A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact, joints, and friction

SUMMARY We present a hard constraint, linear complementarity based, method for the simulation of stiff multibody dynamics with contact, joints and friction. The approach uses a linearization of the modified trapezoidal method, incorporates a Poisson restitution model at collision, and solves only one linear complementarity problem per time step when no collisions are encountered. We prove that, under certain assumptions, the method has order two, a fact that is also demonstrated by our numerical simulations. For the unconstrained (ODE) case, the method achieves second-order convergence and absolute stability while solving only one linear system per step. When we use a special approximation of the Jacobian matrix for the case where the stiff forces originate in springs and dampers attached to two points in the system, the linear complementarity problem can be solved for any value of the time step and numerical simulation demonstrate that the method is stiffly stable. The method was implemented in UMBRA, an industrial-grade virtual prototyping software. Subject Index: 65K10, 90C33. Copyright c 2000 John Wiley & Sons, Ltd.

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

[2]  M. Marques.  Differential Inclusions in Nonsmooth Mechanical Problems , 1993 .

[3]  F. Potra,et al.  On the Numerical Solution of Euler-Lagrange Equations∗ , 1991 .

[4]  Yu Wang,et al.  On the inconsistency of rigid-body frictional planar mechanics , 1988, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

[5]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

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

[7]  Olof Friberg,et al.  Computer aided kinematics and dynamics of mechanical systems, by E. J. Haug, Allyn and Bacon, Boston, 1989. No. of pages: 511. Price: £47.10 (hardback), ISBN 0-205-11669-8; £18.95 (international student edition), ISBN 0-205-12097-0 , 1990 .

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

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

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

[11]  D. Stewart,et al.  Dynamics, Friction, And Complementarity Problems , 1995 .

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

[13]  Jeffrey C. Trinkle,et al.  Toward Sensorless Acquisition of Multiple Contact Points Between Planar Parts , 2005, Multi-point Interaction with Real and Virtual Objects.

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

[15]  Friedrich Pfeiffer,et al.  Multibody Dynamics with Unilateral Contacts , 1996 .

[16]  E. A. Repetto,et al.  Finite element analysis of nonsmooth contact , 1999 .

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

[18]  Florian A. Potra,et al.  Differential-Geometric Techniques for Solving Differential Algebraic Equations , 1990 .

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

[20]  Florian A. Potra,et al.  Potra The Kantorovich Theorem and interior point methods , 2004 .

[21]  George Vanecek,et al.  Building Simulations for Virtual Environments , 1996 .

[22]  J. A. Teixeira de Freitas,et al.  HYBRID-TREFFTZ FINITE ELEMENT FORMULATION FOR SIMULATION OF SINGULAR STRESS FIELDS , 1996 .

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

[24]  R. Tapia,et al.  Optimal Error Bounds for the Newton–Kantorovich Theorem , 1974 .

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

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

[27]  Michael C. Ferris,et al.  Interfaces to PATH 3.0: Design, Implementation and Usage , 1999, Comput. Optim. Appl..

[28]  G. Stavroulakis.  Multibody Dynamics with Unilateral Contacts by Friedrich Pfeiffer and Christoph Glocker, Wiley, New York, 1996 , 1998 .

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

[30]  James F. Cremer,et al.  Isaac: Building Simulations for Virtual Environments , 1995 .

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

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

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

[34]  J. Moreau.  Standard Inelastic Shocks and the Dynamics of Unilateral Constraints , 1985 .

[35]  Vijay Kumar,et al.  Design of part feeding and assembly processes with dynamics , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[36]  王东东,et al.  Computer Methods in Applied Mechanics and Engineering , 2004 .

[37]  Kendall E. Atkinson,et al.  The numerical solution of a non-linear boundary integral equation on smooth surfaces , 1994 .

[38]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.