, Volume 46, Issue 1, pp 113–129 | Cite as

The Gantry-Tau parallel kinematic machine—kinematic and elastodynamic design optimisation

Parallel Manipulators


One of the main advantages of the Gantry-Tau machine is a large accessible workspace/footprint ratio compared to many other parallel machines. The optimal kinematic, elastostatic and elastodynamic design parameters of the machine are still difficult to calculate and this paper introduces an optimisation scheme based on the geometric and functional dependencies to define the workspace and first resonance frequency. This method assumes that each link and universal joint can be described by a mass-spring-damper model and calculates the transfer function from a Cartesian force or torque to Cartesian position or orientation. The evolutionary algorithm based on the complex search method is compared to the gradient-based search function in Matlab integrated optimisation toolbox. Kinematic design obtained by optimisation according to this paper gives a 2D workspace/footprint ratio more than 1.66 and first resonance frequency is more than 50 Hz with components of an existing lab prototype at the University of Agder, Norway.


PKM Evolutionary Design Optimisation Flexible Dynamics Model 


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  1. 1.
    Merlet J-P (2000) Parallel robots. Solid mechanics and its applications, vol 74. Kluwer Academic, Dordrecht MATHCrossRefGoogle Scholar
  2. 2.
    Gough V (1956–1957) Contribution to discussion to papers on research in automobile stability and control and in tire performance. In: Proc auto div institute of mechanical engineers, pp 392–394 Google Scholar
  3. 3.
    Stewart D (1965) A platform with six degrees of freedom. UK Inst Mech Eng Proc 180(15):30–35 Google Scholar
  4. 4.
    Brogårdh T (2000) Design of high performance parallel arm robots for industrial applications. In: Proc of the symp comm the legacy, works, and life of sir Robert Stawell Ball, 100th anniv of a treatise on the theory on the screws. Univ of Cambridge, Cambridge Google Scholar
  5. 5.
    Murray M, Hovland G, Brogårdh T (2008) Singularity-free reconfiguration of the 5-DOF Gantry-Tau parallel kinematic machine. In: Proc 2nd intl workshop on fundamental issues and future research directions for parallel mechanisms and manipulators, Montpellier, Sept 21–22 Google Scholar
  6. 6.
    Dashy A, Yeoy S, Yangz G, Chery I-H (2002) Workspace anal. and singularity rep. of 3-legged parallel manip. In: Proc 7th intl conf in contr, autom, rob and vision, pp 962–967 Google Scholar
  7. 7.
    Tyapin I (2009) Multi-objective design optimisation of a class of parallel kinematic machines. PhD thesis, University of Queensland, Australia Google Scholar
  8. 8.
    Fattah A, Angeles J, Misra A (1995) Dynamics of a 3-dof spatial parallel manipulator with flexible links. In: Proc of the international conference on robotics and automation, pp 627–632 Google Scholar
  9. 9.
    Xi F, Sinatra R, Han W (2001) Effect of leg inertia on dynamics of sliding-leg Hexapods. J Dyn Syst Meas Control 123:265. doi: 10.1115/1.1369600 CrossRefGoogle Scholar
  10. 10.
    Lee J, Geng Z (1993) A dynamic model of a flexible Stewart platform. Comput Struct 48:367 MATHCrossRefGoogle Scholar
  11. 11.
    Hardage D, Weins G (1999) Modal analysis and modeling of a parallel kinematic machine. ASME, Manuf Eng Div 10:857 Google Scholar
  12. 12.
    Zhou Z, Xi F, Mechefske C (2006) Modeling of a fully flexible 3PRS manipulator for vibration analysis. J Mech Des 128:403. doi: 10.1115/1.2167655 CrossRefGoogle Scholar
  13. 13.
    Chen J, Hsu W (2006) Dynamic and compliant characteristics of a cartesian-guided Tripod machine. J Manuf Sci Eng 128:494. doi: 10.1115/1.1954789 CrossRefGoogle Scholar
  14. 14.
    Hovland G, Choux M, Murray M, Brogårdh T (2007) Benchmark of the 3-dof Gantry-Tau parallel kinematic machine. In: IEEE intl conf on robotics and automation, pp 535–542. doi: 10.1109/ROBOT.2007.363042 Google Scholar
  15. 15.
    Dressler I, Robertsson A, Johansson R (2007) Accuracy of kinematic and dynamic models of a Gantry-Tau parallel kinematic robot. In: IEEE int conf on robotics and automation, pp 883–888. doi: 10.1109/ROBOT.2007.363097 Google Scholar
  16. 16.
    Merlet J-P, Daney D (2008) Smart devices and machines for advanced manufacturing. Springer, London Google Scholar
  17. 17.
    Stamper R, Tsai L, Walsh G (1997) Optimisation of a 3 dof translational platform for well-conditioned workspace. In: Proc IEEE intl conf on robotics and automation, pp 3250–3255. doi: 10.1109/ROBOT.1997.606784 Google Scholar
  18. 18.
    Gosselin C, Angeles J (1988) The optimum kinematic design of a planar three-degree-of-freedom parallel manipulator. ASME J Mech Transm Autom Des 110(1):35. doi: 10.1115/1.3258901 CrossRefGoogle Scholar
  19. 19.
    Gosselin C, Angeles J (1989) The optimum kinematic design of a spherical three-degree-of-freedom parallel manipulator. J Mech Transm Autom Des 111:202. doi: 10.1115/1.3258984 CrossRefGoogle Scholar
  20. 20.
    Liu X-J, Wang J (2007) A new methodology for optimal kinematic design of parallel mechanism. J Mech Mach Theory 42:1210. doi: 10.1016/j.mechmachtheory.2006.08.002 MATHCrossRefGoogle Scholar
  21. 21.
    Stock M, Miller K (2003) Optimal kinematic design of spatial parallel manipulators: application of linear delta robot. Trans ASME, J Mech Des 292–301. doi: 10.1115/1.1563632
  22. 22.
    Chablat D, Wenger P (2003) Architecture optimization of 3-dof translational parallel mechanism for machining applications, the orthoglide. IEEE Trans Robot Autom 19(3):403. doi: 10.1109/TRA.2003.810242 CrossRefGoogle Scholar
  23. 23.
    Liu X-J, Wu C, Wang J (2008) A new index for the performance evaluation of parallel manipulators: a study on planar parallel manipulators. In: Proc of the 7th world congress on intelligent control and automation, pp 353–361. doi: 10.1109/WCICA.2008.4592950 CrossRefGoogle Scholar
  24. 24.
    Lou Y, Liu G, Li Z (2008) Randomized optimal design of parallel manipulators. IEEE Trans. Autom. Sci. Eng. 5(2):223. doi: 10.1109/TASE.2007.909446 CrossRefGoogle Scholar
  25. 25.
    Schaffer J (1984) Some experiments in machine learning using vector evaluated genetic algorithms. PhD thesis, Vanderbilt University, Nashville, TN Google Scholar
  26. 26.
    Wang J, Zhang J, Wei X (2006) Evolutionary multi-objective optimization algorithm with preference for mechanical design. Springer, Berlin, pp 497–506 Google Scholar
  27. 27.
    Coello Coello C (2002) Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput Methods Appl Mech Eng 191(11–12):1245–1287. doi: 10.1016/S0045-7825(01)00323-1 MATHCrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Hansen M, Andersen T (2001) A design procedure for actuator control systems using optimization methods. In: Proc of the IEEE 7th Scandinavian international conference on fluid power. Linköping, Sweden, pp 213–221 Google Scholar
  29. 29.
    Hansen M, Andersen T, Mouritsen O (2004) A scheme for handling discrete and continuous design variables in multi criteria design optimization of servo mechanisms. In: Mechatronics and robotics, Aachen, Germany, pp 234–245 Google Scholar
  30. 30.
    Tyapin I, Hovland G (2009) Kinematic and elastostatic design optimisation of the 3-DOF Gantry-Tau parallel kinematic machine. Model Identif Control 30(2):39–56. doi: 10.4173/mic.2009.2.1 CrossRefGoogle Scholar
  31. 31.
    Gosselin C (1990) Determination of the workspace of 6-dof parallel manipulators. ASME J Mech Des 112:331. doi: 10.1115/1.2912612 CrossRefGoogle Scholar
  32. 32.
    Li Y, Kao I (2004) Stiffness control on redundant manipulators: a unique and kinematically consistent solution. In: Intl conf on roborics and automat, pp 3956–3961 Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Lulea University of TechnologyLuleåSweden
  2. 2.University of AgderGrimstadNorway

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