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.

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.

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.

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 addresses the problem of the rolling of a spherical shell with a frame rotating inside, on which rotors are fastened. It is assumed that the center of mass of the entire system is at the geometric center of the shell.
For the rubber rolling model and the classical rolling model it is shown that, if the angular velocities of rotation of the frame and the rotors are constant, then there exists a noninertial coordinate system (attached to the frame) in which the equations of motion do not depend explicitly on time. The resulting equations of motion preserve an analog of the angular momentum vector and are similar in form to the equations for the Chaplygin ball. Thus, the problem reduces to investigating a two-dimensional Poincaré map.
The case of the rubber rolling model is analyzed in detail. Numerical investigation of its Poincaré map shows the existence of chaotic trajectories, including those associated with a strange attractor. In addition, an analysis is made of the case of motion from rest, in which the problem reduces to investigating the vector field on the sphere $S^2$.

This paper is concerned with the problem of optimal path planning for a mobile wheeled robot. Euler elasticas, which ensure minimization of control actions, are considered as optimal trajectories. An algorithm for constructing controls that realizes the motion along the trajectory in the form of an Euler elastica is presented. Problems and special features of the application of this algorithm in practice are discussed. In particular, analysis is made of speedup and deceleration along the elastica, and of the influence of the errors made in manufacturing the mobile robot on the precision with which the prescribed trajectory is followed. Special attention is also given to the problem of forming optimal trajectories of motion along Euler elasticas to a preset point at different angles of orientation. Results of experimental investigations are presented.

This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results.
We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion in a neighborhood of the saddle point. Analysis of three-dimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point.

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.

This paper is concerned with the problem of the interaction of vortex lattices, which is equivalent to the problem of the motion of point vortices on a torus. It is shown that the dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate configurations are found and their stability is investigated. For two vortex lattices it is also shown that, in absolute space, vortices move along closed trajectories except for the case of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero total strength are considered. For three vortices, a reduction to the level set of first integrals is performed. The nonintegrability of this problem is numerically shown. It is demonstrated that the equations of motion of four vortices on a torus admit an invariant manifold which corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices on this invariant manifold and on a fixed level set of first integrals are obtained and their nonintegrability is numerically proved.

This paper presents the results of the study of the dynamics of a real spherical robot of combined type in the case of control using small periodic oscillations. The spherical robot is set in motion by controlled change of the position of the center of mass and by generating variable gyrostatic momentum. We demonstrate how to use small periodic controls for stabilization of the spherical robot during motion. The results of numerical simulation are obtained for various initial conditions and control parameters that ensure a change in the position of the center of mass and a variation of gyrostatic momentum. The problem of the motion of a spherical robot of combined type on a surface that performs flat periodic oscillations is also considered. The results of numerical simulation are obtained for different initial conditions, control actions and parameters of oscillations.

This paper presents experimental investigations of the control algorithm of a highly maneuverable mobile manipulation robot. The kinematics of a mobile manipulation robot, the algorithm of trajectory planning of the mobile robot to the point of object gripping are considered. By realization of the algorithm, the following tasks are solved: solution of the inverse positional task for the mobile manipulation robot; motion planning of the mobile manipulator taking into account the minimization of energy and time consumption per movement. The result of the algorithm is a movement to the point of gripping of the manipulation object; grasping and loading of the object. Experimental investigations of the developed algorithms are given.

The dynamics of a body with a fixed point, variable moments of inertia and internal rotors are considered. A stability analysis of permanent rotations and periodic solutions of the system is carried out. In some simplest cases the stability analysis is reduced to investigating the stability of the zero solution of Hill’s equation. It is shown that by periodically changing the moments of inertia it is possible to stabilize unstable permanent rotations of the system. In addition, stable dynamical regimes can lose stability due to a parametric resonance. It is shown that, as the oscillation frequency of the moments of inertia increases, the dynamics of the system becomes close to an integrable one.

This paper is concerned with the rolling of a homogeneous ball with slipping on a uniformly rotating horizontal plane. We take into account viscous friction forces arising when there is slipping at the contact point. It is shown that, as the coefficient of viscosity tends to infinity, the solution of the generalized problem on each fixed time interval tends to a solution of the corresponding nonholonomic problem.

The motion of a circular cylinder in a fluid in the presence of circulation and external periodic force and torque is studied. It is shown that for a suitable choice of the frequency of external action for motion in an ideal fluid the translational velocity components of the body undergo oscillations with increasing amplitude due to resonance. During motion in a viscous fluid no resonance arises. Explicit integration of the equations of motion has shown that the unbounded propulsion of the body in a viscous fluid is impossible in the absence of external torque. In the general case, the solution of the equations is represented in the form of a multiple series.

For a Chaplygin sleigh moving in the presence of weak friction, we present and investigate two mechanisms of arising acceleration due to oscillations of an internal mass. In certain parameter regions, the mechanism induced by small oscillations determines acceleration which is on average one-directional. The role of friction is that the velocity reached in the process of the acceleration is stabilized at a certain level. The second mechanism is due to the effect of the developing oscillatory parametric instability in the motion of the sleigh. It occurs when the internal oscillating particle is comparable in mass with the main platform and the oscillations are of a sufficiently large amplitude. In the nonholonomic model the magnitude of the parametric oscillations and the level of mean energy achieved by the system turn out to be bounded if the line of the oscillations of the moving particle is displaced from the center of mass; the observed sustained motion is in many cases associated with a chaotic attractor. Then, the motion of the sleigh appears to be similar to the process of two-dimensional random walk on the plane.

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.

The paper is concerned with the problem of stabilizing a spherical robot of combined type during its motion. The focus is on the application of feedback for stabilization of the robot which is an example of an underactuated system. The robot is set in motion by an inter- nal wheeled platform with a rotor placed inside the sphere. The results of experimental investigations for a prototype of the spherical robot are presented.

In this paper, we address the free (uncontrolled) dynamics of a snakeboard consisting of two wheel pairs fastened to a platform. The snakeboard is one of the well-known sports vehicles on which the sportsman executes necessary body movements. From the theoretical point of view, this system is a direct generalization of the classical nonholonomic system of the Chaplygin sleigh. We carry out a topological and qualitative analysis of trajectories of this dynamical system. An important feature of the problem is that the common level set of first integrals is a compact two-dimensional surface of genus 5. We specify conditions under which the reaction forces infinitely increase during motion and the so-called phenomenon of nonholonomic jamming is observed. In this case, the nonholonomic model ceases to work and it is necessary to use more complex mechanical models incorporating sliding, elasticity, etc.

This paper is concerned with a nonholonomic system with parametric excitation—the Chaplygin sleigh with time-varying mass distribution. A detailed analysis is made of the problem of the existence of regimes with unbounded growth of energy (an analogue of Fermi’s acceleration) in the case where excitation is achieved by means of a rotor with variable angular momentum. The existence of trajectories for which the translational velocity of the sleigh increases indefinitely and has the asymptotics $\tau^{\frac{1}{3}}$ is proved. In addition, it is shown that, when viscous friction with a nondegenerate Rayleigh function is added, unbounded speed-up disappears and the trajectories of the reduced system asymptotically tend to a limit cycle.

This paper investigates the rolling without slipping of a homogeneous heavy ball on the surface of a rotating cone in two settings: without dissipation in a nonholonomic setting and with rolling friction torque which is proportional to the angular velocity of the ball. In the nonholonomic setting, the resulting system of five differential equations on the level set of first integrals is reduced to quadratures. A bifurcation analysis of the above system is carried out to determine the possible types of motion. In the second case, it is shown that there are not only trajectories emanating from the lower point of the cone (its vertex), but also trajectories to the vertex of the cone (fall). An analysis of the dependence of the type of terminal motion of the center of mass of the ball on initial conditions is carried out.

This paper discusses two approaches for deriving the equations of motion for a ball that rolls without slipping on the surface of a rotating hyperbolic paraboloid. We analyze two possible methods for defining the surface on which the ball rolls, and show the relationship between the two methods. We describe how the stability of the ball’s rotation at the saddle point depends on the radius of the ball, in the case where the stability analysis is made in dimensionless parameters.

This paper addresses the problem of the motion of a sleigh with a free rotor. It is shown that this system exhibits chaotic and regular motions. The case in which the system is balanced relative to the knife edge is of particular interest because it has an additional integral. In this case, the problem reduces to investigating a vector field on a torus and to classifying singular points on it.

A plane-parallel motion of a circular foil is considered in a fluid with a nonzero constant circulation under the action of external periodic force and torque. Various integrable cases are treated. Conditions for the existence of resonances of two types are found. In the case of resonances of the first type, the phase trajectory of the system and the trajectory of the foil are unbounded. In the case of resonances of the second type, the foil trajectory is unbounded, while the phase trajectory of the system remains bounded during the motion.

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.

We consider the two-body problem on surfaces of constant nonzero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each $q>0$ we show there are two relative equilibria where the masses are separated by a distance $q$. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except $\pi/2$. When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal, there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (‘isosceles RE’) and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At $\pi/2$, the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a five-dimensional phase space and possess one Casimir function.

This paper addresses the problem of the self-propulsion of a smooth body in a fluid by periodic oscillations of the internal rotor and circulation. In the case of zero dissipation and constant circulation, it is shown using methods of KAM theory that the kinetic energy of the system is a bounded function of time. In the case of constant nonzero circulation, the trajectories of the center of mass of the system lie in a bounded region of the plane. The method of expansion by a small parameter is used to approximately construct a solution corresponding to directed motion of a circular foil in the presence of dissipation and variable circulation. Analysis of this approximate solution has shown that a speed-up is possible in the system in the presence of variable circulation and in the absence of resistance to translational motion. It is shown that, in the case of an elliptic foil, directed motion is also possible. To explore the dynamics of the system in the general case, bifurcation diagrams, a chart of dynamical regimes and a chart of the largest Lyapunov exponent are plotted. It is shown that the transition to chaos occurs through a cascade of period-doubling bifurcations.

This paper addresses the problem of controlled motion of the Zhukovskii foil in a viscous fluid due to a periodically oscillating rotor. Equations of motion including the added mass effect, viscous friction and lift force due to circulation are derived. It is shown that only limit cycles corresponding to the direct motion or motion near a circle appear in the system at the standard parameter values. The chart of dynamical regimes, the chart of the largest Lyapunov exponent and a one-parameter bifurcation diagram are calculated. It is shown that strange attractors appear in the system due to a cascade of period-doubling bifurcations.

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 consider the problem of the motion of the Roller Racer.We assume that the angle $\varphi (t)$ between the platforms is a prescribed function of time. We prove that in this case the acceleration of the Roller Racer is unbounded. In this case, as the Roller Racer accelerates, the increase in the constraint reaction forces is also unbounded. Physically this means that, from a certain instant onward, the conditions of the rolling motion of the wheels without slipping are violated. Thus, we consider a model in which, in addition to the nonholonomic constraints, viscous friction force acts at the points of contact of the wheels. For this case we prove that there is no constant acceleration and all trajectories of the reduced system asymptotically tend to a periodic solution.

This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.

This paper is concerned with the problem of three vortices on a sphere $S^2$ and the Lobachevsky plane $L^2$. After reduction, the problem reduces in both cases to investigating a Hamiltonian system with a degenerate quadratic Poisson bracket, which makes it possible to study it using the methods of Poisson geometry. This paper presents a topological classification of types of symplectic leaves depending on the values of Casimir functions and system parameters.

This paper addresses the problem of self-propulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark – Sacker bifurcations and period-doubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermi-like acceleration) is possible.

In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincar´e map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.

A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its dimension is found. The maximal dimension of such subspaces estimates from above the degree of instability of the Hamiltonian system. The stability of typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an equilibrium point. General results are applied to the investigation of linear mechanical systems with gyroscopic forces and finite-dimensional quantum systems.

This paper addresses the problem of plane-parallel motion of the Zhukovskii foil in a viscous fluid. Various motion regimes of the foil are simulated on the basis of a joint numerical solution of the equations of body motion and the Navier – Stokes equations. According to the results of simulation of longitudinal, transverse and rotational motions, the average drag coefficients and added masses are calculated. The values of added masses agree with the results published previously and obtained within the framework of the model of an ideal fluid. It is shown that between the value of circulation determined from numerical experiments, and that determined according to the model of and ideal fluid, there is a correlation with the coefficient $\mathcal{R} = 0.722$. Approximations for the lift force and the moment of the lift force are constructed depending on the translational and angular velocity of motion of the foil. The equations of motion of the Zhukovskii foil in a viscous fluid are written taking into account the found approximations and the drag coefficients. The calculation results based on the proposed mathematical model are in qualitative agreement with the results of joint numerical solution of the equations of body motion and the Navier – Stokes equations.

This paper is concerned with the motion of heavy toroidal bodies in a fluid. For experimental purposes, models of solid tori with a width of 3 cm and external diameters of 10 cm, 12 cm and 15 cm have been fabricated by the method of casting chemically solidifying polyurethane (density 1100 kg/m³). Tracking of the models is performed using the underwater Motion Capture system. This system includes 4 cameras, computer and specialized software. A theoretical description of the motion is given using equations incorporating the influence of inertial forces, friction and circulating motion of a fluid through the hole. Values of the model parameters are selected by means of genetic algorithms to ensure an optimal agreement between experimental and theoretical data.

This article is devoted to the study of self-propulsion of bodies in a fluid by the action of internal mechanisms, without changing the external shape of th e body. The paper presents an overview of theoretical papers that justify the possibility of this displacement in ideal and viscous liquids. A special case of self-propulsion of a rigid body along the surface of a liquid is considered due to the motion of two internal masses along the circles. The paper presents a mathematical model of the motion of a solid body with moving internal masses in a three-dime nsional formulation. This model takes into account the three-dimensional vibrations of the body during motion, which arise under the action of external forces-gravity force, Archimedes force and forces acting on the body, from the side of a viscous fluid. The body is a homogeneous elliptical cylinder with a k eel located along the larger diagonal. Inside the cylinder there are two material point masses moving along the circles. The centers of the circles lie on the smallest diagonal of the ellipse at an equal distance from the center of mass. Equations of motion of the system (a body with two mater ial points, placed in a fluid) are represented as Kirchhoff equations with the addition of external for ces and moments acting on the body. The phenomenological model of viscous friction is quadratic in velocity used to describe the forces of resistance to motion in a fluid. The coefficients of resistance to movement were determ ined experimentally. The forces acting on the keel were determined by numerical modeling of the keel oscillations in a viscous liquid using the Navier – Stokes equations. In this paper, an experimental verification of the proposed mathematical model was carried out. Several series of experiments on self-propulsion of a body in a liquid by means of rotation of internal masses with different speeds of rotation are presented. The dependence of the average propagation velocity, the amplitude of the transverse oscillations as a function of the rotational speed of internal masses is investigated. The obtained experimental data are compared with the results obtai ned within the framework of the proposed mathematical model.

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 this work we investigate the motion of a homogeneous ball rolling without slipping on uniformly rotating horizontal and inclined planes under the action of a constant external force supplemented with the moment of rolling friction, which depends linearly on the angular velocity of the ball. We systematise well-known results and supplement them with the stability analysis of partial solutions of the system. We also perform an experimental investigation whose results support the adequacy of the rolling friction model used. Comparison of numerical and experimental results has shown a good qualitative agreement.

This paper addresses the problem of the Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In this case, the equations of motion admit area integrals, an integral of squared angular momentum and the Jacobi integral, which is a generalization of the energy integral, and possess an invariant measure. After reduction the problem reduces to investigating a three-dimensional Poincaré map that preserves phase volume (with density defined by the invariant measure). We show that in the general case the system’s dynamics is chaotic.

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 book provides an up-to-date overview of results in rigid body dynamics, including material concerned with the analysis of nonintegrability and chaotic behavior in various related problems. The wealth of topics covered makes it a practical reference for researchers and graduate students in mathematics, physics and mechanics.

In this paper, we show that the trajectories of a dynamical system with nonholonomic constraints can satisfy Hamilton’s principle. As the simplest illustration, we consider the problem of a homogeneous ball rolling without slipping on a plane. However, Hamilton’s principle is formulated either for a reduced system or for a system defined in an extended phase space. It is shown that the dynamics of a nonholonomic homogeneous ball can be embedded in a higher-dimensional Hamiltonian phase flow. We give two examples of such an embedding: embedding in the phase flow of a free system and embedding in the phase flow of the corresponding vakonomic system.

This paper presents results of theoretical and experi- mental research explaining the retrograde final-stage rolling of a disk under certain relations between its mass and geometric parameters. Modifying the no-slip model of a rolling disk by including viscous rolling friction provides a qualitative explana- tion for the disk's retrograde motion. At the same time, the simple experiments described in the paper completely reject the aerodynamical drag torque as a key reason for the retro- grade motion of a disk considered, thus disproving some recent hypotheses.

The motion of a body shaped as a triaxial ellipsoid and controlled by the rotation of three internal rotors is studied. It is proved that the motion is controllable with the exception of a few particular cases. Partial solutions whose combinations enable an unbounded motion in any arbitrary direction are constructed.

This is a survey of the main forms of equations of dynamical systems with non-integrable constraints, divided into two large groups. The first group contains systems arising in vakonomic mechanics and optimal control theory, with the equations of motion obtained from the variational principle, and the second contains systems in classical non-holonomic mechanics, when the constraints are ideal and therefore the D’Alembert–Lagrange principle holds.

Nonholonomic mechanical systems arise in the context of many problems of practical significance. A famous model in nonholonomic mechanics is the Chaplygin sleigh. The Chaplygin sleigh is a rigid body with a sharp weightless wheel in contact with the (supporting) surface. The sharp edge of the wheel prevents the wheel from sliding in the direction perpendicular to its plane. This paper is concerned with a Chaplygin sleigh with time-varying mass distribution, which arises due to the motion of a point in the direction transverse to the plane of the knife edge. Equations of motion are obtained from which a closed system of equations with time-periodic coefficients decouples. This system governs the evolution of the translational and angular velocities of the sleigh. It is shown that if the projection of the center of mass of the whole system onto the axis along the knife edge is zero, the translational velocity of the sleigh increases. The trajectory of the point of contact is, as a rule, unbounded.

This paper addresses the dynamics of a disk rolling on an absolutely rough plane. It is proved that the equations of motion have an invariant measure with continuous density only in two cases: a dynamically symmetric disk and a disk with a special mass distribution. In the former case, the equations of motion possess two additional integrals and are integrable by quadratures by the Euler-Jacobi theorem. In the latter case, the absence of additional integrals is shown using a Poincaré map. In both cases, the volume of any domain in phase space (calculated with the help of the density) is preserved by the phase flow. Nonholonomic mechanics is populated with systems both with and without an invariant measure.

This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincar´e map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitch-fork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.

This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.

In this paper we investigate the dynamics of a system that is a generalization of the Chaplygin sleigh to the case of an inhomogeneous nonholonomic constraint. We perform an explicit integration and a sufficiently complete qualitative analysis of the dynamics.

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.

This paper addresses the problem of the inertial motion of a roller racer, which reduces to investigating a dynamical system on a (two-dimensional) torus and to classifying singular points on it. It is shown that the motion of the roller racer in absolute space is asymptotic. A restriction on the system parameters in which this motion is bounded (compact) is presented.

This paper is concerned with the Hess case in the Euler–Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.

In this paper we investigate the dynamics of a system that is a generalization of the Chaplygin sleigh to the case of an inhomogeneous nonholonomic constraint. We perform an explicit integration and a sufficiently complete qualitative analysis of the dynamics.

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.

In this paper the model of rolling of spherical bodies on a plane without slipping is presented taking into account viscous rolling friction. Results of experiments aimed at investigating the influence of friction on the dynamics of rolling motion are presented. The proposed dynamical friction model for spherical bodies is verified and the limits of its applicability are estimated. A method for determining friction coefficients from experimental data is formulated.

This paper presents a comparative analysis of computations of the motion of heavy three-bladed screws in a fluid along with experimental results. Simulation of the motion is performed using the theory of an ideal fluid and the phenomenological model of viscous friction. For experimental purposes, models of three-bladed screws with various configurations and sizes were manufactured by casting from chemically hardening polyurethane. Comparison of calculated and experimental results has shown that the mathematical models considered essentially do not reflect the processes observed in the experiments.

This paper is concerned with the Hess case in the Euler –Poisson equations and with its generalization on the pencil of Poisson brackets. It is shown that in this case the problem reduces to investigating the vector field on a torus and that the graph showing the dependence of the rotation number on parameters has horizontal segments (limit cycles) only for integer values of the rotation number. In addition, an example of a Hamiltonian system is given which possesses an invariant submanifold (similar to the Hess case), but on which the dependence of the rotation number on parameters is a Cantor ladder.

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 two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.

In this paper we consider the controlled motion of a helical body with three blades in an ideal fluid, which is executed by rotating three internal rotors. We set the problem of selecting control actions, which ensure the motion of the body near the predetermined trajectory. To determine controls that guarantee motion near the given curve, we propose methods based on the application of hybrid genetic algorithms (genetic algorithms with real encoding and with additional learning of the leader of the population by a gradient method) and artificial neural networks. The correctness of the operation of the proposed numerical methods is estimated using previously obtained differential equations, which define the law of changing the control actions for the predetermined trajectory.
In the approach based on hybrid genetic algorithms, the initial problem of minimizing the integral functional reduces to minimizing the function of many variables. The given time interval is broken up into small elements, on each of which the control actions are approximated by Lagrangian polynomials of order 2 and 3. When appropriately adjusted, the hybrid genetic algorithms reproduce a solution close to exact. However, the cost of calculation of 1 second of the physical process is about 300 seconds of processor time.
To increase the speed of calculation of control actions, we propose an algorithm based on artificial neural networks. As the input signal the neural network takes the components of the required displacement vector. The node values of the Lagrangian polynomials which approximately describe the control actions return as output signals . The neural network is taught by the well-known back-propagation method. The learning sample is generated using the approach based on hybrid genetic algorithms. The calculation of 1 second of the physical process by means of the neural network requires about 0.004 seconds of processor time, that is, 6 orders faster than the hybrid genetic algorithm. The control calculated by means of the artificial neural network differs from exact control. However, in spite of this difference, it ensures that the predetermined trajectory is followed exactly.

In this paper we study the controlled motion of an arbitrary two-dimensional body in an ideal fluid with a moving internal mass and an internal rotor in the presence of constant circulation around the body. We show that by changing the position of the internal mass and by rotating the rotor, the body can be made to move to a given point, and discuss the influence of nonzero circulation on the motion control. We have found that in the presence of circulation around the body the system cannot be completely stabilized at an arbitrary point of space, but fairly simple controls can be constructed to ensure that the body moves near the given point.

The motion controlled by the rotation of three internal rotors of a body with helical symmetry in an ideal liquid is considered. The problem is to select controls that ensure the displacement of the body with minimum effort. The optimality of particular solutions found earlier is studied.

We consider the movement of Chaplygin sleigh on a plane that is a solid body with imposed nonholonomic constraint, which excludes the possibility of motions transversal to the constraint element (“knife-edge”), and complement the model with an attached mass, periodically oscillating relatively to the main platform of the sleigh. Numerical simulations indicate the occurrence of either unrestricted acceleration of the sleigh, or motions with bounded velocities and momenta, depending on parameters. We note the presence of phenomena characteristic to nonholonomic systems with complex dynamics; in particular, attractors occur responsible for chaotic motions. In addition, quasiperiodic regimes take place similar to those observed in conservative nonlinear dynamics.

In this article, a dynamical model for controlling an omniwheel mobile robot is presented. The proposed model is used to construct an algorithm for calculating control actions for trajectories characterizing the high maneuverability of the mobile robot. A description is given for a prototype of the highly maneuverable robot with four omniwheels, for which an algorithm for setting the coefficients of the PID controller is considered. Experiments on the motion of the robot were conducted at different angles, and the orientation of the platform was preserved. The experimental results are analyzed and statistically assessed.

The experimental stand and the results of investigation of the motion of a brake shoe are described. In the noncritical region, the friction coefficient is determined experimentally. It is shown that its value corresponds to the condition of uniqueness of the solution for construction of this brake shoe. The dynamics observed in the paradoxical-motion region is described.

In this paper, we study the free and controlled motion of an arbitrary two-dimensional body with a moving internal material point through an ideal fluid in presence of constant circulation around the body. We perform bifurcation analysis of free motion (with fixed internal mass). We show that by changing the position of the internal mass the body can be made to move to a specified point. There are a number of control problems associated with the nonzero drift of the body in the case of fixed internal mass.

We consider the controlled motion in an ideal incompressible fluid of a rigid body with moving internal masses and an internal rotor in the presence of circulation of the fluid velocity around the body. The controllability of motion (according to the Rashevskii–Chow theorem) is proved for various combinations of control elements. In the case of zero circulation, we construct explicit controls (gaits) that ensure rotation and rectilinear (on average) motion. In the case of nonzero circulation, we examine the problem of stabilizing the body (compensating the drift) at the end point of the trajectory. We show that the drift can be compensated for if the body is inside a circular domain whose size is defined by the geometry of the body and the value of circulation.

We present the results of theoretical and experimental investigations of the motion of a spherical robot on a plane. The motion is actuated by a platform with omniwheels placed inside the robot. The control of the spherical robot is based on a dynamic model in the nonholonomic statement expressed as equations of motion in quasivelocities with indeterminate coefficients. A number of experiments have been carried out that confirm the adequacy of the dynamic model proposed.

This paper provides a detailed description of various reduction schemes in rigid body dynamics. The analysis of one of such nontrivial reductions makes it possible to put the cases already found in order and to obtain new generalizations of the Kovalevskaya case to $e(3)$. Note that the indicated reduction allows one to obtain in a natural way some singular additive terms that were proposed earlier by D.N. Goryachev.

This paper is concerned with the nonholonomic Suslov problem and its generalization proposed by Chaplygin. The issue of the existence of an invariant measure with singular density (having singularities at some points of phase space) is discussed.

The dynamics of a Painlevé–Appell system consisting of two point masses joined by a weightless rigid rodis studied within two mechanical models, which describe different motion regimes. One of the massescan slide or can be supported at rest on a rough straight line. The boundaries of the region of definitionof each of the models are presented, and the transitions between them are analysed for various frictioncoefficients.

From analysis of time series obtained on the numerical solution of a plane problem on the motion of a body with an elliptic cross section under the action of gravity force in an incompressible viscous fluid, a system of ordinary differential equations approximately describing the dynamics of the body is reconstructed. To this end, coefficients responsible for the added mass, the force caused by the circulation of the velocity field, and the resisting force are found by the least square adjustment. The agreement between the finitedimensional description and the simulation on the basis of the Navier–Stokes equations is illustrated by images of attractors in regular and chaotic modes. The coefficients found make it possible to estimate the actual contribution of different effects to the dynamics of the body.

This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of sub-Riemannian geodesics on $SE(3)$ are obtained.

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.

In this paper we describe the results of experimental investigations of the motion of a screwless underwater robot controlled by rotating internal rotors. We present the results of comparison of the trajectories obtained with the results of numerical simulation using the model of an ideal fluid.

This paper is devoted to an experimental investigation of the motion of a rigid body set in motion by rotating two unbalanced internal masses. The results of experiments confirming the possibility of motion by this method are presented. The dependence of the parameters of motion on the rotational velocity of internal masses is analyzed. The velocity field of the fluid around the moving body is examined.

This paper is concerned with the problem of the integrable behavior of geodesics on homogeneous factors of the Lobachevsky plane with respect to Fuchsian groups (orbifolds). Locally the geodesic equations admit three independent Noether integrals linear in velocities (energy is a quadratic form of these integrals). However, when passing along closed cycles the Noether integrals undergo a linear substitution. Thus, the problem of integrability reduces to the search for functions that are invariant under these substitutions. If a Fuchsian group is Abelian, then there is a first integral linear in the velocity (and independent of the energy integral). Conversely, if a Fuchsian group contains noncommuting hyperbolic or parabolic elements, then the geodesic flow does not admit additional integrals in the form of a rational function of Noether integrals. We stress that this result holds also for noncompact orbifolds, when there is no ergodicity of the geodesic flow (since nonrecurrent geodesics can form a set of positive measure).

This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector $(3, 6, 14)$, the other is defined by two generatrices and growth vector $(2, 3, 5, 8)$. Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.

In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature $S^2$ and $S^3$. We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems.

In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.

This paper is concerned with the dynamics of vortex sources in a deformation flow. The case of two vortex sources is shown to be integrable by quadratures. In addition, the relative equilibria (of the reduced system) are examined in detail and it is shown that in this case the trajectory of vortex sources is an ellipse.

The onset of adiabatic chaos in rigid body dynamics is considered. A comparison of the analytically calculated diffusion coefficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincaré – Zhukovsky problem. The application of Hamiltonian methods to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi’s acceleration).

This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.

This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found. In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.

This paper is concerned with the motion of an unbalanced heavy three-axial ellipsoid in an ideal fluid controlled by rotation of three internal rotors. It is proved that the motion of the body considered is controlled with respect to configuration variables except for some special cases. An explicit control that makes it possible to implement unbounded motion in an arbitrary direction has been calculated. Directions for which control actions are bounded functions of time have been determined.

In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.

In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems.We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.

In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.

This paper is concerned with the process of the free fall of a three-bladed screw in a fluid. The investigation is performed within the framework of theories of an ideal fluid and a viscous fluid. For the case of an ideal fluid the stability of uniformly accelerated rotations (the Steklov solutions) is studied. A phenomenological model of viscous forces and torques is derived for investigation of the motion in a viscous fluid. A chart of Lyapunov exponents and bifucation diagrams are computed. It is shown that, depending on the system parameters, quasiperiodic and chaotic regimes of motion are possible. Transition to chaos occurs through cascade of period-doubling bifurcations.

In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.

In this paper, we consider a system governing the motion of two point vortices in a flow excited by an external acoustic forcing. It is known that the system of two vortices is integrable in the absence of acoustic forcing. However, the addition of the acoustic forcing makes the system much more complex, and the system becomes nonintegrable and loses the phase volume preservation property. The objective of our research is to study chaotic dynamics and typical bifurcations. Numerical analysis has shown that the reversible pitchfork bifurcation is typical. Also, we show that the existence of strange attractors is not characteristic for the system under consideration.

In this paper, we focus on the study of two-dimensional plate dynamics on the Lobachevskii plane $L^2$. First of all, we consider the free motion of such a plate, which is a pseudospherical analog of dynamics of the Euler top, and also present an analog of the Euler–Poisson equations enabling us to study the motion of the body in potential force fields having rotational symmetry. We present a series of integrable cases, having analogs in Euclidean space, for different fields. Moreover, in the paper, a partial qualitative analysis of the dynamics of free motion of a plate under arbitrary initial conditions is made and a number of computer illustrations are presented which show a substantial difference of the motion from the case of Euclidean space. The study undertaken in the present paper leads to interesting physical onsequences, which enable us to detect the influence of curvature on the body dynamics.

This paper is concerned with the problem of a rigid body (tripod) moving with three points in contact with a horizontal plane under the action of dry friction forces. It is shown that the regime of asymptotic motion (final dy- namics) of the tripod can be pure rotation, pure sliding, or sliding and rotation can cease simultaneously, which is determined by the position of the tripod’s supports relative to the radius of inertia. In addition, the dependence of the trajectory of the center of mass on the system parameters is investigated. A comparison is made with the well-known theoretical and experimental studies on the motion of bodies with a flat base

A new integrable problem of nonholonomic mechanics is considered and its mechanical realization is proposed. This problem is a generalization of the well-known problem of А. P. Veselov and L. E. Veselova concerning the rolling motion of the Chaplygin ball in a straight line. Particular cases are found in which integration can be reduced to explicit quadratures.

We consider differential equations with quadratic right-hand sides which admit two quadratic first integrals, one of which is a positive definite quadratic form. We present general conditions under which a linear change of variables reduces this system to some "canonical" form. Under these conditions the system turns out to be nondivergent and is reduced to Hamiltonian form, however, the corresponding linear Lie–Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case the equations are reduced to the classical equations of the Euler top, and in the four-dimensional space the system turns out to be superintegrable and coincides with the Euler–Poincaré equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplication with which the Poisson bracket satisfies the Jacobi identity. In the general case, we prove that there is no reducing multiplier for $n>5$. As an example, we consider a system of Lotka–Volterra type with quadratic right-hand sides, which was studied already by Kovalevskaya from the viewpoint of the conditions for uniqueness of its solutions as functions of complex time.

This is a reply to the comment by V.F. Zhuravlev (see Usp. Fiz. Nauk 185 1337 (2015) [Phys. Usp. 58 (12) (2015)]) on the inadequacy of the nonholonomic model when applied to the rolling of rigid bodies. The model of nonholonomic mechanics is discussed. Using recent results as examples, it is shown that the validity and potential of the nonholonomic model are not inferior to those of other dynamics and friction models.

This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis.

The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.

This paper is an experimental investigation of a round uniform disk rolling on a horizontal surface. Two methods for experimentally determining the loss of contact of the rolling disk from the horizontal surface before its stop are proposed. Results of experiments for disks having different masses and manufactured from different materials are presented. Causes of “microlosses of contact” detected in the processes of motion are discussed.

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.

This paper is concerned with the experimental determination of the added masses of bodies completely or partially immersed in a fluid. The paper presents an experimental setup, a technique of the experiment and an underlying mathematical model. The method of determining the added masses is based on the towing of the body with a given propelling force. It is known (from theory) that the concept of an added mass arises under the assumption concerning the potentiality of flow over the body. In this context, the authors have performed PIV visualization of flows generated by the towed body, and defined a part of the trajectory for which the flow can be considered as potential. For verification of the technique, a number of experiments have been performed to determine the added masses of a spheroid. The measurement results are in agreement with the known reference data. The added masses of a screwless freeboard robot have been defined using the developed technique.

The paper is devoted to the experimental verification of the Andersen–Pesavento–Wang model describing the falling of a heavy plate through a resisting medium. As a main research method the authors have used video filming of a falling plate with PIV measurement of the velocity of surrounding fluid flows. The trajectories of plates and streamlines were determined and oscillation frequencies were estimated using experimental results. A number of experiments for plates of various densities and sizes were performed. The trajectories of plates made of plastic are qualitatively similar to the trajectories predicted by the Andersen–Pesavento–Wang model. However, measured and computed frequencies of oscillations differ significantly. For a plate made of high carbon steel the results of experiments are quantitatively and qualitatively in disagreement with computational results.

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 the problem of the motion of a wheeled vehicle on a plane in the case where one of the wheel pairs is fixed. In addition, the motion of a wheeled vehicle on a plane in the case of two free wheel pairs is considered. A method for obtaining equations of motion for the vehicle with an arbitrary geometry is presented. Possible kinds of motion of the vehicle with a fixed wheel pair are determined.

In this paper, we develop the results obtained by J.Hadamard and G.Hamel concerning the possibility of substituting nonholonomic constraints into the Lagrangian of the system without changing the form of the equations of motion. We formulate the conditions for correctness of such a substitution for a particular case of nonholonomic systems in the simplest and universal form. These conditions are presented in terms of both generalized velocities and quasi-velocities. We also discuss the derivation and reduction of the equations of motion of an arbitrary wheeled vehicle. In particular, we prove the equivalence (up to additional quadratures) of problems of an arbitrary wheeled vehicle and an analogous vehicle whose wheels have been replaced with skates. As examples, we consider the problems of a one-wheeled vehicle and a wheeled vehicle with two rotating wheel pairs.

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 present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems. We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.

This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.

A nonholonomic model of the dynamics of an omniwheel vehicle on a plane and a sphere is considered. A derivation of equations is presented and the dynamics of a free system are investigated. An explicit motion control algorithm for the omniwheel vehicle moving along an arbitrary trajectory is obtained.

This paper deals with the problem of a spherical robot propelled by an internal omniwheel platform and rolling without slipping on a plane. The problem of control of spherical robot motion along an arbitrary trajectory is solved within the framework of a kinematic model and a dynamic model. A number of particular cases of motion are identified, and their stability is investigated. An algorithm for constructing elementary maneuvers (gaits) providing the transition from one steady-state motion to another is presented for the dynamic model. A number of experiments have been carried out confirming the adequacy of the proposed kinematic model.

This paper is a review of the problem of the constructive reduction of nonholonomic systems with symmetries. The connection of reduction with the presence of the simplest tensor invariants (first integrals and symmetry fields) is shown. All theoretical constructions are illustrated by examples encountered in applications. In addition, the paper contains a short historical and critical sketch covering the contribution of various researchers to this problem.

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie –Poisson bracket of rank 2. This Lie – Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.

This paper presents the results of experimental investigations for the rolling of a spherical robot of combined type actuated by an internal wheeled vehicle with rotor on a horizontal plane. The control of spherical robot based on nonholonomic dynamical by means of gaits. We consider the motion of the spherical robot in case of constant control actions, as well as impulse control. A number of experiments have been carried out confirming the importance of rolling friction.

In this paper we consider the problem of motion of a rigid body in an ideal fluid with two material points moving along circular trajectories. The controllability of this system on the zero level set of first integrals is shown. Elementary “gaits” are presented which allow the realization of the body’s motion from one point to another. The existence of obstacles to a controlled motion of the body along an arbitrary trajectory is pointed out.

This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servo-constraint, which implies that the projection of the body’s angular velocity on some body-fixed direction is zero.

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

In this paper we discuss conditions for the existence of the Jacobi integral (that generalizes energy) in systems with inhomogeneous and nonholonomic constraints. As an example, we consider in detail the problem of motion of the Chaplygin sleigh on a rotating plane and the motion of a dynamically symmetric ball on a uniformly rotating surface. In addition, we discuss illustrative mechanical models based on the motion of a homogeneous ball on a rotating table and on the Beltrami surface.

The paper discusses the dynamics of systems with Béghin’s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint — the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) — and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin’s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.

The dynamic model for a spherical robot with an internal omniwheel platform is presented. Equations of motion and first integrals according to the non-holonomic model are given. We consider particular solutions and their stability. The algorithm of control of spherical robot for movement along a given trajectory are presented.

It is well known that in the Béghin– Appel theory servo-constraints are realized using controlled external forces. In this paper an expansion of the Béghin–Appel theory is given in the case where servo-constraints are realized using controlled change of the inertial properties of a dynamical system. The analytical mechanics of dynamical systems with servo-constraints of general form is discussed. The key principle of the approach developed is to appropriately determine virtual displacements of systems with constraints.

We develop the reducing multiplier theory for a special class of nonholonomic dynamical systems and show that the non-linear Poisson brackets naturally obtained in the framework of this approach are all isomorphic to the Lie–Poisson $e$(3)-bracket. As two model examples, we consider the Chaplygin ball problem on the plane and the Veselova system. In particular, we obtain an integrable gyrostatic generalisation of the Veselova system.

This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hinge. Although this system is nonholonomic, it can be represented in Hamiltonian form with a Lie–Poisson bracket of rank two. This Lie–Poisson bracket is used to perform stability analysis of fixed points. In addition, all possible types of integral manifolds are found and a classification of trajectories on them is presented.