Numerical method of working area approximation of the tripod robot taking into account the singularity zones
The article describes the use of optimization algorithms for solving the problem of determining the working area of a tripod robot. The method of approximation of the set of solutions of a system of nonlinear inequalities describing constraints on the geometric parameters of the robot, based on the concept of non-uniform covering, is considered. Based on the method, internal approximations are obtained, defined as a set of boxes. The zones of singularity arising at some positions of the robot are investigated. The conditions for the emergence of singularity zones corresponding to its singularity zones are revealed. The influence of various geometric parameters on the volume of the working area of the robot is analyzed. For the approximation of the working area, the developed algorithm and its modifications with different dimensions of boxes and approaches to the transfer of constraints from the space of input coordinates to the output coordinates are used in connection with the complexity of the computational problem. To implement the algorithms, a software package in the C ++ language has been developed. The results of mathematical modeling are presented. Experimentally verified various dimensions of the grid to calculate the functions, as well as the accuracy of approximation. The obtained results can be used when choosing the geometrical parameters of a tripod robot, providing the boundaries of the working area specified by the technological process, as well as when planning the trajectory, taking into account the circumvention of singularity zones in which the mechanism theoretically controls.
Keywordsparallel robot approximation working area tripod robot singularity zones algorithm
Unable to display preview. Download preview PDF.
This work was supported by the Russian Science Foundation, the agreement number 16-19-00148.
- 1.Kong, H., Gosselin C.M.: Type Synthesis of Parallel Mechanisms. Springer (2007).Google Scholar
- 2.Merlet, J.-P.: Parallel Robots, 2nd ed, Springer (2007).Google Scholar
- 4.Virabyan L.G., Khalapyan S.Y., Kuzmina V.S.: Optimization of the positioning trajectory of planar parallel robot output link. Bulletin of BSTU named after V.G. Shukhov 9, 106–113 (2018).Google Scholar
- 7.Clavel R.: Conception d’un robot parall`ele rapide `a 4 degr´es de libert´e. Ph.D. Thesis, EPFL, Lausanne, n◦ 925 (1991).Google Scholar
- 8.Evtushenko, Y., Posypkin, M., Turkin, A., Rybak, L.: On the dependency problem when approximating a solution set of a system of nonlinear inequalities. Proceedings of the 2018 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering, ElConRus 2018, vol. 2018, pp. 1481-1484 (2018).Google Scholar
- 9.Evtushenko, Y. G., Posypkin, M. A., Rybak, L. A., & Turkin, A. V.: Finding sets of solutions to systems of nonlinear inequalities. In Computational Mathematics and Mathematical Phys-ics 57(8), 1241-1247 (2017).Google Scholar
- 10.Evtushenko, Y., Posypkin, M., Turkin, A., Rybak, L.: The non-uniform covering approach to manipulator workspace assessment. Proceedings of the 2017 IEEE Russia Section Young Researchers in Electrical and Electronic Engineering Conference, ElConRus 2017, pp. 386-389 (2017).Google Scholar
- 14.Bushenkov V.A., Lotov A.V.: Methods and algorithms for linear systems analyzing based on the construction of generalized reachable sets. Computational Mathematics and Mathematical Physics 20 (5), 1130–1141 (1980).Google Scholar
- 15.Voronov E.M., Karpenko A.P., Kozlova O.G., Fedin V.A.: Numerical Methods of Construc-tion of Attainability Domain of Dynamical System. Herald of the Bauman Moscow State Technical University. Series Instrument Engineering 2, 3–20 (2010).Google Scholar
- 16.Evtushenko, Y., Posypkin, M. Effective hull of a set and its approximation. Doklady Mathematics 459 (5), 550–553 (2014).Google Scholar
- 18.Pundru Srinivasa Rao, and Nalluri Mohan Rao: Position Analysis of Spatial 3-RPS Parallel Manipulator. International Journal of Mechanical Engineering and Robotic Research 2 (2), pp. 80- 90 (2013).Google Scholar
- 19.Khalapyan S.Y., Rybak L. A., Malyshev D.I.: Determination of necessary geometric param-eters of tripod robot workspace, taking into account zones of singularity. Adavnces in Engi-neering Reeseach. (Actual Issues of Mechanical Engineering AIME 2017) 133, 648-653 (2017).Google Scholar