Research Papers

Comparison of Selected Methods of Handling Redundant Constraints in Multibody Systems Simulations

[+] Author and Article Information
Marek Wojtyra

e-mail: mwojtyra@meil.pw.edu.pl

Janusz Frączek

e-mail: jfraczek@meil.pw.edu.pl
Institute of Aeronautics and Applied Mechanics,
Warsaw University of Technology,
Nowowiejska 24,
00-665 Warsaw, Poland

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNALOF COMPUTATIONALAND NONLINEAR DYNAMICS. Manuscript received October 24, 2011; final manuscript received June 6, 2012; published online July 23, 2012. Assoc. Editor: Jozsef Kovecses.

J. Comput. Nonlinear Dynam 8(2), 021007 (Jul 23, 2012) (9 pages) Paper No: CND-11-1179; doi: 10.1115/1.4006958 History: Received October 24, 2011; Revised June 06, 2012

When redundant constraints are present in a rigid body mechanism, only selected (if any at all) joint reactions can be determined uniquely, whereas others cannot. Analytic criteria and numerical methods of finding joints with uniquely solvable reactions are available. In this paper, the problem of joint reactions solvability is examined from the point of view of selected numerical methods frequently used for handling redundant constraints in practical simulations. Three different approaches are investigated in the paper: elimination of redundant constraints; pseudoinverse-based calculations; and the augmented Lagrangian formulation. Each method is briefly summarized; the discussion is focused on techniques of handling redundant constraints and on joint reactions calculation. In the case of multibody systems with redundant constraints, the rigid body equations of motion are insufficient to calculate some or all joint reactions. Thus, purely mathematical operations are performed in order to find the reaction solution. In each investigated method, the redundant constraints are treated differently, which—in the case of joints with nonunique reactions—leads to different reaction solutions. As a consequence, reactions reflecting the redundancy handling method rather than physics of the system are calculated. A simple example of each method usage is presented, and calculated joint reactions are examined. The paper points out the origins of nonuniqueness of constraint reactions in each examined approach. Moreover, it is shown that one and the same method may lead to different reaction solutions, provided that input data are prepared differently. Finally, it is demonstrated that—in case of joints with solvable reactions—the obtained solutions are unique, regardless of the method used for redundant constraints handling.

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Garcia de Jalon, J., and Bayo, E., 1994, Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge, Springer-Verlag, Berlin.
Haug, E. J., 1989, Computer Aided Kinematics and Dynamics of Mechanical Systems, Allyn and Bacon, Boston.
Nikravesh, P. E., 1988, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Englewood Cliffs, NJ.
Shabana, A. A., 2010, Computational Dynamics, 3rd ed., Wiley, New York.
Frączek, J., and Wojtyra, M., 2011, “On the Unique Solvability of a Direct Dynamics Problem for Mechanisms With Redundant Constraints and Coulomb Friction in Joints,” Mech. Mach. Theory, 46(3), pp.312–334. [CrossRef]
Wojtyra, M., 2005, “Joint Reaction Forces in Multibody Systems With Redundant Constraints,” Multibody Syst. Dyn., 14(1), pp.23–46. [CrossRef]
Wojtyra, M., 2009, “Joint Reactions in Rigid Body Mechanisms With Dependent Constraints,” Mech. Mach. Theory, 44(12), pp.2265–2278. [CrossRef]
Blajer, W., 2004, “On the Determination of Joint Reactions in Multibody Mechanisms,” ASME J. Mech. Des., 126(2), pp.341–350. [CrossRef]
Udwadia, F. E., and Kalaba, R. E., 1996, Analytical Dynamics: A New Approach, Cambridge University Press, Cambridge, England.
Bayo, E., Garcia de Jalon, J., and Serna, M. A., 1988, “A Modified Lagrangian Formulation for the Dynamic Analysis of Constrained Mechanical Systems,” Comput. Methods Appl. Mech. Eng., 71(2), pp.183–195. [CrossRef]
Bayo, E., and Ledesma, R., 1996, “Augmented Lagrangian and Mass-Orthogonal Projection Methods for Constrained Multibody Dynamics,” Nonlinear Dyn., 9(1–2), pp.113–130. [CrossRef]
Baumgarte, J., 1972, “Stabilization of Constraints and Integrals of Motion in Dynamical Systems,” Comput. Methods Appl. Mech. Eng., 1(1), pp.1–16. [CrossRef]
Müller, A., 2006, “A Conservative Elimination Procedure for Permanently Redundant Closure Constraints in MBS-Models With Relative Coordinates,” Multibody Syst. Dyn., 16(4), pp.309–330. [CrossRef]
Müller, A., 2011, “Semialgebraic Regularization of Kinematic Loop Constraints in Multibody System Models,” ASME J. Comput. Nonlinear Dyn., 6(4), p.041010. [CrossRef]
Wehage, R. A., and Haug, E. J., 1982, “Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems,” ASME J. Mech. Des., 104(1), pp.247–255. [CrossRef]
de Falco, D., Pennestri, E., and Vita, L., 2009, “An Investigation of the Influence of Pseudoinverse Matrix Calculations on Multibody Dynamics Simulations by Means of the Udwadia-Kalaba Formulation,” J. Aerosp. Eng., 22(4), pp.365–372. [CrossRef]
Neto, M. A., and Ambrósio, J., 2003, “Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints,” Multibody Syst. Dyn., 10(1), pp.81–105. [CrossRef]
Lawson, C. L., and Hanson, R. J., 1995, Solving Least Squares Problem, SIAM, Philadelphia.
Garcia de Jalon, J., and Gutiérrez-López, M. D., 2012, “Multibody Dynamics With Redundant Constraints and Singular Mass Matrix,” Proc. of the 2nd Joint International Conference on Multibody System Dynamics, Stuttgart, Germany, May29 – June1 .
Kövecses, J., and Piedbœuf, J.-C., 2003, “A Novel Approach for the Dynamic Analysis and Simulation of Constrained Mechanical Systems,“ Proc. of DETC’03 ASME Design Engineering Technical Conferences, Chicago, IL, Sept.2–6 .
Doty, K. L., Melchiorri, C., and Bonivento, C., 1993, “A Theory of Generalized Inverses Applied to Robotics,” Int. J. Robot. Res., 12(1), pp.1–19. [CrossRef]
Nocedal, J., and Wright, S. J., 2006, Numerical Optimization, 2nd ed., Springer, New York.
Ruzzeh, B., and Kövecses, J., 2011, “A Penalty Formulation for Dynamics Analysis of Redundant Mechanical Systems,” ASME J. Comput. Nonlinear Dyn., 6(2), p.021008. [CrossRef]
Dormand, J. R., and Prince, P. J., 1980, “A Family of Embedded Runge-Kutta Formulae,” J. Comput. Appl. Math., 6(1), pp.19–26. [CrossRef]
Wojtyra, M., and Frączek, J., 2011, “Joint Reactions in Overconstrained Rigid or Flexible Body Mechanisms,” Proc. of the ECCOMAS Thematic Conference Multibody Dynamics2011, Brussels, Belgium, July4–7 .


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Fig. 1

A planar overconstrained mechanism

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Fig. 2

Coordinates of body 5 center of mass versus time

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Fig. 3

Unique joint reaction forces (all methods of handling redundant constraints)

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Fig. 4

Nonunique joint reaction forces (redundant constraints elimination method)

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Fig. 5

Nonunique joint reaction forces (pseudoinverse-based method)

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Fig. 6

Nonunique joint reaction forces (augmented Lagrangian method—different initial approximations)

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Fig. 7

Nonunique joint reaction forces (augmented Lagrangian method—different penalty factors)




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