Prevention unstable conditions in the welding process via robotic technological complexes

The article presents the statement of the problem, models and algorithms for controlling the welding process via robotic technological complexes in conditions of the risk of unstable states. The system dynamics model was used to describe the technological process. As system levels the basic indicators influencing quality of the made production chosen on the basis of experience of operation of the robotic technological complexes are taken. Functional dependences between indicators and their dependences on external factors are determined by approximation of statistical data. The procedure of identification of unstable states for the mathematical model on the example of the Rossler's attractor is developed. An algorithm for preventing the system from falling into unstable states in the process control is proposed


Introduction
When welding steel in robotic technological complexes (RTC) requires a constant assessment of the quality at all stages of the production cycle, including operator control of welding quality during welding, the periodic control by the programmer of the welding parameters, full monitoring of production quality by quality-control (QC) inspector, etc.The lack of control at any stage, for example, due to a deviation in the welding parameters or a insufficient of QC inspectors increases the risk of defective products.Now various systems of quality control of welding in RTC are developed and applied [1][2][3][4][5][6].Mostly they decide the problem of compliance with the welding parameters and the accuracy of positioning of the welding torch.However, in these systems, insufficient attention is paid to the problem of operational control of the welding process in RTC on the criterion of product quality and prevention of unstable states.
These circumstances indicate the relevance of the development and implementation of new mathematical models and algorithms that allow to control the welding process robotic process systems on the criterion of quality of products, as well as to prevent the unstable operation of the RTC.
Taking into account experience of RTC operation, it is possible to define the main indicators of technological process influencing quality of the made production.The highest quality of production is reached at the minimum deviation of values of these indicators from the set values.Then the statement of the problem is as follows: to develop mathematical models and algorithms that allow to find the vector of control actions p(t)∈{P} on the time interval [tP, tF], minimizing the target function: where Xi(t), i = 1,2,…,n is factual quality indicators of the welding process in the RTC; Xi * (t) is the set value of indicator Xi, ωi is weight coefficient of the i-th indicator.When solving the problem of minimizing the function Q(t) , it is important to exclude the system from entering a chaotic state.To do this, we introduce a constraint: where s = {X1(t), X2(t),…, Xn(t)} is the vector of the current system state, s∊ S; S* -set of vectors of possible system states, S* is the subset of state vectors in which the system goes unstable, S*⊂ S, s * j∈S*; ρ -a metric that defines the distance between two states in the space S; ε is the specified constant.

Mathematical model
Due to the complexity of the control object, the use of methods of calculus of variations to solve the problem (1)-( 2) is difficult.Therefore, to describe the relationship between the elements of the welding process in RTC the model of system dynamics was chosen [7][8].
The system dynamics model allows to construct differential equations of the following type for the main phase variables (system levels): where y + (t) is the positive rate of the variable y(t), including all the factors that cause the growth of the variable y(t); y -(t) is the negative rate of variable y, including all factors that cause the decrease of the variable y(t).
The following indicators of the welding process in RTC are identified as system levels: X1 is the number of defective beams per 100 units of production; X2 is the quantity of RTC operators; X3 is the average number of stops of RTC per cycle; X4 is average length of defective welds per 1 unit of production; Х5 is fulfilled works on scheduled maintenance of RTC; X6 is the quantity of programmers; Х7 is the quantity of adjusters of welding equipment; X8 is the quantity of QC inspectors; Х9 is quantity of workshop technologists; Х10 is the number of days of delay in the supply of materials and spare parts for repairing RTC; X11 is average deviation of welding arc voltage; X12 is average current deviation on the engine of the wire feed unit; X13 is the average deviation of the manipulator from the programmed trajectory; X14 is presence in the workplace of the necessary technological documentation; X15 is deflection of shielding gas pressure; X16 is deflection of compressed air pressure; X17 is production plan for a set period, in units of product; X18 is the number of beams adopted by the QC inspectors from the first presentation.
In addition, the model includes external factors that affect the above characteristics: O0 is the number of RTC operators at the beginning of the period; Oin is number of recruited RTC operators for the period; Oout is the number of dismissed RTC operators for the period; Sm is number of production shifts; Rw is the number of RTC involved in the production process; Nst is the number of RTC stops per period; S * is permissible number of RTC stops per welding cycle; Ld is total length of defective weld seams for the period; L* is estimated length of defective weld seams for the period; Mf is the number of completed activities of the scheduled preventive maintenance of RTC; Mp is the number of planned activities of the scheduled preventive maintenance of RTC; P0 is the quantity of programmers at the beginning of the period; Pin is the quantity of recruited programmers for the period; Pout is the quantity of laid-off programmers for the period; R0 is the quantity of adjusters of welding equipment at the beginning of the period; Rin is the quantity of recruited adjusters of welding equipment for the period; Rout is the quantity of dismissed adjusters of welding equipment for the period; C0 is the quantity of QC inspectors at the beginning of the period; Сin is the quantity of recruited QC inspectors for the period; Cout is the quantity of dismissed QC inspectors for the period; T0 is the quantity of workshop technologists at the beginning of the period; Tin is the quantity of recruited workshop technologists for the period; Tout is the quantity of dismissed workshop technologists for the period; Nr is the duration of repair of RTC; Df is the actual delivery time of spare parts and materials for repair of the RTC; Dp is the planned delivery time of spare parts and materials for repair of RTC; ΔU is the average deviation of the welding arc voltage from the nominal value; Δ * U is the permissible deviation of the welding arc voltage from the nominal value; ΔI is the average deviation of the current on the motor of the wire feed unit from the nominal value; Δ * I is the permissible current deviation on the motor of the wire feed unit from the nominal value; ΔT is the average deviation of the manipulator from the programmed trajectory; Δ*T is the permissible deviation of the manipulator from the programmed trajectory; Tdf is the actual number of documents of the technological process; Tdp is the the required number of documents of the technological process; ΔPG is the average deviation of the pressure of shielding gas ; Δ*PG is the permissible deviation of the pressure of shielding gas ; ΔPV is the average deviation of the pressure of compressed air; Δ*PV is the permissible deviation of compressed air pressure; NTP is the number of beams assembled in accordance with the technological process; Nd is the number of beams adopted to QC inspectors from the first presentation for the period; Ab is the number of acts on nonconforming products for the period.
For the variable X1, the differential equation ( 3) has the form: ) Equations for other variables are compiled in a similar way.As a result, the system of equations based on the mathematical model of system dynamics will look as follows: (  (5) where f(Xi) is the functional dependence on indicator Xi, obtained as a result of polynomial approximation of statistical data.
Due to the high dimensionality and nonlinearity, it is difficult to obtain an analytical solution of the system of equations (5) , so the values of indicators Х1, Х2,…, Х18 are determined by its numerical solution.
Further, to solve the problem (1) , it is necessary to find the vector of control actions to minimize the function Q(t).Control actions are implementing in the form of activities plan pj ∈ {P}, j = 1,2 ,.., N: By calculating the values of the Q(t) function for each plan pj ∈ {P} on a given time interval, we can define an action plan which implementation minimizes the target function and is therefore a solution to the problem (1).

Identification of chaotic states
To fulfill the condition (2) it is necessary to develop an algorithm for identification of unstable states.Determine in advance all the possible unstable states is not possible, so we take as a basis one of the most common types of chaos -the Rossler attractor.During the further operation of the system, the set of possible unstable states will be replenished.
Let's define the values of the main indicators of quality X1 (the number of defective beams per 100 units of production), X4 (average length of defective welds per 1 unit of production) and X18 (the number of beams adopted by the QC inspectors from the first presentation), in which the system (5) takes the form of a Rossler's system of equations: