Skip to main content
SpringerLink
Log in

Applicability and generality of the modified Grübler-Kutzbach criterion

  • Published:
Chinese Journal of Mechanical Engineering Submit manuscript

Abstract

A generally applicable criterion for all mechanism mobility has been an active domain in mechanism theory lasting more than 150 years. It is stated that the Modified Grübler-Kutzbach criterion for mobility has been successfully used to solve the mobility of many more kinds of mechanisms, but never before has anyone proven the applicability and generality of the Modified Grübler-Kutzbach criterion in theory. In order to fill the gap, the applicability and generality of the Modified Grübler-Kutzbach Criterion of mechanism mobility is systematically demonstrated. Firstly, the mobility research background and the Modified Grübler-Kutzbach criterion are introduced. Secondly, some new definitions, such as half local freedom, non-common constraint space of a mechanism and common motion space of a mechanism, etc, are given to demonstrate the correctness and broad applicability of the Modified Grübler-Kutzbach criterion. Thirdly, the general applicability of the Modified Grübler-Kutzbach criterion is demonstrated based on screw theory. The mobilities of the classical DELASSUS mechanisms and a modern planar parallel mechanism, are determined through the Modified Grübler-Kutzbach criterion, which are as examples to show the practical application of the Modified Grübler-Kutzbach criterion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. CHEBYCHEV P A. Théorie des mécanismes connus sous le nom de parallélogrammes, 1ére partie[C]//Mémoires présentés à ? Académie impériale des sciences de Saint-Pétersbourg par divers savants, Saint-Pétersbourg, Russie, 1854.

  2. GRÜBLER M. Allgemeine Eigenschaften der Zwangläufigen ebenen kinematischen Ketten: I[J]. Zivilingenieur, 1883, 29: 167–200.

    Google Scholar 

  3. GOGU G. Mobility of mechanisms: a critical review[J]. Mechanism and Machine Theory, 2005, 40(9): 1 068–1 097.

    Article  MathSciNet  Google Scholar 

  4. KONG X, GOSSELIN C M. Mobility analysis of parallel mechanisms based on screw theory and the concept of equivalent serial kinematic chain[C]//ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach, California, USA, September 24–28, 2005: DETC2005-85337.

  5. ZHAO J, ZHOU K, FENG Z. A theory of degrees of freedom for mechanisms[J]. Mechanism and Machine Theory, 2004, 39(6): 621–643.

    Article  MathSciNet  Google Scholar 

  6. HUNT K H. Kinematic geometry of mechanisms[M]. Oxford: Oxford University Press, 1978.

    Google Scholar 

  7. HUANG Z, KONG L F, FANG Y F. Mechanism theory and control of parallel manipulators[M]. Beijing: China Machine Press, 1997.

    Google Scholar 

  8. LIU J F, LI Y W, HUANG Z. Mobility of the Bennett-based linkages[C]//The ASME 2009 International Design Engineering Technical Conferences, San Diego, California, USA, August 30–September 2, 2009: DETC2009-86243.

  9. HUANG Z, LI Q C. General Methodology for Type Synthesis of Lower-Mobility Symmetrical Parallel Manipulators and Several Novel Manipulators[J]. Int. J. Rob Res., 2002, 21(2): 131–146.

    Article  Google Scholar 

  10. HUANG Z, LIU J F, ZENG D X. A General methodology for mobility analysis of mechanisms based on constraint screw theory[J]. Science in China-E, 2009, 52(5): 1 135–1 470.

    Google Scholar 

  11. HUANG Z, LIU J F, Li Q C. Unified methodology for mobility analysis based on screw theory[M] //WANG L, XI J., Smart Devices and Machines for Advanced Manufacturing London: Springer-Verlag, 2008: 49–78.

    Chapter  Google Scholar 

  12. HUANG Z, XIA P. The mobility analysis of some classical mechanism and recent parallel robots[C]// The ASME 2006 International Design Engineering Technical Conferences, Philadelphia, Pennsylvania, USA, September 10–13, 2006: 1 304–1 311.

  13. DAI J S, HUANG Z, LIPKIN H. Mobility of overconstrained parallel mechanisms[J]. ASME J. of Mechanical Design, 2006, 128(1): 220–229

    Article  Google Scholar 

  14. LIU J F, LI Y W, HUANG Z. Mobility analysis of Altmann overconstrained linkages by modified Grübler-Kutzbach criterion[J]. Chinese Journal of Mechanical Engineering, 2011, 24(4): 638–646.

    Article  MathSciNet  Google Scholar 

  15. BALL R. A Treatise on the theory of Screws[M]. England: Cambridge University Press, 1900.

    Google Scholar 

  16. KUTZBACH K. Einzelfragen aus dem gebiet der maschinenteile[J]. Zeitschrift der Verein Deutscher Ingenieur, 1933, 77: 1 168.

    Google Scholar 

  17. TSAI L W. Mechanism design: Enumeration of kinematic structures according to function[M]. Florida, Boca Raton: CRC Press, 2000.

    Google Scholar 

  18. SARRUS P T. Note sur la transformation des mouvements rectilignes alternatifs, en mouvements circulaires; et reciproquement[J]. Academie des sciences, comtes rendus hebdomataires des séances, 1853, 36: 1 0361 038.

    Google Scholar 

  19. DELASSUS Et. Les chaînes articulées fermées etdéformables à quatre membres[J]. Bulletin des sciences mathématiques, 1922, 46: 283304.

    Google Scholar 

  20. WALDRON K J. A study of overconstrained linkage geometry by solution of closure equations-Part I, method of study[J]. Mech. Mach. Theory, 1973, 8(1): 95104.

    Article  Google Scholar 

  21. WALDRON K J. A study of overconstrained linkage geometry by solution of closure equations-Part II, Four-bar linkages with lower pair joints other than screw joints[J]. Mech. Mach. Theory, 1973, 8(2): 233247.

    Article  Google Scholar 

  22. CHEN G, WANG H, ZHAO Y, et al. A kind of kinematically redundant planar paralle manipulator for optimal output accuracy[C]//The ASME International Design Engineering Technical Conferences, San Diego, California, USA, August 30 September 2, 2009: DETC2009-86184.

Download references

Author information

Authors and Affiliations

  1. Hebei Provincial Key Laboratory of Parallel Robot and Mechatronic System, Yanshan University, Qinhuangdao, 066004, China

    Yanwen Li, Jingfang Liu & Zhen Huang

  2. College of Environment and Chemical Engineering, Yanshan University, Qinhuangdao, 066004, China

    Lumin Wang

  3. College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing, 100124, China

    Jingfang Liu

Authors
  1. Yanwen Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lumin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jingfang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhen Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanwen Li.

Additional information

This project is supported by National Natural Science Foundation of China(Grant Nos. 51175446, 50875227), and the Science Supporting Plan of Education Department of Hebei Province, China(Grant No. 2008150)

LI Yanwen, born in 1966, is currently a professor at Yanshan University, China. She received her PhD degree from Yanshan University, China, in 2005. Her research interests include mechachonics engineering, robotics.

WANG Lumin, born in 1964, is currently an associate professor at Yanshan University, China. He received his master degree from Huadong Science and Technology University, China, in 1999. His research interests include chemical science and engineering.

LIU Jingfang, born in 1985, is currently a lecturer at College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, China. She received her PhD degree from Yanshan University, China. Her research interests include theory and control in mechanisms, robotics.

HUANG Zhen, born in 1936, is currently a professor at Yanshan University, China. His research interests include theory and technology in parallel mechanisms, robotics, topological analysis as well as electromechanical integration.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, Y., Wang, L., Liu, J. et al. Applicability and generality of the modified Grübler-Kutzbach criterion. Chin. J. Mech. Eng. 26, 257–263 (2013). https://doi.org/10.3901/CJME.2013.02.257

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3901/CJME.2013.02.257

Key words

Search

Navigation