Identification of Configuration Space Singularities with Local Real Algebraic Geometry

Authors: Marc Diesse, Hochschule Heilbronn

Abstract:

We address the question of identifying the configuration space singularities of linkages, i.e., points where the configuration space is not locally a submanifold of Euclidean space. Because the configuration space cannot be smoothly parameterized at such points, these singularity types have a significantly negative impact on the kinematics of the linkage. It is known that Jacobian methods do not provide sufficient conditions for the existence of CS-singularities. Herein, we present several additional algebraic criteria that provide the sufficient conditions. Further, we use those criteria to analyze certain classes of planar linkages. These examples will also show how the presented criteria can be checked using algorithmic methods.

Keywords: Linkages, configuration space singularities, real algebraic geometry, analytic geometry, computer algebra.

References:

[1] C. Wampler and A. Sommese, “Numerical algebraic geometry and algebraic kinematics,” Acta Numerica, vol. 20, pp. 469–567, 04 2011.
[2] J. Selig, Geometric Fundamental of Robotics, ser. Monographs in Computer Science. Springer, 2005.
[3] A. Mller and D. Zlatanov, Singular Configurations of Mechanisms and Manipulators, 1st ed., ser. CISM International Centre for Mechanical Sciences 589. Springer International Publishing, 2019.
[4] G. Liu, Y. Lou, and Z. Li, “Singularities of parallel manipulators: A geometric treatment,” IEEE Transactions on Robotics and Automation, vol. 19, no. 4, pp. 579–594, 2003.
[5] J. K. F.C. Park, “Singularity analysis of closed kinematic chains,” Journal of Mechanical Design, Vol. 121, 1999.
[6] S. Piipponen, “Singularity analysis of planar linkages,” Multibody System Dynamics, vol. 22, no. 3, pp. 223–243, 2009.
[7] Z. Li and A. M¨uller, “Mechanism singularities revisited from an algebraic viewpoint,” in ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers Digital Collection, 2019.
[8] W. Decker, G.-M. Greuel, G. Pfister, and H. Sch¨onemann, “SINGULAR 4-1-2 — A computer algebra system for polynomial computations,” http://www.singular.uni-kl.de, 2019.
[9] O. Bottema and B. Roth, Theoretical kinematics. Dover, 1990.
[10] M. Diesse, “On local real algebraic geometry and applications to kinematics,” 2019, https://arxiv.org/abs/1907.12134v2.
[11] M. Farber, Invitation to topological robotics. European Mathematical Society, 2008, vol. 8.
[12] G.-M. Greuel and G. Pfister, A Singular introduction to commutative algebra: Second Edition. Springer Science & Business Media, 2012.
[13] D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, ser. Undergraduate Texts in Mathematics. Springer Science & Business Media, 2013.
[14] D. Eisenbud, Commutative Algebra: with a view toward algebraic geometry, ser. Graduate Texts in Mathematics. Springer Science & Business Media, 2013, vol. 150.
[15] A. J. de Jong et al., The Stacks project. https://stacks.math.columbia.edu: University of Columbia, 2018.
[Online]. Available: https://stacks.math.columbia.edu/
[16] G.-M. Greuel, S. Laplagne, and F. Seelisch, “normal.lib A SINGULAR library for computing the normalization of affine rings.”
[17] D. Blanc and N. Shvalb, “Generic singular configurations of linkages,” Topology and its Applications, vol. 159, no. 3, pp. 877–890, 2012.
[18] F. Warner, Foundations of differentiable manifolds and Lie groups, ser. Graduate Texts in Mathematics. Springer Science & Business Media, 2013, vol. 94.
[19] J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry, ser. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer Science & Business Media, 2013, vol. 36.
[20] J. M. Ruiz Sancho, The basic theory of power series. Springer Vieweg, 1993.
[21] J.-J. Risler, “Le th´eor`eme des z´eros en g´eom´etries alg´ebrique et analytique r´eelles,” Bulletin de la Soci´et´e Math´ematique de France, vol. 104, pp. 113–127, 1976.
[22] G. Efroymson, “Local reality on algebraic varieties,” Journal of Algebra, vol. 29, no. 1, pp. 133–142, 1974.