The topic of the article concerns the mathematical theory of the optimal processes and its methods. In particular, the extremely topical aspects of the methodology of the maximum principle and the procedure of its practical use are discussed. The article presents in details the transversality conditions, their role, place and significance in the general procedure for solving of the problems of finding the optimal functions. This work eliminates the existing methodical defect connected with an incomplete use of the transversality conditions. On concrete examples, it shows that the transversality conditions (as one of the necessary conditions of optimality in the form of the maximum principle) are an extremely effective mathematical tool (and even the only one, in some cases) for determination of the characteristic properties, laws and key characteristics (parameters, constants, integrals of motion) of the optimal solutions for the systems of ordinary differential equations (of the dynamic systems). The importance of the topic is explained by the fact that in the overwhelming majority of cases the authors believe that the transversality conditions complicate the problem of search for the optimum functions, instead of simplifying it. However, it is far f/om being so, and it is a standard error, which has become almost typical. In the paper, the authors convincingly demonstrate the following conclusion: for a maximally correct and complete solution of the problems of the optimal control, the transversality conditions, being the necessary conditions for a optimality similarly as the condition of the Hamiltonian maximum and the adjoint system of the differential equations, should be taken into account since the beginning and be included in the system of the equations, which formalize the maximum principle, directly after defining the adjoint variables, formation of the Hamil-tonian and the adjoint system of equations, determination of the properties of the optimal adjoint functions and the control variables. The concrete examples presented in the article confirm the exclusively significant role (quite often critical role) of the transversality conditions in the general algorithm of the procedure for application of the maximum principle.

The paper is devoted to the analysis of the influence of various factors on the dynamic characteristics of the fuzzy logic controller (FLC). In particular, the following factors are considered: the choice of the input and output variables; the number of terms, the turndown and the form of the membership functions of the fuzzy variables, the character of the relation between the space of the antecedents and the space of the consequents (the rule base), the mode of defuzzification and the measures of the sub-conditions significance. A research was carried out in two directions. The first direction consisted in determination of the dependence of FLC static and frequency characteristics on its settings. The second one was an analysis of the frequency characteristics in the fuzzy control systems, the construction of FLC linear model in a form of PD controller and determination of the dependence of its parameters on FLC settings. The conducted research allows us to draw the following conclusions. 1. Selection of the input and output variables and the number of terms should be based on the structure of the control system and on the requirements to its quality, avoiding their unreasonable increase. 2. The logical basis should be chosen at the stage of the system design and should not be changed in the process of its functioning. Thus, the algebraic basis ensures the least nonlinear distortion of FLC characteristics, making its behavior quite predictable. On the other hand, the control surface of FLC, when using the maxmin basis, has a small slope near the origin of the coordinates, which enhances the stabilizing properties of the controller. 3. The turndown and dilatation-concentration degree of the membership functions of the fuzzy variables make it possible to change the system's behavior smoothly in a wide range and can be used for an automatic tuning of FLC during a normal operation. 4. The fuzzy relation between the space of the antecedents and the space of the consequents (the rule base) should be tuned in a test mode before the system operation and should not be changed hereafter. 5. It is advisable to carry out a procedure of defuzzification by the method of the center of g/avity. 6. Measures of the sub-condition significance can be used for an automatic tuning of FLC in cases, when the parameters of the control object in the process of operation do not change by more than one order.

The article is devoted to the theory of Wavelet-transformations with reference to the analysis and synthesis of the dynamic characteristics of the control systems containing an essential nonlinearity of the type of the zone of tolerance, restriction, backlash, etc. of nonlinearity. In the presented work a possibility of application of a device of the theory of Wavelet-transformations for the analysis and synthesis of the dynamic characteristics of the control systems containing essential nonlinearity is considered. A scaling function and wavelet potential function are proposed. The given device has an advantage in comparison with the frequency methods, because it takes into account the processes in the time area. Reactions of the linear systems, including essential nonlinearity, to the influence of a kind of potential function and wavelet are investigated. Errors in the established mode, because of the essential nonlinearities in the tracking system are considered under the influence of the signals representing a Wavelet-number, made of potential functions and wavelet. The condition of stability of the closed-loop system to reaction of the open-loop on the typical influence in the form of a half wave is formulated. The given condition allows us to analyze the stability of the tracking system containing an essential nonlinearity. For this purpose the dependence of the factor of transfer K (j) and the relative delay T₃ (j) for the open-loop system, where parameter j characterizes the width of a half wave, is formed. Application of the theory of Wavelet-transformations for the analysis of the nonlinear systems allows us to investigate the complex systems containing the essential nonlinearity as similarly linear systems. The proposed technique of analysis of the nonlinear systems has an advantage in comparison with the frequency methods, because it considers the processes in the time area and allows us to investigate analytically the errors and stability of the tracking systems.

The paper presents a simulation of a group movement of mobile robots in absence of obstacles in ROS and a kinematics model of the mobile robots used in the simulation. Three types of motion behavior of the mobile robots in a group were considered. First, the authors achieved a synchronization of the mobile robots in a group using the centralized method, which requires a control center, and the decentralized method, in which the mobile robots in a group communicate with each other. Second, the authors simulated movement of four mobile robots in Zhukov forms, when each robot was controlled according to the relative position of the two neighboring robots. Different trajectories of the mobile robots were obtained depending on the selected control parameters. The authors analyzed the relation between the control parameters and the resulting trajectories. Finally, the authors simulated movement of a group of the mobile robots in a convoy type of formation. Three different control algorithms were considered. The authors analyzed and compared the trajectories of the mobile robots moving in the convoy type of formation using three different control algorithms. The advantages and disadvantages of the three control algorithms were presented. The authors discussed a range of problems, which could be encountered when conducting physical experiments, for example, communication between the robots, the process of localization of the mobile robots in a group, and the impact of a sensor noise.

Recent trends in the field of anthropomorphic robotics involve approximation to the motion characteristics of the robot kinematics of a human body. The problem of high energy consumption in implementation of the kinematics of an anthropomorphic robot is partially solved by reduction of the weight of a design. In this regard, of special interest are the developments concerning the study of the stability of the structures and materials capable to withstand the set loads. In this paper a research was done of the stability of the basic structural elements of the lower extremities of Antares anthropomorphic robot under the influence of the external forces, such as straight-line power and twisting. The design of Antares robot employs actuators Dynamixel MX-64T and MX-28, the gears are made of a metal, which allows the robot to move with a given accuracy and a margin of safety. During modeling the assembly of the lower extremities were subjected to simulation of 2 types of loads, those were torque and "direct force" orientated in three directions: vertical (perpendicular to the transverse plane), lateral (normal to the front plane) and lateral (perpendicular to the sagittal plane). The direction of the applied force was selected on the basis of the fact that this force would have its maximal value, when the leg will be in a "sitting" position, therefore, the direction would be directed along the normal to the sagittal plane of the leg. In order to test the leg for twisting a situation was modeled, in which the robot was in the lying position with the inner part of its foot resting on the floor surface. During the study the values of the actuators' torques, intended to move the robot in space, were determined. The moments of rotation of the engine output shaft in a robot leg were determined: ankle, knee, hip joint. The maximal values of the rotational moments, able to withstand the construction of the hip and the thigh of 5 Nm and 5.2 Nm, respectively, were determined. During the simulation it was also found out that raising of the robot from "a sitting position" required 2.4 times less effort than the maximal torque developed by the engine, and the design of the leg could withstand the maximal torque of the motor rotation. On the basis of those data a conclusion was made, that the robot was able to jump in the vertical plane, and later this was proved successfully in real experiments.

The article describes the problems, mathematical models and algorithms for control of the automatic welding technologic processes in the robotic complexes. The problem of a rational control of the process is solved by the criterion used for minimization of the damage caused to the quality by breakdowns. Since a solution by the methods of variations calculus is problematic, the problem of the criterion minimization boils down to development and verification of implementation of a detailed complex action plan for a decrease of the damage due to breakdowns in the technological processes. The action plan was developed based on the causality between the process parameters and learning of the experience by the dispatching personnel. The plan is presented as a directed graph, in which vertices are actions of the plan, and arcs define their relationship and sequence of implementation. The conditions affecting the technological process and implementation of the plan of actions are presented in the form of a production model. For verification of the plan's implementation a logic function was developed in accordance with the graph of the action plan and conditions. By setting different values of the arguments of this function at any time, it is possible to check quickly a possibility of the plan's implementation. The article presents an algorithm for solving of the assignment task at different time intervals by an information-logical scheme. Introduction of the developed models and algorithms in the industrial enterprises using a robotic welding will reduce the damage caused by the defective products.

The paper presents a mathematical model of a motion control system for a vehicle with a chassis of two independent drive wheels. The active wheels are driven by DC motors through the reducers. The vehicle has a wheel speed sensor and a sensor of the lateral deviation from the predetermined route, a signal from which is transmitted to the control unit. The model takes into account the position of the wheeled mobile vehicle in relation to a guide path. It is built with the transfer functions, which allow application of the methods of both classical and modern control theory for the synthesis of the control devices. The model of the vehicle describes three interrelated subsystems, which control the lateral deviation, the longitudinal velocity and the position. Research of each subsystem separately showed theirs drawbacks. The lateral deviation control has a large overshoot and settling time. The longitudinal speed control has an error in steady state. The drawback of the position control is the presence of an overshoot, which demands the reverse engines. These drawbacks can be eliminated by introduction of the control devices into each of the subsystems. Also, the model envisages a trajectory of the vehicle motion in the coordinate plane. Besides, it provides for a model of the trace, defined in the deviations from the straight direction. This model takes into account the presence of the straight sections, the turning sections and the positioning points on the trace. The developed model is useful for a preliminary research of the synthesized control laws of the vehicle motion.

As a rule, modern engineering and manufacturing projects are saturated with electronic equipment for automated monitoring and control, as well as cable communication lines and radio frequency channels for transmission of transactions. During functioning of the intelligent mechatronic systems the number of the communication lines of the frequency ranges is restricted due to application of the CDMA principles. A drawback of this method of channelizing is connected with a high level of the interfering signals in case of an increased quantity of the simultaneously functioning subscribers, whereof signals at the receiving side are highlighted with the hardware and software correlators. Taking this into account, a numeric method was proposed for the accelerated data streams of the bits extraction from the frame bits during the code division multiple access to the transmission of the transactions on the basis of the use of the Walsh function and the intelligent mechatronic systems. The scheme of the code division of the bit streams was analyzed due to introduction of the information source bit streams to the mechatronic system of the intelligent multiplexers at the transmitting channel side and the intelligent bit streams separators at the receiving side. A peculiarity of this code division scheme for the logical channels is that it is fully implemented due to the use of the software as a part of the intelligent mechatronic system. Construction of the software models of the intelligent multiplexer and the intelligent separator proved their high efficiency and stability of their functioning. When up to 256 logical channels are formed in one physical channel, a reliable separation of the information bits is ensured, as well as their diversion by the sources with the allowed delays in the transactions' fulfillment. An essential extra time advantage occurs, because all the separator transactions in the real processor of the bit frames processing take place with the help of short commands of the integer arithmetic.

The topic of the article is an artificial Earth satellite in a circular near-Earth orbit. The satellite possesses a dynamical symmetry and is equipped with an electrodynamic attitude control system based on the usage of Lorentz and magnetic torques. The control torques are ensured by a controllable variation of two electromagnetic parameters of the satellite: the intrinsic magnetic moment and the static moment of charge of the first order. In the satellite's programmed motion the axis of the dynamical symmetry is inclined to the local vertical axis at a constant angle and the satellite slowly rotates around that axis. Such a rotation mode is of great importance in space missions, because it can reduce the temperature gradient effect, which may cause problems with the satellite's functioning. But this mode cannot be realized without a control because of the disturbances, among which the gravitational torque should be mentioned as the most important one. In the present paper the satellite's stabilization in the programmed attitude motion is investigated. The stated problem appears to be more complex than the problem of the uniaxial attitude stabilization of a satellite, and previously it was not treated with the use of the electrodynamic control system. On the basis of the Lyapunov direct method the conditions, under which the electrodynamic control solves the problem, were obtained. A new construction of the Lyapunov function was proposed, and using the function sufficient conditions for the asymptotic stability of the programmed motion were found in an explicit form.

The on-board satellite attitude and orbit control systems (AOCS) equipped with the inertial measurement units (IMU) and star trackers (ST) are the most versatile and reliable. Simultaneous processing of IMU and ST measurements is common for the foreign scientific spacecraft, which makes the basis of the operating principle of both devices. IMU functions in a wide angular rate range and doesn't depend on the external conditions. However, in case the measurement correction by another source of information is not available, the accuracy of the attitude determination might be reduced. The mathematical fusion of IMU and ST measurements allows AOCS to estimate with high accuracy and frequency the attitude and the angular rate parameters. This paper presents a suboptimal algorithm based on reduced Kalman filter and designed for IMU and ST measurements fusion. The Kalman filter state vector was chosen as a three dimensions vector. This vector is the Rodrigues parameters of the attitude between the attitude estimated by ST and the attitude estimated by IMU angular rate integration. Such a state vector has a minimal dimension for that purpose. Also it allows us to estimate the attitude parameters without additional information lag for the stabilization subsystems. Another feature of the algorithm is a different computation frequency of the state vector estimation and gain matrix. The gain matrix and the covariance matrix are 5 times slower than the state vector estimation. This algorithm performs an efficient decrease of the noise equivalent angle. Also this algorithm is not characterized by a computational complexity and significant AOCS computer requirements, which are 2,7 times lower compared with the full Kalman filter. The algorithm can function with various IMU and ST obtained by the state vector provided by the Rodrigues parameters of the attitude between ST attitude and IMU attitude. The algorithm is a part of AOCS software, verified on AOCS test bench. The test bench includes models of SED26 star tracker and GIVUS KIND34-020 IMU. Due to the new algorithm and with this equipment AOCS attitude error is lower than 5 arc. seconds.