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A system of rigid bodies with multiple simultaneous contacts is considered in this paper. The problem is to predict the velocities of the bodies and the frictional forces acting on the simultaneous multi-contacts. This paper presents a numerical method based on an extension of an explicit time-stepping scheme and an application of the differential inclusion process introduced by J. J. Moreau. From the differential kinematic analysis of contacts, we derive a set of transfer equations in the velocity based time-stepping formulation. In applying the Gauss-Seidel iterative scheme, the transfer equations are combined with the Signorini conditions and Coulomb's friction law. The contact forces are properly resolved in each iteration, without resorting to any linearization of the friction cone. Numerical examples of the performance of the proposed method are compared with an acceleration-based scheme using linear complementarity techniques.