Vol. 17, no. 4
Vol. 17, no. 4, 2021
Ligęza A., Żołądek H.
Abstract
We consider the situation where three heavy gravitational bodies form the Lagrange configuration
rotating in a fixed plane and the fourth body of negligible mass moves in this plane. We
present three cases of socalled libration points and we study their stability using linear approximation
and KAM theory. In some situations we prove the Lyapunov stability for generic values
of some parameter of the problem.

Xiong J., Jia Y., Liu C.
Abstract
In this paper, we study the symmetry of a bicycle moving on a flat, level ground. Applying
the Gibbs – Appell equations to the bicycle dynamics, we previously observed that the coefficients
of these equations appeared to depend on the lean and steer angles only, and in one such
equation, a term quadratic in the rear wheel’s angular velocity and a pseudoforce term would
always vanish. These properties indeed arise from the symmetry of the bicycle system. From the
point of view of the geometric mechanics, the bicycle’s configuration space is a trivial principal
fiber bundle whose structure group plays the role of a symmetry group to keep the Lagrangian
and constraint distribution invariant. We analyze the dimension relationship between the space
of admissible velocities and the tangent space to the group orbit, and then employ the reduced
nonholonomic Lagrange – d’Alembert equations to directly prove the previously observed properties
of the bicycle dynamics. We then point out that the Gibbs – Appell equations give the local
representative of the reduced dynamic system on the reduced constraint space, whose relative
equilibria are related to the bicycle’s uniform upright straight or circular motion. Under the full
rank condition of a Jacobian matrix, these relative equilibria are not isolated, but form several
families of oneparameter solutions. Finally, we prove that these relative equilibria are Lyapunov
(but not asymptotically) stable under certain conditions. However, an isolated asymptotically
stable equilibrium may be achieved by restricting the system to an invariant manifold, which is
the level set of the reduced constrained energy.

Burov A. A., Nikonov V. I.
Abstract
The problem of the existence and stability of relative equilibria (libration points) of a uniformly
rotating gravitating body, which is a homogeneous ball with a spherical cavity, is considered.
It is assumed that the rotation is carried out around an axis perpendicular to the axis
of symmetry of the body and passing through its center of mass. The libration points located
inside the cavity are investigated. Families of both isolated and nonisolated relative equilibria
are found. Their stability and bifurcations are investigated. Realms of possible motion are
constructed.

Ivanov A. P.
Abstract
A simple model of a capsule robot is studied. The device moves upon a rough horizontal
plane and consists of a capsule with an embedded motor and an internal moving mass. The motor
generates a harmonic force acting on the bodies. Capsule propulsion is achieved by collisions of
the inner body with the right wall of the shell. There is Coulomb friction between the capsule
and the support, it prevents a possibility of reversal motion. A periodic motion is constructed
such that the robot gains the maximal average velocity.

Mikishanina E. A.
Abstract
This article examines the dynamics of the movement of a wheeled vehicle consisting of two
links (trolleys). The trolleys are articulated by a frame. One wheel pair is fixed on each link.
Periodic excitation is created in the system due to the movement of a pair of masses along the
axis of the first trolley. The center of mass of the second link coincides with the geometric center
of the wheelset. The center of mass of the first link can be shifted along the axis relative to the
geometric center of the wheelset. The movement of point masses does not change the center of
mass of the trolley itself. Based on the joint solution of the Lagrange equations of motion with
undetermined multipliers and time derivatives of nonholonomic coupling equations, a reduced
system of differential equations is obtained, which is generally nonautonomous. A qualitative
analysis of the dynamics of the system is carried out in the absence of periodic excitation and
in the presence of periodic excitation. The article proves the boundedness of the solutions of
the system under study, which gives the boundedness of the linear and angular velocities of
the driving link of the articulated wheeled vehicle. Based on the numerical solution of the
equations of motion, graphs of the desired mechanical parameters and the trajectory of motion
are constructed. In the case of an unbiased center of mass, the solutions of the system can be
periodic, quasiperiodic and asymptotic. In the case of a displaced center of mass, the system
has asymptotic dynamics and the mobile transport system goes into rectilinear uniform motion.

Bardin B. S., Chekina E. A.
Abstract
The orbital stability of pendulumlike oscillations of a heavy rigid body with a fixed point in
the Bobylev – Steklov case is investigated. In particular, a nonlinear study of the orbital stability
is performed for the socalled case of degeneracy, where it is necessary to take into account terms
of order six in the Hamiltonian expansion in a neighborhood of the unperturbed periodic orbit.

Morozov A. I.
Abstract
According to the Nielsen – Thurston classification, the set of homotopy classes of orientationpreserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: $T_1^{}$) periodic homeomorphism; $T_2^{}$) reducible nonperiodic homeomorphism of algebraically finite order; $T_3^{}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_4^{}$) pseudoAnosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are $T_1^{}$, $T_2^{}$, $T_4^{}$ only. Moreover, all representatives of the class $T_4^{}$ have chaotic dynamics, while in each homotopy class of types $T_1^{}$ and $T_2^{}$ there are regular diffeomorphisms, in particular, Morse – Smale diffeomorphisms with a finite number of heteroclinic orbits. The author has found a criterion that allows one to uniquely determine the homotopy type of a Morse – Smale diffeomorphism with a finite number of heteroclinic orbits on a twodimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and nontrivial. It is proved that if the heteroclinic points of a Morse – Smale diffeomorphism are contained only in the trivial domains then such diffeomorphism has the homotopic type $T_1^{}$. The orbit space of nontrivial heteroclinic domains consists of a finite number of twodimensional tori, where the saddle separatrices participating in heteroclinic intersections are projected as transversally intersecting knots. That whether the Morse – Smale diffeomorphisms belong to types $T_1^{}$ or $T_2^{}$ is uniquely determined by the total intersection index of such knots.

Nakamura G., Plaszczynski S., Grammaticos B., Badoual M.
Abstract
We study the effect of an emerging virus mutation on the evolution of an epidemic, inspired
by the appearance of the delta variant of SARSCoV2. We show that if the new variant is
markedly more infective than the existing ones the epidemic can resurge immediately. The
vaccination of the population plays a crucial role in the evolution of the epidemic. When the older
(and more vulnerable) layers of the population are protected, the new infections concern mainly
younger people, resulting in fewer hospitalisations and a reduced stress on the health system. We
study also the effects of vacations, partially effective vaccines and vaccination strategies based
on epidemicawareness. An important finding concerns vaccination deniers: their attitude may
lead to a prolonged wave of epidemic and an increased number of hospital admissions.

Kulathunga G., Devitt D., Fedorenko R., Klimchik A. S.
Abstract
Any obstaclefree path planning algorithm, in general, gives a sequence of waypoints that
connect start and goal positions by a sequence of straight lines, which does not ensure the smoothness
and the dynamic feasibility to maneuver the MAV. Kinodynamicbased motion planning is
one of the ways to impose dynamic feasibility in planning. However, kinodynamic motion planning
is not an optimal solution due to high computational demands for realtime applications.
Thus, we explore path planning followed by kinodynamic smoothing while ensuring the dynamic
feasibility of MAV. The main difference in the proposed technique is not to use kinodynamic
planning when finding a feasible path, but rather to apply kinodynamic smoothing along the
obtained feasible path. We have chosen a geometricbased path planning algorithm “RRT*” as
the path finding algorithm. In the proposed technique, we modified the original RRT* introducing
an adaptive search space and a steering function that helps to increase the consistency
of the planner. Moreover, we propose a multiple RRT* that generates a set of desired paths.
The optimal path from the generated paths is selected based on a cost function. Afterwards, we
apply kinodynamic smoothing that will result in a dynamically feasible as well as obstaclefree
path. Thereafter, a bsplinebased trajectory is generated to maneuver the vehicle autonomously
in unknown environments. Finally, we have tested the proposed technique in various simulated
environments. According to the experiment results, we were able to speed up the path planning
task by 1.3 times when using the proposed multiple RRT* over the original RRT*.

Shaura A. S., Tenenev V. A., Vetchanin E. V.
Abstract
This paper addresses the problem of balancing an inverted pendulum on an omnidirectional
platform in a threedimensional setting. Equations of motion of the platform – pendulum system
in quasivelocities are constructed. To solve the problem of balancing the pendulum by controlling
the motion of the platform, a hybrid genetic algorithm is used. The behavior of the system is
investigated under different initial conditions taking into account a necessary stop of the platform
or the need for continuation of the motion at the end point of the trajectory. It is shown that
the solution of the problem in a twodimensional setting is a particular case of threedimensional
balancing.

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.
