Jump to ContentJump to Main Navigation
Show Summary Details

Archive of Mechanical Engineering

The Journal of Committee on Machine Building of Polish Academy of Sciences

4 Issues per year


SCImago Journal Rank (SJR) 2015: 0.178
Source Normalized Impact per Paper (SNIP) 2015: 0.453
Impact per Publication (IPP) 2015: 0.314

Open Access
Online
ISSN
2300-1895
See all formats and pricing

Implementation of A Geometric Constraint Regularization For Multibody System Models

Implementacja Geometrycznej Regularyzacji Więzów w Układach Wieloczłonowych

Andreas Müller
  • University of Michigan - Shanghai Jiao Tong University Joint Institute, 800 Dong Chuan Road, Shanghai, 200240, P.R. China
  • Email:
Published Online: 2014-08-15 | DOI: https://doi.org/10.2478/meceng-2014-0021

Abstract

Redundant constraints in MBS models severely deteriorate the computational performance and accuracy of any numerical MBS dynamics simulation method. Classically this problem has been addressed by means of numerical decompositions of the constraint Jacobian within numerical integration steps. Such decompositions are computationally expensive. In this paper an elimination method is discussed that only requires a single numerical decomposition within the model preprocessing step rather than during the time integration. It is based on the determination of motion spaces making use of Lie group concepts. The method is able to reduce the set of loop constraints for a large class of technical systems. In any case it always retains a sufficient number of constraints. It is derived for single kinematic loops.

Streszczenie

Nadmiarowe więzy w układach wieloczłonowych (MBS) powaŻnie pogarszajĄ wydajność obliczeniowĄ i dokładność numerycznych metod symulacji systemów MBS. Klasycznym podejściem do rozwiĄzania tego problemu jest numeryczna dekompozycja Jakobianu więzów w kolejnych krokach całkowania cyfrowego. Dekompozycje takie sĄ jednak kosztowne obliczeniowo. W artykule zaprezentowano metodę eliminacji, która wymaga tylko pojedynczej dekompozycji na etapie wstępnego przetwarzania modelu, a nie w trakcie integracji czasowej. Metoda jest oparta na wyznaczaniu przestrzeni ruchu przy wykorzystaniu koncepcji grup Liego. Pozwala ona zredukować zbiór więzów pętli dla szerokiej klasy systemów technicznych, przy czym w kaŻdym przypadku zachowuje ona dostatecznĄ liczbę więzów. Metoda została wyprowadzona i zilustrowana dla pojedynczych pętli kinematycznych.

Key words:: multibody systems; constraints; redundancy; Lie groups; screw systems; numerics; dynamics

References

  • [1] Aghili F.: A unified approach for inverse and direct dynamics of constrained multibody systems based on linear projection operator: Applications to control and simulation, IEEE Trans. Robotics, Vol. 21, No. 5, 2005, pp. 834-849.

  • [2] Arabyan A., Wu F.: An improved formulation for constrained mechanical systems, Multibody System Dynamics, Vol. 2, No.1, 1998, pp. 49-69.

  • [3] Brockett R. W.: Robotic manipulators and the product of exponentials formula, Mathematical Theory of Networks and Systems, Lecture Notes in Control and Information Sciences Vol. 58, 1984, pp 120-129.

  • [4] Carricato M., Parenti-Castelli V.: Singularity-free fully-isotropic translational parallel mechanism, Int. Journ. Robot. Res., Vol. 21, No. 2, 2002, pp. 161-174.

  • [5] García de Jalón J., Gutiérrez-López M. D.: Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraint forces, Multibody Systems Dynamics, Springer, Vol. 30, No. 3, Oct. 2013, pp. 311-341. [Web of Science]

  • [6] Hervé J.M.: Analyse Structurelle des Mécanismes par Groupe des Déplacements, Mech. Mach. Theory, vol. 13, 1978, pp. 437-450. [Crossref]

  • [7] Hervé J.M.: Intrinsic formulation of problems of geometry and kinematics of mechanisms, Mech. Mach. Theory, vol. 17, No. 3, 1982, pp. 179-184. [Crossref]

  • [8] Gogu G.: Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations, Eur. J. Mech. A-Solids, Vol. 23, No. 6, 2004, pp. 1021-1039.

  • [9] Gupta K.C.: Kinematic Analysis of Manipulators Using the Zero Reference Position Description, The International Journal of Robotics Research 1986, Vol. 5, No. 2, 1986. [Crossref]

  • [10] Kim SS., Vanderploeg M.J.: QR Decomposition for state space representation of constraint mechanical dynamical systems, ASME Journal of Mechanisms, Transmissions and Automatic Design, 1986, Vol. 108, pp. 183-188.

  • [11] Kim S., Tsai L.: Evaluation of a cartesian parallel manipulator, in J. Lenarčič, F. Thomas (Eds.): Advances in robot kinematics, 2002.

  • [12] Kong X., Gosselin C.: Type synthesis of linear translational parallel manipulators, in J. Lenarčič, F. Thomas (Eds.): Advances in robot kinematics, 2002.

  • [13] Meijaard J.P.: Applications of the Singular Value Decomposition in dynamics, Computer Methods in applied mechanics and Engineering, Vol. 103, 1993, pp. 161-173.

  • [14] Mukherjee R.M., Anderson K.S.: A Logarithmic Complexity Divide-and-Conquer Algorithm for Multi-flexible Articulated Body Dynamics, J. Comput. Nonlinear Dynam. Vol. 2, No. 1, 2006, pp. 10-21.

  • [15] Müller A., Maisser P.: Lie group formulation of kinematics and dynamics of constrained MBS and its application to analytical mechanics, Multibody System Dynamics, Vol. 9, 2003, pp. 311-352.

  • [16] Müller A.: A conservative elimination procedure for permanently redundant closure constraints in MBS-models with relative coordinates, Multibody Systems Dynamics, Springer, Vol. 16, No. 4, Nov. 2006, pp. 309-330.

  • [17] Müller A.: Semialgebraic Regularization of Kinematic Loop Constraints in Multibody System Models, ASME Trans., Journal of Computational and Nonlinear Dynamics, Vol. 6, No 4, 2011.

  • [18] Murray R.M., Li Z., and Sastry S.S.: A Mathematical Introduction to Robotic Manipulation, CRC Press Boca Raton, 1994.

  • [19] Neto A.M., Ambrosio J.: Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints, Multibody System Dynamics, Vol. 19, No.1, 2003, pp. 311-352.

  • [20] Park F.C.: Computational Aspects of the Product-of-Exponentials Formula for Robot Kinematics, IEEE Trans. Aut. Contr, Vol. 39, No. 3, 1994, 643-647.

  • [21] Ploen S.R.: Geometric algorithms for the dynamics and control of multibody systems, Ph.D. Thesis, Department of Mechanical and Aerospace Engineering, University of California, Irvine, 1997.

  • [22] Ploen S.R., Park F.C.: A Lie group formulation of the dynamics of cooperating robot systems, Rob. and Auton. Sys., Vol. 21, 1997, pp. 279-287.

  • [23] Selig J.: Geometric Fundamentals of Robotics (Monographs in Computer Science Series), Springer-Verlag New York, 2005.

  • [24] Sing R.P., Likings P.W.: Singular Value decomposition for constrained dynamical systems, Journal of Applied Mechanisms, 1985, 52, pp. 943-948.

  • [25] Wojtyra M.: Joint Reaction Forces in Multibody Systems with Redundant Constraints, Multi-body System Dynamics, Vol. 14, No.1, 2005, 14: 23-46.

  • [26] Wojtyra M., Fraczek J.: Solvability of reactions in rigid multibody systems with redundant nonholonomic constraints, Multibody Systems Dynamics, Springer, Vol. 30, No. 2, Aug. 2013, pp. 153-171. [Web of Science]

About the article

Published Online: 2014-08-15


Citation Information: Archive of Mechanical Engineering, ISSN (Online) 2300-1895, DOI: https://doi.org/10.2478/meceng-2014-0021. Export Citation

© 2014 Andreas Müller. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Comments (0)

Please log in or register to comment.
Log in