Nonlinear Dynamics

, Volume 48, Issue 1–2, pp 199–215 | Cite as

Orthogonal Complement Based Divide-and-Conquer Algorithm for constrained multibody systems

Original Article

Abstract

A new algorithm, Orthogonal Complement based Divide-and-Conquer Algorithm (O-DCA), is presented in this paper for calculating the forward dynamics of constrained multi-rigid bodies including topologies involving single or coupled closed kinematic loops. The algorithm is exact and noniterative. The constraints are imposed at the acceleration level by utilizing a kinematic relation between the joint motion subspace (or partial velocities) and its orthogonal complement. Sample test cases indicate excellent constraint satisfaction and robust handling of singular configurations. Since the present algorithm does not use either a reduction or augmentation approach in the traditional sense for imposing the constraints, it does not suffer from the associated problems for systems passing through singular configurations. The computational complexity of the algorithm is expected to be O(n+m) and O(log(n+m)) for serial and parallel implementation, respectively, where n is the number of generalized coordinates and m is the number of independent algebraic constraints.

Keywords

Closed kinematic loops Orthogonal complement Singular configurations Logarithmic computational complexity Divide-and-Conquer 

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References

  1. 1.
    Hooker, W.W., Margulies, G.: The dynamical attitude equations for an n-body satellite. J. Astronaut. Sci. 7(4), 123–128 (1965)MathSciNetGoogle Scholar
  2. 2.
    Luh, J.S.Y., Walker, M.W., Paul, R.P.C.: On-line computational scheme for mechanical manipulators. J. Dynam. Syst. Meas. Control 102, 69–76 (1980)MathSciNetGoogle Scholar
  3. 3.
    Walker, M.W., Orin, D.E.: Efficient dynamic computer simulation of robotic mechanisms. J. Dynam. Syst. Meas. Control 104, 205–211 (1982)MATHGoogle Scholar
  4. 4.
    Rosenthal, D.E., Sherman, M.A.: High performance multibody simulations via symbolic equation manipulation and Kane's method. J. Astronaut. Sci. 34(3), 223–239 (1986)Google Scholar
  5. 5.
    Neilan, P.E.: Efficient computer simulation of motions of multibody systems. PhD Thesis, Stanford University (1986)Google Scholar
  6. 6.
    Vereshchagin, A.F.: Computer simulation of the dynamics of complicated mechanisms of robot-manipulators. Eng. Cybernet. 12(6), 65–70 (1974)Google Scholar
  7. 7.
    Armstrong, W.W.: Recursive solution to the equations of motion of an n-link manipulator. In: Fifth World Congress on the Theory of Machines and Mechanisms, Vol. 2, Montreal, Canada, pp 1342–1346 (1979)Google Scholar
  8. 8.
    Featherstone, R.: The calculation of robotic dynamics using articulated body inertias. Int. J. Rob. Res. 2(1), 13–30 (1983)Google Scholar
  9. 9.
    Brandl, H., Johanni, R., Otter, M.: A very efficient algorithm for the simulation of robots and similar multibody systems without inversion of the mass matrix. In: IFAC/IFIP/IMACS Symposium, Vienna, Austria (1986)Google Scholar
  10. 10.
    Bae, D.S., Haug, E.J.: A recursive formation for constrained mechanical system dynamics: Part I, Open loop systems. Mech. Struct. Machines 15(3), 359–382 (1987)Google Scholar
  11. 11.
    Rosenthal, D.E.: An order n formulation for robotic systems. J. Astronaut. Sci. 38(4), 511–529 (1990)Google Scholar
  12. 12.
    Anderson, K.S.: Recursive derivation of explicit equations of motion for efficient dynamic/control simulation of large multibody systems. PhD Thesis, Stanford University, Stanford, CA(1990)Google Scholar
  13. 13.
    Kreutz-Delgado, K., Jain, A., Rodriguez, G.: Recursive formulation of operational space control. Int. J. Rob. Res. 11(4), 320–328 (1992)Google Scholar
  14. 14.
    Jain, A.: Unified formulation of dynamics for serial rigid multibody systems. J. Guid. Control Dynam. 14(3), 531–542 (1991)MATHGoogle Scholar
  15. 15.
    Featherstone, R.: Robot Dynamics Algorithms. Kluwer Academic, New York (1987)Google Scholar
  16. 16.
    Bae, D.S., Haug, E.J.: A recursive formation for constrained mechanical system dynamics: Part II, Closed loop systems. Mech. Struct. Machines 15(4), 481–506 (1987)Google Scholar
  17. 17.
    Wehage, R.A.: Application of matrix partitioning and recursive projection to O(n) solution of constrained equations of motion. In: ASME Design Engineering Division, DE, Vol. 15-2, pp. 221–230 (1988)Google Scholar
  18. 18.
    Anderson, K.S., Critchley, J.H.: Improved order-n performance algorithm for the simulation of constrained multi-rigid-body systems. Multibody Syst. Dynam. 9, 185–212 (2003)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Critchley, J.H., Anderson, K.S.: A generalized recursive coordinate reduction method for multibody system dynamics. J. Multiscale Comput. Eng. 1(2 & 3), 181–200 (2003)CrossRefGoogle Scholar
  20. 20.
    Lötstedt, P.: On a penalty function method for the simulation of mechanical systems subject to constraints. Report TRITA-NA-7919, Royal Institute of Technology, Stockholm, Sweden (1979)Google Scholar
  21. 21.
    Lötstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42(2), 281–296 (1982)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamic systems. Comput. Method Appl. Mech. Eng. 1, 1–16 (1972)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Baumgarte, J.W.: A new method for stabilization of holonomic contraints. J. Appl. Mech. 50, 869–870 (1983)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Park, K.C., Chiou, J.C.: Stabilization of computational procedures for constrained dynamical systems. J. Guid. Control Dynam. 11(4), 365–370 (1988)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Bayo, E., Garcia de Jalon, J., Serna, M.A.: Modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems. Comput. Methods Appl. Mech. Eng. 71(2), 183–195 (1988)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Anderson, K.S.: An order-n formulation for motion simulation of general constrained multi-rigid-body systems. Comput. Struct. 43(3), 565–572 (1992)CrossRefMATHGoogle Scholar
  27. 27.
    Stejskal, V., Valášek, M.: Kinematics and Dynamics of Machinery. Marcel Dekker, New York (1996)Google Scholar
  28. 28.
    Saha, S.K., Schiehlen, W.O.: Recursive kinematics and dynamics for parallel structured closed-loop multibody systems. Mech. Struct. Machines 29(2), 143–175 (2001)CrossRefGoogle Scholar
  29. 29.
    Fijany, A., Sharf, I., D'Eleuterio, G.M.T.: Parallel O(log n) algorithms for computation of manipulator forward dynamics. IEEE Trans. Rob. Autom. 11(3), 389–400 (1995)CrossRefGoogle Scholar
  30. 30.
    Featherstone, R.: A divide-and-conquer articulated body algorithm for parallel O(log(n)) calculation of rigid body dynamics. Part 1: Basic algorithm. Int. J. Rob. Res. 18(9), 867–875 (1999)CrossRefGoogle Scholar
  31. 31.
    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. Rob. Res. 18(9), 876–892 (1999)CrossRefGoogle Scholar
  32. 32.
    Kim S.S., VanderPloeg M.J.: Generalized and efficient method for dynamic analysis of mechanical systems using velocity transforms. J. Mech. Trans. Autom. Des. 108(2), 1986, 176–182.CrossRefGoogle Scholar
  33. 33.
    Nikravesh P.E.: Systematic reduction of multibody equations to a minimal set. Int. J. Non-Linear Mech. 25(2–3), 143–151 (1990)CrossRefMATHGoogle Scholar
  34. 34.
    Kane T.R., Levinson D.A.: Dynamics: Theory and Application. McGraw-Hill, New York (1985)Google Scholar

Copyright information

© Springer Science + Business Media, B.V. 2006

Authors and Affiliations

  1. 1.Computational Dynamics Laboratory, Department of Mechanical Aerospace and Nuclear EngineeringRensselaer Polytechnic InstituteTroyUSA

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