Vol. 19, no. 4
Vol. 19, no. 4, 2023
Kholostova O. V.
Abstract
This paper is concerned with the motions of a nearautonomous twodegreeoffreedom
Hamiltonian system, $2\pi$periodic in time, in a neighborhood of a trivial equilibrium. It is assumed
that in the autonomous case, in the region where only necessary (which are not sufficient)
conditions for the stability of this equilibrium are satisfied, for some parameter values of the
system one of the frequencies of small linear oscillations is equal to two and the other is equal
to one. An analysis is made of nonlinear oscillations of the system in a neighborhood of this
equilibrium for the parameter values near a resonant point of parameter space. The boundaries
of the parametric resonance regions are constructed which arise in the presence of secondary
resonances in the transformed linear system (the cases of zero frequency and equal frequencies).
The general case and both cases of secondary resonances are considered; in particular, the case
of two zero frequencies is singled out. An analysis is made of resonant periodic motions of the
system that are analytic in integer or fractional powers of the small parameter, and conditions
for their linear stability are obtained. Using KAM theory, two and threefrequency conditionally
periodic motions (with frequencies of different orders in a small parameter) are described.

Alfimov G. L., Lebedev M. E.
Abstract
We consider the equation $u_{xx}^{}u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $u(x)<\xi$, can be put in onetoone correspondence with biinfinite sequences of numbers $n\in \{N,\,\ldots,\,N\}$ (called ``codes'' of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a ``great part'' of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the mapoverperiod (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.

Hoang D., Dudarenko N. A.
Abstract
In this paper, an adaptive compensator for unknown external disturbances for an inverted
pendulum based on the internal model principle is designed. The inverted pendulum is a typical
system that has many applications in social life, such as missile launchers, pendubots, human
walking and segways, and so on. Furthermore, the inverted pendulum is a highorder nonlinear
system, and its parameters are difficult to determine accurately. The physical constraints lead to
the complexity of its control design. Besides, there are some unknown external disturbances that
affect the inverted pendulum when it operates. The designed adaptive compensation ensures the
outputs of the system’s convergence to the desired values while also ensuring a stable system
with variable parameters and unknown disturbances. The simulation results are illustrated and
compared with the linear quadratic regulator (LQR) controller to show the effectiveness of the
proposed compensator.

Rodnikov A. V.
Abstract
The possibility of keeping a spacecraft with a solar sail near an unstable triangular libration
point of a minor planet or a binary asteroid is studied under the assumption that only the
gravitation and the solar radiation influence the spacecraft motion. The case where the solar
sail orientation remains unchanged with respect to the frame of reference of the heliocentric
orbit of the asteroid mass center is considered. This means that the angle between the solar sail
normal and ecliptic, as well as the angle between this normal and the solar rays at the current
point, does not change during the motion. The spacecraft equations of motion are deduced under
assumptions of V.V. Beletsky’s generalized restricted circular problem of three bodies, but taking
into account the Sun radiation. The existence of a manifold of initial conditions for which it is
possible to choose the normal direction that guarantees the spacecraft bounded motion near the
libration point is established. Moreover, the dimension of this manifold coincides with that of
the phase space of the problem at which the libration point belongs to the manifold boundary. In
addition, some proposals for stabilization of the spacecraft motions are formulated for trajectories
beginning in the manifold.

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.

Bardin B. S., Sukhov E. A., Volkov E. V.
Abstract
We consider the planar circular restricted fourbody problem with a small body of negligible
mass moving in the Newtonian gravitational field of three primary bodies, which form a stable
Lagrangian triangle. The small body moves in the same plane with the primaries. We assume
that two of the primaries have equal masses. In this case the small body has three relative
equilibrium positions located on the central bisector of the Lagrangian triangle. In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the socalled local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third and fourthorder resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram. 
Sachkov Y. L., Sachkova E. F.
Abstract
We study the leftinvariant subRiemannian problem on the free nilpotent Lie group of
rank 2 and step 5. We describe some abnormal trajectories and some properties of the set filled
by nice abnormal trajectories starting at the identity of the group.

Gasanov M. V., Gulkanov A. G.
Abstract
This article introduces a mathematical model that utilizes a nonlinear differential equation to
study a range of phenomena such as nonlinear wave processes, and beam deflections. Solving this
equation is challenging due to the presence of moving singular points. The article addresses two
main problems: first, it establishes the existence and uniqueness of the solution of the equation
and, second, it provides precise criteria for determining the existence of a moving singular point.
Additionally, the article presents estimates of the error in the analytical approximate solution
and validates the results through a numerical experiment.

Fam C. A., Nedelchev S.
Abstract
This paper presents a control algorithm designed to compensate for unknown parameters in
mechanical systems, addressing parametric uncertainty in a comprehensive manner. The control
optimization process involves two key stages. Firstly, it estimates the narrow uncertainty bounds
that satisfy parameter constraints, providing a robust foundation. Subsequently, the algorithm
identifies a control strategy that not only ensures uniform boundedness of tracking error but also
adheres to drive constraints, effectively minimizing chattering. The proposed control scheme is
demonstrated through the modeling of a single rigid body with parameter uncertainties. The
algorithm possesses notable strengths such as maximal compensation for parametric uncertainty,
chattering reduction, and consideration of control input constraints. However, it is applicable
for continuous systems and does not explicitly account for uncertainty in the control input.

Demian A. A., Klimchik A. S.
Abstract
This paper discusses the design of an adjustable force compensator for a spherical wrist
dedicated to robot milling and incremental sheet metal forming applications. The design of the
compensator is modular and can be introduced to any existing manipulator design as a single
multibody auxiliary system connected with simple mechanical transmission mechanisms to the
actuators. The paper considers the design of the compensator as an arrangement of elastic
springs mounted on moving pivots. The moving pivots are responsible for adjusting the stiffness
of the wristcompensator coupling. Special attention is given to two compensation schemes in
which the value of the external force can be known or unknown, respectively. The simulation
results show that the analytical derivation of the compensator leads the main actuators to spend
zero effort to support the external force.

Marchuk E. A., Al Badr A., Kalinin Y. V., Maloletov A. V.
Abstract
This paper highlights the role of game theory in specific control tasks of cabledriven parallel
robots. One of the challenges in the modeling of cable systems is the structural nonlinearity of
cables, rather long cables can only be pulled but not pushed. Therefore, the vector of forces
in configuration space must consist of only nonnegative components. Technically, the problem
of distribution of tension forces can be turned into the problem of nonnegative least squares.
Nevertheless, in the current work the game interpretation of the problem of distribution of
tension forces is given. According to the proposed approach, the cables become actors and two
examples of cooperative games are shown, linear production game and voting game. For the
linear production game the resources are the forces in configuration space and the product is
the wrench vector in the operational space of a robot. For the voting game the actors can form
coalitions to reach the most effective composition of the vector of forces in configuration space.
The problem of distribution of forces in the cable system of a robot is divided into two problems:
that of preloading and that of counteraction. The problem of preloading is set as a problem
of nullspace of the Jacobian matrix. The problem of counteraction is set as a problem of
cooperative game. Then the sets of optimal solutions obtained are approximated with a fuzzy
control surface for the problem of preloading, and game solutions are ready to use as is for
the problem of counteraction. The methods have been applied to solve problems of largesized
cabledriven parallel robot, and the results are shown in examples with numerical simulation.

Nedelchev S., Kozlov L., Khusainov R. R., Gaponov I.
Abstract
Adaptive control and parameter estimation have been widely employed in robotics to deal
with parametric uncertainty. However, these techniques may suffer from parameter drift, dependence
on acceleration estimates and conservative requirements for system excitation. To overcome
these limitations, composite adaptation laws can be used. In this paper, we propose an
enhanced composite adaptive control approach for robotic systems that exploits the accelerationfree
momentum dynamics and regressor extensions to offer faster parameter and tracking convergence
while relaxing excitation conditions and providing a clear physical interpretation. The
effectiveness of the proposed approach is validated through experimental evaluation on a 3DoF
robotic leg.
