Abstract: This article reviews the example of using a geometric model of a hypersurface in multidimensional space on the Radishchev blueprint, which reflects interconnection between the four variables in solution of one of the tasks associated with structuring the technological process fulfilled by a welding robot. In structuring technological processes related to welding, it is necessary to solve the optimization task of determining the position of the foundation of welding robot relative to the welded brackets and the axis of the container of cylindrical shape. The welding process requires finding the most optimal welding robot, the geometric model of the kinematic chain of which would move the output link and welding head across all sections of welding seams. As an example, the author reviews the case when the robot is outside the cylindrical container, and welding objects inside and outside. For carrying out the optimization task, the author examines the correlation between the parameters that determine the position of the robot relative to the container, and the minimum possible vertical displacement of the center of the output link based on the graphic optimization model.  The multicomponent system comprised of the four parameters is studied based on the Radishchev blueprint. The author drew the combinations of curves that set the framework of curve lines of the dual level of the hypersurface in the four-dimensional space. For determination of the curves, the author found the combinations of projections of the dots on the Radishchev blueprint. The use of dual level lines on the Radishchev blueprint allowed determining most optimal position of the manipulator mechanism with regards to the cylindrical surface in welding the items for various industrial robots described in this article.

Abstract: With automated planning of the movement of the arm mechanism of an anthropomorphic robot in organized space, there is a need to reduce the time it takes to calculate the trajectory in the space of generalized coordinates. The indicated time significantly depends on the calculation time of the vector of increments of the generalized coordinates at each calculation step in the synthesis of motions along the velocity vector. In the article the geometric studies based on the study of the size and shape of a region in a multidimensional space of generalized speeds specifying the permissible instantaneous states of the arm mechanism of an anthropomorphic robot are carried out. Based on this study a method is proposed that allows one to reduce the time of iterative search of the vector of increments of generalized coordinates. To establish analytical dependences reflecting the relationship between the geometric parameters of the specified area and the generalized coordinates of the arm mechanism, which determine the positions of the configurations, hypersurfaces in four-dimensional space are used. For this, the equations of interpolating polynomials located in four mutually perpendicular planes are used. Based on these four interpolating polynomials, a fourth-order hypersurface equation is obtained that reflects the relationship of geometric and kinematic parameters. The article also presents the results of virtual modeling of the movement of the arm mechanism of an anthropomorphic robot, taking into account the position of the forbidden zone in the system ACAD. The results of calculations using the obtained analytical dependences showed a reduction in the calculation time of test tasks. The conducted studies can be used in the development of intelligent motion control systems of autonomously functioning anthropomorphic robots in an organized environment without the participation of a human operator.