Multibody System Dynamics

, Volume 32, Issue 3, pp 273–298 | Cite as

Dynamics of serial kinematic chains with large number of degrees-of-freedom

  • A. Agarwal
  • S. V. Shah
  • S. Bandyopadhyay
  • S. K. Saha


This paper investigates the dynamic behaviour of serial chains with degrees-of-freedom (DOF) as large as 100,000. A recursive solver called Recursive Dynamic Simulator (ReDySim), based on the Newton–Euler formulation and the Decoupled Natural Orthogonal Complement (DeNOC) matrices, was used to simulate the dynamics of these systems. Planar, as well as spatial motions of chains with moderate- (DOF≤1,000), large- (1,000<DOF≤10,000), and huge- (DOF>10,000) DOF were simulated. The results were validated by several means, such as comparisons with reported results wherever available, results obtained from commercial software, energy checks, etc. The study shows that ReDySim is capable of analysing serial chains of huge-DOF with acceptable numerical accuracy. The scheme is found to be numerically stable as well as computationally efficient, owing to the linear-time computation of the joint accelerations. Numerical studies were conducted to establish the theoretical basis for better performance of the proposed ReDySim solver. With the demonstrated capabilities of ReDySim, it may be found suitable for a large number of applications involving serial systems with huge-DOF.


Dynamics Chains Decomposition Large degrees-of-freedom 



The authors acknowledge the Naren Gupta Chair Professor fund of the last author at IIT Delhi, which was used to support the second author to conduct this research. Anonymous reviewers are also thanked for their constructive comments.


  1. 1.
    Spong, M.W., Vidyasagar, M.: Robot Dynamics and Control. Wiley, New York (1989) Google Scholar
  2. 2.
    Nakamura, Y.: Advanced Robotics: Redundancy and Optimization. Addison Wesley, Reading (1991) Google Scholar
  3. 3.
    Vold, H.I., Karlen, J.P., Thompson, J.M., Farrell, J.D., Eismann, P.H.: A 17 degree of freedom anthropomorphic manipulator. In: Proceedings of NASA Conference on Space Telerobotics, vol. 1, pp. 19–28 (1989) Google Scholar
  4. 4.
    Geradin, M., Cardona, A.: Flexible Multibody Dynamics—A Finite Element Approach. Wiley, New York (2001) Google Scholar
  5. 5.
    Gerstmayr, J., Schoberl, J.: A 3D finite element method for flexible multibody systems. Multibody Syst. Dyn. 15, 309–324 (2006) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Fritzkowski, P., Kamiński, H.: Dynamics of a rope as a rigid multibody system. J. Mech. Mater. Struct. 3(6), 1059–1075 (2008) CrossRefGoogle Scholar
  7. 7.
    Featherstone, R.: An empirical study of the joint space inertia matrix. Int. J. Robot. Res. 23(9), 859–871 (2005) CrossRefGoogle Scholar
  8. 8.
    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. Int. J. Robot. Res. 18, 876–892 (1999) CrossRefGoogle Scholar
  9. 9.
    Yamane, K., Nakamura, Y.: Parallel O(logN) algorithm for dynamics simulation of humanoid robots. In: 6th IEEE-RAS International Conference on Humanoid Robots (2006) Google Scholar
  10. 10.
    Malczyk, P., Fraczek, J.: Cluster computing of mechanisms dynamics using recursive formulation. Multibody Syst. Dyn. 20(2), 177–196 (2008) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Ascher, U.M., Pai, D.K., Cloutier, B.P.: Forward dynamics elimination methods, and formulation stiffness in robot simulation. Int. J. Robot. Res. 16(6), 749–758 (1997) CrossRefGoogle Scholar
  12. 12.
    Tomaszewski, W., Pieranski, P., Geminard, J.: The motion of a freely falling chain tip. Am. J. Phys. 74(9), 776–783 (2006) CrossRefGoogle Scholar
  13. 13.
    Aglietti, G.: Dynamic response of a high-altitude tethered balloon system. J. Aircr. 46(6), 2032–2040 (2009) CrossRefGoogle Scholar
  14. 14.
    Calkin, M.G., March, R.H.: The dynamics of a falling chain: I. Am. J. Phys. 57, 154–157 (1989) CrossRefGoogle Scholar
  15. 15.
    Fritzkowski, P., Kamiński, H.: Dynamics of a rope modelled as a discrete system with extensible members. Comput. Mech. 44, 473–480 (2009) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Fritzkowski, P., Kamiński, H.: Dynamics of a rope modelled as a multi-body system with elastic joints. Comput. Mech. 46(6), 901–909 (2010) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Gatti, C., Perkins, N.: Physical and numerical modeling of the dynamic behaviour of a fly line. J. Sound Vib. 255(3), 555–577 (2002) CrossRefGoogle Scholar
  18. 18.
    Hembree, B., Slegers, N.: Efficient tether dynamic model formulation using recursive rigid-body dynamics. Proc. Inst. Mech. Eng., Proc., Part K, J. Multi-Body Dyn. 224, 353–363 (2010) CrossRefGoogle Scholar
  19. 19.
    Kamman, J., Huston, R.: Multibody dynamics modeling of variable length cable systems. Multibody Syst. Dyn. 5, 211–221 (2001) CrossRefMATHGoogle Scholar
  20. 20.
    Robson, J.M.: The physics of fly casting. Am. J. Phys. 58, 234–240 (1990) CrossRefGoogle Scholar
  21. 21.
    Schagerl, M., Steindl, A., Steiner, W., Troger, H.: On the paradox of the free falling folded chain. Acta Mech. 125, 155–168 (1997) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Wong, C., Yasu, K.: Falling chains. Phys. Rev. Lett. 74(6), 490–496 (2006) Google Scholar
  23. 23.
    Shah, S.V., Saha, S.K., Dutt, J.K.: Dynamic of Tree-Type Robotic Systems. Springer, Dordrecht (2013) CrossRefGoogle Scholar
  24. 24.
    Function Bay Inc.: RecurDyn, Version 7.4 (2009) Google Scholar
  25. 25.
    Strang, G.: Linear Algebra and Its Applications. Harcourt Brace Jovanovich, San Diego (1998) Google Scholar
  26. 26.
    Saha, S.K.: A decomposition of the manipulator inertia matrix. IEEE Trans. Robot. Autom. 13(2), 301–304 (1997) CrossRefGoogle Scholar
  27. 27.
    Saha, S.K.: Analytical expression for the inverted inertia matrix of serial robots. Int. J. Robot. Res. 18(1), 116–124 (1999) Google Scholar
  28. 28.
    MathWorks Inc.: MATLAB, Version 7.4, Release 2009a (2009) Google Scholar
  29. 29.
    Saha, S.K.: Dynamics of serial multibody systems using the decoupled natural orthogonal complement matrices. J. Appl. Mech. 66, 986–996 (1999) CrossRefGoogle Scholar
  30. 30.
    Shah, S.V., Saha, S.K., Dutt, J.K.: Denavit–Hartenberg (DH) parametrization of Euler angles. J. Comput. Nonlinear Dyn. 7(2), 1–10 (2012) CrossRefGoogle Scholar
  31. 31.
    Mohan, A., Saha, S.K.: A recursive, numerically stable, and efficient algorithm for serial robots. Multibody Syst. Dyn. 17(4), 291–319 (2007) CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Weisstein, E.W.: Catenary. MathWorld—a Wolfram Web resource. Accessed 31 March 2012 (2012)
  33. 33.
    Shampine, L.F., Gear, C.W.: A user’s view of solving stiff ordinary differential equations. SIAM Rev. 21(1), 1–17 (1979) CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Lilly, K.W.: Efficient Dynamic Simulation of Robotic Mechanisms. Kluwer Academic, Boston (1993) CrossRefMATHGoogle Scholar
  35. 35.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (2002) CrossRefMATHGoogle Scholar
  36. 36.
    Poole, G., Larry Neal, L.: A geometric analysis of Gaussian elimination. I. Linear Algebra Appl. 149, 249–272 (1991) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • A. Agarwal
    • 1
  • S. V. Shah
    • 2
  • S. Bandyopadhyay
    • 3
  • S. K. Saha
    • 4
  1. 1.Systemantics India Pvt. LtdBangaloreIndia
  2. 2.Robotics Research CentreInternational Institute of Information Technology (IIIT)HyderabadIndia
  3. 3.Department of Engineering DesignIndian Institute of Technology MadrasChennaiIndia
  4. 4.Department of Mechanical EngineeringIndian Institute of Technology DelhiNew DelhiIndia

Personalised recommendations