Skip to main content
SpringerLink
Log in

Graph-based structural analysis of planar mechanisms

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

Kinematic structure of planar mechanisms addresses the study of attributes determined exclusively by the joining pattern among the links forming a mechanism. The system group classification is central to the kinematic structure and consists of determining a sequence of kinematically and statically independent-simple chains which represent a modular basis for the kinematics and force analysis of the mechanism. This article presents a novel graph-based algorithm for structural analysis of planar mechanisms with closed-loop kinematic structure which determines a sequence of modules (Assur groups) representing the topology of the mechanism. The computational complexity analysis and proof of correctness of the implemented algorithm are provided. A case study is presented to illustrate the results of the devised method.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Artobolevski I (1988) Teoría de mecanismos y máquinas. Nauka, Moscú (in Spainsh)

    Google Scholar 

  2. Baránov G (1979) Curso de la teoría de mecanismos y máquinas. MIR, Moscú (in Spanish)

    Google Scholar 

  3. Buśkiewicz J (2006) A method of optimization of solving a kinematic problem with the use of structural analysis algorithm (SAM). Mech Mach Theory 41(7):823–837. doi:10.1016/j.mechmachtheory.2005.10.003

    Article  MathSciNet  MATH  Google Scholar 

  4. Campos A, Guenther R, Martins D (2009) Differential kinematics of parallel manipulators using Assur virtual chains. P I Mech Eng C-J Mech 223(7):1697–1711. doi:10.1243/09544062JMES1156

    Article  Google Scholar 

  5. Chen G, Wang H, Zhao K, Lin Z (2009) Modular calculation of the Jacobian matrix and its application to the performance analyses of a forging robot. Adv Robot 23(10):1261–1279. doi:10.1163/156855309X462574

    Article  Google Scholar 

  6. Durango S, Calle G, Ruiz O (2010) Analytical method for the kinetostatic analysis of the second-class RRR Assur group allowing for friction in the kinematic pairs. J Braz Soc Mech Sci Eng 32(3):200–207. doi:10.1590/S1678-58782010000300002

    Article  Google Scholar 

  7. Galletti C (1979) On the position analysis of Assur’s groups of high class. Meccanica 14(1):6–10. doi:10.1007/BF02134963

    Article  MATH  Google Scholar 

  8. Galletti C (1986) A note on modular approaches to planar linkage kinematic analysis. Mech Mach Theory 21(5):385–391. doi:10.1016/0094-114X(86)90086-8

    Article  MathSciNet  Google Scholar 

  9. Han L, Liao Q, Liang C (2000) Closed-form displacement analysis for a nine-link Barranov truss or a eight-link Assur group. Mech Mach Theory 35(3):379–390. doi:10.1016/S0094-114X(99)00016-6

    Article  MATH  Google Scholar 

  10. Hansen M (1996) A general method for analysis of planar mechanisms using a modular approach. Mech Mach Theory 31(8):1155–1166. doi:10.1016/0094-114X(96)84606-4

    Article  ADS  Google Scholar 

  11. Kolovsky M, Evgrafov A, Semenov Y, Slousch A (2000) Advanced theory of mechanisms and machines. Springer, Berlin

    Book  MATH  Google Scholar 

  12. Mitsi S, Bouzakis K, Mansour G, Popescu I (2003) Position analysis in polynomial form of planar mechanisms with Assur groups of class 3 including revolute and prismatic joints. Mech Mach Theory 38(12):1325–1344. doi:10.1016/S0094-114X(03)00090-9

    Article  MathSciNet  MATH  Google Scholar 

  13. Mitsi S, Bouzakis KD, Mansour G (2004) Position analysis in polynomial form of planar mechanism with an Assur group of class 4 including one prismatic joint. Mech Mach Theory 39(3):237–245. doi:10.1016/S0094-114X(03)00115-0

    Article  MATH  Google Scholar 

  14. Mlynarski T (1996) Position analysis of planar linkages using the method of modification of kinematic units. Mech Mach Theory 31(6):831–838. doi:10.1016/0094-114X(95)00120-N

    Article  Google Scholar 

  15. Mruthyunjaya T (2003) Kinematic structure of mechanisms revisited. Mech Mach Theory 38(4):279–320. doi:10.1016/S0094-114X(02)00120-9

    Article  MathSciNet  MATH  Google Scholar 

  16. Peisakh E (2007) An algorithmic description of the structural synthesis of planar Assur groups. J Mach Manuf Reliab 36(6):505–514. doi:10.3103/S1052618807060015

    Article  Google Scholar 

  17. Popescu I, Marghitu D (2008) Structural design of planar mechanisms with dyads. Multibody Syst Dyn 19(4):407–425. doi:10.1007/s11044-007-9099-6

    Article  MATH  Google Scholar 

  18. Popescu I, Marghitu D, Stoenescu E (2006) Kinematic chains with independent loops and spatial system groups. Arch Appl Mech 75(10):739–754. doi:10.1007/s00419-006-0064-2

    Article  MATH  Google Scholar 

  19. Reich Y, Shai O (2012) The interdisciplinary engineering knowledge genome. Res Eng Des 23(3):251–264. doi:10.1007/s00163-012-0129-x

    Article  Google Scholar 

  20. Saura M, Celdran A, Dopico D, Cuadrado J (2014) Computational structural analysis of planar multibody systems with lower and higher kinematic pairs. Mech Mach Theory 71:79–92. doi:10.1016/j.mechmachtheory.2013.09.003

    Article  Google Scholar 

  21. Servatius B, Shai O, Whiteley W (2010) Combinatorial characterization of the Assur graphs from engineering. Eur J Comb 31(4):1091–1104. doi:10.1016/j.ejc.2009.11.019

    Article  MathSciNet  MATH  Google Scholar 

  22. Tischler C, Samuel A, Hunt K (1995) Kinematic chains for robot hands-II. kinematic constraints, classification, connectivity, and actuation. Mech Mach Theory 30(8):1217–1239. doi:10.1016/0094-114X(95)00044-Y

    Article  Google Scholar 

  23. Wang H, Lin Z, Lai X (2008) Composite modeling method in dynamics of planar mechanical system. Sci China Ser E 51:576–590. doi:10.1007/s11431-008-0035-7

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang H, Eberhard P, Lin Z (2010) Modeling and simulation of closed loop multibody systems with bodies-joints composite modules. Multibody Syst Dyn 24:389–411. doi:10.1007/s11044-010-9208-9

    Article  MathSciNet  MATH  Google Scholar 

  25. Wohlhart K (2010) Position analysis of normal quadrilateral Assur groups. Mech Mach Theory 45(9):1367–1384. doi:10.1016/j.mechmachtheory.2010.03.002

    Article  MATH  Google Scholar 

  26. Zhang Q, Zou H, Guo W (2006) Position analysis of higher-class Assur groups by virtual variable searching and its application in a multifunction domestic sewing machine. Int J Adv Manuf Technol 28:602–609. doi:10.1007/s00170-004-2379-x

    Article  Google Scholar 

Download references

Acknowledgments

The authors wish to acknowledge the technical support by Ing. Ricardo Serrano Salazar during the algorithm’s implementation. Financial support for this work was provided by the Colombian Administrative Department of Science, Technology and Innovation (COLCIENCIAS), and the Colombian National Service of Learning (SENA), Grant 1216-479-22001. The authors gratefully acknowledge this support.

Author information

Authors and Affiliations

  1. Grupo Diseño Mecánico y Desarrollo Industrial, Universidad Autónoma de Manizales, antigua estación del ferrocarril, Manizales, Colombia

    Sebastián Durango

  2. Laboratorio de CAD CAM CAE, Universidad EAFIT, Medellín, Colombia

    Jorge Correa & Oscar E. Ruiz

Authors
  1. Sebastián Durango
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge Correa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oscar E. Ruiz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebastián Durango.

Appendix: Kinematic structure of planar mechanisms with higher pairs

Appendix: Kinematic structure of planar mechanisms with higher pairs

Kinematic and structural analysis of planar mechanisms can be done by replacing higher pairs with lower pairs. Reference [1] establishes the following condition that must be accomplished by the replacement: the equivalent mechanism preserves: a. the degree of freedom of the original one, and, b. the relative movement of all links in the particular position, e.g., the mechanism of Fig. 6a includes a higher pair formed by 2 curves hh and kk. The equivalent mechanism is constructed by tracing a normal nn at the contact point C and determining the curvature centers H and K of the corresponding curves. The equivalent has four links, where the higher pair was replaced by link 3, which forms lower pairs in H and K, Fig. 6b. The degree of freedom is preserved and the mechanism is exclusively formed by lower pairs.

Fig. 6
figure 6

Equivalence of a mechanisms with higher pair. a 1 DOF mechanism with higher pair. b Instantly equivalent mechanism of (a)

Full size image

Once the equivalent mechanism is determined, then it is possible to analyze the kinematic structure, for example, representing the mechanism by a graph and using Algorithm 1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Durango, S., Correa, J. & Ruiz, O.E. Graph-based structural analysis of planar mechanisms. Meccanica 52, 441–455 (2017). https://doi.org/10.1007/s11012-016-0403-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-016-0403-5

Keywords

Search

Navigation