Kinematic Analysis of Parallel Manipulators

  • S. Negri
  • G. Di Bernardo
  • I. Fassi
  • L. Molinari Tosatti
  • G. Bianchi
  • C. R. Boër
Conference paper
Part of the Advanced Manufacturing book series (ADVMANUF)


A method for the kinematic analysis of parallel manipulators, based on Denavit-Hartenberg formalism, is proposed. This formulation allows to model separately each joint-link train (defined as the consecutive structural elements starting from the origin of the reference frame associated with the base, to the origin of the reference frame of the mobile platform through one of the links). This methodology has the main advantage of determining all the generalized variables characterizing the model (in order to evaluate the mechanical limits of the passive joints, their characteristic parameters), and not only those associated with the active joints. In this way, it is possible to model manufacturing and assembly errors in the mechanical components, kinematics errors in the actuators.

Several parallel machinesm according to the adopted classification, have been modeled mapping the stiffness index above the workspace.

The workspace is defined as the Boolean function w: ℜ6 → {0,1} of all the reachable points (x, y, z, ψ, θ, ϕ) considering physical constraints on joints and legs. The set of reachable poses defines the workspace as a connected set of points. The stiffness index was computed considering the scalar value given by the inverse of the condition number k of the Jacobian matrix. This number gives the ratio between the maximum value of the stiffness computed in the direction of the maximum stiffness and the minimum value of stiffness computed in the direction of the minimum stiffness for each point of the closed set of points belonging to the workspace. When 1/k is close to zero it means either that the machine is near a singular configuration or that the stiffness significantly varies along different directions. In both cases, and supposing the manipulator is used for machining operations (i.e.: as a milling machine), the further is 1/k from the unit the worse is the behavior of the machine. This mapping so can be used for a rough evaluation of the usable workspace.


Parallel Manipulator Kinematic Analysis Mobile Platform Stiffness Index Spherical Joint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Stoughton R, Arai T 1993 A Modified Stewart Platform Manipulator with Improved Dexterity. IEEE Transactions on Robotics and Automation 9: 166 - 172CrossRefGoogle Scholar
  2. 2.
    Bhattacharya S, Hatwal H, Ghosh A 1995 On the Optimum Design of Stewart Platform Type Parallel Manipulators. Robotica 13:133: 140CrossRefGoogle Scholar
  3. 3.
    Huang T, Whitehouse T J, Wang J 1998 The Local Dexterity. Optimal Architecture and Design Criteria of Parallel Machine Tools. Annals of the CIRP 47: 347 - 351CrossRefGoogle Scholar
  4. 4.
    Merlet J-P 1990 Les Robots parallèles, Hermès, ParisGoogle Scholar
  5. 5.
    Fichter EF 1986 A Stewart Platform-Based Manipulator: General Theory and Practical Construction. The International Journal of Robotics Research 5:157- 182CrossRefGoogle Scholar
  6. 6.
    Dafaoui El-M, Amirat Y, Pontnau J, Francois C 1998 Analysis and Design of a Six-DOF Parallel Manipulator, Modeling, Singular Configurations, and Workspace. IEEE Transactions on Robotics and Automation 14: 78 - 91CrossRefGoogle Scholar
  7. 7.
    Dasgupta B, Mruthyunjaya T S 1996 A Constructive Predictor-Corrector Algorithm for the Direct Position Kinematic Problem for a General 6-6 Stewart Platform. Mech. Mack Theory 31: 799 - 811CrossRefGoogle Scholar
  8. 8.
    Nanua P, Waldron K J, Murthy V 1990 Direct Kinematic Solution of a Stewart Platform. IEEE Transactions on Robotics and Automation 6: 438 - 443CrossRefGoogle Scholar
  9. 9.
    Gosselin C 1990 Stiffness Mapping for Parallel manipulators. IEEE Transactions on Robotics and Automation 6: 377 - 382CrossRefGoogle Scholar
  10. 10.
    Merlet J-P 1995 Determination of the Orientation Workspace of Parallel Manipulators. Journal of Intelligent and Robotic Systems 13: 143 - 160CrossRefGoogle Scholar
  11. 11.
    Wang Li-Chun T, Hsieh J 1998 Extreme Reaches and Reachable Workspace Analysis of General Parallel Robotic Manipulators. Journal of Robotic Systems 15 (3): 145 - 159zbMATHGoogle Scholar
  12. 12.
    Conti JP, Clinton CM, Zhang G, Wavering AJ 1997 Dynamic Variation of the Workspace of an Octahedral Hexapod Machine Tool During Machining. Technical Research Report, T.R. 97-28, ISRGoogle Scholar
  13. 13.
    Faugere J C, Lazard D 1995 Combinatorial Classes of Parallel Manipulators. Meck Mach. Theory 30: 765 - 776Google Scholar
  14. 14.
    Innocenti C, Parenti Castelli V 1994 Exhaustive Enumeration of Fully Parallel Kinematic Chains. Dynamic Systems and Control 55: 1135 - 1141Google Scholar
  15. 15.
    Patel A J, Ehmann K F 1997 Volumetric Error Analysis of a Stewart Platform- Based Machine Tool. Annals of the CIRP 46: 287 - 290CrossRefGoogle Scholar
  16. 16.
    Wang J, Masory O 1993 On the Accuracy of a Stewart Platform - Part I: The Effect of Manufacturing Tolerances. In: Proceedings 1993 IEEE International Conference on Robotics and Automation, Vol I, pp 114 - 120Google Scholar
  17. 17.
    Masory O, Wang J, Zhuang H 1993 On the Accuracy of a Stewart Platform - Part II: Kinematic Calibration and Compensation. In: Proceedings 1993 IEEE International Conference on Robotics and Automation, Vol I, pp 725 - 731Google Scholar
  18. 18.
    Molinari Tosatti L, Bianchi G, Fassi I, Boer C R, Jovane F 1998 An Integrated Methodology for the Design of Parallel Kinematic Machines (PKM). Annals of the CIRP 47: 341 - 345CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 1999

Authors and Affiliations

  • S. Negri
    • 1
  • G. Di Bernardo
    • 1
  • I. Fassi
    • 1
  • L. Molinari Tosatti
    • 1
  • G. Bianchi
    • 1
  • C. R. Boër
    • 1
  1. 1.ITIA-CNRMilanoItaly

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