Rigid-Body Dynamics with Friction and Impact

Abstract

Rigid-body dynamics with unilateral contact is a good approximation for a wide range of everyday phenomena, from the operation of car brakes to walking to rock slides. It is also of vital importance for simulating robots, virtual reality, and realistic animation. However, correctly modeling rigid-body dynamics with friction is difficult due to a number of discontinuities in the behavior of rigid bodies and the discontinuities inherent in the Coulomb friction law. This is particularly crucial for handling situations with large coefficients of friction, which can result in paradoxical results known at least since Painlevé [C. R. Acad. Sci. Paris, 121 (1895), pp. 112-115]. This single example has been a counterexample and cause of controversy ever since, and only recently have there been rigorous mathematical results that show the existence of solutions to his example. The new mathematical developments in rigid-body dynamics have come from several sources: "sweeping processes" and the measure differential inclusions of Moreau in the 1970s and 1980s, the variational inequality approaches of Duvaut and J.-L. Lions in the 1970s, and the use of complementarity problems to formulate frictional contact problems by Lötstedt in the early 1980s. However, it wasn't until much more recently that these tools were finally able to produce rigorous results about rigid-body dynamics with Coulomb friction and impulses.

GPU-Based Parallel Computing for the Simulation of Complex Multibody Systems with Unilateral and Bilateral Constraints: An Overview

“…The discretization requires the solution of a complementarity problem at each time step, and it has been demonstrated that it converges to the solution to the original differential inclusion for h ! 0 [15,19]. Moreover, the differential inclusion can be solved in terms of vector measures: forces can be impulsive and velocities can have discontinuities, thus supporting also the case of impacts and giving a weak solution to otherwise unsolvable situations like in the Painlevé paradox [20].…”

A State Transition Diagram for Simultaneous Collisions with Application in Billiard Shooting

“…Previous methods either sequence them into two-body collisions [5] by order of normal approach velocity, or set up linear complementarity conditions at all contacts [16,2]. High-speed photographs of such collisions nonetheless show that multiple objects are simultaneously in contact rather than two at a time [17].…”

“…Stewart [17] pointed out that one difficulty with multiple contacts lies in the lack of a continuous impact law. Observations seem to suggest, during simultaneous collisions, the involved objects may have broken and reestablished contacts multiple times.…”

“…1 We will also see that the outcome of impact is affected by the ratios of stiffnesses at the contact points. This concept of "relative stiffness" is cited in [17] as missing from the current impact literature. Section 3 applies the state transition diagram to model billiard shooting with a cue stick.…”

Simulating Multibody Dynamics With Rough Contact Surfaces and Run-in Wear

“…These methods are robust, easily to implement and can be used for dynamical systems with large number of contacts. An important contribution to the time-stepping methods was made by Monteiro-Marques [18], Paoli and Schatzman [21,22], who studied the convergence of the solution, and also by Stewart and Trinkle [26,27], Anitesca and Potra [1] and others.…”