Research Papers

Higher-Order Analysis of Kinematic Singularities of Lower Pair Linkages and Serial Manipulators

[+] Author and Article Information
Andreas Müller

Institute of Robotics,
Johannes Kepler University,
Linz 4040, Austria
e-mail: a.mueller@jku.at

Manuscript received December 27, 2016; final manuscript received October 26, 2017; published online December 22, 2017. Assoc. Editor: Philippe Wenger.

J. Mechanisms Robotics 10(1), 011008 (Dec 22, 2017) (13 pages) Paper No: JMR-16-1390; doi: 10.1115/1.4038528 History: Received December 27, 2016; Revised October 26, 2017

Kinematic singularities of linkages are configurations where the differential mobility changes. Constraint singularities are critical points of the constraint mapping defining the loop closure constraints. Configuration space (c-space) singularities are points where the c-space ceases to be a smooth manifold. These singularity types are not identical and can neither be distinguished nor identified by simply investigating the rank deficiency of the constraint Jacobian (linear dependence of joint screws). C-space singularities are reflected by the c-space geometry. In a previous work, a kinematic tangent cone was introduced as an approximation of the c-space, defined as the set of tangents to smooth curves in c-space. Identification of kinematic singularities amounts to analyze the local geometry of the set of critical points. As a computational means, a kinematic tangent cone to the set of critical points is introduced in terms of Jacobian minors. Closed form expressions for the derivatives of the minors in terms of Lie brackets of joint screws are presented. A computational method is introduced to determine a polynomial system defining the kinematic tangent cone. The paper complements the recently proposed mobility analysis using the tangent cone to the c-space. This allows for identifying c-space and kinematic singularities as long as the solution set of the constraints is a real variety. The introduced approach is directly applicable to the higher-order analysis of forward kinematic singularities of serial manipulators. This is briefly addressed in the paper.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Gosselin, C. M. , and Angeles, J. , 1990, “ Singular Analysis of Closed-Loop Kinematic Chains,” IEEE Trans. Rob. Autom., 6(3), pp. 281–290. [CrossRef]
Zlatanov, D. , Fenton, R. G. , and Benhabib, B. , 1998, “ Identification and Classification of the Singular Configurations of Mechanisms,” Mech. Mach. Theory, 33(6), pp. 743–760. [CrossRef]
Zlatanov, D. , Fenton, R. G. , and Benhabib, B. , 1995, “ A Unifying Framework for Classification and Interpretation of Mechanism Singularities,” ASME J. Mech. Des., 117(4), pp. 566–572. [CrossRef]
Bonev, I. , and Gosselin, C. , 2003, “ Singularity Analysis of 3-DOF Planar Parallel Mechanisms Via Screw Theory,” ASME J. Mech. Des., 125(3), pp. 573–581. [CrossRef]
Gibson, C. G. , and Hunt, K. H. , 1990, “ Geometry of Screw Systems—1,” Mech. Mach. Theory, 25(1), pp. 1–10. [CrossRef]
Gibson, C. G. , and Hunt, K. H. , 1990, “ Geometry of Screw Systems—2,” Mech. Mach. Theory, 25(1), pp. 11–27. [CrossRef]
Hunt, K. H. , 1986, “ Special Configurations of Robot-Arms Via Screw Theory,” Robotica, 4(3), pp. 171–179. [CrossRef]
McCarthy, J. M. , 2000, Geometric Design of Linkages, Springer, New York.
Merlet, J. P. , 1989, “ Singular Configurations of Parallel Manipulators and Grassmann Geometry,” Int. J. Rob. Res., 8(5), pp. 45–56. [CrossRef]
Rico, J. M. , Gallardo, J. , and Duffy, J. , 1995, “ A Determination of Singular Configurations of Serial Non-Redundant Manipulators, and Their Escapement From Singularities Using Lie Products,” Computational Kinematics, J. P. Merlet and B. Ravani , eds., Kluwer, Dordrecht, The Netherlands, pp. 143–152. [CrossRef]
Wolf, A. , Ottaviano, E. , Shohama, M. , and Ceccarelli, M. , 2004, “ Application of Line Geometry and Linear Complex Approximation to Singularity Analysis of the 3-DOF Capaman Parallel Manipulator,” Mech. Mach. Theory, 39(1), pp. 75–95. [CrossRef]
Wang, S. L. , and Waldron, K. J. , 1987, “ A Study of the Singular Configurations of Serial Manipulators,” ASME J. Mech., Trans. Autom. Des., 109(1), pp. 14–20. [CrossRef]
Litvin, F. L. , Zhang, Y. , Parenti Castelli, V. , and Innocenti, C. , 1990, “ Singularities, Configurations and Displacement Functions for Manipulators,” Int. J. Rob. Res., 5(2), pp. 52–65. [CrossRef]
Sugimoto, K. , Duffy, J. , and Hunt, K. H. , 1982, “ Special Configurations of Spatial Mechanisms and Robot Arms,” Mech. Mach. Theory, 17(2), pp. 119–132. [CrossRef]
Wohlhart, K. , 1999, “ Degrees of Shakiness,” Mech. Mach. Theory, 34(7), pp. 1103–1126. [CrossRef]
Chen, C. , 2011, “ The Order of Local Mobility of Mechanisms,” Mech. Mach. Theory, 46(9), pp. 1251–1264. [CrossRef]
Kieffer, J. , 1994, “ Differential Analysis of Bifurcations and Isolated Singularities of Robots and Mechanisms,” IEEE Trans. Rob. Autom., 10(1), pp. 1–10. [CrossRef]
Lerbet, J. , 1998, “ Analytic Geometry and Singularities of Mechanisms,” ZAMM, Z. Angew. Math. Mech., 78(10), pp. 687–694. [CrossRef]
López-Custodio, P. C. , Rico, J. M. , Cervantes, J. , and Sánchez, J. S. , 2017, “ Local Analysis of Helicoid-Helicoid Intersections in Reconfigurable Linkages,” ASME J. Mech. Rob., 9(3), p. 031008. [CrossRef]
Müller, A. , 2016, “ Local Kinematic Analysis of Closed-Loop Linkages- Mobility, Singularities, and Shakiness,” ASME J. Mech. Rob., 8(4), p. 041013. [CrossRef]
Bandyopadhyay, S. , and Ghosal, A. , 2004, “ Analysis of Configuration Space Singularities of Closed-Loop Mechanisms and Parallel Manipulators,” Mech. Mach. Theory, 39(5), pp. 519–544. [CrossRef]
Park, F. C. , and Kim, J. W. , 1999, “ Singularity Analysis of Closed Loop Kinematic Chains,” ASME J. Mech. Des., 121(1), pp. 32–38. [CrossRef]
Brockett, R. W. , 1984, “ Robotic Manipulators and the Product of Exponentials Formula,” Mathematical Theory of Networks and Systems (Lecture Notes in Control and Information Sciences, Vol. 58), pp. 120–129.
Selig, J. , 2005, Geometric Fundamentals of Robotics, Springer, New York.
Zlatanov, D. , Bonev, I. A. , and Gosselin, C. M. , 2002, “ Constraint Singularity as C-Space Singularities,” Advances in Robot Kinematics-Theory and Application, Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 183–192. [CrossRef] [PubMed] [PubMed]
Bochnak, J. , Coste, M. , and Roy, M. , 1998, Real Algebraic Geometry, Springer-Verlag, Berlin. [CrossRef]
Golubitsky, M. , and Guillemin, V. , 1973, Stable Mappings and Their Singularities, Springer, New York. [CrossRef]
Pai, D. K. , and Leu, M. C. , 1992, “ Genericity and Singularities of Robot Manipulators,” IEEE Trans. Rob. Autom, 8(5), pp. 545–559. [CrossRef]
Gibson, C. G. , 1979, Singular Points of Smooth Mappings, Pitman, London.
Whitney, H. , 1965, Local Properties of Analytic Varieties, Differential and Combinatorial Topology: A Symposium in Honor of M. Morse, S. S. Cairns , ed., Princeton University Press, Princeton, NJ.
O'Shea, W. , 2004, “ Limits of Tangent Spaces to Real Surfaces,” Am. J. Math., 126(5), pp. 951–980. [CrossRef]
Müller, A. , 2016, “ Recursive Higher-Order Constraints for Linkages With Lower Linematic Pairs,” Mech. Mach. Theory, 100, pp. 33–43. [CrossRef]
Müller, A. , and Shai, O. , 2017, “ Constraint Graphs for Combinatorial Mobility Determination,” Mech. Mach. Theory, 108, pp. 260–275. [CrossRef]
Müller, A. , 2014, “ Implementation of a Geometric Constraint Regularization for Multibody System Models,” Arch. Mech. Eng., 61(2), pp. 376–383. [CrossRef]
Müller, A. , 2015, “ Representation of the Kinematic Topology of Mechanisms for Kinematic Analysis,” Mech. Sci., 6, pp. 137–146. [CrossRef]
Müller, A. , 2017, “ Topology, Kinematics, and Constraints of Multi-Loop Linkages,” Robotica, accepted.
Müller, A. , 2012, “ On the Manifold Property of the Set of Singularities of Kinematic Mappings: Genericity Conditions,” ASME J. Mech. Rob., 4(1), p. 011006.
Song, C. Y. , and Chen, Y. , 2012, “ A Family of Mixed Double-Goldberg 6R Linkages,” Proc. R. Soc. A, 468(2139), pp. 871–890. [CrossRef]
Kong, X. , and Pfurner, M. , 2015, “ Type Synthesis and Reconfiguration Analysis of a Class of Variable-DOF Single-Loop Mechanisms,” Mech. Mach. Theory, 85, pp. 116–128. [CrossRef]
Connelly, R. , and Servatius, H. , 1994, “ Higher-Order Rigidity–What Is the Proper Definition?,” Discrete Comput. Geom., 11(2), pp. 193–200. [CrossRef]
Tarnai, T. , and Lengyel, A. , 2011, “ A Remarkable Structure of Leonardo and a Higher-Order Infinitesimal Mechanism,” J. Mech. Mater. Struct., 6(1–4), pp. 591–604. [CrossRef]
Donelan, P. , 2007, “ Singularity-Theoretic Methods in Robot Kinematics,” Robotica, 25(6), pp. 641–659. [CrossRef]
Karger, A. , 1996, “ Singularity Analysis of Serial Robot-Manipulators,” ASME J. Mech. Des., 118(4), pp. 520–525. [CrossRef]
Müller, A. , 2014, “ Higher Derivatives of the Kinematic Mapping and Some Applications,” Mech. Mach. Theory, 76, pp. 70–85. [CrossRef]
Cox, D. , Little, J. , and O'Shea, D. , 2007, Ideals, Varieties and Algorithms, 3rd ed., Springer, Berlin. [CrossRef]


Grahic Jump Location
Fig. 1

Decision tree for classifying a point q ∈V as regular or singular, where r=rank J(q). It is presumed that the zero set of the loop constraints is real at q.

Grahic Jump Location
Fig. 2

6R linkage in a rank 4 singularity belonging to L41

Grahic Jump Location
Fig. 3

Underconstrained 7R linkage in the rank 3 singularity q0

Grahic Jump Location
Fig. 4

(a) Planar linkage in the reference configuration where its c-space has a cusp singularity. Bodies are numbered with I,…,VIII and joints with 1,…,10. (b) Topological graph and the γ = 3 FCs.

Grahic Jump Location
Fig. 5

Redundant serial 7R manipulator (KUKA LWR) in the singular configuration q0 with rank 3




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In