Alexander Kilin
Doctor of Physics and Mathematics, Professor
Professor of Department of Theoretical Physics at UdSU
Head of Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles at UdSU
Leading Researcher of the Department of Mathematical Methods of Nonlinear Dynamics at Institute of Mathematics and Mechanics UB RAS
Born: May 31, 1976
In 1997 graduated from Udmurt State University (UdSU).
19972001: research assistant in Laboratory of dynamical Chaos and Nonlinearity, UdSU.
2001: Thesis of Ph.D. (candidate of science). Thesis title: "Computeraided methods in study of nonlinear dynamical systems", UdSU
2002: senior scientist of Laboratory of Dynamical Chaos an Nonlinearity, UdSU.
2004present: Senior scientist of Department of Mathematical Methods in Nonlinear Dynamics, IMM UB RAS;
Scientific secretary of Institute of Computer Science
2009: Doctor in physics and mathematics. Thesis title: "Development of the software package for computer studies of dynamical systems", Moscow Engineering Physics Institute.
since 2010: Head of Laboratory of Dynamical Chaos and Nonlinearity at UdSU
since 2011: Dean of the Faculty of Physics and Energetics at UdSU
Publications:
Kilin A. A., Artemova E. M.
Abstract
This paper addresses the problem of the motion of two point vortices of arbitrary strengths
in an ideal incompressible fluid on a finite flat cylinder. A procedure of reduction to the level set
of an additional first integral is presented. It is shown that, depending on the parameter values,
three types of bifurcation diagrams are possible in the system. A complete bifurcation analysis of
the system is carried out for each of them. Conditions for the orbital stability of generalizations
of von Kármán streets for the problem under study are obtained.

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

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

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

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

Karavaev Y. L., Kilin A. A.
Abstract
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.

Karavaev Y. L., Klekovkin A. V., Kilin A. A.
Abstract
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.

Bizyaev I. A., Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
This paper is concerned with two systems from subRiemannian 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.

Vetchanin E. V., Kilin A. A.
Abstract
This paper is concerned with the motion of an unbalanced heavy threeaxial 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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
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 quasivelocities. 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 onewheeled vehicle and a wheeled vehicle with two rotating wheel pairs.

Kilin A. A., Karavaev Y. L.
Abstract
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.

Kilin A. A., Vetchanin E. V.
Abstract
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.

Karavaev Y. L., Kilin A. A.
Abstract
The dynamic model for a spherical robot with an internal omniwheel platform is presented. Equations of motion and first integrals according to the nonholonomic 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.

Kilin A. A., Bobykin A. D.
Abstract
The problem of motion of a vehicle in the form of a platform with an arbitrary number of Mecanum wheels fastened on it is considered. The controllability of this vehicle is discussed within the framework of the nonholonomic rolling model. An explicit algorithm is presented for calculating the control torques of the motors required to follow an arbitrary trajectory. Examples of controls for executing the simplest maneuvers are given.

Kilin A. A., Karavaev Y. L.
Abstract
The kinematic control model for a spherical robot with an internal omniwheel platform is presented. We consider singularities of control of spherical robot with an unbalanced internal omniwheel platform. The general algorithm of control of spherical robot according to the kinematical quasistatic model and controls for simple trajectories (a straight line and in a circle) are presented. Experimental investigations have been carried out for all introduced control algorithms.

Kilin A. A., Karavaev Y. L., Klekovkin A. V.
Abstract
In this article a kinematic model of the spherical robot is considered, which is set in motion by the internal platform with omniwheels. It has been introduced a description of construction, algorithm of trajectory planning according to developed kinematic model, it has been realized experimental research for typical trajectories: moving along a straight line and moving along a circle.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of a reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.

Bolsinov A. V., Kilin A. A., Kazakov A. O.
Abstract
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.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
In our earlier paper [2] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with noslip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipationfree periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
In the paper we study control of a balanced dynamically nonsymmetric sphere with rotors. The noslip condition at the point of contact is assumed. The algebraic contrability is shown and the control inputs providing motion of the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
We consider the problem of the motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions in which the mutual distances remain finite (we call them pseudoleapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
We consider a nonholonomic model of the dynamics of an omniwheel vehicle on a plane and a sphere. An elementary derivation of equations is presented, the dynamics of a free system is investigated, a relation to control problems is shown.

Borisov A. V., Kilin A. A., Mamaev I. S.
Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support
Abstract
We consider the problem of explicit integration and bifurcation analysis for two systems of nonholonomic mechanics. The first one is the Chaplygin’s problem on noslip rolling of a balanced dynamically nonsymmetrical ball on a horizontal plane. The second problem is on the motion of rigid body in a spherical support. We explicitly integrate this problem by generalizing the transformation which Chaplygin applied to the integration of the problem of the rolling ball at a nonzero constant of areas. We consider the geometric interpretation of this transformation from the viewpoint of a trajectory isomorphism between two systems at different levels of the energy integral. Generalization of this transformation for the case of dynamics in a spherical support allows us to integrate the equations of motion explicitly in quadratures and, in addition, to indicate periodic solutions and analyze their stability. We also show that adding a gyrostat does not lead to the loss of integrability.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored nonholonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of «clandestine» linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.

Borisov A. V., Bolotin S. V., Kilin A. A., Mamaev I. S., Treschev D. V.
Abstract

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
In this paper we develop a new model of nonholonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the threedimensionalpointmap technique, the nonintegrability of the nonholonomic billiard within an ellipse is shown.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
We consider the problems of Hamiltonian representation and integrability of the nonholonomic Suslov system and its generalization suggested by S. A. Chaplygin. These aspects are very important for understanding the dynamics and qualitative analysis of the system. In particular, they are related to the nontrivial asymptotic behaviour (i. e. to some scattering problem). The paper presents a general approach based on the study of the hierarchy of dynamical behaviour of nonholonomic systems.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
We consider the motion of a material point on the surface of a sphere in the field of 2n+1 identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [3], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the onedimensional Nparticle system discussed in the recent paper [13] and show that for the latter system an analogous superintegral can be constructed.

Kilin A. A.
Abstract
3particle systems with a particleinteraction homogeneous potential of degree $α=2$ is considered. A constructive procedure of reduction of the system by 2 degrees of freedom is performed. The nonintegrability of the systems is shown using the Poincare mapping.

Borisov A. V., Kilin A. A., Mamaev I. S.
Abstract
Systems of material points interacting both with one another and with an external field are considered in Euclidean space. For the case of arbitrary binary interaction depending solely on the mutual distance between the bodies, new integrals are found, which form a Galilean momentum vector. A corresponding algebra of integrals constituted by the integrals of momentum, angular momentum, and Galilean momentum is presented. Particle systems with a particleinteraction potential homogeneous of degree $α=2$ are considered. The most general form of the additional integral of motion, which we term the Jacobi integral, is presented for such systems. A new nonlinear algebra of integrals including the Jacobi integral is found. A systematic description is given to a new reduction procedure and possibilities of applying it to dynamics with the aim of lowering the order of Hamiltonian systems. Some new integrable and superintegrable systems generalizing the classical ones are also described. Certain generalizations of the Lagrangian identity for systems with a particleinteraction potential homogeneous of degree $α=2$ are presented. In addition, computational experiments are used to prove the nonintegrability of the Jacobi problem on a plane. 