{"title":"Identification of Configuration Space Singularities with Local Real Algebraic Geometry","authors":"Marc Diesse, Hochschule Heilbronn","volume":173,"journal":"International Journal of Electrical and Computer Engineering","pagesStart":209,"pagesEnd":219,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10012030","abstract":"We address the question of identifying the configuration<br \/>\r\nspace singularities of linkages, i.e., points where the configuration<br \/>\r\nspace is not locally a submanifold of Euclidean space. Because the<br \/>\r\nconfiguration space cannot be smoothly parameterized at such points,<br \/>\r\nthese singularity types have a significantly negative impact on the<br \/>\r\nkinematics of the linkage. It is known that Jacobian methods do not<br \/>\r\nprovide sufficient conditions for the existence of CS-singularities.<br \/>\r\nHerein, we present several additional algebraic criteria that provide<br \/>\r\nthe sufficient conditions. Further, we use those criteria to analyze<br \/>\r\ncertain classes of planar linkages. These examples will also show<br \/>\r\nhow the presented criteria can be checked using algorithmic methods.","references":"[1] C. Wampler and A. Sommese, \u201cNumerical algebraic geometry and\r\nalgebraic kinematics,\u201d Acta Numerica, vol. 20, pp. 469\u2013567, 04 2011.\r\n[2] J. Selig, Geometric Fundamental of Robotics, ser. Monographs in\r\nComputer Science. Springer, 2005.\r\n[3] A. Mller and D. Zlatanov, Singular Configurations of Mechanisms and\r\nManipulators, 1st ed., ser. CISM International Centre for Mechanical\r\nSciences 589. Springer International Publishing, 2019.\r\n[4] G. Liu, Y. Lou, and Z. Li, \u201cSingularities of parallel manipulators: A\r\ngeometric treatment,\u201d IEEE Transactions on Robotics and Automation,\r\nvol. 19, no. 4, pp. 579\u2013594, 2003.\r\n[5] J. K. F.C. Park, \u201cSingularity analysis of closed kinematic chains,\u201d\r\nJournal of Mechanical Design, Vol. 121, 1999.\r\n[6] S. Piipponen, \u201cSingularity analysis of planar linkages,\u201d Multibody\r\nSystem Dynamics, vol. 22, no. 3, pp. 223\u2013243, 2009.\r\n[7] Z. Li and A. M\u00a8uller, \u201cMechanism singularities revisited from an\r\nalgebraic viewpoint,\u201d in ASME 2019 International Design Engineering\r\nTechnical Conferences and Computers and Information in Engineering\r\nConference. American Society of Mechanical Engineers Digital\r\nCollection, 2019. [8] W. Decker, G.-M. Greuel, G. Pfister, and H. Sch\u00a8onemann, \u201cSINGULAR\r\n4-1-2 \u2014 A computer algebra system for polynomial computations,\u201d\r\nhttp:\/\/www.singular.uni-kl.de, 2019.\r\n[9] O. Bottema and B. Roth, Theoretical kinematics. Dover, 1990.\r\n[10] M. Diesse, \u201cOn local real algebraic geometry and applications to\r\nkinematics,\u201d 2019, https:\/\/arxiv.org\/abs\/1907.12134v2.\r\n[11] M. Farber, Invitation to topological robotics. European Mathematical\r\nSociety, 2008, vol. 8.\r\n[12] G.-M. Greuel and G. Pfister, A Singular introduction to commutative\r\nalgebra: Second Edition. Springer Science & Business Media, 2012.\r\n[13] D. Cox, J. Little, and D. O\u2019Shea, Ideals, varieties, and algorithms:\r\nan introduction to computational algebraic geometry and commutative\r\nalgebra, ser. Undergraduate Texts in Mathematics. Springer Science &\r\nBusiness Media, 2013.\r\n[14] D. Eisenbud, Commutative Algebra: with a view toward algebraic\r\ngeometry, ser. Graduate Texts in Mathematics. Springer Science &\r\nBusiness Media, 2013, vol. 150.\r\n[15] A. J. de Jong et al., The Stacks project.\r\nhttps:\/\/stacks.math.columbia.edu: University of Columbia, 2018.\r\n[Online]. Available: https:\/\/stacks.math.columbia.edu\/\r\n[16] G.-M. Greuel, S. Laplagne, and F. Seelisch, \u201cnormal.lib A\r\nSINGULAR library for computing the normalization of affine rings.\u201d\r\n[17] D. Blanc and N. Shvalb, \u201cGeneric singular configurations of linkages,\u201d\r\nTopology and its Applications, vol. 159, no. 3, pp. 877\u2013890, 2012.\r\n[18] F. Warner, Foundations of differentiable manifolds and Lie groups, ser.\r\nGraduate Texts in Mathematics. Springer Science & Business Media,\r\n2013, vol. 94.\r\n[19] J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry, ser.\r\nErgebnisse der Mathematik und ihrer Grenzgebiete. Springer Science\r\n& Business Media, 2013, vol. 36.\r\n[20] J. M. Ruiz Sancho, The basic theory of power series. Springer Vieweg,\r\n1993.\r\n[21] J.-J. Risler, \u201cLe th\u00b4eor`eme des z\u00b4eros en g\u00b4eom\u00b4etries alg\u00b4ebrique et\r\nanalytique r\u00b4eelles,\u201d Bulletin de la Soci\u00b4et\u00b4e Math\u00b4ematique de France, vol.\r\n104, pp. 113\u2013127, 1976.\r\n[22] G. Efroymson, \u201cLocal reality on algebraic varieties,\u201d Journal of Algebra,\r\nvol. 29, no. 1, pp. 133\u2013142, 1974.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 173, 2021"}