Research of relationships between the technical parameters of industrial manipulators

. Industrial manipulators have to become an integral part of modern construction production, playing a key role in the automation and optimization of production processes. When designing and using industrial manipulators, it is important to know the various parameters that interact with each other and affect the overall productivity and efficiency of the production line in which industrial manipulators operate. In the design of the robot process and construction production, it is necessary to know the technical requirements that need to be met by the robot system. Therefore, when designing industrial robots and manipulators, it is necessary to take into account the shape, dimensions, and weight of the building structures with which the industrial robot will work, as well as their physical and mechanical properties. These parameters will determine the general scheme and layout of the robot's components, the design of the clamping device and clamping force, the type of drive and its control principle. This paper is devoted to the identification of regularities between the main technical parameters of reach and lifting capacity of general-purpose industrial manipulators. Regression analysis with the use of hyperbolic and cubic splines is applied. The regression equations were solve based on the minimization of the total squared deviation of the correlation function. The results of the work will be useful for developers of industrial robots and researchers who are engaged in the development of robotic technological lines and productions.


INTRODUCTION
Identifying regularities between the parameters of robots and manipulators is a difficult and important task facing developers of such systems [1][2].This problem is especially important in the development of robotic systems for construction production [3][4].When creating robotic modular systems and when developing drive systems, it is also necessary to know the pressure forces on the drive links in advance [5] depending on the parameters of the robot's executive system.For this to need knows the relationship between the main technical parameters of robots and manipulators.Since research in this direction in robotics is quite scarce, it was suggest investigating the existing designs of industrial manipulators and establishing mathematical dependencies for their further analysis.

PURPOSE OF THE ARTICLE
Investigation of technical parameters of industrial manipulators and determination of the relationships between the parameters of load capacity and reach under consideration.

PRESENTING MAIN MATERIAL
The main technical parameters of industrial robots and manipulators include rated load capacity, reach (reach of the working body), power, number of degrees of mobility, type of workspace [6].
ISSN(online)2709-6149.Mining, constructional, road and melioration machines, 102, 2023, 65-73 The load capacity characteristic defines the largest mass of cargo or technical equipment that a robot can move while ensuring the desired performance and safety [6][7].The reach characterized by the space in which the working body of a robot or manipulator is able to perform the specified functions in accordance with the purpose of the robot [8].Therefore, we note that lifting capacity and reach are the main technical characteristics based on which mechanical robotic systems are developed.
The initial conditions for construction robotization are the procedure for developing technical requirements for the design of a robot or manipulator, taking into account the specifics of construction.For example, when developing construction mechanization tools, it is necessary to take into account the shape, dimensions and weight of building structures, as well as their physical and mechanical properties [4,7].These indicators will determine the general scheme and layout of the robot's components, the type and design of the drive system, and the control principle.All of this indicates that the design of robots and manipulators for construction and installation work will based on the analysis of the technical parameters of robotics means and the technical parameters of construction processes.
In the following, we will consider the methodology for determining the relationships between the technical parameters of the reach and the lifting capacity of stationary manipulators of industrial robots of a typical design with five or six degrees of mobility (Fig. 1) and similar structure.For comparison, the technical parameters of industrial manipulators manufactured by Kawasaki (Japan), ABB (Sweden), and KUKA (Germany) presented in Table 1.
Table 1 shows the data for six manipulators from three manufacturers in order of increasing load capacity.Table 1 shows that KUKA does not indicate power values in the technical characteristics of its robots, so a power comparison for such designs will not be possible.All of the industrial manipulators under consideration have an electric gear drive.
To evaluate the technical qualities of these manipulators, all their designs were compared by the correlations between technical characteristics, which are called comparable criteria.The criterion of the ratio of load capacity Q to reach D shows how much of the cargo mass moved per 1 meter of reach: ( In Fig. 2 shows a histogram with the distribution of the K1 criterion for different designs of the considered manipulators.The histogram below shows how the value of the criterion increases with increasing load capacity for each of the three groups of manipulators К1.The highest values of the K1 criterion are typical for manipulators with a large lifting capacity from 100 to 250 kg.This means that such systems use a more complex drive design that provides significant torque values.
The criterion of the ratio of lifting capacity Q to mass m shows how much of the load mass perceived by a unit of mass of the manipulator's metal structure: ( From the histogram shown in Fig. 3 histogram shows that the designs of manipulators with a large lifting capacity have higher values of the criterion K2 of the ratio of lifting capacity to mass.This may mean that such systems are more massive.It should be note that the designs of the industrial manipulators Kawasaki RS003N and Kawasaki RS007N differ in the considered K2 indicator from similar competing systems, as can be seen in Fig. 3.This explained by the fact that the designs of these robots have a frame structure different from other similar schemes of low load capacity.These robot schemes are built on frames with a higher load capacity, but with a drive of lower traction capacity.To determine the interdependence between the parameters of the manipulators, we will determine the regression equation using the method of least squares [9].The essence of such a technique is to determine a polynomial that will provide the minimum sum of squares of the differences between the actual values of the observed quantity and its calculated value: Let us use the following hyperbola equation as a polynomial [4]: where: a, b is unknown coefficients in the equation. As a variable parameter i x in the function of (4) specify the reach or departure parameter, and as the value of the function i y will be the value of the carrying capacity.
Let us express equality (3) in terms of the hyperbole equation ( 4 To find the unknown coefficients, let's determine the derivatives of these coefficients: (7) To calculate equations ( 6) and ( 7), a table was compile in accordance with the specified form of correlation between departure and carrying capacity data (Table 2).
Taking into account the data from Table 2, equations ( 6) and ( 7) will be as follows: The result was a regression dependence for predicting the lifting capacity parameter depending on the maximum reach of the manipulator, kg: where: L is maximum reach or reach of the manipulator, m.
To determine the reliability of the obtained correlation, we determine the covariance ratio between a given carrying capacity parameter and its predicted value by the Pearson's coefficient, which is equal to the square root of the variance of the studied values [10]: where: Let us determine the average value of the load-carrying capacity for the considered manipulator designs according to their passport values: where:  y the total value of the carrying ca- pacity for all observations; n the total number of observations for a given value.
The results of calculating the parameters of Pearson's coefficient are shows in Table 3.
The reliability of the correlation was assess by N. Leontief's inequality [4]: where: the average error of the correlation ratio.From the given equations it was found that  = 1,33, а  = -0,18.Then: the condition is not fulfilled.
The regression function based on the next cubic polynomial was research: where: a, b, c, k the regression coefficients.The objective function (3) taking into account the given cubic polynomial will be: The derivatives for these coefficients will be as follows: Table 4 shows the calculated data for a given cubic polynomial.
Taking into account the obtained coefficients, a regression equation was obtains for the dependence of the load capacity parameter on the reach for the considered industrial manipulators: For equation ( 20), the variance was also determined and the Pearson test was evaluate.The results of the calculations are shows in the Table. 5.
The Pearson coefficient for this cubic spline (20) is  = 0,682, and the average error of the correlation ratio will be  = 0,12.Then: -the condition is fulfilled.
From the above it follows that the cubic dependence will more accurately reflect the nature of the influence of the studied accessibility factor on the value of the carrying capacity.Fig. 4 shows the graphs on which the dependences of regressions was compare with the studied parameters of departure and carrying capacity for industrial manipulators.

Table 1 .
Technical parameters of industrial manipulators

Table 2 .
Data of the regression function for the hyperbolic polynomial №

Table 3 .
Variance values for the hyperbolic regression function № y

Table 5 .
Variance values for the cubic regression function