TO THE PROBLEM OF IMPROVE POSITIONING PRECISION OF ROBOTIC MANIPULATOR UNDER CONDITIONS OF INCOMPLETE INFORMATION

2019, vol. 19, no. 2, pp. 16–28 16 Introduction A manipulation robot is a mechanical system whose dynamics is described by Lagrange's differential equations. The main difficulties encountered in solving problems of controlling such a mechanical system are due to its high order, non-linearity and the presence of dynamic interaction between various degrees of freedom. In the study of nonlinear continuous and discrete control systems, the main method of analysis with the use of complete models of dynamics is the method of Lyapunov function. It is also used to study the dissipativity of systems when the values of the constant parameters of the system are uncertain. At the same time the main objective is to choose a Lyapunov function that would allow to conduct the analysis effectively enough [1–3]. When solving the problem of synthesis of the control moments of robot manipulator (RM) by this method, a function that reflects the change in the total energy of the dynamic system can be chosen as a Lyapunov function that allows to study the behavior of the phase trajectories of the system in the entire phase space. With this approach, the Lyapunov function is selected from the first integrals of the motion of the system. Along with the task of constructing a model when managing a dynamic system, the task of measuring the state vector of a system with limited noise, which considered in [4–20], is important. Управление в технических системах


Introduction
A manipulation robot is a mechanical system whose dynamics is described by Lagrange's differential equations. The main difficulties encountered in solving problems of controlling such a mechanical system are due to its high order, non-linearity and the presence of dynamic interaction between various degrees of freedom. In the study of nonlinear continuous and discrete control systems, the main method of analysis with the use of complete models of dynamics is the method of Lyapunov function. It is also used to study the dissipativity of systems when the values of the constant parameters of the system are uncertain. At the same time the main objective is to choose a Lyapunov function that would allow to conduct the analysis effectively enough [1][2][3]. When solving the problem of synthesis of the control moments of robot manipulator (RM) by this method, a function that reflects the change in the total energy of the dynamic system can be chosen as a Lyapunov function that allows to study the behavior of the phase trajectories of the system in the entire phase space. With this approach, the Lyapunov function is selected from the first integrals of the motion of the system. Along with the task of constructing a model when managing a dynamic system, the task of measuring the state vector of a system with limited noise, which considered in [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], is important.

Formulation of the problem
The object of the study is a three-link robot manipulator with electric drives, the full model of which dynamics is constructed in the form of Lagrange -Maxwell equations [16,17] and according to the well-known rules [21] is reduced to a non-linear model by the form of the relationship between its input and output parameters, defined by the vector differential I   are introduced as the difference between the current and program values of the phase vector coordinates. It is assumed that the generalized coordinates and velocities of the system are accessible to measurement, and the control forces are subject to restrictions on the norm.
The control is synthesized using the Lyapunov function method in the tensor form of the record, which removes the cumbersomeness of the given expressions.
The Lyapunov function V is constructed as a bundle of first integrals of the perturbed motion of the system. , operating on the degree of freedom  , U  is the control restrained by limitation R is the restricted closed set given from the control resources.
Choosing the controls in the form of 1 with positive constant  , we will obtain The derivative V  of the Lyapunov function is negative definite, and, consequently, the equilibrium position of the system is stable.

Parametrization of control law
The constructed control (4) contains two arbitrary parameters  and  . For their choice, we submit the control law in the form where the tensors 1 K  and 2 K  , whose elements contain unknown parameters, are to be calculated and realized in the controller. Let us take the following objective function as a parameter choice criterion When minimizing the objective function with respect to the required parameters, we shall take the differential equations of the object as constraints in a simplified form: we assume that the RM is in a potential-free field, the mutual inductance factors are negligible, and its dynamics is described by a system of differential equations obtained from the system (1) by linearizing it in the position of the robotic arm with orthogonal arrangement of links. These constrains have the form x Ax BU Cv     . In this equation A is a block matrix whose structure is determined by the metric tensor of the manipulator and the tensors of inductance and mutual inductance of the armature windings and excitation of drive electric motors, Decentralized control (4) makes it easier to solve the problems of control by representing the initial nonlinear system in the form of three subsystems, each of which is subject to mutually uncorrelated disturbances. Given the small dimension of the subsystems, we obtain their equations linearized in the neighborhood of the points of the program trajectory x x u  -are the vectors of the state, control, and disturbances, , A B   are the matrices; , M E C C are the coefficients of proportionality of the torque and EMF resulting from the rotation of the motor armature; L  is the inductance of the armature winding; D  and R  are the coefficients of the viscous friction and the resistance of armature winding, respectively, which determine the dissipation of mechanical and electromagnetic energies, respectively; J  is the moment of inertia of the subsystem brought to the shaft of the electric motor. When setting ratios the task of the subsystem is to find the minimum of the scalar function in the tuning process should be less than zero and as less as possible, thus ensuring the quickest descent in the direction of the minimum Q  . It follows from the (6) that a gradient tuning algorithm can be used to solve the minimization problem, which has the following form of the function under consideration where   is the diagonal matrix of the coefficients of regulating loops.

Control under significant influence of disturbance torques
Among the main reasons for the deterioration of the quality of robot control are unpredictable changes in load moments at the output actuator shafts, the effect of viscous and dry friction forces, the elimination methods of which are difficult to implement, and the intensification of the destabilizing interference of subsystems. These phenomena are difficult to overcome using actuators with only linear feedback on the output variables of the subsystems, especially since the controller gains have design constrains. One of the means of compensation for disturbing factors are signal-type control algorithms that ensure the system's operability at significant rates of change in disturbing factors. The advantages of algorithms include the high speed of adaptive processes and ease of implementation, however, due to the limitation of input signals of electromechanical systems, the adaptability of algorithms is also limited. Accordingly, it is advisable to use them to control the object, when there are disturbing factors with a limited range of their variation and high speed.
Let us construct an algorithm for RM control under a significant influence of disturbance torques using the Lyapunov. We take into account the effect of the dry friction force moments , sp k  is a diagonal matrix, the positive elements of which are chosen based on the requirements imposed on the quality of control, control resources and magnitudes of the disturbance torques. Evaluation of the stability of the movement of the RM in the management (7) gives inequality, which should be guided by the choice of coefficients, based on the condition 0 V   :

Simulation of control law
The structural diagram of the second subsystem of RM is shown in Fig. 1. The presented model realizes the movement along the program trajectory, which determined by the expressions 1 To obtain the measured values of current 2 a I , velocity 2   and position 2  , the Noise blocks that allow generating Gaussian noise, as well as FK units that perform signal filtering, are added to the diagram.   It should be noted that the observed transient processes are conventionally divided into two areas. In the first area, the duration of which is up to 4 seconds, there is a significant change in current due to the start of the engine. The maximum absolute error value (MAO) in this area is 0.14 rad, however, the main program of RM is constructed so that during this period of time there is no interaction with the object. This allows to increase in the specific case under consideration by 60% accuracy relative to the results of applying the gradient tuning method.
When the noise channels Noise1, Noise2, Noise3 are affected, having a deviation of the normal distribution of 1, 0.001 and 0.001, respectively, the MAO value increases. So, when using constant values of coefficients 1 22 K and 2 22 K , improving accuracy by 60% without the influence of noise, the MAO value of the second area of the trajectory will be 0.0286 rad.
However, when using a gradient tuning with signal adjustment, the MAO value of the second trajectory area (Fig. 9) will be only 0.0106 rad.  Fig. 11, the curve of the MAO value at the corresponding points. The smallest MAO value in this case is 0,01 rad that it is less than 6% higher than the result of the gradient tuning method.

Conclusion
This allows to conclude that, despite the less accurate indicators of the gradient tuning method using signal adjustment without regard to the influence of noise compared with the method of selecting and using constant optimal coefficient values, this method works effectively in the conditions of incomplete information and does not require additional calculations of values of the coeffi-