Multi-rigid-body contact problems with Coulomb friction: complementarity and equivalent formulations Roberto Andreani DMA - IMECC - Unicamp andreani@ime.unicamp.br Ana Friedlander DMA - IMECC - Unicamp friedlan@ime.unicamp.br Margarida P. Mello DMA - IMECC - Unicamp margarid@ime.unicamp.br Sandra A. Santos DMA - IMECC - Unicamp sandra@ime.unicamp.br Abstract. This work addresses the problem of predicting the possible accelerations and contact forces of a set of rigid, three- dimensional bodies in contact in the presence of Coulomb friction. Using mathematical programming algorithms, the solution of the contact problem is obtained either by means of a mixed nonlinear complementarity problem (original Coulomb law) or of a linear complementarity problem (pyramidal friction law). Sufficient conditions for obtaining equivalence with formulations as bound- constrained minimization problems are provided. Numerical experiments are presented, using the software BOX-QUACAN, developed by the Optimization group of the Applied Mathematics department of the State University of Campinas. We conclude that the approach is effective for solving the model with friction in both variants, with the approximate pyramidal friction law and with the original Coulomb cone. Keywords. Multi-rigid-body contact problem, Coulomb friction, complementarity problem, bound-constrained minimization 1. Introduction Planning the motion of several rigid bodies in contact is a challenging task. Posed as a nonlinear algebraic differential system (see, for instance, Brenan et all (1996) p. 150), after a time discretization, it can be cast as a sequence of problems, one for each time frame. The papers of Pang and Trinkle (1996) and Trinkle et all (1995, 1997) provide the theory and framework for the single time frame problem, concerned with the computation of contact forces and accelerations of a set of rigid three-dimensional bodies, in the presence of friction. The contact model is formulated in its most general version with the Coulomb friction law. Nevertheless, relaxations (pyramid law) or special cases give rise to simpler problems, namely, linear complementarity (LCP) ones. In the aforementioned papers, only the LCP relaxation is addressed in the numerical experiments. Recently, Tzitzouris (2001), in his thesis work, developed a fully implicit time-stepping scheme for the simulation of the multi-rigid-body contact problem with Coulomb friction. A central feature of the algorithm presented therein is the solver employed at each time step for solving the nonlinear complementarity problems. The Newton-Euler equations governing the motion of the objects, together with the dynamic equations of the manipulator, the Signorini condition, the Coulomb friction law and the requirement that the contact forces be compressive compose a mixed nonlinear complementarity problem, shortly denoted by MNCP, in the following format: . 0 ) v , u ( g and 0 ) v , u ( f u , 0 ) v , u ( f , 0 u that such v and u find , : g and : f Given T m n p m n n m n = = ≥ ≥ ℜ ∈ ℜ ∈ ℜ → ℜ ℜ → ℜ + + (1) The versatility of the MNCP format comprises well-known classes of problems: nonlinear equations, nonlinear programming, nonlinear complementarity problems, and variational inequalities, as well as new problem classes, such as extended linear-quadratic programming and general equilibrium models. Roughly speaking, complementarity problems might be addressed either in a straightforward way, by algorithms specifically developed for this kind of problem, or by the minimization of a merit function created to represent the problem, in the sense that it will be zero only at the solutions of the complementarity problem. Ferris and Pang (1996) and references therein provide further details regarding this classification. In this work we adopt the second approach, casting the MNCP as an equivalent bound-constrained minimization problem. The philosophy behind such a strategy is surveyed by Andreani and Friedlander (2002), with a comprehensive discussion of the important features involved, focusing mainly on variational inequalities and related problems. This paper is organized as follows: the equivalent formulation is stated in Section 2. Specific details on the algorithms used to solve the bound-constrained minimization problem are given in Section 3. The numerical experiments are presented in Section 4. First, a brief description of three study cases from MCPLIB is given, used to