. For this object, equations of motion are presented in the form of Kirchhoff equations for rigid body motion in an ideal fluid, which are supplemented with viscous resistance terms. A prototype of the aquatic robot with an internal rotor is developed. Using this prototype, experimental investigations of motion in a fluid are carried out.

This paper addresses the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that there is no slipping of the sphere as it rolls in the direction of the projection of the symmetry axis onto the supporting plane. It is also assumed that, in the direction perpendicular to the above-mentioned one, the sphere can slip relative to the plane. Examples of realization of the above-mentioned nonholonomic constraint are given. Equations of motion are obtained and their first integrals are found. It is shown that the system under consideration admits a redundant set of first integrals, which makes it possible to perform reduction to a system with one degree of freedom.

In this paper, we study the plane-parallel motion of a circular foil interacting with two vortex pairs in an infinite volume of an ideal fluid. We assumed that the circulation of the velocity of the fluid around the foil was zero. We showed that the equations of motion possess an invariant submanifold such that the foil performed translational motion and the vortices were symmetric relative to the foil’s direction of motion. A qualitative analysis of the motion on this invariant submanifold was made. New relative equilibria were found, a bifurcation diagram was constructed, and a stability analysis is given. In addition, trajectories generalizing Helmholtz leapfrogging were found where the vortices passed alternately through each other, while remaining at a finite distance from the foil.

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.

We investigate the dynamics of particles in a Kerr metric which describes the gravitational field in a neighborhood of a rotating black hole. After elimination of cyclic coordinates this problem reduces to investigating a Hamiltonian system with 2 degrees of freedom. This system possesses an additional Carter integral quadratic in momenta and hence is integrable by the Liouville-Arnold theorem. A bifurcation diagram is constructed and a classification of the types of trajectories of the system is carried out according to the values of first integrals. In particular, it is shown that there are seven different regions of values of first integrals which differ in the topological type of the integral submanifold. Pro-and-retrograde trajectories of particles are found.

A model governing the motion of an aquatic robot with a shell in the form of a symmetrical airfoil NACA0040 is considered. The motion is controlled by periodic oscillations of the rotor. It is numerically shown that for physically admissible values of the control parameters in the phase space of the system, there exists only one limit cycle. The limit cycle that occurs under symmetric control corresponds to the motion of the robot near a straight line. In the case of asymmetric controls, the robot moves near a circle. An algorithm for controlling the course of the robot motion is proposed. This algorithm uses determined limit cycles and transient processes between them.

The dynamics of a system governing the controlled motion of an unbalanced circular foil in the presence of point vortices is considered. The foil motion is controlled by periodically changing the position of the center of mass, the gyrostatic momentum, and the moment of inertia of the system. A derivation of the equations of motion based on Sedov's approach is proposed, the equations of motion are presented in the Hamiltonian form. A periodic perturbation of the known integrable case is considered.

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.

This paper is concerned with the study of permanent rotations of a rigid body rolling without slipping on a horizontal plane (i. e., the velocity of the point of contact of the ellipsoid with the plane is zero). By permanent rotations we will mean motions of a rigid body on a horizontal plane such that the angular velocity of the body remains constant and the point of contact does not change its position. A more detailed analysis is made of permanent rotations of an omnirotational ellipsoid whose characteristic feature is the possibility of permanent rotations about any point of its surface.

In this paper, we consider the dynamics of two interacting point vortex rings in a Bose – Einstein condensate. The existence of an invariant manifold corresponding to vortex rings is proved. Equations of motion on this invariant manifold are obtained for an arbitrary number of rings from an arbitrary number of vortices. A detailed analysis is made of the case of two vortex rings each of which consists of two point vortices where all vortices have same topological charge. For this case, partial solutions are found and a complete bifurcation analysis is carried out. It is shown that, depending on the parameters of the Bose –Einstein condensate, there are three different types of bifurcation diagrams. For each type, typical phase portraits are presented.

Describing the phenomena of the surrounding world is an interesting task that has long attracted the attention of scientists. However, even in seemingly simple phenomena, complex dynamics can be revealed. In particular, leaves on the surface of various bodies of water exhibit complex behavior. This paper addresses an idealized description of the mentioned phenomenon. Namely, the problem of the plane-parallel motion of an unbalanced circular disk moving in a stream of simple structure created by a point source (sink) is considered. Note that using point sources, it is possible to approximately simulate the work of skimmers used for cleaning swimming pools. Equations of coupled motion of the unbalanced circular disk and the point source are derived. It is shown that in the case of a fixed-position source of constant intensity the equations of motion of the disk are Hamiltonian. In addition, in the case of a balanced circular disk the equations of motion are integrable. A bifurcation analysis of the integrable case is carried out. Using a scattering map, it is shown that the equations of motion of the unbalanced disk are nonintegrable. The nonintegrability found here can explain the complex motion of leaves in surface streams of bodies of water.

This paper discusses conditions for the existence of polynomial (in velocities)
first integrals of the equations of motion of mechanical systems in a nonpotential force field
(circulatory systems). These integrals are assumed to be single-valued smooth functions on
the phase space of the system (on the space of the tangent bundle of a smooth configuration
manifold). It is shown that, if the genus of the closed configuration manifold of such a system
with two degrees of freedom is greater than unity, then the equations of motion admit no
nonconstant single-valued polynomial integrals. Examples are given of circulatory systems with
configuration space in the form of a sphere and a torus which have nontrivial polynomial laws
of conservation. Some unsolved problems involved in these phenomena are discussed.

This paper describes the existing designs of spherical robots and reviews studies devoted to investigating their dynamics and to developing algorithms for controlling them. An analysis is also made of the key features and the historical aspects of the development of their designs, in particular, taking into account various areas of application.

This paper is concerned with the controlled motion of a three-link wheeled snake robot propelled by changing the angles between the central and lateral links. The limits on the applicability of the nonholonomic model for the problem of interest are revealed. It is shown that the system under consideration is completely controllable according to the Rashevsky – Chow theorem. Possible types of motion of the system under periodic snake-like controls are presented using Fourier expansions. The relation of the form of the trajectory in the space of controls to the type of motion involved is found. It is shown that, if the trajectory in the space of controls is centrally symmetric, the robot moves with nonzero constant average velocity in some direction.

In this paper we present a method for constructing inhomogeneous velocity fields of an incompressible fluid using expansions in terms of eigenfunctions of the Laplace operator whose weight coefficients are determined from the problem of minimizing the integral of the squared divergence. A number of examples of constructing the velocity fields of plane-parallel and axisymmetric flows are considered. It is shown that the problem of minimizing the integral value of divergence is incorrect and requires regularization. In particular, we apply Tikhonov’s regularization method. The method proposed in this paper can be used to generate different initial conditions in investigating the nonuniqueness of the solution to the Navier – Stokes equations.

This paper is concerned with the motion of an aquatic robot whose body has the form of a sharp-edged foil. The robot is propelled by rotating the internal rotor without shell deformation. The motion of the robot is described by a finite-dimensional mathematical model derived from physical considerations. This model takes into account the effect of added masses and viscous friction. The parameters of the model are calculated from comparison of experimental data and numerical solution to the equations of rigid body motion and the Navier–Stokes equations. The proposed mathematical model is used to define controls implementing straight-line motion, motion in a circle, and motion along a complex trajectory. Experiments for estimation of the efficiency of the model have been conducted.

The paper presents the model of rolling resistance and the application of this model for the control of a pendulum actuated spherical robot on a horizontal plane. Control actions are derived in the form of maneuvers (gaits) which ensure the transition between two steady motions of the system. The experiments confirming the applicability of the model of viscous rolling friction and a method for determining coefficients of rolling resistance from experimental data are presented.

In this paper we presents the results of experimental investigations and simulations of the motion of the Roller Racer. We assume that the angle $\varphi(t)$ between the platforms is a prescribed function of time and that the no-slip condition (nonholonomic constraint) and viscous friction force act at the points of contact of the wheels. The results of theoretical and experimental investigations are compared for various values of the parameters of the control action and design parameters of mobile robot.

We investigate the model of controlledmotion of a pendulum-actuated spherical robot on a horizontal plane, taking rolling resistance into account. We derive equations of motion and obtain partial steady-state solutions.We present algorithms for designing elementary maneuvers (gaits) which ensure transition between two steady motions of the system. These gaits correspond to the acceleration along a straight line and to the turn through a given angle.

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.

This paper investigates the dynamics of a point vortex and a balanced circular foil in an ideal fluid. An explicit reduction to quadratures is performed. A bifurcation diagram is constructed and a classification of the types of integral manifolds is carried out. The stability of critical solutions is studied in which the foil and the vortex move in a circle or in a straight line.

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 addresses the problem of conditions for the existence of conservation laws (first integrals) of circulatory systems which are quadratic in velocities (momenta), when the external forces are nonpotential. Under some conditions the equations of motion are reduced to Hamiltonian form with some symplectic structure and the role of the Hamiltonian is played by a quadratic integral. In some cases the equations are reduced to a conformally Hamiltonian rather than Hamiltonian form. The existence of a quadratic integral and its properties allow conclusions to be drawn on the stability of equilibrium positions of circulatory systems.

In this paper we present a study of the dynamics of a mobile robot with omnidirectional wheels taking into account the reaction forces acting from the plane. The dynamical equations are obtained in the form of Newton – Euler equations. In the course of the study, we formulate structural restrictions on the position and orientation of the omnidirectional wheels and their rollers taking into account the possibility of implementing the omnidirectional motion. We obtain the dependence of reaction forces acting on the wheel from the supporting surface on the parameters defining the trajectory of motion: linear and angular velocities and accelerations, and the curvature of the trajectory of motion. A striking feature of the system considered is that the results obtained can be formulated in terms of elementary geometry.

This paper addresses the problem of balancing an inverted pendulum on an omnidirectional platform in a three-dimensional setting. Equations of motion of the platform – pendulum system in quasi-velocities are constructed. To solve the problem of balancing the pendulum by controlling the motion of the platform, a hybrid genetic algorithm is used. The behavior of the system is investigated under different initial conditions taking into account a necessary stop of the platform or the need for continuation of the motion at the end point of the trajectory. It is shown that the solution of the problem in a two-dimensional setting is a particular case of three-dimensional balancing.

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 nonholonomic systems (the Chaplygin sleigh and the Suslov system) with periodically varying mass distribution. In these examples, the behavior of velocities is described by a system of the form $$ \frac{dv}{d\tau}=f_2(\tau)u^2+f_1(\tau)u+f_0(\tau), \,\, \frac{du}{d\tau}=-uv+g(\tau), $$ where the coefficients are periodic functions of time $\tau$ with the same period. A detailed analysis is made of the problem of the existence of modes of motion for which the system speeds up indefinitely (an analog of Fermi’s acceleration). It is proved that, depending on the choice of coefficients, variable $v$ has the asymptotics $t^{1/k}, \,\, k=1,2,3$. In addition, we show regions of the phase space for which the system, when the trajectories are started from them, is observed to speed up. The proof uses normal forms and averaging in a slightly unusual form since unusual form averaging is performed over a variable that is not fast.
This paper continues a series of studies [1–5] of the dynamics of nonholonomic systems with vary- ing mass distribution due to the prescribed pe- riodic motion of some structural components (ro- tor, point masses etc.). Depending on the choice of the law of variation of mass distribution, such systems generally exhibit a large variety of be- havior, both regular and chaotic. In addition, it turns out that nonholonomic systems are one of the simplest mathematical models of mechan- ical systems exhibiting a phenomenon known as unbounded speedup, which is due to redistribu- tion of internal masses. In this paper, for a cer- tain class of nonholonomic systems (including the Chaplygin sleigh and the Suslov system) we find a criterion which must be satisfied by the periodic variation of mass distribution for the existence of speeding-up trajectories.

This paper addresses the problem of the motion of an unbalanced circular foil and point vortices in an ideal incompressible fluid. Using Bernoulli’s theorem for unsteady potential flow, the force due to the pressure from the fluid on the foil is obtained for an arbitrary vortex motion. A detailed analysis is made of the case of free vortex motion in which a Hamiltonian reduction by symmetries is performed. For the resulting system, relative equilibria corresponding to the motion of an unbalanced foil and a vortex in a circle or in a straight line are found and their stability is investigated. New examples of stationary configurations of a vortex and a foil are given. Using a Poincare map, it is also shown that in the general case of an unbalanced circular foil the reduced system exhibits chaotic trajectories.

The problem of stability of rotating regular vortex $N$-gons (Thomson’s configurations) in a Bose – Einstein condensate in a harmonic trap is considered. A reduction procedure on the level set of the momentum integral is proposed. The dependence of the velocity of rotation $\omega$ of vortex polygon about the center of the trap is obtained as a function of the number of vortices $N$ and the radius of the configuration, $R$. The analysis of the orbital linear and nonlinear stability of the motion of such configurations is carried out. For $N \leqslant 6$, regions of orbital stability of configurations in the parameter space are constructed. It is shown that vortex $N$-gons for $N > 6$ are unstable for any parameters of the system. In this paper, we study the stability of rotating regular vortex $N$-gons in a Bose – Einstein condensate in a harmonic trap. The analysis of the orbital linear and nonlinear stability of motion is carried out. The dependence of the stability of regular vortex $N$-gons on the number of vortices $N$ and the parameters of the system is given.

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.

In this paper, the object of study is a highly maneuverable mobile robot with omni wheels. The paper presents the investigation of the dynamics of omni wheels as part of a mobile platform. The design of a mobile platform with omni wheels is analyzed, taking into account the dynamics of an individual wheel as a part of a mobile platform. The equations of dynamics are presented in the form of Lagrange equations of the second kind with undetermined multipliers. As a result, several conditions of design constraints were identified, under which the omnidirectional motion of the mobile platform is impossible.

This paper describes the design of the underwater robot which is moved by rotating the internal rotor. The design of the robot is described. A finite-dimensional mathematical model describing the robot’s motion is presented. The study of the derived equations of motion is carried out, the influence of the parameters of the control action on the mode of the robot motion is considered.

This paper considers the application of the harmonic balance method for calculating the boundaries of the instability region of the Liouville problem. Explicit equations for the boundaries of the first four instability regions are given.

The work investigates the experimental problem of reparking trailer for a wheeled robot with a trailer. The geometric model with kinematic constraints leads to the sub-Riemannian problem. We solve this problem via nilpotent approximation. The corresponding solution is close to optimal and locally minimizes the total kinetic energy of the driving wheels. A full-scale model is designed in a way to avoid phase constraints usually appearing in trailer systems. We perform 64 different experiments with reparking trailer and obtain the satisfactory accuracy: for one maneuver we repark the trailer with maximum angle error equal to 4 degrees.

Equations governing the motion of vortex $N$-gons in a Bose – Einstein condensate enclosed in a harmonic trap are obtained. It is shown that these equations have two first integrals: the integral of energy and the integral of the moment. The reduction on the fixed level set of the integral of moment is carried out. The equilibrium positions of the system considered are found. Phase portraits of the system of two vortex $N$-gons, each of which consists of two vortices, are constructed.

We consider a nonholonomic system that describes the rolling without slipping of a spherical shell inside which a frame rotates with constant angular velocity (this system is one of the possible generalizations of the problem of the rolling of a Chaplygin sphere). After a suitable scale transformation of the radius of the shell or the mass of the system the equations of motion can be represented as a perturbation of the integrable Euler case in rigid body dynamics. Using this representation, we explicitly calculate a Melnikov integral, which contains an isolated zero under some restrictions on the system parameters. Thereby we prove the absence of an additional integral in this system and the existence of chaotic trajectories. We conclude by presenting numerical experiments that illustrate the system dynamics depending on the behavior of the Melnikov function.

We consider the problem of the stability of rotating regular vortex $N$-gons (Thomson configurations) in a Bose-Einstein condensate in a harmonic trap. The dependence of the rotation velocity $\omega$ of the Thomson configuration around the center of the trap is obtained as a function of the number of vortices $N$ and the radius of the configuration $R$. The analysis of the stability of motion of such configurations in the linear approximation is carried out. For $N \leqslant 6$, regions of orbital stability of configurations in the parameter space are constructed. It is shown that vortex N-gons for $N > 6$ are unstable for any parameters of the system.

We consider a propellerless robot that moves on the surface of a fluid by rotating of the internal rotor. The robot shell has a symmetric shape of NACA 0040 airfoil. The equations of motion are written in the form of classical Kirchhoff equations with terms describing the viscous friction. The control action based on the derived model is proposed. The influences of various model parameters on the robot’s trajectory have been studied.

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.

In this paper we obtain equations of motion for a vortex pair and a circular foil with parametric excitation due to the periodic motion of a material point. Undoubtedly, such problems are, on the one hand, model problems and cannot be used for an exact quantitative description of real trajectories of the system. On the other hand, in many cases such 2D models provide a sufficiently accurate qualitative picture of the dynamics and, due to their simplicity, an estimate of the influence of different parameters. We describe relative equilibria that generalize F¨oppl solutions and collinear configurations when the material point does not move. We show that a stochastic layer forms in the neighborhood of relative equilibria in the case of periodic motion of the foil’s center of mass.

The motion of a circular cylinder in an ideal fluid in the field of a fixed source is considered. It is shown that, when the source has constant strength, the system possesses a momentum integral and an energy integral. Conditions are found under which the equations of motion reduced to the level set of the momentum integral admit an unstable fixed point. This fixed point corresponds to circular motion of the cylinder about the source. A feedback is constructed which ensures stabilization of the above-mentioned fixed point by changing the strength of the source.

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.

The motion of a spherical robot with periodically changing moments of inertia, internal rotors and a displaced center of mass is considered. It is shown that, under some restrictions on the displacement of the center of mass, the system of interest features chaotic dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium point of the ball and of the periodic solution arising in its neighborhood, in the case of periodic rotation of the rotors. It is shown that the lower equilibrium point can become unstable in the case of fixed rotors and periodically changing moments of inertia.

The properties of the Gibbs ensembles of Hamiltonian systems describing the motion along geodesics on a compact configuration manifold are discussed.We introduce weakly ergodic systems for which the time average of functions on the configuration space is constant almost everywhere. Usual ergodic systems are, of course, weakly ergodic, but the converse is not true. A range of questions concerning the equalization of the density and the temperature of a Gibbs ensemble as time increases indefinitely are considered. In addition, the weak ergodicity of a billiard in a rectangular parallelepiped with a partition wall is established.

This paper examines the motion of a balanced spherical robot under the action of periodically changing moments of inertia and gyrostatic momentum. The system of equations of motion is constructed using the model of the rolling of a rubber body (without slipping and twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the equations of motion admit three invariant submanifolds corresponding to plane-parallel motion of the sphere with rotation about the minor, middle and major axes of inertia. The abovementioned motions are quasi-periodic, and for the numerical estimate of their stability charts of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and amplitude of the moments of inertia. It is shown that rotations about the minor and major axes of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In this case, the so-called “Arnol’d tongues” arise in the stability chart. Stabilization of the middle unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It is shown that the nonconservativeness of the system manifests itself in the occurrence of limit cycles, attracting tori and strange attractors in phase space. Numerical calculations show that strange attractors may arise through a cascade of period-doubling bifurcations or after a finite number of torus-doubling bifurcations.

This paper is concerned with the problem of a dynamically symmetric heavy ball rolling without slipping on a cone which rotates uniformly about its symmetry axis. The equations of motion of the system are obtained, partial periodic solutions are found, and their stability is analyzed.

This paper considers the plane-parallel motion of an elliptic foil in a fluid with a nonzero constant circulation under the action of external periodic forces and torque. The existence of the first integral is shown for the case in which there is no external torque and an external force acts along one of the principal axes of the foil. It is shown that, in the general case, in the absence of friction, an extensive stochastic layer is observed for the period advance map. When dissipation is added to the system, strange attractors can arise from the stochastic layer.

The paper is focused to design, simulation and modeling of the compact compliant structures widely used in construction of robotic devices. As the illustrative example it is proposed mechanism for reduction of motion, which enables to improve the accuracy of the positioning system. The physical model is fabricated by 3D printing technology. Its proposed performance characteristics are verified by measurement on the experimental test bed by using laser distance sensors and image sensing/processing technology.

This article is concerned with developing an intelligent system for the control of a wheeled robot. An algorithm for training an artificial neural network for path planning is proposed. The trajectory ensures steering optimal motion from the current position of the mobile robot to a prescribed position taking its orientation into account. The proposed control system consists of two artificial neural networks. One of them serves to specify the position and the size of the obstacle, and the other forms a continuous trajectory to reach it, taking into account the information received, the coordinates, and the orientation at the point of destination. The neural network is trained on the basis of samples obtained by modeling the equations of motion of the wheeled robot which ensure its motion along trajectories in the form of Euler’s elastica.

This paper is concerned with the study of a nonholonomic system with parametric excitation, the Suslov problem with variable gyrostatic momentum. A detailed analysis is made of the problem of the existence of regimes with unbounded growth of energy (an analog of Fermi’s acceleration). The existence of trajectories is shown for which the angular velocity of the rigid body increases indefinitely and has the asymptotics $t^{\frac{1}{2}}$

In this paper, we investigate the motion of a homogeneous heavy ball rolling without slipping on the surface of a rotating cylinder in two settings: a setting without dissipation and a setting with rolling friction torque which is proportional to the angular velocity of the ball. In both caseswe assume that there exists a non-holonomic constraint that corresponds to the condition that there be no slipping at the point of contact. In the first case, the system of five differential equations on the level set of first integrals is reduced to quadratures. To define possible types of motion, we carry out a bifurcation analysis of the reduced system. In the case with friction we show that all trajectories shift on average downward and the ball falls.

This paper addresses a conservative system describing the motion of a smooth body in an ideal fluid under the action of an external periodic torque with nonzero mean and of an external periodic force. It is shown that, in the case where the body is circular in shape, the angular velocity of the body increases indefinitely (linearly in time), and the projection of the phase trajectory onto the plane of translational velocities is attracted to a circle. Asymptotic orbital stability (or asymptotic stability with respect to part of variables) exists in the system. It is shown numerically that, in the case of an elliptic body, the projection of the phase trajectory onto the plane of translational velocities is attracted to an annular region.

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 the motion of the propellerless aquatic robots. There are two models of aquatic robots under consideration that move due to rotation of internal rotors. Mathematical models to describe the motion of the robots are proposed. Experiments with different control actions for fabricated prototypes to verify mathematical models have been conducted.

This paper presents the control algorithm of an omnidirectional mobile robot to implement motion along curvilinear trajectories. The trajectory is defined by the Bezier curves of the third order. An algorithm is proposed to generate control actions for the robot’s motion along a given curvilinear trajectory, taking into account the processes of acceleration and deceleration. The experimental results confirm the applicability of the proposed method.

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 address the problem of the motion of the Roller Racer. We assume that the angle φ(t) between the platforms is a prescribed function of time and that the no-slip condition (nonholonomic constraint) and viscous friction force act at the points of contact of the wheels. In this case, all trajectories of the reduced system asymptotically tend to a periodic solution. In this paper it is shown analytically and experimentally that the chosen control defines periodic trajectories of the attachment point of the platforms on an average along a straight line. We determine the conditions for optimal control when the system moves along a straight line depending on the mass and geometric characteristics of the system and control parameters.

The motion of a spherical robot with periodically changing moments of inertia and gyrostatic momentum is considered. Equations of motion are derived within the framework of the model of "rubber" rolling (without slipping and twisting). The stability of partial solutions of the system is studied numerically. It is shown that the system is nonconservative, and, as a consequence, limit cycles and strange attractors exist in the phase space of the system.

This paper addresses the problem of the motion of a circular foil and point vortices in an ideal fluid. A complete qualitative analysis of the motion of a balanced foil and one vortex is carried out. In particular, periodic solutions are found and their stability is investigated. Using a Poincaré map, it is shown that for an unbalanced foil a reduced system exhibits chaotic trajectories.

The inertial motion of a three-link symmetric wheeled vehicle on a plane is considered. A special case where the system admits an additional first integral of motion and invariant measure with nonsmooth density is found.

The dynamics of a body with a fixed point is considered in the case where the moments of inertia of the system depend periodically on time. A stability of permanent rotations is estimated by a numerical approach. In a neighborhood of permanent rotations the linearization of equations of motion results in Hill's equation, and the stability is determined by the eigenvalues of a monodromy matrix. It is shown that stable rotations may be destabilized by periodically changing the moments of inertia due to parametric resonance.

This paper is devoted to investigations of the motion of an aquatic propeller-less robot. The robot motion implemented by rotation of internal rotor. A simple finite-dimensional mathematical model to describe the motion of the robot is proposed. Experiments with control actions providing the motion along a straight line and a circle have been conducted.

In this paper we consider the controlled motion of a symmetric wheeled three-link vehicle on a plane without slipping. Similar systems have been considered in [1] – [6] . We assume that the vehicle consists of three identical platforms (links). The platforms are connected to each other by joints, and a wheel pair is rigidly fastened to the center of mass of each platform. By a wheel pair we mean two wheels rotating independently on the same axis.