Parametric optimization of the kinematic structure and the movement of the two-link manipulator

doi:10.26565/2304-6201-2020-48-03

Myroslav Demydyuk PhD, Senior Researcher, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics National Academy of Sciences of Ukraine, 3-Б Naukova St., L’viv, 79060, Ukraine https://orcid.org/0000-0002-9719-4512
Vitalii Demydiuk Master’s Degree (System Analysis), Software Engineer, LLC “DevCraft”, (ORIL Inc.,www.oril.co), 3 Syhnivka St., off. 81, L’viv,79040, Ukraine https://orcid.org/0000-0003-3442-2108

DOI: https://doi.org/10.26565/2304-6201-2020-48-03

Keywords: two-link manipulator, transport operation, optimal control, angular coordinate parameterization, nonlinear programming

Abstract

The controlled motion of a two-link manipulator has been studied. The manipulator being under the action of controls (torques at joints) performs a transport operation in the horizontal plane, namely, transfers the cargo from a given initial position to a given final position The speeds of the cargo at the beginning and end of the operation are considered to be zero, and the duration of the movement is given. While preparing the manipulator for executing a given operation, it is possible to change the placement of the joint which connects it to a fixed base. The different angular configurations of manipulator links are available at the beginning and the end of the operation. The problem for aggregate optimization of the base joint coordinates, as well as the boundary configurations of the links of the manipulator and its movements is formulated, providing that the quadratic (over the controls) functional is minimized. The algorithm for building a suboptimal solution of the problem, based on the methodology of parametric optimization is proposed. The angular coordinates of the manipulator are approximated by the sum of a cubic polynomial and a finite trigonometric series (with unknown parameters). The coefficients of the polynomial are found from the initial and final conditions of the transport operation. As a result, the initial problem is reduced to the problem of nonlinear programming with an objective function depending on the coordinates of the base joint, boundary configurations and coefficients of the trigonometric series. The standard numerical procedures for minimization of the functions of many variables have been used to solve the resulting nonlinear programming problem. The results of numerical modeling of two-link manipulator suboptimal motion are described, and the influence of the base joint coordinates on the characteristics of this motion is numerically investigated. The analysis of numerical calculations shows that the position of the base joint and the boundary configurations of a two-link manipulator influence significantly the energy consumption during the transport operation.

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