Time-stepping for three-dimensional rigid body dynamics

Traditional methods for simulating rigid body dynamics involves determining the current contact arrangement (e,g.. each contact is either a "rolling'' or "sliding'' contact). The development of this approach is most clearly Seen in the work of Haug et al. [Mech. Machine Theory 21 (1986) 401-425] and Pfeiffer and Glocker [Multibody Dynamics with Unilateral Contacts (Wiley, 1996)]. However, there has been a controversy about the status of rigid body dynamics as a theory, due to simple problems in the area which do not appear to have solutions; the most famous, if not the earliest is due to Paul Painleve [C.R. Acad. Sci. Paris 121 (1895) 112-115]. Recently, a number of time-stepping methods have been developed to overcome these difficulties. These time-stepping methods use intergrals of the forces over time-steps, rather than the actual forces. This allows impulsive forces without the need for a separate formulation, or special procedures, to cover this case. The newest of these methods are developed in terms of complementary problems. The complementarity problems that define the time-stepping procedure are solvable unlike previous methods for simulating rigid body dynamics with friction. Proof of the existence of solutions to the continuous problem can be shown in the sense of measure differential inclusions in terms of these methods. In this paper, a number of these variants will be discussed, and their essential properties proven. (C) 1999 Elsevier Science B. V. All rights reserved.