We present a differential-algebraic formulation with switching constraints to model the nonsmooth dynamics of robotic systems subject to changing constraints and multiple impacts. The formulation combines a single structurally simple governing equation, a set of switching kinematic constraints, and the plastic impact law, to represent the dynamics of robots that interact with their environment. The main contribution of this formulation is a novel algorithmic impact resolution method which provides an explicit solution to the classical plastic impact law in the case of multiple simultaneous impacts. This method serves as an alternative to prior linear-complementarity-based formulations which offer an implicit impact resolution through iterative calculation. We demonstrate the utility of the proposed method by simulating the locomotion of a planar anthropometric biped.