Modeling and Analysis of Non-unique Behaviors in Multiple Frictional Impacts

Many fundamental challenges in robotics, based in manipulation or locomotion, require making and breaking contact with the environment. Models that address frictional contact must be inherently non-smooth; rigid-body models are especially popular, as they often lead to mathematically and computationally tractable approaches. However, when two or more impacts occur simultaneously, the precise sequencing of impact forces is generally unknown, leading to the potential for multiple possible outcomes. This simultaneity is far from pathological, and occurs in many common robotics applications. In this work, we present an approach to capturing simultaneous frictional impacts, represented as a differential inclusion. Solutions to our model, an extension to multiple contacts of Routh's graphical method, naturally capture the set of potential post-impact velocities. We prove that, under modest conditions, the presented approach is guaranteed to terminate. This is, to the best of our knowledge, the first such guarantee for simultaneous frictional impacts.