
Keyword: "stabilization"
Aleksandr Sergeevich Andreev, Ulyanovsk State University, Doctor of Sciences in Physics and Mathematics, Professor; graduated from the Faculty of Mechanics and Mathematics of Tashkent State University; Head of the Department of Information Security and Control Theory of Ulyanovsk State University; an author of articles, textbooks, and a monograph in the field of stability theory and the motion control of mechanical systems. [email: AndreevAS@ulsu.ru]A. Andreev,
Olga Alekseevna Peregudova, Ulyanovsk State University, Doctor of Sciences in Physics and Mathematics, Associate professor; graduated from the Faculty of Mechanics and Mathematics of Ulyanovsk State University; Professor of the Department of Information Security and Control Theory of Ulyanovsk State University; an author of articles, textbooks, a monograph in the field of the theory of stability and motion control of mechanical systems. [email: peregudovaoa@gmail.com]O. Peregudova


Robust Motion Stabilization of a Mobile Robot with OmnyWheels
The modelling as well as the design and widespread use of wheeled mobile robots in the industrial and social spheres is one of the areas of rapid development of robotics. A sufficiently large class of such robots consists of wheeled robots with rollerbearing or omniwheels. A distinctive feature of the design of such wheels consisting in the fact that the rollers are fixed to them according to a certain scheme allows the robot to move in any direction without a prior turn. This achieves high maneuverability compared to other wheeled carriages. The paper investigates the trajectory tracking control problem of a mobile robot with three omniwheels and with an offset center of mass, i.e. when it is assumed that the center of mass of the robot does not coincide with the geometric center of the platform. Previously, such a problem was not considered. The paper substantiates the control structure that provides tracking of a given trajectory, including taking into account the delay and discreteness of the signal in the feedback. At the same time the control has the property of robustness which consists in the fact that its parameters do not depend directly on the massinertial parameters of the system and the tracked trajectory. The controller is constructed only by using the values of the system parameters bounds. The result has been achieved on the basis of the development of direct Lyapunov method in the study of the stability of nonautonomous systems obtained in the previous papers of the authors. The results of numerical modeling of the problem studied are presented. Wheeled mobile robot, robust control, stabilization, trajectory tracking, Lyapunov functional.



Sections: Mathematical modeling
Subjects: Mathematical modeling. 
Aleksandr Sergeevich Andreev, Ulyanovsk State University, Doctor of Physics and Mathematics, Professor; graduated from the Faculty of Mechanics and Mathematics of Tashkent State University; Head of the Faculty of Mathematics, Information and Aviation Technologies at Ulyanovsk State University; Head of the Department of Information Security and Control Theory; an author of papers, textbooks, and a monograph in the field of stability theory and the motion control of mechanical systems. [email: AndreevAS@ulsu.ru]A. Andreev, Olga Alekseevna Peregudova, Ulyanovsk State University, Doctor of Physics and Mathematics, Associate Professor; graduated from the Faculty of Mechanics and Mathematics of Ulyanovsk State University; Professor at the Department of Information Security and Control Theory of Ulyanovsk State University; an author of articles, textbooks, and a monograph in the field of stability theory and the motion control of mechanical systems. [email: peregudovaoa@sv.ulsu.ru]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 twostep 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 threelink manipulator controlled by three independent DC drives. Mechanical system, stabilization, program motion, actuator dynamics, definite lyapunov function.



Sections: Mathematical modeling
Subjects: Mathematical modeling. 
Aleksandr Sergeevich Andreev, Ulyanovsk State University, Doctor of Physics and Mathematics, Professor; graduated from the Faculty of Mechanics and Mathematics of Tashkent State University; Head of the Faculty of Mathematics, Information and Aviation Technologies at Ulyanovsk State University; Head of the Department of Information Security and Control Theory; an author of papers, textbooks, and a monograph in the field of stability theory and the motion control of mechanical systems. [email: AndreevAS@ulsu.ru]A. Andreev, Olga Alekseevna Peregudova, Ulyanovsk State University, Doctor of Physics and Mathematics, Associate Professor; graduated from the Faculty of Mechanics and Mathematics at Ulyanovsk State University; Professor at the Department of Information Security and Control Theory of Ulyanovsk State University; an author of articles, textbooks, and a monograph in the field of stability theory and the motion control of mechanical systems. [email: peregudovaoa@sv.ulsu.ru]O. Peregudova


Twolink Manipulator Control Synthesis Without Velocity Measuring
The paper investigates the way to treat the problem of twolink 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 nonlocal stabilization of the manipulator nonstationary program motions because of such difficulties like nonlinearity and nonstationarity of the system. The article provides the method of piecewise nonlinear 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 nonstationary motions without system linearization. The article presents the results of numerical simulation that prove obtained theoretical results. Twolink manipulator, stabilization, program motion, control law without measuring velocity, comparison system, lyapunov vector function.



Sections: Mathematical modeling
Subjects: Mathematical modeling. 
Aleksandr Sergeevich Andreev, Ulyanovsk State University, [email: AndreevAS@ulsu.ru]A. Andreev, Stanislav Iurievich Rakov, Ulyanovsk State University, graduated from the Faculty of Mathematics and Information Technologies of Ulyanovsk State University; Junior Researcher of Scientific Research Center of Ulyanovsk State University; an author of articles in the field of the motion control of mechanical systems. [email: rakov.stanislav@gmail.com]S. Rakov


On Control of Twolink Robotic Manipulator on the Base of Picontroller
The article deals with the problem of of twolink robotic manipulator program motions stabilization on a movable base by creating a PIcontroller. 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 proportionalintegral 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 PIcontrol for the various mechanical systems was used. A numerical solution of the resulting system of integraldifferential equations is found. A numerical solution of the received system of integraldifferential equations with abnormal indices is found. The corresponding graphs for the coordinates of the manipulator links proving the theoretical results are built. Twolink manipulator, stabilization, program motion, picontrol, movable base.



Sections: Mathematical modeling
Subjects: Mathematical modeling, Artificial intelligence. 
Olga Alekseevna Peregudova, Ulyanovsk State University, Doctor of Physics and Mathematics, Associate Professor; graduated from the Faculty of Mechanics and Mathematics at Ulyanovsk State University; Professor at the Department of Information Security and Control Theory of Ulyanovsk State University; an author of articles, textbooks, and a monograph in the field of stability theory and the motion control of mechanical systems. [email: peregudovaoa@sv.ulsu.ru]O. Peregudova, Denis Sergeevich Makarov, Ulyanovsk State University, a PostGraduate Student; graduated from the Faculty of Mathematics and Information Technologies of Ulyanovsk State University; Junior Researcher at the Department of Scientific Research of Ulyanovsk State University; an author of articles in the field of the motion control of mechanical systems. [email: prostodenis18@mail.ru]D. Makarov


Control Synthesis for Threelink Manipulator
A stabilization problem of the threelink manipulator program motion by providing continuous nonlinear control with direct and feedback communication is solved in this paper. The manipulator consists of three totally rigid links. The first link is fastened on a horizontal base and can pivot about a vertical axis. The second one is interconnected with the first and the third links by the ideal cylindrical hinges. Relative movements of the second and the third links are performed in a vertical place that pivots about a vertical axis due to the first link motion. Thus, the manipulator can perform threedimensional motion. 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 control moment construction that allows 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 control actions. By applying Lyapunov’s vector functions and the comparison systems the implementation of the built control laws in the stabilization task of the spectrum of the manipulator program motions was proved. The novelty of the results includes solving the stabilization problem of the nonstationary and nonlinear formulation, without using a linearized model, as well as the ability to stabilize not just one but a whole family of program motions. 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 of the manipulator links proving the theoretical results are built. Threelink manipulator, stabilization, program motion, continuous control, comparison system, lyapunov’s vector function.



Sections: Mathematical modeling
Subjects: Mathematical modeling. 
Aleksandr Sergeevich Andreev, Ulyanovsk State University, Doctor of Physics and Mathematics, Professor; graduated from the Faculty of Mechanics and Mathematics of Tashkent State University; Dean of the Faculty of Mathematics and Information Technologies at Ulyanovsk State University; Head of the Department of Information Security and Control Theory of Ulyanovsk State University; an author of articles, textbooks, and a monograph in the field of stability theory and the motion control of mechanical systems. [email: AndreevAS@ulsu.ru]A. Andreev, Ekaterina Alekseevna Kudashova, Ulyanovsk State University, graduated from the Faculty of Mathematics and Information Technologies of Ulyanovsk State University; Junior Researcher at the Department of Scientific Research of Ulyanovsk State University; an author of articles in the field of the motion control of mechanical systems. [email: katherine.kudashova@yandex.ru]E. Kudashova


On Modeling the Control Structure of the Omnidirectional Mobile Robo
Nowadays, requirements to modeling and researching selfdirected robotic systems are extremely high. In order to improve maneuverability and control efficiency, new mobile robots with omniwheels are developed. Such robots are able to move in either direction without turning around. They have these features due to the fact of increase of construction and control rules complexity. Three and fourwheeled mobile robots with rollercarrying wheels have become more widespread. The article deals with the problem of theoretical control establishing to provide arbitrary program motion of threewheeled robots with omniwheels. The computer model of valid control efficiency analysis was developed. For developing this model, .the numerical modeling method that turns the continuous model into the corresponding numerical one was used. Practical application of the introduced stabilizing control algorithm for mechanical systems was demonstrated by the example of threewheeled robot motion stabilization. Mathematical modeling, threewheeled robot, stabilization, control, digitization.



Sections: Mathematical modeling
Subjects: Mathematical modeling, Artificial intelligence. 
Olga Alekseevna Peregudova, Ulyanovsk State University, Doctor of Physics and Mathematics, Associate Professor; graduated from the Faculty of Mechanics and Mathematics at Ulyanovsk State University; Professor at the Department of Information Security and Control Theory of Ulyanovsk State University; an author of articles, textbooks, and a monograph in the field of stability theory and the motion control of mechanical systems. [email: peregudovaoa@sv.ulsu.ru]O. Peregudova, Denis Sergeevich Makarov, Ulyanovsk State University, a postgraduate student; graduated from the Faculty of Mathematics and Information Technologies of Ulyanovsk State University; Junior Researcher at the Department of Scientific Research of Ulyanovsk State University; an author of articles in the field of the motion control of mechanical systems. [email: prostodenis18@mail.ru]D. Makarov


Control Synthesis for Doublelink Manipulator
A stabilization problem of a doubllink 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 vectorfunctions 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 nonstationary 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. Multilink manipulator, stabilization, program motion, relay control, comparison system, lyapunov’s vectorfunction.



Sections: Automated control systems
Subjects: Automated control systems, Mathematical modeling. 

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.



Sections: Theoretical issues of automation of command and control processes
Subjects: Mathematical modeling. 
