ELLIPSOIDAL TONTOR STEP MODEL OF SENSORS FOR AUTOMATED MECHATRONIC SYSTEMS

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.


Introduction
Modern medicine widely uses mechatronic modules in systems for various purposes. So, in this way, under the condition of using integrated sensors, automated diagnostic systems, scanning of human body organs, and robotic surgical complexes are created [1,2].
Integrated sensors have found wide application, in particular in medical mechatronic systems. Integrated sensors are used to measure physiological parameters such as body temperature, pulse, blood pressure, blood oxygen level, and others. 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 [3 -5]. 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 [6].
In general, mechatronic modules and systems with executive bodies have wide applications in various industries where precise control of movement, performance of operations and manipulations is required [6 -8]. They help to improve productivity, efficiency and quality of work in various spheres of life, contribute to automation and the development of new technologies.
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. In this case, we will rely on previous physical and mathematical models [9 -11].
The most appropriate model for determining the spatial displacements of an object is the TONTOR step model [11].
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 [3].
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.

Ellipsoidal TONTOR step model at sensors
In our research [9,10,11], we determined the features of the transition from imaginary spatial coordinates to real ones when measuring the parameters of objects with integrated sensors, which is necessary for specifying the trajectories of movement of mechatronic modules.
Let's use the model created by the authors in the work [11]. In this case, the curvature of real vector [11,Fig. 2,p. 103] is determined by angles 0 ψ and 1 ψ , which are only indirectly related by the torsion coefficient of space. Therefore, let's imagine the problem in a slightly different way. We will assume that distortions in space represent elliptical deformation. In the basics of this theses, situation is considered in [11], where it is shown that the deformation of one half of the step space during the movement of the mechanisms of equipment can be described by a parabolic law.
Thus, let's consider the previous drawing at [11, Fig. 2, p. 103] with the new coordinate system as in (Fig. 1) of this work. Of course, we have a number of simplifications here, but nevertheless the whole situation has a more visual character.
So, if we consider the sum of two parabolas, the branches of which to meet each other are directed, then as a result we get an ellipsoidal geometry of this figure. So, in a general form, we have an ellipse with a central focus, which is located on a plane X(U), O(U), Y(U) at a distance 0 y from the origin of coordinates O(U) . Thus, its description in its current form will be as follows where a, b are the parameters, which defined in [12]. The size of vector     K will be on axis If we transform the equation of the ellipse (1) to the imaginary form, we get By substituting z from equation (2) to equation (3), we have the opportunity to obtain the coordinates of the points 0 M (R) and 1 M (R) Solving (4) with respect to y , we obtain the coordinates of the points 0 M and 1 M , accordingly of the imaginary coordinate system. So, and on the z coordinate The length of the     K ɶ vector will be determined by a known equation ( ) where ( ) The integral (8) cannot be solved in the usual way, because it is a transcendental function. To do this, we expand the integrand into a series using Newton's binomial . (10) To facilitate mathematical operations, let's ignore the last member of the series, considering it equal to [S].
Substituting (9) into (8) and using (10), we obtain the following result We solve each integral, separately using and the following equation (12) After substituting the values from (12), (13) and (14) into (11), we obtain the following result ( ) given certain boundaries In order to determine the angles 0 ψ and 1 ψ , we After the transformation, we get Whence the derivative of the function (16), and therefore the tangent of the angle of inclination will be Similarly, we define the appropriate series of transformations for the M 1 (R) point Using the formula for determining the tangent of the angle of intersection of two straight lines on the plane and equation for the secant line [12], we find the corresponding angles 0 ψ and 1 ψ .
So that will be it -for 0 Finally, consider the radii of curvature, which directly relate to possible distortions. Thus, for both points we will have the following Now consider the realization vectors 0 (R) I and 1 (R) I , which are in the error plane (x, y, z) 0 P = .
Since we have a direction of movement from the point 0 M (R) to 1 M (R) , these vectors will be under the influence of deformation distortions. Thus, making the transition from the imaginary to the real, we always get distortions not less than the value of [S]. So, according to the model of distortions that we are building, the 0 (R) I vector cannot be smaller than [S]. At the same time, the difference between the vectors 0 (R) I and 1 (R) I cannot be less than [S]. In fact, the values of these vectors are determined by the size of the angle γ and the geometric dimensions of the ellipse. If the angle γ is equivalent to the coefficient η (rotation of the space error), then the ratio of the semi-axes of the ellipse is the scaling factor (µ).
Since we considered the simplest case, the distortions are minimal, because this is the first stage of implementation. Under these conditions, for example, the duality changes from the dimensions 0 (R) I to the value b (the semi-minor axis of the ellipse). However, this is the first distortion step that we encounter during the initial stage of obtaining the required step size of the sensor in the mechatronic system in space while performing medical manipulations. For the next stage of the displacement process of this defined object, the real coordinate system becomes imaginary relative to which its real system is developed.
At the same time, the angle γ (Fig. 1) can be both positive and negative depending on the type of process, that is, positive or negative.
In fact, the above is the analytical basis of the considered chain of processes of carrying out spatial movements in the mechatronic system working space from a vector point of view. Each subsequent spatial transition is accompanied by a new angle γ and ellipse in the coordinate 0 y .

Conclusions
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 sensors movement trajectory 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.
Further research in this direction involves the expansion of the proposed TONTOR step model in terms of additional parameters of the sensor movement trajectory during diagnostic and therapeutic manipulations by automated systems.