Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Kong, H., Gosselin C.M.: Type Synthesis of Parallel Mechanisms. Springer (2007).
Merlet, J.-P.: Parallel Robots, 2nd ed, Springer (2007).
Aleshin, A.K., Glazunov, V.A., Rashoyan, G.V., Shai, O.: Analysis of kinematic screws that determine the topology of singular zones of parallel-structure robots. Journal of Machinery Manufacture and Reliability 45(4), 291-296 (2016).
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).
Alizade R.I., Tagiyev N.R., and Duffy J.: A forward and reverse displacement analysis of an in-parallel spherical manipulator. Mechanism and Machine Theory 29(1), 125–137 (1994).
Bulca F., Angeles J., and Zsombor-Murray P.J.: On the workspace determination of spherical serial and platform mechanisms. Mechanism and Machine Theory 34(3), 497–512 (1999).
Clavel R.: Conception d’un robot parall`ele rapide `a 4 degr´es de libert´e. Ph.D. Thesis, EPFL, Lausanne, n◦ 925 (1991).
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).
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).
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).
Lotov A.V., Pospelov A.I.: The modified method of refined bounds for polyhedral approxi-mation of convex polytopes. Computational Mathematics and Mathematical Physics 48 (6), 990–998 (2008).
Kamenev G.K.: Method for polyhedral approximation of a ball with an optimal order of growth of the facet structure cardinality. Computational Mathematics and Mathematical Physics 54 (8), 1235–1248 (2014).
Kamenev G.K.: Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls. Computational Mathematics and Mathematical Physics 56 (5), 756–767 (2016).
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).
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).
Evtushenko, Y., Posypkin, M. Effective hull of a set and its approximation. Doklady Mathematics 459 (5), 550–553 (2014).
Garanzha V.A., Kudryavrseva L.N.: Generation of three-dimensional Delaunay meshes from weakly structured and inconsistent data. Computational Mathematics and Mathemati-cal Physics 52 (3), 499–520 (2012).
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).
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).
Acknowledgments
This work was supported by the Russian Science Foundation, the agreement number 16-19-00148.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Rybak, L., Gaponenko, E., Malyshev, D. (2019). Numerical method of working area approximation of the tripod robot taking into account the singularity zones. In: Uhl, T. (eds) Advances in Mechanism and Machine Science. IFToMM WC 2019. Mechanisms and Machine Science, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-030-20131-9_74
Download citation
DOI: https://doi.org/10.1007/978-3-030-20131-9_74
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-20130-2
Online ISBN: 978-3-030-20131-9
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)