The article defines the relevance of modeling the parameters of spatial location of an abstract object when performing various functions. Thus, modeling makes it possible to determine object movement trajectories both during industrial application in technological processes and when used to create bionic objects, for example, the action of artificial limbs, correction of the movement trajectory of an object with spatial orientation defects. The main goal of research was to substantiate the analytical models of object movement, taking into account spatial coordinate systems, according to which coordinate transformations are carried out during the functioning of an abstract object of various applications. The creation of a sensory complex to compensate for violations of limb functions based on the justification of analytical models of vector field of main systems of the object, characteristic of its vital activity, can solve the possibility of real actions of an abstract biotechnical object in interaction with other objects of the external environment. It is necessary to compare the idealized parameters of vector fields with the real current characteristics of the object under study and to determine the difference as a differential function that corresponds to diagnostic parameters of state of object's limbs trajectory. Or when applied in industrial conditions, errors in the reproduction of the movement trajectory are taken into account. As a result, the study of nature of this functional dependence of the state violation and its restoration in the automated mode of operation of the integrated tool will allow the creation of a computer-integrated hardware solution for the identification of objects that interact by approaching and touching their surfaces. Thus, determining the positioning of TONTOR step in the space of movement of objects and during their interaction provides the possibility of the functioning of each abstract object when performing various types of work. At the same time, it is necessary to significantly develop the base of physical and mathematical models that determine the vector fields of objects in dynamics over a certain time, taking into account the TONTOR step of the phantom and real spaces of the existence and operation of the object. Thus, hardware implementation of this hypothesis increases the accuracy of identification of objects interactions with a human limb, regardless of its condition, and the accuracy of determining their relative location in space.

The conducted theoretical studies provide a basic thesis regarding the creation of a generalized concept of diagnosis of the signs of multiple sclerosis, which should determine the parameters of the distortion of the idealized model of the body system by the real nature of the disease and the state of the object. The work presents an analytical model for determining the onset of the disease using the TONTOR step model. The proposed method provides the doctor with information, excluding the subjective factor, while the results of such information technology significantly increase the accuracy of determining the early stage of the disease. Purpose of work consists in modeling the diagnostic signs of appearance of multiple sclerosis at an early stage of development, and appearance of moment of action's onset of this pathology by analysis methods. With this method of modeling, we have the opportunity to use affine and conformal transformations, which were considered earlier. In this case, the chord should be perceived as a directional vector with location accuracy. A circle is a consequence of an affine transformation. Thus, we have the opportunity to set the coordinates of the points of the trajectory and the circle in an unambiguous correspondence. In addition, since the real motion along the trajectory is characterized by oscillatory processes, the affine transformation of the model can be imagined as an exponential curve. The proposed physical and mathematical information processing models will help determine the main points of creating the principles of operation of technical integrated diagnostic tools. The basis is a vector model for determining the state of biological objects, which will allow information signals to be determined with help of integrated TONTOR sensors and TONTOR step model. As a result, the study of the nature of this functional dependence will provide analytical dependences in digital form, will allow the creation of a computer-integrated hardware solution that will eliminate subjectivity in diagnostics according to the "bad - good" principle.

The article shows peculiarities of the organization of technological processes of processing parts when applying automation of the enterprise and at same time defines main problems associated with production features, technological processes, problems of conservation and rational use of resources. It is shown that these problems are solved by optimizing production processes, in particular, e technological process of manufacturing precise parts of devices. Optimization of production has always been a very relevant issue for many enterprises in various branches of production. This is not surprising, because both the state and private business are always looking for new technologies and possible ways to reduce expenses, increase profits, and solve the problem of general improvement of production efficiency. However, after beginning of the full-scale invasion of Russia on the territory of the sovereign state of Ukraine and the massive destruction of industrial and critical infrastructure, civilian buildings and housing, the issue of effective use of available and surviving resources became extremely urgent. According to official data, as of April 2022, losses of industrial assets amount to $6.7 billion. So, this is why there is a need to optimize production under condition of its automation. On the basis of research on the current state of optimization of automated systems for mechanical processing of parts and basic scientific research on this topic, approaches to the justification of a formalized optimization model aimed at increasing the efficiency of instrument-making production are considered. An optimization solution is proposed, based on use of Lotka-Volterra mathematical model, which will allow optimizing the operation of precision parts manufacturing systems under the condition of production automation. The results of conducted proved the expediency of such bionic approaches to the automation of technological structures of the production enterprise, since the proposed model is mostly used in research of biological processes. So, well-founded analogies made it possible to determine full range of production problems, including logistical problems and personnel problems based on competing models. The created software, namely a web application that performs all the necessary calculations, provides the ability to build graphs that visualize the results of the optimization of automated parts processing unit of enterprise, or the overall system, or its individual structural unit.

Automated monitoring of the presence of such particles present near the main operational means of production or medical equipment with the determination of their trajectories is necessary to improve the efficiency of this equipment and the quality of operations. When performing measurements of the parameters of abstract objects of different origin and properties, for example, at precise parts production, problems of contamination of the surface of the object with discrete particles of another origin are often encountered. It is now known that every abstract entity forms around the area of the presence of solid particles. These solid particles, under the action of interaction forces, have the property to be ordered in space and on the surface of the object. This paper is a result of research and modelling of the interaction of such particles during their shredding and their structural self-organization. Severally consideration is given to the formation of dust layers under the action of coupling forces is reviewed. Models of behaviour of these layers for some typical surface forms of control object are created.

Modern medicine widely uses mechatronic modules in systems of various purposes, such as automated systems of diagnostics, scanning, irradiation. Recently, mechatronic modules in robotic surgical complexes are gaining importance. Thus, with the use of integrated sensors in mechatronic modules, executive manipulators of automated systems, positioning accuracy in defined spatial coordinates can be maintained. This data helps the medical staff to diagnose, monitor the patient's condition and make treatment decisions. The main purpose of sensor support in mechatronic systems is to ensure the accuracy and reliability of the system's functioning. Sensors must be sensitive and stable enough to provide measurements with high accuracy and respond to changes in real time. In addition, similar mechatronic modules are combined with sensors to create bionic limb prostheses, to restore human movement functions in various orthopedic diseases. At the same time, the trajectory of the movement of the executive bodies of the mechatronic medical system, regardless of its purpose, during spatial transformations of searching for the coordinates of a real object, must be determined taking into account possible deformations. Therefore, the accuracy of real-world displacement in space is determined by sensors that measure the parameters of physical objects. Thus, real transformations can be defined by spatial deviations that can be described using an ellipsoidal model. Accuracy, like the strength of the mechatronic module of a robot arm, is a variable value. They depend not only on the number of joints and the mobility of the hinges, but also on the position of the manipulators in space. At one point of coordinates, the module can apply more force than at another. The same is true for positioning accuracy, where the positioning error is greater at some points than at others. Therefore, an important actual problem in the creation of medical robotic systems is to determine the step-by-step movement of such a module in the workspace. Therefore, the purpose of this work is to determine the ellipsoidal TONTOR step model of sensors for automated mechatronic systems, as the motion of the executive manipulators and sensors of the system during transformations from imaginary coordinate space to real space determines trajectory errors. Based on the existing opportunities analyzed in the work application of mechatronic modules in automated medical systems, relevant tasks related to maintaining the positioning accuracy of diagnostic manipulators and sensors are defined. The importance of sensory complexes in measuring various biological parameters that determine the patient's condition is noted. And this involves the application of models of spatial movement of the sensor in the working space of automated equipment, in particular, a robotic mechatronic complex. In the work, it is proposed to use the TONTOR step model to increase the accuracy of the realization of the movement trajectory of the sensors of the mechatronic automated system. The results of creating an ellipsoidal model of the TONTOR step, which most accurately reflects the features of moving an object in space during transformations of the transition to real space, are given.