Engineering Transactions, 67, 3, pp. 333–345, 2019
10.24423/EngTrans.996.20190815

Approaches to the Determination of the Working Area of Parallel Robots and the Analysis of Their Geometric Characteristics

Dmitry MALYSHEV
Belgorod State Technological University named after V.G. Shukhov
Russian Federation

Mikhail POSYPKIN
Dorodnicyn Computing Centre, FRC CSC RAS, Moscow Institute of Physics and Technology
Russian Federation

Larisa RYBAK
Belgorod State Technological University named after V.G. Shukhov
Russian Federation

Alexander USOV
Dorodnicyn Computing Centre, FRC CSC RAS
Russian Federation

The article presents and experimentally confirms two approaches to the problem of determining the working area of parallel robots using the example of a planar robot DexTAR with two degrees of freedom. The proposed approaches are based on the use of constraint equations of coordinates. In the first approach, the original kinematic equations of coordinates in the six-dimensional space (two coordinates describing the position of the output link and four coordinates – the rotation angles of the rods) followed by projecting the solution onto the two-dimensional plane is used. In the second approach, the system of constraint equations is reduced to a system of inequalities describing the coordinates of the output link of the robot, which are solved in a two-dimensional Euclidean space. The results of the computational experiments are given. As an algorithmic basis of the proposed approaches, the method of non-uniform coverings is used, which obtains the external and internal approximation of the solution set of equality/inequality systems with a given accuracy. The approximation is a set of boxes. It is shown that in the first approach, it is more efficient to apply interval estimates that coincide with the extremes of the function on the box, and in the second approach, grid approximation performs better due to multiple occurrences of variables in inequalities.
Keywords: parallel robot; working area; non-uniform coverings; interval analysis; approximation; algorithm; multiple solutions
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References

Tagiyev N.R., Alizade R.I., Duffy J., A forward and reverse displacement analysis of an in-parallel spherical manipulator, Mechanism and Machine Theory, 29(1): 125–137, 1994.

Angeles J., Bulca F., Zsombor-Murray P.J., On the workspace determination of spherical serial and platform mechanisms, Mechanism and Machine Theory, 34(3): 497–512, 1999.

Turkin A., Rybak L., Evtushenko Y., Posypkin M., On the dependency problem when approximating a solution set of a system of nonlinear inequalities, 2018 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus), pp. 1481–1484, 2018, https://doi.org/10.1109/EIConRus.2018.8317377.

Posypkin M., Evtushenko Yu., A deterministic approach to global box-constrained optimization, Optimization Letters, 7(4):819–829, 2013, https://link.springer.com/article/10.1007/s11590-012-0452-1.

Rybak L., Evtushenko Yu., Posypkin M., Turkin A., Approximating a solution set of nonlinear inequalities, Journal of Global Optimization, 71: 1–17, 2017, https://doi.org/10.1007/s10898-017-0576-z.

Sigal I., Evtushenko Yu., Posypkin M., Framework for parallel large-scale global optimization, Computer Science-Research and Development, 23(3–4): 211–215, 2009, https://doi.org/10.1007/s00450-009-0083-7.

Rybak L.A., Turkin A.V., Evtushenko Yu.G., Posypkin M.A., Numerical method for approximating the solution set of a system of non-linear inequalities, International Journal of Open Information Technologies, 4(12): 1–6, 2016, http://injoit.org/index.php/j1/article/view/369.

Evtushenko Yu.G., Numerical methods for finding global extrema (case of a non-uniform mesh), USSR ComputationalMathematics and Mathematical Physics, 11(6): 38–54, 1971, https://doi.org/10.1016/0041-5553(71)90065-6.

Kraynev A.F., Glazunov V.A., Koliskor A.Sh., Spatial Mechanisms of Parallel Structure [in Russian: Prostranstvennye mekhanizmy parallelnoi struktury], Moscow: Nauka, 1991.

Malyshev D.I., Posypkin M.A., Gorchakov A.Yu., Ignatov A.D., Parallel algorithm for approximating the work space of a robot, International Journal of Open Information Technologies, 7(1): 1–7, 2019.

Merlet J.P., Parallel robots, Springer Publishing Company, Incorporated, 2006. https://www.springer.com/kr/book/9781402041327.

Gosselin C., Kun S., Structural synthesis of parallel mechanisms. M.: Fizmatlit, 2012.

Jaulin L., Kieffer M., Didrit O., Walter É.,, Applied interval analysis: with examples in parameter and state estimation, robust control and robotics, Springer Science & Business Media, 2001, https://doi.org/10.1007/978-1-4471-0249-6.

Husty M.L., On the workspace of planar three-legged platforms, World Automation Congress, 3: 339–344, 1996.

Niederreiter H.,. Low-discrepancy and low-dispersion sequences, Journal of Number Theory, 30(1): 51–70, 1988.

Posypkin M., Usov A., Implementation and verification of global optimization benchmark problems, Open Engineering, 7(1): 470–478, ????.

Clavel R., Design of a fast parallel robot at 4 degrees of freedom [in French: Conception d’un robot parallele rapide a 4 degres de liberte], EPFL, Lausanne, 1991. http://dx.doi.org/10.5075/epfl-thesis-925.

Chichvarin A.V., Rybak L.A., Erzhukov V.V., Effective methods for solving problems of kinematics and dynamics of a machine tool with a parallel structure, M.: Fizmatlit, 2011.

Stan S.D., Balan R., Maties V., Optimization of a 2 dof micro parallel robot using genetic algorithms, [in:] Frontiers in Evolutionary Robotics, Hitoshi Iba [Ed.], University of Tokyo, Japan, pp. 465–490, 2007, https://doi.org/10.1109/ICMECH.2007.4280011.

Kuzmina V.S., Virabyan L.G., Khalapyan S.Y., Optimization of the positioning trajectory of the planar parallel robot output link, Bulletin of BSTU named after V.G. Shukhov, 9: 106–113, 2018, http://vestnik_rus.bstu.ru/shared/attachments/184909.




DOI: 10.24423/EngTrans.996.20190815

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