Heterogeneous cyber-physical control systems based on digital twins are in demand by Industry 4.0. In accordance with the contemporary systems engineering methodology, such systems are designed at the level of digital models. The paper proposes approaches to formalization and subsequent automation of solving direct and inverse problems of their design. To unify descriptions of heterogeneous components, we follow a viewpoint-based approach to architecture design recommended by the international standard ISO/IEC/IEEE 42010. Following recent trends, we employ category theory as a mathematical framework for the formal description and solution of design problems. Indeed, category theory is a branch of higher algebra specifically aimed at a unified representation of objects of different nature and relationships between them. The design space of a heterogeneous cyber-physical system is constructed as a subcategory of the multicomma category, the objects of which describe possible system architectures with a fixed structural hierarchy represented from a certain viewpoint as diagrams, and morphisms denote actions associated with the parts selection and replacement during the system design. Direct design problems consist in evaluating the properties of the system as a whole by its architecture and are solved using a universal category-theoretic construction of the colimit of the diagram. The solution of inverse problems that require finding variants of the system architecture, which are (sub-, Pareto-) optimal according to the consumer quality criteria, consists in reconstructing diagrams by their colimit edges. For such reconstruction, optimization algorithms of gradient descent type are reasonable to employ, which navigate along the system design space morphisms calculating the path by means of computer algebra. Typical techniques of assembling cyber-physical systems, such as modular composition and aspect weaving, are described in the language of category theory and illustrated. As an example, we outline the design of energy-efficient robotic production lines represented from the behavior viewpoint as discrete-event simulation models.

The complexity of solving the problem of synthesizing closed-loop control systems that are optimal in terms of rapid response multiply, rapidly increases with increasing order n of the control object — the well-known problem of the "curse of dimension" by R. Bellman. Therefore, for high-order linear objects (n l 4), exact, analytical control algorithms by the rapid response criterion are practically unknown, and for them approximate rapid response control laws are applied. The paper proposes an approach and, on its basis, develops analytical techniques for the synthesis of rapid response high-order systems, which use the idea of transforming the original performance problem to a similar control problem for objects of the first or second order, for which the optimal control algorithms are known. These algorithms underlie the proposed approach to the synthesis of rapid response control systems, which involves finding special functions describing the relationship between the phase coordinates of models of a low-order object (by analogy with the works of A. A. Kolesnikov, they are called aggregated or macrovariables) with the phase coordinates of the original high-order object, as well as the calculation of the parameters of the low-order models used, which provide, in a certain sense, their adequacy to the original object and, accordingly, the rapid response of the synthesized systems. The work distinguishes between the conditional and approximate adequacy of the models of the control object. In the case of conditional adequacy, the parameters of low-order models are found exactly using the eigenvalues and vectors of the original object, and with approximate adequacy, using the least squares method. In the first part of the work, two first-order models (conditionally and approximately adequate) are used, on the basis of which two methods of synthesis of rapid response controllers are developed. In the form of statements 1 and 2, the conditions for the applicability of these techniques are formulated. Accordingly, the main content of the first part of the work is devoted to the results of a comparative analysis of the properties of rapid response controllers obtained by using the proposed synthesis techniques. In the second part of the work, it is proposed to carry out a similar analysis of rapid response control systems synthesized using conditionally and approximately adequate models of a second-order object.

Currently, stereotaxic brain surgery is an actively developing branch of medicine. During these operations, a special needle is inserted into the brain through a hole in the skull. This needle is moved in the brain tissue so that its tip reaches a certain point, after which the necessary medical manipulation is performed (for example, taking a puncture). To ensure accurate positioning of the needle, it is advisable to monitor the process of such operations using the magnetic resonance imaging apparatus. This puts restrictions on the type of actuators that can be used to drive the needle. The paper considers the problem of controlling the penetration of a cylindrical needle into a phantom of the brain where the control force is generated using a piezoelectric drive (PED). To describe the interaction of the needle with the tissue, a phenomenological model is proposed, under which it is assumed that the phantom tissue is a viscoelastic and plastic material, and also demonstrates relaxation properties. When describing forces acting on the lateral surface of the needle from the side of the tissue, the presence of dry friction is taken into account. The proposed model contains a number of parameters that are identified based on experiments carried out at the NCKU (Taiwan). In these experiments, a standard biopsy needle was inserted into a phantom made from agar-agar solution, and the position of the needle and the force acting on it from the tissue were registered. It is shown that the experimental results are in good agreement with the calculations in the context of the model. An algorithm for controlling the needle by setting the frequency of excitation of the probe is proposed. The aim of the control is to introduce the needle to a given depth at a given constant speed, and then hold the needle at this depth. During the process, it is required to avoid overshooting in speed and position. Numerical simulation has been carried out. The effect of the feedback parameters on the nature of the process is investigated. It is shown that the proper choice of parameters allows for avoiding the overshooting.

The solution of task of maintaining the dynamic accuracy of control of working tools of multilink manipulators (MM) when they move along arbitrary spatial trajectories is presented in this paper. In this case, constructive restrictions in all degrees of freedom (DoF) of manipulators and special cases of location of their links are taken into account. In the first part of the paper the features of the new solution to the inverse kinematics problem are considered and special (singular) positions for 6-DoF manipulators with PUMA kinematic schemes are described. These singular positions are characterized by ambiguity in solving inverse kinematics problem of these MM. The presented solution of the inverse kinematics problem takes into account various combinations of generalized coordinates of the MM, which ensure the movement of their working tools to the specified positions with the required spatial orientation. In the second part of the paper a method is considered, which, due to use of an additional (redundant) DoF of the MM, which ensures its movement near work objects, does not allow the manipulator to enter unpredictable in advance singular positions, excludes approaching of working tools to the boundaries of MM working area, as well as some of its DoFs — to restrictions leading to a sharp decrease in the accuracy of the manipulator in the process of performing any technological operations with previously unknown trajectories of movement. Based on this method, a system has been synthesized that makes it possible to automatically generate reference signals for all DoFs of MM, taking into account the current reference positions and orientations of the working tools specified in the absolute coordinate system. As a result, emergency situations are not created and the high quality of the prescribed operations and work is maintained. The results of the performed mathematical simulation confirmed the high efficiency of using the proposed approach to the contour control of the MM.

The paper is devoted to the problem of increasing quality of reproduction of the environment by mobile robot’s surround-view system, operating in the augmented reality mode. A variant of a surround-view system based on the cameras with over-lapping fields of view is being considered. A virtual model has been developed, it includes 3D-CAD models of a mobile robot and surrounding objects, as well as virtual models of cameras. The cross-platform integrated development environment "Unity" was chosen to implement the model. Methods for solving the problem of displaying the surrounding mobile robot space in the "third-person view" mode are determined. A mathematical criterion for assessing the quality of reproduction of the surrounding space is proposed. It is based on the comparison of points obtained from a virtual model with points obtained as a result of projection of images from virtual cameras. To obtain points, ArUco fiducial markers were used, providing an unambiguous comparison of points on the original and synthesized images. The dependence of the value of the objective function of the optimization problem on the projection parameters by the uniform search method are investigated. A method for automatic adaptation of projection parameters using fisheye lenses and stereo vision methods is proposed. Directions for further research are identified.

In modern robotic and mechatronic systems, technologies are in demand that makes it possible to build an optimal trajectory of movement of their actuators. Such technologies are formed by combining navigation methods and building a 3-D map of the surrounding space based on vision systems and are successfully used in robotics and mechatronics. But there is a problem, consisting of a decrease in the accuracy of planning the trajectory of movement, caused by incorrect sections on the map (depth map) due to incorrect determination of the distance to objects. Such defects appear as a result of poor lighting, specular or fine-grained surfaces of objects. This leads to the impossibility of obtaining reliable information about the depth. As a result, the effect of increasing the boundaries of objects (obstacles) appears, and the overlapping of objects makes it impossible to distinguish one object from another. This problem can be solved using image reconstruction methods. The article presents an approach based on a modified algorithm for searching for similar blocks using the concept of quaternions and anisotropic gradient. The analysis of the research results shows that the proposed method allows you to correctly restore the boundaries of objects on the depth map image when reconstructing 3-D scenes, which contributes to an increase in the accuracy of planning the trajectory of motion of the actuators robotic and mechatronic systems.