ISSN 1991-2927
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## Keyword: "program motion"

 A. Andreev, O. Peregudova
 On Motion Control of the Mechanical System on the Basis of Actuator Dynamics The problem of stabilization of the holonomic mechanical system program motion is solved taking into account the actuator dynamics. As it is known, the implementation of control forces and moments for the mechanical systems occurs with the help of actuators (drives), their dynamics affects the motion process. Therefore, the requirement of control implementation precision for modern mechanical systems makes it necessary to take into account the actuator dynamics. The complexity of the problems of constructing the control laws for the mathematical models of mechanical systems with actuators involves the fact that the degrees of freedom of such a system has a higher dimension with respect to the vector of control signals. The paper presents a model of a mechanical system with a drive in the form of a cascade connection of two subsystems: the mechanical one and drives. Herewith, the vector of control for the mechanical subsystem is the state of the subsystem drives. Such representation allows to solve the control problem in the form of a two-step procedure. The first step includes construction of the mechanical subsystem control law in the form of a continuously differentiable time function of time, coordinates and velocities, which carries out the stabilization of the given program motion. After that, on the second step, the relay control law is constructed for a drive subsystem that ensures the asymptotic stability of a stabilizing control law. The specificity of the obtained result involves the use of the definite Lyapunov function, which significantly simplifies the calculations for justification of the relay control law and the terms of its implementation. As an example, the problem on stabilization of a program motion is solved for a space three-link manipulator controlled by three independent DC drives.Mechanical system, stabilization, program motion, actuator dynamics, definite lyapunov function.
 2016_ 4

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 S. Bezglasnyi, V. Krasnikov
 Stabilization of Program Motions of a Single-rotor Gyrostat With a Cavity Filled With Viscous Fluid The asymptotically stable program motions problem of a single-rotor gyrostat with a spherical cavity entirely filled with viscous fluid is studied. The gyrostat is modeled by a system of two connected solid bodies with common axis of rotation. The first body is a carrier that has a cavity filled with highly viscous fluid. The second body is a dynamically symmetric rotor. In the paper, the gyrostat motion equations are constructed in the Lagrange equations form of the second kind. In the equations, the influence of fluid on the motion of the gyrostat is described using the kinematic characteristics of the gyrostat. The program motions realization problem is solved by the active program and stabilizing controls attached to the gyrostat. The 1 Авторы выражают искреннюю благодарность Андрееву А.С. и Перегудовой О.А. за внимание к работе и ряд ценных замечаний.active stabilizing controls are constructed by the principle of feedback. The task is solved on the basis of a direct method of Lyapunov’s functions of stability theory and a method of the limit functions and the limit systems. The results of this paper can be used for designing control systems for objects with cavities filled with fluids.Gyrostat, viscous fluid, program motion, lyapunov’s function, asymptotic stability.
 2016_ 2

Sections: Mathematical modeling

Subjects: Mathematical modeling, Architecture of ship's system.

 A. Andreev, O. Peregudova
 Two-link Manipulator Control Synthesis Without Velocity Measuring The paper investigates the way to treat the problem of two-link manipulator program motion stabilization without velocity measuring. Most familiar strategies for mechanical system motion control including robot manipulators are based on the assumption that both coordinates and velocities of the mechanical system are available to measure. However, certain difficulties relating to practical application arise. These difficulties include, for example, inability of the velocity sensors installation due to various constraints. Another difficulty is connected with the fact that the operation of these sensors involves the problem of noise occurrence and, in consequence, the accuracy of the control problem solution can be decreased. The main approaches to the control problem of mechanical systems (such as manipulators) without measuring velocities apply approximate differentiation of the system coordinates as well as observer construction. These approaches are not fully developed for the problem of non-local stabilization of the manipulator non-stationary program motions because of such difficulties like non-linearity and non-stationarity of the system. The article provides the method of piecewise non-linear continuous control synthesis on the basis of both the construction of the observer and the application of the Lyapunov vector functions method. The novelty of the results consists in constructing the observer which dimension is smaller than the dimension of the system by a factor of two in order to solve the stabilization problem of a wide class of manipulator non-stationary motions without system linearization. The article presents the results of numerical simulation that prove obtained theoretical results.Two-link manipulator, stabilization, program motion, control law without measuring velocity, comparison system, lyapunov vector function.
 2015_ 4

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 A. Andreev, S. Rakov
 On Control of Two-link Robotic Manipulator on the Base of Pi-controller The article deals with the problem of of two-link robotic manipulator program motions stabilization on a movable base by creating a PI-controller. The manipulator consists of two homogeneous links connected by a joint. A moving load is placed in the second link gripper. A movable base makes a translational motion in the horizontal plane. The links of the manipulator are also moving in the horizontal plane. Thus, the manipulator performs planar motion. The manipulator motions are described by the system of Lagrange equations of the second kind. The paper presents a control law carrying out stabilization of the given program motion as a proportional-integral dependence on condition that the base of the manipulator performs predetermined unsteady motion. The problem of program motion stabilization has been solved for the linearized model. For the numerical simulation a new program that allows providing PI-control for the various mechanical systems was used. A numerical solution of the resulting system of integral-differential equations is found. A numerical solution of the received system of integral-differential equations with abnormal indices is found. The corresponding graphs for the coordinates of the manipulator links proving the theoretical results are built.Two-link manipulator, stabilization, program motion, pi-control, movable base.
 2015_ 3

Sections: Mathematical modeling

Subjects: Mathematical modeling, Artificial intelligence.

 O. Peregudova, D. Makarov
 2015_ 2

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 O. Peregudova, D. Makarov
 Control Synthesis for Double-link Manipulator A stabilization problem of a doubl-link manipulator program motion on a fixed base is solved in this paper. Totally rigid manipulator links are interconnected by an ideal cylindrical hinge and via the same hinge the first element is fastened to the base. Thus, the manipulator can perform motions in a vertical plane only. The manipulator motions are described by the system of Lagrange equations of the second kind. The problem on synthesis of the motion control of such system involves the construction of the laws of a control moments change that allow the manipulator to carry out the motion given by the program in real conditions of external and internal disturbances, and the inaccuracy of the model itself. A mathematical model of the manipulator controlled motion is constructed in this paper in case of control actions in the form of continuous and discontinuous functions bounded in modulus. By applying Lyapunov’s vector-functions and the comparison systems we justified the implementation of the built control laws in the stabilization task of the spectrum of the manipulator program motions. The novelty of the results includes solving the stabilization problem of the non-stationary and nonlinear formulation, without changing to a linearized model, as well as the ability to stabilize not just one but a whole family of program motions. With the help of Maple’s mathematical package a numerical solution of the received system of differential equations using both continuous and discontinuous control laws is found. The corresponding graphs for the coordinates and velocities of the manipulator links proving the theoretical results are built.Multi-link manipulator, stabilization, program motion, relay control, comparison system, lyapunov’s vector-function.
 2014_ 4

Sections: Automated control systems

Subjects: Automated control systems, Mathematical modeling.

 S. Bezglasny, M. Khudiakova
 Gyrostat Orientation With Variable Moment of Inertia The article examines a problem of a single-axis and a triaxial orientations of the gyrostat system with inertia moment of carrier wich depend on time (a variable structure). The orientation is researched relatively Koenig’s coordinate system and a random non-inertial coordinate system. The problem is solved analytically by the active control construction applied to the system of bodies on the principle of feedback. The stabilizing control and conditions providing a possibility of desired orientation are obtained. This orientation has a property of an asymptotic stability. The assigned task was being solved on the basis of the Lyapunov’s functions method and a method of limit equations and systems which enable to use the Lyapunov’s functions which possess constant signs derivatives.Gyrostat, program motion, functions with a constant signs, lyapunov's function, asymptotic stability.
 2013_ 2

Sections: Mathematical modeling, calculi of approximations and software systems

Subjects: Mathematical modeling, Electrical engineering and electronics.

 A. Andreev, E. Belikova
 Synthesis of Mechanical System Movement Control The actual paper presents studies in control of general nonlinear mechanical system by means of decomposition into systems of the same degree of freedom. The obtained control law is compared by efficiency with controls ensuring stabilization within infinite time interval. The article suggests an algorithm for solution of task of control decomposition of general mechanical system.Nonlinear mechanical system, control synthesis, decomposition.
 2009_ 4

Sections: Theoretical issues of automation of command and control processes

Subjects: Mathematical modeling.

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