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Journal of Computational and Nonlinear Dynamics | Volume 4 | Issue 4 | Research Paper
Research Papers

A Recursive Hybrid Time-Stepping Scheme for Intermittent Contact in Multi-Rigid-Body Dynamics

[+] Author and Article Information
Kishor D. Bhalerao1

Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180bhalek@rpi.edu

Kurt S. Anderson

Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180anderk5@rpi.edu

Jeffrey C. Trinkle

Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180trink@cs.rpi.edu

1

Corresponding author.

J. Comput. Nonlinear Dynam 4(4), 041010 (Aug 25, 2009) (11 pages) doi:10.1115/1.3192132 History: Received August 22, 2008; Revised October 23, 2008; Published August 25, 2009
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This paper describes a novel method for the modeling of intermittent contact in multi-rigid-body problems. We use a complementarity based time-stepping scheme in Featherstone’s divide and conquer framework to efficiently model the unilateral and bilateral constraints in the system. The time-stepping scheme relies on impulse-based equations and does not require explicit collision detection. A set of complementarity conditions is used to model the interpenetration constraint and a linearized friction cone is used to yield a linear complementarity problem. The divide and conquer framework ensures that the size of the resulting mixed linear complementarity problem is independent of the number of bilateral constraints in the system. This makes the proposed method especially efficient for systems where the number of bilateral constraints is much greater than the number of unilateral constraints. The method is demonstrated by applying it to a falling 3D double pendulum.
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References

1
Lötstedt, P., 1982, “Mechanical Systems of Rigid Bodies Subject to Unilateral Constraints,” SIAM J. Appl. Math., 42 (2), pp. 281–296.  [CrossRef]
2
Pereira, M. S., and Nikravesh, P., 1996, “Impact Dynamics of Multibody Systems With Frictional Contact Using Joint Coordinates and Canonical Equations of Motion,” Nonlinear Dyn., 9 , pp. 53–71.  [CrossRef]
3
Pfeiffer, F., 2003, “The Idea of Complementarity in Multibody Dynamics,” Arch. Appl. Mech., 72 (11–12), pp. 807–816.
4
Trinkle, J., Pang, D., Sudarsky, S., and Lo, G., 1997, “On Dynamic Multi-Rigid-Body Contact Problems With Coulomb Friction,” Z. Angew. Math. Mech., 77 (4), pp. 267–279.  [CrossRef]
5
Barhorst, A., 1998, “On Modelling Variable Structure Dynamics of Hybrid Parameter Multibody Systems,” J. Sound Vib., 209 (4), pp. 571–592.  [CrossRef]
6
Pfeiffer, F., and Glocker, C., 2000, "Multibody Dynamics With Unilateral Constraints" (CISM Courses and Lectures Vol. 421 ), International Centre for Mechanical Sciences, Springer, New York.
7
Pfeiffer, F. G., 2001, “Applications of Unilateral Multibody Dynamics,” Philos. Trans. R. Soc. London, Ser. A, 359 , pp. 2609–2628.  [CrossRef]
8
Moreau, J. J., 1988, “Unilateral Contacts and Dry Friction in Finite Freedom Dynamics,” "Non-Smooth Mechanics and Applications" (CISM Courses and Lectures Vol. 302 ), Springer, Berlin.
9
Baraff, D., 1993, “Issues in Computing Contact Forces for Non-Penetrating Rigid Bodies,” Algorithmica, 10 (2–4), pp. 292–352.  [CrossRef]
10
Stewart, D. E., and Trinkle, J. C., 1996, “An Implicit Time-Stepping Scheme for Rigid Body Dynamics With Inelastic Collisions and Coulomb Friction,” Int. J. Numer. Methods Eng., 39 , pp. 2673–2691.  [CrossRef]
11
Glocker, C., and Pfeiffer, F., 1995, “Multiple Impacts With Friction in Multibody Systems,” Nonlinear Dyn., 7 , pp. 471–497.  [CrossRef]
12
Pfeiffer, F., and Glocker, C., 1996, "Multibody Dynamics With Unilateral Contacts" (Wiley Series in Nonlinear Sciences ), Wiley, New York.
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Pfeiffer, F. G., 2003, “Unilateral Multibody Dynamics,” Int. J. Multiscale Comp. Eng., 1 , pp. 311–326.  [CrossRef]
14
Featherstone, R., 1999, “A Divide-and-Conquer Articulated Body Algorithm for Parallel O(log(n)) Calculation of Rigid Body Dynamics. Part 1: Basic Algorithm,” Int. J. Robot. Res., 18 (9), pp. 867–875.  [CrossRef]
15
Featherstone, R., 1999, “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 (9), pp. 876–892.  [CrossRef]
16
Mukherjee, R. M., and Anderson, K. S., 2007, “Orthogonal Complement Based Divide-and-Conquer Algorithm for Constrained Multibody Systems,” Nonlinear Dyn., 48 (1–2), pp. 199–215.  [CrossRef]
17
Cottle, R. W., Pang, J. S., and Stone, R. E., 1992, "The Linear Complementarity Problem", Academic, New York.
18
Haug, E. J., Wu, S. C., and Yang, S. M., 1986, “Dynamics of Mechanical Systems With Coulomb Friction, Stiction, Impact and Constraint Addition-Deletion—I Theory,” Mech. Mach. Theory, 21 (5), pp. 401–406.  [CrossRef]
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Schwertassek, R., 1994, “Reduction of Multibody Simulation Time by Appropriate Formulation of the Dynamics System Equations,” "Computer-Aided Analysis of Rigid and Flexible Mechanical Systems", NATO ASI, pp. 447–482.
20
Anitescu, M., 2003, “A Fixed Time-Step Approach for Multi-Body Dynamics With Contact and Friction,” International Conference on Intelligent Robots and Systems, Las Vegas, NV, 27–31 Oct. 2003, Vol. 4 , pp. 3725–3731.
21
Mirtich, B., 1996, “Impulse-Based Dynamic Simulation of Rigid Body Systems,” Ph.D. thesis, Department of Electrical Engineering and Computer Science, University of California, Berkeley, Berkeley, CA.
22
Kane, T. R., and Levinson, D. A., 1985, "Dynamics: Theory and Application", McGraw-Hill, New York.
23
Roberson, R. E., and Schwertassek, R., 1988, "Dynamics of Multibody Systems", Springer-Verlag, Berlin.
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Ferris, M. C., and Munson, T. S., 1999, “Interfaces to Path 3.0: Design, Implementation and Usage,” Comput. Optim. Appl., 12 (1–3), pp. 207–227.  [CrossRef]
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Mukherjee, R. M., and Anderson, K. S., 2007, “Efficient Methodology for Multibody Simulations With Discontinuous Changes in System Definition,” Multibody Syst. Dyn., 18 (2), pp. 145–168.  [CrossRef]
26
http://www.autolev.com.
27
Schaechter, D. B., Levinson, D. A., and Kane, T. R., 1982, "AUTOLEV User’s Manual", On-Line Dynamics Inc., Sunnyvale, CA.

Figures

Grahic Jump Location
+Proximal points between two convex bodies
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Proximal points between two convex bodies
Figure 1

Proximal points between two convex bodies

Grahic Jump Location
+Polygonal approximation of the circular friction cone (10)
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Polygonal approximation of the circular friction cone (10)
Figure 2

Polygonal approximation of the circular friction cone (10)

Grahic Jump Location
+The hierarchic assembly and disassembly processes using binary-tree structure
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The hierarchic assembly and disassembly processes using binary-tree structure
Figure 3

The hierarchic assembly and disassembly processes using binary-tree structure

Grahic Jump Location
+Representative bodies of a multibody system
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Representative bodies of a multibody system
Figure 4

Representative bodies of a multibody system

Grahic Jump Location
+Assembly process for a unilateral constraint on body k
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Assembly process for a unilateral constraint on body k
Figure 5

Assembly process for a unilateral constraint on body k

Grahic Jump Location
+Drift in x and y coordinates of the center of mass of the double pendulum for μ=0
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Drift in x and y coordinates of the center of mass of the double pendulum for μ=0
Figure 7

Drift in x and y coordinates of the center of mass of the double pendulum for μ=0

Grahic Jump Location
+Comparison between the time-stepping scheme and AUTOLEV model
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Comparison between the time-stepping scheme and AUTOLEV model
Figure 8

Comparison between the time-stepping scheme and AUTOLEV model

Grahic Jump Location
+x and z coordinates of point P1 for a planar (xz) double pendulum
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x and z coordinates of point P1 for a planar (xz) double pendulum
Figure 9

x and z coordinates of point P1 for a planar (xz) double pendulum

Grahic Jump Location
+Unilateral constraint is defined between the xy plane and point P1 on body A
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Unilateral constraint is defined between the xy plane and point P1 on body A
Figure 6

Unilateral constraint is defined between the xy plane and point P1 on body A

Grahic Jump Location
+Elastic contact between point P1 and xy plane, ϵN=0.7
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Elastic contact between point P1 and xy plane, ϵN=0.7
Figure 10

Elastic contact between point P1 and xy plane, ϵN=0.7

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