Multibody System Dynamics

, Volume 21, Issue 1, pp 1–35 | Cite as

A recursive, numerically stable, and efficient simulation algorithm for serial robots with flexible links

  • Ashish Mohan
  • S. K. SahaEmail author


A methodology for the formulation of dynamic equations of motion of a serial flexible-link manipulator using the decoupled natural orthogonal complement (DeNOC) matrices, introduced elsewhere for rigid bodies, is presented in this paper. First, the Euler Lagrange (EL) equations of motion of the system are written. Then using the equivalence of EL and Newton–Euler (NE) equations, and the DeNOC matrices associated with the velocity constraints of the connecting bodies, the analytical and recursive expressions for the matrices and vectors appearing in the independent dynamic equations of motion are obtained. The analytical expressions allow one to obtain a recursive forward dynamics algorithm not only for rigid body manipulators, as reported earlier, but also for the flexible body manipulators. The proposed simulation algorithm for the flexible link robots is shown to be computationally more efficient and numerically more stable than other algorithms present in the literature. Simulations, using the proposed algorithm, for a two link arm with each link flexible and a Space Shuttle Remote Manipulator System (SSRMS) are presented. Numerical stability aspects of the algorithms are investigated using various criteria, namely, the zero eigenvalue phenomenon, energy drift method, etc. Numerical example of a SSRMS is taken up to show the efficiency and stability of the proposed algorithm. Physical interpretations of many terms associated with dynamic equations of flexible links, namely, the mass matrix of a composite flexible body, inertia wrench of a flexible link, etc. are also presented.


Flexible DeNOC matrices Recursive Simulation Numerical stabile SSRMS 


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  1. 1.
    Angeles, J., Lee, S.K.: The formulation of dynamical equations of holonomic mechanical systems using a natural orthogonal complement. ASME J. Appl. Mech. 55, 243–244 (1988) zbMATHCrossRefGoogle Scholar
  2. 2.
    Ascher, U.M., Pai, D.K., Cloutier, B.P.: Forward dynamics, elimination methods, and formulation stiffness in robot simulation. Int. J. Robotics Res. 16(6), 747–758 (1997) Google Scholar
  3. 3.
    Bauchau, O.A.: Computational scheme for flexible non-linear multibody systems. Multibody Syst. Dyn. 2, 169–225 (1998) zbMATHCrossRefGoogle Scholar
  4. 4.
    Bauchau, O.A.: On the modeling of prismatic joints in flexible multibody systems. Trans. ASME Comput. Methods Appl. Mech. Eng. 181, 87–105 (2000) zbMATHCrossRefGoogle Scholar
  5. 5.
    Bauchau, O.A., Wang, J.: Stability analysis of complex multibody systems. Trans. ASME J. Comput. Nonlinear Dyn. 1, 71–80 (2006) CrossRefGoogle Scholar
  6. 6.
    Baumgarte, J.: Stablization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1, 1–16 (1972) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Book, W.J.: Recursive Lagrangian dynamics of flexible manipulator arms. Int. J. Robotics Res. 3(3), 87–101 (1984) CrossRefGoogle Scholar
  8. 8.
    Cardona, A., Geradin, M.: A beam finite element non-linear theory with finite rotations. Int. J. Numer. Methods Eng. 26(11), 2403–2438 (2005) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Cetinkunt, S., Book, W.J.: Symbolic modeling of flexible manipulators. In: Proc. of IEEE Conf. on Robotics and Automation, pp. 2074–2080 (1987) Google Scholar
  10. 10.
    Chang, B., Nikravesh, P.: An adaptive constraint violation stablisation method for dynamic analysis of mechanical systems. Trans. ASME Appl. Mech. 104, 488–492 (1985) Google Scholar
  11. 11.
    Chedmail, P., Aoustin, Y., Chevallereau, Ch.: Modeling and control of flexible robots. Int. J. Numer. Methods Eng. 32, 1595–1619 (1991) zbMATHCrossRefGoogle Scholar
  12. 12.
    Cloutier, B.P., Pai, D.K., Ascher, U.M.: The formulation stiffness of forward dynamics algorithms and implications for robot simulation. In: Proc. of IEEE Conf. on Robotics and Automation, pp. 2816–2822. Japan, May (1995) Google Scholar
  13. 13.
    Cyril, X.: Dynamics of flexible link manipulators. Dissertation, McGill University, Canada (1988) Google Scholar
  14. 14.
    De Luca, A., Siciliano, B.: Closed-form dynamic model of planar multilink lightweight robots. IEEE Trans. Syst. Man Cybern. 21(4), 826–838 (1991) CrossRefGoogle Scholar
  15. 15.
    D’Eleuterio, G.M.T., Barfooy, T.D.: Just a second, we’d like to go first: a firstorder discretized formulation for structural dynamics. In: Proc. of Fourth Int. Conf. on Dynamics and Controls, pp. 1–24. London (1999) Google Scholar
  16. 16.
    Denavit, J., Hartenberg, R.S.: A kinematic notation for lower-pair mechanisms based on matrices. ASME J. Appl. Mech. 77, 445–450 (1955) Google Scholar
  17. 17.
    Eberhard, P., Schiehlen, W.: Computational dynamics of multibody systems: history, formalisms, and applications. Trans. ASME J. Comput. Nonlinear Dyn. 1, 3–12 (2006) CrossRefGoogle Scholar
  18. 18.
    Ellis, R.E., Ismaeil, O.M., Carmichael, I.H.: Numerical stability of forward dynamics algorithms. In: Proc. of IEEE Conf. on Robotics and Automation, pp. 305–311. France (1992) Google Scholar
  19. 19.
    Haug, E.J.: Computer-Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Boston (1989) Google Scholar
  20. 20.
    Hwang, Y.I.: A new approach for dynamic analysis of flexible manipulator systems. Int. J. Non-Linear Mech. 40, 925–938 (2005) zbMATHCrossRefGoogle Scholar
  21. 21.
    Ider, S.K.: Stability analysis of constraints in flexible multibody systems dynamics. Int. J. Eng. Sci. 28(12), 1277–1290 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Jain, A., Rodriguez, G.: Recursive flexible multibody system dynamics using spatial operators. J. Guid. Controls Dyn. 15, 1453–1466 (1992) zbMATHCrossRefGoogle Scholar
  23. 23.
    Jain, A., Rodriguez, G.: Sensitivity analysis for multibody systems using spatial operators. In: Int. Conf. (VI) on Methods and Models in Automation and Robotics, pp. 30–31. Poland (2000) Google Scholar
  24. 24.
    Jain, A., Rodriguez, G.: Multibody mass matrix sensitivity analysis using spatial operators. Int. J. Multiscale Comput. Eng. 1(2–3) (2003) Google Scholar
  25. 25.
    Kamman, J.W., Huston, R.L.: Dynamics of constrained multibody systems. ASME J. Appl. Mech. 51, 899–903 (1984) zbMATHGoogle Scholar
  26. 26.
    Kane, T.R., Ryan, R.R., Banerjee, A.K.: Dynamics of a cantilever beam attached to a moving base. J. Guid. Control Dyn. 10(2), 139–151 (1987) CrossRefGoogle Scholar
  27. 27.
    Kim, S.S., Haug, E.J.: A recursive formulation for flexible multibody dynamics, Part-I: Open-loop systems. Comput. Methods Appl. Mech. Eng. 71, 293–314 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Li, C.J., Shankar, T.S.: Systematic methods for efficient modeling and dynamics computation of flexible robot manipulator. IEEE Trans. Syst. Man Cybern. 23(1), 77–94 (1993) zbMATHCrossRefGoogle Scholar
  29. 29.
    Martins, J.M., Miguel, A.B., da Costa, J.: Modeling for control of flexible robot manipulators. In: Proc. of Thematic Conf. on Multibody Dynamics. Lisbon, July 1–4, 2003 Google Scholar
  30. 30.
    Meirovitch, L.: Analytical Methods in Vibrations. Macmillan, New York (1967) zbMATHGoogle Scholar
  31. 31.
    Mohan, A., Saha, S.K.: A recursive, numerically stable, and efficient simulation algorithm for serial robots. Multibody Syst. Dyn. 17, 291–319 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Neto, M.A., Ambrosio, J.: Stabilization methods for the integration of DAE in the presence of redundant constraints. Multibody Syst. Dyn. 10(1), 81–105 (2000) CrossRefMathSciNetGoogle Scholar
  33. 33.
    Nikravesh, P.E.: Computer-Aided Analysis of Mechanical Systems. Prentice Hall, New Jersey (1988) Google Scholar
  34. 34.
    Pederson, N.L.: On the formulation of flexible multibody systems with constant mass matrix. Multibody Syst. Dyn. 1, 323–337 (1997) CrossRefGoogle Scholar
  35. 35.
    Saha, S.K.: A decomposition of manipulator inertia matrix. IEEE Trans. Robotics Autom. 13(2), 301–304 (1997) CrossRefGoogle Scholar
  36. 36.
    Saha, S.K.: Dynamic modeling of serial multi-body systems using the decoupled natural orthogonal complement matrices. ASME J. Appl. Mech. 29(2), 986–996 (1999) Google Scholar
  37. 37.
    Saha, S.K.: Analytical expression for the inverted inertia matrix of serial robots. Int. J. Robotics Res. 18(1), 116–124 (1999) Google Scholar
  38. 38.
    Saha, S.K.: Simulation of industrial manipulators based on U D U T decomposition of inertia matrix. Multibody Syst. Dyn. 9, 63–85 (2003) zbMATHCrossRefGoogle Scholar
  39. 39.
    Saha, S.K., Angeles, J.: Dynamics of nonholonomic mechanical systems using a natural orthogonal complement. ASME J. Appl. Mech. 58, 238–243 (1991) zbMATHGoogle Scholar
  40. 40.
    Schiehlen, W.O.: Recent developments in multibody dynamics. J. Mech. Sci. Technol. 19(1), 129–141 (2005) CrossRefMathSciNetGoogle Scholar
  41. 41.
    Shabana, A.A.: Dynamics of flexible bodies using generalized Newton–Euler equation. ASME J. Dyn. Syst. Meas. Control 112(3), 496–503 (1990) zbMATHCrossRefGoogle Scholar
  42. 42.
    Shabana, A.: Flexible multibody dynamics: review of past and recent developments. Multibody Syst. Dyn. 1, 189–222 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, Cambridge (2005) zbMATHGoogle Scholar
  44. 44.
    Sharf, I.: Nonlinear strain measure, shape functions and beam elements for dynamics of flexible beams. Multibody Syst. Dyn. 3, 189–205 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Sharf, I., Damaren, C.: Simulation of flexible link manipulators: basis functions and non-linear terms in motion equations. In: Proc. of IEEE Conf. on Robotics and Automation, pp. 1956–1962. France (1992) Google Scholar
  46. 46.
    Shim, Y.J., Sung, Y.: Stability and four posture control for non-holonomic mobile robots. IEEE Trans. Robotics Autom. 20(1), 148–154 (2004) CrossRefGoogle Scholar
  47. 47.
    Stejskal, V., Valasek, M.: Kinematics and Dynamics of Machinery. M. Dekkar, New York (1996) Google Scholar
  48. 48.
    Strang, G.: Linear Algebra and Its Applications. H.B. Jovanovich Pub., Florida (1980) Google Scholar
  49. 49.
    Theodore, R.J., Ghosal, A.: Comparison of the assumed modes and finite elements modes for flexible multilink manipulators. Int. J. Robotics Res. 14(2), 91–111 (1995) CrossRefGoogle Scholar
  50. 50.
    Thompson, W.T.: Theory of Vibration with Applications. Prentice Hall, London (1988) Google Scholar
  51. 51.
    Usoro, P.B., Nadira, R., Mahil, S.S.: A finite element/Lagrangian approach to modeling light weight flexible manipulators. ASME J. Dyn. Syst. Meas. Control 108, 198–205 (1986) zbMATHGoogle Scholar
  52. 52.
    Walker, M.W., Orin, D.E.: Efficient dynamic computer simulation of robotic mechanisms. ASME J. Dyn. Syst. Meas. Control 104, 205–211 (1982) zbMATHCrossRefGoogle Scholar
  53. 53.
    Wasfy, T.M., Noor, A.K.: Computational strategies for flexible multibody systems. ASME J. Appl. Mech. Rev. 56(6), 553–613 (2003) CrossRefGoogle Scholar
  54. 54.
    Wehage, R.A., Haug, E.J.: Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. ASME J. Mech. Des. 104, 247–255 (1982) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Hi-Tech Robotic Systemz LimitedGurgaonIndia
  2. 2.Department of Mechanical EngineeringIndian Institute of Technology DelhiNew DelhiIndia

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