A Direct Kinematics Problem Solution for the Three-degree-of-freedom Parallel Structure Manipulator Based on Crank Mechanism

Abstract

The paper describes a mechanism representing a kind of mechanisms of parallel kinematics with three degrees of freedom based on the crank mechanism. This mechanism consists of two platforms, namely: the lower fixed and the upper movable. The upper platform is connected to the lower one by six movable elements, three of which are rods attached to the bases by means of spherical joints, and another three have a crank structure.The paper shows an approach to the solution of a direct task of kinematics based on mathematical modeling. The inverse problem of kinematics is formulated as follows: at specified angles of rotation drive (the values of generalized coordinates) to determine the position of the top mobile platform.To solve this problem has been used a mathematical model describing the proposed system. On the basis of the constructed model were made the necessary calculations that allowed us using the values of crank angles connected with the engines to determine the position of the platform in space. To solve the problem we used the method of virtual points to reduce the number of equations and unknowns, which determine the position of the upper platform in space, at a crucial system from eighteen to nine, thus simplifying the solution.To check the solution correctness was carried out numerical experiment. Each generalized coordinate took on values in the range from -30 ° to 30 °; for them a direct positional problem was solved, and its result was inserted, as initial data, in the previous solved and proven inverse problem on the position of the platform under study.The paper presents comparative results of measurements with the calculated values of the generalized coordinates and draws the appropriate conclusions, that this model is in good compliance with the results observed in practice. One of the distinctive features of the proposed approach is that rotation angles of engines are used as the generalized coordinates. This allowed us to state the possibility to use the constructed model for the further development of control systems based on it.The paper shows the possibilities to use the presented solution to determine the working area of the mechanism. Computer simulation methods allowed us to find a working area of the manipulator without restrictions on the physical characteristics of a real model. В работе рассмотрен трехстепенной механизм, являющийся разновидностью механизмов параллельной кинематики, на базе кривошипно-шатунного механизма. Механизм состоит из нижней неподвижной и верхней подвижной платформ. Верхняя платформа соединяется с нижней шестью подвижными элементами, три из которых стержни, прикрепленные к основаниям с помощью сферических шарниров, и три имеют кривошипно-шатунную структуру. Представлен подход к решению прямой задачи о положении. Прямая задача о положении была сформулирована в следующем виде: при заданных углах поворота приводов (значениях обобщенных координат) необходимо определить положение верхней подвижной платформы в пространстве. Использован метод виртуальных точек, что позволило сократить число уравнений и неизвестных, определяющих положение верхней платформы в пространстве с восемнадцати до девяти. Для проверки правильности решения был проведен численный эксперимент. Полученное решение было использовано для нахождения рабочей зоны изучаемого механизма параллельной структуры.