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

## TL;DR

This paper introduces a differential inclusion model for resolving simultaneous frictional impacts in robotics, capturing multiple potential outcomes and guaranteeing termination, addressing a key challenge in impact modeling.

## Contribution

It extends Routh's method to multiple contacts and provides the first termination guarantee for set-valued outcomes in simultaneous impacts.

## Key findings

- Model captures multiple impact outcomes accurately.
- Proves solutions to the impact model always terminate.
- Addresses a fundamental challenge in impact dynamics modeling.

## Abstract

Many fundamental challenges in robotics, based in manipulation or locomotion, require making and breaking contact with the environment. To represent the complexity of frictional contact events, impulsive impact 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 propose an approach for resolving simultaneous frictional impacts, represented as a differential inclusion. Solutions to our model, an extension to multiple contacts of Routh's method, naturally capture the set of potential post-impact velocities.We prove that solutions to the presented model must terminate. This is, to the best of our knowledge, the first such guarantee for set-valued outcomes to simultaneous frictional impacts.