Skip to main content
SpringerLink
Log in

Modular and Recursive Kinematics and Dynamics for Parallel Manipulators

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

Constrained multibody systems typically feature multiple closed kinematic loops that constrain the relative motions and forces within the system. Typically, such systems possess far more articulated degrees-of-freedom (within the chains) than overall end-effector degrees-of-freedom.Thus, actuation of a subset of the articulations creates mixture of active and passive joints within the chain.The presence of such passive joints interferes with the effective modular formulation of the dynamic equations-of-motion in terms of a minimal set of actuator coordinates as well the subsequent recursivesolution for both forward and inverse dynamics applications.

Thus, in this paper, we examine the development of modular and recursive formulations of equations-of-motion in terms of a minimal set of actuated-joint-coordinates for an exactly-actuated parallel manipulators. The 3 RRR planar parallel manipulator, selected to serve as a case-study, is an illustrative example of a multi-loop, multi-degree-of-freedom system with mixtures of active/passive joints. The concept of decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism inherent in parallel mechanisms to develop a dynamics formulation that is both recursive and modular. An algorithmic approach to the development of both forward and inverse dynamics is highlighted. The presented simulation studies highlight the overall good numerical behavior of the developed formulation, both in terms of accuracy and lack of formulation stiffness.

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. Angeles, J., Computer-Aided Analysis of Rigid and Flexible Mechanical Systems, Chapt. on Twist and Wrench Generators and Annihilators. Dordrecht-Boston-London: Kluwer Academic Publishers, 1994.

    Google Scholar 

  2. Angeles, J., Fundamentals of Robotic Mechanical Systems. New York: Springer-Verlag, 2002.

    Google Scholar 

  3. Angeles, J. and Lee, S., ‘The formulation of dynamical equations of holonomic mechanical systems using a natural orthogonal complement’, ASME Journal of Applied Mechanics 55, 1988, 243–244.

    MATH  Google Scholar 

  4. Angeles, J. and Ma, O., ‘Dynamic simulation of n-axis serial robotic manipulators using a natural orthogonal complement’, The International Journal of Robotics Research 7(5), 1988, 32–47.

    Google Scholar 

  5. Armstrong, W., ‘Recursive solution to the equations of motions of an n-link manipulator’, in Proc. 5th World Congress on Theory of Machines and Mechanisms, Montreal, 1979, pp. 1343–1346.

  6. Ascher, U., Pai, D. and Cloutier, B., ‘Forward dynamics, elimination methods, and formulation stiffness in robot simulation’, The International Journal of Robotics Research 16(6), 1997, 749–758.

    Google Scholar 

  7. Ascher, U. and Petzold, L., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Philadelphia: SIAM, 1998.

    MATH  Google Scholar 

  8. Bae, D. and Han, J., ‘A generalized recursive formulation for constrained mechanical system dynamics’, Mechanics of Structures and Machines, An International Journal 27(3), 1999, 293–315.

    Google Scholar 

  9. Bae, D. and Haug, E., ‘A recursive formulation for constrained mechanical system dynamics: Part 2. closed loop systems’, Mechanics of Structures and Machines, An International Journal 15(4), 1987, 481–506.

    Google Scholar 

  10. Balafoutis, C., Patel, R. and Cloutier, B., ‘Efficient modelling and computation of manipulator dynamics using orthogonal cartesian tensors’, IEEE Journal of Robotics and Automation 4, 1988, 665–676.

    Google Scholar 

  11. Blajer, W., ‘A geometrical interpretation and uniform matrix formulation of multibody system dynamics’, Zeitschrift fr Angewandte Mathematik und Mechanik 81(4), 2001, 247–259.

    MATH  MathSciNet  Google Scholar 

  12. Brandl, H., Johanni, R. and Otter, M., ‘A very efficient algorithm for the simulation of robots and similar multibody systems without inversion of the mass matrix’, In Proc. IFAC/IFIP/IMACS International Symposium on Theory of Robots, Vienna, 1986.

  13. Featherstone, R., ‘The calculation of robot dynamics using articulated-body inertias’, The International Journal of Robotics Research 2(1), 1983, 13–30.

    Google Scholar 

  14. Featherstone, R., Robot Dynamics Algorithms. Boston-Dordrecht-Lancaster: Kluwer Academic Publishers, 1987.

    Google Scholar 

  15. Featherstone, R., ‘A divide-and-conquer articulated-body algorithm for parallel O(log(n)) calculation of rigid-body dynamics. Part 2: Trees, loops and accuracy’, The International Journal of Robotics Research 18(9), 1999, 876–892.

    Google Scholar 

  16. García de Jalón, J. and Bayo, E., Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge, New York: Springer-Verlag, 1994.

    Google Scholar 

  17. Geike, T. and McPhee, J., ‘On the automatic generation of inverse dynamic solutions for parallel manipulators’, in Proc. Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators. Quebec City, 2002, pp. 348–358.

  18. Goldenberg, A. and He, X., ‘An algorithm for efficient computation of dynamics of robotic manipulators’. in Proc. Fourth International Conference on Advanced Robotics, Columbus, OH, 1989, pp. 175–188.

  19. Gosselin, C. and Angeles, J., ‘Singularity analysis of closed-loop kinematic chains’, IEEE Transactions on Robotics and Automation 6(3), 1990, 281–290.

    Google Scholar 

  20. Haug, E., Computer Aided Kinematics and Dynamics of Mechanical Systems. Boston: Allyn and Bacon, 1989.

    Google Scholar 

  21. Hunt, K. H., Kinematic Geometry of Mechanisms. Oxford Science Publications, 1990.

  22. Kecskemethy, A., Krupp, T. and Hiller, M., ‘Symbolic processing of multi-loop mechanism dynamics using closed form kinematic solutions’, Multibody System Dynamics 1(1), 1997, 23–45.

    MathSciNet  MATH  Google Scholar 

  23. Kerr, J. and Roth, B., ‘Analysis of multifingered hands’, The International Journal of Robotics Research 4(4), 1986, 3–17.

    Google Scholar 

  24. Khan, W. A., ‘Distributed dynamics of systems with closed kinematic chains’, Master's thesis, Mechanical Engineering, McGill University, Montreal, 2002.

  25. Koivo, A. J. and Bekey, G. A., ‘Report of workshop on coordinated multiple robot manipulators: planning, control, and application’, IEEE Transactions on Robotics and Automation 4, 1988, 91–93.

    Google Scholar 

  26. Kumar, V. and Waldron, K., ‘Force distribution in closed kinematic chains’, IEEE Transactions on Robotics and Automation 4(6), 1988, 657–664.

    Google Scholar 

  27. Luh, J., Walker, M. and Paul, R., ‘On-line computational schemes for mechanical manipulators’, ASME Journal of Dynamic Systems, Measurement and Control 102(2), 1980, 69–76.

    MathSciNet  Google Scholar 

  28. Ma, O. and Angeles, J., ‘Direct kinematics and dynamics of a planar 3-dof parallel manipulator’, in Advances in Design Automation, Vol. 3. Montreal, Quebec, 1989, pp. 313–320.

  29. Mc{M}illan, S. and Orin, D., ‘Efficient computation of articulated-body inertias using successive axial screws’, IEEE Transactions on Robotics and Automation 11, 1995, 606–611.

    Article  Google Scholar 

  30. McMillan, S., Sadayappan, P. and Orin, D. E., ‘Parallel dynamic simulation of multiple manipulator systems: temporal versus spatial methods’, IEEE Transactions on Systems, Man and Cybernetics 24(7), 1994, 982–990.

    Article  Google Scholar 

  31. Merlet, J.-P., Parallel Robots. Dordrecht: Kluwer Academic Publishers, 2000.

    MATH  Google Scholar 

  32. Murray, R., Li, Z. and Sastry, S., A Mathematical Introduction to Robotic Manipulation. Boca Raton, FL: {CRC} Press, 1994.

    MATH  Google Scholar 

  33. Orin, D., Mc{G}hee, R., Vukobratovic, M. and Hartoch, G., ‘Kinematic and kinetic analysis of open-chain linkages utilizing newton-euler methods’, Mathematical Biosciences 43, 1979, 107–130.

    Article  MATH  Google Scholar 

  34. Orin, D. and Walker, M., ‘Efficient dynamic computer simulation of robotic mechanisms’, ASME Journal of Dynamic Systems, Measurement and Control 104, 1982, 205–211.

    MATH  Google Scholar 

  35. Rodriguez, G. and Kreutz-Delgado, K., ‘Spatial operator factorization and inversion of the manipulator mass matrix’, IEEE Transactions on Robotics and Automation 8(1), 1992, 65–76.

    Article  MathSciNet  Google Scholar 

  36. Saha, S. K., ‘A decomposition of the manipulator inertia matrix’, IEEE Transactions on Robotics and Automation 13(2), 1997, 301–304.

    Article  Google Scholar 

  37. Saha, S. K., ‘Analytical expression for the inverted inertia matrix of serial robots’, The International Journal of Robotic Research 18(1), 1999, 20–36.

    Google Scholar 

  38. Saha, S. K., ‘Dynamics of serial multibody systems using the decoupled natural orthogonal complement matrices’, ASME Journal of Applied Mechanics 66, 1999, 986–996.

    Google Scholar 

  39. Saha, S. K. and Angeles, J., ‘Dynamics of nonholonomic mechanical systems using a natural orthogonal complement’, ASME Journal of Applied Mechanics 58, 1991, 238–243.

    MATH  Google Scholar 

  40. Saha, S. K. and Schiehlen, W. O., ‘Recursive kinematics and dynamics for parallel structured closed-loop multibody systems’, Mechanics of Structures and Machines, An International Journal 29(2), 2001, 143–175.

    Google Scholar 

  41. Salisbury, J. K. and Craig, J. J., ‘Articulated hands: Force control and kinematic issues’, The International Journal of Robotics Research 1(1), 1982, 4–17.

    Google Scholar 

  42. Schiehlen, W., ‘Multibody systems and robot dynamics’, in A. Morecki, G. Bianchi and K. Jaworek (eds.), Proc. 8th{CISM}-{IFT}o{MM} Symposium on Theory and Practice of Robot Manipulators. Warsaw, Poland, 1990, pp. 14–21.

  43. Schiehlen, W., Multibody Systems Handbook. Berlin: Springer-Verlag, 1990.

    MATH  Google Scholar 

  44. Shabana, A. A., Computational Dynamics. New York: Wiley, 2001.

    Google Scholar 

  45. Song, S. M. and Waldron, K. J., Machines that Walk. 2 edn., Cambridge, MA: MIT Press, 1989.

    Google Scholar 

  46. Stejskal, V. and Valasek, M., Kinematics and Dynamics of Machinery. New York: Marcel Dekker, 1996.

    Google Scholar 

  47. Stepanenko, Y. and Vukobratovic, M., ‘Dynamics of articulated open-chain active mechanism’, Mathematical Biosciences 28, 1976, 137–170.

    Article  MathSciNet  Google Scholar 

  48. Vereshchagin, A., ‘Computer simulation of the dynamics of complicated mechanisms of robot manipulators’, Engineering Cybernetics 6, 1974, 65–70.

    Google Scholar 

  49. Walker, M. and Orin, D., ‘Efficient dynamic computer simulation of robotic mechanisms’, ASME Journal of Dynamic Systems, Measurement and Control 104, 1982, 205–211.

    Article  MATH  Google Scholar 

  50. Wang, J., Gosselin, C. and Cheng, L., ‘Dynamic modelling and simulation of parallel mechanisms using virtual spring approach’, in Proc. 2000 ASME Design Engineering Technical Conferences. Baltimore, Maryland, 2000, pp. 1–10.

  51. Yiu, Y., Cheng, H., Xiong, Z., Liu, G. and Li, Z., ‘On the dynamics of parallel manipulator’, in Proc. IEEE international Conference on Robotics and Automation. Seoul, Korea, 2001, pp. 3766–3771.

Download references

Author information

Authors and Affiliations

  1. Centre for Intelligent Machines, McGill University, Montreal, QC, Canada

    Waseem Ahmad Khan & Jorge Angeles

  2. Mechanical and Aerospace Engineering, State University of New York at Buffalo, USA

    Venkat N. Krovi

  3. Department of Mechanical Engineering, IIT, Delhi

    Subir Kumar Saha

Authors
  1. Waseem Ahmad Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Venkat N. Krovi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Subir Kumar Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jorge Angeles
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khan, W.A., Krovi, V.N., Saha, S.K. et al. Modular and Recursive Kinematics and Dynamics for Parallel Manipulators. Multibody Syst Dyn 14, 419–455 (2005). https://doi.org/10.1007/s11044-005-1143-9

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-005-1143-9

Keywords

Search

Navigation