  This paper presents existence and uniqueness results for a propagative model
of simultaneous impacts that is guaranteed to conserve energy and momentum in
the case of elastic impacts with extensions to perfectly plastic and inelastic
impacts. A corresponding time-stepping algorithm that guarantees conservation
of continuous energy and discrete momentum is developed, also with extensions
to plastic and inelastic impacts. The model is illustrated in simulation using
billiard balls and a two-dimensional legged robot as examples; the latter is
optimized over geometry and gait parameters to achieve unique simultaneous
impacts.
