Identification of Configuration Space Singularities with Local Real Algebraic Geometry
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.
[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.
[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.
@article{"International Journal of Electrical, Electronic and Communication Sciences:80306", author = "Marc Diesse and Hochschule Heilbronn", title = "Identification of Configuration Space Singularities with Local Real Algebraic Geometry", 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.", volume = "15", number = "5", pages = "140-10", }
