Convergence of a Class of Semi-Implicit Time-Stepping Schemes for Nonsmooth Rigid Multibody Dynamics

In this work we present a framework for the convergence analysis in a measure differential inclusion sense of a class of time-stepping schemes for multibody dynamics with contacts, joints, and friction. This class of methods solves one linear complementarity problem per step and contains the semi-implicit Euler method, as well as trapezoidal-like methods for which second-order convergence was recently proved under certain conditions. By using the concept of a reduced friction cone, the analysis includes, for the first time, a convergence result for the case that includes joints. An unexpected intermediary result is that we are able to define a discrete velocity function of bounded variation, although the natural discrete velocity function produced by our algorithm may have unbounded variation.

[1]  W. L. Jorgensen,et al.  Comparison of simple potential functions for simulating liquid water , 1983 .

[2]  Jong-Shi Pang,et al.  A semi‐implicit time‐stepping model for frictional compliant contact problems , 2004 .

[3]  Roy Featherstone,et al.  Robot Dynamics Algorithms , 1987 .

[4]  Jeffrey C. Trinkle,et al.  Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with coulomb friction , 1996, Math. Program..

[5]  D. Stewart,et al.  AN IMPLICIT TIME-STEPPING SCHEME FOR RIGID BODY DYNAMICS WITH INELASTIC COLLISIONS AND COULOMB FRICTION , 1996 .

[6]  M. Kanat Camlibel,et al.  Conewise Linear Systems: Non-Zenoness and Observability , 2006, SIAM J. Control. Optim..

[7]  Henk Nijmeijer,et al.  Periodic motion and bifurcations induced by the Painlevé paradox , 2002 .

[8]  A. Wallqvist,et al.  Molecular Models of Water: Derivation and Description , 2007 .

[9]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[10]  E. J. Haug,et al.  Computer aided kinematics and dynamics of mechanical systems. Vol. 1: basic methods , 1989 .

[11]  Dirk Helbing,et al.  Simulating dynamical features of escape panic , 2000, Nature.

[12]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[13]  D. Stewart.  Existence of solutions to rigid body dynamics and the Painlevé paradoxes , 1997 .

[14]  M. Anitescu,et al.  Formulating Dynamic Multi-Rigid-Body Contact Problems with Friction as Solvable Linear Complementarity Problems , 1997 .

[15]  Michel Saint Jean,et al.  The non-smooth contact dynamics method , 1999 .

[16]  D. Stewart,et al.  Time-stepping for three-dimensional rigid body dynamics , 1999 .

[17]  Bruce Randall Donald,et al.  On the motion of compliantly-connected rigid bodies in contact. II. A system for analyzing designs for assembly , 1990, Proceedings., IEEE International Conference on Robotics and Automation.

[18]  D. Stewart.  Reformulations of Measure Differential Inclusions and Their Closed Graph Property , 2001 .

[19]  J. Moreau.  Numerical aspects of the sweeping process , 1999 .

[20]  M. Marques,et al.  Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction , 1993 .

[21]  F. Pfeiffer,et al.  Multiple impacts with friction in rigid multibody systems , 1995 .

[22]  B. Brogliato.  Nonsmooth Impact Mechanics: Models, Dynamics and Control , 1996 .

[23]  Mihai Anitescu,et al.  A constraint‐stabilized time‐stepping approach for rigid multibody dynamics with joints, contact and friction , 2004 .

[24]  Pierre E. Dupont,et al.  Analysis of Rigid-Body Dynamic Models for Simulation of Systems With Frictional Contacts , 2001 .

[25]  W. K. Terry,et al.  Advanced Core Design And Fuel Management For Pebble-Bed Reactors , 2004 .

[26]  M. Anitescu,et al.  A Time-stepping Method for Stii Multibody Dynamics with Contact and Friction ‡ , 2022 .

[27]  D. Goulding,et al.  Size selectivity of narrow pores. , 2000, Physical review letters.

[28]  David Baraff,et al.  Issues in computing contact forces for non-penetrating rigid bodies , 1993, Algorithmica.

[29]  D. Stewart.  Convergence of a Time‐Stepping Scheme for Rigid‐Body Dynamics and Resolution of Painlevé's Problem , 1998 .

[30]  Mihai Anitescu,et al.  A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact, joints, and friction , 2006 .

[31]  B. Brogliato,et al.  Numerical simulation of finite dimensional multibody nonsmooth mechanical systems , 2001 .

[32]  Jong-Shi Pang,et al.  Linear Complementarity Systems: Zeno States , 2005, SIAM J. Control. Optim..

[33]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[34]  G. Bartels,et al.  The effect of contact torques on porosity of cohesive powders , 2004, cond-mat/0403110.

[35]  J. Moreau,et al.  Unilateral Contact and Dry Friction in Finite Freedom Dynamics , 1988 .

[36]  J. Trinkle,et al.  On Dynamic Multi‐Rigid‐Body Contact Problems with Coulomb Friction , 1995 .

[37]  G. Bartels,et al.  Pore Stabilization in Cohesive Granular Systems , 2002, cond-mat/0206572.

[38]  Jong-Shi Pang,et al.  Differential variational inequalities , 2008, Math. Program..

[39]  C. Mariotti,et al.  Numerical study of rock and concrete behaviour by discrete element modelling , 2000 .

[40]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[41]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[42]  David E. Stewart,et al.  Rigid-Body Dynamics with Friction and Impact , 2000, SIAM Rev..

[43]  M. Saraniti,et al.  The Simulation of Ionic Charge Transport in Biological Ion Channels: An Introduction to Numerical Methods , 2006 .

[44]  Roux,et al.  Force Distributions in Dense Two-Dimensional Granular Systems. , 1996, Physical review letters.

[45]  M.K. Camlibel,et al.  On the Zeno behavior of linear complementarity systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[46]  Mihai Anitescu,et al.  Optimization-based simulation of nonsmooth rigid multibody dynamics , 2006, Math. Program..

[47]  Patrick G. Xavier,et al.  The Umbra simulation framework as applied to building HLA federates , 2002, Proceedings of the Winter Simulation Conference.

[48]  M. Anitescu,et al.  Formulating Three-Dimensional Contact Dynamics Problems , 1996 .

[49]  Vijay Kumar,et al.  Convergence of Time-Stepping Method for Initial and Boundary-Value Frictional Compliant Contact Problems , 2005, SIAM J. Numer. Anal..