Vol. 19, no. 4

Vol. 19, no. 4, 2023

Kholostova O. V.
This paper is concerned with the motions of a near-autonomous two-degree-of-freedom 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 three-frequency conditionally periodic motions (with frequencies of different orders in a small parameter) are described.
Keywords: Hamiltonian system, multiple parametric resonance, parametric resonance regions, periodic motions, conditionally periodic motions, stability, KAM theory
Citation: Kholostova O. V., On Nonlinear Oscillations of a Near-Autonomous Hamiltonian System in One Case of Integer Nonequal Frequencies, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 447-471
Alfimov G. L.,  Lebedev M. E.
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 one-to-one correspondence with bi-infinite 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 map-over-period (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.
Keywords: Duffing-type equation, periodic coefficients, symbolic dynamics, Smale horseshoe
Citation: Alfimov G. L.,  Lebedev M. E., Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 473-506
Hoang D.,  Dudarenko N. A.
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 high-order 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.
Keywords: adaptive control, unknown external disturbances, inverted pendulum, internal model principle, linear quadratic regulator
Citation: Hoang D.,  Dudarenko N. A., Adaptive Compensation for Unknown External Disturbances for an Inverted Pendulum Based on the Internal Model Principle, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 507-519
Rodnikov A. V.
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.
Keywords: solar sail, libration point, binary asteroid, three-body problem
Citation: Rodnikov A. V., Keeping a Solar Sail near the Triangular Libration Point of a Dumbbell-Shaped or Binary Asteroid, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 521-532
Kilin A. A.,  Ivanova T. B.
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.
Keywords: nonholonomic constraint, first integral, nonintegrability, Poincaré map
Citation: Kilin A. A.,  Ivanova T. B., The Problem of the Rolling Motion of a Dynamically Symmetric Spherical Top with One Nonholonomic Constraint, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 533-543
Bardin B. S.,  Sukhov E. A.,  Volkov E. V.
We consider the planar circular restricted four-body 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 so-called 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 fourth-order resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram.
Keywords: Hamiltonian mechanics, four-body problem, equal masses, periodic motions, orbital stability, symplectic mapping, nonlinear analysis, numerical computation
Citation: Bardin B. S.,  Sukhov E. A.,  Volkov E. V., Nonlinear Orbital Stability of Periodic Motions in the Planar Restricted Four-Body Problem, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 545-557
Sachkov Y. L.,  Sachkova E. F.
We study the left-invariant sub-Riemannian 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.
Keywords: sub-Riemannian geometry, abnormal trajectories, geometric control
Citation: Sachkov Y. L.,  Sachkova E. F., Abnormal Extremals in the Sub-Riemannian Problem with Growth Vector $(2, 3, 5, 8, 14)$, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 559-573
Gasanov M. V.,  Gulkanov A. G.
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.
Keywords: nonlinear differential equations, movable singular point, exact criteria of existence, necessary and sufficient conditions, Cauchy problem
Citation: Gasanov M. V.,  Gulkanov A. G., A Study of a Mathematical Model with a Movable Singular Point in a Fourth-Order Nonlinear Differential Equation, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 575-584
Fam C. A.,  Nedelchev S.
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.
Keywords: optimization, sliding mode control, parametric uncertainty, stability
Citation: Fam C. A.,  Nedelchev S., Optimization Driven Robust Control of Mechanical Systems with Parametric Uncertainties, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 585-597
Demian A. A.,  Klimchik A. S.
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 multi-body 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 wrist-compensator 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.
Keywords: static balancing, force compensation, manipulator design, variable stiffness
Citation: Demian A. A.,  Klimchik A. S., Design of a Robotic Spherical Wrist with Variable Stiffness, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 599-612
Marchuk E. A.,  Al Badr A.,  Kalinin Y. V.,  Maloletov A. V.
This paper highlights the role of game theory in specific control tasks of cable-driven 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 null-space 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 large-sized cable-driven parallel robot, and the results are shown in examples with numerical simulation.
Keywords: cable-driven robot, parallel robot, distribution, null space, cooperative game, fuzzy logic, structural nonlinearity
Citation: Marchuk E. A.,  Al Badr A.,  Kalinin Y. V.,  Maloletov A. V., Cable-Driven Parallel Robot: Distribution of Tension Forces, the Problem of Game Theory, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 613-631
Nedelchev S.,  Kozlov L.,  Khusainov R. R.,  Gaponov I.
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 3-DoF robotic leg.
Keywords: adaptive control, parameter estimation, motion control
Citation: Nedelchev S.,  Kozlov L.,  Khusainov R. R.,  Gaponov I., Enhanced Adaptive Control over Robotic Systems via Generalized Momentum Dynamic Extensions, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 4, pp. 633-646

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