Analysis of the base motion model of industrial robots moving

. Development of mathematical models and algorithms for optimal control of the functioning of industrial robots on a movable base to ensure the accuracy of the trajectory of movement and positioning is considered. An equation for the functioning of industrial robots on a movable base in the implementation of a complex spatial operation obtained and, on its basis, a mathematical model of optimal control is developed.


Introduction
Models of movement and control of industrial robots usually developed taking into account two components of the system: the base and the actuator of industrial robots. Accordingly, the control system of industrial robots also consists of two parts. For the base, the control will be direct (immediate), and for the actuator -feedback control. Analysis of studies shows that there are a number of shortcomings of actuators controlled by a feedback system [1-3]: 1. When the robot moves together with the part in existing models, the gravity of the part is taken into account only when calculating the last link -the gripping element, and when calculating the remaining intermediate links, this factor is not taken into account.
2. The non-linearity of the impact of force factors formed during the movement of industrial robots on the connections is not taken into account; the movements of intermediate links are described by second-order differential equations, as analysis of the literature shows, with constant coefficients. According to practice, these differential equations should take into account the variability of the coefficients.
3. Primary motion error of industrial robots on a movable base is determined using the methods of the logical tree of possibilities and statistical testing. The Logical Possibility Tree is a short and simple method that provides the positional accuracy of the robot based on the distribution laws of discrete primary errors. However, this method has its drawbacks.
In it, the model is not explicit and the decision is based on the knowledge and conclusions of the person who makes the decision. During the manipulation, not all parameters are taken into account, and the logical movements of the links are presented in a tree form. The logical feature tree is more in line with the links that move in sequence. It does not apply to links moving at the same time.
4. The main goal pursued from involving robots in the production process is to accurately and quickly perform all the operations of the process and timely provide the consumer with a quality product or semi-finished product. This is especially noticeable, for example, in machining processes in the mechanical engineering industry. When drawing up the equations of motion of the robot, external forces are taken into account, but internal forces are neglected. So, for example, when a robot moves with a part, it is necessary to take into account both external and internal forces.
Development of ways to control the objects under study, devoid of the above disadvantages, is one of the urgent problems in the field of research.

Statement of the task
It is known that there are currently three generations of industrial robots. The first generation is robots subject to rigid programming, the second generation is robots subject to adaptive control, and the third generation is intelligent robots. This theoretical problem of modeling and controlling the motion of industrial robots is divided into the mechanics of interconnected robots and the control of their motion [4][5].
Mathematical writing of industrial robots with mathematical modeling methods, their formation and substantiation of quality criteria, construction of robot trajectory, development of kinematic and dynamic methods, their analysis and synthesis are the first issues [1]. The main scientific issues of industrial robot control are the research and development of programming methods of robots, the development of algorithms for analyzing and synthesizing the movement of robots, the development of adaptive control algorithms for robots using artificial intelligence, the development of robot sensitivity principles, sensor data. conversion and optimal use are secondary issues [1][2][3].
Whether an industrial robot is complex in a technological process or performing a simple spatial manipulation operation, a number of its characteristics are involved in the process. The main characteristics of an industrial robot are: [2][3][4][5]: Working space of a robot is the space in which the robot's executive mechanism can be located in motion.
Working area of the robot is the space in which the robot's holding device can be located in motion.
Geometric characteristics of the robot's work area are the size of the robot's work area, the cut surface, the linear, angular dimensions, or a set of them.
Basic coordinate system of the robot is a system of coordinates with respect to the transmission of geometric characteristics of the working area of the robot.
Number of moving levels of the robot is the set of the number of free levels of the kinematic chain of the robot's actuator relative to the base coordinates and the number of free levels of the moving device.
Degree of mobility of a robot between positions is the degree of mobility of a robot using a robot to move or traverse a path using a moving device.
Degree of mobility of an industrial robot's actuator is the degree of mobility of a robot's actuator using the movement of a gripping device.
Degree of mobility of the actuator in targeting is the degree of mobility of the robotic actuator in using the grasping device to target.
An industrial robot with six or more links is a complex technical system. Therefore, six or fewer links in the actual design of an industrial robot are involved in the technological process. The smaller the number of links, the simpler the control of the robot. In a very simple industrial robot, the number of links is up to three. Given the complexity of industrial robot movement, it can be divided into the following types [1][2][3]: global -the base is specific to moving robots, the movement of the robot base increases the range of motion; -regional -specific to transport robots, the movement of the links ensures the continuous movement of the material point in the grip device; -local -the robot restricts the movement of the links, directs the positioning device to increase positional accuracy, ie to get the target correctly.
Let's determine the mobility, flexibility, service angle of an industrial robot in the example of a three-link robot [6][7]. Obility of an industrial robot W number of variable generalized coordinates that uniformly determine the position of the space capture device is as follows [7][8]: Basis can be put not only in relation to the basis of the issue of speed for the movable state, but also for the acting mechanism of the industrial robot. When compiling a dynamic equation, the basis of the industrial robot is its excitability, non-excitation, and the appearances of the kinematic pair play an important role [8][9][10].
Dynamic equation of an industrial robot with a fixed base. It is known that the motion of an industrial robot can be represented by three different Lagrange-Euler, Newton-Euler, and D'Alamber equations. These equations have their own conveniences. For example, one is convenient for analytical representation of the motion of an industrial robot, the other is convenient for calculating the coefficients involved in the equation, and the third is convenient for obtaining a numerical solution [9][10][11][12].
Suppose that the motion of an industrial robot is expressed using the Lagrange-Euler equation.
is a dimensional symmetric matrix and is related to the configuration of the industrial robot;

I-for links in form
The motion model of basic moving industrial robots. How the links are connected to each other, i.e. the kinematic pair, plays a key role in performing the complex manipulation operation of the industrial robot's working arm. There are four different views of the kinematic pair [1][2][3][4][5]15].
1st view of the kinematic pair. A place where one side is fastened with a link. 2nd view of the kinematic pair. One-sided fastened link rotation. View 3 of the kinematic pair. Linear migration of the link. View 4 of the kinematic pair. Ball hinge. Let us give the kinematic and dynamic equations of complex spatial motion on these pairs. The appearance of the kinematic pair is less important when constructing a kinematic equation. There can be two cases related to the basis in the kinematics of an industrial robot. The basis of an industrial robot is movable and immovable. The immobility of the robot base is widely covered in [1-5, 14, 16]. According to them, the motion matrix of each link.
. (1) Radius vector representing the position of an industrial robot relative to the base of the grip device r and using the motion matrices of the links, it is possible to give the equation of spatial motion [1][2][3][4][5]: For the moving state of the robot base, the matrix representing the motion of the base is first introduced as follows (2) is the appearance of the formula Using formula (2) it is possible to put the problem of speed on a stationary industrial robot [12][13][14].
In kinematic    , , Euler angles play a key role. Euler angles determine the target when aiming an industrial robotic grip device at an object. These three angles characterize the operation of the three cranes, coinage and ryskanie (search, search) on the grip device. The angles are in Euclidean space OUVW in a rotating system (sometimes called a counting system) [16 -19] Deflection matrix for the first system is as follows: Deflection matrix for the second system is as follows: Deflection matrices for the third system are as follows: M maneuverability of an industrial robot is as follows when the catching device is fixed:

W M
Concept of service angle is introduced in increasing the positional accuracy of the movement of an industrial robot. The service is viewed in three-dimensional space, and the number of free levels can be 6, 7, or more. This is because it is necessary not only to lower the grip device to a given point, but also to direct it to the desired point. The service angle is determined as follows [15][16][17][18][19]: