  Robotic manipulation and locomotion often entail nearly-simultaneous
collisions -- such as heel and toe strikes during a foot step -- with outcomes
that are extremely sensitive to the order in which impacts occur. Robotic
simulators commonly lack the accuracy to predict this ordering, and instead
pick one with a heuristic. This discrepancy degrades performance when
model-based controllers and policies learned in simulation are placed on a real
robot. We reconcile this issue with a set-valued rigid-body model which
generates a broad set of physically reasonable outcomes of simultaneous
frictional impacts. We first extend Routh's impact model to multiple impacts by
reformulating it as a differential inclusion (DI), and show that any solution
will resolve all impacts in finite time. By considering time as a state, we
embed this model into another DI which captures the continuous-time evolution
of rigid body dynamics, and guarantee existence of solutions. We finally cast
simulation of simultaneous impacts as a linear complementarity problem (LCP),
and develop an algorithm for tight approximation of the post-impact velocity
set with probabilistic guarantees. We demonstrate our approach on several
examples drawn from manipulation and legged locomotion.
