, Volume 52, Issue 1–2, pp 441–455 | Cite as

Graph-based structural analysis of planar mechanisms

  • Sebastián DurangoEmail author
  • Jorge Correa
  • Oscar E. Ruiz


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.


Graph Kinematic structure Assur group Structural analysis System group classification 



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.


  1. 1.
    Artobolevski I (1988) Teoría de mecanismos y máquinas. Nauka, Moscú (in Spainsh)Google Scholar
  2. 2.
    Baránov G (1979) Curso de la teoría de mecanismos y máquinas. MIR, Moscú (in Spanish)Google Scholar
  3. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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 CrossRefGoogle Scholar
  5. 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 CrossRefGoogle Scholar
  6. 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 CrossRefGoogle Scholar
  7. 7.
    Galletti C (1979) On the position analysis of Assur’s groups of high class. Meccanica 14(1):6–10. doi: 10.1007/BF02134963 CrossRefzbMATHGoogle Scholar
  8. 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 MathSciNetCrossRefGoogle Scholar
  9. 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 CrossRefzbMATHGoogle Scholar
  10. 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 ADSCrossRefGoogle Scholar
  11. 11.
    Kolovsky M, Evgrafov A, Semenov Y, Slousch A (2000) Advanced theory of mechanisms and machines. Springer, BerlinCrossRefzbMATHGoogle Scholar
  12. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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 CrossRefzbMATHGoogle Scholar
  14. 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 CrossRefGoogle Scholar
  15. 15.
    Mruthyunjaya T (2003) Kinematic structure of mechanisms revisited. Mech Mach Theory 38(4):279–320. doi: 10.1016/S0094-114X(02)00120-9 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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 CrossRefGoogle Scholar
  17. 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 CrossRefzbMATHGoogle Scholar
  18. 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 CrossRefzbMATHGoogle Scholar
  19. 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 CrossRefGoogle Scholar
  20. 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 CrossRefGoogle Scholar
  21. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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 CrossRefGoogle Scholar
  23. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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 CrossRefzbMATHGoogle Scholar
  26. 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 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Grupo Diseño Mecánico y Desarrollo IndustrialUniversidad Autónoma de ManizalesManizalesColombia
  2. 2.Laboratorio de CAD CAM CAEUniversidad EAFITMedellínColombia

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