0
Research Papers

Local Kinematic Analysis of Closed-Loop Linkages—Mobility, Singularities, and Shakiness

[+] Author and Article Information
Andreas Müller

Institute of Robotics,
Johannes Kepler University,
Altenberger Strasse 69,
Linz 4040, Austria
e-mail: a.mueller@jku.at

Manuscript received August 8, 2015; final manuscript received February 9, 2016; published online March 18, 2016. Assoc. Editor: Leila Notash.

J. Mechanisms Robotics 8(4), 041013 (Mar 18, 2016) (11 pages) Paper No: JMR-15-1216; doi: 10.1115/1.4032778 History: Received August 08, 2015; Revised February 09, 2016

The mobility of a linkage is determined by the constraints imposed on its members. The geometric constraints define the configuration space (c-space) variety as the geometric entity in which the finite mobility of a linkage is encoded. The aim of a local kinematic analysis of a linkage is to deduce its finite mobility, in a given configuration, from the local c-space geometry. In this paper, a method for the local analysis is presented adopting the concept of the tangent cone to a variety. The latter is an algebraic variety approximating the c-space. It allows for investigating the mobility in regular as well as singular configurations. The instantaneous mobility is determined by the constraints, rather than by the c-space geometry. Shaky and underconstrained linkages are prominent examples that exhibit a permanently higher instantaneous than finite DOF even in regular configurations. Kinematic singularities, on the other hand, are reflected in a change of the instantaneous DOF. A c-space singularity as a kinematic singularity, but a kinematic singularity may be a regular point of the c-space. The presented method allows to identify c-space singularities. It also reveals the ith-order mobility and allows for a classification of linkages as overconstrained and underconstrained. The method is applicable to general multiloop linkages with lower pairs. It is computationally simple and only involves Lie brackets (screw products) of instantaneous joint screws. The paper also summarizes the relevant kinematic phenomena of linkages.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Husty, M. , Pfurner, M. , Schröcker, H.-P. , and Brunnthaler, K. , 2007, “ Algebraic Methods in Mechanism Analysis and Synthesis,” Robotica, 25(6), pp. 661–675. [CrossRef]
Gogu, G. , 2008, “ Constraint Singularities and the Structural Parameters of Parallel Robots,” Advances in Robot Kinematics: Analysis and Design, J. Lenarcic and P. Wenger , eds., Springer, Amsterdam.
Dai, J. , Huang, Z. , and Lipkin, H. , 2006, “ Mobility of Overconstrained Parallel Mechanisms,” ASME J. Mech. Des., 128(1), pp. 220–229. [CrossRef]
Zhao, J.-S. , Feng, Z.-J. , and Dong, J.-X. , 2006, “ Computation of the Configuration Degree of Freedom of a Spatial Parallel Mechanism by Using Reciprocal Screw Theory,” Mech. Mach. Theory, 41(12), pp. 1486–1504. [CrossRef]
Kong, X. , and Gosselin, C. M. , 2007, Type Synthesis of Parallel Mechanisms, Springer, Berlin.
Baker, J. E. , 1980, “ An Analysis of the Bricard Linkages,” Mech. Mach. Theory, 15(4), pp. 267–286. [CrossRef]
Waldron, K. J. , 1973, “ A Study of Overconstrained Linkage Geometry by Solution of Closure Equations—Part II. Four-Bar Linkages With Lower Pairs Other Than Screw Joints,” Mech. Mach. Theory, 8(2), pp. 233–247. [CrossRef]
Mavroidis, C. , and Roth, B. , 1995, “ Analysis of Overconstrained Mechanisms,” ASME J. Mech. Des., 117(1), pp. 69–74. [CrossRef]
Gogu, G. , 2005, “ Mobility and Spatiality of Parallel Robots Revisited Via the Theory of Linear Transformations,” Eur. J. Mech. A/Solids, 24(4), pp. 690–711. [CrossRef]
Kong, X. , 2014, “ Reconfiguration Analysis of a 3-DOF Parallel Mechanism Using Euler Parameter Quaternions and Algebraic Geometry Method,” Mech. Mach. Theory, 74, pp. 188–201. [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]
Arponen, T. , Piipponen, S. , and Tuomela, J. , 2009, “ Kinematic Analysis of Bricard's Mechanism,” Nonlinear Dyn., 56(1), pp. 85–99. [CrossRef]
Arponen, T. , Müller, A. , Piipponen, S. , and Tuomela, J. , 2014, “ Kinematical Analysis of Overconstrained and Underconstrained Mechanisms by Means of Computational Algebraic Geometry,” Meccanica, 49(4), pp. 843–862. [CrossRef]
Liu, R. , Serré, P. , and Rameau, J. F. , 2013, “ A Tool to Check Mobility Under Parameter Variations in Over-Constrained Mechanisms,” Mech. Mach. Theory, 69, pp. 44–61. [CrossRef]
Wampler, C. W. , and Sommese, A. J. , 2011, “ Numerical Algebraic Geometry and Kinematics,” Acta Numer., 20, pp. 469–567. [CrossRef]
Wampler, C. W. , Hauenstein, J. D. , and Sommese, A. J. , 2011, “ Mechanism Mobility and a Local Dimension Test,” Mech. Mach. Theory, 46(9), pp. 1193–1206. [CrossRef]
Rico, J. M. , Gallardo, J. , and Duffy, J. , 1999, “ Screw Theory and Higher Order Kinematic Analysis of Open Serial and Closed Chains,” Mech. Mach. Theory, 34(4), pp. 559–586. [CrossRef]
Gallardo-Alvarado, J. , and Rico-Martinez, J. M. , 2001, “ Jerk Influence Coefficients, Via Screw Theory, of Closed Chains,” Meccanica, 36(2), pp. 213–228. [CrossRef]
Cervantes-Sánchez, J. J. , Rico-Martínez, J. M. , González-Montiel, G. , and González-Galván, E. J. , 2009, “ The Differential Calculus of Screws: Theory, Geometrical Interpretation, and Applications,” Proc. Inst. Mech. E., Part C: J. Mech. Eng. Sci., 223(6), pp. 1449–1468. [CrossRef]
Müller, A. , 2002, “ Higher Order Local Analysis of Singularities in Parallel Mechanisms,” ASME Paper No. DETC2002/MECH-34258.
Müller, A. , 2014, “ Higher Derivatives of the Kinematic Mapping and Some Applications,” Mech. Mach. Theory, 76, pp. 70–85. [CrossRef]
Müller, A. , “ Higher-Order Constraints for Linkages With Lower Kinematic Pairs,” Mech. Mach. Theory, 100, pp. 33–43.
Lerbet, J. , 1999, “ Analytic Geometry and Singularities of Mechanisms,” Z. Angew. Math. Mech., 78(10b), pp. 687–694.
Whitney, H. , 1965, “Local Properties of Analytic Varieties, Differential and Combinatorial Topology,” A Symposium in Honor of Marston Morse (Princeton Mathematical Series 27), S. S. Cairns, ed., Princeton University Press, Princeton, NJ.
Müller, A. , and Rico, J. M. , 2008, “ Mobility and Higher Order Local Analysis of the Configuration Space of Single-Loop Mechanisms,” Advances in Robot Kinematics, J. J. Lenarcic and P. Wenger , eds., Springer, Amsterdam, pp. 215–224.
Chen, C. , 2011, “ The Order of Local Mobility of Mechanisms,” Mech. Mach. Theory, 46(9), pp. 1251–1264. [CrossRef]
de Bustos, I. F. , Aguirrebeitia, J. , Avilés, R. , and Ansola, R. , 2012, “ Second Order Mobility Analysis of Mechanisms Using Closure Equations,” Meccanica, 47(7), pp. 1695–1704. [CrossRef]
Karger, A. , 1996, “ Singularity Analysis of Serial Robot-Manipulators,” ASME J. Mech. Des., 118(4), pp. 520–525. [CrossRef]
Wohlhart, K. , 1999, “ Degrees of Shakiness,” Mech. Mach. Theory, 34(7), pp. 1103–1126. [CrossRef]
Connelly, R. , and Servatius, H. , 1994, “ Higher-Order Rigidity–What is the Proper Definition?,” Discrete Comput. Geom., 11(2), pp. 193–200. [CrossRef]
Wohlhart, K. , 2010, “ From Higher Degrees of Shakiness to Mobility,” Mech. Mach. Theory, 45(3), pp. 467–476. [CrossRef]
Müller, A. , 2015, “ Representation of the Kinematic Topology of Mechanisms for Kinematic Analysis,” Mech. Sci., 6, pp. 1–10. [CrossRef]
Müller, A. , 2015, “ Local Analysis of Closed-Loop Linkages: Mobility, Singularities, and Shakiness,” ASME Paper No. DETC2015-47485.
Chai, W. H. , and Chen, Y. , “ The Line-Symmetric Octahedral Bricard Linkage and Its Structural Closure,” Mech. Mach. Theory, 45(5), pp. 772–779. [CrossRef]
Wohlhart, K. , 1996, “ Kinematotropic Linkages,” Recent Advances in Robot Kinematics, J. Lenarčič and V. Parent-Castelli , eds., Kluwer, Alphen aan den Rijn, The Netherlands, pp. 359–368.
Golubitsky, M. , and Guillemin, V. , 1973, Stable Mappings and Their Singularities, Springer, New York.
Zlatanov, D. , Bonev, I. A. , and Gosselin, C. M. , 2002, “ Constraint Singularities as C-Space Singularities,” 8th International Symposium on Advances in Robot Kinematics (ARK 2002), Caldes de Malavella, Spain, June 24–28.
Kutznetsov, E. N. , 1991, “ Systems With Infinitesimal Mobility—Part I: Matrix Analysis and First-Order Infinitesimal Mobility,” ASME J. Appl. Mech., 58(2), pp. 513–526. [CrossRef]
Brockett, R. W. , 1984, “ Robotic Manipulators and the Product of Exponentials Formula, Mathematical Theory of Networks and Systems,” Lect. Notes Control Inf. Sci., 58, pp. 120–129.
Selig, J. , 2005, Geometric Fundamentals of Robotics, Monographs in Computer Science Series, Springer-Verlag, New York.
Müller, A. , 2014, “ Derivatives of Screw Systems in Body-Fixed Representation,” Advances in Robot Kinematics (ARK), J. Lenarcic and O. Khatib , eds., Springer, Cham, Switzerland, pp. 123–130.
Rameau, J. F. , and Serré, P. , 2015, “ Computing Mobility Condition Using Groebner Basis,” Mech. Mach. Theory, 91, pp. 21–38. [CrossRef]
Greuel, G. M. , and Pfister, G. , 2012, A Singular Introduction to Commutative Algebra, Springer, Berlin.
Cox, D. , Little, J. , and O'Shea, D. , 2007, Ideals, Varieties and Algorithms, 3rd ed., Springer, Berlin.
Goldberg, M , 1943, “ New Five-Bar and Six-Bar Linkages in Three Dimensions,” Trans. ASME, 65, pp. 649–661.
Baker, J. E. , 1993, “ A Comparative Survey of the Bennett-Based, 6-Revolute Kinematic Loops,” Mech. Mach. Theory, 28(1), pp. 83–96. [CrossRef]
Servatius, B. , Shai, O. , and Whiteley, W. , 2010, “ Geometric Properties of Assur Graphs,” Eur. J. Combinatorics, 31(4), pp. 1105–1120. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

(a) Planar four-bar linkage in singular configuration q0 and (b) a three-dimensional cut of its c-space

Grahic Jump Location
Fig. 2

(a) Three-loop kinematotropic linkage in a singular configuration, (b) in a 1DOF motion mode, and (c) in a 2DOF motion mode

Grahic Jump Location
Fig. 3

Immobile spherical four-bar linkage

Grahic Jump Location
Fig. 4

Planar five-bar linkage in singular configuration q0

Grahic Jump Location
Fig. 5

A 6R Goldberg linkage in a kinematic singularity, which is a regular point of the c-space V

Grahic Jump Location
Fig. 6

6R linkage in a singular configuration q0

Grahic Jump Location
Fig. 7

(a) Two-loop Assur linkage and (b) its topological graph and FCs Λ4 and Λ6

Grahic Jump Location
Fig. 8

(a) Two-loop linkage in singular configuration q0 and (b) its topological graph with FCs Λ1 and Λ7

Grahic Jump Location
Fig. 10

Topological graph and selected FCs Λ1, Λ4, and Λ9 for the three-loop linkage based on a Peaucellier–Lipkin linkage

Grahic Jump Location
Fig. 11

7R-linkage after Ref. [11] in a singular configuration

Grahic Jump Location
Fig. 12

Decision diagram to identify a kinematic singularity with the information available from the higher-order local analysis. The dotted lines indicate that this decision cannot be made with available information.

Tables

Errata

Discussions

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