Implementation and use of the Levenberg-Marquard algorithm in the problems of calibration of robotic manipulators

Abstract

The well-known problem of calibration of an arbitrary robotic manipulator, which is formulated in the most general form, is considered. To solve the direct problem of kinematics, an alternative to the Denavit-Hartenberg method, a universal analytical description of the kinematic scheme, taking into account possible errors in the manufacture and assembly of robot parts, is proposed. At the same time, a universal description of the errors in the orientation of the axes of the articulated joints of the links is proposed. On the basis of such a description, the direct and inverse problem of kinematics of robots as spatial mechanisms can be solved, taking into account the distortions of dimensions, the position of the axes of the joints and the positions of the zeros of the angles of their rotation. The problem of calibration of manipulators is formulated as a problem of the least squares method. Analytical formulas of the objective function of the least squares method for solving the problem are obtained. Expressions for the gradient vector and the Hessian of the objective function for the direct algorithm, Newton-Gauss and Levenberg-Marquardt algorithms are obtained by analytical differentiation using a special computer algebra system KiDyM. The procedures in the C ++ language for calculating the elements of the gradient and hessian are automatically generated. On the example of a projected angular 6-degree robot-manipulator, the results of modeling the solution to the problem of its calibration, that is, determination of 36 unknown angular and linear errors, are presented. A comparison is made of the solution of the calibration problem for simulated 64 and 729 experiments, in which the generalized coordinates - the angles in the joints took the values ±90° and -90°, 0, +90°.