Mathematical modeling of one type of three-link robot manipulator

Determining the parameters and conditions under which, when the robot manipulators are operating, the set of optimality criteria can be achieved is an urgent task. In this regard, the problem arises of mathematical modeling of the executive system of a manipulation robot. The article discusses manipulators having rotational, translational and rotational kinematic pairs of the fifth class. Three possible kinematic schemes for such manipulators are given. To describe the movement of the manipulation robot, generalized coordinates are introduced that uniquely determine its configuration. The angles of relative rotations and linear relative displacements of the links were taken as generalized coordinates. A Cartesian rectangular coordinate system was associated with each link of the manipulator and with the fixed base. The problem of determining the relative position of the links and their position in the inertial space was solved by converting one coordinate system to another. As a result, the basic kinematic relations were found that determine the position, speed, and acceleration of the centers of mass of the links in a fixed coordinate system for robot manipulators having one translational and two rotational degrees of mobility. Based on these relations, the problem of constructing equations of dynamics in the form of Lagrange equations of the second kind was solved. The advantage of this form of recording is the closed form of expressions that determine the dynamics of the system, which makes it possible to correctly take into account the internal forces acting in the system. For each of the manipulators considered, systems of three second-order nonlinear differential equations were obtained that describe their dynamics. Due to the need to assess the load moment occurring on the control drive shaft, external generalized forces were reduced to the corresponding generalized coordinate.